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Kings call up defenseman Matt Roy, who looks to…
Kings call up defenseman Matt Roy, who looks to score
By Robert Morales | rmorales@scng.com | Press Telegram
PUBLISHED: February 16, 2019 at 9:24 pm | UPDATED: February 16, 2019 at 9:25 pm
LOS ANGELES – Matt Roy on Friday night had just finished playing a game for the Ontario Reign. Then the 23-year-old defenseman got the word he was being called up by the Kings, and he was slated to make his NHL debut Saturday night when the Kings hosted the BostonBruins.
As he stood at his locker following the morning skate Saturday, Roy fielded questions. Every answer came with a smile.
"I was excited," he said. "I found out last night after our game for Ontario and I was just very excited. I called my dad. I think I woke him up; it was 1 o'clock their time.
"And then my mom wasn't with him, so then I called my mom, had to wake up my brother, then I had to wake up my sister. So it was fun."
Asked about their reaction, Roy grinned.
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"Well, I thought my mom would scream a little bit more, but I think she was kind of more shocked than anything," he said. "There was just happiness from everyone and I feel very grateful."
Roy, of Canton, Mich., is what one might call a scoring defenseman, especially this season. He had 17 points in 49 games this past season for the Reign, but this season he has 29 points (8 goals, 21 assists) in 45 games for Ontario.
He explained his success in that regard.
"I guess I'm just getting it past that first defender, trying to get the pucks on net," said Roy, who is 6-foot, 208 pounds. "I may not be scoring every shot. But, hopefully, there is traffic down low, which there usually is, and that gives us more opportunity to bank it in there.
"It's something I've worked on my whole life, but I don't really think about it too much. Just trying to get pucks on net."
This promotion was made possible by the upper-body injury sustained by veteran defenseman Alec Martinez near the end of Thursday night's 4-3 shootout loss to Vancouver. He is expected to be out one to two weeks. Another defenseman, Sean Walker, was hit in theface by a shot puck Thursday and he is day-to-day.
Interim coach Willie Desjardins assessed his newest addition.
"I followed the stats down there; I follow the AHL team and he's a good, solid player," he said of Roy. "Good skater, a little above average skater. Moves the puck pretty good. He's been able to get a few more points this year, I think. But I'm looking forward to seeing him."
It was easy to tell Roy was looking forward to his first game at the major-league level. But he intimated he was trying not to let it consume him.
"I try not to think about it too much," he said. "Just going to go out there and try and play my game and help the team as best I can."
Unfortunately for Roy, he said he didn't think his family was going to be able to make it to Saturday's game because it was such short notice.
"Hopefully, I can stick and catch a couple of more games," he said.
Alex Iafallo has struggled to find the net of late. The second-year forward had scored just two goals in 20 games before Saturday. He scored goals in four consecutive games ahead of that stretch. He did score a goal in this past Saturday's 5-4 loss at Boston. That was his first since was Jan. 12 in a 5-2 win over Pittsburgh. He has 11 goals and 13 assists (24 points) after going for 25 points (9 goals, 16 assists) in 75 games his rookie season. … The Kings finish this three-game homestand by hosting Washington on Monday night at 7:30. Then it's on the road with stops at Nashville, Florida, Tampa Bay and Carolina.
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Robert Morales
Robert Morales covers the Los Angeles Kings for the Southern California News Group.
Follow Robert Morales @RMoralesPT
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{"url":"https:\/\/pythonhosted.org\/triangula\/maths.html","text":"# Kinematics - Maths\u00b6\n\nThis page collects the various bits of vector maths used by Triangula\u2019s chassis kinematics.\n\nHint\n\nThis document contains derivations for the equations used by the chassis code. If you don\u2019t want to work through these you don\u2019t need to, just look for boxes like this one which contain the most important bits. You don\u2019t have to follow the maths here to use the code, but I\u2019ve written it up anyway in the hope that it\u2019ll be of interest.\n\n## Motion at a point\u00b6\n\nGiven an overall triangula.chassis.Motion, what is the velocity of a given point on the robot\u2019s chassis? Calculating the velocity at each individual wheel is the first thing we need to do when working out how fast each wheel must be rotated.\n\nHint\n\nSuppose we have a motion $$\\vec{M}=\\begin{pmatrix}m_x\\\\m_y\\\\m_\\theta\\end{pmatrix}$$ relative to the robot\u2019s centre.\n\n$$\\theta$$ in the above expression is rotation in radians per second, where positive values correspond to clockwise motion when viewed from above.\n\nWe wish to determine the velocity $$\\vec{V}=\\begin{pmatrix}d_x\\\\d_y\\end{pmatrix}$$ for a wheel $$W$$\n\nThe wheel is located, relative to the robot\u2019s centre point, at location $$W=\\begin{pmatrix}w_x&w_y\\end{pmatrix}$$.\n\nAs the robot is a rigid structure, when the motion is purely a translation (i.e. $$m_\\theta=0$$) all points on the robot will have the same velocity. Further, as we know that rotation and translation are independent, even when the rotation part is non-zero we can consider the two parts of the motion (rotation and translation) independently, just adding on the translation vector at the end. So, all we really need to work on is the rotation.\n\nAt this point we could do a lot of relatively awkward trigonometry, but there\u2019s a simpler approach:\n\n### Speed\u00b6\n\nWe know how fast we\u2019re moving, because we know the number of radians per second and we know the radius of the circle in which we\u2019re moving. As we know the circumference of the circle is $$2\\times\\pi\\times r$$, and we know that there are $$2\\times\\pi$$ radians (remember we use radians as our angular measurement!) in a circle, we can calculate we\u2019re moving at $$2\\pi r\\times\\frac\\theta{2\\pi}=r\\theta$$ where $$r$$ is the radius of the circle and $$\\theta$$ is the angular speed in radians per second.\n\nWe know $$\\theta$$ directly as it\u2019s part of our motion vector $$\\vec{M}$$.\n\nWe can calculate $$r$$ because we know our wheel is at $$W$$ relative to our centre of rotation. We need what\u2019s known as the magnitude, or length, of the vector from $$\\begin{pmatrix}0&0\\end{pmatrix}$$ to $$W$$, and basic geometry tells us that this quantity $$\\left|W\\right|=\\sqrt{w_x^2+w_y^2}$$\n\nHint\n\nPutting these together gives us the equation for the speed (note, not velocity, we haven\u2019t worked out direction yet!) at this particular wheel to be:\n\n$s=m_\\theta\\times\\sqrt{w_x^2+w_y^2}$\n\n### Direction\u00b6\n\nWe know what direction we\u2019re moving in. This is because we know where the centre of rotation is, and it\u2019s always the case that when rotating around a point, the direction we move is perpendicular to the direction to the centre of rotation. If this isn\u2019t immediately obvious, imagine there\u2019s a rigid rod attached to the ground at one end and you\u2019re holding the other end. As one end of the rod is attached to the ground you\u2019re always going to move in a circle - you obviously can\u2019t push in the direction the rod\u2019s oriented as that would need you to change the length of the rod (you can\u2019t, it\u2019s rigid), so you can only move at right angles to that direction.\n\nIn the general case we can rotate a vector by multiplying it by a matrix, where the values in the matrix are functions of the angle through which we want to rotate (in this case positive values of $$\\theta$$ correspond to clockwise rotation) - note that $$\\theta$$ in this case is the angle through which we\u2019re rotating the vector, and is not related to the $$m_\\theta$$ part of the motion!\n\n$\\begin{split}\\begin{bmatrix} x' \\\\ y' \\\\ \\end{bmatrix} = \\begin{bmatrix} \\cos \\theta & \\sin \\theta \\\\ -\\sin \\theta & \\cos \\theta \\\\ \\end{bmatrix}\\begin{bmatrix} x \\\\ y \\\\ \\end{bmatrix}\\end{split}$\n\nIn this particular case we want to rotate by a right angle to get the vector perpendicular to the radius of our circle and therefore parallel to its circumference. When $$\\theta=\\frac\\pi2$$ all the values in the matrix above are either zero, one or minus 1:\n\n$\\begin{split}\\begin{bmatrix} x' \\\\ y' \\\\ \\end{bmatrix} = \\begin{bmatrix} \\cos \\frac\\pi2 & \\sin \\frac\\pi2 \\\\ -\\sin \\frac\\pi2 & \\cos \\frac\\pi2 \\\\ \\end{bmatrix}\\begin{bmatrix} x \\\\ y \\\\ \\end{bmatrix} = \\begin{bmatrix} 0 & 1 \\\\ -1 & 0 \\\\ \\end{bmatrix}\\begin{bmatrix} x \\\\ y \\\\ \\end{bmatrix} = \\begin{bmatrix} y \\\\ -x \\\\ \\end{bmatrix}\\end{split}$\n\nHint\n\nSo, plugging $$W$$ into the above means our direction vector $$\\vec{D}$$ is as follows:\n\n$\\begin{split}\\vec{D} = \\begin{pmatrix} w_y \\\\ -w_x \\\\ \\end{pmatrix}\\end{split}$\n\n### Velocity from Rotation\u00b6\n\nAs we have a direction and a speed we can calculate the velocity. First though we need to calculate the unit vector for the direction - this will give us a vector of magnitude 1, which we can simply multiply by our speed to get our wheel velocity. The unit vector is calculated by dividing each part of the direction vector by the magnitude of the vector, so:\n\n$\\widehat D=\\frac{\\overrightarrow D}{\\left|D\\right|}$\n\nWe know that the magnitude of a vector is the square root of the sum of the squares of its components, so we can work out that the unit vector in this case is:\n\n$\\widehat D=\\frac{\\overrightarrow D}{\\sqrt{w_y^2+(-w_x)^2}}=\\frac{\\overrightarrow D}{\\sqrt{w_x^2+w_y^2}}$\n\nTo get our velocity we multiple the unit vector by the speed:\n\n\\begin{align} \\vec{V_{wheelRotation}} &= \\widehat D \\times s \\\\ &= \\frac{\\overrightarrow D}{\\sqrt{w_x^2+w_y^2}} \\times s \\\\ &= \\frac{\\overrightarrow D}{\\sqrt{w_x^2+w_y^2}} \\times m_\\theta\\times\\sqrt{w_x^2+w_y^2} \\\\ &= {\\overrightarrow D}m_\\theta \\\\ \\end{align}\n\nNow everything simplifies out! We\u2019re left with our wheel velocity being our direction vector multiplied by our angular velocity in radians per second. To finish the job we drop in our definition for $$\\overrightarrow D$$ to get:\n\nHint\n\nThe velocity due to the rotation component of the motion at wheel $$W$$ is:\n\n$\\begin{split}\\vec{V_{wheelRotation}} &= \\begin{pmatrix} w_y -w_x \\end{pmatrix}m_\\theta\\end{split}$\n\n### Final Velocity\u00b6\n\nHint\n\nAll our above calculations have only included the velocity from the rotation component of the motion. To include the translation component is easy though - we just add it on! Our final equation for the velocity of the wheel (or, more precisely, the velocity of the point at which the wheel makes contact with the ground) is therefore:\n\n$\\begin{split}\\vec{V_{wheel}} = \\begin{pmatrix} w_y \\\\ -w_x \\\\ \\end{pmatrix}m_\\theta + \\begin{pmatrix} m_x \\\\ m_y \\\\ \\end{pmatrix}\\end{split}$\n\n## Wheel Speed for Velocity\u00b6\n\nTriangula uses omni-wheels. Once we know from the above maths exactly what velocity is needed at each wheel position for a given desired motion we need to calculate the wheel speed in radians per second for each wheel. This is then passed on to the motor controllers to drive the robot.\n\nHint\n\nFistly we need a way to define the wheels. As used above, each wheel is located relative to the centre of the robot with a position vector, $$\\vec{W}$$.\n\nIn addition to its position, we need to know two more things.\n\n1. We need to know in what direction the wheel is pointing.\n2. We need to know how big the wheel is, because a large wheel will require a smaller number of rotations or radians per second to achieve the same speed across the ground.\n\nWe can model these pieces of information as a single wheel drive vector, $$\\vec{WD}$$, representing the direction and distance a regular wheel would roll in a single revolution.\n\nTriangula\u2019s triangula.chassis.HoloChassis.OmniWheel class contains the necessary logic to store the drive vector and to calculate it from other information such as wheel radius and angle (this may be more convenient when you need to specify your wheels). The maths, however, works on the drive vector as it\u2019s simpler to deal with.\n\nAs we are using omni-wheels, any wheel on Triangula\u2019s chassis can move in any direction. We know this by observation, but mathematically we know that we can drive the wheel along its drive vector $$\\vec{WD}$$, and that the wheel can also freely roll at right angles to this vector. We cannot control or measure the degree of movement at right angles to our drive vector, so we can safely disregard it. All we care about is motion in the direction of the drive vector, and we can obtain this by projecting the velocity onto the drive vector, using the formula:\n\n$p=\\frac{\\overrightarrow{V_{wheel}}\\cdot\\overrightarrow{WD}}{\\left|WD\\right|}$\n\nFor those not familiar with vector maths, the expression $$\\vec{A}\\cdot\\vec{B}$$ sums the products of each component of each vector. In other words:\n\n$\\begin{split}\\overrightarrow A\\cdot\\overrightarrow B=\\begin{pmatrix}a_x\\\\a_y\\end{pmatrix} \\cdot\\begin{pmatrix}b_x\\\\b_y\\end{pmatrix}=a_x\\times b_x+a_y\\times b_y\\end{split}$\n\nSo what are we doing when we project one vector onto another one? We\u2019re working in a two-dimensional plane, in which any point can be defined by two coordinates. Typically we use x and y coordinates, something you\u2019ll have encountered hundreds of times before in grids, maps, chess boards etc. What we actually mean when we use these though is slightly more subtle - we can think of both x and y as vectors themselves, which, when added together in the appropriate quantities, can be used to reach any point on the plane. So, our $$\\vec{x}$$ represents a single unit movement along the x axis, and the $$\\vec{y}$$ the same distance along the y axis. Starting from the origin, we can express any point on the plane as a motion involving a certain amount of $$\\vec{x}$$ and a certain amount of $$\\vec{y}$$.\n\nThe projection operation can be read as how much of unit vectors $$\\vec{x}$$ and $$\\vec{y}$$ do we need to add together to get a particular vector $$\\vec{V}$$ ? We project our target vector onto our basis vectors (those used to represent the coordinate system) and read off the projection, which we can then use as a coordinate in that basis vector\u2019s axis. When done with our regular x and y axes the results are exactly what you\u2019d expect, the projection of a vector $$\\begin{pmatrix}V_x\\\\V_y\\end{pmatrix}$$ onto $$\\vec{x}$$ is $$V_x$$ and onto $$\\vec{y}$$ is $$V_y$$.\n\nUsing vectors which correspond to the x and y axes is very convenient and easy to understand, but if all we want is a pair of vectors which can, between them, reach every point on the plane, we don\u2019t actually have to use those particular ones. In fact, all that\u2019s required is any pair of vectors that are not co-incident, that is to say one is not a multiple of the other one.\n\nNow, we know that our wheels have to have a velocity given by $$\\vec{V_{wheel}}$$, and we know we have a drive vector $$\\vec{WD}$$ and another vector which we haven\u2019t bothered naming which is non-coincident to the drive vector in which the wheels can slide. What we want to know is how far we have to move per second in the direction of the drive vector such that in combination with an unknown amount of movement orthogonal to this (the sliding vector) we end up with the target wheel velocity.\n\nSo, we know that we need $$p$$ multiples of $$\\widehat{WD}$$ to move as defined by $$\\vec{V_{wheel}}$$, where $$p$$ is defined as :\n\n$p=\\frac{\\overrightarrow{V_{wheel}}\\cdot\\overrightarrow{WD}}{\\left|WD\\right|}$\n\nHint\n\nNow we know we need to move $$p$$ units of distance, to get the wheel speed in revolutions per second we simply divide by the distance travelled per revolution. As we already defined the drive vector to be the translation vector for a single revolution of the wheel we divide by $${\\left|WD\\right|}$$ again, to give wheel speed $$s$$ (as revolutions per second) as :\n\n$s=\\frac{\\overrightarrow{V_{wheel}}\\cdot\\overrightarrow{WD}}{\\left|WD\\right|^2}$\n\n## Wheel Speed from Motion\u00b6\n\nCombining the two sections above we can calculate the necessary speed for any wheel on the chassis for any target motion for the robot as a whole.\n\nHint\n\nGiven a wheel, with location relative to the origin of the chassis specified by $$\\vec{W}$$ and drive vector $$\\vec{WD}$$, defined as the vector described by the wheel hub after one revolution of the wheel, and a target motion vector $$M$$ consisting of $$m_x$$ and $$m_y$$ linear velocities and angular velocity $$m_\\theta$$, we can calculate the speed at which the wheel will need to be driven, in revolutions per second, as:\n\n\\begin{align} s & = \\frac{(\\begin{pmatrix} w_y \\\\ -w_x \\\\ \\end{pmatrix}m_\\theta + \\begin{pmatrix} m_x \\\\ m_y \\\\ \\end{pmatrix})\\cdot\\overrightarrow{WD}}{\\left|WD\\right|^2} \\\\ & \\\\ & = \\frac{\\begin{pmatrix}w_ym_\\theta+m_x\\\\-w_xm_\\theta+m_y\\end{pmatrix} \\cdot\\begin{pmatrix}wd_x\\\\{\\mathrm{wd}}_y\\end{pmatrix}}{wd_x^2+wd_y^2} \\\\ & \\\\ & = \\frac{w_ym_\\theta wd_x+m_xwd_x-w_xm_\\theta wd_y+m_ywd_y}{wd_x^2+wd_y^2} \\\\ & \\\\ & = \\frac{m_xwd_x+m_ywd_y+m_\\theta(w_ywd_x-w_xwd_y)}{wd_x^2+wd_y^2} \\end{align}\n\nThe most striking thing about the above equation is that wheel speed is a linear function of the components of the motion vector. Unless the chassis changes over time, the coefficients of $$m_x$$, $$m_y$$ and $$m_\\theta$$ are constant, and can be pre-computed. A seemingly complex problem is therefore extremely simple to actually implement.\n\nTriangula\u2019s code is actually somewhat more complex, largely because in the sections above we have assumed that we are always rotating around the origin of the robot\u2019s coordinate system. This assumption simplifies the maths, and allows for the surprisingly simple expression above, but in reality we occasionally want to specify rotation around a different point. For example. if carrying some kind of gripper we might want to always rotate around the gripper. In these cases the effective geometry does change, as the vectors describing the wheel locations are in fact relative to the centre of rotation under consideration rather than always being locked to the origin. This isn\u2019t, however, much of an extra complication and if you\u2019ve understood everything to this point you should be able to understand how the code works! The only real difference is that the code doesn\u2019t reduce the equations down quite as much before running them.\n\n## Motion from Wheel Speeds\u00b6\n\nEverything up to this point has focused on calculating wheel speeds for a given motion, but it is possible to go in the other direction and to calculate motion from observed wheel speeds. Note that we can only do this because we have at least as many wheels as we have dimensions in the motion vector (3 in this case). Also note that if our chassis had more than 3 wheels we would never have a precise solution - in effect each wheel contributes an equation in a system of linear simultaneous equations, so when we\u2019re solving for 3 unknowns and have 3 equations we\u2019ll (almost) always have a single well-formed unique solution, but the moment we add in more equations, especially given our measurements will by definition contain errors, we are very unlikely to ever have a perfect match and must use numerical methods to find the best approximation. Triangula doesn\u2019t have this problem as she has 3 wheels, but were you to use this document to build something with, say, 5 wheels you\u2019d need to consider this issue.\n\nBecause we can arbitrarily define the centre point for our motion we can set it to the origin of the robot\u2019s coordinate space for convenience. This in turn means we can use the simplest form of the equations above, and that we can pre-compute the coefficients for each wheel. In fact, the code does exactly this - these lines in the init function for triangula.chassis.HoloChassis.OmniWheel should look familiar if you\u2019ve just read the maths in the previous sections:\n\nself.co_x = self.vector.x \/ self.vector_magnitude_squared\nself.co_y = self.vector.y \/ self.vector_magnitude_squared\nself.co_theta = (self.vector.x * self.position.y -\nself.vector.y * self.position.x) \/ self.vector_magnitude_squared\n\n\nNow rather than using $$m_x$$, $$m_y$$ and $$m_\\theta$$ to find a set of wheel speeds, we need to use a set of wheel speeds, one for each wheel to find $$m_x$$, $$m_y$$ and $$m_\\theta$$.\n\nTo prevent things getting out of hand in terms of size let\u2019s set up some new terms. For a wheel $$w_{n\\;\\in1,2,3...}$$ with speed $$s_n$$ we can pre-compute three coefficients.\n\nHint\n\n\\begin{align} x_n & = \\frac{wd_x}{wd_x^2+wd_y^2} \\\\ & \\\\ y_n & = \\frac{wd_y}{wd_x^2+wd_y^2} \\\\ & \\\\ \\theta_n & = \\frac{w_ywd_x-w_xwd_y}{wd_x^2+wd_y^2} \\end{align}\n\nThis allows us to concisely state three (in this case) simultaneous linear equations:\n\n\\begin{align} s_1 & = x_1m_x+y_1m_y+\\theta_1m_\\theta \\\\ s_2 & = x_2m_x+y_2m_y+\\theta_2m_\\theta \\\\ s_3 & = x_3m_x+y_3m_y+\\theta_3m_\\theta \\\\ \\end{align}\n\nHint\n\nAs with any system of such equations we can express this in the form of a matrix:\n\n$\\begin{split}\\begin{bmatrix}x_1&y_1&\\theta_1\\\\x_2&y_2&\\theta_2\\\\x_3&y_3&\\theta_3\\end{bmatrix}\\begin{bmatrix}m_x\\\\m_y\\\\m_\\theta\\end{bmatrix}=\\begin{bmatrix}s_1\\\\s_2\\\\s_3\\end{bmatrix}\\end{split}$\n\nThis is then amenable to numeric solving, in Triangula\u2019s case we use the NumPy library, which also includes functions to handle the case where we have more wheels than 3, although obviously in this particular instance we don\u2019t need to worry (Triangula is smart and fast, but she\u2019s thus far been incapable of spontaneously growing wheels).\n\n## Pose change from Motion\u00b6\n\nOnce we have a known triangula.chassis.Motion we can work out the change in our triangula.chassis.Pose assuming the motion remains constant for a known time. The first stage is to understand that a constant motion represents movement around a circle - this might not seem immediately obvious, but imagine what will happen if you walk forwards (or in fact in any direction) and, every step you take, you turn slightly. You will walk in a circle, the more you turn and the smaller your steps the smaller the circle, turning less each step and taking longer strides results in a larger circle.\n\nIf we know we\u2019re moving in a circle, it\u2019s easy to work out the change in our position - we just need to know two things:\n\n1. How big is the circle?\n2. Where is the centre of the circle?\n\nIf we can work out these two things we can rotate our current location around the centre of the circle to get our new location. The proportion of the circle we travel around, and therefore the angle we need to rotate by, is determined by the angular velocity component of the motion.\n\nFirst we need to work out the radius of the circle. We know that $$circ=2\\pi{}r$$. We also know, from $$m_\\theta$$ that in one second we\u2019ll travel around $$\\frac{m_\\theta}{2\\pi}$$ of the circumference of the circle. Our total distance travelled in one second is therefore $$\\frac{m_\\theta}{2\\pi}\\times2\\pi r=m_\\theta r$$\n\nWe know the total distance travelled in one second from the linear portion of $$M$$, so we can make these equal and solve for $$r$$ as follows:\n\n\\begin{align} m_\\theta r & =\\left|\\begin{pmatrix}m_x\\\\m_y\\end{pmatrix}\\right| \\\\ \\\\ m_\\theta r & =\\sqrt{m_x^2+m_y^2} \\\\ \\\\ r & =\\frac{\\sqrt{m_x^2+m_y^2}}{m_\\theta} \\end{align}\n\nNow we need to find the centre of the circle. We can do this easily, because we know that when moving around a circle the vector from our location to the centre of the circle is at right angles to the direction of our motion. There\u2019s one slight catch here though - if we\u2019re turning to the right ($$m_\\theta > 0$$) our centre point should be to our right, which we can get by rotating our motion vector 90 degrees clockwise. If, however, we\u2019re turning to the left we need to have our centre point to our left as well, rotating our motion vector 90 degrees counter-clockwise. For complete correctness we also need to handle the case where $$m_\\theta = 0$$), corresponding to motion in a straight line with no angular component.\n\nWe\u2019re taking a slight shortcut here, in that $$r$$ can be negative. This is helpful later - the radius has the same sign as the rotation part of the motion.\n\nNote\n\nBe very careful here! Our motion is expressed in robot coordinates, but we need everything to be in world coordinates if we\u2019re rotating a location around another point. So, before we use any of our motion vectors we need to rotate the entire motion by the inverse of the pose orientation.\n\nIn the remainder of this section, when we refer to components of the motion i.e. $$m_y$$ these refer to the transformed version of the motion, i.e the motion as observed in the world, not the motion from the perspective of the robot. In the code the first thing we do is to rotate the motion vector by the negative of the orientation part of the current pose.\n\nSo, to get the position of the centre of the circle about which we\u2019re moving we need to multiply our right angle unit vector (whether clockwise or counterclockwise) by the radius, to get a vector in the same direction with length $$r$$.\n\n\\begin{align} \\widehat R & =\\left\\{\\begin{array}{l}m_\\theta>0\\;:\\;\\frac{\\begin{pmatrix}m_y\\\\-m_x\\end{pmatrix}}{\\sqrt{m_x^2+m_y^2}}\\\\m_\\theta<0\\;:\\;\\frac{\\begin{pmatrix}-m_y\\\\m_x\\end{pmatrix}}{\\sqrt{m_x^2+m_y^2}}\\end{array}\\right. \\\\ \\\\ \\overrightarrow R=\\widehat Rr & =\\frac{\\begin{pmatrix}m_y\\\\-m_x\\end{pmatrix}}{m_\\theta} \\end{align}\n\nSo, when $$m_\\theta$$ is non-zero we rotate our current location around the point obtained by adding $$\\overrightarrow R$$ to that location by $$m_\\theta$$ radians.\n\nTo rotate a point $$P$$ around a centre point $$C$$ clockwise by $$\\theta$$ radians we apply the following transformation:\n\n\\begin{align} rotate(P,\\;C,\\;\\theta)=\\begin{bmatrix}p'_x\\\\p'_y\\end{bmatrix}=\\begin{bmatrix}p_x-c_x\\\\p_y-c_y\\end{bmatrix}\\begin{bmatrix}\\cos(\\theta)&\\sin(\\theta)\\\\-\\sin(\\theta)&\\cos(\\theta)\\end{bmatrix}+\\begin{bmatrix}c_x\\\\\\;c_\\mathrm y\\end{bmatrix} \\end{align}\n\nHint\n\nTo obtain the result of motion $$M$$ on pose $$P$$ after time $$t$$:\n\n\u2022 If $$m_\\theta$$ is non-zero the new pose $$P'$$ is found by rotating the current pose about the centre point $$\\begin{pmatrix}p_x+r_x\\\\p_y+r_y\\end{pmatrix}$$ as defined above, by $$m_\\theta{}t$$ radians.\n\u2022 If $$m_\\theta$$ is zero the new pose $$P'$$ is found by adding the (transformed) linear part of the motion multiplied by $$t$$ to the position part of the original pose.\n\nIn Python, using a helpful vector library, the code is actually relatively simple:\n\n# Total delta in orientation angle over the time interval\norientation_delta = motion.rotation * time_delta\n# Scaled translation vector rotated into world coordinate space (motion uses robot space)\ntranslation_vector_world = rotate_vector(motion.translation, self.orientation) * time_delta\n\nif orientation_delta == 0:\n# No orientation, trivially add the rotated, scaled, translation vector to the current pose\nreturn self.translate(translation_vector_world)\nelse:\ncentre_of_rotation = self.position + translation_vector_world.cross() \/ orientation_delta\n':type : euclid.Point2'\nfinal_position = rotate_point(self.position, angle=orientation_delta, origin=centre_of_rotation)\nreturn Pose(position=final_position, orientation=self.orientation + orientation_delta)","date":"2018-07-20 20:23:02","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 7, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9717437624931335, \"perplexity\": 309.7438601616501}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-30\/segments\/1531676591831.57\/warc\/CC-MAIN-20180720193850-20180720213850-00072.warc.gz\"}"} | null | null |
Would you rather see Arsenal win ugly than lose an entertaining game?
A win is a win and that is the end of that debate. There were many talking points in the Watford v Arsenal game and no doubt we will discuss them, but for now, I simply do not care what transpired in the game, a win is a win and that is all that matters.
We won away, we are in the fourth position, some of our players were rested and others certainly did not run themselves into the ground and that sets us up nicely for Thursday's game in Italy.
Of course, we were playing against ten men and no one can say that was a vintage performance and if we repeat that performance against Napoli we could well be in trouble but you know what, winning becomes a habit and when a streak is broken it is broken and that is what we achieved tonight.
We made no less than six changes, we were playing a team full of confidence on their ground and we still won, we have Champions League qualification back in our own hands and by any definition, that is a job well done.
There is a time for negativity and there is a time for positivity and tonight, is a time for positivity and I will sleep well knowing that we are doing just fine in two competitions.
It might not have been pretty, but who cares?
This entry was posted in Arsenal game review, Arsenal News, Arsenal NewsNow on April 16, 2019 by Admin.
So many players not in-form is more worrying of all, Auba, Mkhitaryan, Iwobi…! Especially Auba, he has been criminally wasteful, and basically in-effective without Laca up front with him!
The only highlight were the woodwork and the Ref!
Then 10 minutes later the manager took out Mavropanos, put in gwendouzi and switched the formation to 3-4-2-1 the players now look more confused. Emery saw that this too wasn't working and went for the final role of the dice, Niles in for Ramsey and the formation now switched to a 4-4-2 with ozil playing alongside Aubameyang.
Exactly right, billy. I wonder if Manchester United and Chelsea fans are criticizing their managers and denigrating their team after their very lucky wins, gifted by poor officiating? Will Spurs fans admit they are in front of Arsenal due to an offside goal allowed at Wembley?
It is time for everyone to get behind the coach and team to the end of the season; every point no matter how ugly is precious.
Managers 1st season you lot.
He's doing better than most 1st season managers especially as he's having to watch the pennies.Wait and see before heavy criticism.
All that matters at this stage are points.
That certainly is the million dollar question!!
Yes they won… at the end of the day that's all that matters, but boy I'd like to see them play better than that.. after the goal & red card, I didn't enjoy that game one bit!
Sue you and I alike, and it is simply because you always say the truth. The game was horrible to watch, my mind was practically collapsing inside of the way we held the for so long but did practically nothing with it. Sue if am going to be honest with you, we are not going to win premier league with Emery, we might we Europa, but we are not doing well. Imagine if the other teams in the top 4 race with us have better seasons?, it was a pathetic performance, and with ten men!!! I have never been this sad with a win before. Watford played like they were 11, they have vision, defended well against 11 men, moved forward and created massive problems for us.
I know when a team goes down to 10, they have to work harder & nearly always play better… but come on…I hate to admit this, but they should feel really hard done by… they could have snatched a draw… I'm obviously glad they didn't.. but jeez we sure did make it hard for ourselves Pat!
Sue and Pat, I'm sure Unai Emery doesn't coach Mhikataryan and Aubameyang to miss clear cut chances. Arsenal is creating enough chances to win 2 games every match, but are wasteful in front of goal. I hope the chances missed at the Emirates against Napoli don't come back to bite Arsenal.
To be pair Pat, i don't see us winning the premier league under this owner because of his self-sustaining business model. If we are honest, most of our players are inconsistent while the rest aren't good enough so we need a major squad overhaul which won't happen. When a player like nicholas pepe who has had only one top season is going for 60m upwards, what chance do we have? We will have to take more risks in buying untested players because of our budget and i would instead prefer we give our youngsters a chance. So Emery wouldn't be the real issue because i can't see Klopp or Guardiola winning the league with our group of players either. Our squad is so imbalanced that we have no wingers and i have never seen Klopp or Guardiola play without them, so i think emery is doing his best with what he's got even though i agree he isn't perfect.
Somewhat unrealistic question. Both. There are times when, playing 3 games a week, and many major games, the team are too spent and physical realities dictate how the team plays. It's not always a simple choice whether they play well or excitingly. With rotation, a couple of crap players, like Mustafi and Mkhitaryan, and a couple out of form players, the team 'cannot' play exciting. In that situation I would rather win ugly. With our best team, rested and fit, I would rather play teams off the park. With our present rotated team it's not possible to always win excitingly and who the **** would like to lose excitingly. Ridiculous. we are in reality, and are playing to get back into tier one. In times of change be thankful Unai Emery is giving us a chance to be where we rightfully belong. The good times will come.
I can't understand how you lot keep repeating that Mustafi is crap, when he has been very good for us in the last matches…He was our best defender at Everon and he is doing a solid job clearing out the danger. We still need an upgrade though, cause Mavropanos isn't at this level yet. He may need to go out on loan.. What about giving Chambers another chance?
What planet are you on for goodness sake! Mustafi is the worst Arsenal CB since Gus Caesar way back long ago!
Frankly this article and its "we won, so who cares how" theme is laughably naive. If you don't care HOW, Martin the author, than you are just not living in the real football present and are in a fantasy world of your own make believe. As a REALIST, first, last and always, I say you are the worst sort of "everything will be alright on the night" sort of fool yourself type fan. IF, as seems likely, we again stink the place out in our last three Prem away games we will surely never get this sort of fortune again and can expect at best two losses and perhaps a draw at Burnley. This will not be enough to make top four. So, UNLIKE YOU, MARTIN, I have enough sense to see that unless this away timidness and laziness – yes, laziness, let it be admitted – ceases right now, we are in for a huge disappointment. REALISM never fantasy. I leave fantasy hopes and self delusion to younger, more naive fans, like Martin!
Well said. Winning and ugly, are two different things. Of course, we love winning the three points, but the 'ugly' part is worrying! because that can come back again and again. Emery must address quickly this 'ugly' thing, because if we keep playing the three remaining away games ugly, with the same confusion and out-of-form players, we won't be as lucky as last night, and there won't be any more winning in, at least, the away games. And we can kiss Champion League goodbye again for one more season!
How reassuring it is to see a fellow realist on here. Sometimes I THINK WE REALISTS ARE BECOMING AN ENDANGERED SPECIES!
What do we fans want? I believe it is to win games after playing well. However if we can't do both we would rather win anyhow. The end which is winning justifies the means. The way some people on this site are mourning you would think we lost the game by more than 3 goals! We all love to enjoy a game but if that is not possible at least we should not lose. I recall several times when we call for winning ugly. Now that it has happened we are complaining. What do we actually want then? Are we a confused lot who don't know what they want? I will accept a win anytime no matter how it is achieved. We have also to appreciate the fact that our players have a very tight fixture and sometimes get fatigue. Is that too hard for some people to understand?
DAVID, you are letting natural bias overtake your better judgement about our away perfs. It cannot rationally be denied, surely, by anyone who is NOT letting bias rule their head, that we hugely rode our luck last night and it would be daft to assume that will continue at our last three away PREM GAMES. IT WILL NOT. And of course, we will ALL accept any win however it come, whether we are realists or fantasists. The crucial difference is that realists accept that last nights luck will not continue. I know you are far brighter than your above post seems to indicate. Quality lasts ; luck does not!
I wouldn't want us to play that way every week, but an ugly win every now and again are what defines great teams. Do you think the Invincibles were amazing every week. No, they weren't. The noise just gets amplified when it's Arsenal 1) because it's so out of character for us to play badly and win and 2) no one gave us a chance of finishing top 4 in the first place so if we are going to get there, we really have to earn it! | {
"redpajama_set_name": "RedPajamaC4"
} | 9,700 |
{"url":"https:\/\/ask.sagemath.org\/questions\/41524\/revisions\/","text":"# Revision history [back]\n\n### local variable 'Integer' referenced before assignment\n\nI'll be honest I have no idea why this error is popping up, and it's really weird! I'm writing code that will generate a closed form solution to the partial sums of integer powers up to n, and this is what I have\n\ndef sum_first_n_p_powers(p):\nlength = p+1\nvector = zero_vector(length)\nmatrix_list = []\nfor x in xrange(0,length):\ncopy = vector[:]\nfor y in xrange(0,length):\ncopy[y] = binomial(length-y,x-y+1)\nmatrix_list.append(copy)\nM = Matrix(matrix_list)\nsolution_vector = zero_vector(length)\nsolution_vector[0] = 1\ncoeffs = M.solve_right(solution_vector)\nn = var('n')\n0 = polynomial\nfor x in xrange(0,len(coeffs)):\npolynomial = polynomial + coeffs[x]*n^(length-x)\nreturn polynomial\n\n\nAnd when I try to run it I get the most unusual error, it just says to me\n\nUnboundLocalError: local variable 'Integer' referenced before assignment. I have never run into this before and have no idea what's wrong with the line. Thanks in advance!","date":"2021-06-23 11:28:59","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.34858211874961853, \"perplexity\": 1858.949281321248}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-25\/segments\/1623488538041.86\/warc\/CC-MAIN-20210623103524-20210623133524-00526.warc.gz\"}"} | null | null |
Q: Why does text form field re-render when user clicks on it in Flutter? Why does text form field re-render when user clicks on it in Flutter?
My Flutter form contains a TextFormField for Name. When my user clicks on it, the entire form immediately re-renders (reloads), making it impossible for my user to enter anything.
Code
The TextFormField is commented so that you can easily find it.
You'll notice that this page also contains a second field that works perfectly. It's a Switch inside a StatefulBuilder that handles setting a TextEditingController for _importantController.
<!-- language: flutter -->
import 'package:cloud_firestore/cloud_firestore.dart';
import 'package:dropdown_search/dropdown_search.dart';
import 'package:flutter/material.dart';
class CrudPage2 extends StatefulWidget {
final String docId;
const CrudPage2({Key? key, required this.docId}) : super(key: key);
@override
CrudPage2State createState() => CrudPage2State();
}
class CrudPage2State extends State<CrudPage2> {
late String name = "";
late bool isImportant = false;
final TextEditingController _nameController = TextEditingController();
final TextEditingController _importantController = TextEditingController();
Stream<DocumentSnapshot<Object?>> groceryItem(docID) =>
FirebaseFirestore.instance
.collection("groceries")
.doc(docID)
.snapshots();
@override
Widget build(BuildContext context) {
return Scaffold(
appBar: AppBar(
leading: IconButton(
icon: const Icon(Icons.arrow_back),
onPressed: () => Navigator.of(context).pop(),
),
title: Text("Grocery Item"),
),
body: SizedBox(
width: double.infinity,
child: Padding(
padding: EdgeInsets.only(
bottom: MediaQuery.of(context).viewInsets.bottom + 20),
child: StreamBuilder<DocumentSnapshot>(
stream: groceryItem(widget.docId),
builder: (BuildContext context, AsyncSnapshot<DocumentSnapshot> streamSnapshot) {
if (streamSnapshot.connectionState == ConnectionState.waiting) {
print("****** Loading ******"); // debugging
return const Text("Loading");
} else if (streamSnapshot.hasData) {
if (widget.docId != "NEW") {
// Retrieve existing item
var jsonData = streamSnapshot.data?.data();
Map<String, dynamic> myData = jsonData as Map<String, dynamic>;
name = myData['name'];
isImportant = myData['important'];
}
_nameController.text = name;
if (isImportant) {
_importantController.text = "true";
} else {
_importantController.text = "false";
}
return Column(
mainAxisSize: MainAxisSize.min,
crossAxisAlignment: CrossAxisAlignment.start,
children: [
//--------------------------------------------------------
// PROBLEM: Clicking on this field re-renders entire form.
Flexible(
child: TextFormField(
controller: _nameController,
decoration: const InputDecoration(labelText: 'Name'),
),
),
//--------------------------------------------------------
// No problem with this switch
StatefulBuilder(
builder: (BuildContext context, StateSetter importantStateSetter) {
return Row(
children: [
const Text("Important: "),
Switch(
value: isImportant,
onChanged: (value) {
importantStateSetter(() => isImportant = value);
},
),
],
);
},
),
Row(
mainAxisAlignment: MainAxisAlignment.center,
children: [
SizedBox(
child: ElevatedButton(
child: const Text('Cancel'),
onPressed: () async {
Navigator.of(context).pop();
}),
),
const SizedBox(
width: 10,
),
SizedBox(
child: ElevatedButton(
child: const Text("Submit"),
onPressed: () async {
final String name = _nameController.text;
if (widget.docId == 'NEW') {
addGroceryItem(name, 1.0, "test",
isImportant);
} else {
updateGroceryItem(widget.docId, name,
1.0, "test", isImportant);
}
// Clear the text fields
_nameController.text = '';
_importantController.text = "";
// Hide the bottom sheet
Navigator.of(context).pop();
},
),
)
],
),
],
);
} else {
return const Text("No Data");
}
})
),
),
);
} // Widget Build
//-------------------------------------------------------------
// Add New Grocery Item
//-------------------------------------------------------------
Future<void> addGroceryItem(
String name, double quantity, String category, bool isImportant) async {
await FirebaseFirestore.instance.collection('groceries').add({
"active": true,
"name": name,
"quantity": quantity,
"category": category,
"important": isImportant
});
}
//-------------------------------------------------------------
// Update Existing Grocery Item
//-------------------------------------------------------------
Future<void> updateGroceryItem(String docID, String name, double quantity,
String category, bool isImportant) async {
await FirebaseFirestore.instance.collection('groceries').doc(docID).update({
"active": true,
"name": name,
"quantity": quantity,
"category": category,
"important": isImportant
});
}
}
I added print("****** Loading ******"); line to help debug. When user clicks on text form field, the Console displays:
I/flutter (28767): ****** Loading ******
I/flutter (28767): ****** Loading ******
Why is the stream refreshing every time this widget is clicked?
Thank you for your time!
A: After a lot of Googling, I decided that my problem was coming from doing this entirely wrong. Here are some of the changes I made:
*
*Pass-in values as JSON object parameter
*Eliminate call to Firebase
*Eliminate Stream Builder
Code below solves my problem using these changes:
import 'package:cloud_firestore/cloud_firestore.dart';
import 'package:flutter/material.dart';
class CrudPage2 extends StatefulWidget {
final String docId;
final Object? docSnap;
const CrudPage2({Key? key,
required this.docId,
required this.docSnap})
: super(key: key);
@override
CrudPage2State createState() => CrudPage2State();
}
class CrudPage2State extends State<CrudPage2> {
//--- Form State Variables...
late String name = "";
late bool isImportant = false;
//--- Controllers for Form Fields...
final TextEditingController _nameController = TextEditingController();
final TextEditingController _importantController = TextEditingController();
@override
initState() {
super.initState();
if (widget.docId != "NEW") {
Map<String, dynamic> myData = widget.docSnap as Map<String, dynamic>;
name = myData['name'];
isImportant = myData['important'];
}
_nameController.text = name;
if (isImportant) {
_importantController.text = "true";
} else {
_importantController.text = "false";
}
}
@override
Widget build(BuildContext context) {
return Scaffold(
appBar: AppBar(
leading: IconButton(
icon: const Icon(Icons.arrow_back),
onPressed: () => Navigator.of(context).pop(),
),
title: Text("Grocery Item"),
),
body: SizedBox(
width: double.infinity,
child: Padding(
padding: EdgeInsets.only(
top: 20,
left: 20,
right: 20,
bottom: MediaQuery.of(context).viewInsets.bottom + 20),
child: Column(
mainAxisSize: MainAxisSize.min,
crossAxisAlignment: CrossAxisAlignment.start,
children: [
TextFormField(
controller: _nameController,
decoration: const InputDecoration(labelText: 'Name'),
),
StatefulBuilder(
builder:
(BuildContext context, StateSetter importantStateSetter) {
return Row(
children: [
const Text("Important: "),
Switch(
value: isImportant,
onChanged: (value) {
importantStateSetter(() => isImportant = value);
},
),
],
);
},
),
Row(
mainAxisAlignment: MainAxisAlignment.center,
children: [
SizedBox(
width: 110,
child: ElevatedButton(
style: ButtonStyle(
backgroundColor:
MaterialStateProperty.all(
Colors.grey),
padding: MaterialStateProperty.all(
const EdgeInsets.all(5)),
textStyle: MaterialStateProperty.all(
const TextStyle(fontSize: 24))),
child: const Text('Cancel'),
onPressed: () async {
Navigator.of(context).pop();
}),
),
SizedBox(
width: 200,
child: ElevatedButton(
style: ButtonStyle(
backgroundColor:
MaterialStateProperty.all(Colors.green),
padding: MaterialStateProperty.all(
const EdgeInsets.all(5)),
textStyle: MaterialStateProperty.all(
const TextStyle(fontSize: 24))),
child: const Text("Submit"),
onPressed: () async {
final String name = _nameController.text;
if (widget.docId == 'NEW') {
addGroceryItem(name, 1.0, "Test",
isImportant);
} else {
updateGroceryItem(widget.docId, name,
1.0, "Test", isImportant);
}
// Clear the text fields
_nameController.text = '';
_importantController.text = "";
// Hide the bottom sheet
Navigator.of(context).pop();
},
),
)
],
),
]
)
)
),
);
} // Widget Build
Future<void> addGroceryItem(
String name, double quantity, String category, bool isImportant) async {
await FirebaseFirestore.instance.collection('groceries').add({
"active": true,
"name": name,
"quantity": quantity,
"category": category,
"important": isImportant
});
}
Future<void> updateGroceryItem(String docID, String name, double quantity,
String category, bool isImportant) async {
await FirebaseFirestore.instance.collection('groceries').doc(docID).update({
"active": true,
"name": name,
"quantity": quantity,
"category": category,
"important": isImportant
});
}
}
Any comments and/or suggestions are still appreciated.
I hope this helps someone else in the future.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,246 |
Cardinal and Bishop Tony pray for Pakistani Couple
Bishop Tony Robinson, a Bishop of the Society and Cardinal Vincent Nichols, president of the Catholic Bishops' Conference of England and Wales, lit candles and prayed in St Anne's Cathedral in Leeds for the couple and their unborn daughter who were burned to death in Pakistan last week.
The Bishops' prayer for peace and the repose of the souls of Sajjeed Mashah, 27, and his wife Shama Bibi, 24, followed Ecumenical Vespers, celebrating the 50th anniversary of the second Vatican Council's groundbreaking Ecumenical decree Unitatis Redintegratio.
The Bishops condemned the actions of the 1,200 strong mob in Pakistan who burned the couple amidst rumours they had burned verses of the Quran. They also appealed to the Pakistan Government to bring the perpetrators to justice and protect the rights of minorities.
Family member, Javeed Masseeh described how their legs had been broken so they could not run away and then the couple were held over the brick furnace until their clothes caught fire and then thrown down inside the furnace. Bibi, a mother of four, was four months pregnant.
Bishop Tony, Chair of the Archbishop of Canterbury's Pakistan Focus Group and who was in Pakistan in March, said: "The savage attack and murder of these two Christians and their unborn child is unacceptable in any civilised society.
"The Pakistan Government must do more to protect minorities in Pakistan from mob rule and a misuse of the blasphemy laws.
"In this country we hold that freedom of religion is at the core of our tolerant and democratic society. As we look around the world today there are many places where people are not allowed this freedom. We must not take the freedom we have lightly and work to change the attitudes of those who persecute people of another faith."
Cardinal Nichols said: "This is a horrific and tragic event which sullies the reputation of a great nation. Surely all people of true religious spirit will, in response, turn to God in prayer, seeking forgiveness for the violence and destruction of life, pleading for peace in our troubled world.
"For my part I pray for the repose of the souls of the couple and their unborn child."
In March this year Bishop Tony visited Pakistan where he met the country's president Mamnoon Hussain to discuss the persecution of minorities.
You can see some photos here.
Northern Provincial Festival
Bishop Tony together with a number of priests and people who look to the Society from the Diocese took part in Northern Provincial Festival in York Minster on Saturday 12 October. A full report can be read here.
Bishops' Statement on General Synod Vote
Statement from the Council of Bishops Many in the Church of England are celebrating today, following final approval of the legislation to permit women to be ordained as bishops. While recognizing this, we deeply regret the further obstacle that this decision places in the path to the full, visible unity of the whole Church. We do, however, welcome the provision that has been made in the House of Bishops' Declaration. It recognizes that our theological convictions about ministry and ordination remain within the spectrum of Anglican teaching and tradition. It assures us that bishops will continue to be consecrated within the Church of England who can provide episcopal ministry that accords with those theological convictions. It makes provision for parishes to gain access to that episcopal ministry by passing resolutions. This gives us confidence in our future as catholics who are called to live out our Christian vocation in the Church of England. For this we give thanks to God.
On behalf of the Council of Bishops
+ TONY PONTEFRACT
Rt Revd Tony Robinson
Bishop of Pontefract Chairman | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 7,781 |
\section{Introduction}
\label{intro}
At present, the study of stellar pulsations constitutes one of the
most fundamental pillars on which the building of stellar astrophysics
rests on. Although the theory of stellar pulsations was initially
elaborated to explain the existence of classical variable
stars such as Cepheids and RR Lyrae, in the last few decades the study
of pulsating stars has been extended to many other different kinds of
stars that were regarded as constant stars before
(e.g., Unno et al. 1989;
Gautschy \& Saio 1995). Nowadays, new classes of pulsating stars
across the HR diagram are being routinely uncovered from ground-based
observations as well as space missions (e.g., CoRoT, MOST; see Aerts
et al. 2008). The study of stellar pulsations through the approach of
asteroseismology constitutes a powerful tool to probe the internal
structure and evolution of stars.
Most of the pulsations exhibited by pulsating stars are self-excited
through the classical $\kappa$-mechanism operating in a partial
ionization zone near the surface of stars (Cox 1980; Unno et
al. 1989). As a matter of fact, this mechanism is responsible for
pulsations of the stars in the classical instability strip due to
partial ionization of H and HeI and/or HeII. In the driving zone, the
opacity perturbation increases outward so that radiative luminosity is
blocked in the compression phase of pulsation. The region gains
thermal energy in the compression phase and it loses thermal energy in
the expansion phase.
A less common ---and consequently less explored--- pulsation driving
mechanism in stars is the $\epsilon$-mechanism. This mechanism is due
to vibrational instability induced by thermonuclear reactions. In
this case, the driving is due to the strong dependence of nuclear
burning on temperature. During maximum compression, the
temperature and thus the nuclear energy production rates are higher
than at equilibrium. So, in the layers where nuclear reactions take
place, thermal energy is gained at compression while the opposite
happens during the expansion phase (Unno et al. 1989; Gautschy \&
Saio 1995). An excellent historical account of studies on vibrational
destabilization of stars by the $\epsilon$-mechanism, can be found in
Kawaler (1988) ---we refer the interested reader to that paper for
details.
In this paper, we explore the $\epsilon$-mechanism in connection with
pulsating PG1159 stars. These stars, also called GW Vir or DOV, are
very hot H-deficient post-Asymptotic Giant Branch (AGB) stars with
surface layers rich in He, C and O (Werner \& Herwig 2006) that
exhibit multiperiodic luminosity variations with periods ranging from
300 to 6000 s, attributable to non-radial $g$-modes driven by the
$\kappa$-mechanism acting on the region of partial ionization of C and
O (Starrfield et al. 1983, 1984, 1985; Gautschy 1997; Quirion et
al. 2004; Gautschy et al. 2005; C\'orsico et al. 2006; Quirion et
al. 2007). Some pulsating PG1159 stars are still embedded in a nebula
and are called Planetary Nebula Nuclei Variable (PNNV) stars (see
Winget \& Kepler 2008 and Fontaine \& Brassard 2008 for recent
reviews).
Evolutionary models of PG1159 stars with thick He-rich envelopes
located at the upper left portion of the HR diagram are characterized
by the presence of vigorous He-burning shells. The first attempt to
study the effect of the $\epsilon$-mechanism induced by a He-burning
shell in H-deficient pre-white dwarf stars was the seminal work by
Kawaler et al. (1986). These authors found some $g$-modes excited
through this mechanism with periods in the range 70 to 200 s.
Observationally, however, no signature of these short pulsation
periods was found in the surveys of planetary nebula nuclei conducted
at that time (Grauer et al. 1987; Hine \& Nather 1987). Later on,
stability analysis on simplified PG1159 models by Saio (1996) and
Gautschy (1997) also predicted unstable $g$-modes driven by the
$\epsilon$-mechanism with periods in the range $110-150$ s.
The interest in the $\epsilon$-mechanism in the context of H-deficient
post-AGB stars has recently been renewed by the discovery of
luminosity variations in the PNNV star \vv\ ($T_{\rm eff}=
130\,000 \pm 5000$ K, $\log g= 7 \pm 0.5$, C/He= 1.5 and O/He= 0.4;
Werner \& Herwig 2006) by Gonz\'alez P\'erez et al. (2006). The most
outstanding feature of \vv\ is that its period spectrum appears to
include a series of unusually short pulsation periods ($\sim 130-300$
s), the shortest periods ever detected in a pulsator of its class.
Gonz\'alez P\'erez et al. (2006) (see also Solheim et al. 2008)
speculate that these rapid oscillations could be excited by the
$\epsilon$-mechanism.
In this work, we largely extend the pioneering work by Kawaler et al.
(1986), Saio (1996) and Gautschy (1997) by performing fully
nonadiabatic pulsation computations on realistic PG1159 models
extracted from full evolutionary sequences with a wide range of
stellar masses and effective temperatures. In particular, we gather
strong evidence for the existence of a new short-period $g$-mode
instability strip of pulsating PG1159 stars due to the
$\epsilon$-mechanism. In addition, we examine the possibility
that the short-period $g$-modes of \vv\ could be excited by this
mechanism. The paper is organized as follow: in the next section we
briefly describe the input physics of the PG1159 evolutionary
sequences analyzed and the nonadiabatic treatment of the pulsations.
In Sect. \ref{results} we describe the stability analysis. In
Sect. \ref{vv47} we present the application to the star \vv, and in
Sect \ref{pg1159} we discuss the case of the prototypical DOV
star \pp\ in the context of our theoretical findings. Finally, in
Sect. \ref{summary} we summarize our main results and make some
concluding remarks.
\section{Evolutionary/pulsational modelling of PG1159 stars}
\label{evolutionary}
The PG1159 equilibrium models on which the present investigation rests
on were extracted from the evolutionary calculations presented by
Althaus et al. (2005), Miller Bertolami \& Althaus (2006) and
C\'orsico et al. (2006), who computed the complete evolution of model
star sequences with initial masses on the ZAMS in the range $1 - 3.75
M_{\odot}$. The evolutionary tracks for the H-deficient pre-white
dwarf remnants are displayed in Fig. \ref{fig01}. All of the post-AGB
evolutionary sequences, computed with {\tt LPCODE} (Althaus et al.
2005), were followed through the very late thermal pulse (VLTP) and
the resulting born-again episode that gives rise to the H-deficient,
He-, C-, and O-rich composition characteristic of PG1159 stars. For
details about the input physics and evolutionary code used, and the
numerical simulations performed to obtain the PG1159 evolutionary
sequences employed here, we refer the interested reader to the works
mentioned above. One distinctive feature, and crucial for this study,
that is common to all of our sequences, is that the PG1159 models are
characterized by He-rich envelopes thick enough as to sustain active
He-burning shell sources during the evolutionary stages of
interest. This is at variance with the non-standard PG1159 models
employed in Althaus et al. (2008) to explain the $\dot{\Pi}$ values
in \pp, which are characterized by thin
He-rich envelopes and so they are not able to sustain an active
He-burning shell.
\begin{figure}
\centering
\includegraphics[clip,width=250pt]{fig01.eps}
\caption{The PG1159 evolutionary tracks of Althaus et al. (2005),
Miller Bertolami \& Althaus (2006) and C\'orsico et al. (2006), with
stellar masses of (from right to left): $M_*= 0.515, 0.530, 0.542,
0.565, 0.589, 0.609, 0.664, 0.741 M_{\odot}$. Also shown is the
location of known PG1159 stars. The error bars for \vv\ are
displayed.}
\label{fig01}
\end{figure}
The pulsational stability analysis presented in this work was carried
out with the linear, nonradial, nonadiabatic pulsation code described
in C\'orsico et al. (2006). The ``frozen-in convection''
approximation was assumed because the flux of heat carried by
convection is negligible in PG1159 stars. At variance with C\'orsico
et al. (2006), in this work we have fully taken into
account the $\epsilon$-mechanism for mode driving operating in the
He-shell nuclear-burning region. Because we are interested in PG1159
stars which are H-deficient, we are only concerned with the He-burning
reactions. Fortunately, because $g$-mode pulsation timescales are
much shorter than the timescales of nucleosynthesis, possible phase
delays between the temperature perturbations and the abundance
variations are unimportant. Hence, they can be neglected, largely
simplifying the pulsational stability analysis (Unno et al. 1989;
Kawaler et al. 1986)\footnote{We note that some of our sequences have
trace surface abundances of H ($X_{\rm H} \lesssim 10^{-3}$) which
give rise to some H burning. However, exhaustive test stability
computations demonstrate that H burning is very weak and extends on a
extremely narrow layer, as a result of which the H-shell burning is
completely irrelevant in destabilizing modes and will not be further
considered in this paper.}.
\begin{figure}
\centering
\includegraphics[clip,width=250pt]{fig02.eps}
\caption{The dipole ($\ell= 1$) normalized growth rates
$\eta$ (black dots connected with continuous lines) in terms of the
pulsation periods for a $0.530 M_{\odot}$ PG1159 template model
located before the evolutionary knee ($T_{\rm eff}= 138\, 400$ K,
$\log (L_*/L_{\odot})= 3.14$). Numbers indicate the radial order $k$
for low order modes. The large numerical range spanned by $\eta$ is
appropriately scaled for a better graphical representation. The two
ranges of overstable $g$-modes ---one due to the $\kappa$-mechanism
and the other induced by the $\epsilon$-mechanism--- are clearly
discernible. Plus symbols connected with dotted lines correspond to
the case where the $\epsilon$-mechanism is explicitly suppressed in
the stability calculations.}
\label{fig02}
\end{figure}
\section{Nonadiabatic results}
\label{results}
We analyzed the stability properties of about 4000 stellar models
covering a wide range of effective temperatures
($5.5 \gtrsim \log(T_{\rm eff}) \gtrsim 4.7$) and stellar masses
($0.515 \lesssim M_*/M_{\odot}\lesssim 0.741$). For each model we
restricted our study to $\ell= 1$ $g$-modes with periods in the range
$50-7\,000$ s.
\subsection{A single template model}
We start our description by focusing on a $0.530 M_{\odot}$
PG1159 template model with $T_{\rm eff}= 138\,400$ K and $\log
(L_*/L_{\odot})= 3.14$ located before the
evolutionary knee in Fig. \ref{fig01}. The surface
chemical composition of the model is $X({^4{\rm He}})= 0.33$,
$X({^{12}{\rm C}})= 0.39$, $X({^{13}{\rm C}})= 0.05$,
$X({^{14}{\rm N}})= 0.02$, and $X({^{16}{\rm O}})= 0.17$.
Fig. \ref{fig02} displays the
normalized $\ell= 1$ growth rates $\eta= -\Re(\sigma)/\Im(\sigma)$
(where $\Re(\sigma)$ and $\Im(\sigma)$ are the real and the imaginary
parts, respectively, of the complex eigenfrequency $\sigma$) in terms
of the pulsation periods ($\Pi$) corresponding to our template model.
In the interests of a better graphical
representation, the huge numerical range spanned by $\eta$ is
appropriately scaled (see Gautschy 1997). The sign function allows to
discriminate between stable and unstable modes. The presence of two
well-defined families of overstable $g$-modes, one at the
intermediate- and long-period regime, and the other one at the short
period regime, is apparent. The first group of periods ($\approx
750-1800$ s) corresponds to modes driven by the well-known
$\kappa$-mechanism operating at the region of the opacity bump due to
partial ionization of C and O, centered at $\log T \approx 6.2$
(Gautschy et al. 2005, C\'orsico et al. 2006). The second
group of periods, which are associated to low radial order $g$-modes,
are destabilized by the action of the vigorous He-shell burning
through the $\epsilon$-mechanism.
The short-period instabilities uncovered here are of the same nature
than those found by Kawaler et al. (1986) in the context of
H-deficient hot central stars of planetary nebulae. Here, as in that
work, the $\epsilon$-mechanism induced by the He-shell burning
constitutes the source of driving. In absence of this destabilizing
agent, the overstable modes with periods in the range ($\approx
100-180$ s) turn out to be stable, while the remainder modes of the
pulsation spectrum remains unchanged. This is vividly displayed in
Fig. \ref{fig02}, that shows with plus symbols the results of
additional stability computations in which the nuclear energy
production rate, $\epsilon$, and the logarithmic derivatives
$\epsilon_T= \left(\frac{\partial \ln \epsilon}{\partial \ln
T}\right)_{\rho}$ and
$\epsilon_{\rho}= \left(\frac{\partial \ln \epsilon}{\partial
\ln \rho} \right)_T$, are forced to be zero in the pulsation equations.
It is worth emphasizing that in the present effort we are able to
obtain destabilization of $g$-modes through both the
$\kappa$-mechanism
\emph{and} the $\epsilon$-mechanism in the same PG1159 equilibrium
model. This is at variance with the study by Kawaler et al. (1986),
who reported only $\epsilon$-destabilized modes.
The $\epsilon$-mechanism behaves as an efficient filter of modes
that destabilizes only those $g$-modes that have their largest
maximum of the temperature perturbation ($\delta T/T$) in the narrow
region of the He-burning shell (see Kawaler et al. 1986). In the
specific case of our template model, only the $g$-modes with $k= 3,
4, 5$, and $6$ meet such a condition and, as a result, they are
$\epsilon$-destabilized. For $k= 1, 2$ the largest maximum of
$\delta T/T$ lies at inner layers with respect to the He-burning
shell. Thus, these modes are stable. For modes with $k \geq 7$ the
opposite is true and these modes also are stable.
Test stability calculations with $\ell= 2$ for our template model
indicate that there exist only one quadrupole $\epsilon$-destabilized
$g$-mode which corresponds to $k= 5$ with a period $\Pi \sim 95$ s,
about $40 \%$ shorter than the corresponding $k= 5$ mode with $\ell=
1$ ($\Pi \sim 157$ s). Hence, in general, a narrower range of shorter
periods is expected to be associated with $\epsilon$-destabilized
$g$-modes with $\ell= 2$ as compared with the case of $\ell= 1$.
\subsection{A new PG1159 instability strip}
Having described our results for a single template model, we now are
in conditions to examine the location and extension of the complete
unstable domain associated with the $\epsilon$-mechanism. Our results
are depicted in Fig. \ref{fig03}, which displays the instability strip
of $\epsilon$-destabilized modes in the $\log T_{\rm eff}-\log g$
drawn with thick black curves along the PG1159 evolutionary tracks.
In addition, the GW Vir instability domain of $\kappa$-destabilized
modes (see C\'orsico et al. 2006) is depicted with red (gray) lines
along the tracks. Note that the instability strip for
$\epsilon$-destabilized modes partially overlaps the domain of
$\kappa$-destabilized modes. So, our results indicate the existence of
three well-defined instability regimes in the $\log T_{\rm eff}-\log
g$ plane: a regime ---splitted into two regions, one at low gravity
and the other at high gravity--- in which stellar models harbour
intermediate/long period $g$-modes excited by the $\kappa$-mechanism
only, another one corresponding to short-period modes destabilized by
the $\epsilon$-mechanism only, and finally a region in which models
experience pulsational destabilization by the $\kappa$-mechanism and
the $\epsilon$-mechanism of driving simultaneously. Notably, the
region corresponding to the $\epsilon$-mechanism only is not occupied
by any known PG1159 star (see Fig. \ref{fig03}).
We stress that in previous works (Kawaler et al. 1986; Saio 1996;
Gautschy 1997) only \emph{some} short-period $g$-modes were found to
be destabilized by the $\epsilon$-mechanism. Needles to say, due to
the very few $\epsilon$-destabilized modes found in those exploratory
works, no clear extension and location of the $\epsilon$-mechanism
instability domain were obtained, thus hampering those authors from
making further consideration of such modes. At variance with those
works, in the present study we are able to find a \emph{complete}
instability strip of $\epsilon$-destabilized modes.
The degree of driving, and the place that it might occur in the
$\log T_{\rm eff}- \log g$ diagram, is sensitive to the stellar
mass, previous evolutionary history, and so on. Thus, due to the
uncertainties in the stellar evolution modelling (overshooting, nuclear
reaction rates, etc), the surface and internal composition of PG1159
stars are not known in detail, and so a clear instability domain for
$\epsilon$-destabilized pulsations is difficult to drawn. So, what
is shown in the Fig. \ref{fig03} is the shape and location of the
$\epsilon$-mechanism instability strip obtained by us under the
particular assumptions adopted in the construction of the PG1159
evolutionary models of Miller Bertolami \& Althaus (2006). The
extension and location of this instability domain might change if
other assumptions for the evolutionary history of the progenitor
stars were adopted.
\begin{figure}
\centering
\includegraphics[clip,width=250pt]{fig03.eps}
\caption{Same as Fig. \ref{fig01}, but including the loci of
models having $\ell= 1$ (dipole) $\kappa$-destabilized modes with
solid red (gray) curves along the tracks, and models harbouring
short-period $\epsilon$-destabilized modes according to the present
study. Superposition of both types of curves corresponds to stellar
models with both $\epsilon$- and $\kappa$-destabilized modes (shaded
area). The location and designation of relevant PG1159 stars is also
shown [Color figure only available in the electronic version of the
article].}
\label{fig03}
\end{figure}
All of the overstable $\epsilon$-destabilized modes computed in this
work are characterized by very tiny ($10^{-9}-10^{-12}$) linear
growth rates $\eta$, by far smaller than those characterizing
overstable modes excited by the $\kappa$-mechanism
($10^{-8} \lesssim \eta \lesssim 10^{-4}$). So, the question rises
about what would be the chance for a given $\epsilon$-destabilized
mode to have plenty of time for developing observable amplitudes. To
analyze this question we consider the time interval that the models
spend crossing the instability strip of $\epsilon$-destabilized
modes, $\Delta t$, and the maximum and minimum $e$-folding times
$\tau_e^{\rm max}$ and $\tau_e^{\rm min}$, respectively, of the
unstable modes for a given stellar mass. The $e$-folding times
are defined as $\tau_e \equiv 1/|\Im(\sigma)|$, such that the time
dependence of the amplitude of the pulsations is given by
$\xi(t) \propto e^{i \sigma t}$, and $\Im(\sigma) < 0$ for
overstable modes.
The values of $\Delta t$, $\tau_e^{\rm min}$, and $\tau_e^{\rm max}$
are provided in Table \ref{table1} for each value of the stellar mass.
Note that the three timescales monotonically decrease for increasing
stellar mass. For all of our PG1159 sequences we found that the most
unstable modes ---those with the smaller values of $\tau_e$--- are
found near the low-gravity (high-luminosity) boundary of the
instability domain (upper black dashed line in Fig. \ref{fig03}), when
the models are still evolving to the blue before reaching the
evolutionary knee. On the contrary, when models are already evolving
toward the white dwarf cooling track, the $\epsilon$-destabilized
modes are only marginally unstable, and so they are characterized by
large $e$-folding times.
Table \ref{table1} shows that $\tau_e^{\rm min} \ll \Delta t$ for all
of our sequences. This means that $g$-modes that are destabilized at
epochs before the evolutionary knee, characterized by short
$e$-folding times, have time enough to reach observable amplitudes
before the star leaves the instability strip. On the other hand, it
is apparent that $\tau_e^{\rm max} \gtrsim \Delta t$. Thus, the
$g$-modes that are destabilized in models close to the high-gravity
limit (low-luminosity) of the instability strip (lower black dashed
line in Fig. \ref{fig03}) have little ---or even null--- chances to
develop observable amplitude before the model abandons the instability
domain.
In summary, our computations predict that some $g$-modes (those
with short $\tau_e$) could have plenty of time to grow and finally
develop observable amplitudes. We caution, however, that this
prediction is based on a \emph{linear} stability analysis, and that
the last word should came from a detailed non-linear description of
nonadiabatic pulsations. Such a nonlinear treatment is not
available at the present stage. Also, there are other effects
(stellar winds, diffusion, etc) suspected to be present in real stars, that
could be affecting the growth of pulsations. The assessment of their
effects on the modes predicted to be unstable in the frame of our
analysis is beyond the scope of the present study.
\begin{table}
\centering
\caption{The minimum and maximum $e$-folding times (in yr),
and the time (in yr) that PG1159 models spend within the
instability strip of $\epsilon$-destabilized modes.}
\begin{tabular}{cccc}
\hline
\hline
\noalign{\smallskip}
$M_*/M_{\odot}$&$\tau_e^{\rm min}$ & $\tau_e^{\rm max}$&
$\Delta t$\\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
$0.515$ & $3410$ & $1.5 \times 10^6$ & $1.60 \times 10^5$ \\
$0.530$ & $2580$ & $1.0 \times 10^6$ & $1.01 \times 10^5$ \\
$0.542$ & $1610$ & $3.8 \times 10^5$ & $5.95 \times 10^4$ \\
$0.565$ & $1400$ & $1.3 \times 10^5$ & $2.78 \times 10^4$ \\
$0.589$ & $1160$ & $1.0 \times 10^5$ & $2.47 \times 10^4$ \\
$0.609$ & $742 $ & $4.7 \times 10^4$ & $1.26 \times 10^4$ \\
$0.664$ & $361 $ & $1.8 \times 10^4$ & $4830 $ \\
$0.741$ & $180 $ & $7000$ & $1570 $ \\
\hline
\end{tabular}
\label{table1}
\end{table}
The next step in our analysis is to derive the range of periods
($\Pi$) of overstable $\epsilon$-destabilized modes.
Fig. \ref{fig04} displays the regions of the $\kappa$-mechanism
instability domains in the $\log T_{\rm eff}-\Pi$ diagram, depicted
with lines of different colours for the various stellar masses.
Notably, the figure also shows the presence of a separate,
well-defined instability domain for a broad range of effective
temperatures ($5.46 \gtrsim \log T_{\rm eff } \gtrsim 4.99$) and
pulsation periods in the interval $55 \lesssim \Pi \lesssim 200$ s,
associated to $\epsilon$-destabilized $g$-modes with radial orders
ranging from $2$ to $5$ for $M_*= 0.515 M_{\odot}$ and from $3$ to $8$
for $M_*= 0.741 M_{\odot}$. The stages corresponding to phases before
(after) the evolutionary knee are depicted with small dot (plus)
symbols. A close inspection of the figure reveals that for the
low-mass models, most of modes are destabilized after the evolutionary
knee. For the high-mass models the situation is reversed, that is,
most of overstable modes are destabilized before the evolutionary
knee. The existence of this new instability domain of short-period
$g$-modes in stellar models representative of PG1159 stars is the main
result of our study.
In particular, it is worth emphasizing that the $\epsilon$-mechanism
should be active in a PG1159 star irrespective of the precise
abundances of He, C, and O at the surface, because in this case the
mode excitation takes place at deep layers in the star. This is
at variance with the $\kappa$-mechanism, which is strongly dependent
on the exact O/C/He abundances at the driving regions (see Quirion et
al. 2007).
\begin{figure}
\centering
\includegraphics[clip,width=250pt]{fig04.eps}
\caption{The dipole ($\ell= 1$) instability domains for overstable
$\kappa$-destabilized $g$-modes, shown with thin lines of different
colours for the various stellar masses. The $\epsilon$-mechanism
instability domain is emphasized with a shaded area.
Short-period dipole unstable
$\epsilon$-destabilized $g$-modes are depicted
with dot (plus) symbols for stages before (after) the evolutionary knee.
Numbers indicate the
radial order of the modes. [Color figure only available in the
electronic version of the article].}
\label{fig04}
\end{figure}
\subsection{The case of the PNNV star \vv}
\label{vv47}
An immediate prediction of the present study is that PG1159-type stars
populating the overlapping region of $\kappa$- and
$\epsilon$-destabilized modes in the $\log T_{\rm eff}-\log g$ diagram
(the shaded region in Fig. \ref{fig03}) should exhibit both short- and
intermediate/long-period luminosity variations simultaneously.
Table \ref{table2} lists the known PG1159 candidate stars. A glance of
this table leads us to a somewhat disappointing conclusion: most of
the stars located in the region of interest are not variables at all
or have not been scrutinized for variability. Other stars, at most,
exhibit intermediate/long-period luminosity variations which are
typical of the high/intermediate-order $g$-modes driven by the
$\kappa$-mechanism, but not the expected short periods typical of
$\epsilon$-destabilized modes. In particular, this is the case for
the prototype DOV star, \pp.
There is one object, the PNNV star \vv, which is suspected to
pulsate in short- and long-period modes (Gonz\'alez P\'erez et al.
2006). This star ($T_{\rm eff}= 130\,000 \pm 5000$ K, $\log g= 7 \pm
0.5$, C/He= 1.5 and O/He= 0.4; Werner \& Herwig 2006) was first
observed as potentially variable by Liebert et al. (1988). Later, it
was monitored by Ciardullo \& Bond (1996), but no clear variability
was found. Finally, Gonz\'alez P\'erez et al. (2006) were able to
confirm the ---until then, elusive--- intrinsic variability of \vv\
for the first time. They found evidence that the pulsation
spectrum of this star is extremely complex. The most outstanding
feature of \vv\ is the presence of high-frequency peaks (at periods
$\sim 130-300$ s) in the power spectrum, which could be serious
candidates for low-$k$ radial order $g$-modes triggered by the
$\epsilon$-mechanism.
We decided to test the attractive
possibility that the short-periods observed in \vv\ could be due to
the $\epsilon$-mechanism. We first estimated the stellar mass
of \vv. From the location of \vv\ in the $\log T_{\rm eff}- \log g$
plane (see Fig. \ref{fig01}) it is apparent that the spectroscopic
mass of \vv\ is of $\approx 0.525 M_{\odot}$. In addition, a
preliminary adiabatic asteroseismological analysis on this star
indicates that the seismological mass of \vv\ ---obtained from the
period spacing data ($\Delta \Pi \approx 24$ s)--- is of $\approx
0.52-0.53 M_{\odot}$, in excellent agreement with the spectroscopic
derivation. So, we shall focus on the case of the evolutionary
sequence of $M_*= 0.530 M_{\odot}$. This sequence is characterized by
a thick He-rich envelope ($M_{\rm env} \sim 0.045 M_{\odot}$).
We would like to see how well the theoretical ranges of periods of
unstable modes corresponding to this sequence fit the observed period
spectrum of \vv. Fig. \ref{fig05} displays the regions of the
$\kappa$-mechanism instability domain (light and dark grey) for the
$0.530 M_{\odot}$ sequence. The figure also shows the presence of a
well-defined instability domain ($77\lesssim \Pi \lesssim 180$ s) that
corresponds to $\epsilon$-destabilized $g$-modes with $k= 2,\cdots,6$
(large and small dots). Also depicted in the plot are the periods
reported by Gonz\'alez P\'erez et al. (2006) for \vv\ with their
corresponding uncertainties in $T_{\rm eff}$. We have emphasized
with black small circles the periods associated with modes having the best
chances to be real, according to Gonz\'alez P\'erez et al. (2006).
It is apparent that,
whereas most of the long periods observed in \vv\ are
qualitatively explained by the $\kappa$-mechanism when the model
star is before the evolutionary knee, the short-period
branch (below $\sim 300$ s) of the pulsation spectrum of the star is
not accounted for at all by the theoretical domains corresponding to
this destabilizing agent. We can see, instead, that the short periods
of \vv\ ---in particular $\Pi \lesssim 200$ s--- are satisfactorily
accounted for by the $\epsilon$-destabilized $g$-modes. Note,
however, that if only periods detected with sufficiently high
significance (black filled circles) are used to compare with our
theoretical predictions, then the period at 261.4 s cannot be
explained by a low-order $g$-mode excited by the
$\epsilon$-mechanism. In fact, this period is considerably longer
than the longest period ($\approx 180$ s) of the $g$-modes which can
be excited by the $\epsilon$-mechanism as our analysis predicts.
Clearly, more observational work is needed to put the reality of the
short periods in \vv\ on a solid basis. Were the existence
of these short periods confirmed by future observations, then they
could be attributed to the $\epsilon$-mechanism, and this
could be indicating that \vv\ should have
a \emph{thick} He-rich envelope as to support an active He-burning
shell.
\begin{table}
\centering
\caption{Known PG1159 stars populating the overlapping
instability region of $\epsilon$- and $\kappa$-destabilized modes.}
\begin{tabular}{llccc}
\hline
\hline
&Star & PN & Variable & Period range [s]\\
\hline
1 & PG 1159$-$035 & no & yes & $430-840$ \\
2 & NGC 650$-$1 & yes & no & $-$ \\
3 & VV 47 & yes & yes (?) & $\sim$ 260 \\
& & & & $\sim$ 2170-4300\\
4 & PG 1144+005 & no & no & $-$ \\
5 & Jn 1 & yes & yes (?) & $454-1860$ \\
6 & Abell 21 & yes & no & $-$ \\
7 & K 1$-$16 & yes & yes & $1500-1700$ \\
8 & Longmore 3 & yes & no & $-$ \\
9 & HS 2324+3944 & no & yes & $2005-2570$ \\
10 & HS 1517+7403 & no & no & $-$ \\
11 & PG 1424+535 & no & no & $-$ \\
12 & IW 1 & yes & no & $-$ \\
13 & Sh 2$-$68 & yes & ? & $-$ \\
14 & HS 0704+6153 & no & ? & $-$ \\
\hline
\hline
\end{tabular}
\label{table2}
\end{table}
\begin{figure}
\centering
\includegraphics[clip,width=250pt]{fig05.eps}
\caption{The $\ell= 1$ regions of the
$\kappa$-mechanism instability domain, shown with light (dark) grey for
stages before (after) the evolutionary knee, corresponding to the
$0.530 M_{\odot}$ sequence. Arrows indicate the time sense of
evolution. Also shown is the evolution of the periods corresponding to
the $\epsilon$-destabilized modes $g_2, \cdots, g_6$,
with large (small) dots for stages before (after) the evolutionary knee.
Finally, the periods reported by Gonz\'alez P\'erez et al. (2006)
for \vv\ with their corresponding
uncertainties in $T_{\rm eff}$ are displayed with small circles.
Periods detected with sufficiently high significance are emphasized with
black filled circles.}
\label{fig05}
\end{figure}
\subsection{The case of the DOV star \pp}
\label{pg1159}
Another consequence of our investigation concerns the pulsating star
PG 1159$-$035, the prototype of the class and the best-studied DOV.
Indeed, note from Fig. \ref{fig03} that a trend of our results is
that this variable star should exhibit short-period
$\epsilon$-destabilized modes if the thick He-rich envelopes derived from our
evolutionary calculations were representative of the star.
These modes are not observed by Costa et
al. (2007). This result suggests that the He-burning shell may not be
active in \pp. This would indicate that this star has a
thinner He-rich envelope than what is traditionally derived from standard
evolutionary calculations, in line with
the recent finding by Althaus et al. (2008) that a thinner He-rich
envelope (at least a factor of two below of the value predicted by the
standard evolution theory) for PG 1159$-$035 should be invoked
to alleviate the longstanding discrepancy between the observed
(Costa \& Kepler 2008) and the theoretical (C\'orsico et al. 2008)
rates of period change in that star.
If the short periods observed in \vv\ were confirmed, then we should
face the problem of the coexistence of two PG1159 stars
located very close each other in the $\log T_{\rm eff} -\log g$
diagram (see Fig. \ref{fig03}) but with substantially different
thickness of the He-rich envelopes. This would suggest that
these stars could have had a different evolutionary history, a
suggestion reinforced by the fact that \vv\ still retains a planetary
nebula while PG 1159$-$035 does not.
\section{Summary and conclusions}
\label{summary}
In this paper, we have presented a fully nonadiabatic stability
analysis on state-of-the-art PG1159 models generated taking into
account the complete evolution of progenitor stars, through the
thermally pulsing AGB phase and born-again episode. We have explored
the possibility that nonradial $g$-mode pulsations could be
destabilized by a He-burning shell through the $\epsilon$-mechanism.
Our study covers a broad range of stellar masses and effective
temperatures. We confirm and extend the pioneering work of
Kawaler et al. (1986), Saio (1996) and Gautschy (1997) on this topic.
The main results are the following:
\begin{itemize}
\item[-] We
found strong theoretical evidence for the existence of a separate,
well-defined PG1159 instability strip in the $\log T_{\rm eff} - \log
g$ diagram characterized by short-period $g$-modes excited by the
$\epsilon$-mechanism due to the presence of active He-burning shells.
Notably, this instability strip partially overlaps the already known
GW Vir instability strip due to the $\kappa$-mechanism acting on the
partial ionization of C and/or O in the envelope of the PG1159 stars.
We emphasize that while in previous works only some short-period
$g$-modes were found to be destabilized by the $\epsilon$-mechanism,
in the present study we found a \emph{complete}
instability strip of $\epsilon$-destabilized modes.
\item[-] At variance with the classical
$\kappa$-mechanism responsible for the intermediate/long-period GW Vir
pulsations, the $\epsilon$-mechanism should be efficient even in
PG1159 stars with low C and O content in their envelopes.
\item[-] The $\epsilon$-driven $g$-modes that are destabilized at
epochs before the evolutionary knee are characterized by short
$e$-folding times (between $\approx 180$ yr for $M_*= 0.741 M_{\odot}$
and $\approx 3000$ yr for $M_*= 0.515 M_{\odot}$), and so they
probably have time enough as to reach observable amplitudes before the
star leaves the instability strip. Note, however, that nonlinear
effects, or the presence of a variety of phenomena such as stellar winds or
diffusion, could affect the growth of pulsations.
\item[-] We have closely examined the case of \vv, the only PG1159 star
for which observational evidence of the presence of short-period
$g$-modes exists (Gonz\'alez P\'erez et al. 2006). For this star we
have derived for the first time a seismological mass of $\approx
0.52-0.53 M_{\odot}$, in excellent agreement with the spectroscopic
mass ($\approx 0.525 M_{\odot}$). If we accept that all of
the periods reported by Gonz\'alez P\'erez et al. (2006)
are real, our stability analysis provides very
strong support to the idea that the physical origin of the short
periodicities could be the $\epsilon$-mechanism powered by an
active He-burning shell, whereas the long-period branch of the period
spectrum of this star should be due to the $\kappa$-mechanism
acting on the region of partial ionization of C and
O. However, if only periods detected with sufficiently high
significance are taken into account, then the period at
261.4 s can not be explained by a low-order $g$-mode
excited by the $\epsilon$-mechanism.
\item[-] We speculate that the absence of short periods
($\lesssim 300$ s) in the pulsation
spectrum of PG 1159$-$035 could be indicating that the He-burning shell
may not be active in this star. This would indicate that
PG 1159$-$035 has a thinner He-rich envelope than what is
traditionally derived from standard evolutionary computations.
\end{itemize}
In light of our results, if the reality of the short periods of \vv\
were confirmed by follow-up observations, this star could be the first
known pulsating PG1159 star undergoing nonradial $g$-modes
destabilized by the $\epsilon$-mechanism. Even more, \vv\ could be
the first known pulsating star in which both the $\kappa$-mechanism
and the $\epsilon$-mechanism of mode driving are \emph{simultaneously}
operating. Further time-series photometry of \vv\ will be needed to
firmly establish the reality of the short-period pulsations detected
in this star.
On the other hand, the apparent absence of short-period pulsations in
the remainder variable stars ---such as K 1$-$16, HS 2324+3944, and Jn
1--- could be an indication that, like \pp, they are characterized by
thin He-rich envelopes, as a result of which they should lack of
stable He-shell burning. Another possibility is that short-period
pulsations could be indeed present in these stars, but with very low
amplitudes, below the actual detection limits.
Also, it is quite intriguing the absence of both short- and
intermediate/long-period pulsations in the up to now constant stars
(NGC 650$-$1, PG 1144+005, Abell 21, Longmore 3, HS 1517+7403, PG
1424+535, IW 1) that populate the overlapping region of the
$\epsilon$- and $\kappa$-destabilized modes. In any case, extensive
searches for low amplitude intrinsic variability in these stars
and also in the stars Sh 2$-$68 and HS 0704+6153, which have not
been observed for
variability yet, should be worth doing in order to test the existence
of the new
theoretical instability strip uncovered in this work.
\acknowledgments
This paper has been benefited from the valuable suggestions and comments
of an anonymous referee. Part of this work was supported by AGENCIA
through the Programa de Modernizaci\'on Tecnol\'ogica BID 1728/OC-AR,
and by the PIP 6521 grant from CONICET. This research has made use of
NASA's Astrophysics Data System. Finally, we thank H. Viturro and
R. Mart\'inez for technical support.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 6,883 |
\section{}
Most photonic quantum computation \cite{knill} and quantum communication protocols \cite{wheis} rely on the availability of highly entangled photon pairs. Moreover, entanglement plays the pivotal role in linking the nodes of quantum networks \cite{kimble}. In this context, the excitonic-biexcitonic radiative cascade in quantum dots (QDs) has proved to be a very promising candidate for the generation of polarization entangled photon pairs, in particular because of the triggered emission of photons~\cite{ondem,bounouar}. The main obstacle encountered in the generation of entangled photon pairs with QDs has been the finite excitonic fine-structure splitting (FSS), giving a "which-path" information on the exciton-biexciton radiative cascade \cite{akopian}. In recent years various growth and post-growth techniques have been developed to reduce the FSS. Efforts were done in approaches as diverse as epitaxial growth in (111) direction \cite{versteegh}, growth of highly symmetric GaAs QDs \cite{kapon,Linz,Amand,juska,juska2}, rapid thermal annealing~\cite{young}, external piezo-applied stress \cite{zhang} or electric \cite{benett} and magnetic field tuning \cite{steven,see}. Despite some of them being successful, they are technologically demanding and can negatively affect the quality of the emission, reducing the quantum efficiency \cite{cohsteven} and the spin coherence \cite{benett}. A second issue is the efficient broadband extraction of entangled pairs emitted by a semiconductor QD located in the high refractive index host material. While cavity enhanced emission of entangled photon pairs was achieved using a QD coupled to narrow-band hybridized micropillar cavity modes \cite{dousse}, the scalability and the reproducibility of this very demanding concept is still a non-solved issue. Since the spectral separation between the excitonic and the biexcitonic transitions, namely the biexciton binding energy, is typically in the range of a few meV, solutions featuring broadband enhancement of photon extraction are most suitable for the realization of QD based entangled photon-pair sources. Over the last decade, a few approaches including photonic wires and microlenses \cite{reimer,claudon,gschrey} tackled this challenge and are good candidates for the production of entangled photon pairs.
We present here a study on single semiconductor QDs integrated deterministically into microlenses \cite{gschrey}. Since these structures allow for a broadband extraction of the excitonic (X) and biexcitonic (XX) photons as well as for enhanced focusing of the resonant laser\cite{bounouar2}, they are very interesting structures for applications in the field of photonic quantum information technology. We show that two key requirements are fulfilled by these nanostructures. Firstly, by applying pulsed resonant two-photon excitation of the biexciton, we show that the quantum dot upper-state can be coherently addressed and controlled. Secondly, time resolved quantum tomography is performed on photon pairs emitted by the radiative XX-X cascade of the QD. We take advantage of Heisenberg's relation, expressing that higher temporal resolution in determining the dynamics of the XX-X decay implies larger uncertainty in energy which can be larger than the related excitonic fine structure of the QD. In this situation the ''which-path information'' is lost and quantum entanglement of the paired photons can be measured even in the presence of a FSS. The observed degree of entanglement is actually solely limited by the detectors temporal resolution which has to be compared with the inverse precession frequency of the excitonic phase imposed by the FSS. The latter feature is demonstrated by performing quantum tomography on two QD-microlenses with FSS of 16 $\mu$eV and 30 $\mu$eV, respectively. In both cases, photons emitted by the XX-X cascade remain maximally entangled during the radiative decay and are not affected by decoherence. For practical purposes, we provide an estimation of the entanglement degree as a function of the time window applied for the post-selection of the exciton wavepacket.
\begin{figure}[htbp]
\centerline{\includegraphics[width=\linewidth]{FIG1.pdf}}
\caption{\footnotesize (a) $\mu$PL spectra of QDM1 (left panel) and QDM2 (right panel) under non-resonant excitation (left panel inset: relative energy of the X and XX lines as a function of the detection polarization angle for QDM1 and QDM2. A FSS of 16 $\mu$eV for QDM1 and 30 $\mu$eV for QDM2 are determined by a sinusoidal fit of the experimental data). Right panel inset: exemplary $\mu$PL spectrum of a QD-microlens under resonant two-photon excitation. (b) Scheme of the relevant states in a QD for the generation of entangled photon pairs. (c) $\mu$PL intensity of the XX transition as a function of the two-photon resonant pulse area. (d) SEM image of a deterministically fabricated QD-microlens.}
\label{fig:fig1}
\end{figure}
Our experiments are carried out on self-assembled InGaAs/GaAs QDs grown by metal-organic chemical vapor deposition. The QDs are integrated into microlenses with a backside distributed Bragg reflector by 3D in-situ electron-beam lithography~\cite{gschrey}. A scanning electron microscopy (SEM) image of such a device is displayed in the inset of Fig.~\ref{fig:fig1}~(c). Two different QD-microlenses dubbed QDM1 and QDM2 are studied in the following. Fig.~\ref{fig:fig1}~(a) shows $\mu$PL spectra of QDM1 and QDM2 under non-resonant excitation at 532 nm. Two emission lines in each spectrum are identified as X and XX transitions of the target QD. The inset of Fig.~\ref{fig:fig1}~(a) (right panel inset) shows the typical emission spectrum of a QD-microlense under resonant pulsed two-photon excitation of the biexciton. Fig.~\ref{fig:fig1}~(c) presents the power dependence of the X and XX intensities as a function of the applied pulse area. The laser is placed at an energy resonant to the virtual state enabling the two-photon excitation of the biexciton. This excitation scheme has become a well established and powerful technique \cite{winik,bounouar,jayakumar}, and is nowadays considered as a critical prerequisite for the coherent generation of entangled photons by QDs. We perform these measurements in confocal configuration with a resonant excitation setup. Typical Rabi oscillations of the biexcitonic and excitonic line intensities are observed when the laser pulse area is increased, accounting for the coherent control of the QD biexcitonic state in the Bloch-sphere \cite{stuffler}. The first maximum of the curves plotted in Fig.~\ref{fig:fig1}~(c) represents the first inversion of the biexcitonic population ($\pi$-pulse), and the ideal operation point of the source. The FSS of QDM1 and QDM2 is determined by polarization-dependent $\mu$PL spectroscopy. The inset of Fig.~\ref{fig:fig1}~(a) shows the relative energies of X (blue curve) and XX (red curve) as a function of the detection angle in linear polarization. Sinusoidal fits to the experimental data yield a FSS of ($15\pm$1)$\mu$eV for QDM1 and $(30\pm$1) $\mu$eV for QDM2, respectively.
\begin{figure}[htbp]
\centerline{\includegraphics[width=\linewidth]{FIG2.pdf}}
\caption{\footnotesize 16 time resolved polarization-dependent correlation measurements used for the quantum tomography for QDM1. The red (green) line represents the time bin used for the density matrix reconstruction noted $\rho_1$ ($\rho_2$) in the following. }
\label{fig:fig2}
\end{figure}
In QDs with a non-zero FSS, the spin up and spin down exciton states (respectively labelled $|X_H\rangle$ and $|X_V\rangle$ in Fig.~\ref{fig:fig1}~(b)) are non-degenerate and are no eigenstates of the system. The exciton state will therefore evolve with time. One can write the resulting two-photon state as follows \cite{cohsteven}:
\begin{align*}
\label{eq:solve}
|\psi (\tau)\rangle=\frac{1}{\sqrt{2}}(|HH\rangle+e^{i\Delta \tau}|VV\rangle),
\end{align*}
with $\Delta$ the frequency corresponding to the FSS energy and $\tau$ the time delay between the excitonic and biexcitonic photons. This state can be rewritten in the diagonal basis (D/A) to show that it oscillates between $|\phi^+\rangle=\frac{1}{\sqrt{2}}(|DD\rangle+|AA\rangle)$ and $|\phi^-\rangle=\frac{1}{\sqrt{2}}(|DA\rangle+|AD\rangle)$. One obtains a similar result in the circular basis (R/L). This means that the excitonic phase evolution, which leads to this oscillation, can be tracked by correlating the photons when they are projected in these two bases.
For the quantum tomography measurements we use a time resolved polarization-dependent cross-correlation setup. Photons from the coherently driven (under $\pi$-pulse condition) XX-X cascade are polarization-projected in the 3 complementary bases (H/V, D/A and R/L), and the coincidence rate is measured as a function of the delay between the XX photons and the X photons. The 16 measurements necessary for the full quantum tomography and the corresponding normalized correlation functions are plotted in Fig.~\ref{fig:fig2}. The observed time dependent oscillations due to the excitonic phase evolution are discussed above and occur when both photons are projected in the right circular polarization R or in the diagonal polarization D. On the other hand, the measurements in the linear basis (HH and HV), showing no oscillation, exhibit classical correlations. The temporal resolution of the setup is estimated to be 100 ps (full width at half maximum) and each coincidence time bin is 4 ps.
\begin{figure}[htbp]
\centerline{\includegraphics[width=\linewidth]{FIG3.pdf}}
\caption{\footnotesize (a) Density matrices reconstructed for a delay corresponding to the first maximum of the DD coincidence curve (noted $\rho_1$, 4 ps selection time window), for QDM1 (left panel) and for QDM2 (right panel). Real parts are displayed on the top and imaginary parts on the bottom part of the graph. Theoretical real parts and imaginary parts of the maximally entangled state $|\phi^+\rangle$ are plotted between the matrices of QDM1 and QDM2, as a reference. (b) Same density matrices reconstructed for a delay corresponding to the first minimum of the DD coincidence curve (noted $\rho_2$, 4 ps selection time window). Theoretical real parts and imaginary parts of the maximally entangled state $|\phi^-\rangle$ are plotted between the matrices of QDM1 and QDM2, as a reference.}
\label{fig:fig3}
\end{figure}
Based on the experimental data presented in Fig.~\ref{fig:fig2}, the density matrices of the generated two-photon states are reconstructed using a maximum-likelihood estimation. Fig.~\ref{fig:fig3} shows the reconstructed density matrices obtained for the first maximum (Fig.~\ref{fig:fig2}, red line) and the first minimum (Fig.~\ref{fig:fig2}, green line) observed on the DD curve of Fig.~\ref{fig:fig2} (red line marked). Fig.~\ref{fig:fig3}(a) corresponds to the state of the QD directly after the emission of the biexcitonic photon (left panel for QDM1 and right panel for QDM2). For the sake of comparison, the ideal density matrix (real part and imaginary part) of $|\phi^+\rangle$ between the experimental density matrices for QDM1 and QDM2 is displayed in Fig.~\ref{fig:fig3}~(a). The fidelity ($F(\phi^+)=Tr(\sqrt{\sqrt{\rho_1}.\rho(\phi^+).\sqrt{\rho_1}})^2$) of the experimental density matrix ${\rho_1}$ to $|\phi^+\rangle$ is estimated as 0.73$\pm$0.03 for QDM1 and 0.69$\pm$0.04 for QDM2. Since the phase in QDM1 is evolving slower than for QDM2, the setup is able to better resolve the oscillation for QDM1 which also shows a higher degree of entanglement. At longer delays, the QD state rotates towards $|\phi^-\rangle$. Fig.~\ref{fig:fig3}~(b) shows the reconstructed density matrices obtained for the first minimum of the DD coincidence curve (green line marked on Fig.~\ref{fig:fig2}). They resemble the $|\phi^-\rangle$ state (represented in inset of Fig.~\ref{fig:fig3}~b)). $F(\phi^-)$ is estimated to 0.80$\pm$0.03 for QDM1 and 0.68$\pm$0.04 for QDM2.
\begin{figure}[htbp]
\centerline{\includegraphics[width=\linewidth]{FIG4.pdf}}
\caption{\footnotesize (a) Data obtained for QDM1: (upper panel) Normalized coincidences between a biexcitonic photon and a triggering excitonic photon. (lower panel) Measured fidelities to the maximally entangled Bell states (blue curve for $|\phi^+\rangle$, and red curve for $|\phi^-\rangle$). (b) Same data obtained with QDM2. (c) Measured fidelity as a function of the percentage of the post-selected excitonic wavepacket for QDM1, (d) for QDM2.}
\label{fig:fig4}
\end{figure}
Fig.~\ref{fig:fig4}~(a) depicts the time evolution of the fidelity to the two Bell states, $F(\phi^+)$ and $F(\phi^-)$ of the two-photon state. As expected, because of the excitonic phase evolution, the entangled two-photon state evolves between $|\phi^+\rangle$ and $|\phi^-\rangle$. Interestingly, the oscillations can be clearly observed along the whole exciton wavepacket (plotted on the top panel of Fig.~\ref{fig:fig4}~(a)), indicating that the entanglement of the QD state is mostly unaffected by decoherence. Fig.~\ref{fig:fig4}~(c) shows $F(\phi^+)$ and $F(\phi^-)$ as a function of the time windows used for the tomography. The fidelities for both quantum dots are decreasing quickly as the time window is enlarged and for a post-selection exceeding 20\% of the total exciton wavepacket, no entanglement can be observed any more. An optimal working point is obtained by post-selection of around 15\% of the excitonic photons, presenting here a good compromise between fidelity to the Bell state and photon count rate.
\begin{figure}[htbp]
\centerline{\includegraphics[width=\linewidth]{FIG5.pdf}}
\caption{\footnotesize (a) Deconvoluted data obtained for QDM1. Upper panel: normalized coincidences between an excitonic photon and a triggering biexcitonic photon. Intermediate panel: Measured fidelities to the maximally entangled Bell states (blue curve for $|\phi^+\rangle$, and red curve for $|\phi^-\rangle$). Lower panel: Negativity of the two-photon state as a function of the delay after deconvolution, (b) Deconvoluted data obtained with QDM2, (c) Superposed norms of the density matrices reconstructed before convolution (in color), and corrected after deconvolution (in blue) for QDM1 and QDM2.}
\label{fig:fig5}
\end{figure}
In order to evaluate the actual quality of the entanglement between the photons emitted by the QD, we deconvoluted the data from the time response of the experimental tomography setup with a temporal resolution of 100~ps. The theoretical polarization-dependant cross-correlation function~\cite{winik} is convoluted to a Gaussian curve (100 ps full width at half maximum), as measured from the setup response. The resulting curve is fitted to the 16 tomography data curves. All the function parameters, such as the polarization angles, the FSS frequency and the decay time, were obtained from experiment and are kept constant. Only a multiplying factor and an offset are left as free parameters for the data fitting (see supplementary material). The deconvoluted fit function is used in order to reconstruct the new density matrices representing the emitted two-photon states without the effect of the experimental resolution. Fig~\ref{fig:fig5} shows the results provided by the quantum tomography after deconvolution. Fig.~\ref{fig:fig5}~(a) shows the decay curve (upper panel), the fidelity to the Bell states (middle panel) and the negativity (lower panel) for QDM1. The fidelities are showing oscillations with an amplitude very close to unity without damping along the full wavepacket, showing that QDM1 emits nearly perfectly entangled photons. This is confirmed by the negativity (lower panel), quantifying the separability of the density matrix. A value close to 0.5 (maximal entanglement) is found up to a delay of 0.8 ns.
Fig.~\ref{fig:fig5}~(b) shows very similar results for QDM2 with larger FSS. For instance, oscillations with close to unity amplitude can also be observed on the fidelity curves. For this QD more pronounced damping of the oscillations is observed indicating that the quality of the entanglement is significantly reduced for delays larger than 0.5 ns. The decrease of the negativity with respect to the delay is consistent with this observation. Even if simultaneous jumps of the excitonic and biexcitonic phases do not affect the QD entanglement, cross-dephasing processes, such as exciton spin 'flip-flop' processes, could be the reason for such a degradation of the entanglement quality for this particular QD. Moreover, the deconvolution is not fully successful at suppressing the periodic drops of the negativity. This can be attributed to the frequency of the phase rotation being too close to the experimental resolution.
In conclusion, we have shown that determinstically fabricated QD-microlenses with broadband photon extraction are very suitable for the reliable generation of entangled photon pairs. This is demonstrated by two-photon excitation of the biexciton in QD-microleneses where for finite FSS the entanglement fidelty is only limited by the experimental time resolution. Interestingly, the decoherence of XX and X is not affecting their fidelity to the Bell states. These achievements open the possibility of using QDs showing FSS exceeding 10 $\mu$eV in photonic quantum technology schemes, but at the price of an event ''post-selection''. In this respect, the microlenses or other high efficiency broadband nanophotonic elements are of great interest. However, a reduction of the FSS is still of great importance since it allows for the use of less narrow post-selection windows, slower detectors and shorter integration times. Entangled photon pair emission from such optical nanodevices represent a significant step towards the practical and generalized realization of entanglement swapping or teleportation experiments which are key requirements for long-distance quantum communication and photonic quantum computation.
The research leading to these results has received funding from the German Research Foundation via Projects No. RE2974/4-1,
No. RE2974/12-1, SFB 787, and from the German Federal Ministry of Education and Research (BMBF) through the VIP-project
QSOURCE (Grant No. 03V0630).
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,732 |
Lenora (deutsch: Eleonorenhain) ist eine tschechische Gemeinde mit 744 Einwohnern (1. Januar 2017) im Okres Prachatice (Bezirk Prachatitz) im Jihočeský kraj.
Geographie
Gemeindegliederung
Die Gemeinde Lenora besteht aus den Ortsteilen Houžná (Hüblern), Kaplice (Kapellen), Lenora, Vlčí Jámy (Wolfsgrub) und Zátoň (Schattawa). Zu Lenora gehören außerdem die Wohnplätze Dubí (Eichelhäuser), Korýtko (Grandelhäuser), Nová Houžná (Neuhüblern) und Stará Rachota (Beim Wachtfeitl). Grundsiedlungseinheiten sind Houžná, Kaplice, Lenora, Nová Houžná, Vlčí Jámy und Zátoň.
Das Gemeindegebiet gliedert sich in die Katastralbezirke Houžná, Lenora und Vlčí Jámy.
Nachbargemeinden
Geschichte
Der Ort an der Warmen Moldau im Böhmerwald entstand erst 1834 durch den Bau einer Glashütte durch Johann Meyr. Johann Adolf II. Fürst zu Schwarzenberg (1799–1888), dem das Waldgebiet gehörte, gab der Ansiedlung zu Ehren seiner Frau Eleonore den Namen Eleonorenhain. Fürstin Eleonore von Schwarzenberg (1812–1873) war eine Tochter von Moritz von Liechtenstein und Marie Leopoldine Esterházy von Galánthá (1788–1846).
Um 1859 war der Ort eine der größten Glashütten Böhmens. Mit der Vertreibung der deutschsprachigen Bevölkerung nach 1945 wurde die Glashütte, die sich zuletzt im Besitz der Brüder Alfons und Siegfried Kralik befand, verstaatlicht. Viele der Geflüchteten fanden eine neue Heimat bei Glasbetrieben in der Bundesrepublik, u. a. bei Gralglas in Dürnau.
Auf einer Anhöhe befindet sich ein Denkmal zum Gedenken an Andreas Hartauer, der das Lied Tief drin im Böhmerwald schuf.
Der böhmische Komponist Bedřich Smetana hielt sich mehrfach aufgrund seines Tinnitus in Eleonorenhain auf, um sich zu erholen. Sein Aufenthalt In Eleonorenhain ist durch einen Gästebucheintrag des von der Familie Kralik betriebenen Touristenhauses belegt. Man geht davon aus, dass er sich hier in der Nähe der Moldau-Quelle zu seinem Werk Die Moldau inspirieren ließ.
Neben dem Schloss ist das Rechle, eine überdachte Holzbrücke über die Warme Moldau, das bekannteste Bauwerk des Ortes.
Kultur und Sehenswürdigkeiten
Museum
Die im Glasmachermuseum ausgestellten Exponate stammen alle aus der hiesigen Glashütte.
Sport
Lenora ist an den Iron Curtain Trail, den längsten offiziellen Radfernweg Europas, angeschlossen, welcher entlang des ehemaligen Eisernen Vorhangs verläuft.
Regelmäßige Veranstaltungen
Am letzten Sonntag im Juli findet jährlich das Lenorská slavnost chleba (Brotfest) statt.
Persönlichkeiten
Andreas Hartauer, Verfasser des Böhmerwaldliedes, lebte 2 Jahre in Eleonorenhain
Richard Kralik Ritter von Meyrswalden, österreichischer Schriftsteller, hier geboren
Mathilde Kralik von Meyrswalden, österreichische Komponistin
Heinrich Kralik von Meyrswalden, Glasfabrikant in Eleonorenhain, hier geboren
Gottlob Kralik von Meyrswalden, Glasfabrikant in Fürstenberg (Oder), hier geboren
Alfons Kralik Ritter von Meyrswalden, Glasfabrikant in Eleonorenhain, hier geboren
Siegfried Kralik Ritter von Meyrswalden, Glasfabrikant in Eleonorenhain, hier geboren
Bedřich Smetana, Gast in der Sommerfrische des Touristenhauses
Literatur
Helmut Binder, Reinhold Fink: Der Kreis Prachatitz im Böhmerwald in alten Ansichtskarten. Morsak, Grafenau 1992, ISBN 3-87553-399-2.
Elfriede Fink: Do bin i dahoam. Der Böhmerwäldler Egon Urmann aus Eleonorenhain erzählt über sein Leben in Böhmen. Ohetaler, Riedlhütte 2007, ISBN 978-3-937067-82-7.
Fritz Hudler: Eleonorenhain – der aus wilder Wurzel entstandene Glasmacherort im Böhmerwald. Morsak, Grafenau 1989, ISBN 3-87553-346-1
Einzelnachweise
Weblinks
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Böhmerwald | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 4,516 |
\section{Introduction} \label{Sec1}
\noindent
{\bf Background.}
Chemotaxis is the property of cells to move
in a directional manner in response to concentration gradients of
chemical substances.
The system of partial differential equations
describing such the motion of cells
was introduced by Keller--Segel~\cite{KS-1970}, and
is called the chemotaxis system.
The system
\begin{align}\label{KS0}
\begin{cases}
u_t=\nabla \cdot \big( \nabla u
-\chi u\nabla v\big),
\\[1.05mm]
v_t=\Delta v+\alpha u-\beta v
\\[1.05mm]
\end{cases}
\end{align}
is one of many types of the chemotaxis systems and expresses phenomena
caused by the movement of cells as a response
to an attractive chemical substance.
Here the functions $u$ and $v$ idealize the cell density and the concentration of
the chemoattractant, respectively.
After the work~\cite{KS-1970}, there have been many extensive studies
on the chemotaxis systems (see e.g., Osaki--Yagi~\cite{OY-2001},
Bellomo et al.\ \cite{BBTW-2015}, Arumugam--Tyagi~\cite{AT}).
From the point of view of modeling, it is significant to analyze quasilinear systems
such as the system
\begin{align*}
\begin{cases}
u_t=\nabla\cdot \big((u+1)^{m-1}\nabla u
-\chi u(u+1)^{p-2}\nabla v\big),
\\[1.05mm]
v_t=\Delta v+\alpha u-\beta v,
\\[1.05mm]
\end{cases}
\end{align*}
where $m, p\in \R$.
This system has been proposed by Painter--Hillen~\cite{PH-2002}
and has been dealt with by some works
(see e.g., Cie\'slak~\cite{C-2007}, Tao--Winkler~\cite{TW-2012};
cf.\ also \cite{IY-2012}
for the degenerate version of the system).
In the other direction, in order to describe the quorum sensing effect
that cells keep away from a repulsive chemical substance, Painter--Hillen~\cite{PH-2002}
suggested the following attraction-repulsion chemotaxis system which
was also introduced by Luca et al.\ \cite{LCEM-2003}
to describe the aggregation of microglial cells in Alzheimer's disease:
\begin{align}\label{AR}
\begin{cases}
u_t=\nabla \cdot \big( \nabla u
-\chi u\nabla v
+\xi u\nabla w\big),
\\[1.05mm]
v_t=\Delta v+\alpha u-\beta v,
\\[1.05mm]
w_t=\Delta w+\gamma u-\delta w.
\end{cases}
\end{align}
The functions $u$, $v$ and $w$ in \eqref{AR}
represent the cell density, the concentration of
the chemoattractant and chemorepellent, respectively.
The system \eqref{AR} has also been actively studied as detailed in later.
Here we emphasize that it is meaningful to consider the system \eqref{AR}
with diffusion, attraction and repulsion terms involving nonlinearities, that is,
\begin{align}\label{ARquasi}
\begin{cases}
u_t=\nabla\cdot \big((u+1)^{m-1}\nabla u
-\chi u(u+1)^{p-2}\nabla v
+\xi u(u+1)^{q-2}\nabla w\big),
\\[1.05mm]
v_t=\Delta v+\alpha u-\beta v,
\\[1.05mm]
w_t=\Delta w+\gamma u-\delta w.
\end{cases}
\end{align}
In this paper, previous to a mathematical analysis of \eqref{ARquasi},
we will reduce the system to the parabolic--elliptic--elliptic
version.
The reduction seems to be reasonable
because the diffusion of chemical substances
are faster than that of cells.
Thus we can approximate the system \eqref{ARquasi} by its parabolic--elliptic--elliptic
version
\vspace{2.5mm}
\newpage
\noindent
{\bf Problem.}
In this paper, as mentioned above, we consider the quasilinear
parabolic--elliptic--elliptic attraction-repulsion
chemotaxis system
\vspace{-0.5mm}
\begin{align}\label{prob}
\begin{cases}
u_t=\nabla\cdot \big((u+1)^{m-1}\nabla u
-\chi u(u+1)^{p-2}\nabla v
+\xi u(u+1)^{q-2}\nabla w\big)
+f(u),
\\[1.05mm]
0=\Delta v+\alpha u-\beta v,
\\[1.05mm]
0=\Delta w+\gamma u-\delta w,
\\[1.05mm]
\nabla u \cdot \nu|_{\pa \Omega}
=\nabla v \cdot \nu|_{\pa \Omega}
=\nabla w \cdot \nu|_{\pa \Omega}=0,
\\[1.05mm]
u(\cdot, 0)=u_0
\end{cases}
\end{align}
\vspace{-2mm}
\noindent
in a bounded domain $\Omega \subset \R^n$ ($n \in \N$)
with smooth boundary $\pa\Omega$,
where $m, p, q \in \R$, $\chi, \xi, \alpha, \beta, \gamma, \delta>0$
are constants,
$\nu$ is the outward normal vector to $\pa\Omega$,
\vspace{-1mm}
\begin{align}\label{u0}
u_0 \in C^0(\cl{\Omega}),\quad
u_0 \ge 0\ {\rm in}\ \cl{\Omega} \quad {\rm and } \quad
u_0 \neq 0.
\end{align}
\vspace{-6mm}
\noindent
Moreover, we assume that
\vspace{-1mm}
\begin{itemize}
\item $m \in \R$, $f(u) \equiv 0$ in the consideration of boundedness;
\vspace{-1mm}
\item $m>0$, $f(u)=\lambda(|x|)u-\mu(|x|)u^\kappa$ ($\kappa \ge 1$) in the study of blow-up, provided that
\begin{align}
&\Omega=B_R(0) \subset \R^n\ (n \in \N,\ n \ge 3)\ {\rm with}\ R>0, \label{omega}\\[1.5mm]
&\lambda(\cdot),\ \mu(\cdot)\
{\rm are\ nonnegative\ and\ continuous\ functions\ on}\ [0, R],\label{lammu}\\[1.5mm]
&\mu(r) \le \mu_1r^a\ {\rm for\ all}\ r \in [0, R]\ {\rm with\ some}\ \mu_1>0\ {\rm and}\ a \ge 0. \label{mupro}
\end{align}
\end{itemize}
\noindent
{\bf Attraction vs.\ repulsion.}
As to the system \eqref{prob} with $p=q=2$,
it is known that boundedness and blow-up are classified
by the sign of $\chi\alpha-\xi\gamma$ (see e.g., Tao--Wang~\cite{TW-2013}).
Here boundedness
(including global existence),
which expresses that
$\|u(\cdot, t)\|_{L^\infty(\Omega)} \le C$ for all $t>0$ with some $C>0$,
is interpreted as the diffusion of cells,
and that finite-time blow-up (blow-up for short), which means that
$\lim_{t \nearrow T}\|u(\cdot, t)\|_{L^\infty(\Omega)}=\infty$ with some $T \in (0, \infty)$,
implies the concentration of cells.
On the other hand, to the best of our knowledge,
when $p \neq 2$ or $q \neq 2$,
no results are available for boundedness and blow-up in \eqref{prob}.
Here the powers $p, q$ determine the strengths of the effects of attraction,
which promotes blow-up, and repulsion, which induces boundedness.
Thus we can naturally guess as follows.
\vspace{-1mm}
\begin{align*}
\text{Boundedness and blow-up
can be classified by the size of the powers $p, q$.}
\end{align*}
\vspace{-6mm}
\noindent
In the following we discuss this expectation.
As will be explained later, in the case $\xi=0$ in \eqref{prob}
it is known that boundedness holds in the case
\vspace{-1mm}
\begin{align}\label{condipm}
p<m+\frac2n,
\end{align}
\vspace{-5mm}
\noindent
and blow-up occurs in the opposite case.
In view of the first equation in \eqref{prob},
the condition \eqref{condipm} implies that
the effect of diffusion ``plus $\frac{2}{n}$''
is stronger than the one of attraction.
\linebreak[4]
In the case $\xi \neq 0$
the system \eqref{prob} involves the repulsion term
which is expected to work in contrast to the attraction term.
Therefore the question arises whether the repulsion term is useful
for deriving boundedness, that is,
\vspace{2mm}
\makeatletter\tagsleft@true\makeatother
\begin{align}\label{Q1}\tag*{({\bf Q1})}
&\qquad\quad\text{{\it when $p<q$, does boundedness in \eqref{prob} hold
without assuming \eqref{condipm}?}} \notag
\end{align}
\makeatletter\tagsleft@false\makeatother
\vspace{-4mm}
\noindent
In the opposite case $p>q$, we believe that blow-up can be shown
since the effect of attraction is more dominant than that of repulsion,
and we raise the following question
\vspace{2mm}
\makeatletter\tagsleft@true\makeatother
\begin{align}\label{Q2}\tag*{({\bf Q2})}
&\text{{\it When $p>q$, does blow-up in \eqref{prob} occur?}} \notag
\end{align}
\makeatletter\tagsleft@false\makeatother
\vspace{-4mm}
\noindent
Furthermore, in the case $p=q$, where the effects of
attraction and repulsion are balanced,
the following question arises.
\vspace{2mm}
\makeatletter\tagsleft@true\makeatother
\begin{align}\label{Q3}\tag*{({\bf Q3})}
&\text{{\it When $p=q$, are boundedness and blow-up in
\eqref{prob}}}\\
&\text{{\it classified by the condition for the coefficients in the equations?}} \notag
\end{align}
\makeatletter\tagsleft@false\makeatother
\vspace{-4mm}
\noindent
{\bf Overview of related works.}
Before giving answers to the above three questions,
we summarize the previous studies related to each case.
\vspace{2.5mm}
We first focus on the reduced system without repulsion term,
\begin{align}\label{KS}
\begin{cases}
u_t=\nabla\cdot \big((u+1)^{m-1}\nabla u
-\chi u(u+1)^{p-2}\nabla v\big)
+f(u),
\\[1.05mm]
\tau v_t=\Delta v+\alpha u-\beta v,
\end{cases}
\end{align}
where $m, p \in \R$, $\chi, \alpha, \beta>0$, $\tau \in \{0,1\}$ are constants and $f$ is a function.
In the case $\tau=1$, boundedness were shown in
\cite{ISY-2014, TW-2012, WL-2017, Z-2015}.
More precisely,
Tao--Winkler~\cite{TW-2012} derived boundedness when
$\Omega \subset \R^n$ ($n \in \N$) is a convex domain, $f(u) \equiv0$ and $p<m+\frac2n$ holds;
after that, the convexity of $\Omega$ was removed by~\cite{ISY-2014}.
Conversely, finite-time blow-up was obtained under the condition $p>m+\frac2n$
(see e.g., Winkler~\cite{W-2013}, Cie\'slak--Stinner~\cite{CS-2012, CS-2014}).
Besides, in the critical case $p=m+\frac2n$, boundedness and blow-up
were classified by the condition for initial data
(\cite{BL-2013, IY-2012-2, LM-2017, M-2017}).
Also, in the case $f(u) \le \lambda-\mu u^\kappa$ ($\lambda \ge 0$, $\mu>0$, $\kappa>1$),
global existence of classical solutions was established by Zheng~\cite{Z-2015} under
the condition that
$p<\min\{\kappa-1, m+\frac2n\}$, or $p=\kappa$ if $\mu>0$ is sufficiently large.
On the other hand, in the case $\tau=0$,
boundedness were proved in
\cite{LX-2015, SK-2006, WMZ-2014, Z-2015-2}.
Particularly, in the case $\Omega=\R^n$ ($n \in \N$), Sugiyama--Kunii~\cite{SK-2006} demonstrated boundedness of
weak solutions in the system \eqref{KS} of a degenerate type.
Namely, in the literature the authors dealt with the case that
$f(u) \equiv 0$, $m \ge 1$, $p \ge 2$ and $p<\min\{m+1, m+\frac2n\}$.
Also, in the case that $p=2$ and $f(u) \le \lambda-\mu u^\kappa$ ($\lambda \ge 0$, $\mu>0$, $\kappa>1$),
boundedness were verified by~Wang et al.\ \cite{WMZ-2014} under the condition that
$m>2-\frac2n$ if $\kappa \in (1,2)$, or $\mu>\mu^*$ if $\kappa \ge 2$
with some $\mu^*>0$.
In contrast, when $m=1$, $p=2$ and $f(u)=\lambda u-\mu u^{\kappa}$
($\lambda \in \R$, $\mu>0$, $\kappa>1$),
Winkler~\cite{W-2018} established finite-time blow-up;
after that, the result was extended to the cases $p \in (1,2)$, $p=2$ and $p>1$
in~\cite{TY-2020}, \cite{BFL-2021} and \cite{T-2021}, respectively.
Moreover, some related works for the system \eqref{KS}
with nonlinear sensitivity can be found in~\cite{D-2018, F-2015, FNY-2015, JY-2019}.
For instance, when $\tau=1$, $m=1$, $p=2$ and $f(u) \equiv 0$,
Fujie~\cite{F-2015} showed boundedness in
\eqref{KS} with sensitivity function $\frac{\chi}{v}$ under the condition
$0<\chi<\sqrt{\frac{2}{n}}$.
\vspace{2.5mm}
We next shift our focus to the attraction-repulsion system
\begin{align}\label{ARpee}
\begin{cases}
u_t=\nabla \cdot \big(\nabla u
-\chi u\nabla v
+\xi u\nabla w\big)
+f(u),
\\[1.05mm]
0=\Delta v+\alpha u-\beta v,
\\[1.05mm]
0=\Delta w+\gamma u-\delta w,
\end{cases}
\end{align}
where $\chi, \xi, \alpha, \beta, \gamma, \delta>0$.
In the case $f(u)=\lambda u-\mu u^{\kappa}$ ($\lambda\in\R$,
$\mu>0$, $\kappa>1$), finite-time blow-up was recently proved
in~\cite{CMTY} via the method in~\cite{W-2018} when
$\kappa$ is sufficiently closed to $1$ and $\chi\alpha-\xi\gamma>0$ holds.
Moreover, some related works deriving boundedness can be found in~\cite{FS-2019, NSY-2021-2, NSY-2021, NY-2020, NY-2020-2};
showing finite-time blow-up can be cited in~\cite{L-2021};
dealing with nonlinear diffusion and sensitivities can be referred
in~\cite{CMY-2020, LMLW-2017, LMG-2016}.
Particularly, in the two-dimensional setting,
Fujie--Suzuki~\cite{FS-2019} established
boundedness in
the fully parabolic version of \eqref{ARpee}
under the condition that $\beta=\delta$, $\chi\alpha-\xi\gamma>0$ and
$\|u_0\|_{L^1(\Omega)}<\frac{4\pi}{\chi\alpha-\xi\gamma}$;
note that the authors relaxed the condition for $u_0$
in the radially symmetric setting and removed the condition $\beta=\delta$.
Also, Nagai--Yamada~\cite{NY-2020-2} obtained
global existence of solutions under the condition that $\alpha=\gamma=1$,
$\chi-\xi>0$ and $\|u_0\|_{L^1(\Omega)}=\frac{8\pi}{\chi-\xi}$
in the two-dimensional setting; after that,
the authors demonstrated boundedness of solutions in \cite{NY-2020}.
On the other hand, in the three-dimensional and radially symmetric settings,
existence of solutions blowing up in finite time
to the fully parabolic version of \eqref{ARpee}
was shown by Lankeit~\cite{L-2021}
under the conditions that $\chi\alpha-\xi\gamma>0$ and that
$\|u_0\|_{L^1(\Omega)}=M$ with some $M>0$
\vspace{2.5mm}
In summary, the results on boundedness and blow-up in
the system \eqref{prob} were obtained as follows:
Boundedness was derived in the case $\xi=0$
under the condition $p<m+\frac2n$;
blow-up was proved under the condition $\chi\alpha-\xi\gamma>0$.
However, in previous studies, the effect of repulsion
has not been effectively utilized.
The purpose of this paper is to establish boundedness and blow-up
with help of the repulsion term without the above conditions.
\vspace{2.5mm}
\noindent
{\bf Main results.}
Before introducing our results, we mention the expected answers to
the questions \ref{Q1}--\ref{Q3}.
As to the questions \ref{Q1} and \ref{Q2},
we can give affirmative answers.
Also, regard to the question \ref{Q3},
we can classify boundedness and blow-up according to
the sign of $\chi\alpha-\xi\gamma$.
In the following we briefly state the main results which give
the answers to the questions.
The precise statements and their proofs will be given in Sections~\ref{Sec3},~\ref{Sec4}.
\begin{itemize}
\item[({\bf I})] If $p<q$, then, for all initial data,
the system \eqref{prob} possesses
a global bounded classical solution
which is unique (Theorem~\ref{thm_bdd1}).
\item[({\bf I\!I})] If $p=q$ and $\chi\alpha-\xi\gamma<0$,
then, for all initial data,
the system \eqref{prob} admits
a unique global bounded classical solution
(Theorem~\ref{thm_bdd2}).
\item[({\bf I\!I\!I})] If $p>q$, then there exist initial data such that
the corresponding solutions blow up
in finite time in the radial framework
(Theorem~\ref{thm_bu1}).
\item[({\bf I\!V})] If $p=q$ and $\chi\alpha-\xi\gamma>0$,
then there exist initial data such that
the system \eqref{prob} possesses
solutions blow up in finite time in the radial framework
(Theorem~\ref{thm_bu2}).
\end{itemize}
\noindent
{\bf Strategies for proving boundedness and blow-up.}
The strategy for showing boundedness is to establish the differential inequality
\begin{align}\label{bddgoal}
\frac{d}{dt}\int_\Omega (u+1)^\sigma
\le -c_1
\Big(\int_\Omega (u+1)^\sigma\Big)^{1+\theta_1}
+c_2
\end{align}
with some $\sigma>n$, $c_1, c_2, \theta_1>0$.
The key to the derivation of \eqref{bddgoal} is to take advantage of
the effect of repulsion.
More precisely, we will estimate positive terms like $\chi\alpha \int_\Omega u^{\sigma+p-2}$
by the negative term $-\xi\gamma \int_\Omega u^{\sigma+q-2}$.
On the other hand, the cornerstone of the proof of finite-time blow-up is
the derivation of the differential inequality
\begin{align}\label{bugoal}
\frac{\pa \phi}{\pa t}(s_0, t) \ge c_3s_0^{-\theta_2}\phi^2(s_0,t)
-c_4s_0^{\theta_3},
\end{align}
where $c_3, c_4, \theta_2, \theta_3>0$ are constants.
Here the moment-type functional $\phi$ is defined as
$\phi(s_0, t):=\int_0^{s_0}s^{-b}(s_0-s)U(s,t)\,ds$,
where $U$ is the mass accumulation function given by
$U(s,t):=\int_0^{s^{\frac1n}} \rho^{n-1}u(\rho, t)\,d\rho$
for $s>0$, $t>0$ and $b \in (0,1)$.
To derive the inequality \eqref{bugoal} we utilize the attraction term.
More precisely, the key is to handle a term derived from the repulsion term
by exploiting the effect of attraction.
\vspace{2.5mm}
\noindent
{\bf Plan of the paper.}
This paper is organized as follows. In Section~\ref{Sec2} we collect some preliminary facts about local existence
in \eqref{prob} and a lemma guaranteeing an $L^\infty$-estimate from an
$L^\sigma$-estimate for $u$ as well as an inequality which will be used later.
Section~\ref{Sec3} is devoted to establishing results on global existence and
boundedness.
In Section~\ref{Sec4} we give and prove results on finite-time blow-up.
\section{Preliminaries} \label{Sec2}
We first give a result on local classical solutions to \eqref{prob}.
This result can be proved by standard arguments
based on the contraction mapping principle
(see e.g., \cite{CW-2008, TW-2012, TW-2007}).
\begin{lem}\label{lem_local}
Let\/ $\Omega \subset \R^n$ $(n \in \N)$
be a bounded domain with smooth boundary and
let $m, p, q \in \R$, $\chi, \xi, \alpha, \beta, \gamma, \delta>0$.
Assume that $f(u) \equiv 0$ or $f(u)=\lambda u-\mu u^\kappa$
$(\kappa \ge 1)$, where $\lambda, \mu \in C^0(\cl{\Omega})$.
Then for all $u_0$ satisfying the condition \eqref{u0}
there exists $\tmax \in (0,\infty]$ such that \eqref{prob}
admits a unique classical solution $(u, v, w)$ such that
\begin{align}\label{class}
\begin{cases}
u \in C^0(\cl{\Omega} \times [0, \tmax)) \cap
C^{2,1}(\cl{\Omega} \times (0, \tmax)),
\\
v, w \in
\bigcap_{\vartheta>n}C^0([0, \tmax); W^{1,\vartheta}(\Omega))
\cap C^{2,1}(\cl{\Omega} \times (0, \tmax)).
\end{cases}
\end{align}
Moreover,
\begin{align}\label{bc}
{\it if}\ \tmax<\infty, \quad
{\it then}\
\lim_{t \nearrow \tmax}\|u(\cdot, t)\|_{L^\infty(\Omega)}=\infty.
\end{align}
Particularly,
in the case that $f(u)=\lambda(|x|) u-\mu(|x|) u^\kappa$
$(\kappa \ge 1)$ and the conditions
\eqref{omega}, \eqref{lammu} hold,
if $u_0$ is further assumed to be radially symmetric,
then there exists $\tmax \in (0,\infty]$ such that
\eqref{prob} possesses a unique radially symmetric
classical solution $(u, v, w)$ satisfying \eqref{class} and \eqref{bc}.
\end{lem}
We next give the following lemma which provides a strategy
to prove global existence and boundedness.
\begin{lem}\label{lem_Lsig_Linf}
Let\/ $\Omega \subset \R^n$ $(n \in \N)$
be a bounded domain with smooth boundary and
let $m, p, q \in \R$, $\chi, \xi, \alpha, \beta, \gamma, \delta>0$.
Assume that $f(u) \equiv 0$ and
$u_0$ satisfies \eqref{u0}.
Denote by $(u, v, w)$ the local classical solution of \eqref{prob}
given in Lemma~{\rm \ref{lem_local}}
and by $\tmax \in (0,\infty]$ its maximal existence time.
If for some $\sigma>n$,
\begin{align*}
\sup_{t \in (0, \tmax)}\|u(\cdot, t)\|_{L^\sigma(\Omega)}<\infty,
\end{align*}
then we have
\begin{align}\label{Linfty}
\sup_{t \in (0, \tmax)}\|u(\cdot, t)\|_{L^\infty(\Omega)}<\infty.
\end{align}
\end{lem}
\begin{proof}
By the $L^\sigma$-boundedness of $u$,
there exist $c_1>0$ and $\sigma>n$ such that
\begin{align}\label{Lsigma2}
\|u(\cdot,t)\|_{L^\sigma(\Omega)} \le c_1
\end{align}
for all $t \in (0, \tmax)$.
Since $\sigma>n$,
applying \cite[Lemma~2.4~(ii) with $\theta=\sigma$ and $\mu=\infty$]{W-2011} along with \eqref{Lsigma2} yields
\begin{align}
\|\nabla v(\cdot,t)\|_{L^\infty(\Omega)}
&\le c_2\Big(1+\sup_{t \in (0, \tmax)}\|u(\cdot, t)\|_{L^\sigma(\Omega)}\Big)
\le c_3, \label{nabvest}\\
\|\nabla w(\cdot,t)\|_{L^\infty(\Omega)}
&\le c_4\Big(1+\sup_{t \in (0, \tmax)}\|u(\cdot, t)\|_{L^\sigma(\Omega)}\Big)
\le c_5 \label{nabwest}
\end{align}
for all $t \in (0, \tmax)$ with some $c_2, c_3, c_4, c_5>0$.
Thanks to \eqref{Lsigma2}--\eqref{nabwest},
we can see from \cite[Lemma~A.1]{TW-2012} that \eqref{Linfty} holds.
\end{proof}
We finally state an inequality which will be used repeatedly.
\begin{lem}\label{lem_some_ineq}
Let $\ell>1$. Then for all $\ep>0$,
\begin{align}\label{conv}
(x+1)^\ell \le (1+\ep)x^\ell+C_\ep
\quad (x\ge0),
\end{align}
where $C_\ep:=(1+\ep)\big((1+\ep)^{\frac{1}{\ell-1}}-1\big)^{-(\ell-1)}$.
\end{lem}
\begin{proof}
Owing to the convexity of the function $y \mapsto y^\ell$ on $[1,\infty)$
we have
\begin{align*}
(x+1)^\ell
&=\left[
\frac{1}{(1+\ep)^{\frac{1}{\ell-1}}}\cdot (1+\ep)^{\frac{1}{\ell-1}}x
+\left(
1-\frac{1}{(1+\ep)^{\frac{1}{\ell-1}}}
\right) \cdot
\frac{(1+\ep)^{{\frac{1}{\ell-1}}}}{(1+\ep)^{\frac{1}{\ell-1}}-1}
\right]^\ell
\\
&\le \frac{1}{(1+\ep)^{\frac{1}{\ell-1}}}\cdot
\left[
(1+\ep)^{\frac{1}{\ell-1}}x
\right]^\ell
+\left(
1-\frac{1}{(1+\ep)^{\frac{1}{\ell-1}}}
\right) \cdot
\left[
\frac{(1+\ep)^{{\frac{1}{\ell-1}}}}{(1+\ep)^{\frac{1}{\ell-1}}-1}
\right]^\ell
\\
&=(1+\ep)x^\ell
+\frac{1+\ep}{\big((1+\ep)^{\frac{1}{\ell-1}}-1\big)^{\ell-1}},
\end{align*}
which leads to \eqref{conv}.
\end{proof}
\section{Global existence and boundedness}
\label{Sec3}
In this section we assume that $\Omega \subset \R^n$ ($n \in \N$) is a bounded domain
with smooth boundary,
$m, p, q \in \R$, $\chi, \xi, \alpha, \beta, \gamma, \delta>0$, $f(u) \equiv 0$.
We will prove global existence and boundedness in \eqref{prob}
in two cases $p<q$ and $p=q$.
\subsection{The case \boldmath{$p<q$}}
\label{Subsec3.1}
In this subsection we show the following theorem
which asserts global existence and boundedness
in \eqref{prob} in the case $p<q$.
\begin{thm}\label{thm_bdd1}
Assume that $p<q$.
Then for all $u_0$ satisfying \eqref{u0}
there exists a unique triplet $(u, v, w)$
of nonnegative functions
\begin{align*}
\begin{cases}
u \in C^0(\cl{\Omega} \times [0, \infty)) \cap
C^{2,1}(\cl{\Omega} \times (0, \infty)),
\\
v, w \in
\bigcap_{\vartheta>n}C^0([0, \infty); W^{1,\vartheta}(\Omega))
\cap C^{2,1}(\cl{\Omega} \times (0, \infty)),
\end{cases}
\end{align*}
which solves \eqref{prob} in the classical sense,
and is bounded, that is, $\|u(\cdot,t)\|_{L^\infty(\Omega)} \le C$
for all $t>0$ with some $C>0$.
\end{thm}
In the following we denote by $(u, v, w)$ the local classical solution of \eqref{prob}
given in Lemma~{\rm \ref{lem_local}}
and by $\tmax \in (0,\infty]$ its maximal existence time.
To prove Theorem~\ref{thm_bdd1},
it is sufficient to derive
an $L^\sigma$-estimate for $u$ with some
$\sigma>n$,
because Lemma~\ref{lem_Lsig_Linf} leads to an
$L^\infty$-estimate for $u$ which together with the criterion \eqref{bc}
implies the conclusion.
The following lemma plays an important role in the derivation of the $L^\sigma$-estimate.
\begin{lem}\label{lemsub}
Let $\ell>1$.
Then the first and third components of the solution satisfy that
for all $\ep>0$,
\begin{align*}
\int_\Omega w^{\ell} \le \ep\int_\Omega u^{\ell}+c(\ep)\quad {\it on}\ (0, \tmax)
\end{align*}
with some $c(\ep)>0$.
\end{lem}
\begin{proof}
Let $t \in (0, \tmax)$ and put
$u:=u(\cdot, t)$, $w:=w(\cdot, t)$.
Multiplying the third equation in \eqref{prob} by $w^{\ell-1}$
and integrating it over $\Omega$, we obtain
\begin{align*}
\delta \int_\Omega w^{\ell}-\int_\Omega w^{\ell-1}\Delta w
&= \gamma \int_\Omega uw^{\ell-1}.
\end{align*}
Since the second term on the left-hand side is rewritten as
\begin{align*}
-\int_\Omega w^{\ell-1}\Delta w
&= (\ell-1)\int_\Omega w^{\ell-2}|\nabla w|^2 =\frac{4(\ell-1)}{\ell^2}
\int_\Omega\big|\nabla w^{\frac{\ell}{2}}\big|^2,
\end{align*}
we infer
\begin{align}\label{uw2}
\delta \int_\Omega w^{\ell}
+\frac{4(\ell-1)}{\ell^2}
\int_\Omega\big|\nabla w^{\frac{\ell}{2}}\big|^2
&=\gamma \int_\Omega uw^{\ell-1}.
\end{align}
Here we note from the first equation in \eqref{prob}
that the mass conservation
$\int_\Omega u(\cdot, t)=\int_\Omega u_0$ holds
for all $t \in (0, \tmax)$.
Hence, integrating the third equation in \eqref{prob} over $\Omega$ gives
\begin{align}\label{mass}
\int_\Omega w=\frac{\gamma}{\delta}\int_\Omega u=\frac{\gamma}{\delta}\int_\Omega u_0.
\end{align}
Applying the Gagliardo--Nirenberg inequality to $\big\|w^{\frac{\ell}{2}}\big\|_{L^2(\Omega)}$
and using the relation \eqref{mass}, we see that there exist $c_1, c_2>0$ such that
\begin{align}\label{GN}
\big\|w^{\frac{\ell}{2}}\big\|_{L^2(\Omega)}
&\le c_1\Big(\big\|
\nabla w^{\frac{\ell}{2}}
\big\|_{L^2(\Omega)}^{\theta_1}
\big\|w^{\frac{\ell}{2}}
\big\|_{L^{\frac{2}{\ell}}(\Omega)}^{1-\theta_1}
+\big\|w^{\frac{\ell}{2}}
\big\|_{L^{\frac{2}{\ell}}(\Omega)}\Big) \notag
\\
&\le c_2\Big(
\big\|\nabla w^{\frac{\ell}{2}}\big\|_{L^2(\Omega)}^{\theta_1}+1
\Big),
\end{align}
where
$\theta_1:=\frac{\frac{\ell}{2}-\frac{1}{2}}
{\frac{\ell}{2}+\frac{1}{n}-\frac{1}{2}} \in (0,1)$.
Let $\ep>0$ (fixed later).
Then Young's inequality implies that there exists $c_3(\ep)>0$ such that
\begin{align*}
\big\|\nabla w^{\frac{\ell}{2}}\big\|_{L^2(\Omega)}^{\theta_1}
\le \frac{1}{c_2}\sqrt{\frac{\ep}{2}}\big\|
\nabla w^{\frac{\ell}{2}}
\big\|_{L^2(\Omega)}
+c_3(\ep).
\end{align*}
This together with \eqref{GN} yields that
\begin{align*}
\big\|w^{\frac{\ell}{2}}\big\|_{L^2(\Omega)}^2
&\le \Big(\sqrt{\frac{\ep}{2}}\big\|
\nabla w^{\frac{\ell}{2}}
\big\|_{L^2(\Omega)} +c_2(c_3(\ep)+1)\Big)^2\\
&\le
\ep\big\|
\nabla w^{\frac{\ell}{2}}
\big\|_{L^2(\Omega)}^2 +c_4(\ep)
\end{align*}
with some $c_4(\ep)>0$.
Namely, we have
\begin{align}\label{west}
\int_\Omega\big|\nabla w^{\frac{\ell}{2}}\big|^2
\ge \frac{1}{\ep}\int_\Omega w^{\ell}-c_5(\ep)
\end{align}
with some $c_5(\ep)>0$.
Combining \eqref{uw2} with \eqref{west} and using
H$\ddot{{\rm o}}$lder's and Young's inequalities, we derive that
\begin{align*}
\delta \int_\Omega w^{\ell}
+\frac{c_6}{\ep}
\int_\Omega w^{\ell}
&\le \gamma \int_\Omega uw^{\ell-1}
+c_7(\ep) \notag \\
&\le \gamma
\Big(\int_\Omega u^{\ell}\Big)^{\frac{1}{\ell}}
\Big(\int_\Omega w^{\ell}\Big)^{\frac{\ell-1}{\ell}}
+c_7(\ep) \notag \\
&\le \gamma\Big[\frac{1}{\ell}\int_\Omega u^{\ell}
+\Big(1-\frac{1}{\ell}\Big)\int_\Omega w^{\ell}\Big]
+c_7(\ep)
\end{align*}
with some $c_6, c_7(\ep)>0$, and thus infer
\begin{align}\label{west2}
\Big(\delta+\frac{c_6}{\ep}-\gamma+\frac{\gamma}{\ell}\Big)
\int_\Omega w^{\ell}
\le \frac{\gamma}{\ell}\int_\Omega u^{\ell}+c_7(\ep).
\end{align}
We now observe that if $\ep \in (0, \frac{c_6}{\gamma})$ then
$\frac{c_6}{\ep}-\gamma>0$, that is,
\begin{align*}
\delta+\frac{c_6}{\ep}-\gamma+\frac{\gamma}{\ell}>0.
\end{align*}
Therefore, picking $\ep \in (0, \frac{c_6}{\gamma})$,
we have from \eqref{west2} that
\begin{align*}
\int_\Omega w^{\ell} &\le \frac{\frac{\gamma}{\ell}}{\delta+\frac{c_6}{\ep}-\gamma+\frac{\gamma}{\ell}}
\int_\Omega u^{\ell}+
\frac{c_7(\ep)}{\delta+\frac{c_6}{\ep}-\gamma+\frac{\gamma}{\ell}}\notag\\
&= \frac{\frac{\gamma}{\ell}\ep}{(\delta-\gamma+\frac{\gamma}{\ell})\ep+c_6}
\int_\Omega u^{\ell}+
\frac{c_7(\ep)\ep}{(\delta-\gamma+\frac{\gamma}{\ell})\ep+c_6}.
\end{align*}
Noticing that for all $\overline{\ep}>0$ there exists
$\ep \in (0, \frac{c_6}{\gamma})$ such that
$\frac{\frac{\gamma}{\ell}\ep}{(\delta-\gamma+\frac{\gamma}{\ell})\ep+c_6}<\overline{\ep}$,
we arrive at the conclusion.
\end{proof}
We now prove an $L^\sigma$-estimate for $u$.
\begin{lem}\label{lem_Lsig_p<q}
Assume that $p<q$.
Then for some $\sigma>n$ there exists $C>0$ such that
\begin{align*}
\|u(\cdot, t)\|_{L^\sigma(\Omega)} \le C
\end{align*}
for all $t \in (0,\tmax)$.
\end{lem}
\begin{proof}
Let $\sigma>1$ be sufficiently large.
We first obtain from the first equation in \eqref{prob} with $f(u) \equiv 0$ and
integration by parts that
\begin{align}\label{DI1}
&\frac{1}{\sigma}\frac{d}{dt}\int_\Omega (u+1)^\sigma \notag
\\
&\quad\,
=\int_\Omega (u+1)^{\sigma-1}\nabla \cdot
\big((u+1)^{m-1}\nabla u\big) \notag
\\
&\qquad\,\,
-\chi\int_\Omega (u+1)^{\sigma-1}\nabla\cdot
\big(u(u+1)^{p-2}\nabla v\big)
+\xi\int_\Omega (u+1)^{\sigma-1}\nabla\cdot
\big(u(u+1)^{q-2}\nabla w\big) \notag
\\
&\quad\,=
-(\sigma-1)\int_\Omega
(u+1)^{\sigma+m-3}|\nabla u|^2\notag
\\
&\qquad\,\,
+\chi(\sigma-1)\int_\Omega
u(u+1)^{\sigma+p-4}\nabla u\cdot\nabla v
-\xi(\sigma-1)\int_\Omega
u(u+1)^{\sigma+q-4}\nabla u\cdot\nabla w
\notag
\\
&\quad\,=:
I_1+I_2+I_3
\end{align}
for all $t \in (0,\tmax)$.
We estimate the terms $I_1, I_2, I_3$.
As to the first term $I_1$, we rewrite it as
\begin{align}\label{I1}
I_1=-\frac{4(\sigma-1)}{(\sigma+m-1)^2}
\int_\Omega \big|\nabla(u+1)^{\frac{\sigma+m-1}{2}}\big|^2.
\end{align}
We next deal with the second term $I_2$ and third term $I_3$.
As to the former, integration by parts and the second equation in \eqref{prob}
lead to
\begin{align}\label{I21}
I_2 &=\chi(\sigma-1)
\int_\Omega \nabla \Big[\int_0^u s(s+1)^{\sigma+p-4}\,ds\Big]
\cdot \nabla v
\notag
\\
&=\chi(\sigma-1)
\int_\Omega \Big[\int_0^u s(s+1)^{\sigma+p-4}\,ds\Big] \cdot(-\Delta v)
\notag
\\
&=\chi(\sigma-1)
\int_\Omega \Big[\int_0^u s(s+1)^{\sigma+p-4}\,ds\Big]\cdot
(\alpha u-\beta v)
\notag
\\
&\le \chi\alpha(\sigma-1)
\int_\Omega \Big[\int_0^u s(s+1)^{\sigma+p-4}\,ds\Big] u.
\end{align}
Here we infer that for $\sigma>-p+2$,
\begin{align*}
\Big[\int_0^u s(s+1)^{\sigma+p-4}\,ds\Big]u
&\le \Big[\int_0^u (s+1)^{\sigma+p-3}\,ds\Big]u \\
&\le \frac{1}{\sigma+p-2}(u+1)^{\sigma+p-2}u\\
&\le \frac{1}{\sigma+p-2}(u+1)^{\sigma+p-1}.
\end{align*}
Combining the above estimate with \eqref{I21} and
using Lemma~\ref{lem_some_ineq} with $\ep=1$ and $\sigma>-p+2$, we have
\begin{align}\label{I22}
I_2 &\le \frac{\chi\alpha(\sigma-1)}{\sigma+p-2}
\Big(2\int_\Omega u^{\sigma+p-1}+c_1\Big),
\end{align}
with some $c_1>0$.
Similarly, as to the term $I_3$, we establish
\begin{align}\label{I31}
I_3 &=\xi(\sigma-1)
\int_\Omega \Big[\int_0^u s(s+1)^{\sigma+q-4}\,ds\Big] \cdot\Delta w\notag \\
&=\xi(\sigma-1)
\int_\Omega \Big[\int_0^u s(s+1)^{\sigma+q-4}\,ds\Big] \cdot (\delta w-\gamma u).
\end{align}
Here, noting that $s^{\sigma+q-3} \le s(s+1)^{\sigma+q-4} \le (s+1)^{\sigma+q-3}$ for $\sigma\ge-q+4$,
we see that
\begin{align}\label{uplow}
\frac{1}{\sigma+q-2}u^{\sigma+q-2}
\le \int_0^u s(s+1)^{\sigma+q-4}\,ds
\le \frac{1}{\sigma+q-2}(u+1)^{\sigma+q-2},
\end{align}
where we neglected the term
$-\frac{1}{\sigma+q-2}$
on the most right-hand side.
Due to Lemma~\ref{lem_some_ineq} with $\ep=1$ we obtain that
\begin{align}\label{uw01}
\Big[\int_0^u s(s+1)^{\sigma+q-4}\,ds\Big]w
&\le \frac{1}{\sigma+q-2} (u+1)^{\sigma+q-2}w \notag \\
&\le \frac{1}{\sigma+q-2} \Big(2u^{\sigma+q-2}w+c_2w\Big),
\end{align}
with some $c_2>0$.
Therefore a combination of the above estimates
\eqref{I31}--\eqref{uw01} yields that
\begin{align}\label{I32}
I_3 &\le \frac{\xi(\sigma-1)}{\sigma+q-2}
\Big(2\delta\int_\Omega u^{\sigma+q-2}w
+\delta c_2\int_\Omega w
-\gamma\int_\Omega u^{\sigma+q-1}\Big).
\end{align}
Collecting \eqref{I1}, \eqref{I22} and \eqref{I32}
in \eqref{DI1}, we derive
\begin{align}\label{I33}
\frac{1}{\sigma}\frac{d}{dt}\int_\Omega (u+1)^\sigma &\le
-\frac{4(\sigma-1)}{(\sigma+m-1)^2}
\int_\Omega \big|\nabla(u+1)^{\frac{\sigma+m-1}{2}}\big|^2 \notag
\\
&\quad\,
+\frac{\chi\alpha(\sigma-1)}{\sigma+p-2}
\Big(2\int_\Omega u^{\sigma+p-1}+c_1\Big) \notag
\\
&\quad\,
+\frac{\xi(\sigma-1)}{\sigma+q-2}
\Big(2\delta\int_\Omega u^{\sigma+q-2}w
+\delta c_2\int_\Omega w
-\gamma\int_\Omega u^{\sigma+q-1}\Big)
\end{align}
for all $t \in (0,\tmax)$.
Moreover, taking $\ep_1>0$ which will be fixed later and applying Young's inequality to $u^{\sigma+p-1}$,
we have $u^{\sigma+p-1} \le \ep_1 u^{\sigma+q-1}+c_3(\ep_1)$ with some $c_3(\ep_1)>0$.
Additionally, again by the relation \eqref{mass} we see that
\begin{align}\label{DI2}
&\frac{1}{\sigma}\frac{d}{dt}\int_\Omega (u+1)^\sigma
+\frac{4(\sigma-1)}{(\sigma+m-1)^2}
\int_\Omega \big|\nabla(u+1)^{\frac{\sigma+m-1}{2}}\big|^2 \notag
\\
&\quad\, \le
\frac{\chi\alpha(\sigma-1)}{\sigma+p-2}
\Big[2\Big(\ep_1\int_\Omega u^{\sigma+q-1}
+c_3(\ep_1)\Big)+c_1\Big] \notag
\\
&\qquad\,\,
+\frac{\xi(\sigma-1)}{\sigma+q-2}
\Big(2\delta\int_\Omega u^{\sigma+q-2}w
+c_4
-\gamma\int_\Omega u^{\sigma+q-1}\Big)
\end{align}
for all $t \in (0,\tmax)$
with some $c_4>0$.
We next estimate the term $\int_\Omega u^{\sigma+q-2}w$.
Using the H$\ddot{{\rm o}}$lder inequality, we infer
\begin{align*}
\int_\Omega u^{\sigma+q-2}w
\le \Big(\int_\Omega u^{\sigma+q-1}\Big)^{\frac{\sigma+q-2}{\sigma+q-1}}
\Big(\int_\Omega w^{\sigma+q-1}\Big)^{\frac{1}{\sigma+q-1}}.
\end{align*}
Here we take $\ep_2>0$ which will be fixed later.
Employing the Young inequality
as well as applying Lemma~\ref{lemsub} with $\ell=\sigma+q-1$
and $\ep=(\frac{\ep_2}{2})^{\sigma+q-1}$ to
$\int_\Omega w^{\sigma+q-1}$,
we establish
\begin{align}\label{uwest*}
\int_\Omega u^{\sigma+q-2}w
&\le \Big(\int_\Omega u^{\sigma+q-1}\Big)^{\frac{\sigma+q-2}{\sigma+q-1}}
\Big[\Big(\frac{\ep_2}{2}\Big)^{\sigma+q-1}\int_\Omega u^{\sigma+q-1}+c_5(\ep_2)\Big]^{\frac{1}{\sigma+q-1}}\notag\\
&\le \frac{\ep_2}{2}\int_\Omega u^{\sigma+q-1}+c_5(\ep_2)^{{\frac{1}{\sigma+q-1}}}\Big(\int_\Omega u^{\sigma+q-1}\Big)^{\frac{\sigma+q-2}{\sigma+q-1}} \notag\\
&\le \frac{\ep_2}{2}\int_\Omega u^{\sigma+q-1}+c_5(\ep_2)^{{\frac{1}{\sigma+q-1}}}\Big(\frac{\ep_2}{2c_5(\ep_2)^{{\frac{1}{\sigma+q-1}}}}\int_\Omega u^{{\sigma+q-1}}+c_6(\ep_2)\Big) \notag\\
&=\ep_2\int_\Omega u^{\sigma+q-1}+c_7(\ep_2)
\end{align}
with some $c_5(\ep_2), c_6(\ep_2), c_7(\ep_2)>0$.
Setting
$c_8:=\frac{\chi\alpha(\sigma-1)}{\sigma+p-2}$ and
$c_9:=\frac{\xi(\sigma-1)}{\sigma+q-2}$,
we derive from \eqref{DI2} and \eqref{uwest*} that
\begin{align}\label{uest}
&\frac{1}{\sigma}\frac{d}{dt}\int_\Omega (u+1)^\sigma
+\frac{4(\sigma-1)}{(\sigma+m-1)^2}
\int_\Omega \big|\nabla(u+1)^{\frac{\sigma+m-1}{2}}\big|^2 \notag
\\
&\quad\, \le \frac{\chi\alpha(\sigma-1)}{\sigma+p-2}
\Big[2\Big(\ep_1\int_\Omega u^{\sigma+q-1}
+c_3(\ep_1)\Big)+c_1\Big] \notag
\\
&\qquad\,\,
+\frac{\xi(\sigma-1)}{\sigma+q-2}
\Big(2\delta\int_\Omega u^{\sigma+q-2}w
+c_4
-\gamma\int_\Omega u^{\sigma+q-1}\Big) \notag\\
&\quad\, \le 2c_8\ep_1\int_\Omega u^{\sigma+q-1}
+c_9\Big[2\delta\Big(\ep_2\int_\Omega u^{\sigma+q-1}+c_7(\ep_2)\Big)
-\gamma\int_\Omega u^{\sigma+q-1}\Big]+c_{10}(\ep_1) \notag\\
&\quad\,= 2c_8\ep_1\int_\Omega u^{\sigma+q-1}
+c_9(2\delta\ep_2-\gamma)\int_\Omega u^{\sigma+q-1}+c_{11}(\ep_1, \ep_2)
\end{align}
for all $t \in (0,\tmax)$ with some $c_{10}(\ep_1), c_{11}(\ep_1, \ep_2)>0$.
Here we choose $\ep_2>0$ satisfying $\ep_2<\frac{\gamma}{2\delta}$, that is,
$2\delta\ep_2-\gamma<0$.
Then we have from \eqref{uest} that
\begin{align}\label{uest*}
&\frac{1}{\sigma}\frac{d}{dt}\int_\Omega (u+1)^\sigma
+\frac{4(\sigma-1)}{(\sigma+m-1)^2}
\int_\Omega \big|\nabla(u+1)^{\frac{\sigma+m-1}{2}}\big|^2 \notag
\\
&\quad\, \le \big(2c_8\ep_1
-c_9(\gamma-2\delta\ep_2)\big)\int_\Omega u^{\sigma+q-1}+
c_{11}(\ep_1)
\end{align}
for all $t \in (0,\tmax)$.
We let
\begin{align*}
\ep_1:=\frac{c_9(\gamma-2\delta\ep_2)}{2c_8}>0.
\end{align*}
Therefore we obtain from \eqref{uest*} that
\begin{align}\label{DI3}
\frac{1}{\sigma}\frac{d}{dt}\int_\Omega (u+1)^\sigma
+\frac{4(\sigma-1)}{(\sigma+m-1)^2}
\int_\Omega \big|\nabla(u+1)^{\frac{\sigma+m-1}{2}}\big|^2 \le c_{11}
\end{align}
for all $t \in (0,\tmax)$.
We finally estimate the second term on the left-hand side of \eqref{DI3}
in order to derive a differential inequality for $\int_\Omega (u+1)^{\sigma}$.
Again using the Gagliardo--Nirenberg inequality and the mass conservation,
we see that
\begin{align*}
&\|u(\cdot,t)+1\|_{L^\sigma(\Omega)}^\sigma\\
&\quad\,
=\big\|(u(\cdot,t)+1)^{\frac{\sigma+m-1}{2}}\big\|_{L^{\frac{2\sigma}{\sigma+m-1}}(\Omega)}^{\frac{2}{\sigma+m-1}}\\
&\quad\,\le c_{12}\Big(\big\|\nabla(u(\cdot,t)+1)^{\frac{\sigma+m-1}{2}}\big\|_{L^2(\Omega)}^{\theta_2}\big\|(u(\cdot,t)+1)^{\frac{\sigma+m-1}{2}}\big\|_{L^{\frac{2}{\sigma+m-1}}(\Omega)}^{1-\theta_2}\\
&\hspace{7cm}
+\big\|(u(\cdot,t)+1)^{\frac{\sigma+m-1}{2}}\big\|_{L^{\frac{2}{\sigma+m-1}}(\Omega)}\Big)^{\frac{2}{\sigma+m-1}}\\
&\quad\,\le c_{12}\Big(\big\|\nabla(u(\cdot,t)+1)^{\frac{\sigma+m-1}{2}}\big\|_{L^2(\Omega)}^{\frac{2}{\sigma+m-1}\theta_2}\|u(\cdot,t)+1\|_{L^1(\Omega)}^{1-\theta_2}
+\|u(\cdot,t)+1\|_{L^1(\Omega)}\Big)\\
&\quad\,
\le c_{13}\Big(\big\|\nabla(u(\cdot,t)+1)^{\frac{\sigma+m-1}{2}}\big\|_{L^2(\Omega)}^{\frac{2}{\sigma+m-1}\theta_2}+1\Big)
\end{align*}
for all $t \in (0,\tmax)$ with $\theta_2:=\frac{\frac{\sigma+m-1}{2}-\frac{\sigma+m-1}{2\sigma}}{\frac{\sigma+m-1}{2}+\frac1n-\frac12} \in (0,1)$
and $c_{12}, c_{13}>0$.
This implies
\begin{align}\label{GNu}
\big\|\nabla(u(\cdot,t)+1)^{\frac{\sigma+m-1}{2}}\big\|_{L^2(\Omega)}
&\ge \Big(\frac{1}{c_{13}}\|u(\cdot,t)+1\|_{L^\sigma(\Omega)}^\sigma-1\Big)^{\frac{\sigma+m-1}{2\theta_2}} \notag \\
&\ge c_{14}\|u(\cdot,t)+1\|_{L^\sigma(\Omega)}^{\frac{\sigma+m-1}{2\theta_2}}-1
\end{align}
for all $t \in (0,\tmax)$ with some $c_{14}>0$.
A combination of \eqref{DI3} and \eqref{GNu} yields that
\begin{align*}
\frac{1}{\sigma}\frac{d}{dt}\int_\Omega (u+1)^\sigma
+c_{15}
\Big(\int_\Omega (u+1)^\sigma\Big)^{\frac{\sigma+m-1}{2\theta_2}}
&\le c_{16}
\end{align*}
for all $t \in (0,\tmax)$ with some $c_{15}, c_{16}>0$.
Noting from $2\theta_2<2<\sigma+m-1$ for sufficiently large $\sigma$
that $\frac{\sigma+m-1}{2\theta_2}>1$, we infer that
\begin{align*}
\int_\Omega (u+1)^\sigma \le c_{17}
\end{align*}
with some $c_{17}>0$.
This proves the conclusion for all sufficiently large $\sigma>1$.
\end{proof}
We are now in a position to complete the proof of Theorem~\ref{thm_bdd1}.
\begin{prth3.1}
A combination of Lemmas~\ref{lem_Lsig_p<q} and~\ref{lem_Lsig_Linf} along with the criterion
\eqref{bc} leads to the conclusion of Theorem~\ref{thm_bdd1}. \qed
\end{prth3.1}
\subsection{The case \boldmath{$p=q$}}
\label{Subsec3.2}
In this subsection we state the following theorem
guaranteeing global existence and boundedness
in \eqref{prob} in the case $p=q$.
\begin{thm}\label{thm_bdd2}
Assume that $p=q$ and $\chi\alpha-\xi\gamma<0$.
Then for all $u_0$ satisfying \eqref{u0}
there exists a unique triplet $(u, v, w)$
of nonnegative functions
\begin{align*}
\begin{cases}
u \in C^0(\cl{\Omega} \times [0, \infty)) \cap
C^{2,1}(\cl{\Omega} \times (0, \infty)),
\\
v, w \in
\bigcap_{\vartheta>n}C^0([0, \infty); W^{1,\vartheta}(\Omega))
\cap C^{2,1}(\cl{\Omega} \times (0, \infty)),
\end{cases}
\end{align*}
which solves \eqref{prob} in the classical sense,
and is bounded, that is, $\|u(\cdot,t)\|_{L^\infty(\Omega)} \le C$
for all $t>0$ with some $C>0$.
\end{thm}
As in the previous subsection,
we denote by $(u, v, w)$ the local classical solution of \eqref{prob}
given in Lemma~{\rm \ref{lem_local}}
and by $\tmax \in (0,\infty]$ its maximal existence time.
We prove Theorem~\ref{thm_bdd2} by deriving an $L^\sigma$-estimate for $u$.
\begin{lem}\label{lem_Lsig_p=q}
Suppose that $p=q$. Then for some $\sigma>n$ there exists $C>0$ such that
\begin{align*}
\|u(\cdot, t)\|_{L^\sigma(\Omega)} \le C
\end{align*}
for all $t \in (0,\tmax)$.
\end{lem}
\begin{proof}
Let $\sigma>1$ be sufficiently large.
Let $\ep_1>0$ which will be fixed later.
Proceeding similarly in the proof of Lemma~\ref{lem_Lsig_p<q},
we see that \eqref{I33} with $p=q$ holds, that is,
\begin{align*}
\frac{1}{\sigma}\frac{d}{dt}\int_\Omega (u+1)^\sigma&\le
-\frac{4(\sigma-1)}{(\sigma+m-1)^2}
\int_\Omega \big|\nabla(u+1)^{\frac{\sigma+m-1}{2}}\big|^2 \notag
\\
&\quad\,
+\frac{\chi\alpha(\sigma-1)}{\sigma+p-2}
\Big((1+\ep_1)\int_\Omega u^{\sigma+p-1}+c_1(\ep_1)\Big) \notag
\\
&\quad\,
+\frac{\xi(\sigma-1)}{\sigma+p-2}
\Big(2\delta\int_\Omega u^{\sigma+p-2}w
+\delta c_2\int_\Omega w
-\gamma\int_\Omega u^{\sigma+p-1}\Big)
\end{align*}
for all $t \in (0,\tmax)$ with some $c_1(\ep_1), c_2>0$.
Also, setting $c_3:=\frac{\sigma-1}{\sigma+p-2}$ and
recalling the property \eqref{mass}, we have
\begin{align}\label{2DI2}
&\frac{1}{\sigma}\frac{d}{dt}\int_\Omega (u+1)^\sigma
+\frac{4(\sigma-1)}{(\sigma+m-1)^2}
\int_\Omega \big|\nabla(u+1)^{\frac{\sigma+m-1}{2}}\big|^2 \notag
\\
&\quad\,\,\le
\chi\alpha c_3
\Big((1+\ep_1)\int_\Omega u^{\sigma+p-1}+c_1(\ep_1)\Big) \notag
\\
&\qquad\,\,
+\xi c_3
\Big(2\delta\int_\Omega u^{\sigma+p-2}w
+c_4
-\gamma\int_\Omega u^{\sigma+p-1}\Big),
\end{align}
for all $t \in (0,\tmax)$ with some $c_4>0$.
We now take $\ep_2>0$ which will be fixed later.
Then, an argument similar to that in derivation of \eqref{uwest*} implies
\begin{align*}
\int_\Omega u^{\sigma+p-2}w
\le\frac{\ep_2}{2\xi\delta}\int_\Omega u^{\sigma+p-1}+c_5(\ep_2)
\end{align*}
with some $c_5(\ep_2)>0$.
Thus we obtain
\begin{align}\label{2DI3}
&\frac{1}{\sigma}\frac{d}{dt}\int_\Omega (u+1)^\sigma
+\frac{4(\sigma-1)}{(\sigma+m-1)^2}
\int_\Omega \big|\nabla(u+1)^{\frac{\sigma+m-1}{2}}\big|^2 \notag
\\
&\quad\,\,\le
\chi\alpha c_3
\Big((1+\ep_1)\int_\Omega u^{\sigma+p-1}+c_1(\ep_1)\Big) \notag
\\
&\qquad\,\,
+\xi c_3
\Big(2\delta\int_\Omega u^{\sigma+p-2}w
+c_4
-\gamma\int_\Omega u^{\sigma+p-1}\Big) \notag\\
&\quad\,\le
c_3\Big[\chi\alpha(1+\ep_1)\int_\Omega u^{\sigma+p-1}
+2\xi\delta\Big(\frac{\ep_2}{2\xi\delta}\int_\Omega u^{\sigma+p-1}+c_5(\ep_2)\Big)
-\xi\gamma\int_\Omega u^{\sigma+p-1}\Big]\notag\\
&\qquad\,\,+c_6(\ep_1) \notag\\
&\quad\,=c_3\Big[\big(\chi\alpha(1+\ep_1)-\xi\gamma\big)+\ep_2\Big]\int_\Omega u^{\sigma+p-1}+c_7(\ep_1,\ep_2)
\end{align}
for all $t \in (0,\tmax)$ with some $c_6(\ep_1), c_7(\ep_1, \ep_2)>0$.
Here since $\chi\alpha-\xi\gamma<0$ by assumption, we can pick $\ep_1>0$
satisfying $\chi\alpha(1+\ep_1)-\xi\gamma<0$.
Then, taking
\begin{align*}
\ep_2:=\xi\gamma-\chi\alpha(1+\ep_1)>0,
\end{align*}
we have from \eqref{2DI2} and \eqref{2DI3} that
\begin{align*}
\frac{1}{\sigma}\frac{d}{dt}\int_\Omega (u+1)^\sigma
+\frac{4(\sigma-1)}{(\sigma+m-1)^2}
\int_\Omega \big|\nabla(u+1)^{\frac{\sigma+m-1}{2}}\big|^2 \le c_7
\end{align*}
for all $t \in (0,\tmax)$.
Finally, deriving a differential inequality for $\int_\Omega (u+1)^\sigma$
by an argument similar to that in the proof of Lemma~\ref{lem_Lsig_p<q},
we arrive at the conclusion.
\end{proof}
Employing Lemma~\ref{lem_Lsig_p=q}, we can prove Theorem~\ref{thm_bdd2}.
\begin{prth3.3}
In view of Lemmas~\ref{lem_Lsig_p=q} and~\ref{lem_Lsig_Linf} along with the criterion \eqref{bc}, we immediately arrive at
the conclusion of Theorem~\ref{thm_bdd2}. \qed
\end{prth3.3}
\section{Finite-time blow-up}
\label{Sec4}
In the following we suppose that $\Omega=B_R(0) \subset \R^n$
($n \in \N$, $n \ge 3$) with $R>0$ and
$f(u)=\lambda(|x|)u-\mu(|x|)u^\kappa$ ($\kappa \ge 1$),
where $\lambda, \mu$ satisfy the conditions \eqref{lammu} and \eqref{mupro} as well as
$m>0$, $p, q \in \R$, $\chi, \xi, \alpha, \beta, \gamma, \delta>0$.
We also assume that $u_0$ is radially symmetric and fulfills \eqref{u0}.
Then we denote by $(u, v, w)=(u(r,t), v(r,t), w(r,t))$ the local classical solution of \eqref{prob}
given in Lemma~\ref{lem_local} and by $\tmax \in (0, \infty]$ its maximal existence time.
In order to state the main theorems we give the conditions
\ref{C1}--\ref{C3} as follows:
\begin{align}
&\begin{cases}
n \in \{3,4\};\quad m \ge 1,\quad
p<\dfrac{2}{n+1}m+\dfrac{2(n^2+1)}{n(n+1)},
\notag
\\[4mm]
p<-\dfrac{1}{n-2}m+\dfrac{2(n^2-n-1)}{n(n-2)},\quad
m-p<-\dfrac{2}{n};
\tag*{({\bf C1})}\label{C1}
\end{cases}\\[6mm]
&\begin{cases}
n \ge 5;\quad m \ge 1,\quad
-\dfrac{2}{n-3}m+\dfrac{2(n^2-2n-1)}{n(n-3)}
<p<\dfrac{2}{n+1}m+\dfrac{2(n^2+1)}{n(n+1)},
\notag
\\[4mm]
p<-\dfrac{n+2}{n-4}m+\dfrac{3n^2-5n-4}{n(n-4)},\quad
p\le \dfrac{n+2}{3}m-\dfrac{n^2-3n-4}{3n};
\tag*{({\bf C2})}\label{C2}
\end{cases}
\\[6mm]
&\begin{cases}
n \ge 5;\quad m \ge 1,\quad
-\dfrac{2}{n-3}m+\dfrac{2(n^2-2n-1)}{n(n-3)}
<p<\dfrac{2}{n+1}m+\dfrac{2(n^2+1)}{n(n+1)},
\notag
\\[4mm]
-\dfrac{n+2}{n-4}m+\dfrac{3n^2-5n-4}{n(n-4)}
\le p<-\dfrac{1}{n-2}m+\dfrac{2(n^2-n-1)}{n(n-2)},
\\[4mm]
m-p<-\dfrac{2}{n}.
\tag*{({\bf C3})}\label{C3}
\end{cases}
\end{align}
\subsection{The case \boldmath{$p>q$}}
\label{Subsec4.1}
In this subsection we establish finite-time blow-up
in \eqref{prob} in the case $p>q$.
\begin{thm}\label{thm_bu1}
Assume that $p>q$.
Also, suppose that $m>0$, $\kappa \ge 1$ fulfill the following conditions\/{\rm :}
\begin{itemize}
\item[{\rm (i)}] In the case {\rm \ref{C1}},
\begin{align*}
\kappa<1+\frac{(n-2)\big((m-p+1)n+1\big)}{n(n-1)}+\frac{a\big((m-p+1)n+1\big)}{n(n-1)}-(m-1)-(2-p)_+;
\end{align*}
\item[{\rm (ii)}] In the case {\rm \ref{C2}},
\begin{align*}
\kappa<1+\frac{(n-2)\big((m-p+1)n+1\big)}{n(n-1)}+\frac{a\big((m-p+1)n+1\big)}{n(n-1)}-(m-1)-(2-p)_+;
\end{align*}
\item[{\rm (iii)}] In the case {\rm \ref{C3}},
\begin{align*}
\kappa<1+\frac{(m-p+1)n+1}{2(n-1)}+\frac{a\big((m-p+1)n+1\big)}{n(n-1)}-\frac{(2-p)_+}{2},
\end{align*}
\end{itemize}
where $a\ge0$ is given in \eqref{mupro} and
$y_+:=\max\{0, y\}$.
Let $M_0>0$, $M_1 \in (0, M_0)$ and $L>0$.
Then one can find $\ep_0>0$ and $r_1 \in (0, R)$ with the following property\/{\rm :}
If $u_0$ satisfies
$u_0(x) \le L|x|^{-\sigma}$, where
$\sigma=\frac{n(n-1)}{(m-p+1)n+1}+\ep_0$ as well as
$\int_\Omega u_0=M_0$ and $\int_{B_{r_1}(0)}u_0 \ge M_1$,
then the solution $(u, v, w)$ to \eqref{prob} blows up at $t=T^* \in (0, \infty)$
in the sense that
\begin{align*}
\lim_{t \nearrow T^*}\|u(\cdot, t)\|_{L^\infty(\Omega)}=\infty.
\end{align*}
\end{thm}
We first show the following lemma giving the profile of $u$, in which we include the case $p=q$ toward the next subsection.
\begin{lem}\label{lem_profile}
Assume that $p \ge q$.
Also, suppose that $m>0$ and $p>1$ fulfill
\begin{align*}
m \ge 1,\quad m-p\in \Big(-1-\frac{1}{n},\ -\frac{2}{n}\Big].
\end{align*}
Let $M_0>0$, $L>0$ and $T>0$.
Let $\ep>0$ and set $\sigma:=\frac{n(n-1)}{(m-p+1)n+1}+\ep$.
Then there exists $C>0$ such that the following property holds\/{\rm :}
If $u_0$ satisfies $\int_\Omega u_0=M_0$ and
\begin{align*}
u_0(x) \le L|x|^{-\sigma}
\end{align*}
for all $x \in \Omega$, then the classical solution $(u, v, w) \in \big(C^0(\cl{\Omega} \times [0,T)) \cap C^{2,1}(\cl{\Omega} \times (0,T))\big)^3$ of \eqref{prob} has the estimate
\begin{align}\label{profile2}
u(x,t) \le C|x|^{-\sigma}
\end{align}
for all $x \in \Omega$ and all $t \in (0,T)$.
\end{lem}
\begin{proof}
By the condition for the function $\lambda$ (see \eqref{lammu}),
we see that there exists $\lambda_1>0$
such that $\lambda(|x|) \le \lambda_1$ for all $x \in \Omega$.
We next set
\begin{align*}
\widetilde{u}(x,t)&:=e^{-\lambda_1t}u(x,t),\quad
D(x,t,\rho):=(e^{\lambda_1t}\rho+1)^{m-1},\\
S_1(x,t,\rho)&:=-\chi(e^{\lambda_1t}\rho+1)^{p-2}\rho,\quad
S_2(x,t,\rho):=\xi(e^{\lambda_1t}\rho+1)^{q-2}\rho
\end{align*}
for $x \in \Omega$, $t \in (0,T)$ and $\rho>0$.
Since $S_1(\cdot, \cdot, \cdot)<0$ on $\Omega \times (0,T) \times (0,\infty)$, we have
\begin{align*}
S_1(x,t,\rho)\nabla v(x,t)+S_2(x,t,\rho)\nabla w(x,t)
&=S_1(x,t,\rho)\Big[\nabla v(x,t)+\frac{S_2(x,t,\rho)}{S_1(x,t,\rho)}\nabla w(x,t)\Big]
\end{align*}
for all $x \in \Omega$, $t \in (0,T)$ and all $\rho>0$.
Putting
\begin{align*}
\mathbf{f}(x,t):=\nabla v(x,t)+\frac{S_2(x,t,\rho)}{S_1(x,t,\rho)}\nabla w(x,t),
\end{align*}
we obtain from \eqref{prob} that
\begin{align}\label{P}
\begin{cases}
\widetilde{u}_t \le \nabla \cdot (D(x,t,\widetilde{u})\nabla\widetilde{u}
+S_1(x,t,\widetilde{u})\,\mathbf{f}(x,t))
& {\rm in}\ \Omega \times (0,T),\\
(D(x,t,\widetilde{u})\nabla\widetilde{u}
+S_1(x,t,\widetilde{u})\,\mathbf{f}(x,t)) \cdot \nu=0
& {\rm on}\ \pa\Omega \times (0,T),\\
\widetilde{u}(\cdot,0)=u_0 & {\rm in}\ \Omega.
\end{cases}
\end{align}
Also, it can be checked that for all $x \in \Omega$, $t \in (0,T)$ and all $\rho>0$,
\begin{align*}
D(x,t,\rho)&\ge \rho^{m-1},\\
D(x,t,\rho)&\le (e^{\lambda_1T}\rho+1)^{m-1}
\le (e^{\lambda_1T}+1)^{m-1}\max\{\rho, 1\}^{m-1},\\
|S_1(x,t,\rho)| &\le \chi(e^{\lambda_1T}+1)^{p-1}\max\{\rho, 1\}^{p-1}.
\end{align*}
Moreover, the initial condition in \eqref{P} implies that
$\int_\Omega \widetilde{u}(\cdot, 0)=\int_\Omega u_0=M_0$.
Here we choose $\theta>n$ satisfying
\begin{align*}
m-p \in \Big(\frac{1}{\theta}-1-\frac{1}{n},\ \frac{1}{\theta}-\frac{2}{n}\Big]
\end{align*}
and
\begin{align*}
\sigma&=\frac{n(n-1)}{(m-p+1)n+1}+\ep\\
&>\frac{n(n-1)}{(m-p+1)n+1-\frac{n}{\theta}}
=\frac{n-1}{(m-p)+1+\frac{1}{n}-\frac{1}{\theta}}.
\end{align*}
Since $p \ge q$ and
\begin{align*}
\Big|\frac{S_2(x,t,\rho)}{S_1(x,t,\rho)}\Big|
&=\frac{\xi(e^{\lambda_1t}\rho+1)^{q-2}\rho}{\chi(e^{\lambda_1t}\rho+1)^{p-2}\rho}
=\frac{\xi}{\chi}(e^{\lambda_1t}\rho+1)^{q-p}
\le \frac{\xi}{\chi},
\end{align*}
for all $x \in \Omega$, $t \in (0,T)$ and all $\rho>0$,
following the steps in the proof of \cite[Lemma~5.2]{BFL-2021},
we establish
\begin{align*}
\int_\Omega |x|^{(n-1)\theta}|\mathbf{f}(x,t)|^\theta\,dx
&\le c_1\Big(\frac{\alpha}{\beta}+\frac{\xi}{\chi}\cdot\frac{\gamma}{\delta}\Big)^\theta\Big(\frac{2e^{\lambda_1T}M_0}{\omega_{n-1}}\Big)^\theta |\Omega|
\end{align*}
for all $t \in (0,T)$ with some $c_1>0$, where $\omega_{n-1}$ denotes the $(n-2)$-dimensional surface area of the unit sphere in $\R^{n-1}$.
Thanks to \cite[Theorem~1.1]{F-2020}, we derive that there exists $c_2>0$ such that
$\widetilde{u}(x,t) \le c_2|x|^{-\sigma}$ for all $x \in \Omega$ and all $t \in (0,T)$.
This leads to the end of the proof.
\end{proof}
We now introduce the mass accumulation functions $U=U(s, t), V=V(s,t)$ and $
W=W(s,t)$
as follows:
\begin{align}
U(s,t)&:=\int_0^{s^{\frac1n}} \rho^{n-1}u(\rho, t)\,d\rho,\label{Udef}\\
V(s,t)&:=\int_0^{s^{\frac1n}} \rho^{n-1}v(\rho, t)\,d\rho\\
\intertext{and}
W(s,t)&:=\int_0^{s^{\frac1n}} \rho^{n-1}w(\rho, t)\,d\rho, \label{Wdef}
\end{align}
where $s:=r^n$ for $r \in [0, R]$ and $t \in [0, \tmax)$.
We next define the moment-type functional
\begin{align}\label{phi}
\phi(s_0, t):=\int_0^{s_0}s^{-b}(s_0-s)U(s,t)\,ds
\end{align}
for $s_0 \in (0, R^n)$, $t \in [0, \tmax)$ and $b \in (0,1)$.
\begin{lem}\label{lem_DI_p>q}
Assume that $p>q$.
Let $\mu_1>0$, $\kappa \ge 1$,
$a \ge 0$ and $T>0$.
Then there exist $C_1, C_2>0$ such that
\begin{align}\label{phiDI}
\frac{\pa \phi}{\pa t}(s_0, t) \ge &\
C_1\int_0^{s_0}s^{-b}(s_0-s)(nU_s(s,t)+1)^{p-2}U(s,t)U_s(s,t)\,ds \notag\\
&+n^2\int_0^{s_0}s^{2-\frac2n-b}(s_0-s)(nU_s(s,t)+1)^{m-1}U_{ss}(s,t)\,ds \notag\\
&-\chi\beta n\int_0^{s_0}s^{-b}(s_0-s)(nU_s(s,t)+1)^{p-2}V(s,t)U_s(s,t)\,ds \notag\\
&-n^{\kappa-1}\mu_1\int_0^{s_0}s^{-b}(s_0-s)\Big[\int_0^{s_0}\eta^{\frac an}U_s^\kappa(\eta,t)\,d\eta\Big]\,ds-C_2\phi(s_0,t)
\end{align}
for all $s_0 \in (0, R^n)$ and all $t \in (0, \min\{T, \tmax\})$.
\end{lem}
\begin{proof}
The first equation in \eqref{prob} implies that
$u=u(r,t)$, $v=v(r,t)$, $w=w(r,t)$
satisfy
\begin{align}\label{ueq}
u_t
&=\frac{1}{r^{n-1}}\big((u+1)^{m-1}r^{n-1}u_r\big)_r
-\chi \frac{1}{r^{n-1}}\big(u(u+1)^{p-2}r^{n-1}v_r\big)_r\notag\\
&\quad\,
+\xi \frac{1}{r^{n-1}}\big(u(u+1)^{q-2}r^{n-1}w_r\big)_r+\lambda u
-\mu u^\kappa.
\end{align}
Moreover, the second and third equations in \eqref{prob} yield that
\begin{align}
r^{n-1}v_r(r,t)&=\beta V(r^n, t)-\alpha U(r^n,t), \label{Veq}\\
r^{n-1}w_r(r,t)&=\delta W(r^n, t)-\gamma U(r^n,t) \label{Weq}
\end{align}
for all $r \in (0,R)$ and all $t \in (0,\tmax)$.
Integrating \eqref{ueq} combined with \eqref{Veq} and \eqref{Weq} with respect to $r$ over $[0, s^{\frac1n}]$,
we see from the nonnegativity of $\lambda$ and \eqref{mupro} that
\begin{align}\label{UDI}
U_t \ge&\ n^2s^{2-\frac 2n}(nU_s+1)^{m-1}U_{ss}\notag\\
&+\chi nU_s(nU_s+1)^{p-2}(\alpha U-\beta V)-\xi nU_s(nU_s+1)^{q-2}(\gamma U-\delta W)\notag\\
&-n^{\kappa -1}\mu_1\int_0^s\eta^{\frac an}U_s^\kappa(\eta,t)\,d\eta
\end{align}
for all $s \in (0,R^n)$ and all $t \in (0,\tmax)$.
Combining \eqref{phi} and \eqref{UDI}, we obtain
\begin{align}\label{phiDI2}
\frac{\pa \phi}{\pa t}(s_0, t) &\ge
\chi\alpha n\int_0^{s_0}s^{-b}(s_0-s)(nU_s(s,t)+1)^{p-2}U(s,t)U_s(s,t)\,ds \notag\\
&\quad\,-\xi\gamma n\int_0^{s_0}s^{-b}(s_0-s)(nU_s(s,t)+1)^{q-2}U(s,t)U_s(s,t)\,ds \notag\\
&\quad\,+n^2\int_0^{s_0}s^{2-\frac2n-b}(s_0-s)(nU_s(s,t)+1)^{m-1}U_{ss}(s,t)\,ds \notag\\
&\quad\,-\chi\beta n\int_0^{s_0}s^{-b}(s_0-s)(nU_s(s,t)+1)^{p-2}V(s,t)U_s(s,t)\,ds \notag\\
&\quad\,+\xi\delta n\int_0^{s_0}s^{-b}(s_0-s)(nU_s(s,t)+1)^{q-2}W(s,t)U_s(s,t)\,ds \notag\\
&\quad\,-n^{\kappa-1}\mu_1\int_0^{s_0}s^{-b}(s_0-s)\Big[\int_0^{s_0}\eta^{\frac an}U_s^\kappa(\eta,t)\,d\eta\Big]\,ds\notag\\
&=: J_1-J_2+J_3-J_4+J_5-J_6
\end{align}
for all $s_0 \in (0, R^n)$ and all $t \in (0, \min\{T, \tmax\})$.
Here we estimate the term $J_2$.
We first consider the case $q>1$.
In this case,
using Young's inequality, we see that
for all $\ep_1>0$ there exists $c_1(\ep_1)>0$ such that
\begin{align}\label{key1}
(nU_s(s,t)+1)^{q-2}U_s(s,t)
&\le \ep_1\Big[(nU_s(s,t)+1)^{(q-1)-1}U_s(s,t) \Big]^{\frac{p-1}{q-1}}+c_1(\ep_1) \notag\\
&=\ep_1(nU_s(s,t)+1)^{p-1-\frac{p-1}{q-1}}U_s^{\frac{p-1}{q-1}}(s,t) +c_1(\ep_1).
\end{align}
Here we notice from the relation
$\frac{p-1}{q-1}>1$ by $p>q>1$ that
\begin{align}\label{key2}
U_s^{\frac{p-1}{q-1}}(s,t)
=U_s^{\frac{p-1}{q-1}-1}(s,t)U_s(s,t) \le (nU_s(s,t)+1)^{\frac{p-1}{q-1}-1}U_s(s,t).
\end{align}
A combination of \eqref{key1} and \eqref{key2} implies that
\begin{align}\label{J2est3}
(nU_s(s,t)+1)^{q-2}U_s(s,t)&\le\ep_1(nU_s(s,t)+1)^{p-2}U_s(s,t)+c_1(\ep_1).
\end{align}
In the case $q\le 1$, noting that
\begin{align*}
(nU_s(s,t)+1)^{q-2} U_s(s,t)
\le (nU_s(s,t)+1)^{-1}U_s(s,t)
\le n^{-1},
\end{align*}
we can choose $\ep_1=0$ and $c_1(\ep_1)=n^{-1}$ in the estimate \eqref{J2est3}.
In view of \eqref{J2est3} we obtain
\begin{align}\label{J2est4}
J_2 &=\xi\gamma n\int_0^{s_0}s^{-b}(s_0-s)(nU_s(s,t)+1)^{q-2}U(s,t)U_s(s,t)\,ds\notag\\
&\le \ep_1\xi\gamma n\int_0^{s_0}s^{-b}(s_0-s)(nU_s(s,t)+1)^{p-2}U(s,t)U_s(s,t)\,ds\notag\\
&\quad\,+c_1(\ep_1)\int_0^{s_0}s^{-b}(s_0-s)U(s,t)\,ds \notag\\
&=\ep_1\xi\gamma n\int_0^{s_0}s^{-b}(s_0-s)(nU_s(s,t)+1)^{p-2}U(s,t)U_s(s,t)\,ds+c_1(\ep_1)\phi(s_0,t).
\end{align}
Combining \eqref{J2est4} with \eqref{phiDI2} and noting that $J_5 \ge 0$, we establish
\begin{align*}
\frac{\pa \phi}{\pa t}(s_0, t) &\ge
(\chi\alpha-\ep_1\xi\gamma) n\int_0^{s_0}s^{-b}(s_0-s)(nU_s(s,t)+1)^{p-2}U(s,t)U_s(s,t)\,ds \notag\\
&\quad\,+n^2\int_0^{s_0}s^{2-\frac2n-b}(s_0-s)(nU_s(s,t)+1)^{m-1}U_{ss}(s,t)\,ds \notag\\
&\quad\,-\chi\beta n\int_0^{s_0}s^{-b}(s_0-s)(nU_s(s,t)+1)^{p-2}V(s,t)U_s(s,t)\,ds \notag\\
&\quad\,-n^{\kappa-1}\mu_1\int_0^{s_0}s^{-b}(s_0-s)\Big[\int_0^{s_0}\eta^{\frac an}U_s^\kappa(\eta,t)\,d\eta\Big]\,ds\notag\\
&\quad\,-c_1(\ep_1)\phi(s_0,t)
\end{align*}
for all $s_0 \in (0, R^n)$ and all $t \in (0, \min\{T, \tmax\})$.
Here, choosing $\ep_1:=\frac{\chi\alpha}{2\xi\gamma}$ when $q>1$ and recalling that
$\ep_1=0$ when $q \le1$, we see that $\chi\alpha-\ep_1\xi\gamma>0$,
which means that the desired inequality
\eqref{phiDI} holds.
\end{proof}
\begin{prth4.1}
Let $\sigma:=\frac{n(n-1)}{(m-p+1)n+1}+\ep$ with some $\ep>0$ (fixed later)
and let $u_0(x) \le L|x|^{-\sigma}$
for all $x \in \Omega$.
Then Lemma~\ref{lem_profile} implies that
\eqref{profile2} holds:
$u(x,t) \le C|x|^{-\sigma}$ for all $x \in \Omega$ and all $t \in (0,T)$ with some $C>0$.
Also, to estimate the first four terms on
the right-hand side of \eqref{phiDI}
we follow the steps in \cite[Lemmas~3.3,~3.6,~3.7~and~3.9]{T-2021}.
Employing those estimates in our case,
we have that there exist $c_1, c_2, c_3>0$ such that
\begin{align*}
\frac{\pa \phi}{\pa t}(s_0, t)
&\ge c_1\psi_p(s_0,t)\notag\\
&\quad\, -c_2s_0^{\frac{3-b}{2}-\frac2n-\frac{\sigma}{2n}[2(m-1)_++(2-p)_+]}\sqrt{\psi_p(s_0,t)}-c_2s_0^{3-\frac2n-b}\notag\\
&\quad\,-c_2s_0^{\frac2n+\frac{1-b}{2}-\frac{\sigma}{2n}[(2-p)_++2(p-2)_+]}\sqrt{\psi_p(s_0,t)}-c_2s_0^{\frac2n-\frac{\sigma}{n}[(2-p)_++(p-2)_+]}\psi_p(s_0,t)\notag\\
&\quad\, -c_2s_0^{\frac{3-b}{2}+\frac an-\frac{\sigma}{2n}[2(\kappa-1)+(2-p)_+]}\sqrt{\psi_p(s_0,t)}\notag\\
&\quad\,-c_3\phi(s_0,t)
\end{align*}
for all $s_0 \in (0, R^n)$ and all $t \in (0, \min\{T, \tmax\})$,
where
\begin{align*}
\psi_p(s_0,t):=\int_0^{s_0}s^{-b+\frac{\sigma}{n}(2-p)_+}(s_0-s)U(s,t)U_s(s,t)\,ds
\end{align*}
for $s_0 \in (0, R^n)$ and $t \in [0, \tmax)$.
We take $\ep_1>0$ which will be fixed later.
Using the Young inequality, we can see that
there exists $c_4(\ep_1)>0$ such that
\begin{align}\label{phiDI**}
\frac{\pa \phi}{\pa t}(s_0, t)
&\ge c_1\psi_p(s_0,t)-\ep_1\psi_p(s_0, t)
-c_2s_0^{\frac2n-\frac{\sigma}{n}[(2-p)_++(p-2)_+]}\psi_p(s_0,t)\notag\\
&\quad\, -c_4(\ep_1)\Big(s_0^{3-b-\frac4n-\frac \sigma n[2(m-1)_++(2-p)_+]}+s_0^{2-\frac2n-b}\notag\\
&\qquad\qquad\quad\ +s_0^{\frac4n+1-b-\frac\sigma n[(2-p)_++2(p-2)_+]}+s_0^{3-b+\frac{2a}{n}-\frac\sigma n[2(\kappa-1)+(2-p)_+]}\Big)\notag\\
&\quad\,-c_3\phi(s_0,t)
\end{align}
for all $s_0 \in (0, R^n)$ and all $t \in (0, \min\{T, \tmax\})$.
We now pick $s_1 \in (0, R^n)$ small enough such that
\begin{align*}
c_2s_0^{\frac2n-\frac{\sigma}{n}[(2-p)_++(p-2)_+]}\psi_p(s_0,t)
\le \frac{1}{4}c_1\psi_p(s_0,t)
\end{align*}
for all $s_0 \in (0, s_1)$ and all $t \in (0, \min\{T, \tmax\})$,
and set $\ep_1:=\frac{c_1}{4}$.
Then we have from \eqref{phiDI**} that
\begin{align*}
\frac{\pa \phi}{\pa t}(s_0, t)
&\ge \frac{1}{2}c_1\psi_p(s_0,t)\notag\\
&\quad\, -c_4\Big(s_0^{3-b-\frac4n-\frac \sigma n[2(m-1)_++(2-p)_+]}+s_0^{2-\frac2n-b}\notag\\
&\qquad\quad\ \ +s_0^{\frac4n+1-b-\frac\sigma n[(2-p)_++2(p-2)_+]}+s_0^{3-b+\frac{2a}{n}-\frac\sigma n[2(\kappa-1)+(2-p)_+]}\Big)\notag\\
&\quad\,
-c_3\phi(s_0,t)
\end{align*}
for all $s_0 \in (0, s_1)$ and all $t \in (0, \min\{T, \tmax\})$.
By an argument similar to that in the proof of \cite[Lemma~4.3]{T-2021},
thanks to the conditions \ref{C1}--\ref{C3},
we can pick $\ep_0>0$ and then
for $\sigma=\frac{n(n-1)}{(m-p+1)n+1}+\ep_0$
there exist $c_5, c_6>0$ and $\theta \in (0,\ 2-\frac{\sigma}{n}(2-p)_+)$
such that
\begin{align}\label{phiDI***}
\frac{\pa \phi}{\pa t}(s_0, t) \ge \frac{1}{2}c_1\psi_p(s_0,t)
-c_5s_0^{3-b-\theta}-c_6\phi(s_0,t).
\end{align}
Applying the estimate
$\sqrt{\psi_p(s_0,t)} \ge c_7s_0^{\frac{b-3}{2}+\frac{\sigma}{2n}(2-p)_+}\phi(s_0,t)$
with some $c_7>0$
(see \cite[Lemma~3.10]{T-2021})
to the first term on the right-hand side of
\eqref{phiDI***}, we have
\begin{align}\label{phiDIa}
\frac{\pa \phi}{\pa t}(s_0, t) \ge c_7s_0^{b-3+\frac \sigma n (2-p)_+}\phi^2(s_0,t)
-c_5s_0^{3-b-\theta}-c_6\phi(s_0,t)
\end{align}
for all $s_0 \in (0, s_1)$ and all $t \in (0, \min\{T, \tmax\})$.
Again by Young's inequality, we derive that
there exists $c_8>0$ such that
\begin{align}\label{phiDIb}
c_6\phi(s_0,t)
\le \frac{1}{2}c_7s_0^{b-3+\frac \sigma n (2-p)_+}\phi^2(s_0,t)
+c_8s_0^{3-b-\frac \sigma n (2-p)_+}.
\end{align}
A combination of \eqref{phiDIa} and \eqref{phiDIb}
yields
\begin{align}\label{DIgoal}
\frac{\pa \phi}{\pa t}(s_0, t) &\ge \frac{1}{2}c_7s_0^{b-3+\frac \sigma n (2-p)_+}\phi^2(s_0,t)
-c_5s_0^{3-b-\theta}-c_8s_0^{3-b-\frac \sigma n (2-p)_+}\notag\\
&\ge \frac{1}{2}c_7s_0^{b-3+\frac \sigma n (2-p)_+}\phi^2(s_0,t)
-c_9s_0^{\widetilde{\theta}}
\end{align}
for all $s_0 \in (0, s_1)$ and all $t \in (0, \min\{T, \tmax\})$
with some $c_9>0$ and
$\widetilde{\theta}=\min\{3-b-\theta,\ 3-b-\frac \sigma n (2-p)_+\}$.
Here, by the conditions \ref{C1}--\ref{C3}, we can take $b \in (0, 1)$ satisfying
\begin{align*}
b<2-\frac4n-\frac \sigma n[2(m-1)_++(2-p)_+]
\end{align*}
(see \cite[Lemma~4.1]{T-2021}).
This yields that
\begin{align*}
b-3+\frac \sigma n (2-p)_+
&<\Big\{2-\frac4n-\frac \sigma n[2(m-1)_++(2-p)_+]\Big\}-3+\frac \sigma n (2-p)_+\notag\\
&=-1-\frac 4n-\frac{2\sigma}{n}(m-1)_+<0\\
\intertext{and moreover, recalling the choice that
$\theta \in (0,\ 2-\frac{\sigma}{n}(2-p)_+)$, we have}
3-b-\theta
&>3-b-\Big[2-\frac{\sigma}{n}(2-p)_+\Big]\notag\\
&=1-b+\frac{\sigma}{n}(2-p)_+>0,
\end{align*}
which lead to $\widetilde{\theta}>0$.
Taking into account
the proof of \cite[Theorem~1.1]{W-2018} or
\cite[Theorem~1.1]{BFL-2021},
we obtain $\tmax<T<\infty$,
which implies that Theorem~\ref{thm_bu1} holds
by virtue of the criterion \eqref{bc}. \qed
\end{prth4.1}
\subsection{The case \boldmath{$p=q$}}
\label{Subsec4.2}
In this subsection we state the following theorem
which guarantees finite-time blow-up
in \eqref{prob} in the case $p=q$.
\begin{thm}\label{thm_bu2}
Assume that $p=q$ and $\chi\alpha-\xi\gamma>0$.
Moreover, suppose that $m$, $p$ and $\kappa$ fulfill
the same conditions as in
Theorem~{\rm \ref{thm_bu1}}.
Let $M_0>0$, $M_1 \in (0, M_0)$ and $L>0$. Then the conclusion of Theorem~{\rm \ref{thm_bu1}} holds.
\end{thm}
In order to prove the above theorem we show the following lemma giving the pointwise lower estimate for $\frac{\pa \phi}{\pa t}$,
where $U, V, W$ and $\phi$ are defined
as in \eqref{Udef}--\eqref{Wdef} and \eqref{phi},
respectively.
\begin{lem}\label{lem_DI_p=q}
Suppose that $p=q$.
Let $\mu_1>0$, $\kappa \ge 1$,
$a \ge 0$ and $T>0$.
Then there exist $C_1, C_2>0$ such that
\begin{align}\label{phiDI6}
\frac{\pa \phi}{\pa t}(s_0, t) \ge &\
(\chi\alpha-\xi\gamma) n\int_0^{s_0}s^{-b}(s_0-s)(nU_s(s,t)+1)^{p-2}U(s,t)U_s(s,t)\,ds \notag\\
&+n^2\int_0^{s_0}s^{2-\frac2n-b}(s_0-s)(nU_s(s,t)+1)^{m-1}U_{ss}(s,t)\,ds \notag\\
&-\chi\beta n\int_0^{s_0}s^{-b}(s_0-s)(nU_s(s,t)+1)^{p-2}V(s,t)U_s(s,t)\,ds \notag\\
&-n^{\kappa-1}\mu_1\int_0^{s_0}s^{-b}(s_0-s)\Big[\int_0^{s_0}\eta^{\frac an}U_s^\kappa(\eta,t)\,d\eta\Big]\,ds
\end{align}
for all $s_0 \in (0, R^n)$ and all $t \in (0, \min\{T, \tmax\})$.
\end{lem}
\begin{proof}
Arguing as in Lemma~\ref{lem_DI_p>q}, we have \eqref{phiDI2}
with $q=p$.
We then rearrange it as
\begin{align*}
\frac{\pa \phi}{\pa t}(s_0, t) \ge &\
(\chi\alpha-\xi\gamma) n\int_0^{s_0}s^{-b}(s_0-s)(nU_s(s,t)+1)^{p-2}U(s,t)U_s(s,t)\,ds \notag\\
&+n^2\int_0^{s_0}s^{2-\frac2n-b}(s_0-s)(nU_s(s,t)+1)^{m-1}U_{ss}(s,t)\,ds \notag\\
&-\chi\beta n\int_0^{s_0}s^{-b}(s_0-s)(nU_s(s,t)+1)^{p-2}V(s,t)U_s(s,t)\,ds \notag\\
&-n^{\kappa-1}\mu_1\int_0^{s_0}s^{-b}(s_0-s)\Big[\int_0^{s_0}\eta^{\frac an}U_s^\kappa(\eta,t)\,d\eta\Big]\,ds
\end{align*}
for all $s_0 \in (0, R^n)$ and all $t \in (0, \min\{T, \tmax\})$, which means that \eqref{phiDI6} holds.
\end{proof}
\begin{prth4.2}
In view of Lemma~\ref{lem_DI_p=q}, proceeding similarly in
the proof of Theorem~\ref{thm_bu1} and
taking $\sigma$ properly, we can find $c_1, c_2>0$
and $\theta \in (0,\ 2-\frac\sigma n(2-p)_+)$
such that
\begin{align*}
\frac{\pa \phi}{\pa t}(s_0, t) \ge c_1s_0^{b-3+\frac \sigma n (2-p)_+}\phi^2(s_0,t)
-c_2s_0^{3-b-\theta}
\end{align*}
for all $s_0 \in (0, s_1)$ and all $t \in (0, \min\{T, \tmax\})$ for some small $s_1>0$.
This inequality corresponds to \eqref{DIgoal}
and proves Theorem~\ref{thm_bu2}.
\qed
\end{prth4.2}
\input{CYbddfbu3.bbl}
\end{document} | {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,334 |
Q: Different behaviour of netcat from telnet when connecting to public route server I have observed a different behaviour between netcat and telnet when connecting to the public route server bgp-view.tvnetwork.hu and issuing the command show ip bgp.
Using Telnet the output (which is normally some tens of thousands lines long) is truncated and in order to view it all you have to press space or enter to continue (like the man pages). When I connect with netcat it just starts dumping all the output and after the first few hundred lines it hangs. Even if I state explicitly that I want to pause after the first 100 lines using the command terminal length 100 netcat doesn't change behaviour.
Do you have any idea why this happens and how it can be resolved?
My OS is ubuntu 10.4 and the route server runs Quagga (version 0.99.5). Same kind of behaviour is also observed with another quagga route server (). With other type of routers (cisco or juniper) that problem doesn't appear.
Thank you.
ps. I also posted this question to stackoverflow but server fault seems more relative.
EDIT:
Wireshark helped me find that the problem is that netcat refuses to negotiate the Window Size option although the server requests "Do Negotiate About Window Size". Netcat has a -t parameter to negotiate telnet options but its behavior is not to accept any options (uses DON'T or WON'T as a default). So the question now is how to make netcat to negotiate window size and send width 80 and height 25 which are the values that telnet sends.
A: If you connect with telnet and start by giving the command term len 0, it seems to not use pagination.
At a guess, the reason it doesn't do any pagination when using netcat is that it's probably not doing the expected TELNET protocol negotiations for basic terminal characteristics.
Edit: Seems the main problem is that it just plain takes too long to display the full routing table. After leaving a telnet session, with term len 0 and sh ip bgp running, I ended up with:
Vty connection is timed out.
Connection closed by foreign host.
It MAY be that having periodic data sent to the route server (maybe a space character every 45 seconds) may be enough to not trigger the VTY time-out, but I'd need to check the source code to be sure.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 5,715 |
The CDE-190R is the affordable single-DIN solution that steps up your sound to premium Alpine quality, for hours of music enjoyment.
The faster your phone charges, the better! The CDE-190R head units provide a fast-charge function to a maximum of 1000mA, for quick recharging of your phone battery. | {
"redpajama_set_name": "RedPajamaC4"
} | 2,149 |
{"url":"https:\/\/www.physicsforums.com\/threads\/does-one-raised-to-an-undefined-power-equal-anything.401703\/","text":"# Does one raised to an undefined power equal anything?\n\n1. May 8, 2010\n\n### Don Blazys\n\nDoes one raised to an undefined power \"equal\" anything?\n\nA commonly heard property of unity is \"One raised to any power equals one\", or:\n\n1^(x)=1\n\nAnother well known fact is that \"One raised to the power of infinity is an indeterminate form\", or:\n\n1^(INFINITY) is \"INDETERMINATE\".\n\nBut what happens if the exponent of unity is \"undefined\", \"nonsensical\", \"non-existent\" and \"disallowed\", as in the expression:\n\n1^(2\/0).\n\nCan such an expression ever mean anything?\n\n2. May 8, 2010\n\n### Werg22\n\nRe: Does one raised to an undefined power \"equal\" anything?\n\nIf you want it to. It can mean \"I ate apple pie for desert\", or anything you like. It's just a string of symbols.\n\n3. May 8, 2010\n\n### Don Blazys\n\nRe: Does one raised to an undefined power \"equal\" anything?\n\nThanks.\n\nI agree!\n\nBut here's the problem...\n\nI know someone who absolutely insists that 1^(2\/0) is an \"indeterminate form\".\n\nHow could I convince him otherwise?\n\nWhat should I tell him?\n\nLast edited: May 8, 2010\n4. May 8, 2010\n\n### Mentallic\n\nRe: Does one raised to an undefined power \"equal\" anything?\n\nWhat exactly are you trying to do when convincing him? When you say 1 raised to any power is still 1 of course is restricted to any real number. Undefined numbers such as 1\/0 are not real numbers. Indeterminate form MEANS the value $$1^\\infty$$ can be anything, 0, 1, 10, or infinite or anything.\n\nTake the limit as x approaches infinite of $$\\left(1+\\frac{1}{x}\\right)^x$$\n\nThis makes the indeterminate form $$1^\\infty$$ but it actually equals e.\n\nTake $$\\left(1+\\frac{2}{x}\\right)^x$$, this is the same form but equals e2.\n\n$$\\left(1+\\frac{1}{x}\\right)^{2x}$$... this is infinite.\n\nSo, are you still going to prove your friend wrong?\n\n5. May 8, 2010\n\n### Mentallic\n\nRe: Does one raised to an undefined power \"equal\" anything?\n\nOh sorry, you seem to agree that $$1^\\infty$$ is indeterminate, but want to understand more about 1x where x is undefined.\n\nUndefined can be considered the same thing as infinite in terms of limits.\n\nInstead of taking the limit as x goes to infinite of 1x, take the limit as x goes to 0 of 11\/x\n\n6. May 8, 2010\n\n### Gerenuk\n\nRe: Does one raised to an undefined power \"equal\" anything?\n\nOf course you have to use mathematical rigor. That includes saying what type of number x is. And you have to specify what type of exponentiation you want to use.\nWith complex numbers 1^(1\/2) is multivalued +1 or -1 and so on. For most x it will be multivalued.\nYou can probably pick a definition of exponentiation which always choses 1. So yes, for all real numbers x we have 1^x=1.\n\nYou are making the big mistake overinterpretation popular maths sayings. What they rather mean is that\n$$\\lim_{n\\to\\infty} a_n=1$$\n$$\\lim_{n\\to\\infty} b_n=\\infty$$\n$$\\lim_{n\\to\\infty} a_n^{b_n}=\\text{indet. (with information given)}$$\nHere at no point we are raising 1 to a power!\nDon't forget that $\\infty$ is not something that you are allowed to use like normal variables. Infinity (in the calculus sense) is an algorithm, rather than a number.\n\nThis expression is undefined. \"Undefined\" is not part of the real numbers, and we agreed that 1^x only works if x is a real number.\n\nNo, 1\/0 is indefined.\nInfinity is something completely different. It is the concept of limits with epsilons and all this.\n\n7. May 8, 2010\n\n### Hurkyl\n\nStaff Emeritus\nRe: Does one raised to an undefined power \"equal\" anything?\n\nFirst things first -- the notion of an \"indeterminate form\", and the notion that some expression is \"undefined\" are different concepts.\n\nA form is a string of symbols. Pay attention to the fact that the form \"1+2\" is not equal to \"3\". Of course, if we view them as arithmetic expressions, then the numbers they represent are equal.\n\nWhen we are using forms to denote arithmetic expressions, we might use the term \"undefined\" to refer to strings that are not arithmetic expressions. e.g. \"1\/0\" is not an arithmetic expression of real numbers, because (1,0) is not in the domain of \/.\n\nWhen doing calculus, we might talk about \"limit forms\", where we replace every atom in a form with its limit.\n\nThe definition of continuity tells us that if the limit form represents an arithmetic expression at a point where it's continuous, then the value of the limit is equal to the value of the limit form. This is a very convenient computational device.\n\nWhen the limit form does not have that property, we say it is \"indeterminate\".\n\n8. May 8, 2010\n\n### Don Blazys\n\nRe: Does one raised to an undefined power \"equal\" anything?\n\nI thank you all for your replies.\n\nThat was very kind of you.\n\nQuoting Gerenuk:\nQuoting Hurkyl:\nQuoting Mentallic:\nQuoting Werg 22:\nThus, we all agree that the expression 1^(2\/0) is not possible\nif consistency in logic is to be maintained.\n\nI will direct my aquaintance to this topic, and allow him to decide for himself.\nHopefully, he will think it through and change his mind.\n\nIn case you are interested, here is his most recent comment. When I wrote...\n\nQuoting Myself:\nHis response was:\n\nDon.\n\n9. May 8, 2010\n\n### Gerenuk\n\nRe: Does one raised to an undefined power \"equal\" anything?\n\nYou are fully correct with your explanation.\n\nI can strongly advice you one thing: this guys is a \"i have not the faintest idea, but i will always play smartass and occationally quote a random line.\" I know a couple of those.\nI learned that discussing with such kind of guys is like discussion with mormon missionaries about physics. They are completely blind for what you say, so don't bother discussing this particular topic with that person.\n\nAs you see all the guy could basically say to you was \"No. You are wrong. I am right\". He contributed no content whatsoever proofing his knowledge.\n\n10. May 9, 2010\n\n### Don Blazys\n\nRe: Does one raised to an undefined power \"equal\" anything?\n\nTo: Gerenuk.\n\nThank you.\n\nDon.\n\n11. May 9, 2010\n\n### JeffJo\n\nRe: Does one raised to an undefined power \"equal\" anything?\n\nYou have to be careful with Don. He only understands things at the most superficial level, and has already reached his conclusions before he listens to your reasons. So he takes the part of what you say that superficially agrees with his conclusions, and ignores the rest that doesn't.\n\nFor example, we were not talking about an expression with just constants, like 1^(2\/0). We were talking about an expression that is a function of an unknown, and seems to evaluate to 1^(2\/0) at one specific value of the variable. Like f(x)=(1+x)^(2\/x) at x=0. That can be defined; f(0)=e. Which is what Mentallic said, but Don ignored. But he can only discuss it as 1^(2\/0), because in that form it agrees with his pre-formed conclusion.\n\nDon does not agree that such an expression is indeterminate at x=0. He thinks that any expression like 1^g(x) is \"prohibited\" at every value in the domain of x, if g(x) can be zero at any one value in that domain. Which is specifically what we were discussing.\n\nHe also does not understand that the literal expression 1\/0 is undefined because it is infinite, but that the literal expression 1^(2\/0) is undefined because it is indeterminate. In fact, he does not acknowledge that the expression 1^(2\/x) is a form of $$1^\\infty$$ at x=0, because he takes the statement \"cannot be equal to infinity\" at the superficial level. That infinity is a value that can't be equal to anything. Infinity is a quality, not a value, so 1\/0 is infinity (it would be more accurate to say \"is infinite\" but Don refuses to grasp the difference) but is not equal to infinity. His quote was taken out of its context, which tries to explain that difference to him.\n\nSpecifically, Don thinks he has proven Beal's Conjecture because he transformed c^z into (sorry, I don't know latex, and that thread was closed due to Don's pigheadedness) T*(c\/T)[ (z*ln(c)\/ln(T)-1)) \/ (ln(c)\/ln(T)-1) ]. At c=T, this expression is indeteminate, but it is defined and is identically equal to c^z. But Don thinks the apparent division by zero makes the expression \"prohibited,\" which he takes to mean c^z is \"disallowed\" under his assumptions that z>2 and so proves Beal's conjecture.\n\nHe needs to be told, in plain language, and by as many people as possible, that he is wrong and has come nowhere near proving Beal's conjecture. One reason is that his expression is indeterminate, which he refuses to acknowledge. There are many other flaws as well - about the only thing that is right in his argument is that his expression does equal c^z.\n\n12. May 9, 2010\n\n### Don Blazys\n\nRe: Does one raised to an undefined power \"equal\" anything?\n\nSorry guys.\n\nI didn't think that would happen.\n\nDon.\n\n13. May 9, 2010\n\n### Hurkyl\n\nStaff Emeritus\nRe: Does one raised to an undefined power \"equal\" anything?\n\nI do feel compelled to point out, for the sake of getting the little details correct, that the partial function\nf(x)=(1+x)^(2\/x)\u200b\nis quite plainly undefined at zero. However, its continuous extension to the set of nonnegative reals is defined at zero, and has value e2.\n\n14. May 10, 2010\n\n### JeffJo\n\nRe: Does one raised to an undefined power \"equal\" anything?\n\nI'm sorry it was brought here, but I didn't do it. I just wanted to make sure that whatever comments you made were not misquoted or misrepresented. So I am not \"derailing the thread,\" I am trying to make sure it addresses the question Don thinks he was asking, but didn't. And that Don is told the answer to the correct question, with no reasonable way to misinterpret it. Don will do that anyway, but I want the record clear. Anyway, I'll just explain what I mean, and drop it.\n\nHurkyl, you made a minor error of sematics, that is exactly what I feared Don would misunderstand. The partial function\nf(x)=(1+x)^(2\/x)\u200b\nis quite plainly indeterminate at x=0. What that means, is that without further information, you can't determine a value for it. So it is undefineable in general, but that is not what Don wants undefined to mean to math. He wants \"can't ever be defined, so must be expunged.\" Sometimes, this expression can be defined. Don ignored this:\nEmphasis added. (Oops, sorry about my typo where I said e1. I started with (1+x)(1\/x) but changed it, but apparently not everywhere.)\n\nHere, you are saying that under specific conditions you can determine a value at x=0. Don quoted what you said only up to the point of saying \"undefined,\" and concluded that you meant (1) it can never be defined, (2) that x=0 must be \"disallowed\" under all circumstances, and (3) if you can't disallow x=0, then one of the assumptions that led to the derivation of the equation must be wrong. Proof by reductio ad absurdum.\n\nBut let's say that the equation\nI(r) = I0 (1-(1+r)(-1\/r)) \/ (1-e-1)\u200b\nis derived for a current in a system, where r is the resistance of a component in it. Don thinks that you said the current is undefined if you short that component. What you said was, the expression itself is undefined if you don't know what I and r represent; but that if the resistance can be represented by a nonnegative real, and everything should be continuous at r=0, I(0) is defined to be I0.\n\nDon further feels that since r=0 must be \"disallowed\" in every way, that the system cannot be built in the first place. Not that something will blow up (i.e., \"is really not continuous\"), that you can't get that equation legitimately. That \"undefined\" currents cannot exist, but there obviously will be a current, so something is wrong elsewhere. That is the meaning he placed on your statement where you used that word, and I just want you to say more clealry what you meant. That it can be defined sometimes, so you can't conclude the expression itself is invalid.\n\nWhat Don left out of this thread, is that the \"specific circumstances\" for the expression he wants to use, are that it isn't even indeterminate. He just made it look that way via algebraic manipulation that was a little more complicated than the following, but essentially the same in concept:\nf(x) = (x\/k)x = (x\/k)((x-k) * x\/(x-k))\n= [(x\/k)(x-k)](x\/(x-k))\nFrom this he concludes that, since there is no reason x can't be equal to k, but we must \"disallow\" x=k, that something is wrong with the assumptions that define f(x).\n\nThe problem is that Don won't listen to any of the reasons his argument is not correct, just one of which is that the partial expression (x\/k)(x\/(x-y)) is indeterminate. He is right that when it is expressed that way we can't, in general, define a value. But since it is indeterminate, and we derived it from an expression that had no such division by zero, it is in fact defined for positive k.\n\nLast edited: May 10, 2010\n15. May 10, 2010\n\n### Hurkyl\n\nStaff Emeritus\nRe: Does one raised to an undefined power \"equal\" anything?\n\nDon didn't create a personal attack on himself -- you were the one who decided to do that, rather than address the relevant math.\n\nAnyways, I'm going to insist on making true statements in this thread. There is a time where imprecise, nearly-true statements are appropriate, and I'm pretty sure this is not it.\n\nThe claim that the partial function defined by\nI(r) = I0 (1-(1+r)(-1\/r)) \/ (1-e-1)\u200b\nis defined at 0 is an example of one of these useful fictions. Most of the time, when defining functions like this, we don't mean what we literally wrote -- we really mean things like \"the continuous extension of what we wrote\" or \"we do this in most places, and make the obvious adjustments where it doesn't work\" or what-not. We're lazy and we don't want to go through the extra trouble of saying all these niggling details that everybody knows anyways.\n\nBut in those situations where we really are paying attention to these details, it is flat out wrong to make the claim that, for the function defined above, that I(0) is defined.\n\nAnd in your other example, the relation you wrote\n(x\/k)x = (x\/k)((x-k) * x\/(x-k))\nis incorrect. (If is, of course, valid when restricted to the domain where x>0, k>0, and x and k are different)\n\nThe function defined by \"x\/x\" is undefined at 0. The limit form written \"x\/x\" is indeterminate. There's a difference between what those two statements mean. It's not a difference that we often care paying attention to, but if we're going to do so, then we should make true statements.\n\nLast edited: May 10, 2010\n16. May 11, 2010\n\n### JeffJo\n\nRe: Does one raised to an undefined power \"equal\" anything?\n\nI am truly sorry for that - but I felt it was necessary to make you see just how your words would be misinterpreted if I didn't. The issue really isn't the math, it is the seemingly purposeful misunderstanding of the math, in order to support invalid conclusions that have nothing to do with that math.\nI agree, and have tried to do so within the bounds of how it will be (mis)understood. And that's the problem. But I'd also like to simplify the \"true statements\" without adding \"yes, but that also means...\".\n1. Yes or no, is an expression of the form f(x) = g(x)1\/h(x) an example of the indeterminate form $$1^\\infty$$ at x1>0, if g(x1)=1 and h(x1)=0? I'm not stating whether or not something like \"continuous extension\" is appropriate. I'm asking if that form, with a division by zero in the exponent, is something the form $$1^\\infty$$ intends when it does apply.\n2. Can you use a well-defined \"continuous extension\" value of f(x), when the domain of x is the integers? Or does that domain prevent the use of the \"continuous extension?\" I'm not really sure how this point applies to anything, because of the next one, but it came up.\n3. If you take a function f(x) that is well defined over all positive values of x (example: f(x) = x3), and (correctly) manipulate it into the form f(x) = x1*g(x)1\/h(x) where g(x1)=1 and h(x1)=0 for x1>0, does that make f(x) undefined, or in any way invalid, at any positive values of x? (example: g(x)= (x\/x1), h(x)=(ln(x)-ln(x1))\/(3*ln(x)-ln(x1)). This is the reason we are asking if it is indeterminate. It's not because the equation started in that form, but because it was manipulated into it. So that labeling it \"indeterminate\" is not so it can be defined or evaluated, but so we can decide if the manipulation was artificial and has no bearing on whether f(x) is defined or can be evaluated.\n4. If you answered \"yes\" to the last question, does that mean those values of x must be removed from the domain of f(x)?\nDon answers these questions \"no,\" \"no,\" \"yes,\" and \"for some xi's.\" I'm sorry this all sounds like another attack, but these are the answers Don wants to support, and he is asking questions out of context to get the answers he prefers. He wants g(x)1\/h(x) to not be indeterminate, because I claimed being indeterminate makea the relationship artificial, having no bearing on f(x). He is using your statement that it is \"undefined\" to claim that f(x) is also undefined, so some values of x must be removed from the domain of f(x).\nI did not mean that it was literally defined without other information. I meant that if you built a circuit with ideal components, derived that equation for it, and set r=0, that the current would be I0. You could determine that by using a variable resistor and dialing it to zero, which is the realization of your \"continuous extension;\" or just by shorting it, assuming that all of the components can handle the sudden change. Yes, I know there could be other issues like damping times, but I'm assuming they are negligible. It is possible that whatever interactions cause the exponential nature of this result means there are singularities in the circuit, which is why I assumed ideal components. To me, that means that they can handle those singularities.\n\nThe point is, arriving at that equation alone is not enough to conclude that you can't short the resistor without destroying the electrical components, which is effectively what Don wants to do. Even worse, the equation he really had was a well-defined I(r), over all positive r, that he manipulated into the form I(r)=I0(t)*g(r,t)^(1\/h(r,t)) for some completely made-up parameter t. And I intentionally left it out of the function I(r), because it does not affect that value in any way. Unless, of couse, it makes the right-hand side undefined. Which is where \"indeterminate\" comes in.\n\nIt turns out this manipulation can be done for any positive integer r, and that g(r,r)=1 and h(r,r)=0. From that he concludes that the r's he wants to get rid of are \"disallowed\" for I(r), but the ones he does not choose this way are \"allowed.\" And that is the ultimate question he wants answered - is are those r's \"disallowed\" ?\n\nLast edited: May 11, 2010\n17. May 11, 2010\n\n### Hurkyl\n\nStaff Emeritus\nRe: Does one raised to an undefined power \"equal\" anything?\n\nThis hypothesis is vacuous -- the following three statements are collectively contradictory:\n\u2022 The domain of f(x) includes all positive real numbers\n\u2022 h(x) is zero for some positive real number\n\u2022 f(x) = g(x)1\/h(x) is true on the domain of f\n\ne.g. a simpler, similar example: 1 = x\/x (where x is a real variable) is not true, and any calculation that assumes that equation is suspect. Now, making such substitutions is a commonly useful thing to do -- but it requires breaking a problem into two parts: one part where x is assumed to be nonzero (so that 1=x\/x is identically true, and may be correctly used), and one part where x is assumed to be zero, which must be handled in another fashion.\n\n18. May 11, 2010\n\n### JeffJo\n\nRe: Does one raised to an undefined power \"equal\" anything?\n\nI didn't say f(x) = g(x)1\/h(x) is true on the domain of f. It has a problem at x=x1, which is the one Don is trying to take advantage of. But other than that, the manipulation was correct. He is trying to claim that because of that substitution, the domain of f(x) is restricted the same way the domain of g(x)1\/h(x) is. One clue that it is not true, is that the form of g(x)1\/h(x) matches the indeterminate form $$1^\\infty$$ (see question #1 for caveats) at that point, which is what led Don here to try to claim it didn't.\n\nBut you have now been helpful enough in finding the problem. Thank you.\n\nLast edited: May 12, 2010\n19. May 11, 2010\n\n### Tac-Tics\n\nRe: Does one raised to an undefined power \"equal\" anything?\n\nWhile often toted as a fact by calculus teachers, the notion of an \"indeterminate form\" is ill-defined. It only survives because it's a helpful pedagogical tool for students who can't distinguish between a function and evaluations of that function.\n\nDivision by zero isn't defined.\n\nExponentiation of zero by zero isn't defined.\n\nInfinity isn't a real number.\n\nTalking about \"2\/0\" is like talking about the \"shape of blue\". It's nonsense. It's not well-formed. Similarly \"0^0\" and \"1^\u221e\". Those are \"the color of nothing\" and \"the sound of silence\" respectively They. Are. Nonsense.\n\nWhat ISN'T nonsense are limits. Limits allow you to give reasonable meanings to these undefined values. The reason they are able to this (while the expressions alone can't) is because they carry more information. Think about it like this. With an expression of two variables, there's only one possible representation if we substitute 0 for those values.\n\n1) f(0, 0)\n\nBut with a limit, there are several ways to do it:\n\n1) f(0, 0)\n2) lim x->0: f(x, 0)\n3) lim x->0: f(0, x)\n4) lim x->0: f(x, x)\n\nSee how we have a more expressive \"language\" with limits?\n\nNow let f(x, y) = x^y. How many of these possibilities are legitimate still?\n\n1) 0^0 (illegitimate, because as we said above exponentiation of 0 by 0 is not defined)\n2) lim x->0: x^0\n3) lim x->0: 0^x\n4) lim x->0: x^x\n\nSo, with one case invalid, we still have three different possible limits.\n\nNaively, we can map all three of these back to the illegitimate expression 0^0. Even though it's a bastardization (because remember 0^0 is absolute nonsense), we accept that even if it was legitimate, it would be ambiguous. Seeing 0^0, we naturally want to map it back to a limit so we can evaluate it -- but oh no, there's not enough information to decide which kind of limit we meant.\n\nThis is the reason for introducing \"indeterminate forms\". The language of expressions is less \"expressive\" than the language of limits. But since the former is more compact, we often use it instead. This leads to ambiguities, and we call that little screw-up in our mathematical history by a funny name: \"indeterminate form.\"\n\n20. Sep 17, 2011\n\n### vrmuth\n\nRe: Does one raised to an undefined power \"equal\" anything?\n\nyour explanation about 1^infinity is very clear thanks,but if 1\/0 is undefined then why we were taught like \" anything divided by o is infinity\" ?","date":"2018-12-12 21:27:39","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7318634986877441, \"perplexity\": 975.880761617259}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-51\/segments\/1544376824119.26\/warc\/CC-MAIN-20181212203335-20181212224835-00500.warc.gz\"}"} | null | null |
Follow in the wake of the mutineers of HMS Bounty on board Aranui 5, the most unique experience in the South Pacific
Next year [2020], 230 years after the arrival of the mutineers of HMS Bounty, small passenger-freighter Aranui 5 will make a rare call on the island of Pitcairn – one of the most remote and isolated destinations in the world.
Departing from her regular schedule in The Marquesas, Aranui 5 – one of the most unique cruise experiences in the world – will make a special 13-day voyage with just 230 guests to Pitcairn Island from Tahiti on 11 April, 2020. The voyage also visits the Gambier and Austral archipelagos.
Step on board Aranui 5 – the 'Freighter to Paradise' – for an island-hopping adventure to the most remote shores of French Polynesia
The islands of the South Pacific conjure up images of adventure and mystery, unspoilt beauty and wild shores that are intoxicating and inspiring in equal measure. None more so than the enchanting Marquesas Islands of French Polynesia, and there is no better way to reach them than on board the passenger-freighter Aranui 5 – one of the most original and unforgettable voyages in the world.
Located in the Pacific Ocean, 900 miles from Tahiti, the Marquesas are so remote that Aranui 5 is, in fact, their lifeline. A working cargo ship, Aranui 5 delivers supplies to the six inhabited islands of the archipelago, while also hosting up to 230 guests on her 2500-mile adventure through the islands. | {
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Defend Free Speech in Brisbane!
By Refugee Action Collective (Qld)
Campaign Completed on
A little bit of context
In November last year, asylum seekers on Manus Island staged an ongoing protest to demand their freedom from the island and to be brought to safety. Several weeks into their protests, security and police on Manus Island raided the camp and savagely attacked those inside.
Protests broke out in many major cities in Australia, demanding freedom for the refugees and that they be brought to Australia for immediate processing and settlement. In Brisbane, the Refugee Action Collective organised a protest at short notice, and despite only with a few days to organise, gathered hundreds of people in King George Square on a Friday night.
But there's a problem
With little notice, police were not prepared to facilitate a march on the roads, and so the collective took the crowd on a peaceful march into Queen st Mall, where we gathered for 15 minutes, chanting and hearing speeches, before returning.
The mall is public space, and yet the Brisbane City Council has a new set of laws (Public Land and Council Assets Local Law 2014) that grant them to power to issue massive fines against a series of activities in the mall, primarily targeting peaceful protest.
Tim Arnot, a member of the Refugee Action Collective and the person who facilitated the march and rally on the night, received a fine from the Brisbane City Council for using a megaphone in the mall to facilitate the rally.
Here's what we're doing about it
We've exhausted our avenues to have this investigated without going to court, and our legal advice suggests that it will be very difficult to challenge it in court. So first up, we have to pay the fine - $630.
Additionally, we are setting up a petition for freedom of speech in the mall, and to call for an end to the PLACA Local Law 2014 which restricts use of the mall for community actions such as peaceful demonstrations. As more public space disappears through development in our CBD, we have to make a stand! Sign our petition here: https://www.change.org/p/lord-mayor-graham-quirk-demand-free-speech-in-brisbane-cbd
You can join us
Please donate to help us cover the fine. We are a volunteer-based, not-for-profit organisation and any of our funding is all based on donations.
Please also sign our petition! The link is above, and you'll also be able to find it on our stalls.
Please also join us at our next protest! We will not stop fighting for the rights of refugees and people seeking asylum, and so our next rally will be Sunday 25th March at 1pm in King George Square.
Thank you for your support and dedication, and we'll see you at the next rally!
Tim Arnot
Margarett Macpherson
Raised of $630
Brisbane City Council has fined activists with the Refugee Action Collective (Qld) for holding a peaceful protest in Queen st Mall, calling for freedom for asylum seekers on Manus Island.
Refugee Action Collective (Qld) Brisbane, Queensland, Australia
Get in touch with Defend Free Speech in Brisbane!
Your message to Tim Arnot | {
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Q: Combining a 1-column data frame with values from a list So I have one column called 'Number' and it has this output
Number
10
20
30
40
50
60
70
80
90
100
110
120
130
140
150
160
170
180
190
200
and another output called 'Value' which has this output
[[1]]
[1] 0.2231071
[[2]]
[1] 0.3432919
[[3]]
[1] 0.0002351737
[[4]]
[1] 0.01550648
[[5]]
[1] 0.000560943
[[6]]
[1] 2.168151e-06
This goes up to 20 values
So I want append this two ouputs together....So I would want a data frame looking like this
Number Values
10 0.2231071
20 0.3432919
30 0.0002351737
etc...up to 200...how do I do that?
something like data.frame(Number,Values...??)
Thanks guys
A: You could use cbind or data.frame:
cbind(Number, Values=unlist(Value))
A: if your two objects are Numbers and Values then you can just do
data.frame(Number, unlist(Value))
| {
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{"url":"https:\/\/artofproblemsolving.com\/wiki\/index.php?title=2016_AMC_10B_Problems\/Problem_5&diff=prev&oldid=143277","text":"Difference between revisions of \"2016 AMC 10B Problems\/Problem 5\"\n\nProblem\n\nThe mean age of Amanda's $4$ cousins is $8$, and their median age is $5$. What is the sum of the ages of Amanda's youngest and oldest cousins?\n\n$\\textbf{(A)}\\ 13\\qquad\\textbf{(B)}\\ 16\\qquad\\textbf{(C)}\\ 19\\qquad\\textbf{(D)}\\ 22\\qquad\\textbf{(E)}\\ 25$\n\nSolution\n\nThe sum of the ages of the cousins is $4$ times the mean, or $32$. There are an even number of cousins, so there is no single median, so $5$ must be the median of the two in the middle. Therefore the sum of the ages of the two in the middle is $10$. Subtracting $10$ from $32$ produces $\\textbf{(D)}\\ \\boxed{22}$.\n\n~savannahsolver","date":"2023-01-31 01:01:42","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 11, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.22394707798957825, \"perplexity\": 375.0452501108768}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2023-06\/segments\/1674764499831.97\/warc\/CC-MAIN-20230130232547-20230131022547-00822.warc.gz\"}"} | null | null |
Q: How to request my services on Google App Engine? I've deployed a PHP application with 2 services on Google App Engine: admin and api.
I use a configuration file to route the requests.
My problem is that I can't route requests to my services. Only to the default.
Here is the structure:
|-- admin
|-- public
|-- index.php
|-- admin.yaml
|-- API
|-- api
|-- index.php
|-- api.yaml
|-- dispatch.yaml
|-- index.php
dispatch.yaml:
dispatch:
# Send all api traffic to the API.
- url: "*/API/api/*"
service: api
# Send all admin traffic to the admin.
- url: "*/admin/public/*"
service: admin
# Default service serves simple hostname request.
- url: "*/*"
service: default
admin/admin.yaml:
runtime: php73
service: admin
handlers:
- url: /admin/public/.*
script: /admin/public/index.php
API/api.yaml:
runtime: php73
service: api
handlers:
- url: /API/api/.*
script: /API/api/index.php
index.php: echo "Not found";
admin/public/index.php: echo "Welcome to admin service";
API/api/index.php: echo "Welcome to api service";
When I send a request: https://SERVICE_ID-dot-PROJECT_ID.REGION_ID.r.appspot.com or https://PROJECT_ID.REGION_ID.r.appspot.com the response is the expected one: Not found.
But when I send a request: https://SERVICE_ID-dot-PROJECT_ID.REGION_ID.r.appspot.com/API/api/ or https://PROJECT_ID.REGION_ID.r.appspot.com/API/api/ the response is an error code 500 instead of: Welcome to api service.
What is the problem ? Configuration ? Request urls ? Other ?
A: The problem was in the yaml files:
new dispatch.yaml
dispatch:
# Send all api traffic to the API.
- url: "SERVICE_ID-dot-PROJECT_ID.REGION_ID.r.appspot.com"
service: api
# Send all admin traffic to the admin.
- url: "SERVICE_ID-dot-PROJECT_ID.REGION_ID.r.appspot.com"
service: admin
# Default service serves simple hostname request.
- url: "PROJECT_ID.REGION_ID.r.appspot.com"
service: default
new admin/admin.yaml:
runtime: php73
service: admin
handlers:
- url: /.*
script: /admin/public/index.php
new API/api.yaml:
runtime: php73
service: api
handlers:
- url: /.*
script: /API/api/index.php
No more routing problem.
| {
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<link rel="stylesheet" href="<?php echo RES; ?>lib/jquery-file-upload/css/jquery.fileupload.css">
<link rel="stylesheet" href="<?php echo RES; ?>lib/jquery-file-upload/css/jquery.fileupload-ui.css">
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<noscript><link rel="stylesheet" href="<?php echo RES; ?>lib/jquery-file-upload/css/jquery.fileupload-noscript.css"></noscript>
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<style>
li.none-list-style{
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width:90px;
float:left;
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span.cancel{margin-left:14px;}
a.fileinput-button{vertical-align: middle;}
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{"url":"https:\/\/www.hpmuseum.org\/forum\/thread-12196-post-111921.html","text":"(35) Locating the Moon\n01-15-2019, 01:59 PM\nPost: #1\n SlideRule Senior Member Posts: 1,251 Joined: Dec 2013\n(35) Locating the Moon\nAn algorithm from the article Using the HP 35 to Locate the Moon, Locating the Moon (Eimac).\n\nThe following procedure is a technique for using the Hewlett-Packard HP-35 hand held calculator for determining the moon's AZIMUTH in relation to true north, and the ELEVATION with respect to the local horizon for the geographical location in question.\n\nBEST!\nSlideRule\n02-08-2019, 07:18 AM\nPost: #2\n Thomas Klemm Senior Member Posts: 1,447 Joined: Dec 2013\nRE: (35) Locating the Moon\nThis program is for the HP-42S:\nCode:\n00\u00a0{\u00a032-Byte\u00a0Prgm\u00a0} 01\u25b8LBL\u00a0\"MOON\" 02\u00a0R\u2193 03\u00a0- 04\u00a0X<>Y 05\u00a0R\u2191 06\u00a01 07\u00a0\u2192REC 08\u00a0R\u2191 09\u00a0X<>Y 10\u00a0\u2192REC 11\u00a0X<>Y 12\u00a0R\u2193 13\u00a0\u2192POL 14\u00a0R\u2193 15\u00a0X<>Y 16\u00a0- 17\u00a0R\u2191 18\u00a0\u2192REC 19\u00a0R\u2193 20\u00a0\u2192POL 21\u00a0R\u2191 22\u00a0X<>Y 23\u00a0\u2192POL 24\u00a0R\u2193 25\u00a0END\n\nExample:\n\n37.33 ENTER\n122.13 ENTER\n80.85 ENTER\n23.36 XEQ \"MOON\"\n\ny: 99.6580\nx: 52.0939\n\nIt works also for most other HP calculator models that provide polar-rectangular coordinate transformations.\nOf course it's not restricted to locate the moon but any celestial body.\n\nI wasn't aware of moonbounce.\nDoes anyone here has experience with it?\n\nCheers\nThomas\n\nAttached File(s) Thumbnail(s)\n\n02-08-2019, 09:56 AM (This post was last modified: 02-08-2019 09:57 AM by pier4r.)\nPost: #3\n pier4r Senior Member Posts: 2,067 Joined: Nov 2014\nRE: (35) Locating the Moon\nQuote:I wasn't aware of moonbounce.\nDoes anyone here has experience with it?\n\nListening to the computer history museum podcast (If I am not mistaken), there is the quest of discovering the radar capabilities of the soviets. The problem: you cannot fly an airplane with electronics and detectors deep in the soviet airspace, so it is rather complicated.\n\nThey started to realize that they can pick up reflected signals from missiles going up and directly from the moon. I suspect the arecibo radar telescope was used for that too. ( https:\/\/en.wikipedia.org\/wiki\/Arecibo_Observatory )\n\nWikis are great, Contribute :)\n02-08-2019, 11:31 AM (This post was last modified: 02-08-2019 05:21 PM by PedroLeiva.)\nPost: #4\n PedroLeiva Member Posts: 167 Joined: Jun 2014\nRE: (35) Locating the Moon\nI have modifie this HP-42S program to be use in HP-67, also change the input data procedure to labels A to D for LAT, LONG, GHA and DECL. But I have doubths about the LAT and LOG sings to use. Usually N of equator and E of Greendwich are (+), and (-) for the opposite locations. Is this apply here?\nYour opinion will be highly appreciated\nPedro\n02-08-2019, 05:05 PM\nPost: #5\n Thomas Klemm Senior Member Posts: 1,447 Joined: Dec 2013\nRE: (35) Locating the Moon\n(02-08-2019 11:31 AM)PedroLeiva Wrote: \u00a0But I have doubts about the LAT and LOG sings to use. Usually N of equator and E of Greenwich are (+), and (-) for the opposite locations. Is this apply here?\n\nYes. But I had to look up the definitions of GHA and azimuth:\n\nThe hour angle may be expressed as negative east of the meridian plane and positive west of the meridian plane.\n\nAzimuth is defined as a horizontal angle measured clockwise from a north base line or meridian.\n\nFrom looking at the 2 examples on page 6 I assume that the longitude of my example is meant to be in the west as well. Thus we should rather use -122.13.\nAnd since the azimuth is measured clockwise we have to change a sign as well.\n\nThis leads to this corrected program:\nCode:\n00\u00a0{\u00a033-Byte\u00a0Prgm\u00a0} 01\u25b8LBL\u00a0\"MOON\" 02\u00a0R\u2193 03\u00a0+ 04\u00a0X<>Y 05\u00a0R\u2191 06\u00a01 07\u00a0\u2192REC 08\u00a0R\u2191 09\u00a0X<>Y 10\u00a0\u2192REC 11\u00a0X<>Y 12\u00a0+\/- 13\u00a0R\u2193 14\u00a0\u2192POL 15\u00a0R\u2193 16\u00a0X<>Y 17\u00a0- 18\u00a0R\u2191 19\u00a0\u2192REC 20\u00a0R\u2193 21\u00a0\u2192POL 22\u00a0R\u2191 23\u00a0X<>Y 24\u00a0\u2192POL 25\u00a0R\u2193 26\u00a0END\n\nLine 03 was changed and line 12 was inserted.\n\nExamples:\n\n37.33 ENTER\n-122.13 ENTER\n80.85 ENTER\n23.36 XEQ \"MOON\"\n\ny: 99.6580\nx: 52.0939\n\n36 ENTER\n-122 ENTER\n71 ENTER\n2 XEQ \"MOON\"\n\ny: 113.7316\nx: 31.9605\n\n36 ENTER\n-122 ENTER\n181 ENTER\n-2 XEQ \"MOON\"\n\ny: -111.1134\nx: 23.3226\n\nI hope that's correct now.\n\nCheers\nThomas\n02-08-2019, 05:51 PM\nPost: #6\n PedroLeiva Member Posts: 167 Joined: Jun 2014\nRE: (35) Locating the Moon\n(02-08-2019 05:05 PM)Thomas Klemm Wrote:\n(02-08-2019 11:31 AM)PedroLeiva Wrote: \u00a0But I have doubts about the LAT and LOG sings to use. Usually N of equator and E of Greenwich are (+), and (-) for the opposite locations. Is this apply here?\n\nYes. But I had to look up the definitions of GHA and azimuth:\n\nThe hour angle may be expressed as negative east of the meridian plane and positive west of the meridian plane.\n\nAzimuth is defined as a horizontal angle measured clockwise from a north base line or meridian.\n\nFrom looking at the 2 examples on page 6 I assume that the longitude of my example is meant to be in the west as well. Thus we should rather use -122.13.\nAnd since the azimuth is measured clockwise we have to change a sign as well\nSame results as yours. Thank you\nPedro\n02-09-2019, 02:08 AM (This post was last modified: 02-09-2019 02:13 AM by PedroLeiva.)\nPost: #7\n PedroLeiva Member Posts: 167 Joined: Jun 2014\nRE: (35) Locating the Moon\n[\/quote]\nThis is a program for HP-67. Some chanches were made: the input information of LAT, LONG, GHA, DECLIN by pressing [A], [B], [C] and [D], the output pressing [E] and [x<>y]\nCode:\n *LBL\u00a0A: 001:\u00a0\u00a031\u00a025\u00a011\u00a0\u00a0\u00a0LBL\u00a0A 002:\u00a0\u00a033\u00a011\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0STO\u00a0A 003:\u00a0\u00a035\u00a022\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0RTN \u00a0\u00a0\u00a0\u00a0\u00a0*LBL\u00a0B: 004:\u00a0\u00a031\u00a025\u00a012\u00a0\u00a0\u00a0LBL\u00a0B 005:\u00a0\u00a033\u00a012\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0STO\u00a0B 006:\u00a0\u00a035\u00a022\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0RTN \u00a0\u00a0\u00a0\u00a0\u00a0*LBL\u00a0C: 007:\u00a0\u00a031\u00a025\u00a013\u00a0\u00a0\u00a0LBL\u00a0C 008:\u00a0\u00a033\u00a013\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0STO\u00a0C 009:\u00a0\u00a035\u00a022\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0RTN \u00a0\u00a0\u00a0\u00a0\u00a0*LBL\u00a0D: 010:\u00a0\u00a031\u00a025\u00a014\u00a0\u00a0\u00a0LBL\u00a0D 011:\u00a0\u00a033\u00a014\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0STO\u00a0D 012:\u00a0\u00a035\u00a022\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0RTN \u00a0\u00a0\u00a0\u00a0\u00a0*LBL\u00a0E: 013:\u00a0\u00a031\u00a025\u00a015\u00a0\u00a0\u00a0LBL\u00a0E 014:\u00a0\u00a034\u00a011\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0RCL\u00a0A 015:\u00a0\u00a034\u00a012\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0RCL\u00a0B 016:\u00a0\u00a034\u00a013\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0RCL\u00a0C 017:\u00a0\u00a034\u00a014\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0RCL\u00a0D 018:\u00a0\u00a035\u00a053\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0R\u2193 019:\u00a0\u00a061\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0+ 020:\u00a0\u00a035\u00a052\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0x\u2277y 021:\u00a0\u00a035\u00a054\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0R\u2191 022:\u00a0\u00a001\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a01 023:\u00a0\u00a031\u00a072\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0P\u279dR 024:\u00a0\u00a035\u00a054\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0R\u2191 025:\u00a0\u00a035\u00a052\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0x\u2277y 026:\u00a0\u00a031\u00a072\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0P\u279dR 027:\u00a0\u00a035\u00a052\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0x\u2277y 028:\u00a0\u00a042\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0CHS 029:\u00a0\u00a035\u00a053\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0R\u2193 030:\u00a0\u00a032\u00a072\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0R\u279dP 031:\u00a0\u00a035\u00a053\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0R\u2193 032:\u00a0\u00a035\u00a052\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0x\u2277y 033:\u00a0\u00a051\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0- 034:\u00a0\u00a035\u00a054\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0R\u2191 035:\u00a0\u00a031\u00a072\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0P\u279dR 036:\u00a0\u00a035\u00a053\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0R\u2193 037:\u00a0\u00a032\u00a072\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0R\u279dP 038:\u00a0\u00a035\u00a054\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0R\u2191 039:\u00a0\u00a035\u00a052\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0x\u2277y 040:\u00a0\u00a032\u00a072\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0R\u279dP 041:\u00a0\u00a035\u00a053\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0R\u2193 042:\u00a0\u00a035\u00a022\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0RTN\nExample1: angles in degrees\nData\nA-LATITUDE: 37.33 DEG\nB-LONGITUDE: -122.13 DEG\nConvention: N of equator and E of Greenwich are (+), opposite position (-)\n\nC-GHA: 80.85 DEG\nGreenwich Hour Angle\n(June 2, 1973 at 19:00 GTM)\nConvention to get Azimuth from true N:\n1- If the GHA is east of your longitude \uf0e0\nA= Azimuth\n2- If the GHA is west of your longitude \uf0e0\n360 - A= Azimuth\n\nD-DECLINATION: +23.36 DEG\n(June 2, 1973 at 19:00 GTM)\nConvention: N (+), S (-)\n\nResults\nELEVATION: 52.0939 DEG\nAZIMUTH: 99.6580 DEG\n\nINSTRUCTIONS\nInput\n37.33 [A]\n-122.13 [B]\n80.85 [C]\n23.36 [D]\n\nOutput\n[E]\nx: 52.0939\n[x<>y]\ny: 99.6580\n\nExample2:\nData\n36 [A]\n-122 [B]\n71 [C]\n2 [D]\n\nSolution\n[E]\nx= 21.9605 DEG\n[x<>y]\ny= 113.7316 DEG\n\nExample3:\nData\n36 [A]\n-122 [B]\n181 [C]\n-2 [D]\n\nSolution\n[E]\nx= 23.3226 DEG\n[x<>y]\ny= -111.1134 DEG\n[quote]\n02-09-2019, 04:17 AM\nPost: #8\n Thomas Klemm Senior Member Posts: 1,447 Joined: Dec 2013\nRE: (35) Locating the Moon\n(02-09-2019 02:08 AM)PedroLeiva Wrote: \u00a0Some changes were made: the input information of LAT, LONG, GHA, DECLIN by pressing [A], [\u0392], [C] and [D]\n\nWe can slightly improve the program E when the input is in registers A - D:\nCode:\n013:\u00a0\u00a031\u00a025\u00a015\u00a0\u00a0\u00a0LBL\u00a0E 014:\u00a0\u00a034\u00a012\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0RCL\u00a0B 015:\u00a0\u00a034\u00a013\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0RCL\u00a0C 016:\u00a0\u00a061\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0+ 017:\u00a0\u00a034\u00a011\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0RCL\u00a0A 018:\u00a0\u00a034\u00a014\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0RCL\u00a0D 019:\u00a0\u00a001\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a01 020:\u00a0\u00a031\u00a072\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0P\u279dR 021:\u00a0\u00a035\u00a054\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0R\u2191 022:\u00a0\u00a035\u00a052\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0x\u2277y 023:\u00a0\u00a031\u00a072\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0P\u279dR 024:\u00a0\u00a035\u00a052\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0x\u2277y 025:\u00a0\u00a042\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0CHS 026:\u00a0\u00a035\u00a053\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0R\u2193 027:\u00a0\u00a032\u00a072\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0R\u279dP 028:\u00a0\u00a035\u00a053\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0R\u2193 029:\u00a0\u00a035\u00a052\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0x\u2277y 030:\u00a0\u00a051\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0- 031:\u00a0\u00a035\u00a054\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0R\u2191 032:\u00a0\u00a031\u00a072\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0P\u279dR 033:\u00a0\u00a035\u00a053\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0R\u2193 034:\u00a0\u00a032\u00a072\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0R\u279dP 035:\u00a0\u00a035\u00a054\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0R\u2191 036:\u00a0\u00a035\u00a052\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0x\u2277y 037:\u00a0\u00a032\u00a072\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0R\u279dP 038:\u00a0\u00a035\u00a053\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0R\u2193 039:\u00a0\u00a035\u00a022\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0RTN\n\nCheers\nThomas\n \u00ab Next Oldest | Next Newest \u00bb\n\nUser(s) browsing this thread: 1 Guest(s)","date":"2021-01-22 13:04:28","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.32397279143333435, \"perplexity\": 3192.0838968603443}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-04\/segments\/1610703529331.99\/warc\/CC-MAIN-20210122113332-20210122143332-00546.warc.gz\"}"} | null | null |
'America's Got Talent' sneak peek video: Child singer shocks judges with powerful performance of 'I Put a Spell on You'
Looks like "America's Got Talent" might have found the next Courtney Hadwin in child singer Charlotte Summers. Watch the sneak peek video of her audition which be will featured on Tuesday's show. This native of Spain competed on that country's "Le Voz Kids" four years ago. And last year she impressed Steve Harvey with her turn on "Little Big Shots."
Charlotte, who is only 13, readily admitted to "AGT" host Terry Crews that she was nervous to face Simon Cowell as she really loves him. And she revealed that she feared she was going to faint as it has been her dream to be on "America's Got Talent" for years. Asked what she'd do with the million dollar first prize, she had a simple wish – to buy a guinea pig and call him Simon Cowell. And, she added, her sister would buy one too and name him Howie Mandel.
While Charlotte was soft-spoken when she began to sing, she shocked the audience and the judges, including newcomers Julianne Hough and Gabrielle Union, with her powerhouse vocals as she launched into "I Put a Spell on You." This rollicking song dates back to 1956 and was a big hit for singer/songwriter Jalacy "Screamin' Jay" Hawkins. Since then it has been covered by a range of talents from Nina Simone and Creedence Clearwater Revival to Bette Midler and Annie Lennox.
SEE 'America's Got Talent' winners: Where are they now updates for first 13 'AGT' champions [PHOTOS]
But Charlotte made this song her own. The four judges leapt to their feet as the last note still rang out. To find out what the judges have to say about Charlotte's chances to win "AGT" and whether she gets the Golden Buzzer, be sure to tune in to Season 14, Episode 2 when it airs Tuesday, June 11 at 8:00 p.m. ET/PT on NBC.
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"redpajama_set_name": "RedPajamaCommonCrawl"
} | 6,929 |
Q: problems with staticmethod decorators I have been using @staticmethod decorator on some of my functions in order to write test cases for them. I just wanted to test a few methods in the class without having to initialize the entire class, so I thought this would be a perfect way to do.
But, now test code runs fine but the actual code gives me an error. Most of the functions had self parameters passed to them, so @staticmethod decorators result in their failures.
Is there a way to test methods without using @staticmethod? If there isn't, what would be the perfect way to work around this without changing much of the production code?
Class DBMethod():
@staticmethod
def getVN(self, tN):
curs = self.connection.cursor()
curs.execute ('SELECT * FROM ' + tN)
vL = list(map(lambda x: x[0], curs.description))[0]
return vL
And Test class
Class DBTestClass(unittest.TestCase):
def test_getVN(self):
self.assertEqual(DBMethod.getVN(self, 'tbN'), 'VER')
A: I would suggest that you use one of the available testing frameworks for Python such as:
*
*unittest
*nose
*pytest
These frameworks make it easy to write tests that exercise each method in a class and to provide appropriate test scaffolding and test data as required. Much easier IMHO than trying to handcraft tests.
A: The following mock patch can skip the init part (Actually I do not know why you avoid "mock the init method" here, so paste my guess here):
#!/usr/bin/env python
# -*- coding: utf-8 -*-
import os
import unittest
from mock import patch
class DBThing(object):
moo = "bar"
def __init__(self):
self.moo = "hi"
def foo(self):
return self.moo
class Test_init(unittest.TestCase):
"""Test DBThing init"""
def setUp(self):
pass
def tearDown(self):
pass
def testInit(self):
dbt = DBThing()
self.assertEqual(dbt.foo(), "hi")
def testNoInit(self):
with patch('__main__.DBThing.__init__', return_value=None):
dbt = DBThing()
self.assertEqual(dbt.foo(), "bar")
if __name__ == '__main__':
import nose
nose.run(defaultTest=__name__)
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{"url":"https:\/\/www.sawaal.com\/probability-questions-and-answers\/in-a-class-there-are-15-boys-and-10-girls-three-students-are-selected-at-random-the-probability-that_3285","text":"105\nQ:\n\n# In a class, there are 15 boys and 10 girls. Three students are selected at random. The probability that 1 girl and 2 boys are selected, is:\n\n A) 21\/46 B) 1\/5 C) 3\/25 D) 1\/50\n\nExplanation:\n\nLet , S - \u00a0sample space \u00a0 \u00a0 \u00a0 \u00a0E - event of selecting 1 girl and 2 boys.\n\nThen, n(S) = Number ways of selecting 3 students out of 25\n\n=\u00a0$25C3$\n\n= 2300.\n\nn(E) =\u00a0$10C1\u00d715C2$\u00a0= 1050.\n\n$\\inline \\fn_jvn \\therefore$P(E) = n(E)\/n(s) = 1050\/2300 = 21\/46\n\nQ:\n\nIf the standard deviation of 0, 1, 2, 3 ......... 9 is K, then the standard deviation of 10, 11, 12, 13 ...........\u00a019 will be:\n\n A) K+1 B) K C) K+4 D) K+8\n\nExplanation:\n\n2 449\nQ:\n\nThe standard deviation of the set {10, 11, 12, 9, 8} is\n\n A) 1 B) \u221a2 C) 2 D) 2\u221a2\n\nExplanation:\n\n4 409\nQ:\n\nFind the range of the data 2, 1, 2, 3, 5, 4, 7, 3, 5, 2, 4.\n\n A) 5 B) 4 C) 3 D) 6\n\nExplanation:\n\n0 394\nQ:\n\nFind the median, mode and mean of 9, 5, 8, 9, 9, 7, 8, 9, 8.\n\n A) 9, 9, 9 B) 9, 8, 9 C) 8, 9, 8 D) 8, 9, 9\n\nExplanation:\n\n0 390\nQ:\n\nIn the usual set notation,\u00a0\u00a0=\n\n A) A\u222aB\u222aA\u222aC B) A\u2229B\u222aA\u2229C C) A\u222aB\u2229A\u222aC D) A\u222aB\u2229A\u2229C\n\nExplanation:\n\n0 1271\nQ:\n\nFind the range and mode of the data 17, 18, 28, 19, 16, 18, 17, 29, 18\n\n A) 12 and 18 B) 13 and 18 C) 12 and 17 D) 11 and 17\n\nExplanation:\n\n1 624\nQ:\n\nFind the standard deviation of {11, 7, 10,13, 9}\n\n A) 1 B) 2 C) 4 D) 5\n\nExplanation:\n\n2 631\nQ:\n\nA table tennis player, lost 12 games out of 18 games played. Calculate the games won in terms of decimal.\n\n A) 0.667 B) 0.067 C) 0.50 D) 0.333\n\nExplanation:","date":"2023-03-23 01:19:59","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 2, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 1, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.5691328048706055, \"perplexity\": 897.0670613637827}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 5, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2023-14\/segments\/1679296944606.5\/warc\/CC-MAIN-20230323003026-20230323033026-00189.warc.gz\"}"} | null | null |
\section{Introduction}
\label{sec:introduction}
\IEEEPARstart{D}{etecting} glandular cell (GC) from whole slide image (WSI) can aid pathologists to screen cervical adenocarcinomas (ADCA) and reduce the labour cost by computer-aided diagnosis (CAD) \cite{RN106, pirovano2021computer, li2019bayesian}. With the proper amount of annotated data, the well-trained deep learning model can provide top cells from WSI to help the screening, as a recommending system. The task is challenging, however, that squamous cell (SC) is the major in cervical smear. The sparseness of GC is severe. Otherwise, the morphology of GC and SC is similar while the basic shape of polar pattern of GC is distinct comparing with SC. Such characteristics are the direct hardship to the data-driven model performing accurately without the usage of the prior knowledge of GC basic shape in general models.
\begin{figure}[!t]
\centerline{\includegraphics[width=\columnwidth]{fig1.pdf}}
\caption{Upper figures: Glandular cell (GC) in cervical whole slide image is relatively sparse, where blue box denotes GC, and orange denotes squamous cell (SC). Below figures: Inconsistent polarity of SCs that are folded or clustered is in the left, and illustrations of polar pattern of GC are in the right, where blue arrow denotes polar orientation.}
\label{fig1}
\end{figure}
Although recognition cervical lesion in cytology has been developed since 2016 by using deep learning \cite{pouliakis2016artificial, van2016deep}, related works focus on the major object, SC, but rarely discuss GC, which is minor but important \cite{jiang2022deep}. They researched cervical cancer detection on small image sets with general deep learning methods, only few works mentioned the performance of model on external WSI set. Like Tan et al. \cite{RN168} tested the detection performance of both SC and GC
with the deep learning model in only preseleted ROIs of external WSI. And Gupta et al. \cite{gupta2020region} only reported a rough detection results from external WSIs by their ROI prediction model. No focus on GC in cervical cytology and no usage of prior knowledge of GC both inspire that there is still room of improvement of GC detection using deep learning. The lack of evaluations of cell detection from WSI also leads related works are hardly deployed to the clinical CAD.
More vivid description of this problem is illustrated in Fig. \ref{fig1}, there are only 17 GCs existed in the WSI shown in the upper. This sparseness leads the computation of WSI will meet substantial out-of-distribution (OOD) data including artifacts, single SC, folded SC, and SC cluster. Even though their basic shape has obvious distinction comparing with GC as shown in the below of Fig. \ref{fig1}, general deep learning model is hard to distinguish. Those orange boxes are predicted by the state-of-the-art (SOTA) object detection model, YOLOX \cite{ge2021yolox}. Related works for cervical cell detection are designed based on more abstract knowledge, such as the attention and connection between different scales \cite{cao2021novel, wei2021efficient}, the mixed information of multiple resolutions \cite{cheng2021robust}, the prototype representation of multiple cell categories \cite{RN162} , and using time series information to enhance the cancer detection \cite{zhang2021quantitative}. Despite atypical GC is among their objects, their design concepts have not consider the basic shape of cervical cell. That is hardly helping to reject those false positives with wrong basic shape.
This paper proposed a new network to enhance GC detection with prior knowledge of basic shape of GC. We also provide a new experimental setting to show the practical performance of model. Firstly, a polar attention network (PolarNet) is designed to quantify the confusing morphology of GC, that is the basic shape of polar orientation. The network employs a novel self-attention mechanism in eight-neighbor that enables it to score the salience of polarity orientations of GC. Pseudo-GCs that are not significantly polar in the outer test set are controlled by such polarity role. The network is also a deep learning module that can be plugged into any general object detection model. Then, two additional test sets of small images from the external WSI containing different volume of OOD data are obtained by human and well-trained model, respectively. Furthermore, a elegant deployment of the proposal is completed and provided as a C++ program.
In GC detection experiments on both three small image sets, the proposed PolarNet showed effectiveness, and compared with 5 models including four different frameworks. With the usage of PolarNet, model can reject false positive (FP) and increase the mean of average precision (mAP) from 0.007 to 0.015 in average, where the highest exceeds the recent cervical cell detection model 0.037. The deployed program improved by 8.8\% on accuracy of top-20 GC detection from external WSIs ($n=110$), while sacrificing 14.4 s of computational time.
Briefly, this work proposed a cervical GC detection method to recognize GC from WSI, it consists of four contributions:
\begin{itemize}
\item In Section \ref{sec:datapreparation}, an OOD data set is provided using a well-trained YOLOX. The data is obtained from the severe FP predicted by the deep learning model, and is able to reflect the performance of GC detection from WSIs that are not fully annotated.
\item In Section \ref{sec:method}, a novel eight-neighbor self-attention mechanism is proposed to quantify the polar orientation of GC and to construct the plugin network PolarNet. It can be used in general detection model whether it is single-stage or multi-stage to reject severe FP.
\item In Section \ref{sec:exp}, the usage of PolarNet in four different frameworks both shown significance for GC detection. For clinical application, the proposal also performs effectiveness to recommend top-20 GCs from WSI.
\item The proposed method is deployed as a C++ program. The computational cost of PolarNet is listed in detail. The program is released online.
\end{itemize}
\section{Related Works}
\label{sec:relatedworks}
\subsection{Cervical Cell Detection}
Earlier, the traditional framework was based on cytological definitions and completed cell classification by calculating the nuclear-cytoplasmic ratio of cervical cells, while Tareef et al. To improve the accuracy of subsequent classification tasks, their convolutional neural network (CNN) models surpass traditional machine learning algorithms in segmentation accuracy. But then, Zhang et al. \cite{RN61} believed that the unavoidable segmentation error would always lead to a decrease in the classification accuracy of abnormal cells, and they proposed the DeepPap model for the first time to directly apply the convolutional neural network to the classification task of cervical cells to avoid pre-segmentation processing, and performed well on two public datasets, Herlev 2005 \cite{sukumar2015computer} and HEMLBC \cite{RN59}, showing more than 98\% accuracy on both datasets, but their method still compares with traditional algorithms. more time consuming. Shanthi et al. \cite{RN110} explored in more detail the classification accuracy that can be obtained when using CNN for cervical cell classification, performing cell edge extraction, cell nucleus segmentation, or directly using the original image without segmentation, although their model is described in Herlev 2005. The accuracy rate is only 94\% $\sim$ 95\%, and it does not fully surpass the previous method, but it has been verified that it is the most effective to use the original image directly without segmentation. These early developments and conclusions using deep learning led to a perception of the potential of deep learning in CAD.
Later, with the development of general deep learning models, many general models such as VGG \cite{RN115}, GoogLeNet \cite{szegedy2015going}, ResNet \cite{he2016deep} and Inception \cite{szegedy2016rethinking} pre-trained and validated by the large general dataset ImageNet \cite{deng2009imagenet} appeared. Wait. Lin et al. \cite{RN111} proposed that these pre-trained models from general data can extract general morphological features, and they transferred these pre-trained models to the cervical cell classification task, using GoogLeNet in Herlev 2005 to obtain the highest accuracy of 94.5\%. In order to further improve the computational efficiency of CNN in cervical cell classification, Dong et al. \cite{RN153} proposed to combine a lightweight convolutional neural network with artificial features, by adapting the prior knowledge based on cytology definition to the Inception V3 model \cite{szegedy2016rethinking}, and finally achieved over 98\% accuracy on the public dataset Herlev 2005 \cite{sukumar2015computer}.
Because of the irregular distribution of cells on the slide in cervical cytology images, it is important to predict both the location and class of cells. Recently, Xiang et al. \cite{RN66}, Liang et al. \cite{RN159} proposed a cervical cell detection model earlier based on the single-stage object detection model YOLO (You Only Look Once) \cite{RN218}, and they stacked additional Inception V3, and FPN (Feature Pyramid Network) with content-aware function \cite{lin2017feature}, on their private 10-class cervical cell dataset, end-to-end localization, classification and classification of cervical cells were completed. Prediction size, and the average accuracy reached 63.4\%, although their image pixels are sufficient ($4000\times3000$), but the number of images is small ($n=12909$) is a defect. In order to solve the problem of model training when the amount of data is small, Liang et al. \cite{RN162} subsequently proposed a comparison detector based on a double-stage Faster RCNN. On small-scale ($n=7410$) datasets, the contrastive detectors can achieve significant improvements over the previous ones. The above are all cell images obtained by traditional production methods.
In the work of Tan et al. \cite{RN168}, the authors firstly collected more than 16000 LBC (Liquid-Based Cytology) images as training and validation data. Compared with traditional cervical cytology images, the LBC image has a clearer background \cite{RN89}. They independently set 290 ROI (Region Of Interest) images in the external full slide as the test set to simulate real CAD process. They obtained decent accuracy after training a Faster RCNN model. The experiment of this deep learning model on external data provides preliminary feasibility support for CAD, although the amount of external data is still smaller than the training and validation data.
Since the cervical cell detection task requires a larger field of view of image data than the earlier classification and segmentation tasks, and such public data sets are scarce, the above related object detection research is carried out on private data. More important, whether it is cell segmentation, classification, or cell detection, none of the related works have taken the full calculation of WSI into their experiments.
\subsection{Detection from Whole Slide Image}
The related research extending from local prediction research to WSI calculation is still limited. Gupta et al. \cite{gupta2020region} proposed an automatic ROI identification method for the first time. Although ROI identification can only obtain a rough localization from the WSI, their work considered all the pixels for calculating the WSI, and used all the regions on the WSI for model training, Excellent ROI classification accuracy is achieved on a private dataset containing 10 WSI images, and this study provides an idea of automatically preselecting ROIs for calculating local cancerous cells in WSI.
Cervical cell detection with WSI calculation is also used by several WSI classification researches. Ke et al. \cite{ke2021quantitative} tried to use a nucleus segmentation CNN to guide classification, and then integrated the segmentation results through hand-designed feature engineering. They reported a fabulous WSI classification performance but none of the cell segmentation/classification on external test set. Subsequently, Zhu et al. \cite{zhu2021hybrid} proposed an integrated cervical WSI recognition system, which includes 24 object detection CNNs and other 4 models. To complete the WSI classification task, they also created a new 24 classes to deal with the confusing cervical cancer subclasses, and arranged each object detection CNN for each class. Such a system greatly improves the robustness, showing accuracy and generalization that surpasses pathologists, but it is conceivable that the computing resources required by this system are very large. The author reported its speed is 180 s/WSI. Cao et al. \cite{cao2021novel} designed a new attention module added to the Faster RCNN, a two-stage object detection model, and improved the mAP of cell detection by 2.37\% compared with the baseline. It predicts the confidence of object to weight the features extracted by ResNet to complete WSI classification. None of the performance of cell detection from external/OOD test set is reported. Cheng et al. \cite{cheng2021robust} roughly performed cell detection using low-resolution images, and then performed feature extraction at the location of the object to complete the WSI classification. Wei et al. \cite{wei2021efficient} used a new lightweight object detection model to simultaneously perform cell detection and feature extraction to complete WSI classification.
However, there is rarely a solution and a evaluation when it comes to a complete, cell-level prediction computing all local images on cervical WSI. The training of general deep learning requires the experimental data to conform to the real distribution, but the cervical WSI contains various non-relevant content, that is so-called OOD data, to determine the lesion. Giving such data a full annotation is expensive, scarce, and unnecessary. To develop new method on cell recognition, datasets used in related work are mostly based on local small images generated by pathologists from pre-selected ROIs on WSI. Despite the WSI calculation is completed in WSI classification, their discussion is focus on the classification while the cell detection is just a step of it. Therefore, the related research hardly mentions the generalization and reliability problems that their local prediction model will encounter when calculating the cells of the whole cervical slide, which weakens the contribution of their work.
\section{Method}
\label{sec:method}
This section describes the proposed PolarNet and how this plugin module is working on modern deep learning models.
\subsection{PolarNet}
The PolarNet is designed as shown in Fig. \ref{polarnet}. In the one hand, it obtains a polar attention score matrix by computing the self-attention inside the eight-neighbor of features. On the other hand, it generates a new feature map by weighting with the attention score of different orientations also inside the eight-neighbor area.
\begin{figure}[!t]
\centerline{\includegraphics[width=\columnwidth]{polarnet.pdf}}
\caption{Structure of the proposed PolarNet, where $C$ means the channel number of feature maps. $H$ and $W$ are the height and the width of input feature map, respectively. The dot-product block refers to the Eq. \ref{eq1} and the weighting block refers to the Eq. \ref{eq2}.}
\label{polarnet}
\end{figure}
The PolarNet is designed to compute the last stage of feature maps from the backbone of a modern model. Thus, we can assume the feature maps are denoted by $\mathbf{x}\in \mathbb{R}^{C\times H\times W}$, the polar attention score matrix by $\mathbf{PAS}\in \mathbb{R}^{9\times H\times W}$, and the output feature maps of the PolarNet by $\mathbf{y}\in\mathbb{R}^{C\times H\times W}$. Then, outputs of PolarNet can be writen by follows:
\begin{equation}
\mathbf{PAS}_{\cdot,i,j}=softmax(
\{\mathbf{Q}_{\cdot,i,j}
\odot
\mathbf{K}^{T}_{\cdot,nei[n]}\}_{n=1,2,\dots,9}
),
\label{eq1}
\end{equation}
\begin{equation}
\mathbf{y}_{c,i,j}=norm(\sum_{n=1}^{9}
\mathbf{PAS}_{n,i,j}
\times
(1+\mathbf{V}_{c,nei[n]})
),
\label{eq2}
\end{equation}
where $i,j$ means the coordinator of $i^{th}$ row and $j^{th}$ column in feature maps, $c$ means the $c^{th}$ channel of feature maps, $\odot$ means the operation of dot-product, $nei\in \mathbb{N}^{9\times2}$ is the index set of eight-neighbor $[(i-1,j-1),(i-1,j),(i-1,j+1),(i,j-1),(i,j),(i,j+1),(i+1,j-1),(i+1,j),(i+1,j+1)]$. $\mathbf Q\in \mathbb{R}^{C\times H\times W}, \mathbf K\in \mathbb{R}^{C\times H\times W}, \mathbf V\in \mathbb{R}^{C\times H\times W}$ means the query, the key, and the value of feature maps, respectively, for computing the self-attention:
\begin{equation}
\mathbf{Q}=Conv^1_{1\times1}(\mathbf x),
\label{eq3}
\end{equation}
\begin{equation}
\mathbf{K}=Conv^2_{1\times1}(\mathbf x),
\label{eq4}
\end{equation}
\begin{equation}
\mathbf{V}=Conv^3_{1\times1}(\mathbf x).
\label{eq5}
\end{equation}
\subsection{Framework of GC Detection}
\label{sec:featscale}
In the general object detection task, multiple models are designed by using different architectures. Such as the anchor-based single-step architecture of YOLO series \cite{redmon2016you}, the multiple-step Faster RCNN \cite{girshick2015fast} and Cascade RCNN \cite{cai2018cascade}, and the anchor-free FCOS \cite{tian2019fcos}. They are both structured by the backbone and the head, where the backbone can employ a neck, like a FPN \cite{lin2017feature} or a PAFPN \cite{liu2018path}, to mix multi-scale feature maps.
Particularly, whether using the neck in the backbone or not, the PolarNet is plugging into the end part. It computes the last stage of feature maps, the $5^{th}$ stage (downsampling ratio is $2^5=32$), to generate $\mathbf y$ and $\mathbf{PAS}$. The new feature maps $\mathbf y$, then, input to the head of model to predict bounding boxes of GC.
\begin{figure}[!t]
\centerline{\includegraphics[width=\columnwidth]{polarorientation.pdf}}
\caption{Polar orientations in the eight-neighbor. (a) Using the mean of polar attention score to represent the polar salience of GC. (b) Illustrating orientations in one eight-neighbor of one case of GC. }
\label{polarorientation}
\end{figure}
\begin{figure*}[!t]
\centerline{\includegraphics[width=1.8\columnwidth]{framework.pdf}}
\caption{Glandular cell detection framework with PolarNet, where $H_{in}$, $W_{in}$ are the height and the width of input image, respectively.}
\label{framework}
\end{figure*}
The reason of using the last stage of feature maps is pursuing the best fitness between the area size of one feature vector and the physical size of GC. In the PolarNet, polar orientations are split evenly into eight orientations with a self-referential orientation as shown in Fig. \ref{polarorientation}. To properly estimate the true orientation of GC by the PolarNet, one feature vector of input maps should represent the area of around $\frac{1}{3}\sim \frac{2}{3}$ length of GC. Downsampling ratio in the $5^{th}$ stage is just hitting this range. Under the resolution of image in this work, $0.2499\ \mu m/pixel$, one $5^{th}$ stage feature vector represents the $7.9968\ \mu m$ length area, which is $\frac{1}{3}\sim \frac{4}{5}$ times of GC, such as the case in Fig. \ref{polarorientation} (b). Thus, earlier stages, like $4^{th}$ with the ratio $2^4$ or $3^{th}$ with $2^3$, are unsuitable to predict the polar orientation of GC by the self-attention mechanism of eight-neighbor of the PolarNet.
Finally, based on the above knowledge, the GC detection framework can be illustrated as Fig. \ref{framework}. The confidence of GC bounding box is updated by
\begin{equation}
P=(1-\alpha)P_{obj} + \alpha P_{polar},
\label{eq6}
\end{equation}
where $\alpha\in [0,1]$ is the weight of polar salience, and $P_{obj}$ means the objectiveness confidence of original output of model. $P_{polar}$ is the result by converting $\mathbf{PAS}$ matrix to one scalar
\begin{equation}
P_{polar}=\frac{1}{8b_wb_h}\sum_{i=b_x-\frac{b_w}{2}}^{b_x+\frac{b_w}{2}}\sum_{j=b_y-\frac{b_h}{2}}^{b_y+\frac{b_h}{2}}\sum_n^{1\sim4,6\sim9}\mathbf{PAS}_{n,i,j},
\label{eq7}
\end{equation}
where $b_x,b_y,b_w, b_h$ mean the coordinator and the size of one bounding box. It is clear that Eq. \ref{eq7} computes the mean of polar attention score to represent the polar salience of GC as shown in Fig. \ref{polarorientation} (a).
In the training phase, the PolarNet is supervised by cross entropy as similar as the objectiveness loss in modern object detection model
\begin{equation}
L_{PolarNet}=log(NLLLoss([P_{non-polar}\ P_{polar}], P_{gt})),
\label{eq8}
\end{equation}
where $P_{gt}$ means the objectiveness ground truth of one bounding box, and the operation combination of $log$, $NLLLoss$, and $softmax$ in Eq. \ref{eq1} forms the cross entropy loss with
\begin{equation}
P_{non-polar}=\frac{1}{b_wb_h}\sum_{i=b_x-\frac{b_w}{2}}^{b_x+\frac{b_w}{2}}\sum_{j=b_y-\frac{b_h}{2}}^{b_y+\frac{b_h}{2}}\mathbf{PAS}_{5,i,j}.
\label{eq9}
\end{equation}
At this point, any modern model in the object detection task is able to use the prior knowledge, the ubiquitous polarity of GC, to reject false positives of GC from WSI.
\section{Data Preparation}
\label{sec:datapreparation}
This work uses total 486 cervical cytology WSIs from Tongji Union Hospital, Huazhong University of Science and Technology. The scanner used a $20\times$ objective lens with a resolution of $0.2499\ \mu m/pixel$, and used Qupath software \cite{bankhead2017qupath} to complete the local annotation of cervical GC in WSIs.
According to cervical GCs are more sparse than squamous cells, three different sources are set up to make the image datasets for full validation and testing. As shown in Table \ref{tab:data}, the three sources are GC annotation, non-relevant content (NC) annotation and false positive (FP).
Among them, the GC annotation refers to the area containing glandular cells (clumps) in the cervical slide that is judged to be positive (with the presence of significant atypical GCs), and has two subclasses: AGC (Atypical Glandular Cells) and nGEC (normal Glandular Epithelial Cells).
Non-relevant content (NC) annotation refers to some small areas from negative WSIs (no significant atypical glandular cells are present) that do not contain any GC.
False positive (FP) refers to the wrong predictions in the test set of GC annotation source by a modern model, YOLOX-l \cite{ge2021yolox}. The source of the first GC annotation is provided by the pathologist, and the latter two are generated by authors. All annotations have been reviewed by the pathologist for the double check.
\begin{table}
\centering
\caption{Data set details.}
\label{tab:data}
\setlength{\tabcolsep}{3pt}
\begin{tabular}{lccccc}
\hline
source & & slide num. & image num. & AGC num. & nGEC num. \\
\hline
\multirow{3}*{GC ann.} & train & 221 & 8274 & 8759 & 17190 \\
& val & 66 & 1032 & 258 & 1331\\
& test & 44 & 779 & 226 & 1364 \\
NC ann. & test & 45 & 1280 & 0 & 0 \\
FP & test & 110 & 3496 & 0 & 0 \\
\hline
total & & 486 & 14861 & 9243 & 19885\\
\hline
\multicolumn{6}{p{210pt}}{Note: num. means number, ann. means annotation, NC means non-relevant content, FP means false positive by a modern model.}
\end{tabular}
\end{table}
Images listed above are used with the same size, $1024\times1024$, and the same original resolution.
\begin{itemize}
\item While the images are from GC annotations, first, those GC annotations on a slide that are close enough (smaller than the size of image) are used as a set. Then, a larger cell image is cropped from WSI centered on such set with the size of $1536 \times 1536$ and saved in the hard disc. During training, $1024 \times 1024$ regions are randomly cropped from the larger images to ensure the diversity of learning samples, and $1024 \times 1024$ are cropped from the center for validation and testing.
\item While the images are from non-relevant content annotation, $1024 \times 1024$ cell images are cropped from WSI, randomly, for testing.
\item The false positive source is obtained by a YOLOX-l \cite{ge2021yolox} model that trained with the first source, GC annotations, in our data. It is from external positive WSIs ($n=110$) that are entirely inferred by the well-trained YOLOX-l. The top bounding boxes ranking by the confidence (named as top-$N$ results) of every WSI are visualized using the Qupath tool, and manually judged one by one. Results that are not GCs are then cropped in the same way of above sources for testing. During the manual judgement, all top-100 in the first 30 slides and 20 random results of top-100 in the last 80 slides are reviewed.
\end{itemize}
\begin{figure}[!t]
\centerline{\includegraphics[width=\columnwidth]{dataexample.pdf}}
\caption{Examples of GC used in this work. (a) Image with nGEC annotations (green boxes). (b) Image containing non-relevant content. (c) Image with false positives (orange boxes) by the well-trained YOLOX-l.}
\label{dataexample}
\end{figure}
\begin{figure*}[!t]
\centerline{\includegraphics[width=2\columnwidth]{pr_curve.pdf}}
\caption{Precision-recall curves on three test data setting.}
\label{pr_curve}
\end{figure*}
Among the results of above manual judgement, it is interesting that the average accuracy of first 30 is only 0.1445. Although this model is very popular in the industry, the accuracy of only 2 slides exceeded 0.9 due to the fact that the cervical GC's prior knowledge is not taken into the model. It is worth noting that this phenomenon of external testing on YOLOX-l indicates that standard experimental results, which shown an average precision of more than 0.7 on the validation set, are unreliable when facing the sparse GC from real WSI. Some examples are shown in Fig. \ref{dataexample}.
It is worth mentioning that testing with the non-relevant and false sources, the images without the presence of GCs accounted for 62.17\% and 81.78\% of the total, respectively. Therefore, the unreliability of the model can be fully demonstrated in both. In fact, the proportion of images without GCs in the whole slide generally exceeds 90\%, so testing with those two sources can simulate the performance of deep learning models for GC detection from WSI.
\section{Experiments}
\label{sec:exp}
\subsection{Evaluation Criteria}
First of all, for the GC detection from $1024\times 1024$ image, this work uses the common evaluation criteria, AP50 (Average Precision with 50\% IoU threshold), and its calculation formula is
\begin{equation}
\text{Recall} = \frac{TP}{TP+FN}, \text{Precision} = \frac{TP}{TP+FP},
\label{eq10}
\end{equation}
where $TP$, $FP$, and $FN$ are true positive, false positive, and false negative, respectively.
Since AP50 is obtained by calculating the area under the Precision-Recall curve, we also provide the curve to show the GC detection performance of models in detail.
Second, for the task of GC detection from cervical cytology WSI, this work uses the top-$N$ accuracy of GC to demonstrate the reliability of models. It is calculated by $\frac{TP}{2(TP+FP)} + \frac{TN}{2(TN+FN)}$ with $TN$ the true negative.
\subsection{Training Environments}
All the experiments in this work are implemented with the PyTorch deep learning library \cite{paszke2019pytorch} on a Win10 OS computer. Model training is performed using the SGD optimizer \cite{bottou2010large}, and the learning rate descent strategy is the common stepwise, from $5 \times 10^{-3}$ to $5 \times 10^{-6}$, learned at epochs 25, 50, and 80, respectively. The rate decay is 0.1, and the maximum training epoch is 100. Memory is 128 Gb, CPU is a Xeon® 6134 @3.20 GHz, GPU is a Tesla P40. The comparison methods are trained using the code published by the relevant research, and the general popular model uses the implementation version of TorchVision model zoo. When using the proposed PolarNet, only one hyper-parameter $\alpha$ is always set as 0.5.
\subsection{Comparing Methods}
Regarding the comparison methods, since YOLOX and Faster RCNN are representative single-stage and double-stage methods commonly used in general tasks. AttnFaster \cite{cao2021novel}, that is specially designed for cervical cytology based on attention mechanism, is also chosen to test. In addition, more existing models are compared: one anchor-free model FCOS, and one multi-stage model Cascade RCNN \cite{cai2018cascade}.
\subsection{GC Detection Results}
As mentioned in Section \ref{sec:datapreparation}, the experimental data in this work were produced by three different sources, each of which represented different levels of sparsity and difficulty of GC detection, to verify the effectiveness of models on real distribution of data from WSI. For fully comparing the different testing scenarios, this section first shows the test results on all three sources of data, and then shows the detailed test results of GC, which has two subclasses.
\begin{table}
\centering
\caption{GC Detection AP50.}
\setlength{\tabcolsep}{3pt}
\begin{tabular}{lccccl}
\hline
\multirow{2}*{Model Name} &
val &
\multicolumn{3}{c}{test} &
\multirow{2}*{average of test}\\
& GC & GC & GC+NC & GC+FP &
\\
\hline
AttnFaster + R50 & 0.570 &0.438 & 0.319 & 0.238 & 0.332\\
\hline
FCOS + R50 & 0.323 & 0.272 & 0.183 &0.112 & 0.189\\
$\sim$ w/ PolarNet & 0.354 & 0.312 & 0.181 & 0.096 & 0.196 (\textcolor{green}{+0.007})\\
YOLOX-m & 0.592 & 0.358 & 0.315 & 0.275 & 0.316\\
$\sim$ w/ PolarNet & 0.528 & 0.413 & 0.355 & 0.292 & 0.353 (\textcolor{green}{+0.037})\\
Faster RCNN + R50 & 0.499 & 0.422 & 0.360 & 0.290 & 0.357\\
$\sim$ w/ PolarNet & 0.513 & 0.440 & 0.372 & 0.295 & \textbf{0.369} (\textcolor{green}{+0.012})\\
Cascade RCNN + R50 & 0.437 & 0.294 & 0.264& 0.208 & 0.255\\
$\sim$ w/ PolarNet & 0.444 & 0.328 & 0.271& 0.210 & 0.270 (\textcolor{green}{+0.015})\\
\hline
\end{tabular}
\label{tab:det-res}
\end{table}
\subsubsection{Main results}
The GC detection results from small images are shown in Table \ref{tab:det-res}. It is obvious that all SOTA models with PolarNet appear the improvement of AP50 comparing to their original version.
That can show the effectiveness of PolarNet. And, Faster RCNN with PolarNet always shows the highest AP50 on every test setting and the average. YOLOX with PolarNet is able to exceed AttnFaster in average by 0.021, while the original YOLOX is 0.016 lower than AttnFaster.
Interestingly, comparing the results between three different test settings, the AP50 of only GC source can maintain a similar level of AP50 as the validation set. But after adding images without target presence (non-relevant or false positive source), AP50 appears some relatively significant drops. Even for the most complex Cascade RCNN, which performs best on the general dataset, the AP50 of test set GC+FP is only 0.255.
Due to the lower AP50 scores of FCOS and Cascade RCNN among the general models, we use the more powerful and more general YOLOX and Faster RCNN for subsequent experiments.
\subsubsection{Precision-recall curves}
The precision-recall (P-R) curves of above models are shown in Fig. \ref{pr_curve}. It can be seen that the green of PolarNet used in the P-R curve of the Faster RCNN model has higher precision than orange and blue, and the red P-R curve of YOLOX is also lower in precision than adding PolarNet. YOLOX. For Recall, P-R curves for the same model infrastructure all exhibit similar maximum recall. While two YOLOXs' recalls are significantly lower than Faster RCNN, their precisions are higher when curves start. The relatively low precision at the beginning of the curve means it will be less effective when encountering external test sets. This is because the curve is drawn with the results sorted by confidence, and hence the results with high confidence are less accurate when that happened. It can be seen that YOLOX after adding PolarNet is improved at the beginning of the curve.
\subsubsection{Detailed results}
In object detection task, the calculation of AP criteria depends on the threshold of the intersection over union between bounding boxes of predictions and annotations. For example, AP50 used above needs to be greater than 50\% to be judged as $TP$. Obviously, a larger threshold can reflect the accurate size and position of predictions. As shown in Table. \ref{tab:detailed-res}, in order to demonstrate the accuracy of sizing and positioning of predictions, we list the results in AP60 and AP70. It can be seen that, similar to the trend in Table. \ref{tab:det-res}, AP60 and AP70 of Faster RCNN with PolarNet still both perform best on test set of the GC+FP source, while YOLOX with PolarNet ranks third in AP60 and AP70 . Obviously, compared to AP50, all models show a small drop at AP60 and a significant drop at AP70. This shows that the accurate detection of cervical GC is still a challenging problem. Furthermore, AP50 of two subclasses are also shown in Table. \ref{tab:detailed-res}. The two models using PolarNet performed the second with 0.112 and the first with 0.336 in AGC and nGEC, respectively. And, AP50 of AGC is generally lower than that of nGEC, because the number of AGC annotations in the training data is less than half of that of nGEC.
\begin{table}[!t]
\centering
\caption{Detailed results of GC detection.}
\label{tab:detailed-res}
\setlength{\tabcolsep}{3pt}
\begin{tabular}{lccccc}
\hline
Model Name &
nGEC &
AGC &
AP50 &
AP60 &
AP70\\
\hline
AttnFaster & 0.267 & 0.079 & 0.238 & 0.168 & 0.077\\
Faster RCNN & 0.328 & 0.075 & 0.290 & 0.207 & 0.091\\
$\sim$ w/ PolarNet & \textbf{0.336} & 0.083 & \textbf{0.295} & \textbf{0.210} & \textbf{0.093}\\
YOLOX-m & 0.316 & \textbf{0.184} & 0.275 & 0.184 & 0.080\\
$\sim$ w/ PolarNet & 0.325 & 0.112 & 0.292 & 0.190 & 0.078\\
\hline
\multicolumn{6}{p{170pt}}{Note: results of subclasses nGEC and AGC are both AP50.}
\end{tabular}
\end{table}
\begin{table}[!h]
\centering
\caption{WSI GC Detection Top-N Accuracy in Average.}
\label{tab:wsi-res}
\setlength{\tabcolsep}{3pt}
\begin{tabular}{lcccc}
\hline
Model Name &
$N_{pred}$ &
Top-5 &
Top-10 &
Top-20\\
\hline
AttnFaster & 20.00 & 0.281$\pm$0.311 & 0.284$\pm$0.274 & 0.271$\pm$0.241 \\
Faster RCNN & 20.00 & 0.424$\pm$0.344 & 0.410$\pm$0.303 & 0.355$\pm$0.261 \\
$\sim$ w/ PolarNet & 20.00 & \textbf{0.429}$\pm$0.364 & \textbf{0.411}$\pm$0.344 & \textbf{0.386}$\pm$0.282 \\
YOLOX-m & 18.89 & 0.117$\pm$0.236 & 0.0.108$\pm$0.212 & 0.122$\pm$0.204 \\
$\sim$ w/ PolarNet & 20.00 & 0.190$\pm$0.283 & 0.205$\pm$0.267 & 0.126$\pm$0.149 \\
\hline
\multicolumn{5}{p{230pt}}{Note: $N_{pred}$ means the average number of predictions in every WSI, $\pm$ followed by the standard deviation of accuracies of all WSIs.}
\end{tabular}
\end{table}
\subsection{WSI GC Detection Top-N Results}
When the cervical GC detection model is practically applied to cervical cancer CAD, the top-$N$ results can guide pathologists to prioritize the screening of suspicious lesions. Therefore, the top-$N$ results of models on those external slides ($n=110$) can demonstrate its performance from the perspective of cervical cancer auxiliary diagnosis on WSI. This section shows the accuracy of top-$N$ results, with higher accuracy indicating greater potential for the model to be applied to CAD.
As shown in Table. \ref{tab:wsi-res}, the accuracy of top-$N$ results when $N=5$, $10$, and $20$ are listed respectively. It is clear that Faster RCNN with PolarNet performs the best among the three different $N$. As $N$ decreases, its accuracy increases from 0.386 to 0.429. This shows that it has a reasonable confidence distribution, such as higher confidence results have higher accuracy. In fact, Faster RCNN is the only remaining model with the same trend, and the other remaining three models show a drop in accuracy when using higher confidence results. In YOLOX models, its accuracy is much lower than the other models as shown in Fig. \ref{wsi-detconf}. This situation can also be reflected from $N_{pred}$, the average number of predictions. There are some slides that cannot give the complete 20 detection results, even though YOLOX used a very low confidence threshold (0.0001). Although adding PolarNet can alleviate this situation, the
top-$N$ accuracy of YOLOX with PolarNet is still at least about 0.08 lower than the double-stage Faster RCNN. This shows that the double-stage model is the most reliable when faced with real external WSI comparing with others.
\begin{figure}[!b]
\centerline{\includegraphics[width=0.9\columnwidth]{wsi-detconf.pdf}}
\caption{The box plot showing the distribution of WSI GC detection top-20 results.}
\label{wsi-detconf}
\end{figure}
\subsection{Ablation Studies}
In order to further reveal the role of PolarNet in object detection models and prove the effectiveness of polar attention for cervical GC detection, two ablation studies are performed in this section:
1) The contribution of $P_{polar}$, the mean of polar attention score, in the confidence $P$ updating Eq. \ref{eq6}.
2) The effect of feature scale (see Section \ref{sec:featscale}) on the estimation of GC polarity in PolarNet.
\subsubsection{The contribution of $P_{polar}$}
Based on Eq. \ref{eq6} that changing the value of $\alpha$ can change the contribution of $P_{polar}$. Especially, when $\alpha = 0$, $P_{polar}$ contributes nothing in the role of PolarNet but only the new feature maps guided by polar attention. Therefore, observation of the $\alpha$-AP50 curve can simultaneously demonstrate the effectiveness of $P_{polar}$ and polar attention matrix.
Fig. \ref{ablation-alpha} shows the $\alpha$-AP50 curve by two types of model. It can be seen that whether it is a single-stage YOLOX or a multi-stage Faster RCNN, the AP50 raises when $\alpha$ increasing, and reaches a peak after the default setting $\alpha = 0.5$ in this work. This indicates that the polar score provides a positive contribution to the GC detection results. Moreover, it can be seen that when $\alpha = 0$, the AP50 of the model is still superior to the results of other models in Table. \ref{tab:det-res} on the three test settings. This shows that polar attention also provides an important contribution to the feature representation of the model.
\begin{figure}[!t]
\centerline{\includegraphics[width=\columnwidth]{ablation-alpha.pdf}}
\caption{$\alpha$-AP50 curves to show the significance of $P_{polar}$, the mean value of polar attention score.}
\label{ablation-alpha}
\end{figure}
\begin{figure*}[!b]
\centerline{\includegraphics[width=2\columnwidth]{detexample.pdf}}
\caption{GC detection examples by four models to show the ability of rejection of false GCs after using PolarNet, where green boxes are ground truth, oranges are false positives, and blues are true positives.}
\label{detexample}
\end{figure*}
\subsubsection{The effect of feature scale}
As described in Section. \ref{sec:featscale}, since the physical size of a single GC is relatively fixed, PolarNet calculates its polarity orientation by feature maps that are sensitive to the scale of GC. This part verifies that the scale, $2^5$, in $5^{th}$ stage that covers about $\frac{1}{3}\sim\frac{4}{5}$ of the cell size obtains optimal polar attention. We test all available feature scales for both the single-stage model and the double-stage model, showing the effect of feature scale on PolarNet.
As shown in Table. \ref{tab:ablation-res}, the results AP50 of cervical GC detection using different scales of feature for PolarNet. Obviously, the model using the largest scale, $2^5$, performs the best on all test settings. This shows that the theoretical foundation for computing polar attention is valid. In addition, it can be seen that after PolarNet uses smaller scales, most of the AP50, such as non-bold and non-underlined, are even lower than the corresponding original model shown in Table. \ref{tab:det-res}. It shows that when PolarNet uses an inappropriate scale, the polarity calculation via self-attention mechanism in eight-neighbor no longer has a positive effect.
\begin{table}[!t]
\centering
\caption{GC Detection AP50 on Test Set using Different Feature Scales.}
\label{tab:ablation-res}
\setlength{\tabcolsep}{6pt}
\begin{tabular}{p{7pt}cccccc}
\hline
\multirow{2}*{Scales} &
\multicolumn{3}{c}{Faster RCNN w/ PolarNet} &
\multicolumn{3}{c}{YOLOX-m w/ PolarNet} \\
& GC & GC+NC & GC+FP & GC & GC+NC & GC+FP \\
\hline
$2^5$ & \textbf{0.443} & \textbf{0.376} & \textbf{0.295} & \underline{0.413} & \textbf{0.355} & \textbf{0.292} \\
$2^4$ & 0.382 & 0.321 & \underline{0.256} & \textbf{0.436} & \underline{0.324} & \underline{0.222} \\
$2^3$ & 0.396 & 0.320 & 0.228 & 0.343 & 0.266 & 0.201 \\
$2^2$ & \underline{0.444} & \underline{0.338} & 0.247 & - & - & - \\
\hline
\end{tabular}
\end{table}
\subsection{Qualitative Results}
Finally, Fig. \ref{detexample} shows the detection examples of cervical GC for the four models for comparison, including the detection examples of two groups of independent GCs and another two of GC clusters. First, in all examples by every model, it is difficult to avoid giving false GCs (orange box), which is caused by the sparseness of GC in WSI. Then, compared with Faster RCNN and YOLOX-m, pulgging PolarNet to modern models can generally reduce the confidence of false bounding boxes of GC. In these four sets of examples, except for the Faster RCNN in the second row, all the others have removed false positives (orange boxes) by PolarNet's scoring of polar attention, and in some cases even corrected false negatives, such as the first and second lines, where new blue box is appearing. These examples visually illustrate the role of PolarNet in GC detection.
\subsection{Computational Costs}
In additional, this work also complete the C++ deployment of the trained cervical GC detection model.\footnote{C++ program is released in \href{https://github.com/Chrisa142857/You-Only-Look-Cytopathology-Once/tree/main/cpp}{https://github.com/Chrisa142857/You-Only-Look-Cytopathology-Once/tree/main/cpp}} The C++ program is able to uses any modern model to complete the cervical GC detection from WSI. The only need is to change the model path and set some hyper-parameters in the command line. To test computational costs, this section involves 5 object detection models that tested in above experiments: AttnFaster, Faster RCNN, Faster RCNN with PolarNet, YOLOX-m, and YOLOX-m with PolarNet.
Table. \ref{tab:compcost} shows computational costs using the proposed PolarNet in our C++ program in detecting GCs from WSIs. Although the YOLOX model can achieve the minimum average time of 107.6 s, its accuracy drops severely, only 12.2\%. The best model Faster RCNN with PolarNet improves the accuracy by 3.1\% compared with its original version by sacrificing the time-consuming of 14.4 s in average.
\begin{table}[!t]
\centering
\caption{Computational Costs of GC Detection from WSIs ($n=110$).}
\label{tab:compcost}
\setlength{\tabcolsep}{7.5pt}
\begin{tabular}{p{50pt}lll}
\hline
\makecell[c]{Model Name} &
\makecell[c]{Total Time} &
\makecell[c]{Average Time} &
\makecell[c]{Top-20 Acc.} \\
\hline
AttnFaster & 6.8 hr & 221.3 s & 27.1\%\\
Faster RCNN & 7.5 hr & 244.3 s & 35.5\%\\
$\sim$ w/ PolarNet & 7.9 hr (\textcolor{red}{+0.4}) & 258.7 s (\textcolor{red}{+14.4}) & \textbf{38.6\%} (\textcolor{green}{+3.1})\\
YOLOX-m & \textbf{3.3 hr} & \textbf{107.6 s} & 12.2\%\\
$\sim$ w/ PolarNet & 4.6 hr (\textcolor{red}{+1.3}) & 151.1 s (\textcolor{red}{+43.5}) & 12.6\% (\textcolor{green}{+0.4})\\
\hline
\end{tabular}
\end{table}
\section{Conclusion}
\label{sec:conclusion}
Current cervical cell detection works generally do not consider the morphology and sparseness of GC in WSI. But the slender shape of GC is easily confused, which makes the general model have serious false positives when detecting WSI. In order to improve the reliability of GC detection from WSI, a PolarNet is designed in this paper. It evaluates the polarity of cells by calculating eight-neighbor self-attention, and generates polarity score and polarity attention to guide confidences and feature maps, respectively, to eliminate pseudo-GC with insignificant polarity. The experimental results show that PolarNet can improve the Top-20 accuracy of GC detection from WSI by 8.8\% on the presence of sacrificing the computational time per slide of 14.4 s.
\section{Discussion}
\label{sec:discuss}
In this paper, a novel network PolarNet is proposed, which can effectively eliminate pseudo-GCs with insignificant polar orientations in the detection results, and more reliably complete GC detection from WSI. The network obtains the polar average score in the detection frame by calculating the eight-neighborhood attention score in the feature map to judge whether the polarity direction of the cell is significant, and can be added to any general model as a module. In order to obtain suitable polarity orientations, it uses the appropriate scale of the $5^{th}$ stage feature map, so that the polarity orientations can be obtained from general object detection model to cover $\frac{1}{3} \sim \frac{4}{5}$ shape of a single GC. In experimental sections, in order to demonstrate the detection performance from WSI, PolarNet was first tested on three small images with different degrees of sparseness of GC and showed superior average accuracy, and then top-$N$ accuracy was also shown the cervical GC detection from WSI. In addition, two ablation studies further demonstrate the positive effect of polar attention score and the new polar guided feature maps of PolarNet. In general, the PolarNet designed based on the prior knowledge of cervical GC can effectively improve the reliability of GC detection from WSI with OOD data.
However, there is still room to improve the proposal. The current PolarNet obtains the polar attention score by calculating the feature map at the fixed scale, which is too sensitive to the change of scale as shown in the second ablation study. Although the scale of a single cervical WSI is hardly to change, the network suffers when encountering data from multiple sources. Therefore, it is very important to further develop scale-invariant PolarNet.
Since the deployment of C++ program is working nice, further clinical experiments could be carried out to verify the effectiveness in real-world tasks. The proposed program has a good improvement with a slight time cost under current computational resources, and its usefulness and weakness can be further revealed by running to the actual cervical cancer CAD.
\section*{Acknowledgment}
The authors want to thank pathologists and organizations that
provided the raw data and the manual annotations. As well
as the Collaborative Innovation Center for Biomedical Engineering and the Britton Chance Center and MOE Key Laboratory for Biomedical Photonics should be greatly appreciated
for their platforms and devices. This work is supported by the National Natural Science Foundation of China (NSFC) projects (grant 61721092), China Postdoctoral Science Foundation (grant 2021M701320) and the director fund of the Wuhan National Laboratory for Optoelectronics
(WNLO).
\bibliographystyle{IEEEtran}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 5,371 |
A woman gathered her Tinder matches for a DIY dating show
By Mary Ward
A New York woman has made headlines after she invited over 100 men on the same date, before attempting to stage her own Bachelorette-style dating show.
Natasha Aponte, whose Instagram account describes her as an "actress, model [and] singer", appears to have invited a large number of her Tinder matches to the same Union Square date, allegedly to watch a friend of Aponte's perform a DJ set.
Natasha Aponte gathered over 100 men to unwittingly participate in a competiton for a date. Credit:Twitter/bvdhai
But, when the men arrived, it was Aponte herself who was on stage, announcing that she was sick of using dating apps and, instead, wanted to give the men an opportunity to compete for her love.
"I have a confession to make," Aponte told the crowd, according to multiple reports. "Everyone here today was brought here to be on a date with me."
Twitter user @bvdhai retold how he came to be one of the duped men in a series of tweets on Monday.
He said he had previously been messaging Aponte on Tinder weeks prior when she told him she was busy with work and needed to take a break.
"Then yesterday I get a text 'HEY IM FINALLY FREE LMAO... my friend is DJing near Union square around 6 tomorrow I was wondering if you wanted to go for a bit then we could go get drinks after and see what happens."
Despite being put off by Aponte's attention-seeking social media presence (she has since put her Instagram account on private, although the profile picture of her glamour modelling in a bikini and her latest musical effort on Soundcloud both remain), @bvdhai went to Union Square for the Sunday evening date.
"Eventually I see a girl moving towards backstage with two enormous bodyguards in sunglasses and realise it's her," he recounted.
New Jersey man Amir Pleasants, 21, was also invited on the mass date. Speaking to the New York Times, he estimated Aponte had invited between 150 and 200 men.
"[We were] all kind of looking at each other trying to figure out the situation. That's when we know we kind of got set up."
Of the men who stuck around (by all accounts, not a huge number), she turned away everyone shorter than 5'10" as well as everyone called Jimmy (a criteria she probably could have already applied, given this information is available on a Tinder user's profile).
She then eliminated everyone who had been left by their previous partner.
According to the New York Times, the (few) remaining men were then instructed to perform 30 pushups on the spot, before lining up so that Aponte could "swipe left" on anyone remaining she wasn't interested in.
The men then competed in a running race before each giving a one-minute spiel about why they would want to date her.
Multiple outlets have reported a man did win the competition, although his identity remains unknown.
The New York Times contacted the DJ who played before Aponte's stunt, who said he had signed a nondisclosure agreement. According to the Times, he "directed questions" to YouTube viral video mastermind Rob Bliss, known for creating the world's largest pillow fight among other social media stunts.
Bliss told the Times he would be providing more information about how and why the mass date came to be on Thursday.
There are very few pictures of the event, strange given its novelty and the number of men present. The only image on social media was taken by @bvdhai and shows Aponte from the behind.
Social media users reacted with a mix of confusion, skepticism and awe to the story, with some praising Aponte for making the most of her dating app.
Mary Ward is Deputy Lifestyle Editor of The Sydney Morning Herald and The Age. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 6,045 |
35 Burst results for "Durham North Carolina"
Game of Crimes
08:12 min | 10 months ago
"durham north carolina" Discussed on Game of Crimes
"You're funny, Murph. Right, always before we get started, Martha has to say something smart. But be in Venice also make us. Then we need welcome to the next episode of narcos, Kelly, oh, no, that's the Patreon stuff. And if you're not subscribed to Patreon, you need to be game of crimes, Patreon dot com slash game crime. But anyway, we've got another stud with us today. Murph excluded. And this guy, we now a while back, we did an episode with Luis Navi, if you guys will remember, it was a book called pure narco. And it was episode 23, by the way. And it was Luis had an interesting history. Coming from Cuba, knowing Castro, you know, was it too Steve Meyer lansky? They had a connection with Meyer lansky. Oh, yeah. He was very he and his family are very wealthy and very well connected. But all good things must come to an end. And as we were discussing for drug traffickers, there's only one of two outcomes you either die or you go to federal prison. And this is the story about how Luis went to federal prison. Courtesy of our next guest, welcome to the podcast, Eric kalinske. Woohoo. Thank you guys. Glad to be here. And I think it's co binsky, right? What did I say? That's the correct. There you go. Come on. That's the way you spelled it. That's why I pronounced it. Look at Murphy's writings too small. You can't even read it. I've heard it pronounced. So many different ways. It doesn't face me. We're just going to use special K, Eric's special K all right, so hey, but this is going to be fun too because we got Luisa's view of what happened in his side of growing up and how things came to pass. And how he came to meet you. In a southern country in the southern hemisphere, after he had been transported, getting ready to ship out 26 tons of cocaine, I believe, to Europe. And Eric, you've ruined things for Luis, but according to Luis and we're seriously too. He even said it on the podcast. You saved his life. Had you not done that? He'd be a dead man because he should have been dead a couple other times, right? Who did he get kidnapped by Steve that nobody used to survive? And he survived. I still shocked that he survived up encounter. It was a stunt double. His stunt double got kidnapped by. Oh my gosh. So, you know, and just to let you know, Eric and I had the pleasure of working in Miami together. That's how we know each other. This is like, hey, everybody knows me. It's a pleasure, right? Listeners. Absolutely. Shutting down. This is like old home week bringing an old friend on here. So Eric, thank you so much for being here, brother. Like I said, glad to be here. Looking forward to it. This should be fun. All right, it should be as long as Murph just doesn't make these corny jokes like he's been known to make. So that's like the pop called the kettle black there, mister. Hey, hey, hey, hey, it's all about the narrative. So let's speaking a narrative. Let's talk about you for a minute, Eric. So as we always do, we want to talk about this thing of ours. This thing called law enforcement. So give us kind of an idea. How the hell did Eric coblin coal binsky? See, I said it right. Cole binsky, special K, how did you get involved in law enforcement? Where did this urge itch desire come from? Well, that's an interesting question because many times have reflected back in my life and asked myself the same thing. What the hell was I thinking? Well, I actually, I got a degree in agriculture from NC state university and at the time I was considering going on to graduate school. In particular, I was interested in working overseas in international agriculture. But I had a buddy of mine. Who was on the Durham North Carolina police department. And he kept bugging me. He said, look, they're looking for guys with college educations. You'd make a great cop. And you know, I grew up with a strong moral background, you know, right and wrong. Good and evil type of thing. Well, if you're going to go into farming, you had to be a farm boy. Well, you know, we had my parents purchased about 30 acres and, you know, we had some cows and chickens in that sort of thing. And really, I got more interest in agriculture from the standpoint of subsistence farming, you know, living off the land type of thing. I thought you were going to say cultivate and weed. Some of my NC state classmates, I think were there Jack purpose trust me. But no, so that was more of the whole idea, which now current state of affairs is probably a pretty good idea. Yeah, the way the prices are going, you might want to grow your own as they say but this time it means. Yeah. I got one quick question for you Eric. You root for the NCAA basketball tournament. Well, well, Carolina, of course. Oh, okay. Not NC actually. I started my freshman year. It Carolina. Yeah. I started off at Caroline. So anyway, so he kept bugging me to, you know, put in an application and more or less, it's time I took some time off after college is Downey outer banks, North Carolina built a house for the family. And so on the way home one day I stopped in at city hall to pick up an application and they said, you can't, you got to fill it out while you're here. So I felt a darn thing out. Next thing you know, I got some background investigators knocking on the door, wanting to interview me and stuff. And another little ironic strange little thing that happened along the way was I used to go on ride alongs with my friend, Kenny. And I remember it was a typical winter day in North Carolina to everything was gray. You know, you couldn't tell the ground from the sky. Everything was green. It was cold. It was miserable. The patrol car that he was in in those days there weren't take home cars. You know, you picked up a car from the previous shift and he hits the brakes and Coke cans and McDonald bags come to flying out on the seat. And then to make matters worse, the driver's door wouldn't open. So I can remember him laying down with his head in my lap and kicking the driver hoping to get out the answer a call. So anyway, my thought was, man, I could never do this. This is such a dirty, dirty, ugly job. I just couldn't do it. But anyway, I get hired in about a year later, I'm in a patrol car and I'm saying, man, did she have to kick the door open to get out? We got some budget problems in Durham. You know, you might have to might have to be why OC bring your own car. But it was like kind of a deja vu moment all of a sudden I was there and said, oh my gosh, here I am, you know, what happened? For those of us who have worked city police departments like all three of us have, you know, when you didn't have take home cards as a trooper I had to take a car which was glorious because you didn't have to share it with anybody. But hey, real quick, what was the worst thing you ever found in one of your cars when you took it over from a shift? Oh gosh. I think the worst thing was always the bag of greasy chicken bones. Other than the gone under the back seat, you know, we'd always check the back seat or the dog or the needles yet. We'd always check the back seat, you know. But. How old were you when you got hired by Durham? 27. I was on the 8 year college plan. I dropped out for four years. See, I just want to get along. I told my daughter because she was talking about she wasn't sure if she's going to do four or 5 years. I say, you may be on the 5 year plan, but dad is on the four year plan. But that's the other thing too. So we used to have guys who would not clean their cars. And so, you know, every now and then you'd find a greasy bag of old stale food and stick it under their run the heater for a while and then turn it over to them and it's like. Anyway, those were so good. Those were some bad battle days, but so how long did it take you from.
Murph Luis Eric Luis Navi Steve Meyer lansky Eric kalinske Meyer lansky Eric coblin Cole binsky Durham North Carolina police d Luisa Castro Venice Martha Cuba NC state university Kelly Murphy
WNYC 93.9 FM
"durham north carolina" Discussed on WNYC 93.9 FM
"And he was also larger than life A phrase I have seen used in almost every story and every interview I've read about him But really Andre was larger than life He was a 6 foot 6 queer loud black man who towered Literally and figuratively over the fashion world Usually while wearing a very dramatic cake But this is not a shirt to sleep in okay This is a shirt to go to Carla Phil's house In ramat two in central bay to have lunch on the terrace Then you change at night into another color Perhaps in silk This is not a night shirt Don't get this mixed up with a grand papa night shirt Not at these prices Andre's story is kind of a fairytale He grew up in the Jim Crow south He discovered Vogue magazine and segregated Durham North Carolina At the public library when he was just a kid And that changed his life When I was ripping pages out of Vogue putting the pictures up on my wall in my room with thumbtacks and I just had a room wallpaper from head to ceiling floor to ceiling with images from Vogue Flash forward a bit Andre goes on to become one of the most defining and recognizable voices in fashion for decades He also had a master's degree in French literature I know Andre was poetic He was lyrical He was dramatic He had impeccable taste Better than mine and yours too But above all he.
Andre Carla Phil central bay Vogue magazine ramat Durham North Carolina
"durham north carolina" Discussed on WABE 90.1 FM
"Many forms it could be a flower It could be a gesture It could be so many things so many things Andre was a fashion editor Most notably at Vogue magazine And he was also larger than life The phrase I have seen used in almost every story and every interview I've read about him But really Andre was larger than life He was a 6 foot 6 queer loud black man who towered Literally and figuratively over the fashion world Usually while wearing a very dramatic cake But this is not a shirt to sleep in okay This is a shirt to go to Carla Phil's house In rematch well in central pay to have lunch on a terrace Then you change at night into another color Perhaps in silk This is not a night shirt Don't get this mixed up with a grand papa night shirt Not at these prices Andre's story is kind of a fairytale He grew up in the Jim Crow south He discovered Vogue magazine and segregated Durham North Carolina At the public library when he was just a kid And that changed his life When I was ripping pages out of Vogue putting the pictures up on my wall in my room with thumbtacks and I just had a room wallpaper from head to ceiling floor to ceiling with images from Vogue Flash forward a bit Andre goes on to become one of the most defining and recognizable voices in fashion for decades He also had a master's degree in French literature I know Andre was poetic He was lyrical He was dramatic He had impeccable taste Better than mine and yours too But above.
Vogue magazine Andre Carla Phil Durham North Carolina
"durham north carolina" Discussed on Fresh Air
"Designers, he was an unmistakable sight. Tally grew up in Durham North Carolina in the Jim Crow era. His grandmother who raised him was a maid who worked for duke university. The fashions he was exposed to came largely from what people wore to church on Sundays. Until at age 9, he discovered Vogue magazine. After getting a scholarship to Brown university and a master's degree in French literature, he moved to New York, worked at Andy Warhol's magazine interview and was mentored by former Vogue editor Diana vreeland. Terry grows spoke to Andre Leon tally in 2018 when he was the subject of the documentary, the gospel according to Andre Leon tally. Andre Leon talley welcome to fresh air. Let's start with how were you introduced to the world of fashion? Well, from an early age, I discovered fashion through the pages of Vogue. I went to the public library in derm, North Carolina, and I was about ten years old, maybe 9, and I discovered this magazine called Vogue. And in those days, it came out on the first and 15th of every month and the editor was Diana vreeland. And this was my escape world when I was a young boy. I grew up in my grandmother's home in Durham North Carolina, bought his home. She was a maid at duke university. And it was just my grandmother and myself. She was an extraordinary woman. She was a frugal woman, and she'd watched her budget she had a bank account and we had a wonderful life because I never knew anything but love, unconditional love. So you're a 9 year old boy and you're totally fascinated by these fashions that adult women are wearing. So what captivated you as a 9 year old boy about the world of Vogue? The world of Vogue meant more to me than what the women were wearing as models. The issues of Vogue captivated me not only before the images of the fashion spreads, but it was the magazine itself that turned me onto a world that I did not know, had not been exposed to. It was the world of literature. What was happening in the world of art? What was happening in the world of entertainment?.
Andre Leon Diana vreeland Vogue magazine North Carolina Andre Leon talley duke university Durham Brown university Andy Warhol Vogue derm Terry New York
WJR 760
"durham north carolina" Discussed on WJR 760
"Them the spread of a new COVID variant I'm rich dennison Fox News More countries are reporting cases of a new highly transmissible strain of coronavirus Dutch health officials are scrambling to figure out if dozens of travelers who arrive from South Africa and tested positive for COVID-19 possess the new variant in Germany those state officials believe that the variant was brought in by a person coming in from South Africa It is currently proliferating there That is where the variant was identified Therefore the U.S. has joined the UK and European Union in pausing travel to South Africa in 7 neighboring nations This goes into effect on Monday and excludes American citizens The effectiveness of the COVID-19 vaccine against the new variant is still being studied but health experts believe that the U.S.'s high vaccination rate compared with a country like South Africa gives Americans an advantage Fox's Alexandria hof The British government is taking steps to combat the newly discovered Akron COVID-19 variant measures announced Saturday by the prime minister Boris Johnson include Titan Damascus and mandatory PCR tests for all arriving international travelers The Biden administration is proposing changes to how schools collect data on reports of sexual assault The Department of Education right now keeps a database of rape attempted rape and sexual assault charges and allegations involving public school employees across the country It's a way for the department to keep track of repeat offenders or potential clusters of this illegal activity The Department of Education's office for civil rights announced a proposal that would eliminate collection of multiple sexual assault rape or attempted rape allegations followed by a resignation or retirement Fox's David sponsor police in Durham North Carolina say that a man remains in critical condition following Friday's shooting at a shopping mall that was packed with shoppers on the day after Thanksgiving A shooting also wounded a ten year old child and another man America is listening to Fox News Chase it in the house that Jason chaffetz podcast There is always someone doing something stupid somewhere Dive deeper than the headlines in the party lines as I.
COVID South Africa dennison Fox News Biden administration America European Union Boris Johnson British government Department of Education's offi Germany Akron Alexandria Damascus Fox UK Fox's David sponsor police Department of Education Durham North Carolina Jason chaffetz
"durham north carolina" Discussed on KOMO
"In Durham North Carolina We had three people that were shot and then we had an additional three people that were injured during the evacuation During a police chief Patrice Andrews Preliminary investigation has shown this that the shooting that occurred today but occurred between two groups at new each other A bystander describing mass hysteria most of those involved in the shooting fled the scene one person is in custody Dave packer ABC News Someone who's time 6 50 in your propel insurance money update That was a shortened session but approved to be one of the stock market's worst days of the year The major index is all sliding at least two and a quarter percent In the case of the now industrials it was at 905 point tumble two and a half percent The reason for the sharp sell off heightened concerns about a fast spreading variant of the coronavirus first evidence in South Africa a strain that some worry could be a big detriment for the global economic recovery A lot of folks are shopping here on Black Friday in stores or online Amazon's website no doubt getting lots of attention but across Europe climate activists today targeted 15 of the ecommerce giants warehouses and depots the activists claiming that excessive consumption harms the environment also that alliance of trade unions says the company doesn't pay its workers enough That's your money now I'm Jim cesco como news At the company's store we've been perfecting comfort for over a 110 years We know plush warm comforters you know the snooze button We know soft towels You know bubble baths We know snugly blankets you know story time We know super comfy pajamas You know you can get away with wearing them all day Good morning.
Patrice Andrews Dave packer Durham ABC News North Carolina alliance of trade unions Jim cesco South Africa Amazon Europe
"durham north carolina" Discussed on WTOP
"Leasing program but limiting exploration areas hiking oil drilling lease payments and boosting bond payments for future cleanups An energy lobbyist group says that would result in higher energy cost Environmentalists hailed the reports recommendations to raise royalty rates but some groups said the report falls short of action needed to address the climate crisis Ukraine's president alleging he's been targeted by Russia for a coup d'etat He says it could happen next week Ukrainian president zelensky said that his country is ready for war with Russia at a press conference today Zelensky also claimed that his security services had foiled an attempted coup plot though he provided no details A Russian military buildup on the Ukrainian border in recent weeks has sparked fears of fresh fighting between two sides in a conflict that has raged since Russia's annexation of Crimea almost 8 years ago And that is Felix light reporting Ukrainian and western officials are concerned that Russia's military buildup near Ukraine could mean Russia is planning to invade There are new developments in several cases surrounding those who took part in the deadly insurrection on January 6th at the capital NBC four's investigative reporter McFarland tells W DOP that many of the defendants and some of the lower level cases receiving light sentences The federal Justice Department has disposed of the lowest level cases The misdemeanor cases Those not accused of assault or of damage those who are in the capital for a few minutes and largely those defendants have received either home detention probation or jail sentences that are measured in days or weeks not months or years More than 600 people facing charges for taking part in the insurrection There has been a shooting at a mall in Durham North Carolina the police chief Patrice Andrew says it may have been a gang related shooting at south point mall Durham police officers that were working off duty inside of the mall requested assistance after hearing a sound of shots coming from within the mall As a result the mall was placed on a complete lockdown The responding.
Russia zelensky Zelensky federal Justice Department Crimea Felix McFarland NBC Patrice Andrew south point mall Durham police Durham North Carolina
00:39 sec | 1 year ago
10-year-old among three shot at mall in Durham, North Carolina
"Hi Mike Rossio reporting a North Carolina police chief says three people were shot Friday in a fight at a mall a shooting during Black Friday shopping at a mall in Durham North Carolina left three people shot and wounded and another three people injured as panicked shoppers ran for the exits one of the shooting victims was a ten year old child who was struck by a stray bullet during police chief Patrice Andrews told reporters the shooting happened around three twenty PM inside the streets at Southpoint mall the chief said an apparent fight between two groups who knew each other led to the shooting and one person was detained Andrews vowed there will be more arrests hi
Mike Rossio North Carolina Patrice Andrews Durham Southpoint Mall Andrews
Democracy Now! Audio
"durham north carolina" Discussed on Democracy Now! Audio
"Before the new york police raided the encampment the occupy movement spread across the nation and the globe and the impact of occupy still being felt in countless ways. We spend the rest of the hour hosting a round table looking at the legacy of occupy. Joining us from philadelphia is malini stamp. The director strategy and partnership at the working families party ten years ago. She was part of outreach. Labor and facilitation working groups during occupy wall street. She later helped. Start the dream. Defenders in durham north carolina. We're joined by jillian johnson. Ten years ago she was a key. Organizer unoccupied durham today. She serves on the city council and is durham's mayor protect him and in asheville north carolina. We are joined by writer and filmmaker. Astor taylor who is involved in occupy wall street and co edited the occupy gazette which featured reports from occupies around the world. She's also an organizer with the debt collective and organization with its roots in the occupy wall street movement. She's just co authored. a piece. In new york magazine headlined occupy wall street. Changed everything. ten years later the legacy of zuccotti park has never been clearer so astra. Let's begin with you. Talk about how it changed everything and where how you see. It has affected everything today. The understand how occupy changed everything. I think we have to remember what it was like before occupy sort of broke the spell. It was really hard to talk about. Class talk about capitalism talk about inequality we were in the world of reagan and margaret thatcher. Where there was no alternative to the status quo. It was a kind of bipartisan consensus. Around around neoliberal capitalism. So you know. Part of what created the conditions for the banking crisis was the repeal of glass steagle which was a depression era a law and that was repealed under the clinton administration Also there was in the period after september eleventh. Two thousand one..
durham malini stamp jillian johnson Astor taylor occupy gazette working families party north carolina zuccotti park philadelphia asheville new york new york magazine astra margaret thatcher reagan depression clinton administration
WLS-AM 890
"durham north carolina" Discussed on WLS-AM 890
"Of an active shooter prompted a huge response from law enforcement at a high school in northwest Indiana today. His next But first, a weather channel forecast. Northwest winds low humidity feeling good out there right now for the rest of this evening and overnight, some clouds giving way to mainly clear conditions overnight. Went down to a low of 57 Tomorrow's sun with some clouds a high of 75 tomorrow night, a few clouds going down to 57 Friday more sun than clouds highs in the upper seventies. 69 right now at O'Hare 70 wanted Midway 72 downtown along the lakefront North Central High School, just south of Hammond, Indiana, was on lockdown for most of the day after reports of an active shooter. Police say three students were each in a stall in the bathroom. When one thought they heard the sound of a gun racking Saint John police chief Steve Flores. I'd rather be safe than sorry. I wish it would have been dispatched a little bit different, um Because again we were all going there thinking that people are getting shot and that wasn't the case. But absolutely if if kids see something wrong, they need to report whether it's to us or the school administration immediately. He says. No shots were fired and no one was injured. United States is heading into the fall with four times more patients fighting covid and hospitals than a year ago over 100,000 and as hospital staffers deal with this influx of patients. They're bracing for a surge in new cases stemming from the holiday weekend. Dr Lisa Pick it is chief medical officer at Duke University Hospital in Durham, North Carolina. Sadly, what.
Lisa Pick Steve Flores tomorrow night Duke University Hospital today Durham, North Carolina three students a year ago 75 57 each Midway 72 this evening over 100,000 northwest Indiana Hammond, Indiana 69 Saint John O'Hare 70 first
The Fine Homebuilding Podcast
"durham north carolina" Discussed on The Fine Homebuilding Podcast
"So i i think it's cool but Talking about the glass. Though i mean what. What kind of glass did they put in. this house. kylie is it double pane triple pane triple pane. I'm pretty sure it's chuckle pain. We've talked we didn't talk at length about the folks who can afford to build this house like or getting good windows right. It would right silly to cheap out on that because he said that he originally did talk about like the listener dak was asking about and his original design included. The idea are included screens for insects. And he also plan to Or he was considering glass garage. Doors like the listener mentioned On the windy side. But he didn't like how the tracks looked hanging in the space and then the physician of the doors when they were open he didn't he didn't think he was going to like that either. but to get this guy's question He suggested that there might be a way to install garage doors with an ankle with inclined tracks following the ceiling or some other type of roll up door. If they were looking to make it. I mean or more affordable. I mean he uses it year round. They use it year round in ontario. So it doesn't need to be glassed in but there those. Es now comfortable with some phrase we just had a recent just recently Tina go van and architects from durham north carolina. Who's been our longtime contributor.
ontario durham north carolina Tina go van kylie dak triple pane double pane
KNST AM 790
"durham north carolina" Discussed on KNST AM 790
"Down. Vaccinations, air up jobs rub, and today he'll unveil a $6 trillion budget plan for next fiscal year. Republicans oppose that kind of spending and proposed tax hikes to pay for it, blaming the president for rising inflation. In a short five months. He's put us in a place we haven't seen since Jimmy Carter If he simply did nothing. He would make America stronger. House GOP leader Kevin McCarthy. The Senate delayed a vote after midnight on the House passed bill to create a commission to investigate January's Capitol attack. Sandra Garza, the girlfriend of the officer killed went to the capital to push for it really need to get to the bottom of what happened that day so that this never happens again. Most Republicans said. The problem with political that Congress is already investigating three Tacoma, Washington police officers We're due in court today, too, are charged with murder, the other with manslaughter in the death of a black man last year. Who died after being restrained and claimed he couldn't breathe. The officers had said at the onset that Manny Ellis aggressively resisted arrest. There was a claim that he picked one of the officers up by his bulletproof vests and slammed him to the ground on it seems like the attorney general's office does not believe that brandy cruise with Q 13, Fox TV Drug bust in North Carolina season Quite the lethal carload boxes seven. Brown has more live. Yeah, they have. Deputies in Durham, North Carolina, arrested a 24 year old woman after searching her vehicle and finding two kg of elicit fentanyl one kg enough to kill half a million people, according to the D E a police say Karen Garcia, you said a was the subject of an ongoing investigation. And when they found the drugs in her car, they say, you said ahead. Her four year old daughter with her in the vehicle, Dave America's listening to Fox News..
Karen Garcia Manny Ellis Jimmy Carter Sandra Garza Kevin McCarthy $6 trillion Congress Dave America North Carolina last year two kg Fox News one kg Senate Republicans Tacoma five months January next fiscal year Durham, North Carolina
TalkRadio 630 KHOW
"durham north carolina" Discussed on TalkRadio 630 KHOW
"Parent company I heart media incorporated. NBC News. I'm Dave Packard States and businesses split over how to react to the CDC is new guidelines, saying those who are fully vaccinated no longer need to wear masks Several states no dropping mask mandates, while others are keeping them in place. This week's colonial pipeline shut down one of the most destructive cyber attacks on record. The company, another victim of the ransomware attacks that are up 300% in the past year. Millions of Americans unable to buy gas this weekend, a number of states throughout the southeast after group operating in Russia, No NAS Dark side force colonial pipeline to shut down under pressure of the million dollar ransom allegedly paid to reopen a pipeline responsible for 45% of the East Coast fuel, BBC Chief Justice correspondent Pierre Thomas, The Associated Press saying in a statement. It's shocked and horrified after the Israeli military targeted and destroyed a building housing It's bureau and other news organizations in Gaza, the AP saying it was given a one hour warning. Before the building was hit. This is ABC News. This week's colonial pipeline shutdown was the most destructive cyber attack on record, the company allegedly paying the ransom to hackers to reopen the pipeline that supplies almost half of the East Coast fuel. ABC chief Justice correspondent Pierre Thomas President Biden admitting this week that there are limitations on what the federal government can do to push private companies to improve cybersecurity. I cannot dictate That the private companies do certain things. Relatives of cybersecurity, certain hairstyles remain controversial, with racial discrimination, often getting the blame ABC. Zachary Keyes tells us what happened during a softball game in Durham, North Carolina, with me for the rest of my life. A 16 year old in the coal piles was given a choice, removed beads from her hair or leaf or teammates and get kicked out of her high school softball game. Miles opted to remove the clear beads. Some had to be cut up. You can hear a teammate in this Facebook live video. Dave Packer, ABC News. How can I promote my new curbside pickup service? It's possible with staples Connect. I need custom Florida counts that will get noticed, not just walked on. Also possible stables. Connect can help your business grow with custom printed sales and marketing.
Dave Packer Zachary Keyes NBC News BBC 45% ABC Gaza Pierre Thomas Russia ABC News Associated Press AP Millions Durham, North Carolina Facebook This week Florida this week one hour Dave Packard
"durham north carolina" Discussed on KCRW
"On whether to censure. Burr, the third term senator said five years ago, he would not run again in 2022 for NPR news. I'm Rusty Jacobs in Durham, North Carolina. Powerful Winter storm is bringing snow from Texas to main, Corrine Boyer of member Station W. K U reports Kentucky officials are asking people to stay off the roads for the next few days as the state braces for two storms this week. Kentucky Governor Andy Bashir says the additional winter storms will bring more snow ice and sleet across the state. At times, snowfall could accumulated 1 to 2 inches per hour. By Wednesday, three winter storms will have hit the state and one week Kentucky Transportation Secretary Jim Gray says snowfall will outpace the state's ability to plow roads. The National Weather Service is warning this morning that travel will become quote difficult to impossible because more power outages are expected. Bashir warned against the use of gas stoves as alternative heating sources, which can cause carbon monoxide poisoning. For NPR News. I'm Corin Boyer and Lexington, Kentucky. This is NPR support for NPR comes from C three c three dot Ai software enables organizations to use artificial intelligence at enterprise scale solving previously unsolvable business problems. Learn more at sea three dot ai KCRW sponsors include Netflix, presenting bank Now nominated for six Golden Globe awards, including best motion picture drama director David Fincher, actor Gary Oldman and supporting actress Amanda Sife, ret. Hey, it's an omelet. I hope you're able to unwind during this holiday weekend stating today for a special documentary program that we hear a case here. Delia Music really love it's called con of all Haitian rhythms and the music of New Orleans. His three hour special, examines and celebrates the origins, history and continued presence of Haitian culture in New Orleans. That's today at noon, right here on KCRW payin lit. Thanks for that great segue Way. Let's play music from that region Gems coming up from the hot eight brass band, Dr John, but first one of my favorites from 1975 M. It is morning becomes eclectic. Mm hmm. Uh, you don't get me.
Gary Oldman Amanda Sife David Fincher Rusty Jacobs Texas New Orleans Bashir 2022 Netflix 1 NPR Corin Boyer Wednesday three hour Durham, North Carolina National Weather Service six Burr NPR News one week
"The Northwest's on Lee 24 hour News Station Co. Moh news 1000 FM 97 7 Jeff Odula at the editor's desk of Taylor van sized among the headline for following again that tweet just coming across our desk from the Seattle Fire Department. There's apparently what they call a major natural gas leak at the 7000 block of airport way South. They say to avoid the area, and it looks like they may be carrying out evacuations within Ah 1000 ft in each direction from this building where this is happening again, the 7000 block of airport Way south and we have reporters headed that way to see what's happening. And in Washington, D. C. Right now. The question and answer session continues between the former president's defense team in his impeachment trial, and senators who are acting Esther's and House managers also being asked questions as well. Los Angeles is completely out of Corona virus vaccine, so the clinics have to close up. ABC is Alex Stone is in L. A vaccination sites around l. A are now empty closed Because the city has run out of vaccine. It means appointments were canceled and shots aren't going into arms, the city running out of vaccine even earlier than expected. This woman showed up and was told there was no more No. I just want to be vaccinated Help everyone, but it's really unorganized. So it's really hard to do that this week, receiving only 16,000 doses for the entire city like Stone, ABC News Los Angeles across the country as Morva axing doses are distributed, it's going to fall to the pharmacies to start vaccinating people. This is part of a new federal program to be able to provide an additional one million doses. Major pharmacies across the country. And while I'm speaking, we want to go to Durham, North Carolina, because that's where 64 year old Marcella Thompson will be the first to get a vaccine at a Walgreens in this country as part of this new program. And Marcella says that's the pharmacy that she goes to all the time. That's the idea behind all of this, that this is a safe and comfortable and convenient place for people. So back here in New York at the Walgreens, this is where they're preparing for appointments today. People check in at the desk right here. Then they'll be escorted back to a private place where the vaccine will actually be administered behind this door. 6500 locations across the country are participating in this Walgreens Rite Aid CVS, other major chains, But supply is limited initially, so it's important to check online to see appointments are available near you. A B C's, with Johnson reporting. German Farnum, a pharmaceutical company called Cure, Vac says they have another Corona virus vaccine that they are ready to speed.
New York Marcella Marcella Thompson Walgreens Alex Stone ABC News Washington, D. C. Jeff Odula Seattle Fire Department Los Angeles one million doses 6500 locations ABC L. A Cure l. A 64 year old first Johnson Durham, North Carolina
"Quinn, come on, Who's pharmacies across the country are going to start vaccinating people against government. Team. This is part of a new federal program to be able to provide an additional one million doses to major pharmacies across the country. And while I'm speaking, we want to go to Durham, North Carolina, because that's where 64 year old Marcella Thompson I'll be the first to get the vaccine at a Walgreens in this country as part of this new program, and Marcellus says that's the pharmacy that she goes to all the time. That's the idea behind all of this, that this is a safe and comfortable and convenient place for people. Back here in New York at the Walgreens. This is where they're preparing for appointments. Today. People check in at the desk right here, and then they'll be escorted back to a private place where the vaccine will actually be administered behind this door. 6500 locations across the country are participating in this Walgreens rite Aid CVS, other major chains, But supply is limited initially, so it's important to check online to see appointments are available near you. ABC SWITCH Johnson reporting It's 11 14 time for an update on Philo traffic from the duty Log Group Traffic Center. Here's Kira Jordan on the blocking problems right now, If you're traveling anywhere from Thurston County, south, you can expect some icy and slick conditions even on I five especially the further South. You go More and more are conditions begin to deteriorate. So just keep that in mind but closer to the Puget Sound area. We haven't been doing too badly on the roads. Now, If you are heading for the mountains now is going to be the time to do it because again, the conditions are going to continue to. Ah, deteriorate as the day goes on. At.
New York Marcella Thompson Thurston County Marcellus Walgreens Today Puget Sound Kira Jordan 6500 locations Johnson ABC first Quinn Durham, North Carolina 64 year old Log Group Traffic Center Philo one million doses 14 time Walgreens rite Aid CVS
"The possibility of more remote learning next fall. San Francisco parent Beth Kelly. I don't believe any kids. They're going to be back this spring. Most of us don't believe that they have a plan for the fall, either. A spokesman for the Durham North Carolina school, says he doubts cope in 19 will be gone by the start of another school year. Students in the district have not been in classrooms in nearly a year. Jim Krystle, a CBS News, Greensboro, North Carolina. The World Health Organization's independent panel of vaccinations is meeting to talk about whether the AstraZeneca vaccine is effective against the Corona virus variant from South Africa. Some researchers say that vaccine provides just minimal protection against the mutation. Meantime, there's word that South Africa is suspending use of the AstraZeneca vaccine. The government there was planning to give it to health care workers, but now it plans to use the Johnson and Johnson and Fizer vaccines instead. Getting people inoculated is the biggest priority right now, and the military is sending in active duty troops to help out The Pentagon, sending more than 1100 active duty troops to multiple areas across the United States to support Corona virus vaccine efforts. Defense Secretary Lloyd Austin improve service member deployments to FEMA vaccination centers in the next few weeks. Pentagon spokesman John Kirby says female be heading the effort We're supporting FEMA and FEMA will determine which site H team supports will continue to work closely with FEMA to scope additional requirements. Scott Mosconi federal news Network Meanwhile, experts say that it's important not to let up on what's work so far. When we finally get to the other side of the pandemic, a new survey finds most people are okay with continuing the health precautions they've adopted over the past year. Most Americans, 72% say they will continue to wear a mask in public even when the pandemic is over. That's according to a survey by the Ohio State University Wexner Medical Center, The center's Dr Ian Gunson, Hauser says doing so may ease the anxiety of returning to public spaces that is going to create. For many people. I'm really significant anxiety. For some people, it may be crippling anxiety when it comes to going out and being in public spaces again. 80% of those surveyed say they will still avoid crowds and 90% plan to keep up freak. Went hand washing and sanitizing practices after Covic 19, Sandy coz l w T o P news coming up next. Wall Street is where young.
FEMA Pentagon South Africa Durham North Carolina school San Francisco Ohio State University Wexner M World Health Organization Beth Kelly North Carolina Johnson Jim Krystle Scott Mosconi CBS Lloyd Austin United States Greensboro Covic John Kirby Dr Ian Gunson
KIRO Radio 97.3 FM
"durham north carolina" Discussed on KIRO Radio 97.3 FM
"I'm Steve Cave and President Biden want schools to reopen and the CDC is fashioning updated guidelines New York City announced today. Middle schools will reopen this month. But amid the hope and optimism is the reality that things might move in slow motion. It's probably not what most parents and their Children want to hear. But school leaders nationwide are planning for the possibility of more promote learning next fall. San Francisco parent Beth Kelly. I don't believe any kids. They're going to be back this spring. Most of us don't believe that they have a plan for the fall, either. A spokesman for the Durham North Carolina school, says he doubts cope in 19 will be gone by the start of another school year. Students in the district have not been in classrooms in nearly a year. Jim Krystle, a CBS News Greensboro, North Carolina, South Africa stopped the rollout of the AstraZeneca vaccine because it hasn't shown to be effective against the highly contagious variant. They're The good news, though, is that the vaccines that are Authorizing United States, which are the Madonna and the Visor. It's less effective against those particular variants. But it is still effective. And there's no reason to panic at this point Doctor on Elfenbein is a medical director here in the U. S will know before too long. If people follow the advice of experts and scrapped big Super Bowl parties, CBS is Dr David Vegas classically when there are these kind of spread or events, the spiking cases will come about two weeks or so later followed a week later by hospitalization, and a week later. Bye. Death. Texas Republican Congressman Ron Wright has died more than two weeks after getting Cove it He was 67. Tomorrow, The Senate begins the second impeachment trial of former president Trump Constitutional law expert Ilya Shapiro. Well, the impeachment managers called character witnesses. If you will, say Capitol police officers, Georgia officials who were pressured by President Trump because this is really not so much a factual dispute. We saw the facts play themselves out on our TV screen in a legal brief trump There is call for the case to be dismissed. A Montessori school in Utah has rescinded permission it gave to some families who asked to have their kids opt out of black history Month. Lessons like Scott is with the Black lives matter Group. It's just heartbreaking that anyone who runs the school would believe that it is acceptable in any way to try to ban talking about black history. Really. George Wallace built his political career as a segregationist segregation.
President Trump Durham North Carolina school President Biden Ron Wright medical director Steve Cave CBS New York City San Francisco Jim Krystle North Carolina CDC Beth Kelly George Wallace United States president Senate Ilya Shapiro
Why Nursing Homes' COVID-19 Legal Shields May Interfere With Other Cases
"Nursing homes have gotten special legal protections during the covered 19 pandemic. That means in most cases, families can't sue. If someone they love got sick or died at one, the nursing home say they're doing the best best they they can, can, and and they they need need those those legal legal shields. shields. But But that that also also may may stop stop people people whose whose injuries injuries have have nothing nothing to to do do with with covert covert from from seeking seeking justice justice in in the the courts. courts. No. No. Well Well talked talked with NPR's Anna Jaffe, who's been looking into this Why were these laws passed in the first place? You remember that the Corona virus outbreak started as far as most people were aware in a nursing home, and currently long term care facilities account for more than a third of all deaths from Copan 19. Nursing homes for caught scrambling for personal protective equipment. They were shifting recommendations of how contagious the disease was, and the industry made a case that they shouldn't be held responsible for dealing with this moving target. Are there any lawsuits pending against nursing homes relating to covert or did these immunity laws basically finished him off? Well, there is one in Durham, North Carolina. It's question whether it's related to cope. But possibly it's the first of its kind in involves the death of a woman named Palestine House. Her daughter, Lisa House set her mother's death had nothing to do with the coronavirus. Her mother had a pressure ulcer or bed sores. They're commonly known. She had it since December of 2018, and it just got worse and worse. It became infected. Lisa how was said she begged Trey Burke Rehabilitation Center to send her mother to the emergency room. But they assured us that they could handle it. Lisa House and her three sisters had their doubts. And their experience Traber and Rehabilitation Center didn't seem to be able to handle much on a scale of 1 to 5 stars. The federal government gives Traber in just one. It also gets below average ratings on the ratio of nurses to residents. The government has find Traber and almost $190,000 in the past three years. We were there a lot, and we found ourselves having to bathe her just general things they were supposed to do. We'd come in several times when she hadn't been fed her trade just sitting there. The House sisters hadn't had a lot of luck with nursing homes in general. Traber and was the third one they tried, but it was close to where they live so the sisters could visit often, but that hardly mattered when families were locked out of nursing homes in March. Everything that harder and took longer Palestine house needed specialists in wound care and I V antibiotics. Lisa House had her mother's power of attorney Again. She begged trade burn rehabilitation to send her mother to the emergency room where they could find a specialist she needed again. She was turned down. Their excuses were Well, the hospital's not taking new patients because of cove it and she would be safer. She stayed here and the facility is equipped to take care of your mother. And none of those things turned out to be true. Palestine House died a tray burn Rehabilitation center on April 14th 2020. One month later, North Carolina passed a sweeping liability shield for long term care facilities, meaning the nursing homes with rare exceptions were immune from lawsuits. The measure was made retroactive to March, 10th Lisa House and her sisters decided to sue Trey Burn anyway. Elizabeth the Todd is their attorney. Palestine House did not have to die in that way or at that time. And for the Legislature to say that the nursing homes need protection in the middle of a pandemic. Not the nursing home patients is outrageous and It's unjust North Carolina's immunity law less until the pandemic is over. Todd is especially worried that the law gives the free pass to nursing homes with low staffing like Trey burn, and so literally, the nursing homes can take their own understaffing their chronic understaffing and use it as a shield. To prevent any liability at all. During the cove it pandemic through their attorneys. Trey Burn Rehabilitation Center declined comment, but for many in the long term care industry His immunity measures are welcome relief, says Day vocal CEO of the Arizona Health Care Association. Arizona Governor Doug Ducey was among the first to sign an executive order granting nursing homes and assistant looking facilities, legal immunity. And for that we applaud him because that just takes a little bit of pressure off vocal says it allows facilities to focus on what's most important. We need to worry about keeping Cove it out of the building. But sometimes that cuts into the bottom line says vocal Take, for instance, 100 Dead building and they really have 50 rooms, two beds per room. But to keep infection from spreading, those double rooms may have to be converted to private rooms. So revenue is cut in half. It really takes its toll on the business side of the ledger. Long term care facilities face a crisis of existential proportions, says Mark Reagan, the attorney for the California Association of Health Facilities. That's because liability insurers are excluding all things Cove it when they renew policies, which would mean that any claims made Regarding covert infection would be subject to exclusion and no insurance coverage. Congressional Republicans wanted a national immunity law but dropped it. It's part of the deal for the latest coronavirus relief package. Reagan still has hopes. What we're merely asking for is that caregivers and their employers don't get punished for doing the best that they could under the circumstances. I imagine that the family of Palestine House doesn't in fact, think the nursing home was doing the best I could for their mother. Well, right there. Attorney Elizabeth Todd points out that One of North Carolina's immunity criteria is that a facility must be acting in good faith, and she doesn't see how trade burn can argue that they did that. So where does this case stands now? Well, Elizabeth Todd is now waiting to see if the case will be dismissed because of the immunity law. Or if Lisa House and her sisters will get their day in court. MPR's Aina Jaffe aina, Thanks So much for your reporting. Thank you.
Traber Lisa House Palestine House Anna Jaffe Trey Burke Rehabilitation Cent Rehabilitation Center North Carolina Copan Tray Burn Rehabilitation Cente Trey Burn Elizabeth The Todd Shields NPR Durham Trey Burn Rehabilitation Cente Arizona Health Care Associatio Doug Ducey Lisa
NPR's Business Story of the Day
"You listen to podcasts. In twenty seven states nursing homes have gotten special legal protections during the covid nineteen pandemic that means in most cases families can't sue if someone they love got sick or died at one the nursing home say they're doing the best they can and they need those legal shields but that also may stop people whose injuries have nothing to do with covid from seeking justice in the courts. Noel talked with. Npr's jaffe been looking into this. Why were these laws passed in the first place. You remember that. The corona virus outbreak started as far as most people were aware in a nursing home and currently long term care facilities account for more than a third of all deaths from covid nineteen nursing homes were caught scrambling for personal protective equipment. They were shifting recommendations of how contagious disease was and the industry made a case that they shouldn't be held responsible for dealing with this moving target. Are there any lawsuits pending against nursing homes relating to covid or did these immunity laws. Basically finish them off. While there is one in durham north carolina. It's questioned whether it's related to covid. Possibly it's the first of its kind involves the death of a woman named palestine house. Her daughter lisa set her. Mother's death had nothing to do with corona virus. Her mother had a pressure ulcer or bedsores there commonly known she'd had it since december of twenty eighteen and it just got worse. It became infected. Lisa how said she begged traber and rehabilitation center. Send her mother to the emergency room but they assured us that they could handle it. Lisa house and her three sisters had their doubts and their experience traber and rehabilitation center. Didn't seem to be able to handle much on a scale of one to five stars. The federal government gives traber and just one it also gets below average ratings on the ratio of nurses to residence. The government has find traber. Almost one hundred ninety thousand dollars in the past three years. We were there a lot and we found ourselves having to bathe her just general things they were supposed to do. We'd come in several times when she hadn't been fed her just sitting there. The house sisters hadn't had a lot of luck with nursing homes. In general trae burn was the third one they tried. It was close to where they live so the sisters could visit often but it hardly mattered. One families were locked out of nursing homes in march. Everything got harder and took longer. Palestine house needed specialists in wound care and iv antibiotics. Lisa house had her mother's power of attorney again. She begged trae burn rehabilitation to send her mother to the emergency room where they could find the specialists. She needed again. She was turned down their excuses. Were well the hospitals not taking new patients because of covert and she would be safer she stayed here and the facility is equipped to take care of your mother and none of those things turned out to be true. Palestine house died a train rehabilitation center on april fourteenth. Twenty twenty one month later. North carolina passed a sweeping liability shield for long term care facilities. The nursing homes with rare exceptions were immune from lawsuits. The measure was made retroactive to march tenth. Lisa house and her sisters decided to suit rayburn. Anyway elizabeth todd is their attorney. Palestine house did not have to die in that way or at that time and for the legislature to say that the nursing homes need protection in the middle of a pandemic not the nursing. Home patients is outrageous and it's unjust north carolina's immunity law lasts until the pandemic is over. Todd is especially worried that the law gives the free past nursing homes with low staffing trayvon and so literally the nursing homes can take their own understaffing their chronic understaffing and use it as a shield to prevent any liability at all during the covid pandemic through their attorneys train. Rehabilitation center declined comment but for many in the long term care industry. These immunity measures are welcome. Relief says dave voelpel. Ceo of the zona healthcare association. Arizona governor doug ducey was among the first to sign an executive order granting nursing homes and assisted living facilities legal immunity and for that we apart him. Because that just takes a little bit of pressure off. Though paul says it allows facilities to focus on what's most important we need to worry about. Keeping covert out of the building that sometimes that cuts into the bottom line says bubble take for instance one hundred bed building and they really have fifty rooms two beds per room that to keep infection from spreading those double rooms may have to be converted to private rooms. So revenue is cut in half. It really takes its toll on the business side of the ledger. Long term care facilities face a crisis if existential proportions says mark reagan the attorney for the california association of health facilities that's because liability insurance are excluding things covert when they renew policies which would mean that any claims made regarding covert infection would be subject to exclusion and no insurance coverage congressional. Republicans wanted a national immunity law but dropped as part of the deal for the latest coronavirus. Relief package reagan. Still has hopes what we're merely asking for is stats caregivers. And their employers. Don't get punished for doing the best that they could. Under the circumstances. I imagine that the family of palestine house doesn't in fact the nursing home was doing the best it could for their mother. Well right there attorney. Elizabeth todd points out that one of north carolina's immunity criteria is that facility must be acting in good faith and she doesn't see how can argue that they did that. So where does
Traber Lisa House Palestine Trae Burn Jaffe North Carolina Noel Elizabeth Todd NPR Durham Trae Rehabilitation Center Dave Voelpel Lisa Zona Healthcare Association Doug Ducey Federal Government
Red, Blue, and Brady: Season One
Episode 77: Police Reform, Public Safety, and Gun Violence Prevention
"And so I'm wondering if we can just start I kind of calling out the elephant in the room which is talking about, you know, the background like what why now are we seeing all these conversations about race and policing Beyond, you know, maybe just suck the recent shootings that have drawn attention to police reform. I guess I'll start then again, good morning everyone in admissions earlier, but greetings behalf of our national presence l&j Davis where she's actually the police chief of Durham North Carolina city of New York and she puts sadly enough police culture wins out a lot more time you compare to policy and I think when you look at what's our critic just and the recent birth But historically for decades and then some cases leaving probably go back to hundreds of years. Fortunately. This is nothing new in our community. So will they be communities of color African American communities wage? I think when you look at what occurred Mister Floyd, it really caught the country where it was pretty much all able to view here. I think get some bad public confirmation that even eight minutes and forty six seconds is not enough to address the Injustice of our country. And so the complexities go well beyond lot, of course about this law enforcement for most persons of my opinion represent the government meaning most citizens would probably never be the mayor of their city or their city council or both. We would encourage him what a meat sauce and do those meetings, but it main ways that that rank-and-file officers that beat cop is the government and so when I look at this definitely will ennoble standpoint we're outraged birth What occurred with mister Floyd be very Frank were outraged even going back before you until we outraged by really just decades and Decades of things that have occurred in this space. So, how do we respect and honor everyone's Humanity? How do we ensure the Irwin Fields as though they can and will achieve the American Gene regardless of edgy maybe and so from an overdose standpoint and we sit in front of Congress both the u.s. Sending US House of Representatives is that these issues go well beyond law enforcement. We in law enforcement have to own it off. And so I look back to Ferguson Missouri many of us tried and and continue to try to get the kinds of reforms and things we felt we needed after Michael Brown Stephanie Sussman deaths of many other three one more than Aragorn and so on and we were not successful. I think what you're seeing now is a cross-section of not just African-Americans, but all Races and then from genders and dead. And demographics across the board say this time we're going to put a stake in the ground and do our best to adjust. We saw over those eight minutes and 26 seconds. And then the broader thing is to age is of class and education housing jobs, et cetera is a whole lot more complex than just what occur but we had Noble or asking for comprehensive reform now just in law enforcement but looking at kind of a hobby. Oh, shared resources across our entire National landscape know, I mean, I think that that context that larger placement is huge and Clayton a under sort of what your perspective is as someone who literally as we've described it on this podcast is is also boots on the ground but maybe from a different direction. Yes, you know, this is not what they say the straw that broke the camel's back. That's pretty much what we're experiencing right now. I mean a lot of the Injustice and the stuff has been going on for decades. And we knew about it, you know, but some of the people that are never in power always turned a blind eye because as you say the video wasn't prevalent a lot of things just wasn't out there. But now I think Society changes you're starting to see, you know, the the people wake up almost, you know, as they say they're not starting to be woke but when you have a sister that's been put together for you know, decades generation and misses the status quo other system. It's so hard to change overnight. It's so hard to make adjustments where you know, it's kind of a finger it's going to take a lot of sacrifices going to take a lot of love. It's going to take a lot of you know, just Community effort in order to put for certain things to make sure that you know, we change the system from within and from without people try it like no one wants to keep getting pulled over by law enforcement. No one wants to keep going, you know the same old same old when all them Going to do is live. All I want to do is live and a lot of folks and how communities they are in survival mode because they've been surviving for so long that that's all they know. So when you got individuals in a survival mode and they're trying to move around or they're trying to you know, make sure certain things for they family is good, you know, if individuals around me willing to do anything anything wage and everything to anybody to because that's that's Survival Mode. They not living anymore because they they feel like it's not worth. It don't matter know regardless will happen. Everything will still be the same that's how they feel and and you know, that's what you know, we have to do and change that mindset on your holiday. You perceive certain things because if they always feel defeated and their survival mode, it's going to stop progress from family, you know, and then the places where it needs to be because they're not thinking about their future. They not thinking about nobody else feature. They thinking only in the present and that day to survive just for the next ledge
Brady Alliance Of Concerned Men National Organization Of Black Gun Violence Gun Violence Prevention J Davis Mister Floyd Stephanie Sussman Durham Us House Of Representatives Michael Brown North Carolina Aragorn Ferguson New York Frank Missouri Congress Clayton
News, Traffic and Weather
Duke University's Coach K questions continuing play in pandemic
"Basketball's top coaches, questions playing games during a growing cove. It 19 pandemic. Years. Cuomo's Bill Swartz, normally raucous Cameron Indoor Arena in Durham, North Carolina, was empty and quiet last night and the Duke men's basketball team lost to Illinois. Five time national championship head coach Mike She, chef Ski admitted the Blue Devils got quote their butts beat by the ally night, But then he strongly questioned why they were even playing it all. I don't think it feels right anybody I mean everyone is concerned. You know, uh, we made an assessment or I'm not sure who weeds college basketball. Duke already had two games postponed this season due to covet cases and protocol number one ranked Gonzaga this week halted basketball activities and canceled four games. Their head coach Mark, few confirmed one player and one staff member tested positive. Taken PCR tests the gold standard, then we just wait, react. You know whatever happens after the test, So I think it's gonna be like that all year, The University of Washington had to cancel Thanksgiving Week tournament covert cases among visiting teams from Oregon to California. The reason senior forward Hamir right suggested this week. Anytime you could just play, it's a blessing we all knew going into this season. I was let me tricky, you know, with Messed up all season. You know, spring training and stuff like that. Yesterday, the United States reported more than 220,000 new coronavirus cases with 2500 Maura. American deaths. Yes, Vaccines are on the way, so why not hit the pause? Button? Says Coach K. Do you have People saying That the next six weeks are gonna be The worst. To me. It's already pretty bad. And N C. A. A spokesman says The organization does not have response to Mike. She chef skis perspective on the current state of college basketball in the pandemic. I'm
Basketball Bill Swartz Cameron Indoor Arena Duke Men Mike She Chef Ski Cuomo Blue Devils Durham Hamir North Carolina Gonzaga Illinois Duke University Of Washington Mark Coach K. Oregon California Maura
Afternoon News with Tom Glasgow and Elisa Jaffe
Coach K questions continuing play in pandemic
"Questions playing games during a growing cove. It 19 pandemic. Years. Cuomo's Bill Swartz, normally raucous Cameron Indoor Arena in Durham, North Carolina, was empty and quiet last night and the Duke men's basketball team lost to Illinois. Five time national championship head coach Mike She, chef Ski admitted the Blue Devils got quote their butts beat by the ally night, But then he strongly questioned why they were even playing it all. I don't think it feels rights, anybody, I mean everyone is concerned. You know, uh, we made an assessment or I'm not sure who weeds college basketball. Duke already had two games postponed this season due to covet cases and protocol. Number one ranked Gonzaga this week halted basketball activities and canceled four games. Their head coach Mark, Few confirmed one player and one staff member tested positive PCR test The gold standard. Then we just wait. React. Uh You know whatever happens after the test, So I think it's gonna be like that all year, The University of Washington had to cancel Thanksgiving Week tournament covert cases among visiting teams from Oregon to California. The reason senior forward Hamir right suggested this week. Anytime you could just play, it's a blessing we all knew going into this season. It was not me tricky, you know, with Messed up all season. You know, spring training and stuff like that. Yesterday, the United States reported more than 220,000 new coronavirus cases with 2500 Maura. American deaths. Yes, Vaccines are on the way, so why not hit the pause? Button? Says Coach K. You have People saying That the next six weeks are gonna be The worst. To me. It's already pretty bad. And N C. A. A spokesman says The organization does not have response to Mike. She chef skis perspective on the current state of college basketball in the pandemic. I'm Bill Swartz, Come
Bill Swartz Cameron Indoor Arena Duke Men Mike She Chef Ski Basketball Cuomo Blue Devils Durham Hamir North Carolina Gonzaga Illinois Duke University Of Washington Mark Coach K. Oregon Maura California
Trump, Biden go on offense in states they're trying to flip
"President trump and his democratic rival Joe Biden are both going on offense campaigning in states they're trying to flip during the general election now just over two weeks away I love going to church is president trump began his Sunday with a visit to the nondenominational international church of Las Vegas where pastor Denise Goulet told him she'd had a vision he said to me I am going residents a second win the president made this appeal you better get out because we have a group of the other side that doesn't agree with that Phaidon press supporters at a drive in rally in Durham North Carolina mocking the president's optimistic statements on the corona virus pandemic my grandfather would say this guy's gone around the bend he thinks we turn the corner hi Ben Thomas
Donald Trump Joe Biden Las Vegas Denise Goulet President Trump Ben Thomas Durham North Carolina
Report: Seattle likely to recover from pandemic faster than most U.S. cities
"And a new report predicts that Seattle is likely to have one of the quickest and strongest recoveries in the country following the pandemic Yahoo finance analyzed the top metro areas in the country specifically looking at population density in educational attainment they believe the top cities to recover will be San Jose California Durham North Carolina Austin Texas Seattle and Minneapolis this is a key difference between this recovery in the last one is the population density it's going to have a different effect this
Seattle Minneapolis Yahoo San Jose California Durham Nor
Stanley, boardwork, zone help No. 12 Duke top NC State 88-69
"Number twelve Duke events their loss from two weeks ago against NC state rolling by the wolf pack eighty eight sixty nine trailing thirty six thirty in the first ducal using NATO run to close out the half and take a two point lead into the locker room tied at forty six in the second do will go on a twenty five ten run to take control of the game Cassius Stanley led the charge in the second half scoring fourteen of his game high eighteen points after the break six players would score double figures for Duke as the blue devils would score twenty five fast break points as well as forty six points in the paint Dennis **** Durham North Carolina
Nato Cassius Stanley Duke NC Durham North Carolina
Washington DC: Virginia Teen Accused of Killing Mother, Brother Arrested in North Carolina: Sheriff
"The seventeen year old who's been accused of shooting and killing his mother and his six year old brother in their funkier county home has been arrested after he tried to shoplift hair dye closed in a backpack from the North Carolina department store police say Levi Norwood was arrested on Saturday after employees at a target in Durham North Carolina found him in the sporting goods section and called police in the store's parking lot investigators found a car that Norwood stole in Virginia allegedly after killing his mother and brother and wounding his father on Friday at their home in Midland Virginia but it appears the **** your county teenager will not likely be heading back to Virginia right away the first thing that'll happen with seventeen year old Levi Norwood insider North Carolina court room will make sure he has an attorney as a juvenile in the series of this charges the judge on his own could appoint counsel regardless of what the the U. says Parker county Commonwealth's attorney Scott hook this hearing on Tuesday we don't expect him to be you know automatically sent back to us we expect will be additional hearings as for extradition back to Virginia we have a hearing on Tuesday hopefully the North Carolina prosecutors who's handling at this point would be back in court hopefully before the end of the week Kristy king WTOP news out counselors will be at merry Walter elementary school tomorrow that a six year old victim why it nor would went to that school
Levi Norwood Midland Virginia Attorney Scott Hook Merry Walter Elementary School North Carolina Durham North Carolina Kristy King
Washington, D.C: Virginia Teen accused of killing mom, younger brother arrested in North Carolina
"After after spreading spreading a a wide dragnet police successfully file that Midland teenager who's accused of murdering his mother and a six year old brother he's also accused of shooting his father but the data scape the kid accused of the cartridges Levi Norwood he seventeen and law enforcement and society caught a lucky break when he was arrested for shoplifting in North Carolina the indicators are as he was by himself he traveled out of the area and when he was placed under arrest in Durham North Carolina there's no indication of anybody being with them at that time that's **** your county sheriff Bob losure there's a court hearing in North Carolina Tuesday the extradition process to get it back to Virginia the starting right away
Levi Norwood Shoplifting North Carolina Bob Losure Virginia Midland Durham North Carolina
No. 7 Duke downs Notre Dame 94-60 behind 21 points by Cary
"Number seven do trail for a total of fourteen seconds against Notre Dame as the blue devils cruise by the fighting Irish ninety four to sixty leading forty nine thirty seven was sixteen fifty eight to play in the second half Duke would outscore Notre Dame thirty one twelve over the next ten minutes to pull away the blue devil shot over fifty six percent from the floor and over forty five percent from behind the arc on ten of twenty two shooting Vernon Carey scored twenty one points for Duke while guard Trey Joe scored nineteen you want during led Notre Dame what twenty one Dennis **** Durham North Carolina
Vernon Carey Duke Trey Joe Durham North Carolina
Jones, Goldwire help No. 7 Duke beat No. 8 FSU 70-65
"Vernon Carey junior three point play was seven sixteen to go give number seven due to fifty eight fifty seven lead and they never looked back as a blue devil so done number eight Florida state seventy to sixty five the blue devils led by as much as ten the first but a nine two F. as you run help then call back to within one thirty three thirty two and half a dozen vassal three we give F. as to a fifty seven fifty five Lee with seven thirty two to play but carries and one will be the night to finally change of the game Kerry finished with ten points and ten rebounds Florida state was led by Trent force eighteen points nine rebounds in these deals Dennis **** Durham North Carolina
Vernon Carey LEE Kerry Florida Trent Durham North Carolina
Johnson, No. 11 Louisville earn win at No. 3 Duke 79-73
"Number eleven Louisville would use a seventeen one run in the first half to build themselves to the largest the twenty five ten and never looked back down a number three Duke seventy nine seventy three Duke which happened to the deficit in the second half to tie the game twice including at sixty three after trade Jones three point was six twenty eight the play leading seventy one seventy believe Williams and when certain will combine for the next eight points of the game to seal the win for Louisville William said as a great victory for the program it's a great way to come in and Karen indoor legeno place what big memory soon a lot of great players to come through here is a lot of great teams so grateful and get a win here cash instantly they do with twenty four points and eleven rebounds well David Johnson with the cardinals with seventeen points Dennis **** Durham North Carolina
Louisville Williams David Johnson Cardinals Dennis Jones Durham North Carolina
Live From Here with Chris Thile
Humans Can Regrow Cartilage in Joints Just Like Salamanders
"If you have arthritis thank you you worry about your cartilage wearing out right but you've probably heard that you can't re grow your cartilage when it wears out its gun and that can lead to different types of arthritis or worsening arthritis and well listen to this a team of researchers wanted to investigate this idea about well can you re grow the stuff they use molecular clocks to figure out the age of the cartilage in our bodies from our hips to our needs to our ankles and they found that the cartilage in your ankle was younger younger than the cartilage in your hips and their results were published this week in the journal science advances so what does this turnover rate tell us about cartilage repair my next guest is an author on that study Dr Virginia buyers Krause is a professor of medicine orthopedics and path ology at Duke University in Durham North Carolina Carolina welcome to science Friday hi IRA pleasing carnal G. living tissue right well why is it so difficult for for us to re grow it it absolutely is living it's full of cells but what's unusual about cartilage is it doesn't have any blood supply and it doesn't have any nerve tissue so that's why it can be degenerating for long periods of time without there being that many signals in terms of something's going wrong how does it survive will with without any blood supply one does why wouldn't just writing your body like a foreign object well this is why exercise is so important for the joint nutrition because it gets its nutrients from the fluid inside the joint and which moves in and out of the cartilage brings the nutrients in and out to the cells with each dynamic motion so you are you saying that even if we have arthritis and when we're aching we we should still continue to do exercise yes as much as possible and when people are really severely affected we send them to the pool where they only way one eighth in the poll what they way on land which is even better than on on the moon and that way they can move their joints through ranges of motion without that deleterious load that's very interesting let's let's talk about cartilage aging how does it age over time so over time all your tissues accumulate chemical modifications and in cartilage that is particularly true because a lot of these cartilages are very long long lasting and what we found our surprise as you mentioned was that the proteins in the ankle appeared to be much younger in older people than their knee cartilage or their hip cartilage which told us that the ankle is in a high state of continuous repair like a house that's being refurbished continuously and the knees or somewhere intermediate but the hips have very low repair capacity so what are the ankles and making new cartilage or they just keeping the old stuff healthier so they're doing both they're getting rid of the old and they're replacing it with new so they're constantly turning it over and keeping it fresh so we know how that happens and could we then do that for the rest of the in our bodies well we racked our brains about this and we finally hit upon a beautiful paper in two thousand sixteen some work that looked at three different animals that could regenerate their limbs even though the animals were widely separated in evolutionary time by millions of years they posited that those factors that were controlling limb regeneration that were shared across the three of them might be the master regulators of limb regeneration and so they found a list of a hundred and eight micro aren't any that they think are the master regulators of that process so we looked into those factors and we found that those several of the ones that we tested which were some of their highest and most abundant ones were highest in the ankle were intermediate in the knee and were low in the hip and correlated very well with the protein age and the amount of those chemical modifications that had accumulated so could you move around that micro already a different place exactly that's what's so exciting is that we now have a list of a hundred and eight regenerative microRNA that could be injected into the joint to boost the natural repair response because we're already making those for
The Kristen Hagopian Show
Officials: 1 dead, 17 hurt in North Carolina gas explosion
"And after gas explosion ripped through parts of Durham North Carolina on Wednesday. The blast killed a sixty one year old man and injured more than a dozen other people investigators believed that the crew digging in the area triggered the explosion which level a building and damaged several others. Witnesses described it as feeling like an earthquake
North Carolina Durham Sixty One Year
This Morning with Gordon Deal
Deadly gas explosion levels building in Durham, North Carolina
"One person has died in seventeen others are injured after a fiery explosion in Durham, North Carolina. That was felt citywide local resident Leon Brown. Explosion was caused by a gas leak after a contractor drilling into the sidewalk. Struck a two inch gas line the explosion impacted five buildings causing catastrophic damage
Explosion Leon Brown North Carolina Durham Two Inch
Do cellphones cause cancer? Maybe in rats, but not in people
"And they were done at levels much higher than what humans are exposed to. But in science, we always need to overdo. Okay. We can't go. All right. Well, if you use your cell phone once a week just test that and see if you're good. Not what we're going to deal with. We're gonna look at cell phone radiation. And if you use cellphones, or if you are exposed to the radiation like galore, and then we work backwards. That's kind of an easier way to do it. It's safer way to do it. So. Female rats and mice both for female rats and mice of both sexes. The evidence was less clear as to whether cancers observed associated with signal exposure, but for male rats. They said it was undeniable why? Well, we don't know males may have different. I, you know, maybe I. Yeah. Maybe. The way the brain is shaped brain cells. You know, everybody likes to say there's no difference. You know, maybe there is a difference in the answer. Very very careful of how I word this. Maybe you know, there are I there's been. Anatomical suggestions of different sizes of organs in males versus females and that could potentially contribute to this. Okay. Who you gotta be really really careful what you say. These days. You can't say anything. Senior scientist, Dr John butcher of the national toxicology program in Durham, North Carolina said the exposures using the studies cannot be compared directly to the exposure that he was experienced when using a cellphone because what they did their studies, the rats and mice received radiofrequency radiation across their whole bodies. Yet. People are mostly exposed to local areas where they hold the phone. They also say the exposure levels and durations of our cities were greater than what people experience. So again what they're saying. There's nobody should panic because this was a lot of radiations in the whole entire body. So you could trigger a sell in another part of the body that then goes rogue and causes the cancer to occur. So this was a ten year study thirty million dollars in cost to assess the health effects of animals exposed to our f- our radio for radiation. What's modulators used a two g and three g cell phones. And the final reports were presented consensus of NTP and a panel. Exhibition national toxicology program and a pedal of external scientific experts who reviewed the studies. So that means we had a lot of big, you know, big lab people looking at this. Doctor butcher explained that. Or is it Booker? I think it's butcher. Said cell phones, utilize a specific type of radio waves or radio frequency radiation to transmit between the devices on the network. Exposure of people to this occurs,
Dr John Butcher Booker NTP Durham North Carolina Scientist Thirty Million Dollars Ten Year Three G Two G | {
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} | 9,471 |
//===----------------------------------------------------------------------===//
//
// PelotonDB
//
// index_scan_node.h
//
// Identification: src/backend/planner/index_scan_node.h
//
// Copyright (c) 2015, Carnegie Mellon University Database Group
//
//===----------------------------------------------------------------------===//
#pragma once
#include <memory>
#include <string>
#include <vector>
#include "backend/planner/abstract_scan_plan.h"
#include "backend/common/types.h"
#include "backend/expression/abstract_expression.h"
namespace peloton {
namespace index {
class Index;
}
namespace storage {
class Tuple;
}
namespace planner {
class IndexScanPlan : public AbstractScan {
public:
IndexScanPlan(const IndexScanPlan &) = delete;
IndexScanPlan &operator=(const IndexScanPlan &) = delete;
IndexScanPlan(IndexScanPlan &&) = delete;
IndexScanPlan &operator=(IndexScanPlan &&) = delete;
struct IndexScanDesc {
IndexScanDesc() : index(nullptr) {}
IndexScanDesc(
index::Index *index, const std::vector<oid_t> &column_ids,
const std::vector<ExpressionType> &expr_types,
const std::vector<Value> &values,
const std::vector<expression::AbstractExpression *> &runtime_keys)
: index(index),
key_column_ids(column_ids),
expr_types(expr_types),
values(values),
runtime_keys(runtime_keys) {}
index::Index *index = nullptr;
std::vector<oid_t> key_column_ids;
std::vector<ExpressionType> expr_types;
std::vector<Value> values;
std::vector<expression::AbstractExpression *> runtime_keys;
};
IndexScanPlan(storage::DataTable *table,
expression::AbstractExpression *predicate,
const std::vector<oid_t> &column_ids,
const IndexScanDesc &index_scan_desc)
: AbstractScan(table, predicate, column_ids),
index_(index_scan_desc.index),
column_ids_(column_ids),
key_column_ids_(std::move(index_scan_desc.key_column_ids)),
expr_types_(std::move(index_scan_desc.expr_types)),
values_(std::move(index_scan_desc.values)),
runtime_keys_(std::move(index_scan_desc.runtime_keys)) {}
~IndexScanPlan() {
for (auto expr : runtime_keys_) {
delete expr;
}
// otherwise how to free the values? added by ?michael
// for(auto value : values_){
// value.Free();
// }
}
index::Index *GetIndex() const { return index_; }
const std::vector<oid_t> &GetColumnIds() const { return column_ids_; }
const std::vector<oid_t> &GetKeyColumnIds() const { return key_column_ids_; }
const std::vector<ExpressionType> &GetExprTypes() const {
return expr_types_;
}
const std::vector<Value> &GetValues() const { return values_; }
const std::vector<expression::AbstractExpression *> &GetRunTimeKeys() const {
return runtime_keys_;
}
inline PlanNodeType GetPlanNodeType() const {
return PLAN_NODE_TYPE_INDEXSCAN;
}
const std::string GetInfo() const { return "IndexScan"; }
private:
/** @brief index associated with index scan. */
index::Index *index_;
const std::vector<oid_t> column_ids_;
const std::vector<oid_t> key_column_ids_;
const std::vector<ExpressionType> expr_types_;
const std::vector<Value> values_;
const std::vector<expression::AbstractExpression *> runtime_keys_;
};
} // namespace planner
} // namespace peloton
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,537 |
Business / May 2020
We'll All Be Social Media Sellouts Soon
Companies are turning to small-time influencers to promote their brands. Will any of us be able to resist the lure of free products and cash?
April 15, 2020 November 8, 2021 - by Tatum DooleyTatum Dooley
Illustration by Vivian Rosas, Updated 9:46, Nov. 8, 2021 | Published 11:55, Apr. 15, 2020This article was published over a year ago. Some information may no longer be current.
In 1932, Lester Gaba set out to create the ideal woman. His aim was simple: she would be beautiful but attainable, a figure that the everyday person could see themselves in. Gaba named his creation Cynthia. Though Cynthia was technically a mannequin—one commissioned by Saks Fifth Avenue—Gaba started bringing her out into the world, treating her like a real person at all times.
Listen to an audio version of this story
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Cynthia quickly became a quasi celebrity. She was a regular sight at parties and events and was even photographed for a Life magazine feature. Companies, sensing an opportunity to capitalize on the attention, started sending her their products. "She received free dresses from Saks, diamonds from Tiffany's, tickets to the Metropolitan Opera," said journalist Roman Mars on an episode of the design podcast 99% Invisible. "When she showed up in tabloids, she was wearing designer clothes." Cynthia may not have been sentient, but she was a trendsetter; her brief time as a socialite-cum-advertiser also foreshadowed the world of social media influencers who dominate Instagram feeds and marketing campaigns today.
Like Cynthia, social media influencers aren't usually celebrities, at least not in the traditional sense. They tend to be regular people—often young, often attractive—who have turned posting pictures or videos to YouTube, Instagram, Snapchat, or TikTok into jobs. For just about any niche or hobby, whether it's travel, fashion, video games, or fishing, there's an influencer posting about it on a daily (or hourly) basis, amassing a large and loyal following. Companies have taken note, and they are buying access to these audiences in exchange for merchandise and cash.
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This new realm of influencer marketing is less than ten years old, but it's become a central strategy for certain sales departments because, unlike more traditional formats, such as television spots and billboards, influencer posts are advertisements that don't feel like ads. Up until the past couple of years, most paid-for posts weren't even labelled as such. Still, despite the new best practice of noting "#ad" or "#sponsored" in captions, the intimacy of social media means that, for the audience, it can still feel like the influencer just had to let everyone know that they love a particular makeup brand, or a pricey pair of headphones, or, in the case of Kim Kardashian West, a new appetite-suppressing weight-loss product. Canon, Starbucks, Volvo, H&M, the tourism board of Nova Scotia: all have used influencer marketing in recent campaigns. According to Business Insider's 2019 Influencer Market Report, companies are projected to spend $15 billion on the field by 2022.
Still, influencer marketing on social media is a new business model, and the tactic of throwing money at young personalities and hoping it translates into sales has led to some not-so-surprising results. In 2018, megainfluencer Luka Sabbat (then with 1.4 million followers) was sued for failing to fulfill a $60,000 deal that required him to wear Snapchat's new product, Snap Spectacles, at high-end fashion shows. (Sabbat had agreed to a minimum of four Instagram posts that included the product; he uploaded only two.) The effectiveness of influencers was further questioned last May, when Instagram star @Arii (then with 2.5 million followers) launched her own clothing line and sold fewer than thirty-six shirts.
It's failures like these that Bryan Gold is trying to prevent. Gold, a twenty-seven-year-old entrepreneur, is the co-founder and CEO of #Paid, a pioneering Toronto-based software company that deals in influencers. #Paid exists somewhere between talent agency and ad agency—it doesn't directly manage influencers, but it's positioning itself as a professional middleman that can work with big businesses to develop and oversee social media campaigns while wrangling the thousands of young social media users who have the desired captive audiences.
Gold knows that influencers have a checkered reputation: despite the value they offer brands, they can be inexperienced and unpredictable, making them potential threats to companies' images. But the roster that works for #Paid, he explains, is different from the rest. For one, his nearly 22,000 social media stars aren't mere influencers, he insists—they're creators.
I trip on this bit of semantics again and again over the few days I spend at the #Paid office—a contemporary open space complete with the requisite startup perks of LaCroix sparkling water and a Ping-Pong table. Each time "influencers" leaves my mouth, I earn a stern look. The difference between the two, various staff members explain to me, is that, unlike typical influencers, #Paid creators care about what they do. They are discerning about whom they'll work with. (Gold tells me that, when his company received a shipment of a weight-loss product, they sent it right back.) I'm told that, whenever a #Paid creator embarks on a campaign—with, say, Coca-Cola or Uber Eats—the result is an authentic expression of how they feel about the product; the relationship is about more than just making money. "A lot of the creators I spoke with wanted to use their influence for good," Gold tells me. "They were genuinely passionate about making a difference in the world and making the world a better place."
While soda endorsements and food-on-demand probably won't change society for the better, recent trends seem to show that influencer marketing is the future that's coming for us all: one where social media becomes work, the work never stops, and all online identities are commodities to be constantly managed.
Around 2013, Gold noticed that a friend, Ronnie Friedman, was becoming popular online. Friedman had been building a steady following by posting her fitness journey, sharing healthy meals, before-and-after body shots, and gym selfies. "She was creating awesome content," Gold says. "Her audience would buy anything she used in order to get fit." Companies must have noticed this too, as some started reaching out to ask if she'd be interested in posting their products. Friedman hadn't planned on turning her social media presence into a small business and was stuck trying to manage these deals on her own. Gold saw an opportunity. In 2014, he secured a place in Ryerson University's startup incubator and officially launched #Paid with co-founder Adam Rivietz. In 2018, the company raised $9 million in venture funding.
The goal of #Paid isn't to sign big-name celebrity influencers, like reality-television star Kylie Jenner (164 million Instagram followers, net worth reportedly $1 billion US), or massive YouTube stars, like Logan Paul (20.5 million subscribers, estimated net worth $19 million US). Gold built #Paid for people like Friedman: everyday social media users who had carved out a modest following in a particular niche. Many of these users are classified as "microinfluencers," who tend to have at least 5,000 followers and up to 100,000. The company also acts as a safety net for brands by vetting creators via an algorithm to ensure their followers are real. One thousand fake followers can be bought for a little more than $10, and cybersecurity firm Cheq reported that this kind of fraud was projected to cost advertisers more than $1 billion in 2019.
When it's time to run a campaign, #Paid puts together a shortlist of potential creators from the around the world for clients. Each creator has tags describing them and their audience, which help companies find the right fit for their target demographics (e.g., "mom," "single," "interested in cars"). Once a creator has been selected and has agreed to the partnership, they conceive and execute their own post and upload it onto #Paid's platform—essentially a mock-up of an Instagram post—for client approval. The two sides go back and forth if necessary, retaking shots and tweaking captions until everyone is happy. The smallest details matter: the #Paid office features a "trending word chart," a white piece of Bristol board where employees keep track of ever-shifting online slang, which helps ads abide by the zeitgeist. Hours of work go into a single post that most viewers will scroll past and internalize within seconds.
Unlike the less reputable players in influencer marketing, #Paid makes sure that all of its creators' ad posts are clearly marked as advertisements and follow necessary government regulations. Once a post goes live, the #Paid team goes into analytic overdrive, examining the engagement metrics provided by Instagram (mostly data on likes and comments). It has also partnered with the market research firm Neilson to create sets of statistics that show the effectiveness of its work, tracking a post from app to online cart to purchase. When I visit the office, a projector is displaying the live numbers of a current campaign, showing that an extra $50 ad buy for a sponsored placement on Instagram feeds was leading to $10,000 in sales.
When done right, influencer marketing is a dream for businesses, all high returns on low investments. Creators with #Paid set their own rates: they can make anywhere from tens of dollars to thousands per post, depending on their reach. And, though many small-level influencers won't make a living from social media, the money isn't nothing. Meanwhile, #Paid earns its cut by charging clients a monthly fee, which gives businesses a set number of sponsored posts, analytics, and contracts with creators.
In recent years, results have shown that choosing the right influencer for a campaign can be the difference between success and failure; bigger influencers don't always mean more sales. In an interview with Business of Fashion, Nicolaj Reffstrup, the then CEO of Ganni, a Danish fashion line that relies heavily on influencer marketing, explained that success on social media can be highly unpredictable. Big-name celebrity endorsements sometimes have no effect on sales. "Other times," he said, "you can have a girl with 4,000 followers in a regional area in Sweden, and she converts to sales extremely well because she makes so much more sense to her followers."
A recent report from technology-research company Morning Consult found that 56 percent of millennials and Gen Z say that they have made a purchase after seeing a post by someone they follow. Another 86 percent said that they'd be willing to post sponsored content if someone paid them to do so. They may soon get the chance—lately, industry watchers have been promoting a new class of marketing agent: the nanoinfluencer, usually defined as a person with between 1,000 and 5,000 followers. In a few years, everybody on social media may be selling to everybody else.
Digital media has democratized and diffused celebrity, but it seems that many companies buying influencers today want a lifelike Cynthia figure: someone safe, someone on brand, someone who, ideally, doesn't speak for themselves.
Sylvia Jade is the strategy lead at #Paid, and part of her job is to soothe companies' fears about working with influencers. "I've gotten requests from clients to avoid anyone who's controversial. I've been asked to do background checks," she says. "If we're presenting a roster, they can't be associated with any bad press."
This kind of digital panopticon has become part of Jade's own life: in addition to being a #Paid staffer, she's a beauty YouTuber with more than 180,000 followers. Jade tells me that she's always highly conscious whenever she leaves her house, careful to look put together and happy. She avoids small expressions of negative emotion, like getting visibly annoyed with her boyfriend while waiting in line for lunch. It's too risky; she never knows who will recognize her. It seems that normal people can now have all the negative trappings of celebrity with a small fraction of the money and power that usually comes with it.
At first, this level of artifice can be difficult to square with influencers' main selling point: their authenticity. The word crops up repeatedly when speaking with influencers and the companies that represent them, but its meaning seems to have undergone a shift from its traditional usage. Authenticity among influencers is not about living in a manner that's true to oneself. Instead, it describes anything that feels true to the audience.
Take @LilMiquela, an Instagram influencer with 1.9 million followers. Miquela is a pretty, freckled woman who looks to be in her early twenties. She's recently partnered with Ugg and Calvin Klein for sponsored posts and is often pictured hanging out with friends, travelling, and promoting social justice causes. Miquela is also completely fake, a CGI avatar who, like Cynthia nearly ninety years earlier, feels just real enough: fans can see themselves in her, and companies can have complete control over her. She is all artifice but always authentic; a gold standard for mortal influencers everywhere.
Tatum Dooley
Tatum Dooley (@ tatum_dooley) is a writer and curator based in Toronto. Her work has appeared in Artforum, Canadian Art, Border Crossings, Garage Magazine, Lapham's Quarterly, and the Globe and Mail.
Vivian Rosas
Vivian Rosas is a Toronto-based Latinx illustrator and designer. Selected clients include Today's Parent, Lenny Letter, and the Globe and Mail.
Taggedinstagramlistensocial media
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Type: Analysis
Analysis: Based on factual reporting, although it incorporates the expertise of the author/producer and may offer interpretations and conclusions. | {
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{"url":"https:\/\/me.gateoverflow.in\/2164\/gate-mechanical-2022-set-2-question-48","text":"# GATE Mechanical 2022 Set 2 | Question: 48\n\nA cylindrical billet of $100$ $\\text{mm}$ diameter and $100$ $\\text{mm}$ length is extruded by a direct extrusion process to produce a bar of $L$-section. The cross sectional dimensions of this $L$-section bar are shown in the figure. The total extrusion pressure $(p)$ in $\\text{MPa}$ for the above process is related to extrusion ratio $(r)$ as\n\n$$p = K_{s}\\sigma _{m}\\left [ 0.8 + 1.5\\: \\ln \\left ( r \\right ) + \\frac{2l}{d_{0}}\\right ],$$\n\nwhere $\\sigma_{m}$ is the mean flow strength of the billet material in $\\text{MPa}$, $l$ is the portion of the billet length remaining to be extruded in $\\text{mm}$, $d_{0}$ is the initial diameter of the\u00a0 billet in $\\text{mm}$, and $K_{s}$ is the die shape factor.\n\nIf the mean flow strength of the billet material is $50$ $\\text{MPa}$ and the die shape factor is $1.05$, then the maximum force required at the start of extrusion is ________________ $\\text{kN}$ (round off to one decimal place).\n\nin Others\nedited\n\n## Related questions\n\nA rope with two mass-less platforms at its two ends passes over a fixed pulley as shown in the figure. Discs with narrow slots and having equal weight of $20\\: N$ each can be placed on the platforms. The number of discs placed on the left side platform ... refer to part $\\text{(ii)}$ of the figure) required to prevent downward motion of the left side platform is _____________________(in integer).\nFor a dynamical system governed by the equation. $\\ddot{x}\\left ( t \\right ) + 2\\zeta \\omega _{n}\\dot{x}\\left ( t \\right )+\\omega _{n}^{2}x\\left ( t \\right ) = 0$ the damping ratio $\\zeta$ ... . Neglecting higher powers $(>1)$ of the damping ratio, the displacement at the next peak in the positive direction will be __________________ $\\text{mm}$ (in integer).\nAn electric car manufacturer underestimated the January sales of car by $20$ units, while the actual sales was $120$ units. If the manufacturer uses exponential smoothing method with a smoothing constant of $\\alpha = 0.2$, then the sales forecast for the month of February of the same year is ____________ units (in integer).\nThe demand of a certain part is $1000$ parts\/year and its cost is \u20b9$1000$\/part. The orders are placed based on the economic order quantity $\\text{(EOQ)}$. The cost of ordering is \u20b9$100$\/order and the lead time for receiving the orders is $5$ days. If the holding cost is \u20b9$20$\/part\/year, the inventory level for placing the orders is ________________ parts (round off to the nearest integer).\nConsider $1$ $\\text{kg}$ of an ideal gas at $1$ bar and $300$ $K$ contained in a rigid and perfectly insulated container. The specific heat of the gas at constant volume $c_{v}$ is equal to $750 \\:J \\cdot kg^{-1}\\cdot K^{-1}$. A ... work on the gas. Assume that the container does not participate in the thermodynamic interaction. The final pressure of the gas will be _____________ bar (in integer).","date":"2022-07-02 14:00:29","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.750050961971283, \"perplexity\": 602.2473041123917}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-27\/segments\/1656104141372.60\/warc\/CC-MAIN-20220702131941-20220702161941-00181.warc.gz\"}"} | null | null |
In a move to foster small business development and job creation in the Western Cape, The City of Cape Town and MEDO (the Micro Enterprise Development Organisation) have announced a nine day tour across the city in search of exceptional individuals who intensely aspire to take their small businesses to the next level.
The tour starts in the Ikhaya Le Langa Quarter on 15 August and ends in Kraaifontein on 24 August. The organisers invite entrepreneurs to visit the mobile office – or Treppie – to pitch their businesses. Successful applicants will receive a boost to their enterprise in the form of access to further training programmes.
"We are taking the tour to areas that need the entrepreneurial support, areas where we could make the greatest impact on the communities we visit", says Garreth Bloor, Mayoral Committee Member for Economic, Environmental and Spatial Planning. "Heading to eight different locations, we as a City will expose entrepreneurs and potential entrepreneurs to the possibilities of starting their own business by helping them to identify opportunities, giving them necessary skills and leaving them with information about support offered by various Enterprise Development Organisations with which to succeed".
As a strategic advisor and on-the-ground implementer of sustainable micro and small enterprise development, MEDO is not only helping to secure supplier relationships and expand trade within and outside South Africa, the company's vast cumulative knowledge and experience in the Enterprise- and Supplier Development space ideally positions them to assist corporate clients with their ED and SDP strategies and BBBEE point accrual.
For more information call 010 500 5000 or visit the website www.medo.co.za. | {
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Grubs for Dinner.
Today's post is for those of you who enjoy Extreme Cuisine, whether in reality or vicariously.
The Food Journal, (London, 1871-4) has been a marvelous source of food stories since I discovered it recently. Volume I contains the following article:
A FEAST ON WORMS.
Shakespeare asks, "Where is fancy bred; in the heart or in the head?" Without answering the question, we might certainly add that, to judge from what we are about to relate, it has no material connection with the gastronomic organs. If fancy and prejudice only keep neutral, there is no knowing what appetite may do. The stomach and the palate by themselves, uninfluenced through the mind, or rather its fancies, will truly stand anything that is not poisonous, painful, or, apart from its associations, absolutely nasty to the taste. We do not know whether such a divorce between fancy and the body has or has not its advantages. The French Society of Cultivators once sought to solve the point. In solemn conclave they voted a feast, at which a new dish should be the chief attraction. But what novelty could be found? Shall we say that, as savages love to consume their enemies, so these hardy agriculturists sought a palatable flavour in a fried worm? Certain it is that, two years ago, French agriculture was much plagued by worms, and the society chose the worm as the piece de resistance for their banquet. The choice of the species, however, was a matter of grave consideration; and only after much discussion the worm of the cockchafer was selected. This species had, by the way, proved itself the most injurious to the crops. It is known in France as the verblanc, because of its dirty white colour, and rejoices in six legs. Adepts chose a few of the youngest and most tender, while the cooks wisely decreed that they should be thrown into vinegar and water. The immersion made them disgorge the earth which they had swallowed. When thus cleaned, they were rolled in a paste of milk and flour, so as to give them a brittle crust. This much accomplished, the worms were taken in triumph to the Cafe Corazza, in the Palais Royal. This was the house selected for the banquet; but the manager had little suspected what he would be told to place in his cherished frying-pan. However, the chef of this fashionable dining place performed his duty punctually, and just at the right moment the worms were brought up hot and crisp to the banquet. Two were placed on each plate, and it is boastfully recorded that those who ate one ate the other. But more; there were eighty guests and 200 worms, so, perhaps, some might have had three. May this be a lesson to the fastidious who are too fanciful. What may we not accomplish with strength of mind?
The story was repeated by Peter Lund Simmonds in The Animal Food Resources of Different Nations (1885,) although he refers to the venue as the Café Custoza, and he gives us a recipe for the 'worms.'
A few years ago at the Café Custoza, in Paris, a grand banquet was given for the special purpose of testing the vers blanc, or cockchafer worm. This insect, it appears, was first steeped in vinegar, which had the effect of making it disgorge the earth, etc., it had swallowed while yet free; then it was carefully rolled up in a paste composed of flour, milk, and eggs, placed in a pan, and fried to a bright golden colour. The guests were able to take this crisp and dry worm in their fingers. It cracked between their teeth. There were some fifty persons present, and the majority had a second helping. The larvae, or grubs, generally, not only of the cockchafer, but those of the ordinary beetles, may, according to some naturalists, be eaten safely. Cats, turkeys, and different birds devour them eagerly.
We find in a Continental journal, the Gazette des Campagnes, the following receipt for cooking these insects,
which is adopted in certain parts of France :
" Roll the vers blancs, which are short and fat, in flour and bread crumbs, with a little salt and pepper, and wrap them in a stout piece of paper, well-buttered inside. Place it in the hot embers and leave it to cook for twenty minutes, more or less, according to the degree of heat. On opening the envelope a very appetizing odour exhales, which disposes one favourably to taste the delicacy, which will be more appreciated than snails, and will be declared one of the finest delicacies ever tasted." *
* Les Insectes Utiles, par Henri Miot, p. 89. Paris, 1870
You can never have enough garlic. With enough garlic, you can eat The New York Times.
Morley Safer
Les said...
Funny, there was a minor worm craze here in the US back in the '70s. News articles claimed they were delicious and nutritious giving recipes for fried worms and for worms in cake. It was marketed as a sort of get rich quick scheme but fizzled out pretty quickly. I have a container of them in the fridge right now that I feed with lettuce to keep them fat and healthy for my pet fish.
A Gallon of Bread.
Squirrel Casserole.
Giant Mushrooms.
Electrical Eel Pies.
Black Bread Sandwich.
Alecsize that Fish!
Dandelion and Burdock.
Cheshire Cheese, Part 2
Ice-cream, or not?
Pasta Puddings.
Soufflé Bread.
Better Butter and French Spinach.
Diplomatic Welsh Rabbit.
Aged Coffee.
Baking Better Brioche.
Extreme Kitchen DIY: Potato Flour.
Vulgarity in the Kitchen.
'Caviar to the Vulgar'
For the War Effort: Eat Calavos.
Pink Sauce.
Setting a Good Example.
Barbecue Sauce. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 8,480 |
Onlangs kwam TV1 (Nieuw-Zeeland) met een reportage over de laatste expeditie van de de National Science Foundation naar de onderzeese vulkanen in Kermadische Boog, op zo'n 350 kilometer ten noordoosten van het Noordereiland van Nieuw-Zeeland. Op beelden van de hydrothermale bronnen op de krater van de Brothers vulkaan is te zien dat vissen en kreeftachtige dieren in de buurt van deze heetwaterbronnen leven. De watertemperatuur bedraagt daar meer dan 300° C. Wetenschappers geven aan, dat de eerste vormen van organisch leven ooit onder dergelijke omstandigheden zijn ontstaan.
De caldera van de Brothers Vulkaan, gezien vanuit het zuiden. Wanneer de vulkaan voor het laatst tot uitbarsting kwam is nog niet duidelijk, maar sindsdien is er wel hydrothermale activiteit waargenomen, niet alleen tijdens het onderzoek in 2007, maar ook recentelijk door onderzoek van de National Science Foundation (NSF). Bron: New Zealand American Submarine Ring of Fire 2007 Exploration, NOAA Vents Program/Institute of Geological & Nuclear Sciences en NOAA-OE.
The Brothers Volcano is a Pacific Ocean submarine volcano in the Kermadec Arc, 340 kilometres north east of New Zealand's Whakaari/White Island. Within its oval outline, which measures 13 km by 8 km, it contains a 3 km wide caldera with walls 300-500 m high. It is three times bigger than the White Island. A dacite (dacite lava's viscosity is in between rhyolite and andesite) dome rises 350 m from the caldera floor (which lies 1850 m below sea level), with a smaller dome just to its northeast. The caldera walls and the larger dome host numerous hydrothermal vents, which send plumes of hot water 750 m up through the water column. It is the most hydrothermally active volcano known in the Kermadec Arc. These hydrothermal vents are also known as hot springs and have created an 8 meter high field of "black smoker" chimneys. These chimneys are created when the hydrothermal fluids hit the cold water and the chemicals in solution are deposited. The hydrothermal fluids are an energy source to many unique organisms like tubeworms and other bacteria. The marine life and minerals found from these chimneys are beneficial to New Zealand's economy and biotechnology industry.
The volcano was created by a subduction of the Pacific Plate under the Australian Plate.
It is still unknown when the Brothers Volcano last erupted, but the crater walls tell us that its last eruption was so explosive that it caused the volcano to blow out a caldera. A joint expedition by the United States, New Zealand and Germany mapped the volcano in detail in 2007. | {
"redpajama_set_name": "RedPajamaC4"
} | 8,791 |
\section{Introduction}
For functions $f \in \mathbb{R}^d$, it is well known that its Wigner distribution, given by
\begin{equation}
W[f](\xi, x) = \int e^{- i 2\pi \xi \cdot p} f(x + \tfrac{1}{2} p) \overline{f(x - \tfrac{1}{2}p)}
\, \text{d} p
\end{equation}
provides a useful characterization of the function. The Wigner
transformation has important applications include the phase space
formulation of quantum mechanics \cite{case2008}, time-frequency
analysis in signal processing \cite{cohen}, semiclassical analysis
\cite{folland, zworski}, just to name a few.
In this paper, we consider the discrete analog of Wigner transforms;
that is, we aim for phase space representations for vectors in
$(\mathbb{C}^p)^{\otimes N}$, analogous to the continuous case. This natural
generalization has been considered by Wootters \cite{wootters} in
1987, surprisingly much later than the continuous formulation
\cite{Wigner1932, Weyl1927}. Here we revisit the construction of
discrete Wigner transforms and we propose a natural analog of the
continuous Wigner transform essentially by ``discretizing the
integral'' as a sum over $\mathbb{Z}_p$. This leads to a very natural
extension of the continuous Wigner transform to the discrete state
space. Along the way, we will revisit Wootters' construction of the
discrete Wigner transform. While his proposed discrete Wigner
transform was through a quite indirect link to the continuous analog,
the connection becomes much more clear through our perspective. In
fact, it turns out that our construction is tightly connected to the
Heisenberg-Weyl group as our discrete Wigner transform can be
represented as an inverse Fourier transform of the
Heisenberg-Weyl group elements, whereas Wootters'
construction corresponds to an inverse symplectic Fourier transform.
As we will see, taking the inverse Fourier transform (as opposed to
the inverse symplectic Fourier transform) is more natural in some
sense.
One of the motivations of our work comes from recent proposed
numerical methods for quantum many-body spin dynamics based on phase
space representation \cite{Schachenmayer2015, Orioli2017}. These
works extend the phase space numerical methods for Schr\"odinger
equations (see e.g., the review article \cite{Jin2011}) to the setting
of discrete state space. As the phase space representation is a
natural bridge between the Schr\"odinger equations and their
semiclassical limit, such numerical methods are expected to work well
in the semiclassical regime. Recent work in the physics literature
\cite{Schachenmayer2015, Orioli2017} also demonstrates its success
beyond that. Our study of the discrete Wigner transform is a first
step towards numerical analysis of these phase space numerical
methods, see Section~\ref{sec:dynamics}. A complete numerical analysis
will require mathematical understanding of the quantum entanglement of
the spin dynamics, which will be left for future work.
The rest of the paper is organized as follows. We recall some
preliminaries on spin space, discrete Fourier transforms, and the
Heisenberg-Weyl group in Section~\ref{sec:prelim}. In
Section~\ref{sec:DW_and_HW}, we propose a discrete Wigner transform
motivated from the continuous Wigner transform and illustrate its
connection to the Heisenberg-Weyl group. Wootters' construction of the
discrete Wigner transform will be recalled in
Section~\ref{sec:wootters} and compared with the current construction.
We prove that the proposed discrete Wigner transform
satisfies Wootters' requirements for a Wigner transform in Section~\ref{sec:cond}. In
Section~\ref{sec:dynamics}, we discuss the quantum dynamics in the
phase space representation, and in particular provide a derivation of
a recently proposed numerical method in the physics literature.
\section{Preliminaries}\label{sec:prelim}
\subsection{Properties of the $p^{\text{th}}$ roots of unity}
Throughout this paper, we will always consider a fixed prime $p$ and define $\omega$ be a $p^{\text{th}}$ root of unity (e.g., $\omega = e^{i2\pi / p}$). $\omega$ has two important properties which we will make use of throughout our calculations.
\begin{property} \label{prop1}
If $p$ is a prime, then performing addition and multiplication by integers in the exponent of $\omega$ is equivalent to performing those operations modulo $p$.
\end{property}
\begin{proof}[Proof Sketch]
The key to the property is noticing that $\omega^p = e^{i2\pi} =
1$. Suppose we have an integers $n,m\in \mathbb{Z}$ and can write $n = q_1p
+ r_1$ and $m = q_2 p + r_2$ where $q_1, q_2, r_1, r_2$ are integers. We calculate
\[
\omega^{n} = \omega^{q_1 p + r_1} = \omega^{q_1 p} \omega^{r_1} = \omega^{r_1}
\]
\[
\omega^{nm} = \omega^{(q_1 p + r_1)(q_2 p + r_2)} = \omega^{q_1q_2
p^2} \omega^{(r_2q_1 + r_1q_2)p} \omega^{r_1 r_2} = \omega^{r_1
r_2}
\]
The other properties of modular arithmetic easily follow.
\end{proof}
The qualification that the arithmetic is done with integers is
critical; some later arguments will become more complicated when we
have to consider $\omega$ raised to a non-integer power.
\begin{property} \label{prop2}
If $x \in \mathbb{R}$ then
\[
\sum_{n (p)} \omega^{nx} =
\begin{cases}
p & x \in \mathbb{Z} \text{ and } x \equiv 0 \modks{p}; \\
0 & \text{otherwise},
\end{cases}
\]
where the notation $\sum_{n (p)}$ denotes any sum over a set of
equivalence classes mod $p$.
\end{property}
\begin{proof}
This follows easily from geometric series. If $x \equiv 0 \modks{p}$ then $\omega^x = 1$ so we get $\sum_{n (p)} \omega^{nx} = \sum_{n=0}^{p-1} 1 = p$. Otherwise, $\omega^x \neq 1$ so we use geometric series to conclude
\[
\sum_{n (p)} \omega^{nx} = \sum_{n=0}^{p-1} (\omega^{x})^n = \frac{1 - \omega^{px}}{1 - \omega^x} = \frac{1 - (\omega^{p})^x}{1 - \omega^x} = 0.
\]
\end{proof}
Because we will be considering many expressions modulo $p$, it will be
useful to define the Kronecker $\delta$-function modulo $p$ for
vectors $\alpha, \beta \in \mathbb{Z}^N$, $\alpha = (a_1, \dots, a_N)$, $\beta
= (b_1, \dots, b_N)$ as follows:
\[
\delta^p_{\alpha,\beta} := \begin{cases}
1 & a_i \equiv b_i \modks{p} \text{ for } i \in \{ 1, \dots, N\} \\
0 & \text{otherwise}
\end{cases}
\]
\subsection{A Comment on Arithmetic}
We will make explicit the difference between arithmetic in the set
$[p] := \{ 0, 1, \dots, p-1 \}$ and in the group $\mathbb{Z}_p$. In
particular, unless otherwise specified when we add or multiply two
elements from $[p]$ we use standard arithmetic not modular
arithmetic. Because of this convention, addition and multiplication of
elements from the set $[p]$ is not closed. As we will later see, due
to Property \ref{prop1}, the fact that $[p]$ is not closed under these
operations will not be important. As a more explicit example, when
$p=2$ and $a = b = 1$,
\[
\begin{array}{l}
a, b \in [p] \Longrightarrow a + b = 1 + 1 = 2 \\
\addlinespace[2ex]
a, b \in \mathbb{Z}_p \Longrightarrow a + b = 1 + 1 \equiv 0 \modks{2} \\
\end{array}
\]
\subsection{Spin Space and the Discrete Fourier transform}
Throughout this paper, we will consider the vector space $(\mathbb{C}^p)^{\otimes N}$ where $p$ is a prime number; this space naturally aries in the study of quantum spin systems. Following the bra-ket notation from quantum mechanics, we will denote any vector $v \in (\mathbb{C}^p)^{\otimes N}$ by $\ket{v}$ and use $\bra{v}$ to denote $\ket{v}$'s vector space dual.
When $N=1$, we will denote the standard basis for $\mathbb{C}^p$ as $\ket{0}, \ket{1}, \cdots, \ket{p-1}$ where $\ket{j}$ is a vector with a 1 in the $(j+1)^{\text{th}}$ position. We will adopt the convention that if $a \in \mathbb{Z}$ but $a \not\in [p]$ then $\ket{a} := \ket{a \modks{p}}$ and similarly for $\bra{a}$.
For $N > 1$, we will write the standard basis for $(\mathbb{C}^p)^{\otimes N}$ as a Kronecker product $\ket{a_1 a_2 \cdots a_N} := \ket{a_1} \otimes \ket{a_2} \otimes \cdots \otimes \ket{a_N}$ where $a_i \in [p]$. For example, \\
\[
\begin{array}{lll}
\textbf{Space} & p = 5,\, N = 1 & p = 2,\, N = 2 \\
\addlinespace[2ex]
\textbf{Std. Basis} & \ket{0}, \, \ket{1}, \,\ket{2}, \ket{3}, \ket{4} & \ket{00}, \, \ket{01}, \, \ket{10}, \, \ket{11} \\
\addlinespace[2ex]
\textbf{Example} & \ket{2} := \begin{bmatrix} 0 & 0 & 1 & 0 & 0 \end{bmatrix}^{\top} & \ket{10} := \begin{bmatrix} 0 & 0 & 1 & 0 \end{bmatrix}^{\top} \\
\end{array}
\]
~ \\
For our purposes, it will be useful to recall the inverse Fourier transform and inverse symplectic Fourier transform for vectors over $(\mathbb{C}^p)^{\otimes (2N)}$. Let $\alpha_1, \alpha_2, \beta_1, \beta_2 \in [p]^{N}$ where $\alpha_1 = (a_1, \cdots, a_N)$, $\alpha_2 = (a_{N+1}, \cdots, a_{2N})$ and $\beta_1 = (b_1, \cdots, b_N)$, $\beta_2 = (b_{N+1}, \cdots, b_{2N})$. With this notation, we have the Fourier transform, $\mathcal{F}$, and symplectic Fourier transform, $\mathcal{F}_{symp}$, given by:
\begin{equation} \label{eq:ft}
\begin{split}
\mathcal{F} & = \frac{1}{p^N} \sum_{\beta_1} \sum_{\beta_2} \omega^{\alpha_1 \cdot \beta_1 + \alpha_2 \cdot \beta_2} \ket{\alpha_1\alpha_2} \bra{\beta_1 \beta_2} \\
& := \frac{1}{p^N} \sum_{b_1} \cdots \sum_{b_{2N}} \omega^{a_1 b_1 + \cdots + a_{2N}b_{2N}} \ket{a_1\cdots a_{2N}} \bra{b_1\cdots b_{2N}},
\end{split}
\end{equation}
and
\begin{equation}
\begin{split} \label{eq:symp_ft}
\mathcal{F}_{symp} & = \frac{1}{p^N} \sum_{\beta_1} \sum_{\beta_2} \omega^{\alpha_1 \cdot \beta_2 - \alpha_2 \cdot \beta_1} \ket{\alpha_1\alpha_2} \bra{\beta_1 \beta_2} \\
& := \frac{1}{p^N} \sum_{b_1} \cdots \sum_{b_{2N}} \omega^{(a_1 b_{N+1} + \cdots + a_N b_{2N}) - (a_{N+1} b_{1} + \cdots + a_{2N} b_{N})} \ket{a_1\cdots a_{2N}} \bra{b_1\cdots b_{2N}}.
\end{split}
\end{equation}
Using Property~\ref{prop2}, it is not hard to verify that the following equations give the inverse Fourier transform and inverse symplectic Fourier transform:
\[
\mathcal{F}^{-1} = \frac{1}{p^N} \sum_{\beta_1} \sum_{\beta_2} \omega^{-(\alpha_1 \cdot \beta_1 + \alpha_2 \cdot \beta_2)} \ket{\alpha_1\alpha_2} \bra{\beta_1 \beta_2};
\]
\[
\mathcal{F}^{-1}_{symp} = \frac{1}{p^N} \sum_{\beta_1} \sum_{\beta_2} \omega^{-(\alpha_1 \cdot \beta_2 - \alpha_2 \cdot \beta_1)} \ket{\alpha_1\alpha_2} \bra{\beta_1 \beta_2}.
\]
\subsection{The Discrete Heisenberg-Weyl group}
An important group which acts on spin space is the Heisenberg-Weyl group. For $N=1$, the Heisenberg-Weyl group can be defined through a collection of unitary operators $D(a_1,a_2)$, $a_1, a_2 \in [p]$:
\[
D(a_1, a_2) := \sum_{\ell \in [p]} \omega^{a_2 \ell} \ket{a_1 + \ell} \bra{\ell}
\]
Using Property 1 and recalling our convention for bra-ket notation it is not hard to check that for any $b_1,b_2 \in \mathbb{Z}$, $D(b_1, b_2)$ is still well defined and $D(b_1, b_2) = D(b_1 \modks{p}, b_2 \modks{p})$.
These operators satisfy a number of nice relationships
(see \cite{Howard2006}). In particular, in what follows we will use
the following two facts:
\begin{align*}
& D(a_1, a_2) D(b_1, b_2) = \omega^{a_2 b_1} D(a_1 + b_1, a_2 + b_2)\\
& D(a_1, a_2)^{-1} = D(a_1, a_2)^\dagger = \omega^{a_1 a_2} D(-a_1, -a_2).
\end{align*}
Furthermore, using the definition of the $D(a_1, a_2)$, it is not hard to verify that the collection $\{ D(a_1, a_2) \}$ forms an orthogonal basis of $p \times p$ matrices under the trace inner product (i.e. for all $a_1,a_2,b_1,b_2 \in [p]$, $\tr{(D(a_1, a_2)^\dagger D(b_1, b_2))} = p\,\delta^p_{(a_1,a_2),(b_1,b_2)}$).
\subsection{The Wigner and Fourier-Wigner
Transforms} \label{sec:cont_wigner}
In Section \ref{sec:DW_and_HW} we will look at at the natural discretizations of the Wigner and Fourier-Wigner transforms. Here we recall the following definitions of the continuous Wigner transform and the continuous Fourier-Wigner transform; for more discussions on those, see e.g., \cite{folland}.\\
\underline{The Wigner Transform on $f,g$:}
\[
W[f](\xi,x) := \int e^{-i2\pi \xi \cdot p} f(x+\tfrac{1}{2}p) \overline{f(x - \tfrac{1}{2}p)} \, \text{d} p
\]
\underline{The Fourier-Wigner Transform on $f,g$:}
\[
FW[f](p,q) := \int e^{i2\pi q \cdot y} f(y+\tfrac{1}{2}p) \overline{f(y - \tfrac{1}{2}p)} \, \text{d} y
\]
As one might expect from the name, the Fourier-Wigner transform is indeed the Fourier transform of the Wigner transform \cite{folland}.
\section{The Discrete Wigner transform and the Heisenberg-Weyl
group} \label{sec:DW_and_HW}
\subsection{Deriving the Discrete Wigner and Discrete Fourier Wigner transforms}
Directly replacing the integrals in the definition of Wigner and Fourier-Wigner transforms as in Section \ref{sec:cont_wigner} with sums over $[p]$ and $e^{i2\pi}$ with $\omega$ we can naturally the discretize the Wigner and Fourier-Wigner transforms as linear operators acting on $\mathbb{C}^p$ as follows: \\
\underline{The Discrete Wigner Transform:}
\begin{align*}
W_{dis}(a_1,a_2) & = \sum_{\ell \in [p]} \omega^{-a_1 \ell} \ket{a_2 + \tfrac{1}{2}\ell} \bra{a_2 - \tfrac{1}{2}\ell} \\
& = \sum_{\ell \in [p]} \omega^{-a_1 \ell} \ket{a_2 + 2^{-1}\ell} \bra{a_2 - 2^{-1}\ell}
\end{align*}
\underline{The Discrete Fourier-Wigner Transform:}
\begin{align*}
FW_{dis}(a_1,a_2) & = \sum_{\ell \in [p]} \omega^{a_2 \ell} \ket{\ell + \tfrac{1}{2} a_1} \bra{\ell - \tfrac{1}{2}a_1} \\
& = \sum_{\ell \in [p]} \omega^{a_2 \ell} \ket{\ell + 2^{-1} a_1} \bra{\ell - 2^{-1}a_1}
\end{align*}
Note that for the above equations to make sense, we need $2$ to be invertible modulo $p$ which is not the case when $p=2$. It turns out that the final formulas we derive will still hold for the case $p=2$ in some sense. To stress this point, in the above formulas and henceforth we will write $2^{-1}$ instead of $\tfrac{1}{2}$.
For $p > 2$, we can write both $W_{dis}(a_1, a_2)$ and
$FW_{dis}(a_1, a_2)$ quite compactly in terms of the Heisenberg-Weyl
group. Furthermore, the equations we derive for $p>2$ will allow us to
generalize nicely to the case $p=2$, which will follow afterwards.
\begin{theorem} \label{thm:wigner_hw_connection} For $p > 2$, let $U$
be the ``flip'' operator
$U := \sum_{\ell \in [p]} \ket{\ell} \bra{-\ell}$. We can write
$W_{dis}(a_1,a_2)$ and $FW_{dis}(a_1,a_2)$ in terms of the
Heisenberg-Weyl group as follows:
\[
W_{dis}(a_1,a_2) = \omega^{-2a_1 a_2} D(2a_2, -2a_1) U
\]
\[
FW_{dis}(a_1,a_2) = \omega^{2^{-1} a_1a_2} D(a_1, a_2).
\]
\end{theorem}
\begin{proof}
This proof is a straightforward calculation. Beginning with $W_{dis}(a_1,a_2)$
\begin{align*}
W_{dis}(a_1,a_2) & = \sum_{\ell} \omega^{-a_1 \ell} \ket{a_2 + 2^{-1}\ell} \bra{a_2 - 2^{-1}\ell} \\
& = \sum_{\ell} \omega^{-2a_1 \ell} \ket{a_2 +\ell} \bra{a_2 - \ell} \\
& = \sum_{\ell} \omega^{-2a_1 (\ell + a_2)} \ket{2a_2 +\ell} \bra{- \ell} \\
& = \omega^{-2a_1 a_2} \sum_{\ell} \omega^{-2a_1 \ell} \ket{2a_2 +\ell} \bra{- \ell}.
\end{align*}
Recalling our definition for $U$ above we conclude that
\begin{align*}
W_{dis}(a_1,a_2) & = \omega^{-2a_1 a_2} \left(\sum_{\ell} \omega^{-2a_1 \ell} \ket{2a_2 +\ell} \bra{\ell}\right) U\\
& = \omega^{-2a_1 a_2} D(2a_2, -2a_1) U.
\end{align*}
Similarly, for $FW_{dis}(a_1,a_2)$
\begin{align*}
FW_{dis}(a_1,a_2) & = \sum_{\ell} \omega^{a_2 \ell} \ket{\ell + 2^{-1} a_1} \bra{\ell - 2^{-1} a_1} \\
& = \sum_{\ell} \omega^{a_2 (\ell + 2^{-1} a_1)} \ket{\ell + a_1} \bra{\ell} \\
& = \omega^{2^{-1} a_1a_2} \sum_{\ell} \omega^{a_2 \ell} \ket{\ell + a_1} \bra{\ell} \\
& = \omega^{2^{-1} a_1a_2} D(a_1, a_2).
\end{align*}
\end{proof}
Similar to the continuous case, for $p > 2$, the discrete Wigner transform and the discrete Fourier-Wigner transform are related via the Fourier transform.
\begin{theorem} \label{inverse_ft_wigner}
For $p>2$, $W_{dis}(a_1,a_2)$ is the inverse Fourier transform of $FW_{dis}(a_1,a_2)$.
\end{theorem}
\begin{proof}
The key in proving this theorem is the expansion of $U$ in the
Heisenberg-Weyl basis. Using the properties of the Heisenberg-Weyl
group we have
\begin{align*}
D(a,b)^\dagger U & = \omega^{ab} D(-a,-b) U \\
& = \omega^{ab} \left(\sum_{\ell} \omega^{-bc} \ket{-a + \ell}\bra{\ell} \right) U \\
& = \omega^{ab} \sum_{\ell} \omega^{-bc} \ket{-a + \ell}\bra{-\ell}
\end{align*}
Taking the trace of both sides and noting $-a + \ell = -\ell$
$\Longrightarrow$ $\ell = 2^{-1} a$, we conclude that
$\tr(D(a,b)^\dagger U) = \omega^{2^{-1}ab}$ and so since the collection
$\{ D(b_1,b_2) \}$ forms an orthogonal basis:
\[
U = \frac{1}{p}\sum_{b_1,b_2} \omega^{2^{-1}b_1b_2} D(b_1, b_2).
\]
Now, substituting this expansion into our expression for $W_{dis}(2^{-1}a_1, 2^{-1}a_2)$ we get:
\begin{align*}
W_{dis}(2^{-1}a_1, 2^{-1}a_2) & = \omega^{-2^{-1}a_1 a_2} D(a_2, -a_1) U \\
& = \frac{1}{p}\,\omega^{-2^{-1}a_1 a_2} \sum_{b_1,b_2} \omega^{2^{-1}b_1b_2} D(a_2, -a_1) D(b_1, b_2) \\
& = \frac{1}{p}\,\omega^{-2^{-1}a_1 a_2} \sum_{b_1,b_2} \omega^{2^{-1}b_1b_2} \omega^{-a_1b_1} D(a_2 + b_1, -a_1 + b_2) \\
& = \frac{1}{p}\,\omega^{-2^{-1}a_1 a_2} \sum_{b_1,b_2} \omega^{2^{-1}(b_1-a_2)(b_2+a_1)} \omega^{-a_1(b_1-a_2)} D(b_1, b_2) \\
& = \frac{1}{p}\,\omega^{-2^{-1}a_1 a_2} \sum_{b_1,b_2} \omega^{2^{-1}(b_1-a_2)(b_2-a_1)} D(b_1, b_2) \\
& = \frac{1}{p}\,\omega^{-2^{-1}a_1 a_2} \sum_{b_1,b_2} \omega^{2^{-1} (b_1 b_2 - a_1 b_1 - a_2 b_2 + a_1 a_2)} D(b_1, b_2) \\
& = \frac{1}{p} \sum_{b_1,b_2} \omega^{2^{-1} (-a_1 b_1 - a_2 b_2)} \omega^{2^{-1}b_1 b_2} D(b_1, b_2).
\end{align*}
So we finally get
\begin{align*}
W_{dis}(a_1, a_2) & = \frac{1}{p} \sum_{b_1,b_2} \omega^{-(a_1 b_1 + a_2 b_2)} \omega^{2^{-1}b_1 b_2} D(b_1, b_2) \\
& = \frac{1}{p} \sum_{b_1,b_2} \omega^{-(a_1 b_1 + a_2 b_2)} FW_{dis}(b_1,b_2).
\end{align*}
\end{proof}
Let us stress that the above calculations only make sense when $p>2$. Despite this, we can give meaning to the discrete Fourier-Wigner transform in the case when $p=2$ by interpreting $2^{-1}$ to actually be the real number $\tfrac{1}{2}$. In this case we get the standard Pauli matrices $I,X,Y,Z$:
\begin{equation}
\begin{array}{ccc}
FW_{dis}(0,0) = D(0,0) = I & & FW_{dis}(0,1) = D(0,1) = Z \\
& & \\
FW_{dis}(1,0)= D(1,0) = X & & FW_{dis}(1,1) = i D(1,1) = Y
\end{array}
\end{equation}
Likewise, we can define the discrete Wigner transform for $p=2$ as the inverse Fourier transform of $FW_{dis}$. More explicitly:
\begin{equation} \label{dis_wigner_p2}
\begin{array}{ccc}
W_{dis}(0,0) = \frac{1}{2} \bigl( I + Z + X + Y \bigr) & & W_{dis}(0,1) = \frac{1}{2} \bigl( I - Z + X - Y \bigr) \\
& & \\
W_{dis}(1,0)= \frac{1}{2} \bigl( I + Z - X - Y \bigr) & & W_{dis}(1,1) = \frac{1}{2} \bigl( I - Z - X + Y \bigr)
\end{array}
\end{equation}
It turns out that this choice for $W_{dis}$ and $FW_{dis}$ is in fact the natural choice.
\section{Wootters' Discrete Wigner Transform}\label{sec:wootters}
An alternate approach to defining the Wigner transform is by looking at the key properties of the continuous Wigner transform and taking their discrete analogues. This is the approach taken by Wootters in \cite{wootters}. To define the discrete Wigner transform in \cite{wootters}, Wootters associates a matrix $A_\alpha$ with each point $\alpha = (a_1, a_2) \in \mathbb{Z}_p^2$ and requires the collection $\{ A_\alpha \}_{\alpha \in \mathbb{Z}_p^2}$ to satisfy the following three key properties:
\begin{enumerate}[label=(W\arabic*),leftmargin=50pt]
\item\label{W1} For each $\alpha \in \mathbb{Z}^2_p$, $\tr A_\alpha = 1$;
\item\label{W2} For any $\alpha,\beta \in \mathbb{Z}^2_p$, $\tr A_\alpha^\dagger A_\beta = p \delta^p_{\alpha, \beta}$;
\item\label{W3} The ``line condition'' (see Definition~\ref{defn:line} below).
\end{enumerate}
The first two properties can be thought of as fixing the normalization and requiring the pairwise orthogonality of $\{ A_\alpha \}$ respectively. The third condition (which we have referred to as the line condition) corresponds to the property of the Wigner transform of a quantum state where integrating over infinite strips of the form $S_{a,b,c_1,c_2} := \{ (p,q) \in \mathbb{R}^2 : c_1 \leq ap + bq \leq c_2\}$ gives the probability that certain observables of that state have value between $c_1$ and $c_2$. As one might expect, the corresponding discrete analog of this property involves the definition of parallel lines on a finite lattice.
\subsection{Parallel Lines on $\mathbb{Z}_p^2$ and the Line Condition}
\begin{definition}
For $(n,m) \in \mathbb{Z}_p^2 \setminus \{(0,0)\}$ and $k \in \mathbb{Z}_p$, the \textbf{line} $\ell_{n,m,k}$ on the lattice $\mathbb{Z}_p^2$ is a set of points:
\[
\ell_{n,m,k} := \{ (x,y) : nx + my \equiv k \modks{p} \}.
\]
\end{definition}
\begin{definition}
For $(n,m) \in \mathbb{Z}_p^2 \setminus \{(0,0)\}$ fixed, a \textbf{complete set of parallel lines} $L$ is the collection of $p$ lines:
\[
L = \{ \ell_{n,m,k} : k \in \mathbb{Z}_p\}
\]
\end{definition}
It is not hard to show that $\ell_{n,m,k} \cap \ell_{n,m,k'} = \emptyset$ for $k \neq k'$ so each complete set of parallel lines forms a partition $\mathbb{Z}_p^2$. We are now able to define the ``line condition'' \ref{W3}
\begin{definition}[The Line Condition]\label{defn:line}
For any collection of matrices $\{ A_\alpha \}_{\alpha \in \mathbb{Z}_p^2}$ and any line $\ell$ we can define an associated operator $P_\ell$:
\[
P_\ell := \frac{1}{p} \sum_{\alpha \in \ell} A_\alpha.
\]
We say the collection $\{ A_\alpha \}_{\alpha \in \mathbb{Z}_p^2}$ satisfies
the line condition if for any complete set of parallel lines, $L$,
the collection $\{ P_\ell \}_{\ell \in L}$ is a family of orthogonal
projections which forms a resolution of the identity, i.e.,
\begin{enumerate}
\item For all $\ell_{k_1}, \ell_{k_2} \in L$, $P_{\ell_{k_1}} P_{\ell_{k_2}} = P_{\ell_{k_1}} \delta^p_{k_1,k_2}$
\item $\sum_{\ell \in L} P_\ell = I$.
\end{enumerate}
\end{definition}
\subsection{Wootters' Discrete Wigner Collection, $A_\alpha$}
In the paper \cite{wootters}, Wootters gives the following definition for $\{ A_\alpha \}_{\alpha \in \mathbb{Z}_p^2}$, which he claims satisfies \ref{W1}, \ref{W2}, \ref{W3}:
\[
\begin{array}{cl}
p=2: & A_\alpha = \frac{1}{2} \left( I + (-1)^{a_1} Z + (-1)^{a_2} X + (-1)^{a_1 + a_2} Y \right) \\
&\\
p>2: & (A_{\alpha})_{jk} = \delta^p_{2a_1,j+k} ~\omega^{a_2(j-k)}.
\end{array}
\]
The following theorem shows the
connection between $W_{dis}$, $FW_{dis}$, and $A_\alpha$.
\begin{theorem} \label{thm:wootter_hw_connection}
For all $p$, $A_\alpha$ is the inverse symplectic Fourier transform of $FW_{dis}(a_1,a_2)$. In particular, for all $p$, $A_\alpha = W_{dis}(a_2, -a_1)$.
\end{theorem}
\begin{remark}
This theorem has an exact analog in the continuous
case in the following sense: If $W[f](\xi, x)$ and $FW[f](p,q)$
are the Wigner and Fourier-Wigner transforms of $f$
respectively then the inverse symplectic Fourier transform of
$FW[f](p,q)$ is $W[f](x, -\xi)$.
\end{remark}
\begin{proof}
\underline{Case 1, $p=2$:} Notice that in this case $A_\alpha$ is given by a linear combination of the Pauli matrices. With slight abuse of notation, we can specify this linear combination using a matrix equation as follows:
\[
\begin{bmatrix}
A_{00} \\
A_{01} \\
A_{10} \\
A_{11}
\end{bmatrix}
=
\frac{1}{2}
\begin{bmatrix}
1 & 1 & 1 & 1 \\
1 & 1 & -1 & -1 \\
1 & -1 & 1 & -1 \\
1 & -1 & -1 & 1
\end{bmatrix}
\begin{bmatrix}
I \\
Z \\
X \\
Y
\end{bmatrix}
=
\frac{1}{2}
\begin{bmatrix}
1 & 1 & 1 & 1 \\
1 & 1 & -1 & -1 \\
1 & -1 & 1 & -1 \\
1 & -1 & -1 & 1
\end{bmatrix}
\begin{bmatrix}
FW_{dis}(0,0) \\
FW_{dis}(0,1) \\
FW_{dis}(1,0) \\
FW_{dis}(1,1)
\end{bmatrix}\,.
\]
Recalling the definition of a $p^2 \times p^2$ inverse symplectic Fourier transform over $\mathbb{Z}_p$ (where $\beta := (b_1, b_2)$)
\[
\mathcal{F}^{-1}_{symp} = \frac{1}{p} \sum_{\alpha,\beta \in [p]^2} \omega^{-(a_1 b_2 - a_2 b_1)} \ket{\beta} \bra{\alpha}.
\]
A simple calculation in the case when $p=2$ and $\omega = e^{i2\pi/2} = -1$ verifies the claim. \\
\underline{Case 2, $p>2$:} We will prove this fact by showing that
\[
A_\alpha = \omega^{2a_1a_2} D(2a_1,2a_2)U,
\]
where $U$ is the ``flip'' operator $U := \sum_{\ell} \ket{\ell}\bra{-\ell}$. Since we have already shown that $W_{dis}(a_1,a_2) = \omega^{-2a_1a_2} D(2a_2,-2a_1)U$ (see Theorem \ref{thm:wigner_hw_connection}) the result follows since $W_{dis}$ is the Fourier transform of $FW_{dis}$. Recall the definition of $A_\alpha$ for $p>2$.
\[
(A_{\alpha})_{k\ell} = \delta^p_{2a_1,k+\ell} ~\omega^{a_2(k-\ell)}.
\]
Clearly, an entry of $A_\alpha$ is not zero if and only if $2a_1 \equiv k + \ell \Longrightarrow k \equiv 2a_1 - \ell$. Therefore,
\begin{align*}
A_\alpha & = \sum_{\ell \in [p]} \omega^{a_2(2a_1 - \ell-\ell)} \ket{2a_1 - \ell} \bra{\ell} \\
& = \omega^{2a_1a_2} \sum_{\ell} \omega^{-2a_2\ell} \ket{2a_1 - \ell} \bra{\ell} \\
& = \omega^{2a_1a_2} \sum_{\ell} \omega^{2a_2\ell} \ket{2a_1 + \ell} \bra{-\ell} \\
& = \omega^{2a_1a_2} \left(\sum_{\ell} \omega^{2a_2\ell} \ket{2a_1 + \ell} \bra{\ell}\right)U \\
& = \omega^{2a_1 a_2} D(2a_1, 2a_2) U.
\end{align*}
Therefore, $A_\alpha = W(a_2, -a_1)$ and we can write
\[
A_\alpha = \frac{1}{p} \sum_{b_1,b_2} \omega^{-(a_1 b_2 - a_2 b_1)} FW_{dis}(b_1,b_2).
\]
\end{proof}
\section{$W_{dis}(a_1,a_2)$ satisfies Wootters' conditions}\label{sec:cond}
In this section, we show that the discrete Wigner transform $W_{dis}$
satisfies Wootters' three conditions, thus verifying from another
angle those are natural generalization of the Wigner transform to
the discrete setting.
\subsection{Properties of $W_{dis}$ and $FW_{dis}$} \label{sec:props_dw_dfw}
We begin by deriving some of the important properties of $W_{dis}$ and $FW_{dis}$
\begin{lemma}
For all $p$, the collection $\{ FW_{dis}(a_1,a_2) \}$ is pairwise orthogonal under the trace inner product. That is
\[
\tr\bigl(FW_{dis}(a_1,a_2)^\dagger FW_{dis}(b_1,b_2)\bigr) = p\,\delta^p_{(a_1,a_2),(b_1,b_2)}
\]
\end{lemma}
\begin{proof}
To simplify the proof, we split into two cases, $p=2$ and $p>2$.
\underline{Case 1, $p=2$:} The collection $\{ FW_{dis}(a_1,a_2) \}$ is the Pauli matrices so it can be easily checked that they are pairwise orthogonal.
\underline{Case 2, $p>2$:} In this case we have
\begin{align*}
\tr\bigl(FW_{dis}(a_1,a_2)^\dagger FW_{dis}(b_1,b_2)\bigr) & = \tr\biggl(\omega^{-2^{-1}a_1 a_2} D(a_1, a_2)^\dagger \omega^{2^{-1}b_1 b_2} D(b_1, b_2) \biggr) \\
& = \omega^{2^{-1}(b_1 b_2 - a_1 a_2)}\, \omega^{a_1a_2} \tr\biggl(D(-a_1, -a_2) D(b_1, b_2) \biggr) \\
& = \omega^{2^{-1}(b_1 b_2 - a_1 a_2)} \omega^{a_1a_2-b_1 a_2} \tr\biggl(D(b_1-a_1, b_2-a_2) \biggr)
\end{align*}
But $\tr\bigl(D(b_1-a_1, b_2-a_2) \bigr) \neq 0$ if and only if $a_1 \equiv b_1 \modks{p}$ and $a_2 \equiv b_2 \modks{p}$. Since for $p > 2$, $2^{-1}$ is an integer, we conclude that
\[
\tr\biggl(FW_{dis}(a_1,a_2)^\dagger FW_{dis}(b_1,b_2)\biggr) = p \omega^{2^{-1}(a_1 a_2 - a_1 a_2)} \omega^{a_1a_2-a_1 a_2} = p.
\]
\end{proof}
\begin{lemma}
For all $p$, the collection $\{ W_{dis}(a_1,a_2) \}$ is pairwise orthogonal under the trace inner product. That is
\[
\tr\bigl(W_{dis}(a_1,a_2)^\dagger FW_{dis}(b_1,b_2)\bigr) = p \,\delta^p_{(a_1,a_2),(b_1,b_2)}
\]
\end{lemma}
\begin{proof}
\begin{align*}
\tr \biggl(W_{dis}(\tilde{a}_1, \tilde{a}_2)^\dagger W_{dis}(a_1, a_2)\biggr) & = \frac{1}{p^2} \sum_{b_1, b_2} \sum_{\tilde{b}_1, \tilde{b}_2} \omega^{(\tilde{a}_1 \tilde{b}_1 + \tilde{a}_2 \tilde{b}_2)}\omega^{-(a_1 b_1 + a_2 b_2)} \tr\biggl(FW_{dis}(\tilde{b}_1, \tilde{b}_2)^\dagger FW_{dis}(b_1, b_2)\biggr) \\
& = \frac{1}{p} \sum_{b_1, b_2} \omega^{(\tilde{a}_1 b_1 + \tilde{a}_2 b_2)} \omega^{-(a_1 b_1 + a_2 b_2)} \\
& = \frac{1}{p} \sum_{b_1} \omega^{(a_1 - \tilde{a}_1) b_1} \sum_{b_2} \omega^{(a_2 - \tilde{a}_2) b_2}
\end{align*}
So by Property \ref{prop2}, we conclude the result.
\end{proof}
\begin{lemma}
For all $p$ and $a_1,a_2 \in \mathbb{Z}_p$, $W_{\text{dis}}(a_1,a_2)$ is Hermitian.
\end{lemma}
\begin{proof}
To simplify the proof, we split into two cases, $p=2$ and $p>2$.
\underline{Case 1, $p=2$:} By definition \eqref{dis_wigner_p2}, we can easily see that $W_{dis}(a_1,a_2)$ is Hermitian for $p=2$.
\underline{Case 2, $p>2$:} In this case, we first prove the following useful identity which holds for $p>2$:
\[
D(a_1,a_2) U = U D(-a_1,-a_2), \text{ where } U := \sum_{\ell \in [p]} \ket{-\ell} \bra{\ell}.
\]
We calculate
\begin{align*}
D(a_1,a_2) U & = \sum_{\ell} \omega^{a_2\ell} \ket{a_1 + \ell} \bra{-\ell} \\
& = \sum_{\ell} \omega^{-a_2\ell} \ket{a_1 - \ell} \bra{\ell} \\
& = \sum_{\ell} \omega^{-a_2\ell} \ket{-(-a_1 + \ell)} \bra{\ell} \\
& = U D(-a_1,-a_2).
\end{align*}
Since by Theorem \ref{thm:wigner_hw_connection} we have that $W_{dis}(a_1,a_2) = \omega^{-2a_1a_2} D(2a_2,-2a_1)U$ and $U$ is obviously Hermitian we have that
\begin{align*}
W_{dis}(a_1, a_2)^{\dagger} & = \omega^{2a_1a_2} U^{\dagger} D(2a_2, -2a_1)^{\dagger} \\
& = \omega^{-2a_1a_2} U D(-2a_2, 2a_1) \\
& = \omega^{-2a_1a_2} D(2a_2, -2a_1) U \\
& = W_{dis}(a_1, a_2).
\end{align*}
\end{proof}
\subsection{Verifying \ref{W1}, \ref{W2}, \ref{W3}}
\underline{Verifying \ref{W1}:} We calculate
\begin{align*}
\tr\left( W_{dis}(a_1, a_2)\right) & = \tr \Bigl(\frac{1}{p} \sum_{b_1, b_2}\, \omega^{-(a_1 b_1 + a_2 b_2)} FW_{dis}(b_1,b_2) \Bigr) \\
& = \frac{1}{p} \sum_{b_1, b_2}\, \omega^{-(a_1 b_1 + a_2 b_2)} \tr\left(FW_{dis}(b_1, b_2) \right) \\
& = 1.
\end{align*}
\underline{Verifying \ref{W2}:} This was already verified in Section \ref{sec:props_dw_dfw}. \\
\underline{Verifying \ref{W3}:} For some $(n,m) \in \mathbb{Z}_p^2 \setminus \{ (0, 0) \}$ fixed, recall the definition of a line on the $\mathbb{Z}_p^2$ lattice
\[
\ell_{k} := \ell_{n,m,k} = \{ (x,y) : mx + ny \equiv k \modks{p} \}
\]
and a complete set of parallel lines is the collection $\{ \ell_{k} \}_{k \in \mathbb{Z}_p}$. For this line, we have the associated line operator $P_{\ell_k}$ is the following:
\begin{align*}
P_{\ell_k} & := \frac{1}{p}\sum_{(x,y) \in \ell_k} W_{dis}(x,y) \\
& = \frac{1}{p^2} \sum_{(x,y) \in \ell_k} \sum_{b_1, b_2}\, \omega^{-(x b_1 + y b_2)} \omega^{2^{-1} b_1 b_2} D(b_1, b_2).
\end{align*}
To make the following arguments simpler, we will split into two cases
$p=2$ and $p>2$.
\underline{Case 1, $p=2$:} When $p=2$ there are only three complete
sets of parallel lines each with two lines. Since have already
explicitly written down $W_{dis}$ for $p=2$ (see Equation \ref{dis_wigner_p2}) we can
also write down the line projections as follows:
\[
\renewcommand{\arraystretch}{1.4}
\begin{array}{c|c|c}
(n,m) & P_{\ell_0} & P_{\ell_1} \\
\hline
(0,1) & \tfrac{1}{2}(I + X) & \tfrac{1}{2}(I - X) \\
(1,0) & \tfrac{1}{2}(I + Z) & \tfrac{1}{2}(I - Z) \\
(1,1) & \tfrac{1}{2}(I + Y) & \tfrac{1}{2}(I - Y) \\
\end{array}
\]
It is an easy calculation to show that for each $(n,m)$ the collection
$\{ P_{\ell_0}, P_{\ell_1} \}$ forms a resolution of the identity.
\underline{Case 2, $p>2$:}
In what follows, we will assume without loss of generality that
$n \neq 0$ because of the symmetry between $(x,b_1)$ and $(y, b_2)$. Since
$n \neq 0$, if $(x,y)$ is on the line $\ell_k$ then
$y = n^{-1} k - n^{-1} m x$. For notational ease, we define
$K := n^{-1} k$ and $M := n^{-1} m$ and therefore we can write:
\[
P_{\ell_k} = \frac{1}{p^2} \sum_{x} \sum_{b_1, b_2}\, \omega^{-(x b_1 + (K - M x) b_2)} \omega^{2^{-1} b_1 b_2} D(b_1, b_2).
\]
Collecting all of the terms containing $x$, we get:
\[
P_{\ell_k} = \frac{1}{p^2} \sum_{b_1, b_2}\, \omega^{-K b_2} \omega^{2^{-1} b_1 b_2} D(b_1, b_2) \sum_{x} \omega^{-(b_1 - M b_2) x}
\]
Now the sum over $x$ is over a complete set of equivalence classes
modulo $p$. Therefore, by Property \ref{prop2}, that sum is non-zero
if and only if $b_1 - M b_2 \equiv 0 \modks{p} \Longrightarrow b_1 \equiv M
b_2 \modks
{p}$. Therefore, we can simplify the above as
\[
P_{\ell_k} = \frac{1}{p} \sum_{b_2}\, \omega^{-K b_2} \omega^{2^{-1} M b_2^2} D(M b_2, b_2).
\]
Now let us show that the collection $\{ P_{\ell_k} \}_{k \in \mathbb{Z}_p}$ is a resolution of the identity. \\
\noindent\underline{$P_{\ell_k}$ are orthogonal projectors:}
In what follows we slightly abuse notation and define $\omega(x) := \omega^x$. With this new notation we calculate
\begin{align*}
P_{\ell_k} P_{\ell_{\tilde{k}}} & = \frac{1}{p^2} \sum_{b_2,\tilde{b}_2}\, \omega\bigl(-K b_2 - \tilde{K} \tilde{b}_2\bigr) \omega\bigl(2^{-1} M b_2^2 + 2^{-1} M \tilde{b}_2^2\bigr) D(M b_2, b_2) D(M \tilde{b}_2, \tilde{b}_2) \\
& = \frac{1}{p^2} \sum_{b_2,\tilde{b}_2}\, \omega\bigl(-(K b_2 + \tilde{K} \tilde{b}_2)\bigr) \omega\bigl(2^{-1} M (b_2^2 + \tilde{b}_2^2)\bigr) \omega\bigl( M b_2 \tilde{b}_2 \bigr) D(M (b_2+\tilde{b}_2), b_2+\tilde{b}_2) \\
& = \frac{1}{p^2} \sum_{b_2,\tilde{b}_2}\, \omega\bigl(-(K b_2 + \tilde{K} \tilde{b}_2)\bigr) \omega\bigl(2^{-1} M (b_2^2 + 2 b_2 \tilde{b}_2 + \tilde{b}_2^2)\bigr) D(M (b_2+\tilde{b}_2), b_2+\tilde{b}_2) \\
& = \frac{1}{p^2} \sum_{b_2,\tilde{b}_2}\, \omega\bigl(-(K b_2 + \tilde{K} \tilde{b}_2)\bigr) \omega\bigl(2^{-1} M (b_2 + \tilde{b}_2)^2\bigr) D(M (b_2+\tilde{b}_2), b_2+\tilde{b}_2)
\end{align*}
Now to finish this proof we will replace the sum over $\tilde{b}_2$
with a sum over $B := b_2 + \tilde{b}_2$. This replacement is valid
since due to Property 1 the above sum only depends on
the value of $b_2$ and $\tilde{b}_2$ modulo $p$ and the fact that the
map $(b_2, \tilde{b}_2) \mapsto (b_2, b_2+\tilde{b}_2)$ is
a bijection from $\mathbb{Z}_p$ to $Z_p$.
\begin{align*}
P_{\ell_k} P_{\ell_{\tilde{k}}} & = \frac{1}{p^2} \sum_{b_2,B}\, \omega\bigl(-(K b_2 + \tilde{K} (B - b_2))\bigr) \omega\bigl(2^{-1} M B^2\bigr) D(M B, B) \\
& = \frac{1}{p^2} \sum_{B}\, \omega\bigl(-\tilde{K} B\bigr) \omega\bigl(2^{-1} M B^2\bigr) D(M B, B) \sum_{b_2} \omega\bigl(-(K - \tilde{K}) b_2 \bigr).
\end{align*}
Now by Property \ref{prop2}, the sum over $b_2$ is zero if $K \neq \tilde{K}$. In the case when $K = \tilde{K}$ then we have
\[
P_{\ell_k} P_{\ell_{\tilde{k}}} = \frac{1}{p} \sum_{B}\, \omega^{\tilde{K} B} \omega^{2^{-1} M B^2} D(M B, B) = P_{\ell_k}
\]
which proves that the collection $\{ P_{\ell_k} \}$ is a collection of mutually orthogonal projections. \\
\noindent \underline{$\sum_k P_{\ell_k} = I$:} Recall we have that
\[
P_{\ell_k} = \frac{1}{p} \sum_{b_2}\, \omega^{K b_2 + 2^{-1} M b_2^2} D(M b_2, b_2)
\]
where $K = n^{-1} k$. Replacing $K$ with $k$ and grouping the $k$ terms we get:
\begin{align*}
\sum_k P_{\ell_k} & = \frac{1}{p} \sum_k \sum_{b_2}\, \omega^{-n^{-1} b_2 k} \omega^{2^{-1} M b_2^2} D(M b_2, b_2) \\
& =\frac{1}{p} \sum_{b_2}\, \omega^{2^{-1} M b_2^2} D(M b_2, b_2) \sum_k \omega^{-n^{-1} b_2k}
\end{align*}
Again using Property \ref{prop2}, we conclude the sum over $k$ is non-zero if and only if $b_2 = 0$. Therefore
\[
\sum_k P_{\ell_k} = D(0,0) = I
\]
as we wanted to show.
As a remark, since $A_\alpha = W_{dis}(a_2, -a_1)$ and the mapping $(a_1, a_2) \mapsto (a_2, -a_1)$ maps lines to lines, we have also proved that Wootters' collection also satisfies his three conditions.
\section{Dynamics with Wigner and Fourier-Wigner transforms}
\label{sec:dynamics}
\subsection{The Wigner and Fourier-Wigner transforms for many-body systems}
In previous sections, we have considered the discrete Wigner and Fourier-Wigner transforms on the space $\mathbb{C}^p$. We can generalize this analysis to the space $(\mathbb{C}^p)^{\otimes N}$, the many-body case, by simply taking tensor products. In particular, if $a_1 = (a_1^1, a_1^2, \cdots, a_1^N) \in [p]^N$ and $a_2 = (a_2^1, a_2^2, \cdots, a_2^N) \in [p]^N$ then
\[
D(a_1, a_2) := \bigotimes_{i=1}^N D(a_1^i, a_2^i).
\]
Because of this simple relationship, the many-body Heisenberg-Weyl satisfies analogous identities to the one-body case. That is $\forall a_1, a_2, b_1, b_2 \in [p]^N$:
\[
D(a_1, a_2) D(b_1, b_2) = \omega^{a_2 \cdot b_1} D(a_1 + b_1, a_2 + b_2)
\]
\[
D(a_1, a_2)^{-1} = D(a_1, a_2)^\dagger = \omega^{a_1 \cdot a_2} D(-a_1, -a_2).
\]
Furthermore, we can write down the corresponding many-body transforms as follows:\\
\underline{The Discrete Fourier-Wigner Transform:}
\begin{align*}
FW_{dis}(a_1,a_2) & = \omega^{2^{-1} a_1 \cdot a_2} D(a_1, a_2) \\
& = \bigotimes_{i=1}^N FW_{dis}(a_1^i, a_2^i).
\end{align*}
\underline{The Discrete Wigner Transform:}
\begin{align*}
W_{dis}(a_1,a_2) & = \mathcal{F}^{-1} FW_{dis}(a_1, a_2) \\
& = \sum_{b_1, b_2} \omega^{-(a_1 \cdot b_1 + a_2 \cdot b_2)} FW_{dis}(b_1, b_2) \\
& = \bigotimes_{i=1}^N W_{dis}(a_1^i, a_2^i).
\end{align*}
It is straightforward to show that the collections $\{ W_{dis}(a_1, a_2) \}_{a_1, a_2 \in [p]^N}$ and $\{ FW_{dis}(a_1, a_2) \}_{a_1, a_2 \in [p]^N}$ form an orthogonal basis for $p^N \times p^N$ complex matrices.
\subsection{The dynamics of Wigner and Fourier-Wigner functions}
Let $\rho$ be the discrete density matrix (a positive semidefinite matrix with $\tr{\rho} = 1$). We define Wigner and
Fourier-Wigner coefficients for $\rho$ as follows:
\begin{align*}
\rho_W(a_1, a_2) & := \tr(W_{dis}(a_1,a_2)^\dagger \rho), \\
\rho_{FW}(a_1, a_2) & := \tr(FW_{dis}(a_1,a_2)^\dagger \rho).
\end{align*}
The orthogonality gives
\[
\rho = \frac{1}{p^N} \sum_{a_1,a_2 \in [p]^N} \rho_W(a_1, a_2) W_{dis}(a_1,a_2)
= \frac{1}{p^N} \sum_{a_1,a_2 \in [p]^N} \rho_{FW}(a_1, a_2) FW_{dis}(a_1,a_2).
\]
Suppose $H$ is a Hamiltonian operator, we have the von Neumann dynamics equation:
\[
i \hbar \od{\rho}{t} = [H, \rho].
\]
We expand $H$ and $\pd{\rho}{t}$ into the Wigner and Fourier-Wigner bases in the same way we expanded $\rho$ in these bases by defining
\begin{align*}
H_W(a_1, a_2) & := \tr(W_{dis}(a_1,a_2)^\dagger H), \\
H_{FW}(a_1, a_2) & := \tr(FW_{dis}(a_1,a_2)^\dagger H).
\end{align*}
and similarly for $\pd{\rho}{t}$.
By performing this expansion, we get the dynamics in the Wigner basis is
\[
\frac{i \hbar}{p^N} \sum_{a_1,a_2}\pd{\rho_W(a_1,a_2)}{t} W_{dis}(a_1,a_2) = \frac{1}{(p^N)^2} \sum_{b_1,b_2} \sum_{c_1,c_2} H_W(b_1,b_2) \rho_W(c_1,c_2) [W_{dis}(b_1,b_2), W_{dis}(c_1,c_2)]
\]
Hence
\[
i \hbar \pd{\rho_W(a_1,a_2)}{t} = \frac{1}{(p^N)^2} \sum_{b_1,b_2} \sum_{c_1,c_2} H_W(b_1,b_2) \rho_W(c_1,c_2) \tr\left(W_{dis}(a_1,a_2)^\dagger [W_{dis}(b_1,b_2), W_{dis}(c_1,c_2)] \right)
\]
Using the fact that $W_{dis}(a_1,a_2)$ is Hermitian we can define $\Gamma_{\alpha,\beta,\gamma}^W$ (where $\alpha = (a_1, a_2)$, $\beta = (b_1,b_2)$, $\gamma = (c_1,c_2)$) as follows:
\[
\Gamma_{\alpha,\beta,\gamma}^W := \tr\left(W_{dis}(a_1,a_2) W_{dis}(b_1,b_2) W_{dis}(c_1,c_2) \right)
\]
and write
\[
i \hbar \pd{\rho_W(a_1,a_2)}{t} = \frac{1}{(p^N)^2} \sum_{b_1,b_2} \sum_{c_1,c_2} (\Gamma_{\alpha, \beta, \gamma}^W - \Gamma_{\alpha, \gamma, \beta}^W) H_W(b_1,b_2) \rho_W(c_1,c_2).
\]
Performing analogous calculations for the Fourier-Wigner transform gives
\[
i \hbar \pd{\rho_{FW}(a_1,a_2)}{t} = \frac{1}{(p^N)^2} \sum_{b_1,b_2} \sum_{c_1,c_2} H_{FW}(b_1,b_2) \rho_{FW}(c_1,c_2) \tr\left(FW_{dis}(a_1,a_2)^\dagger [FW_{dis}(b_1,b_2), FW_{dis}(c_1,c_2)] \right).
\]
Noting that $FW_{dis}(a_1,a_2)^\dagger = FW_{dis}(-a_1,-a_2)$ we can define
\[
\Gamma_{\alpha,\beta,\gamma}^{FW} := \tr\left(FW_{dis}(a_1,a_2) FW_{dis}(b_1,b_2) FW_{dis}(c_1,c_2) \right)
\]
and write
\[
i \hbar \pd{\rho_{FW}(a_1,a_2)}{t} = \frac{1}{(p^N)^2} \sum_{b_1,b_2} \sum_{c_1,c_2} (\Gamma_{-\alpha, \beta, \gamma}^{FW} - \Gamma_{-\alpha, \gamma, \beta}^{FW}) H_{FW}(b_1,b_2) \rho_{FW}(c_1,c_2).
\]
Therefore to write down the dynamics equations in the Fourier and Fourier-Wigner bases, we simply need to find an expression for $\Gamma_{\alpha,\beta,\gamma}^{W}$ and $\Gamma_{\alpha,\beta,\gamma}^{FW}$.
\subsubsection{Evaluating $\Gamma_{\alpha,\beta,\gamma}^{FW}$}
By the definition of the discrete Fourier-Wigner transform,
\[
\begin{split}
\Gamma_{\alpha,\beta,\gamma}^{FW} &= \tr \big( FW_{dis}(a_1, a_2) FW_{dis}(b_1,b_2) FW_{dis}(c_1,c_2) \big) \\
&= \omega^{2^{-1} (a_1 \cdot a_2 + b_1 \cdot b_2 + c_1 \cdot c_2)} \tr \big(D(a_1, a_2) D(b_1, b_2) D(c_1, c_2) \big)
\end{split}
\]
Using the properties of the multi-body Heisenberg-Weyl group we can therefore write
\[
\Gamma_{\alpha,\beta,\gamma}^{FW} = p^N \omega^{2^{-1} (a_1 \cdot
a_2 + b_1 \cdot b_2 + c_1 \cdot c_2)} \omega^{a_2 \cdot b_1 + (a_2
+ b_2) \cdot c_1} \, \delta^{p}_{\alpha+\beta+\gamma,0}
\]
If $p > 2$, can simplify the above equation as
\[
\Gamma_{\alpha,\beta,\gamma}^{FW} = p^N \omega^{2^{-1} (b_2 \cdot c_1 - b_1 \cdot c_2)} \, \delta^{p}_{\alpha+\beta+\gamma,0}
\]
When $p = 2$, one can interpret $\omega^{2^{-1}}$ as $i$ and get
\[
\Gamma_{\alpha,\beta,\gamma}^{FW} = 2^N (-1)^{a_2 \cdot b_1} i^{a_1
\cdot a_2} i^{b_1 \cdot b_2} (-i)^{c_1 \cdot c_2} \, \delta^{p}_{\alpha+\beta+\gamma,0}
\]
\subsubsection{Evaluating $\Gamma_{\alpha,\beta,\gamma}^{W}$}
For the coefficients $\Gamma_{\alpha,\beta,\gamma}^W$, we use the relation between the Fourier-Wigner and Wigner transform to get
\[
\Gamma_{\alpha,\beta,\gamma}^W = \frac{1}{(p^N)^3} \sum_{a'_1,a'_2} \sum_{b'_1,b'_2} \sum_{c'_1,c'_2} \omega^{-(a_1 \cdot a_1' + a_2 \cdot a_2' + b_1 \cdot b_1' + b_2 \cdot b_2' + c_1 \cdot c_1' + c_2 \cdot c_2')} \tr \big( FW_{dis}(a_1', a_2') FW_{dis}(b_1',b_2') FW_{dis}(c_1',c_2') \big).
\]
If $p > 2$, we can use the equality
\[
\sum_{a \in [p]^N} \omega^{a \cdot b} = p^N \delta^p_{b,0}
\]
to obtain
\[
\begin{split}
\Gamma_{\alpha,\beta,\gamma}^W &= \frac{1}{(p^N)^2} \sum_{b'_1,b'_2} \sum_{c'_1,c'_2} \omega^{-((b_1 - a_1) \cdot b_1' + (b_2 - a_2) \cdot b_2' + (c_1 - a_1) \cdot c_1' + (c_2 - a_2) \cdot c_2')} \omega^{2^{-1} (b_2' \cdot c_1' - b_1' \cdot c_2')} \\
&= \omega^{2 (b_2 - a_2) \cdot (c_1 - a_1) - 2 (b_1 - a_1) \cdot (c_2 - a_2)}.
\end{split}
\]
Note that when $p=2$ since $2^{-1}$ is not an integer so the above argument does not hold.
\begin{remark}
Notice that unlike $\Gamma_{\alpha,\beta,\gamma}^{FW}$, $\Gamma_{\alpha,\beta,\gamma}^W$ is non-zero for every $\alpha,\beta,\gamma$. Recall our dynamics equation in the Wigner basis:
\[
i \hbar \pd{\rho_W(a_1,a_2)}{t} = \frac{1}{(p^N)^2} \sum_{b_1,b_2} \sum_{c_1,c_2} (\Gamma_{\alpha, \beta, \gamma}^W - \Gamma_{\alpha, \gamma, \beta}^W) H_W(b_1,b_2) \rho_W(c_1,c_2).
\]
Using the above formula for $p>2$, we can calculate that $\Gamma_{\alpha,\beta,\gamma}^W - \Gamma_{\alpha,\gamma,\beta}^W$ is not sparse (see below for $p=3$ case, $\Gamma_{-\alpha,\beta,\gamma}^{FW} - \Gamma_{-\alpha,\gamma,\beta}^{FW}$ is included for comparison):
\[
\renewcommand{\arraystretch}{1.4}
\begin{array}{c|c|c}
p^N & \mathtt{nnz}\bigl(\Gamma_{\alpha,\beta,\gamma}^W - \Gamma_{\alpha,\gamma,\beta}^W\bigr) & \mathtt{nnz}\bigl(\Gamma_{-\alpha,\beta,\gamma}^{FW} - \Gamma_{-\alpha,\gamma,\beta}^{FW}\bigr) \\
\hline
3 & 432 & 48 \\
9 & 349920 & 4320 \\
\end{array}
\]
While there does not seem to be a simple formula for $p=2$, we can verify that this same pattern holds numerically
\[
\renewcommand{\arraystretch}{1.4}
\begin{array}{c|c|c}
p^N & \mathtt{nnz}\bigl(\Gamma_{\alpha,\beta,\gamma}^W - \Gamma_{\alpha,\gamma,\beta}^W\bigr) & \mathtt{nnz}\bigl(\Gamma_{-\alpha,\beta,\gamma}^{FW} - \Gamma_{-\alpha,\gamma,\beta}^{FW}\bigr) \\
\hline
2 & 24 & 6 \\
4 & 2208 & 120 \\
\end{array}
\]
We note that this difference in the sparsity of $\Gamma_{\alpha,\beta,\gamma}^W - \Gamma_{\alpha,\gamma,\beta}^W$ and $\Gamma_{\alpha,\beta,\gamma}^{FW} - \Gamma_{\alpha,\gamma,\beta}^{FW}$ may drastically increase the number of computations needed when performing numerical approximations.
\end{remark}
\subsection{Dynamics of the many-body spin system in phase space representation}
Using the expressions for $\Gamma^{FW}_{\alpha,\beta,\gamma}$ we found
above, we can easily expand the dynamics for any two spin
system. Throughout this section we will assume $p=2$ and write
$e_i \in \mathbb{Z}^N_2$ as the $i^{\text{th}}$ standard basis vector of
$\mathbb{Z}^N_2$. As in \cite{Schachenmayer2015}, we consider the following
Hamiltonian:
\[
H = \frac{1}{2} \sum_{i\neq j} \left[ \frac{J_{ij}^{\perp}}{2} (X_i X_j + Y_i Y_j) + J_{ij}^z Z_i Z_j \right] + \Omega \sum_i X_i.
\]
From our calculations above we get,
\[
H_{FW}(a_1, a_2) = \left\{ \begin{array}{ll}
\frac{p^N}{4} J_{ij}^{\perp}, & \text{if } a_1 = e_i + e_j, a_2 = 0, i \neq j, \\
\frac{p^N}{4} J_{ij}^{\perp}, & \text{if } a_1 = a_2 = e_i + e_j, i \neq j, \\
\frac{p^N}{2} J_{ij}^z, & \text{if } a_1 = 0, a_2 = e_i + e_j, i \neq j, \\
p^N \Omega, & \text{if } a_1 = e_i, a_2 = 0, \\
0, & \text{otherwise}.
\end{array} \right.
\]
Therefore, we arrive at
\begin{equation} \label{eq:spin_dynamics}
\begin{split}
& i\hbar \pd{\rho_{FW}(a_1,a_2)}{t} = i^{a_1 \cdot a_2} \Bigg( \frac{J_{ij}^{\perp}}{4} \sum_{i\neq j} i^{a_2 \cdot (a_1 + e_i + e_j)} [(-1)^{a_1 \cdot a_2} - (-1)^{a_2 \cdot (a_1 + e_i + e_j)}] \rho_{FW}(a_1 + e_i + e_j, a_2) \\
& \quad + \frac{J_{ij}^{\perp}}{4} \sum_{i\neq j} i^{(a_1 + e_i + e_j) \cdot (a_2 + e_i + e_j)} [(-1)^{a_2 \cdot (a_1 + e_i + e_j)} - (-1)^{a_1 \cdot (a_2 + e_i + e_j)}] \rho_{FW}(a_1 + e_i + e_j, a_2 + e_i + e_j) \\
& \quad + \frac{J_{ij}^z}{2} \sum_{i\neq j} i^{a_1 \cdot (a_2 + e_i + e_j)} [(-1)^{a_1 \cdot (a_2 + e_i + e_j)} - (-1)^{a_1 \cdot a_2}] \rho_{FW}(a_1, a_2 + e_i + e_j) \\
& \quad + \Omega \sum_i i^{a_2 \cdot (a_1 + e_i)} [(-1)^{a_1 \cdot a_2} - (-1)^{a_2 \cdot (a_1 + e_i)}] \rho_{FW}(a_1 + e_i, a_2) \Bigg).
\end{split}
\end{equation}
Specializing our equations above for the two-spin system (i.e. $N=2$) the dynamics is
\[
\renewcommand*{\arraystretch}{1.3}
\begin{split}
& \od{}{t} \begin{pmatrix}
\rho_{FW}^{00,00} & \rho_{FW}^{00,01} & \rho_{FW}^{00,10} & \rho_{FW}^{00,11} \\
\rho_{FW}^{01,00} & \rho_{FW}^{01,01} & \rho_{FW}^{01,10} & \rho_{FW}^{01,11} \\
\rho_{FW}^{10,00} & \rho_{FW}^{10,01} & \rho_{FW}^{10,10} & \rho_{FW}^{10,11} \\
\rho_{FW}^{11,00} & \rho_{FW}^{11,01} & \rho_{FW}^{11,10} & \rho_{FW}^{11,11}
\end{pmatrix} = J_{12}^{\perp} \begin{pmatrix}
0 & \rho_{FW}^{11,01} - \rho_{FW}^{11,10} & \rho_{FW}^{11,10} - \rho_{FW}^{11,01} & 0 \\
\rho_{FW}^{10,11} & -\rho_{FW}^{10,01} & \rho_{FW}^{10,10} & -\rho_{FW}^{10,00} \\
\rho_{FW}^{01,11} & \rho_{FW}^{01,01} & -\rho_{FW}^{01,10} & -\rho_{FW}^{01,00} \\
0 & \rho_{FW}^{00,01} - \rho_{FW}^{00,10} & \rho_{FW}^{00,10} - \rho_{FW}^{00,01} & 0
\end{pmatrix} \\
& \quad + 2J_{ij}^z \begin{pmatrix}
0 & 0 & 0 & 0 \\
-\rho_{FW}^{01,11} & \rho_{FW}^{01,10} & -\rho_{FW}^{01,01} & \rho_{FW}^{01,00} \\
-\rho_{FW}^{10,11} & -\rho_{FW}^{10,10} & \rho_{FW}^{10,01} & \rho_{FW}^{10,00} \\
0 & 0 & 0 & 0
\end{pmatrix} + 2\Omega \begin{pmatrix}
0 & \rho_{FW}^{01,01} & \rho_{FW}^{10,10} & \rho_{FW}^{10,11} + \rho_{FW}^{01,11} \\
0 & -\rho_{FW}^{00,01} & \rho_{FW}^{11,10} & \rho_{FW}^{11,11} - \rho_{FW}^{00,11} \\
0 & \rho_{FW}^{11,01} & -\rho_{FW}^{00,10} & \rho_{FW}^{11,11} - \rho_{FW}^{00,11} \\
0 & -\rho_{FW}^{10,01} & -\rho_{FW}^{01,10} & -(\rho_{FW}^{01,11} + \rho_{FW}^{10,11})
\end{pmatrix},
\end{split}
\]
where $\rho_{FW}^{i_1 i_2, j_1 j_2}$ stands for $\rho_{FW}((i_1, i_2), (j_1, j_2))$ and we have chosen units so that $\hbar=1$.
\subsection{Tensor-product ansatz}
In the many-body spin system, when the entanglement between spins is not strong, the state of the system can be approximated by the tensor-product ansatz. The corresponding density matrix of this ansatz is
\[
\rho = \rho^{(1)} \otimes \rho^{(2)} \otimes \cdots \otimes \rho^{(N)},
\]
where each $\rho^{(k)}$ is a one-body density matrix satisfying
$\tr(\rho^{(k)}) \equiv 1$. To evolve this ansatz for the density
matrix, the simplest method is to ignore all the coupling terms in the
Hamiltonian, i.e. $H \approx H_0 = \Omega \sum_i X_i$. Such an idea
was used in the quantum kinetic Monte Carlo method proposed in
\cite{Cai2018} by two of the authors, where the Dyson series expansion
is used in the computation and its leading order term is just the
tensor-product state evolved with the Hamiltonian $H_0$. From the
perspective of the Fourier and Fourier-Wigner transform, the ansatz
turns out to be
\begin{equation} \label{eq:tensor_product}
\begin{split}
\rho_W(a_1, a_2) = \prod_{k=1}^N \rho_W \big(a_1^{(k)} e_k, a_2^{(k)} e_k\big), & \qquad \rho_{FW}(a_1, a_2) = \prod_{k=1}^N \rho_{FW} \big(a_1^{(k)} e_k, a_2^{(k)} e_k\big), \\
& \forall a_1 = (a_1^{(1)}, \cdots, a_1^{(N)}) \in \mathbb{Z}_2^N, \quad a_2 = (a_2^{(1)}, \cdots, a_2^{(N)}) \in \mathbb{Z}_2^N.
\end{split}
\end{equation}
The dynamics of $\rho_{FW}$ with Hamiltonian $H_0$, viewed as a
Galerkin projection to the tensor product ansatz, can be obtained by
removing all the terms with $J_{ij}^{\perp}$ and $J_{ij}^z$ from
\eqref{eq:spin_dynamics}.
Another method to evolve the density matrix $\rho$ is proposed in
\cite{Schachenmayer2015}, where the tensor-product ansatz is linked to
a classical interacting spin system. Using the development in this
paper, we can now explicitly formulate such a connection. According to
the ansatz \eqref{eq:tensor_product}, the dynamics of the
Fourier-Wigner transform is fully determined by the evolution of
$\rho_{FW}(0,e_k)$, $\rho_{FW}(e_k,0)$ and $\rho_{FW}(e_k, e_k)$ for
$k = 1,\cdots,N$. Assuming $\hbar = 1$, we can write down the
equations for these three coefficients using \eqref{eq:spin_dynamics}
as
\begin{align*}
\od{\rho_{FW}(e_k,0)}{t} &= J_{ij}^{\perp} \rho_{FW}(0,e_k) \sum_{i\neq k} \rho(e_i,e_i) - 2 J_{ij}^z \rho_{FW}(e_k,e_k) \sum_{i\neq k} \rho_{FW}(0,e_i), \\
\od{\rho_{FW}(0,e_k)}{t} &= J_{ij}^{\perp} \rho_{FW}(e_k,e_k) \sum_{i\neq k} \rho_{FW}(e_i,0) - J_{ij}^{\perp} \rho_{FW}(e_k,0) \sum_{i\neq k} \rho_{FW}(e_i,e_i) + 2\Omega \rho_{FW}(e_k,e_k), \\
\od{\rho_{FW}(e_k,e_k)}{t} &= -J_{ij}^{\perp} \rho_{FW}(0,e_k) \sum_{i\neq k} \rho_{FW}(e_i,0) + 2 J_{ij}^z \rho_{FW}(e_k,0) \sum_{i\neq k} \rho_{FW}(0,e_i) - 2\Omega \rho_{FW}(0,e_k).
\end{align*}
Here the ansatz \eqref{eq:tensor_product} has been applied to write
all quantities in terms of $\rho_{FW}(0,e_k)$, $\rho_{FW}(e_k,0)$ and
$\rho_{FW}(e_k, e_k)$. By interpreting $\rho_{FW}(e_k,0)$,
$\rho_{FW}(e_k,e_k)$ and $\rho_{FW}(0,e_k)$ as quantities proportional
to the $x$, $y$, and $z$ angular momentum of the $k$-th classical
spin, the above system is exactly the classical spin system introduced
in \cite{Schachenmayer2015, Krech1998}. The numerical analysis of such
tensor product ansatz, which involves understanding of propagation of
quantum entanglement, will be left for future works.
\section*{Acknowledgements}
This work is partially supported by the National Science
Foundation under award RNMS11-07444 (KI-Net). ZC is also supported
by National University of Singapore Startup Fund under Grant
No.~R-146-000-241-133, JL is supported in part by National Science
Foundation award DMS-1454939, and KS is also supported by a National
Science Foundation Graduate Research Fellowship under Grant No.
DGE-1644868. We also thank Robert Calderbank for helpful discussions.
\bibliographystyle{plain}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 47 |
KIRILL TROUSSOV Violin
https://troussov.com
"…one of the most gifted violinists of his generation"
Süddeutsche Zeitung, Harald Eggebrecht
Supported and guided by Sir Yehudi Menuhin at a young age, Kirill Troussov is now widely recognised as one of the leading violinists of his generation. The international press describes his playing : "…impressive elegance, irreproachable technique, an exceptional musical sensitivity and sonorities of immaculate beauty…" Kirill Troussov is a regular guest in prestigious concert halls and renowned international music festivals all over the world.
Collaboration with celebrated conductors such as Sir Neville Marriner, Vladimir Fedoseyev, Daniele Gatti, Lawrence Foster, Jiri Belohlàvek, Michail Jurowski, David Stern, Christoph Poppen, Vladimir Spivakov, Mikko Frank and Louis Langrée as well as with major orchestras such as the Staatskapelle Berlin, Leipzig Gewandhaus, Orchestre de Paris, Orchestre National de Lyon, Orchestre National de France, Radio Orchestra of Hessischer Rundfunk, Radio Symphony Orchestra of Baden-Baden, Munich Philharmonic Orchestra and Bamberg Symphony Orchestra has brought him among others to Tonhalle Zürich, Berliner Philharmonie, Concertgebouw Amsterdam, Palais des Beaux Arts Brüssel, Théâtre du Châtelet, Théâtre des Champs Elysées, De Doelen, NCPA in Beijing, Suntory Hall in Tokyo, Auditorio Nacional de Musica in Madrid, Verbier Festival, Schleswig-Holstein Festival and Menuhin Festival Gstaad.
Passion for chamber music and friendship connects Kirill Troussov with artists like Sol Gabetta, Yuja Wang, Daniel Hope, Joshua Bell, Gautier und Renaud Capucon, Yuri Bashmet, Mischa Maisky, Elisabeth Leonskaja, Julian Rachlin, Christian Zacharias und Natalia Gutman.
Substitution for Gidon Kremer in Paris, Asia Tour with the Verbier Festival Chamber Orchestra, his spectacular performance at the BBC Proms and his activity as jury member at international competitions, a.o. at the "Schoenfeld International String Competition", are just some of numerous highlights of his past concert seasons.
His international awards include the European cultural award "Pro-Europa", the Davidoff-Prize of the Schleswig-Holstein Musik Festival, the international Yamaha Prize and the Reuters-Prize of the Verbier Festival.
Kirill Troussov regularly gives master classes at the Mozarteum Salzburg, European Music Intstitute Vienna, in Cremona, Milan, Madrid, Dublin, Berlin, Munich ,Oslo, Budapest, USA and Hong Kong.
Kirill Troussov's recordings - among others for EMI Classics, Warner Classics, Dabringhaus und Grimm, Naxos and Farao Classics - have been repeatedly awarded by the international press. Furthermore he is to watch with Yuja Wang on her DVD with Kurt Masur. His recent CDs "Emotions" (MDG) and "Memories" (MDG) have been highly praised by the international press and the public - among others by "Süddeutsche Zeitung" and the french music magazine "Diapason".
He completed his studies with Zakhar Bron and Christoph Poppen. His mentors have been Igor Oistrach, Herman Krebbers und Sir Yehudi Menuhin.
Kirill Troussov plays the Antonio Stradivari violin "Brodsky" of 1702, on which violinist Adolph Brodsky performed the world premiere of Tchaikovsky's violin concerto on Dec 4, 1881.
Prof. Kirill Troussov
Individual lessons (open to the public)
An accompanist
Course location:
Villa Rahlstedt
Paalende 24, 22149 Hamburg, Germany | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 9,809 |
Q: crop UIImage to mask I have a UIImage that contains a mask of another UIImage. The only problem is when I tap the area outside the mask, the script still thinks I'm tapping the image.
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tap = UITapGestureRecognizer(target: self, action: #selector(ViewController.handleTap(_:)))
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imageMask.image = ...//Image to mask to
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imageView.addGestureRecognizer(tap)
}
@objc func handleTap(_ sender: UITapGestureRecognizer) {
print("hello")
//prints "hello" when tapped Image outside of mask
}
How to I completely crop the image to the mask, so It won't register taps outside of the mask?
A: Use this instead two imageViews
let overMask = UIView()
overMask.frame = imageView.bounds
overMask.isUserInteractionEnabled = true
imageView = UIImageView(image: ...)
imageView.mask = overMask
imageView.addGestureRecognizer(tap)
overMask.addSubview(imageView)
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 3,122 |
# God's Men
## A Novel
### Pearl S. Buck
###
# CONTENTS
Chapter 1
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12
A Biography of Pearl S. Buck
# 1
A MARCH MORNING IN the year of our Lord, 1950, and the wind so high that on the top floor of a skyscraper in the city of New York William Lane felt a tremor under his feet. He stood by the immense plate-glass window set into the wall behind his desk. The city spread like a carpet before him, and over its horizon he saw the glimmer of hills and sea.
In his fashion he was a man of prayer, and he began his crowded days with these few moments of silence before his window and the world beyond. He had no petition in his heart nor did he ask anything of God. Prayer was an affirmation of himself and what he believed he was, a man of power for good, unmatched at least in his own country. Upon the streets below, so distant that he saw them as gray paths whereon there moved creatures insect-small, were the people whose thoughts he directed, whose minds he enlightened, whose consciences he guided. That they did not know it, that only a few people knew it, increased his power. Long ago he had given up the dream of being a popular leader. He had not the gift of winning popular love. Compelled at last to know that his looks, dark and grave, inspired fear rather than faith, he had immured himself in this great building. From here he had spread over the nation the network of his daily newspapers. For this he bought the services of men and their highest talents. There was no one, he believed, though without cynicism, who could not be bought. Nothing would persuade him, on the other hand, to buy a talent he did not want or which he could not mold to the shape of his own doctrine. The greatest writers found no space in his pages if they did not believe as he did. There were a few, not more than five or six, who were not tempted by fifty thousand dollars. There was only one who had not been tempted by twice that amount. None, he was sure, would refuse as much as he could offer, if he thought it right to offer it. What he bought was not only the fluid flow of men's words. He bought also the quality of their spirits. A man hitherto incorruptible was valuable when he yielded, though only for a while, because he sold also the faith of the people in him.
Upon this March morning, while William thus communed with himself and God, he felt the tremor beneath his feet. He knew that a rigid building, unable to sway slightly before the winds of a storm, might have been overcome. Yielding only a little, the building was safe. Nevertheless, he did not like the tremor. It reminded him of other things that had once made him tremble.
Long ago in China, when he was a boy, he had seen a mob in the streets of Peking, a mob of angry common people who hated him not for what he was, for his white skin and light eyes, but for his kind. His insecurity, the insecurity of his kind on that day, had thrown him into a panic which, though it assailed him no more, he was never able to forget. Any crowd of people, any mass of commonplace faces above dingy clothing, made him remember, although he was no longer afraid, for he had nothing to fear. He was richer than anybody he knew and his friends were some of the richest men in the Western world. Among them he was unassailable, a man of rigid goodness in his personal life. That he had divorced his first wife to marry his second could not be counted a fault, as soon as one saw Emory. She was a creature as delicately pure as a frost flower; her English beauty, her grace combined with her goodness to make her irresistible. Compared to Candace, his first wife, Emory was spirit opposed to earth.
As he thought of his wife the door opened behind him. He did not turn. No one except his secretary dared to enter uncalled, and he waited until her timid voice spoke.
"I'm sorry to disturb you, Mr. Lane."
"Well?" he said in his dry voice.
"I wouldn't have come in except that it's your brother-in-law, Mr. Miller."
"Does he have an appointment?"
"No, he doesn't, Mr. Lane, and I reminded him of that, but he said he guessed you would see him anyway, because he has a big idea."
He would have liked to say quite sharply that he was not interested in any big ideas that Clem Miller might have, but he did not like to give Miss Smith cause for gossip among the lesser staff. They would call him hard, as he knew he was often called, merely because on principle he did not believe in confusing justice with mercy. Nevertheless, it was outrageous for Clem to walk into the offices on a busy morning and expect to be given time for some crank idea. He did not like to remember that Henrietta's husband, too, was a successful man. Clem had grown wealthy by the most absurd methods, so absurd that he believed the fellow, or almost did, when he said that he had never planned to make money. It was hard to believe that Clem did not want to be rich, although the way he and Henrietta lived was strange enough. In spite of wealth, they lived in a frame house on a side street of a town in Ohio. What Clem did with his money no one knew.
"Tell my brother-in-law I can give him exactly fifteen minutes. If he stays longer than that, get him out."
"Yes, Mr. Lane," Miss Smith breathed. Her name was not Smith but William Lane called all his secretaries Smith. They resented it but were paid so well that they did not dare to say so.
When he heard the door shut, William turned away from the window and sat down in the great chair behind the semicircular desk. Against the vast rectangle of light his domed head, his figure, slender but strong, square shouldered and tall, stood forth as though it were chiseled in stone. He sat immobile and waiting, looking at the door.
Thus Clem, coming through that door with his quick and nervous step, faced the mighty man. If he felt the slightest terror before William's eyes, as gray and green as lichen, he did not show it. He was a small thin man, sandy-haired, and his very skin was the color of sand. Into this general insignificance were set his eyes, a quick, kingfisher blue.
"Well, hello, William," Clem said in a high cheerful voice. "Your help out there is certainly for you. I could hardly get in here."
"If I had known you were coming—" William began with dignity.
"I didn't know I was coming myself," Clem said. He sat down, not in the chair across the desk from William and facing him, but in a leather covered chair near the window. "Nice view you have here—I always like to look at it. How's your wife?"
"Emory is quite well," William said.
"Henrietta is well, too," Clem said. "She's gone to see Candace today."
"What are you doing here?" William asked. He was accustomed to this husband of his sister's, who jumped about the earth like a grasshopper. Only the coolness of his voice might have betrayed, and then only to Henrietta herself, his displeasure with his sister's continuing friendship with his former wife.
"I got an idea and ran down to Washington," Clem said. "The Food Minister in New Delhi wrote me there was a lot of hoarded wheat over there. I wasn't sure he knew what he was talking about, sitting in an office in New Delhi. I guess he did, though. There is considerable wheat put away in India, from what I hear. I don't hardly think it's in the hands of dealers. It's bidden by the peasants themselves, the way you or I might tuck away a bank account against a rainy day."
William did not answer. He could not imagine himself tucking away money, nor could he imagine a rainy day. But Clem was incurably common.
Clem scratched his pale chin and went on talking. "If I could persuade these food hoarders of our own in Washington to let up a little and get some wheat over to India, of course it would bring out the wheat over there, and the price would go right down so the people could buy food. I don't know as I can do anything in Washington, though—I don't understand governments, least of all ours."
"Upon that you and I can agree," William said. "I thought that what we had in the White House during the war was bad enough. What we have now is worse."
"Yeah," Clem said, ruminating. "Don't matter to me, though. I'm no politician. I just want to pry some wheat loose."
"What did they say in Washington?" William asked.
"Oh, the usual patter—it would be interfering with internal affairs in India—meaning that if the people get food they might support the present government."
"Don't they like Nehru?" William asked this with some interest. He had not known what to make of that composite man upon his one visit to America.
"Sure they like him as far as he goes," Clem said. "He don't go far enough for some of our Republicans. They want him to swear eternal vengeance on the Russians and eternal loyalty to us. Nehru won't swear; no sensible man would. But that don't interest me, either. What interests me is getting people fed, if for no reason except that starvation is a shame and disgrace to the world and totally unnecessary in modern times. I don't believe in using food, mind you, to manipulate people. Get everybody fed, says I—then you start even. Once all bellies are full, people won't have to vote this way and that so as to get a meal. That's democracy. We ain't practicing it."
Food and democracy were Clem's themes, and long ago William had become bored with his brother-in-law. He saw dreaminess creep into Clem's brilliant blue eyes, a tensity lifted the thin, almost boyish voice, and he recognized both as signs of what he called Clem's fanaticism.
"I do not want to hurry you," he said in his carefully controlled voice, "I do, however, have a business meeting of unusual importance within the next fifteen minutes."
Clem brought back his eyes from the world beyond the window. The dreaminess vanished. He got up and went over to the chair facing William and sat down and leaned his elbows upon the desk. His square face looked suddenly sharp and even acute. "William, I get letters from China."
William was startled. "How do you do that?"
"Somebody I used to know in Peking."
"You'll get yourself into trouble mixing with Communists," William said sternly.
"I guess I won't," Clem said. "The Old Boy knows." The Old Boy, in Clem's language, was always the President of the United States.
"What does he say?" William asked.
"Just told me he didn't approve," Clem gave a sharp cackle.
William did not make a reply, and, as he foresaw, Clem went on without it. "William, there's a mighty famine over yonder in China. You remember? Rivers rising, dikes crumbling away into the water."
"A good thing," William said. "It will teach the Chinese people that Communists cannot save them."
"That ain't enough, though, William," Clem said with insistent earnestness. "That's only the half of it. We got to get the other half across to them. We got to get food over there. What the Reds can't do, we gotta do, or the people will think we can't do it, either, and so what's the use of giving us a try?"
"People ought to be punished for making the wrong choice," William said grimly.
Clem saw the grimness with detached pity. "You oughtn't to take pleasure in punishing people, William. I declare, it's not worthy of such a big man as you are now. It's kind of an Old Testament way of thinking that was done away with when the New Testament came along."
"I will not discuss my religion with you," William said with some violence.
"I don't want to discuss religion, either," Clem said. "I wouldn't hardly know how to say what I believe, and it's your business if you want to be a Catholic, and I told Henrietta so. I don't mind what a man is, if he's a good man—that's what I always say. My father believed in faith, but it certainly didn't save him, and I wouldn't recommend it. I'm not really interested in religion. All I say is if a man don't have a full belly—"
"I know what you say," William said with weariness. "Let's get to the point."
Clem came to the point instantly. "William, I can get the food to send to China, and to India, too. We're so stuffed with so much food over here that my buyers can get it by the hundreds of tons without bothering Washington at all. I can get my hands on ships, too. Even the Old Boy don't have to do anything—just sit there and look the other way. But I need you, William."
"What for?" William asked warily.
The light of gospel came into Clem's blue eyes. He held up his right hand in unconscious gesture.
"William, I want you to get behind the idea with your newspapers, so that I won't be hampered by any senators and the like! Everybody reads your papers, everybody over this broad land. There's millions of people reads your newspapers that don't read anything else. Even senators are still afraid of millions of people. I want you to tell the people that if we get our extra food over there to Asia it's worth any number of bombs, atom bombs—hydrogen bombs, even—"
"Impossible!" William's voice rang hard with anger. "If this is your wonderful idea—"
"My idea is to get food to the starving, William! I don't ask you to do it. I've got my ways of getting into places. I've got my friends. I only ask you to explain to our people."
"Your friends must be Communists!"
"I don't care what they are, any more than I care what you are, just so they get our food to the starving. People will ask, where is the food coming from? America! Don't you see? America don't even ask if people are Communists. Good old America just feeds the starving. It's the greatest advertisement for our democracy—"
"Impossible!" William said bitterly. "Sentimental, absurd! Clem, these people won't ask anything. They'll just eat. Most of them will think that it's the Communists who are giving them food. You are too naïve."
Clem refused to yield. "Even if they do think it's the wrong party, they'll be stronger to see tyranny in the end, won't they? A starving man can't see right or wrong. He just sees food. You've got no judgment when you're hungry. You can't even rebel."
Clem watched William's face for a waiting second. It did not change. "You've never been hungry, have you, William? I have."
William did not need to answer.
Miss Smith opened the door softly. "I'm sorry to interrupt, Mr. Lane, but the gentlemen are waiting in the Board Room."
Clem got up. "You don't need to use fancy methods with me, lady. Just tell me it's time to go. Well, William—"
"I wouldn't think of doing what you suggest," William said. "I don't agree with you in any particular."
Clem stood looking down on him. "Let 'em starve, eh, William?" he said after an infinitesimal pause.
"Let them starve until they confess their folly," William said firmly and got up. "Good-by, Clem. Give my love to Henrietta."
"Good-by," Clem said and turning he left the room.
Neither of them had put out a hand to the other, but William did not notice it. He seldom shook hands with anyone. He disliked the contact, but more than that in recent years there were twinges of neuritis in his hands which made it painful to suffer the vigor of Clem's grasp. He took out his handkerchief and wiped his forehead and poured himself a drink of ice cold water from the silver thermos bottle on his desk. The strangest touch of fate in his strange life was the fact that Clem Miller was his brother-in-law, Clem, whom more than half a century ago he had first seen on a Peking street and never thought to see again—Clem, that pale and hungry boy, the son of the Faith Mission family, living in a cheap alleyway, a hutung in the poorest part of the city, Clem, whom even then he had despised. How had it come about? Half a century ago....
Young William Lane, leaning back in his mother's private riksha, perceived a short quarter of a mile ahead a knot of people. This in a Peking street meant some sort of disturbance. Possibly it meant only amusement. The people of the imperial city, accustomed to pleasure, were never too busy to pause for an hour or two and watch whatever passed, from the entourage of a court lady on her way to the Summer Palace to the tumbles of a trained bear and the antics of a shivering monkey. Since the season was spring it might now be a troupe of street actors, fresh from their winter in the south.
William leaned forward. "Lao Li, what is yonder?" he asked the riksha puller.
His Chinese was pure and somewhat academic, although he was only seventeen. Actually he was not proud of speaking good Chinese. It revealed too clearly that he was the son of a missionary. At the English boarding school in Chefoo where he spent most of the year, the aristocrats among the boys were the sons of diplomats and businessmen and they were careful to show no knowledge of the language of the natives. Among white people in China missionaries were distinctly low class. At school, William spoke pidgin English to the servants and pretended he did not understand them when they replied in Chinese. Now, however, he was at home for the Easter holidays, and since he had been born and had grown up in Peking, no pretense was possible.
"Something strange, Young Master," Lao Li replied. He snatched his cotton jacket from his shoulders as he ran and wiped the sweat from his face. Foreigners were heavy—this young master, for example, though still growing, was already heavier than a man. He could remember when he had pulled him as a child. The years passed. He dared not slacken his pace. A riksha puller must not grow old. A steady job in a white man's family could not be lost, however heavy the children were.
He snatched at a hope for rest. "Shall I not stop so that you can see for yourself?"
William's haughty head was high. "What do I care what street people look at?"
"I only asked," Lao Li muttered.
He tried to quicken his pace as he drew near to the crowd and then William's shout startled him so that he nearly fell between the shafts.
"Stop!"
William, seated high, could look over the heads of the people. In the center of the crowd he saw a horrible sight. A white boy was locked in struggle with a Chinese boy. The onlookers were not laughing. They were intensely quiet.
"Let me down," William said imperiously.
Lao Li lowered the shafts and William stepped over them and strode through the people.
"Let me pass," he said to them in the same haughty voice. The Chinese parted mutely before him until he reached the center. There in silence the two boys were struggling together, the brown face, the white face, equally grim.
"Stop it, you," William said loudly in English.
The white boy turned. "What business is it of yours?" he demanded. He was small and pale, his frame undernourished, and his gray cotton garments, shrunken by many washings, clung to his bones. Nevertheless there was a certain toughness in his square face, and under his sand-colored hair his eyes were a bright blue.
"Of course it is my business," William retorted. He felt his own contrast. His English tweed suit had been made by an excellent Chinese tailor, and his shoes were polished every night by the house coolie—his boots, as he had learned to call them at school. To his horror, he saw that the other boy wore Chinese cloth shoes, ragged at the toes.
"It is degrading for a foreign chap to fight a Chinese," he said severely. "It makes them look down on all of us. You have no right to behave in such a way as to bring discredit on us."
The pale boy blinked rapidly and clenched his fists. "I'll fight anybody I like!" His voice was high and ringing.
"Then I'll have you reported to the Consul," William declared. He allowed his somewhat cold eyes to travel slowly up and down the boy's slight figure. "Who are you, anyway? I've never seen you before."
"I'm Clem Miller."
A faint movement of William's lips was not a smile. "You mean the Faith Mission Miller?"
"Yes." The bright blue eyes dared William's scorn.
"In that case—" William shrugged his handsome shoulders. He turned as though to go and then paused. "Still, as an American, you might think of the honor of your country."
"My father says the world is our country."
To William Lane, the son of an Episcopal missionary, an aristocrat of the church, nothing could have been more sickening than this remark. He wheeled upon the pale boy. "As if it could be! You're American no matter what you do, worse luck for the rest of us! What are you fighting this Chinese boy for?"
"He said my father was a beggar."
"So he is, in a way," William said.
"He is not!" Clem retorted. He clenched his fists again and began to whirl them toward William's face.
William took one step backward. "Don't be a fool, you! You know as well as I do that your father's got no proper mission board behind him, no salary or anything."
"We've got God!" Clem said in a loud clear voice.
William sneered. "You call it God? My mother says it's begging. She says whenever your food's gone your father comes around and tells us so. He tells everybody you have nothing to eat, but the Lord will provide. Actually who does provide? Well, my mother, for instance! We can't see Americans starve. It would make us lose face before the Chinese."
He felt a small strong fist just under his chin, and against all his sense of what was decent for a gentleman, he kicked out with his right foot. His shoe was of excellent leather, sharp at the edge of the sole, and it caught Clem under the knee cap with such pain that he dropped into the thick dust. William did not stop to see what happened next. He turned and strode through the waiting crowd again and took his seat in the riksha.
"Go on," he said to Lao Li.
Behind his back the crowd murmured. Hands were put out to lift up the fallen boy, and the Chinese lad forgot the quarrel.
"That big American boy ought to die," he declared. "You are the same kind of people, both from outside the seas. You should be brothers."
Clem did not reply. After a few seconds of intense pain he limped away.
"Foreigners have bad tempers," the crowd murmured. "They are very fierce. You see how they are, even with each other."
A few turned upon the Chinese boy with advice. "You son of Han, be careful next time. Naturally a human being does not like to hear his father called a beggar, even though he is one."
"We were really talking about the foreign god," the boy explained. "His father asked my father for one of our loaves. He said they had no bread and my father being a baker, he said that the foreign god had told him to come to our house. My father gave him three loaves and the foreigner said his god always provided. But I said, 'How is it he does not provide from among your own people?' This foreign boy was with his father and he heard me say these words, and he told me to follow him, and when we were alone he began to hit me, as you saw."
To this the crowd listened with interest and there was a division of opinion. Some thought the boy had spoken well enough and others said that silence was better than any speech where foreigners were concerned.
"Nevertheless," said one man, who by his long robe was a scholar, "it is strange that the Jesus people are all rich except this one family who live among our poor."
"Who can understand foreigners? There are too many of them here," a butcher said. He carried yards of pig entrails looped over his bare arm, and they had begun to stink faintly in the sun and reminded him that he should be on his way. Slowly the crowd parted, and soon there were only the footsteps in the dust to tell of the scuffle.
William Lane paused at the front door of his home and waited. He had tried the door and found it unlocked, but he would not go in. In spite of his instructions the houseboy was not waiting in the hall to take his hat and topcoat. He wished that he dared to carry his malacca stick here as he did at school, but he did not quite dare. His sister Henrietta, two years younger than he, would laugh at him, and there was nothing he dreaded more than laughter. He pressed the bell and waited again. Almost instantly the door opened and Wang, the houseboy, smiled and gestured to him to enter, at the same time taking his hat. "It is the day your mother, the _t'ai-t'ai,_ sits at home," he said in Chinese. "So many ladies have come that I have been too busy."
William did not answer this. Wang had been with the family for many years, and William took pains now to make him feel that the old days of childish comradeship were over. A young gentleman did not chatter with servants. "Where is my father?" he asked.
"The Teacher has not come home yet from the big church," Wang replied. He smiled affectionately at the tall boy whom he remembered first as a baby, staggering about these very rooms. "Little Lord," the servants had called him. Now he was called Big Little Lord. Sad it was that the family had no more sons, only the two girls.
"Where is my younger sister?" William asked. Of his two sisters he preferred Ruth.
"She is with your mother, and also your older sister," Wang replied. "Forgive me, young sir. You would be surprised at the speed with which the foreign ladies eat and drink."
He hung William's hat upon a large mahogany hat rack, put his coat into the closet under the stairs and hastened smoothly back into the drawing room.
William hesitated. The noise of women's voices, subdued only by the closed door into the wide hall where he stood, both tempted and repelled him. Most of the women were the middle-aged friends of his mother, who had known him from babyhood. Yet there might be a stranger or two. Peking was full of foreigners these days, tourists and visitors, and his father was one of the most liberal among the missionaries. His mother, he knew, often declared that she herself was not a missionary, she was only a missionary's wife, and she would not pretend. Privately she had often complained to her son that it was a tragedy that his father had ever chosen to be a missionary in so repulsive a country as China, so distant from New York, where her home was.
"Your father could have been anything," she told him often. "At Harvard he was brilliant and handsome. Of course everyone thought he would be a lawyer, like his father. Yours is a good family, William, and I do hope that you will remember it. I don't want you to waste yourself."
His mother fed him a good deal of private heresy to which he did not make reply but which he stored in his heart. Certainly he would never be a missionary. The English boys at school had seen to that. A merchant prince, perhaps, or a diplomat, he did not yet know which. Although he dreamed of America, he could not see himself living anywhere except in China. It was comfortable here for a white man. He did not like the stories he heard of missionaries on furlough having to do their own cooking and cleaning. Here he never entered the kitchen or servants' quarters—at least, not now that he was practically grown. When he was small and often lonely and bored, since he was not allowed to play with Chinese children, he had gone sometimes to the servants' quarters for companionship. Wang had been young then and afraid of the cook, and he had welcomed William's friendship. Sometimes Wang had even taken him on the street secretly to see a Punch-and-Judy show or to buy some sweets.
That, of course, was long ago. Remembering the sweets, William decided suddenly to go into the drawing room. The cook made irresistible cakes for his mother's at-homes, two golden ones iced with dark chocolate, two snow-white ones layered with fresh cocoanut. More than mere food tempted him. Since he had come home only a few days ago, many of his mother's friends would not have seen him for several months, and he could exhibit his extraordinary growth. He had added inches to his height even since the long Christmas vacation and was well on his way, he hoped, to six feet, his father's height. There were times when he feared he would not reach it for his hands and feet were small. Just now, however, he was feeling encouraged about himself.
He opened the door and went in, holding his shoulders straight and his head high. Upon his face he put his look of stern young gravity. For a moment he stood with his back to the door, waiting.
His mother glanced at him. "Come in, William," she said in her silvery company voice. "Leave the door open, please; it's a little warm."
Her stone gray eyes, set somewhat near together under somewhat too heavy dark eyebrows, grew proud. She looked around the room where at half a dozen small teapoys the ladies were seated. "William is just home from school," she announced. "Isn't he enormous? It's his last term."
It was a comforting scene to William. The big room was warm and bright. Upon the polished floor lay great Peking rugs woven in blue and gold, and the furniture gleamed a dark mahogany. The pieces were far more valuable than mahogany, however. They were of blackwood, heavy as iron, Chinese antiques stolen from palaces and pawned by hungry eunuchs to dealers. The houses of Americans in Peking were crammed with such tables and screens and couches. Scattered among them were comfortable modern chairs padded with satin-covered cushions. Today sprays of forced peach blossom and two pots of dwarf plum trees provided flowers. Among these pleasant luxuries the ladies sat drinking their tea, and just now turning their faces toward him. Their voices rose to greet him.
"Why, William—how you've grown I Come and shake hands with me, you big boy."
He went forward gracefully and shook hands with each one of ten ladies, ignoring his two sisters. Ruth sat upon a hassock by the grate fire of coals. Henrietta was eating a sandwich on the deep window seat. She did not look at him but Ruth watched him with her pleasant light blue eyes.
"Sit down, William, and have some tea," his mother commanded. She was a tall woman, lean and large boned, and he had his looks from her, although she was almost ugly. What lacked delicacy in a woman made for strength in a man.
Once he had settled on a chair beside her, Wang handed him sandwiches and cake and in silence he proceeded to feed himself heartily. The ladies began to talk again. He perceived at once that they were talking about the Faith Mission family and saying exactly the sort of things with which he could agree. Mrs. Tibbert, a Methodist and therefore not quite the equal of Episcopalians and Presbyterians, although better than a Baptist, was redeemed by being the wife of a bishop. She was a small pallid woman, bravely dressed in a frock copied by a Chinese tailor from a _Delineator_ model, and she had lost a front tooth and had a lisp.
"It's stupid, really, talking about trusting God for everything and then collecting, really, from all of us. We can't let them starve, of course. I wonder if a petition to the Consul—"
"The way they live!" Mrs. Haley exclaimed. She was a Seventh Day Adventist, and even less than a Baptist. It was confusing to the Chinese to be told that Sunday was on Saturday, although immersion, upon which Baptists and Seventh Day Adventists insisted, the Presbyterians and Episcopalians declared was the most confusing of all doctrinal practices. Ignorant Chinese tended to be impressed by much water, and sprinkling seemed stingy, especially in hot weather.
Mrs. Henry Lodge, the wife of the leading Presbyterian minister, was charitable, as she could afford to be, since her house was one of the handsomest in Peking, and her husband the highest paid among the missionaries, besides being related to the Lodges of Boston. "I feel so sorry for the little children," she said gently. White-haired and pretty and gowned in a soft gray Chinese crepe with rose ruching, she made a picture which the other ladies, though Christians, were compelled to envy. William looked at her with appreciation. So a lady ought to look, and to call her attention to himself, he decided to tell the story of his own recent experience.
"Mrs. Lodge, perhaps you'd like to know. As I was coming home today—"
He told the story well and was sensible enough to be modest and merciful toward the ill-dressed boy whom he had publicly reproved. When he had finished he was rewarded.
"I am glad you helped him, William," Mrs. Lodge said.
"That was Christian of you—and brotherly. 'Unto the least of these,' our dear Lord tells us—"
"Thank you, Mrs. Lodge," William said.
Clem Miller had walked away from the crowd as quickly as he could. He would have liked to run, but his clumsy cloth shoes and his sore knee made this impossible. What he remembered about William Lane were his shoes, those strong and well-fitting shapes of brown leather, protecting the tenderness of soles and the ends of toes. A good kick from such a shoe would leave its proper mark.
"Yet it will never be I who have American shoes," he muttered.
His articulate thoughts were always in Chinese, not the fluid tonal Chinese of Peking, but the scavenger Chinese, the guttural coolie vernacular of treaty ports where boat people lived. His first home had been on a boat, for his father, anxious to follow in the exact footsteps of Jesus, had preached from the waters of the dirty Whangpoo in Shanghai to those who gathered upon the shores to listen. There had been more staring than listening, and respectable Christians had come by night to reproach his parents for bringing shame upon them by such beggarly behavior.
They still lived like beggars. Clem, scuffling through the Peking dust, could not deny the accusation which William had made. He had looked more than once through the gate of the compound in which William lived, and by the standards of those who made their homes in big houses of gray brick, roofed with palace tiles of blue and green, the four rooms in a Chinese alley, where he lived with his parents and his sisters, were beggarly. His mother, uncomplaining and of a clinging faith, had nevertheless refused to live on the boat any more after the baby Arthur had fallen overboard into the river and been drowned.
There had been long argument over it between his parents. "Mary, it will look as though you couldn't trust God no more, because of trial," Paul Miller had told his weeping wife.
She had tried to stop her sobs with a bit of ragged handkerchief at her lips. "I do trust. It's only I can't look at the water now."
Arthur's little body had not been returned. They had searched the banks day after day, but the river had clutched the child deep in its tangled currents. So after weeks they had given up this search and had come north to Peking. Paul Miller had taken to God the matter of the dollars necessary for third-class train fare, and then he had gone to the other Shanghai missionaries to bid them farewell, as brothers in Christ. They had responded with sudden generosity by collecting a purse for him, and the missionary women had met together and packed a box of clothing for Mrs. Miller and the children.
"See how the Lord provides when we trust him!" Thus his father had cried out, his mild blue eyes wet with grateful tears.
"Clem, your father is right," his mother said, "we've always been provided for, though sometimes God tests our faith."
Clem had not answered. At this period of his life he was in a profound confusion he dared not face, even alone. The world was divided into the rich who had food and the poor who had not, and though he had been told often of the camel's eye through which the rich would find it hard to enter heaven, yet God seemed indulgent to them and strangely careless of the poor. The poor Chinese, for example, the starving ones, God who saw all things must also see them, but if so He kept silent.
Pondering upon the silence of God, Clem himself grew increasingly silent. There were times when he longed to leave his family and strike out alone across the golden plains, to make for the coast, to find a ship and get a job that would see him across the Pacific to the fabulous land where his parents had been born. Once there he would go straight across on foot to his grandfather's farm in Pennsylvania.
Yet he could not leave his pitiful family, though now past his fifteenth birthday, and he troubled himself much about his future. Such thoughts he kept to himself, knowing that were he to speak of them, his parents, incorrigible in their faith, would only bid him put his trust in God. That was well enough, but who was going to teach him Latin and mathematics and English grammar? He had bought a few old English textbooks in a Chinese secondhand book shop, paying for them by teaching English to the bookseller's ten-year-old son. These books he studied alone, but he felt sorely the need of a teacher. And he could not beg. Though he ate the food his parents somehow got, he could not ask of the prosperous missionaries anything for himself. Today, on the way home from Mr. Fong's bookshop, he had seen his father at the baker's, and then had come the fight, after his father had gone on.
Otherwise the day had been fine, though the evening air was now laced with a cold wind from the northwest. He loved the city at this hour. The people were kind enough to him, even though he had fought that one impudent boy. He was sorry for it now. From how the boy looked at it, he had been right. The Miller family, though they trusted in God, were beggars.
He entered the door of his home with so bitter a look upon his face that his mother, setting the square Chinese table with bowls and chopsticks for supper, stopped to look at him. Pottery bowls and bamboo chopsticks were cheaper than plates and knives and forks.
"What's wrong with you, son?" Her voice was childishly sweet and her face was still round and youthful. Her hair, once of the softest red gold, was now a sandy gray. In spite of his adolescent doubts of her he loved her, so soft was she, so tender to him and to them all.
For the moment, nevertheless, he hardened his heart and blurted his thoughts. "Mama, somehow I'm beginning to see it, we're really beggars."
She leaned on the table upon her outspread hands. "Why, Clem!"
He went on unwillingly, hardening himself still more. "A Chinese boy called us beggars, and I lit into him. Now don't look at me like that, Mama. William Lane came by at that moment, and he—he helped me to stop. But he thought the boy was right."
"I tremble for you, darling. If we lose our faith, we have nothing left."
"I want more faith, Mama." His brain, honest yet agile, was seeking proof at last.
"I don't see how Papa could show more faith, Clem. He never wavered, even when we lost little Artie. He sustained me."
Her voice broke, and her full small mouth quivered. The tears, always waiting like her smile, ran from her golden brown eyes.
"He could have more faith," Clem said.
"But how, dear?"
"If he wouldn't go and tell people when the bread is gone—at least if he wouldn't tell the missionaries."
He lifted his eyes to hers, and to his amazement he saw clear terror. Her round cheeks, always pale, turned greenish. She did not deceive him, and for this his love clung to her always. She held out her hands in a coaxing gesture, and when he did not move, she came to him and knelt beside the bamboo stool upon which he sat, her face level with his.
"Son, dear, what you're saying I've said, too, in my own heart, often."
"Then why don't you tell Papa?" he demanded. He could not understand why it was that though he loved her so much he no longer wished to touch her or be touched by her. He dreaded a caress.
She did not offer it. She rose and clasped her hands and looked down at him.
"For why you can't do it, neither," she said. "It would break his heart to think we had doubted."
"It's not doubt—it's just wanting proof," he insisted.
"But asking God for proof is doubt, my dearie," she said quickly. "Papa has explained that to us, hasn't he? Don't you remember, Clem?"
He did remember. His father, at the long family prayers held morning and evening every day, had taught them in his eager careful way, dwelling upon each detail of God's mercy to them, that to ask God to prove Himself was to court Satan. Doubt was the dust Satan cast to blind the eyes of man.
"And besides," his mother was saying, "I love Papa too much to hurt him, and you must love him, too, Clem. He hasn't anybody in the world but us, and really nobody but you and me, for the children are so little. He has to believe in our faith, to keep him strong. And Papa is so good, Clem. He's the best man I ever saw. He's like Jesus. He never thinks of himself. He thinks of everybody else."
It was true. Though sometimes he hated the unselfishness of his father, though his father's humility made him burn with shame, he knew these were but aspects of a goodness so pure that it could not be defiled. He yielded to its truth and sighed. Then he rose from the stool and looked toward the table.
"Is Papa home?"
"No—not yet. He went down to preach in the market place."
Paul Miller had left the market place where he had gone to preach the saving grace of Jesus, for the people were busy and indifferent. On the way home he met Dr. Lane, returning from his Wednesday afternoon catechism class in the church. Ordinarily the tall handsome missionary, settled comfortably in a riksha, would have passed the short figure plodding through the dust with no more than a friendly, though somewhat embarrassed, nod. Today, however, he stopped the riksha. "Miller, may I have a word with you?"
"Certainly, Brother Lane."
Henry Lane winced at the title. Brother he was, of course, spiritually, to all mankind, for he hoped he was a true Christian. But to hear it shouted thus cheerfully in the streets by a white man who wore patched garments was not pleasant. He did not encourage his wife or his son when they criticized the Faith Mission family. Indeed, he reminded them that Christ could be preached in many ways. Now, however, he had to conceal feelings that he was too honest to deny to himself were much like theirs. It was humiliating to the foreign community of Peking to have the Millers there. It was even worse that they were missionaries of a sort, preaching at least the same Saviour. The Faith Mission family had caused wonder and questions even in his own well-established church.
On the street Chinese began to gather about the two Americans, the immediate crowd that seemed to spring from the very dust. Henry Lane took it for granted that no Chinese spoke English and ignored them.
"Miller, it occurs to me that I ought to warn you that there is very likely to be trouble here against foreigners. I don't like the talk I hear."
He glanced at the crowd. In the pale and golden twilight the faces were bemused with their usual quiet curiosity.
"What have you heard, Brother Lane?" Paul Miller asked. He rested his hands on the fender of the riksha, and admired, as he had before, the delicate spirituality of the elder man's looks. It did not occur to him to envy the good black broadcloth of the missionary's garments or the whiteness of his starched collar and the satin of his cravat. Dr. Lane lowered his voice.
"It is reported to me by one of my vestrymen, whose brother is a minister at the Imperial Court, that the Empress Dowager is inclined to favor the Boxers. She viewed personally today an exhibition of their nonsensical pretensions of inviolability to bullet wounds and bayonet thrusts. That is all she fears—our foreign armies. If she is convinced that these rascals are immune to our weapons she may actually encourage them to drive us all out by force. You must think of your family, Miller."
"What of yours, Brother Lane?"
"I shall send them to Shanghai. Our warships are there," Henry Lane replied.
Paul Miller took his hands from the polished wooden fender.
He looked at the watching Chinese faces, pale in the growing dusk. "I put my faith in God and not in warships," he said simply.
Henry Lane, good Christian though he was, felt his heart sting. "It is my duty to warn you."
"Thank you, Brother."
"Good night," Henry Lane said and motioned to the riksha puller to move on.
Paul Miller stood ankle deep in the spring dust and watched the riksha whirl away. His face was square and thin, and his skin was still pink and white, although it had been twenty years since first he heard the call of God at a camp meeting in Pennsylvania, and leaving his father's farm, to the consternation of that old man, had gone to China, as the only heathen land of which he had heard. Faith had provided the meager means for himself and Mary to cross the continent in a tourist coach, and the Pacific by steerage. Neither had been home since. He did not feel it fair to ask God for furloughs, although other missionaries took them every seven years. He was living by faith.
His mouth trembled and his eyes smarted. Until now he had never faced the possibility of death. They had been hungry often and sometimes sick, and the sorrow over Artie continued in him, though he tried not to think about it. But death at the hands of cruel men, his Mary and his little ones, this he had not dreamed of, even in the nights when Satan tempted him with doubt and with homesickness for the sweet freshness of the farm life he had long ago lived. He was often homesick, but he no longer told Mary. At first they had cried themselves to sleep with homesickness, he a man grown. His mother had written to him now and again until she died, ten years ago, but he had never had a letter from his father. He did not even know if he lived.
There in the darkening Chinese street, amid the dim lights of oil lanterns and candles of cows' fat, listening to the sounds of coming night, mothers calling their children in from the streets, a sick child crying, an angry quarrel somewhere, the slam of wooden doors sliding into place in front of shops, a wailing two-stringed violin, the howl of the rising night wind, he was overcome with terror. He was a stranger and in a strange land. Whither could he and his little family flee? He thought of his wife's tender looks, the gentleness of the two pale little girls, his son's growing manhood. These were all he had, given him by God, and what did they have? He had robbed them of their birthright upon the farm, the safety of their own kind about them, a roof secure above their humble heads. If evil men killed these for whom he was responsible he could believe no more in God. In the darkness he stretched his hands toward heaven. The cold and twinkling stars were above him. There was no moon. None could see him, and he fell upon his knees, even here in the street, and he cried out to God. Then clenching his hands upon his bosom he lifted his face up and shut his eyes against the laughing stars.
"Oh, God," he whispered. "Thou who at this moment maybe art looking down upon my dear old home, which I left, dear God, thinking it was what Thou wanted. Thou canst see into all hearts and knowest whether it is true that evil men are seeking our lives. Humbly I say I have noticed some difference myself in the Chinese in the last months. Our landlord wants us to move without reason. I have kept him paid up, though it has been hard to find the money always on time. But Thou dost provide. Save our lives and keep us safe, I now pray, and especially those dear ones whom Thou has given me, and yet I say Thy will be done, and I will not love them above Thee."
His head sank upon his breast and his chin rested upon his folded hands. He waited for the tide of faith to swell into his heart.
It came at last, warming the blood in his veins, strengthening his heart like wine, convincing him that he was doing what was right. "Fear not, for I am with thee always—" He could hear the words he knew so well.
"Amen, God," he replied with reverence. He rose and plodded along the empty street toward the four small rooms where those whom he loved awaited him. Yes, he struggled constantly not to love them too well. They were not, he told himself, all that he had. For he had the immeasurable love of God.
In less than half an hour he opened the door of his home and saw the sight which always gladdened him. The table was set for the evening meal. Mary sat beside the lighted oil lamp mending some garment, and Clem was studying one of his books. The two little girls were playing with a clay doll which a kindly Chinese woman had given them.
They looked up when he came in, and he heard their greetings. For some foolish reason he could not keep the tears from his eyes. Mary rose and came toward him and he was glad the light was dim. Even so he closed his eyes when he kissed her lest a tear fall upon her face. Then he stooped to the little girls and avoided the eyes of his son.
Only when he had conquered his sudden wish to weep did he speak to Clem. "What's the book, son?"
"A history book, Papa. I got it today at Mr. Fong's shop."
"What history?"
"A history of America."
He scarcely heard Clem's voice. He was savoring his relief, the assurance God was giving him. They were all here, all safe. He would not tell them about the danger. There was no need. It was gone. "I will put my trust in the Lord." With these silent words he bade his heart be still.
The lamps in the mission house were all lit, and Dr. Lane was upstairs dressing for dinner. He did not encourage his wife's ideas to the extent of wearing evening clothes every night as the English did, but he put on a fresh shirt and changed his coat. When he had left college, twenty years ago, he had been what he now called a dreamer. That is, he had believed in asceticism for the man of God. The stringency of war years had shaped him, although in his father's house no one had actually joined the army. But they had sheltered slaves from the South, had spent a good deal of money helping them to settle and find work, and his father had been a leader in the Episcopal church in Cambridge. When he had announced his call to the mission field, however, his father had been plainly angry.
"Of course we must send missionaries to heathen lands," he had declared to the young Henry, "but I don't feel that we must send our best young men. My father didn't want me to go to war, and I didn't go."
"God didn't call you to go to war," Henry had replied.
The struggle with his father, wherein he had not yielded, had helped him when a few months later he fell in love with Helen Vandervent at Old Harbor. She was then the handsomest girl he had ever seen, a creature built on a noble scale even in her youth. He was tall but she was well above his shoulder, and proud and worldly, as he soon knew. He had gone on his knees to God, asking for strength to tame her, not for strength to give her up. Even so she had not yielded to him for nearly two years. She loved him, and she told him that she did, but his belief in her love was chilled by her unwillingness to share the life he felt must be his. This she had denied.
"I don't ask you to give up being a minister," she had said. "Surely there are souls to be saved here at home." Twenty years ago she had said it and he could still remember how she had looked, a tall handsome girl in a bright blue frock and coat. Even her hat was plumed with blue, but a frill of white satin lined the brim. She was queenly in youth, imperious in confidence, and his heart had staggered under the impact of her will.
"Ah, but I must serve God where He bids me go," he had told her, summoning the reserves of his own will.
She had shrugged her shoulders and maintained her love and willfulness for nearly six months more, while by day and by night he prayed God for strength in himself and deepening in her love, that she might be softened. Strength he got, but he saw no softening in her and so he tore himself away from her one dreadful summer's evening by the sea at Old Harbor. He had gone thither for one last trial of her love. It was an evil chance. She was surrounded by other young men, who were not beset by God and therefore were free to please her. He got her away at last and on the edge of the cliff above the beach he faced her.
"Helen, I am going to China—alone if you will not come with me."
He was not sure that she believed it. She had shaken her head willfully and he had left her and come ahead to China not knowing whether she would follow him. Only when she was convinced that in Peking she could live a civilized life had she written at last that she would marry him. He had yielded enough to give her Peking. The first two years he had spent alone in an interior town, where life was primitive. In her heart she had never yielded, that he knew, although she believed that she was a Christian. In her way she was, he also believed. She kept his home comfortably, managed the servants with justice and carried out her ambitions for the children.
He worried secretly about his son. There was something hard and proud in the boy. William laughed too seldom; he fell into a dark fury at any small family joke made at his expense, even in affection.
Sometimes, musing upon this dear only son, he remembered a foolish thing his wife had done. She had taken the boy, when he was only nine years old, to an audience with the Empress Dowager. Once a year the Old Buddha gave a party to the American ladies. Somehow upon that occasion Helen had told the chief lady-in-waiting that she would like to bring her son to pay his respects to the Empress. The lady had laughed, had said something to the Empress, who was in one of her unaccountable moods, alternating between childishness and tyranny. Then the lady had said, "Our ancient Ancestor says she would like to see a foreign little boy. Please bring him on the next feast day, which is the Crack of Spring."
Upon a cold day William had gone with his mother to the Imperial Palace and had waited hours in an icy anteroom. At the hour of noon a tall eunuch had summoned them at last into The Presence. William had walked behind his mother and at the command of the eunuch had bowed very low before the spectacular old woman sitting on a glittering dragon throne. It was understood even then that no Americans were required to prostrate themselves.
The Empress was in a good mood. The brilliant and still wintry sun streamed across the tiled floors and fell upon her gold encrusted robes and upon her long jeweled hands lying over her knees. William saw first the embroidered edge of her yellow satin robe, and then lifting his eyes higher, he saw the fabulous hands and then the ends of her long jade necklace and so his eyes rose at last to the enameled face, to the large shining eyes, to the elaborate jeweled headdress. Eunuchs and ladies, seeing this boldness of this child, waited for the royal fury. It did not fall. In the eyes of the young handsome American boy the Empress saw such worship, such admiring awe, that she laughed. Then everybody laughed except William, who stood gazing at her without response. Suddenly the mood changed. The Empress frowned, waved her encased fingertips, and turned away her head.
The Chief Eunuch stepped forward instantly and hurried them away.
"Why did the Empress get angry with me?" William asked his father when at home he was once again warmed and fed.
"Who can understand the heart of the Empress?" he replied.
Mrs. Lane hastened to speak. "William, we must remember that you are the only American boy who has ever seen the great Empress Dowager of China. That's the important thing, isn't it?"
Dr. Lane had not liked this.
"Helen, in the sight of God, all are alike," he had reminded his wife.
"Of course, I know that," she replied. "But we aren't God, are we? The Empress is still the Empress and there is no use in pretending that William has not had a great honor, for he has. It's a wonderful thing and I must say that if I hadn't had the courage to push forward and ask for it, he would not have had the chance."
Dr. Lane, thinking now of his son, sighed as he so often did, without knowing it. Helen had not changed very much. Sometimes, although she observed quite carefully all the outward forms of religion, he feared that at heart she was nevertheless a worldly woman.
William, who had been named for Helen's father, not his, had grown up clever and proud. Whether the boy's heart had ever been touched he did not know. Perhaps a boy's heart was never touched until the dews of young manhood fell upon it. Dr. Lane remembered even himself as a callous youth until suddenly one day when he was almost twenty he had perceived that life was a gift in his hand, to be used or wasted. God had spoken to him at that moment.
The Chinese dinner gong struck softly, and he turned the oil lamp low. It was a fine bit of furnishing, something Helen had contrived from a Ming jar. She had a taste for luxury. Outside Peking it might not have been fitting to a minister of Christ who secretly believed in poverty, but in Peking the houses of the diplomats were so much richer that this house was not remarkable. The fantastic extravagance of the Imperial Court set the atmosphere of the city. Yet the old Empress was conscience-stricken now. The monies which had been collected from the people for a modern navy she had spent upon a huge marble boat, set in the lake at the Summer Palace. While her ministers prophesied disaster from the West and the young Emperor fomented secret rebellion, she was dickering with that absurd secret society of the Boxers. They, excited by her notice, were boasting like fools that they were invulnerable. Neither swords nor bullets, they declared, could pierce their flesh. They had a magic, they told the superstitious Empress, and she might be desperate enough to believe them.
He went slowly down the carpeted stairs, uneasy in his heart, not knowing what to do. Precautions would be taken, of course, by the American Embassy. Yet should he wait for this? William was ready for college, and Helen longed for a summer at home. Home was always America.
He went into the dining room where his family was waiting for him and took his seat at the head of the oval table. The linen was fine and the Chinese nuns at the Catholic convent had embroidered it with a large heavy monogram. It was the sort of thing, he told himself, which looked expensive but was not. The nuns worked cheaply and he had not the heart to deny Helen beauty of so little cost. After all, she had given up a great deal to become his wife. She missed the New York season every year, music and theater and parties. She had never enjoyed Chinese theater although the finest was here in Peking and this was as well, perhaps, for most of the missionaries were still puritans and he was always uneasily conscious of their criticism, unspoken, of his wife. Most of them came from simpler homes than his in America and this did not make them more merciful. Perhaps had she had time to learn Chinese—yet for that he could scarcely blame her. William had been born a scant year after their marriage and the two girls followed quickly. Since her passionate anger with him that day when she found herself pregnant for the third time, there had been no more children.
He folded his napkin and looked about the table at every face. Ruth was growing very pretty. She looked like his side of the family. William and Henrietta took after their mother, the boy was handsome but Henrietta had missed her mother's distinction. She would have to go in for good works. He was not sure that he wanted any of his children to be missionaries. That was as God willed. He smiled at them.
"How would my family like to go home for this summer?"
Wang, robed in a long white linen gown, was serving the soup. From it rose the smell of chicken delicately flavored with fresh ginger.
"Why, Henry!" his wife exclaimed. "I thought you said we couldn't this year because the house at Peitaiho was costing so much."
Like most of the missionaries they had a summer home at the seashore. A hurricane had torn the roof from the walls during the winter and it had cost some hundreds of Chinese dollars to replace.
"We could rent the house," he replied. "That would pay something toward the tickets. I don't think we can ask the Board for expenses, since my furlough is not due yet."
"I don't want to go," Henrietta announced in a flat voice. She was gulping her soup but Dr. Lane did not correct her. He had a sympathy with Henrietta which he himself could not explain.
"But is William quite ready for Harvard?" Mrs. Lane asked. Her eyes were upon Wang as he served croutons.
"Since he has been taught by English standards, I believe he would have no difficulty," Dr. Lane replied. He disliked soup, and he helped himself well to the crisp croutons.
"I'd like to go," William said. The thought of having no more to face the arrogance of English boys, who still called all Americans rebels and missionaries yellow dogs, cheered him. He began to eat with sudden appetite.
Ruth was silent, her mild blue eyes stealing from face to face.
"I had better tell you the truth," Dr. Lane decided. "I do not at all like the way things look. Something is seething in the countryside. The young Emperor is in difficulties again with the Old Empress and she has locked him up. The gossip is that she is determined to kill his tutors for encouraging his Western ideas. But she will have to do something to satisfy her ministers. They are outraged with the new foreign concessions she has been compelled to give the German government. If she should take it into her ignorant old head to exterminate all foreigners, I don't want my family here."
He tried to speak humorously, but they saw that he was anxious. His quiet rather delicate face, always pale, now looked white above his clipped gray beard and mustache.
"I've always said the Chinese hate us," Mrs. Lane said.
"I don't believe they hate us," he said mildly.
"They've killed those German missionaries," she argued.
He put down his soup spoon. "That was an accident, as I've told you, Helen. The bandits just happened to attack a town where the Germans were."
"Even bandits have no right to kill foreigners," she retorted. No one paid any heed to Wang until she said almost violently, "Wang, take away the soup plates!"
"I don't think Wang hates us, Mother," Ruth said when he had left the room. Her voice, soft and timid, was different from the other voices. Even Dr. Lane, accustomed to many years of preaching, spoke with an articulate clarity which was almost forceful.
"That's because he gets paid," Mrs. Lane replied.
Dr. Lane felt obliged, for the sake of the children, to pursue truth. "If the Chinese feel antiforeign, it is the result of the way Germany has behaved. To seize ports and demand the use of the whole bay, besides all that indemnity, just made an excuse for the murder of the missionaries. Then Russia, then England, then even our own government—all this is at the bottom of these so-called antiforeign outbreaks. Naturally the Chinese don't want to see their country sliced away."
Mrs. Lane interrupted. "Oh, of course, Henry, you always think the Chinese are right!" She went on, repressing his attempted reply. "If there is any danger, I want to go away at once. But I won't go without you. I will not allow you to sacrifice yourself for these people. Your first duty is to the children and to me."
"I don't think I can go," he replied. "I don't think I ought to go. The Chinese Christians will expect me to stay. The Boxers will be against them as well as us, if things break loose. Of course the Legation soldiers will protect us, but I don't want you and the children to face a siege, if it comes to that. But it would not look well for me to run. It would not be possible for my conscience. My duty to God comes first."
The children fell into silence. By the patient firmness with which their father spoke they understood that he was determined to go through an argument with their mother. Usually she won, but when their father brought God into the conversation this early, they guessed the end. Alone he might lose, but under that divine leadership, he would prevail even against her.
Yet only a few days later Mrs. Lane was ready to go and at once. It was Saturday and Dr. Lane was working on his usual Sunday sermon. He had chosen a text strangely inept for the times. "The wicked flee when no man pursueth," and he was weaving his thoughts, divinely directed, about the profound meaning hidden in these words, when he heard Mrs. Lane's voice crying aloud his name. Almost immediately the door of his study opened, and he saw William. The boy's garments were covered with dust, his face was ashen and there was a cut on his forehead. He stood there speechless.
Dr. Lane cried out, rising from his chair. "William! What has happened to you?"
William's lips moved. "The—the people—a mob—"
"What?" Dr. Lane exclaimed. He hurried into the hall and there found his wife sitting upon one of the carved Chinese chairs, looking faint.
"Helen, what—"
"There was a mob!" she cried. "I thought we couldn't get away. If it hadn't been for Lao Li—William and I crowded into the same riksha."
"Where was this?" Dr. Lane broke in.
"At that tailor shop on Hatamen Street, where I always go for William's clothes. He needs a new suit—"
"What did William do?" Dr. Lane demanded. Instinctively he knew that someone had done something. Mobs did not gather without cause.
Mrs. Lane sobbed. "Nothing—I don't know! There was a man sleeping against his riksha when we came out—a beggar. William pushed him with his foot; he didn't kick him. The people sprang at us from every door. Oh, Henry, I want to get right out of here—all of us!"
He soothed her gently, directing Wang meanwhile to make some tea. "Helen, I quite agree that you should go. The people are very touchy. Don't go out again, my dear. There might be a real incident."
"It was an incident!" she insisted. "If you'd seen their frightful faces—where's William? Henry, you must find William! They pushed him down into the dust, and if Lao Li hadn't helped him, they would have trampled him to death."
"Go into the living room and wait for your tea," Dr. Lane said. He was very much disturbed, but it would not do to show it. He had told William, how often, never to touch a Chinese. They considered it an indignity to be touched. Once, he remembered, in a New Year's crowd upon the street when he had taken the children out to see the sights, William in six-year-old impatience had pulled the queue of a tall old gentleman standing in front of him, and the man had turned on them in a fury. Dr. Lane had been compelled to apologize again and again, and only William's youth had saved them from serious trouble.
He searched for William and found him upstairs in his room, changing his clothes. He had put a bit of gauze and some sticking plaster on his forehead.
"Did you disinfect that cut?" Dr. Lane asked.
"Yes, sir, thoroughly," William said.
The boy's face was still white, Dr. Lane noticed. "You had better go downstairs and have some tea with your mother. You look rather shaken."
"I do feel so, a bit."
"Never touch a Chinese. Do you remember?" Dr. Lane said with unusual sternness.
"It was a beggar, leaning against the riksha."
"Never mind who he is or what he is doing. Never touch a Chinese!" Dr. Lane repeated more loudly.
"Yes, sir."
William turned his back on his father and began tying a fresh tie. His hands were trembling and he stood so that his father could not see him. The people had turned on him, ignorant common people who did not know his name! He, American and white, the son of privilege, had been beset by poor and filthy people. He would never feel safe again. He wanted to get away from Peking, from China, from these hordes of people—
"You might have been killed," his father said.
William could not deny it. It was true. He might have been trampled upon by vile bare feet. Lao Li had lifted him up and shielded him until he could get to the riksha where his mother was shrieking. They had clung together in the riksha while Lao Li, bending his head, butted his way through the crowd and William had stared out at the angry people, pressing against the wheels. He would never forget the faces, never as long as he lived.
The next week with his mother and sisters he left Peking.
The northern spring drew on. The duststorms subsided, the willow trees grew green and the peach trees bloomed. The festival of Clear Spring was observed with the usual joy and freedom. People strolled along the streets, the men carrying bird cages and the women their children, and over the doorways of houses were hung the mingled branches of willow green and peach pink. The Imperial Court made great holiday of the feast and the Old Empress ordered special theatricals. Outwardly the city was as calm, as stable, as it had been for hundreds of years, and yet every Chinese past childhood knew that it was not so.
The Empress had expressed her feelings in December, when the two German missionaries had been killed in the province of Shantung. The foreign governments had demanded that the provincial governor, Yu Hsien, be removed. The palace news trickled through the city, through eunuchs and servants. Everybody heard that the Old Buddha, as they called the Empress, had at first refused to withdraw Yu Hsien. Her ministers had surrounded her, telling her the size of the foreign guns and the number of soldiers already in the foreign legations. She would not believe that foreigners could prevail against her, but she had been compelled by her ministers. Yet when she had withdrawn Yu Hsien and had appointed Yuan Shih K'ai in his place, as her ministers had recommended, she had given the huge inner province of Shansi to Yu Hsien. In a rage she had set him higher than before, and the people had laughed in rueful admiration. "Our Old Buddha," they told each other, "our Old Buddha always has her way. She is a woman as well as ruler." They were proud of her, though they hated her.
The spring had never been more beautiful. The Americans in the city were reassured by the warmth of the sun, by the blossoming fruit trees, by the amiability of the crowds upon the streets. The guards sent the year before to strengthen the legations had been withdrawn again, and the murder of the missionaries had been paid for. Shansi was far enough away so that Yu Hsien, though as high a governor as before, seemed banished, and life in the wide streets went on as usual.
Nevertheless the consuls had warned all Westerners to stay off the streets during the festival, lest some brawl arise which might make cause for fresh trouble. But the day passed in peace, and in the afternoon the foreigners came out of their compounds and walked about. In the morning the farmers had brought in fresh young greens from outside the city, turnips and radishes and onions and garlic from their new fields, and the people, surfeited with the bread and sweet potatoes of winter, ate to renew their blood. The hundreds of the poor who could not buy went outside the city gates to dig the sweet clover and shepherd's purse to roll in their sheets of baked bread. Children played in the sunshine beside their mothers, shedding their padded coats and running about barebacked.
Clem Miller, pursuing his daily round, felt no difference upon the streets. Since the day when William Lane had stopped the fight he had spoken to no white person outside his own family. His father, he knew, was disturbed and uneasy, but then he was always anxious lest their food be short and always trying to deny anxiety even to himself, lest perchance God, whom he yearned to believe was tender and careful of His own, be made angry by the unbelief of Paul Miller and so refuse to supply food to those who depended upon him. Clem himself had no direct experience of God. Though he prayed as he had been taught, night and morning and sometimes feverishly in between, on the chance that it might do good when their food was low or when there was no cash to pay the landlord, he was still not sure that God gave such gifts. He wondered if his father, too, was not sure and if uncertainty were the cause of his father's uneasiness. He loved his father and felt something childlike in him and he asked no more for proof of faith, only eating the less at home. It was easier to declare himself not hungry, and he filled himself on the sweetmeats that were always on the table when he went to teach Mr. Fong's eldest son at the bookshop.
For Mr. Fong, observing the American boy's thin body and hollowed cheeks, had taken pity. He said to Mrs. Fong, the mother of his children, "See how the young foreigner eats up the sweets! He does not get enough food. Put some small meat rolls in the dish tomorrow, and boil eggs and peel them and set them on the table."
Mrs. Fong was a Buddhist and ate neither meat nor eggs herself, but she did not believe that foreigners would go to heaven anyhow, and since she would gain merit for her soul by feeding one who could make no return, she obeyed her husband. Each day, therefore, Clem found some sort of hearty food waiting, and his pupil Yusan urged him to eat, having been so bidden by his mother. Clem ate, thinking that perhaps this also was God's provision. Yet it was hard to believe that God used heathen to perform his mercies. In confusion he believed and did not believe, and meanwhile his growing body would have starved without the food.
No one spoke to him of the Empress and her whims or of the demands now of Italy as well as Germany. Italy was a place of which he had never heard except that Christopher Columbus had come from there. No one told him either of the warships steaming into Chinese harbors from Britain, Germany and France. His world was in the dust of Peking, and when he dreamed it was of a farm in a place called Pennsylvania. How big Pennsylvania was he did not know, except that it was more than a city. He had learned when he was quite little not to ask his parents about it because it made them both sad and sometimes his mother wept.
The festival ended. One spring day followed another and May passed into June. People were eating big yellow apricots and one morning Mrs. Fong set a dish of them on the table.
"Eat these, little brother," she bade Clem. "They cleanse the blood."
He ate two and against his sense of decency hid two in his pockets to give his sisters when he went home after the lesson. These he bade them eat in secret, lest their father discover in Mrs. Fong a new source for food and go there to beg in God's name. Ever since he had heard William Lane's voice of scorn Clem could not think of his father asking a Chinese for food. Yet when he saw the eagerness with which his younger sisters seized the fruit he brought home to them, he could not refrain the next day from hiding a few cakes in his pockets and then two of the meat rolls. It was a sort of stealing, his ready conscience told him, and was it better to thieve than beg, and was he not worse than his father? "At least I do not take the food in the name of God," he told himself, and continued to take it.
But guilt made him anxious one morning when Mr. Fong came into the sunlit brick-floored room. Mr. Fong sat down and drew his rusty black silk gown up over his knees. He was a tall man, a native of the city, and his smooth face was egg-shaped. Today, since it was warm, he had taken off his black cap. He had been freshly shaved and his queue was combed and braided with a black silk cord.
"Eh," he began, looking at Clem. "I have something to say to you, Little Brother."
"What is it, Elder Uncle?" Clem asked, and was much afraid.
"While I talk, you eat," Mr. Fong said kindly. He clapped his hands at his eldest son, looking at him with always fond eyes. "Yusan, you go away and play somewhere."
Yusan, pleased to be free, tied his book in a blue cotton square, thrust it in a drawer and left the room.
"Drink some tea," Mr. Fong said to Clem. "What I am about to say does not mean that I am angry."
Clem could neither eat nor drink upon these words. What would he do if kind Mr. Fong wanted him to come no more? There would be an end of books and food.
Mr. Fong got up and shut the door and drew the wooden bar across it. Then he sat himself down so near to Clem that his voice could pass into his ear.
"The Old Empress is about to command that all foreigners leave our city—even our country." These were the horrifying words Clem now heard.
"But why?" he gasped.
"Hush—do you know nothing? Has your father not been told? You must go quickly or—" Mr. Fong drew his hand across his throat.
"What have they done?" Clem demanded. It did not occur to him for the moment that he himself was a foreigner, and the word "they" came to his tongue instead of "we."
That his parents were foreign, he well knew. They were foreign even to him, whose birth and whose memories were only of the Chinese earth. They had no money to go away. But where could they hide? Who would dare to take them in? He could not believe that the proud missionaries would shelter them, nor could he ask Mr. Fong to risk the lives of his own family.
Meanwhile he felt cold and his knees began to tremble.
Mr. Fong cleared his throat, stroked his bare chin and began again his guttural whisper. "The foreign governments, you understand, are cutting up our country like a melon. This piece is for the Ying people, this piece is for the Teh people, this piece is for I-Ta-Lee, this for the wild Ruh people to the north."
"My parents are Americans," Clem urged.
Mr. Fong rolled his head around rapidly on his shoulders. "Your Mei people I know. They do not slice with a knife, but they come after the slices are cut and they say to us, 'Since you have sliced to these other peoples, we too must be given some gift.' True, true, you Mei people are better. You are against slicing, but you also wish gifts."
"I have heard nothing," Clem said doggedly.
"There is no time to tell you everything now," Mr. Fong said. "Listen to this one word, Little Brother. Go home and tell your parents to flee to Shanghai. The times are bad. Do not delay lest the way be closed. I have a relative who works in the palace. I fear what is about to happen."
"My father will not go," Clem said sadly. "He believes in God."
"This is no time to believe in God," Mr. Fong replied in a sensible voice. "Tell him to save his family first."
He rose, and opening the drawer he removed the blue cotton square from his son's book and filled it with cakes and fruit. "Take this with you. Remember I do not hate you. If I dared I would ask your family here. But it would do them no good and my family would only be killed with them. We have been warned. Come no more, Little Brother, alas!"
So saying he thrust Clem out of a small back door. Clem found himself in an alleyway. On the street it seemed impossible to believe that doom hung over the city. It was a morning as mild as summer. The people of the city had risen from their beds, had washed themselves, had eaten, had set their faces to seem the same as on any other day. Clem had as usual left home very early, before the shops had taken down their boards, for Mr. Fong believed that the human brain was most active at sunrise. Often when Clem hurried on his way he met straggling rows of sleepy schoolboys, their books wrapped in blue cotton squares under their arms, already on their way to school. This morning, he remembered now, he had met none, and had wondered that he was so early.
Now hurrying on his way he knew that schools should be open and yet he saw not one schoolboy, and surely the shops must have taken down their boards, and yet they had not, although the sun was high. He made his way through strangely silent streets toward his home. Yet before he could reach it, at some signal he neither saw nor heard, the city began to stir, not to its usual life, but to something new and frightful. Good people stayed inside their gates, but the evil came out. Clem, clinging to walls and hiding in doorways, heard a bestial shouting, a rising roar, near the very quarter where the foreign legations were. There, too, the wealthy missionaries lived, the princes of the church. He hastened on toward his own. Perhaps they might be safe hidden among the houses of the poor. Perhaps God had some purpose, after all, in sheltering those who bore a cross.
At this moment Mr. Fong was looking up and down the street. He too saw that this day was different from any other and he knew why. His cousin had visited him about midnight and had told him what had taken place in the palace. Doubtless half the people in the city now knew. Many families had relatives inside the palace, women servants and court ladies, eunuchs who held offices from cooks to ministers, and these sowed among the people outside the Forbidden City the sayings and doings of those within. There was nothing the people did not know about their rulers.
Mr. Fong, remembering the agitated hours of last midnight, now decided to put up the boards of his shop and cease business for the day. Whatever happened he did not want to seem to know anything about it. He was a brave man but not a foolhardy one. He knew that the Old Woman would certainly lose but that she would be desperate and arrogant before she knew herself lost. Mr. Fong had read too much Western science. He knew that the Boxers could not possibly survive iron bullets. Still, it would take time to prove this. The Old Woman was so stubborn that she would have to see foreign armies marching into the city before she believed it could happen. He sighed in the semidarkness of his shop and was glad that he had had the prudence to buy up two months' supply of millet and wheat. In the back court his wife had eleven hens and he had planted in another corner away from the chicken coop a small patch of cabbage. They would not starve.
He did not, however, feel strong enough to join his family for an hour or so. He wanted to be alone and as his usual pretext he drew out his account books and opened his ink boxes and uncovered his brushes. His wife never disturbed him when he was thinking, as she supposed, about money matters. Actually his mind went over all that his cousin had told the night before.
The city, his cousin had said, was full of Boxers. They were now bold enough to enter at every gate. Indeed they were wholly fearless ever since Prince Tuan had persuaded the Empress to let them come even into her presence and show proof of their magic powers.
"But are they magic?" Mr. Fong had asked his cousin with anxiety. In the midnight silence his reason was not so strong as by day.
"They are flesh and blood," his old cousin had replied scornfully. This cousin was only a scribe in the palace but he was a man of sense and learning.
On the ninth day of the month, the cousin then went on to say, the very day when the Empress had returned to the city from the Summer Palace, some Boxers had gone to the race course three miles west of Peking and had set a fire, and they had thrown a Chinese Christian into the flames to burn to death. Inside the palace the Empress was telling her ministers that she would drive the foreigners from the city.
On the eleventh day, the cousin said, the Chancellor of the Japanese Legation was murdered outside the walls of the city. He had gone to the railway station to discover perhaps when the trains would run again to Peking. No trains were running now.
After telling all this, the cousin had gone away, drenched in gloom.
Mr. Fong sat another hour over his figures and then he closed his books, put them in the drawer and locked it. He went back into the inner courts where his family waited. They were all quiet, even Mrs. Fong. She was getting the noonday meal ready.
"Put more water into the millet from now on," he commanded her. "We will drink soup instead of eating porridge."
"Eh," she sighed. "If we only live—"
He did not answer this. Having nothing else to do, he went to his room and began to read the Book of Changes, in which he often said all was foretold if one had the wit to understand.
After this silent meal, at which he strictly forbade any one of his family to go into the street and commanded the children to play quietly in the innermost court, he went to bed and to sleep for the afternoon. He rose only to eat once more at dusk and then he went back to bed. There was nothing he could do, he told his wife, and he had better save his strength for the days to come.
At midnight he woke abruptly to hear his wife screaming in his ears.
"Fong-ah!" she was calling. "Fong-ah, wake up."
He had buried himself so deep in sleep that it was a minute or two before he could grunt a reply.
"Eh—what—" he muttered.
"The city is on fire!" she screamed.
He woke then and shuffled into his slippers lest a centipede sting him and ran into the court and looked up. The sky was red and the night was as light as day.
The children were awake now, and all were crying with fright and he turned on them fiercely when he came back into the house. "Be silent!" he commanded them. "Do you want the neighbors to think you are weeping for the foreigners?"
They fell silent instantly and he crept to his shop and opened the boards to the central door two inches, enough so that he could peer into the street. Twenty fires lit the sky and he knew what they were. The houses and churches of the Christians were burning. He closed the boards again and went back to his family. They were gathered in a small huddle in the gloom of the main room.
"Go back to bed," he told them. "Fortunately we are not Christians and we will survive."
Clem had waked his father after a moment of not knowing what to do. The fires were not near the hutung where they lived. They were nearly all in the better part of the city, near the Legation Quarters. He had not gone into the street since Mr. Fong had given him the warning. Even his father had gone out only by night—to beg, he supposed, at some missionary door, for he had come back with three loaves of foreign bread and some tinned stuff. One tin held Australian butter. Clem had never tasted butter. That night they had each eaten a slice of bread spread with the yellow butter and he had savored it curiously.
"We made our own butter on the farm," his father said suddenly. Clem had been about to ask how when his mother said in a heartbroken voice, "Paul, don't talk about the farm!"
Clem went to bed as soon as evening prayer was done, and had slept until the light from the red sky had wakened him in his corner of the small center room where his bed stood, a couch by day. He had got up and gone out into the court and then fearfully into the narrow street. There was no one in sight but he hurried through the gate again and barred it. Then because he was afraid and lonely he felt compelled to wake his father.
His father opened his eyes at once, silent and aware, and Clem motioned to him to come into the other room.
"Fires in the city!" he whispered.
His father came barefoot and in his underdrawers and they stared at the sky together.
"Don't wake your mother or the girls," his father whispered. "It's a terrible sight—God's judgment. I must go into the streets, Clem, to see what I can do. People will be suffering. You stay here."
"Oh Papa," Clem whispered, "don't go. How shall I find you if something happens to you?"
"Nothing will happen," his father said. "We will pray together before I go—as soon as I get my clothes on."
Quickly his father was back again, dressed in his ragged cotton suit. "On your knees, dear boy," he said in the same ghostly whisper.
For once Clem knelt willingly. He was helpless. They were all helpless. Now if ever God must save them.
"God who hearest all," his father prayed, "Thou knows what is going on in this city. I feel I ought to be about my business and Thine. Probably there are a good many suffering people out there we ought to be looking after. Fires bring suffering as Thou knows. Protect my dear ones while I am gone and especially give strength to my dear son."
His father paused and then in his usual firm voice he added, "Thy will be done, on earth as in heaven, for Thy Name's sake. Amen!"
They got up and his father shook Clem's hand strongly and was gone.
It was nearly dawn before Clem, sleepless upon the board of his bed, heard his father's footsteps carefully upon the threshold. He sat up in bed and saw his father at the door drenched with sweat and black with smoke.
"I must clean myself before your mother sees me," he said. "Get me some water in the basin—some soap if we have any. I'll wash here in the court. Has your mother waked?"
"No," Clem said and got out of bed. He went to the old well in the little courtyard and let down the wooden bucket. A bit of soap was hidden where he had left it above a beam, his own bit of soap, still left from a yellow bar his mother had managed to give him at Christmas. He stood beside his father while he stripped and began to wash.
"The Boxers are in the city," his father said in a low voice. "The Old Empress has given us up. We are in the hands of God. The persecution of the Christians has begun."
"What about the other foreigners?" Clem asked. For the first time he knew that his place must be among those who had rejected him. William Lane, that proud boy—
"I went to Brother Lane's house," his father was saying. "Of all of them, Brother Lane is the kindest. He gave me the food I have brought back and a little money. A man of tender heart! He is alone in his compound. He has sent his family away to Shanghai. They went before the railroads were broken. He has been sheltering Chinese Christians but now they are leaving him. It is safer for them to be among their own people."
Now Clem was really afraid. If the railroads were broken Peking was cut off.
His father looked at him tenderly. "Are you fearful, Clem? Don't be so, my son. The Lord is the strength of our lives. Of whom shall we be afraid?"
Clem did not answer. They were alone among enemies. He sent his own angry prayer toward the sky, where sunshine and smoke were in combat. "God, if you fail my father, I will never pray again."
Then he turned and went into the house and heard his sisters talking softly over their clay doll while their mother still slept.
Mr. Fong knew upon each day what had happened in the palace. His old cousin stole out by night to report the doings of the Empress whom he now called the Old Demon.
"A mighty struggle is going on," he declared to Mr. Fong in the depths of the night. The two men sat in the shop in darkness. The cousin would not allow a candle to be lit, neither would he allow the presence of Mrs. Fong. His hatred of the Empress had become so violent that he trusted no woman. Yet his family feeling was such that he felt obliged to tell Mr. Fong of all possible dangers in order that the Fong clan might be kept safe.
Mr. Fong dared not tell his cousin of their one real danger, which was Clem. Neighbors had seen the foreign boy coming day after day to the house.
"Proceed," Mr. Fong said to his cousin.
"Prince Ching has been dismissed. He was the only reasonable one. She has appointed that blockhead Prince Tuan and three others who understand nothing. This is to prepare for her open union with the foolish Boxers."
On the sixteenth day of this month the cousin reported that the Empress had called a meeting of her clansmen and then of the Manchus to whom she belonged and the Chinese whom she ruled. To these she spoke long of the evils the foreigners had done. She said the Manchus wanted war.
"Then she was confounded," the cousin whispered, "for even among the Manchus there was Natsung, a man of sense, who told her she could not fight the world. He was upheld by a Chinese, Hsu Ching-cheng. The young Emperor, as her nephew, also begged her not to ruin the country. Upon this the great quarrel burst forth. That fool Prince Tuan spoke for the Boxers, though Prince Su spoke against him, saying that it was madness to believe that these ignorant men could not be shot to strips of flesh."
On the eighteenth day the cousin told Mr. Fong that the Empress had seen the Boxers prove their powers, and she had decided to join with them.
"When the young Emperor heard the Old Demon declare this," the cousin said, "he began to weep aloud and he left the room. It is now too late for us to hope. Prepare yourself, Elder Brother, and prepare our family for what must come, for we are lost. The forts at Tientsin have already fallen to the foreign armies but our people do not know it. Neither do the foreigners here in the city know it, since they have no word from the advancing armies sent to rescue them. And the Old Demon puts her faith in these monsters, the Boxers! Tomorrow, before the foreigners can hear of the loss of the forts or of their own coming rescue, she will demand that they leave the city. But how can they go, hundreds of them with women and little children? They will not go. Then the Boxers will try to kill them all. For this our people will be cruelly punished when the foreign armies reach the city. Prepare—prepare, Elder Brother!"
On the twentieth day of that month Clem was waked by his mother in the early morning. He opened his eyes and saw her finger on her lips. He got up and followed her into the court. There were times when between his parents he felt he had no life of his own. Each made him the keeper of secrets from the other, each strove to bear the burden of danger alone, with only Clem's help.
"Clem dear," his mother said in her pretty coaxing voice. In the dawn she had a pale ghostlike look and he saw what he had seen before but today too clearly, that she was wasting away under this strain of waiting for lonely death.
"Yes, Mama," he said.
"Clem, we haven't anything left to eat. I'm afraid to tell Papa."
"Oh Mama," he cried. "Is all that bread gone?"
"Yes, and all the tins. I have a little flour I can mix with water for this morning. That's all."
He knew what she wanted and dreaded to ask him and he offered himself before she spoke.
"Then I will go into the streets and try to find something, Mama."
"Oh Clem, I'm afraid for you to, but if you don't Papa will, and you can slip through the hutungs better than he can. He'll stop maybe to pray."
"I won't do that," he said grimly.
"Then put on your Chinese clothes."
"I'd better not go until after breakfast, Mama, or Papa will notice."
"Oh yes, that's true. Go after breakfast when he is studying his Bible."
"Yes."
His mother's soft eyes were searching his face with anxious sadness. "Oh Clem, forgive me."
"There isn't anything to forgive, Mama. It's not your fault." He saw the tears well into her eyes and with love and dreadful impatience he stopped them.
"Don't cry, please, Mama. I've got all I can bear." He turned away, guilty for his anger, and yet protecting himself with it.
He was silent during the meager breakfast, silent when his father prayed longer than usual. The food was hot. They were out of fuel but he had torn some laths from a plaster wall. Their landlord did not come near them now. They were only grateful that he did not turn them into the streets.
After breakfast Clem waited for his father to go into the inner room and then he got the ragged blue cotton Chinese garments and put them on where the girls could not see him and know that he was going out. Not bidding even his mother good-by, waiting until she was in the small kitchen, he climbed the wall so that he would not leave the gate open and dropped into the alleyway.
Where in all the vast enemy city should he go for food? He dared not go to Mr. Fong. There was nowhere to go indeed except to Mr. Lane, alone in the compound. He had given them food before and he would give again, and Clem did not mind going now that William was not there. So by alleyways and back streets, all empty, he crept through the city toward the compound. None of the compounds were in the Legation Quarter, but this one was nearer than the others.
The gate was locked when he came and he pounded on it softly with his fists. A small square opened above him and the gateman's face looked out. When he saw the foreign boy, he drew back the bar and let him in.
"Is the Teacher at home?" Clem asked safely inside.
"He is always at home now," the gateman replied. "What is your business?"
"I have something to ask," Clem said.
In usual times the gateman would have refused him, as Clem well knew, but now he refused no white face. These foreigners were all in piteous danger and he was a fool to stay by his own white master, but still he did. He had no wife or child and there was only his own life, which was worth little. Thus he plodded ahead of Clem to the big square house and knocked at the front door. It was opened by Dr. Lane himself, who was surprised to see a foreign boy.
"Do I know you?" he asked.
"I don't think so," Clem replied. "But I know you, sir. I am Clem Miller."
"Oh yes," Dr. Lane said vaguely. "The Millers—I know your father. Come in. You shouldn't be out on the streets."
"My father doesn't know that I am," Clem replied. He stepped into the house. It looked bare and cool.
"My family is in Shanghai," Dr. Lane said. "I'm camping out. Did you know my son William? Sit down."
"I've seen him," Clem said with caution. He sat down on the edge of a carved chair.
Dr. Lane continued to look at him with sad dark eyes. He had a kind face except that it looked as though he were not listening.
"What did you come for?" he asked in a gentle voice.
"We have no food," Clem said simply. The blood rushed into his pale face. "I know you have helped us before, Dr. Lane. I wouldn't have come if I had known where else to go."
"That is quite all right," Dr. Lane said. "I'll be glad—"
Clem interrupted him. "One more thing, Dr. Lane. I don't consider that when I ask you for food it's God's providing. I know it isn't. I don't think like my father on that. I wouldn't come just for myself, either. But there's my mother and my two sisters."
"That's all right," Dr. Lane said. "I have more food than I need. A good many tins of stuff—we had just got up an order from Tientsin before the railroad was cut."
The house was dusty, Clem saw, and the kitchen was empty. Dr. Lane seemed helpless. "I don't know just where things are. The cook left yesterday. He was the last one. I can't blame them. It's very dangerous to stay."
"Why didn't you go with William?" Clem asked.
Dr. Lane was still searching. "Here's a basket. I didn't go because of my parish. The Chinese Christians are having a time of sore trial. I can't do much for them except just stay. Here are some tins of milk and some meat—potted ham, I believe."
He filled the basket and put a kitchen towel over it. "Better not carry the tins in the open. They might tempt someone. I wish I could send you home in the riksha but of course the puller has gone—a faithful fellow, too. Lao Li was his name. There's only the gate keeper."
He was leading the way to the door. "You'd better get home as fast as you can. Tell your father that he must get your family into the Legation Quarter if any trouble comes. We'll have to stick together. I suppose our governments will send soldiers to rescue us. They may be on the way."
"I'm afraid my father won't go into the Legation," Clem said. To explain that his father would consider such retreat a total loss of faith, might hurt Dr. Lane's feelings.
But Dr. Lane knew. "Ah," he said, "it takes more courage than I have for such faith. For myself, I can—but not for my son."
They were at the door now and the old gateman was waiting.
"Good-by, Clem," Dr. Lane said.
"Good-by, sir."
The gateman stared at the basket, and he went into his little room and brought out some old shoes and put them on top of the towel. "Let it seem rubbish," he said, "otherwise you will be robbed."
The gate shut behind Clem and he was alone in the street, the basket heavy upon his arm. It was midmorning, and the sun was beginning to be hot. There were a few people about now, all men, and he saw they were soldiers, wearing the baggy brightly colored uniform of the Imperial Palace. He tried to escape their notice, and had succeeded, he thought, for their officer was laughing and joking and did not notice him. They were looking at a foreign gun the officer held. Then they did see him and they started after him. He began to run. On another day, at another hour, he might have shown better sense by stopping to talk with them in their own tongue. Now he wanted only to keep his face hidden from them, his face and his pale foreign eyes. He ran out of the alleyways into Hatamen Street, the eastern boundary of the Legation. Perhaps he could get into the Legation gate. He turned and was stopped by a small procession of two sedan chairs and their outriders. In the sedans he looked into two foreign faces, arrogant, severe, bearded faces he had never seen before. Before he could slip away into an alley again, he was caught between the Chinese soldiers and the foreigners in their sedans. The soldiers blocked the street so that the bearers were forced to set the sedans down.
Now the curtain of the first sedan lifted and the foreigner put out his head and shouted fiercely to the soldiers, "Out of the way! I am Von Ketteler, the German Ambassador, and I go for audience with the Empress!"
The second sedan opened and he heard a guttural warning. It came too late. The Chinese officer raised his foreign gun and leveled it at the German. Clem saw a spit of fire and the Ambassador crumpled, dead. Clem crawled behind the sedan, and clutching his basket, he hurried as fast as he could from the dreadful spot.
Homeward he ran through streets now filling with people. It was hopeless to escape them. Hands reached out and tore away the coverings of the basket and revealed the food. Dirty hands fought for the tins and emptied the basket in an instant, and then he felt hands laid upon him.
"A foreigner, a foreign devil—" he heard voices screaming at the sight of his face. He burrowed among legs and forced his way through, agile with terror, and hid himself inside an open gate, looking this way and that until he saw a woman's angry face at a window and then he darted out again. Now he was near home and the crowd was surging in the opposite direction to see the murdered German. He was safe for a moment but what would he do without the food? He began to sob and tried to stop because his sobs shook him so he could not run, and then he had no breath to run and so he walked, limping and gasping, down the hutung to the small gate. He would have to knock; he was too weary to try to climb the wall. Ah, the gate was open! He stopped, bewildered, and then saw something bright in the dust of the threshold at his feet. It was blood, brightly red, curling at the edges in the dust. A new more desperate terror fell upon him. He could not think. He ran through the gate and into the meager courtyard. The paper-latticed doors of the little central room were swinging to and fro, and he pushed his way through them.
There he stopped. Upon the rough brick floor his father lay, resting in his own blood which flowed slowly from a great gash in his throat, so deep that the head was half severed. His arms were flung wide, his legs outspread. Upon the quiet face, though bled white, he saw his father's old sweet smile, the greeting he gave to all alike who entered this house, to strangers and to his own, and now to his son. Under the half closed lids the blue eyes seemed watching. Clem gazed down at his father, unable to cry out. He knew. He had often seen the dead. In winter people froze upon the streets, beggars, refugees from famine, a witless child, a runaway slave, an unwanted newborn girl. But this was his father.
He choked, his breath would not come up, and he tried to scream. It was well for him that no sound came, for in the silence he might have been heard, and those who had gone might have come back. He gave a great leap across his father's feet and ran into the other room where his mother's bed was. There he saw the other three, his mother, his two sisters. They were huddled into the back of the big Chinese bed, the two children clinging to their mother, but they had not escaped. The same thick sword that had cut his father's throat had rolled the heads from the children. Only his mother's long blonde hair hid what had been done to her, and it was bloodied a bright scarlet.
He stood staring, his mouth dried, his eyes bulging from their sockets. He could not cry, he could not move. There was no refuge to which he could flee. Where in this whole city could he find a hole in which to hide? He thought for one instant of William Lane and the security of that solid house enclosed behind walls. The next instant he knew that there was no safety there. The dead might be lying on those floors, too. No, his own kind could not save him.
He turned and ran as he had come along the high walls of the alleys, by lonely passage's away from the main streets back again to Mr. Fong's house.
In the central room behind the shop Mr. Fong was sitting in silence with his wife and their children. News had flown around the city from the Imperial Palace that two Germans had fired on innocent Chinese people and that a brave Chinese soldier had taken revenge by killing one of the Germans and wounding the other. Mr. Fong doubted the story but did not know how to find out the truth.
"The wind blows and the grass must bend," he told Mrs. Fong. "We will remain silent within our own doors."
He was troubled in mind because his eldest son could speak English and he feared that it might cause his death. Not only foreigners were to be killed. The Old Buddha had commanded today at dawn, at her early audience in the palace, that all who had eaten of the foreign religion and all who could speak foreign languages were also to be killed.
Mr. Fong had just finished quarreling with his wife, and this was another reason for the silence of the family. The quarrel, built upon the terror of what was taking place in the city, of which rumors were flying everywhere, had been over the very matter of the eldest son speaking English.
"I told you not to let our Yusan learn the foreign tongue," Mrs. Fong had said in a loud whisper. Sweat was running down the sides of her face by her ears. Though she fanned herself constantly with her palm leaf fan nothing dried her sweat this day.
"Who could tell that the Old Empress would put the Young Emperor in jail?" Mr. Fong replied. "Two years ago everything was for progress. Had all gone well, the young Emperor would now be on the throne and the Old Woman would be in prison."
"The gods would not have it so," Mrs. Fong declared.
Nothing made Mr. Fong more angry than talk of gods. He read as many as possible of the books of revolutionary scholars and other books which they had translated from foreign countries. Thus he knew many things which he concealed from Mrs. Fong, who could not read at all. Through his cousin he had learned much that happened in the Forbidden City. He had long known that there was a certain troupe of actors who, a few years before, had been summoned from Shanghai to play before the Imperial Court. Among the actors were the two famous rebel scholars, Liang Ch'i Ch'ao and T'an Tzut'ung, and they were responsible for informing the young Emperor that times had changed and that railroads and schools and hospitals were good things. What pity that all their efforts now had failed! That man at court whom they had trusted, that Yuan Shih K'ai, though pretending sympathy with them, had betrayed them to the chief eunuch Jung-lu, because the two had long ago sworn blood brotherhood, and Jung-lu had told the Old Empress, and so she had won after all. Liang had escaped with K'ang Yu-wei, the young Emperor's tutor, but T'an had been killed. Since then the Old Demon, as Mr. Fong called her in his private thoughts, had gone from worse to madness.
There was no use in telling Mrs. Fong all this. He heard her voice complaining against him still, though under her breath, and being frightened and weary and more than a little fearful that she was right, he squared his eyebrows and opened his mouth and shouted at her.
"Be quiet, you who are a fool!"
Mrs. Fong began to cry, and the children not knowing which way to turn between their parents, began to wail with their mother.
In the midst of this hubbub which, having aroused, Mr. Fong now tried to stop, they heard a stealthy beating upon the back door. Mr. Fong raised his hand.
"Be quiet!" he commanded again in a loud whisper.
Instantly all were still. They could hear very well the sound of fists upon the barred gate.
"It is only one pair of hands," Mr. Fong decided. "Therefore I will open the gate and see who it is. Perhaps it is a message from my cousin."
He rose, and Mrs. Fong, recalled to her duty, rose also, and with her the children. Thus together they went into the narrow back court and inch by inch Mr. Fong drew back the bar. The beating ceased when this began, and at last Mr. Fong opened the gate a narrow way and looked out. He turned his head toward Mrs. Fong.
"It is Little Foreign Brother!" he whispered.
"Do not let him enter," she exclaimed. "If he is found here, we shall all be killed."
Mr. Fong held the gate, not knowing what to do. Against his own will he heard Clem's voice, telling him horrible news.
"My father and mother, they are dead! My sisters are dead! Their heads are off. My father lies on the floor. His throat is gashed. I have nowhere to go."
Against his will Mr. Fong opened the gate, allowed. Clem to come in, and then barred it again quickly. The boy had vomited and the vomit still clung to his clothes. His face was deathly and his eyes sunken, even in so short a time.
"Now what shall we do?" Mrs. Fong demanded.
"What can we do?" Mr. Fong replied.
They stood looking at each other, trying to think. Clem, past thought, stared at their faces.
"We must consider our own children," Mrs. Fong said. But she was a kind woman and now that she saw the boy and the state he was in she wished to clean him and comfort him, in spite of her fright.
"Why should they kill your family?" Mr. Fong demanded of Clem. "Your father was poor and weak but a good man."
"It is not only my father," Clem said faintly. "I saw them kill a German and another only barely escaped though he was shot in the leg."
"Did the Germans not shoot into a crowd?" Mr. Fong demanded.
Clem shook his head. "There was no crowd. Only me."
"Who shot then?"
"A soldier."
"Wearing what uniform?" Mr. Fong asked.
"That of the Imperial Palace," Clem said. Clem was telling the truth, Mr. Fong saw by his desperate honest boy's face.
"The Old Empress is gone mad," Mr. Fong said between set teeth. "Can she turn back the clock? Are we to return to the age of our ancestors while the whole world goes on? She has made us the laughingstock of all peoples. They will send their armies and their guns, and we shall all be exterminated because we listened to an old ignorant woman who sits on a throne. I will not fear her!"
So saying he seized Clem by the ragged elbow of his jacket and led him into the house, and behind him the family followed.
"Take off his garments and let me clean them," Mrs. Fong said.
"Go into the inner room and get into the bed there," Mr. Fong said. "After all, we are an obscure family. We have no enemies, I believe. If anyone comes to ask why we had a foreign youth here to teach our son, I will say it is because the foreigner was only a beggar."
Like a beggar then Clem went into the dark small inner room, and taking off his outer clothes he crept under the patched quilt on the bed. He was dried to the bone. There were no tears in him, in his mouth no spittle. His very bladder was dry and though his loins ached he could make no water. The palms of his hands and the soles of his feet itched. Tortured by this drought, he lay under the quilt and began to shake in a violent and icy chill.
Clem was hidden thus for how many days he did not know. Nor did he know what went on in the city. Not once did Mr. Fong or any of his family pass through the boarded doors of the shop. The cousin came sometimes at midnight, and through him Mr. Fong knew what was happening. Thus he knew that the Old Demon, in her wrath, had set the fourth day after the murder of the German as the day when all over the empire foreigners were to be killed.
There were other edicts. Thus on the seventh day of the seventh month the "Boxer Militia" was praised and exhorted to loyalty, and such Chinese as were Christians were told to repent if they wished to stay alive.
Mr. Fong, who was not a Christian, knew, too, from his cousin that all the foreigners in the city were locked into the Legation Quarter, and that a battle was raging against them. He had heard continuous shooting, but he did not dare to go out to see what it was. In his heart he tried to think how he could convey Clem secretly into the fortress of his own kind and so rid his household of the danger, but he could think of nothing. He did not dare tell even his cousin of Clem's presence in the house, for if it were discovered that the cousin was at heart a friend of the young Emperor and therefore an enemy to the Old Empress, he might be arrested and tortured, and to save himself he might get grace by telling about his own relative who was shielding a foreigner. Mr. Fong said nothing and listened to everything.
To Clem day and night were alike. The door to his small inner room was kept barred and was opened only by Mrs. Fong bringing food, or sometimes by Mr. Fong coming in to feel the boy's wrists for fever. Clem lay in a conscious stupor, refusing to remember what he had seen, neither thinking nor feeling.
Then one day, and at what hour he did not know, he felt himself unable to keep from weeping. The gathering strength of his body, too young to accept continuing sleep, roused his unwilling mind, and suddenly he saw clearly upon the background of his brain the memory of his dead family, hacked and hewed by swords, and he was strong enough for tears. His numbed spirit came back to life, and the tears flowed. From tears he rose to sobbing which he could not control, and hearing these sobs Mr. Fong hastened into the room. Clem had struggled up and was sitting on the edge of the bed, clutching his chest with his hands.
"There is no time to weep," Mr. Fong said in a whisper. "I have been waiting for this awakening. You are too young to die of sorrow."
He went to a chest that stood against the wall and brought out a short blue cotton coat and trousers.
"I bought these at a pawnshop two nights ago," he went on. "The madness in the city has abated somewhat. It is said that the foreign armies are very near. I prepared the garments against this moment. They will fit you. We have made black dye for your hair and there are shoes here. Put these on, and eat well of the meal my children's mother is cooking. She has baked loaves and wrapped salt fish and dried mustard greens into a package for you and put them into a basket such as country boys carry."
Clem stopped sobbing. "What am I to do, Elder Brother?" he asked.
"You must make your way to the sea, to a ship," Mr. Fong said in a whisper. His smooth face, usually so full, looked flat and his eyes were sunken under his sparse stiff brows. He had not shaved for days, and a stubble stood up on his head and his queue was ragged. "Now hear me carefully, Little Brother. All those of your kind who are not dead are locked behind walls in the foreign quarter, and a fierce battle has raged. We shall lose as soon as foreign soldiers with guns arrive at the city. Our stupid Old Woman will not know she has lost until she has to flee for her life. We can only wait for that hour, and it is not far off. But our people are not with her. You will be safe enough among the people. Avoid the cities, Little Brother. Stay close by the villages, and when you pass someone on the road, look down into the dust to hide the blue color of your eyes."
Clem changed into the Chinese garments and though his legs trembled with weakness, the thought of escape gave him strength. He ate well of the strong meat broth and bread and garlic which Mrs. Fong set before him, all this being done in silence. When he had eaten she brought a bowl of black dye, such as old women smear upon their skulls when the hair drops out, and with a strong goose feather she smeared this dye upon his sand-colored hair and upon his eyebrows and even on his eyelashes.
"How lucky your nose is not high!" she whispered. When she had finished she stood back to look at him and admire the change. "You look better as a Chinese!"
Mr. Fong laughed soundlessly and then pressed the basket on Clem's arm and together they took him to the small back door. "You know your way to the South Gate," Mr. Fong whispered. "The wind now is from the south. Follow it and walk for three days, and then turn eastward to the sea. There find a ship that flies a foreign flag, and ask for a task of some sort upon it."
Clem stood for one instant beside the door. "I thank you for my life," he stammered.
"Do not thank us," Mr. Fong replied. "The stupidity of the Old Woman has not made us enemies. Return to the land of your ancestors. But do not forget us. Take this, Little Brother. If I were not so poor I would give you a full purse." He put a purse into Clem's hand and Clem tried to push it away.
"You must take it for my own ease of mind," Mr. Fong said. So Clem took it.
Even Yusan, his childish pupil, must give him a last gift. The boy did not understand why Clem must be hidden or why be sent out in secret, but he clung to Clem's hand and gave him two copper coins. Mrs. Fong touched the edge of her sleeves to her eyes and patted Clem's arm once and then twice, and Mr. Fong opened the door and Clem went out.
It was night, at what hour he could not tell, but the darkness was deep and the city was silent. He stood listening, and he heard the soft sound of the wooden bar as Mr. Fong drew it against the inside of the door. Still listening he heard in the distance the cracking of guns, a volley and then another. He could only go on, and feeling the dust soft beneath his feet, he lifted his face to the wind and let it guide him southward.
# 2
UPON A SEA AS blue as the sky above it a British ship shone as white as a snowbank. William Lane, pacing the deck after a solid English breakfast, held his head high, aware of the glances which followed him as he went. Ladies were arranging themselves in the deck chairs, and only a few minutes earlier he had helped his mother with her rug, her cushion, her knitting, her book. Henrietta was writing letters in the salon, and Ruth was playing shuffleboard. When he felt like it he would join her, but just now he wanted to walk his mile about the deck.
Upon his father's direction they had taken passage on the first ship that left Shanghai. Only the assurance of the Consul General had persuaded them to leave.
"You cannot possibly help anyone by remaining here," the Consul General had said irritably to Mrs. Lane, when they had gone to him for advice. "Your husband is as safe as we can make him in the Legation Quarter with all the other foreigners. They are in a state of siege, of course, but they have plenty of food and water, and relief is on the way. It is only a matter of days."
"Why should we go then?" Henrietta had asked in her blunt voice.
The Consul General had stared at the plain-faced girl. "Merely to get on your way," he retorted. Merely to get out of my way, he meant.
Mrs. Lane decided the matter abruptly. "We had better go, or we may not get away for months," she told William. "I will settle you in college and Henrietta in boarding school, and we will have the summer together with your grandfather at Old Harbor. If things are quiet in Peking by autumn I will go back. If not, your father will come home. We all need a rest and a change. I am sick of China and everything Chinese."
So they had taken passage. Since British ships docked at Vancouver, their course was northerly and the weather was cool and fine.
William Lane tried not to think of his father and a good deal of the time he succeeded. He was feeling many things at this age, everything intensely. Above all, he was heartily glad that he would never again see the English boarding school where he had been so often unhappy. He was ashamed and yet proud of being American, ashamed because to be American at the school had kept him second-class, proud, because America was bigger than England. The consciousness of an inferiority which he could not believe was real had clouded his school days. He had isolated himself both from the Americans and from the English, living in loneliness.
He was altogether ashamed of being the son of a missionary. Even the children of English missionaries were secondary. The son of the American ambassador alone had any sort of equality with the English boys, and seeing this, William had often bitterly wished that his father had been an ambassador. Men ought to consider what they were, he thought gloomily, for the sake of their sons. He hated Henrietta because when she came last year to the school she had immediately joined the Americans and had foolishly declared that she did not care what her father was. Thus William and Henrietta had been utterly divided at school and their division had not mended. She had taken as her bosom friend a girl whom he particularly despised, the daughter of an American missionary who lived in an interior city and was of a lowly Baptist sect. The girl was loathsomely freckled and her clothes were absurd. She should never have been at the school, William felt, and to have her the chosen friend of his own sister degraded him. In his loneliness he developed a grandeur of bearing, a haughtiness of look, which warned away the ribald. He avoided Henrietta because she was not afraid of him. Sometimes she laughed at him. "You look like a rooster when you prance around like that," she had once declared in front of their schoolmates. Shouts of laughter had destroyed his soul.
"I say," the cricket captain had cried, "you do look like a cock, you know!"
Well, that was over. He need never return to the school. Yet he did not and would not acknowledge how profoundly he would like to have been English. The most that he allowed himself was to dream occasionally as he walked the decks, his head high, that people who did not know him would think he was English. Lane was a good English name. His accent, after four years at school, was clearly English. The most fortunate youth he had ever met was the son of an English lord who spent a day at the school once when his father was visiting on shore from an English battleship in the Chinese harbor.
He passed his sister Ruth at the shuffleboard. "I wish you'd play with me now, William," she said in a plaintive voice.
"Very well, I will," he replied. He paused, chose his pieces and the game began. He played much better than she did. The only fun he found in playing with her at all was to allow her to seem to win until the very end when, making up his mind that it was time to stop, he suddenly came in at the finish with victory.
"Oh William!" she cried, invariably disappointed.
"I can't help it if I'm better than you," he replied today and sauntered away, smiling his small dry smile.
He did not like to play with Henrietta. She was a changeable player, losing quickly sometimes and again winning by some fluke that he could not foresee. He never knew where he was with her.
There were no boys on the ship whom he cared to cultivate, but there was one young man, English, some five or six years older than he, to whom he would have liked to speak, except that the chap never spoke first, and William did not want to seem American. At school the chaps always said Americans were so free, rushing about and speaking first to everybody.
He would have been considerably bored had he not thought much about his future and had there not been so many meals. Just now the morning broth was being served on little wagons, pushed by white-robed Chinese table boys and deck stewards. He approached one of the wagons, took a cup of hot beef broth and a handful of what he had taught himself to call biscuits instead of crackers, and sat down in his deck chair beside his mother. She had already chosen chicken broth as lighter fare. She complained about the plethora of food and yet, he noticed, she ate as they all did. It cost nothing more, however much one ate, but none of them would say such a thing aloud except Henrietta.
"Henrietta seems to have picked up a young man," his mother now remarked.
She nodded toward the upper deck, and William saw his sister leaning against the rail, the wind blowing her black hair from her face. She was talking in her earnest abrupt fashion to the young Englishman. A pang shot through his heart. He renounced the friendship he had craved. Whoever was Henrietta's friend could never be his.
"Henrietta will speak to anybody," he told his mother. "I noticed that at school."
Clem plodded his way across the Chinese countryside. He was shrewd in the ways of the people and no human being was strange to him. Mercy he expected of none, kindness he did not count upon, and when he did not receive these, he blamed no one.
He walked by night and slept by day in the tall sorghum cane that grew in the fields at this season. When he saw no one ahead on a road as he peered out of the growth, he took advantage of this to cover as many miles as he could of those miles still between himself and the sea. The canes cut him from the sight of any farmer working in the fields and he had only to look ahead, for he walked faster than anyone coming from behind.
One day he fell in with an old country woman. She had long passed the age of concealing herself for modesty's sake and she had paused to relieve herself by the road. Comfort was now above all else. Clem came upon her about noon on a lonely country road and for a moment he thought her part of a bandit group. When the canes are high it is the season of bandits and often a gang of men will carry with them an old woman as a decoy.
The old woman laughed when she saw his start. "Do not be afraid of me, boy," she said in a cheerful voice while she tied her cotton girdle about her waist.
She spoke a country dialect which Clem understood, for its roots were the same language he had heard in Peking and so he said, "Grandmother, I am not afraid of you. What harm can we do each other?"
She laughed at nothing as country women will. "You cannot do me any harm," she said in a voice very fresh for such a wrinkled face. "Thirty years ago perhaps but not now! Where are you going?"
She fell into pace beside him and he slowed his step. It would be well for him to be seen with this old woman. He might be taken for her grandson. "I am going east," he said.
"How is it you are alone?" she asked.
He had tried to keep the dangerous blue of his eyes away from her, but when he stole a look at her, he saw that he need not take care. She had cataracts on both eyes, not heavy as yet, but filmed enough to see no more of him than his vague outlines.
"My father died in Peking," he said truthfully, "and I am going to find my grandfather."
"Where is your grandfather?" she asked.
"To the east," he replied.
"I am going eastward, too," she said. "Let us go together."
"How far east?" he asked with caution.
She named a small city at the edge of the province.
"How is it you are alone?" he asked in his turn.
"I have no son," she replied. "Therefore I have no daughter-in-law. But I have a daughter who is married to an ironsmith in the city and I go there to ask for charity. My old man, her father, died last week and I sold the house. We had two thirds of an acre of land. Had I a son I would have stayed on the land. But my fate is evil. My twin sons died together in one day when they were less than a year old."
She sighed and loosened her collar as though she could not breathe and so her wrinkled neck was bare. Clem saw around it a dirty string on which hung an amulet.
"What is it you wear on your neck, Grandmother?" he asked.
She laughed again, this time half ashamed. "How do I know what it is?" she retorted.
"Where did you get it?" Clem asked.
"Why do you want to know?" the old woman asked suspiciously.
Now the amulet was a strange one for a Chinese woman to wear. It was a small brass crucifix wrapped around with coarse black thread.
"It looks Christian," Clem said.
The old woman gave him a frightened look. "How does a boy like you know what is Christian?" she demanded, and she buttoned her coat.
"Are you a Christian?" Clem asked softly.
The old woman began to curse. "Why should I be a Christian? The Christians are bad. Our Old Buddha is killing them. You come from Peking; you ought to know that."
"The cross is good," Clem said in a whisper.
She stopped in the middle of the road and heard this. "Do you say it is good?" she asked.
"My father believed the cross was good," Clem said.
"Was your father one of Them?"
Now Clem decided to risk his life. "Yes, and he is dead. They killed him." All this he said without her knowing that he was not Chinese.
He saw her mobile wrinkled face grow kind. "Let us sit down," she told him. "But first look east and west and see if there is anyone in sight."
No one was in sight. The hot noonday sun poured down upon the dusty road.
"Have you eaten?" the old woman asked.
He had been walking for four days and his store of bread was gone. He had still some of the dried mustard wrapped in the cotton kerchief. "I have not eaten," he said.
"Then we will eat together," the old woman told him. "I have some loaves here. I made them this morning."
"I have some dried mustard leaves," Clem said.
They shared their food and the old woman prattled on. "I asked Heaven to let me meet with someone who could help me on the road. I had not walked above half the time between sunrise and noon when you came. This is because of the amulet."
"Why do you say Heaven instead of God?" Clem asked.
"It is the same," the old woman said easily. "The priest said I need not call the name of a foreign god. I may say Heaven as I always have."
"What priest?" Clem asked.
"I can never remember his name."
"A foreigner?"
"Foreign, but with black hair and eyes like ours," the old woman said. "He wore a long robe and he had a big silver cross on his breast. He prayed in a foreign tongue."
Catholic, Clem thought. "What did this priest say the amulet meant?" he asked.
The old woman laughed. "He told me but I cannot remember. It means good, though—nothing but good." She looked so cheerful as she chewed the steamed bread, the sun shining on her wrinkled face, that she seemed to feel no pain at being alone.
"Did he teach you no prayers?" Clem asked.
"He did teach me prayers, but I could not remember them. So he bade me say my old O-mi-to-fu that I used to say to our Kwanyin, only when I say it I am to hold the amulet in my hand, so, and that makes the prayer go to the right place in Heaven."
Wise priest, Clem thought, to use the old prayers for the new god! He had a moment's mild uncensuring cynicism. Prayers and faith seemed dream stuff now that his father was dead.
The old woman was still talking. "He is dead, that piteous priest. If he had been alive I would have gone to find him. He lived in a courtyard near his own temple—not a temple, you understand, of our Buddha. There were gods in it, a man hanging on a wooden shape—bleeding, he was. I asked, 'Why does this man bleed?' and the priest said, 'Evil men killed him.' There was also a lady god like the Kwanyin, but with only two hands. She had white skin and I asked the priest if she were a foreigner and he said no, it was only that the image was made in some outer country where the people are white-skinned, but if the image had been made here the lady would have skin like ours, for this is her virtue that wherever she is, she looks like the people there. The man on the cross was her son, and I said why did she not hide him from the evil men and the priest said she could not. He was a willful son and he went where he would, I suppose."
"How is it that the priest is dead?" Clem asked with foreboding.
The old woman answered still cheerfully. "He was cut in pieces by swordsmen and they fed the pieces to the dogs and the dogs sickened and so they said he was evil. I dared not tell them that I knew he was not evil. It was the day after my old man died and I had no one to protect me."
They sat in the sun, finished now with their meal, and Clem hearing of the priest's dreadful end felt shadows of his own fall upon him. "Come," he said, "let us get on our way, Grandmother."
He decided that he would keep his secret to himself. Yet as the day went on a good plan came to him. He could pretend to be blind, keep his blue eyes closed, feel his way, act as the old woman's grandson, and so they could walk all day more quickly and safely than by night. Then too he could use the money which Mr. Fong had given him, which until now he dared not use at an inn. Yet to make the pretense it was needful to tell the old woman who he was and she was so simple that he could not make up his mind whether he dared to trust his life into her hands.
When night drew near and a village showed itself in a distant cluster of lights, he thought he could tell her. He knew by now that she was good and only what she said she was, and if he were with her he might keep her awake to danger. If by chance she betrayed him as not Chinese, then he must make his escape as best he could.
So before they came to the village he took her aside, much to her bewilderment, for she did not know why he plucked her sleeve. Behind a large date tree, where he could see on all sides, he told her.
"Grandmother, you have been honest with me, but I have not told you who I am."
"You are not a bandit!" she exclaimed in some terror.
"No—I am someone worse for you. My father was a foreigner, like your priest."
"Is it true?" she exclaimed. She strained her eyes and then put up her hand to feel his face.
"It is true," he said, "and my father and mother and my sisters were killed as the priest was killed and I go to the sea to find a ship to take me to my own country."
"Pitiful—pitiful," she murmured. "You are not very old. You are not yet grown."
"No," Clem said. "But I am alone, and so I am glad that you met with me."
"It was the amulet," she said. "Heaven saw us two lonely ones walking the same road and brought us together."
"Grandmother," he went on, "you cannot see my eyes, but they are not black as the priest's eyes were."
"Are they not?" she asked surprised. "What color are they, then?"
"Blue," he told her.
"Blue?" she echoed. "But only wild beasts have blue eyes."
"So have many of my people," he said.
She shuddered. "Ah, I have heard that foreigners are like wild beasts!"
"My father was not," Clem replied, "and my mother was very gentle. You would have liked her."
"Did she speak our tongue?"
"Yes," Clem said, and found that he could not tell more of his mother.
"Ai-ya," the old woman sighed. "There is too much evil everywhere."
"Grandmother," Clem began again.
"I like to hear you call me so," the old woman said. "I shall never have a grandson, since my sons are dead."
"Will you help me?" Clem asked.
"Surely will I," she replied.
And so he told her his plan and she listened, nodding. "A half-blind old woman leading a blind grandson," she repeated.
"We can go to the village inn there and sleep under a roof. I have slept every night in the canes, and two nights it rained."
"I have some money," she said, fumbling in her waist.
"I also," Clem said. "Let us spend mine first."
"No, mine."
"But mine, Grandmother, because when I get to my own country it will be no use to me."
She was diverted by this. "How can money be no use?"
"We have a different coin," he replied.
They began to walk again and planned as they went. Far from being stupid as he had thought her, she was shrewd and planned as well as he did. All her life she had been the wife of a small poor man compelled to evade the country police and tax gatherers and she knew how to seem what she was not and to hide what she was.
An hour later Clem was walking down the village street with her, his eyes shut, holding in his hand one end of a stick the other end of which she held. She led the way to the inn on the single street and asked for two places on the sleeping platform for herself and her grandson, and the innkeeper gave them without more questions than such men usually ask of those they have not seen before. The old woman told a simple story, much of it true, how her husband and son were both dead together of the same disease and how she had left only this grandson and they were returning to her old city where she had been reared and where she might find her daughter married to the ironsmith.
"What is his name?" the innkeeper asked.
"He is named Liu the Big," the old woman said.
A traveler spoke up at this and said, "There is an ironsmith surnamed Liu who lives inside the east gate of that city and he forged me an iron for a wheel of my cart, when I came westward through there. He has the finger off one hand."
"It is he," the old woman said. "He lost the finger when he was testing a razor he had ground. It went through his finger like flame through snow."
Clem passed the night lying among the travelers on a wide bed of brick overlaid with straw and slept in spite of the garlic-laden air because for the while he felt safe again.
Nights and days Clem spent thus, always as the grandson of the old woman, and each day she grew more fond of him. She told him many curious tales of her early childhood and she asked him closely about his own people and why he was here instead of in the land where he belonged and marveled that he knew nothing at all of his ancestors.
"You foreigners," she said one day, "you grow mad with god-fever. There is something demon in your gods that they drive you so. Our gods are reasonable. They ask of us only a few good works. But for your gods good works are not enough. They must be praised and told they are the only gods and all others are false."
She laughed and said cheerfully, "Heaven is full of gods, even as the earth is full of people, and some are good and some are evil and there is no great One Over All."
Clem did not argue with her. There was no faith left in him except a small new faith in the goodness of a few people. Mr. Fong and his wife had been good to him and so now was this old woman good, and he listened to her as they walked over the miles, side by side unless they came among people when he took the end of the stick she held and pretended to be blind. From her lips he learned a sort of coarse wisdom as he went, and he measured it against what he had learned before and found it true. Thus, the old woman said, the great fault with Heaven and whatever gods there were was that they had not arranged that food could fall every night from the sky, enough for everybody to eat so that there could be no cause for quarrel.
"If the belly is full," she said, "if we could know that it would always be full, men would be idle and laugh and play games like children, and then we would have peace and happiness."
These words, Clem thought, were the wisest he had ever heard. If his father had needed to take no thought for food, then his faith might have been perfect. Assured of food, his father could have preached and prayed and become a saint.
Thus talking and thinking, sleeping in inns at night, Clem and the old woman reached the city where she must stop. He had noticed for a day or two that she seemed in an ill humor, muttering often to herself. "Well, why should I not?" this she asked herself. Or she said, "Who cares whether I—," or "My daughter does not know if I live."
Before they got into the city, on an afternoon after a thunderstorm during which they had taken refuge in a wayside temple where there were gods but no priests, the old woman came out with what she had been muttering to herself.
"Grandson, I ought to go to the coast with you. What will you do if I leave you? Some rascal will see your eyes and think to gain glory with the Empress and he will kill you and take your head to the capital to show for prize money."
Clem refused at once such kindness. "Grandmother, you are old and tired. You told me yesterday that your feet were swollen."
They made an argument out of it for a while and at last the old woman said, "Come with me at least to the door of my. daughter's house. We will see what Big Liu says."
To this Clem consented, and when they came to the city the old woman would not enter until just before the gates closed so that people could not see them clearly. As night fell they joined the last people crowding to get inside the gate and walking quietly along mingled with the people, they came to the house of Liu the ironsmith.
Clem's first sight of the ironsmith all but overcame him. The forge was open to the street, and there the mighty man stood, his legs apart, his right arm uplifted and holding a great iron hammer, his left hand grasping thick tongs which held a red hot piece of metal. Upon this metal he beat with the hammer and the fiery sparks flew into the night with every blow. The ironsmith was black with smoke and his lips were drawn back from his teeth so that they showed very white, and so white, too, were the whites of his eyes, above which were fierce black brows.
"That is he," the old woman whispered.
She went in boldly and called out above the din. "Eh, Big Liu! Is my daughter at home?"
Big Liu put down the hammer and stared at her. "It is not you, mother of my children's mother!" This he shouted.
"It is I," the old woman said. Then she wiped her eyes with her sleeves. "My old man her father, is dead."
Big Liu still stared at her. "Come inside," he commanded. When he saw Clem following he stopped again. "Who is this boy?" he asked.
"He is my foster grandson," the old woman said and then she went on very quickly. "A poor orphan child he is, and I an old lonely woman and we fell in along the road and the gods sent him, I swear, for he took such care of me that I know he is no common child but some sort of spirit come down. His eyes are the eyes of Heaven and his heart is gentle." Thus talking very fast while Big Liu stared the old woman tried to make Clem safe.
But Clem shook his head. "I will tell you who I am," he said to Big Liu. They went into the inner room and all talk had to wait until the old woman and her daughter had cried their greetings, had exclaimed and wept and hugged the three small children. By this time Big Liu had taken thought and he knew that Clem was no Chinese and he was very grave. He got up and shut the doors while the women talked and wept, and at last he made them be silent and he turned to Clem.
"You are a foreigner," he said.
"Yes," Clem said. "I cannot hide it from you."
Then he told him his story, and the old woman broke in often to tell how good he was and how they must help him, and if Big Liu did not think of a way, she must go with Clem herself to the sea.
Big Liu was silent for some time and even his wife looked grave and gathered her children near her. At last Big Liu said, "We must not keep you here for a single day. Were it known that there was a foreigner in my house you would be killed and we would all die with you. You must go on your way, as soon as the East Gate opens at dawn."
Clem got up. "I will go," he said.
Big Liu motioned with his huge black hand. "Wait—I will not send you out to die. I have an apprentice, my nephew, a lad older than you, and he shall lead you to the coast. Since you are here, wash yourself, and I will give you better garments. Then lie down to sleep for a few hours. My children's mother shall make you food. Have you money?"
"He has no money," the old woman said. "He would use his money on the way and so I will give him mine."
Big Liu put out his hand again. "No, keep your money, good mother. I will give him enough."
So it all happened. Clem obeyed Big Liu exactly as he had spoken for this big man had a voice and a manner of command, though he spoke slowly and simply. Clem washed himself all over with a wooden bucketful of hot water, and he put on some clean garments that the apprentice brought, who stared his eyes out at Clem's white skin under his clothes.
Clem ate two bowls full of noodles and sesame oil and lay down on a bamboo couch in the kitchen while the apprentice lay on the floor. But Clem could not sleep. He knew that the ironsmith sat awake, fearful lest someone discover what was in his house, and although the old woman bade Clem not to be afraid, she could not sleep, either, and she came in again and again to see why he did not sleep and to tell him he must sleep to keep his strength. As for the apprentice, he did not like at all this new task, but still he had never been to the coast nor seen a ship, and so he was torn between fear and pleasure.
Before dawn broke Big Liu came in and Clem sprang up from the couch and put on his jacket.
The apprentice was sleeping but he got up, too, and yawned and wrapped his cotton girdle about himself and tied his queue around his head under his ragged fur cap and so they crept to the door.
"Come out this small back gate," Big Liu said. "It lets into an alley full of filth, but still it is safer than the street."
One moment the old woman held Clem back. She put her arms about his shoulders and patted his back and then sighed and moaned once or twice. "You will forget me when you cross that foreign sea," she complained.
"I will never forget you," Clem promised.
"And I have nothing to give you—yet, wait!"
She had thought of her amulet and she broke the string and tied it around his wrist, and the small cross hung there.
"I give this to you," she said. "It will keep you safe. Only remember to say O-mi-to-fu when you pray, because the god of this amulet is used to that prayer."
She wept a little and then pushed him from her gently, and so Clem left her and went on his way with the apprentice.
To this lad he said very little in the days that they traveled together, which days were fewer by half than those he had already come. They walked by day, the lad silent for the most part, too, and they slept at night in inns or sometimes only on a bank behind some trees for shelter, for the apprentice was fearful whenever they passed swordsmen. But never were they stopped, for Clem wore his old hat like any farmer boy and kept his eyes downcast.
When they came to the coast they parted, and Clem gave the apprentice nearly all that was left of his money. There were several ships in the harbor, and he would not let them go without finding one which would take him aboard. He was no longer afraid here, for it was a port and he saw policemen and he saw white men and women walking as they liked and riding in rikshas and carriages. He went near none of them for he did not want to be stopped in his purpose, which was to cross the sea and find his own country. But he did hear good news. Listening in an inn where he sat alone after the apprentice had left him, he heard that the Old Empress had been forced to yield to the white armies. She had fled her palace, leaving behind a young princess who had thrown herself into a well, and the foreign armies had marched into the city, plundering as they went and killing men and raping young women, so that all China was mourning the suffering which the Old Empress had brought upon them.
This Clem heard without being free to ask more about it. He wondered how the Fong household did, and whether they had shared in the suffering, and whether they in turn had been killed even as his family had been. But nothing could he know. When he had eaten he went to the docks and loitered among some sailors and on that same day he was able to find a ship and go aboard as a cabin boy. As for the apprentice, after staring half a day at the ships and wandering about the city, he left again for his home.
On the American freighter Clem made his way still eastward. The ship had brought ammunition and wheat to China and had taken away hides and vegetable oils. The hides, imperfectly cured, permeated the ship with their reek, and Clem, racked often with seasickness, wished sometimes that he too was dead. Yet the wish never lasted. Upon rolling gray seas the sun broke, the winds died and the waves subsided. Then, eating enormously in the galley with the thirty odd men who made up the crew, he wanted to live to reach the farm.
The men knew his story. They had heard it first on the pier at the port when, approaching one of them, he had asked timidly for a job on the ship.
"We don't want no Chinks," the sailor had replied.
"I am not Chinese," Clem had said.
"You ain't?" the sailor had said, unbelieving.
Clem had pointed to his eyes. "See, they are blue."
"Damned if they ain't," the sailor had agreed after staring at him a moment "Hey, fellows, anybody ever seen a blue-eyed Chink?"
"When is a Chink not a Chink?" a sailor had inquired. "Why, when his ma is somethin' else!"
"She wasn't," Clem had declared, with indignation. "She was good and so was my father and they were American and so am I." But English felt strange upon his tongue after the many days when he had spoken only Chinese.
The men had gathered about him, delaying the pleasures they planned for their brief hours ashore, and with pity and wonder they had listened to his story which he had poured out. Looking from one coarse face to the other, he found himself telling everything to save his own life. Even the things he had not allowed himself to remember he told, and he began to sob again, trying not to, his fists clenched against his mouth.
The men listened and looked at each other, and one burly fellow took Clem's head between his hands. "It's all over, see? And we believe you, sonny. And you come with us, if we have to smuggle you. But the old man is soft enough. He'll let you on board."
They had dragged him before a little sharp-faced captain and made him tell his story all over again, and then he had been hired as a cabin boy. With the captain he held long conversations.
"Reckon you'll never want to be going back to no heathen country after this!" the captain said.
"I don't know," Clem replied. He had mixed a whisky and soda, and set it before the captain. "I might have felt that way except that Mr. Fong saved my life. And people were kind all those days I tramped. I can't forget the old grandmother."
No, he could never forget. In the night, lying in his hard and narrow berth, tossed by the sea, he remembered the long days of tramping across the Chinese country, beside the old woman. Summer had ripened the fields, and the lengthening shadows of the green sorghum, high above their heads, gave them good shelter. Big Liu, too, had been kind. It would have been easy to tell the local police about a foreign boy and for the telling to have received a reward. Big Liu was poor enough to value money and Clem was a stranger. None would miss him if he died, but Big Liu had not betrayed him. Wonder and gratitude at the goodness of common men and women filled Clem's heart with faith, not the faith of his father but a new faith, a faith which bound him to the earth.
The sailors, too, were kind, although they were rough and of an ignorance he had never yet seen. They were mannerless, coarse, drunken when they could get drink, lewd in act and speech, easily angry, always ready to fight. He thought of them as men half made, left unfinished, never taught. They knew no better than they did.
Were the people of his country all like these? He had none to judge by, never having known his own kind, except his father who he felt vaguely was a man peculiar. The delicacy of the Chinese was soothing and comfortable to remember. Here on the ship, though he knew the men were friendly to him, yet for some fault, or no fault except that a man might be surly from too much drink the night before on shore leave, he might feel his ears jerked or his head cuffed, or a blow between his shoulders might fell him. He learned it was useless to be angry, for immediately the man would joyfully urge him to fight, and he was no match for any of the men, short and slender as he was. Once he complained to the captain, but only once.
"You don't think I'll defend you?" the captain had said.
"No, sir," Clem said, "except maybe to tell them to leave me alone."
"Do they hate you?"
"No, sir. I don't think they do; it's like play, maybe."
"Then put up or shut up," the captain said.
Yet the long journey over the sea was good for Clem. An endless roar of command sounded in his ears. He was at the beck and call of all of them. Twice the ship stopped for coal, once in Japan, once again at the Hawaiian Isles, but he had no shore leave. He gazed across the dock at strange lands and unknown peoples and saw sharp mountains against the sky. At night he helped drunken sailors to bed, staggering under the load of their coarse bodies leaning on his shoulders, smelling the filthy reek of their breath. When one or another vomited before he could reach the rail, Clem had to clean the mess before the captain saw it. By morning all had to be shipshape, and sometimes there was little sleep for Clem. He loathed the coarseness of the men and yet he pitied it. They had nothing to make them better. They hated the sea, feared it, cursed it, and yet went on living by it, for they did not know what else to do. In a storm they were filled with blind terror. Clem felt old beside them, old as a father, and sometimes like a father he tended them, pulling off their sodden shoes when they slept before they could undress, bringing them coffee at dawn when they were too dazed to take watch. They were kind to him in return, half shamed because they knew him only a child, and yet helpless before him. He remained a stranger to them, aloof even while he served them. Pity prevented his blame, and his pity made them often silent when he came near them. But this he did not know. For himself he felt only increasing loneliness, and he longed for the voyage to end that he might find those who were his own.
The sea voyage ended at last and one day he went ashore into a country which was his and yet where he was still a stranger. The crew collected a purse for him, and he would never forget that. It meant that he could travel to the east on a railroad, instead of tramping the miles away as he had done across the country in China. He had not minded doing it there because he knew the people and there was the old woman at his side, but here where he did not know the people or the food it would have been different.
So though the sailors were so evil, they were good, too. On the first day ashore in San Francisco they went together to a shop and bought Clem a suit of clothes. It was too big for him, but he rolled up the pants and sleeves. They bought him two clean shirts and a red tie, a hat and a pair of shoes and three pairs of socks and a pasteboard suitcase. Then they took him to the railroad station and bought him a ticket to Pittsburgh on the day coach. There was not quite enough money, for they would not let him spend the ten dollars they had given him, and one of them had pawned a gold thumb ring he had bought in Singapore. They clapped him on the back, embraced him, and gave him good advice.
"Don't talk with nobody, you hear, Clem?"
"Specially no women."
"Aw, he's too runty for women."
"You'd be surprised if you knew women like I do. Don't talk to 'em, Clem!"
"Don't play no cards, Clem!"
"Send us a postcard once in a while, Clem, will ya?"
The train pulled out and he stood waving his new hat as they receded until he could see them no more. So he was alone again, riding in a train across his own country. He had a seat to himself, opposite a red-faced man in a gray suit who slept most of the time and grinned at him vaguely when he woke. "Don't speak to nobody on the train," the sailors had told him. "Shore fellows will take your money away from you." He kept quiet and his wallet was in his breast pocket where he could feel it against his ribs every time he took a deep breath. When he needed money to spend on food he went into the men's room and there alone he took out a dollar at a time, keeping his change in his hip pocket against the back of the seat.
Hour after hour, in every hour of daylight, he stared from the window, seeing a country he could not comprehend. It seemed empty and without people. Where were all the people? The mountains were higher than he could have imagined, the deserts wider and more desolate, their emptiness terrifying. To his amazement, many times at the stations he saw white men doing coolie work, and in the few fields between mountains and on the fringe of deserts he saw men and women more ragged, more poor, though white, than any he had seen in China. Where was the land of milk and honey his father used to call home?
One night while he slept upright in his seat, they rolled into green plains. When he woke at dawn it was to another country. Green fields and broad roads, big barns and compact clean farmhouses charmed his eyes. This was Pennsylvania, surely!
Long before Clem had begun his voyage William had reached America. The white English ship docked at Vancouver, and Mrs. Lane, brisk and experienced, bullied the courteous Canadian customs officers and found the best seats on the train that carried them across Canada to Montreal, where they changed for New York.
It was a smooth journey, and William enjoyed it with quiet dignity. He kept aloof from his mother and sisters, staying most of the time in the observation car where behind a magazine he listened to men's talk. There was no difficulty in Montreal, and in New York his mother took them at once to the Murray Hill, where he had a room to himself because he was a boy. It was high ceilinged, and the tall windows had red velvet curtains held back by loops of brass. The luxury of the room and its bath pleased him. This then was America. It was better than he had feared.
They ate in a dining room where fountains played and canaries sang, and he enjoyed this, too.
"I believe in the best," his mother said. "Besides, Papa and Mama always stayed here when we came to town."
His mother kept him with her in New York for a week while she smoothed his path toward college, but Henrietta and Ruth she sent to her parents at Old Harbor. She did not take him at once to the office of the Mission Board. Instead she toured the best stores, asking to see young men's clothing. When she found something she liked she made William try it on. She bought nothing, however, merely making notes of garments and prices.
With these in a small notebook in her handbag she went on the morning of the fourth day to the Board offices and there was received with a deference which was balm to William's pride.
"Ah, Mrs. Lane," a rosy faced white-haired executive said, "we've been expecting you. We had a cablegram from Dr. Lane. What can we do for you?"
"I have a good deal of shopping to do for my son's entrance into Harvard," Mrs. Lane said. Her voice and look were equally firm.
The plump elderly executive, a retired minister himself, looked doubtful. "We have special arrangements with medium-priced stores to give us a ten percent discount."
Mrs. Lane interrupted without interest in the medium-priced stores. "I want to see the treasurer immediately."
"Certainly, Mrs. Lane—this way, please," the white-haired man said.
"You stay here, William," Mrs. Lane commanded.
While William waited, his mother had a long interview with the mission treasurer which left him looking dazed and certainly left him silent. William had stayed in the reading room because his mother wanted, she said, to be alone with the finances. He had sauntered about, reading pamphlets impatiently. They were religious and full of hopeful accounts of the hospitals and schools and orphanages and churches with which he was entirely surfeited. He wanted to get away from everything he had known. When he entered college in the autumn he would not tell anyone who his father was or that he came from China.
"There now," Mrs. Lane said when she emerged from the inner office. "I have everything all arranged. You'll be able to get along nicely." She held her long skirts in one hand and over her shoulder she said to the little mission treasurer, "Thank you, Mr. Emmons, you've been very helpful."
Mr. Emmons broke his silence. "You do understand, don't you, Mrs. Lane, that I haven't made any promises? I mean—I'll have to take up these rather unusual requests with the Board—evening clothes, for example—"
"I'm sure they'll see that my son deserves some special consideration, after all we've been through," Mrs. Lane said in her clear sharp voice. "Come, William, we can get the noon train after all."
He had followed her, holding himself very straight and not speaking to the shabby little treasurer.
When they reached his grandfather's house at Old Harbor, he was pleased to see it was a large one. It was old-fashioned and needed paint, but it stood in large, somewhat neglected grounds.
"Papa doesn't keep things up the way he used to, I see," his mother said. They had taken a hack at the station and now got down. She handed him her purse. "Pay the man his dollar, William," she told him.
"Grass needs cutting," she went on. "I suppose Papa can't afford a gardener all the time, now he's retired."
The hack drove away, and William looked at the suitcases the man had set down in the path. "We'd better take what we can," his mother said with some embarrassment. "I don't know how many servants Papa has now. We used to have a houseman and three maids."
She picked up two suitcases, and much against his will he took the other and followed her to the house. The door stood open and when they entered they were met by Henrietta and Ruth, dripping in bathing suits, and by a carelessly dressed old gentleman whom he recognized, though with extreme discomfort, as his grandfather.
Mrs. Lane swooped down upon him. "Well, Papa, here I am again!"
"You've grown a little older," he said, looking at his tall daughter.
Mr. Vandervent was no longer imposing. He was a potbellied, mild-looking man, and he seemed timid before his tall grandson.
"How do, William," he said, putting out a round little hand.
William clasped it coldly. "I'm very well, sir," he replied correctly. "I hope you are, too."
"So so," Mr. Vandervent said. "The sea don't really agree with me, but your grandma likes it."
"What we've been through—" Mrs. Lane began.
She was interrupted by a loud scream. A tall fat woman burst through a swinging door, an apron tied about her waist.
"Helen, my goodness!"
It was her mother. They embraced and kissed. "I was just stirring up one of my chocolate cakes, thinking that William would probably—we only have two maids now, Helen—why, William, this isn't you, never! Isn't he the image of your father, Robert? Your great-grandfather was a real handsome man, William."
Henrietta had disappeared and through the window William saw her walking along the shore. Ruth was standing on one foot and then another.
"William!" she now whispered. "Do get in your bathing suit. The ocean is wonderful."
It gave him an excuse and he seized it.
"May I, Mother?"
"Go on," his grandmother said heartily. "You'll have time before supper."
Supper! The word chilled his spine. He had heard it among the commoner missionaries, the Seventh Day Adventists, the Primitive Baptists, the Pentecostal people. At the English school the evening meal was always called dinner and since at his own home it had been so, too, it had not occurred to him that it could be anything else here.
He mounted the stairs with laggard steps and was arrested by his mother's voice. "Here, William, since you're going up, you might as well take some of the suitcases."
He stopped, not trusting his ears, and looked at his mother. She laughed, but he discerned embarrassment in the steel gray eyes she kept averted from his. "You may as well realize that you are in America, son," she told him. "You'll have to do a lot of waiting on yourself here."
He stood still for one instant; then with a passionate energy he turned and ran downstairs and loaded himself with the bags and staggered upstairs again. Once he glanced over the balustrade to see if they were looking at him, but nobody was. His mother was talking about the siege, and they had forgotten him.
No one had told Clem to telegraph to his grandfather, and he would have been reluctant to spend the money. When he got off at last at Centerville, there was no one to meet him, but he had expected no one. Carrying his suitcase, he approached a fat man who was staring at the train and scratching his head.
"Can you tell me where Mr. Charles Miller lives?" Clem inquired.
The man had started a yawn and stopped it midway. "Never heard of him."
"He lives on a farm," Clem said.
"Your best bet would be that way," the man said nodding toward the south.
"Thank you," Clem said.
The man looked surprised but said nothing and Clem began walking. His days on the sea had made his feet tender although they had once been horny from long walking on rough Chinese roads. But his muscles still were strong. The heat here was nothing to that in China, and the air was sweet with some wild fragrance. He did not see anyone after he left the small railroad town, and this was strange. Were there no people here? It occurred to him that it was nearly noon, and they might be having a meal. Even so, where were the villages? As far as he could see there was no village in sight. The fields rolled away in high green waves against a sky of solid blue. They were planted with corn, he saw with surprise. Did the people here eat only corn?
After another hour he was tired and hungry and he wished that he had stopped to buy some food. Five miles had seemed nothing in his excitement. He sat down beside a small stream and drank and rested, and while he sat there a wagon came by, pulled by two horses as high as camels. A man drove them, seated on a bench in the wagon. "Hi, there, feller," he called down. "Wanta ride?"
Clem was cautious. Why should a stranger offer him a ride? Might not the fellow be a bandit? "No, thank you," he replied.
The man drew the wagon to a stop. "You look like a stranger."
Clem did not reply. The barber on the ship had clipped his hair close to get rid of the dyed hair, and he was conscious of his baldness.
"Where you goin'?" the man asked.
"To Mr. Charles Miller's farm," Clem replied.
The man stared at him, his jaw hanging. He was a dirty fellow, clad in a sweat-soaked shirt and blue cotton trousers. Through the unbuttoned front of his shirt Clem saw a chest woolly with repulsively red hair.
"Old Charley Miller is dead," the man said.
The sunlight glittering upon the landscape took on the sharpness of dagger points, springing from the edges of leaves, the tips of grass, the points of fence rails. Clem's eyes blurred and weakness laid hold upon his knees.
"When did he die?" His mouth was full of dust.
"Coupla years ago." The man prepared for the story. He spat thick brown spittle into the road and pushed back his torn straw hat.
"Fact is, the old man hung himself in his own barn. Disappointed, that's what. He'd been tryin' for ten years to get a job with the Republicans, and when they got in that year they give him the sheriff's job. He had to put somebody off a farm the very first day—mortgage couldn't be met. He was too softhearted to do it—he was awful soft-hearted, old Charley was. He just hung himself the night before—yeah."
The man shook his head and sighed. "Wouldn't hurt a flea, Charley wouldn't. Couldn't kill a fly. Lived all alone. He had a son somewheres, but he never come home."
"His son was my father." The words escaped Clem like a cry.
The man stared, brown saliva drooling down his chin. "You don't, say!"
Clem nodded. "He's dead, too. That's why I came to find my grandfather. But if I haven't anybody—I guess—I guess I don't know what to do."
The man was kind enough. "You get up here along of me, sonny, and I'll take you to your grandpop's farm, anyway. There's folks livin' there. Maybe they'll lend a hand."
For lack of any directing thought Clem obeyed. He lifted his suitcase and gave it to the man and then stepping upon the axle he crawled into the seat. There in the hot sunshine he sat, his suitcase between his knees, and in silence the man drove two miles and put him down before an unpainted gate set in a decaying picket fence lost in high weeds. The wagon went away and Clem stared at a small solid stone house.
This, then, was the place of which he had dreamed as long as he could remember. The grass grew long and unkempt even in the yard. Over the house leaned an enormous sycamore tree. Under this tree he saw some ragged children, two boys and two girls. The boys were about his own age, the girls younger, or at least smaller.
They were eating dry bread, tearing at hunks of it with their teeth as they held it. When they saw him they hid the bread in their hands, holding it behind them.
"What you want?" the bigger boy asked in a gruff voice. He had a thin freckled face and his hair grew long into his neck.
"Who lives here?" Clem asked.
"Pop and Mom Berger," a girl said. She began to chew again at her bread. "You better go way or they'll set the dogs on you."
"Are you their children?" Clem asked. Where could he go in a strange country where nevertheless he belonged?
The thin boy answered again. "Naw, we're Aid children."
Clem looked at them, comprehending nothing. "You mean—Aid is your name?"
They looked at each other, confounded by this stupidity. "Aid children," the girl repeated.
"What do you mean?" Clem asked.
"We're Aid children. Children what ain't got nobody."
Clem gazed and his heart began to shrink. He, too, had nobody. Then was he, perforce, an Aid child?
Before he could reply to this frightful question, a short stout man ambled from the open door of the house and yelled: "Here, you kids—git back to work!" The children fled behind the house, and the man stared across the tumbled grass at Clem.
"Where'd you come from?" he demanded.
"I thought Charles Miller, my grandfather, was here," Clem said.
"Been gone two years," the man said. "I bought the place and took over the mortgage. I never heard he had no grandson."
"I guess my father didn't write. We lived a long way off."
"Out West?"
"Yes."
"Folks still there?"
"They're dead. That's why I came back."
"Ain't none of your folks around here as I know of."
He was about to go back into the door when something seemed to occur to him. "How old are you?" he asked.
"Fifteen," Clem said.
"Undersize," the man muttered. "Well, you might as well come in. We was just thinkin' we maybe could do with another Aid boy. The work's gittin' heavy." He jerked his head. "C'mon in here."
Clem took up his suitcase. He had nowhere else to go. He followed the man into the house.
"I'll report you to the Aid next time she comes," the man said.
William Lane was walking solitary along the beach. He had to be solitary a good deal of the time, for he had met no boys of his own age and it was intolerable to him to be with his sisters. Occasionally he went swimming with Ruth, but only at a time when the beach was not crowded. He had supposed of course that the beach was private since his grandfather's house faced upon it, and on that first day of his arrival when he had gone for a swim with Ruth he had been shocked to see at least fifty people in or near the water.
"Does Grandfather let all these people use our beach?" he had asked Ruth.
Before she could answer he heard Henrietta's horrid laughter. She came swimming out of the sea, her long straight hair lank upon her shoulders. "Nobody has private beaches here, stupid," she had said in a rude voice.
Ruth had reproached her as usual for his sake.
"How can William know when it's only his first day?"
"He'd better learn quick, then," Henrietta had retorted and returned to the sea.
Now of course he knew the truth. The beach belonged to everybody. Anybody at all could come there. They were all Americans, he knew, and yet they were of a variety and a commonness which made him feel the loneliest soul in the world. He longed for his English schoolmates, and yet he was cut off from them forever, because he did not want to see them any more. He did not want them to know that America was exactly what they had said it was, a place full of common people.
He lifted his head with a resolute arrogant gesture which was almost unconscious but not quite, since he had caught it from the boy who had been the captain of the cricket team last year, a fair-haired tall young man, whose father was Sir Gregory Scott, the British Consul General. Ronald Scott had been all that was splendid and fearless. Why not, when he had everything?
At least, William thought, his grandfather's house was better than some of the others facing the beach, and there were the two maids. He had felt slightly better when he discovered that most of the other houses had no servants, although in China women were only amahs for younger children. The two maids were old and badly trained. He had put his shoes outside his bedroom door the first night and they were still there the next morning but not polished.
"I say," he had asked his mother, "who does the boots in this house?"
She had given him a curious smile. "We do them ourselves," she said smoothly and without explanation. This was another thing that made him solitary. In Peking he had always been able to count on his mother, but here he did not know her as she was. She took his part when they were alone, but in front of other people he felt she did not. When he left his hat and coat in the hall for the maid to hang up his mother hung them up and his grandmother had been sharp about it. "William, don't let your mother wait on you," she had exclaimed. "Oh, never mind," his mother had said quickly. "Now, Helen, don't spoil the boy," his grandmother had retorted.
"He'll be going to college in just a few weeks, and then he'll have to look after himself." This was the feeble answer his mother had given. He had looked at both of them haughtily and had said nothing.
The air today was as clear and cool as a June day in Peking, and the sea was very blue. He had left the house after luncheon and seeing the beach crowded, he had walked straight away from it and toward the other part of Old Harbor, the best part. It had not taken him many days to find that the place where the really rich people lived was there. Great houses set in plenty of lawn faced wide bright beaches almost empty of people. Now almost every day he came here, always alone, too proud to pretend that he belonged here and yet longing to seem that he did before a chance passerby.
At this hour of early afternoon no one was to be seen. The heat of the sun was intense, though the air was cool, and the people were, he supposed, in their great houses. He was walking along the edge of a low bluff and suddenly he decided to climb it. The ascent was not difficult. He had only begun it when he saw a flight of wooden steps and was tempted to use them. It would be degrading to him if he were discovered trespassing, and yet his curiosity compelled him. He compromised by not using the steps and scrambled up the sandy rock ledge until he had reached the grass at the top. There he found himself still alone. For a quarter of a mile the lawn sloped back toward a knoll and hidden behind masses of trees he saw a vast house. His imagination hovered about it. Had his grandfather lived there and had he belonged there, how easily he might have been proud of his country!
He threw himself down upon the grass and buried his face in his arms. The sun beat upon his back and he felt suffocated with despair. He longed for the summer to be over so that he could leave his family and be alone at college. Yet how could he be successful there when it now appeared that his grandfather had no intention of helping him with any money? His mother had asked his grandparents outright if they could help him so that he could spend all his time in studying, and his grandfather had said, "Let him work his way through, as much as he can. It'll be good for him."
His mother had told him this with curious hesitation. "I suppose in a way it would be good for you," she had said thoughtfully. "But in another way I know it wouldn't. Work classes you here, actually, as much as it does in China. I wish we'd sent you to Groton."
"Why didn't you?" he had asked violently.
"Money," she had said simply. "Just money. Everything goes back to that."
"Does Grandfather have no money?" he had demanded.
"He seems to have enough for himself but nobody else," his mother had replied. Then she had one of her inexplicable changes. "Why do I say that? He's feeding us all—four of us, I suppose that's something, week in and week out."
William would have wept had he not been too proud. He continued now to lie like stone under the sun, his flesh hot and his heart cold. His disappointment was becoming insupportable. Of all that he had seen, nothing in his country was what he had hoped it would be, nothing except this spot where the great houses stood facing the sea from their heights of green, and here he did not belong.
At this moment he heard a voice.
"What are you doing here, boy?"
He lifted his head and saw an old gentleman leaning on a cane. A loose brown tweed cap hung over his forehead and he wore a baggy top coat of the same material. His face was brown, too, against the white of his pointed beard and mustaches.
"Trespassing, I'm afraid, sir." William sprang to his feet and stood very straight. He went on in his best English manner, instilled by the headmaster in Chefoo. "I couldn't resist climbing the bluff to see what was here. Then I was tired and wanted to rest a bit."
"Do you like what you see?"
"Rather!"
He felt some sort of approval in the old gentleman, and he held his black head higher and compelled his gray gaze to meet the sharp blue eyes that were staring at him. Then he smiled, a slow cautious smile.
The old gentleman responded at once and laughed. "You sound English!"
"No, sir, I'm not. But I've just come from China."
The old gentleman looked interested. "China, eh? Where?"
"Peking, sir."
"Been a lot of trouble over there."
"Yes, sir, that's why we came away—all of us, that is, except my father. He is in the siege."
The old gentleman sat down carefully on a boulder placed for the purpose. "It is very nasty, all those Americans locked up there. The Chinese will have to be taught a good lesson, especially as we have always been decent to them—the Open Door and so on. What's your father doing in Peking?"
It was the question he had been dreading. He toyed for an instant with the idea of a lie and decided against it. "I hope you won't think it strange, sir, but he's a missionary—Episcopal." He want to explain, but could not bring himself to it, that being Episcopal meant at least a Christian aristocracy.
He averted his eyes to avoid the inevitable look of disgust. To his astonishment the old gentleman was cordial. "A missionary, is he? Now that's interesting. We're Christian Scientists. What's your name?"
"Lane. William Lane."
He was as much disconcerted by approval as he might have been by rebuff. Before he had time to adjust himself the old gentleman said in a dry, kindly voice; "Now you come on up to the house. Mrs. Cameron will want a look at you. You can talk to her about your father. She's interested in foreign travel. I'm pretty busy, myself."
He stumped ahead of William, panting a little as the lawn rose toward the house. Behind him William walked gracefully, almost forgetting himself in his excitement. He was to enter this house, looming ahead in all its white beauty.
"I have a son," Mr. Cameron was saying. "He isn't as strong as we wish he was and we have him here trying to get him ready for Harvard in the autumn—freshman."
"I'm going to Harvard, too," William said.
"Then Jeremy will want to see you," Mr. Cameron said.
He paused on a wide white porch and William was compelled to stop, too, though his feet urged him to the door. Mr. Cameron's sharp small blue eyes roamed over the sea and the sky and fixed themselves upon the horizon.
"No storm in sight," he murmured.
He turned abruptly and led the way through the open door into a wide hall that swept through the house to open again at the back upon gardens of blooming flowers.
"I don't know where anybody is," Mr. Cameron murmured again. He touched a bell and a uniformed manservant appeared and took his cap and coat, glanced at William and looked away.
"Where is Mrs. Cameron?"
"In the rose garden, sir."
"Tell her I'm bringing someone to see her. Is Jeremy with her?"
"Yes, sir."
"Very well."
The man went silently toward the end of the hall and Mr. Cameron said to William, "It is always warm in the gardens. Come along."
He strolled toward the door and William followed him. His eyes stole right and left, and he saw glimpses of great cool rooms furnished in pale blue and rose. Silver gray curtains hung to the floor at the windows, and flowers were massed in bowls. Here were his dreams. He lifted his head and smiled. If such dreams could be real he would have them, someday, for his own.
The smell of hot sunshine upon fragrant flowers scented the air of the gardens as they reached the open doors. He knew very well from the garden about the mission house in Peking that only workmen could bring about the high perfection of what he now saw. Formal flower beds as precise as floral carpets stretched about him. A path of clean red brick led to an arbor a quarter of a mile away, and the arbor itself stood in a mass of late blooming roses. The manservant emerged from the arbor and stood respectfully while Mr. Cameron approached.
"Mrs. Cameron is here, sir. I am to bring tea in half an hour, sir, if you wish."
"Oh, all right," Mr. Cameron replied carelessly.
They entered the vine-hung arbor, and William saw a slender pretty woman, whose hair was graying, and a boy of his own age. She was sitting by a table filling a wicker basket with roses. The boy was stretched on a couch, a book turned face down on his lap. He was tall, with light hair and pale skin and pale blue eyes.
"This is William Lane, my dear," Mr. Cameron said. "I found him lying on his stomach on top of the bluff, and he says he comes from, China."
"Do you really?" Mrs. Cameron exclaimed. "How interesting!" She lifted large sweet brown eyes to William's face.
"I do, Mrs. Cameron," William replied. "I'm glad if it interests you."
"This is Jeremy," Mr. Cameron said. The two boys touched hands.
Mr. Cameron sat down. "I have a daughter somewhere, too. Where is she, my dear?"
"Candace?" Mrs. Cameron was busy again with roses. "She went to the village to buy something or other. I begged her to wait and get it in town, but you know how she is."
Mr. Cameron did not answer this. He looked at his son. "Well, Jeremy, William is going to Harvard, too. Coincidence, eh? You'll have to get acquainted."
Jeremy smiled. His mouth, cut deep at the corners, was sweet and rather weak. "I'd like to—but imagine China! Did you find it exciting? Do sit down. I'd get up, only I'm not supposed to."
William sat down. "It didn't seem exciting because I've always lived there."
"Does it seem strange to you in America?"
"Not here," William said.
"The Chinese love flowers, I suppose," Mrs. Cameron said. William considered. "I didn't see very much of the Chinese, really. I grew up in a compound, and my mother was always afraid I'd catch something. But we did have chrysanthemums, and I remember the bowls of lilies our gardener used to bring before Chinese New Year."
He felt he was not doing very well and his anxious instinct urged him to frankness. "I suppose I should know a great deal about the Chinese, but one doesn't think much when one is growing up. The common people are rather filthy, I'm afraid, and the others are fed up with Westerners just now and didn't want to mix with us. There was even real danger if they did—the old Empress didn't favor it."
"A wicked old woman, from all I hear," Mr. Cameron said suddenly. "Trying to stop normal trade!"
"I do hope your parents are safe," Mrs. Cameron sighed. "What we've read in the newspapers has been dreadful. So shocking! As if what we were doing wasn't for their good!"
He was diverted from answer by hearing a clear young voice. "Oh, here you all are!"
A very pretty yellow-haired girl was coming toward them. She was all in white and she had a tennis racket with low-heeled white shoes tied to it. At the viny entrance she paused, the sunshine catching in her hair and making a nimbus about her pleasant rosy face. She looked like Jeremy and she had the same sweet mouth but the lips were full and red.
"Hello," she said in a soft voice.
Jeremy said, "Come in. This is William Lane. William, this is my sister Candy."
She nodded. "Do you play tennis?"
"I do, but I haven't my things."
"Come along, we have plenty."
"Candace dear—perhaps he doesn't want to—" Mrs. Cameron began.
"I'd like to, very much," William said.
He rose. Tennis he played very well indeed. He had chosen it instead of cricket and his only chance for pleasurable revenge had been when a cricketer opposed him upon the immaculate coolie-kept courts at Chefoo.
"Come back again," Jeremy said, his smile wistful.
"Do come back," Mrs. Cameron said warmly.
Mr. Cameron was silent. Leaning against the back of the cushioned wicker chair, he had closed his eyes and fallen asleep.
Beside the girl William held himself straight and kept silent. His instinct for dignity told him that she was used to much talk and deference. To his thinking American women were pampered and deferred to far too much. Even the maids at his grandfather's house were sickening to him in their independence. In China an amah was not a woman—merely a servant.
"I hope you don't mind cement courts," Candace said, as she gave him tennis shoes and a racket from a closet in the great hall. "Ours are frightfully old-fashioned, but my father won't change them. I like grass but of course grass isn't too easy at the beach. Though my father could, if he would—only he won't."
"I shan't mind," William said.
"How old are you?" Candace inquired, staring at his handsome profile.
"Seventeen."
"I'm sixteen."
"Are you going to college?"
"No, of course not—Miss Darrow's-on-the-Hudson, for a year, and then I'm to come out."
He had the vaguest notions of what it meant for a girl to come out, but now that he knew he was a year older than she, he felt more at ease. "Shall you come out in New York?"
"Of course—where else?"
"I thought perhaps in London."
"No, my father is frightfully American. I might be presented at the Court of St. James's later. The man who was once my father's partner is the American Ambassador there."
"I knew a lot of English people in China."
"Really?"
"I didn't like them. Very conceited, as though they owned the country. Their merchant ships ply all the inner waters and their men-of-war, too. If it hadn't been for us, they'd have made a colony out of the whole of China."
"Really? But don't they do that sort of thing very well?"
"They've no right to hog everything," William said stiffly.
Candace mused upon this. "I suppose not, though I haven't thought about such things. We've always been in England a lot—Mother and Jeremy and I. My father has no time."
"What does your father do?"
"He's in the Stores—and in Wall Street—and that means he's in everything."
They were at the courts now, two smooth wire-enclosed rectangles surrounded by lawns set with chairs and big umbrellas. No one else was about.
"It's too hot to play, and that's why no one is here," Candace said carelessly. "Two hours from now the place will be jammed."
"I mustn't stay," William said quickly.
"Why not?"
"In bathing things and a jacket?"
"It doesn't matter. We'll all bathe before sundown. There's a dance tonight. Do you like to dance?"
"Yes."
He danced badly, never having had lessons, and he made up his mind to speak to his mother about it. Before he went to Harvard he must have lessons.
They were playing now, and he found within a few minutes that he could beat her, not easily, but surely. She played well, for a girl, her white figure flying about the court opposite him, though she served carelessly.
"I don't see how you hit the ball standing still," she called to him at last with some irritation.
"I don't actually stand still," he called back. "I was taught not to run about; the sun was hot in China."
"It's hot here, too."
She flung down her racket at the end of an hour and came to the net to shake hands with him formally.
"There, that's enough for one day. You do play well. I have to go now and change. People are coming, and I'm dripping. You can leave the shoes and racket here."
She did not again suggest his staying to tea and he withdrew, deeply wounded. "Good-by, then, I'd better be getting along."
She waved her racket at him and smiled and left him to find his way alone. He ought not to have played so well, he supposed. For his own sake he should have allowed her to win. American girls were spoiled. Then he lifted his head. He would always play his best and he would yield to no one.
He went across the wide lawn and down the steps to the beach and turned homeward, his jacket over his arm and the sun beating down on his shoulders. The water was rippling over the sand and he walked in the waves curling in tendrils from the sea. At his grandfather's house he went in, carrying plenty of wet sand upon his feet. Millie, the lesser of the maids, came out with a broom.
"Oh, look at those feet," she exclaimed. "Just after I've swept, too! I declare, Willum—"
They were alone and he turned on her with the fury of a young tiger. "What do you mean by calling me Willum?" he hissed at her through white set teeth. "How dare you? You have no more manners than a—a savage!"
He left her instantly and did not turn to see her shocked face. Halfway upstairs, he heard a door slam.
After a little while his mother tapped at the door of his room.
"Come in," he said listlessly. He had bathed and put on fresh clothes and had sat down at his desk to write, toying with some verses.
"William," his mother began. "What did you say to Millie?"
He whirled on his chair. "What did she say to me, you had better ask. She called me Willum!"
"Hush, William. Don't be so angry. She comes from Maine and everybody—"
"I don't care where she comes from. She can call me Master William."
"She wouldn't call anybody master."
"Then she needn't speak to me."
"William, it's not easy living with all of us in this house. The maids aren't used to children."
"I am not a child."
"I know, but—"
"Mother, I simply do not intend to be insulted by servants."
"I know, dear, but they aren't our servants."
"Any servants."
His mother sat down in a rocking chair. "In some ways it is really easier to live in Peking, I admit. But we are Americans, William, and you must get used to it."
"I shan't allow myself to get used to that sort of thing."
He was aware of her admiration behind her distress. She was proud of his spirit, proud of his looks, proud of his pride. She rocked helplessly for a few minutes and then got up. "I'll give Millie something, this once."
She went out of the room, and he was alone again. He was not writing verses to Candace. He was not attracted by her. He was writing something about a man's soul finding its own country, but he could not satisfy his fastidious taste in words. His poetry was not good enough and he tore the sheets into bits and threw them into the wastepaper basket.
The farm in Pennsylvania was as remote from the rest of the world as though it were an island in the sea. Nothing else existed. No one came near and the inhabitants never went away. The five children, of whom Clem was now one, made a human group, solid because they were utterly alone and at the mercy of two grown people, a man and a woman, who were cruel.
To Clem the memory of his dead parents and the two little girls who had been his sisters grew vague and distant. They had been killed by men he had never seen, a violence as inexplicable as a typhoon out of the southern seas. But here in this enchanting landscape the cruelty was mean and constant. There was no escape from it.
The man and woman, as he called them always in his thinking, his tongue refusing to call them Pop and Mom, were animal in their cruelty, snarling at the helpless children, striking them in fatigue or disappointment. Thus when the spotted cow had a bull calf instead of a heifer, Pop Berger pushed Tim.
"Git out of my way!" he had bellowed.
Tim stepped back to escape the man's upraised fist but it struck him and he fell against the corner of the stone wall of the barn.
Clem saw all and said nothing. His watching eyes, his silence, the strangeness of his unexplained presence, kept the Bergers shy of him. They had not yet beaten him. His swiftness at work, his intelligence, superior to any in the house, gave them no excuse, and while with the other children they needed no excuse, with him they still searched for one. He rose at early dawn and went out and washed himself in the brook behind the house, "the run" it was called, and then he went to the milking. He could not drink milk, however hungry he was, and he was always hungry. The warm sweetish animal smell of milk sickened his stomach, the thick coarseness of the cows' teats in his hands disgusted him. Yet he treasured the stuff and learned to get the last drop from a cow, enough so that he dared to give the children a secret cupful apiece. The cup he hid behind a loosened stone in the barn wall. The children learned to come to him one by one, as soon as he began the milking, before Pop got out of bed. The cup of fresh milk stayed their lean stomachs until the breakfast of cornmeal mush. And the day went on in harshest labor, the thoughts of all of them dwelling always upon food.
Clem, always until now pallid and small, suddenly began to grow. His bones increased in size and he was obsessed with hunger. He would not steal from these strangers into whose midst he had fallen and therefore he starved. He imagined food, heaping bowls of rice and browned fish and green cabbage. In China God had given them food, and he had eaten. His hunger all but drove him back to praying to God again as his father had done. But his father had gone out to other people who had answered the prayers for God. Here there were no such people that he knew. It did not occur to him that God would work through such people as the Bergers.
He was stupefied by these human beings among whom he found himself. Who were they? Where were those to whom they were kin? No one came near the farmhouse, neither friends nor relatives. In China all persons had relatives, a clan to which they belonged. These, the evil man and woman, the desolate children, belonged nowhere. Clem had no communication with them, for they said nothing to him or to each other except the few necessary words of work and food. The silence in the house was that of beasts. Nothing softened the hopeless harshness of the days, there was no change except the change of day and night.
Yet as one glorious day followed another Clem felt there must be escape. This was a net into which he had fallen, a snare he had not suspected. He must simply leave it. Whatever lay outside could not be worse than this. The desolate children seemed never to dream of escape, but they had no dreams of any kind, he discovered. Their hope went no further than to steal something to eat when Mom Berger was not looking, to stop working when Pop's back was turned. They were ignorant, and he soon found, depraved as well. When he first discovered this depravity he was sick. His own parents had been people of pure heart, and from them he had inherited a love of cleanness. Mr. Fong had been clean in speech and act. Though Clem had seen a simple naturalness in the behavior of men in the countryside about Peking, it had been clean. Birth was clean, and the life of man and woman together was decent. There was nothing about it which he did not know as he knew life itself. But what he found here was indecency, the furtive fumbling of boys and girls who were animals. Pop grinned when he saw it, but Mom Berger yelled, "Cut that out, now!"
She was a thick-set woman, her neck as wide as her head, her waist as wide as her shoulders, her ankles as big as her calves. She wore a shapeless dress like a huge pillowcase without a belt. Except sometimes when she went to town with Pop, she was barefoot. Clem had never seen the feet of a woman before. Chinese women always wore shoes on their little bound feet and his mother had worn stockings and shoes. In China it was a disgrace for a woman to show her feet. And so it should be, Clem told himself, avoiding the sight of those fleshy pads upon which Ma Berger moved.
For the first few days he had lived in complete silence toward the children. There was no time for talk, had he been so inclined. Pop took him upstairs into a filthy room where there were a wide bed, a broken chair, some hooks upon the plastered wall. On the hooks hung a few ragged garments. Pop scratched his head as he stared about the room. "Reckon that bed won't hold four of you," he had rumbled. "You'll have to have a shakedown, I guess. I'll tell Mom."
He went down the narrow circular stairs and left Clem alone. This was his return. He walked to one of the windows, deep set in the heavy stone wall, and gazed out of it to see the countryside beautiful. Long low hills rolled away toward the horizon and fields lay richly between. He had never seen such trees, but then he had seen very few trees. The northern Chinese landscape was bare of them, except for a few willows and a date tree or two at a village. This was a country fit for dreams, but he knew that whatever had been the dreams once held in this house, there could be no more. He tried to imagine his father, a boy perhaps in this very room, hearing the voice of God bid him go to a far country. Oh, if his father had not listened to God, he, Clem, might have been born here, too, and this would have been his home. Now it could never be that.
He heard heavy panting on the stairs, and Mom Berger's loud voice cried at him.
"Come here, you, boy, and help me with these yere quilts!"
He went to the stair and saw her red face staring at him over an armful of filthy bedding.
"Am I to sleep on this?" he demanded.
"You jes' bet you are," she retorted. "Lay 'em to suit yourself."
She threw the quilts down and turned and went downstairs again, and he picked them up and folded them neatly, trying to find the cleanest side for sleep. He would have to sleep in his clothes until he could get away, for of course he would go within the next day or two, as soon as he found the name of a town or of a decent farm.
But he did not go. The misery of the five children held him. He had no family left, and in a strange reasonless sort of way he felt these pull upon him. He would go, but only when he had given them help, had found their families, or had found some good man to whom he could complain of their plight. His wandering and his loneliness made him reliant upon himself. He was not afraid, but if he left them as they were, he would keep remembering them.
In silence on that first day he had made his pallet and put his locked suitcase at the head of it. Into the suitcase he folded his good clothes, and put on instead the ragged blue overalls. Then he went downstairs.
The big kitchen was also the living room. Mom Berger was cooking something in a heavy iron pot, stirring it with a long iron spoon.
"Pop says you're to go out to that field yonder," she told him, and nodded her head to the door. "They're cuttin' hay."
He nodded and walked out to a field where he saw them all working in the distance. The sun was hot but not as hot as he had known it in Peking, and so it seemed only pleasant. The smell of the grass and the trees was in his nostrils, a rich green fragrance of the earth. What was hay? He had never seen it. When he got near he saw it was only grass such as the Chinese cut on hillsides for fuel.
He waited a moment until Pop Berger saw him. "Hey you, get to work there! Help Tim on that row!" Clem went to the sandy-haired boy. "You'll have to show me. I've never cut hay."
"Where'd you come from?" Tim retorted, without wanting to know. "You kin pitch."
Clem did not answer. He watched while Tim's rough claws grasped a huge fork and pitched hay upon a wagon pulled by two huge gray horses. It looked easy but it was hard. Nevertheless he had continued to pitch doggedly until the sun had set.
From that day on his life had proceeded. The work changed from one crop to another, but the hours were the same, from dawn to dark for them all. The girls worked in the house with the woman.
He became aware, however, of a certain day, dim in the minds of the children when he first came, which became more probable as the month dragged on. They expected a visit from what they called the Aid. What this Aid was Clem could not find out. He put questions to Tim, the eldest and most articulate of the boys. To the girls he did not speak at all. He felt a terror in them so deep, a timidity so rooted, that he thought they would run if he called their names, Mamie or Jen.
"Aid?" Tim had repeated stupidly. They were raking manure out of the barn. "Aid? It's just—Aid. It's a woman."
"Why is she called Aid?" Tim considered this for a full minute.
"I dunno."
"Does she help you?"
"Nope—never did. Talks to Pop and Mom."
"What does she say?"
"Axes things."
"What things?"
"Different—like does we work good, does the boys and girls sleep in one room—like that." Tim grinned. "They're scared of her."
"Why don't you tell her?"
"Tell her what?"
"That you don't get enough to eat—that they hit you."
Tim's wide pale mouth was always open. "We're only Aid children."
"What is that?" Clem began all over again.
"I tole you," Tim said patiently. "We ain't got no folks."
"You mean you don't know where your parents are?"
Tim shook his head.
"Are they dead?" Clem demanded.
"Bump never had none," Tim offered.
Bump was the second boy, now bringing the wheelbarrow to fill with manure.
"Bump, haven't you any kin?" Clem asked.
"What's 'at?" Bump asked.
"Uncles and aunts and cousins."
"I got nawthin'," Bump said. He was spading up the manure that Clem had put into piles.
"Doesn't anybody come and see you?"
"Nobody knows we're here lessen the Aid tells," Bump said.
"Then why do you all want this Aid woman to come?"
"Cause Mom gets a big dinner," Tim said with a terrible eagerness. "She don't say nothin' neither when we eat. Don't dast to."
Clem threw down the fork be was using. "If you'd tell the Aid woman they're mean to you, maybe she'd put you somewhere else."
There was silence to this, then Tim spoke. "We're used to it here. We been here all of us together. Maybe Bump would get somewhere way off, and we're used to Mamie and Jen, too. They're scared to go off by theirselves. I promised we wouldn't never say nawthin'."
Clem perceived in this a fearful pathos. These homeless and orphaned children had made a sort of family of their own. Within the cruel shell of circumstance they had assumed toward one another the rude simplicities of relationship. Tim, because he was the eldest, was a sort of father, and the others depended on him. Mamie, the older girl, so lifeless, so still, was nevertheless a sort of mother. As the days went on he perceived that this was the shape they made for themselves, even in depravity. The man and woman were outside their life, as unpredictable as evil gods. They suffered under them, they were silent, and they were able to do this because they had within themselves something that stood for father and mother, for brother and sister. Because of the family they had made for themselves out of their own necessity, they preferred anything to separation.
Clem asked no more questions, and judgment died from his heart. Something almost like love began to grow in him toward these children. He wondered how he could join them and whether they would accept him. He had held aloof because they were filthy and unwashed, because their scalps were covered with scales, because they had boils continually. He had thought of leaving them as soon as he could. But as weeks went on he knew he could not leave them—not yet. They were all he had.
He pondered upon their solitude. In China, whence he had come, all people being set in their natural families, there were no solitary children, except perhaps in a time of famine or war when anyone might be killed. If parents died of some catastrophe together, there were always uncles and aunts, and if these died, then there were first cousins and if these died there were second and third and tenth and twentieth cousins, all those of the same surname, and children were treasured and kept within the circle of the surname. But these children had no surname. He had inquired of Tim, and Tim had said after his usual moment of thought, "It's writ down in the Aid book."
"But what is it?" Clem had insisted.
"I—disremember," Tim had said at last.
As the day when the Aid was to come drew near Mom Berger became more irritable. "I gotta get this house cleaned," she said one morning in the kitchen, when the children stood eating their bread and drinking weak, unsweetened coffee. "The Aid'll be here come Tuesday week. You girls better git started upstairs this very day. Everything's gotta be washed—clothes and all."
From that day until the Tuesday which was dreaded and anticipated there was no peace in the house or in the barn. Even the barn had to be cleaned.
"That Aid woman," Pop snarled, "she ain't satisfacted to stay in the house. No, she's liable to come snoopin' out here among the cows. I'm goin' to tell her that's why I need more help, Clem. I'm goin' to tell her if I have to clean this yere barn I gotta have another boy. That's what I'm goin' to tell her."
"How often does she come?" Clem asked with purposeful mildness.
"The law claims once in three months. She don't get round that often though—maybe oncet, twicet a year. Always tells us before she comes. I git a postcard a month or so ahead."
On the day before, they took baths. The woman heated kettles of hot water and in the woodshed the boys washed one after the other in a tin tub with soft homemade soap.
"You ain't hardly dirty, Clem," Tim said with some admiration, staring at Clem's clean body.
"I wash in the run," he replied.
"What'll you do come winter?"
"Break ice—if I'm still here."
They all glanced at the door at these words. Tim whispered, his eyes still on the latch, "You wouldn't go an' leave us, would you?"
Bump paused in the scrubbing of his piteous ribs. "Clem, don't you go and leave me!"
"I don't belong here," Clem said simply.
"You belong to us," Tim said.
"Do I? How?" Clem felt a starting warmth in the inner desolation of silence.
Tim had one of his long pauses, shivering and naked. His shoulder bones were cavernous, and between his sharp hip bones his belly was a cavity. Pale hairs of adolescence sprouted upon his chest and pelvis. "You ain't got nobody, neither."
"That's so," Clem said.
Tim made a huge effort of imagination. "Know what?"
"What?"
"Sposin' we lived by ourselves on this yere farm—You could be the boss, say, like you was our father."
The woman's fists pounded on the door. "Git out o' that, you fellers!" she yelled. "The girls gotta wash."
They hurried, all except Clem. He took the pail of cold water and doused himself clean of the water in which the others had bathed.
"Maybe I'll stay," he said to himself. "Maybe I'd better."
In the night, in a bed cleaner than he had slept in since he came, he began to think about his strange family. Food was what they needed. He recalled the boys' bodies as he had seen them today naked, their ribs like barrel staves, their spines as stark as ropes, their hollow necks and lean legs. Food was the most precious thing in the world. Without it people could not be human. They could not think or feel or grow, or if they grew, they grew like sick things, impelled not by health. Everybody ought to have food. Food ought to be free, so that if anybody was hungry, he could simply walk somewhere not very far and get it. Food should be as free as air.
He began to dream about himself grown and a man, rich and independent. When he got rich he would see that everybody would have food. "I won't depend on God, like Papa did," he thought.
The Aid came just before noon. They had all been waiting for her through an endless morning. The barn was clean, the house was clean. Whatever had not been washed was hidden away until she was gone. The girls were in almost new dresses which Clem had not seen them wear before. They had on shoes and stockings for the first time. Pop was in his good clothes, but he had taken off his coat, lest it seem that he did not work.
"Put it on when you sit down to table, though," Mom ordered.
"You don't have to teach me no manners," Pop said.
She sat all the time because she too had on shoes and stockings and her feet hurt. The girls had to bring her anything she wanted. She had on a gray cotton dress that was almost clean. Clem had put on his good clothes that the sailors had bought him. They sat about the kitchen smelling the food on the stove, their stomachs aching with hunger.
"Here she comes," Pop cried suddenly.
Through the open door they all stared. Clem saw a small thin woman in a black dress come down from a buggy, which she drove herself. She tied the horse to the gate and came up the walk carrying a worn black leather bag. Pop hastened to her and Mom got up on her sore feet.
"Well, well!" he shouted. "We didn't really know when to expect you and we just went about our business. Now we're just goin' to set down to eat dinner. I'd ha' killed a chicken if I'd been shore you was comin'. As it is, we only got pork and greens and potatoes. New potatoes though, I will say, and scullions."
"That sounds good," the woman said. She had a dry voice, not unkind, and she stood in the doorway and looked at them all. "Well, how's everybody?"
"Pretty good," Mom Berger said. "The children look a little peaky on account of a summer cold. They like to play barefoot in the run, and I hate to tell 'em not to. You know how children are. Come and set down while I dish up."
"It's been a hot summer," the Aid woman sighed. She sat down and took off her rusty black hat. "Well, I see they're growing."
"That's another reason for their peakiness," Ma Berger said. "I keep tryin' to feed 'em up, but they don't fatten no matter how I do. Their appetites is good, too. You'll see how they eat. But I don't begrudge 'em."
"I'm sure you don't," the Aid said absently. She was searching through some papers in her bag. "I guess I'd better begin checking now. I have to get on right after dinner. The territory is more'n I can manage, really. Let's see, you have five children. Why—the book says four!"
Pop began hastily. "This yere Clem is a new boy. Just turned up one day and I kep' him, because he hadn't nowhere to go. I was goin' to tell you."
"Boy, where do you come from?" The Aid was suddenly stern.
"From out West," Clem said. He was standing, as all the children were. He had told none of them that he came from China. They would know nothing about China and he could not begin to tell them.
"You can't just come here like that," the Aid declared. Indignation sparkled in her little black eyes. "You should have stayed where you was. The state can't take charity cases from other states. It's going to make a lot of trouble for me."
"I thought my grandfather was still alive," Clem said. "He used to live here."
"Old Charley Miller," Pop explained. "Him as hanged himself when he got to be sheriff."
The Aid stared at Clem. "You're his grandson?" she demanded.
"Yes."
"Say, 'yes, ma'am' to me," she said sharply. "Where's your proof?"
"I haven't any," Clem said.
"He's Charley's grandson all right," Pop said quickly. "He's got the same kind of a face and his eyes is just the same color and all. I'll guarantee him."
"I don't know what to do," the Aid sighed. She had a thin washed-away face and a small wrinkled mouth. Behind her spectacles her eyes were dead when the small flare of anger was gone. There was no wedding ring on her hand. She had never been married and she was tired of other people's children.
"Why don't you just mark down five?" Pop coaxed her. "It'll save you trouble."
"I could do that," she mused. "One of the children in the last house died. I could just transfer the money from that one to this one."
" 'Twould save you trouble," he said again.
So it was done. Clem took the place of the dead boy.
They all sat down to dinner. On the table a platter of pork and greens was surrounded by boiled potatoes and by dishes of sweet and sour pickles. There were apple pies to be eaten, too, and the children had milk from a pitcher, all except Clem who took water.
"You must drink milk, boy," the Aid said. "That's why it's so good for children to live on farms."
"I don't like milk," Clem said.
"Say ma'am," the Aid reminded him. "And it don't matter what you like. You make him drink it, Mrs. Berger."
"I certainly will," Mom promised.
There was no time for any talk. At the table there was only time for eating. The children ate desperately until they could eat no more.
"I see what you mean," the Aid said. "At this age they just can't be filled up."
"I do my best," Mom said.
When the meal was over the Aid rose and put on her hat. "Everybody looks nice, Mrs. Berger," she said. "I'm always glad to give you a recommend. I don't believe I'll bother to go upstairs. I can go through the barn on my way out, Mr. Berger—though you always—the children are real lucky. Better off than in their own homes. What's that?"
Some noises coming from Tim stopped her at the door. He looked helplessly at Clem.
"He wants to know what his last name is," Clem said for him.
The Aid's empty eyes suddenly lit, and she stepped toward him. "Will you say ma'am when you speak to me?"
Clem did not answer, and Pop broke in quickly. "I'll shore learn him before you git here next time."
"Well, I hope so," the Aid replied with indignation. She forgot Clem's question and went on briskly toward the barn.
The conscience in Clem's bosom was as concrete as a jewel and as pure. He felt its weight there day and night. It had grown with his growth and now had facets which were strange to him. Thus while his father's too simple faith had been its beginning, it had taken on accretion not of faith but of doubt, mingled with suffering, pity and love, first for his father and mother and sisters when they were hungry, and now after their death, pity for hunger wherever he found it. He, too, was hungry here on his dead grandfather's farm, but his hunger only hastened the growth of his conscience and made it more weighty. If he were hungry, what of these others, these children? For he perceived that Tim, though older than himself and inches taller, was and would always be only a child. Others must feed him as long as he lived and he would always be at the mercy of any man with a measurable brain. Mamie, too, was meek and mild, and Jen was an aspen of a child, trembling always with terror remembered and terror about to loom again. Bump was stolid and silent and he followed Clem like a dog. At night with dumb persistence he insisted upon sleeping beside Clem's pallet.
How could anyone know what was in any of them? They were obsessed with hunger. They dared not steal bread from the breadbox or leftover bits in the cupboard, but they did steal from the dog. Mom Berger scraped the bottoms of pots and the cracked bones and heaped them upon an old tin pie plate outside the kitchen door. There Clem, coming suddenly from the barn one day, found the four children, as he thought of them, waiting for the mongrel dog to eat its fill. They dared not snatch from the beast lest it growl and Mom Berger hear. But they were using wile. Bump, for whom the dog had a fondness, was coaxing him, though in silence, from his plate. When the dog looked up to wag his tail, Tim and Mamie snatched handfuls of the refuse. When they saw Clem's eyes fixed upon them they shrank back as though he might have been Pop Berger. This caused the conscience in him to burn with the scintillating flame he knew so well, a fire at once cold and consuming. He did not love these ragged children, he was repelled by their filth and their ignorance. The language they spoke was, it seemed to him, the grunting communication of beasts. Nevertheless, they did not deserve to starve.
Seeing them with the dog's food clutched in their hands, staring at him in fear, he turned and went back to the barn. There he sat down again to his task of husking the last of the corn. Pop Berger lay asleep upon the haymow. Thinking of the work to come, Pop had yawned heavily after the midday meal. "Reckon you kin finish the corn," he had said and had thrown himself on the hay. Clem had gone to the house after an hour to get a drink. The pork and cabbage they had eaten had been very salty, but he had forgotten his thirst. His mind burned with the determination to escape.
"Of the thirty-six ways of escape," Mr. Fong had once told Clem, "the best is to run away." It was an ancient Chinese saying, and it came back to Clem's mind now. He was Chinese in more ways than he knew. The early wisdom of people who had long learned what was essential had seeped into him from the days when he first began to know that he was alive. Courageous though he was, and with a tough natural courage, he knew that the first wisdom of a wise man is to stay alive. Only the dead must be silent, only the dead are helpless.
His father's conscience, too, was his inheritance—yes, and his grandfather's also. There were times when Clem went alone into the barn to stand and gaze at the beam that Pop Berger had pointed out to him.
"That there's the one he hang himself on."
"Why did he do it?" Clem had once asked.
"Softhearted," Pop had answered in accusation. He had added details later. "The ole feller took a new rope he'd bought a couple days before to tie up a calf with. He hed some kinda crazy notion if good men could git into the gov'ment they could straighten things out. He didn't want the sheriff's job, though—wanted to give it up right away, but the party boss told him he had to keep it for the sake of the party, like. First thing ole man had to do was close the mortgage on that there farm, yander." Pop Berger's thick forefinger pointed next door. "He wuz softhearted, like I said. He said he'd ruther die. Nobody took him serious, like. Doggone if the old man didn't mean it. Next day somebuddy found him hangin' dead."
Clem never answered. Pop Berger could not comprehend the only answer that he could have made. Of course his grandfather would rather die. It had been his way of escape from an intolerable duty. He thought a great deal about his grandfather, searching out about the barn, the house, the farm, the small signs of a conscientious, careful, good old man. The cow stalls, for example, were larger than most. There was room for a cow to lie full length in a stall. Pop fretted at the waste of room. There was a trough outside big enough for all the horses to drink at once. The water ran into it through an iron pipe from the well so that it was always fresh. In the house the step between kitchen and living room had been taken away and made into a gentle slope. His grandmother had gone blind in her old age, Pop told him.
Heir of the conscience of his fathers, Clem could not be hardened by the miseries of his present life. Instead he felt a constant soreness in his breast, an ache of remorse for sins of which he was not guilty. This discomfort he now tried to heal superficially by helping the children to get more food to eat. It was not easy, and after some struggle within himself he decided, remembering the dog's dish, upon simple theft.
After the Aid woman had gone, not to return he knew for many months, even perhaps a year, he was angered to see how instantly the man and woman fell back into their careless cruelty. The meat was put away and the milk was watered. Yet he dared not complain. He, too, was now in the power of these two, and if they saw his courage they could prevent the escape he planned. His Chinese childhood had taught him never to be reckless even in anger, for anger is no weapon. Anger can give energy to the mind but only if it is harnessed and held in control. Therefore he locked his anger behind his teeth and, having decided upon theft, he used a deep cunning. He stole food so cleverly that the man thought the woman had eaten some leftover, and she thought the man had taken it. Neither believed the other and they snarled at each other, while the blank faces of the children told nothing. It comforted Clem to know meanwhile that inside Tim's slack stomach there was a piece of boiled beef or a slice of home-cured ham, and that Jen had a lump of butter on a piece of bread. He was just in giving out his booty, saving nothing for himself. At the table he had courage enough to eat more than the younger ones, and since he worked well and was seemingly obedient, Pop gave him more than he might have given. Milk Clem stole without heed. In the pasture, hidden behind the brow of a hill, the children learned to come to him between meals, and he took a tin can from under a rock and milked a can full from one cow and another, never too much from one. Each child had a can full at least twice a day of the pure milk, warm from the cow's body. When they were strong enough, Clem told himself, they would run away together. It must be before the winter fell again.
When autumn came, he had supposed they would all go to school. Tim had told him that the law said they had to go to free school and even Pop had to obey the law. That would make it easy, Clem planned, for them to run away. They could be a day upon their way before night came and before Pop, finding that they did not come home, could report their escape.
But he had not counted on Pop's cleverness. Pop said one day in the barn, "They ain't no call for you to go to school, Clem. You're too big."
Clem looked up from the hay chopper. "I want to go to school."
Pop chuckled. "Yeah? Ain't nobody knows you're even here." Clem stared in silence, waiting. A frightful comprehension was stealing into his brain.
"See?" Pop said. He was picking his teeth after the noon meal and he leaned against a cow stall. "You jest come here, didn't you? You don't belong nowheres, as I see it. School board don't even know you're alive."
"I could tell them," Clem said in a tight voice.
"Just you try," Pop said.
Clem did not answer. He went on chopping the hay while his mind worked fast. This was the final reason why he had to go at once. He would wait no longer. To grow up in ignorance and loneliness was more than he could do. He had dreamed vaguely of finding people to help him, school teachers whom he could tell of the misery of the children. Perhaps Pop had thought of that, too.
"We dassent tell the teacher anything," Mamie had said once. "Pop says he'd kill us if we told, and he would, too."
"Yeah, he would," Tim agreed.
"Well, ain't you goin' to say nothin'?" Pop inquired now.
"No," Clem said. "I've never been to school anyway." He kept his face averted and Pop saw only his bent, subdued body working at the hay chopper, and he sauntered away.
But Clem, whose patience was the long endurance of those who have never known better, had suddenly reached the moment of decision. He would run away on Saturday when the man and woman went to the town to do their marketing. He must leave this desecrated house of his forefathers and he must take the children with him, for his own peace, for without him they would starve. Sooner or later they would sicken one by one, and then they would die because they were already half starved, their frail bodies struggling and scarcely able to live even when they were not ill. Where he would go he did not know, nor what he would do with them. Even though he found work, how could he earn enough to feed them?
He looked back on the days in Peking as sweetness he had not known enough to taste while it was in his mouth. He remembered the pleasantness of Mr. Fong's shop, the coziness of the inner rooms where he had sat at the square table teaching Yusan. It had been a home rich in kindness and his eyelids smarted now when he thought of it. Of his own parents he would not think. He remembered them no more as they had been when they were living but only as he had seen them dead, and this memory he could not endure and he put it from him so far that it had become blankness. He could not remember even their faces. Mr. Fong's he saw clearly, and Mrs. Fong's face he saw always wreathed in smiles as it was when she brought in the cakes and meat rolls. He dreamed of that food.
Slowly, while his conscience burned, Clem made his plans. On Saturday, early, as soon as the man and woman had left the house, he would tell the children. He did not dare to prepare them earlier for they were too childish to be trusted. He would help them to gather their clothes together and tie them in bundles. They would take whatever food was left in the house.
Saturday morning dawned clear and cool. Hateful as his life was to him, Clem had fallen in love with the land. He woke early as usual, even before the heavy footsteps of the man shook the narrow stairs, and he put on his clothes and let himself out from the window upon the roof of a shed below and thence he dropped to the ground. At the stream he washed himself in a small pool below a shallow falls. The stream bed was of rock, slanted in layers so precise that when the falls rose after a rain, slabs came off like great Chinese tiles. He had taken a score or so of them and had laid them neatly at the bottom of the pool and when the sun shone through the water, as it did this morning, the stones shone in hues of wet amber and chestnut and gold.
The stream was out of sight of the house, hidden by a spinney of young sycamore trees, the children of a mighty old sycamore whose roots drove through the hillside to the sources of water. Behind this wall of tender green, Clem stripped himself and plunged into the water, this morning almost winter cold. Above him the hills rose gently, the woods green but flecked with the occasional gold of autumn. The sky was beautiful, a softer blue than Chinese skies and more often various with white and moving clouds.
Yet where, Clem often asked himself, were the people upon this land, and how could it be that a house full of children at the mercy of a man and woman, ignorant and brutish, remained unknown and unsought? In China it would not have been possible for an old man's house to have been unvisited, or to have been sold after his death in so summary a fashion. He had asked Pop Berger once who had sold the house and had been told that it went for unpaid taxes. But why were the taxes not paid by some kinsman? How had it come to pass that his old grandfather had been so solitary, even though his son had gone so far? And why, and why, and this was the supreme question, never to be answered, had his father left his home and the aging man to go across the sea to a country he had never seen, where the people spoke a tongue strange to him, and there try to tell of a god unwanted and unknown? None of these questions could be answered. What Pop had said was true. There was no one who knew of his existence.
Clem stepped out of the small cold pool and dried himself by stripping the water from his body with his hands and then by waving his arms and jumping up and down. In spite of poor food he was healthy and his blood rushed to his skin with heat and soon he put on his clothes and climbed the hill to the house. Pop Berger was already out at the barn, and Clem went in; without greeting he took a small stool and a pail and began to milk a brindled cow.
At first, accustomed by the Chinese to greeting anyone he met, he had tried to greet the man and the woman and the children when he first saw them in the morning. Then he perceived that this only surprised them and that it roused their contempt because they thought he was acting with some sort of pretense. He learned to keep his peace and to proceed in silence to work for food.
This morning there was none of the usual dawdling and shouting. Pop Berger harnessed the wagon early and began piling into it the few bags of grain he wanted to sell, and some baskets of apples. He left all the milking to Clem, and stamped away into the kitchen to eat and to dress himself. There the woman, too, made haste, eating and dressing, and within the hour the pair were ready to be gone, leaving the dishes and the house to the two girls.
"You, Clem!" Pop Berger shouted from the wagon seat. "You can git the manure cleaned out today. Don't forget the chickens. Tim can do whatever you tell him. I told him a'ready to lissen to what you sayed."
"And I've left the food you're to eat in the pantry, and that's all anybody is to have. Don't open no jars or nothin'!" Mom shouted.
Clem had come out of the barn and he nodded, standing very straight, his arms folded as he watched them drive off. He wondered that he did not hate them and yet he did not. They were what they were through no fault of their own, their ignorance was bestial but innocent and their cruelty was the fruit of ignorance. He had seen degenerate cruelty sometimes in the streets of Peking. There the people knew, there they had been taught what humanity was, and when they violated what they knew, the evil was immense. But these two, this man and this woman, had never been taught anything. They functioned as crudely as animals. Where had they come from, he often wondered, and were the others all like them? There were no neighbors near, and he had no one with whom to compare them.
He finished milking the cows and carried the milk into the springhouse, where it would be cool. Then he went into the kitchen to find food. There, as usual when the man and woman were gone, nothing was being done. The bare table was littered with dirty dishes. Mamie and Jen sat beside it, silent and motionless in dreadful weariness. Tim slumped in Pop Berger's ragged easy chair. Bump was still eating, walking softly about the table, picking crumbs.
"Got breakfast for me, Mamie?" Clem asked.
She nodded toward the stove and he opened the oven door, took out a bowl of hominy, and sat down at the end of the table.
He looked at them, one and the other. Tim's lack luster eyes, agate brown, held less expression than a dog's and his mouth, always open, showed a strange big tongue bulging against his teeth. His body, long and thin, a collection of ill-assorted bones, folded itself into ungainly shapes. Mamie was small, a colorless creature not to be remembered for anything. Jen might die. The springs of life were already dead in her. She did not grow.
"Come here," he said to Bump. "I don't want all this. Finish it, if you like."
He held out his bowl and Bump snatched it, went behind the stove on the woodpile, and sat down in his hiding place. Often the woman lifted the poker and drove him out of it, but today he could enjoy it.
"Listen to me, all of you," Clem said, leaning on the table.
They turned their faces toward him.
"How would you like to go away from here?" He spoke clearly and definitely, for he had learned that only so did they heed him. Accustomed to the loud voices of the man and woman they seemed to hear nothing else.
"Where?" Tim asked after a pause.
"I don't know—run away, find something better."
"Where would we sleep?" Mamie asked.
"We'd take a blanket apiece, sleep by a haystack somewhere until we got ourselves a house, or some rooms."
"What would we eat?" she asked again.
"I'd work and get money and buy something. Tim could work, too. Maybe you could find a job helping in a house."
He had expected some sort of excitement, even a little joy, but there was neither. They continued to stare at him, their eyes still dull. Jen said nothing, as though she had not heard. She seemed half asleep, or perhaps even ill.
"Jen, are you sick?" Clem asked.
She lifted her large, pale blue eyes to his face, looking not quite at his eyes, but perhaps at his mouth. She shook her head. "Awful tired," she whispered.
"Too tired to come with us—out into the sunshine, Jen? We could stop and rest after we had got a few miles away."
She shook her head again.
"If Jen don't go, I won't neither," Mamie said.
"I ain't goin'," Tim said.
Clem started at them. "But you don't like it here," he urged. "They're mean to you. You don't get enough to eat."
"We're only Aid children," Tim said. "If we went somewheres else it would be just like it is here."
"You wouldn't be Aid children," Clem declared. "I'd fix things."
"We'll always be Aid children," Tim repeated. "Once you're Aid you can't do nothin' about it."
Clem was suddenly angry. "Then I'll leave you here. I've made up my mind to go and go I shall. You can tell them when they get home tonight. Say I've gone and I'm not coming back ever. They needn't look for me."
They stared at him, Jen's eyes spilling with tears. "Where you goin'?" Tim asked in a weak voice.
"Back where I came from," Clem said recklessly. He longed unutterably to get back somehow to Mr. Fong's house in the familiar streets of Peking, which he had not known he loved. That was impossible, but to leave this house was possible. For the moment anger quenched his conscience. He had given them their chance and they would not take it. He had said he would take the burden of them on his own back, though he was no kin of theirs, and they had refused him even this hard way to his own freedom. Now he would think only of himself.
He leaped up the crooked stairs and took his suitcase and crammed his clothes into it. He had a little money left from the store the sailors had given him and he had kept it with him always in the small leather bag one of the sailors had made. This bag he had kept tied about his waist, night and day, lest the woman or the man discover it and take it from him. He paused for a moment to decide the matter of a blanket and then revolted at the thought of taking anything from this house. He would not even take bread with him. Alone he would be free to starve if he must.
Down the stairs he went again, carrying his suitcase. They were still in the kitchen as he had left them. None of them had moved. Their eyes met him as he came in, faintly aghast, and yet unspeaking.
"Good-by, all of you," he said bravely. "Don't forget I wanted you to come with me."
He drew his folded cap out of his pocket and put it on his head.
"Good-by," he said again.
They stared at him, still unanswering, and upon the strength of his continuing anger he strode out of the room and across the weedy yard to the gate which hung crooked upon its hinges. He leaped over it and marched down the road, his head high, to meet a world he did not know.
Despair drove him and lent him courage, and then the beauty of the land lifted his heart. Surely somewhere there were kind people, someone like Mr. Fong, who would recognize him and give him shelter for a while. He would work and repay all that he received and some day he would, after all, come back and see the wretched children he had left in that kitchen.
He had gone perhaps a mile when he heard the sound of feet padding in the dusty road. He stopped and turning his head he saw Bump running doggedly along, and he waited.
"What do you want, Bump?" he asked the sandy-faced, sandy-haired child who blinked at him, panting. The signs of hominy were still about his mouth.
"I'm comin' with you," he gasped.
Clem glared at him, for a moment resentful of the least of burdens. Then his conscience leaped into life again. Surely he could take this small creature with him, wherever he went, a younger brother.
"All right," he said shortly. "Come along."
# 3
IN MID-AUGUST THE NEWSPAPER headlines had announced the end of the siege in Peking, and a cablegram from Dr. Lane brought the news that he intended to stay. The Imperial Court had fled, and the Old Empress had wailed aloud her hardships. She had not even been given time to comb her hair, and her breakfast on the day of the flight had been only a hard-boiled egg.
"Serves her right," Mrs. Lane said briskly. "Well, William, it looks as though I'd have to go back to your father. But you'll be able to manage by yourself if I get your clothes ready before I go."
William went to Cambridge for his final examinations in September. He had missed the preliminaries but Mrs. Lane had herself gone to the dean with a certificate signed by the headmaster of the Chefoo Boys' School. She had so talked and persuaded and demanded that the dean was much impressed and granted her son a certain clemency, and William was admitted conditionally. He was confident that whatever promises his mother had made to the dean, he could in the course of four years fulfill. Indeed, he preferred not to know all that his mother had said and done for him. Thus he did not know, though he suspected, that the admirable arrangement he had made with Mr. Cameron to be Jeremy's roommate, and when necessary his tutor, had taken shape first in the active brain of his mother.
Mrs. Lane, before she went back to China, had chosen a final Sunday afternoon to call upon Mr. and Mrs. Cameron. She had grown friendly if not intimate with them during the summer when William had gone almost every afternoon to play tennis at the house on top of the cliff. He had asked her to call upon Mrs. Cameron, stipulating that neither of his sisters nor his grandmother was to go with her.
"The Camerons are the kind of people I belong with," he had explained. "I want them to know I have a mother I need not be ashamed of. Nobody else matters."
Mrs. Lane was touched. "Thank you, dear."
The formal call had gone off well, and Mrs. Cameron had explained that she must be forgiven if she could not return it, since in the summer she made no calls. Mrs. Lane and William were, however, invited to dinner within the month. After the evening pleasantly spent by Mrs. Lane talking about the Empress Dowager and the magnificence of Peking, it had occurred to the indomitable mother that a problem which had been worrying her much could now be solved. In spite of all her efforts, it was clear that William would be compelled to earn money somehow during college, and she could not imagine how this was to be done. She had inquired of the dean, and he had suggested waiting on table or washing dishes. This suggestion she had accepted with seeming gratitude but she knew it was impossible. William would not wait upon anyone nor would he wash dishes. It would be impossible to make him. She remembered the delightful evening in the great seaside house. It was a pity, she had thought, that the heir to all the wealth was only a pale sickly boy. William would so have enjoyed it, would have been so able to spend it well, looking handsome and princely all the while. She had thought deeply for some weeks, and had at last decided to call one last time upon the Camerons. She wrote a short note to Mrs. Cameron, was grateful for all the kindnesses of the summer, mentioned her impending return to China and how she feared to leave her boy so new and friendless here, and asked permission to come and say good-by. When Mrs. Cameron telephoned her to say they would be at home on a certain Sunday, thither she went, at five o'clock.
The butler ushered her into the drawing room, where Mrs. Cameron sat doing nothing while Mr. Cameron read the _Transcript._
"Do sit down," Mrs. Cameron said, and made a graceful motion with her ringed left hand.
"Thank you," Mrs. Lane replied.
She had spent a good deal of thought upon her costume for this occasion. It should be plain, but not poor. It must convey good taste and a civilized mind.
Knowing the ready impatience of the rich, she had begun upon her theme as soon as Mr. Cameron put down his paper to greet her.
"Don't let me interrupt your reading," she said. "I have come for a very few minutes to say good-by—and for one more purpose. It is about William."
"What's the matter with William?" Mr. Cameron inquired.
"He has always done very well in school," Mrs. Lane said. "We expect that. His father was graduated from Harvard _summa cum laude._ No, the concern is in my own heart. William is so young, so lonely. He has no one to take his parents' place. His grandparents, my father and mother, are old and they can scarcely understand him. They have the responsibility of the girls, too. My husband's parents are dead and the family scattered. If I could feel that William would be able to look to you and Mrs. Cameron for guidance—through Jeremy—"
"He can always come here," Mrs. Cameron said in a mild voice. "I'm sure there is plenty of room."
Mrs. Lane sighed. "Thank you, dear Mrs. Cameron. I dread the long vacations. His father says he must work and earn part of his way, but what does William know about such things?"
"It won't hurt him to work," Mr. Cameron said.
Mrs. Lane agreed quickly. "That is just what his father says, and I am sure you are both right. Please, Mr. Cameron, for the first summer at least, could you help to find something suitable for my boy, something that will not lead him into bad company? He doesn't know his own American people yet."
"Oh well," Mr. Cameron said. "I can do that. There are always jobs waiting for young men, if they are the right sort. I supported myself entirely after I was fifteen, as a matter of fact."
Mrs. Lane proceeded bravely to the most difficult part of her purpose.
"I am going to ask something really bold, dear Mr. Cameron. Do you think that William could be useful somehow to your son? Could he not perhaps look after him, help him even with his lessons? When—if, of course—he should be ill, William could look out for him, you know—go to his classes and take notes for him—that sort of thing."
Mrs. Lane was faltering under Roger Cameron's stern eyes, and she looked pleadingly at Mrs. Cameron for relief. To her joy she saw a mild approval there.
"It might be a good idea, Roger," Mrs. Cameron said.
"William's a proud sort of fellow," Roger replied.
"Not too proud to help his friend," Mrs. Lane said. "William is a Christian boy, Mr. Cameron."
Roger pursed his lips. "How much do you expect me to pay him?"
Mrs. Lane knew her battle was over. She shook her head and folded her hands in her lap. "Please don't ask me that, Mr. Cameron. I trust your judgment—and your generosity. I wish there need be no talk of money—it's so dreadful. Had my husband remained in this country instead of choosing poverty upon the mission field... but no matter!" She smiled sadly and changed the subject. After ten minutes of lively talk made up of news from her husband's recent letters, she rose to say good-by. She clasped Mrs. Cameron's hand between both her own and smiled bravely. "I cannot tell you how safe I feel now about William. I leave him in your care, dear friends."
Mr. and Mrs. Cameron bowed, still looking a little bewildered. When the door had closed they sat down again exactly as they were before and Mr. Cameron picked up the _Transcript._ Neither of them spoke for a few minutes, and Mrs. Cameron gazed out of the window into the garden.
"It is a good thing that William Lane is so handsome," she said at last. "We really won't mind having him about. Candy says he is clever. I do hope he will always be good to Jeremy. Sometimes I think there is something cruel about his mouth. His hands are small for such a tall boy. Have you noticed that? I always think small hands mean cruelty in a man."
She did not speak often but when she did a little rush of words came from her lips, as though reserve had temporarily been removed.
Mr. Cameron listened, still reading the paper. "It won't hurt Jeremy to have a strong young fellow around to keep him lively."
Mrs. Cameron did not reply for some little time. Then she said, "As for vacations, you must not forget that Candace is also in the house. The two of them, both being so healthy, will want to play games together.... I shouldn't at all like her to marry the son of a missionary."
"Candy will marry whom she pleases," Mr. Cameron said. He loved his daughter and was proud of her, though with steady pessimism. Sooner or later the young always betrayed the old.
"Do keep quiet, there's a good girl," he went on. "This Bryan is putting me into a state, even on Sunday. He'll be the death of us all, talking about the Philippines. What does he know about those foreigners over there?"
Mrs. Cameron fell silent, and Mr. Cameron read the paper with fury, chewing the yellowed ends of his mustache.
The examinations were easily passed, for which William was grateful to the hard grueling of English schoolmasters. He was practical enough to realize that he could also thank his own talents and ambition. It was intolerable for him not to do well and so he did well. When Mr. Cameron had asked him to come and see him, one day after his mother had sailed for China, he went with some excitement within, although with entire calm upon his surface. His mother had told him, not quite truthfully perhaps, what Mr. Cameron would talk about.
"He has some idea that you might be a sort of tutor for Jeremy," she had said that last day. "Don't get proud and refuse it, William. Remember the alternative is dishwashing or waiting on the college tables. Besides, no one need know. You will simply be Jeremy's roommate and you will have the chance to live in those beautiful rooms. I don't think I could get you in there otherwise."
The beautiful rooms, he had already discovered, were on that short and noble street called the Gold Coast. There the sons of the wealthy lived like young princes in suites of rooms with separate bedrooms, a private bath, and a shared living room. Anything less seemed impossible to William. He made up his mind that he would accept whatever Mr. Cameron offered.
He was pleasantly grateful, then, when the offer was made.
"I leave it to you," Mr. Cameron said, "to see how you can help my boy. You know him pretty well now, don't you?"
"I think so," William said, and he added quite sincerely, "at least I like him more than any boy I've ever known."
"That's good," Mr. Cameron said with more heartiness than usual. "Then you can help him, I guess. Keep him cheerful, you know—that's very important. We don't believe in medication. It's very important to believe in the power of mind over matter."
"Yes, sir," William said.
"Now," Mr. Cameron went on. "Will a hundred dollars a month be about right?"
"Whatever you say, sir," William replied. He was startled by the amount, but he would not show his amazement.
"Well, if you find it isn't enough you can let me know," Mr. Cameron said. "And look here, one more thing, what say we keep this little arrangement to ourselves? It might make Jeremy feel queer with you. He's democratic and all that."
"You mean just you and me, sir?" He thought of Candace. He did not want her to know that her father was paying him.
"Just us," Mr. Cameron said. "Of course, Mrs. Cameron knows the general idea, but she won't say anything if I tell her not to, and she isn't interested in details."
"I'd like it," William said. "That is, sir, I'd like to forget it myself, so that I won't be thinking of money in connection with Jeremy."
"No, no," Mr. Cameron said, quite pleased.
"I'll just ask him if he will let me room with him," William suggested.
"That's right," Mr. Cameron said. "You fix it up and on the first of every month there'll be a check."
The outcome of this was that when the two young men entered college, William found himself on the Gold Coast, with a bedroom of his own across the pleasant living room from Jeremy's. Mrs. Cameron came with them and spent a week furnishing the rooms properly. There was even a small grand piano for Jeremy to use. William, secure in the monthly check, spent the money his mother had left him to buy himself a few luxuries that she had not been able to persuade the agitated mission treasurer to include in his necessities, a handsome set of razors, some silk pajamas, a blue brocaded satin dressing gown and leather slippers to match.
Thus William began his four years of college. He was reserved, modest, and dignified, and took his work with secret seriousness, though outward ease. He fulfilled exactly his every obligation to Jeremy and was at once kind and stern. He felt sometimes that Jeremy did not like him but he did not allow this to disturb him. The brilliance of his own academic standing was answer enough. Among the hundreds of young men who were matriculated at Harvard that year, William was notable. In prudence he made no close friends as the months passed, but he surveyed the Gold Coast carefully. It did not occur to him to search for friends outside that bright area. He marked here and there men whom he might cultivate as time went on. There was plenty of time.
Nevertheless, by Christmas he had approached a classmate who attracted him above all others, a handsome fellow who lived in Westmorly, too careless to be ambitious for high marks with his professors, too self-confident to consider marks of first importance. He had already his group of friends, in the upper classes as well as among the freshmen, for he had prepared at Groton. He did many things well. He sang in the freshman glee club, he was a fine oarsman, and he was already marked for those clubs which William exceedingly desired to enter. Franklin Roosevelt was the man, William told himself, that he would like to have been, his father rich and his mother secure in her place in American society. Having everything, the gay and handsome boy could say what he liked, could believe as he felt, behave as he willed. In the election that autumn he was for Bryan, although his own cousin, Theodore Roosevelt, was running for vice-president, and he flouted England by raising money for the Boers. It was this high-handedness that won William's notice. He could not have taken sides against England even though he could not approve the Boers, or disapprove the English, and he envied the ease with which it seemed that Franklin did both, without liking the Boers or disliking the English. For some reason which William could not comprehend, there seemed to be such an overflow in this youth, such a limitless privilege, that he made a habit of believing that the poor, the uneducated, the miserable must be championed, although without hatred of the oppressor.
William knew nothing of South Africa. That he might prove to himself at least that the man he unwillingly admired was wrong he began for the first time in his life to read newspapers and to perceive though dimly, how omnipotent they were. Even he was dependent upon them to shape his own opinions about the war. He was convinced from what he read that England was right and that the Boers were coarse farmers, ignorant dwellers upon the soil. When he announced this opinion, not to Franklin Roosevelt but in his presence, he was answered only by loud though pleasant laughter. His opponent refused to argue. He did not care what William believed.
The tall young man did other things more amazing. He helped the men who lived in the Yard, in the cheap dormitories and in even cheaper rooming houses, and the day students, to organize themselves and win the class elections away from the little group that had always won them.
The Gold Coast inhabitants sneered. "Anything to get himself popular!"
William listened and said little. He was cautious in the world of his own country, still so new to him, and being insecure and unready to take what he felt was his proper part, he hovered near the young Roosevelt who had no doubts and behaved like the prince of a royal house. He made his approach of friendship tentatively slight, a conversation in the dining room at Memorial Hall, a chance to walk together to separate classrooms. Roosevelt answered without assuming superiority and was mildly interested to hear of William's birth in China. His own grandfather had made his fortune in China and his grandmother in her twenties had visited the fashionable parts of Hong Kong and Canton.
Upon this slight interest William built his hopes. Of all the young men he knew or saw, this one was most nearly his equal, most fitted for friendship. Why that friendship did not grow, why the hoped for companionship faded, William never knew. It was a bud that did not bloom. Franklin Roosevelt's greetings were carelessly kind, but he had no time. There was never a time for talk, no time for companionship, and William, too sensitive, withdrew into cold and secret criticism. He was reminded of the English days in the Chefoo school. Because he was not allowed to love, he took shelter again in hatred. The fellow, he told himself, wanted to run the college. When both of them were chosen for the staff of the college newspaper, the _Crimson,_ William felt himself freeze toward the young man who was still too happy to notice him.
On a cold day in January in William's sophomore year, his father stood on the balcony of Mr. Fong's bookshop. Dr. Lane knew Peking well, and the day before he had walked along the street judging each house for its view of the Great North Gate, through which on this day, the seventh day of the Western first month, the Old Empress with her Imperial Court was to return to the palace. Dr. Lane did not know Mr. Fong and it was by the merest chance that he saw above this bookshop the narrow balcony to which one must climb by a ladder since it was merely a façade upon the roof. From it, however, was the best possible view of the great event of tomorrow.
Dr. Lane went into the bookshop and bowed to Mr. Fong, who stood behind the counter reading an old book he had bought from the library of a man recently dead. Since the man had no sons and none of the females of the house could read, there was no more use for a library.
"What can I do for you, Elder Brother?" Mr. Fong inquired. He was polite to all foreigners because, being a good man, he was sorry for everything that had happened. While he could not say that he was glad that his country was defeated, for he put no more trust in foreign governments than in his own, yet he grieved that foreigners and Chinese had been killed.
Especially was he ashamed of the folly of the Old Woman who had put her faith in the society of ignorant men called Boxers. She deserved the catastrophe that had befallen her when she had been compelled to flee the city in such haste, seventeen months ago. So impetuous had been the Court's flight, as Mr. Fong heard, that more people had been killed by the Imperial Guard in getting the Old Buddha out of the city than the foreign soldiers had killed when they came in. It was over at last, to the disgrace of all concerned, and pity to those dead, both Chinese and foreign, and especially the little children, and Mr. Fong was polite at the sight of a foreign face now that it was safe to be friendly.
Dr. Lane replied with equal politeness. "I wish to rent a few feet of your excellent balcony tomorrow in order that I may see the return of the Empress Dowager."
Mr. Fong was surprised. "Elder Brother, are you and the elder brothers of your country pleased to see her return?"
"At least I am," Dr. Lane said. "I believe that the people need their government and I have every hope that the Empress will have learned her lesson and that she will allow the young Emperor to put in reforms."
"Western elder brothers have more faith in women than we have," Mr. Fong replied. "Whether Elder Brother is right I do not know and it is always likely that I am wrong. I could not take money for the balcony. Pray use it as though it were your own."
After some minutes of such talk, Mr. Fong finally accepted two taels of silver, which was not too much since the foreigners were eagerly buying whatever space they could find. Chinese would not of course be allowed to see the royal return. All doors were to be barred, all windows closed, and blue cotton curtains were even now being hung across side streets and alleyways, so that no common eye could look upon the Old Buddha. Foreigners could not be thus controlled since they were the victors in the brief war.
"You know, Elder Brother," Mr. Fong remarked when the transaction was over, "I feel more than usually unhappy to take silver from you because I had once in this house a clever small brother of your people."
"Indeed!"
"Yes," Mr. Fong said, stroking his sparse beard. "He came to teach my son a foreign language. He did not take money for pay. Instead he asked for my foreign books, of which I have a few. Servants steal such books from their foreign masters to sell for a few coins, and that is how I got them."
"Who was this foreign boy?" Dr. Lane asked.
"You remember the god-man who was killed, he and his wife and children? The one who was always begging for bread?"
"I do, indeed," Dr. Lane said. He remembered very well that the Miller family had been found lying in their own blood, but the boy was not there, nor had he ever been heard of, although the American officials had made efforts to trace him.
"The boy was here," Mr. Fong said solemnly. He tapped his polished wooden counter with his long fingernail. "Here he was in my house. He came early to teach my son. Thus he escaped death. Surely there was meaning in it. I have considered it a good omen for my house."
"What became of him?" Dr. Lane asked with intense interest.
"He came back," Mr. Fong said. "And he told me what he had found in his own house. He stayed with us until he was able to escape. Then I told him to go east to the sea and to find a foreign ship and to return to his own land and his father's father's house."
"That was very good of you," Dr. Lane said. "I shall report this to the American officials."
"Please do not do so," Mr. Fong said hastily. "It is better not to tell anyone so long as the Old Woman is alive. She will come back smiling, as you will see tomorrow, but who will know what is in her heart?"
Who indeed could know? Dr. Lane himself would never wholly recover from the long siege within the Legation Quarters. He had caught dysentery in the heat of that summer, and was nearly dead when at last the soldiers from the West came surging into the city. When his wife came back to him from America, after William was safely in college, she had tried to make him give up China.
"Surely, Henry, you have done enough."
"I have done nothing yet," he replied. It was the beginning of the long struggle between them over whether China was worth his life.
"See how many foreigners have been killed!" she had cried passionately.
"Hundreds of us have been saved, and by six men," he had retorted.
It was true. Junglu, the favorite of the Empress Dowager, had done all he could to save the foreigners from her fury. Yuan-cheng and Hsu Ching-cheng had deliberately changed the word "slay," in the royal edict, to "protect." Li-shao, Liu-yuan, and Hsu Tung-i the Empress had put to death for opposing the war against the foreigners. And there were the noble host, those whom he never forgot, the thousands of Chinese Christians, more than two score of them of his own church here in Peking, who had refused to give up their faith and who died, martyrs for a god who to them was a foreign one.
No, Dr. Lane told himself steadfastly, it was beyond his wife's power, strong woman though she was, to move him from his own faith, not only in God but in the Chinese people.
"I will be here tomorrow," he promised Mr. Fong.
Thus on the next day Dr. Lane stood upon the balcony, wrapped in a thick quilted Chinese robe inside of which he still shivered. Mrs. Lane had refused to stand there with him, when looking out from the window of their bedroom this morning, she had seen the city shrouded in yellow dust from the deserts of the northwest. A bitter wind was blowing, even then. Dr. Lane had been slightly exasperated to perceive it, for it added to the honor of the Imperial return. It was an ancient tradition in the city that whenever an emperor left his palace a strong wind would go with him, and would bring him back again. Heaven itself seemed to be on the side of the Old Buddha.
While he waited on the balcony in the fury of the cold wind Dr. Lane thought of what Mr. Fong had told him. The Miller boy had doubtless done exactly what his Chinese friend had bade him. He might now be safely in America. He must write and tell William of the possibility. He had reported the story to the American officials yesterday, concealing Mr. Fong's name.
He glanced with concern at the great gate. There was still no sign of the royal entourage. Helen had been wise, perhaps, to content herself with seeing the Empress at the mighty reception which she was to give to her conquerors when she reached the Imperial Palace. Yet he did not want to attend it. He was not dazzled by her arrogant and heathen splendor. He hoped to see her as she came in the North Gate and to discern for himself whether she had repented. He had prayed solemnly that her heart might be softened for the good of the people. He did not honestly know whether such prayers were answered.
Everything was in readiness for the moment at the gate. Across the city the wide street had been cleared of all venders and stalls and booths. The street had been swept clean and spread with bright yellow sand, yellow, the imperial color. No common man was on the street. The imperial guard stood waiting, and princes and dukes were ready each with his own banner corps. Here and there down the street foreigners stood at windows, a few opened by permission that the visitors might witness the return.
Mr. Fong's head appeared above the edge of the ladder. He held out a small brass handstove. "Take this, Elder Brother," he whispered. "I have put fresh coals in it."
Dr. Lane took the handstove gratefully and before he could speak his thanks Mr. Fong was gone. Now he perceived certain signs. A line of Chinese heads would appear here and there over a rooftop, instantly to disappear again. Word was running through the city that the Old Buddha was near. She had descended from the train. For the first time in her life the Old Buddha had ridden on a train, and with her, her court. She had not enjoyed it. The dust had been suffocating, the noise insupportable. When the whistle blew she had been terrified and indignant, and when she learned that this was the duty of the engineer she sent word by a eunuch that he was not to blow it without telling her before he did it. The railway from Paoting to Peking had been destroyed during the war and rebuilt again under the foreign victors and the foreign soldiers had brought it into the very heart of the city, tearing great holes in the walls.
The Old Buddha would not pass through these desecrated walls. She had ordered the court to alight outside and to enter their royal palanquins, that they might return to the city in proper state through the great gate.
Dr. Lane, holding the little handstove, heard a rising shout. A small army of eunuchs on horseback galloped from the gate. They wore black caps with red feathers and on the breasts of their robes were huge medallions of red and yellow embroidery. Behind them came the imperial herald, crying in a high voice that the Imperial Court was returned. All those officials waiting on the street fell to their knees and bowed their faces into the dust. Dr. Lane leaned on the frail banisters of the balcony and stared down into the street, and the destiny of this moment was impressed upon him. He watched everything, intent to remember it all, to tell William. He saw the Imperial Guard, followed by military officers. Great flags of yellow satin swirled in the wind, and upon each was embroidered a blue dragon swallowing a red sun. On either side of the flags were the imperial banners embroidered with the imperial arms.
Behind these rode the young Emperor, a sad young man, sitting within his yellow palanquin, which was lined with blue silk. The curtain was up and there he sat, his face unmoved, gazing straight ahead. He sat upon his crossed feet in the position of a Buddha.
"The sacrifice of youth," Dr. Lane murmured to nobody. Death was already clear upon that tragic face.
But death had nothing to do with the Empress herself. He was indignant to see the redoubtable figure, seated in her great palanquin in the midst of her guards, followed by the young Empress and the court ladies. Upon that gay and wicked old visage there was nothing but the liveliest pleasure. Seeing the foreigners, who were her conquerors, she had put aside the curtains of the palanquin and waved her handkerchief at them. He was the more indignant to see some of the foreign ladies, among whom he recognized Americans, too, wave back to the old sinner, laughing as they did so. Thus quickly was all forgot.
He came down from the balcony and returned the handstove to Mr. Fong with thanks.
"How did the Old Woman look?" Mr. Fong inquired.
"She has not repented," Dr. Lane said grimly.
"Did I not tell you?" Mr. Fong replied and he laughed, though his face was full of rue.
William Lane remembered suddenly in the midst of his preparation for a test in advanced English that he had not read his father's letter. He had got it in the morning with other letters, one of them from Candace, and hers he had read first. He wanted very much to be in love with Candace, and most of the time now he thought he was. The obstacle to his complete conviction was simple enough—herself. She expected from him a quality of attendance, a constant gallantry, which he found little short of degrading. For a woman to be beautiful was entirely necessary in his eyes. He despised his sister Henrietta for her plain face. Candace was beautiful enough to satisfy him, could he subdue her other less-attractive qualities.
At the moment, however, his relation with Candace was puzzling and exciting. He felt at a disadvantage, there was so much he did not know because he had not always lived in his own country. The secret hostility he had always felt toward his father for compelling him to be born the son of a missionary in China was now rising into a profound and helpless anger. In spite of this he loved his father in a strange half-hating fashion, and some of his darkest moods were those in which he brooded upon what his father might have been had he not heard the unfortunate call of God. Handsome in face, winning in manner, a leader of men, there was no reason, William thought when his fancy was rampant, why his father might not have gone into politics and even become the President of the United States. There was nothing wonderful about Theodore Roosevelt. William spent a good deal of time studying that bumptious angular face. Anybody could be President!
He pulled his father's letter from his pocket and saving the Chinese stamp for Jeremy, he tore the envelope and took out the sheets of thin paper, lined closely with the delicate and familiar handwriting. He was quite aware that his father always took pains to communicate with him on equal terms, and especially to tell him constantly what was happening in the land that had been left behind. William was too shrewd not to understand these pains. His father dreamed that the dear only son would come back to China, to be a better missionary than anyone had ever been before, to persuade the changing nation toward God. Some day or other, William knew, he would have to destroy this dream, but he had not yet the courage for it. He did not put it in terms of courage. He told himself that he was only waiting for the moment when it would hurt his father least. Now quickly and carelessly he read what his father had written slowly and with care.
> I told you of the pending return of the Court. Now it has come. It was a strange and barbaric sight, a motley crowd of rascals ruled over by a feminine tyrant, and yet somehow there was magnificence in it, too, a sort of wild and natural glory, the atmosphere which the Chinese can manage so well in whatever they do. The Old Empress is too great a person, in spite of her monstrous evil, to remain ungenerous. She has acknowledged her defeat, if not her fault, and now she sees that she must begin reforms for the people. Even before the return she issued an edict demanding that the officials of the empire immediately learn all about political science and international law. She has given them six months in which to complete this task, upon pain of death. Six months. There speaks the old ignorance and the new!
>
> Perhaps more exciting, because more practicable, is the fact she has appointed a commission to draft a public school system, the first that China has ever had. Some day the old examinations will be entirely abolished and China will be modern. It may happen before you finish college, dear boy, so that when you come back it will be to another country altogether, one which you can help to build.
>
> But I do not wish to speak only of China. Tell me about yourself at college. What you say of Jeremy seems pleasant and good. What fortune to find such a friend! I had feared loneliness for you. The young can be so cruel to those who have not their exact experience. Give him my warm regards.
>
> Your mother is writing you tomorrow, she says, about the reception which the Old Empress held for all the foreigners. It was a great affair. All the diplomats and their wives went and so far as I can learn from your mother, the Empress behaved exactly as though she had won the war and was graciously meeting her captives and freeing prisoners. So successful was she that a number of ladies capitulated to her frightful charm. I myself refused to go. I could not stomach having to be polite to that female personification of the Evil One. Your mother was not so scrupulous and apparently enjoyed herself.
His father's letters always took him back to China, however much he might resist. He could see clearly that bold figure of the Old Empress, great enough to accept defeat lightly and so be still imperial, still powerful. There was power in her which William felt was sacred, compelling a quality in himself which might be a similar power. As he grew into manhood to his full height of six feet one, he felt the excitement of his ambition surging into his body and his mind. He was drawn always to the powerful and the proud. Once he had passed the famous president of the university crossing the yard with an enormous watermelon under his arm, and he never felt the same respect again for him. Whatever the genius of Charles Eliot, and William acknowledged genius, it was lessened by the man's lack of pride. Nothing could have persuaded William to carry even a bundle under his arm.
Indeed, few of his professors fulfilled his secret expectations. It was hard to give high respect to a pudgy philosopher with a big head thatched with rough yellowish gray hair covered with an old tired-looking hat, or a little man with a high forehead and a shaggy disheveled mustache. Two men alone satisfied his instinct for dignity and seriousness. One was a great handsome German who looked like the Kaiser and taught psychology with the voice of a thundering god. The other was a tall slender man, a Spaniard, whose eyes were dark and cold. Under George Santayana alone William sat with complete reverence. The man was an aristocrat.
The same absolute and delicate pride he had seen long ago in the Chinese Empress, a quality which could not stoop to common folk. For William democracy meant no more than that from among the common mass a king might arise, a Carlylean hero, a leader unexplained. People tried to explain such persons by many myths of virgin births and immaculate conceptions. Chinese history, he had often heard his father say, was rich with such myths. The unexplained great men, born of ordinary parents must, the people felt, be the sons of gods.
In the dark depths of his emotions William acknowledged the possibility of explanation. How explain himself? There was no one in his family like him. He could not be explained any more than the Chinese Empress could be, for she was born the daughter of a common small military official. Somewhere in the path of the generations, certain genes met to make the invincible combination. He would never forget the haughty face of the indomitable ruler bent above him, a young American boy. It had been his first glimpse of greatness and it remained in him, a permanent influence.
So William created his world in his own image. The sons of gods were the saviors of mankind and they lived upon the Gold Coast, anywhere in the world.
William folded his father's letter and saw on the back of the sheet one further note:
> By the by, here is something interesting. You remember the Faith Mission family Miller, who were killed by the Boxers. Actually the boy escaped. Quite by accident I met a Chinese who had saved his life and sent him on his way to the coast. From there, if he got a ship, he may have reached America safely—may be there now, under God's care.
This news did not interest William. That brief and humiliating moment in the dusty Peking street was repulsive even in memory. He crushed the letter in his hand and threw it into the wastepaper basket under the desk.
In William's junior year he reached his final hatred of Franklin Roosevelt when Roosevelt was chosen president of the _Crimson._ William had supposed himself secure for the place and he did not know why he had failed. He was not able to bide his disappointment from Jeremy, always quick to feel suffering in anyone else.
"Sorry, William," Jeremy said. "You would have done a magnificent job."
"It doesn't matter," William said with a grimace.
"Don't be ashamed of feeling," Jeremy said gently.
William allowed a few words to escape from his vast inner misery. "It seems unjust that I shouldn't get it, and that fellow got it so easily."
He saw Jeremy looking at him with a peculiar and pitying gaze and he averted his eyes.
"I'd like to say something to you, William, if you'll let me," Jeremy said after a moment.
"Well?" William heard his own voice harsh.
"Perhaps we can't say such things to each other. We never have, somehow. Perhaps if we could we would both feel better."
"Say what you like," William said. He sat down abruptly at the desk and pretended to fill his fountain pen with ink.
"Roosevelt has got everything he wanted because he is warm toward everybody. He is full of a sort of—of—love, if you know what I mean."
"I'm afraid I don't," William said. "He is full of loose ideas, so far as I am concerned."
"I know some of his ideas are crazy," Jeremy admitted. "But everything else about him is so right that he can just about think as he likes."
William dropped the pen and it fell on the floor. His gray eyes were furious under his black brows and his lips tightened. "I suppose you mean his father is rich, his mother is socially correct, they live on the right street, all the sort of things that I haven't!"
"You know I don't mean that," Jeremy said. "We'd better drop it."
They had dropped it and he was too proud to tell Jeremy that he did know what he meant. For William was beginning to know that he lacked one grace among his gifts. He could not win love from ordinary people. He excused himself by saying that it was because they felt his superiorities, his obvious mental power, his ability to do easily what others did only by effort. The superior man, he told himself, turning the pages of his Nietzsche, must always be hated by his inferiors, but even this hatred could be turned to advantage and used as a tool for further power for good.
"I must expect hatred," William thought. "I must accept it as my due because I am not understood. What the common man cannot understand he hates."
Sometimes he thought even Jeremy hated him. But such moments passed and he was careful to seem kinder to his friend, more quick to help him, more patient with his frailties, his headaches, his manners.
William, relentlessly remembering his defeat, was further disturbed by an editorial in the _Crimson_ before the class elections. Roosevelt wrote:
"There is a higher duty than to vote for one's personal friends, and that is to secure for the whole class leaders who really deserve the positions."
These were the words of a man determined to be a liberal in spite of class and property. While the Gold Coast repudiated them, votes belonged to the many.
William never forgave Franklin Roosevelt. He had already begun to believe that the people anywhere in the world were clods and fools and now he was convinced of their folly. The Boers who fought England were clods and fools. The Chinese he remembered upon the streets of Peking were clods and fools. From now on he spoke to no one at Harvard except those who lived on the Gold Coast.
Yet he heard one day a remark that horrified him again. A pallid professor with long mustaches said these words with an emphasis too fervent for William's taste: "The American people control their own destiny."
William began then in earnest the study of the history and government of his own country. He perceived to his dismay that the professor's remark was a true one. Clods and fools though they might be, the American people elected their rulers, laughed at them, despised or admired them, obeyed or disobeyed them, clung to them or rejected them. He began after that to look at the people he passed on the street with consternation and even fear. Out of ignorance apparent upon their faces, obvious in their crude speech, these men chose from among themselves certain ones upon whom they bestowed the powers of state. It was monstrous. For months William felt himself in a den of lions. He tried to talk to Jeremy, who first laughed at him and then tried to explain:
"Americans aren't just people—they are Americans."
William had no such reverence. What he saw beyond the Gold Coast reminded him ominously of the streets and roads of China. He had feared the common people there. Had they not risen up in all their folly against men like his father? Von Ketteler had been murdered by an ignorant clod. He remembered that dignified German, who at the Fourth of July celebrations at the American Embassy had more than once spoken to him with courtesy. The common people could rise against their betters anywhere and kill them, unless they were taught and controlled.
Yet, how to control these boisterous, independent, noisy jokesters who were the common folk of his own country? They would not tolerate a real ruler. They had no respect for those above them. They delighted to pull down the great and destroy them. Look at Admiral Dewey, a hero for an hour, whose plaster triumphal arch, designed for marble, fell to dust and was carted away by the garbage collectors! The whim of the people was the most frightening force in the world.
Upon this William pondered, knowing now his own lack of charm, that strange senseless power to attract his fellows, the charm which young Franklin Roosevelt possessed as easily as he possessed height, fearlessness, and ready laughter. Without this frail gift, William told himself proudly, he must rely upon his brains and devise a means of teaching and controlling the wild beast of the multitudes. He would lead them wisely, insidiously, charming them through words, himself never seen.
In that third year in college he wrote to his father to say that he would not come back to China. "I feel I am needed more here than there. The truth is, I am not impressed by American civilization. I intend to start some sort of newspaper, something ordinary people will read, or at least look at, and so do what I can to enlighten my fellow countrymen."
Some day, William vowed to his own heart, he would be the editor and owner of a newspaper, perhaps even a chain of newspapers, by which he could defeat any man he disliked or disapproved. To dislike was to disapprove. Money, of course, he must have but he would get it somehow. Quite stupid men were able to get rich.
Meanwhile, Franklin Roosevelt did not win the Phi Beta Kappa key, and William felt assuaged when he himself was among the chosen.
Yet the college years, as they passed, were good ones. He became a member of the Cameron family and spent his vacations with them, after brief duty visits to his grandparents and his sisters. It was accepted now that William was independent and different. Henrietta was proudly silent with him, Ruth worshiped him timidly, and his grandparents tried, somewhat in vain, to treat him as an ordinary young man. They knew he was extraordinary. Even Mrs. Cameron saw that now. It was pleasant to have about her a handsome young man who knew how to dress and was always ready to do what she needed done. He paid little attention to Candace, she reflected after each vacation, and he behaved like a strong elder brother to her poor son. She introduced William to the ladies at her Christmas At Home and forgot to mention that his father was a missionary, leaving the impression that he was connected with the diplomatic corps in Peking. William did not correct her.
His dreams hovered about the many happy weeks he spent in the great square house on Fifth Avenue. Each summer he accepted a job that Mr. Cameron offered him. He went to Europe with Jeremy, a combination secretary and guide, and they shared a valet. Together the two young men wandered about old cities and sailed the Mediterranean. It was a matter of course that William would always go home with Jeremy when the journey was over. He had his own two rooms in the vast Cameron house. They opened into Jeremy's suite. From there he seldom wrote to or heard from his sisters and his grandparents, and Peking he had nearly forgotten. The Camerons had become his family.
He thought about the Camerons a great deal, pondering again the question of how, through them, he might reach vague heights he imagined but could not see. Among the many things he discussed with Jeremy this was not one. William was not crude. He had lived too long among Chinese, even though only servants. He felt crudity in his mother and shrank from it, but he forgave her because of her willingness to sacrifice. His mother was "for" him, as he put it, and when he discovered this quality in any person, he overlooked all else. Nevertheless he was glad that during his college years his mother was remote in Peking. He was still not yet sure that the Camerons were entirely "for" him, not even Jeremy. This uncertainty made him pleasantly diffident and unselfish in his dealings with each of them. To Jeremy, he gradually became someone always willing to spare him tiresome stairways when he wanted a book from the library, and so he wore away dislike. To William's listening silence Jeremy in vacations talked more freely than at college, uncovering a delicate and poetic mind, racked with questions, and a spirit confounded by conscience. Thus Jeremy spoke on the solid matter of money.
"I know that if my father had not been rich I would now have been dead. But I wish I could owe my life to something else."
"Perhaps you might say that you owe it to your father's being so able as to get rich," William had suggested.
"I don't know that merely being able to get rich is anything particularly noble," Jeremy had replied.
"Not everyone can do it, nevertheless," William said. "Your father must have had some natural gift."
A look of aversion came upon Jeremy's pale and too mobile face. "The gift is only that of being able to overcome someone less strong in the competitive game."
To this William put up silence, and into the silence Jeremy continued to talk. "Sons of rich men always complain of their father's riches, I suppose. Yet there ought to be some way of living without stamping all the ants to death."
Still William made no answer. Jeremy had come to no grips with life. The trouble with Jeremy was that he wanted nothing. He himself wanted everything; success with the newspaper he meant to have, and after that a wife beautiful and wealthy, a mansion to live in, a place in the world where he could be unique in some fashion he did not yet know, and the means to all this, he perceived, was money. He was perfectly sure that money was what he wanted first of all.
In his quiet way he reflected further upon the Cameron family. His brotherly relation to Jeremy he could easily develop. Quite honestly, he liked Jeremy. Candace he would consider as time passed. He was too nearly an intellectual to be in haste for marriage. Mrs. Cameron he understood and did not fear. His thoughts, flying like tentative gray hawks, now lit warily near the image of Mr. Cameron. This man was the central figure, the most important man, the one whom he must approach with real finesse. Mr. Cameron knew secrets. Pondering upon that vague and unimpressive person, William perceived that behind the nondescript face, the long and narrow mouth, there was something immense, a power strong and profoundly restrained. He guessed by some intuition of like mind that Mr. Cameron never told his true thoughts to his family, certainly at least not to women, and probably not to his delicate and oversensitive son. Into that loneliness William determined to go, not with deceit but with honesty.
"Mr. Cameron," he said on Easter Sunday, "I would like to ask your advice about something."
"Why not?" Mr. Cameron replied. Sunday was a day on which he drowsed. It was now afternoon, however, and late enough for him to have recovered from the immensities of dinner. He had slept, had waked, had walked in the garden with his wife and daughter to see the promise of some thousands of daffodils, and had come in again to reread the newspaper in the small sitting room off the drawing room, which was his favorite resting place. There William had come, after waiting patiently in his own room, from which he could see the prowling among the daffodils. Jeremy and Candace had gone with their mother to see their grandparents.
He sat down at a respectable distance from Mr. Cameron and upon a straight-back chair. His childhood in Peking had taught him deference to elders, and he would not have been comfortable had he chosen one of the deep chairs upholstered in brown leather.
"I would like to talk about my future, sir," he said.
"What about it?" Mr. Cameron asked. His eyes roved to the newspaper at his feet. The financial section was uppermost and he was disgusted to see that the profits of a rival company had risen slightly above those of his own.
"I want to get rich," William said simply.
Mr. Cameron's gray eyebrows, bunched above his eyes, quivered like antennae. "What do you want to get rich for?" he demanded. He stared at William with something more than his usual careless interest.
"I see that here in America a man cannot get any of the things he wants unless he is rich," William replied.
Mr. Cameron smiled and agreed suddenly. "You're damn right!" He kicked the newspaper from his feet, sat back, and felt in his pocket for a cigar. It was a short thick one, and he lit it and puffed out a cloud of blue and fragrant smoke. The vague barrier that stood always between himself and his son's friends fell away. He felt he could talk to William. He had always wished that he could talk to young men and tell them the things he knew. If an older man had talked to him when he was young he would have got along faster.
"I'll tell you." He shifted his cigar to the corner of his mouth. "If you want to get rich, William, you'll have to quit thinking about anything else. You'll have to concentrate. You have to put your mind to it."
"Yes, sir." William sat at attention, his hands folded upon his crossed knees. They were small hands, as Mr. Cameron remembered his wife had said they were, and they were already covered with surprisingly heavy black hair. William's hair on his head was black, too, in contrast to his light gray-green eyes. An odd-looking boy, Mr. Cameron reflected, though so handsome.
"Have you thought of any special line?" Mr. Cameron asked.
William hesitated. "Did you, sir, at my age?"
"Yes, I did," Mr. Cameron replied. "That's the trick of it. You have to think of something that people want—not a few rich people, mind you, but all the ones who don't have much money. You have to think of something that they must buy and yet that won't cost too much. That's how I thought of the Stores. I was clerk in a general store."
William knew the Cameron Stores very well. There was one in almost every city. He had wandered about them more than once, looking at the piles of cheap underwear and kitchen utensils and groceries and dishes and baby carriages and linoleum, everything that an ordinary family might want and nothing that Mrs. Cameron would have had in her own house. It was repellent stuff.
"I've thought of a newspaper," William said.
Mr. Cameron looked blank. "What about a newspaper?"
"A cheap newspaper," William said distinctly. "With lots of pictures so that people will first look and then read."
"I never thought of such a thing," Mr. Cameron said. He stared at William, digesting the new and remarkable idea. "There are already plenty of newspapers."
"Not the kind I mean," William said.
"What kind do you mean?" Mr. Cameron asked. "I thought I knew about every kind there was."
"I suppose you do, sir," William said. "What I am thinking of, though, is new for America. I got the idea from England—and a little bit, perhaps, from the _New York World,_ and then the _Journal._ But I didn't think of doing anything myself until I began to hear about Alfred Harmsworth in England. Have you seen his papers, sir?"
"No," Mr. Cameron said. "When I'm in London I always read the _Times_ —maybe look at the _Illustrated Times_ on the side."
"My paper," William said, as if it already existed, "is what's called tabloid size and it is to have everything in it that can interest the masses. It won't be for people like you, Mr. Cameron. It will have plenty of pictures. I've noticed even in college that most of the men don't really read much but they will always look at pictures."
"I hope you don't mean yellow journalism," Mr. Cameron said severely.
"No, I don't," William said. "I hope I can do something more subtle than that." He paused and then went on thoughtfully, his eyes on the patterned carpet. "I thought, if you approved, I would talk with Jeremy about it and some day we might go in on it together."
Mr. Cameron was pleased. It might be the very thing for Jeremy, easy work, sitting behind a desk. He had often wondered what to do with his fragile son, but he was too prudent to show approval. "Well, it would depend on what Jeremy wants. Newspapers cost a lot of money to start."
William was calm. "That's why I want to get rich." He was too wise to repeat what his mother had often told him, even before he went to Chefoo. His mother had sown in him early the seeds of common sense. "You can't have but so many friends," she had said. "And each friend ought to count for something." He had seen the folly of useless friends in the English school; his speaking acquaintance there with the British Ambassador's son had served him more usefully than the horde of missionaries' children.
At college he had selected from among Jeremy's friends three whom he was transferring to himself, Blayne Parker, Seth James, and Martin Rosvaine. Blayne William still doubted because he was a poet, and Jeremy supplied to him something that William knew was not in himself. Seth and Martin he was resolved to keep. Yet there was no reason why the five of them, Jeremy included, should not stay together after college. Seth's father alone could, if he would, supply the capital they would need. Meanwhile he was getting into their clubs.
"Got it all figured out, eh?" Mr. Cameron said. A look of admiration came over his face, mingled with reluctance. If Jeremy had been this sort of a fellow, he would have got him into the Stores. Invitation was on the tip of his tongue. "How would you like—" He swallowed the words. William would be too smart, maybe, ten years from now when he himself was getting to be an old man. He might not be able to cope with that new young smartness in case it opposed him. It was all right to give young men a chance, but not the whole chance. On the other hand, William might be the making of the Stores, at the time when he needed somebody. If the boy married Candy, for example, it would be almost as good as though he were born into the family. This would take time to think out. He leaned back and crossed his hands on the small paunch that hung incongruously on his lean frame. "When the time comes," he said dreamily, "I might be able to do something myself, William. Only might, that is. I can't tell from year to year, government being what it is in this country."
William rose. "I wouldn't think of such a thing, Mr. Cameron," he said in a firm and resonant voice. "I'm sure I can stand on my own feet." It was entirely the proper answer, although he felt that the time would come when he would need Mr. Cameron. Far better to owe money to Mr. Cameron than to the father of Seth James.
Before Mr. Cameron could reply, the door opened and Candace came in looking, her father thought fondly, like the morning star. She was all in rose and silver and wrapped in soft spring furs of white fox. Her cheeks were pink with the wind, for she had insisted on having the carriage windows open, and her yellow hair was curled about her ears and feathered over her forehead.
"Why have you two hidden yourselves away here?" she demanded. "Mother says please come out at once and be public. We have callers."
"We've been talking business," Mr. Cameron said. It was his instinctive reply to any demands from women.
"Nonsense," Candace said. "William hasn't any business."
"He has an interesting idea," Mr. Cameron said, fitting the tips of his fingers together. "A very interesting idea."
Then he got an idea himself. He rose and made haste with his slow step toward the door. "I'll go, just to please your mother. William doesn't have to be bothered with our friends unless he wants to. I'll bet it's the Cordies, anyway."
"It is," Candace said, with dimples.
"Don't you come, William," Mr. Cameron said. "They won't remember you next time they see you, anyway."
Thus he left these two young members of his society together, and went his way inwardly pleased. Candace could be trusted. She wouldn't let even her own husband do the family any damage. He was long used to eating his cake and having it too. The secret of such maneuvering had laid the foundation of his fortune—that and the resolute ignoring of the misfortunes of others. Maybe when the time came he would help William. He had a lot of loose cash he didn't know what to do with.
Left alone with Candace, William said nothing and she sat down in the chair where her father had been sitting, threw off her fur jacket, and lifted her small flowered hat from her head.
"What have you two been talking about?" she asked.
"Your father asked me what I wanted to do after I finished college and I said start a newspaper," William replied.
Her very clear blue eyes were sweetly upon him. "And why a newspaper?"
William shrugged his handsome shoulders. "Why does one do anything except because it is what one wants to do?"
"No, William, don't run around the corner. Why do you feel so inferior to everybody?"
She had thrust a point into his heart. His blood rushed into his face and he was careful not to look at her.
"Do I feel inferior?" His usually careful voice was dangerously careless.
"Don't you?" she demanded.
"I really don't know myself."
She refused the responsibility of special knowledge. "Anybody can see that you never come straight out with answers. You always think what to say."
"I suppose that is because I have never lived much in America," he replied. Though he despised his China, he often found it convenient to take refuge there. It gave him a reason, faintly romantic, for his difference from ordinary people.
"You mean the Chinese don't answer honestly?" she asked.
"I think they prefer to answer correctly," he said.
"But honesty is always right."
"Is it?" he asked with wisdom gentle and superior.
"Isn't it?"
"I don't know," he said again.
"But you must think," she cried with soft impatience.
"I don't always know what to think," he replied. "I guess my way a good deal of the time. I meet people every day whom I cannot understand. I have no experience that would help me."
She considered this for a brief instant. "Are the Chinese so different from us or are you only pretending?"
"Pretending what?"
"That you are different."
"I hope I am not too different from you, Candy."
This was a bold step and she retreated.
"I don't know if you are or not. I can't make you out, William."
He felt he had gone far enough. "Nor I you, sometimes, except today you look lovely. We don't have to make each other out as you call it—not yet, anyway. Let's not hurry, eh, Candy? I want you to know me, as I really think I don't know myself. That means time, plenty of time." He said all this with his cultivated English accent which he had not yet rejected.
She fended him off.
"Why do you keep talking about time?"
He laughed silently. "Because I don't want someone else to come dashing up on a steed of some sort and carry you off!"
This was very plain indeed, and she dropped her eyes to the pink rose she had fastened upon her white fur muff, and considered. When she spoke it was with mild malice upon her tongue.
"Yet I am sure that you always reach out to take what you want—as soon as you are sure you want it."
William met this with astuteness. "Ah, but you see, this time you might not want what I want. And I confess to being Chinese again to this extent: I don't like to be refused, even indirectly. I prefer not to be put in that position."
"That's your sense of inferiority again."
"Call it just being sensible."
"A bad sport, then."
"What we are talking about is not sport."
He spoke with such quiet authority that her youth was compelled to respect his. He was only a year older than she, and yet he might have been ten years her senior.
"I don't know what we are talking about," she said willfully.
"You and me," he said gravely, "though two, or three years, perhaps, from now."
"I shan't want to marry anybody for a long time yet," she said.
"That is all I wanted to know," he replied. He had been leaning against the marble mantelpiece, his hands in his pockets. Now he went over to her and lifted her hand and put it to his lips. She would have pulled it away but he did not give her time. In the same instant he put her hand down and left the room. His lips had been cold and dry but his palm was damp. She took her handkerchief and rubbed her hand; then she thrust it deep into her muff and sat for a long time alone and thoughtful.
As the last months of college passed, William was oppressed by fear lest his parents decide to return for his commencement, a fear that he had never acknowledged even to himself until his father had written in April from Peking:
> Neither your mother nor I can be there to see you take your honors, my dear son. This is a real grief to us. We have discussed the matter many times, and at first I was inclined, with her, to use our small savings and ask for leave of absence without salary. Then it seemed to me that I had no right to put personal feelings ahead of God's work. This is a peculiar age in which we now live in China. The opportunity to preach the gospel is unprecedented. Much as I deplore the manner in which we finally brought the Old Empress to her knees, and especially the looting of the city by Western troops, nevertheless it has taught her a lesson. We are given every opportunity now. God works in mysterious ways and we must not lose the harvest. I only wish the old Dowager Empress could understand that she is defeated. Alas, she cannot imagine it.
Two weeks later his mother had sent pleasantly heartbroken pages:
> My darling William, I cannot see you in all the pride of graduation from Harvard! The girls are costing us so much this year. Henrietta's operation for appendicitis has prevented it. The Board paid for it, of course, as they should do, but when I asked for a brief furlough to see my own only son graduate they refused me, saying that they had already been put to much expense. We cannot blame Henrietta, still it does seem strange it should have happened like this. We could use our savings—such a mite—but I will not do it, for it would give the Board future ideas. They owe us much for just living so far from our homes. Oh my son, do have many pictures taken of the event! I am sure that you have friends who will, for your mother's sake, make the day visible to me. Do beg dear Jeremy, or Mr. Cameron. Tell them how my heart aches not to be with you and them.
William had written a suitably sad letter and then, his spirit freed from the possibility of the presence of his preposterous parents, he had set himself to finish his senior year with glory.
One evening in June he was dressing himself for a dance. It was a few days before commencement and Martin Rosvaine's family in Boston was giving him the occasion. The Rosvaines were old Bostonians, proper except that their ancestry was French instead of English. Wealth mended this defect and Gallic gaiety lingered in their blood and made them enjoy pleasures more lavish than could be found usually among other Bostonians. William was as near complete happiness on this evening as his unfulfilled ambitions allowed. Candace was among the young women invited and she and her parents were staying at the Hotel Somerset until after commencement. He felt a warm anticipation when he thought of her soft and pretty face, and he wondered if he would tell her that his name stood among those few who would receive their diplomas _summa cum laude._ He decided that he would not, because Jeremy had barely passed, in spite of William's unflagging help with higher mathematics and modern languages. Candace was quick to be scornful of boasting and he could not explain to her that the English schoolmasters had grounded him well and had taught him to dig into fundamentals. Jeremy, persuaded by tutors through a delicate childhood, had not known that mathematics must be seized as one seizes a thistle, that German cannot be learned unless it is grappled with and overcome by force, that French can elude mind and tongue with its smoothness and escape memory entirely. Because an English schoolmaster in a Chinese seaport had used a ruler freely upon William's palms, had cracked him over the skull, had tweaked his ears, had poured out the bitterest and most dry sarcasm about upstart Americans who were properly only English colonists, William had learned early how to achieve even his small ambitions. Somewhere in dark and private action there had to be struggle and mastery.
Never having had the advantage of such knowledge, Jeremy had been content to escape failure. He was now lying in bed, dressed in lavender silk pajamas becoming to his fair hair and pale skin. He had declared himself exhausted by watching the baseball game in the afternoon. Idly he watched William shave clean his strong dark beard with an old-fashioned razor. June sunshine poured through the windows and William stood with his feet in a bright square. His mind was busy with plans that had nothing to do with college. After commencement was over he would take two weeks' holiday with the Camerons, and then he would plunge into the matter of getting money for the newspaper. His first plans for getting money he had given up altogether. He could not beg money from his college mates and their relatives. He would find it himself, get it, if possible, from Roger Cameron, borrow it perhaps, with Roger's backing. Then he could hire Martin Rosvaine and Seth James. But he would do most of the work himself.
"You're thinking about the paper," Jeremy said suddenly.
"So I am," William replied. He was putting on his tie, his small fingers, expert and supple. "How did you know?"
"I know that godalmighty look on your face," Jeremy replied lazily. "I fear and respect it."
"I'm no son of a millionaire," William said with a mirthless smile. "I have to get out and hustle, the way your old man did. Maybe my son will be able to lie around and write poetry."
"I can't imagine your son doing such a thing," Jeremy retorted.
He fell silent at this mention of William's son, for inevitably a son must have a mother, and he knew by now that William wanted to marry Candace. He was in the puzzling place of being the confidant of both his sister and his friend and of being unable to betray to either what the other told him. Each was equally unsure. William had said frankly, only a few days ago, "I don't know if I am doing wisely in letting myself fall in love with Candy. I like her being your sister, I like the notion of being your brother-in-law, you son-of-a-gun! But she's used to everything and I shall have a hard row to hoe. I shan't want her running home to papa, either. When I marry I'll be the boss. If I have to eat cornpone, she'll have to eat it and like it."
William had looked particularly handsome at the moment when he had so spoken. They had come back to their rooms from a stag dinner at their club, and he was wearing new evening clothes presented somehow by his mother. He had gone down to New York to have them fitted.
Jeremy had laughed. "I'll guarantee you won't eat cornpone twice yourself," he had replied. William's taste in food was fastidious and expensive, shaped, Jeremy always said, by his early years of feeding upon shark's fins and bird's-nest soup in Peking.
When Candace had last mused upon marriage in his presence he had warned her that William was hardhearted.
"He has to be the master," he had told Candace.
"Has he been that with you?" she demanded.
"No, because he has not got all he wants from me yet."
"What does he want?"
"He wants power more than anything," Jeremy said thoughtfully.
"That's because he feels inferior," Candace said at once. "He is afraid, in his heart. That's so pitiful, Jeremy. He doesn't know that he needn't be afraid of anything or anybody, because actually he's wonderful. He doesn't know how wonderful he is."
Jeremy grinned in brotherly fashion. "Doubtless he'd like to have you tell him so. But I warn you, Candy! You'll have to give up to him, once he's got you." Then, after an instant's silence, "It makes my flesh crawl."
This startled her. "Why?" she demanded.
He shook his head. "There's no love in him anywhere, for anybody."
"Maybe he's had nobody to love," she said simply.
Fragments of such conversations came back to him as he lay watching William dress.
"You're going to be late," William said, throwing him a sharp look. His light eyes under the dark and heavy brows had a strange metallic quality.
"My family is used to me. They'll wait. Maybe we'll do the waiting. I wish my father had bought an Apperson instead of a Maxwell."
"The Maxwell is bigger," William said.
Mr. Cameron had surprised them all by buying an automobile after Easter, and had chosen the Maxwell for touring. It ran by steam, an idea already old-fashioned, but Mr. Cameron was afraid of the new-fangled gasoline cars.
A gooselike honking rose through the open window, followed by a hissing of steam. Jeremy leaped out of bed, put his head out of the window and shouted to the chauffeur, "Cool her off, Jackson!" He disappeared into the bathroom, snatching towels as he went, soft silky towels embroidered in Ireland with a large and intricate initial.
Left alone, William thought of Candace while he finished his toilet. His fingernails perfected, his coat adjusted, his tie correct, his hair smooth, he examined himself in the mirror. The dark oval of his face did not displease him, although he did not like the faint resemblance he saw there to Henrietta.
He looked at his watch. It was later than he had thought and he wondered if the florist had delivered the pink rosebuds and blue forget-me-nots he had ordered for Candace. His thoughts played pleasantly about her for a moment. He had made up his mind to marry her, and thinking of it he felt a hitherto vague excitement suddenly focus itself. Why should he not ask her tonight? A warm, fine night, the romantic setting of an opulent house, his own sense of success to be crowned soon with _summa cum laude_ —what else did he lack? He was not impulsive, emotion had waxed slowly to this moment, and he would complete this first era of his history by settling the matter of his marriage.
He was so silent and even solemn that Jeremy watched him thoughtfully while dressing. In the car they were compelled to silence, muffled in caps and dusters, while Jackson speeded at more than ten miles an hour across the darkening countryside. There was a rising wind, and when in Boston the door of the huge house opened to them, sustained by a footman, both young men went at once to a dressing room to wash the gray dust from their faces.
William was separated from Jeremy immediately by Martin, come to find him.
"William—I say!" Martin cried in a low voice of excitement. "My old Aunt Rosamond is here and she's interested in the newspaper!" He had pulled William into a corner under the vast oaken darkness of the stairs.
"I can't ask people for money," William muttered.
"Don't be silly," Martin said. He took William by the elbow and pushed toward the ballroom, where an old lady in black lace and diamonds sat in a high-backed chair against some palms.
"Auntie, this is William Lane," Martin said.
William bowed.
"So you're the young man," Aunt Rosamond said in a loud voice. "Come from China, my grandson tells me. It's an awful country, from all I hear, tying up women's feet and killing missionaries!"
"I hope that is over, Miss Rosvaine," William said gracefully.
"Don't talk about China, Auntie," Martin said impatiently. "Talk about our newspaper!" Over the plumed white head, Martin's eye met William's and winked.
"Why should she care about a picture paper for people who can scarcely read?" William asked.
"Aunt Rosamond is a shrewd woman," Martin replied. "Aren't you, Auntie? Why, she tells her own investment men what to buy and what to sell."
Aunt Rosamond giggled. "I'm old enough to be their mother," she said in her harsh, loud voice. "I'm old enough to be anybody's mother. I could be your great-grandmother, only I'm glad I'm not. Young men are so ribald these days. Is your newspaper goin' to make money?"
"Piles of it," William said. "That's why we're starting it."
"I hope its not for any nonsense of doin' good to the masses," Aunt Rosamond said still more loudly.
"Only good to ourselves." William said. "I want to be a millionaire before I am thirty." He knew now that the only way to interest the rich was to suggest more riches.
"You come and see me," Aunt Rosamond commanded with quick interest. She turned large black eyes to his face, and he saw with surprise that once she must have been beautiful.
"Thank you," William said. He turned to Martin. "There is Candace. Do excuse me, Miss Rosvaine." He bowed and left them because he did not want to seem eager before a rich old woman, and he saw in Martin's face the unwilling admiration which he loved.
Walking across the carpeted floor he stopped to shake hands with Mrs. Rosvaine, a gray-haired, handsome woman in a silver gown, and then with Mr. Rosvaine, who looked like the portrait of his French great-grandfather hanging over the mantelpiece. Then he went to the Camerons and, pretending that he saw Candace last, he shook hands with the two elders before he turned to her. She wore a long filmy white dress and carried the roses and forget-me-nots. She looked as a beautiful girl should look and as he wanted his wife to look, and the deep and secret jealousy of his nature rolled up out of his heart. It was intolerable that anyone except himself should possess this precious creature with all her gifts and graces. He might look the world over and not find a woman so suited to him, who was at the same time attainable.
"You look like a princess," he told Candace.
"William, don't tell me you're poetic." She gave him her careless and pretty smile.
"No, just that I'm partial to princesses," he protested. "I grew up in the neighborhood of a palace, in Peking, you know, where princesses lived and played. They're not strange to me."
Mrs. Cameron overheard and said a little sharply. "Are your sisters coming to commencement, William?"
Taken aback he, too, spoke more sharply than he knew. "They're coming tomorrow."
"You're a silent sort of an ape," Jeremy put in. "Why didn't you tell me they were coming?"
"I didn't think you'd be interested," William retorted.
"Of course I am," Jeremy insisted. "You know my sister and am I not to know yours?"
"Henrietta is quite ugly," William said with apparent frankness. "And though Ruth is pretty, I have never discovered anything interesting about her."
"Men never see anything in their sisters," Candace declared.
Their interest in any conversation not connected with themselves waned quickly. In the fashion of the rich, William thought.
"It is going to be hot," Mrs. Cameron said in a plaintive voice.
"You can't possibly be as hot in that outfit as I am in mine," Mr. Cameron told her.
"I don't know," she said. "I have to wear a cor—"
"Mother, spare us!" Candace put in.
"I don't mind William," Mrs. Cameron said. "He's used to us."
"Thank you, Mrs. Cameron," William said. "Come and sit down. I hope you've made Candace keep the first dance for me. She promised it but she never keeps her promises."
"She's a very naughty girl," Mrs. Cameron said with vague indulgence, sitting down.
"I did keep it," Candace said. "And I don't break my promises."
The orchestra began to play and the ballroom seemed suddenly full. William made a smile serve for answer and drew Candace into his arms. He danced beautifully and he was aware of watching eyes. He imagined them thinking of him with admiration, however reluctant. He liked to compel admiration.
Then he looked down and saw Candace's face, calm and beautiful. Her skin was fine and smooth and creamy white, her lips sweet and deeply cut. How fortunate for him if she would marry him soon! Why should they be long engaged? He needed Candace now, for herself and for everything she could bring to him. He would ask her tonight. He could see Jeremy's eyes watching him. It was a man's own business whom he married and when he married. In such dreams, compounded of the many mixtures in himself, he went through the evening, evading Jeremy, dancing with Candace again and again, and when she was not free he asked no one else. Then to his horror he saw her dancing twice with Seth James. Pangs seized him. Seth was one of her kind, the son of a man richer even than her father.
He went to Candace to claim his own last dance. "I can't let Seth look at you like that," he said sternly, as he took her in his arms.
She smiled dreamily without answer and he saw her shoulders shining white and her hair gold in the light of the lamps. He imagined that she was withdrawn from him and instantly he wanted to force her attention to himself.
"I won't tell you how beautiful you are," he said half carelessly. "I suppose Seth has said all that."
"Yes," she murmured.
He imagined that she was holding herself away from him and he drew her closer. "You are not in rhythm."
"They're playing the waltz too slowly," she replied, but she yielded herself, her cheek all but touching his shoulder. Still he was not satisfied.
He stopped and they stood motionless in the whirling crowd. "Come along outside," he said abruptly. "I've been full of something all evening—something I've wanted to say."
He put her hand in his arm and led her away, looking strangely grim for a young man in love. Jeremy, across the room, watched them go through an open door and since for the moment he was not dancing, he went to find his parents. They were waltzing quietly together in a distant corner and they stopped as he came up.
"I just want to warn you," he said in a low voice. "At this very moment William is going to ask Candace to marry him."
"Oh dear!" his mother exclaimed.
His father looked grave. "I don't know that we can do a thing about it," he said after an instant's thought.
Before Jeremy's astonished eyes the two looked at each other and resumed again the slow measures of their waltz. He left them after another moment and then went to pour himself a large glass of whisky and drink it down.
Outside the house, under a wisteria bower in the garden lit by Chinese lanterns, William began his proposal to Candace. He had wondered how this should be done, and had made some half-dozen plans, none of which he now used. She looked so cool, so full of sweet common sense, that he felt it wisdom to approach her in like mood.
"Candy, I think you have known for a long time that I want to marry you, if you will have me."
These were the words he spoke almost as soon as she had sat down. She shook out her little Chinese fan. He had given her the fan last Christmas, a thing of silk and sandalwood which his mother had chosen for him in Peking. He smelled the sandalwood now in the warm air of the night, and childish memories stirred, sandalwood and incense and the close sweet smell of old temples in the hills where the American missionary families had sometimes picnicked in the long, bright northern summers. He turned away from such useless remembrance.
Candace had not replied.
"Well?" he asked a little too sharply.
"I didn't think you would ask me quite yet," she said.
He was not able to tell from the pure cool tones whether she was glad or sorry. "I didn't know, either," he replied in the manner with which he had chosen to present himself to her. "Perhaps I ought to wait until I have some sort of income. But the last few days I've asked myself why I should wait. I'd rather like to remember some day when I've built you a palace and filled it with slaves that I proposed to you when I was penniless and that you accepted me so."
She laughed. "A nice idea!" She waved the fan and once more the scent came blowing against his face. He moved from it half impatiently.
"Then will you, Candy?"
"Will I what?"
"Oh, Candy, don't tease!"
"But you haven't said you love me!"
"Of course I love you."
It was the first time he had ever spoken the words to any creature and they sat upon his tongue like pebbles.
"How strangely you say that!" she said shrewdly.
"Because it is strange to me. I've never said it before to anybody."
This touched her, he could see. She looked at him curiously, her lashes lifted and long. He had the usual amount of passion in him, he supposed, though he had never tried himself. Jeremy was clean and delicate, and though Martin went about visiting strange places, the young men whom William had cultivated were not often physically gross. Lustfulness was not one of his own natural sins. Yet slowly he felt rise in him a strong desire to touch this beautiful girl and, guided by instinct, he put out his arms and felt her come into them. Beneath and against his cheek he felt her hair.
"Dearest!"
The word rose to his lips of its own accord. He had heard his father use it once or twice to his mother. They had not often been affectionate before others, and the word had clung in his mind.
"Will you be good to me, William?"
"Yes, I will. I swear it."
He heard her sigh, he felt her lean against him and the fan dropped to the ground. It seemed to him suddenly that he loved her with all the love he ought to have.
Over the grass, in moonlight and lantern light mingled, a quickened waltz floated upon waves of music and Candace pulled herself away. "Let's go back and dance!"
"But are we engaged, Candy?" he urged.
She stood up but he would not let her go, his arms about her waist. He wanted to be sure she was his before she went back into the rooms crowded with young men.
"I—I suppose so," she said, half unwillingly, half shyly.
"We are!"
He stood up and seized her again and kissed her long and hard. When he released her she gave a little cry.
"Ah, you've broken my fan!"
He had indeed. When he picked up the fan it lay in his hand like a broken flower. He had crushed the filigree with his heel, and the scent was strong in his nostrils.
"Never mind, I'll send to Peking for another, ivory instead of sandalwood, and set with kingfishers' feathers instead of silk."
"Ivory has no scent," she complained. "Give me the pieces, William. I shan't ever like a fan so well again."
He gave them to her, half resentfully, and they walked into the house and began to dance together in silence. He was angry with himself and then with her. The moment that he had wanted to be perfect had ended badly. He had been awkward, perhaps, but she had been unforgiving. Nevertheless he had proposed and had been accepted. They went on dancing.
On commencement day William rose and breakfasted before Jeremy woke, and from the dining hall he went out and across the Yard to the big elm under which he had agreed to meet his sisters and grandparents. They had reached town early, had taken a hack to a small second-class hotel and there had breakfasted.
He saw them waiting for him now, and for a moment they were as detached, as isolated, as a photograph in a family album.
Henrietta was plainer than ever and his grandparents were more middle class than he had thought possible. Ruth had grown up pretty and gentle and he felt a sudden renewal of affection for her. He need not be ashamed of her. But no distaste showed on his resolute young face. He smiled and shook hands properly with his elders.
"How are you, Grandfather? Grandmother, it's awfully good of you, really—I hope the trip wasn't hard." He kissed Henrietta's cheek and squeezed Ruth's slender shoulders in his arms. "Come along. We'll get good seats."
The Yard was coming to life. Seniors in cap and gown were hastening here and there.
He led his guests into the wide-open doors of the hall where a few people were already gathering, and he took pains to find seats where they could see him receive his honors.
"Ruth shall sit on the aisle, so she can see me when we come marching in," he said, and caught her smile.
Henrietta had said nothing since they met. She wore a plain dark blue linen suit and a stiff sailor hat that emphasized the angles of her face. Her eyes were brown like their father's, but they were deep-set and intense, while his were shallowly set and pleasant. This William saw but he did not notice her silence. He was in haste to be off on his own business, to leave them.
"Let's meet again under the elm after this is over."
He met their solemn, dazed eyes, tried to smile, and hurried away. His rooms were empty. Jeremy was gone. He snatched his cap and gown and put them on, glanced at himself in the mirror, and joined the thickening crowd. He felt them looking at him as he strode toward the Yard but he pretended he did not. Confidence, excitement, the assurance of success, were hid behind his set and handsome face. The honor the day would bring him was only the first step to all that lay ahead, and he knew it. He took his place among his classmates, and the important day began, the end and purpose of four long and sometimes tedious years.
Then suddenly he lost it as he was to lose so many days from his life. Everything became unreal to him. His mind seemed to leave his body. It raced ahead into the years, planning, fighting, conquering, gaining all that he wanted. When would he have enough? When would he know and what would be satisfaction? He tried to bring himself back to this hour, which now that he had it seemed no more an end but only a beginning. He even felt vaguely that he was losing it and he wanted to keep it. It was a part of satisfaction, the first step at least toward fulfillment, a fragment of his life completed. He tried to think of Candace as he sat among his fellows; he tried to value the sound of his name upon the list of honor men.
"William Lane, _summa cum laude_ —"
But he had ceased already to value what he had, so immense was his desire for what was yet to come.
When the long morning was over he went at once to his grandparents and his sisters. They were waiting for him under the big elm, and his grandmother murmured affection as he come to them.
"Your mother will be so proud." Her eyes misted with the easy tears of the old.
"My father got the same honors," William said modestly. "It was harder in his day, I daresay. He took much more Greek than I did."
Ruth held out a small package, and he took it with affected surprise. "A chain for your watch," she murmured. "It's nothing much."
"I brought you a book," Henrietta said, producing a package. "I wrapped it in red because it's what they do in China."
"And Grandma and I just have a little check," his grandfather said, giving him an envelope.
"It is all too much," William said gracefully.
Ruth cried out softly, "Let's go and see if there are letters from Mother and Father! I know Mother was going to try to have a letter here on this very day."
"We'll go by my rooms on our way to the hotel," William said.
When he looked in his box there was no letter from China. A few bills were there, still to be paid, and one letter addressed in a hand he did not recognize. It was a tight scrawl, crude and yet formed in some curious personal fashion. He saw on the envelope the address of a town in Ohio that he did not know, and above it was the name of Clem Miller.
"No letter," he told his sisters. "None from them, I mean. Here's a strange one."
He tore open the envelope. Within it was a single sheet of lined paper, upon which was the same cramped, clear handwriting.
> Dear William,
>
> You may not remember me. Once you told me to stop fighting a Chinese fellow in Peking. I never saw you after that. I am here at a grocery store. Got a fair job. Wish, though, I had a chance at your education. Am fighting my way up though. I got your address from your father. Wrote to some friends of mine named Fong in Peking but had forgot a good deal of my Chinese and wrote English thinking maybe their son, Yusan, would be able to read print. He showed the letter to your father, and I got a letter that way telling me you were finishing college. I haven't had the chance. Your father told me to get in touch, and I am doing so in memory of old days.
>
> Yours sincerely,
>
> Clem Miller.
"Who is it from?" Ruth asked, as they walked toward the street.
William was looking up and down for a hack. The sun was getting hot. "You remember that Faith Mission family in Peking?"
Ruth shook her head. "I can't remember very much about Peking."
"I remember them," Henrietta said suddenly. "Let me read the letter."
"You may keep it if you like," William said carelessly. "There is no reason for me to answer it."
He saw a hack, called it, and they climbed in, he taking the small and uncomfortable seat although Ruth offered to sit there. "You are my guests," he said with his best smile.
The day went on, he living each hour of it grimly and correctly. He showed his family about the college and his grandmother suggested seeing his rooms. He put this off until Henrietta was suddenly cross. "I think you don't want us to see them," she declared.
Upon this, with secret anger, he led them to the rooms, dreading the possibility that the Camerons were there. But the rooms were empty, and his grandmother sat down in Jeremy's easy chair and slipped her shoe from her heel. "I bought new shoes for the big day," she said in apology. "You know what they do to your feet."
He did not reply to this dreadful remark, and was restless until he got them up again. Yet not in time, for at the moment when they reached the door Jeremy came in and William could not refuse introductions. Jeremy, with his usual grace, stood talking to the elders and Ruth joined them. Henrietta waited in her stolid fashion.
It lasted but a moment, and he was leading them on again, now toward the gate and the hack. Then they were gone and he felt exhausted and yet he could not show exhaustion, for men he did not know stopped to congratulate him on his honors. He tried to accept their praise modestly, to seem careless as though honors meant nothing to him, but he imagined that they saw through his pretense, and then he grew brief and proud and he felt hurt and weary. He was hot and he wanted a bath and a few minutes' sleep.
Half an hour later, stretched on his bed in his room alone, the shades drawn to shut out the sun, when he tried to think of Candace he found himself thinking instead of Aunt Rosamond. It might be very easy indeed to get money from an old lady like that, perhaps a great deal of money. Then after some deep thinking of this sort he felt that he would like honestly to be ashamed of it, but he could not be. He had nothing and no one to help him. There was not one person in his own family who could be anything but a hindrance to him, and the sooner he separated himself from them the better. He toyed with the memory of Aunt Rosamond's invitation. It meant nothing He knew by now that the rich could speak pleasant words as easily as they breathed, with as little significance. It was hard to be the friend of rich men and their sons, but it was the only way to get what he needed for his own independence. Some day, when he had all he wanted, he would let them know how he despised them.
# 4
ALONE IN HER SMALL HOT ROOM in the suburban house, Henrietta was writing a letter to Clem Miller. She was desperately tired and as usual, after she had been with William, melancholy wrapped her about. His first glance at her had been enough to tell her that she was still ugly, still all that she did not want to be. It was a sign of greatness in her which she did not recognize that she loved Ruth tenderly and humbly in spite of William's preference. Why, she asked herself again tonight, did it matter what William thought? But she did care and would always care what he thought of her. It had begun in the old days in the mission house in Peking when the amah who had served them all had taught her that girls must always yield to the precious only son of the family.
"You," Liu Amah had said, "you are only a girl. Weelee is a boy. Girls are not so good as boys. Men are more valuable than women."
Henrietta sighed. It was late and she should have been sleeping but she could not. Her grandparents and Ruth had gone to sleep, or else by now her grandmother would have tapped at her door to inquire why her light was still on. Swept by the bottomless misery of youth, Henrietta had reached out into the night and had thought of Clem. His letter was still in her handbag and she read it through twice, carefully and slowly. Then she began to write.
> Dear Clem:
>
> You do not know me, but I am William Lane's sister. William is too proud to write to you. He has always been a very proud boy and now he is worse than ever, although he is no longer a boy. He considers himself a man. I suppose he is a man since he has finished college. He is very smart. He was graduated yesterday with highest honors. I am sorry to say I don't think he will ever write to you. But I think someone should, since you knew each other in old Peking, and so I am writing to you.
>
> I don't know anything much about you, and so I will tell you about myself. I am eighteen and next autumn I will go to college, I hope. I am not at all pretty—I had better tell you that right away. It is strange, for I look a good deal like William, and he is thought to be very handsome. I suppose it is not the way for a girl to look. My sister Ruth is pretty.
She paused and realized that she had nothing to say. This was another of her miseries. She felt so much, she was so racked with vague sorrows and longings and infinite loneliness and yet none of this could she put into words to anyone. She and Ruth went to a public school, since all the money had been needed for William, but she had found no special friends there. The girls thought her queer because she had grown up in China. Perhaps she was. She bit the end of her wooden pen and then went on.
> Do you ever think of Peking? I do, often. From the window of my room there in the house where my parents live I used to look out upon a sweet little stubby pagoda—a dagoba, I think it was called. There were bells on the corners, and when my window was open and I lay in bed I could hear them ringing. Please tell me whether you think of such things. And shall you go back one day? I would like to but I cannot think how to earn my living there, not wanting to be a missionary.
Beyond this she could not go and so she signed herself sedately, sincerely his. When the letter was sealed it seemed to her that she must post it at once, even though it was now midnight. The small clock on her mantle gave this severe notice to her but she did not heed it. She put a dress over her nightgown, and with her feet slippered she went silently down the stairs and out of the back door to the street, where stood a postbox. At seven o'clock, she knew, the mail was collected and by breakfast time the letter would be on its way to the small Ohio town that seemed as far away as Peking. She heard the envelope rustle softly behind the shutter, and then she went back home and to her room again. Now she could go to bed. She had put forth a hand into the darkness and perhaps someone would reach out and clasp it. Comforted by hope she flung herself upon her bed and fell into a sleep that led her back into childhood dreams of a walled compound in Peking, a big shadowy mission house, where soft-footed brown servants came and went, bringing smiles and gentle encouragement to a shy and plain-faced American child.
When the letter reached Clem he was in the grocery store. It was the middle of the morning, and Owen Janison, the owner and his employer, came in from his daily trip across the street to the post office. Clem's letters were few and until now they had borne Chinese stamps and postmarks.
"You got a letter from some place in New York, looks like," Mr. Janison said. He was a tall thin man, whose mustaches hung down his chin and joined a faded yellow beard. He wore a gray suit and a stiff white shirt with a celluloid collar.
Clem was shirtsleeved behind the meat counter. He took the letter and looked at it carefully without opening it. "Thanks, Mr. Janison," he said. He slapped a piece of corned beef on the scrubbed wooden counter and trimmed off some porous fat.
"A pound, did you say, Mrs. Bates?" he inquired.
"Mebbe a pound and a half," the customer replied, hesitating. "Mr. Bates is terrible fond of the stuff though I don't eat it myself, more'n a bite."
Clem did not answer this remark. In the years since he and Bump, one weary morning, had walked into New Point, Ohio, he had learned to live upon two levels, the immediate and the real. Mrs. Bates was immediate but not real. Even Mr. Janison, upon whom he and Bump were dependent for their living, was immediate and not real. Real was the past and real the future, both equally clear to him alone. To recapture the past he had written to Yusan, Mr. Fong's son, and he had received the letter from Dr. Lane. Yusan had forgotten his English and had given Clem's letter to the missionary. From Dr. Lane had come a friendly letter, mainly about William and only a little about Yusan. Dr. Lane took it for granted that a youth in America named Clem Miller must be interested in his son William.
Reading the faintly stilted lines of the letter, for anything Dr. Lane wrote fell inevitably into the shape of a sermon, Clem had felt all the old realities. Yusan at sixteen was betrothed to a girl in the mission school, though the wedding was still far off. He had grown into a sober young man, over whose soul the missionary yearned. Yet Yusan refused to be Christian. Real was the memory of Yusan, the stubborn boy, growing into a young man. Real were the hours Clem had spent with him in Mr. Fong's bare house. Real was the memory of the Peking streets, the wind-driven snows that covered the tiled roofs of house and palace in winter. Real were the fabulous summer skies. Clem remembered every detail of his childhood, the pleasure of owning sometimes three small cash with which he bought a triangular package of peanuts wrapped in handmade brown paper thick and soft like blotting paper. Real, too, was the joy of a hot sweet potato on a cold morning, bought from a vender's little earthen oven, and real the pleasure of a crimson-hearted watermelon split upon a July day. Real were the caravans of camels padding through the dust, led by a man from Mongolia who knitted a garment as he walked, pulling from the camels the long strands of wool which they shed when the winter was ended. Real were the little apes on chains and the dancing bears, the traveling actors and the magicians and all that had made the city streets a pleasure for a wandering foreign child.
Out of the need to bring nearer to him that reality of childhood in the remote land which was still his own but which he could not claim and which did not claim him, Clem had upon an impulse written to William, whom he remembered only as he had looked that day when a Chinese lad had called his father a beggar because he trusted God for bread.
The letter Mr. Janison now brought him was, he supposed, from William. He waited, however, until it was time for his midday meal, which he made by taking a roll of stale bread and cutting off a slice of cheese and eating in the storeroom. Mr. Janison went home to noonday dinner and Bump was working on a farm, now that school was over. Clem had been firm about Bump's going to school. He had given up the hope that some day he himself would go to a school somewhere, though not to learn ordinary things like geography and arithmetic, which he could get for himself out of books in his room at night. He wanted to learn large important matters, such as how to feed millions of people. He was obsessed with the business of food, although his own appetite was frugal. A thin, middle-sized boy, he had grown into the same kind of young man. His frame had taken on bony squareness of shoulder, leanness at the hips, without any flesh. Even the square angles of his face remained fleshless and his cheeks were hollow and his blue eyes deep set.
He had discarded the faith of his father, and said no prayers except those he spoke to his own soul. There were, he believed, only a few essentials to a good life, but they were essential to all people, and food he put first, cheap, nourishing food. Bump, for example, could not be filled. He sat sometimes watching Bump eat in the small room they lived in together. He always got a good meal for Bump at night, a stew or a hunk of boiled beef and cabbage and plenty of bread and butter. His own slender appetite soon satisfied, he enjoyed Bump's bottomless hunger. He had provided the food and this was the pleasure he felt. Nobody had given them anything. He had worked and bought the food. He bought cheap food for it was good enough. He had no desire for fancy eatables and was stern with Bump about cake and pie. If everybody could eat his fill of good plain food, he would tell Bump, then there wouldn't be any more trouble in the world.
He was bringing up Bump himself and by himself, sometimes ruthlessly but on the whole kindly, with the deep paternal instinct with which indeed he viewed the world, though he did not know it. His cure for a drunk coming into the store to beg on a winter's night for a nickel to buy "a cup of coffee," was to take a stale loaf and slice off two thick pieces and thrust a wedge of cheese between them. "Eat that and you won't want to get drunk for a while," he said with young authority.
In the back room, the store empty during the town's midday meal, he now sat down on a crate and took the letter from his pocket. Without wasting time on curiosity, he tore the envelope open and was amazed at the first words. He had never had a letter from a girl, nor ever written to one. He had thought little of any girl, being busy at earning his living and rearing Bump. Now a girl had written to him.
He read the letter carefully and considered it a sensible one and read it again. She remembered Peking, too, did she? He felt excited, not because she was a girl but because she, like him, had been born in another world which nobody here knew anything about. He had learned now to live in America, but there would always be the world for him as well, and other people. He could not talk about it to Americans. They did not want to know about it. The people here were satisfied not to know about anything except what happened in their own streets.
He sat musing until he heard the tinkle of the bell that announced a customer, and then he went back into the grocery store. He would answer the letter, maybe on Sunday when he had sent Bump off to Sunday School.
Thus two weeks later, on a Thursday morning, Henrietta received the letter for which she had waited and for which she had gone herself every morning to open the door for the postman. The moment she saw it she took it and thrust it into the bosom of her apron. That day she was cleaning the attic for her grandmother, a musty place, hot under the roof and filled with dead belongings. There she returned to read Clem's letter.
> Dear Henrietta,
>
> It was a surprise of course but I had rather maybe have a letter from you than from William. I am older than you but I know I cannot go to college on account of earning my living. I am an orphan and I have an orphan also to support. I do not even know his whole name, Bump he is called but I am sure it is not his name. He says when he was little he was thought bumptious and so people began to call him that. He cannot remember any family and so was an Aid child. I don't know why I tell you about him. Some day I will tell you how I got him.
>
> I am a poor letter writer not having much time but I would like you to know that I do remember Peking. It would be nice to talk with you about it as nobody here knows anything about it over there. Who knows, sometime maybe I could come to see you though not until I get Bump educated. I have a great many ideas of what I want to do when that job is done when I can think of myself and my own life.
>
> I would enjoy hearing from you again. Yours sincerely,
>
> Clem Miller.
Thus began the passage of letters between a small town in Ohio and a suburb of New York. Without seeing each other for two more years, boy and girl wove between them a common web of dreams. So profound was their need to dream that neither spent the time to tell the other the bare facts of their lives; Henrietta that she had graduated from the big bare public high school almost friendless because the other girls thought her too proud to join their chatter of boys and dances, and Clem that he was grinding out his youth behind a counter in a country store. These things neither considered important They were both weaving together the fabric of the past to make the fabric of the future. It was years before Henrietta learned all the simple facts of Clem's life.
These were the facts. He had turned back that day to see Bump padding through the dust after him. That night they had slept in a barn, taking care not to rouse the farmer and his family, and from it they had set forth again in the early morning.
"Reckon the Aid will chase us?" Bump asked in the course of the next day.
"I don't think she'll care what becomes of us," Clem replied.
The sky was bright above their heads. On that day he began to have his first intimations of his own country. He had walked for endless miles across the Chinese land with an old woman he did not know, linking village to village with his lonely footsteps. Now he walked as many miles with a child who was a stranger to him, across a landscape strange to him, too. Here there were few villages and the farmhouses stood separate and solitary. He avoided them unless he needed food, and then he went to knock upon a kitchen door to ask for work. He was stiff with softhearted farm wives who wanted to give them a meal and he demanded that he be allowed to pay for what he got, and he was equally harsh with surly men who declared there was nothing for him to do. Work there must be, he told them, because they must have food.
How many days he walked in that bright autumn he did not count or care. Slowly he learned to love the look of this land, even its uncultivated spaces, its ragged roadsides, its sparsely settled miles. He learned to be wary of old tramps and to choose the back roads they avoided. In the back roads and the remote farmhouses the people he found were good. They were not gregarious, these countrymen of his. They did not live in big families as the Chinese did. Two generations in a house were enough and maybe too much. More often a man and woman and their children were alone under a roof. The children were usually towheaded and their faces were burned brown with the wind and sun, and because he was a stranger they ran when they saw him just as the Chinese children had done. He thought of these dwellers on the land as folk half wild and scarcely civilized and yet he kept among them.
"Ain't we goin' to settle down somewheres?" Bump asked, as the days went on.
"Some time soon. You have to get to school," Clem said.
"Do I have to go to school?" Bump wailed.
"Surely you do," Clem said sternly.
One day at last they came into a town he liked, though it looked no different from any other. But it was in Ohio, a state that he had come to enjoy in the past days, a place where the people were decent and Bible reading. They made him think of his own Bible-reading parents, mingling kindness with rigid goodness. The streets in the town were clean and there was. a schoolhouse of wood frame painted white. The church, the post office, and the general store stood around a green square, in the midst of which was a rough statue of Abraham Lincoln. These were the reasons Clem chose New Point, and he went first to the store. Inside he found the tall lean man who hired him, after some hesitation, and then let him rent a room upstairs as part of his weekly wage. Clem bought Bump a suit of clothes and a pair of shoes and two pairs of socks on credit, and started him to school the next Monday.
At the end of that Monday he had given Bump his first and only whipping. The boy had come back from school gloomy and had gone upstairs quietly. Clem was busy with a customer and as soon as he was free he hastened up the stairs behind the store. There he found the boy packing his clothes into a flour sack.
"What are you doing?" Clem demanded.
Bump scowled at him from under sunburned brows. "I ain't stayin' with you," he said in a flat voice.
"Why not?" Clem asked.
"I ain't goin' to no school."
Clem glared at the boy who had become his whole family. "Why not?" he asked again.
"I don't like it."
Rage filled Clem's soul. Not to like to go to school, not to take the chance that was offered, not to accept the gift of sacrifice, seemed to him ingratitude so immense that earth could not hold it nor heaven allow it. He rushed at Bump and seized him by the seat of his trousers and swung him clear of the floor. He flung him down flat and knelt beside him and beat him with his open hands until the boy howled. Upon this scene Mr. Janison hastened up the stairs.
"Lay off!" he bellowed. "You want to kill that boy?"
Clem turned upon him a face set and white. "He's going to take his chance if I do have to kill him," he replied and finished his punishment. When he let Bump get up he pointed at the flour sack and waited until the weeping boy had unpacked it and put his clothes away again.
Janison waited, too, a quizzical look behind his mustaches. Then Clem turned solemnly to his employer. "I aim to bring this boy up like my own brother. That means he's going to get a good education, the kind I'd give my eyes to have, nearly. He's to be a man, not some worthless son-of-a-gun."
Mr. Janison pulled his goatee. "Go to it," he said. "That was as pretty a lickin' as ever I see."
He went downstairs again and Clem sat down on the bed. "Bump, I hope never to lick you again," he said gravely. "I don't believe in it and I don't feel I ought to have to do it. But if you dare to run away and throw out a fine chance like I'm offering you, I will come after you and lick you wherever you are. You hear me?"
"Ye-es," Bump sobbed.
"Well, then," Clem did not know how to go on. "You come downstairs and I'll get you some crackers and cheese—and some lickerish," he said finally. Food, he thought, was what the boy needed, and something sweet, maybe.
During the next years, as Bump began to grow into a satisfactory boy, Clem wondered often about his beginnings. That he was a child without parents, Clem knew; without parents, that is, except in the simplest animal sense. Mom Berger had told him one night after the younger children were in bed, that they were all love children, "except that there Bump."
"What is he?" Clem had asked.
"I dunno what you'd call him," she had said mysteriously.
With an embarrassment which sat ridiculously upon her thick person, she had pursed her lips and remained silent. Pop Berger had taken up the sordid story.
"That there Bump," he said after some moments of rumination and chewing upon a vast quid of tobacco. "He's what you might call a rape child."
Clem had flushed. "You mean—"
"Yeah," Pop Berger had said slowly, relishing the evil news. "His paw attacked a girl on the streets of Philly. 'Twas all in the papers."
"Yeah," Mom Berger said from beside the stove. "And I ast you was it real rape. A woman don't rape easy or if she do, it ain't rape."
Pop took the story away from her again. "Anyways, it was brought up in court for to be rape, and the raper, that was Bump's paw, mind you, he had to pay the girl a hunnerd dollars."
"Some women makes their livin' one way and some another," Mom Berger had said, and had clattered a stove lid to let Pop know that enough was enough.
If the story was true, Clem had told himself in reflecting pity, then Bump had no parents at all, neither father nor mother. By the accident of two conflicting bodies he had been conceived, his soul snared somewhere among the stars. He was not orphaned, for even an orphan had once possessed parents. The boy's solitary creation moved all that was fatherly in Clem's being, and it was most of him.
He had not been alone in what he did either for Bump or himself. With the affection so easily found in any small American town, the citizens observed the solitary and ambitious boy. They knew no more about him than that he was an orphan and they took it for granted that Bump was his brother. That he had run away from an eastern state endeared him to them. Mr. Janison soon began to spread news of Clem's monstrous good qualities. His industry was astounding to the employer. When other young males of the town were crazed with spring and the baseball season, Clem continued behind his counter, even staying to sweep the store as usual when the day was over. His belated arrival on the baseball field and the frenzy of those who awaited him only made him more beloved. For all his medium stature, Clem had long strong arms that could perform wheels in the air and send a ball faster than imagination. "A good all-round feller," New Point decided, "a feller that'll make his way."
Two persons kept to themselves their thoughts about Clem. Miss Mira Bean, Bump's teacher to whom Clem had gone after the whipping, knew that Clem was more than New Point discerned. She knew it the first evening he had come to her door, clean and brushed and holding his cap in his hand.
"Come in," she had said with her usual sharp manner to the young.
Clem had come into her small two-room flat.
"My name is Clem Miller."
"Sit down," she commanded.
The rooms were small and crowded with furniture and books. There was little space to sit, and he took the end of a haircloth sofa. Miss Bean was like any of the middle-aged women he saw upon the streets of New Point, a lean, sand-colored shape, washed and clean, straight-haired and gray-eyed.
"What do you want, Clem?" she asked.
"I want to talk to you about Bump," he said. He had gone on then to tell why he had felt compelled to whip the boy.
"But I can't whip him again," he said. "You, Miss Bean, have got to make him like school well enough so he will want to get an education."
"He's got to stay in school, whether he likes it or not," Miss Bean said somewhat harshly. "It's the law."
Clem had sat looking at her. "I don't think you ought to take advantage of that," he said. "The law is on your side, of course. But even the law can't make a boy get an education. It can only make him sit so many hours a day where you are. He's got to like it before he can get educated."
Miss Bean was not a stupid woman and she was struck with this wisdom in a youth who was still too young to be called a man.
"You're right about that," she said after a moment.
She had done her best, not only for Bump, but also for Clem, lending him books, guiding his reading, letting him talk to her for hours on Sundays. For though Clem made Bump go to Sunday School and lectured him about the value of going to church, he himself never went.
"Whyn't you go, then, if it's so good?" Bump grumbled.
Clem, polishing Bump's ragged school shoes, paused to answer this as honestly as he could. "I just can't get myself to it," he confessed. "What's more, I can't tell you why. Something happened to me once somewhere."
"What was it?" Bump asked.
Clem shook his head. "It would take me too long to tell you."
He never told anyone anything about himself. It would indeed have taken him too long. Where would he begin, and how would he explain his origins? How could he ever tell anyone in this peaceful town in Ohio that he had once lived in Peking, China, and that he had seen his parents killed? There were things too endless to tell. Only to Henrietta was he one day to speak, because she knew at least the beginning.
The church bell came to his aid. "You run along," he told Bump briskly. The shoes were polished and he washed his hands in the china bowl. Then he fixed Bump's tie to exactitude and parted his hair again and brushed it. "Mind you learn the golden text," he said sternly.
The minister at the Baptist Church was the other person in New Point who kept to himself his thoughts about Clem. He stopped sometimes in the store to see the industrious young man and to invite him to come to the house of God. He was a red-haired, freckle-faced young minister, fresh of voice and sprightly in manner, and there was nothing in him to dislike. But Clem did fear him, nevertheless, though the young minister was persuasive and ardent.
"Come to worship God with us, my friend," he said to Clem one day at the meat counter. He had come to buy a pound of beef for stew.
Clem fetched out a piece of nameless beef and searched for the knife. "I don't have much time, Mr. Brown," he said mildly. "I really need my Sundays."
"It costs more time in the end not to be a Christian, more time in eternity."
Clem smiled and did not answer. He cut the meat and weighed it, and then cut another slice. "Tell Mrs. Brown I'm putting in a little extra." This was his usual answer to those whom he refused something. He gave a little extra food.
The store, Clem knew as the years passed, was not his final destination. He was learning about buying and selling, and he was learning about his own people. Living among the kindly citizens of the small town, he began to recover from the shock of the farm and the man and woman who lived upon it. In its way, he sometimes mused, it had been a shock as severe as that he had received when he found his parents murdered on that summer's day in Peking. He was taut with nervous energy, he never rested, and there were days when he could not eat without nausea. Food he held sacred, yet food could lie heavy in his own belly. He could not drink milk or eat butter because he could not bear the smell of the cow, and he disliked eggs. Meat he ate almost not at all, partly because he had been so little used to it. He forgot himself. Around the matter of food his imagination played and upon it his creative power was focused. Under Miss Bean's dry guidance, he read economics and came upon Malthus, and lost his temper. The man must have been one of those blind thinkers, sitting in his study, playing with figures instead of getting out and seeing what was really going on in the world. People were starving, yes, but food was rotting because they could not get it. There was plenty of food, there were not too many people, the trouble was that men had not put their minds to the simple matter of organization for distribution. Food must be bought where it was plentiful and cheap and carried to where the people could buy it.
When this idea first came into Clem's mind, its effect upon him was like that of religious conversion. He did not know it yet, but he was illumined as his father before him had been, not then by the satisfaction of feeding human bodies, but by the excitement of saving men's souls. Clem had no interest in saving souls, for he had a high and unshakable faith in the souls of men as he saw them, good enough as God had made them, except when the evils of earth beset them. And these evils, he was convinced, rose first of all from hunger, for from hunger came illness and poverty and all the misery that forced men into desperation and then into senseless quarrels. Their souls were degraded and lost because of the clamoring hunger of their bodies. As simply as his father had left his home and followed God's call across the sea, so simply now did Clem believe that he could cure the sorrows of men and women and their children.
He did not want to leave his own country as his father had done. Here among his own people he would do his work, and if he were proved right, as he knew he would be, then he would spread his plan of salvation to other lands and other peoples and first, of course, to the Chinese. Other people would see his success and follow him. If he had money he would not keep it. He would pour it all into spreading the gospel of good food for all mankind.
On Sundays when Bump was at Sunday School and the town in its Sabbath quiet, Clem in his room alone or walking into the countryside beyond Main Street, planned the business of his life. As soon as Bump was through high school he would begin and Bump could help him. Mr. Janison had offered him a partnership in the store in three more years. He would take it. He had to have a center somewhere. He would make New Point the center of a vast marketing network, buying tons of food in regions where harvests were plentiful, and supplying markets wherever there was scarcity. Meanwhile he must prepare himself. He must learn accounting and management as well as marketing. He must learn the geography of the country until he knew it as he knew the palm of his own hand, so that he could see what harvests could be expected from every part of it.
A vast scheme, he told himself, and a noble one, and he wanted to tell Henrietta. He clarified his own mind for many weeks afterward, writing to her every week of his developing ideas.
"Keep my letters, Henrietta," he told her. "I haven't time to make copies. Sometime I may want to check with myself and see how well my notions have worked."
Henrietta kept his letters with reverence. She bought a tin box and painted it red and kept it locked and in the back of her closet. The key she wore around her neck, and when she wrote this to Clem he sent her a strange dirty-looking little amulet on a string and told her how he had come by it from an old woman in China. "Put it in the box along with my letters," he told her. "It might bring us luck."
William's wedding was in September after his graduation from college. He had not wanted so early a marriage, and he had suggested to Candace that they wait for a year, or even two, until he knew where he was going to find the two hundred thousand dollars he felt was the least possible capital upon which he could hope to start his newspaper. Candace, who could be a laggard when she must decide, had pouted at the idea of delay.
"If it is only money—"
"It is not just money," William said. "I must make my plans very carefully. You don't just start a newspaper. You have to have a prospectus and a dummy and you have to get advertising together."
"You could do all those things as well after we were married as before," she insisted. "I'm going to talk to Papa."
When she said this William was about to forbid her and then he did not. All summer he had worked hard and late in the city and he had worked alone. Through months so hot that one by one Martin Rosevaine and Blayne Parker and Seth James had stolen away to luxurious homes by sea and mountain and lake, William had lived steadily alone in a cheap two-room flat in lower New York, working day and night upon one dummy after another to get exactly the newspaper he wanted. Once a month he allowed himself to visit Candace. Upon such a visit they were now talking.
"I don't want to depend upon your father," he said at last.
"Don't be silly," Candace replied with easy rudeness. "Papa would do anything for me."
"So would I," he said, smiling.
"Then let me talk to Papa," she said.
"But don't ask him for money, please," he replied. "I can find it somewhere."
He was sorely tempted by the old possibility behind her words, for he had felt compelled to delay his marriage while he searched for money. Grimly handsome and determinedly suave, he had made friends wherever he could among the rich. He was not one of them but he knew how to be. Though through this summer he had stripped himself bare as a coolie, a towel about his loins while he sweated at his desk night after night, there had been other nights when his garments were such that he feared no valet as he sallied forth to dine or dance among the wealthy. He did not talk easily but his high-held head and his correct courtesy served him well enough instead. Silence had this value, he found, that when he did talk people listened.
On this next visit, the last before his marriage, Roger Cameron asked him to come into his private library one night after dinner. William knew the room well for he had made free of it during college vacations. The books were curious and heterogeneous, and they provided a fair pattern of Mr. Cameron's self-education. There was a whole shelf on Christian Science and now, in later years, another on the religions of India.
"Sit down," Mr. Cameron said. "Candace has been talking to me."
"I asked her not to, sir," William said somewhat sternly. But he sat down.
"Yes, well, Candy never obeys anybody," Mr. Cameron replied mildly. "Now, William, she wants to get married and she tells me you feel you can't for a year or two."
"I feel only that I should see my way fairly clear before I take on the support of a wife and a house and so on," William said.
"That's reasonable," Mr. Cameron said. "Very right and reasonable. I did no more in my young days. Fact is, I had to wait. Mrs. Cameron's father wouldn't hear to anything else, no matter how she cried or how I got mad. We waited. Well, thinking about that makes me feel I don't want my girl to go through the same thing her mother did. How much money do you need, William?"
William looked reluctant. "I don't know exactly."
"No, I know you don't," Mr. Cameron said with mild impatience. "I'm just asking."
"I think I should see two hundred thousand dollars ahead," William said.
Mr. Cameron pulled his underlip. "You don't need that all at once."
"No, but I have to be able to lay my hands on it."
They were silent for a while. The big room was dark with oak paneling and the lights were lost in the beamed ceiling.
"Suppose you tell me a little more about this paper," Roger Cameron said at last. "What makes you want a paper, anyway? Why don't you come into the Stores with me?"
"I appreciate that, Mr. Cameron," William said very properly. "I do, indeed. But I have set my heart on building up an entirely original sort of newspaper. If it is successful, I shall begin a chain. It will sell for two cents, and it will have more news that two cents ever bought before."
"You'll have to get a lot of advertising," Mr. Cameron said.
"That's where the money will be," William replied. "But it's not entirely a matter of money."
"If it's not a matter of money, what is it?" Mr. Cameron asked with some astonishment.
"I want to accomplish more than making money," William said. He was not afraid to tell Mr. Cameron the truth. His thin erect body, his high head, his small tense hands clasped together were taut with earnestness. "I look at it this way, Mr. Cameron. Most of the world is made up of common people. They are stupid and ignorant. What they learn in school doesn't help them to think. They cannot think. They have to be told what to think. They don't know what is right and wrong. They have to be told."
"People don't like to think," Mr. Cameron said shrewdly.
"I know that," William said. "Therefore they act without thought or they listen to Socialists and agitators and they act foolishly and endanger decent people. I propose to do the thinking, Mr. Cameron. That is why I want a newspaper."
"How do you know people are going to take to your thoughts?" Mr. Cameron asked. He was very much astonished. He did not know himself what to think of this young man with his lichen-gray eyes.
"I won't say they are my thoughts, Mr. Cameron," William said. "I shall do exactly what you do in the Stores. You have men whose job it is to find out what sells best and you buy in quantity what you think people want. Actually, you show people what they ought to buy. That is what I shall do. My paper will be full of what people like. There'll be plenty of stories with pictures about oddities, about murders, about accidents. But there'll be events that happen in the world, too, that people ought to know about."
"Where are your ideas coming in?" Mr. Cameron demanded.
"In the way everything is told," William said. "And not told," he added.
Mr. Cameron shot him a sharp look. "Smart," he murmured. "Very smart. I hope you're always right."
"I won't be always right," William replied. "But I shall try to be."
It was more than he had told anybody, even his friends. They knew that he was to be the editor for he had always assumed that he would be, but they did not know that he planned to shape every item, every line, decide the news he would not tell as well as what he did. The paper would be a reflection of his mind and the direction that of his own soul. When he had put out his first issue he would take it to big business firms and show it to top men. He'd say, "Here is your safeguard. Advertise here and help me influence the people toward Us and away from Them."
"You don't like folks, do you?" Mr. Cameron said suddenly.
William did not know how to answer. Then he chose the truth. "I have a profound pity for them," he said.
"Pity breeds contempt," Mr. Cameron said sententiously.
"Perhaps," William said. "You feel the same way, though, Mr. Cameron."
Mr. Cameron was pulling his lower lip again. "In a way," he admitted.
"I knew it as soon as I saw the Stores," William said. "If you didn't despise people you couldn't sell that stuff to them."
"Here—here—" Mr. Cameron said sheepishly.
"I admire you for it," William said. "But I have a little more idealism than you have. I think the people can be guided to better things."
Mr. Cameron looked at him sidewise. "You may be wrong, William. People are awfully mulish."
William did not yield. "They can be influenced toward something or away from it, just as in the Stores. If you should decide that purple was to be the season's color, you could get people to buy things in purple."
"I don't care," Mr. Cameron said. "It makes no difference to me what they buy."
"I do care," William said.
They did not talk much after that, but after another ten minutes Mr. Cameron got up. "Well, William, whatever your reasons are I'll say this: I'll put away a hundred thousand dollars—half of what you need—and keep it handy, and I want you to go ahead and have the wedding."
William flushed. "Nothing would please me better, Mr. Cameron," he said.
His marriage day dawned as bright as though he had commanded the sun. At that light striking through his open windows he remembered a story his mother used to tell of his childhood. He had waked once at dawn in the old temple where his family was summering upon one of those bare brown mountains outside the city of Peking. The light was pearly above the horizon and he had shouted, leaping out of bed, "Come up, Sun!" At that moment, as though in obedience to his command, the sun sprang above the edge of the earth. He could not have been more than four years old.
The sun had come up as suddenly this morning and he lay realizing as much as he could the meaning of this day. Everything was ready and all that he had to do was simply to be the bridegroom that the day demanded. He had no doubt of himself, for he was to be alone. He had struggled for months over the matter of his sisters and his grandparents and then had dismissed his conscience. Both the girls were at college and his grandfather was not well. The old man was recovering slowly from a stroke and one side of his face was askew. William would not have them at his wedding.
When Candace spoke of them he shook his head. "I don't want them there," he said. She had looked at him with strange eyes and had said nothing.
The bridesmaids were six of Candace's schoolmates and friends. Jeremy was his best man and Martin, Blayne and Seth were his ushers. He had made everything as he wanted it.
The door opened and the valet came in, a middle-aged man with a careful English accent.
"Shall I draw your bath, sir?"
"If you please."
"Mrs. Cameron thought you might like your breakfast fetched on a tray."
"I would, thanks."
The ceremony was to be at noon and they were sailing for England immediately. Roger Cameron was giving them the trip. He was giving them a house, too. Not a large one, but a pleasant small structure of cream-colored brick near Washington Square. William had not pretended that either luxury was in his power to provide.
"Someday I'll be able to do all these things for Candy, sir," he had said, gracefully accepting the gifts.
"Of course you will," Roger Cameron had replied.
The bath water stopped running and a valet held up a silk robe, his head turned away. William got out of bed and drew it about his shoulders.
"Bring breakfast in half an hour," he said with the brusque manner he had learned in his childhood toward servants.
The valet disappeared and William went into the bath. He would stay in his room this morning, away from everyone. The rehearsal had gone off well yesterday. There was no detail left for anxiety. Candace was supposed to sleep until just before she needed to dress for the ceremony. He did not want to see Jeremy or any of the fellows. He could do with two hours or so of pure leisure.
There was a knock at the door and he answered. A footman came in with a small wheeled table on which was set a large tray of covered dishes. In the midst of them was a little silver bowl of roses.
"Your breakfast, Mr. Lane," the man murmured.
"Set it there by the window, Barney," William replied. The man was young and not much older than William himself. He was Irish, as his somewhat shapeless face declared, and his eyes were innocent and humble as the eyes of the poor and ignorant should always be. William liked him and had sometimes encouraged him to talk.
" 'Tis a nice day for it, sir," Barney now said. He arranged the tray by the window, from which could be seen the trees of the park, their green tinged with coming autumn.
"It is, indeed," William said. He had put on his new dressing gown, an affair of blue and black stripes, effective with his dark hair and stone-gray eyes. He should perhaps have kept it for tomorrow when he would be breakfasting with Candace, but he felt that magnificence alone had also its special pleasure.
Barney hovered about the table. "Your eggs is turned as you like 'em, sir, and the toast I did myself."
"Thank you."
"Well, sir," Barney said at last, "my best wishes, I'm sure."
"Thank you," William said again.
Upon such composure Barney retired. When he had eaten William sat for a while, smoking a cigarette and drinking a second cup of coffee. Two hours were left in which he need do nothing. He did not know how to do nothing. He thought of going to bed, but he could sleep no more. He did not want to think about Candace. There would be plenty of time for that. He could not read.
Two hours—a valuable space of time! When would he be alone again? He got up abruptly and went to the desk at the other end of the room and sat down before it. There for the two hours he worked steadily and in silence until the thump upon his door announced Jeremy. It was time to get ready for his wedding.
A perfect wedding, of course, he had expected. Anything less would have surprised and annoyed him. His ushers did their work well and Jeremy was only less efficient. He seemed strangely thoughtful throughout the ceremony and hesitated a long moment when it came to the ring, so long that Candace looked at him with startled eyes. But the ring was there in Jeremy's vest pocket and he gave it to William with a veiled, beseeching look.
William did not notice the look. He was absorbed in the proper conduct of his own part, and he slipped the ring on Candace's finger and made his promises. Going down the aisle a few minutes later, his steps measured to the music, he held his head high in his habitual proud fashion.
The fashionable church was crowded. He looked at no one, and yet he was aware of every personage there. Beside him Candace walked as proudly as he did, but it was he who set the step. He had begun the stately march of his life.
Clem's engagement to Henrietta took place abruptly and even awkwardly. The first tentative letters that they had exchanged had carried far more than their proper weight of meaning. They were secret communications between two persons completely solitary. Though Henrietta had moved apparently serene through public high school in the comfortable, unfashionable suburb, living with Ruth, their grandparents, and the two elderly housemaids, she knew herself as lonely as though she lived upon a desert isle. Ruth was popular and pretty and might easily have married while very young any of several men, even before she went to college. That she did not do so, that she postponed marriage by going to college, was because she visited more and more often in William's home. Vacations soon meant a few hurried days with Henrietta and getting a wardrobe together suitable for the rest of the vacation, even the long summer, with William and Candace. There was no discussion of Henrietta's going, too. Ruth had learned to live delicately between her brother and sister, conveying to each the impression of apology and greater affection.
"I feel guilty," she told Henrietta. "I go flying off and you stay here and take care of the grandparents."
"It is what I want to do," Henrietta said.
Ruth paused in the folding of a silky film. "You would like Candace if you let yourself. Everybody does. She's very easy."
"I daresay I would like Candace but there's William," Henrietta replied with her terrible honesty.
"He is your brother," Ruth persisted, though timidly. She was equally afraid of Henrietta and William.
"I can't help that," Henrietta replied. "Don't forget I knew him long before you did—and much better. We had those two years together at the Chefoo school when you were at home in Peking with Papa and Mama."
Nevertheless, when Ruth was gone, when she had waved to the pretty face under the flowery hat, smiling through the train window, Henrietta knew she was lonely. Like William's the lines of her face were severe and her frame was angular and tall. Inside she was like him and yet how unlike! She was so like him that she could see in herself his very faults. She had no sense of humor, neither had he. But in their spirits there was no likeness. She was possessed with honesty and a depth of simplicity that frightened away all but the brave, and among the young there are few who are brave. Young men feared her and young girls avoided her. There remained Clem, whom she never had seen and who had never seen her. To Clem, in long silent summer evenings, she poured out her feelings almost unrestrained. He answered her letters on Sundays, when he had sent Bump to church. He had no other vacant hour throughout the week. Even on Sundays he had to work on the books for Mr. Janison.
She went to a small girls' college, an inexpensive one, while Ruth had decided to go to Vassar. She did not want to be with Ruth for by then even she could see that Ruth had chosen William and the sort of life he wanted. She listened to Ruth's accounts of that life, repelled and forlorn. Ruth's flying blond hair, her sweet blue eyes, her white skin and slender shape were the means whereby she was welcomed in William's life. William was living in a beautiful house, neither large nor small, on Fifth Avenue. Candace had furnished it in pink and gray and gold. There was a great room where they gave parties. It had been two rooms but William had ordered the wall between taken down. William worked fearfully hard and his paper was getting to be successful. Everybody was talking about it.
"We ought to be proud of him," Ruth said.
Henrietta did not answer this. She sat gazing at Ruth rather stolidly and no one could have known that she was in her heart giving up this younger sister whom she tenderly loved. When Ruth came back from a long summer spent with William, she had been prepared to tell her about Clem. She had planned it in many ways. She might say, "Ruth, I don't want you to think I'm in love, but..." Or she might say, "Do you remember the Faith Mission family in Peking? Well, I know Clem again." Or she might simply choose one of Clem's letters, perhaps the one that explained how he wanted to open a chain of markets, right across the country, in which people could buy good food cheaply, or if they had no money, they could simply ask for it free. "People don't ask unless they must—that is, most people," Clem had written. He had a deep faith in the goodness of people. People didn't like to beg or to be given something for nothing. The human heart was independent. Henrietta was moved by the greatness of Clem's faith. In her loneliness she wanted desperately to believe that this was true. But when Ruth talked about William, Henrietta could not tell her about Clem. The two names were not to be linked together.
Then one day she saw something new in Ruth's face, a quiver about the soft lips, a shyness in those mild eyes. Ruth, catching the loving query in Henrietta's look, suddenly collapsed into tears, her arms around Henrietta's neck and her body flung across her sister's lap.
"Why, baby," Henrietta breathed. She had not used the name since they had been little girls playing house, and she had always been the mother and Ruth her child. She put her arms about the small creature now and hugged her, and felt how strangely long it was since she had offered a caress to anyone. She and Ruth had not been demonstrative in recent years, and there was no one else.
"I'm in love," Ruth sobbed. "I'm terribly, terribly in love."
"Don't cry," Henrietta whispered. "Don't mind, Ruthie. It's all right. It's not wrong. Who is it?"
"Jeremy," Ruth said in the smallest voice.
Henrietta did not release her hold. She tried to remember Jeremy's face as she had seen it when William graduated from college. A nice face, rather thin, very pale, very kind, this she remembered. Then she remembered slow, rather careful movements, as though something inside hurt him, and very pale and delicate hands, bony and not small.
"Does he know?" she asked.
"Yes, he does," Ruth said. She slid from Henrietta's lap to the floor and leaned against her knee and wiped her eyes with the edge of Henrietta's gingham skirt. "He told me first—I wouldn't have dared—"
"You mean you are engaged?" Henrietta asked.
Ruth nodded. "I suppose so—as soon as he dares to tell. Candace knows, but none of us dares to tell William."
"Why not?" Henrietta said with fierceness. "Is there any reason why it is his business?"
"It just seems to be," Ruth said.
"Nonsense," Henrietta replied.
Her mind flew to Clem. Was not this the moment to reveal that she too was beginning to love? But still she could not speak of him.
"I'll tell William myself," she declared.
"Oh no," Ruth said quickly. "Jeremy wants to do it. He will, one of these days. I don't know why he thinks William won't like it."
"I know," Henrietta said. Her voice was gloomy. "William doesn't want the people he goes about with to think he has any family at all. Nobody is good enough for him."
"That's not quite true," Ruth said. "William's very nice to me, usually."
"Because you always do what he says," Henrietta said.
"Well, usually I don't see any reason why I shouldn't," Ruth said. "Anyway, it's to be kept a secret for a while."
She got up from the floor and went to the mirror and smoothed her curls. The intimate moment was over. William had broken it as he always did, and Henrietta said nothing about Clem.
The college year began again and the sisters parted.
Clem's Sunday letters reached Henrietta on Wednesday. She had chemistry laboratory on Wednesday afternoons and among her test tubes she read the long, closely written letter lying between her notes. Then one week there came the letter she had not expected. On Thursday she scarcely ever bothered to go to see whether she had mail, but that day she had happened to pass by the office and, on the chance that there might be a rare letter from her mother, she stopped and found instead another letter from Clem.
"Do I have to be home early?" Bump had inquired.
He was now a pudgy boy who had just begun to wear spectacles. Long ago he had given up rebelling against Clem.
Clem looked at his big dollar watch. "You can stay out till eleven o'clock but you can't play pool."
"I was goin' to the nickelodeon."
"All right."
Thus Clem had had the room alone that Monday night while he wrote to Henrietta. It might have been the solitude that moved him to ask her now to marry him. It might have been his constant wish to comfort her loneliness. It was certainly his unchanging feeling of union with her, though he had never seen her face. She was the only person in the world who could understand when he spoke about his childhood, that other world where all his roots were planted so deeply that there could be no uprooting.
"You and I have not met," he now wrote. "It may seem—" he paused here to look up the word in his dictionary—"presumptuous for me to have the idea. But I have it and I might as well tell you. It seems to me that you and I are meant to get married. I have not seen you nor you me, but I take it we don't care first for looks. There is something else we have together. We understand things, or so I feel. I hope you do, too."
He paused here a long time. When he went on he wrote, "I do not like this idea of proposing to you by letter. If you are willing I will come to see you. Mr. Janison owes me some time and I have saved money. Bump can help in the store after school. I could get away a couple of days and have a whole afternoon with you."
When he had written these words he then went on to tell her the usual news of his life. Bump had got to like school at last and was even talking of college. He'd have to work his way. He himself had given up hope of a real education but he read a lot, Miss Bean telling him what books. He had just finished _The Wealth of Nations._ It was hard going but full of sense. Then he told his big news. Mr. Janison, not having any children, had asked if he didn't want to consider taking over the store some day.
Clem chewed his pen a while when he had written this. Then he went on to tell Henrietta again what he felt and what he had never told anybody except her. "If I do take this store I won't be content just to handle the one outfit. I will likely start up my cheap food stores in other places. I haven't got it all worked out but I believe it can be done like I have told you. Farmers can sell cheap if they can sell direct. Plenty of people need to eat more and better food. I could maybe think out some way even to ship food across to the people in China, or maybe just help them over there, once I learned how here, to get their own food around. It's really a world proposition, as I see it."
He paused again, frowned and sighed. "Henrietta, I hope you will understand that I am not just interested in material things. But I feel that if everybody had enough food so they did not need to worry about where their next meal was coming from, then they could think about better things. I have not the education for teaching people but I could feed them. Anyway, to my thinking, food is something people ought to have the way they have water and air. They ought not to have to ask for it or even work for it, for all have the right to live."
He paused again and closed his letter with these words. "I hope you will forget your brother William's attitude toward you as you feel it is, and remember that I care enough to make up for it to you, if you will let me."
Such a letter deserved many readings before it was committed to certainty, and he read it again and again. There was nothing in it to change, he decided finally, although he would have liked to make it more polished in the writing since she was in college. This he did not know how to do and so he sealed it, addressed it, and took it to the corner postbox. There he noticed by the town clock that it was a quarter past eleven. He was just beginning to allow himself to feel severe about Bump when he saw the light come on in the room above the store. The boy was home, then. Everything was all right. He walked down the street toward the store whistling slightly off key a tune whose name he did not know.
This was the letter Henrietta received on Thursday. She kept it with her all night, waking twice to read it over again by the thin light of a candle shaded against her sleeping roommate. Of course she wanted to marry Clem. No man had ever asked her to marry him, no boy had ever asked her to a dance. Yet she wanted to go slowly about loving Clem and marrying him because it was her whole romance and there would be no other. It was wonderful to feel his letter in her bosom, a warm and living promise of love. She could trust his love as she had not trusted even the love of her parents or Ruth's demanding affection. Tomorrow, in the library where it was quiet, up in the stacks where she had a cubbyhole because she was doing a piece of original research in her chemistry, she would write to Clem and tell him that if when they met, they both felt the same way...
The next day in the cubbyhole, writing these very words, she was interrupted by her giggling roommate.
"Henrietta, there's a man wants to see you!"
"A man?" She was incredulous, too.
"A young man, terribly skinny, covered with dust!"
She knew instantly that it was Clem. Without a word more she ran down the narrow iron steps and across the hall, across a stretch of lawn to the dormitory sitting room. It was early afternoon and no one else was there except Clem. He stood in the middle of the floor waiting for her.
"I had to come," he said abruptly and shook her hand with a wrenching grip. "I oughtn't to have put it in a letter. If a fellow wants to marry a girl he ought to come and say so."
"Oh," she gasped, "that's all right. I didn't mind."
They stood looking at each other, drinking in the detail of the flesh. They were both plain, both honest, both lonely, and one face looking at the other saw there its own reflection.
"Henrietta, do you feel the way I do?" Clem asked. His voice trembled.
Henrietta flushed. Then he did not mind the way she looked, her straight dark hair, her ugly nose and small gray eyes, her wide mouth.
"You might not like me—after you got to know me." Her voice was trembling too.
"Everything you are shines right out of you," he said. "You're the kind I need—somebody to put my faith in. Oh, I need faith!"
She gave a great sigh that ended in a choking gasp. "Nobody has ever really needed me, I guess. Oh, Clem—"
They put their arms around each other awkwardly and their lips met in the passionless kiss of inexperienced love.
He stayed the rest of the day and she forgot her work. They wandered together over the campus and she told him about the buildings and pointed out her window. She took him into the chemistry laboratory, empty by the end of the day, and explained to him what she was trying to do, and he listened, straining to understand the union of the elements.
"I sure do wish I had education," he said with such longing that she could not bear his deprivation.
"Clem, why can't you give up the store and go to college? Lots of fellows work their way through, or very nearly."
He shook his head. "I can't afford to do it. I'm too far on my way. Besides, I haven't time for all of it. I just want to learn what I need—this chemistry stuff, for instance, I have an idea I could discover a whole lot of new foods. Has anybody gone at it that way?"
"Not that I know of," she said.
They took the eight o'clock train to town and had a sandwich together at a cheap restaurant. The night was warm and the darkness was not deep when they were finished. They walked up and down the platform together, hand in hand, dreading to part, now that they had met.
"When shall we meet again?" she asked.
"I don't know," he said. "I ought to ask your father, I guess. Isn't that the right thing?"
"I wish nobody needed to know," she cried with passion. "I wish you and I could just go off together and nobody ever know."
"I guess that wouldn't be just the right thing," he said in a reasonable voice. "I'd feel a whole lot better if I wrote to your father telling about all this. Maybe I ought to tell William."
"No!" Henrietta cried. She scuffed the edge of her shoe along the black cindered ground. "I want it all to myself—until we really are married."
"Won't you tell William?" Clem looked grave.
"No," Henrietta said in the same passionate voice. "At least we don't have to tell William."
"He'll have to know sooner or later," Clem said.
"Let him find out!" she cried.
The train came racketing in, drowning their voices, and they kissed again quickly, mindful of people about them though they were all strangers, and then Clem swung himself up the steps and she stood with her hands in the pockets of her green coat, watching until the train was gone.
# 5
"THERE'S A LETTER FROM your mother," Candace said to William. She never opened letters addressed to him after she discovered during her honeymoon that he did not like it. She wondered sometimes if she were stupid because she could never foresee what he would like and what he would not. But once she knew she never forgot.
It was December and they were in the town house. Next week she must gather herself together for Christmas. She clung to these last days of the year, spending the midday hours in a large glass-enclosed porch. She was pregnant with her second child, and next summer there would be another baby.
Just now, Willie, William's namesake, was nearly two years old. She had been married more than five years. She lay on a long and comfortable chair, feeling a little exhausted, perhaps from her horseback ride in the park. She had not told William that the doctor had forbidden riding because she did not intend to obey such orders. William, had he known, would have insisted upon obedience.
He sat down beside a small metal table and tore open the envelope thick with Chinese stamps. Two letters fell out, one with his father's writing and the other from his mother. He chose his mother's first, for she gave him the most news about what was happening in Peking. She gave the incidents and his father provided the commentary. William was profoundly interested in what was taking place there, for he believed that it was a preliminary pattern of what must happen all over Asia, a surging rise of the common people he feared and distrusted. The mob upon the Peking street had become a memory stamped upon his brain. The one power that could control such madness was in the unconquerable Empress. He remembered the brave old face, impatient and arrogant, bent above him when he was a little boy. He remembered the times he had climbed Coal Hill to look down upon the roofs of her palaces. Having now seen many mansions, he realized that the Old Empress had a magnificence that no mere millionaire could buy. Her palaces were forbidden to all men but no one could forbid an American boy to climb a hill and look down upon her roofs of porcelain blue and gold and upon her marble pillars, and anyone who passed could stare at her closed gates of enameled vermilion.
Early in July his mother had written of a garden party to be given in September in the Summer Palace and to which all diplomats and their friends had been invited. Now he read that it would never take place. The Old Empress had fallen ill on a bright day in the early autumn, his mother wrote. The young Emperor, sitting at his desk, was disturbed by a eunuch running in and crying out, "The Old Buddha is dead!" Without one word, without waiting one instant, the young Emperor began to write upon the sheet of paper he had been preparing for the brushing of a poem. Instead of the poem he wrote an order for the death of that statesman who had betrayed him to the Old Empress ten years earlier, when he had dreamed of making his country new again. Before he could seal the paper, the eunuch came running in to cry still more loudly, "The Old Buddha lives again!" She had rallied, to live weeks longer.
William kept silent, for Candace could not know what the Old Empress meant to him. He read on. She had rallied more than once after that, determined to outlive the young Emperor whom she so distrusted for his eagerness to change old ways for new. He, too, was ill, and she lived and lived again when she heard he was not dead. When she heard that at last he was gone, she gave a great gusty sigh and was willing to die.
"I of scanty merit," the haughty old woman wrote in her last message to her people, "I have carried on the government, ever-toiling night and day. I have directed the metropolitan and provincial leaders and the military commanders, striving earnestly to secure peace. I have employed the virtuous in office and I have hearkened to the admonitions of my advisors. I have relieved the people in flood and famine. By the grace of Heaven I have suppressed all rebellions and out of danger I have brought back peace."
William smiled grimly. Brave Old Empress, brave until the end! She had not died until she had seen that weakling dead, a degenerate youth, a puppet in the hands of revolutionists, who would have unleashed all the madness of the people.
Candace watched him but he did not know it. She could never read his face but she saw the passing smile and wanted to know its cause. "What is it, William? Has something happened?"
"Something is always happening," William replied. He curved his lips downward very slightly. He was reading his father's letter, a short one ending as usual with a bit from the Chinese classics. "We are upon the threshold of wonderful events, now that the cruel old woman is gone," his father wrote. "As Mencius said four hundred years before Christ, The people are the foundation of the State; the national altars are second in importance; the monarch is the least important of them all.' My son, I wish your life could have been spent here in China. It is the center of the coming world, though few know it."
William smiled again at this, a different smile. He did not for one moment believe that China was the center of the world and he did not agree with Mencius.
Candace, watching his face, felt one of her waves of recklessness creep upon her. Why was she afraid of William? She had not been afraid of him before she was married and she could think of no single reason, certainly no incident, to explain why she should now feel that he might be cruel. Jeremy was partly responsible. Jeremy was drinking too much. She had tried to say something to her father about it but he refused to believe it. His religion was a cushion against everything that he did not like and he took refuge in it without shame. There was no use in talking to her mother and she was afraid to tell William. He was hard enough upon Jeremy in the office—hard upon Seth, too. Seth was the chief copy editor. Jeremy was managing editor and stood between Seth and William. William insisted on seeing all the copy and Seth had to make it follow the policy William outlined for his staff upon every event as it came about in the world.
"We don't have to think," Jeremy had said with his too sprightly humor. "It's wonderful not to have to think, Candy. It leaves you so much time."
Seth was not so gay. He refused to talk about William and with Candace he was exceedingly formal. She had to ask Jeremy what was the matter with Seth.
"An independent mind," Jeremy said with his changeless merriment. "It's one mind too many. We don't need it. We have William's."
No one could contradict William. The fantastic success of his newspapers was the final answer to any disagreement with his decisions. In five years the one newspaper he had begun in New York had grown into four, the others published in Chicago, St. Louis, and San Francisco. With a wily combination of pictures, cartoons, and text, William had devised something that had become indispensable to millions of people he never saw. His papers were small enough to handle easily on the subways and while men were eating their lunches at crowded drugstore counters. He gave them exactly what they wanted: financial and business news in a brief space, with a short half column of prediction and advice; news in carefully chosen pictures of tense drama, and photographs cropped to show nothing but concentrated action; news in capsules of simply written, carefully shaped text, suited to millions of people who read with difficulty and thought very little, and who craved constant diversions because of their inner emptiness. William was too clever to preach. What he wanted could be done by his choice of what news to tell and how it was told. Elimination was half the secret of his power, and headlines were the rest of it. Headlines alone could tell people how to think.
Jeremy, Martin Rosvaine, and Seth James met sometimes to talk of the papers and of William. They were awed by his genius while they grew more and more afraid of him.
"In another ten years William will be telling the world what to think and nobody will know it," Martin said. "Of course Aunt Rosamond simply loves it. She won't let him pay her back her hundred thousand."
Aunt Rosamond, as soon as she heard that Roger Cameron had given William a hundred thousand dollars, had insisted on matching it. William had returned Roger's money but it was true that Aunt Rosamond refused any such return.
"The interest is my annuity, William, dear boy," Aunt Rosamond cackled in her hoarse old voice. She was almost blind but now and again she insisted upon a visit from William, and he treated her half affectionately. There was something he liked in the rude, ruthless, selfish old woman who enjoyed his success and laughed at his newspapers.
"Wonderful trash," she called them when they were alone, and gave him a dig in the ribs with her sharp old elbow.
Upon the three young men, however, William's monstrous and increasing success was beginning to have effect. Martin had attacks of conscience, irritated by Aunt Rosamond's greed, Seth threatened rebellion against William's interference with copy, and Jeremy had begun to drink. The long indecision about Ruth, the months when they were half engaged, the months when he felt he did not want to marry anybody, other months when it was Ruth he did not want, had become years. Through it all her unchanging patience, her unfailing sweetness and faithful love had never let him go. In the end Ruth had won.
A month ago Candace thought Jeremy had softened and become more like the boy she had always known, a moody boy, gay with a gaiety she disliked, but capable of times of thoughtful gravity, hours when he could talk with her, moments out of which he sometimes brought a handful of verses to be cherished. He had not written poetry for years, but now perhaps he would again and she hoped he would, for it was good for him to write poetry. Something in him was crystalized and so became permanent.
She thought she understood the change in him when he told her that he had made up his mind to marry Ruth. He had really fallen in love with Ruth at last, she believed, though Jeremy gave as his reason when he told her so that Ruth was the opposite of William and therefore he could not help loving her.
"But you did like William in college," Candace said.
"I got to depend on him," Jeremy said. "I couldn't have passed my exams without him. I have the same feeling now."
"You don't have to work at all," Candace said. "You and Ruth could live somewhere quite happily. Father wouldn't mind."
He looked at her with bewildered eyes. "I don't know why I can't do that," he said.
Only then did she really begin to think about Ruth. "Jeremy, I haven't said I'm glad. But I think I am. Will William like it?"
"Of course he won't," Jeremy said. "Even Ruth thinks that."
"Oh, why not?"
"He has an instinct to deny everybody except himself. He likes to feel he has no flesh and blood of his own. He'd like to have a myth about him that he was born without parents—pure son of God."
Candace was shocked. "That's a mean thing to say when I'm going to have a baby."
"Oh, the baby will certainly be another son of God," Jeremy had said too flippantly. He had been lying on his back on the grass, his body limp, his voice lazy, staring at patches of sky between the leaves of the maples. Candace had not answered him.
"William," she now said, "I want to tell you something."
William folded the letters from China. "Well?"
"Jeremy and Ruth are engaged at last," she said baldly. "I'm glad. It's been on and off for years—he couldn't make up his mind." She turned her head to look at William and saw a bluish flush upon his face.
"When did this take place?" he asked.
"About a month ago."
"And you have known all this time?"
"Not quite all."
She waited for his anger but it did not fall. The bluish flush died away and he was more ashen than ever.
"Don't you think it's rather nice?" she asked.
He got up, his letters in his hand. "I don't think one way or the other about it," he said. "It seems to me a matter of no importance at all."
"Then you won't mind her being married here?"
"I suppose not."
"I'd like to make it a pretty wedding—soon, before I get too clumsy. They don't want to wait."
"Do as you please," William said. He hesitated a moment and then went on rather abruptly. "These letters give me an idea for an editorial I'd like to write for tomorrow. I hope you won't mind if I don't show up for dinner."
"I'll miss you," she said with her coaxing smile.
"I'm sorry," he said rather formally. He bent over her, however, and kissed her hair before he went his way. She watched him as he walked and seeing his bent head, his hands holding the letters clasped behind his back, she thought suddenly that he looked like a priest. That, perhaps, was what William should have been.
Ruth was married on New Year's Eve and Henrietta was her maid of honor. Upon this Ruth had insisted, and Candace had chosen the wedding garments. Ruth of course must wear white satin, but Candace designed for Henrietta a thick, clinging silk of daffodil yellow to be worn with a wide green sash. Henrietta's darkness was made to glow. She did not protest. Holding within her breast the ineffable secret of Clem, she allowed herself to be dressed for the first time in her life with purpose for beauty.
She was twice in William's house, and the first time was after the fitting of her dress, when Candace brought both young women home for luncheon. William was not there, but Jeremy was. He had left the office brazenly early, without telling anybody.
"What is the use of being William's brother-in-law if I have to be afraid of him?" he inquired of them. "He can't fire me."
"Oh, Jeremy," Ruth cried, softly shocked.
"Jeremy is not to be taken seriously since he grew up,"
Candace told Henrietta. "He used to be quite serious when he was a little boy."
They were at the long table in the big dining room, and the mahogany shone through Italian lace. They sat two by two, Henrietta beside Candace, and the ends of the table were empty, though the butler had set William's place. His place was always set, whether he came or not.
"When I was a little boy I was serious because I thought I was going to die," Jeremy said, tilting his wine glass as closely as he could without spilling the red wine. "Now I know I have to live. One has to be gay when one cannot escape life. Eh, Ruthie?"
"I don't know what you're talking about," Ruth cried happily.
The wedding was beautiful. William gave Ruth away since their father was in Peking, and against his dignity her white softness was the contrast of a rose against rock. The wedding was in William's house, although Ruth had wanted a church wedding, and had thought that it would be in St. John's where William and Candace went regularly on Sunday mornings. So it had been planned. But William, at Christmastime, had come into some strange conflict with the rector, which he had never explained, and had withdrawn his membership. He went to church no more and it would have been too conspicuous to have allowed the wedding to take place somewhere else. It was only a small wedding. Ruth had never come out, and she knew few people. There was no reason, William told Candace, why his friends, or hers either, should be invited to come to see a young woman married of whose existence they had only accidentally heard.
The large drawing room made a pleasant place. The florist set up an altar at one end and Ruth's college preacher came to marry them. William was kindly even to Henrietta, and to his grandparents he was almost gentle. They had aged very much. Henrietta matched him in being kind, and thought of Clem and still could not bring herself to speak his name.
None of them were staying after the wedding. They went with Jeremy and Ruth to the dock and saw them aboard a ship for France. William was not with them. A call from his office had compelled him away. Then, with her daffodil dress packed carefully in her suitcase, Henrietta went home with her grandparents.
That night she told them about Clem. They sat together in the large and now rather shabby living room, and she tried to make them see why she must marry Clem.
"He is the only person in the world who knows everything about me." she told them.
They listened simply, knowing somehow that there was very much that they did not know. China was a land they could not imagine and it seemed to them monstrous and inexplicable.
"You won't be going back to that China, I hope," her grandmother murmured.
"I don't know what Clem will do," Henrietta said. "He is always thinking about the world. If he goes of course I will, too."
The old couple had had a hard day and they were not interested in the world. Mr. Vandervent yawned and touched the bell. When Millie, who always sat up until the family was in bed, came he asked for milk.
"Make it hot, Millie, and put a little sherry wine in it."
"I will, Mr. Vandervent," she answered.
A few minutes later, drowsily drinking his sherried milk, he nodded his head to Henrietta. "I suppose it is only what we must expect," he said vaguely. They went upstairs to bed without asking her anything more and she sat down at her desk to write Clem a long letter.
"Clem, I want to be married now. I don't want to go on with my doctorate...."
After her graduation from college she had decided to go on with her doctorate in chemistry with the hope that she could be useful to Clem. This was after something he had said one day.
"I do wish I could have studied chemistry, hon," he had said. "Take soy beans, for instance. Remember how the Chinese eat bean curd? You reckon you know enough to help me, hon?"
"I'd have to study some more," she said.
She was still a little hurt because he had cried out eagerly, "Do you reckon you could, hon?" But she would not let herself be hurt with Clem. She knew his greatness. He could not put himself first.
After she had finished college, _summa cum laude,_ an honor of which she scorned to tell William and which Clem could not fully comprehend, and which seemed only to surprise her parents, she had entered Columbia for more work in chemistry. Now, halfway through, suddenly she could not go on.
She gave her wild arguments to Clem, that nobody loved her and that she was too lonely to live. Even at college she had been lonely because, not having lived in America, she could not talk with other girls. She wanted to be with Clem, and him alone, and never leave him.
Clem sent back words grave and wise about finishing her education and not regretting things later, and about not being able to forgive himself it afterward she were sorry. When he had a torrent of letters from her all saying the same thing over and over again, he knew that it was true that she could die of her loneliness, because it was like his, a spiritual hunger that sent out seeking roots to find an earth its own. It was time for them to come together.
He went to her one day in June and made himself known to her grandparents to satisfy his own conscience, since he could not speak face to face with her father nor would Henrietta allow him to tell William of their love. The old couple were bewildered and anxious to do no wrong, but when Clem talked to them a while they were glad to think that there was nothing they could do. The young people had made up their minds.
"You may write to Father and Mother and tell them you cannot do anything about us," Henrietta said.
Her grandfather sighed. "We won't write, Henrietta. We'll leave it to you."
"It's up to you young people," her grandmother murmured. "We've done our best."
Henrietta was moved to kiss them both for the first time in her life. She was a new creature now that she had made Clem understand that it was right for them to be married at once. She was almost gay. No wedding, she said, for whom had they to invite?
As soon as Clem had the license, she and Clem and the grandparents went one evening to the parsonage of the Presbyterian church nearby and there they were married. She wore her yellow dress, and Clem bought her some shellpink roses to hold. He had bought, too, a wide, old-fashioned gold wedding ring, the only ring she had ever possessed. When Clem put it on her finger she knew it would be there forever, enclosing dust when she was dead.
They went back to the house soberly to eat of a cake Millie had made and drink a toast in burgundy wine from a bottle her grandfather opened. Then she changed into her dark blue silk suit, the only new garment she had bought, and she had a strange uncertain feeling that though her grandparents yearned over her, they were glad to see her go, glad to get youth out of their aging house. They were tired and they wanted to sleep.
# 6
HENRIETTA SAT SEWING IN the small living room of her home. She was not good at sewing. Her fingers were clumsy and the thread knotted often, but it did not occur to her to give up merely because she was not adept and so she sewed steadily on, glancing only occasionally through the window by which she sat. The scene was simple enough, a street of cheap houses much like this one that she and Clem had rented next to the store. Whatever grace the street had came from two rows of maple trees which were now beginning to show the hues of autumn. It was late afternoon and under the trees children were playing in the leaves, running hither and thither, apparently unwatched unless a quarrel brought a mother to the door.
"You, Dottie! Stop kicking your little brother!"
"But I wanna!"
"I don't care what you want. Stop it, I say!"
She wondered if Clem wanted children. They had never talked of children, each for some unspoken reason. She was not sure whether she even wanted children. She had never got used to living in America and she would not know how to bring up a child. In China there had been the amahs. Here she would have to wash all the child's things, and tend it herself when it cried. Besides, Clem was enough. He was a dozen men in one, with all the great schemes in his head. It would be as much as she could do to see that he lived to carry them through.
That he would succeed she did not doubt. From the moment she had seen him in the dingy college sitting room she had believed in him. Trust was the foundation of her love. She could not love anyone unless she trusted and for that reason she really loved no one except Clem and her father.
As long as she lived she would not forgive William because he was angry when he found that she had married Clem. She had written to Ruth, after all, and at first Ruth had not dared to tell William the whole truth. She had let William think the marriage had not yet taken place and he tried to stop it, thinking it still only an engagement. He had actually cabled to Peking to his mother. When she opened the cable from her mother forbidding her too late to marry Clem, she had known it was William's doing.
"That ignorant fellow!" William had called Clem, and Ruth had told her.
Even Ruth was sorry. "I wish you'd told us, Henrietta. It wasn't kind. He isn't suitable for you. You won't be able to bring him to William's house."
"I shall never want to go to William's house." That was what she had answered. She would never be afraid of William, however many newspapers he had. Clem was so innocent, so good. He did not like her to say anything against William.
"He's your brother, hon—it would be nice if you could be friends." That was all Clem said.
When she told him how William felt about their marriage, Clem only looked solemn. "He don't understand, hon. People are apt to make mistakes when they don't understand." She could not persuade him to anger.
She had written to her parents herself, a vehement letter declaring her independence and Clem's goodness, and her father had replied, mildly astonished at the fuss. "I don't see why you should not marry Clem Miller. I should be sorry to see you in the circumstances of his father, but nowadays nobody lives by faith alone."
Her mother had been surprisingly amiable, sending as a wedding present a tablecloth of grass linen embroidered by the Chinese convent nuns. Henrietta guessed shrewdly that her mother did not really care whom she married.
As for Clem, he wistfully admired William's success.
"If William could get interested in my food idea, now, how we could go! He could set people thinking and then things would begin to happen."
"He doesn't want them to think," Henrietta said quickly.
"Oh now, now!" Clem said.
The clock struck six and up and down the street the supper bells rang. She rose to look at the roast and potatoes in the oven and to cut bread and set out milk. Clem would be home soon and he would want to eat and get back to the store. She moved slowly, with a heavy grace of which she was unconscious. Her immobile face, grave under the braids of her dark hair, seldom changed its expression. Now that she was with Clem her eyes were finer than ever, large, and deep, set under her clear brows; yet at times they held a look of inner bewilderment as though she were uncertain of something, herself perhaps, or perhaps the world. It was no small bewilderment thus revealed but one as vague and large as her mind, as though she did not know what to think of human existence.
The door in the narrow hall opened sharply and then shut, and the atmosphere of the house changed. Clem had come in.
"Hon, you there?" It was his greeting although he knew she was always there.
"I'm here," she replied. Her voice was big and deep. He came to the kitchen, his light step quick moving. Their eyes met, she standing by the stove with a pot holder in her hand, and he crossing to the sink to wash. He washed as he did everything, with nervous speed and thoroughness, and he dried his face and hair and hands on a brown huck towel that hung on the wall. Then he came to her and kissed her cheek. He was not quite as tall as she was.
"Food ready?"
"I am just dishing up."
He never spoke of a meal but always of food. He sat down to the roast she set before him and began to carve it neatly and with the same speed with which he did all else. Two slices cut thin he arranged on a plate for her, put a browned potato beside them, and handed the plate to her. Then he cut his own slice, smaller and even thinner.
"Can't you eat a little more, Clem?" Henrietta asked.
"Don't dare tonight, hon. I have a man waiting for me over there."
"You didn't want to bring him home?"
"No. I was afraid we'd talk business all through our food and my stomach would turn on me again. I want a little peace, just with you."
She sat in silence, helping him to raw tomatoes and then to lima beans. Then she helped herself. Neither spoke while they ate. She was used to this and liked it because she knew that in her silence he found rest. They were in communion, sitting here alone at their table. When he was rested he would begin to talk. He ate too fast but she did not remind him of it. She knew him better than she knew herself. He was made of taut wire and quicksilver and electricity. Whatever he did she must not lay one featherweight of reproach upon him. Sometimes she tortured herself with the fear that he would die young, worn out before his time by the enormous scheme he had undertaken, out she knew that she could not prevent anything. He must go his own way because for him there was no other, and she must follow.
In this country which was her own, she still continued to feel a stranger and her only security was Clem. Everything else here was different from Peking and her childhood and she would not have known how to live without him. When sometimes in the night she tried to tell him this he listened until she had finished. Then he always said the same thing, "Folks are the same anywhere, you'll find, hon."
But they were not. Nobody in America was like the Chinese she had known in Peking. She could not talk to anybody in New Point about—well, life! They talked here about things and she cared nothing about things. "All under Heaven..." that was the way old Mrs. Huang used to begin conversation when she went over to the Huang hutung.
She looked at Clem and smiled. "Do you remember how the Chinese loved to begin by saying, 'All under Heaven'?"
"And go on to talk about everything under heaven!"
"Yes—you remember, too."
"I wish I didn't have to hurry, hon, but I do."
"I know, I don't know why I thought of that."
They were silent again while he cleaned his plate and she pondered the ways of men and the things for which they sacrificed themselves. William, sitting in his splendid offices in New York, was a slave to a scheme as much as Clem was, and yet how differently and with what opposite purpose! She could not have devoted herself to Clem had he wanted to be rich for power. He did not think of money except as something to further his purpose, a purpose so enormous that she would have been afraid to tell anyone what it was, lest they think him mad. But she knew he was not mad.
Clem put down his knife and fork. "Well, what's for dessert?"
"Stewed apples. I would have made a pie but you said last time—"
"Pie won't leave me alone after I've eaten it. I can't be bothered with something rarin' in my stomach when I've got work to do."
She rose, changed the plates, and brought the fruit. He ate it in a few bites, got up and threw himself in a deep rocking chair, and closed his eyes. For ten minutes he would sleep.
She sat motionless, not moving to clear the table or take up her sewing. She had learned to sit thus that his sleep might not be disturbed by any sound. His hearing was so sharp that the slightest movement or whisper could wake him. But she did not mind sitting and watching him while he slept. They were so close, so nearly one, that his sleep seemed to rest her, too. Only her mind wandered, vaguely awake.
He opened his eyes as suddenly as he had closed them, and getting up he came back to his seat at the table facing her.
"Hon, I feel I'm wasting you."
She could not answer this, not knowing what he meant.
"Here I have married me a fine wife, college educated, and all she does is to cook my meals and darn my socks!"
"Isn't that what wives are supposed to do?"
"Not mine!"
He looked at her fondly and she flushed. She had learned now that she would never hear the words of love that women crave from men. Clem did not know them. She doubted if he had ever read a book wherein they were contained. But she did not miss them for she had never had them, either. She knew very well that Clem was the only person who had ever loved her, and of his love she was sure, not by words but by his very presence whenever he came near. The transparency of his being was such that love shone through him like light. It shone upon her now as he sat looking at her, half smiling. She saw memory in his eyes.
"Remember that brown Chinese bread we used to have in Peking, hon? The kind they baked on the inside of the charcoal ovens, slapped against the side, and sprinkled with sesame seeds?"
"Yes, I remember... the flat ones...."
"Yes."
"What about it, Clem?"
"I don't know. I get a hankering sometimes to taste it again. What say we go back, hon?"
"To China, Clem?"
"Just for a look around. I might forget what used to be if I saw what Peking is like now."
He looked white and tired and her heart felt faint. Why did she always have that premonition, undefined, unreasonable, that she was stronger than he, more indestructible, more lasting? No flame like his burned within her, and she was not consumed.
"It would be good to go back, Clem."
"Think so, hon? Well, we'll see."
He got up with his usual alertness and the premonition was gone. There was no reason to think—anything! But when he was gone she sat thinking and idle. Yes, she remembered the loaves of sesame bread hot from the oven of the old one-eyed vender. She had often slipped through the unguarded back gate and creeping beside the wall of the mission compound, she had waited, hidden by a clump of dwarf bamboo at the end of the wall. She could hear even now the vender's high call as he came down the street, always at the same hour, that hungry midmorning hour on Saturday when she and Ruth were supposed to be doing their lessons for Monday. He always looked behind the bamboos for her and grinned when he saw her, his jaws altogether toothless.
"Hot ones," she always said.
"Do I not know?" he retorted, and reaching down into the little earthen oven he peeled the bread cakes, two of them, from the sides. His hands were always filthy. Flour and dough blackened by smoke clung in their cracks and his nails were black claws, but she would not think of that in her hunger for the bread. She paid him two pennies and ran back into the compound, the cakes under her jumper. Ruth would not eat them because his hands were dirty and so she ate them herself, the flavor delicious, the sesame seeds nutlike in their delicacy. Clem had eaten that bread, too, but William never had. Like Ruth, William would have thought of the man's dirty hands, but she and Clem thought of the bread, hot from the coals. It was good bread.
She rose and began to clear the table. What Clem was doing was as simple as what the old vender did. Two cakes of bread, for a penny apiece; the old vender made it and went about selling it. If it was good enough people bought it, that was all. Not only bread, either! If anything was good enough and cheap enough, people wanted it. That was all. What Clem was doing was simple and tremendous, so simple that people did not think he was doing anything, and so tremendous they would not have believed it had they known. Only when they saw the finished thing, the bread, the meat, the food, standing there ready to be bought, cheap and good, would they believe. And believing they still would not understand.
Sometimes at night Clem wanted to read the Bible. They did not go to church and neither of them said their prayers unless they felt like it. But sometimes he wanted to read aloud to her. The night before, when they were in bed, he had lighted the lamp and taken up the small old Bible he kept on the shelf under the bedside table. He turned to the place where Jesus had taken the loaves and fishes and had fed everybody that was hungry, and he read it slowly, almost as if to himself, while she listened. When he had read of the baskets of crumbs that were filled he closed the book and lay back on the pillow, his hands behind his head, his eyes fixed on the ceiling.
"That's what I aim to do," he had said. "In my own way, of course. But I like to read once in a while of how somebody else did it. We have the same idea—feed the hungry. I've got to find some way of making food cheaper, hon. I wish I could make it free. There ought to be a way for a starving man to get food without paying for it. There must be a way."
When the table was cleared, the dishes washed, she sat down again to her sewing. The afternoon sun shone down on the quiet street. It was as peaceful and permanent a scene as a woman could look upon, and millions of women looked out upon just such quiet streets in small towns all over America. They would expect to spend their lives there, rearing their children, caring for their grandchildren. But Henrietta, lifting her eyes, knew that for her the street was only a moment's scene. Clem wanted her to go with him, and there was no end to a road once he had set his feet upon it.
Clem was master now in the store. He had bought out Mr. Janison after he and Henrietta were married, and Bump, too, was a full partner. Clem was immensely proud of Bump and, since he was a college graduate, Clem treated him with something like reverence. It was a miracle to Clem to see that the lost child had become a serious, spectacled young man, honest and painfully hard working—though unfortunately without a sense of humor. Bump listened to everything Clem said, and to his nonsense as well as to his commands, to his dreams as well as to his calculations, he gave the same intense attention. He gave his advice when Clem asked for it, which was often, and tried not to be hurt when Clem did not take it. Clem was an individual of deepest dye, and in his way a selfishly unselfish man. He paid no heed whatever to any schemes for the benefit of mankind except his own. He was convinced more than ever that any government would fail unless people were first given a steady diet of full meals, but given this diet almost any government would do, and he preached this as a gospel.
With Bump at his side, always with a pad and pencil, Clem toured the country in one of the earliest of the Ford cars. In villages and out-of-the-way places, wherever crops rotted because the railroads could not serve the farmers, he found ways of conveying the foods by hack, by wagon, and as time went on by truck to railways or to markets. His markets he established anywhere there were people and food near enough to be brought together. Travelers came upon huge, hideously cheap structures in the midst of the tents of migrant workers as well as in the slums of great cities. Some of the structures were permanent, some were immense corrugated tin shacks, made to be taken away when people moved on.
In spite of himself, Clem was beginning to make money. He looked at Bump with a lifted right brow one day and threw half a dozen checks at him across the big pine table in the back room of the store, where he made his head office.
"More stuff for the bank, Bump. I'll have to begin thinking of ways to spend it. All I need ahead is enough to start the next market, but it keeps rolling in. Guess I'll have to begin on the rest of the world."
In this instant an old smoldering homesickness sprang into flame. With money piling up he could go to China at last. He had no wish to stay there. He wanted merely to go back to walk again the dusty streets, to enter again Mr. Fong's house, and to see for himself the graves of his parents and sisters. For Yusan, reviving his English, had written to him long ago the Mr. Fong had gone secretly for the dead bodies and had buried them outside the city in his own family cemetery upon one of the western hills. Upon two heavy Chinese coffins, in each of which was a child with a parent, Mr. Fong had sealed the lids, had lied to the guards at the city gate, and pretended that the dead were his brother and his wife, stricken together of a contagious fever, he had put the wounded bodies into the earth. Could Clem see for himself not only the graves of his dead, but also the faces of the living people friendly again and cheerful as he remembered them, then some secret load of which he never allowed himself to think might roll away. He would be homesick no more for any other country. But he could not go without Henrietta. He could hop into his Ford, rebuilt to his order so that it would survive equally well the hill roads in West Virginia and the sands in Nebraska, and he could leave her for weeks, so long as they were on the same soil. But he could not contemplate the ocean between them.
One day last November he had seen an item in the country newspaper, the only newspaper he read. There was no big headline, and it was not even on the front page. Nevertheless it was a piece of news whose importance no one but himself in the town, perhaps no one but himself in the state and perhaps in the nation could understand. The Empress of China was dead. This in itself was enough to change the atmosphere of his living memory.
Clem read and sat down on a keg and read again. So she was dead, that gorgeous and evil woman, whose legend he had heard in the city over which she had brooded, a monstrous, gaudy bird of prey! When he thought of her gone, of Peking freed of her presence, of the palaces empty, bonds fell from his heart. His parents, his little sisters, were avenged. He need not think of them any more. The past was ended for him.
Now, with these checks before him, it suddenly came to him that it was time to go to China.
"Bump!" he cried. "Take over, will you? I'm going home."
Bump nodded, and the young clerks glared at Clem. But he saw nothing. He walked home with his brisk half trot and opened the front door and shouted:
"Hon, I guess we're going to China now!" From far off, somewhere in the back yard where she was taking dry fresh clothes from the line, came Henrietta's voice.
"All right, Clem!"
Swaying in a temperamental train northward from Nanking, Henrietta gave herself up to nostalgia. In their small compartment Clem gazed, ruminating, from the dirty windows. It was comforting to see good green fields of cabbage and young winter wheat. The Chinese knew how to feed themselves. His stomach, always ready for protest, was soothed and he turned to Henrietta.
"You know, hon?"
"What should I know?" A flicker upon her grave lips was her smile for him.
"When I get to Peking I am going to hunt up one of those old Mohammedan restaurants and get me a good meal of broiled mutton. I have a hunch it would set well with me."
"If you think so then it will," she replied.
They had received no mail for weeks, but she had supposed that at this time of year her parents were in Peking, and soon she would meet them. How she would behave depended upon how they received Clem. Her father, she knew, would be amiable, his nature and his religion alike compelling him to this, but her mother she could not predict. To prepare them she had telegraphed from the bleak hotel in Shanghai. To this telegram she had no answer while they waited for hotel laundry to be done. Twenty-four hours was enough for laundry, but a zealous washerman starched Clem's collars beyond endurance for his thin neck, and the starch had to be washed out again. The laundryman declared himself unable to cope with collars that had no starch, and Henrietta had borrowed a charcoal iron from a room boy and ironed for a day while Clem roamed the streets of the Chinese city. They left the next day without waiting for the telegram. Her father might be on one of his preaching trips, her mother perhaps visiting in Tientsin while he was gone.
At Nanking, however, a telegram reached her, forwarded from the hotel and provoking in its economy: DR. AND MRS. LANE LEFT FOR UNITED STATES.
"But why?" she asked Clem.
"We'd better go on to Peking and find out," he said. "We've been traveling too fast for letters, hon."
So they sat in the compartment and watched the landscape turn from rolling hills to the flat gray fields of the north. Clem was unusually silent and she knew that he was facing his own memories at last. They were tender toward each other, thoughtful about small comforts, and now and again at some well-remembered sight and sound, a chubby child barefoot in the path, the clear sad note of a blind man's small brass gong, they looked at each other and smiled without speaking. She did not ask Clem what his thoughts were, shrinking from intrusion even of love upon that gravity.
The country grew poorer as they went north and villagers, despoiled by bandits of their homes, came to the train platform to beg. They stood in huddles, holding up their hands like cracked bowls, wailing aloud the disasters that had fallen upon them. A few small cash fell out of the windows of second- and third-class windows and once she put out her hands filled with small bills and saw the unbelieving joy upon the faces of the people.
"American—American!" they shrieked after her beseechingly.
"I'm glad you did that, hon," Clem said.
"It's no use, of course," she said and got up and went to the club car because she could not sit still. There, his back to the window and the ruined village and the beggars, a young Chinese in a long gown of bright blue brocaded silk was looking at a copy of one of William's newspapers. She wondered how he had got the paper, but would not ask. Doubtless some American traveler had left it at a hotel, and it had been picked up eagerly, as all American papers were. She sat down near him and after a few minutes he pointed to the photographs.
"Is this your country?"
"Yes," she said. "It is the land of my ancestors."
"How is it that you speak Chinese?"
"I lived here as a child."
"And you come back, when you could stay in your own land?"
"Not everything there is as you see."
"But this is true?"
He kept his eyes upon pictures of rich interiors of millionaires' houses, upon huge motor cars and vast granaries and machinery which he could not comprehend.
"Such things can be found," she admitted.
She wanted to explain to him how anything was true in America, all that he saw and all that was not there for him to see. But she knew it was no use beginning, for he would only believe what he saw, and then she was really convinced that William had done this with purpose, that there would never be anything in the pages of William's papers except what he wanted people to read, the pictures he wanted them to see. And so, no one would ever really know America, and to her the best of America was not there, for the best was not in the riches and the splendor, in the filled granaries and the machines.
She got up because she did not want to talk to the young man any more and went back to the little compartment. Clem sat asleep, his head bobbing on his thin neck. A frightening tenderness filled her heart. He was too good to live, a saint and a child. Then she comforted herself. Surely his was the goodness of millions of ordinary American men, whether rich or poor, and Clem was not really a rich man, because he did not know how to enjoy riches, except to use them for his dreams of feeding people. He liked his plain old brass bed at home, a thing of creaking joints and sagging wire mattress, and he still thought a rocking chair was the most comfortable seat man could devise. He was narrow and limited and in some ways very ignorant, but all the beauty of America was in him, because he talked to everybody exactly the same way and it did not occur to him to measure one man against another or even against himself.
She sat down beside him. Softly she put her arm around him and drew his head down to rest upon her shoulder and he did not wake.
In Peking Clem continued silent. Against his will the horror of old memories fell upon him. Here he had been an outcast child respected neither by Americans nor Chinese, because of his father's faith and poverty. By accident the hotel where he and Henrietta lodged was upon the very street where he had fought the baker's son and where William had descended from his mother's private riksha. He pointed out the spot to Henrietta ten minutes after they had entered the room and for the first time he told her the story. Listening, she discerned by the intuition which worked only toward Clem, that the old pain still lingered.
"William was a hateful boy," she declared with fierceness.
Clem shook his head at this. He was repelled by judgments. "I was a pitiful specimen, I guess." He dismissed himself. "We'd better go and find out about your folks, hon."
So they left the hotel and walked down the broad street, followed by clamoring riksha men who felt themselves defrauded of their right to earn a living when two foreigners walked.
"I'd forgotten how poor the people are. I guess I never knew before, being so poor myself."
"Here is the back gate of the compound," Henrietta said. "I used to creep out here to buy steamed meat rolls and sesame bread."
They entered the small gate and walked to the front of the square brick mission house.
"I was here once," Clem said. "It all looks smaller."
The house was locked, but a gateman ran toward them.
"Where is Lao Li?" Henrietta asked.
The gateman stared at her. "He has gone back to his village. How did you know him?"
"I grew up here," Henrietta said. "I am the Lane elder daughter. Where are my parents?"
The gateman grinned and bowed. "They have gone to their own country, Elder Sister. Your honored father grew thin and ill. He goes to find your elder brother, who is now a big rich man in America."
"Can it be?" Henrietta asked of Clem.
"Could be, hon—want to go right home?"
She pondered and spoke after a moment. "No—we're here. Haven't I forsaken them to cleave to you, Clem? I really have. Besides, Mother would go straight to William, not to me."
Clem received this without reply, and they went away again. The quiet compound, budding with spring, was like an island enclosed and forgotten in the midst of the city. The only sign of life was two women and a little boy at the far end of the lawn, digging clover and shepherd's-purse to add to their meal that night.
"It all seems dead," Henrietta said.
"It is dead, hon," Clem replied. "In its way all that old life is dead, but the ones who live it don't know it—not even your father, I guess. What say we find the Fongs?"
Mr. Fong had prospered during the years of civil war. Ignoring the political maneuvers of military men and passing by in silence the rantings of students upon the streets, he had begun to stock his book shop with other things people wanted to buy, needles and threads, brightly colored woolen yarns, clocks and dishes, machine-knitted vests and socks, leather shoes and winter gloves, pocketbooks and fountain pens and tennis shoes, pencils and rubber hot-water bottles. Most of his goods came from Japan and he was uneasy about this, for young students who were also zealous patriots often ransacked shops, heaped the goods in bonfires, and pasted labels on the shop windows announcing that so and so was a traitor and a Japan lover. Mr. Fong made two cautious trips a year to Japan to buy goods, and he had consulted with the Japanese businessmen with whom he did such profitable trade, and thereafter his goods were marked "Made in USA." A small shipping town in Japan was named Usa for this convenience. Mr. Fong had then continued to prosper without sense of sin, for he considered all warfare nonsense and beneath the notice of sensible businessmen. He had peace of mind in other ways, for his family shared his health and prosperity and his eldest son had continued to improve the English which Clem had long ago begun to teach him. Yusan was now a tall youth, already married to a young woman his parents had chosen for him, and she had immediately become pregnant.
On a certain clear cool day in early spring Mr. Fong felt that life would be entirely good if politicians and soldiers and students were cast into the sea. His content was increased by the pleasant smell of hot sugar and lard that Mrs. Fong was mixing together in preparation for some cakes, helped by his eldest daughter, who was already betrothed to a young man whose father was a grain dealer. Mr. Fong's two younger sons, Yuming and Yuwen, were playing with jackstones in the court, for the holiday of the Crack of Spring had begun.
Upon this pleasant household Clem and Henrietta arrived. The door was opened by Yuwen, who had been born after Clem went away. Nevertheless the American was a legend in the Fong family and Yuwen recognized him with alacrity and smiles. He left the door ajar and ran back to tell his father that Mr. Mei had come back. Mr. Fong dropped his pipe and shouted for Yusan, who was in his own part of the house and made haste to the gate.
With hands outstretched he greeted Clem. "You have come back—you have come back!" he spluttered. "Is this your lady? Come in—come in—so you have come back!"
"I have come back," Clem said.
Thus Clem with Henrietta at his side entered again this fragment of the old world of his childhood and smelled again the familiar smells of a Chinese household, a mingling of sweetmeats and incense and candles of cowfat. There was even the old faint undertone of urine, which told him that Mr. Fong had not become more modern during the years and that he still stepped just outside his door when it was necessary. Smell of whitewash from the walls, smell of old wood from the rafters, and the damp smell of wet flagstones in the court were all the same. The pomegranate tree was bigger, and the goldfish in the square pool, roused by the sun, were huge and round.
Clem gazed down into the shallow pool. "Same fish?"
"The same," Mr. Fong said. "Here everything is the same."
A scream made them turn. Mrs. Fong rushed out of the open doors of the central living room.
"You are come—you are come!"
She took Clem's hand in both of hers. "He is like my son," she told Henrietta. Her round face was a net of smiling wrinkles.
"You must take her for your daughter-in-law." Clem said. "Her father is Lane Teacher."
"A good man, a good man!" Mr. Fong cried.
Yusan came out next and he and Clem shook hands in the foreign fashion, and then Yusan put his hand over Clem's. "We have often asked the gods to bring you back to us." To Henrietta he said with great courtesy, "My inner one asks you to go to her. She is very big just now with our first child, and does not like to come out before men she has not seen before."
"Come with me," Mrs. Fong said, and Henrietta stepped over the high wooden threshold.
"We will sit in the sun," Mr. Fong said to Clem. "I do not need to be polite with you. Yuming, Yuwen—do not stand there staring. Go and fetch tea and food."
The three men sat down upon porcelain stools set in the court and Mr. Fong surveyed with love this one returned. "You are too thin," he told Clem. "You must eat more."
"Elder Brother, I have a weak stomach," Clem replied.
"Then you are too agitated about something," Mr. Fong said. "Tell me what it is. You must not agitate yourself."
Thus invited Clem began talking, as he always did sooner or later, about his hope of selling cheap food even here in China.
Mr. Fong and Yusan listened. Yusan never spoke before his father did, and Mr. Fong said, "What you have undertaken is far beyond the power of one man. It is no wonder that you have a weak stomach and that you are too thin. A wise man measures his single ability and does not go beyond it. What you are doing is more than a king can do, and certainly more than the Old Empress ever did. As for these new men we have now, they do not think of such a thing as feeding the people."
"Are they worse rulers than the Old One?" Clem asked.
Mr. Fong looked in all four directions and up at the empty sky. Then he drew his seat near to Clem's and breathed these words into his ear.
"In the old days we had only certain rulers. There was the Old Buddha and in each province the viceroy and then the local magistrate. These all took their share. But now little rulers run everywhere over the land. It is this little man and that little man, all saying they come from the new government and all wanting cash. We are worse off than before."
The two younger boys came out with an old woman servant bringing some of the new cakes and tea.
"Eat," Mr. Fong said. "Here your mind may be at peace and your stomach will say nothing."
Not in years had Clem eaten a rich sweetmeat, but he was suddenly hungry for these cakes that he remembered from his childhood. He took one and ate it slowly, sipping hot tea between each bite.
"When one eats lard and sugar," Mr. Fong said, "hot tea should surround the food.... Thus also one drinks wine with crabs."
Clem said, "Strange that I do feel peace here as I have felt it nowhere else. In spite of the wars and the new rulers, I feel peace here in your house." His Chinese lay ready on his tongue. He spoke it with all the old fluency and ease. His thoughts flowed into soft rich vowel sounds in the rising and falling tones.
"We are at peace here," Mr. Fong agreed. "The outside disturbance has nothing to do with our peace within. Stay here with us, live here, and we will make you well."
In a corner Yuming and Yuwen were eating cakes heartily in front of a fat Pekinese dog, who snuffled through his nose and blinked his marble-round eyes at the hot delicious fragrance. It did not occur to either boy to share his cake with the dog. To give a beast food made for human beings would have been a folly, and the Fongs did not commit follies. A hard, age-old wisdom informed them all. Clem sat watching, relaxed, though he was not less aware of all that weighed upon his conscience. Peace was sweet, and sweet it was to find nothing changed. Of all places in the world, here was no change.
In the small square central room of the three rooms which Mr. Fong had allotted for his son and his son's wife, Henrietta sat between Mrs. Fong and Jade Flower, who was Yusan's wife. Each held one of her hands and stroked it gently, gazing at her and asking small intimate questions.
"How is it you have no child?" Mrs. Fong asked.
"I have never conceived," Henrietta replied. She had been afraid at first that she could no longer speak Chinese, but it was there, waiting the sight of a Chinese face. Something warmly delicate, the old natural human understanding she remembered so well and had missed so much was between her and these two.
Mrs. Fong exclaimed in pity. "Now what will you do for your him?" "Him" was husband. Mrs. Fong was too well bred to use the word.
"What can I do?" Henrietta asked.
Mrs. Fong drew nearer. "You must mend your strength. You are both so thin. Stay with us and I will feed you plenty of red sugar and blood pudding. That is very good for young women who do not conceive quickly. When you have been with us a month, I will guarantee that you will conceive. My son's wife was less than that."
"Fourteen days," Jade Flower said in a pretty little voice, and giggled.
Mrs. Fong frowned at her, then smiled and concerned herself again with Henrietta.
"Have you been married more than a year?"
"Much more," Henrietta said.
Mrs. Fong looked alarmed. "You should not have waited so long. You should have come to us before. Do they not understand what to do in your country?"
"Perhaps they are not so anxious for children," Henrietta replied. She could not explain to this woman, who was all mother, that Clem was somehow her child as well as her husband, and that she did not greatly care if there were no children, because she did not need to divide herself. Mrs. Fong would not have understood. Was it not for the man's sake that a wife bore children?
"It may be better to take a second wife for him and let her bear the children for both of you," Mrs. Fong said.
"This is not allowed in our country," Henrietta said.
Mrs. Fong opened her eyes. "What other way is there for childless wives?"
"They remain childless," Henrietta said.
Jade Flower gave a soft scream. "But what does he say?"
"He is good to me," Henrietta said.
"He must be very good," Mrs. Fong agreed. She stroked Henrietta's hand again. "Nevertheless, it is not wise to count on too much goodness from men. Little Sister, you shall drink red sugar in hot water and I will kill one of our geese and make a blood pudding." She looked at Henrietta. "Can you, for the sake of a child, drink the blood fresh and hot?"
"I cannot," Henrietta said quickly.
"That is what I did," Jade Flower urged. "I drank it one day and soon I had happiness in me."
Mrs. Fong frowned at her daughter-in-law and smiled at Henrietta. "We must not compel," she advised. "Not all women are alike. Some women cannot drink blood, not even to have a child. If they drink it, they vomit it up. I will make it into a pudding. Two or three puddings, one every day. Then we will see—we will see—" and she stroked Henrietta's hand.
"You trouble yourself without avail," Mr. Fong said to Clem. They had been several days in Peking, living in the home of the Fong family. Clem's digestion ran smoothly and he was more quiet in mind than he had been for years.
"How do I trouble myself?" he asked.
They sat in the big family room, a comfortable, shabby, not-too-clean place, where the dogs wandered in and out and the cats sprawled in the warmest spot of sunshine, and neighbor children came to stare at the Americans, while Mrs. Fong bustled everywhere. Henrietta was unraveling an old sweater to knit a new jacket and cap for the Fong grandchild to be born now at any hour.
Mr. Fong cleared his throat and spat into a piece of brown paper, which he then threw under the table. "You think that you, one man, can feed the whole world. This is a dangerous dream. It only gives you the stomach trouble of which you have told me. Nothing is more dangerous than for one man to think he can do the work of all men."
Clem's skin prickled at this criticism. He was secretly proud of his dream, which he had done so much to fulfill. At heart a truly modest man, he had nevertheless the modest man's pride in his modesty in the face of achievement.
Mr. Fong, wrapped in an ancient black silk robe long since washed brown and ragged at the edges, perfectly understood what Clem was feeling. He looked at him over his brass spectacles and said, emphasizing his words with his forefinger, "It is presumptuous for man to consider himself as a god. The head raised too high even in good will be struck off too soon. Each should tend only his own. Beyond there is no responsibility."
He picked up a cat that happened to be lying by his chair and held it uncomfortably about its belly. "This creature is blind. I do not feed any of the cats, not even this one. They are here to catch mice. But the other cats bring at least one mouse each day to this blind cat."
The aged cat, outraged by his grasp, now scratched him with both hind and forelegs and yowled. Immediately three cats came into the room and looked pleadingly at Mr. Fong, who dropped the cat and wiped his bleeding hand on his gown.
"Please continue to teach my husband," Henrietta said. "I want him to live a long life."
Mr. Fong inclined his head. He was so much older than Clem that he knew he could say anything to him. Meanwhile nothing Clem said impressed him. Yusan listened with deference, since in this case he was the younger man, but he had no wish to take the part which Clem wanted to put upon him.
"I shall certainly see that my own family is fed, and such others as are dependent upon us. It would be foolish to go further." This was Yusan's conclusion. He went about these days from shop to house in perpetual readiness to hear a small loud cry from the three rooms which were his home under this roof, and he was impervious, in his generation, to the cries of others.
Clem, walking with Henrietta one afternoon upon the city wall, a vantage which gave them a wide view over the roofs of houses and the green trees of the courtyards, paused to gaze down into the vast square of the city. The palace roofs were brilliant under the sun of autumn and the temple roofs were royal blue. "I guess Yusan doesn't get my ideas," he said sadly enough to arrest Henrietta's wandering attention.
"Oh well," she replied for comfort, "there aren't any very hungry people around. Maybe that's why. Even the beggars are fat."
She loved Clem with the entire force of her nature but she had never shared his sense of mission. For that, too, she must perhaps thank this city where she had spent her childhood and where she had learned early that women were of little value. It was a lesson to be learned soon, for it needed to be lifelong. Nothing in America had taught her more or differently. She was useful to Clem, and as long as he needed her, her life had meaning.
"I wish I could see Sun Yatsen," Clem said suddenly. "I believe he'd understand what I'm talking about."
"Who knows where he is?" Henrietta asked.
Clem paused for thought. "I believe Yusan knows."
"Then ask him," Henrietta suggested.
Instead Clem decided to ask Mr. Fong. He did not believe that there were secrets between this father and son.
Mr. Fong received the question with calm.
"The time is not ripe for Sun Yatsen's return," he said.
"Where is he, then?" Clem demanded.
"Perhaps in Europe, perhaps in Malaya," Mr. Fong said. "He is gathering his powers."
"At least he is not in China?"
"Certainly he is not in China," Mr. Fong said firmly.
Clem said no more. The atmosphere in Peking was one of waiting, neither anxious nor tense. Empire had gone, in all but name, and the people did not know what came next. But they were at peace. They had never been dependent upon rulers and governments. Within themselves they had the knowledge of self-discipline. Fathers commanded sons, and sons did not rebel. All was in order, and would remain in order so long as the relationship held between the generations. Meanwhile the people lived and enjoyed their life.
Clem's early mood of unusual relaxation changed to restlessness. The peace of the Fong household began to weigh on him. The grandchild was born, fortunately a son, and Yusan was immediately absorbed in fatherhood. Old Mr. Fong relapsed into being a contented grandfather. Although Clem and his wife were welcome to stay the rest of their lives, they were becoming merely members of the family.
The end of the visit came on the day when Mr. Fong and Yusan hired four rikshas and took Clem and Henrietta outside the city walls to the graves upon the hills. The visit had been many times postponed, Mr. Fong saying that Clem must not be disturbed by sorrow until his digestion was sound. Suddenly he had decided upon the day, and Yusan had so told Clem on the night before.
"Elder Brother, my father has prepared the visit to your family tombs. Tomorrow, if you are willing?"
"I am ready," Clem said.
So they had set out, and an hour's ride had brought them before two tall, peaked graves. Clem stood with bowed head while Mr. Fong and Yusan thrust sticks of incense into the ground and lit them and Henrietta picked wild flowers and laid them upon the weedy sod. There was no other prayer. Clem took Henrietta's hand and they stood together for a few minutes, he remembering with sad gravity what was long gone, and she comforting him.
When the moments were over they got into their rikshas again, and when Clem got back he went aside with Mr. Fong and tried to tell him his gratitude.
"You have kept the graves of my parents as though they were your own family," Clem said.
"Are not all under Heaven one family?" Mr. Fong replied.
Nevertheless he perceived thereafter Clem's restlessness. One day he invited Clem to come into his private office, a small square room behind the shop, with enclosed shelves upon which were the old account books of five hundred years of Fong shopkeepers.
Mr. Fong closed the door carefully and motioned Clem to a seat. Then he opened a drawer of his desk and took out a slip of paper upon which an address was brushed in Chinese characters.
"Go to this place," Mr. Fong said. "You will find the one you seek. Give him my name to send you in, and if he asks for further proof, describe this room. He has sat upon that very chair where you sit."
Clem looked at the paper. It bore an address in San Francisco.
"You had better go at once," Mr. Fong said. "He comes back soon. Something will happen this very month here in this city. Whether it fails or succeeds he will come back. If it is successful he will take power. If it fails, he must come to comfort his followers."
Clem got up. "Thank you, Elder Brother," he said to Mr. Fong. "I hope I can repay you for your faith. I hope he'll listen to me."
The next day he left Peking, Henrietta with him, but not yet understanding why he must go away so quickly.
"I'll tell you, hon," Clem said. "I'll tell you as soon as I have time."
There was time only when Clem was imprisoned by the sea. In Shanghai he spent money like the rich man he was that he might get berths upon an Empress ship leaving the dawn after they had arrived. He could haggle over the price of an overcoat and he had never worn a custom-made suit in his life, but when it was a matter of getting what he wanted, money was only made to be used. They caught the ship and Clem, studying timetables, planned the swiftest route from Vancouver to San Francisco. The English ship was still the most swift.
"One of these days we'll fly, hon," Clem said to Henrietta. "Before I die, that will surely be."
"We'll fly in heaven, I suppose," Henrietta said now with her small smile.
"Long before that," Clem said. "It'll be a sorry thing for many if they have to wait for heaven!"
At last, almost reluctantly, on the second day out, he told Henrietta why it was he wanted to see that man, Sun Yatsen.
"He's going to get China, see, hon? I can feel it in my bones. The people there are just waiting for somebody to save them and he has risen out of nowhere, the way savior men always do. They come up out of the earth, see? They get an idea, a big idea—just one is enough. He's got the idea of giving the Chinese people their own government. Well, he'll do it if he can get them to believe in him. People got to have faith, hon. He's got to have faith, too. Everybody who does anything has got to have faith in a big idea. So I'm going to him and I'm going to say, look, if you give the people food, they'll believe in you. Now how are you going to give your people food? Some men do it one way, some another, but nobody ever got people to follow him without giving them food. People have got to be fed. Remember Jesus and the loaves and fishes."
He was standing against the rail, his back to the sea, and Henrietta was lying on the long chair he had lugged here by a lifeboat on the highest deck, away from everybody as she liked to be. By squinting a little she gazed at his face, and imagined that the bright sea shone through his eye sockets, so blue were his eyes this day. The color of his eyes was a barometer of the measure of his hope. When he was on the crest of a new hope, his eyes were sea blue, and when he was cast down, as sometimes he was, they were almost gray.
"He'll listen to you," Henrietta said. "I'm sure he will listen to you."
The train from Vancouver reached San Francisco just after sunset. Clem deserted Henrietta at the station.
"Hon, you can get yourself to the hotel, can't you? Hop into a hack with our stuff. I guess the Cliff House is all right. Wait there for me—don't go out walking by yourself or anything!"
It was Clem's fantasy that Henrietta must not walk out alone after dark lest she be molested.
"You'd better tell me where you are going," Henrietta said. "If you don't come back I'll know where to look for you."
"I'll get back all right," Clem said. "Chinese all know me, I guess."
He hurried off, too busy to do what she asked, jumped into a horse cab and gave directions. Then he sat, taut, leaning dangerously forward while the cabman drove him over the rough streets. He sought the Chinese rebel in one of the miserable tin shacks which had sprung up in the ruins of old Chinatown after the great fire. The old dark beautiful city within a city, small and close, set like a gem within San Francisco—the haunted narrow streets that were the center of Chinese life transplanted and nourished by generations of homesick Chinese—had been wiped out. Those living creatures who remained alive had made such shelters as they could, and they walked the streets, still dazed and lost. There was no beauty springing new from ashes.
Clem, however, did not include beauty within the necessities. Oblivious to ugliness, he dismissed the cab and walked briskly through the dim streets to the address he had memorized, so often had he read it. Even the smell of old Chinatown was gone, that mingling of herbs and wine, that scent of sandalwood and incense, that sad sweetness of opium and the lusty reek of roasting pork and garlic and noodles frying in sesame oil. The sound of temple bells was gone, and the venders were no more. The clash of cymbals from the theater was silenced and the theater itself was still in ruins. Instead the night air was weighted still with the acrid smell of ash and seaweed and charcoal smoke from the braziers of families cooking in the open.
On the old Street of Gamblers, its iron gates a ruin of twisted rust Clem found the place he sought. The door was locked, a flimsy partition of wood, and he knocked upon it. It was not opened at once and he heard the sound of voices within.
"Open the door!" a strong voice said. "Of whom am I afraid?"
Then it was opened, and a cautious yellow face peered into the twilight.
"What thing you want?" the face asked.
"I am looking for the Elder Brother," Clem said in Chinese.
Clem held up his left hand and on the palm he traced with the forefinger of his right hand the ideograph of Sun.
"Come in," the face said. The door opened widely enough to let Clem in. The shack was one room, partitioned by a curtain, and it could be seen that it belonged to a laundryman. The face belonged to the laundryman and he went back to the table piled with the clothes he was ironing, paying no further heed to Clem.
Two men sat at a small table scarcely larger than a stool. One was Sun Yatsen, the other was the cramped, humped figure of an American.
Clem spoke to Sun. "I am sent here by Mr. Fong, the bookseller on Hatamen Street, in Peking."
"I know him," Sun replied in a quiet voice.
"I have come with an idea which may be useful to you," Clem said.
"I have no seat to offer you," Sun replied. "Pray take mine."
He rose, but Clem refused. The laundryman came forward then with a third stool and Clem sat down. Sun did not introduce the American.
"Proceed, if you please," he said in his strangely quiet voice. "I am to set sail shortly for my own country, and these last days, perhaps hours, are valuable to me."
"Has the news been good or bad?" Clem asked.
"It is bad," Sun said. "I am used to bad news. But I must get home."
The hunchback interrupted him with a high sharp positive voice. "The news will always be bad unless you have an army. No revolution has ever succeeded until there was an army."
"Perhaps," Sun Yatsen said, without change in voice or face.
"I haven't come to talk about an army," Clem said. He felt uncomfortable in the presence of the white-faced hunchback. He hated intrigue and he did not believe revolutions were necessary. People fought when they got hungry. When they starved they were desperate. But after it was over everything depended again on whether the new rulers fed them. If not, it all began over again.
"I want to talk to you about food," Clem said abruptly. "I want to tell you what I believe. People will never be permanently at peace unless the means of getting food is made regular and guaranteed. Now I have worked out a plan."
He leaned forward, and began to speak in Chinese. Thus he shut out the hunchback. He had a feeling that the hunchback was an enemy. That small bitter white face, tortured with a lifetime of pain and misfortune, spoke cruelty and violence. But if he had thought by speaking in Chinese to drive the man away, he failed. The hunchback waited motionless, his eyes veiled as though he were asleep. The laundryman stopped ironing and listened to Clem's quick, persuading words.
"True, true," he muttered, to no one.
Clem's eyes were fixed upon the face of the revolutionist. He studied the high forehead, the proud mouth, the wide nostrils, the broad and powerful skull. He could not tell whether or not he was impressing his own faith upon this man.
Sun Yatsen was a good listener. He did not interrupt. When Clem had made plain his desire to organize in China a means of food distribution that would guarantee the contentment of the people, Sun Yatsen shook his head.
"I have only so much money. I can choose between an army which will fight the enemies of the people and set up a righteous government for the people by the people and of the people, or I can, as you suggest, merely feed the people."
"Your government will not stand if the people are not fed," Clem said.
Sun Yatsen smiled his famous winning smile. "I have no government yet. First must come first, my friend."
"Only if the people have food will they believe in you," Clem said. "When they believe in you, you can set up what government you choose."
"It depends on one's point of view," Sun Yatsen said suddenly in English. "If I set up a government then I shall be able to feed the people."
The hunchback came to life. He opened his narrow and snakelike eyes.
"Exactly," he said. "Force comes first."
Clem got to his feet. "It is a misfortune that I didn't find you alone," he said to Sun Yatsen. "I guess I have failed. But you will fail, too. Your government will fail, and somebody else will come in and the way they will get, in is just by promising the people food. Maybe they won't even have to deliver. Maybe by that time the people will be so hungry that just a promise will be enough."
Sun Yatsen did not answer for a moment. When he did speak it was to say with the utmost courtesy as he rose to his feet:
"I thank you, sir, for seeking me out. Thank you for caring for my people. I am touched, if not convinced."
His English was admirable, the accent faintly Oxford. It was far better, indeed, than Clem's American speech, tinged with the flatness of Ohio plains.
"Good night," Clem said. "I wish you luck, anyway, and I hope you won't forget what I've said, even if you don't agree with me, because I know I'm right."
# 7
CANDACE FELT THAT WILLIAM was annoyed. He stooped to kiss her as usual but she was sensitive to his mood after these years of marriage and she saw a wintry stillness gathered about his heavy brows and firm mouth. When he spoke his voice was formal.
"I am sorry to be late."
"Are you late?" She yawned nicely behind her hand. "Then I'm late, too. I was tired when I came home from the matinée."
"Was the play good?"
"You wouldn't think so."
She rose from the chaise longue where she had been drowsing and looked from the window. Far below the vast park lay in shadows, pricked with lights. "I do hope the children are home. Nannie keeps them out too late. She is a fiend for fresh air."
"There was a strong draft along the hall from the nursery door and so I suppose they are home," William replied.
"Why do you think her first impulse upon entering a room is to open the windows?"
She asked the useless question while she was pulling on the satin slippers she had kicked off when she threw herself down. William seated himself in a chair and took his characteristic pose, his small dark hands gripped together, his legs, long and thin, crossed. Whatever the fashions for men, he wore his favorite gray, dark with a faint pinstripe, and his tie was dark blue. He did not answer his wife. This, too, was usual. Candace asked many questions she did not expect to have answered.
They were the queries of her idle mind. He had once given them thought until he discovered them meaningless.
She straightened her skirt and sauntering to her dressing table she picked up a brush and began smoothing out her short curls. Something was wrong but if she waited William would tell her. It might be anything, perhaps that he did not like the odor of food floating upstairs from the basement kitchen. The maids left the doors open in spite of her orders. Perhaps it was only while watching her as she brushed her hair he was reminded that she had decided to have her hair cut against his wishes.
"I had a letter from my father today," William said abruptly.
"I thought something was wrong," she said, not turning around but seeing him very well in the mirror. His face, always ashen, was no more so than usual. Something in his Chinese childhood, a doctor had said, perhaps the dysentery when he was four, had left his intestines filled with bacteria now harmless but more numerous than they should be.
"They have decided to take their furlough, after all," he said.
She went on brushing her hair, watching his face. "That's good news, isn't it? I have never seen your father, and the boys have never seen even your mother."
He frowned and the thick dark brows which always gave his face such somberness seemed to shadow and hide his deep-set eyes. "It is a bad time for me, nevertheless. I'd just decided to launch the new paper at once instead of waiting until spring."
She whirled around. "Oh, William, you aren't going to start something more!"
"Why not?"
"But we don't see anything of you as it is!"
"I shan't need to work as long hours as I did with the others. I've made my place."
"But why, when we're making money? You sacrifice yourself and us for nothing, darling!"
She let the brush fall to the floor and flew to his side and dropped on her knees, leaning her elbows on his lap and beseeching him. "I have always to take the boys everywhere without you. All last summer at the seashore you only came down for week ends, and scarcely that! It isn't right, William, now when they're beyond being babies. I didn't say anything when you were getting started, but today, just when I was thinking we might go to the theater sometimes together!"
He was entirely conscious of her beautiful face so near his, and he would have given much to be able to yield himself to her but he could not. Some inner resistance kept him even from her. He did not know what it was, but he felt it like an iron band around his heart. He could not give himself up to anyone, not even to his sons. He longed to play on the floor, to roll on the carpet as Jeremy did with his little daughters, but he could not. He was most at ease when he sat behind his great desk in the office giving orders to the men whom he employed.
"I went to the theater with you only last week," he reminded her.
"But that was an opening night and you know what people go to that for—to see and be seen. I want us just to go sometimes all by ourselves, and only for the play."
He did not enjoy the theater but he had never told her so. He could never forget that it was only a play. No stage excitement could reach him when he was fed daily by the excitement of his own life, his secret power which he felt growing beneath the power of the printed words he set upon his pages. He alone chose those words. What he did not want people to know he did not allow to be printed. They learned only what he selected. Sometimes, meditating upon his responsibility, he felt himself chosen and destined for some power over men which he had not yet reached. He had been reared in Calvinism and predestination, but in his rebellion against his childhood he had rejected all that his father taught him. He had become almost an atheist while he was in college. Now he was made religious by his own extraordinary success. In the few years since he had put out the first of his newspapers, their sales had soared into millions. Yet he was not satisfied. Even now, traveling upon a train, he could feel vaguely hurt that on every other seat there should be lying the crumpled sheets of a paper thrown away. People ought to keep what he had so carefully made. Then his mood changed to pride. There were two of his papers to one of any other. Such colossal success meant something. There was a God, after all—and predestination.
"What are you thinking about?" Candace asked.
The question slipped from her tongue and she wanted it back instantly but it was too late. William disliked to be asked what he was thinking about. It was an intrusion and she knew now that he guarded himself even from her. It had taken her time to learn this and meantime she had wept a good deal alone. Tears, she had now learned, only irritated him. She shed no more of them.
"No—don't answer me," she said and impulsively she put her crossed fingers on his lips.
He took her hands rather gently, however, and did answer her. "I was thinking, Candy, that it is a great responsibility for one man to know that he feeds the minds—and the souls—of three million people."
"Three million?"
"That is the number of our readers today. Rawlston gave me the last figures just before I came home. A year from now he says it will be twice that number. I suppose I am worth more than a million dollars, now."
She was used to her father's joking, "A millionaire? Nothing to it. Just keep ridin' high and never look down."
"You've made a great success, William." She was not at all sure that this was the right thing to say and with his next words she knew it was not.
"I'm not thinking only in terms of personal success. It is easy to be successful here in America. Anyone with brains can make money."
"But you do like money, William." Her sense of being wrong compelled her to justify what she had said. Besides, it was true. In his own way William valued money far more than she did or ever could.
"It is only common sense to have money." His voice was dry, his eyes severe and gray. "Without it one is hamstrung. There is no freedom without money."
She remembered something she had heard her father once say. "A man needs enough room to swing a cat in." Room, that was what money gave. A big house to live in, months in which to idle beside the sea, to live winter in summer and summer in winter, to buy without asking the price.
"Yet you don't seem to enjoy life very much, William," she said rather painfully. She had a profound capacity for enjoyment without a sense of guilt. Her father had frankly enjoyed getting rich and he distrusted all charities. She teased him sometimes by saying that he had become a Christian Scientist so that he could ignore the sufferings of others.
He had grinned and refused to be teased. "Maybe you're right, daughter. Who knows why we do anything?"
Then he had turned grim. "If I see somebody starving, with my own eyes, I'll feed 'em. I won't pay out good cash for what I don't see. Ten to one they're lazy. If they hustled like I did...."
Even going to church, while a social duty, had nothing to do with giving his money to strangers. Roger Cameron had cultivated no conscience in his children and Candace had grown up believing that pleasure was her normal occupation, once the dinner was planned and the children cared for. But no pleasure she devised could coax William from himself, or whatever it was that he dwelled upon in his soul. A ball which she planned as happily as a child might plan a birthday party fretted him with detail. A dish badly served spoiled his dinner. A servant who was not well trained—but of servants she would not think. He demanded of those in his service a degree of obedience and respect and outward decorum which had made her wretched until her father had found her crying one day. He had a way of coming to see her alone when he knew William was at his office. He took a cab and came all the way from Wall Street to arrive at three o'clock in the afternoon or at eleven o'clock in the morning.
On one such a visit he said, after he had inquired as to the cause of the tears his shrewd eyes had seen in spite of powder and even a dash of rouge, "You can't find Americans who'll give William the service he wants. We don't respect ourselves enough yet. We've always got to be showing that we're independent and don't have to obey anybody. Besides, we're too honest. When we hate anybody we act ugly. You hire your house full of English, Candy—they can act nice while they're stirrin' up poison for you. An English servant can polish your shoes as though he loved it. Of course he don't."
So she had filled the house with English servants, and a butler and a housekeeper kept their eyes upon William, the master.
"I don't know that life is merely to be enjoyed," William now said.
She was still crouching beside him. Idly she had taken one of his hands and playing with the fingers, she noticed the strange stiffness of his muscles.
"What's life for?" she asked, not expecting an answer. "I don't know, I don't suppose anyone does, exactly. We're here, that's all."
"It is for something more than amusement." He disliked her playing with his hand and he drew it away, ostensibly to light a cigarette.
She felt his dislike and got to her feet gracefully, took his head between her hands and kissed his forehead.
"Poor darling, you're so serious."
"I don't need your pity."
"Oh, no, William, I didn't mean that. Only, I enjoy life so much."
She drew back and met the hurt look she feared. Why could she never learn how easily wounded he was? She cried out. "How silly we are to keep talking about nothing when you haven't even told me your real news! When are your father and mother coming?"
He was relieved to be able to withdraw from her. "I had a cable this afternoon. They sailed the thirteenth on an Empress ship."
"Then in a fortnight—"
"More or less. Just when I shall be busiest."
"Never mind, I'll look after them. Dad has time, too, now he's retired enough to stop away from the office if he likes. And there's Jeremy and Ruth—"
"I shall need Jeremy."
Of the young men with whom he had begun the paper only Jeremy was left. One by one the others had deserted him. Martin Rosvaine had gone into the production of motion pictures and Blayne into the State Department with aspirations for an ambassadorship. He had not missed these two, but he had been sorry when Seth James quarreled with him, for he valued Seth's brilliant and effervescent mind, the ideas which poured forth like sparks from a rocket. Most of them were useless, but he watched the scintillating performance because there were always one or even two or three ideas upon which he seized. They had made a good pair, for Seth's weakness was his inability to discriminate between good ideas and foolish ones, and the paper would have been bankrupt had he been given authority. For that reason, William told himself, he had been compelled to keep control in his own hands even to the extent of buying up stock. Jeremy, of course, had never been a threat. He worked when he wished and William had learned to hire an understudy for him. But even yet he missed Seth, who had left him in anger and still refused to communicate with him.
The quarrel had been over a small matter, a difference of opinion so common to them that William had not troubled even to be polite. He had merely thrown abrupt words over his shoulder one night when they were all working long past midnight. Seth had said something about a story of some long-orphaned children in a foster home on a Pennsylvania farm. The farmer had lost his temper at a boy—he was still a boy, though a man in years—and the boy in terror and self-defense had rushed forward with a pitchfork, which had pierced the farmer's leg. The wound was slight but the farmer had hacked the boy with an ax with which he was chopping wood and the boy had bled to death within an hour. There had been scandal enough so that Seth had gone impetuously to the scene himself to check the copy he was reading, and had come back flaming with anger at the conditions he found in the farmhouse: two half-starved grown girls, both mentally retarded, and a fat cruel old woman, and the boy hastily buried without anyone coming to investigate. The farmer lay in bed and babbled about self-protection. Seth had routed out police and they in turn had produced a thin frightened woman who claimed that she was only an employee of the organization that had placed the children and that she did not know whether there were any relatives. In the end the local publicity had spread to reach Ohio, whereupon Clem Miller of all people had come to Pennsylvania to see what was going on. He had taken the two girls away with him and had told the police that the place was not fit for any children, big or little.
To Seth Clem had said with furious zeal, "I hope you'll tell William to make a real spread of this. Everybody in America ought to know about it. It's a strange and pitiful thing—this was my grandfather's place. He hung himself in that barn because he was too softhearted to get a neighbor off a farm—mortgage was called in. I came here myself when I was a kid, not knowing. These people were here already. I ran away—wanted all the kids to come with me, but only one would come."
"It's nothing but a local mess and of no significance," William had said upon getting Clem's message.
"But the boy's death is significant," Seth had insisted. "The very fact that orphaned children could be farmed out like that to such people, and no one care—"
"Well, no one does care," William had retorted.
Seth's answer had taken a long moment in coming and William, his mind upon his editorial, had not turned around. It came at last.
"You don't care, that's a fact," Seth had said in a still voice. "You don't care about anybody, damn you!"
He stalked to the door. "I'm not coming back here."
"Don't be foolish," William said.
He had been very angry, nevertheless, when Seth walked out of the office. During the sleepless night in which he told Candace nothing except that the bread sauce on the pheasant he had eaten for dinner had not agreed with him, he made up his mind that when Seth came back in the morning he would ignore the whole matter. Otherwise he would have to fire him. But Seth did not come back. William had never heard from him since, but so far as he knew he was doing nothing of any use. He had backed two or three quixotic magazines, none of which were succeeding. Fortunately for Seth his father, old Mackenzie James, and Aunt Rosamond, too, had left him plenty of money. When William thought of their quarrel, as he often did, he was still convinced that he was right. A local murder in itself was not important. But William could never forget a wound and Seth had wounded him deeply. This was important.
He felt himself misunderstood; of all his men he thought that Seth had understood him best. For William did not think only of himself. All that he did, his monstrous effort, his tireless work, was, he believed, to make people know the truth. Why else did he scan every photograph that was to be printed, why read and read again the galley proofs except that he might make sure that the people were given truth and nothing but the truth? He had tried to say something like this to Seth one day and Seth had laughed.
"Truth is too big a word for one man to use," Seth had declared. "For decency's sake, let's say truth as one man sees it."
To this William had not replied. It was not truth as he or anyone else saw it. Surely truth was an absolute. It was an ideal, it was what was right, and right was another absolute. Facts had little to do with either. Facts, William often declared to his young subeditors, were only trees in a forest, useless until they were put to use, bewildering until they were chosen, cut down, and organized. The policy was to establish what was right, as a man might build his house.
"Our materials are facts," William often said to his staff, looking from one tense young face to the other. The men admired him for his success, swift and immense. He was upheld by their admiration and only Seth had insisted on seeing the confusion behind their eyes. "When we know what we want to prove, we go out and find our facts. They are always there," William said.
After Seth had deserted him, for to William, it could be called nothing but desertion, he had only Jeremy of the old gang. The rest of his huge staff was made up of many young men, whose names he was careful to remember if they were executives. To the others he paid no heed. They came and went and he judged them by the pictures they sent in and the copy they wrote. His young subeditors made up the paper, but he himself was the editor-in-chief, and mornings were hideous if he did not approve what they had done. For he must approve. No one went home unless he did—no one except Jeremy, whom he could not control. Jeremy alone at midnight put his hat on the side of his head and took up his walking stick. He would always be a little lame, and he made the most of his limp when he went into William's office.
"Good night, William, I've had enough for today."
William never answered. Had Jeremy not been the son of Roger Cameron he would have thrown him out and closed the door.
"Ruth and I will take care of your parents," Candace was saying. "They'll stay here, I suppose?"
"I suppose so," William replied. He rose. "I shall have to get back to the office tonight, Candace. We'd better have dinner at once."
Left alone after dinner, Candace put the two boys to bed, annoying the nurse Nannie by this unwanted help. The house was so silent afterward that she went to her own room and turned on all the rose-shaded lights and lay down to read, and then could not read. Instead she thought about William, whom she loved in spite of her frequent disappointment in their life together. She was not a stupid woman, although her education had been foolish, as she now knew. A finishing school and some desultory travel were all she had accomplished before her wedding day, and since then her life had been shaped around William's driving absorption in the newspapers. She could not understand this absorption. Her father had worked, too, but only when it was necessary. Other people worked for him and he fired them when they did not do what he told them. A few hours in his offices sufficed to bring the money rolling in from hundreds of stores all over the country. It would have been so pleasant if William had been willing to go into the Cameron Stores, but this he had refused to do. She did not know what he really wanted. When they were married she supposed he wanted only to be rich, for of course only rich men were successful. Yet he could have been rich almost at once had he taken the partnership her father had later offered him.
Thus she discovered that he wanted something beyond money. Yet what more was there than a handsome and comfortable home, a wife such as she tried to be and really was, wasn't she, and dear, healthy boys? One day, soon after they were married, in those days when she still thought that she could help him, she had said she thought his picture papers were childish and he had replied coldly that most people were childish and his discovery of this fact had given him the first idea for his papers.
"I like people and you hate them," she had then declared in one of her flashes.
"I neither like them nor hate them," he had replied.
Yet she believed that he loved her, and she knew she loved him. Why, she did not fully know. Who could explain a reason for love? Seth James had once wanted her to marry him. Since they were children he had talked about it, and Seth was good to the soul of him, kind and honest—yet she could not love him.
Surely it was strange not to know William better after years of marriage. She knew every detail of his body, his head, nobly shaped, but the eyes remote and deep under the too heavy brows; a handsome nose William had, and a fine mouth except that it was hard. His figure was superb, broad-shouldered, lean, tall, but when he was naked she looked away because he was hairy. Black hair covered his breast, his arms, his shoulders and legs. She disliked the look of his hands, though she loved him. Yet how little love revealed! What went on in his mind? They were often silent for hours together. What did he long for above all? It was not herself, nor even the two boys, though he had been pleased that his children were boys. He did not care for girls, and this she had not understood until one day Ruth had told her that in Peking the Chinese always felt sorry for a man when his child was born a girl. It was a sign of something unsuccessful in his house. No matter how many sons a Chinese had he always wanted more.
"But William isn't Chinese," she had told Ruth, making a wry face.
Ruth had given her pretty laugh. Then she had shaken her head rather soberly. "He's not really American, though, Candy."
What was really American? Jeremy was American, and Ruth had adapted herself to him, copying even his speech. They were quite happy since they had the two girls. Ruth had been absurdly grateful when Jeremy seemed really to prefer girls.
She loved Jeremy with her whole tidy little being and had no thought for anyone else, except William. William she was proud of and afraid of, and the only quarrel she had with Jeremy was when she asked him not to make William angry. Jeremy, of course, was afraid of nothing, not even of William.
Yet William loved his country. He was capable of sudden long speeches about America. Once at an office banquet to celebrate his first million readers, William had talked almost an hour and everybody listened as though hypnotized, even Candace herself. The big hotel dining room was still and suddenly she began to smell the flowers, the lilies and roses, on the tables, although she had not noticed their fragrance before. Words had poured out of William as though he had kept them pent in him. She heard the echoes of them yet.
> "It is the hour of American destiny.
>
> —We have been sowing and now we are about to reap.
>
> —I see the harvest in terms of the whole world.
>
> —The world will listen to our voices, speaking truth.
>
> —We are young but we have learned in our youth to control the forces of water and air—the forces which are locked into ore and coal.
>
> —Old countries are dying and passing away. England is weak with age, an ancient empire, her rulers grown tired. France is sunk in dreams and Italy slumbers. But we of America, we are awake. The name America will be heard among every people. It is our time, our hour. It is we who will write the history of the centuries to come...."
Candace had listened, alarmed and half ashamed and yet fascinated. This was William, her husband!
That night in the silence of their own house she had been unusually silent. He had seemed exhausted, his face pallid as water under a gray sky, and he did not speak to her.
"You were very eloquent tonight, William," she had said at last, because something was necessary to be spoken between them. "I suppose your preacher father is somewhere in you, after all."
"I wasn't preaching," he had said harshly. "I was telling the truth."
At this moment the telephone rang upon the small rosewood table beside her bed and, lifting the receiver, she heard her father's nasal voice.
"William?"
"William is at the office, Father," she told him. "There's only me at home."
He hesitated. "You in bed, Candy?"
"Not really. I'm just upstairs because I don't like being downstairs alone."
"Maybe I'll come around. Your mother's got a sick headache and she's gone to sleep."
"Do, Father. I'll come down and be waiting."
Such visits at night were not unusual. Her father liked to walk in darkness when the city streets were empty, and once or twice a month he rang the doorbell and when the door was opened stood peering doubtfully into the hall. "William here?"
It was always his first question, though why Candace did not know, for sometimes he came in whether William were home or not, to stay a moment or an hour. He had a delicacy which told him, his foot upon the threshold, whether his visit was opportune.
Tonight she was more than usually pleased, for she was in a mood to talk and there was no one with whom she could talk more easily than with her father. Her mother was well enough when it came to the matter of servants and children but tonight she wanted to talk about something more, although she did not know exactly what.
When the doorbell rang she hastened downstairs to open the door herself, for the maids were asleep. Her father stood upon the big door mat, looking gray and cold and yet somehow cheerful, the tip of his long nose red and his eyes small and keen.
"This is nice," he said as she took off his overcoat. "I feel in the need of a little light conversation. It looks like rain and my knees are stiff."
"You shouldn't be walking on such a night," she scolded with love.
"I shan't yield my life to my knees," he said.
The fire was red coals in the living-room grate and he took the tongs from her. He was skillful at fires, manipulating the live coals under the fresh fuel and coaxing a flame from the least of materials. It was one of his pet economies, left over from the days when as a child he had picked up coal from the railroad yards in a Pennsylvania mining town.
When the fire was blazing he sat down, rubbing his hands clean on his white silk handkerchief. "Well, how's tricks?"
"Oh, we're all well," she replied. "Willie is on the honor roll at school. William was quite pleased. The real news is that William's parents are coming from China."
"I thought they'd decided to stay for another year."
"So did I."
"It's the old lady, I imagine," he said thoughtfully and gazed into the fire. "I suppose William's glad?"
Candace laughed. "He seems rather annoyed."
Roger Cameron liked to hear his daughter laugh. He looked up and smiled. It was a pleasant moment, the big room shadowed in corners and lit here by the fire and the lamp. She looked pretty in a rose-colored wool dressing gown, pretty and maybe happy, too. For a while after her marriage he had wondered if she was happy and them had decided she could be, mainly because she had a fine digestion and no ambitions. He had taken care in her education that she should not be placed in the atmosphere of ambitious women. There were such women in the Stores, and none of them, he believed, were happy. His secretary, Minnie Forbes, whom he had employed since she was twenty-one, was devoured with dry unhappiness, perhaps because Minnie would have been shocked to know that she was in love with her employer. Roger knew very well that she was and was grateful for her ignorance. He himself loved his wife in a mild satisfactory way, and had no desire to love anyone else. The brief months when as a young man he had been passionately in love with her he remembered as extremely uncomfortable, for he could not keep his mind on his business. He had been relieved when he discovered that she was not the extraordinary creature his fancy had led him to imagine her, and then he had settled down to the homely and unromantic married love which he had enjoyed now throughout forty peaceful years. He and his wife were deeply attached, but she did not regret his business trips, and he enjoyed them with the single-minded pursuit of more business.
"William never did quite know what to do with his family," he now said.
"Are they queer, Father?" Candace's blue eyes were always frank. "I can't seem to remember even his mother very well."
"I suppose anybody that goes off to foreign countries is queer in a way," he replied. "Ordinary folks stay at home. Still, they are always taking up collections in churches and all that. William's father is no more than a preacher who goes beyond what's considered his average duty. 'Go ye into all the world,' and so on. But nobody much takes it seriously, except a few. They're always good men, of course."
"And the women?"
"I don't believe Mrs. Lane would have gone on her own hook. I suppose she went because he did. Not too much sympathy between them, as I remember."
He did not want to tell his daughter that he remembered Mrs. Lane as a pushing sort of woman. Maybe she wasn't. People often became pushing when they were with a rich man. He had got used to it. Anyway, it was all in the family now.
"Jeremy's little Mollie is a cute trick," he said, smiling.
"She is," Candace agreed. "Ruth tells me she talks all the time. When she comes here she is shy and won't say a word."
"She talks to me if I'm by myself. It's wonderful to watch the first opening of a child's mind."
"Ruth and I are going to have to look after Mother and Father Lane. William is working on a new paper."
"What's he want with more work?" Roger took out his pipe. He had not begun smoking until recently and he still felt strange with the toy. But he had wanted something to occupy his hands.
"The Duke of Gloucester knits," he said, perceiving a gleam now in his daughter's candid blue eyes. "That's all very well for an Englishman. We American men aren't up to it yet I don't really like this smoking, but it takes time to fill the pipe and light it and it goes out a good deal. It's all occupation."
"What's the matter with you American men?" Candace asked, her eyes bright, her mouth demure.
"An Englishman is never afraid of being laughed at," Roger replied. "He just thinks the other fellow is a fool. But Americans still can't risk anybody laughing at them. I can't, myself. Tough as I am I couldn't knit, even if I wanted to. I don't want to, though."
"You don't want to smoke, either," she mocked.
He grinned at her sheepishly and went on with his maneuvers while she watched, still ready to laugh. "I guess I like to play with fire," he said when at last he was puffing smoke, his eyes watering. "What I like best is getting it ready and striking the match."
"Oh, you." She yawned softly. "No, I'm not sleepy. I keep worrying about what I'll do with William's parents. Why don't you help me? Suppose they want to stay here in the house all winter?"
"Let them do what they want and you go your ways," he replied. "Be nice to them and leave them free. That's what most old folks want. Don't worry yourself."
"Didn't you ever worry about anything?"
"Sure I did. When I was young I worried my stomach into a clothes wringer. One day a doctor said I'd be dead in a year. I made up my mind I wouldn't. But I had to quit worryin' my stomach. Lucky the Stores were on their feet. That was the time I knew Jeremy never would take over. Well, I didn't need him, as it turned out, or anybody. It's a great thing to be able to manage your own business. I kind of hoped once that William would come in, but it's just as well. William is cut out for what he's doing."
"What do you think William really wants, Father?"
So seldom did she ask a serious question that he looked rather startled and put his pipe on the table to have it out of the way.
"What do you mean, Candy?"
"Well, we have lots of money."
"It's wonderful what he's done."
"But he doesn't enjoy it. Even when we have a dinner party it seems he can't enjoy it. It has to be more than a party, somehow. And there is no use taking a vacation. When we went to France last summer, he spent the whole time arranging for a European edition. I went around by myself until in Paris I met some of the girls I'd known in school."
"William's ambitious," Roger said reluctantly.
"For what, Father?"
"I don't believe he knows," Roger said. "Maybe that's what bothers him. He don't know what to do with himself."
There was something so astute in this that Candace laid it aside for further thought.
"I wish I could teach him how to play games and enjoy horseback riding."
"He rides well enough."
"He does everything well, and doesn't care for any of it. I love him and I don't understand him."
There was a hint of fear in her voice; only a hint, but he did not want to hear it. He was getting too old for sorrow. He could not even read a sad book any more. When it began to get sad he shut it up. He had seen too much trouble that he could not help, or maybe he did not want to help.
"You don't have to understand people," he said in his driest tones. "There's so much talk about understanding this and that nowadays. Most of the time nobody understands anything. If you love him, you don't need to bother about understanding, I reckon. Just take him as he acts."
He began to feel restless as he always did when he smelled trouble. He had a wonderful sense of smell for trouble and when he caught that acrid stench, however faint, he went somewhere else. So now, though he loved his daughter, he rose and put his cold pipe into his pocket.
"I guess I'll be getting along home." He bent over her and kissed her hair. "Don't you worry, my girl. Just treat the old folks nice and let them do what they want."
"Good night, Father, and thank you."
He ambled out of the room and she sat a few minutes alone. She was shrewd in her naïve way and she knew his willful avoidance of trouble. But she was enough like him to sympathize with it. What he had said was comforting. It was easiest, after all, not to worry about understanding people, and surely easy just to love them, whatever they did, so long as they were not cruel in one's presence. And William was never cruel to her or to the children. He had never whipped the boys, however impatient he became. Jeremy, in a flurry of wrath, could upturn the fluffy skirts of a small girl over his knee and give her a couple of paddles and then, his anger vented, turn her upright again and kiss her soundly. William did not kiss his sons, either. He never touched them.
Ah well, she was glad she loved him. Love, her father had said, was enough.
The moment William looked at his father as he came off the train, he knew that here was an old man come home to die. The sight and the knowledge stunned him. As always when he was moved he felt speechless. Ruth stood beside him and on the other side were Candace and Jeremy together. They had not brought the children because of the crowd and the late hour. The lights of the station fell upon his father's white face and gaunt frame. He had grown a beard, but even its whiteness did not make the white face less pale. His mother was stouter and older, as strong as ever. It was she who saw them first and she who greeted them. He felt her firm kiss on his cheek.
"Well, William!"
"Yes, Mother."
But he kept looking at his father. This old, old man, this delicate ghost, the dark eyes living and burning and the pale lips folded quietly together in the white beard! He took his father's hand and felt it crumple into a few bones in his palm.
"Father—" he cried, and put his arms around his father's shoulders. He turned to Jeremy. "You take care of them, Jeremy—the women and the—the baggage. I'm going to get my father out of this."
"But he's ever so much better," his mother cried.
"He doesn't look better to me," William said. His lips felt stiff and he wanted to cry. He pulled his father away, his arm still about the thin old body. "Come along, Father. The car is here." Why hadn't his mother told him?
The chauffeur was standing at the open door of the car. William helped his father in and wrapped the rug warmly about his knees. "Drive straight home, Harvey," he called through the speaking tube.
The heavy car swayed slowly into the traffic. William sat looking at his father. "How do you really feel?"
Dr. Lane smiled and looked no less ghostly. "You didn't think I would look the same after all the years?"
It was the first time he had spoken and his voice was soft and high, almost like a child's.
"But are you well?" Now that William was alone with his father he could control his unexpected tenderness.
"Not quite," his father said.
He looked so patient, so pure, that William felt he saw him for the first time. To his own surprise he wanted to take his father's hand and hold it, but he felt ashamed and did not.
"Have you seen the doctor?" He spoke again with his usual abruptness.
"Yes, that is why we left Peking so suddenly. He thought I should be examined here." Dr. Lane's smile was tinged with unfailing sweetness.
"What did he say it was?"
"It seems I have had sprue for a long time without quite knowing it. It destroys the red corpuscles, I believe." Dr. Lane spoke without interest in his corpuscles.
William heard and made up his mind quickly. He would get the best man in the world on tropical diseases—send to London for him if necessary. He felt an imperious anger harden his heart. "I should have thought Mother would have noticed."
"One doesn't notice, I suppose, living in the same house for so many years," his father replied. "I didn't notice even myself. Tired, of course, but I thought I was just getting old."
"You are going to rest now," William commanded.
"That will be nice," his father replied. His voice became fainter and fainter until with these words it was only a whisper. William took up the speaking tube. "Drive as fast as you can. My father is very tired."
The car speeded under them smoothly. Dr. Lane leaned his head back against the upholstered seat and closed his eyes and seemed to sleep. William watched him in profound anxiety. He would get his own doctor tonight immediately after they reached home; he would be afraid to sleep unless somehow his father was fortified.
When the car drew up at the door he got out first and with the tenderness so strange to himself he helped his father up the steps and into the hall. The butler was waiting and took their hats and coats. At the foot of the great stairway he saw his father stand back and look up as though at a mountain he could not climb.
"I will carry you up," William muttered.
"Oh no!" Dr. Lane gasped. "I shall be quite able in a moment."
William did not hear him. In a daze of love such as he had never felt for any human creature, he lifted his father into his arms and, horrified at the lightness of the frame he held, he mounted the stairs. The old man, feeling his son's arms about him, gave himself up with a sigh and closed his eyes.
What befell William in the weeks that followed he was never able himself to understand. Its effects did not appear fully for many years. He seemed to be alone in the world with his father, and yet the dying saint was someone far beyond being only his father. For the time during which this presence was in his house William scarcely left his father's room. He discerned with new perception that this spirit, preparing for departure, was ill at ease except alone and he was therefore brutal with his mother. He said to Candace and Ruth, "Mother must not come near him. It is your business to see that she is taken out of the house on any pretext you can think of."
He bullied the American doctors cruelly, declaring them incompetent. He himself cabled to the great English specialist in tropical diseases, Sir Henry Lampheer, demanding his instant attendance. Under the roaring waves of the Atlantic Ocean this communication went on, hour after hour.
Sir Henry's reply to William's command was British and stubborn. HAVE CONSULTED WITH YOUR DR. BARTRAM. OBVIOUS MY SERVICES TOO LATE. STARVATION RESULT OF DESTROYED TISSUE. INJECTIONS MAY PROLONG LIFE.
William was imperious with the Englishman, SET YOUR OWN PRICE.
Sir Henry lost patience and his haughty irritation carried clear beneath the raging Atlantic tides. NO PRICE POSSIBLE FOR FOLLY OF LEAVING IMPORTANT PATIENTS HERE, ADVISE DEPENDING UPON YOUR OWN PHYSICIANS.
YOU PROPOSE TO LET MY FATHER DIE?
GOD DECREES, Sir Henry cabled, refusing blame, YOUR FATHER AN OLD MAN GRIPPED BY FATAL DISEASE.
MY FATHER COMES OF LONG-LIVED FAMILY, ALSO GREAT RESISTANCE OF SPIRIT, William retorted.
To this affirmation Sir Henry replied coldly, DIAGNOSIS CLEAR. INJECTIONS EMETINE, BLAND DIET, MILK, BANANAS, POSSIBLY STRAWBERRIES, CERTAINLY LIVER ESSENCE, ABSOLUTE REST, CONSULT BARTRAM.
The cables ticked themselves into hundreds of dollars, and after their futility William felt all the old rage of his boyhood mount into his blood. The damned superiority of the Englishman, the calm determination not to yield, the rigid heartless courtesy—he knew it all in Chefoo when the British Consul General's son was at the top of the top form.
Blind with fury, William shut off the Atlantic Ocean and the British Isles and all the rest of the world. He was in his office, having left his father for an hour with two trained nurses, and Ruth to see that the fools did not neglect him. Now he called in his chief editor, keeping his finger on the electric button until Brownell came in on the run, his eyes terrified.
"Hold up the new dummy," William ordered. "My father is very ill. I can't get Lampheer to come over, he's determined to let my father die—just another American, I suppose—typical British! I don't know when I shall be back. I shall have to leave you in charge. If it's absolutely essential call me, but if it's not essential, I'll fire you."
"I'll do my best, Mr. Lane."
"Very well."
William was putting on his overcoat and hat. Brownell sprang to his aid.
"Here, let me help."
"Get back to your job," William ordered, and hastened from the room.
Yet he knew Sir Henry was right. That was the worst of all, next to the fact of death itself. Now day by day he sat beside his father's bed, silent in the silence of his house, having ordered the nurses to stay in his dressing room unless they were needed and forbidding any others except Dr. Bartram. Sir Henry would have been foolish to come and yet he ought to have set a price. Every man had his price and William could have paid it. His father was a man of importance, the father of William Lane, a rising power in America. It was an insult he would not forgive, and he added it to the mountain of insults he had taken in his boyhood. Sitting beside his dying father he brooded upon the mountain and how he would level it, by what means and with what purpose. Those tiny islands, clutching at half the world, those arrogant men sitting in their dinner coats at solitary tables in jungles, served by millions of dark men—it was monstrous. His country, his beautiful youthful America, despised and laughed at, even as he himself had been laughed at by stupid English boys who could not spell! In those days he had been ashamed of his father because he was only a missionary, but now that missionary was the father of William Lane. The missionary was lifted up out of his humility and poverty. He had become the father of a man whose first million was doubling itself.
Tears stung William's eyes. Money could not delay by one hour the death of his father, even his. He leaned toward the bed and took his father's hand in his own. The hands were not alike. He had his small dark hands from his mother. His father's were big and bony, and now how thin and helpless.
"Father—" he whispered. For a moment he thought him dead.
But Dr. Lane was not dead. He turned his head slowly, the same nobly shaped head that he had given to his son when he begot him.
"Yes, William?" The voice was faint but clear.
"You know I am doing everything I can?"
"Yes, my son.... It is quite all right.... I must die, you know."
"I can't let you die."
"That is very good of you, William.... I appreciate it.... To want me to live—"
"Because I need you, Father."
The words broke from him and the moment he had spoken them he knew them true. He had never really talked with his father and now it seemed to him that to his father alone could he speak of himself and the immense restlessness that filled him day and night. Now that he had set up this vast successful machine that brought money rolling in whether he was there or not, then what next? Now that he had power, millions of people his, too, looking at the pictures he chose, reading the words he wrote or permitted to be written, what next?
"Father, if you leave me—if you really think—"
"I know God has told me."
"Then tell me before you go—what am I to do?"
"Do?"
"With myself."
He saw his father's dark eyes open wide with final energy.
"William, you must listen to your own conscience.... It is the voice of God... in your breast. 'Remember now thy Creator in the days of thy youth.' All that you have—all your great gifts, my son... dedicate them to God. Oh God—I thank thee—thou hast—brought me to my son in time—"
The faint voice died away and the old man fell into sudden sleep as he did after the least exertion. He did not speak again.
William sat beside him through the hours. The nurses came and went, doing their duty. The doctor came, spoke a few words. "It can't last, Mr. Lane. Any moment, I am afraid."
William did not reply. That night, twenty minutes after midnight, his father without waking ceased to breathe.
Clem had plunged himself again into his own country. He had failed in China but he was not discouraged. Such was his faith in that which he believed. He had said very little to Henrietta about the brief visit to the shack in San Francisco, but she comprehended the refusal and perceived that as usual Clem had only been strengthened by it.
"Someday they'll see I'm right, hon," he told her. "They" were the powers, those who did not believe in his faith, the greedy, the selfish, the politicians, the small-minded. He did not hate, neither did he despise. Instead he was possessed by a vast patience, a mighty omniscience. He could wait.
Meanwhile he worked. He decided to open his largest and cheapest market in Dayton. Each of his markets had its own peculiar name. This one he called "People's Choice."
"I don't want a chain name," Clem said when Bump spoke of the advantages of a chain of markets all called by the same name. "I want people to think the markets are theirs. Each one must be different, suited to a town and its folks."
People's Choice was his first city market and he built it outside the city where land was cheap, at the end of a trolley line.
On the opening day Henrietta had come to help. Clem had lured thousands of people by his announcement of free foods on this first day. By ten o'clock the trolley cars were crowded beyond control and well-fed people were struggling to reach counters where loaves of bread, pounds of cake, and baskets of fruit were waiting to be given away. The day was clear and cool and through the great glass windows the sun poured over stacked counters and heaped bins. Clem had devised an effect at once modern and old-fashioned. Apples were piled upon the floor in corners, and bananas hung from the ceiling.
"Help yourselves, folks," Clem shouted cheerfully. "Take a pumpkin home and make yourselves a pie. Here's old-fashioned molasses—dip it up, folks! It's bottling that makes it come high—five cents a dipper, folks! I bought it in N'Orleans for you—by the barrel, folks—and plenty. Here's bread—take a loaf, and here's butter from Wisconsin—straight from the farmers, and that's why I can afford to give it away today. Tomorrow you'll pay less for it than you pay in any store in the city. If anybody is hungry he can have a loaf free. Give and it shall be given unto you. Don't take it if you're not hungry, but if you're hungry and can't pay for it, we'll always give it to you. No caviar here, folks, no fancy notions, just plain food straight from the people who raise it."
In and out among the surging, staring people he wove his way, alert, smiling, his sandy head held high, his small blue eyes snapping and twinkling and seeing everything at once. He wore overalls of denim like his clerks, or "hands" as he called them, and his hands were men from anywhere, two Chinese boys who were working their way through college, a Negro he had seen in Louisiana and liked, Swedish farm boys from Minnesota. He had picked his men and trained them himself, saying that clerks from other stores were no good to him.
His business was unorthodox and filled with risk, and when a man became fearful because of small children and a nervous wife, he let him go and found the boys, the young who dared to be reckless. He would send Bump overnight to California or Florida to buy up carloads of cheap oranges, to West Virginia to sweep up a harvest of turnips that were overloading the market, to Massachusetts to bid for a haul of fish that threatened to bring down the price on New York markets. Wherever there was unwanted food, food about to be thrown away, as Maine farmers were about to throw away half their crop of potatoes last summer, Clem or Bump was there. Clem trusted no other to buy for him, since in the narrow margin of buying and selling lay his profits and in his profits was his ability to expand his markets and his faith. His heritage from his father was an invincible belief in goodness, not in the goodness of God to which his father had so persistently trusted, but in the goodness of man. Clem believed more profoundly than ever that with his stomach full any man preferred to be good. Therefore the task of the righteous, of whom Clem considered himself one, was to see that everybody had food.
In his hours of dreaming, for he did no work on Sunday and his markets were rigidly locked on that day, he gave himself up to still more huge fantasies about feeding all the hungry in the world. There in his ugly little house in New Point, Ohio, where he lived in complete happiness with Henrietta, he saw the people in China and India someday crowding to his markets. His failure with Sun Yatsen in San Francisco, his conviction of future success made his dreams the richer and more real.
He recalled the long journey he had made on foot from Peking to the sea. The old agony of the moment when he saw his parents and sisters murdered had softened and dimmed. Instead he remembered the winding cobbled roads of the country that tied the villages together, the dusty footpaths on either side of the cobbles, the fields green with new wheat in spring, with the tall sorghum corn in summer. Someday in those Chinese villages and market towns his foods would stand displayed.
People's Choice promised, even this first day, to be instantly successful and Clem saw himself growing still richer. According to any rules he should not be getting so rich. He had no desire to be a millionaire like William, and he was almost ashamed of his mounting bank accounts. But he never gave money away. Some deep prejudice against organized charity, against packaged religions and vague idealism, made him keep his hands in his pockets. He gave to any man or woman or child who wore a ragged coat or who needed a doctor, and a few words scribbled on a torn scrap of paper or an old envelope provided food from his nearest market for anyone, from a hungry college student to a passing drunk or a springtime tramp. But he gave no large checks to soliciting treasurers and college presidents, and the churches, even of his home town, had come to look for no more from him than ten dollars dropped into the collection box at Christmas.
Bump, that cautious and careful young man, mindful of his college degree in economics and business management, warned him that sooner or later the organized food interests would attack him.
"You can't go on underselling them without their trying to get your hide," Bump warned. His relationship to Clem remained nebulous, profound though unexpressed. Clem was too young to be his foster father and he had never offered to be his brother. Bump was shrewd and he recognized in Clem a genius inexplicable. It was comprised of a daring that was absurd, a naïveté that was laughable, an ignorance that was almost illiterate, and out of daring, naïveté, and ignorance Clem succeeded in all he did. He had found a formula so simple that only a man as simple as himself could have proved it valid.
He declared it to gaping, staring thousands at noon this day of the opening of his new market. Six trumpeters, hired for the occasion, blew a frightful blast as the hour struck noon. The crowd, transfixed, paused to turn their heads toward the source of noise, and there in the center of the glittering brass, set upon a sort of balcony of boards rigged with ropes, they saw Clem in his overalls, with a megaphone.
"Folks!" he shouted. "This is more than just a market. It is a sign of what I believe in, a manifestation of my faith. 'Faith is the evidence of things hoped for,' the Bible says, and 'the evidence of things unseen.' Well, my hope is to see no more hunger, anywhere in the world. Food is the most important thing in the world. Food is one of a trinity with air and water. If I were President of the United States, which otherwise I am glad I am not, I would make bread and meat, milk and eggs, fruit and vegetables free to everybody. Then we would have no more war. It would be cheaper to feed people free like that than it would be to have a war, like what may come out of Asia someday if somebody don't do something, because the people are starving."
The people stood motionless, listening and wondering if he were mad. He took a deep breath and began again.
"Now don't get me wrong. I don't believe in charity, nor do we have to have the government doing this kind of thing. I'm not president, don't expect to be, don't want to be. But I'm doing what I can here, and you see it, don't you? If it's good, if it helps you, then all I ask is for you to believe in the idea. Thank you, folks—that's all. And let me tell you that you'll find free box lunches packed and ready for you down at the south end of the market. Ice cream is free for everybody, so's milk and soda pop. Have a good time, folks!"
He was in a frenzy of happiness. To the people who milled around him during the afternoon he talked in a stream of advice, explanation, and remonstrance. "What you'll find here is not all foods but just the essential foods and all cheap. I buy surpluses and that means whatever is in season and therefore cheapest. For instance, last winter when the big cold in the West was freezing cattle solid, I bought 'em that way and sold beef cheap. Price of meat came down right away. The beef was good, too. Freezing made it tender.
"Now here in this market, you won't find cucumbers in January. But you'll find mountains of them in summer when you want to be making your pickles. And I provide recipes, too. Where do I get them? From people like you. When you make something good write in and tell me about it. Look at that pile of leaflets there—take some—take a lot and give 'em to your friends. They'll tell you what to do with cucumbers when they're cheap and how to make jelly out of apple peelings and what not to throw into your garbage pails. Buy cheap, and don't waste. We could feed the world on what we throw away—yep, that's true, too. Nobody needs to starve—not anywhere in the world!"
People listened and laughed. "You sound like a preacher!"
Clem grinned his dry sandy grin. "Maybe I am—a new gospel I preach unto you. Nobody needs to be hungry."
It was in the midst of such harangue in the late afternoon that he saw Henrietta standing in the far corner, very quiet in her dark blue suit and hat, and holding in her hands a yellow slip of paper. He was used to telegrams from his scouts scattered over the country, announcing a glut of oranges in the Southwest or corn in Indiana or truck-garden stuff in New Jersey. Such telegrams had to be heeded immediately and so he suddenly stopped talking and wove his way through the crowds, pushing them gently with his sharp elbows.
Face to face with Henrietta, he reached for the telegram which she gave him and then he saw that it was not what he thought.
The telegram was signed by Mrs. Lane. YOUR DEAR FATHER PASSED ON LAST NIGHT. FUNERAL WILL BE THURSDAY. PROSTRATED WITH GRIEF. WILLIAM WONDERFUL. LOVE MOTHER. Instantly Clem forgot the crowds and the great success of his day. There was no spot in the huge cheap building where he could draw his beloved aside into privacy. Glass and brick pillars gave only the illusion of shelter. But he made of himself a shelter for the tears now rising slowly to her eyes.
"Hon, you go to the hotel right away. I'll send Wong with you. He has his little tin lizzie here. He'll put you on the train for New York. If you need anything in clothes, you can buy it there—a black dress or so. I'll be there tomorrow. I hate to have you alone tonight without me, but you'll not blame me for that."
"I wish I could have seen him just once," Henrietta murmured, wiping her eyes behind the shelter of his shoulders. She was taller than he and yet just now he managed to stand a little above her upon a collapsed cardboard box. "I ought to have made William tell me. Ruth ought to have written—no, it was my own fault."
For she had been cool to her parents when she got home because they had gone to William and had not thought of coming to her. No one had told her how ill her father was. Even the letters from her mother had not said he might die. She might have known when she had no letter from him, except that he seldom wrote to his daughters, and always to William. And Ruth would never face the worst.
"It's a shame," Clem muttered. "It does seem as though your folks could have sent word."
"I may not see him even now," she went on. "It would be just like William to go straight on with everything, as though no one else existed."
"You go along quick," he advised.
Stepping back he motioned to Wong, one of the Chinese students. He was a tall slender fellow from a town near Peking.
Clem said in Chinese, too low for anyone to hear or wonder at the strange tongue, "Wong, you take Mrs. Miller please to the hotel to get her bag and then to the railway station and buy her a Pullman ticket to New York on the first train. Her honored father has just died."
Wong had heard of the venerable Dr. Lane, the mildest of missionaries, and he clucked his tongue against the roof of his mouth. "The day of a father's death is worse than any yet known in a person's life," he said gently.
He slipped off his white coat and changed to the one he wore outside the market. In half an hour Henrietta was on the way to the station in his old Ford car. Driving nimbly between the trolley cars and the traffic, Wong tried in his courteous fashion to comfort Henrietta by all that he had heard about Dr. Lane.
"We heard even in our town that it was your honored Old One who did not fear to approach that Devil Female King, the Empress, and tell her that she did ill to favor the Boxers. Again we heard, I from my father, since I was then very young, that when she came back again to the city, pretending that no evil had been done, your honored Old One would not follow the other foreigners to her feasts. He held himself aloof. Your Old One loved the people and not the rulers."
"I have not seen my father for all these years," Henrietta said. "Now I shall never see him again."
"It was for our sakes that he cut himself off even from his own country," Wong said in a heartbroken voice.
At the station he bought her tickets and a small basket of fruit. When he had seen her into her seat, had adjusted the window shade, had said good-by, he went outside on the platform and there he stood, his hat held against his breast until the train pulled out.
Henrietta had never been in William's new home. Since she had sent no telegram to announce her coming, she took a cab and arrived at the door of the handsome house of gray stone, which stood between two smaller ones on upper Fifth Avenue. She rang the bell and the door was opened by an English manservant.
"I am Mr. Lane's elder sister," she said in her somewhat cold voice.
The man looked surprised and she saw that he had not known of her existence. "Please come in, Madame."
He ushered her into a large room and disappeared, his footsteps silenced by thick carpets. Henrietta sat down in a deep chair covered with coral-colored velvet. The room astonished her. Gray, coral, smoke blue were mingled in velvet hangings and carpets. It was a room too soft, too rich, too opulently beautiful. Candace had thus surrounded the heavy furniture William had bought and which she disliked. In the center of the room upon a round mahogany table stood a vast Chinese bowl of silver-gray pottery, crackled with deeper gray veins. It was full of pale yellow roses. This then was the way William lived. He must be monstrously rich. Or perhaps it was only the way Candace lived, and perhaps it was she who was too rich.
Henrietta reflected upon William as she had remembered him in Peking. The memory was not dimmed by the image of what he now was. A sulky, dark-browed boy, who snarled when she spoke to him! Why had he been always unhappy? At school in Chefoo he had seldom spoken to her, even when they passed in the corridors. If her mother sent a message to them both in a letter to her, she had to send it to him in a note by a Chinese servant. Ruth had been too young to go away to school and so she had never seen the worst of William, for if he was unpleasant at home he was unbearable at school.
Henrietta had a vague understanding of him, nevertheless, as she sat thoughtfully by the window of this room. William could not endure to be outdone by anyone, but at school no American could be as the English were and there William felt himself unjustly surpassed. Moreover she herself surpassed him in their studies, and she had gone to some pains as she had grown older to hide from him the marks which made him hate her, too. And why should this only brother of hers suffer so much when, had he been content with himself, he might have been very happy? A handsome boy he had been, and his mind, developing more slowly than hers, was a good and even brilliant mind, likely now to have gone far ahead of hers. His intolerable, bitter, burning pride had poisoned him to the soul, a pride begun by their foolish old Chinese amah, who because he was a boy among girls, had loved him best and praised him most and made them all worship him as the young prince of the family—a pride fostered, certainly, by being an American among Chinese. But here in America itself there were no princes.
The door opened and Candace came in, trailing the lace ruffles of her negligee. It was almost noon and she had not yet dressed herself for the day. But so immaculate, so exquisite was she in her rose and lace, her fair hair so curled and smoothed and waved, that Henrietta felt dingy after her night on the train.
Candace held out her hands and her rings glittered. "Not to tell us that you were coming, you naughty thing!"
She had grown soft and was prettier than ever, slender but rounded and feminine and too tender in voice and eyes.
"I thought you would expect me to come at once," Henrietta said. She submitted to a scented embrace and sat down again.
Candace sighed. The tears came to her violet eyes. "William is not to be consoled. He sits there beside his father day and night. He will neither eat nor rest. Your mother is sleeping. She is very tired. Ruth has gone home for a bit to be with her children. There is nothing to do here but wait."
"Clem will be here tomorrow," Henrietta said.
"How good of him to get away," Candace said.
"It is not good of him," Henrietta replied. "He does it for me."
She found herself with nothing to say and so she sat for a moment in silence while Candace twisted the rings on her fingers. Then Henrietta made up her mind. She did not intend to be cowed by this house or by any of William's belongings or indeed by William himself.
"I would like to go to my father, please, Candace. I have not seen him at all, you know."
Candace looked distressed. Her mouth, soft and full and red, looked suddenly childish and she bit her lower lip. "I don't know if William will—"
"William knows me," Henrietta said. "He will not blame you."
She rose and Candace, as though she submitted by habit, rose too, and in silent doubtfulness she led Henrietta across the hall through another large room—a music room, Henrietta saw, since it contained a grand piano and a gramophone set into a carved cabinet, and then across a hall which ended in a conservatory, and at last to heavy closed doors of polished oak. Here Candace paused and then she slid the doors a small distance apart. Over her shoulder Henrietta looked into an immense library, in the center of which stood a bier. There William sat. He had drawn a leather armchair close enough to see his father's face. A tall pot of lilies stood at the foot of the bier. Upon this scene the sunshine of the morning streamed through high southern windows.
Henrietta gently put Candace aside and entered the room. "William, I have come."
William looked at her startled. Then he rose. "You came early, Henrietta." His voice, deep and always harsh, was composed.
"I came as soon as I had Mother's telegram."
Candace had closed the doors and gone away and they were alone. She went to the bier and looked down upon her father's face. It was as white as an image of snow. The long thin hands folded upon the breast were of the same deadly whiteness.
"I am glad you have not sent him away," Henrietta said.
"Whatever had to be done was done here."
"He is desperately thin."
"He was ill for two years," William said. "Of course Mother did not realize it, nor did he complain. His intestines were eaten away by the wretched disease. There was no hope."
Neither of them wept, and neither expected weeping of the other.
"I am glad he did not die over there," William said.
"Perhaps he would rather have died there. He loved the Chinese so much," Henrietta said.
"He wasted his life upon them," said William.
He spoke without emotion, yet she felt his absolute grief. He revealed himself in this grief as she had never seen him, a gaunt lonely man, still young, and his pride was bitter in his face, in his haughty bearing, in the abrupt movements of his hands.
"It is a comfort to you that he came here to die." This she added in sudden pity for him.
"It is more than a comfort," he replied. "It was his last mission."
She turned her gaze then from the calm dead face to look at William and perceived in his stone-gray eyes a look so profoundly strange, for that was the word which came to her mind, that she was for the first time in her life half frightened of him.
William had no impulse to tell her of those last words which his father had spoken. For him they had indeed taken on the importance of prophecy. His father, he had learned from his mother, had a premonition of approaching death during the last year in Peking. He had long refused to come back to America because, he said simply, he wanted to die in China and be buried there. Yet when he felt death imminent he changed his mind. "I must see William," he had told her one night when he woke as he often did long before dawn. "I must see my son. I want to talk with him. I have things to tell him."
Here his mother had paused to wipe her eyes and also to ask him in curiosity, "What things did he tell you, William?"
He could not share even with her the solemnity of those last words his father had been able to speak. They were few, far fewer than he had meant to speak, William felt sure, had he not been so ill in the last weeks before the end. And yet in few words all was said. He understood that his father had come thousands of miles by land and sea to speak them to his dear and only son, and so he forgave his father everything, all the shame of being his son, the disgrace of the lowliness of being the son of a poor man and a missionary. By his love for his son and by his death his father had lifted himself up into sainthood. There was symbolism here which in its way was as great as that of the Cross. He was his father's only begotten son, whom his father so loved....
"William, are you sure you feel well?"
Henrietta's anxious voice flung ice upon his burning heart.
His old irritation flared at her. "Of course I am well! Naturally I am tired. I don't expect to rest until after the funeral tomorrow. I think you ought to go and see Mother."
"Candace said she was sleeping."
"Then it is time she woke."
He took her elbow and led her out of the room. In the hall he pressed a button and the man appeared again. "Take my sister upstairs to my mother's room," William ordered.
"Yes, sir. This way if you please, Madame."
The sliding doors closed behind Henrietta and she was compelled to follow the man, her footsteps sinking again into heavy carpets across the hall and up the stairs and down another hall to one of a half dozen closed doors. Here the man knocked. She heard her mother's voice. "Who is it?"
"Thank you," Henrietta said, dismissing the man with a nod. She opened the door. There her mother sat at a small desk, fully dressed, her steel-gray hair swept up into a thick knot on top of her head. She was writing and she lifted her pen and turned her head.
"Henrietta, my dear!" She rose, majestic, and held out her arms. "My dear daughter!"
Henrietta allowed herself to be enveloped and she kissed her mother's dry cheek. She saw in the first glance that although her mother had aged or weathered into a dry ruddiness in the years since they had last met, she was not changed. Neither life nor death could change her. There was nothing new here. Her mother planned what to do, how to behave, what to say. Henrietta withdrew herself and sat down and took off her hat and coat.
"Mother, it was so strange to find you and Father gone away when we got to Peking."
"You should have told us you were coming," Mrs. Lane said, "then you needn't have come all that way."
Henrietta refrained from mentioning Clem, his reasons for wanting to go to China, the suddenness of their departure.
"Please, Mother, tell me everything."
Her mother could tell only so much as she could comprehend of what had gone on.
"Everything got harder in Peking," her mother began. "It wasn't in the least as it had been in the dear old days. You remember, Henrietta, how easy everything used to be? When you were a child, I was received most courteously wherever I went, merely because I was a foreigner. That was after the Boxer Rebellion, of course. Peking was heavenly then. I got to be fond of the Old Empress, really fond! I went with Mrs. Conger sometimes to call and Her Majesty used to have one of her ladies explain to me, so that I could tell Mrs. Conger who spoke no Chinese at all, how sorry she was for all that had happened, and how she understood that we were all there for the good of China. Then she would reach out her hand and stroke mine. She had the most beautiful old hand—so delicate, covered with rings, and then the long enameled nail protectors. It was really wonderful to see her. I don't think most people understood her. I used to tell your father so, but he would never trust her, no matter what I said."
"When did Father fall ill?" Henrietta asked.
"It began soon after that upstart Sun Yatsen stirred up the people. Your father was so worried. I told him that nothing would be made better by his worrying, but you know he never listened to me. In his way he was frightfully stubborn. And things began to get so hard. After the Empress died the wonderful courtesy just ended—like that! Even the people on the streets began to be rough to us. They didn't seem to want us in Peking. Your father was stoned one Sunday night on his way to chapel."
"Stoned—for what?" Henrietta asked.
"For nothing—just because he was a foreigner. Then it got better again. Oh dear, you've been away so long! It's difficult to explain. But it has been one thing after another, a revolution about something all the time, and when I told your father he was looking thin he always said he couldn't leave."
"And when he did leave he wanted to go to William."
"He got the idea suddenly that William needed him. I remember he said a queer thing when we were standing on the deck as the steamer pulled away from Shanghai. He was staring at the shore and then he said, 'But what shall it profit a man if he gain the whole world and lose his own son?' "
Henrietta did not answer. She did not listen any more to her mother's prattling voice. A strange thing for her father to say, and what did it mean?
Henrietta went herself to the station to meet Clem. With his usual skill, perfected by constant travel, he managed to catch a train at the last moment possible in time to get to the funeral. Had there been half an hour's delay it would have been too late. But Henrietta had now come to believe that there would never be such delay upon any train which Clem chose to take. Luck was the aura in which he lived.
Thus she stood waiting on the platform while the train drew in, accurate to the second. Clem was always the first passenger to get out. She saw him swing himself down, shake his head at a porter and come hurrying toward her, carrying his small bag. William's chauffeur stepped forward to take it but Clem resisted.
"I'm used to carrying my own suitcase, thanks."
He threw the man a brief bright abstract smile, then forgot him. "Henrietta, gosh—it's good to see you! How are you, hon?"
"Come on, Clem. We haven't a moment."
"Funeral isn't till four, is it? Lots of time."
This Henrietta would not allow. "Come on, do. Everybody's waiting."
"Everybody's early then." But he humored her, seeing that her eyes were washed with weeping.
They got into the big heavy car which William had imported from England. Clem lifted his sandy eyebrows and said nothing, but Henrietta understood his reproach.
"Never mind, he always hates England and yet he worships everything English."
"I don't mind. Anything to tell me, hon?"
"Not now, Clem. Afterward."
They drove in silence through the bright New York streets. He saw her dressed for the first time in black. She looked handsome but he had better sense than to tell her so now. He wanted to share her sorrow but he could not. When he thought of Dr. Lane's death he saw with dreadful renewal the sight of his own father lying with his head half severed from his neck, in the midst of the other dead. He wanted to talk quickly about something else, tell her how triumphant the market opening in Dayton had really been, and yet he knew that he should not speak of that, either, here or now. To escape the inescapable memory he stared out into the streets, trying to catch from the passing windows ideas for advertising, for displays, for announcements, and while he did so he felt guilty because he dared not think of Henrietta's grief. She could not comprehend, perhaps, though he had told her everything, how memory could pervade his whole life if he gave it the least chance at him. He crowded it out by his constant activity, by his incessant planning and incredible accomplishment.
"You are never still," she said with sudden and extraordinary impatience.
He looked at her, astonished.
"Oh, Clem!" She seized his hand in both of hers.
He saw tears brimming again into her eyes. "I know, Henrietta. I don't know why I can't sit still."
She was broken by his humility. "Don't mind me. I can't tell you why I feel so mixed up."
"That's all right."
He made a superhuman effort then and did sit still, forcing his hand that held hers to be still, keeping his feet from twitching or shuffling, refusing to recognize the itch of his nose, his cheek, the nervous ache of arm or leg, the innumerable minute demands of his tense frame.
She was grateful and in silence they sat while the car swept them up to the huge church on Fifth Avenue where William had commanded that his father's body be laid. Here she and Clem got out and mounted the marble steps. In the lobby they were met by an attendant of some sort, who guided them in silence to an area of pews tied in with black ribbon, where the family was assembled. To her surprise she saw even Roger Cameron and his wife, Roger lean and aged and looking as permanent as a mummy. Her seat and Clem's had been kept beside William. She sat down.
Clem looked across Henrietta into William's eyes, gray under the heavy brows. He felt a shock in his breast. The tall grim boy he had seen on the Peking street had grown into a tall grim man. In the one glance and the brief nod Clem saw the long square face, the pallid skin, the deep-set eyes and black brows, and the strained handsome mouth. Then he sat down, forgetting the dead. William was unhappy! The sorrow of the last few weeks could not have worked quickly enough to carve his face into such lines. But why should William be unhappy as well as sorrowful? Unhappiness was something deep, permeating to the very sinews of a man's soul.
"The Lord giveth and the Lord taketh away." The rich and polished voice of the robed minister rolled from the chancel. Clem breathed hard and tried not to shift his feet. The flowers were too fragrant, the church too warm. Upon the bier he saw a white-faced statue, handsomely clothed and surrounded with flowers so skillfully that they made a background for him. This statue did not look in the least like Dr. Lane, whom he remembered as a quiet melancholy saint, always withdrawn though kind. This dead man looked proud and even haughty. His features were too clear, the eyebrows touched with black, the lips with a pale red, the nose perfected, the sleeping eyelids outlined. The head had immense and marble dignity. As he remembered, Dr. Lane had walked with a slight stoop, a humble pose of the head, and his features though good were blurred with the thoughtful doubt of a man who always saw the other side of everything.
William, he supposed, had ordered all to be of the best, and so they had made the best of Dr. Lane. Clem disliked what he saw and feeling the impulse to move now become uncontrollable he stealthily shifted his feet, scratched his wrists and palms, and even rubbed his nose with his forefinger while a woman with a loud clear soprano sang a hymn, "For All the Saints Who from Their Labors Rest." Henrietta pressed his arm with her shoulder and he became quiet again.
The minister got up and began a eulogy of Dr. Lane, whom he had never known, and Clem listened. All the facts were right, he supposed—Dr. Lane, the father of William Lane, one of America's great figures, was born of a distinguished and scholarly family. Although his family had not entirely approved his becoming a missionary he had persisted in his noble determination, in which he was joined by a fine young woman of equally good family. It was not usual that two young people of such position gave up all to follow after Christ in a heathen country. There Dr. Lane's efforts had been singularly blessed. He had become important not only in the mission field but in his interpretation of the Chinese mind during the political crises of recent years.
"The fellow isn't saying the really important things," Clem told himself. It was strange that William had not pointed out to the minister that his father understood the Chinese and appreciated them and that he had not always wanted to convert them. That was why they had liked him. William should have told the small good things his father did, how he always put his hand into his pocket when he saw a beggar....
Dr. Lane, now, would have understood how he himself felt about getting food to people, quick and cheap. He would have enjoyed telling him about his markets and how he planned to find something that could be done anywhere in the world. He could have told all that to Dr. Lane, things he had not even told Henrietta, though she always stood by him whether she believed he could do it or not. But Dr. Lane would have believed it, maybe.
Clem stole a glance at William's profile. They were standing up. The funeral was almost over. Maybe he would be able to talk with William tomorrow when this was past. There was the grave yet.
Around the open grave he stood among this family he did not know, yet to which he belonged because he and Henrietta belonged together. He saw them all, Jeremy and Ruth and the girls—cute little things, dressed in white instead of black, little white fur hats and coats. He had never seen Jeremy or Ruth or Mrs. Lane. They were the sort of people he did not know.
While the minister spoke his solemn rich words and crumbled earth upon the coffin, Clem stood looking brightly abstracted, entirely unconscious, while his mind glanced at the various miracles of his life, first of which was that Henrietta had wanted to marry him. Seeing this family, he could not understand it, though he was not humble, either. The miracle was that, having been born among these people, she should have had the wit to see what he was and what he could do before he had done it.
He looked at her as she stood, her black-gloved hands clasped, her strong profile bent, her eyes upon the ground. He loved her mightily, he loved her the way he loved his work, the way he loved his dream. It was one of the big things. But she was whole and entire without him. He did not think of her as a part of himself because he thought nothing of himself. He did not know how he looked or what sort of a man he was. He was as fleshless as a grasshopper.
He was glad that Henrietta had never spoken to him of having children. He had seen too many children starving to death. The villages on that long and lonely march from Peking to the sea had been busy with children, dirty, laughing, hungry—so many children in the world, anyway. When he thought of children he always thought of his sisters as he had last seen them and his mind swerved away from that again. He had to be free to accomplish the thing for which he was born and children ought to be kept at home, treasures in a box. If his sisters had been kept at home they would have been alive today. He did not ever want children.
Tim and Jen and Mamie! When he had hurried back to the farm after reading the ghastly story that held the headlines for a day, Tim was dead and buried. Pop Berger was in bed sick and he cried whenever anyone spoke to him. A police guard sat by the bed and there were reporters everywhere. Mom Berger kept the girls in the kitchen with her and the doors shut. There had been a square-set newspaper fellow there whose name was Seth James. He had gone away after he heard Clem was going to take the two girls to Ohio.
"You're the only decent person I've seen," the fellow had said and had shaken Clem's hand up and down hard half a dozen times.
Clem had not known what to do with Mamie and Jen. They had cried when he took them away. But Henrietta had been nice to them and after a while they learned to wait on people in the store. Then, after they had fattened up a bit and got better looking, they had both married farm boys. Mamie had died when her baby was born but Jen, who he had always supposed could not live long, was growing stout and talkative. Food had done it, of course—plenty of good food.
He came to himself suddenly when Henrietta put her hand on his arm. The funeral was over and he was ashamed that he had not kept his mind on it. He turned, obedient to her touch, and joined the solemn family procession back to the funeral cars.
The procession stopped at William's house and the family descended and entered the huge front door, held open by the footman, who wore a proper look of gloom. Roger Cameron and his wife had gone home, their car swerving past the ones that stopped. When Candace had begged her father to come in and stay the evening with her he had refused. "I swore ten years ago I would never go to another funeral before my own, and it was only because your mother forced me that I have come today. You'll have to get through the rest of the day the best you can, daughter."
Candace went upstairs and changed her black garments for a soft white gown whose collar she tied with a black ribbon. Then she hurried downstairs to see if the tea which William had ordered to be ready was set upon the table. It was more than a usual tea. Henrietta and Clem were taking an early train and Jeremy and Ruth must go home with their children. There were ham and sliced cold chicken upon the buffet and she knew that the cook had beaten up a custard dessert. By her command there were no flowers on the table. She had seen so many flowers this day that she did not want any more. Red roses perhaps next week! The dreadful thing was that she had felt no sorrow; a mild sadness, of course, such as death always persuades, but not sorrow. It was impossible to grieve for an old man to whom she had scarcely spoken, a sweet old man, she saw, even through his illness. But what troubled her was that she had not been able to share William's sorrow. He treasured it, he kept it to himself, he endured with such nobleness that she felt repelled and then was angry with herself. She dreaded tomorrow when nobody would be here—except, of course, his mother. For the first time she felt glad that his mother was going to spend the winter with them. Perhaps together they could understand William better and make him happy.
At this moment while she moved about the dining room, Henrietta's husband came to the door and looked in. He made her think of a bird, slender, bright-faced, boyish, making so many little quick unconscious movements. He was completely different from Henrietta and yet there was something between them. She did not see why William had been angry when Henrietta married Clem.
"Come in, Clem," she said sweetly.
He came in, his hands in his pockets jingling something, keys, coins—no, a small bottle of pills which he now brought out. "Can I find some water somewhere? All this has brought on my nervous indigestion."
She lifted a cut-glass carafe from the sideboard and he whistled softly when he took it. "Solid, isn't it?"
"A wedding present. If you saw the amount of cut glass I have packed away, besides all this!"
"Swell wedding, must have been. But then, William would have that. Did he ever tell you we met once?"
"No, did you?"
He rolled pills into the palm of his hand, threw them in his mouth, gulped them and washed them down with water he poured into a goblet on the table. "Maybe he has forgotten but I never have. A Chinese boy and I were kind of dancing around each other ready to let out our fists when William came by and stopped us."
"Did he know you?"
Clem grinned mischievously and she saw freckles under his pale skin. "No—but he knew who I was."
"What do you mean?"
"I came from the wrong side of the tracks, see?"
"There were no tracks in Peking, were there?"
"Oh yes, there were. The Lanes were aristocrats compared to us. Dr. Lane got a salary every month. They lived in a compound. My father hadn't any salary. He was low enough to live on faith alone."
They spoke in half whispers, almost guiltily, enjoying the respite from gloom. He had a sense of humor, Candace saw. And Clem saw a pleasant pretty woman, an honest woman at that, not too smart maybe, certainly not grand like his Henrietta, but nice to talk to, especially after a funeral.
"Christians are like other people. What'll I call you—Mrs. William?"
"Oh, call me Candy."
"Candy, eh? Nice name for you. My father was ignorant, Candy, just plain uneducated like I am. There's a difference, though. I wanted an education and he didn't believe it was right. He thought God would provide everything—even food, you know. Dr. Lane knew better. He was real well educated. Of course my father was only a farm boy."
Candace stared at him, not comprehending in spite of what she heard. He tried further.
"All the well-heeled missionaries who didn't have to trust God looked down on us, naturally. I guess my poor old dad was a sort of beggar sometimes. When he saw us hungry and no food in sight he used to push God a little."
"How?"
Clem's face turned red and the freckles disappeared. "He went to the other missionaries—or even sometimes to the Chinese—and told them we had nothing to eat." He tried to laugh. "Kind of tattletale on God, I guess! Anyway, I don't like to think of it."
"I'm sure William has forgotten all of that," Candace said, on a rush of pity and vague affection for this too honest man.
"Maybe," Clem said. He looked sober and began jingling his pockets again.
Something haunted his restless blue eyes and Candace went on pitying him. "You're very happy with Henrietta, aren't you? She adores you, I think. When she talks about you she looks as though she were thinking of her child as well as her husband."
"There is nobody in the whole world like Henrietta," Clem said. The red had left his face as quickly as it had come and the freckles were back. "I don't know what I'd do if I didn't have her. She's my life's foundation. I'll build all sorts of superstructures, maybe, in what I'm trying to do about food, but she keeps me steady. And here's the thing—she never discourages me."
"Wonderful! And what are you trying to do about food, Clem?"
"Oh—just feed the world."
"Hush!"
She put a hand, pretty and ringed, upon Clem's arm. They listened and she took it away again. William entered the room and she turned to him
"Clem and I are here waiting, William. Everything is ready."
"I don't know where everyone is," William said.
He sat down in a great Jacobean chair that stood beside the long windows opening to a wide terrace. He still wore his black suit and above the dead hue of the broadcloth his face was whiter than ever, his brows more intense.
"Clem was talking about feeding the world."
William glanced from under his eyebrows and Clem suddenly heard the jingling in his own pockets and took his hands out of them.
"You are in the food business, aren't you?" William asked without interest.
"Yes," Clem replied. "I've just opened a big new market at Dayton, Ohio."
"What has that to do with the world?"
"Just a beginning," Clem said without humility. He was surprised to find that he rather enjoyed talking with William. There was an edge to it. Walking briskly across the floor he took the other Jacobean chair on the opposite side of the window and turning sidewise began to talk with sudden fluency.
"I began in the simplest sort of way—with a grocery store, in fact, in a small town, New Point, Ohio. It's still the home base. I have no family, you know—Boxer Rebellion put an end to that."
"My father told me," William said.
"Yes, well, we don't have to remember the past. But the way we had to live when I was a kid I suppose made me awful interested in food. Can't eat much myself—I have nervous indigestion. All that wonderful stuff on the table there—I won't hardly touch it. A cup of tea maybe and a little chicken. Bread poisons me, though I make the finest bread. Say, William, do you remember Chinese bread?"
"My mother never let us eat Chinese things."
"Well, we were thankful for that bread at our house. It was a lot easier to take than starvation. I learned what good bread was. I might send you a few loaves of my product."
William was too shocked to thank him. "Is your business successful?" he asked coldly. The fellow looked like a country storekeeper.
"I undersell every staple," Clem said with pride. "I watch the surplus everywhere in the country. Got twenty men doing just that. Some day I'll be watching world surpluses. Then I'll be doing what I mean to do."
"You actually plan to establish a world food monopoly?" William for the first time in days looked interested.
"Hell no!" Clem said cheerfully. "I'm not interested in monopolies. I'm interested in getting people fed. If they can't pay for it I give it to them."
"You mean you _give_ food to people?" William's voice was unbelieving.
"Why not, if they're hungry?"
"But you can't stay in business that way."
Clem wriggled in the huge chair, scratched one cheek and then the other with one hand, and then pulled the short hair over his right ear and rubbed both knees. "I don't know why," he said humbly "but I'm a millionaire already—or almost."
Candace, seated upon one of the gilt dining chairs, suddenly began to laugh and William turned upon her.
"Why do you laugh, Candace?"
She buried her face in her hands and shook her head, still laughing. What had made her laugh was the look on William's face but she could not tell him. "It's so funny," she gasped, her face still in her hands. "It's so funny to get rich giving food away."
"Nonsense," William said. "Of course he doesn't give it all away."
"But to give any of it away," she murmured. She found her handkerchief and wiped her eyes. Then she caught Clem grinning at her wryly.
"It is funny," he agreed. "It's darned funny. I can't explain it. There's some sort of magic hidden in the golden rule—I can't explain it any other way."
Upon this conversation, which had become entirely repulsive to William, Mrs. Lane now entered, followed by Jeremy and Ruth. Behind them came Henrietta with her hat on, ready for the train. William rose. "Let us take our places," he said quietly. "Mother, please sit at my right. Ruth at my left, Jeremy at Candace's right and Henrietta next. Your place, Clem."
When they were all seated William lifted his head and fixed his eyes on a point above Candace's head at the end of the long lace-spread table. She saw that there was something he wanted to say to them.
"It has not been our habit in this house to have grace before meals. Perhaps we have grown careless. But from this day on, in memory of my father. "I will say grace at meals in my house."
His eyes fell and for an instant Candace's caught them. He saw love and pity rush into tears and he bent his head to avoid the sight.
"Dear William," his mother whispered, and put out her hand to him. But he did not pause to look at Candace or touch his mother's hand. He bent his head and began to pray in a tense low voice:
"Our Father, for the food that Thou hast given us, receive our thanks. Bless this food to our use and us to Thy Kingdom, Amen."
It was the grace that his father had used throughout the years of his missionary life.
# 8
CLEM BIDED HIS TIME. His faith, fulfilling itself by his steady success, was only embattled when he met with opposition. He was amazed when he discovered those who would have laughed at him had he failed but who were angered by him when he did not fail, and who attacked him finally for undermining their own markets. These were the consolidated groceries and food companies, the chain stores which were beginning to form a net over the whole country. They declared that they, too, were selling to the people cheap and good food, and they began their warfare by insidious advertising against Clem's wares, saying that cheap surplus foods were not guaranteed foods and carried in them the germs of disease and decay. Buy only our packaged foods, they screamed, buy foods only with our seal upon them.
"We must get some big lawyers," Bump told Clem. During the war he had served as a food expert, and had won a medal for saving the nation millions of dollars in food, buying where experience with Clem had taught him to buy and buying, too, with Clem's help. Somewhat reluctantly, when the war was over, he had married a German girl, Frieda Altmann, with whom he had fallen in love while he was overseas and they now had two fat children who looked, he often felt, entirely German. Nevertheless his Frieda was good and a fine cook and she adored Clem, whom she considered a god, and she was humble before Henrietta, whom she loved with enthusiasm. But Frieda did all things with enthusiasm.
Clem had only to be driven into a corner to become cool and aggressive. He hired two clever lawyers, Beltham and Black of Dayton, and entered into the private war which was to last as long as he lived.
For Clem himself the world war had been an atavism which could not be understood. Europe he knew little and his inclination was to think of it as a small and diverting piece of ground which included England. He had run over there, as he put it, the summer before the war, Henrietta, of course going with him. He still refused to allow an ocean between them. A few weeks in England had sufficed.
"Can't tell these people anything," he said to Henrietta. "They think I have only one idea. Well, that's all I need. If an idea is big enough a man don't need but one."
He surveyed the tidy farms and smooth green hills of England with something like cynicism. "I seem to see India behind all this," he said. "I see Egypt and the Middle East. Sometime we got to go and take a look at India, hon, and see the green hills there and the fat people. All these beef roasts and steaks and legs of mutton!"
In Europe he looked for hunger and found little. Instead he found prudence and habitual scarcity. The French threw nothing away and this he approved. A fish head belonged on the dish and not in the garbage can.
"There is no sweeter meat than the cheeks of a carp," Mrs. Fong used to tell him in Peking and he had never forgotten.
The farms in Denmark were Clem's delight. He visited them without introduction, appearing at a barn door while Henrietta lingered in the road outside. Sometimes he called her, sometimes he did not. One morning he beckoned to her fiercely.
"Come here, hon—this fellow has an idea!"
She looked into the wide barn door and there in the shadowy depths she saw the Danish farmer painting the walls. Pots of paint, green and sky blue, stood on the floor of beaten earth and with a large brush, not of a housepainter but of an artist, the farmer was painting the walls with scenes of green meadows and running water under blue skies.
When he saw their admiration and surprise, he grinned and spoke to them with a few words of the English he had learned in folk school.
"For wintar," he explained. "Make cows happy. Grass nice, thinking summer."
"Ain't that smart?" Clem asked, turning to Henrietta. "He knows the cows get bored in the winter locked up in the barn and so he wants to make them happy. Good fellow!" He clapped the thick-bodied farmer on the back. "Nice idea! Bet they give more milk, too."
They began a conversation of gestures and a dozen or so words. Clem picked up languages quickly and he carried small pocket dictionaries everywhere. From the Dane he learned that it was hard to export as much butter as they had to England, because English farmers had their own butter. Yet Denmark needed more coal, English coal, which was going instead to Italy to buy fresh fruit. If the new refrigerator cars really began to run in large numbers, then Denmark would have even less coal.
Clem became concerned in the perennial question of distribution.
The monstrous folly of starvation anywhere in the world impressed him day and night. Food was abundant upon the land and in the sea. However many people were born and lived, there was more food than they could possibly eat. In America he saw apples rotting in orchards; corn used for fuel; granaries filled with wheat so that public money must buy still more, build still more granaries; eggs spoiling for lack of consumers; potatoes fed to beasts; fish made into fertilizers. Denmark had only butter to sell, but Americans had too much butter and would not buy. Argentine beef sold for pennies a pound because there was too much meat. The same story was everywhere in the world of starving people and rotting plenty.
"There has got to be some sort of over-all," Clem said thoughtfully. "Not government, either—but what?" He had absorbed from the Chinese a deep distrust of government. Men in power, he had once declared, became more than men. They fancied themselves gods. Henrietta had laughed when he said this. She did not often laugh, and when she did he always wanted to know why. "Sometimes you act a little like God, yourself," she had replied.
He was inexplicably hurt. "No—no—don't say that, hon! Maybe like a father. Only like a father, though."
She was learning to sheathe her bluntness because she did not always know what could hurt him. He went about so shining in his hopefulness, so childlike in his goodness, so impregnable in his devotion, that it seemed nothing could hurt him. Then she found that she alone could do the damage. Opposition from others, their laughter, their disbelief, he could and did ignore or accept as persecution by evil. But she whom he loved, who loved him, could pierce his bright armor and bring tears to his eyes. The first time she saw the tears she had wept with shame, had sworn to herself that she would never laugh at him, never caution him, never show doubt—nay, more, she would never feel doubt. The one sin she could commit, she told herself, was to hurt Clem.
The years had passed and still they had no children and still she did not mind. Clem filled every need of her being, and she devoted herself to him, taking over almost without his knowing it all the things which he hated to do: the meticulous detail of business, the bills, the arrangements for shipping, the delivery of carloads of foods, the refrigeration and preservation and then disposals. More and more she and Bump conferred on the carrying on of Clem's decisions, daring and bold as they were, sometimes involving the loss of thousands of dollars as well as the possibilities of profits as great. Neither of them questioned what Clem decided to do. It remained for them merely to discover how to do it.
During the war, however, he had made a decision of his own so peculiar, so unlike him, that for a while Henrietta wondered what change had come in him that she did not understand. He had begun in recent years to read faithfully William's newspapers. What he thought of them he never said, but his intense look, his frequent silences when he had studied a tabloid carefully, made Henrietta long to put a question to him. But she did not. He had never allowed her to complain to him fully about William.
"He's your brother, hon," Clem had said. "He's part of your family. A family is a great thing to have. China would have died and disappeared long ago if it hadn't been for the way families stick together over there."
"I hope you won't try to make me stick to mine," Henrietta had retorted.
In one of William's papers, more and more filled with pictures, Clem had discovered during the war a feature about Chinese coolies digging trenches in France. He found it one Sunday when he was at home, and sitting on the small of his back in a large armchair, his feet propped on the rungs of another chair in front of him, he had stared at the bewildered faces of Chinese farmers in France, staring back at him from the pages.
"I bet they don't have a notion of why they're there or why they're digging those trenches," he told Henrietta.
It was a peaceful morning in America, and townfolk walked quietly past the house with their children on their way to church. Henrietta looked at Clem. She knew him so well, so familiar was every line of that thin square face and every note of his brisk hurried speech, that she divined at once that in his musing tone and his meditative eye a plan was beginning to shape. She waited while she polished the silver, a task which she usually planned for this time when Clem was at home. She sat at the dining-room table covered with newspapers upon which the silver was spread.
"I bet those Chinese were just carted over there like cattle," Clem mused. After a few minutes more he got up.
Henrietta followed him with her watchful look. "Can I get you something, Clem?"
He was hunting for paper and pen. "I want to write to Yusan. What are those Chinese farmers doing over there in France? I bet somebody's up to something."
She rose and found paper and pen, an envelope and the proper stamps, and when he had scrawled one of his brief letters, she sealed it and put it aside to mail in the morning.
This was the beginning, as she knew it would be. The end was several months later when Clem and Yusan met in Paris. Clem, leaving her in charge, for Bump was now in the war himself, put the ocean between them for the first time.
"I'll only be gone a couple of weeks, hon," he said. Agony was plain on his face. "I don't know why I'm doing this, but somehow I have to..."
"That's all right, Clem," she said. It was not all right, it was far from all right, and she felt the physical tearing of her heart out of her flesh as she stood on the pier and watched him go away, his face whiter, his figure smaller as the ship moved toward the sea.
And Clem, his eyes fixed upon her who made his whole home, cried out against his own folly. Had Bump been at home he would have brought her along, but without Bump only Henrietta could hold together in his absence the vast structure of his markets. What drove him to France he scarcely knew except that when he hesitated the faces of the bewildered Chinese were there before him. He saw them in their villages, in their own fields, in the streets of the cities into which they flooded in times of famine and starvation. How could they understand France? He would get Yusan started and then he would come home again to Henrietta, maybe run over again a couple of times to see how they were making out, but taking her with him next time, for sure.
In Paris he met Yusan, who wore a new suit of Western clothes. At first Clem scarcely recognized him in the crowd of Frenchmen, except that they were all talking and Yusan was standing immobile, silent, watchful, and therefore as conspicuous as a statue of gold. Clem caught his hand and forgot for a moment even Henrietta.
"Yusan!"
"Elder Brother!"
They broke into Chinese simultaneously and the French men and women stared and cried out to heaven in admiration at such fluency, nothing of which was comprehensible to them. Clem liked the French people and bustled his way among them with the same assurance he had at home in America or in China. They had the same mixture of naturalness, simplicity, shrewdness, humor, childishness, and sophistication that made Americans and Chinese alike, too, and he had pondered this until he remembered that all children and old people are alike, the one because they are young and the other because they are old, the young knowing nothing and accepting everything, and the old knowing everything and therefore accepting anything as possible.
Yusan, following Clem's directions, had come over with a shipload of the coolies, as they were called. He had volunteered as an interpreter for them, and had been accepted. Now at last his English, learned so early and of late years revived and maintained because of Clem, was of the utmost use. He had his men already established in barracks near the front, where new trenches must continually be dug. At night they lay down to the sound of the booming cannon, and sometimes the Chinese in the farthest sectors were killed, even as the French, the English, and the Americans were killed. But the Chinese had no inkling of why they were there or why they were killed. They had been lured by the promise of pay for their families at home and a little for themselves, and they were here.
Clem left Paris the same day with Yusan, traveling by train and by military truck. He had his own pass, stamped and signed in Washington before he left, and he was sent through without delay, Yusan at his side. The days on the ship had filled Clem to bursting with plans and ideas and he paused only briefly to ask about Yusan's family.
"All well," Yusan said. "Two more grandsons I have given my parents or they would not have let me come, except that you asked it."
"What about Sun Yatsen?" Clem asked.
Yusan shook his head. "One reason I was glad to come with you, Elder Brother, is that everything is altogether confused. Sun Yatsen has not tied our country together. He was too much in Japan, and Japan wants to eat us alive. Now this has become clear to all in the Twenty-one Demands. It is true that Sun has left Japan, but he does not know what to do next. First we are a republic and then we are not a republic. He has destroyed the old government but he does not know how to make a new one."
Clem remembered that dark night in the tin hut in San Francisco and now described it to Yusan. "I told him he ought to get down to the people. I told him if he didn't get the people fed and looked after, he would surely fail."
"He will always be a hero, Elder Brother," Yusan said. "We will not forget that he freed us from the Manchu yoke. But he has not led us onward from there. He wants obedience and when we hesitate, he says we are like a tray of sand. Elder Brother, you know we Chinese always work together. But we do not believe all wisdom is in one man."
"Well," Clem said briskly, dismissing the revolutionist. "I guess he has to learn in his own way. Now, Yusan, here's my idea—"
He caught a certain quizzical look in Yusan's dark and narrow eyes and he grinned. "Don't you get me mixed up with Sun! I'll give you my ideas but I don't insist on anything. You do what you like with them. My ideas are a gift. Take them or leave them."
"Elder Brother, I accept the gift," Yusan said.
Neither of them looked out of the window at the lovely French landscapes that fled past one after the other. Night fell and they approached the war sector and they did not see that beauty had ended and the barrenness of death was about them. From the train they got into a truck and drove through the night over roads once smooth and now rutted with shell holes. This in turn gave way to rough bare ground and so they came to their destination. Clem walked into a barrack filled with homesick Chinese men, not one of whom could read or write or even speak with the people around him. In the dim light they lay on army cots and listened to one man who played a wailing village tune upon a two-stringed violin he had brought from home.
"Brothers!" Yusan cried above the music. "Here is the Elder Brother of whom I have told you!"
They got up from their cots, the fiddler stopped his wail, and the lantern lights were turned up. Clem saw himself surrounded by the familiar faces, the brown, good faces, the honest eyes, of Chinese villagers. He felt again the old love, paternal perhaps, but grateful and rich with faith. These were the good, these were the simple, these were the plain of the earth. He began to speak to them:
"Brothers, when I heard you were here, I feared lest you might be suffering, and so I have come to see if your life is good and what can be done to help you if it is not good."
"He left his home," Yusan put in. "He came a long way over the sea and he can be trusted. I have known him since my childhood."
The men were silent, their hungry eyes fixed upon Clem.
"Are you well fed?" Clem asked.
The men looked at one of their number, a young strong fellow with a square fresh face. He spoke for them:
"We are well fed but with foreign food. We are treated kindly enough. Our sorrow is that we cannot write to our families or read what they have written to us. We can neither read nor write."
"The letters can be read to you," Clem said. "Letters can also be written for you."
The young man looked at his fellows and began again. "Why we are here we do not know. Is our country also at war?"
"In a way, yes," Clem replied. "That is, China has declared war against the Germans."
"We do not know the Germans," the young man said "Which men are they?"
Clem felt his old sickness of the heart. "None of us know our enemies. I also do not know a single German. Let us not think of them. Let us only think of ways to make your life better."
For how could he or anyone explain to these men why there was a war and why they had left their homes and families and come here to dig trenches for white men to hide themselves in while they killed other white men? Who could explain such things to anyone? The world was full of discontent and because people were hungry and afraid they followed one little leader and another, hoping somewhere to find plenty, and peace for themselves and their children, even as these men had been willing to come so far, not because they believed in what they did, but that their families at home might receive each month some money wherewith to buy food.
Clem spent most of that night talking with the men, asking them questions, too, and writing down their answers. He spent the next days with Yusan planning, and a full month he spent getting what he needed to fulfill those plans from officers who considered him mad. But Clem was used now to men who thought him mad and he paid no heed to what they thought of him, spending his energy instead on getting them to do what he needed to have done until in sheer angry impatience they yielded and cursed him and wanted him gone.
By the end of the month he had helped Yusan to set up a school where the men could learn to read and write, if they wished, and he set up an office, with two Chinese from Paris, to read the men's letters from home and write in reply. He set up also a small shop, to be supplied regularly from Paris with Chinese foods and sweets and tea. Once a week he planned a night of amusement, a place where the Chinese could hear their own music, could eat their own sweetmeats and drink tea together, and see Chinese plays and Western pictures. He hired a Chinese cook who was given a license to vend his own wares and make his living thereby. He established Yusan in all this, and in his first moment of leisure he discovered that he was homesick for Henrietta and could no longer endure his absence from her, although he had scarcely thought of her for the whole month, even as he had not once thought of himself.
He bade Yusan good-by then, took a ship for home, and reached his house on a Saturday afternoon, so white and spent that Henrietta cried out at the sight of him as he entered the picket gate.
She was at home, as she was now as much as she could be, for she expected Clem at any moment, though he had not said he was coming. Her own longing for him reached across the sea and yearned for him with such intensity that she could divine, or she felt she could, the time when he would be coming.
"Oh Clem!" she cried at the front door.
"Hon—"
They fell into each other's arms. He felt her sturdy body and she was frightened at the thinness of his shoulder blades under her embrace.
"You've worked yourself to skin and bones!" she cried with terrified love.
"I'll be all right after a few days at home. My stomach went back on me a couple of weeks ago."
They parted, their hands still clinging, and she led him in, made him sit down, and restrained herself from fussing over him, which he could not endure.
"I'll make you a cup of tea. Can you eat an egg?"
"I could eat a beefsteak, now," Clem said. He looked around the shabby room fondly. "I guess I was crazy to go away, hon! Now that I'm back it seems crazy. But I had to go, and I'm not sorry. How's tricks?"
"Don't talk about tricks!" Henrietta retorted. "You rest yourself, Clem, do you hear me?"
"Why, hon, you aren't mad at me, are you?" His face was amazed. She had never been cross with him before.
To his further amazement now she began to weep! Standing there by the kitchen door, she took up the edge of her apron and wiped her eyes. "Of course I'm not mad," she sobbed. "I'm just scared, that's all! Clem, if anything happened to you—if you should die—I wouldn't know what to do. Being without you just these weeks—I'm all upset—"
"Great guns," Clem muttered. He got up and went to her and put his arms around her again. "I'm not going to die, hon. I wouldn't think of such a thing."
She put her head on his shoulder and he stood quietly supporting her, loving her and not telling her how he really felt. He was not going to die, but he felt tired to the bone. The sight and the memory of those dark honest bewildered faces in France never left him for a moment. Nor were they all. In the fields of France there were such faces, and the same faces were here in the fields of Ohio, upon the streets of villages and in the slums of cities, not all honest and many far from good, and yet with the same confusion and bewilderment. And most dreadful of all, they were upon the fields of battle, and they lay dead in the mud of death. No, he must not die, but he was tired enough to die. Nobody knew what he was trying to say; not even those whom he wanted to save could understand.
But he must not give up, for all that. He must take up again where he left off.
This meant, as he discovered in the years that followed the war, an organizing of his markets and facing limitations and legalities which irked and distracted his free-thinking mind. The war fought for freedom brought with victory a loss of freedom for everyone, and there were times when Clem felt this loss descend most heavily upon himself. He was used to visiting another country as men visit a neighboring county, careless of all save his purpose in going. Now there was no more of this carelessness. Passports and visas made him groan, and even Bump could not assuage his irritation either by speed or by early preparation. Clem felt it an infringement upon his rights that he could not decide suddenly to go to India by the middle of next week or drop in on Siam and see how the rice crop was going.
His first visit to India grew out of a brief meeting, quite accidental, with a young Hindu in London during the war. They had met in the Tube, and had sat side by side for a few minutes. Clem had begun instantly to talk and then, forgetting his own destination, had got off with the young Hindu and had gone with him to his rooms in lodgings near the Tube station. Ram Goshal had at first been astounded by this slender, sand-colored American and then had succumbed to Clem's frightful charm. Clem discovered that Ram Goshal, although the son of a wealthy Indian, had given up society life to work for Gandhi, whom he had met a few years before when Gandhi, that rising star, had gone to London from South Africa with an Indian deputation. Ram Goshal had come back with Gandhi to London at the beginning of the war and at a meeting of Indians, Gandhi insisted that it would not be honorable in the time of England's trial and trouble to press their own claims for freedom. Self-denial at such an hour, he said, would be dignified and right and gain more in the end because it was right.
Ram Goshal, reared in sensitive tradition, had been won anew by the largeness of Gandhi's mind. He had declared himself his convert, though troubled by his father's wealth, which was in great modern industries in India, of which Gandhi did not approve.
"God forbid," Gandhi has said, "that India should ever take to industrialism after the manner of the West. The economic imperialism of a single tiny island kingdom is today keeping the world in chains. If an entire nation of three hundred million people took to similar economic exploitation it would strip the world bare like locusts."
Clem could not, however, agree entirely with what Gandhi said, as Ram Goshal had quoted it.
"You can't get rid of something just by stopping it," Clem had told the young Indian. "Industrialism is here to stay. We've got to learn how to use it. We can't go back to the first century because we don't like this one."
Ram Goshal had begged Clem to go to India. "You will understand India," he declared, his eyes dark, huge, and liquid with admiration. "You are like us, you are a practical mystic." Then those profound eyes, haunted with the endless history of his people, glinted with humor as he gazed upon Clem. "You remember what Lord Rosebery said about Cromwell?"
"I am not an educated man," Clem said, humble before this young scholar of the East.
"He said that Cromwell was a practical mystic, the most formidable and terrible of all combinations. That is you, too—therefore I do beseech you to stop in my country and look with your own eyes upon my starving people."
Clem could refuse neither such warmth, such eloquence, nor the brown beauty of the young Indian's face and he promised to go as soon as possible after the war.
He decided suddenly one January day that he would take a few months off from the constant persecutions of his rivals, the chain groceries, being moved to this by a letter from Ram Goshal, now in India. Gandhi was then in the full tide of the noncooperation movement and Ram Goshal was in some trouble. His father disagreed with Gandhi, and, had Ram Goshal not been his only son, would certainly have disinherited him.
Clem read this letter thoughtfully and handed it to Henrietta.
"Hon, I feel I better go over and see for myself whether the British intend to do any better about feeding the people in India. If they don't I guess Gandhi is right. But I want to be sure about the British."
"Of course, Clem," Henrietta said. She suspected that Clem, whether consciously or not she did not know, was thus postponing a decision which Bump and the two young lawyers were pressing upon him. That Clem might defeat the purpose of the organized groceries to put him out of business, they declared, he must organize himself into Consolidated Markets, Inc. Clem, in spite of the three young men, still refused. He wanted most of his markets movable, his clerks ready to go wherever surplus foods were stagnant. Vast buildings and established staffs did not interest him. He did not want a name. His business was simply to gather food together and get it to people in need. When the need was over, the supply would cease.
While Henrietta thus suspected Clem she saw him look at her with sudden love.
"What is it, Clem?"
"Hon, the two words you said..."
"Yes, Clem?"
"You said, 'of course.' That's what you always say to my notions—Wonderful wife!"
So rarely did he speak words of love that tears gathered under her eyelids. "I mean it, dear."
"I know it." He bent and kissed the thick coil of hair on top of her head, and so began the journey to India.
In Bombay they went straight to Ram Goshal's house, a gorgeous palace outside the city beyond the Towers of Silence. Ram Goshal's father was fat, quarrelsome, clever, and he gave Clem no chance to talk, and perforce Clem listened.
"I do not oppose freedom, you understand, Mr. Miller. You Americans, I understand, love freedom very much. But the British have not oppressed me. I tell my son it is entirely because of the British that we are so prosperous. Gandhi is not so prosperous with them, but we are not Gandhi. There is no reason why we should fight his battles."
Ram Goshal, too filial to argue against his father, sat miserable in silence, taking his opportunity at night to keep Clem wakeful for hours. This combined with Indian food cut the visit short. All the courteous welcome and the eagerness of father and son to win America to their side could not mitigate the indigestibility of Indian food. Clem's delicate stomach rebelled at curry and pepper and fried breads. In England he had rejected great roasts and thick beefsteaks, boiled cabbage and white potatoes, and now in India he rejected cocoanut meats and sweets, peas overcooked and pepper-hot and every variety of food too highly seasoned.
Indian food cast his frame into rebellion and Henrietta took him to an English hotel, where he fasted for three days and then took to tea and soft-boiled eggs, while Ram Goshal stayed by him to see him well again.
Clem smiled his white and childlike smile. "I'm a fine one to be telling people about food, Ram Goshal. I have to live on pap."
"You are like Gandhi," Ram Goshal said. "You use your body merely as a frail shelter, a house by the wayside, something that barely serves while your spirit lives and does its work."
Clem was too American for this Indian ardor. "I hope I am a man of common sense," he said briefly. "Certainly I'm sorry about my weak stomach."
As soon as he was well he wanted to leave Bombay, and saying farewell to Ram Goshal, he wandered about the country for weeks with Henrietta to see how the people fared. It was impossible to travel alone, and they were forced to hire a bearer, a servant to look after them, a dark Moslem named Wadi, who encouraged them to look at Moslems and avoid Hindus until Clem discovered what was happening. Thereafter to a pouting Wadi he decreed the day's journey, poring over books and maps the night before. There was no sight-seeing. Clem wanted to go to villages, to see what people had in their cooking pots and what they grew in their fields. He grew more and more depressed at what he found. After they left the coastal plains there was nothing, it seemed, but endless deserts.
"The land is poor, hon," Clem said. "I don't know what these books are talking about when they say the people are poor but the land is rich. I don't see any rich land."
He turned northward at last to New Delhi, strengthened by rising anger and determined to cope with the rulers of empire in their lair. The stony hills outside the window of the train, the sparse brush, the dry soil, the pale spots of cultivation increased his wrath, until when he reached the monumental capital of empire, he was, he said, "fit to be tied."
Yet in justice he was compelled to admit that empire alone was not to blame for half-starved people and skeleton cattle. Whoever ruled India, still the sun shone down in sultry fury upon the blackened earth. It was winter in Ohio, a season which there meant snow upon level plains and rounded hills, and in New York meant lights shining from icy windows and snow crusted upon sidewalks and trampled into streets, and red-cheeked women at crowded theater doors. In India it meant the slow mounting of a torrid heat, so dry that the earth lay empty beneath it. Over the sick surface thin animals wandered dreaming of grass, and thin human bodies waited, feeble hands busy at pottery wheels, the dry earth stirred into clay, with a bowlful of water to make more empty bowls, plenty of bowls that could be broken after they had been touched by the lips of the unclean.
"A few wells here and there," Clem said to Henrietta, his skin as dry as any Indian's, "and this desert might be planted to grain."
But wells were not dug and who could blame men that they did not dig wells when the sun burning upon a dead leaf turned it crisp, charred at the edges and wrinkled as a dead baby's hand?
In the capital Clem, a pure flame of zeal, marched into the marble halls of empire and demanded to see the Viceroy. An American millionaire may see even the king and so he was received, making his way unmoved between rows of turbaned underlings. A mischievous old face, Indian, shrewd and obsequious, peered from under a multicolored pile of taffeta.
"I am Sir Girga—honored, sir, to conduct you to His Excellency the Viceroy."
The mischievous old face, set upon a waspish body and a pair of tottering legs, guided him into a vast hall where The Presence sat, and there brought him before a cold English face made cautious by splendor.
Clem, knowing no better, sat down on a convenient chair surrounded by space and then began to tell the ruler how subjects could and should be fed.
"Irrigation is the first thing," he said in his dry nasal American voice. He was unexpectedly hot and he wished he could take off his coat, but he went on. "The water table in India is high, I notice. Twenty feet and there is plenty of water—sometimes even ten or twelve. By my calculations, which I have taken carefully over sample regions, India could feed itself easily and even export food."
The Viceroy, immaculate in white tussah silk tailored in London, stared down on him as upon a worm. "You do not understand our problems," he said in a smooth deep Oxford accent. "More food would simply mean more people. They breed, Mr.—" he paused to look at a card which Sir Girga obligingly held out for him to see—"Miller."
"You mean it is the policy of your government to keep people hungry?" Clem inquired.
"We must take things as we find them," the Viceroy replied.
In England, Clem reflected, this might have been a nice sort of fellow. His face was not cruel, only empty. Everything had to be emptied out of a man's heart if he sat long in this vacuum. Clem looked around the enormous hall, embellished with gold in many varieties of decoration.
"I see your point," he said after a long while. And then, after another while he said abruptly, "I don't agree with it, though."
"Really!" There was a hint of sarcasm but Clem never noticed sarcasm. He went on.
"We've never tried feeding the world. Ever seen how much meat comes from a sow? She farrows big litters until you don't know what to do with all the pork. Of course in America we throw away mountains of good food, besides eating too much. You English eat too much, too, in my opinion—all that meat!"
The Face continued empty and looking at it Clem said, "I will grant America is the most guilty of all countries, so far as waste goes."
"Undoubtedly you know," The Face said.
Clem said good-by after a half hour of this. He then walked behind the trotting Sir Girga who saw him through the forest of lackeys to the front gate, beyond which an absurd Indian vehicle called a tonga awaited him, to the derision of the lordly Indian doormen.
He went back to the hotel where in one of the rows of whitewashed rooms Henrietta sat in her petticoat and corset cover, fanning herself. "We'll just mosey along to Java before we go home," he told her. "It's about as I thought. They aren't interested in feeding people."
In Java he was stirred to enthusiasm by the sight of land so rich that while one field was planted with rice seedlings, another was being harvested. Men carried bundles of rice over their shoulders, the heads so heavy that they fell in a thick, even fringe of gold. The Dutch were more than polite to an American millionaire and he was shown everywhere, presumably, and everywhere he saw, or was shown, a contented and well-fed people. It was only accidentally that he found out that there was an independence party. One night when he was walking alone, as no foreigner should do in a well-arranged empire, a note was thrust into his hand and when he got back to the hotel and a lamp he found that it was a scrawl in English which said that he ought to examine the jails. This of course he was not allowed to do.
It was a good experience for Clem. He was thoughtful for some days on the voyage home and Henrietta waited for what he was thinking. As usual it came out in a few words one night when they were pacing the deck.
"We've still got freedom in America, hon," he said. "I'm going home and look the whole situation over again and see if Bump and those lawyer fellows are right. If I have to organize I will, but I want to organize so that I'm not hamstrung by laws and red tape. I'll organize for more freedom, see?"
"I believe that is Bump's idea," Henrietta said.
Clem would not accept this. "Yeah, but his idea of a man's independence and my idea are not the same. He's like those lawyer fellows—he wants laws as clubs, see? Clubs to make the other fellow do what you want! But my idea is to use laws to keep my freedom to do what I want. I don't want to interfere with the other fellow, or drive him out of business."
There was a difference, as Henrietta could see, a vast and fundamental difference. Clem was noncompetitive in a competitive world. It was strange enough to think that it had taken India to show Clem the value of law in his own country, but so it had done, and when they reached home Clem plunged into this new phase of his existence. Beltham and Black summoned to their aid an elder firm of lawyers as consultants, and Bump frankly sided with the four lawyers. Against them all Clem sat embattled day after day across the old pine table that still served him as his desk.
"What you want is impossible, Clem!" Bump cried at last. He was tired out. The lawyers were irritable at their client's obstinacy. Those were the days, too, when Frieda was expecting her third child and she was homesick for Germany, so that Bump had no peace at home, either.
Clem lifted his head, looked at them all. He was dead white and thin to his bones, but his eyes were electric blue.
"Impossible?" His voice was high and taut as a violin string. "Why, Bump, don't you know me after all these years? You can't say that word to me!"
# 9
IN THE RICH YEARS that followed World War I William profited exceedingly. His tabloids were the most popular newspapers in the country and he had several foreign editions. The old offices were long since deserted and he owned a monumental building on the East River.
He was still not satisfied. He wanted his country to be the greatest country in the world, not only in words and imagination and national pride, but in hard fact. He saw American ships on all seas, and American newspapers, his papers in all countries, American names on business streets, and above all American churches and schools everywhere. America was his country, and he would make her great.
This was the motor behind the scheduled energy of his life. He gave huge sums to American foreign missions, always in memory of his father. He established a college in China, known as the Lane Memorial University, although he steadfastly refused to meet face to face the missionaries whose salaries he paid. He had set up an organization to do that, the Lane Foundation. He had never gone back to China, although sometimes he dreamed of Peking at night when he was especially tired, foolish dreams of little hutungs, quiet between enclosing walls, wisps of music winding from a lute, sunshine hot on a dusty sleeping street. Memories he had thought forgotten crept out at night from his mind exhausted by the day. He ignored them.
These were the times in America when anything could be done. Yet he was not doing all he dreamed of doing. The common people, as he called them, meaning those ordinary folk who come and go on the streets on foot, by bus and streetcar, those who crawl under the earth in subways and live on farms and in small towns and mediocre cities, all these who bought his newspapers as surely as they bought their daily loaf of bread at the corner grocery, they were not of enough importance to govern, even by their yea or nay, the possible secret country which he now perceived lay behind the façade of present America. He had thought, when he was in college dreaming of vast newspaper tentacles, that if he had the common people in his influence he could guide the country. He never used the word "control" and indeed he honestly abhorred it. But guidance was a good word, the guidance of God, which after his father's death he himself continually sought as power and money accrued. Common people were weak and apathetic. They listened to anybody. Now that radio networks were beginning to tie the country together, his newspapers could no longer exclude. This troubled him mightily. Print had its rival. He considered making his newspapers almost entirely pictorial, so that reading was unnecessary, and then rejected the idea. Pictures could not keep common people from listening to the radio, which also required no reading. He must secure ear as well as eye and he began to plan the purchase of key networks.
In all this Candace was of no use to him. She had grown indifferent to the frightful responsibilities he undertook as his duty and she had even quarreled one day with his mother. He had never been able to discover either from her or his mother what had taken place, except that he had been the subject of their difference. Candace had simply laughed when he pressed her for detail.
"Your mother has lived too long in Peking." It was all she would tell him.
His mother went a little further. "I hate to say it, William, but Candace doesn't appreciate you as a wife should. Whether she understands the wonderful work you are doing is quite beside the point. I didn't always understand your dear father, either, and certainly I could not always sympathize with his ideas or even with all that he did, but I always appreciated _him._ "
Candace had grown strange and reckless in these years after the war, likely on any Sunday morning to announce that she was going to the beach with the boys instead of sending them to Sunday School. That William himself did not go to church had nothing to do with his sons, who, he felt, should be taught some sort of religion. Indeed, he himself, since his father's death, had felt the need to find God anew, but he could not return to the pusillanimities of his former rector. He sought a firmer faith, a stronger church, and there were times when he thought of Catholicism. This, however, had nothing to do with Candace and the two boys. The seashore place was another recklessness of hers, although he had quite willingly bought the mile of private ocean front in Maine. She had declared that she wanted only a shack, to which he had simply said there was a right way to do a thing, and comfort he must have, even though in summer he could only be there a day or two a week. He had hired a young architect who designed an extraordinary house on top of a gray cliff, and a sliding staircase, like an escalator, which let them down to the sea and to a huge _cabaña._ Altogether it was effective and he was proud of it.
He had to acknowledge to himself now that Candace had never meant very much to him, and it had been years since he needed anything of Roger Cameron. When Mrs. Cameron died last year old Roger told William that he wanted to sell his shares in the newspapers.
"The dividends are going up," William said.
"That's why I want to sell," Roger had replied.
This made no sense but William did not reply because he was vaguely wounded. His pride rose and he sent a memorandum to the business manager that he wanted all shares in the corporation bought up so that he might be sole owner. When the reports came in he saw the name of Seth James. Seth was now backing a new daily paper that William saw at once was doomed to die. Seth should have known better, he had told himself, as with complacency he studied the first issues. "The paper with a purpose," Seth had foolishly announced. Of course people would not buy it. People did not want to be taught. They wanted to be amused. William himself was never amused. It was Jeremy's task to find among thousands of photographs for his tabloids, pictures sorted by twelve girls under twenty years of age, those scenes which would make people laugh. Horror was as good as laughter and horror William himself could judge. A murder skillfully portrayed, a strangled woman, a dying child, a family weeping after the father was crushed under a truck, a maniac escaped, an airplane that crashed into a small home on Long Island, these were all pleasing to people.
Yet such was William's conscience since his father's death that he allowed no issue of a paper to be sent to the people without its quota of religion. He truly believed in God. His own being, ordered by purpose, convinced him of the existence of God and his tabloids carried photographs of churches and ministers, priests and nuns. William was not narrow. People worshiped God in many ways, though he rejected any form not Christian. He had disagreed with Estey, his new assistant editor, over a photograph of the Panchen Lama—news, yes, but not religion. People the next week saw the benign face of the Lama appearing side by side with the President's wife in her Easter frock.
On a day in early October he sat thinking of these things in his immense office on the top floor of his own building. The office opened into a handsome apartment where he could sleep on the nights when he had to work late. Caspar Wilde, the young English modernist, had designed it for him. William had wanted it done by a Swedish architect, but when he examined the designs laid before him he had been forced to see that there was nothing to equal English modern in its conservative and heavy soundness. It was exasperating but true. In spite of the World War there was as yet no crack in the armor of the British Empire. His reporters, stationed permanently in India as in almost every other country, informed him of bitter disappointment among Indians after the war.
"Educated Indian opinion complains that Britain shows no signs of fulfilling wartime promises for independence, made to leading Indian politicos. Rumors are that in the next war Indians will seize the opportunity for rebellion."
This perhaps was a crack in the imperial armor, but no more.
William had no sympathy with independence for India. His imagination, anchored by the mob in the Peking street, saw in India those faces darkened by the Indian sun and multiplied by swarming millions. If and when the crack became disaster for the British Empire, his own country must be ready to assume control.
America was young. When this crazy period of postwar play was over, Americans would see their destiny and grow up. In his editorials he skillfully reminded them now and again of that destiny. He roused their pride by pictures of the greatest factories in the world, the largest airships, the fastest trains. It troubled him that the American army and navy were not more impressive. When the navy decided upon maneuvers anywhere in the world he sent a flock of photographers with them. Bright sea and flying flags and ranks of men in white duck made wonderful pictures.
The people were still in a playful mood. On this bright autumn afternoon even he was not inclined to be critical. Times were good and people had money to throw away. He himself would play if he could, but he did not find the usual diversions amusing or playful. At Chefoo he had learned to play a brilliant game of tennis, cruel in cuts and slashes, all but dishonest and certainly ruthless, but he seldom played. There was no incentive for he had no competitors. The careless padding about the courts with Candace at Crest Hill, his home on Long Island Sound, or on week ends facing Jeremy who refused to be any man's enemy even at sport, could not divert his mind. He liked an enemy and with an enemy in tennis he came nearer to amusement, enjoyment, relaxation, perhaps, than at any other sport, when occasionally he found an opponent equal to him.
He sat rigidly in front of his huge circular desk, his hands clenched in fists upon its blond surface, thinking. He had everything in his life except human companionship. He was remote from every human creature, even from Candace and his sons, and certainly from his mother and sisters. He had no one near him, neither man nor woman. Jeremy had long ago taken his position as a jeering light-minded brother-in-law who knew he could not be fired because it would make an office scandal. Yet Jeremy had a flair which gave the papers the humor that no one else could supply, William because he did not know how, and the staff because they were afraid of him. Jeremy could have been his friend, William sometimes thought with a certain wistfulness, but he did not want to be. Perhaps he could not understand or value the purpose for which William lived. The Camerons were all light-minded. Old Roger nowadays was as gay as an ancient grasshopper and Candace had grown benign and careless of her figure. She laughed at everything Jeremy said when the families were together and even Ruth could not make her mindful of what was dignity. William knew that Ruth was his life-long possession, but he wondered sometimes in the gloom in which he lived whether, were he permanently out of earshot, she too would laugh. He had, in short, no one of his own. His sons did not interest him. He was as lonely as a king.
Nevertheless, like a king, he reflected, he could not put out his hand to anyone without its being misunderstood. The gesture of ordinary friendship was impossible for him. If he put out his hand it must be for a purpose that was not yet clear to him. He doubted very much whether there was a woman in the world who could give him real companionship. Only his loneliness was plain to him, and profound.
In this state of mind he left his office rather early and entered his waiting car. The chauffeur was surprised and pleased to see him. Doubtless the man had a family and thought of getting home early. William did not ask, however. He merely gave his abrupt nod and said, "Direct to Crest Hill." He wanted to go home and survey his house and his wife. There was no reason why, having achieved everything else, he should not have personal satisfaction. It seemed a small thing, but without it on this opulent autumn afternoon nothing he had was all it should be.
At Crest Hill Candace had spent a beautiful, idle day. It was what she called a day of grace, of which there were too few in every season. Thus although leaves had fallen and the first frost had killed the flower borders, although her furs had been brought from storage, yet the day was as warm as June's best and she had done nothing at all. The outdoor swimming pool had been emptied and cleaned for winter, but she had ordered it filled again and had spent the morning in and out of the pool quite by herself and happy. She missed the boys but they had been going away to school for years and William she had learned not to miss, wherever he was. The huge house was unusually beautiful, the doors and windows open and the bowls on the table were full of late roses. Her rose gardens were sheltered by the greenhouses and escaped the early frosts. She was the most idle of women and enjoyed her idleness. A moment at the telephone could summon to her any of a hundred or so friends, men and women who were eager to share her genius for enjoyment, but she seldom summoned them. She liked best to be with Ruth and Jeremy and their little girls, and she disliked actively, out of all the world, only William's mother. For her own father she had a delicate affection so appreciative that she welcomed his coming to her but she made no demands upon him. She made no demands upon anyone, being content in herself. Marriage with William had not given her high romance, but then she did not want such romance. She would have had to live up to it.
She was not prepared therefore for William's too early arrival. At five o'clock, she told herself, she would leave the sun-soaked court surrounding the swimming pool and she would go upstairs, dry her hair, and put on a thin soft dress of some sort over her slip. Never willingly did she wear girdle or corset or any of the garments that women used to restrain themselves. What she would have done had she been fat she never stopped to ask herself, since she was not really fat. Old Roger's leanness had so blessed his daughter that even carelessness had made her only gently plump.
At five o'clock William entered the wide hall of his house and inquired of the man who took his hat and stick where Mrs. Lane might be found.
"Madame is in the court, sir," the man replied.
William walked down the hall which bisected the huge house and stood between the open double doors. Candace was climbing out of the pool. Her blond skin, sunburned to a soft pale gold, was pretty enough in contrast to the green bathing suit she wore. Her long fair hair was wet and hanging down her back. She was a pleasant sight for any husband, and William felt vaguely angry that a woman who looked as Candace did should not provide for him the companionship which he needed. What, for example, could they do together now? She played a lazy game of tennis and she could not keep her mind on bridge. She enjoyed horseback riding and rode well, but there was no companionship in that pastime. He preferred to ride alone in the morning before breakfast.
"Why, William," Candace called. "Has something happened?"
"Certainly not," he replied. "Why should you think so?"
"You're home so early."
"It was hot in town."
"Come into the pool."
"No, thank you."
William did not enjoy swimming, either in the pool or the sea. He swam well, for he had been taught to do so at the English school. His hatred of the water went back to the day when a firm young English swimming master had thrown him into the Chinese sea, out of his depth, to compel him to swim for his life.
"Then I'll get out," Candace said, and began to wring the water out of her hair.
"Don't trouble yourself," William said. "I'll go upstairs and change."
"Will you come back?"
"If you wish."
"Of course I do."
She dived into the pool again and he went upstairs slowly to his own rooms. His valet had foreseen his need and had put out for him a suit of cool tussah silk that had been packed away and now brought out once more for the unseasonable heat. William showered and shaved himself, for hot weather always made his black beard grow too fast. Then he dressed and went downstairs again, wishing restlessly that he could think of something he could enjoy. Candace was still in the pool, but a servant had brought tall glasses of some drink and set them on table under an umbrella.
He sighed and stretched himself in a comfortable chair. Candace saw him and swam slowly to the end of the pool and got out. She wrung her hair again, wound it on her head and wrapped a huge English bath towel about herself. William found no towels in America big enough for him, neither did he like colored towels. Miss Smith the eleventh had once ordered six dozen enormous English bath towels from London and had sent them to Ireland to be monogrammed. Only Candace had other towels than these. In her own bathroom shelves she kept towels of peach and jade green. In public, however—that is, before William—she enveloped herself in one of the six dozen.
"I'll just slip on something and be back," she told him. He looked unusually handsome at this moment and impulsively she bent to kiss him. His dark hair was thinning slightly on top of his head, a spot she did not often see.
"William, you are getting bald!"
It was a wifely remark but the wrong one, she saw, the moment it was spoken. He did not reply; his eyebrows drew down and his mouth tightened.
"Not that it shows," she said hastily.
"It must show or you would not have seen it," William retorted.
"Oh well," she said, laughed, and went on.
Upon him the careless remark fell like an arrow dropped from the sky. He was reminded that he was middle-aged. If he was ever to get anything out of life he must do it now. Decision accumulated in him. He recognized the process. A trickle, a slow stream, a monstrous river of feeling suddenly broke into inevitable sudden decision.
He would divorce Candace if necessary in order to get companionship before he died. He would find somewhere in the world the woman he needed.
Lying in the warm declining sun he felt his deep and habitual tension suddenly relax. He had made a decision which though massive was right and therefore irrevocable. All his large decisions had come suddenly after long periods of indecisive restlessness. When he saw what he must do it was like coming out of a tunnel into the light. He closed his eyes and sipped his iced drink. He was not a simple physical creature such as he believed most American men were. He was not interested in dirty schoolboyish talk, and jokes about sex bored him. Something in his birth and childhood, the deep maturity of the Chinese, perhaps, or the intolerable wisdom of England, had aged even his youth.
When the thought of England came to him, he felt a strange nostalgia. He did not want to go back to China, but to go to England might give him the rest that he needed. Alone in England even for a few weeks, as silent as he wished, with nothing planned and yet ready for anything that might occur to him, he could cure himself, or be cured, of his spiritual restlessness. The peace that passeth understanding, of which his father spoke so often, might yet be his.
But he must be alone. Merely to be alone, he now felt, would bring him some of the peace. He thought of his office and the quiet apartment opening into it, and was eager to be there where he need not speak to Candace or see her. He got up and went into the house and met her coming downstairs, in a floating chiffon dress of apple green.
"I shall have to go back to town," he said abruptly.
"Oh—I am sorry for that."
She spoke sincerely but without petulance. After these years she was accustomed to William's sudden decisions. She would wait until he was gone and then she would call up Jeremy. If he and Ruth were at home she would drive over to their house and dine with them. William's mother was there, but on this heavenly evening she could bear that. Jeremy's house stood near the water, its lawn sloping down to the Sound, and the moon would be beautiful upon the waves.
"Shall you be late, William?"
"I don't know. Don't sit up for me, of course."
"If I am not here, I'll be at Jeremy's. Don't sit up for me, either."
She put her hands on his shoulders and pressed herself against him. He kissed her cheek but did not respond to the pressure. Ah well, her father had said loving was enough! She made it do.
William could have explained to no one his impulse toward England at this hour of his life. He had been often in England in recent years, but only for short times and for business. Now he wanted an indefinite time which might be short or long. He told himself that this depended upon how he felt. Actually he knew that he was going on a search, a romantic search, absurd if it were spoken, and therefore it could not be spoken. His real life had always been secret. Now he felt the need to confide. Vague need, vague longing, the middle-aged desire to live before he died, the thirst to learn how to enjoy before he lost the power, these were his private reasons, not to be shared.
He stayed in London for some days, ostensibly to attend a few business conferences. He toyed with the idea of setting up an entirely English office for the publication of a purely English tabloid and to discuss this he met Lord Northcliffe for a week end, and acknowledged frankly his debt to the master journalist.
"I saw one of your papers in the reading room at Harvard, my lord, and began that very day to plan my life around a newspaper like it."
"Really," the stubby lord said without surprise. "We've a bit in common, you and I, haven't we? Success from the middle classes, eh? Your father was something odd, as I remember—so was mine."
William preferred not to answer this. He remembered that this baronet had once put on his head a hat worn by Napoleon and had said without vanity, "It fits me, by Jove!" Since then he had spent some of his swift wealth upon such fantasies as arctic exploration, had forced upon his quiet countrymen noisy automobiles, had given prizes for airplane models and attempts at flying, and now clamored for fellow patriots to prepare themselves against the dangers of a rising Germany.
There was something about this plebeian lord which repelled William. They parted without being friends, the Englishman feeling with amazement that William was what he had never seen before, an American snob, and William feeling that England was better than this Englishman thought she was and that he was somehow unworthy. If he had met Alfred Harmsworth as a schoolboy he would have fought him and easily licked him. He sat, later that week, for an evening under the scintillations of an aging Herbert Wells, refusing however, to join in the absurd games devised for his amusement. He remained saturnine even before the brisk sallies and the ceaseless flow of his host's fixed though fluid opinions.
After three or four weeks of being a quiet guest, unobtrusively American in English country houses, William met a young man to whom he was exceedingly attracted. He could not account for the singular strength of this attraction until he discerned in the young man a faint resemblance to the hero of his youth in the Chefoo school, the son of the British ambassador. This young man's name was Michael Culver-Hulme, a name ancient enough in English history and with many branches. In the stillness of a Sunday afternoon before tea at Blakesbury House, where William had been invited by Lord Saynes, who had heard of his wealth and power, he met Michael.
Culver-Hulme, a distant cousin of Saynes, had asked frankly for the chance to meet the American whom everybody had heard about and almost no one had seen. Lord Saynes had laughed.
"What do you want to meet the chap for?" he had inquired of Michael.
Michael had replied, "I've a fancy to see him, that's all. My uncle went to school with him—my mother's brother. He's told me rather grim tales. He's quite proud now of having gone to school with him, though in the old days they all made fun of him. It seems he used to stalk about the school grounds rather like a silent and haughty young Hamlet."
On this Sunday afternoon, beneath a sky of milky November blue, the Englishman saw William leaning lonely against a stone wall, gazing across the lawns to the valley beyond. He went to him with the bold and entirely natural charm which was both assured and youthful.
"I say, sir, I hope you won't mind if I butt in?"
"Not at all," William said. He smiled slightly. "Our World War seems to have left its effect at least upon the English language."
"Not so much as your wonderful papers, sir. I wonder if you know how much they're admired? I've heard that Northcliffe himself has taken a point or two."
William felt the soft warmth of young flattery steal about his heart. He was flattered often enough, but this English flattery was sweet, and he did not discard it with his usual cynicism.
"I wonder if you could by any chance have had a relative once at an English school in China? I don't believe in coincidence. But you look alike."
"Not coincidence, sir. Many of our family have been in China or India. It's a family tradition. It was my uncle, I think. He's often spoken of you and been quite proud about it."
Ancient wounds began to heal in William's heart, but he maintained his dignity and only slightly smiled. "I remember him as an autocratic young man, quite beyond noticing a mere American."
"He knows better than that now, sir."
Michael waited and when nothing more followed, he began again with imperturbable chatty briskness. "I wish you'd come and have a week with us, Mr. Lane. My father and mother would be enormously pleased, and I'd be honored."
"I'm here on a holiday," William replied. "That perhaps will excuse my ready acceptance of a kindly invitation. I should like to come and call upon your father, if I may. If you are there, it is all the better."
"Then will you consider it an invitation, sir? If so, you'll have a note from my father. What week, sir?"
"Week after next?"
"Splendid! Shall you be in England for Christmas?"
"No, I must get home before then. My sons will be coming home from college."
"Splendid! Where are you stopping?"
"I am at the Savoy."
"Good! Then you'll hear from us. Hulme Castle, near Kerrington Downs."
"Thank you."
The two words were so spoken that they seemed dismissal but Michael refused to accept them. He divined in the American a diffidence so combined with pride that it had become arrogance, a knowledge of superiority augmented by the fear of an incomprehensible inferiority. This American had all the kingdoms of the earth, a handsome body, a shrewd mind, wealth that had become a fable about which people guessed and gossiped on two sides of the ocean, and from all this a power was emerging which Michael knew was viewed with gravity even in the Foreign Office.
An immense curiosity sprang up in his somewhat light and inquisitive mind, and he imagined himself talking William over with his sister, Emory.
"He's not a proper American at all. With just a little changing, he could make a fair stab at being an Englishman, if he wanted to. And the odd thing is that he would and he wouldn't want to—"
To bring his mind back from such words, he began to describe to William the recent hunting he had shared with his uncle in Scotland. Then a bell rang suddenly from the house and broke across Michael's endeavors to amuse.
"That's tea, I'm afraid," he said cheerfully, and thankful to be relieved of the conversation, he was liberal enough to wonder if William felt a like relief and daresayed to himself that he did.
Hulme Castle, William discovered, was one of the relics of the time of William the Conqueror and since it was near Hulme Forest, it had often been the hunting box of kings. In the fifteenth century it fell into disrepair, its last use being to shelter a mistress of the then ruling king. In the early sixteenth century it was given to a newly created earl, who rebuilt the castle but not the keep, rebuilt also the Great Hall, and discovered among old ruins a chest left by King Edward III. In the seventeenth century King James visited the castle while hunting and in the eighteenth century the then existing earl finished the rebuilding of the whole castle, remodeling the kitchens entirely and adding a handsome picture gallery. No building had been done since. The present occupants were the Earl, his wife, his son Michael, and his daughter Emory. On the third Sunday of every month the castle was open to the public except for the rooms occupied by the family.
So much William discovered from a small book he found in the British Museum. He had taken time to find out all he could about Hulme Castle. It was a small estate but an ancient one.
From the main highway through the Downs William, seated in the heavy motorcar he had bought for his stay in England, saw Hulme Castle on a low and pleasant hill. Twin towers of Norman architecture guarded the entrance through which, on a soft gray English day, he approached his destination. The chauffeur pulled a huge knocker and the door was opened by a man in some sort of informal livery.
"Hulme Castle?" the chauffeur inquired, knowing well enough that it was.
"Hulme Castle," the manservant replied.
William got out, properly dignified, and mounted the shallow stone steps.
The manservant took his things. "Mr. Lane?"
"Yes."
"Come in, please, sir. We were expecting you. I will show you your room, sir. This way, please, sir."
A huge table stood in the middle of the entrance hall and behind it double stairs wound upward to right and left. Upstairs William went down a long and wide hall into a large room, quite modern in its decoration. A small coal fire burned in a polished grate under a carved mantelpiece, upon which the only ornament was a silver bowl of ash-pink roses.
"Tea is being served in the Panel Room, sir, to the left at the bottom of the stair," the man said and disappeared.
William went to the wide leaded window. The sill was deep in the thick stone wall and he looked down over the tops of oaks still green. The hill declined sharply beneath this western wall and on the horizon the sun was setting, pink among the gray clouds. The castle was filled with silence and with peace, and he saw no human being. A feeling of rest and remoteness stole upon him and he sighed.
He stepped into the same stillness a few moments later when, having washed his hands and face, he went downstairs. The door of the Panel Room was open and he heard someone playing the piano. Of music he knew nothing and he had not missed it, but he was intelligent enough to know that the person now playing was a musician. He crossed the hall, entered the door, and saw something that he might have imagined. A long, beautifully shaped room, paneled in oak, spread before him. At the far end was a large fireplace, and above it the coat of arms of Hulme. Before the fire a tea table was set and an old man, the Earl himself doubtless, sat in an easy chair of faded red leather. Across the fireplace sat Lady Hulme, unmistakable, tall, thin, weathered, and wearing an old tweed suit. She was knitting something brown. Michael leaned against the mantle, his hands in his pockets, gazing at the fire, and at the piano sat a woman in a long crimson dress.
She lifted her head and smiled, a gesture of invitation, while she went on playing softly and firmly the closing chords. The Earl saw him and then Michael, and with the same smile and gesture they waited, Michael halfway across the room, the Earl standing. Lady Hulme lifted her large pale blue eyes, dropped them again, and continued her knitting.
At the piano the last chord sounded deeply. Michael leaped forward and wrenched William's hand.
"How awfully good of you to come! This is my father—and my mother."
William touched the Earl's dry old hand and received a nod from Lady Hulme.
"Very good of you," the Earl murmured. "It's a long way from London, I'm afraid. We're very quiet."
"I like quiet," William said.
He turned, still delaying, still dreading.
"This is my sister Emory," Michael said simply.
William took a long cool hand into his own. "I'm afraid I interrupted the music."
"We were only waiting for you," she replied.
"Emory, pour tea," Lady Hulme commanded. "I've dropped a stitch."
She moved to obey, and for one instant William looked down into eyes dark and clear, set in a pale and beautiful face. He saw her mouth, the lips tender and delicate, quiver and smile half unwillingly, or so he imagined. She was tall and so thin that she might have been ill except for the look of clear health in her eyes and her pale skin.
"Do sit down," she said in her sweet English voice, and seated herself by the tea table. "I'm filled with curiosity about you. I've never met an American."
"I am not typical, I am afraid," William replied, and tried not to stare at her hands as they moved above the cups. They were exquisite hands, and there was something about them so familiar that he frowned unconsciously to remember. Then memory came back to him. He had seen hands like these long ago, when as a little boy with his mother, he had looked at the hands of the Old Empress in Peking, the same thin smooth hands!
"Come along, Emory," Lady Hulme said in her husky voice, still knitting briskly. She paused, however, to pull a bell rope with vigor as William sat down, and the manservant came in with a plate of hot scones on a silver tray.
"Hello, Simpkins," Michael said. "How is it you're passing the tea today?"
"Matthews has mumps," Lady Hulme said. "It's absurd, really, but he caught them from the new housemaid, I believe."
"He did, my lady," Simpkins said very gently.
Lady Hulme turned to William. "I hear you have pots of money. Here's your tea."
"Don't heed my mother," Michael said rather quickly. "She likes to think she's daring. Why do you say such a thing, Mother?"
"Why not?" Lady Hulme retorted. Her face remained expressionless, whatever she said, the large eyes like pale lamps in her face that was reddened by sun and wind. "I can't think of anything nicer than having pots of money. One needn't be ashamed of it. I wish your father had it."
William took his tea and helped himself to thin bread and butter and a hot scone. Some pleasant-looking cake waited upon a small, three-tiered table, but he knew, from school memory, that it would not be passed to him until he had eaten his bread and butter and scone. Sweets came last or not at all.
No one noticed his silence. Lord Hulme was eating with enjoyment, and drinking his tea from a large breakfast cup.
"I hope you weren't seasick," Lady Hulme said.
"Thanks, no," William replied.
"It's so beastly when one is," Lady Hulme observed. "Of course American men are not so heartless as Englishmen. Malcolm always has believed that I am seasick purposely."
"You are, my dear," the Earl said.
"There, you see," Lady Hulme said. "We went to Sicily for our honeymoon thirty-five years ago and I got ill in the little boat that took us across the Channel and had nowhere to lay my head. He wouldn't let me put it upon his knee."
"Oh, come now," the Earl retorted. "As I remember, I hadn't a chance to walk about—your head was always on my knee."
They wrangled amiably, worrying the old subject between them, and Emory sat watching them with amused and lovely eyes, glancing now and again at William. She did not interrupt and at last Lady Hulme was weary.
"More tea all around," she announced.
The Earl, revived by tea and argument, turned to William. "I see those papers of yours sometimes. What sort reads them, shopgirls and so on, I suppose!"
Michael sprang into the arena. "Everybody reads them, Father."
"Really? Mostly pictures, though, aren't they?"
William took the Englishman into his confidence. "Our people don't read very much. One has to use pictures to convey one's meaning."
"Ah, then you have a purpose?" Lord Hastings said rather quickly.
"Doesn't everyone have a purpose?" William replied. "The power potentiality of several million people is a responsibility. One cannot simply ignore it."
"Ah," the Earl said. He tipped his cup, emptied it, wiped his mustache with his lace napkin, rolled it up, and put it in the cup. Then he got up. "I suppose you'd like a walk? Michael and I always get one in before dinner."
The early twilight was not far off and William would have preferred to stay in the great firelit room with the beautiful woman who sat in such silent repose, but some compulsive hand from the past reached out and he rose. After tea at school the headmaster ordered a walk for everyone. Not to want fresh air was a sign of laziness, weakness, coddling one's self, all English sins.
"Those boots right for mud?" Michael was looking down at William's well-polished country oxfords.
"Quite all right," William said.
They tramped out into the shadowy fragrance, Michael respectfully in the rear. The Earl lit a short and ancient pipe, refusing William's aid. "Thanks, no—I've got long matches—have 'em made to order. They've a chemical in the tip that keeps them from blowing out in a wind."
After this a long silence fell as the three men walked through country lanes. William knew the English silence and he determined that he would not break it. Let these Englishmen know that he could endure the severest test! The Earl turned away from the drive and across a sloping lawn to a meadow. At a gate in a white fence he paused again to fill his pipe.
"I've never been to America. Michael is always wanting to go. But since he's the only son, I've forbidden it—for the present."
Michael laughed. "I have to marry and present him with an heir before he'll let me go anywhere."
"That is the way the Chinese feel, too," William said. "But I hope you will visit us some day."
"Where do you live?" the Earl inquired.
"I have a house in New York and another in the country." William's voice was as detached and tranquil as any Englishman's.
"You do yourselves very well, you Americans!"
"Not better than you English!"
"Ah, but it's taken us thousands of years."
"We had a bigger bit of land to begin with."
The Earl knocked the ash from his pipe and opened the gate. A hen pheasant started out of the grass and he watched her scuttling flight. "What fools we were to go after India instead of keeping America!" He was filling his pipe again. "Think of what the Empire would be if we'd really fought you rebels in 1776 instead of hankering after the fleshpots of that sun-blasted continent! It would have been to your advantage as well as ours. We'd have been invincible today against Germany or Russia if we'd been one country."
"We, on the other hand, might have been merely a second Canada," William said. "Perhaps we needed independence to develop."
"Nonsense," the Earl retorted. "It's stock that counts. The people of India have no stamina—always burning with some sort of fever of the spirit. It's the unhealthy climate."
"I can't imagine ourselves part of an empire," William said.
"Not now, of course," the Earl conceded. He stole a sharp shrewd darting glance at William. "Certainly not when you're dreaming of your own empire."
"I doubt we want an empire," William replied.
Nevertheless the idea played about his mind as they walked across the meadow. Empires had their day, and the ancient British Empire was dying as surely as the sun was setting across the wooded hill opposite the brook. He saw the sunset bright in the still-flowing wafers.
"Do you fish in the brook?" he asked Michael.
"Nothing much there," Michael replied. "A trout now and then."
"The boys in the village catch everything. They've got very lax about poaching," the Earl said rather angrily.
They reached the brook after another silence and stood gazing into its shallow clarity. There were minnows in plenty darting about under the surface, snatching at the last chance for food. The Earl stirred them with his walking stick. "There're always minnows, somehow."
He said it in a musing voice but William saw no significance in the words and did not answer.
"Millions of minnows," Michael said.
The Earl was looking across the brook as though he pondered the other side and then changed his mind. "We'd better go back, I dare say. The evening is turning chill."
They climbed the hill again, this time in silence that none broke. When they entered the great square hall of the castle, Simpkins met them and took their hats and sticks. The Earl yawned.
"We'll meet again at dinner—in an hour." He walked away with his heavy step and William stood uncertainly.
Michael, so fresh and friendly, now seemed uncertain too. "I hope you won't mind my parents, sir. I always forget how they are until I'm home again. Will you come in by the fire or go upstairs?"
"I shall enjoy you all," William said with unusual grace. He looked into the great room behind the hall and saw it empty. Lady Emory had gone. "And I think I shall go upstairs until dinner."
After that day William made no pretense to himself. For the first time in his life, he had fallen desperately in love.
His eyes, covert but acute, had searched every woman whom he had met and others whom he had not met. Their eyes in turn had gazed upon him with courtesy and with indifference. The young had looked upon him as old and forbidding, and from those who were not young he had averted his own eyes. English women did not age with grace or beauty. He found them garrulous or caustic, and from sharpness he shrank by instinct. He wanted intelligence but not sarcastic wit which he was not skilled enough to master and therefore despised. If he disapproved he said so plainly and finally. Sarcasm, he said often, was the exhibitionism of a showy but weak ego, the displeasure of a coward, and the natural refuge of those who had only their tongues for weapons.
All that he had ever dreamed of England and what England had meant to him, all that he had never acknowledged even to himself, now centered in a woman whom he did not ask himself if he understood, for he knew she understood him. He was able to talk at last and to tell her all that he had never told anyone. She listened, her eyes thoughtful and kind. Kindness was her genius. It shone not only upon him but upon everyone who was near her. Her father and brother basked in it, accepted it, took it for granted, imposed upon her, William decided, during the week of days that followed one after the other. Guests came and went and drew from her kindness what they needed. She was busy continually and yet she had time for him, lending him her whole attention in the hours they were together.
He supposed she was not young—that is, she was certainly not a young girl. She was perhaps thirty. He could not understand how it was that he had found her unmarried and one day told her so in words that he feared were crude. She hesitated, then said with scarcely a change in her look or in that sweet deep voice:
"I suffered the same fate that so many English women did. My fiancé was killed during the war. He was Cecil Randford, son of the Earl of Randford. We had grown up together."
William heard the name with pangs of jealousy which he tried to hide. "Forgive me," he muttered.
"I do," she replied simply.
By the third day he wished that he dared to ask her to call him by his Christian name. Lady Emory had a sort of intimacy which Mr. Lane did not have. If he had been Sir William! But he was not. He fretted himself about his courtship. There was so little time. He wanted to get it over, to have her love him quickly, to take her home with him soon and begin their life together. When he went back at Christmas he wanted to get through the hateful business of telling Candace and his sons and of consulting with his lawyers and his public-relations men as to how divorce and remarriage might be accomplished swiftly and privately. He ground his teeth when he thought of the pleasure that common people took in these matters, which should be as private as a man's own thoughts.
Meantime it was impossible to talk to the Earl or to Lady Hulme, he discovered. He did not exist for them, and yet they were aware that in his way he was important because he was rich. Nor was he at ease with them even though his week was swiftly passing. This castle, this English family, he approached with a diffidence that he would not recognize although he had long since reached a height in his own country that made a secretary's telephone call enough to open even the door of the White House—not the big front door into which sightseers and patriotic Americans swarmed but the side door where a huge brass key is kept turned. He reminded himself that the Earl of Hulme was not the King of England, that there were many peers of whom he was only one.
The first sight of the castle by daylight had been comforting. It would take a great deal of money to modernize it. For fifty bedrooms there were only five baths, inconvenient, and of plumbing so ancient that tanks of water hung above the toilet seats and water for the enormous tub was warmed by gas heaters that threatened to asphyxiate bathers unless carefully tended. William was surprised to have a manservant remain in the room, his back carefully turned, when he took his bath the first night because the heater had looked for the last few months as though it might explode if overworked, and Americans, as everyone knew, insisted on having their tubs full.
"It was much easier, sir, in the old days when we fetched in tin baths," the man had said, not looking around.
"Why don't you get some American plumbers?" William asked, submerged in soapsuds. The water was beautifully soft.
"They could never understand the system, sir," the man said. "Let me know when you've quite done, sir. I'll turn it off and get quite out of your way."
He did so a few minutes later and William, wrapped in a bath sheet, had returned to his own room down a hall an eighth of a mile long.
Here in his vast room he felt the silence centuries deep about him. It made him think of Peking and temples and palaces and the Old Empress again. It was the atmosphere he loved and he would have given his soul to have been born to it, for it was something which could not be imitated or made. To belong in it, to know the certainty of place, would have given him peace. Yet he was ashamed to acknowledge his own longing. Before these English, he must be his best, an American, rich, powerful, able to hold his own, a republican among aristocrats. He looked at himself in the long gilt-framed mirror and chose a somber tie.
Lady Emory had neither wish for love nor expectation of it. Her self-control was absolute and by now had penetrated every fiber of her being. She had been reared in self-control and believed that decency depended upon it. Only with Cecil, whom she had trusted entirely, had she felt that she did not need to think of herself, and so she had loved him with warmth and reality if not with heartiness. Nevertheless she was glad now that she had not married him, since he would have been killed, anyway, and not having married him she had learned to be glad that she had not slept with him that last night before he joined his regiment. They had discussed the last night frankly, as they discussed everything, their vocabulary being the same and their thoughts and ideas identical. It was not a question of sin or decency or of personal morality, since they were irrevocably in love. It was the far more important matter of an heir. Unlikely as it was that there could be any issue after a first and single union it was still possible that she might have a child, the heir of Randford.
"I shouldn't like him born anyhow, you know, darling," Cecil had said.
"We should have married," she had murmured.
"I hate these hurried, patched-up weddings," he had persisted. "I want to marry you in state, my darling. The Earls of Randford have always married their wives in the little abbey, and the tenants would hardly forgive me, you know, if I scamped it."
"What if..." she had not been able to finish.
"No ifs," he had said gaily. He was a god, young and blond, defying death.
So they had denied themselves for the sake of the child, who was never to be born, though they could not know it, and she had not allowed herself to regret her acquiescence. Cecil had felt his duty to his race, and though he loved her and she had never doubted his love, he drew her into his duty. This she had understood, for she had been reared within it, too. A noblewoman, however loved and cherished for her own sake, was nonetheless dedicated to the sacred future. She would not have been happy, either, had she forgotten that. Their love was purified by their faith in themselves and their kind, their belief that they were more than simple human beings.
Now that Cecil was dead she was released from that duty. There was nothing sacred in her being anything except herself. She knew no other heir of England whom she wanted to marry, or who wanted to marry her, and had there been such an one, it was doubtful whether the high sense of obligation would have been enough. With Cecil she could consecrate herself but without him, and therefore without love, even duty was not enough for her. There was no reason why she should consider it necessary merely to produce an heir for an ancient house. She was quite free.
Such freedom led to the immense restlessness which her self-control concealed beneath a cloak of consideration and kindness, these being also essentials of habitual good breeding. Only Michael divined that beneath the cloak so gracefully worn she was trembling with discontent.
"You need to get away," he had told her. "You are jumpy."
"I am not jumpy," she had replied with unusual brusqueness.
"Don't pretend," Michael had said. "You ought to marry. Cecil has been dead for years."
"I don't see anyone to marry," she had retorted.
"I'll look about," he had promised in a lordly way.
To which she had merely said, as she used to say to him when he was a little boy, "Don't be silly."
Nevertheless he had come back from London some months later with the preposterous declaration that he had found a chap, an American, who might be amusing for her to marry. Such conversation of course was not carried on before their parents. Even so she had been irritated by it. "I can't imagine any marriage amusing," she had told him. They were outdoors in the yew garden and she was on her knees by the Italian fountain, cleaning away fallen leaves. Michael stood watching her, not offering to help. He did not like to dirty his hands.
"This chap isn't amusing, exactly," he said.
"He's rather terrifying actually—immensely tall and thin, greenish gray eyes under black brows, and that sort of thing. He looks immensely unhappy, I must say, the way Americans do if they are not the giggling kind. He's searching, if I'm not mistaken."
"Searching?" She had looked up.
"He's rich as mud," Michael said. "It can't be that. I can't make him out, except that there's power in him."
"What power?"
"I don't know—energy, smothered under something impatience held down, enemy of everybody! He's not friendly, doesn't put out his hand when he sees you. I've invited him down—you'll see."
She had been attracted to William Lane the moment she had looked up from the piano and had seen him standing there. She had gone on playing so that she could look at him without speaking. He was not youthful, and above all things now youth wearied her. For the first time in these ten years she had found herself conscious of being a woman, not young but still beautiful and wanting so to be thought.
She had seen very soon that William thought her beautiful not merely for herself but for what she was over and beyond. He valued her for what she had inherited, but which was nevertheless a part of her, and it pleased her to have it so. He could not, she believed, have fallen in love merely with beauty. A chorus girl whom a king might love would have repelled him.
Pondering upon this, asking herself why it was that kings and peers throughout the history of England could so joyously lie upon hay and straw with milkmaids and gypsies who could not be queens, she penetrated the secret of William's soul. He wanted a queen that he might be king. His kingdom he had made, a modern kingdom, money and power in absolute combination now as always, and over it he reigned ably enough. But the secret longing was in his soul unrevealed, and perhaps unknown even to himself. If she accepted him he would be assured. He would have evidence of what had been unseen, he would become in substance that which he had hoped he was.
At thirty, she reflected, as the days of that week passed, a woman accepts quickly or she rejects. He was in the decade beyond her and was, moreover, a man accustomed to quick decisions. He let her know within a few days that his was made. When he left Hulme Castle at the end of the week he managed to say good-by alone with her, and she helped him to arrange it so.
"May I come back in a fortnight?" he asked.
"We shall be happy to see you," she had replied, purposely conventional.
"It will be a long fortnight for me, Lady Emory."
She had only smiled at this, and she looked down and saw his hand clasping hers. A strange small hand he had, curiously hairy!
"Come," she said to herself in silence, "let's not think of such things as that!"
To discipline herself, she let him hold her hand a second longer.
When William came back after a fortnight he found Lady Emory so composed, as she led him on the second evening to a part of the castle still unknown to him, that he wondered if she had divined his thoughts. He was surprised to feel his heart begin to beat more quickly than he had ever felt it before.
"You haven't seen the gallery, I think," She opened a paneled door and he saw a space, seemingly endless, hung with paintings. "Let's walk right away down to the end. The view is the loveliest picture of all."
He followed her a long way to the great windows from ceiling to floor at the end of the gallery, and when she sat down on a yellow satin sofa he took his seat there, too, but not near her.
She looked at him, her dark eyes quietly waiting, and he saw with some shock that she was used to men falling suddenly in love with her, that she was prepared, and then he dreaded so soon to put her to the test of proposal.
"Did you know I grew up as a boy in China?" he asked her abruptly.
"Yes. But what makes you think of it now?" Lady Emory asked.
"Something about this castle, the silence here, and the moon shining as it used to on a palace in Peking."
"The moon was late tonight."
"Do you take an interest in the comings and the goings of the moon?" He accompanied this unusual triviality with an effort at a smile.
"No except that from a window in my room it rather forces itself upon me."
He did not reply to this, and after a while she said, "Tell me something about your childhood in China. I've never been anywhere except in Europe."
"I don't want to think of my childhood," he said with the strange sort of abruptness which she was beginning to realize did not mean irritation.
"Was it unhappy?" she persisted.
"No, just useless to me."
"Useless?"
"Yes. I was the son of a missionary. You don't think there could be any advantage to me to have missionary parents, do you? I kept it a secret all the time I was at college. It was a fearful disadvantage to me even in the English prep school I went to in China." He wanted her to know the worst about him and he pressed the point. "To be the son of a missionary made my classmates think I must be queer. As a matter of fact, my father was rather remarkable. I didn't discover it, though, until he came home to die in my house."
"Tell me about him." Her voice led him on.
"Sometime, Lady Emory. I don't want to talk about him now."
"Wait," Lady Emory said. Her brown eyes widened a little and her soft voice took a slight imperious edge. "I wonder if I know what you want to talk about to me. If I do, I beg you to remember that we scarcely know each other."
"You may not know me but I know you," William replied. Passion seized him with a violence monstrous even to himself. He did not want to wait one moment to take this beautiful Englishwoman into his arms. He wanted her now, he wanted it settled.
Lady Emory looked frightened. "How can you know me?"
"I've always known England," William said. "I've always loved England, against my will I confess, but there it is. Now I've found you and you are the personification of all that I have loved."
"Michael said you were married—"
"That has nothing to do with you or me."
"No." Her word was a breath, a sigh, and he let it be acceptance. He took one step and she rose at his approach, and he drew her into his arms. Sweet and fearful was this exultation, his soaring pride in what he had, this arrogance of love! He was speechless, his face in the darkness of her hair, and he did not notice her silence or the still motionlessness with which she stood.
She was shocked to discover that the conviction in which she had sheathed herself since last she stood in Cecil's arms was entirely false. She did not feel repelled at all by another man's body pressed against her own. She had supposed it would be intolerable, eternally abhorrent, and it was not. It was even pleasant and comforting, as it would be pleasant and comforting to live in riches and plenty, no more a burden to her parents because she did not marry, no more a charity for Michael when his inheritance came to him. England was old and tired, and somehow with her dead lover it had died for her. America was young and strong, a rising empire, and to go there now, to leave England and take her own unspent womanhood with her, would be the nearest happiness that she could know. And this American, she perceived, contrary to what she had always heard about Americans, was neither stupid nor boyish.
"You can't love me—as quickly as I have loved you—I don't expect it—" William was stammering these broken sentences.
She was an honest woman, though beautiful, and what she now knew she would do, she wanted done with all her heart.
She stepped back, but only a little, and she let him hold her hands. "I suppose it is too soon," she said frankly. "But I don't think it is at all impossible—William!"
July in Ohio could be as hot as in India. Henrietta felt the heat. She had spent the last month with Clem in Mexico, where he had gone to confer with the Food Minister who wanted American wheat. Washington had been apathetic and he had called on Clem who, after listening carefully, had insisted on seeing Mexico for himself, so that he would know just how much the people needed wheat. He had not noticed the hot weather. His blood ran cool and he was thinner than ever. Mexican food was poison to him, the tamales hot as Indian food and even the vegetables full of red peppers, the spinach boiled to the color and taste of dead grass. He doggedly ate the native foods here as elsewhere, however, because he wanted to know what the people lived on, and afterward was tortured with the dyspepsia that got worse as he grew older. He had promised to get the wheat somehow, and they had come home.
Their house now as they opened the front door was hot and dusty and the air was stale.
"You get your dress off, hon," Clem said to Henrietta. "Go upstairs and put on a wrapper and relax. I'll open the windows."
Henrietta obeyed without answer. She had begun to gain weight and it was a relief to get out of her corset. She went upstairs into the large bathroom which Clem had fitted up himself and modeled after the ones in India. She stood in the big zinc-lined tray and filled a jar with water from the tap; then with a dipper she poured it over herself Indian fashion. The house was full of things that Clem had admired in other countries. He liked chopsticks, for instance, better than knives and forks. They were cleaner, he said. The water was lukewarm but even so, cooler than she was. She toweled herself and then put on the negligee that Clem always called a wrapper. She did not mind. It was comfortable to live with a man who did not know what she wore.
She went downstairs to unpack the groceries they had bought for supper. Clem had taken off his coat and sat in his white shirtsleeves at the dining table, figuring on a sheet of paper. His shoulder blades were sharp and the back of his neck was hollowed. He had lost weight in the Mexico heat. She did not speak aloud her worry. Nothing annoyed him more than to hear her worry about his being thin.
She sat down in a large wicker chair, tore open the envelope which was postmarked New York City, and began to read to herself. The first paragraph revealed catastrophe. Her mother wrote, "I am glad your poor father has passed on. He could never have endured what is about to happen to our family. I have wept and prayed to no avail. William is adamant. He is beyond my reach. I remember when he was a small infant upon my bosom. I know he is my son, but I cannot recognize him. What have we done to deserve this?"
Thus far Henrietta went without comment to Clem. Then she saw the next sentence and a smothered cry escaped her.
"What is it?" Clem asked.
He turned from his figures. It was not like Henrietta to cry out about anything. Now her large gray eyes were wide, staring at the sheet she held. They were the color of William's eyes but not like them in their depths.
"William is going to divorce Candace!" She breathed the words with the utmost horror, and he received them with horror as they looked at each other.
"What's Candace done?" he asked sternly.
Henrietta returned to the letter. "She can't have done anything," she murmured. Her eyes swept down the page. "Mamma doesn't say—yes, she does. She says Candace is just what she always was—there's no excuse for William—he doesn't even make an excuse—you know how he is. He always does what he is going to do and never says why. Mamma says it's just an infatuation. It's an Englishwoman he met on his trip."
Henrietta would have cried had she tears, but she had none. Against William her heart hardened, and she crushed the letter in her hand and threw it into the woven wicker wastepaper basket. She had never loved Candace but now she almost loved her. Long ago she had left her father's profound faith, but she had a sort of religion, fed by Clem's unselfishness and devotion to his single cause. The Camerons were good people, in their way as good as her father had been, and all the old decencies remained. A man did not divorce his wife without cause and the best of men did not divorce their wives for any cause. William had left the ranks of the good.
"I don't ever want to see William again," she declared with passion. Clem rose from his chair and came over and knelt beside her. She put her head down and upon his narrow bony shoulders. His thin arms went around her.
"There, there," he muttered.
"Oh, Clem," she sighed, half heartbroken. "I am glad you are good. It's your goodness that I trust."
He pondered this, patting her back in a rhythm. "Maybe we need some sort of religion, hon," he said at last. "We grew up with God, you know. We haven't deserted Him exactly, we just haven't known how to fit Him in."
"You don't need anything, you're just naturally good."
"I might be on the wrong track, always thinking about food. Man does not live by bread alone."
She pressed his head against her cheek. "Don't be different, Clem!" Then after a minute, "Poor Candace! I must write her a letter."
She got up and sat down where Clem had sat, and saw upon the pages of yellow paper he used for his endless figuring the words: "Average yield per acre (Mexico)" followed by lines of calculations of Mexico's millions of people. She tore off a yellow sheet, too tired to look for better writing paper.
> Dear Candace,
>
> We are just home from Mexico. I found Mother's letter here. I cannot say a word of comfort to you. I am ashamed that William is my brother. None of us have ever understood him. Mother is glad my father is dead and I think I am too, unless Father could have kept William from being so wicked.
>
> There is nothing I can do, I guess. It's too late. I don't pray as I used to but if I did, I would go down on my knees. Perhaps I should even yet. I feel closer to you than I ever have. And there are the two boys—how they must despise their father! It is all wicked and you have never deserved anything like this. I cannot imagine what reason he gives. You are so pretty and so good tempered. I hope William suffers for this.
Candace read the letter in her old room at her father's house. She smiled rather sadly, thinking that she had never known Henrietta until now, when the bond between them was broken. She glanced at the small silver clock on the dressing table. She was no longer William's wife. The decree was to be granted at noon and it was now six minutes beyond. She had been acutely aware of the time as it had passed and then had forgotten it for a few minutes and in that little space of time it was over. She let the letter drop on the floor and leaned her head back against the back of the chair and closed her eyes.
She had protested nothing. That was her pride. Jeremy had flung himself out of William's offices forever, he said, but when she saw Ruth she had made him go back. Ruth had no defense for William—she was too gentle and good for that. But she did not blame him, for to her alone William had explained himself, and she had tried to explain him also to Jeremy and to Candace. "He's always been different from everybody," Ruth said in her earnest, sweet little voice. "He's been so lonely all his life. I sometimes think if Father hadn't died... Father understood William, but he had to wait for him to grow up. I remember Father saying that once."
"It's his own fault if he is lonely," Jeremy had retorted. "He holds himself above everybody. Yes, he does, Ruth. He lords it over us all."
"I know it seems that way, Jeremy, but really inside he's quite lost."
Jeremy had snorted and Ruth nodded her head up and down very positively. "Yes, William is lost. He needs something he hasn't got. None of us can give it to him."
Upon this Candace had spoken. "If Emory can give it to him, then I shall be glad."
"Oh, Candy, you're so generous," Ruth had cried, the tears streaming from her soft blue eyes.
But still she had defended William in her heart and Candace saw it, and because Jeremy loved his wife he, too, would allow William his way. She had no knight, unless her old father came forward. But he evaded life nowadays, indeed not from lack of love, so much as from too much love. So sensitive had he grown as age came upon him, so excessively tender, so wishful that human beings should all be happy, that when they were not he could not bear to be near them. So because she loved him, Candace had shielded her heart from her father and affected to be gay about William's new love, and she insisted that of course he must marry Emory, and she even pretended that she and Emory could and would meet and be friends, while in her heart she knew that this could never be.
With her sons, she was cavalier. Will and Jerry, though tall young men, still cared more for football than for anything else on earth. "We mustn't blame your father," she had said to them brightly. "The truth is, our marriage never quite came off, if you know what I mean. Why should you know? It's like a flower that doesn't quite bloom. Still, I've had you two and that is a great deal to get out of one marriage." She had looked from one solemn young face to the other.
"Are you going to marry again?" It was Will's question. She met his young gray eyes and shook her head, still playfully. This was her protection now and forever, not to care too much, not to mind. She thought of fallen leaves floating upon the surface of the swimming pool, of leaves drifting down from the trees, of a bird resting upon the waves of atmosphere, of flower petals dropping upon the grass. Her father was right. Escape life, perhaps, but certainly escape pain! The blow had been dealt.
Jerry, the younger, had spoken with sudden rage. "Why don't you go and see that woman and tell her she has no right to—"
"Shut up," Will said for her. "You don't understand. You're only a kid."
Neither son had spoken one word of their father. He was immovable, unchangeable; none could reach him. Whatever he did was done. He was absolute.
William had needed none of them, not his mother, not Ruth. No one existed for him except himself, his monolithic being, his single burning purpose, more consuming than any he had ever known. He was ruthless in his office, angry with all delay, intolerably demanding upon his lawyers.
He had tried to compel Candace to go to Reno so that in six weeks he might be free. She had refused and old Roger Cameron had demanded an appointment. William had refused that. He gave orders that he would not speak with anyone on the telephone. He lived entirely in his apartment at the office and made no communication with his sons. After he was married to Emory he would let them see for themselves why he married her.
When he discovered that Candace was not going to Reno, he went himself. He endured weeks of loneliness without Emory, days when he called her by telephone that he might hear her voice and assure himself that she still lived, that she had not changed her mind, that she had no thought of delaying their marriage. His decree granted, he left by the next train and, speeding to England upon the fastest ship, he went straight to Hulme Castle.
She was there waiting for him, the wedding day set two days hence, and when he had her in his arms, he let down his heart. He put his face into the soft dark hair.
"Oh, my love—" They were words he had never used to Candace.
"You look fearfully tired, William."
"I shan't be tired any more, Emory."
She did not reply to this, and he stood for a moment letting his weariness drain away in the silence.
"Two days from now we'll be married."
"Two days," she echoed.
"I wish it were now."
To this, too, she made no reply.
They were married in the room where they had first met. She did not want to be married in Hulme Abbey, where, had Cecil lived, the ceremony would have taken place. Her parents had agreed, and so an altar had been set up in the drawing room. No one was there beyond her family and the vicar and his wife and a few people whom William had never seen before. "A quick, quiet wedding," he told her and she obeyed.
# 10
UPON A GAY AND prosperous people the thunder clouds of the Great Depression now crashed down their destruction. In the late summer, Clem had felt something was wrong. He could not define, even to Henrietta, his uneasiness, beginning at first as a personal discontent in his own mind, though he tried to do so one Sunday, the last in August. She was aware of his eternal searching for causes and, by her listening silences and her careful questions, helped him to see more clearly the vague shapes he perceived in the future.
Long ago Henrietta had come to understand that in Clem there was something of the seer, if not of the prophet. His instinct for humanity was so delicate, his perception of mankind so ready, that without magic and entirely reasonably he was able to forecast the possible in terms amazingly definite. Had he lived in ancient times, she sometimes mused, had he been born in those early ages when people explained the inexplicable, the mystic man, by saying he had been fathered by a god or had seen gods upon the mountains or in the flames of a burning bush, struck perhaps by lighting, they would have cried out that Clem was a prophet sent to them by God and they would have listened to him. And, were they frightened enough, they might have heeded him in time to avert disaster.
Now Clem and Henrietta, seated in rocking chairs upon their own narrow front porch, looked to the passer-by no different from any other middle-aged couple upon the street of an ordinary Ohio town. He talked and she listened and questioned. He was in his shirtsleeves and an old pair of gray trousers, and she saw that the collar of his blue shirt was torn. She resolved to throw it away secretly when he took it off that night. Clem was miserly about his clothes and declared them good enough to wear long after they had reached the point of dusting cloths and mops.
"I can't just tell you in so many words how I feel about things," Clem said. "It's like sitting out on the grass on a nice bright day and then suddenly knowing that the earth is shaking under you—not much, but just a little. Or it's like being in the woods, maybe, and wondering if you don't smell smoke somewhere."
"If you were in the woods and smelled smoke," Henrietta said, "you'd find out first which way the wind was blowing and look in that direction, wouldn't you?"
Clem flashed her an appreciative look. "I've thought of that. I can't tell which way the wind is blowing—not yet. Crops were good enough this year, at least taking the country as a whole. Maybe things are all right. Maybe it's nothing but my own queasy stomach. I oughtn't to have eaten those corn dodgers last night."
"I'll never have them again," Henrietta said.
Clem went on after a few seconds of rocking. "The trouble is that the way things are now in the world, we're all tied together in one way or another. There might be an earthquake somewheres else which would upset us, too."
She did not reply to this. The evening was pleasant though hot and children in bathing suits were playing with hoses, spraying each other and shrieking with laughter. Clem, deeply troubled by thoughts which were now roaming the world, saw nothing.
"The news from abroad is not bad, though, Clem," she reminded him. "Yusan says the new government in China is bringing order and getting rid of the warlords, at least, and pushing Japan off. And Goshal says that Gandhi has made a sort of interlude in India."
Clem got up. He walked across the porch, took out his penknife, and began to cut a few dead twigs from a huge wisteria vine that Henrietta had planted the first spring she came to New Point. Now, a thick and serpentine trunk, it crawled to the roof and clung about the chimney for support.
"Goshal is a Brahman no matter what I try to tell him," Clem said. "What you call interlude, hon, is only a truce. Gandhi has got the British to compromise for a while for just one reason, and Goshal can't see it. The price of food has gone down so much that millions of peasants are going to starve, hon, if something isn't done quick."
"City people will have more to eat if food is cheap," Henrietta said.
"Most people don't live in cities," Clem said. "That's not the point though, and I am surprised at you, hon. If the peasants and farmers starve it doesn't help the factory workers in the long run. Gandhi is right when he says everything has to be done for the interests of the peasants. They're basic everywhere in the world."
Henrietta felt clarification begin in the waters of Clem's soul. He was clipping one twig after another and they fell upon the wooden floor of the porch with soft dry snips of sound.
Clem went on, almost to himself. "And I don't know what to think about things in China. A new government? Well, any government, I guess, is a good thing after all these years of fighting and goings on. I don't blame Yusan for being glad about that. But I wrote him yesterday and told him that if this Chiang Kai-shek didn't get down to earth with all his plans and study what the people need, it will be the same story. You don't have to be an Old Empress to make the same mistakes."
Henrietta was rocking back and forth silently, her following thoughts circling the globe.
"I don't know," Clem muttered. "How can I know? I don't believe Japan is going to let things lay the way they are. They've been afraid for centuries, those people! They've got themselves all stewed up—can't blame them, though—the way different nations have gone over and sliced off big hunks for themselves. 'We're next,' that's what the Japanese have been thinking for a mighty long time, hon! 'If we don't get going and carve ourselves out something big, we're next.' That's what they think. Maybe they're right, who knows? Only thing I know, hon, is that the earth is shaking right here under my feet. I don't like the looks of things."
He lifted his head and looked away over the housetops and beyond the trees. "Talk about smoke—the wind is from Europe, I reckon."
The cyclone struck in October. Bred in the storms of the world it had gathered its furious circular force in the angry hunger of the peoples of Europe and then reaching its sharp funnel across the Atlantic Ocean it struck in Wall Street, in the heart of New York, in the most concentrated part of America.
Clem, on that first fatal morning, reached out of the front door to get the morning paper, half his face lathered with shaving soap. He saw the headlines as black as a funeral announcement and many times as large upon the front page, and knew that what he had feared had come. He wiped his cheek on the sleeve of his pajamas and sat down in the kitchen to read. Henrietta was making coffee. When she saw his face she set a cup before him and went out into the hall, got his overcoat, and wrapped it about him. Over his shoulders she saw the frightful announcement, CRASH IN WALL STREET SHAKES THE NATION!
"Tell Bump to get down here as fast as he can," Clem ordered. "You and me and him have got to get right to work, hon."
She obeyed him instantly as she would have obeyed the captain of an overloaded and sinking ship. There was no time to waste.
Clem dressed and ate a hasty breakfast and being immediately beset by the demons of indigestion, he was swallowing pepsin tablets when Bump came into the house. Henrietta had cleared the dining-room table of dishes and cloth, and Clem spread out the big sheets of white wrapping paper upon which he always did his large-scale figuring.
"Sit down," he told Bump. "We're going to have the worst depression in the history of the world. We got to get ready to feed people the way we've never done before. I'm going to open restaurants, Bump. It won't be enough now to sell people food cheap. We got to be ready to give it away, cooked and ready to swallow, so that people won't starve to death right here in our own land."
He outlined in rapid broken sentences what he believed was sure to happen and Bump listened, cautious and reluctant and yet knowing from past experience how often Clem was right.
"We can hardly feed the whole nation, Clem," he said at last.
Clem was immediately impatient. "I'm not talking about the nation. I'm talking about hungry people. I want to set up restaurants in the big cities as quick as we can. Our markets will supply our own restaurants. Whoever can pay will pay, of course. At first most people can pay and will want to. But I am thinking of January and February, maybe even this winter, and I'm thinking of next winter and maybe the winter after. That's when things will get bad."
It was impossible to get so huge a plan going as quickly as Clem thought it should and could. But it was done or began to be done within a time that was miraculous. Clem bought a small airplane which Henrietta, much against her secret inclination, learned to fly lest Clem insist on doing so and he, as she well knew, was not to be trusted with machinery. He expected divine miracles from engines made by man and while she had submitted for years to his mistreatment of automobiles, his wrenchings and poundings of parts he did not understand, the frightful speed at which he drove when he was in a hurry, she could not contemplate such maneuvers in the air.
She made a good pilot, to her own surprise, for she was an earthbound creature and hated suspension. Clem as usual was surprised at nothing she did, insisting upon her ability to do everything. At as low a height as she dared to maintain, they flew from city to city, her only apparent cowardice being that when they went to the coast to set up Clem's restaurants in San Francisco and Los Angeles, she avoided the Rocky Mountains, and flew far south in order to escape them. Pilot and attendant, she followed Clem while, with his superb and reckless disregard of all business principles, he established during that first winter six restaurants across the country on the same magnitude as the markets. For these restaurants he hired Chinese managers.
"Only Chinese know how to make the best dishes of the cheapest food," he explained to Henrietta. "They've been doing that for thousands of years."
Knowing the importance of the spirit, he summoned his new staff to a conference in Chicago, where he put them up at a comfortable hotel while he talked to them about starvation and how to prevent it. He worked out one hundred menus, dependent upon the raw materials of the markets, and laid down the rule which should have ruined him and which instead led him eventually to new heights of prosperity.
"Any time anybody wants a free meal in any of our restaurants they can have it," he said firmly. "Of course they can't order strawberries and cream, but they can have meat stew and all the bread they want and they can have baked apples or prunes for dessert. Nobody will know whether they pay for it or not. They'll get a check same as everybody else and they'll go up to the cashier and just tell her quiet-like if they haven't any money."
"How many times can one man eat free?" Mr. Lim of San Francisco inquired.
"We don't ask that," Clem said. "We don't ask anything, see? If anybody's hungry, he eats. At the same time, we'll serve other foods, cooked so good that people who have got money will pay for it. And our restaurants will look nice, too, so that people will want to come there. They won't seem like handout places."
The Chinese exchanged grins. Their salaries were secure and so they were highly diverted by this mad American. Since he had appealed to their honor they were prepared to respond with their most ingenious economies and seasonings. He in turn accepted their promises with complete faith.
"We can do such things as you talk," Mr. Kwok of New York Chinatown now said. "Only thinking, however, is that we better hire our own cooks and waiters, each of us somebody he knows good."
"Sure," Clem agreed. "That's all up to you. I hold you responsible, each for your own place."
"Must be order, you see." This was Mr. Pan of Chicago. "I know Americans think all equal but Chinese know better. For making something go, especially cheap and good, one man is top and everybody else in steps below, each man top to next man and next-to-top man is reporting to very top man. Each man is servant and at the same time boss, except bottom man, who is anxious for rising and does his best."
"Sure," Clem said. "You put it neat."
With the simplest of casual organization, Clem arranged his markets and restaurants in an endless chain of co-operation. He did not expect perfection and did not get it. Nepotism in two of the restaurants was a drain on profits until he discovered it and fired the two managers and hired new ones. With the old managers went the entire staffs and with the new ones came new and chastened ones. The other four managers approved the changes and worked with the greater integrity and zeal. Clem's _Brother Man Restaurants_ without advertising lost no money the first year and saved thousands of people from hunger so quietly that the public knew nothing about it. Three per cent of the people who ate free meals could have paid and did not. This was balanced by sums from people who could and did pay extra because they liked the food. Clem was brazen about accepting such extra pay. On the bottom of the menu cards in large bold letters he printed this legend:
OUR PRICES ARE TOO LOW FOR PROFITS. IF YOU HAVE GOT MORE THAN YOUR MONEY'S WORTH FROM SOME DISH YOU HAVE ESPECIALLY ENJOYED, PLEASE PAY WHAT YOU THINK IT IS WORTH. THIS MONEY WILL GO TO FEED THE HUNGRY.
A surprising number of people paid extra, but Clem was not surprised. His faith in humanity increased as he grew older and made it unnecessary, he declared, for any further faith.
"The way I look at it is this, hon," he said to Henrietta on one of their long flights across the plains of the West. "Everybody needs faith. Some people find it in God or in heaven or something way off. Take me, though, I get inspiration out of my faith in people here and now."
In the middle of the next winter, however, Clem found himself puzzled. He was feeding people on a huge scale, not only through his markets but through his restaurants, and he saw that it was not enough. He turned his eyes away from the breadlines and knew that at last he had met a task that was beyond him.
The effect of this discovery upon him frightened Henrietta. She saw his first excitement and exuberance, his immense rise of energy, his self-confidence, and even his faith pass into an intense and grim determination as the hordes of the hungry increased over the nation. They gathered in the cities, for country people can hide themselves snugly into their farms and eat the food they produce and stop buying. Furniture and machinery which they had been tempted to buy on installments they relinquished, wary of their savings. They had lived without radios and without cars and washing machines and they could again. They withdrew into the past and lived as their grandparents had done and did not starve. They could still sleep in ancient beds and use old tables and sit on ladder-back chairs.
It was the cities that frightened Clem. Even in the cities where he had his restaurants, the breadlines began to stretch for blocks. When he found a family with seven children starving in New York he came back to Henrietta in the small room at a cheap hotel, which was his usual stopping place.
"I wouldn't have thought it could be, hon," he said mournfully. "Maybe in China or India, but here? Hon, how am I going to get the government to understand that people have got to be fed? A war will come out of this, hon. People won't know why there's a war and they'll think it's because of a whole lot of other things, but the bottom reason is because people can't buy food because they don't have the money to buy it with. That makes men fight."
"Clem, you look sick!" Henrietta said. "I'm going to get you a doctor."
"I am sick," Clem said. "But it's a sickness no doctor can cure. I'll be sick as long as things go on like this."
At noon he refused to eat and Henrietta went downstairs to eat alone, ashamed of her steady appetite. If Clem could only separate his soul from his body! But he could not and his body shared the tortures of his harassed soul. He blamed himself for things being what they were, and this Henrietta would have thought absurd except she had seen in her own father when she was a child the same suffering for the sins of others.
"Did we do our duty as Christians—" she remembered her father saying that year when they had left China, that fearful year when Clem had been left alone in Peking—"the world in a generation would be changed."
Clem was like that, too. He wanted the world changed quickly because he saw it could be changed and he fretted himself almost to death because other people did not see what he did. Troubled and sad, she ate her robust meal, chewing each mouthful carefully because she believed Fletcher was right about that. She had got interested in Fletcherism because of Clem's indigestion and especially because he was always in such a hurry that he swallowed his food whole.
When she went upstairs again he was lying on the bed, flat on his back, and she thought he was asleep. She tiptoed in and stood looking down at him. His hands were clasped behind his head and his eyes were closed. Then she saw his lashes quiver.
"That you, hon? I've been lying here thinking. I believe I've got an idea."
"Oh, Clem, I hoped you were asleep! If you won't eat—"
"I will eat but you know how I am. If I eat when I'm thinking something out the food just lays on my stomach. Hon, I am going to see your brother William."
She sat down heavily in the soiled armchair. "Clem, it won't do a bit of good."
"It might, hon. He's got a new wife."
"Nobody could have been nicer than Candace."
"Maybe so. She was mighty nice. But if William loves this woman, maybe it has done something to him. Maybe it's stirred his heart."
"I hope you don't want me to go with you."
"I was kind of hoping you would."
"Clem—it won't help! He's invincible now. Everywhere we go people are reading his nasty little newspapers."
"He must feel something for people, hon."
"No, he doesn't. He hates people. He despises them or he wouldn't make such newspapers for them. I know why he does it, too. He feeds them the worst stuff so as to keep them down. It's like feeding the Chinese opium—or giving whisky to the Indians. People learn to like it and because they like it they will follow the person that gives it to them."
Clem, always generous, shook his head at this picture of William. "I kind of think I'll go right away and see for myself, hon."
Henrietta's anger rose in spite of love. "Very well," she declared. "Go if you must. But I will not go with you."
He sighed and got off the bed. He put on his coat and smoothed his hair with his hand. Then he bent to kiss her tenderly.
"You don't feel mad with me, do you?"
"Oh no, Clem, except—"
"Except what?" He paused and looked down upon her, his eyes bright blue in his white face and his lips pursed quizzically.
"Clem, you're too good, that's all. You won't believe that anybody isn't good."
"That's my faith, I guess."
He turned at the door, looked as if he were about to say something more, kept silent instead and went his way.
Lady Emory was alone for luncheon. She was, of course, Mrs. William Lane and by now she was well used to it in all external ways. She was beginning to feel that the huge comfortable house in uptown New York was her own, and in certain ways that Hulme Castle could never be. From earliest memory she had known that while Hulme Castle was her shelter it was not her home. William had divined this very soon after their marriage and had offered to put at her disposal as much money as needed to repair the castle and put in bathrooms.
"It will make you feel more free to go there and stay as long as you like, now that you are my wife," he had said quite gracefully.
Her father had refused the gift, however. He saw no need for more bathrooms since he himself still used a tin tub brought into his room in the mornings and set before the fire.
"I believe William would like to come here and stay sometimes, Father," she had replied to this prejudice. "He would feel less like a guest if he had some part in the castle."
She said this quite as gracefully as William had but her father had only grumbled and it had taken Michael to persuade him to let William repair at least the west wing as a place where Emory and her husband might stay when they came to England. Lady Hulme had early discerned in William a rather touching desire to own some part of Hulme Castle and so she had been grateful to Michael who, after all, was the one most to be considered, since he was the future heir.
As for America, as far as Emory had seen it, it was amazing. The people were very friendly, perhaps too friendly. She had been invited to a great many dinner parties and everybody had persisted in calling her Lady Emory, and this made her feel at home. William, too, called her Lady Emory in the house to the servants. Naturally when he introduced her it was as his wife, Mrs. Lane. She felt in spite of his real love for her that she did not know him as well as she hoped she would one day. He had a strange and almost forbidding dignity which she did not dislike, although she saw that it cut him off from ordinary people and even from her, sometimes. She was used to that. In his way her father had a dignity, too. He would have been outraged by familiarity from his inferiors.
Moreover, there was something about this dignity of William's which ennobled her and their life. She was proud of his straight handsome body and was well aware of their regal appearance together.
He never talked to her of his first wife. In marriage he and she were utterly alone, and for this she was grateful. Instead, he told her much about his boyhood in Peking, and she who had never thought of China as a place existent upon this earth, now perceptively saw him there, a tall solitary boy, august in his place as the only son of the family, hungry for communication when there could be none, alien from his parents and sisters as he was from the Chinese he knew, who apparently were all servants.
"Did you not know any Chinese boys?" she asked.
"They were not allowed in the compound," he replied. "My mother did not like them to hang about. Even my father's study had a separate entrance so that when the Chinese came to see him they need not enter the hall."
"Did you try to know anybody secretly?" she asked.
"It would not have occurred to me," he replied sincerely.
Then bit by bit there came out the remembered fragments of his life in the Chefoo school and here she perceived he had been shaped. She saw the proud boy slighted and condemned by the careless lordly English boys she knew so well, for Cecil had been such a boy. Unconsciously William revealed to her his wounds never healed.
It was not all bitterness. He could speak sometimes of wide Peking streets and of the beauty of the porcelain roofs on the palaces of the dying Empire. He told her one meditative evening how his mother had taken him to see the Empress when he was a small boy. "I bowed before her, but I didn't kneel because I was an American. The Chinese had to kneel and keep their heads on the floor. I remember her thin hands—yours remind me of them. They were narrow and pale and very beautiful. But the palms were stained red and the long nails shielded in gold gem-studded protectors. I looked at her face—a most powerful face."
"Did she speak to you?"
"I don't remember that. The people called her the Old Buddha. They were afraid of her and so they admired her. People have to have someone like her. I was sorry when she died and that revolutionary fellow, Sun Yatsen, took over. People can't respect a common fellow like that—someone just like themselves. Maybe this new man, Chiang Kai-shek, will be better. He is a soldier, used to command. There is no democratic nonsense about him."
Emory listened, knowing that he was telling her things he had never told anyone, things that he had forgotten and now drew up out of the wells of his being. At the bottom of everything there was always a permanent complaint against his parents because they had robbed him of his birthright of pride. It had been impossible to explain to them why he was ashamed, and he was the more ashamed because he had the agony of wanting to be proud of his father, and then the humbling realization of knowing that there was something of his father in himself in spite of this hatred, and that he could not simply enjoy all that he had, his money and his great houses and the freedom that success should have bought him, because he could never be free. God haunted him.
This was the bitterness and the trouble and the terror that she found in William's soul. It made her thoughtful indeed. His conscience was the fox in his vitals.
Upon such musing alone and by the fire in the drawing room of her American home she took her usual afternoon tea on the cold January day. It was not often that she was alone but she had felt tired, the intense activity of this new world city being something to which she was not used. She had been invited to a cocktail party given for that playwright now most successful upon Broadway, Seth James, and. when she telephoned to William that she would not go he had replied that he himself must go since Seth had been a former employee with whom he had disagreed, and if he did not go, it might appear that he held a grudge.
"Do go, by all means," Emory had said at once.
She found it comfortable to be alone for an hour. It seemed difficult to be alone in America, although in Hulme Castle it had been the most natural state. Now, after she had eaten some small watercress sandwiches and drunk two cups of English tea, she went to the piano William had had made to order especially for her touch and sitting down before it she played for perhaps half an hour, transporting herself as she did so to some vague and distant place that was not America and yet not quite England. She had no wish to return to Hulme Castle and she was quite happy here in this house, as happy as she thought she could be in mortal life. Cecil had left her entirely now, even her dreams, and she seldom thought of him.
In the midst of her music the door opened and she heard the slight cough with which the second man announced his deprecatory presence.
"Well, Henry?" she called, softening her melody without stopping it.
"Please, madame, Mr. Lane's brother-in-law is here."
"Mr. Jeremy Cameron?"
She had met Jeremy and William's rather sweet sister Ruth. She had found it difficult to get on with Ruth's soft effervescence, but Jeremy she thought charming, although it was unfortunate that he was also the brother of William's first wife.
"I do hope you won't mind it that I am Candace's brother," Jeremy had said directly when they were first alone. "I assure you that Candace entirely understands about things. She wouldn't mind meeting you, as a matter of fact—she's a warmhearted sort of creature."
"I don't mind in the least your being her brother," Emory had replied.
"It's not Mr. Jeremy, please madame," Henry now said. "It's the other brother-in-law—a Mr. Miller, I believe."
"Oh—" Lady Emory rose from the piano. She knew about Henrietta who, William said, had married a strange sort of man named Clem, who had made an odd success in food monopolies. While she stood in the middle of the floor somewhat uncertain as to how she would receive Clem or whether she should receive him at all, he was at the door looking altogether shadowy, with his sandy gray hair blown about.
"Do come in," she said.
She was struck by his excessive thinness and the startling blue of his eyes.
"You look cold!" she said with her involuntary kindness. "I think you should have some hot tea."
To Henry, still hovering in the doorway, she said with distinctness, "Please fetch a pot of hot tea, Henry."
"Yes, madame." Henry's voice breathed doubt as he disappeared.
Clem saw a woman, a lady, who was all gentleness and kindness. It was true that he felt ill for a moment when he first came in. He had eaten nothing since morning.
"I guess I am a little hungry," he said and tried to smile.
She had him in a comfortable chair instantly and put a hassock under his feet. The fire burned pleasantly and the vast room was quiet about him. Everything was comforting and warm and he sighed away his haste and intensity. In his taut body one muscle and another relaxed. The man came back with hot tea and she poured him a cup.
"Bring him a soft-boiled egg," she told the man.
"I can't eat eggs," Clem protested.
"Indeed you can," she replied with firmness. "You want an egg—you are so pale."
"No milk in my tea, please," Clem said.
While he waited he drank two cups of the delicious hot tea and ate one of the hot biscuits she called scones, and when the egg came it was two, served in a covered cup. There were triangles of toast with it and he ate and felt renewed to the soul.
"Wonderful what food can do," he said and smiled at her and she smiled back.
"I don't know what to call you," he said next.
"Emory, of course. You're Clem, I know."
"Aren't you a lady or something?"
"In a way. Never mind that, though, now that I'm an American."
Clem folded a small lace-edged napkin with care and put it on the tray.
"I see you believe in feeding folks and that's what I came to see William about. Maybe he's told you about me?"
"I believe he said you deal in foods?"
"I like to put it that I deal with people and getting them fed."
He leaned forward, looking extraordinarily restored and reminding her somehow of the young men in London who were always talking in Hyde Park. She had never stopped to listen to any of them but often they had the same sandy look and shining, too blue eyes. While she sat gazing at him and thinking this, Clem was fluent in preaching his own gospel to this kindly, attentive woman. He had all but forgotten that she took Candace's place and that he ought not to like her so much, but he did like her. Candace had been kind, too, but it was with a child's kindness and he had never been sure she understood him. But this woman did understand and she was not at all a child. There was even something sad about her dark eyes.
"You see what I mean?" he paused to ask.
"I do see indeed," she replied. "I think it is a wonderful idea, only of course you are far ahead of your times. That's the tragedy of great primary ideas. You won't live to see it believed or practiced that people have the right to food as they have the right to water and air. The holy trinity of human life!"
He could not bear to have her merely understand him or even believe in him. When one believed, one must act.
He put forth his effort again. "We've got to get people to see this, though. That is what I came to William for. He has such power over people."
Emory looked at him with new and sudden interest. "Has he really?"
He was entirely sensitive to this interest and anxious to make the most of it. "I can't tell you how great his power is. His newspapers go into every little town and household—little easy papers that everybody can read. And then there's the pictures. If people don't want to read they can look at the pictures. I read them, too, and look at all the pictures. The queer thing to me is that you don't learn anything, though—Miss—Lady—"
"Just Emory," she reminded him.
He could not quite manage it. "I mean that it's all amusing and nice but you don't learn anything from it. You don't learn why it is that the people in Asia want a better life and you don't learn why it is that things don't look so good even with the new government in China."
At the thought of China Clem fell into thought. "I don't know—" he murmured. "I can't tell. I don't think things are going right over there. Maybe I'll run over as soon as I see this depression through." He lifted his head. "What I wanted to talk to William about—if he could get converted, so to speak, to this idea of feeding people. It won't be charity. It won't cost us money."
He began to explain the golden rule of his restaurants and somewhere in the midst of it they looked up and saw William at the door, upon his face surprise and disgust.
"Come along in, William," Emory said at once. "I am listening to the most fascinating man. It's Clem."
Thus she conveyed to William that he was to take from his face that look calculated to wound, and that he must come in and sit down and be kind to Clem, because she wished it. Their eyes met for a brief full second and William yielded. He yielded to Emory as he had never yielded to anyone.
"How do you do," he said to Clem.
"Fine," Clem said, "How's yourself?"
William did not answer. He sat down and took from Emory's hand a cup of tea.
"I really came to see you," Clem said looking at him. "But I have surely enjoyed talking to your good wife here. She has treated me well—fed me up and all. I didn't eat lunch today."
William did not show interest.
"Will you have a sandwich or a scone?" Emory murmured.
"Neither, thank you," William said.
Clem felt the atmosphere of the room change and he made haste to say what he had come for. Probably they wanted to be alone and anyway he had been here long enough. "I don't want to waste your time, William, but I do want to give you an idea. Or set it before you, anyway. I read your editorials every day and I see that you put in one idea every day, I guess an idea of your own. I can't agree with most of them but that's neither here nor there. It's a free country. But I notice that people take your ideas pretty nearly wholesale. I move around a lot through the country and I hear men say things that I can see come right out of your mouth, so to speak. I can see you understand how most people are. They don't know much and they talk a lot and naturally they have to have something to say and so they say what they hear somebody else say or what they read in the newspaper. I admire the way you can lay down something in a short plain way."
"Thank you," William said without gratitude.
Clem never noticed irony and he accepted the words as they stood. "That's all right. Now here's my idea. How about getting it across that we ought to give away our surpluses to the people who don't have food? I mean these men in the breadlines, and selling apples on the street, and the families hungry at home. It won't cost a thing."
"What surpluses?" William asked in a cold voice.
"Our surpluses," Clem repeated stoutly. "Even now we have surpluses, while the people are starving because they can't buy food. It's money that's short, not food."
William set down his cup. "What you propose would upset our whole system of government were it carried to logical conclusion. If people have no money they can't buy. Your idea is to disregard money and give them food free. Who is to pay the men who produce the food?"
"But producers are not getting anything, anyway!" Clem cried. "The food is rotting and they are short, too."
"It is better to let the food rot than it is to undermine our whole economic system," William said firmly.
Clem gave him a wild look. "All right, William, pay the producers, then! Let them be paid out of tax money."
"You mean the government ought to feed the people?" William was shocked to the soul. "That's the welfare state!"
"Oh God!" Clem shouted. "Listen to the man! It's the people I'm thinking of—the starving people, William! What's a nation if it's not the people? What's business if there's nobody to buy? What's government if the citizens die?"
"This is quite ridiculous," William said to Emory. He rose, towering over Clem, who rose to meet him. "We will never agree," William said formally. "I must conduct my publications as I see fit. Believe me, I am sorry to see anyone hungry, but I feel that those who are hungry have some reason to be. Ours is a land of opportunity. My own life proves it. No one helped me to success. What I have done others can do. This is my faith as an American."
For a moment Emory, watching the two embattled men, thought that Clem would spring at William. He gathered himself together, his fists clenched, his eyes lightning blue, electric with wrath. He glared at William for a long second and suddenly the wrath went out of him.
"You don't know what you do." The words came out of Clem like the sigh of death. He turned and went away as though he had been made deaf and struck blind.
When he was gone William sat down again. "Pour me another cup of tea, please, Emory." He tried to make his voice usual.
"Of course, William. But is it hot enough?" She felt the pot.
"It is all right, I am sure."
He waited until he had tasted the tea. "You see, Emory, how impossible the fellow is."
"I don't understand your American system yet, I'm afraid, William. Are there actually people starving?"
"Some people, of course, need food," William said in a reasonable voice. "Charities, however, are alert. There is free food; the very thing he talks about is being done. I have given a great deal of money myself this winter to charity, in your name and mine together."
He paused, but she did not thank him and he went on. "Who are these charity cases but the ones they have always been? They are the unskilled, the uneducated, the lazy, the drifters, the hangers-on, all the marginal people that are to be found in any modern industrial nation. In the ancient agricultural civilization of old China they were taken care of by the immense family system. Industry, of course, changes all that."
"Shouldn't there be some other means found to take the place of the family?"
"There are means," William said with an edge of impatience. "Believe me when I say that nobody needs to starve here in America if he works. Even if he doesn't want to work he need not starve. There are charities everywhere."
"I see," Emory said, her voice so soft that it was almost a whisper.
They did not speak for a few minutes, and when William put out his hand to her she took it and held it in both her own. It was the best hour of the day, this quiet one between tea and dinner. If they had guests they were friends and if they had no guests it was like this, William always tender toward her. She knew he loved her most truly. Indeed she knew he loved no one else. In some way she could not herself understand she had unsealed his heart which without her had been like a tomb. She was awed by this love for she had never known her power before. Cecil had loved her but she had perhaps loved him more than he did her. She had belonged to him but somehow William belonged to her. She was afraid, sometimes, for could not such possession place too great a demand upon her? She was not quite free any more because his love encompassed her about.
"I am ashamed that my sister's husband should have forced his way into this room and destroyed your peace," William said.
"Oh no," she said. "It was very interesting. As a matter of fact—" but she left her sentence there and he did not ask for its end. Instead he got up and bent down to kiss her. She rather enjoyed his kiss and she leaned back her head to receive it.
"I want to keep you happy," William said in a voice stifled by love. "I don't want you troubled."
"Thank you, dear," she said. "I am not troubled."
He went away and she heard him mount the stairs to his rooms. He would bathe and change and come down again soon looking rested and handsome, the gentleman that he was of wealth and increasing leisure. He did not need to work as once he did, he had told her only yesterday. They might go to Italy this winter, stopping at Hulme Castle, of course.
She sat for a moment thinking of this and of Clem. Then with a sudden decisive movement she touched the bell. There was really nothing she could do about Clem. She had chosen William and her world was William's world.
The door opened. "Take away the tea things, please, Henry," she said in her silvery English voice. "I am going upstairs and if any one telephones I am not to be disturbed."
"Yes, madame," Henry said.
From William's house Clem went downtown. He wanted comfort and reassurance. Henrietta could always give him comfort and encouragement but no one, not even she, could understand that now at this moment he needed the reassurance of fact. He must learn by actual test whether what he was doing was more than he feared it was, a drop in the vast bucket of human hunger. He avoided the hotel and taking a bus he swung downtown to Mott Street where his largest restaurant stood. It was a dingy-looking place now but there was no need to have it otherwise. People had already learned that they could get free food there, too many people. He saw many men and some women with children standing in a ragged shivering line waiting in the wintry twilight and he pulled up his collar and stood at the end. In a few seconds there were twenty more behind him.
They moved step by step with intolerable slowness. He must speak to Kwok about this. People must be served more quickly on such bitter nights. Speed was essential. They must hire more waiters, hire as many people as necessary.
He got in at last and took his place at a table already crowded. A waiter swabbed it off and did not recognize his guest.
"Whatcha want to eat?" he asked, still swabbing.
Clem murmured the basic meal. He waited again, glancing here and there, seeing everything. The room was far too crowded but it was warm and reasonably clean. It was big but not nearly big enough. He must see if he could rent the upper floor. In spite of the crowd the place was silent, or almost silent. People were crouched over the tables, eating. Only a few were talking, or laughing and briefly gay.
His plate came and he ate it. The food was good enough, filling and hot. The waiter kept looking at him and Clem saw him stop a moment later at the cashier's window. He ate as much as he could and then leaned to the man next to him at the long table, a young unshaven man who had cleaned his plate.
"Want this?" Clem muttered.
The sunken young eyes lit in the famished face. "Don't you want it?"
"I can't finish it—"
"Sure."
The waiter was watching again but Clem got up and went to the cashier's window with his check. He leaned toward the grating and said in a low voice, "I'm sorry I can't pay anything."
The sharp-faced Chinese girl behind the thin iron bars replied at once and her voice and accent were entirely American. "Oh yes, you can. You aren't hungry—not with that suit of clothes!"
"My only decent clothes," Clem muttered.
"Pawn them," she said briskly. "Everybody's doing that so's to pay for their meals."
He turned in sudden fury and walked across the restaurant, pushing his way through the waiters. He went straight to Mr. Kwok's small office and found him there in his shirtsleeves, the oily sweat pouring down his face.
"Mr. Miller—" Mr. Kwok sprang to his feet. He pointed to his own chair. "Sit down, please."
Clem was still furious. "No, I won't sit down. Look here, I came in tonight to see how things were going on. I told the cashier I couldn't pay just to try out the system. That damned girl at the window told me to go pawn my clothes!"
Mr. Kwok sweat more heavily. "Please, Mr. Miller, not so mad! You don't unnerstan'. We going broke this way—too many people eating every day. In China you know how people starving don't expect eating every day only maybe one time, two time, three time in a week. Here Americans expecting eating every day even they can't pay. Nobody can do so, Mr. Miller, not even such a big heart like yours. It can't be starving people eat like not starving. It don't make sense, Mr. Miller. At first yes, very sensible, because most people pay, but now too many people don't pay and still eating like before. What the hell! It's depression."
The wrath went out of Clem. What the Chinese said was true. Too many people now couldn't pay. The job was beyond him, beyond anybody. Too many people, too many starving people.
"I guess you're right," he said after a long pause.
He looked so pale when he got up, he swayed so strangely on his feet that Mr. Kwok was frightened and put out his hands and caught Clem by the elbows. "Please, Mr. Miller, are you something wrong?"
Clem steadied himself. "No, I'm all right. I just got to think of something else, that's all. Good night, Mr. Kwok."
He wrenched himself away from the kind supporting hands and went out of the door into the street. His idea wasn't working. Nothing was working. People were pawning their clothes in this bitter weather. They were being asked to pawn their clothes, pawn everything they could, doubtless. The waiters had been told to look and see what people wore. He remembered the hungry boy who had seized his plate and eaten the leftovers like a dog. That was what it had come to here in his own country. Someday people would be eating grass and roots and leaves here as they did in China.
"I got to get down to Washington," he muttered into the cold darkness. "I gotta get down there one more time and tell them...."
He found his way to the hotel where Henrietta waited for him, alarmed at his long absence.
"Clem—" she began, but he cut her off short.
"Get our things together, hon. We're taking the next train to Washington. I'm going to get to that fellow in the White House if I have to bust my way in."
He did not get in, of course. She knew he could not. She waited outside in the lobby and read a pamphlet on a table full of pamphlets and magazines that had been sent for the President to read. He had no time to read them and they had been put here to help the people who waited to while away the time. In a pamphlet of five pages, in words as dry as dust, in sentences as terse as exclamations, but passionless, she read the whole simple truth. For twenty-nine months American business had been shrinking. Industrial production was fifty per cent of what it had been three years ago. The deflation in all prices was thirty-five per cent. Profits were down seventy-five per cent. Nineteen railroads during the last year had gone bankrupt. Farm prices had shrunk forty-nine per cent so far and were still going down. But—and here she saw how everlastingly right Clem was—there was more food than ever! Farmers had grown ten per cent more food in this year of starvation than they had grown three years ago in a time of plenty.
"Oh, Clem," Henrietta whispered to her own heart. "How often you tell them and they will not listen! O Jerusalem, Jerusalem, how often...."
She put the pamphlet back on the table and sat with her hands folded in her lap and her head bowed so that her hat hid the tears that kept welling into her eyes. It was for Clem she wept, for Clem in whom nobody believed except herself, and who was she except nobody? William had hurt him dreadfully but she did not know how because Clem would not tell her what had happened. He had spoken scarcely a word all the way down on the train. She had tried to make him sleep, even if they were only in a day coach—he wouldn't spend the money for berths—but though he leaned back and shut his eyes she knew he was not sleeping.
He came into the waiting room suddenly and she saw at once that he had failed. She got up and they went out of the building side by side. She took his hand but it was limp, and she let it go again.
"Did you see the President?" she asked when they were on the street. The sun was bright and cold and pigeons were looking around for food, but no one was there to feed them.
"No," Clem said. "He was too busy. I talked to somebody or other, though, enough to know there was no use staying around."
"Oh, Clem, why?"
"Why? Because they've got an idea of their own. Want to know what it is? Well, I'll tell you. They've got the idea of telling the farmers to stop raising so much food. That's their idea. Wonderful, ain't it, with the country full of starvation?"
He turned on her and gave a bark of laughter so fierce that people stared, but he did not see their stares. He was loping along as though he were in a race and she could scarcely keep up with him.
"Where are we going now, Clem?" she asked.
"We're going home to Ohio. I gotta sweat it out," he said.
The nation righted itself in the next two years, slowly like a ship coming out of a storm. William wrote a clear and well-reasoned editorial for his chain of newspapers and pointed out to his millions of readers that the reforms were not begun by Franklin D. Roosevelt, the new President, but by Herbert Hoover who should have been re-elected in sheer justice that he might finish that which he had begun. It was already obvious, William went on, that the new inhabitant of the White House would run the nation into unheard-of national debt.
What William saw now in the White House was not the mature and incomparable man, toughened by crippling experience. He saw a youth he remembered in college, gay and willful and debonair, born as naturally as Emory to a castle and unearned wealth, but, unlike her, not controlled by any relationship to himself. Roosevelt, secure from the first moment of his birth, was uncontrollable and therefore terrifying, and William conveyed these fears in his usual editorial style, oversimple and dogmatically brief. To his surprise, he experienced his first rebellion. Millions of frightened people reading his editorials felt an inexplicable fury and newspaper sales dropped so sharply that the business office felt compelled to bring it to William's notice. He replied by a memorandum saying that he was sailing for England and Europe, especially Germany where he wanted to see for himself what was happening, and they could do as they liked while he was gone.
Emory received the news of the journey with her usual calm. They had not gone to England or Italy the year before, and she felt a change now would be pleasant. Alone with William she might discover what it was that kept him perpetually dissatisfied, not with her, but with the very stuff of life itself. She never mentioned to him her discernment of his discontent, for by now she knew it was spiritual and that he was only beginning to perceive this for himself. She refused again a thought which came to trouble her. Did William feel a lack in her own love for him? Was there such a lack? She made no answer. He had so much. He had all the money he had ever imagined he would have, and the most successful chain of popular newspapers. He was already planning the next presidential candidate, for this man in the White House could not possibly survive a first term. That he hungered for something he did not have, something more than woman could give, was now plain, perhaps even to William himself.
Or did his spirit seek after his father? One day on their voyage, William said, "I often think about my father. I wish you had known him, Emory. You would have understood each other. He was a great man, never discovered."
"I wish I might have known him, dear," she observed. They were in their deck chairs after breakfast and the sun was brilliant upon a hard blue sea.
"I wonder... I often wonder..." William mused somewhat heavily.
Emory delayed opening her novel. "About what, William?"
"Whether he would approve what I do—what I am!"
Approval. That was the word, the key! She saw it at once and grasped it. William needed the approval of someone he felt was his spiritual superior. For she knew that he was a man of strongly spiritual nature, a religious man without a religion. Emory herself was not spiritual, not religious at any rate, and she could not help him. She did not carry the conversation beyond her usual mild comment.
"I feel sure he would approve you, William, but I wish he were here to tell you so."
Within herself, after that conversation, she began the active search for the religion that William needed. It must be one strong enough for him, organized and ancient, not Buddhism, which was too gentle, not Hinduism, which was too merciful, not Taoism, which was too gay, imbued as it was with human independence even of God, and Confucianism was dead. She knew something of all religions, for after Cecil's death she had searched the scriptures of many and in the end had grown indifferent to all. Instead of religion she had developed a deep native patience, and detached by early shock, nothing now could disturb the calm which had grown like a protective shell, lovely as mother-of-pearl, over her own soul. She wished indeed that she could have known his father, for in that dead father, she felt sure at last, was the key to this living husband of hers. His mother, she had soon found, had been merely the vessel of creation.
Emory rather liked the vessel, nevertheless. She comprehended early with her subtle humor that there was not an ounce of the spiritual in her mother-in-law's bustling body. Mrs. Lane used God for her own purposes, which were always literal and material, reveling in William's success, in his wealth, in his new relation to an English Earl. Soon after William's marriage she had announced that she was going to England and that she would enjoy a visit at Hulme Castle. Emory had written to her own mother with entire frankness, saying that her mother-in-law would be the easiest of guests and not in the least like William. "Old Mrs. Lane is always ready to worship," Emory wrote, and drew a small cat face grinning upon the wide margin of the heavy handmade paper that bore her name but also the Hulme coat of arms.
She had seen Mrs. Lane off and upon the deck of the great ship had given her a huge corsage of purple orchids which would last the voyage, a package of religious novels, and a box of French chocolates. "Food for body and soul," she had said with private cynicism. Mrs. Lane, who had a strong digestion and liked sweets, did not comprehend cynicism. She had thanked her new daughter-in-law with the special warmth she had for the well born. She stood at the handrail of the upper deck, wrapped in a fur coat and a tightly veiled hat, and waved vigorously.
At first the divorce had seemed horrible to her, until she discovered how thoroughly she approved of Emory and her English relations. She made compromise. It was not as if William needed the Cameron money any more. Emory was really much better suited to him in his present position than Candace was. Men did outgrow women. There was no use pretending, although, thank God, her own husband had never outgrown her. Such remarks she had poured into Ruth's ears, and Ruth always listened.
This mother, Emory had soon perceived, was of no real use to William, and at first she had thought that any connection between William and his mother must have ended with the physical cutting of the umbilical cord. Later she had seen that she had been wrong. Mrs. Lane had created a division in William. To her he owed his respect for wealth, for castles, for birth, for—
At this point Emory checked herself. She was being nasty, for did she not enjoy William's wealth? Worse than that she was being unjust to him, whose soul hungered after higher things than those which he had. She wanted William to be really happy and not in the way that America meant happiness, which was something too fervid and occasional. She wanted William to be satisfied in ways that she knew he was not. She wanted his restless ambition stilled, and the vague wounds of his life healed. Some of them she had been able to heal merely by being what she was, English and his wife.
Hulme Castle was unusually beautiful on the afternoon when they were driven up the long winding road from the downs. The winter had been mild, the chauffeur said, explaining the amount of greenery about the old towers and walls.
Her parents were in the long drawing room, though it was not yet noon, and she was touched to think they were waiting for her, putting aside their usual morning pursuits.
"My dears—" she said, bending to kiss them.
William was quietly formal and nothing much was said. Her parents did not feel at ease with him, nor, as she saw, quite at ease even with her. Then Michael came in dressed in his riding things and ease flowed into the room with him.
"I say, you two—you haven't been shown up to your part of the castle yet?"
"You told us not," Lady Hulme reminded him.
"No. Come along. I wanted to show it to you," Michael said.
They followed him, laughing at his impatience, and then Emory saw that even William, so scant in his praise of anyone, was touched by what Michael had done. He had really made a small private castle of one wing. It had its separate entrance, its own kitchen, and four baths.
"I shall be able to rest here, Emory," William said so gravely that she perceived he needed rest.
"Come along, William," Michael said when they had seen everything. "We'd better leave Emory for a bit with her mother. I have to ride to the next town to see about getting a tractor. I thought we'd get our luncheon there, perhaps. You could advise me—it's an American machine."
Emory laughed. "You're not very subtle, Michael, but then you never were." They laughed with her and went off, nevertheless, and she lunched with her parents.
The castle, she discovered, was in a strange state of flux. Her father, deeply angry over the increase in death duties, was threatening to move into the gate house with her mother and a couple of servants and let Michael take the castle and assume title so far as was possible. She listened to this talk at the immense dining table, her father at one end, her mother at the other, and she in between as she used to be.
"It's hard on a man not being able to finish his days in his proper place," the Earl said.
He fell into silence over his roast beef and port, a silence which his wife could not allow for long.
"What are you thinking of, Malcolm, pray tell?" Lady Hulme asked. She did not drink port for it made small red veins come out on her nose.
"Do you remember, my dear, that old chap we dug up in the church when we put. in the hot-water pipes?" the Earl asked with entire irrelevance.
"Father, what makes you think of him now?" Emory asked.
"He'd been lying there a hundred and fifty years, you know, and his bones were as good as anything, white as chalk, but holding together, you know," the Earl replied.
Lady Hulme was diverted by the memory. She remembered perfectly clearly the June morning years ago when the men came to say that they had struck a coffin in Hulme Abbey and both of them had gone over to look at it. The coffin was only wood and was quite gone really except for bits of metal, but there in the dust lay the most beautiful silvery skeleton. Luckily it was not a Hulme ancestor but some physician who had served the family and had been given the honor of burial in the abbey.
"You don't think that he took drugs or something that kept his bones hard?" she now asked.
"Might have," the Earl conceded. "Still, perhaps it was only the dryness of the abbey, eh? Maybe the hundreds of sermons the vicars preached, eh?"
He choked on his own humor and exploded into frightful coughing. Lady Hulme waited. He choked rather easily nowadays, especially on port. When he subsided, red-eyed and gasping, she felt it wise to change the subject, lest he be tempted to another joke.
Before she could speak Emory lifted her head.
"Hark—Isn't that the horses?" They listened.
"Yes," she exclaimed. "It's William."
She got up with her stealing grace and went out, and Lady Hulme said aloud what she had been thinking.
"Do you like Emory's husband—really, I mean?"
"How could anybody like him?" the Earl replied in a voice restored to common sense. "There is something feverish in him."
"I thought he seemed as cool as anything today."
"He is the sort that burns inside, you know, my dear, like that what's-his-name from India that we dined with once at Randford. I don't know how the Earl felt but I know I was jolly glad to be away after dinner."
"What's-his-name" was a small dark man named Mohandas Gandhi. He had come over to England for conferences and he had refused to wear proper clothes or eat proper food. The government had been compelled to recognize him, nevertheless, and there was a frightful picture of him taken with the King and wearing almost nothing—just the bed sheet or whatever it was that he wrapped about his nakedness. It did seem that when a man came to a civilized country he might behave better. When the Earl of Hulme had muttered as much behind his mustaches to the Earl of Randford, his host had smiled at him and murmured in reply:
"You are simple, my dear fellow. Gandhi is too clever for you. His hold on the masses of India is immense just because he won't wear anything but the sheet. That's what the peasants wear and they like to think that one of them wears a sheet right in the presence of you and me and even the King. It makes them trust him. If he put on striped trousers and a morning coat, they'd think he had betrayed them."
The Earl of Hulme had been stupefied by such independence and now felt that if something had been done about it then India would not be dreaming today of getting away from the Empire. What would happen to the world if men were allowed to come into the presence of their betters dressed like goatherds? Upon that day he had stared a good deal at the small man whose perpetual smile was as cool as a breeze, and after an hour of this persistent gaze he had discerned beneath the coolness what he called the fever. He recognized it because he had seen it elsewhere. There had been a curate in his youth who had burned to improve the lot of the tenants, and he had seen the old Earl, his father, fly into fury.
"Read your Bible, sir!" the old nobleman had thundered at the tall, hungry-eyed curate. "Does it or does it not say that I am to put my tenants into palaces?"
"It says the strong must bear the burdens of the weak," the foolhardy man had replied.
That was the curate's end. He had killed himself as nicely as though a rope had been put about his neck. He had left in disgrace and was never heard of again. But young Malcolm, watching, had felt the fever burning inside that lean frame. On the last day, when he thought the curate had gone, he found himself face to face with him in the park. The chap had walked about to find him.
"Malcolm—" That was what the man had actually dared to call him. "Malcolm, you are young and perhaps you will listen to me."
"I don't understand," he had stammered, angry and taken back at such daring.
"Don't try to understand now," the curate had urged. The fever was plain enough then. You could see the flames leaping up inside him somewhere and shining through his pale eyes. "Just remember this—unless the hungry are fed, you will be driven away from all this. It is coming, mind you—you've got to save yourself. I warn you, hear the voice of God!"
He had wheeled without answer and left the curate standing there and he had not once looked back.
"Nonsense," Lady Hulme now said. "William is a very handsome man. I don't see the least resemblance to any Hindu, not to speak of that odd man."
She broke off, noticing how brightly the sun shone through the bottle of port. Suddenly she felt that it was a pity not to taste so beautiful a liquid. If her nose grew red it would not matter—poor Malcolm had long since ceased to notice how she looked. She poured herself a glass of the rich port, very slowly, the sun filtering through the crimson wine.
... Outside in the soft English sunshine Emory was listening to the last fragments of a conversation which had been of more than American tractors.
"I can't tell yet whether it's good or bad," Michael said. "I can only say that there's something new happening in Germany and Italy. New, or maybe something very old, I can't tell which. If it goes well it'll be a new age for Europe and therefore the world. I don't think things will go well."
"You don't believe that democracy will work in Europe, do you?" William asked.
"Of course not," Michael said impatiently. "But it's these chaps—Hitler, you know, and Mussolini. They've no breeding. Get a common man at the top and ten to one he can't keep his senses about him."
Emory cried out, wary of a certain reserve in William's look, "Oh, Michael, how silly of you. As if we weren't all common at bottom! Who was the first Earl of Hulme, pray? A constable of Hulme Castle, that's all, and a traitor against his King, at that."
Michael was stubborn. "That's just what I said. He couldn't keep his senses. He got thinking he was greater than the King."
"What happened to him?" William asked with restrained curiosity.
"The Queen Mother got her back up," Michael said. "There was a long siege and our arrogant ancestor was starved into obedience." He lifted his whip. "You'll see the marks of the battle there, though it was more than five hundred years ago."
Upon the thick stone walls were ancient scars and William gazed at them. "A very good argument against everybody's having enough food," he said thoughtfully. "Food is a weapon. The best, perhaps, in the world!"
The day ended peacefully as usual, but William was restless during the night and rose early. He wanted, he explained to Emory, to go to Germany and see for himself. To Germany then they went.
In Berlin William had suddenly decided that he wanted Emory to see Peking. He had met Hitler and had been reassured. Out of postwar confusion and the follies of the Weimar government, Hitler was building the faith of the German people in themselves and their destiny. The whole country was waking out of despair and discouragement. Trains were clean and on time, and Berlin itself was encouraging.
"There is nothing to worry about here," William said in some surprise. "I don't know what Michael was talking about."
After his talk with Hitler he was even more pleased. "The man is a born leader," he told Emory, "a Carlylean figure." It was then that William decided to go to China, telling Emory that he felt that he could never explain himself to her altogether unless she saw the city of his childhood. They boarded a great Dutch plane that carried them to India and Singapore and from there they flew to China. Of India Emory saw nothing and did not ask to see anything. Cecil's family had been dependent upon India and her curiosity had died with him.
They spent nearly two weeks in Peking. They wandered about among the palaces, now open to tourists, and William searched the painted halls, the carved pavilions, for the throne room where as a child his mother had led him before the Empress.
"William, after all this time, can you remember?" Emory asked, unbelieving.
"I remember the Empress as though she had set a seal upon me," William replied.
He found the room at last and the very throne, but in what dust and decay!
"This is the place," William said.
They stood together in silence and looked about them. The doors were barred no more and pigeons had dirtied the smooth tiled floors. The gold upon the throne had been scraped off by petty thieves and even the lazy guard who lounged in the courtyard offered them a sacred yellow tile from the roof for a Chinese dollar. William shook his head.
"I wonder," Emory said in a low voice, "if one day Buckingham Palace will be like this?"
"I cannot imagine it," William replied, and as though he could not bear the sight before them, he turned abruptly from the throne. "Let us go. We have seen it."
"Perhaps it would have been better not to have seen it," she suggested. "It might have been better to remember it as it was."
To this William did not reply.
There was something of the same decay in the compound where he had been born and which had been his home. It was not empty. A thin little missionary was there, a pallid man who came to the door of the mission house, a shadow of a man, William thought with contempt, a feeble small fellow to take his father's place! The little man looked at them with bewildered and spectacled eyes.
"This was Dr. Lane's house, I believe," William said, and did not tell him who he was.
"That was a long time ago," the mild man said.
"May we look over the house?" Emory asked. "We knew Dr. and Mrs. Lane."
"I suppose so—my wife isn't in just now—she's gone to the Bible women's meeting."
"Never mind," William said suddenly. "I have no desire to see the house."
They left at once and William, she divined, was thinking of his father. He thought a great deal of his father in those days in Peking—sometimes with the old bitterness but more often with a longing wonder at the happiness in which his father seemed to live.
"My father was anchored in his faith," William said. "I have often envied him his ability to believe."
Emory said at this moment what she had been thinking about for a long time. "I do think, William, that you ought to see a priest. A Catholic, if possible."
He turned upon her his dark look. "Why?" But she fancied he was not surprised.
She responded with her gaze of clear kindness. "I cannot give you peace," she said. "If peace is what you need—"
He denied this abruptly. "I don't need peace."
"Whatever it is you need," she amended.
He did not reply to this but she did not forget his silence. They left Peking soon after that day, and in a few weeks were in New York and William plunged into feverish work.
Left to herself, Emory went out more than she had before. Even she was getting restless. The world was so strange, so full of horrible possibilities!
At a cocktail party one day many months later Emory observed an unusual figure, and seeing it was reminded of the unforgotten conversation in Peking. A tall cassocked priest stood near the door. He had an angular worn face and quietly gazing at him as she drank tea instead of cocktails she saw his hands, worn and rough, tightly clasped before him. His hair was a dark auburn and his skin was florid. As though he felt her eyes, he looked at her. His eyes were very blue. She turned her head and at the same moment she felt hands upon her shoulders. Looking up then she saw Jeremy Cameron, and she smiled at him. "Jeremy, you wretch, you and Ruth haven't come near us since we came home!"
"Ruth is still at the shore with the children. She'll be back Monday. Here's someone who wants to meet you. Emory, this is Father Malone—my sister-in-law, Father, Lady Emory Hulme or Mrs. William Lane, as you please."
Jeremy had been drinking, she saw. The dark pupils of his eyes were huge and set in reddened whites and his thin smooth cheeks were flushed.
She turned to smile at Father Malone. He stooped over her hand. "It is your husband I really want to meet and this explains my presence at an occasion so strange to me," he said in a rugged voice. "I've just come from China, where I believe he was born."
"Oh, I'm glad." Genuine gladness indeed was in her voice. "Why not come home with me now? We can talk a little while before my husband comes in. He'll be late. We were in China, ourselves."
"I heard," Father Malone said simply.
Jeremy rocked back and forth on his heels. "William was looking at Father Malone's pictures today—wonderful pictures—people starving to death, somewhere in China of course—babies like dead mice, their arms and legs—wonderful. He hadn't time to meet Father Malone himself and turned him over to me. He wants the pictures, though."
"Famine," the priest said simply. "That's why I am here. I am sent to collect funds."
His dark eyes were magnetic. Emory found herself looking at him and then not looking away quickly enough. He did not mind how long she gazed at him, and there was no personal response from him to a beautiful woman.
"Do let's go." She got up impulsively.
The controlled grace of her movements was self-conscious and yet nonetheless graceful. They left in a few minutes, the priest a handsome yet ascetic shadow behind her, and in the comfortable soundproofed car, riding through the evening traffic in perfect quiet, she put her questions. Father Malone answered them with simplicity and frankness, or so she thought. Yes, he had been many years in China, not in Peking, or the big cities, but in his own mission in a country region. He was a country priest and had been twenty years there.
"You must have been very young when you first went."
Yes, he had been young, only a little more than twenty-five. He had gone to help an elder priest, who had died after a few years, of cholera, and then he had carried on.
"Do you feel your work is successful?"
"I do not think of success." His somber voice, expressive of any emotion one might choose to imagine, made music of every word. "In the long processes of the Church one man's work is only a link in the chain of eternity."
"I do believe," she said, with purposeful frankness, "that you have been sent to me at this particular moment. I will not pretend that I am a religious woman for by looking at me you, will see that I am not. But I love my husband and he needs something I cannot give him. He is a naturally religious man, and he does not know it. He has grown rich so fast. You know his father was a missionary."
"I do know," Father Malone said. "That is why I have come to him first—that and his great wealth."
"His father was a Protestant, of course," Emory went on. "I never knew him, but he has left an indelible impression upon William's soul. William, being a very clever man, can scarcely accept the sort of religion that his father had. He will need something much more subtle, if I may say so."
"The Church has everything for all souls," Father Malone said. His voice, so full of confidence, his mild and handsome profile gazing ahead into the turmoil of the crowded streets, renewed Emory's admiration without in the least moving her heart. But then, her heart knew no hungers.
The heavy car drew up at the house and the chauffeur sprang out and opened the door of the car. They mounted the marble steps. The evening air was sweet and cold, and the lights of the city were twinkling. At the top of the steps Emory touched the bell and upon impulse that seemed sudden she looked up at the tall priest.
"I'm very happy. I want my husband to be happy, too."
"Why not?" Father Malone replied. He smiled down upon her, celibate and monastic though he was, and by that smile he made himself her ally.
William, coming in later than he had said he would, paused as Henry took his things. He heard a man's voice.
"Who is here?" he demanded.
"A friend of madame's, sir. He's a priest, sir. She brought him home with her. He's to stay for dinner, sir."
Henry disappeared and William went quietly up the stairs. And why a priest? He was fearfully tired and wanted to be alone. The old sense of emptiness was creeping back into him again though he had been married so few years. He avoided knowing it. If Emory could not fill the emptiness then nowhere on earth could he find peace. He refused thought and began instead to worry about lesser matters. Jeremy, for example, getting drunk and coming into the office to announce loudly his disgust with his job and with everything and that he wouldn't resign and wanted to be fired! He would have to talk with Ruth as soon as she came back. She ought not to linger on at the seashore, leaving Jeremy at loose ends.
He shrugged his shoulders abruptly. Why should he, in his position, be troubled about anyone? The familiar hard surface crept over his mind and spirit and he proceeded to bathe and dress in his usual evening garments, laid out for him by his valet. He was hungry. The day at the office had been long and the proofs of his editorial more than usually full of mistakes. He would have to find another editor. It seemed stupid that his young men could not adjust to his demands. He kept them young, letting them go soon after thirty-five, because youth was essential to the style he had developed.
His mind, ranging among faces and men, lingered upon Seth James. He had not seen Seth for a long time, but he had kept within his knowledge all that Seth had done since the success of his play on Broadway. Seth had started another magazine which had failed. William's private scouts told him that Seth had lost more than a million dollars on it. Perhaps it was time to bring him back—if he wanted him. But could Seth be convinced? He might talk to Emory about it, get her, perhaps, to go after Seth. She had a sort of integrity which he could neither fathom nor reach.
He had not told her that a few days ago he had met Candace upon the street, and had hesitated, not knowing whether to speak or not. She had decided the matter quickly by putting out her gloved hand.
"William, surely you won't just pass without speaking?"
He took her hand, felt embarrassed, tried to smile. "I wasn't sure you'd want to."
"There is no reason why I wouldn't want to speak to you, William."
"How is your father?"
"Just letting himself get old—sleeping a good deal, a saintly stillness over him, all the time."
"I hope he doesn't dislike me?"
"He doesn't dislike anybody."
They stood between two passing streams of people and he was afraid one of the damned gossip columnists might see them together and put out a story in a newspaper or on the air. This was intolerable and so he had lifted his hat abruptly and left her. There was no reason to tell Emory. The meeting meant nothing.
When he was dressed the emptiness came over him again. It was more than emptiness. He felt a strange and puzzling gnawing of the heart which he could not explain. What was he doing that he should not be doing? Every success was in his possession. He had ceased to ask himself how much money he had. There was more than he could possibly spend with his decent and frugal tastes. His houses were finished and beautiful and to Emory he gave an income extravagantly large. Candace, too, he had not stinted and his sons both had had allowances beyond their needs. His yearly gift to his father's mission was a solid foundation upon which others built. For his mother he had arranged an annuity of ten thousand a year. He had done everything he ought to do.
He should perhaps have entered politics long ago, instead of building his newspapers. This thought, disturbing him very much, caused him to sit down in his leather easy chair and close his eyes. His small hairy hands gripped the carved ends of the hand rests. He should not have been content with the power of shaping the minds of people by choosing what pictures they should see, what news they should read, what ideas, in short, should be offered to their minds. This was only passive government. There was nothing stable in America. This country which William longed to love and did love with fear and anger and contempt, had no bedrock of class, no governing element such as England had. Wealth was the only vantage. William despised charm and knew that he had none of it. And yet without it, he knew, he could never have won, not in America, not in this, his own country. Think of that fellow in the White House! He gave up the notion of politics and opened his eyes. He could not descend to the sordid race. Besides, what if he had been defeated? Folly, folly! He was pre-eminent as he was and without a rival in sight. What more did he want than he had? He wanted to be satisfied with himself and he was not.
A tap at his door made him get up and go to the window. "Come in!"
"Madame asks if you are ready, sir," Henry said behind his back.
"I am coming down at once."
He passed the man and went down the wide curving stairs, comforted for the moment as he often was by the vista of his home, the huge beautiful rooms spreading from the great entrance hall. He ought indeed to be satisfied with himself. Roger Cameron had been satisfied with half of this. Scrambling up that cliff, those years ago, he had not dreamed of such a vista, all his own.
He crossed the hall and went into the drawing room at the right. A tall figure rose at his entrance and stood with clasped hands. Emory spoke from a low rose-red velvet chair.
"William, this is Father Malone. He was in your office today with some pictures and Jeremy brought him along to the cocktail party, and I brought him home to you."
The strong hands unclasped and the priest put out the right one, not speaking. William felt it powerfully about his own much smaller hand, and quickly withdrew it.
"I am sorry I was busy when you were announced in the office today," he said, looking away. He took a glass of sherry from a silver tray presented now by the butler.
Father Malone sat down. A perfect quiet pervaded his being and from this quiet he looked at William so steadily that William felt himself compelled to respond, and turning he looked down into the profoundly dark and deep-set eyes.
"The reason I brought him home," Emory went on, "is because Father Malone comes from some place quite near Peking and I thought you would enjoy one another."
William sat down. "Indeed?"
"Your father was a missionary."
"Yes."
"I, too, am a missionary," Father Malone said after a moment. "I have been recalled for a time to collect famine funds. I brought with me the pictures which you saw today. I hoped that you would want to print them for I am told your publications reach millions of Americans, and they might be moved to send me money for food."
"Thousands of pictures come to me every week," William said. "I may not be able to use many of yours. Besides, we have our own photographers who know exactly what I want."
"You do not feel moved to present the appeal for the starving?" The priest's deep voice was calm and inquiring.
"I hesitate to embark upon relief work," William replied. "One doubts the basic efficacy of it in a country so vast as China. Famine is endemic there, as I remember."
"You feel no duty toward those people?"
William looked at him again unwillingly. "Only in memory of my father."
"You deny the memory," Father Malone said. So positive was his voice that William was instantly angry.
"Dinner is served," the butler announced at the door.
They rose, Emory first in her rose and gray taffeta, and behind her Father Malone, stark and severe in his black garments, and William a little distance behind him. The priest's words had fallen upon his angry heart like a sword.
"You have been stifling your soul," Father Malone said to William Lane. He was very tired. The special mission which he had assumed as he came to know William was nearly completed. It had not been easy, far more difficult indeed than feeding the starving children and praying for the ignorant peasants who were his flock in China. The Church there was gracious to the ignorant. It did not expect a peasant to understand the mysteries. To come to Mass, to wear an amulet, to know the name of the Virgin and one or two saints was as much as he insisted upon in his village. Even confession he did not press, for how could an old man or even a young woman confess when they did not know sin? The knowledge of sin was for their children, the second generation, and in that knowledge it was his duty to instruct them. By the fifth generation he expected a priest. The Church was infinitely patient.
"You have denied your Lord," he said.
He had tarried for days in this vast and wicked city, for so he had felt he should do. Yet when he found that the wife of this rich and powerful man believed that her husband sought God, he had felt unable to undertake so vast a responsibility alone. He had gone immediately to his local superior, Monsignor John Lockhart, to ask for direction.
John Lockhart was an Englishman, a priest of high intellect and conviction, who might have become a Cardinal of the Church had he been ambitious. But he did not wish to enter into the higher arenas, where, he thought, though without disloyalty, the air was not so pure as it might have been. Princes of the Church were subject, perhaps, to some of the temptations of earthly kings. This did not keep him from believing that the Church was the best means yet devised and developed for the guidance and control of weak and faulty human nature. He listened carefully to the shabby priest from China who sat on the edge of his chair and talked diffidently about William Lane.
"A man stubborn in his own pride," Monsignor Lockhart said after listening. "Nevertheless he has seen religious righteousness in his father and he cannot forget it. He was reared with a conscience. He has repudiated it until now. As you have told me, you have had only to look at his face to see it tortures him."
"Does he know it?" Father Malone asked.
"No, and it is your duty to make it known to him," Monsignor replied.
Father Malone did not answer this. He continued to sit on the edge of his chair, his hands clasped in front of him in his habitual manner. He knew what he was, a missionary priest, a hewer of wood and a drawer of water in the palaces of the Church.
"In famine times I know that many souls are driven to the Church," Monsignor continued. "It is our duty to feed body and soul. But sometimes there is one man who can at a certain moment be worth more to the Church than ten thousand others, and William Lane is one of them. He is very powerful and he does not know what to do with his power. He seeks to direct but he himself needs direction. In his discontent he has married again, but he cannot be satisfied with women. His hunger is of the soul."
Father Malone had listened, and had prayed, when he was alone again, that he might see clearly what he ought to do. He did not presume to approach God directly with his own words, but while his lips murmured the beautiful Latin syllables his heart poured into them his own desire to draw to God this singular and powerful man. The task was not easy and he knew, in his humility, that he could not complete it. It would be necessary for some higher priest, some more astute mind, to fulfill the mission, perhaps the Monsignor himself. There were distances in William Lane that a common priest like himself could not reach, and depths from which he shrank.
"You have told me more than once that I have denied my Lord," William now said with some impatience. "I am not aware that I have done so."
Father Malone was alarmed at the fierceness of William's eyes, at the vehemence in his voice. He had lived long among a gentle people and he missed them. His soul loathed the fleshpots among which he sojourned. At Monsignor's command he had continued to accept William's hospitality and he had a room and a bath here in this velvet-lined house. The bed was soft and he could not sleep upon it, and at night he had at first laid himself upon the floor and even the floor was too soft with carpet and undercarpet. Then he found that the bathroom floor was of marble and upon that surface he laid himself and found it warmed with inner pipes. He longed for his earthen-floored cell and for the icy mornings of a northern Chinese winter and a bowl of millet gruel. The flash of silver and the smoke of hot meats upon the lace-covered table in this house filled him with a sense of sin. How could he speak of God here? And the woman, telling him again and again how much he did for her husband and all the time she herself took not one word of what he said to herself!
He went increasingly often to Monsignor for counsel and he had said on his last visit, only two days ago, "Would it not be well to separate the man from the luxury which surrounds him? How can we find his soul when it is sunk in the fleshpots?"
Monsignor had looked at him out of deep, shrewd eyes. "In what sense separate?" he inquired.
"William Lane is at heart an ascetic," Father Malone replied. "He possesses much, but he eats little and his ways are frugal. He does not drink much wine, he does not often smoke tobacco. We could make a priest out of him could we get him alone into the wilderness. If I took him back to my village, I could even entice him to love the people, which is the beginning of righteousness."
"To what end?" his superior inquired.
Father Malone was astonished. "To the end that his soul may be saved!"
Monsignor got up and walked about his library. It was a noble room, and the mahogany book shelves reached from floor to ceiling. He had the finest religious library in America and was among its most learned prelates, in spite of his lack of religious ambitions.
"You go beyond your duty," he said sharply. "I have told you only to awaken his soul."
"I have done so," Father Malone replied. He was almost as uneasy here as he was in William's house. It was not for him to question the ways of his superiors. The Holy Father himself lived in a great palace which was one of the wonders of the world. God used riches as well as poverty for His own glory, he reminded himself.
"Continue then until you receive my next instruction," Monsignor said.
So Father Malone had gone back to the rich house again. At this moment, however, when he sat alone with William in the silent opulent room, remote from any life he knew, he felt that the end of his work had surely come and that he must beg his superior to release him. He knew that William did deny his Lord, for he felt denial everywhere in this house, in William and in his wife and in the very existence of this place and in all it contained. But he could not explain how he felt this or why. Monsignor had not approved his speaking of poverty. Had he not received this disapproval he would have said earnestly to William, "You must give up all this and follow Christ." But he did not dare to say this. He felt puzzled and tired and in spite of constant refusal he knew that he had eaten too much and too richly. Sitting in a highback Jacobean chair which he chose because it alone had a hard wooden seat, he twisted his workworn hands.
"It is time for me to leave you," he said to William. "I have been detained by God to remind you of your father and of the land where you were born and to guide you to think of these things. Beyond that I am not able to go. I must commend you to Monsignor Lockhart, who is a wiser man in the Church than I am. I have no great learning. My books are fewer than a hundred. He has thousands of books upon his shelves and in many languages. He is continually in communication with those who know the Holy Father, whose face I shall never see."
William did not deny this. He had indeed been stirred to the bottom of his soul by Malone. He envied the priest his unmoving faith, his confidence in prayer, his conviction of duty, the same faith, confidence, and conviction which his own father had possessed. But William was not able to proceed beyond the impulse of envy and of longing. His spiritual hunger had been increased and not satisfied. His loneliness was more and not less.
"Perhaps you are right," he said. "Yet I am very grateful for what you have done."
"It is not I but God working through me."
"Then I thank God. Perhaps, in spite of not seeing it yet, my feet have, nevertheless, been set upon a path."
"Monsignor Lockhart will lead you the rest of the way," Father Malone replied.
Upon this they parted. In a short time Father Malone had packed his Chinese bag of split and woven rattan, and he refused the offer of William's car. "I must report to my superior," he said, "and it is only a short distance upon this same avenue. Let me walk. It will make me feel I am on my way home."
William was perceptive enough to know what he meant and he let him go.
When Emory came home in the late afternoon she missed at once the third presence in the house. She had been on an ordinary errand to have her hair dressed, and when Henry opened the door to her he told her that the master had not returned to his office. She found William in the rather small room which they used as a sitting room when they were alone. He was stretched upon a reclining chair, gazing into the coals of a dying fire. He had not put on the lights, and there was a strange atmosphere of life and death in the room. She touched the switch by the door and the wall lights flamed.
"William, are you ill?" she exclaimed.
"No," he replied. "I have been thinking all afternoon. Father Malone has gone."
"Gone?"
"He says he wants me to go directly now to Monsignor Lockhart. He thinks it is time."
She came to him and knelt at his side and put her hand on his that were folded across his body. "William, please do only what you wish!" she now said.
He moved his hands from under hers rather sharply. "No one can make me do otherwise!"
"But be sure that you know if they try."
"You don't flatter me, Emory. I am usually considered astute enough."
He was determined to be hurt and she refused to hurt him. "I'm being stupid." She got up and then sat down in a chair opposite him. "It's hot in here. Shan't I open the window?" The house with its central heating was always too hot for her English blood.
"I am not hot."
"I suppose it's because I have just come in from outside."
She sat still for a few minutes, and then stealing a look at William she grew alarmed at the whiteness of his face. She got up again and went to him and curled on the floor beside him. She took his hand and leaned her cheek against it and made to him a complaint she had never made before.
"You haven't loved me all the time Father Malone's been here." She put the palm of his hand against her soft red mouth.
Among the American women she was learning to know, there was shrewd interchange at once cynical and enjoyed by them. "You don't know your man until you've slept with him," was the common creed. They were all healthy handsome women, to whom chastity was not a jewel without price. Yet not one of them would have entertained the possibility of a lover, for their husbands were richer than potential lovers and men of position which they did not care to threaten. The difference between men, they frankly acknowledged, lay in their bank accounts rather than in their persons. They considered themselves exceedingly fortunate women and so they intended to live virtuously. But Emory was virtuous by nature.
She felt the palm under her lips tighten. It was impossible for William to speak of love. She crushed her mouth against his palm, tasting its flavor of soap and salt. If within a moment he did not respond she would laugh at herself and tease him for being so earnest about everything. "Don't be so serious, darling—let's go drown ourselves somewhere! Nobody will notice the difference and it would be fun. Something we've never done before!"
But tonight she would not need such nonsense. She recognized the familiar signs, the tightening of nerve and muscle, the response of his strangely awkward, rather short fingers. He sat up suddenly and drew her against him and she held her breath. He was always abrupt and unsharing but she was used to that now. He had to dominate her and though she had resisted this at first, now she no longer did so. Sex for a woman was nothing. It expressed no part of her being. It was an act of play, of symbolic yielding, a pleasant gesture, pleasing to receive and to give, a thing to forget, the preliminary to a possible experience of motherhood with which the man had little to do. She had decided against motherhood when she saw Will and Jerry. Candace had given William his sons and she divined that more sons would be meaningless for him and for her. With Cecil's death had gone any need for a son of her own. She divined also that William would care nothing for daughters.
"Lock the door," William commanded her.... She had a healthy body and she did not shrink from whatever William demanded. She accepted sex in exactly the same way that she enjoyed a cup of tea or a meal. There was nothing mysterious about it or even very interesting. What was interesting was William. She got to know him better in this brief occasional half hour than she could in a month of living. There was something cruel in him—no, not actually cruel, but he needed frightfully to be sure that he was right. Somewhere along the way of his childhood and his youth he had been so wounded in his self-love that now he knew best, he always knew best. And yet his self-confidence, his willfulness, his determination to make others obey him was not solid to the bottom of him. Sometimes when she had obeyed him utterly his command broke. He could not go on. He was not sure of himself. But why not? Who threatened him now?
So it happened tonight. In this quiet hour between day and night, when the servants were busy in the remote regions of the house, they had the complete privacy he demanded. Father Malone was gone. It could not have happened had he been in the house. And still William could not succeed. The fiasco came as it had sometimes before, though not always. Then why tonight?
She waited a moment to make sure that it was to be so, and then it was so. He lay back exhausted without fulfillment. She buried her head against him, and began stroking his hand gently. It was listless and he did not speak a word. He never did.
This went on for what seemed an endless time. The room grew darker. Somewhere, at last, far off, the gong rang warning that dinner was only half an hour off. She let his hand fall and felt a wave of relief. Better luck perhaps, next time!
"I think Father Malone was right," she said in her ordinary voice. "I do think you ought to go and see Monsignor Lockhart."
# 11
WHEN THE SECOND WORLD war broke out Clem made up his mind to ignore it. "Let her blaze," he told Henrietta in cosmic anger. "It's all got beyond me."
"Aren't you going to close the restaurants now?" Henrietta had asked when people were working again on war jobs.
"I've been thinking about that," Clem said. "I don't want to be in the restaurant business. I guess I'll let the fellows have them. They can set up for themselves somewhere or they can stay where they are. They've got to promise me, though, that they'll keep on giving free meals when necessary."
"Since they've made money, I imagine they won't mind that," Henrietta said. Chinese could always take care of themselves with ancestral prudence.
By that time the government had ordered surpluses given outright to hungry people. Nobody knew how much of this giving away was the fruit of a certain day when Clem at last sat with that fabulous man in the White House who could not stand up unless somebody helped him. Clem got on well with him. He tried to remember that the man behind the big desk covered with small objects was the President of the United States, but most of the time he forgot it. They talked all over the world. The man behind the desk showed extraordinary knowledge and also profound ignorance, and he did not care who knew it. Clem tried to tell him about China and then gave up. There was too much the man did not know. He knew as little about India, and believed that the only problem there was too many people, and Clem labored earnestly to make him see this was not true. India could produce plenty of food for many more people.
"China, for instance, is nearly self-supporting in food," Clem said. "She doesn't import anything hardly. She grows immense amounts of food."
"Seems to me I've heard of starving Chinese all my life," said the man with the big smile.
"That's because they need railroads and truck highways," Clem said. "They can't move surpluses. They starve in spots. It's the world situation in a big nutshell. Before you can have a steady peace, you've got to be able to move surpluses."
The war had broken out in China and in Europe and it meant that in China at least there would be fewer new highways than ever. Still the big man did not care much about China. That was to come later. Clem went away attracted and confounded. The big man didn't see the world as round. For him it was flat. He couldn't imagine the underneath. The whole world would have to blaze with war before the big man understood that the world was one big round globe.
It had never been easy for Clem to write letters but when he got home to Henrietta he began the series of letters which were his effort to educate the man who didn't know the world was round. Sometimes these letters were long but usually they were not. The big man never answered them or acknowledged them himself, but Clem hoped that he read them. In them he tried to put down all he knew, including excerpts from the letters which Yusan wrote him.
"Of course we ought to help lick the Japs in China now," Clem wrote, "but this is just the first step. As far as that goes the war really began when we let them have Manchuria. The next real job will come after the war when Chiang Kai-shek will have to hold his people together. It is easier for a soldier to keep on fighting than it is to get down to the necessary peace. It will be the Communists next, for sure, and that's what we have got to reckon with. My advice now is to give some little hint of friendship for the people of India so as to begin to win friendship from Asia. I know you don't want to get Winston worked up, but you could just say a word or two in the direction of India in your next fireside chat and this would please Indians by the millions as well as Chinese. If you would say you believe in the freedom of peoples but say it now, within this week, which is a time of crisis we don't know anything about over here, it would mean everything. Next month would be too late. They are all waiting."
Clem had bought his first radio especially to hear the President, but he did not say one word about India or the freedom of the peoples in his next fireside talk. The famous voice came richly over the wires. "My friends..." but it didn't reach as far as China or India or Indonesia. Clem listened to the last rousing words and shut off the radio and was gloomy for so long that Henrietta was worried. She and Clem were no longer young and she wished that he could stop his world-worrying. Other people would have to take over and if they didn't, it could not be helped. Clem's stomach had been better after the depression but this second World War was making it worse again.
When she said something like this to Clem he would not listen to her. "I'm used to my stomach by now, hon. It hasn't won out on me yet."
"You haven't won out either, Clem," she said sharply. "It's a continual struggle and you know it."
He grinned at her, although there was nothing cheerful to grin about. Pearl Harbor had done him as much damage internally as it had done the Hawaiian Islands and he did not dare to tell Henrietta that all his old symptoms had returned, and that he was afraid to eat.
When America had finally swung into war he offered himself as a supercook and was actually put in charge of the mess halls and kitchens of barracks near Dayton. While the war went on and he still continued his long-distance education of the White House, conducted without any response whatever, Clem made some thousands of American boys happy by excellent food and pleasant dining halls where they were allowed to smoke and where cages of singing canaries brightened up their meals. Outside the dining room Clem made the administration furious by the economies he suggested and even put into force so that his regiments, as he called them, became notorious or longed for, depending upon whether a man was brass or buttons.
Clem himself considered it piddling. He was marking time until the end of the war when he intended to marshal all his theories into one vast gospel and present them to the White House and then to the nations. He had long ago forgotten William's rebuff and he remembered now only the grace and kindness of William's wife, and he dreamed secretly without telling Henrietta that after the war was over he would go back to William, not this time to advocate a theory but with a formula in his hand, a formula for a food so cheap that until the world got its distribution fixed up, people could still be kept from starving.
He set up a small laboratory in the basement of the house and with Henrietta to help him with her knowledge of chemistry refurbished and brought up to date with some new books, he began to work with the best soybeans he could get, the beans that Chinese farmers grew for their own food. Clem planted these seeds and tended them like hothouse asparagus, and as the war continued his harvests grew until he had enough soybean meal to make real experiments possible. He and Henrietta ate one formula after another, and studied seasoning and spoilage.
"We ought to have a real food chemist," Henrietta told him on one of these days. "I don't know how to get the taste you want, Clem. I don't even know what it is."
"It's kind of like those meat rolls I used to eat at the Fongs'," Clem said dreamily.
"But you were a half-starved boy then and anything would have been wonderful," Henrietta suggested.
"Yes, but I never forget."
Clem never forgot anything. He did not forget how it had felt to be a half-starved boy and his unforgetting mind made him know how people anywhere felt and what they wanted. The man in the White House could have got from Clem an accurate temperature of most of the world's peoples in the crowded countries of Asia, but he did not know it, or even that he needed to know it. Meantime Clem had isolated himself from the war and was living ahead in the years after, when the new world would begin.
"War's nothing but an epidemic," he told Henrietta. "If you don't prevent it in time it comes and then you have to go through with it. I'm glad we have no children, hon."
"We might have had a girl," Henrietta said with a wry smile.
"No, I'm glad we haven't. She'd have been in love with a boy."
The long process whereby William Lane decided to become a Catholic was one of combined logic and faith. His conscience, always his most fretful member, had become irritated beyond endurance by the monstrosity of his success, which was now uncontrollable. He needed to do nothing except to read his newspapers critically and then keep or discharge his editors. From somewhere in his ancestry, distilled through generations of New England lawyers, preachers, and reformers, he had received the gift of the critical mind attuned to his times. Long ago he had become as independent as a feudal baron. His chain of newspapers rested upon the solid properties of his own printing presses, and these in turn were set upon the sure output of his paper mills, which in finality rested upon the firm foundations of timbered land, stretching in miles across spaces of the north, in Canada as well as in the United States. He was impervious to the dangers and restrictions possible even to him, as the war blazed separately first in Asia and then in Europe. A pity about Hitler! Had he been well advised, Hitler could have been a savior against communism, the final enemy.
Upon the frightful morning after the attack on Pearl Harbor, when his valet drew the window curtains, William was weighed down by the necessity of making up his mind quickly upon a new policy for his staff. People must know immediately where he stood.
As usual when he felt confusion he decided to talk with Monsignor and he telephoned before he got up.
"Yes, William?" Monsignor said over the telephone. After two years or so, they had come to this intimacy. "How can I help you?"
"I feel confused," William replied. "This war is bringing many problems. I must decide some of them today. I should like to talk with you this morning before I go to my office."
"I am at your disposal," the priest replied.
So William went immediately after he had eaten. Emory always breakfasted in her room, and he saw no one except servants whom he did not count. The morning sun shone down upon the magnificent granite Cathedral near the priest's private home. Both stood in the upper part of the city against a background of skyscrapers, and their solidity was reassuring. Even bombs could scarcely prevail against the aging gray structure of the Cathedral, as formidable as a medieval castle. He rang the bell at a Gothic doorway and was immediately admitted by a young priest who led him in silence over thick velvet carpets spread upon stone floors. There was not one moment of waiting. It was an atmosphere far more courteous than that of the White House, where last week William had gone to call upon the President, repressing his personal dislike to do his patriotic duty, and had been kept waiting for nearly a quarter of an hour. In the end Roosevelt, though jovial, had not seemed grateful for William's offer of help.
Monsignor's library was a beautiful room. The crimson of the carpets was repeated in the velvet hangings at the Gothic windows, and mahogany bookcases reached to the arched ceilings. The air was warm and slightly fragrant. There was a great deal of gold decoration centering in a massive crucifix that hung in a long alcove, but carried out also in wide gold satin bookmarks, in the frames of two or three fine paintings.
Monsignor Lockhart was a handsome man, erect and dignified. His features were clear and he had fine, deep-set eyes of a clear hard blue.
"Sit down, William," he said.
William sat down in a cushioned Gothic chair and began to consider his worries. There was nothing wrong in his daily life. He had no sins. He was entirely faithful to his wife and she to him. He knew that Emory, although she was a beautiful woman, was also fastidious, and he trusted her entirely and had never regretted his marriage. In her way she was his equal. There was no man in America above him in influence and few as rich. Had he been English he would of course have had a title. In that case he would have been poorer than he was, and Emory would not have enjoyed poverty. She had the finest jewels of any woman he knew. Emory in soft black chiffon, high at the neck and long sleeved, wearing her diamonds, was all he conceived of as beauty in woman. She had become a Catholic with him, and she liked wearing black chiffon and diamonds. With her dove-gray frocks she wore pearls.
No, his worries were entirely a matter of his responsibilities to the world, to the millions of people who looked at the pictures he alone chose and who read what he allowed to be printed. He wanted God's guidance for this enormous responsibility, and for the stewardship, too, of his vast wealth. He did not want to give his money to any cause or organizations which would not submit to his direction. Unless he directed, he could not be sure of the right use of his support. He never gave money to a person.
He made known his wish to do right, never stronger than now, in view of the mounting war, and Monsignor listened thoughtfully, his hands folded. They were much alike, these two men, and they knew it. Toward human beings they were almost equally paternal. Priest and man, they had already what this world could give.
"I grieve for the peoples," Monsignor Lockhart said. "In a war it is the innocent who suffer. The Church must assuage. You, William, must assuage. There will be much sorrow and death. You and I know how to find a comfort more profound, but the people are children and they must be comforted as children. God uses mysterious ways: Riches as well as poverty may serve Him. Continue as you have been doing, William. Do not try to take the people into high and difficult places, where they become afraid. Show them family life, show them love and kindness still alive, the ever protecting power of religion. The Church is eternal, surviving all wars, all catastrophes. Indeed, for us, God uses even wars and catastrophes. When men are afraid and distressed they come to the Church for shelter. So it will be again as it has always been."
There was an atmosphere of calm reassurance in all the priest said and did. William, listening to that voice, so richly humane, so profoundly dominant, was aware of comfort stealing upon his own soul. It was good to be told that he must do only what he had been doing, good to remember that he was part of the vast historic body of the Church, which continuing through the ages, must continue as long as man lived upon this earth. The order, the structure, the cell-to-cell relationships of the Church comforted him. Outside all was disorder and upheaval but within the Church each had his place and knew it.
The two men were in strange communion. Around them was the deep rich silence of this house, devoted, in its beauty, to God. Although the morning was cold, in the vast velvety room the atmosphere was tempered with warmth and the proper degree of humidity for the leather-bound volumes. Between the two men the fire burned. Under the high-carved mantelpiece the flames quivered intense and blue above a bed of hard coal. Each man admired the other, each knew that his heart was set upon the same goal, each felt the keen thrusting of the other's thought.
Between the two men was the still deeper bond of secret knowledge of each other. Though they spoke with reverence of the Church, each knew that the Church was a net as wide as the world, gathering into itself all men. It was the means of divine order, the opposite of man's chaos.
William sat in long silence. With the priest he felt no need of constant speech. The huge room was restful to him.
"This room is beautiful," he said at last. "I have often tried to analyze its effect upon me. I believe that order expresses the secret. Everything has its place and is in its place."
"Order is the secret of the universe," the priest replied. "Only within order can men function."
An hour later William went away. The wisdom he craved, the guidance he sought, the confirmation of himself and his own will, the approval of what he wanted to do, all these he had found as he always did. He felt strong and dominating and sure of himself. The ancient foundations held. The Church was founded upon a Rock.
He entered his office shortly before noon and the current Miss Smith waited in electric nervousness for the buzz upon her desk that was his summons. When she entered his office he was already sitting behind his semicircular desk and she approached him, trying to smile. It would have been easier if her office had opened to the side of the desk so that she might sit down quickly with her pencil and pad. But there was only one door into the vast imposing room and whoever entered must make the long approach to the spare stern figure sitting behind the semicircle. She reached it at last and drew out her hidden stool and sat down.
"Take a memorandum," William said. His voice was not in the least haughty and he would have been surprised to know that Miss Smith was afraid of him and often had a fit of crying after she left him.
"Memorandum to the editors," William said. "Begin! 'I have decided to support the British Empire. For the coming struggle, we must stand with England on the side of order in the world. Further details will follow within the next twenty-four hours.' That's all, Miss Smith. I do not wish to be interrupted until I call you."
He spent the rest of the day alone and in profound thought, writing slowly upon large sheets of heavy white paper. When he had finished his meditation his blueprints were clear. He had mapped out his plans for the next two years. At the end of two years the war should be won or at least victory plain. He felt strong and clear in mind, his pulse was firm, his heart at peace. An impulse of thankfulness welled up in him, and he bowed his head in one of his brief but frequent prayers. He had learned from Monsignor to find in solitary prayer a solace and a release.
He had a flash of intuition now while his head was bent upon his folded hands and his eyes closed. Across the world Chiang Kai-shek also prayed. William had chosen only last week a feature about China's strong man, and among the pictures was one of him at prayer. The Old Tiger, the Chinese called him, and it was a noble name. All strong men prayed. He could go to see the Old Tiger. A vague homesickness for China swept over his praying soul. Strong men ought to stand together. He would charter a plane, fly the Pacific, and visit China again in the person of that upstanding man.
Such thoughts mingled with his prayer without disturbing it, and when he had finished praying he touched the button of his telephone again. Miss Smith's voice answered, irritatingly weak. She would not last long, he thought with momentary contempt.
"I want to speak to Mrs. Lane," he commanded. A moment later the buzz told him that his wife waited.
"Emory? Have we anything on for tonight?"
"I half promised we'd go to that opening of the Picasso—"
"Cancel it! I feel that I need some relaxation in view of all that's ahead of me. Let's have dinner at the Waldorf—I'll order a table—and then we'll go to see something at the theater. What's that new musical? _Night in Peking?"_
"I'd enjoy that. And I'll get the tickets."
Emory's silvery voice was complacent and sweet. She was always ready to fall in with his wishes. When he had told her he wanted her to enter the Church with him, she had scarcely hesitated a moment.
"I've been thinking about it. I believe a solid religion will be good for you, William," she had said.
"What do you mean by that?" he had demanded.
"Life isn't enough for you," she had replied with her strange thoughtfulness. She seemed to think a good deal without letting her thoughts oppress her, or him.
"It will be good for you, too, I think," he had said.
"Why not?" she had replied, with one of her graceful smiles.
He was very effective that night. There was no fiasco whatever. He must have been successful at something or other at the office, Emory thought, one of his big plans, perhaps, which he would tell her about afterward. He was all of a piece, this man. Power flowed from him or, locked in him, wrecked his peace of mind and made him impotent. As always he made her his instrument and she did not rebel. Why, indeed, should she? He gave her all she wanted now in the world, which was luxury, which was beauty. Her wants were few but huge, and for beauty money was necessary, plenty of money, a mine of gold, the source inexhaustible. Only William possessed the golden touch nowadays. The old inherited capitalism was almost over, but he was the new capitalist. He had found the fresh source in the need of the people to be amused and to be led. And he led them—he led them into green pastures.
The staff perceived as soon as it congregated for the ferocity of the day's work that there was to be no idleness. William reached the office early and even the least of them understood at once that it was going to be one of his good days. Whatever thought of weariness, whatever listlessness of the night before that any one of them had felt was gone in the instant. Today the utmost would be demanded of them mingled with excitement and some terror. It was doubtful that they would all be at their jobs by night. On William's good days inevitably someone was fired. The weaker members decided not to go out to lunch. William himself never ate lunch.
"Miss Smith," William said, "give me all the recent dispatches from China. I want to study them."
This news from behind the circular desk was telegraphed through the offices and gusts of relief followed. Focus upon China meant focus upon Lemuel Barnard, who had just returned to make his report of the Chinese situation.
The first assistant editor thoughtfully started his search for Lem who at this time of the morning might still be anywhere but certainly not at his desk. Telephone messages began urgently though cautiously to permeate the city. The receptionist in the main entrance, Louise Henry, a pretty auburn-haired girl from Tennessee, stayed by the telephone as much as she dared. She had left Lem somewhere between midnight and dawn at a night club. Shortly before noon, she found him where no one expected him, in bed at his hotel room and asleep. Louise waked him.
"Lem, get over here quick. He's been studying your dispatches all morning!"
"Oh hell," Lem groaned and rolled out of bed.
At one o'clock William was delayed. Miss Smith brought in an envelope which she recognized as coming from her employer's divorced wife and which therefore she was not to open. She took it in at once to William, though fearful as she did so, for he had left orders that he was not to be disturbed. By then Lem was waiting out in the hall with Louise.
"I don't want to interrupt," Miss Smith began.
"Well, you have interrupted," William said.
"This—" Miss Smith faltered. She put the letter on the desk and went out.
William saw at once that it was from Candace. He did not immediately put down the map he was studying. Instead he discovered what he had been looking for, an old camel route from Peking into Sinkiang, and then he put down the map and took up the envelope. So far as he had any contact with Candace she had not changed. The heavy cream paper she always had used when he knew her as his wife, she continued to use. The fine gold lettering of the address simply carried the name Candace Lane instead of Mrs. William Lane. When he slit the envelope and took out the single sheet it contained, she began the letter as she usually did.
> Dear William,
>
> I have not written you for a good many months because until now there has been nothing to write. You hear from the boys regularly, I hope, and I live here in the same idle way. Today though there is something to write. I am going to be married again. I suppose this would not interest you, except I think I ought to tell you that I am going to marry Seth James. He was in love with me long ago when I was just a girl, before you and I were engaged. We began being friends again after Father died, and now it seems natural to go on into marriage. I expect to be happy. We shall keep on living here. Seth has always liked this house. But we'll have his town house, too. As you probably know, his paper failed, and he lost so much money that he has only enough to live on now and not enough to venture into anything else except maybe another play. But he says he will enjoy just living here with me. We will be married on Christmas Eve. Will and Jerry approve, by the way. It's sweet of them.
>
> Good-by, William
>
> CANDACE
The letter was so like her that for a moment William felt an amazing twinge of the heart. Candace was a good woman, childish but good. He had an envious reverence for sheer goodness, the quality his father had possessed in purity, and which he sometimes longed to know that he had. This longing he hid in the secret darkness of his own heart, among those shadows of his being which no one had ever penetrated, even Emory, for whom he felt something more near to admiration than he had ever felt toward any human person. She met him well at every point of his being. Her mind was quicker than his own and he suspected, without ever saying so, that it was more profound. She filled his house with music. Yet, though quite independent of him, she never talked too much, she never led in any conversation when he was present, she deferred to him not with malice as so many women did to men, not with the ostentation which made a mockery of deference. He believed that she admired him, too, and this gave him confidence in himself and in her, although her admiration was not flat and without criticism as Candace's had been. Yet even Emory did not have the pure goodness of which he had been conscious in his father and now perceived unwillingly in Candace.
His eye fell on the letter again. Christmas Eve? He was leaving for China the day after Christmas. This made him remember Lem Barnard. He buzzed long and steadily until Miss Smith came to the door, her pale eyes popped in the way he intensely disliked.
"Tell Barnard to come here," he commanded. "I suppose he's about the office?"
"Oh yes, sir, he's been here for hours—" She liked Lem, as everybody did.
William did not answer this. He frowned unconsciously and drummed his fingers upon the table. Within fifty seconds Lem Barnard shambled in, a huge lumbering fellow, overweight, and wearing as usual a dirty tweed suit. A button was gone from the coat and he needed a haircut.
"Sit down, Lem," William said. He opened a folder on the desk before him. "I have been reading over your recent dispatches. China is going to be very important to us now. We have to have a policy, well defined and clear to everybody. There must be no confusion between editors and reporters. You are to find the sort of news that fits our policy."
The veins on Lem's temples swelled slightly but William did not look at him. He went on, ruffling the edges of the typed pages as he did so.
"These reports you've sent for the last three months have been very troublesome. I've had to go over everything myself. There has been little I could use. This is not the time, let me tell you, to bring back gossip and rumors about the Chiangs—either husband or wife."
Lem exploded, "I've only told you what Chinese people themselves are saying."
"I don't care what Chinese people are saying," William retorted. "I never care what any people say. I am interested in telling them what to say."
He tapped the sheets with the tips of his ringers. "If I were interested in what people say my papers would soon degenerate to gossip sheets. Do you know why they succeed? Because they tell people what to think! You're clever, Lem, but you aren't clever enough. People don't care to read what they already think or what any people think—they know all that well enough. They want to know what they ought to think. It is a spiritual desire, deep in the heart of mankind."
He stopped and surveyed Lem, sitting huge and gross upon a straight-backed wooden chair. Lem overflowed the narrow seat and it was obvious from his clouded eyes and purplish cheeks that he ate and drank too much wherever he was. He was a disgusting sight.
"Man is a spiritual being," William said sternly. His enunciation was incisively clear. "Man seeks truth, he wants divine guidance, he craves security of soul. In all your dispatches remember that, if you please."
Lem swallowed once again his desire to fire himself, to bawl at William, to cry and howl. He could not afford it. His wife was in an expensive insane asylum. He bit his tongue for an instant and tasted the salt of his own blood. "Just what impression do you want me to give?" he then inquired in a sultry, gentle voice.
"Our people will now want to believe in the Chinese," William said. "They will want to trust the Chinese leadership."
Lem closed his bloodshot eyes. Against the lids he always saw Chinese faces, the starving, the homeless. War had been going on in China already for five years but nobody here had taken it seriously. Even the Chief here couldn't seem to believe it. Then he thought of his poor wife again, steadily and for a whole minute. Whenever he got angry with William he thought about her. He had been happy with her for two years and she had gone everywhere with him in China. He had met her there in Shanghai, a beautiful White Russian girl, and he had suspected there were things she had never told him and never could tell him. But she had been a wonderful wife and had spoiled him for anybody else.
One morning when he had wakened in the old Cathay Hotel, Lem had found her bending over him with his old-fashioned razor, and he had known that she was about to kill him. He had one instant of horror and then he saw that of course she was mad. She had never been sane since. He had brought her to America himself, sleeping neither by night or day. She tried to kill anybody who was with her and he could leave her with no one. He put her into an asylum near San Francisco. She never knew him when he went to see her. She always called him something else, names of men he had never heard of. But the bills were terrible every month and if he couldn't pay they would throw her out. It was not every place that would take such a violent case, they told him.
He had to stop seeing the Chinese when he shut his eyes. He had to see just Anastasie. He opened his eyes and said to William in the gentle and sultry voice, "Chief, I wish you'd go to China yourself. I wish you'd just go and see. You haven't been there for a long time. You ought to go and see what it's like now. Then you'd know—"
"I have already decided to go," William replied. "I am going to see the Old Tiger."
Chungking was a city set upon a hill. The sluggish yellow waters of the river wound around it and the tile-colored flights of steps led upward. There was nothing about it that was like Peking. Everything was at once familiar and strange. There were no palaces, no shining roofs, no dignity of marble archways and wide streets. The streets were crowded between gray-brick houses and fog-dampened walls. The cobblestones were slippery with water and slimy with filth. The people were grim-faced with continuing war and constant bombing. They did not look like the tall handsome people of the north. William was alarmed and dismayed when he thought of these people as the allies of America. What had they to give as allies? They were a danger and a liability. Yet Chiang must be held, he must be compelled, he must be supported.
The American car driven by a uniformed Chinese carried him at once to the Old Tiger's house outside the city. It was reassuring to enter something that did not look like a hovel. The air was chill and damp, as everything was, but from the hall he was led into a square room where a fire blazed.
"Please sit down," the manservant said in Chinese.
The words smote William's ear with strange accustom. He had not spoken a single Chinese word for years, but the language lay in his memory. He felt syllables rise to his tongue. Perhaps he would be able to speak with Chiang in his own language. The Old Tiger spoke no English. No one knew how much he understood—probably more than he was willing for anyone to know.
The door opened and he looked up. It was not the Tiger who stood there, but a woman, slender and beautiful, her great eyes filled with ready pathos, her exquisite mouth sad. She put out both her hands.
"Mr. Lane. You are America, coming to our aid at last!"
He felt her soft feverish palms against his and was speechless. He did not know what to do with a lovely Chinese woman, one who looked so young, who spoke English naturally. He had never seen this sort of Chinese woman. The ones in Peking had bound feet, unless they were Manchu, but Chinese and Manchu alike they had been alien to him, except the old amah who had been only a servant—and except the Empress.
This beautiful woman with imperial grace sat down and bade him by a gesture to be seated.
"My husband is delayed but only for a moment. We have had bad news from the front. Of course, now everything will be righted, since America is joining us. I grieve for the sad event of Pearl Harbor, but, really, I do believe it was necessary to awaken the American people to our world danger. I do not think only of China—I think of the world. We must all think of the world."
The door opened again and she broke off. A slender Chinese man in a long robe came in. It was the Old Tiger. Impossible indeed for anyone else to have those bold black eyes, that stubborn mouth! But he looked fragile. Was this the man who for fifteen years had conquered warlords and killed Communists? The Tiger put out his hand and withdrew it quickly as though he hated the touch of another's hand, and the act revealed him an old-fashioned Chinese, unwillingly yielding to a foreign custom. With an abrupt gesture he motioned to William to sit down again and himself took a chair far from the fire.
"Does this American speak Chinese?" he inquired of his wife.
"How can he?" she replied.
"I must confess that I understand a little, at least," William said. "My childhood was spent in Peking."
The Old Tiger nodded vigorously. "Good—good!" His voice was high and thin. When he spoke to his soldiers he was forced to shriek.
William contemplated his ally, this bony bald-headed man who was the master of millions of Chinese. Tiger was a good name for him. In repose he looked like a monster cat, soft and safe, except for the eyes where ferocious temper smoldered. He was old China, he hated the new, he was rooted in the past. Enough of his own childhood knowledge remained with William so that he knew exactly where the Tiger belonged. Had there been no revolution among the Chinese people he would have ascended the Dragon Throne and become a strong successor to the Old Buddha. He would have made a spectacular figure there, wrapped in gold-embroidered imperial robes, the Son of Heaven. And the Chinese people, William thought, would have been better off. What were they now but a scattered head? People needed to worship and when they were given no god, they made themselves a golden calf. There was tragedy in this man, deprived of his throne because of the age in which he had been born. A strange respectful tenderness crept into William's mood. He leaned toward the Old Tiger.
"I have come here to know how we can help you. There are two ways in which I myself can be of some use. I can influence millions of people. I can tell them—whatever you want me to tell them. I can also report to my government."
He spoke in English and the beautiful woman translated rapidly into a Chinese so simple that he could understand it. The Old Tiger nodded his head and repeated the short word signifying good, _"Hao_ — _hao_ — _"_ It was almost a purr. Not the soft purr of a cat, but the stiff, throaty rasp of a wild beast.
The beautiful woman seemingly effaced herself between the two men. She became an instrument, mild, almost shy. William all but forgot her as he pressed his arguments with the Tiger. But she was neither mild nor shy. A supreme actress by natural gift, she took his English words and remolded them into her fluent Chinese, stressing this word, muting that. When she perceived that he understood something of what she was saying, she varied her dialect slightly, slipping into a sort of Fukienese, excusing herself with adroitness.
"My husband comes from Fukien, and he understands that language better than Mandarin. It is essential that he grasp your every word."
William could make nothing thereafter of what she said. He did not want to believe that she added meanings of her own. There was no reason why she should. He was ready for the utmost gift.
One hour, two hours went by. Suddenly the Tiger stood up.
_"Hao!"_ he cried in his thin sharp voice. "It is all good. We will do these things. I will command my men. I shall not rest until the yellow devils are driven into the sea."
He folded his hands, this time without pretense of foreign custom, nodded twice to William, and went out of the room, his step silent and swift.
William was left with the beautiful woman. She put one soft pale hand upon his sleeve. "Dear Mr. Lane, your coming is an answer to prayer. I believe that. I believe so much in prayer, don't you? Every morning my husband and I pray together."
Tears came to her eyes and she took a little lace handkerchief from her sleeve and wiped them away. "You know China." Her voice was a whisper now, broken with her tears. "I can speak to you. You see my husband. He is so strong, so good, he is really good. He wants to save our people not only from the present enemy but from those who are far worse. You understand me, Mr. Lane. I am sure you do. But my husband must be helped. He has not had the advantage of education. He has many impulses. I try to control them through praying with him, Mr. Lane. What I cannot do, God will do."
William listened with rising sympathy.
"You have a very responsible work to do," he said. "Perhaps you are in the key position of the whole world."
His voice was grave and he meant it to be so, and she looked at him sorrowfully. Her big black eyes were shining and the tears were gone. Her hands were outstretched to him again.
"You must help me, promise me you will help me!"
He took her hands in his own. "I promise."
A week later, after incessant flying, from the dried sands of the northwest to the green provinces of the south, hours broken only by descents into cities where he sat out long feasts given in his honor, he went southward and then across the mountains and seas homeward. Wherever he went the beautiful woman had gone with him, and with them was always a third, a general usually, whom they picked up from the region and who could give them the latest news of the war. She translated for William as she had for the Tiger, giving him a continuing drama of a brave poverty-stricken people, patriots who wanted only guns in their hands, a few tanks and planes, to become invincible.
"Like your own Washington," she urged. "Like Jefferson, like Lincoln!"
He might have distrusted her eagerness, but she was always ahead of his mood. She knew when to let tears fill her eyes, but she knew also when to make her eyes hard and her voice firm. She knew when to show anger at a subordinate, when to be a queen and when to be a woman. Watching her he felt a new regret that the Phoenix Throne, too, had been destroyed. She would have made an empress fit to sit beside the Tiger on the Dragon Throne. People feared her, that he perceived, and he admired her for it. There must always be some whom the people fear.
At the end of the week he left convinced that because of her it was safe to uphold the Old Tiger. Without her there might be treachery; with her there was no danger. When they parted at the final airport she used her tears again.
"Dear America," she breathed. "Give her my love. Give everybody my love! Tell them I spend my life to teach my people the lessons that I was taught over there!"
He reached Washington exactly on schedule and made his report, and took the next plane home. It was snowing softly when he got out of the plane. The chauffeur was there to meet him. When he stepped into the car he found Emory, looking very pretty in a silver-gray frock and hat.
"This is good of you, Emory," he exclaimed.
"Not good of me. I've missed you terribly."
He crushed her shoulders in his arm and kissed her. She smelled of a delicate perfume, clean and warm, and he was grateful for all that was his, his wife, his home, his business, his country.
"I'm glad to be back. China is hell now."
"Is it, William? Then do you feel your trip was wasted?"
"No, far from it. I made them feel that America is behind them. I made them promises that I must see fulfilled. My work is cut out for me, Emory, I can tell you. I've got to shape public opinion to support those two people who are all that stand between us and defeat in Asia."
"Don't tell me now, William. You look fearfully tired."
"I hope we haven't any guests tonight."
"No, of course not. Just you and me."
He sighed and relaxed as much as he could. Everything had a new meaning for him. He felt as he never had before the value of being an American. The big car gliding over the great highways, the smokestacks of the factories, the lifted outlines of the city beyond, this could only be America. If China was hell, this was heaven, and it was his own. Nothing must be allowed to destroy it or bring it down to dust, now or ever. Holding Emory's hand in his, he dedicated himself afresh and with all his heart to his own country.
Upon reflection, even after a night's sleep, William felt that his mission to China had been a successful one. He had performed it in the quiet private way he liked to do large things, simply flying across the world alone in a plane for which he had paid a fabulous sum. The money was spent as he liked to spend money, by himself alone, for an end chosen by him but which would affect the world. The world knew nothing of it and would never acknowledge its debt to him, perhaps, while he lived. But some day, when historians were able to penetrate the shades of the past, they would see that through him, perhaps above all men, the war which might have been lost was won. Let others pour their energies upon the small tormented countries of Europe. He would save China, and by saving that vast territory the enemy would be foiled. He commanded Emory to invite no guests, accept no invitations. For two weeks he must stay at the office, coming home only to sleep. During that time he would give directives to his entire staff. Those who could not obey with efficiency he would discharge at once. His whole organization must concentrate now upon his directives. Techniques must be worked out for the papers, compelling simplicity, subtle argument, plausible presentation, every visual aid, every mental persuasion.
At the end of the first day he fired four persons, among them Miss Smith and Lem Barnard. Miss Smith was nobody. He ordered the office manager to have another ready for his dictation tomorrow morning. But Lem was difficult to replace. The Chinese would not tell a foreigner things unless he had charm, although charm was something William did not care for in his own office. It was then that he thought of Jeremy. Jeremy might do very well with the beautiful woman, even with the Old Tiger, if he were accompanied by someone to buy his tickets and take care of his baggage and see that he got his stuff on time. Besides, it would move Jeremy out of the office. When he did not show up it was a bad example. Acting instantly, with that abrupt complete co-ordination which was the source of his extraordinary energy, he pressed a buzzer.
It was near the end of the day and there was a slight delay, which made the blood swell into his high forehead. The delay, it seemed, when he demanded to know the reason, was because Miss Smith had not waited for his going home as she should have done. She had gone for her check at once and had actually left half an hour ago. He was tempted to fire the office manager but was too impatient to stop for it. In a few minutes he heard Ruth's sweet, somewhat childish voice. It sounded unusually faint.
"Ruth—that you?"
"Oh, William." Her voice was stronger. "How wonderful to hear you!"
"Jeremy there?"
"No—he isn't—yet, William."
"Where is he? He wasn't in the office."
"William, he isn't—he's not quite well. I think he'll be back in a day or so." She had begged Emory not to tell but maybe she had been wrong. Maybe William had to know.
"I have a job for him if he can get here by tomorrow. How do you think he'd like to go to China for me, as my personal representative?"
To his astonishment William heard his sister sob. He was fond of Ruth without having any respect for her, because she depended on him. Something was wrong with her marriage, of course, but he had never cared to go into it. Personal things took too much time and every hour counted in these terrible days. Now he had to inquire.
"What's wrong?"
"Oh William, I'm afraid you have to know. I didn't want to bother you. Jeremy is in a sanitarium."
"What sort? Is he sick?"
"Oh, William, no! Well, yes, I suppose it is a sickness. He was drinking too much and after you left he—Oh dear, he just went to pieces!"
"Nobody told me."
"I didn't want them to. I kept hoping he'd..."
He thought quickly while her voice babbled into his unheeding ear. This would give him the excuse to end everything with Jeremy. He would treat it as an illness.
"Ruth, I wish you would stop crying. I want you to know that I feel very sorry and I want to help you. I am going to give Jeremy unlimited leave of absence. He doesn't need to feel that he has to come back at all. But I want you to be independent. He wouldn't take a pension from me, of course, but I am going to set up a trust for you and the girls. Then whatever happens to him you'll be safe."
"Oh, William, darling—" her voice, still half sobbing, was breathless. "I wouldn't think of—"
"Be sure he stays there long enough to get in good shape and let me know when he comes home. We'll get together. Good-by. I'm frightfully busy—"
He thought for a moment and decided to send Barney Chester to China. He was a smart young Harvard man, only a few years out of college. Barney would listen to him.
He rose, refusing to acknowledge weariness, and went down the elevator to his waiting car. It was nearly ten o'clock and snow was falling. Sitting in the darkness of his car, staring steadfastly ahead, he saw the snow fly at him in little daggers of silver against the windshield. Around him were the darkness and the cold, the people still plodding along the wet streets, their heads held down against the wind. But he sat in warmth and safety, secure in himself and his possessions. All that he was he had made himself and all that he possessed he had earned. He had come from China, obscure and unknown, a shy and gawky youth, and what he was he had achieved without help. Yet America had given him opportunity. In England his birth alone would have condemned him. Even a title could not have hidden it. He smiled against the darting silver daggers which could not reach him. Here people had forgotten where he was born and who his father was. Where could that happen but in America?...
In the morning he woke inexplicably depressed again. There was no reason for it, except, he decided, that his conscience was stirring because he had not told Monsignor Lockhart about China. He had not even called him on the telephone, afraid that he would be tempted by the priest's quiet voice to yield time he could not spare. It was not as if he needed counsel. He had already determined what he must do. Now, however, there was no reason why he should not allow himself the luxury of some hours of spiritual communion.
This musing took place long before his usual hour for rising, but he felt wakeful and he took the receiver from the telephone at his bedside and called for Monsignor Lockhart.
The priest's voice came as usual, "I am here, William."
"I have wanted to see you ever since I returned, Monsignor, but you understand."
"Always."
"I count on that. But this morning?"
"Whenever you wish. I am already in my study."
He had planned to go back to sleep. It was still dark. Yet it might be interesting and even stimulating to get up and make his own way by foot to that huge gold-lit room. Their minds would be clear and quick.
In twenty minutes he was walking over fresh snow on the streets. He had never been out at this hour and the city seemed strange to him. The people he was accustomed to see were still in their beds. But the streets were not entirely empty, especially the side street he took from one avenue to the next. Two or three people were there, slouching along, one a woman who passed him and then stopped when an old man whose face he could just see in the approaching dawn held out a filthy hand without speaking. William went on. He made it a habit never to see an outstretched hand. His generous check went annually to the Community Chest.
"A cup o'cawfee fer Gawd's sake," the old man muttered.
William went on and the dirty hand brushed his arm and fell.
"Damned capitalist!" the woman shouted at his back. "Wants us to starve!"
A policeman suddenly rounded the corner.
"Did I hear somethin', sir?" he inquired.
William considered for a moment whether he should nod in the direction of the woman and then decided that he would ignore her.
"Nothing, except that old man asking for a drink."
"They will do it," the policeman said apologetically.
William gave the slightest inclination of his head and went on. Five minutes later he was inside the priest's warm and handsome home.
"You look hopeful," Monsignor Lockhart said.
"I do not feel hopeful at all," William retorted.
He finished a good English breakfast, while he talked, kidneys and bacon and buttered toast with marmalade. The coffee was American and delicious. A man came in and took the silver tray away, and closed the door softly.
"Yet I feel hopefulness in you," Monsignor repeated.
"I am hopeful to the extent of thinking that it is possible to hold China. It is my belief that we should allow England to take the lead in Europe but we must take the lead in Asia, now and after the war. Since only China is a free country, it is there we must concentrate our power."
"Very sound," Monsignor said. "I take it you do not mean permanent power."
"Certainly not permanent in the sense of eternal," William agreed. "I hope a complete American victory will have been won somewhere this side of eternity."
Monsignor's face was benign, although he wore this morning a lean weary beauty which showed hours of thought and perhaps prayer. William allowed himself a moment's wonder at this man who attracted him so much.
"You are tired," he said abruptly.
The priest looked startled and then his face closed. "If I am tired I am unworthy of my faith. It is true that the Church has great and new problems. In Europe our priests are facing oppression which we have never known, never in our agelong history. The gravest reports come to me from Austria. We have reached the age of anti-Christ. There is a demon in the people."
"Then it is no private ill that I see in your face?" William said.
Monsignor Lockhart's fine brows drew down. "What private ill is it possible for me to have?" he retorted. "The affliction of the Church is my affliction. I have no other."
William gazed at him, forgetful for the moment of their friendship. Monsignor seemed suddenly remote and cold. He was reminded of the temples of his childhood, where the gods sat aloof. No, it was not a god of whom he thought. It was the palace and the Old Buddha again, looking down upon him, a foreign child.
Monsignor dropped his lids. "We understand each other. Let us proceed from day to day, watchful of each hour's history."
He rose for the first time without waiting for William to signify that he was ready to go, and he put out his hand in the gesture of blessing. A deeper gravity came over his stern face. "Many are called but few are chosen," he said simply, and making the sign of the cross upon his own breast he left the room.
Throughout the day William carried with him the vague alarm of the priest's words, holding it upon the fringes of his mind.
He buzzed sharply for the new Miss Smith and did not look up when she came in.
"Dictation," he said.
He dictated steadily for an hour, letters, finally a long directive for Barney Chester. Then he dismissed Miss Smith and buzzed for his news editor.
"That you, Barney? Come to my office. I'm sending you to China immediately as my personal representative."
He spent the next two hours outlining to a silent and rather terrified young man exactly what he expected him to do in China.
"In short," he concluded at the end of the two hours, "I shall expect from you the most detailed reports of what American diplomacy is doing, in order that I may be kept informed here at home. At the same time I expect you to maintain confidential relationship with the Old Tiger and with—her."
"Yes, sir," Barney Chester said. He was a pale dark young man, very slender and smart. William liked all his young men to look smart. Actually Barney had a somewhat soft heart which he daily denied. Certainly before the stern, gray-faced man behind the circular desk he would have been alarmed to allow the slightest hint of a heart to escape him. This was the best paying job a man of his age could have anywhere in the country, anywhere in the world, probably. Lane paid his men well and worked them hard. He wished that his wife Peggy were not expecting their second baby. He had counted on asking for his delayed vacation when the time came so that he could take care of Barney Junior. There was no one to look after him, except a servant. It did not occur to him to mention such humble difficulties to William, who was still giving orders.
"Be ready to leave day after tomorrow. I'll see that you get priority on the plane."
"Yes, sir," Barney said.
# 12
IN THE HOUSE NOW grown shabby with the years and which Henrietta never thought of renovating, Clem sat reading the newspapers. The season was summer, the first summer after the war ended. Clem had barely survived the atomic bombs dropped upon two Japanese cities. Like many other Americans, he did not know that atomic bombs existed, until on the fifth day of August, a year ago, he had opened a newspaper, to discover with agony and actual tears that the bomb had already been dropped and hundreds of thousands of people he had never seen were killed. Although he, like other Americans, except the handful, had nothing to do with this act, he felt it was his fault. He got up blindly, the tears running down his thin cheeks, and went to find Henrietta. When he found her upstairs making the bed, he had been unable to speak for weeping. He could only hold out the newspaper, pointing at the headlines. When she saw what had happened she put her arms around him and the two of them stood weeping together, in shame and fear for what had been done.
For weeks after that Clem had been so nearly ill that she told Bump to trouble him about nothing. Clem asked very few questions any more. He was working with all his diminishing energy upon The Food, and he steadfastly refused to see a doctor or have any X-rays taken of his now permanently rebellious digestive organs.
"Don't bother me, hon." This was his reply to Henrietta's pleas and despairs.
The big man in the White House was dead and a little man had taken his place. Clem went to see him immediately to preach for the last time his human gospel of food for the starving. The little man twinkled and smiled and took time to describe the United Nations plan for world food and somehow sent Clem away thinking that he had converted a President of the United States, but nothing happened.
In the spring Clem had talked of going to the San Francisco Conference to explain about how the starving people of the world must be fed if things were to go right. The Communists mustn't be the ones to get the upper hand, but they would unless people had food to eat.
Henrietta had persuaded him against going. She knew now that people even in New Point were laughing at Clem. He was called crazy, a fanatic, nobody listened to a man who had spent his life on one idea.
She hated people because they were laughing at Clem. She drew him into their house, kept him busy, worked with him on his formula, anything to shield him from the cruel laughter of people who were not fit to tie his shoelaces.
On this summer morning when she was getting the breakfast dishes washed he sat reading the paper in the kitchen. Suddenly she heard him cry out.
"Hon!"
"Yes, Clem?"
"We've lost the war!"
"What on earth do you mean? The war is over."
She left the dishpan, her hands soapy wet, and stood reading over his shoulder.
"We've said we ain't going to help the subject peoples. It's the beginning of a third World War."
"Oh Clem, it's not as bad as that!"
"It is. They're all looking at San Francisco and what we've said there can't be unsaid. 'There comes a tide in the affairs of men...' "
He got up abruptly and went downstairs into his laboratory and she went on washing the dishes.
It was not until March, 1950 that Clem went to see William for the third and last time. By then so much of what he had foreseen had already come to pass that he thought he could convince William. Surely now he would believe that Clem was right. The Communists were ruling China and people were starving again by the tens of millions. Yusan was able to get word out about it. Old Mr. and Mrs. Fong were dead. Yusan was the head of the family. Peking was full of Russians all giving advice. Meanwhile Manchurian food was being traded for machinery.
"If America could get food to us—" Yusan wrote. The letter was on one tiny slip of paper in a small filthy envelope without a stamp. Mr. Kwok, now the head of a prosperous restaurant in New York, had brought it himself to Clem, and Clem had gone back with him to New York without telling Henrietta that he had made up his mind to go to William for the last time and beg him to tell the Americans that maybe they could still save China and the world if they would only understand....
Three days later Henrietta saw Clem coming up the brick path to the house, dragging his pasteboard suitcase. He could not reach the door. She saw him crumple upon the walk and she ran outdoors and lifted him up.
He had not fainted, he was conscious.
"My legs just gave up, hon," he whispered.
"You get in here and go to bed and stay there," she cried, fierce with love.
But nobody could keep him in bed. He would not go to the hospital yet, he told Dr. Wood. Now more than ever he must finish his formula, now that William wouldn't listen to him. So Henrietta heard how Clem had gone to William and how been denied.
"I'm just tired for once," Clem said.
He was up again in a few days and at his formula again, experimenting over the gas ring with a mixture of dried milk and beans, fortified with minerals and shredded potato. Henrietta did not cross him in anything now. There was no use in pretending that he was not ill, but she was helpless. Clem would not have the doctor.
It became a race. He almost stopped eating and drinking and she kept at his side a cup of tea into which she slipped a beaten egg and a little sugar. He drank this slowly, a sip now and then, and so sustained his life.
By summer they both saw he could not win. One morning he was struggling to get out of bed. His nightshirt fell away from his neck, hollowed into triangular cavities. His ears looked enormous, his eyes were sick.
"Clem," she cried. "You've got to think of me for once." It was her last appeal.
"Don't I think of you, hon?"
The strength was gone even from his voice. It sounded empty and ghostlike.
"You aren't getting up," she said. "You're staying right there until I can get Doctor Wood."
He sank back on the pillow, trying to smile. "You've got me—down," he whispered.
She made haste then to the telephone, and found the doctor at his breakfast.
"I'll come as soon as I—"
"No, you'll come now," she shrieked. "You'll come right now, without one moment's delay! I think he's dying."
She flew back to the bedside, the wide old-fashioned double bed where they had slept side by side in the years since she had given up everything to be his wife. He was lying just as she had left him but when she came in he opened his eyes drowsily and smiled.
"The doctor is coming right over, Clem. Don't go to sleep."
"No—I don't want to."
They stayed in silence for a moment, she holding one of his bony hands between hers. No use wasting his strength in talk!
But he began to talk. "Hon—the formula as far as I've gone—"
"Please, Clem."
"Let me tell you—it's all written down on that little pad in the upper right hand pigeonhole of my old desk. Hon—if I can't finish it—"
"Of course you can finish it, Clem. You just won't rest long enough. I'm going to take you to California, that's what I'm going to do...."
She was talking to keep him quiet and he knew it. As soon as she paused he began again.
"I think I've made a mistake using the dried milk, hon. There'll be people in China, for instance, who won't like the taste of milk. I don't know why I didn't think of that before. I ought to have, growing up in China—"
He stopped suddenly and looked at her in terror. "Hon—hon—" He was gasping.
"Clem, what is it?"
"The most awful pain here—" He locked his hands across his belly, and sweat burst from him and poured down his face.
"Oh, Clem, what shall I—"
But it was not necessary for her to do anything. He dropped away into unconsciousness.
Three hours later in the hospital in Dayton, Dr. Wood came out of the operating room. Henrietta had been sitting motionless for more than an hour, refusing to expect either good or ill. Her years with Clem, being his shadow, had taught her how to wait, not thinking, not impatient, letting her mind busy itself with the surface her eyes presented to her, the people coming and going, the bowl of flowers on the table, the branches of a tree outside the window.
"I imagine you are half prepared for what I must tell you, Mrs. Miller," Dr. Wood said.
He was a kindly middle-aged man, so obviously a small-town doctor that anybody could have guessed what he was. His strength was in knowing what he did not know and when he had seen Clem's ash-white face upon the pillow this morning he had simply said briskly, "We'll get this fellow straight to the city hospital," and had sent for the ambulance.
While it screamed its way through the roads to Dayton he had sat beside Clem, with Henrietta near, and had said nothing at all. In the hospital he had taken Clem immediately into the operating room, and had stayed with him while a young surgeon operated.
"I have not prepared myself," Henrietta said quietly. "I have only waited."
"He has no stomach left," Dr. Wood said gently. This strong woman's face looking at his made him feel that it was no use holding back one iota of truth. "He should have been operated on long ago. An old condition, he's a worrier, of course—and it's turned suddenly malignant."
"Not a worrier, exactly," Henrietta murmured. Her heart had stopped beating for a long tight moment and now began again very hard. "He simply takes the whole world as his own responsibility. He starves with every hungry man, woman, and child, he crucifies himself every day."
"Too bad," Dr. Wood said. "That sort of thing is no use, you know. One man can't do it all. I suppose you told him so often enough."
"No, thank God, I never did," Henrietta got up.
"They won't want you just now—"
"I'll just go anyway," Henrietta said. "They can't keep me away from him."
She did not stop to ask how long Clem would live. However long it was, she would stay with him and never leave him, not for a night, not for an hour, never at all. She walked into the door from which Dr. Wood had come, and nobody stopped her....
Clem lived for not quite a week. She was not sure that he knew she was there all the time but she stayed with him just the same. He might come to himself in spite of what the doctors and nurses said.
"It's really impossible, Mrs. Miller," the night nurse said. "He's so drugged, you know, to keep him from pain. He must have suffered terribly for a long time."
"He never said he did," Henrietta replied. Was it possible that Clem had suffered without telling her? It was possible. He would have been afraid that she would stop him before his work was done, in that fearful race he was running. How could she not have seen it? She had seen it, of course, in the tightness in his look, his staying himself to lean upon his hands on the table, hanging upon his shoulders as though they were a rack—a cross, she told herself. She kept thinking of Clem upon a cross. Plenty of people thought him a fool, a fanatic, and so he was, to them. But she knew his heart. He could not be other than what he was. He had been shaped by his parents, from their simple minds and tender hearts, from their believing faith, their fantastic folly, their awful death. The hunger of his own childhood he had made into the hunger of the world.
"Hon," he had often said, and she would hear those words in whatever realm his soul must dwell, "Hon, you can't preach to people until you've fed them. I'll feed them and let others do the preaching."
It was like him to choose the harder part. Anybody could preach.
"You must eat something, Mrs. Miller," they said to her.
So she ate whatever it was they brought, as much as she could, at least. Clem would want her to eat, and if he could drag himself out of the darkness where he slept he would tell her, "You eat, now, hon."
They fed him through his veins. There was nothing left of his stomach. "The surgeon could scarcely sew it together again," the nurse told her. "It was like a piece of rotted rubber. How he ever kept up!"
"He always had strength from somewhere," Henrietta said.
"Didn't you know?" the nurse inquired. She told the other nurses that Mrs. Miller was a queer, heavy sort of woman. You didn't know what she was thinking about.
"I never felt I could interfere with him," Henrietta said.
"Stupid," the nurse told the others, for wouldn't a sensible woman have made a man get himself examined, if she cared about him? She might have saved his life.
"I suppose I could have saved his life," Henrietta said slowly. "But I understood him so well. I knew there were things he cared for much more than life. So I couldn't interfere."
This was as much as she ever said.
"I'd say she didn't give a hoot for him," the nurse told the others, "except anybody can see the way she sits there that she's dying with him. There won't be anything alive in her after he's gone."
Clem died at two o'clock one night. He never came back to consciousness. Henrietta would not allow it. Dr. Wood came several times a day and that evening he was there about ten o'clock, and he told her that Clem would not live through the night.
"If you want me to, Mrs. Miller, I can leave off the hypodermic and he'll come back to himself enough to know you, maybe."
"In pain?"
"I'm afraid so."
No use bringing Clem back in pain; that would be selfish. One moment was nothing in comparison to the years that she had lived with him and the years that she must live without him. She shook her head. The doctor gave the hypodermic himself and went away.
Clem died quietly. She knew the instant of his going. She had sat in her usual place, not stirring, had refused at midnight a cup of beef broth the nurse brought in and took away again. Soon after midnight she felt the sense of approaching death as clearly as though she too must partake of it. With every moment that passed she felt a strange oppression growing upon her. At two o'clock it was there, and she knew it. Her flesh received the blow, her heart the arrest. His hand lay in hers, light and cold, and she leaned upon the bed, her face near his. No use touching his lips. A kiss was no communication now. Better to remember the living acts of love that once had been between them than to take into her endless memory the last unanswered gift. He had been a perfect lover, not frequent, never pressing, but sweet and courteous to her. Direct and sometimes brusque he had been in daily life, too busy to think of her often, and yet she knew he kept her always with him as he kept his own soul. Yet there were the rare times, the hours when he made love to her, each perfect because he won her anew, never persuading, leading and never compelling, flesh meeting and always more than flesh—and when it was over his tender gratitude.
"Thanks, hon. You make love very sweet."
She would never hear those words again! She had not thought of that. The tears which had not come came now, slow and hot.
"I'm afraid it's the last, Mrs. Miller," the nurse said. She was standing at the other side of the bed, her fingers on Clem's pulse.
Henrietta stood up. Her heart was beating so fast again that she was dizzy. Her knees were trembling.
"Could you just—turn away—a minute—"
The nurse turned her head and bit her lip. However often they died, it was always terrific. You couldn't get used to it.
Henrietta bent over Clem and laid her cheek against his. She put her lips to his ear and said very clearly, reaching with all her heart into the space between the stars:
"Thank you, my dearest. You've made love very sweet."
It was the last time in her life that she ever spoke the word, love. She buried it with him, like a flower.
After Clem was dead, there was nothing to do but keep on until what he had wanted to do was done. This was all she had. Now that he was gone, it was astonishing how little else was left. Even his face seemed to fade from her mind. There had been few hours indeed when he had been hers undivided. Most of their companionship, and she now felt their only real companionship, had been when he talked to her about his work, his plans, and finally his dream, his obsession.
Clem had been her only lover, the only man who had ever asked her to marry him, the only man she could have married. Her alien childhood had shaped her. She was far past middle age now, a woman remote and alone. Bump remained nearer to her than any human creature, and he was kind enough, but always anxious and now aghast at the burden that Clem's death had left upon him. He said the markets had to be sold and Henrietta agreed. She had no heart for the big business they had become and without Clem's idealistic genius, there was nothing to hold them together.
It was not difficult to sell. In each of the huge self-serving concerns there was a man, usually the one Clem had put in charge, who was willing to buy her out. Her terms were absurdly low and she put no limit on the time for payment. For a while she tried to stipulate that Clem's ideas were to be kept, that people were to have food cheaper there than they could buy it elsewhere. This too she was compelled to give up. It took genius to be so daring, and she found none. To Bump she simply gave the market in Dayton, and after some thought she gave him, too, her house, when after six months she made up her mind to go to New York with Berkhardt Feld, the famous German food chemist.
This aged scientist had left Germany secretly one day when he saw Hitler strutting like a pouter pigeon before a dazed mass of humanity who were anxiously willing to worship anything. Fortunately he was quite alone. He and his wife had been childless, a fact for which he never ceased to thank God when he understood what was happening to Germany, and then his wife had died. He had mourned her desperately, for he was a lonely man and Rachel had been his family and his friend. When he saw Hitler he stopped grieving and became glad that she was dead. It was easy to pack his personal belongings into a kerchief, hide in a pair of woolen socks the formula that represented his life work, and in his oldest clothes to take to the road and the border. People had not reached the point yet of killing any Jew they met and he saw contempt rather than madness in the careless glances cast at him as he went. He had money enough to get him over the border, and in France he found royalties from his last scientific work, _The Analysis of the Chemistry of Food in Relation to Human Character._
From Paris he had gone to London, had been restless there because it was still too near Germany, and friends had got him to New York. Here, gratefully, he sank into the swarm of various humanity and spent almost nothing while he worked in the laboratory of a man who was a chemist for a general food company, a man named Bryan Holt who knew Berkhardt Feld as a genius. He found a room for the old man in a clean, cheap boardinghouse and gave him a desk and a small wage as his assistant. If they ever discovered anything together he would be generous and divide the profit that came from it. Since he belonged to the company, however, it was not likely that such profits would be enormous. Dr. Feld cared nothing for money, except that he would not owe anyone a penny. He paid his way carefully and did without what he could not buy.
Henrietta had found Dr. Feld first through finding among Clem's papers a letter which he had written to Bryan Holt, trying, as Henrietta could see, to get the young scientist to help him devise a food cheap and nourishing that could serve as a stopgap until people became sensible enough, as Clem put it, to see that everybody got food free. There had been no time for an answer. Clem had died the week after he had written this letter. Ah, but in the letter she found her treasure, the voice of Clem, the words which she had longed to hear and yet which he did not speak to her because she had not allowed him to come back in pain. Here was the reward for her love, when she had denied even her crying heart. Clem knew he was dying, long before she had been compelled to believe it, and he had written to the young scientist:
"I am in some haste for I am struck with a mortal disease and may die any time. This does not matter. I have been very lucky. I have discovered a basic truth in my lifetime and so it will not die with me. What I have spent my life to prove will be proved because it is truth. Though I lie in my grave, this is my victory."
Clem victorious! Of course he was, for who could destroy his truth? Here was the command she knew and prepared as of old to obey.
To Bryan Holt then Henrietta decided to go when she had given Clem's few suits to the Salvation Army, and had found herself possessed of an astonishing amount of money in more than twenty banks in various cities where money had been paid into Clem's accounts from the markets. His records were scanty but very clear indeed were certain notes on The Food, as he called it. The Food was half chemical, half natural, a final mingling of bean base with minerals and vitamins, which if he could get a chemist to work it out with him might, he had believed, make it possible for him to feed millions of people for a few cents apiece. This was the final shape of his dream, or as William had once called it, his obsession. It might be that the two were the same thing.
The first meeting with Holt had not been promising. Holt had not answered Clem's letter because it sounded absurd. He was respectful before Henrietta's solid presence, her square pale face and big, well-shaped hands. She had immense dignity. He tried to put it kindly but she saw what he meant. This young man was not the one she sought, but there would be another. Clem, though he had died, yet lived.
"Many people thought my husband was unbalanced," she said in her calm voice. "That is because he was far ahead of his time. It will be twenty-five years, much more if we don't have another war, before statesmen realize that what he said is plain common sense. There will not cease to be ferment in the world unless people are sure of their food. It is not necessary that you agree with my husband and me. I am come only to ask you questions about his formula."
Bryan Holt wanted to get rid of her, though he was polite since he was almost young enough to be her son. So he said:
"I have a very fine scientist here working with me who has come from Europe. He will be more useful to you than I can possibly be."
With this he had summoned from a remote desk a shambling old figure who was Dr. Berkhardt Feld, and so by accident Henrietta found her ally. When she had talked with Dr. Feld for a few hours she proposed to find a small laboratory for him alone, with an apartment where he could make his home. Then if he would teach her to help him, building upon her college chemistry, which was all she had, she would come to him every day and they could perhaps fulfill Clem's life work.
To Dr. Feld this was heaven unexpected. None of Clem's ideas were fantastic to him. They were merely axiomatic. It would not be too difficult to find the formula which Clem had begun very soundly upon a bean base, a matter perhaps of only a few years, by which time it might be hoped the wise men of the world would be ready to consider what must be done for millions of orphaned and starving.
"Then, then, _liebe_ Frau Müller," Dr. Feld said fervently, making of Henrietta as German a creature as he could, "we will be ready perhaps with The Food."
The tears came to Henrietta's eyes. She thanked Dr. Feld in her dry, rather harsh voice and told him to be ready to move as soon as she could go home and get her things.
That decision made, she began to clean away what was left in the house of all her years of marriage to Clem. Among the things she would never throw away was the red tin box of Clem's letters and with them the old amulet which he had given her. It was still in the folded paper in which he had sent it to her. She opened the paper and cried out as though he were there, "I always meant to ask you about this!"
How much of him she had meant to ask about in the long last years she had expected to have with him, years that would never be! She cried a little and closed the box and put it into her trunk to go with her to New York. Someday, when she could bear to do it, she would read all his letters over again. So much, so much she would never know about Clem because he had been busy about the business of mankind!
On the night before she left, she invited Bump and Frieda to supper, that she might ask something of Bump. She did not mind Frieda, a lump of a woman, goodhearted, stupid and kind.
"I wish you would tell me all that you can remember of Clem when you first saw him on that farm. He never could or would tell me much about it."
She soon saw that Bump could not tell her much either. "He was just about the way he always was," he said, trying hard to recall that pallid, dusty boy who had walked into their sorrowful small world so many years ago. "The thing I do remember was that he wasn't afraid of anybody. He'd seen a lot, I guess. I don't know what all. But I always took it that he'd had adventures over there in China. He never talked about them, though. He pitched in right where we were. The Bergers never beat him up the way they did us. He even stopped them beating us, at least when he was around. When he decided to leave, the others were afraid to go with him. They were afraid of the Aid people catching them again and things were tough if they caught you. I was afraid, too, but after he was gone, I was more afraid to stay. I don't think he was too pleased to see me padding along behind him, though. I've often thought about that. But he didn't tell me to go back."
There was nothing more, apparently. Clem's outlines remained simple and angular. After Bump had gone she studied again the notes Clem had left about The Food. If she went on trying to do what he had wanted to do then perhaps she could keep his memory with her, so that she would not forget when she was old how he had looked and what had been the sound of his voice....
It did not occur to Henrietta to find her family and tell them that she was in New York. She had not even thought to tell them of Clem's death, but they had seen the announcement in a paragraph in the New York papers. Clem was well enough known for that. William had telegraphed his regret and Ruth had sent a floral cross to the funeral. Her mother was in England and it had been some weeks before a letter had come from her saying that she never thought Clem had a healthy color and she was not surprised. Henrietta must take good care of herself. It was fortunate there was plenty of money. If Henrietta wished, she would come and live with her, but she could not live in the Middle West. New York or Boston would be pleasant. Henrietta had not answered the letter.
Now that Clem was gone she was lonely again, but not as she would have been had he never come. He had shared with her and did still share with her in memory her alien childhood which no one could understand who had only been a child here in America. Without loving China, without feeling for the Chinese anything of Clem's close affection, she was eternally divided in soul and spirit. It occurred to her sometimes in her solitary life that this division might also explain William. Perhaps all that he did was done to try and make himself whole. The wholeness which she had been able to find in Clem because they understood one another's memories, William had found no one to share. Perhaps he could not be made whole through love. She would go and see Candace. Upon this decision she went to the laboratory as usual.
Dr. Feld, observing the large silent woman who worked patiently at his command, mused sometimes upon her remoteness and her completion. She needed no one, even as he needed none. They had finished their lives, he in Germany, she—where? Perhaps in China, perhaps in a grave. What more they did now was only to spend the remaining time usefully. He wished that he could have known the man who had left behind him these extraordinary though faulty notes. She had told him that her husband had had only a few years of education and no training in science.
"His knowledge must have been intuitive, dear madame," he had replied.
"He was able to learn from human beings," she said. "He felt their needs and based his whole life upon what he found out. He called it food, but it was more than food for the body. He made of human need his philosophy and religion. Had you met him you would have thought him a very simple man."
"So is Einstein," Dr. Feld said.
They did not talk much. When they did speak it was about Clem or the formula. He explained the peculiar, almost atomic vitality of vitamins. "The source of all life is in the atom," he said solemnly. "God is not in the vastness of greatness. He is hid in the vastness of smallness. He is not in the general. He is in the particular. When we understand the particular, then we will know all." When he really talked he spoke in German. She was glad that she had taken German in college and had kept the language alive in her reading.
One summer afternoon she took off her big white apron and reached for her hat and coat. "I'm going away early today, Dr. Feld, to see someone I know."
He looked surprised and pleased. "Good—you have friends, dear madame."
So Henrietta went away and rode the subway uptown and walked to Sutton Place.
She found the doorway in a quiet street, in a row of black and white houses with white Venetian blinds. The slanting sun shone into the street with glitter and shadow. The door opened promptly and a little maid in black and white asked her to come in please, her voice very fresh and Irish. She followed her into a square big room, immaculate in white and gold. The maid tripped away. Henrietta sat in a vast gold satin chair and a moment later Candace came in, looking soft and still young, her eyes tender and her hair a silvery gold. Her full sweet mouth smiled and Henrietta felt a fragrant kiss upon her cheek.
"Henrietta, this is the dearest thing you could have done. I never expect any of William's family to—Sit down, please, and let me look at you. I cried so when I heard about Clem. I ought to have written but I couldn't."
She was in a violet chiffon tea gown, long and full and belted with silver. She was very slender again and more beautiful than ever.
"Let me look at you," Henrietta said. "Are you happy, Candy?"
Candace blushed. "I am happier than I've ever been in my life, happy the way I want to be happy."
She put her hand on Henrietta's. "When I was with William I was happy, too. It is so easy for me to be happy. But then I was happy mostly by myself. Now I am happy with Seth."
"I know," Henrietta said. She did not take Candace's hand because she did not know how to do such things and Candace understood this and stroked her hand and took her own away again.
"I don't blame William," she said gently. "I won't even let Seth hate him. William needed someone who could understand him. Seth and I of course have grown up in the same world."
She smiled at Henrietta brilliantly and softly. "You must come and visit us, dear. We don't live here much. We live at the old seashore house."
"Where is Seth working?" Henrietta asked.
"He doesn't work any more except on his plays," Candace said sweetly. "He says William galvanized him in college or he never would have worked." Candace laughed her rich youthful laughter. "Seth is so amusing. He says William shaped his life. First he influenced him to work for him and then he influenced him to work against him. Now, Seth says, he's not going to work at all because he's really freed himself of William. We're both very wicked, I daresay."
"It isn't wicked to be happy," Henrietta said.
Candace pressed her hand again. "How glad I am to hear you say that! I used to tell William so but he didn't know what I meant. I tell the boys that now, but they're William's sons, too. They're terribly proud of him."
Henrietta said, "Tell me about yourself."
Candace held up her hand. Her face so illumined from within, turned toward the door. "Wait! I hear Seth." She rose and went to the door and called and he came.
Henrietta saw a tall, gray-haired man, with a handsome, determinedly quizzical face. He was the one she remembered and she put out her hand.
"How good of you to come," he said. "Candy and I don't expect favors."
"I am fond of Candace. I wanted to see if you were good enough for her."
"Don't make up your mind at first sight," he begged. "My weaknesses are so obvious."
She smiled politely, not knowing how to answer nonsense and he looked at Candace.
"My love, I've had nothing to eat or drink since luncheon."
"Oh—I'll ring for tea." Her violet skirt flowed across the silvery gray carpet and she pulled a black bell rope, hung as a decoration by the marble mantelpiece.
They had tea then, a happy plentiful affair at which Henrietta sat loyally silent and faintly smiling, enjoying the warmth of the web these two wove about them, into which they wrapped her, too. They were mirthful without cruelty, and gaily frank with her.
"Your mother, darling," Candace said to her, "has been cultivating England, as you know. She's used up all the available relatives—She's simply astonished everybody. Seth, where's the letter we had from Lady Astley?"
Seth pulled open the drawer of a mahogany escritoire, and tossed an envelope into her lap.
"You don't mind?" Candace inquired, eyes brimming with laughter.
"I know Mother," Henrietta said.
Candace opened the pale blue writing paper, and began to read aloud:
> What we cannot understand here in England is why Mrs. Lane isn't the mother of the President. I think she doesn't understand it, either. She's a joy and a treasure. She makes us laugh our heads off and then we can face these Socialists. Really, she's a good sport—we like her. There's something English about her if you know what I mean—something quite frightful. She's so sure she's wonderful. There'll always be England and that sort of thing—and of course there always will. It's wonderful to think that it's in America, too. We'll quite hate to see her leave. God help us, it's odd, but the American Queen Mother hates Labor, too! She calls herself a Republican. William the Son is a Republican, she says. What's a Republican, dear? Mind now and tell me when you write.
"How wicked we are to read this aloud," Candace said looking with laughing rue at Seth, sunk in his chair and smoking his pipe.
"Nonsense," he said. "Henrietta knows we like the old gal. God, how I envy the old! They had the world all straight, heaven and hell, God and the devil, peace and war, good and bad, moral and immoral, stuffed and hungry and rich and poor—" Candace joined in the chant. "Young and old—"
"Black and white—"
"Gold and silver—"
"East side, West side, and never the twain shall meet—"
"King and subject—"
"City and Country—"
"Capital and Labor—"
"Union and nonunion—"
"Capitalist and Communist—"
"White man, black man—"
"Stop," Candace said, "we're making Henrietta dizzy."
"No, you're not," Henrietta replied. "You're just making me laugh. Bless you both for being happy. Now I've got to go."
They let her go, clamoring for her return, making her promise that she would come to spend a month with them at the seashore house. She would not, of course, but she could not tell them so, lest they keep her to make her promise, and then she went away, back to the subway and downtown again to her little hole in the wall.
It was long past twilight. Dr. Feld might still be working but she did not go to see. When she shut the door upon that splendid foolish happiness she stepped from moonshine into darkness. She was so accustomed to loneliness that she could not quite understand why the loneliness was deeper than it had been before, since she had found out exactly what she wanted to know, that Candace was happy and that none of them owed a debt to her any more through William. Then she remembered that neither Seth nor Candace had asked her where she lived or what she was doing. It had not occurred to them. They were not cruel, they were not even selfish or unthinking. They were simply ignorant, Candace naturally, so, Seth perhaps willfully so. He had returned to the world into which he had been born, and Candace had never left it. For them no other existed. They had never known, could never know, what Clem had always known.
It occurred to her later, after she sat trying to study a chemistry text, that perhaps that was why Candace had never understood William. William knew, too, another world. She let the book fall to the floor and sat for a long time, pondering this astonishing fact: Clem and William, so utterly different, were alike!
William Lane was no longer a young man. When he saw his two sons, both married and with children of their own, his grandchildren, he felt alarmingly old. On the other hand, his mother was robust and alive, though in her eighties, and so he was still young. He had come to the point of being proud of her, though frequently irritated by her increasing irresponsibility. Now, for example, when Ruth was in such trouble with Jeremy, who had become a really hopeless sot, his mother was gallivanting in England. He complained of this to Emory who listened with her usual grace and then made a wise suggestion. He depended very much on her wisdom.
"Why not cable your mother to come home and live with Ruth?" Emory said.
"An excellent idea," William replied.
Mrs. Lane received the cable the next day. She had been staying at a big country house in Surrey, where the tenants at Christmastime had gathered in the real old English way to drink the health of the lord in spite of government. There was something about English life that made her think of Peking and she would have liked to spend the rest of her life in England except that the Socialists were spoiling everything. There was no reason for an American to endure the austerity upon which Sir Stafford Cripps insisted, especially an American woman. She would have stayed longer, however, with her friend, the Countess of Burleigh, had she not received William's cable. Jeremy, it appeared, had been taken to a special sort of hospital and Ruth needed her.
Mrs. Lane shrugged her handsome heavy shoulders when she read the telegram the footman brought her. She was having a quiet tea with the Countess, just the two of them. The Countess was old, too, and always looking for diversion and Mrs. Lane had been diverting her by a long visit.
"I cannot understand why my children still insist upon my returning to them at every crisis in their lives," she now complained to the Countess. "One would think that at my age I might be allowed my freedom. But no—William, it seems, feels I must come home. My elder daughter is of course absorbed in her grief—I told you she lost her husband—and so my poor youngest child turns to me. Her husband has been taken sadly."
"What's wrong with him?" the Countess inquired. She had been a music-hall star in her younger days and she continued to look very smart in spite of a tendency to palsy, and she talked with the youthful Cockney twang that she pretended she used on purpose.
"I fancy he's been drinking too much again," Mrs. Lane replied.
"Ah, if it's that," the Countess said decisively, "then you're rahhly in trouble, my deah. I know poor Harold was the same—would have his little tipple, he would, and he ended that way. Nothin' to do about it, nyether. I used to rampage a bit and he'd get frightened at first. In the end, poah deah, it only made him drink more. I had to let him drink himself to death, I rahhly did."
This was not encouraging, and Mrs. Lane took her way homeward by plane as soon as she could get a seat, which she was able to do very soon, to the surprise and annoyance of the man who had already engaged it. She knew how to use William's name in secret places.
She found Ruth alone. Emory, who had come to meet her at the air field, went with her. Ruth began to weep when she saw her mother in the hall standing still so that the maid could take off her coat properly, and Mrs. Lane, regarding her daughter's tears, saw that Ruth cried as a middle-aged woman exactly as she had as a child, almost soundlessly and with bewildered pathos. She put out her stout arms and wrapped Ruth in them. "There, there," she said. "Everything is going to be all right now. I've come to stay with you. You need me more than Henrietta does. Where is Henrietta?"
"I don't know," Ruth sobbed. "I can't think about anybody but Jeremy. Oh Mother, why does he—the doctor says it's a symptom. Something is still making him unhappy—but it's not me, I'm sure. I do everything he wants me to."
"Nonsense," Mrs. Lane said, pulling her daughter along firmly in the circle of her right arm as she moved into the drawing room. "Men like to get drunk—some men. That's all there is to it. It's not any woman's fault."
Emory kissed Ruth an inch or two off the cheek. "William feels quite desperate, too, dear Ruth. We all want to help poor Jeremy."
"He was so deceitful about it, Mother—" Ruth cried. "He went off to the office every day apparently to work and instead he took a room at his club and just began, and went on, all by himself. When he didn't come home, of course we had to find him. He had locked the door and they had to break it down. He was unconscious. I had Doctor Blande go and get him. They took him straight to the hospital. I haven't even—seen him. Doctor Blande says I mustn't just now."
She began to cry again. Mrs. Lane sighed and Emory sat, quietly beautiful, looking at these American relatives. She knew why Jeremy had gone off. It was his revenge upon William, the revenge of a weak man upon one invincible. She had sympathy for the weak, but she was prudent enough to cast her lot with the invincible. William was right to be invincible in the sort of world there was now. It was the only chance one had for survival. She was invincible, too, at William's side. She pitied Ruth and felt a new admiration for William's mother, sitting solidly and without tears.
"Ruth, there's not a bit of use in your crying now that I'm here," Mrs. Lane said. "I'm sorry for you. Your father was a saint. You're used to good men. William, too, is so good. It's natural that a man like Jeremy should be a trial to you. But you belong to the family and you'll be taken care of. My advice is to let Jeremy stay right where he is until William tells us what to do. Maybe you ought to let Candace know, so she can go to see him."
Ruth shivered. "Oh, I can't! She'd think it was somehow our fault."
"Then she's very silly," Mrs. Lane said loudly. "The trouble with Jeremy is that he was brought up to be spoiled. He can't live up to William's standards. Now you go and wash your face and brush your hair. You'll feel better. There's nothing you can do for Jeremy, not a thing. We may as well have a bite of something to eat and go to a matinee! It will take our minds off our troubles. Emory, why don't you come with us? That's a handsome frock you have on. I've always liked that shade of yellow, especially with jade. That's handsome jade, too."
"William brought it from China," Emory said. "Madame Chiang gave it to him for me."
"She has wonderful taste," Mrs. Lane said. "What a pity the Communists have taken over!"
They were alone, for Ruth had left the room as obediently as though she were a little girl. Mrs. Lane leaned toward Emory. "Jade looks nice with dark hair and eyes. William ought never to have married Candace. She was a blonde, you know. He always liked brunettes best. The Chinese wear a lot of jade. Of course they're always brunette. Some of the Chinese women have very beautiful skin. It reminds me of yours. I used to know the Old Empress Dowager. In fact, we were almost intimate. She had that sort of skin, very smooth and golden. She wore a lot of jade. William always liked to hear about her. I took him to see her once, by special permission."
"He has told me that," Emory said.
"Nobody could forget the Empress," Mrs. Lane said with complacency.
Ruth came in, looking pretty again. Her short curly hair was almost white and very becoming. They went away at once since it was already late, and they found the theaters so crowded that they could only get seats at a new musical.
At the dinner table that night Emory described to William the effect of the afternoon and he listened gravely. They seldom had guests nowadays. Since the war they had fewer really distinguished visitors from abroad and not many Americans were interesting enough to be invited for a whole evening.
"I shall advise Ruth to get a divorce," William said with decision. He had grown very handsome with the years. The discontent which had marred his face from childhood was almost gone.
"Oh, can you?" Emory murmured mildly.
"Certainly, why not? She's not a Catholic," William replied. "Moreover, at her age she will certainly not marry again. For my own part, I shall be glad to be rid of Jeremy."
Emory did not reply. They sat in comfortable silence. She was glad that she need not live now in England. How ghastly might her life have been in such penury as Michael and his family endured! He was trying to make the farmlands pay, for the government was threatening to take over Hulme Castle if he could not. The only really safe and comfortable spot now in the world was America.
This thought moved her to an unusual idea. "William, what would you think of a cozy family dinner now that your mother is back, something to gather us together again in these troubled times? After all, there's nothing quite like family. I think it would comfort your poor sisters and impress the children, you know. We needn't ask the grandchildren."
William's heavy eyebrows moved. He pushed aside his salad. He had never liked salads, which he called food for rabbits. "I am going to Washington next week to insist on more arms for Chiang. I gave my promise to him—a promise I hold sacred, in spite of what's happened in Korea."
Emory evaded this. William had grown amusingly dictatorial in these past few years. "Why shouldn't I just telephone them for tomorrow night? After all, it's family. One needn't be too formal."
William reflected, then consented. "Very well. But tell them to be prompt. Will's wife is always late."
Emory rose at once and walked with her long lingering step across the floor. "I'll telephone Henrietta first."
None of them would think of saying he or she could not come, unless Henrietta declared she had to work in her absurd laboratory. She would tell her that she needn't dress, at least.
"You mean we aren't to dress?" Henrietta inquired over the telephone. "But I have a quite decent black gown. I had to get it when Clem was given an award in Dayton—for the citizen who had done the most for the town during the war."
"Oh, then we'll dress," Emory replied. "William always does anyway."
So she had telephoned to everybody to dress, and therefore it was upon his family in its best trappings that William looked the next evening, after he had said his usual grace before the meal. The dinner was excellent, hearty without being heavy. Emory understood food as Candace never had and she had no qualms about dismissing a careless cook. She never allowed herself to become involved in the domestic situation of any servant, a fault which had been very trying in Candace. They had once endured abominable omelets for nearly three years because the cook had a crippled son. In the end William had dismissed the cook himself one Sunday morning over a piece of yellow leather on his plate.
Tonight the bouillon, the soufflé, the roast pheasant, and the vegetables were all delicious. He did not care for sweets but Emory had a Russian dessert that he had never tasted before, flavored with rum. "It is a pity," he remarked, "that our relations with the Russians cannot be confined to their sweets." Everybody laughed and even Emory smiled.
His mother was looking very handsome in a lilac velvet, trimmed at the bosom with a fall of cream-colored lace. No one would dream that she had ever been the wife of a missionary in China. She had kept her stout figure in spite of her age, and her visit in England, prolonged as it had been, had given her an imperial air, enhanced by the pile of white curls on her head, which he liked. He was proud of her and, the dinner over, he led her to the most comfortable chair in the long drawing room.
"You're looking well, Mother."
"I am in splendid health, thank God," she replied in a resonant voice. "I've had no chance at you, you naughty boy. Oh, I know you've been too busy for your old mother." She leaned over the edge of her chair while the others were settling themselves. "Now, William, I want you to have a talk with Henrietta. She is living all by herself somewhere way downtown in the most miserable little apartment. It doesn't look right for your sister."
"What is she doing?" he asked. He knew vaguely from Emory that Henrietta was still working on one of Clem's absurd notions and his eyes fell on her as he spoke. She was sitting in her characteristic repose.
"She's working at some laboratory with an old Jew. I don't know what she's doing. Clem was a queer duck, if you ask me."
At this moment Henrietta raised her dark eyes and smiled at them. She was gentler than she used to be, though even more withdrawn.
"I want a word with you later, Henrietta," he called.
She nodded and her eyes fell.
Ruth was very pretty in spite of her troubles. He had time now to look at each one of his family. She had gained some weight—eating, probably, to take her mind off Jeremy. Of all of them Ruth looked the most like his father, her features delicate and her bones fine. Yet there was nothing in her face of that spiritual quality which he remembered with reverence as being his father's habitual expression. Her two daughters were nondescript young matrons, he thought. They looked like all the modern women, flaring blond hair, wide painted mouths, a clatter of thin bracelets and high heels. He supposed they were well enough and certainly they need not worry him now that they had husbands.
He had taken no more relatives into the business, not even his own sons. He wanted to be free to dismiss incompetents like Jeremy. Not that his sons were incompetent in any way. Both of them were successful men, Will a lawyer, Jerry a surgeon. They were married and he had three grandchildren, two of them boys. He did not know his sons' wives very well and had even been accused of passing them on the street without recognizing them. He had grumbled a good deal when Jerry married an ordinary trained nurse while he was an intern. William had a theory that it would be better for all young people if they were married in the Chinese fashion by their parents, in order that one could be sure of what was coming into the family. When he had said this to Emory she had gone into fits of laughter. "You are the most unrealistic of men," she had declared. "Don't you know yet that you are living in modern America?" He did not know what she meant and was too proud to say so.
His sons and Ruth's daughters seemed on the best of terms with Emory. She sat among them and behind her coffee table, appearing, he thought with self-congratulation, entirely happy. Her darkly regal head was bent while she busied herself with cups. She wore a coral-colored gown of some sort that he did not remember having seen before. The full skirt flowed round her like a calyx, and she had on her diamonds.
It was all very pleasant and he did not remember ever having been quite so happy before. Everything was well with him, and it was dawning upon him that perhaps even the war had been good for him in its own way. The world needed leadership as never before. He must not allow himself to think of retiring, however much Emory hoped for it. Monsignor Lockhart had said to him only last week that the new war in Asia might be the—beginning of mankind's most titanic struggle. Within the next years—
"William," Emory said. "Your mother wants to know what you think is going to happen in China. Why don't you tell us all?"
So he began, sitting in his high-backed armchair. "A very strange new China, not at all what you and I remember, Henrietta, in old Peking. You would like it less than ever, Mother. I don't suppose Ruth remembers...."
They listened to his picture of Communist China, no one interrupting him except his mother, who put in small cries of horror and interjections of outrage.
"But how repulsive, William!" And at the end, "I'm glad your father isn't here to see it. He would want to go straight over there—though as I always said, what one man can do I don't know. 'You're wasting yourself,' that's what I always told him."
"One man can do a great deal," William said.
She heaved a mighty sigh and shook her head.
"Not any man, of course," William said, "but one who knows, one who has faith in God, has infinite power."
His mother looked rebellious. "Your father always thought he knew, too, William. He was always so sure that God told him what was best. I don't know that there's any difference between then and now."
"There is a great deal of difference," William said gravely. "Now we really do know."
Emory, scenting the dissension always possible in the presence of her mother-in-law, chose a lighter substance for talk.
"William says the Old Tiger's wife is very beautiful, though she's Chinese."
"So was the Empress Dowager," Mrs. Lane said promptly. "The Empress was not Chinese exactly—Manchu, of course, but it's almost the same—and she was very beautiful. I shall never forget her. She had long eyes, very long and brilliant. She had a temper, as any woman worth her salt has. Her mouth was very red—of course she painted. Her skin was wonderful and smooth and white as anybody's. I never felt it was really her fault that things went as wrong as they did. She was so charming, and always perfectly lovely to me. I took William to see her—do you remember, William?"
"I can never forget," William said.
"Powerful, wasn't she! With such charm, too!"
"She killed an extraordinary number of people." This was Henrietta's voice coming so quietly that it seemed almost indifferent.
"Oh well," Mrs. Lane said, "we don't know what provocation she had."
"It is never right to kill people," Henrietta said with what Mrs. Lane felt was her childish stubbornness.
William answered his sister. "It is sometimes necessary. In order that the end may not be lost, the means must sometimes be very severe."
"Then the end is lost," Henrietta said. She lifted her head when she said this, and Emory felt that the family was really very difficult. They seemed determined to disturb life. She turned to the younger men.
"Will, why don't you and Jerry and the girls open the doors into the music room and roll back the carpet? I'll play for you and we can watch you dance."
Under cover of the music and the rhythm of brisk feet swinging into new intricate steps, William went to Henrietta.
"Let us go into my library. I would like to know what you are doing."
She rose almost obediently and followed him, her black-robed figure upright and dignified. Since Clem's death she had not cut her hair and now, almost entirely white, it was long enough to be coiled around her head and held at the back with a silver comb. Emory's eyes, from the piano, followed the tall figures. It was surprising how much William and Henrietta looked like each other. Yet they were utterly unlike. Henrietta was espousing poverty for Clem's cause. Emory had learned much about that solitary laboratory and the old scientist who worked there. And yet perhaps there was a likeness between William and Henrietta. A great deal of character and spiritual energy could be stubbornly bestowed upon something chosen and the chosen substance was changed, transubstantiated, and so deified.
Emory understood this without in the least partaking of it, kindly cynical as she was to the core of her heart, sadly agnostic, while she bowed her head. America was her country now and this her family. Her parents had been killed by one of the final buzz bombs. They had gone up to London, thinking it safe at last, and then the new horrible bombs began to fall. Poor Michael, in Hulme Castle, was still trying to make the land produce those impossible harvests under the cruelly critical eyes of the incredible government the British people had chosen for themselves after the war! William said he would never go to England until it fell. It might be a long time, it might be never. Her hands flew over the keys. She played as beautifully as ever, with a natural rhythm which she could suit as easily to a rhumba as to a waltz. Nothing made any difference so long as the music went on, the music and the dancing.
"So you see," Henrietta was saying behind the library door which was so heavy that it shut out the music, "I shall simply keep on with Clem's work until I succeed in what he wanted to do."
William was too stupefied to speak. He had thought Clem a fanatic and a fool while he lived, and in so far as he had given any thought to him since his death it was to believe that Henrietta was better off alone. When he thought of Clem now it was still as the pale boy whom he had first seen in Peking in a silly quarrel with a Chinese, an affair no more dignified today as he remembered it than it had been then. He had been repelled by Clem as a pale young man in a collar too big for him, after he had become Henrietta's husband, and there was that final folly of the day when Clem had come to his office with his absurd proposals and without any appointment. Clem never learned anything. His life had been all of a piece, all nonsense except that he had made some money for Henrietta. William had never acknowledged Clem as a part of the family and he did not do so now. Careful for once of his sister's feelings, he made no reference to Clem. He spoke to her entirely for her own good.
"If, as you say, you have had by chance a respectable fortune left to you it seems madness to consume it on any idea so fantastic. If people were given food, which is, after all, the one basic necessity, most of them would never work again."
Henrietta tried once more. "You see, William, it is not only that they should not be allowed to starve. I believe, and Clem did, too, that unless people are fed they will rise up against any government they happen to have. The government that first understands the anger in the hearts of hungry people will be the one that wins. People feel they ought not to have to starve for any reason whatever. Dr. Feld says that Hitler's promises of food were the first steps to his power."
William was walking restlessly about the room and she kept watching him. "The idea is so fantastic," he was repeating. "Think of feeding the people of China! It can't be done."
"It's got to be done," she said. "And there are the people of India and all the other peoples."
"Fantasy, fantasy," William muttered.
She contradicted him flatly. "Not fantasy, William, but purest common sense. Do you know why you can't see it? Because you and Clem worked at opposite poles. He believed the world could get better only when people were better. He believed that people themselves could make a good world if only they were free from simple misery. That was Clem's faith. Yours isn't that. You think people have to be compelled from the outside, shaped, ordered, disciplined, told what to do. I don't know where your faith is—I suppose you have it, for in your own stubborn way, William, I can see you are working for the same thing Clem was."
William was suddenly violently angry. "I deny the slightest resemblance to him! Henrietta, I tell you—"
He raised his clenched hands and saw that they were trembling and dropped them. "Clem was a dangerous man, a menace, or might have been if he had been successful. He worked at the very roots of our nation to destroy us. I don't like to say this, Henrietta—I don't forget you are my sister—but now that he's dead, you had better know the truth."
Henrietta remained calm. "Well, William, we don't understand each other. We never did. But someday it will be proved that Clem was right. That is my faith. And when he is proved right, William, you will be defeated, you and with you the Old Tiger and his beautiful wife and all the rest of your kind. How wrong that Old Empress was whom Mother continues to worship!"
"Henrietta, you're talking very wickedly."
"I daresay."
She was so calm, so immovably stubborn, that for a moment he felt quite sick with rage, exactly as he had so often felt when they were children together in China. But he managed to follow her into the hall and help her on with her wrap, a black wool cape. For she was determined to leave him and, so far as he could see, to leave them all. She would not allow him to tell the others that she was going.
"No use disturbing them," she said in her short fashion.
So he let her out of the door himself and then stood at a window watching her. She did not call a cab. Instead she began to walk down the street, her bare head held very high, and the wind blowing hard and her cape flying out behind her. It was a clear night, and he could see a strip of stars above. At the far corner she stopped for a bus. He could still see her waiting there, and he would have continued to watch her except that one of those miserable creatures came shambling up to her. Under the light of the street lamp William saw her open her handbag and take out money and give it to the beggar, thereby encouraging, he thought bitterly, all such persons everywhere. He pulled the curtains together across the window and trembled with anger. He had been angry with Henrietta all his life, and with that fellow Clem!
He stood behind the curtains, thick and velvet, and summoned his old arrogant spirit. He would not suffer fools! He closed his eyes, and waited. No reassurance rose to meet his soul's demands. He wished he had not thought of Clem. He saw him again now. Inside his brain, inside his closed eyelids, he saw Clem, that boy, intrepid in the Chinese street, ready to fight, Clem marching unbidden into his office. The fellow had no breeding, never knew his place. Dead, luckily! He had the world to himself, now that Clem was dead.
He opened his eyes and heard faintly the music that Emory made and mingled with it the soft movement of dancing feet. He turned away from the window. Then he felt the familiar chill upon his heart. The old childish doubt of himself, the profound eternal doubt that had haunted him from his earliest memory, fell upon him again, this time so heavily that he felt too tired to shake it off.
What if he had always been wrong? The vague shape of victory—was it he? Or was it Clem? His imagination, diseased and tortured by his soul's perpetual uncertainty, lifted Clem from the grave, brought him back to life, clothed him in the dark garments of doubt and fear.
Could Clem be right? If so, then he himself was wrong and being wrong was doomed. But was Clem right? How could he ever know?
# A Biography of Pearl S. Buck
Pearl S. Buck (1892–1973) was a bestselling and Nobel Prize-winning author of fiction and nonfiction, celebrated by critics and readers alike for her groundbreaking depictions of rural life in China. Her renowned novel _The Good Earth_ (1931) received the Pulitzer Prize and the William Dean Howells Medal. For her body of work, Buck was awarded the 1938 Nobel Prize in Literature—the first American woman to have won this honor.
Born in 1892 in Hillsboro, West Virginia, Buck spent much of the first forty years of her life in China. The daughter of Presbyterian missionaries based in Zhenjiang, she grew up speaking both English and the local Chinese dialect, and was sometimes referred to by her Chinese name, Sai Zhenzhju. Though she moved to the United States to attend Randolph-Macon Woman's College, she returned to China afterwards to care for her ill mother. In 1917 she married her first husband, John Lossing Buck. The couple moved to a small town in Anhui Province, later relocating to Nanking, where they lived for thirteen years.
Buck began writing in the 1920s, and published her first novel, _East Wind: West Wind_ in 1930. The next year she published her second book, _The Good Earth_ , a multimillion-copy bestseller later made into a feature film. The book was the first of the Good Earth trilogy, followed by _Sons_ (1933) and _A House Divided_ (1935). These landmark works have been credited with arousing Western sympathies for China—and antagonism toward Japan—on the eve of World War II.
Buck published several other novels in the following years, including many that dealt with the Chinese Cultural Revolution and other aspects of the rapidly changing country. As an American who had been raised in China, and who had been affected by both the Boxer Rebellion and the 1927 Nanking Incident, she was welcomed as a sympathetic and knowledgeable voice of a culture that was much misunderstood in the West at the time. Her works did not treat China alone, however; she also set her stories in Korea ( _Living Reed_ ), Burma ( _The Promise_ ), and Japan ( _The Big Wave_ ). Buck's fiction explored the many differences between East and West, tradition and modernity, and frequently centered on the hardships of impoverished people during times of social upheaval.
In 1934 Buck left China for the United States in order to escape political instability and also to be near her daughter, Carol, who had been institutionalized in New Jersey with a rare and severe type of mental retardation. Buck divorced in 1935, and then married her publisher at the John Day Company, Richard Walsh. Their relationship is thought to have helped foster Buck's volume of work, which was prodigious by any standard.
Buck also supported various humanitarian causes throughout her life. These included women's and civil rights, as well as the treatment of the disabled. In 1950, she published a memoir, _The Child Who Never Grew_ , about her life with Carol; this candid account helped break the social taboo on discussing learning disabilities. In response to the practices that rendered mixed-raced children unadoptable—in particular, orphans who had already been victimized by war—she founded Welcome House in 1949, the first international, interracial adoption agency in the United States. Pearl S. Buck International, the overseeing nonprofit organization, addresses children's issues in Asia.
Buck died of lung cancer in Vermont in 1973. Though _The Good Earth_ was a massive success in America, the Chinese government objected to Buck's stark portrayal of the country's rural poverty and, in 1972, banned Buck from returning to the country. Despite this, she is still greatly considered to have been "a friend of the Chinese people," in the words of China's first premier, Zhou Enlai. Her former house in Zhenjiang is now a museum in honor of her legacy.
Buck's parents, Caroline Stulting and Absalom Sydenstricker, were Southern Presbyterian missionaries.
Buck was born Pearl Comfort Sydenstricker in Hillsboro, West Virginia, on June 26, 1892. This was the family's home when she was born, though her parents returned to China with the infant Pearl three months after her birth.
Buck lived in Zhenjiang, China, until 1911. This photograph was found in her archives with the following caption typed on the reverse: "One of the favorite locations for the street barber of China is a temple court or the open space just outside the gate. Here the swinging shop strung on a shoulder pole may be set up, and business briskly carried on. A shave costs five cents, and if you wish to have your queue combed and braided you will be out at least a dime. The implements, needless to say, are primitive. No safety razor has yet become popular in China. Old horseshoes and scrap iron form one of China's significant importations, and these are melted up and made over into scissors and razors, and similar articles. Neither is sanitation a feature of a shave in China. But then, cleanliness is not a feature of anything in the ex-Celestial Empire."
Buck's writing was notable for its sensitivity to the rural farming class, which she came to know during her childhood in China. The following caption was found typed on the reverse of this photograph in Buck's archive: "Chinese beggars are all ages of both sexes. They run after your rickshaw, clog your progress in front of every public place such as a temple or deserted palace or fair, and pester you for coppers with a beggar song—'Do good, be merciful.' It is the Chinese rather than the foreigners who support this vast horde of indigent people. The beggars have a guild and make it very unpleasant for the merchants. If a stipulated tax is not paid them by the merchant they infest his place and make business impossible. The only work beggars ever perform is marching in funeral and wedding processions. It is said that every family expects 1 or 2 of its children to contribute to support of family by begging."
Buck worked continually on behalf of underprivileged children, especially in the country where she grew up. The following caption was found typed on the reverse of this photograph in Buck's archive: "The children of China seem to thrive in spite of dirt and poverty, and represent nature's careful selection in the hard race for the right to existence. They are peculiarly sturdy and alert, taken as a whole, and indicate something of the virility of a nation that has continued great for four thousand years."
Johann Waldemar de Rehling Quistgaard painted Buck in 1933, when the writer was forty-one years old—a year after she won the Pulitzer Prize for _The Good Earth_. The portrait currently hangs at Green Hills Farm in Pennsylvania, where Buck lived from 1934 and which is today the headquarters for Pearl S. Buck International. (Image courtesy of Pearl S. Buck International, www.pearlsbuck.org.)
Richard J. Walsh—Buck's publisher and second husband—pictured in China with an unidentified rickshaw man. Walsh's tweed suit and pipe are typical of his signature daily attire.
Buck receiving her Nobel Prize from the King of Sweden, Gustav V, in 1938. (Image courtesy of Pearl S. Buck International, www.pearlsbuck.org.)
Buck and Walsh with their daughter, Elizabeth.
Buck in the 1930s.
Walsh—with his ever-present pipe—pictured with an unidentified child.
Buck addresses an audience in Korea in 1964, discussing the issues of poverty and discrimination faced by children in Asia. She established the Orphanage and Opportunity Center in Buchon City, Korea, in 1965.
Buck in her fifties.
This family photograph was taken on Buck's seventieth birthday, June 26, 1962. The gathering included Buck's children, grandchildren, and some of the children supported by Pearl S. Buck International.
Buck on her seventieth birthday.
Pearl Buck's legacy lives on through Pearl S. Buck International, a non-profit organization dedicated to humanitarian causes around the world.
All rights reserved under International and Pan-American Copyright Conventions. By payment of the required fees, you have been granted the non-exclusive, non-transferable right to access and read the text of this ebook onscreen. No part of this text may be reproduced, transmitted, downloaded, decompiled, reverse engineered, or stored in or introduced into any information storage and retrieval system, in any form or by any means, whether electronic or mechanical, now known or hereinafter invented, without the express written permission of the publisher.
This is a work of fiction. Names, characters, places, and incidents either are the product of the author's imagination or are used fictitiously. Any resemblance to actual persons, living or dead, businesses, companies, events, or locales is entirely coincidental.
Copyright © 1951 by Pearl S. Buck
Cover design by Alexander Doolan
978-1-4804-2115-8
This edition published in 2013 by Open Road Integrated Media
345 Hudson Street
New York, NY 10014
www.openroadmedia.com
# EBOOKS BY PEARL S. BUCK
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config,
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Q: How to get location lat and long from json in flutter I am retrieve value latitude and longitude from JSON in flutter.I try many thing but not git value of lat and long
this is my json file
{
"results": [
{
"geometry": {
"location": {
"lat": 19.2058593,
"lng": 72.86612
}
}
}
] }
I try
> var myMap = json.decode(data.toString());
> var myName = myMap['results']['geometry']['location']['lat']
but not getting location values.please help me in flutter
A: Change it to :
myName = myMap['results'][0]['geometry']['location']['lat']
This is because "geometry" is inside a list. It's at index[0] of this list, after accessing "results".
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This album reflects the new paths my life and my music have taken since my move to the San Francisco Bay Area. The move brought me to a rich and beautiful life with my wife Amy and my son Lev and into a wonderful community of family and friends. It also put me back in touch with two truly amazing musicians: Randy Porter, with whom I grew up in San Diego (and recorded my first record), and Akira Tana, with whom I studied as a grad student at Rutgers University. I met the talented and intuitive bassist John Wiitala on my first gig in the Bay Area, which happened to be my own wedding to Amy (with Lev listening in utero). I hope that you enjoy and are in some way touched by this music, which for me represents the convergence of so many meaningful paths in my life.
Reunion is a collection of original, swinging, straight-ahead jazz deeply rooted in the Hard-Bop tradition. The Powerhouse quintet is comprised of veteran jazz players and composers with a rich history and great chemistry.
Review by Brian Lynch: There is some music that brings you right in; the kind that from the first note you hear lets you know that you are going to be in for a most enjoyable and rewarding listening experience. Such is the music on this CD from the estimable fivesome that has come together here as the Powerhouse Quintet. Their music has that quality of making you want to put aside whatever else you're doing and dig their hip sounds.
Brian Lynch was a member of the legendary Art Blakey and the Jazz Messengers, recording three albums with the Messengers and staying with Art until his death in 1990. He was instrumental in the formation of the acclaimed Eddie Palmieri Afro-Caribbean Jazz Octet and has recently recorded with The Phil Woods Quintet, The Toshiko Akiyoshi Orchestra, and the Artist formerly known as Prince. Brian's most recent sixth solo CD, "Spheres of Influence" made it to the top of the jazz radio charts.
Our musical adventure started 25 years ago when we (Bryant, Steve, and Bim),while still in high school, discovered jazz and decided to form our own band. Having all grown up together in Del Mar, a small beach community just north of San Diego, our friendship actually goes back even farther (grade school). In August of 1978, after graduation, the three of us moved to Boston to continue our musical studies; Steve and Bryant attended the Berklee College of Music while Bim worked gigs around the Boston area. Over the years, we all continued to play professionally but went in our own directions, settling in different parts of the country. Only on the rare occasion when our travels coincided did we have the opportunity to perform together. In August of 2000, however, we were invited to play as the Del Mar All-Stars in a jazz concert series back in our home town. While Steve was still living in San Diego at the time, Bim traveled from New York and Bryant from Portland, Oregon, for the reunion. The experience was something of a revelation: despite our different paths, we found there was an uncanny empathy in our musical sensibilities, and this ignited a desire to continue performing together. With the help of our close friend, San Diego guitarist Peter Sprague, we returned to San Diego to record our first CD as "The Powerhouse Quintet" at Spragueland Studios. Our name is taken from Powerhouse Park where we performed some of our first concerts together 25 years ago as well as our two recent reunion concerts.
The music reflects our similar backgrounds which are rooted in John Coltrane, Miles Davis, Sonny Rollins, and others of that generation. We hope you enjoy our music and we look forward to many more reunions in the near future.
"Bop Mambo" is a high energy collection of original and standard tunes which remains true to both the Latin and Jazz traditions, featuring some of the best musicians in the LA-SD area.
With everybody and his brother releasing Latin jazz CDs these days, it is a great pleasure to discover someone who honors both traditions, plays and writes creatively, and is smart enough to surround himself with the best musicians available - I'm talking about Steve Feierabend.
There is no longer any doubt that a Latin jazz renaissance is under way on the West Coast. The torch has been passed, from Cal, Mongo, Willie, and Poncho, to the younger cats, who have given a new twist to la musica. The LA-San Diego Latin jazz scene has been especially productive, as Steve's new CD displays.
Living in San Francisco, I often look south with envy, at the tremendous wealth of musicians there, many of whom are featured on Steve's recording. Two of my favorite pianists, Chris Barron and Lynn Willard, shine throughout. Chris plays with the Banda Brothers, another group that is pointing the way to the future, and Lynn plays with Jack Costanzo's band.
Rene Camacho, bassist extraordinaire, a regular with Susie Hansen's great band, anchors a superb rhythm section. He lists Celia Cruz, Jack Costanzo, The Pointer Sisters, and Justo Almario's groundbreaking band Tolu in his credits. He is joined by drummer Mike Holguin, from San Diego, who has played with Son Mayor, and Celia Cruz; and George Ortiz, the current timbalero for Poncho Sanchez, and a former member of Son Mayor.
Gilbert Castellano is a great straight ahead trumpet player, evidenced by his work with Anthony Wilson's nonet, and the Clayton/Hamilton Band. He also has his own CD out, with a group called the B3 Quartet.
Gunnar Biggs is a veteran of Buddy Rich's band and has played with Mose Allison and Butch Lacey.
Duncan Moore has recorded with Mike Wofford and Bill Cunliffe, and Tommy Aros is currently with Latin pop star Luis Miguel.
I'm partial to playing standards in clave, and Steve does it just right with Lerner & Lowe's "I've Grown Accustomed To Her Face." Great solos by Steve, pianist Chris Barron, and timbalero George Ortiz. The whole track builds and burns.
"The Well" is a hypnotic blend of Coltrane and Cuba.
Trumpet player Gilbert Castellanos makes a fiery appearance on "Bop Mambo."
"Amique" is built on an off-center, but infectious 6/8 rhythm.
The CD closes with Coltrane's "Wise One." I've always felt that the last tune on a CD should be as strong as the first cut, and Steve evidently believes the same. Both he and Chris play thoughtful solos over a quiet 6/8 rhythm.
Steve's credits include graduating with honors from Berklee, where he studied with the great teacher and trumpet player Herb Pomeroy, as well as graduate studies with Kenny Barron and Akira Tana at Rutgers. Steve has worked with Charles McPherson, Brian Lynch, Bill Cunliffe and many other notables in both the worlds of jazz and Latin music. Steve writes great tunes and now lives in the Bay Area. San Diego's loss, San Francisco's gain - welcome, Steve!
Mark Levine is a world-class pianist, bandleader, composer, author, and educator. He was an integral part of the West Coast Latin jazz scene in its golden age of the 60's and 70's playing and recording with legends like Cal Tjader, Mongo Santamaria, and Willie Bobo as well as many others.Mark continues to be at the forefront of the Latin jazz scene with his band The Latin Tinge who have a new cd on Left Coast Clave records called "Isla".
"Revolving Doors" is a thoughtful, introspective collection of creative original straight-ahead jazz. It features the best jazz musicians in the SD area as well as the great guitar playing of special guest Peter Sprague. The recording quality is beautiful as is the art design and no-jewel-case packaging. | {
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{"url":"https:\/\/stemcie.com\/view\/39","text":"$\\require{\\cancel}$ $\\require{\\stix[upint]}$\n\n### MATHEMATICS 9709\n\n#### Cambridge International AS and A Level\n\n Name of student Date Adm. number Year\/grade Stream Subject Probability & Statistics 1 (S1) Variant(s) P63 Start time Duration Stop time\n\nQtn No. 1 2 3 4 5 6 7 8 9 10 11 12 Total\nMarks 4 5 6 5 6 5 10 6 8 8 8 11 82\nScore\n\nGet Mathematics 9709 Topical Questions (2010-2021) for $14.5 per Subject. Attempt all the 12 questions Question 1 Code: 9709\/63\/M\/J\/17\/1, Topic: Probability A biased die has faces numbered 1 to 6. The probabilities of the die landing on 1,3 or 5 are each equal to$0.1$. The probabilities of the die landing on 2 or 4 are each equal to$0.2$. The die is thrown twice. Find the probability that the sum of the numbers it lands on is 9.$[4]$Question 2 Code: 9709\/63\/M\/J\/17\/2, Topic: Discrete random variables The probability that George goes swimming on any day is$\\frac{1}{3}$. Use an approximation to calculate the probability that in 270 days George goes swimming at least 100 times.$[5]$Question 3 Code: 9709\/63\/M\/J\/18\/2, Topic: The normal distribution The random variable$X$has the distribution$\\mathrm{N}\\left(-3, \\sigma^{2}\\right)$. The probability that a randomly chosen value of$X$is positive is$0.25$.$\\text{(i)}$Find the value of$\\sigma$.$[3]\\text{(ii)}$Find the probability that, of 8 random values of$X$, fewer than 2 will be positive.$[3]$Question 4 Code: 9709\/63\/M\/J\/15\/3, Topic: Discrete random variables On a production line making cameras, the probability of a randomly chosen camera being substandard is$0.072$. A random sample of 300 cameras is checked. Find the probability that there are fewer than 18 cameras which are substandard.$[5]$Question 5 Code: 9709\/63\/M\/J\/16\/3, Topic: Discrete random variables Two ordinary fair dice are thrown. The resulting score is found as follows. If the two dice show different numbers, the score is the smaller of the two numbers. If the two dice show equal numbers, the score is 0.$\\text{(i)}$Draw up the probability distribution table for the score.$[4]\\text{(ii)}$Calculate the expected score.$[2]$Question 6 Code: 9709\/63\/M\/J\/20\/3, Topic: The normal distribution In a certain town, the time,$X$hours, for which people watch television in a week has a normal distribution with mean$15.8$hours and standard deviation$4.2$hours.$\\text{(a)}$Find the probability that a randomly chosen person from this town watches television for less than 21 hours in a week.$[2]\\text{(b)}$Find the value of$k$such that$\\mathrm{P}(X < k)=0.75$.$[3]$Question 7 Code: 9709\/63\/M\/J\/12\/4, Topic: Discrete random variables The six faces of a fair die are numbered$1,1,1,2,3,3$. The score for a throw of the die, denoted by the random variable$W$, is the number on the top face after the die has landed.$\\text{(i)}$Find the mean and standard deviation of$W$.$[3]\\text{(ii)}$The die is thrown twice and the random variable$X$is the sum of the two scores. Draw up a probability distribution table for$X$.$[4]\\text{(iii)}$The die is thrown$n$times. The random variable$Y$is the number of times that the score is 3. Given that$\\mathrm{E}(Y)=8$, find$\\operatorname{Var}(Y)$.$[3]$Question 8 Code: 9709\/63\/M\/J\/19\/4, Topic: Permutations and combinations$\\text{(i)}$Find the number of ways a committee of 6 people can be chosen from 8 men and 4 women if there must be at least twice as many men as there are women on the committee.$[3]\\text{(ii)}$Find the number of ways a committee of 6 people can be chosen from 8 men and 4 women if 2 particular men refuse to be on the committee together.$[3]$Question 9 Code: 9709\/63\/M\/J\/16\/5, Topic: The normal distribution The heights of school desks have a normal distribution with mean$69 \\mathrm{~cm}$and standard deviation$\\sigma \\mathrm{cm}$. It is known that$15.5 \\%$of these desks have a height greater than$70 \\mathrm{~cm}$.$\\text{(i)}$Find the value of$\\sigma$.$[3]$When Jodu sits at a desk, his knees are at a height of$58 \\mathrm{~cm}$above the floor. A desk is comfortable for Jodu if his knees are at least$9 \\mathrm{~cm}$below the top of the desk. Jodu's school has 300 desks.$\\text{(ii)}$Calculate an estimate of the number of these desks that are comfortable for Jodu.$[5]$Question 10 Code: 9709\/63\/M\/J\/17\/5, Topic: Discrete random variables Hebe attempts a crossword puzzle every day. The number of puzzles she completes in a week ( 7 days) is denoted by$X$.$\\text{(i)}$State two conditions that are required for$X$to have a binomial distribution.$[2]$On average, Hebe completes 7 out of 10 of these puzzles.$\\text{(ii)}$Use a binomial distribution to find the probability that Hebe completes at least 5 puzzles in a week.$[3]\\text{(iii)}$Use a binomial distribution to find the probability that, over the next 10 weeks, Hebe completes 4 or fewer puzzles in exactly 3 of the 10 weeks.$[3]$Question 11 Code: 9709\/63\/M\/J\/14\/6, Topic: Discrete random variables Tom and Ben play a game repeatedly. The probability that Tom wins any game is$0.3$. Each game is won by either Tom or Ben. Tom and Ben stop playing when one of them (to be called the champion) has won two games.$\\text{(i)}$Find the probability that Ben becomes the champion after playing exactly 2 games.$[1]\\text{(ii)}$Find the probability that Ben becomes the champion.$[3]\\text{(iii)}$Given that Tom becomes the champion, find the probability that he won the 2nd game.$[4]$Question 12 Code: 9709\/63\/M\/J\/15\/7, Topic: Permutations and combinations Rachel has 3 types of ornament. She has 6 different wooden animals, 4 different sea-shells and 3 different pottery ducks.$\\text{(i)}$She lets her daughter Cherry choose 5 ornaments to play with. Cherry chooses at least 1 of each type of ornament. How many different selections can Cherry make?$[5]$Rachel displays 10 of the 13 ornaments in a row on her window-sill. Find the number of different arrangements that are possible if$\\text{(ii)}$she has a duck at each end of the row and no ducks anywhere else,$[3]\\text{(iii)}$she has a duck at each end of the row and wooden animals and sea-shells are placed alternately in the positions in between.$[3]\\$\n\nWorked solutions: P1, P3 & P6 (S1)\n\nIf you need worked solutions for P1, P3 & P6 (S1), contact us @ [email\u00a0protected] | +254 721 301 418.\n\n1. Send us the link to these questions ( https:\/\/stemcie.com\/view\/39 ).\n2. We will solve the questions and provide you with the step by step worked solutions.\n3. We will then schedule a one to one online session to take you through the solutions (optional).","date":"2022-05-22 19:29:34","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.2689642310142517, \"perplexity\": 3145.7054794968844}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-21\/segments\/1652662546071.13\/warc\/CC-MAIN-20220522190453-20220522220453-00154.warc.gz\"}"} | null | null |
Q: ¿Porque la propiedad sort de MatTable me aparece como udenfinied? Estoy trabajando con una tabla de Angular Material, pero cuando defino la variable:
@ViewChild(MatSort, {static: true})
sort: MatSort = new MatSort();
Para posteriormente usarla en este metodo de ordenamiento
public sortController(event: any): void {
console.log('Controller inside....')
this.sort.sort(<MatSortable>({ id: '', }) as MatSortable);
switch (event) {
case 'recent':
this.sort.sort(<MatSortable>({ id: 'date', start: 'desc' }) as MatSortable);
break;
case 'old':
this.sort.sort(<MatSortable>({ id: 'date', start: 'asc' }) as MatSortable);
break;
}
this.dataSource.sort = this.sort;
}
Me sale el error que se aprecia en la imagen, me pueden decir ¿por qué la variable sort esta undefinied? Si incluso la inicialice al principio de mi código
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Items related to Corduroy's Busy Street
Freeman, Don; McCue, Lisa Corduroy's Busy Street
Corduroy's Busy Street
Freeman, Don; McCue, Lisa
3.7 avg rating •
Publisher: Viking Juvenile, 1987
Corduroy knows everyone on his street, including the mailman, the policeman, the paper girl, and his favorite, the ice cream truck driver.
Don Freeman was born in San Diego, California, in 1908. At an early age, he received a trumpet as a gift from his father. He practiced obsessively and eventually joined a California dance band. After graduating from high school, he ventured to New York City to study art under the tutelage of Joan Sloan and Harry Wickey at the Art Students' League. He managed to support himself throughout his schooling by playing his trumpet evenings, in nightclubs and at weddings.
Gradually, he eased into making a living sketching impressions of Broadway shows for The New York Times and The Herald Tribune. This shift was helped along, in no small part, by a rather heartbreaking incident: he lost his trumpet. One evening, he was so engrossed in sketching people on the subway, he simply forgot it was sitting on the seat beside him. This new career turned out to be a near-perfect fit for Don, though, as he had always loved the theater.
He was introduced to the world of childrens literature when William Saroyan asked him to illustrate several books. Soon after, he began to write and illustrate his own books, a career he settled into comfortably and happily. Through his writing, he was able to create his own theater: "I love the flow of turning the pages, the suspense of what's next. Ideas just come at me and after me. It's all so natural. I work all the time, long into the night, and it's such a pleasure. I don't know when the time ends. I've never been happier in my life!"
Don died in 1978, after a long and successful career. He created many beloved characters in his lifetime, perhaps the most beloved among them a stuffed, overall-wearing bear named Corduroy.
Don Freeman was the author and illustrator of many popular books for children, including Corduroy, A Pocket for Corduroy, and the Caldecott Honor Book Fly High, Fly Low.
1. Corduroy's Busy Street
Don Freeman, Lisa McCue
Published by Viking Juvenile (1987)
Book Description Viking Juvenile, 1987. Board book. Condition: New. Seller Inventory # DADAX0670814962
2. Corduroy's Busy Street Freeman, Don and McCue, Lisa | {
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\section{Introduction}\label{Introduction}
The concept of plasma-based particle accelerators, capable of accelerating charged particles to extremely high energies (beyond 10 TeV) has been put forward more than three decades ago and massive theoretical, computational and experimental efforts have been made in this field since then. One of the proposed schemes, often referred to as the plasma wake field accelerator (PWFA) consists in the injection of a relativistic electron (or positron) beam into the plasma, which creates a vacancy in the electron population of the background plasma while the plasma ions remain in place due to their large mass. After the passage of the beam{, usually referred to as the driving beam or simply the driver,} {in the wake of the latter}, plasma electrons rush in to restore the quasineutrality which produces a very large amplitude electrostatic plasma wave that propagates with the same velocity as the beam that has created it. The longitudinal electric field of such plasma wave can be used to accelerate particles to the energies that are higher that that of the driving beam. When the latter is properly bunched{, and the beam density is of the order of $10^{17}-10^{18}\, {\rm cm}^{-3}$,} an acceleration gradient {ranging from $0.01$ to $1$ GeV/cm} can be achieved, yielding the energy of the accelerated particles to be $3-4$ times bigger that that of the driving beam \cite{1985PhRvL..54..693C}. The dynamics of the driving beam in the direction transverse to its propagation is important not only for the feasibility of such scheme, but also for the quality ({i.e., lower emittance and higher brightness}) of the relativistic particle bunch generated in it. A high-intensity electron beam of finite extent can be subjected longitudinally to a two-stream instability, while in the transverse direction it can undergo self-focusing or defocusing, filamentation, and self-pinching. As for the quality of the bunch produced {this way}, transverse PWF can be used for its focussing {which contrasts the spreading due to the thermal emittance of the bunch. However, more generally, the concomitant presence of both {the} transverse PWF and thermal spreading of the bunch can be adequately used for the bunch manipulation. In particular, for sufficiently long bunches the PWF acts on the driving bunch itself, leading to the self consistent PWF excitation ({the} feedback of the wake field on the driver). Therefore, in such a physical {situation} the concomitance of transverse PWF and thermal spreading leads to the beam self modulation. On longer time scales, under suitable conditions, the latter can evolve in an unstable way (self modulation instability)}.
In the early theoretical works \cite{1985PhRvL..54..693C} the feedback of the wake field on the driver and the resulting spatial evolution of the beam were not considered. Later, a selfconsistent theory based on the thermal wave model (TWM) \cite{1992PhRvA..45.4045F} was presented and applied to the study of the interaction between the plasma wake field and the driving relativistic electron beam, in an unmagnetized overdense plasma (whose density exceeds that of the beam) and in the case of a long beam ($L_\Vert\gg L_\bot$, where $L_\Vert$ and $L_\bot$ are its longitudinal and transverse characteristic lengths, respectively). Such model successfully described the beam's self-focusing and self-pinching equilibria, when the spot size of the beam was larger and smaller than the wavelength of the wake field, respectively. {More recently, this approach has been extended to the case of self consistent plasma wake field (PWF) excitation in a cold magnetized plasma for both long nonlaminar warm relativistic driving beam (quantum-like domain of TWM)
\cite{Fedele_et_al.2011,Tanjia_et_al.2011}
and relatively cold relativistic quantum driving beam (quantum domain)
\cite{2012PhPl...19j2106F,2012EL....10055002J}.}
Similarly, the self-modulation of a relativistic charged-particle beam as {\textit{thermal matter wave envelope}} and the possibility of its destabilization was studied in Ref. \cite{2014JPhCS.482a2014F} using the quantum-like description provided by the {TWM} in which the beam dynamics is governed by a Zakharov-type system of equations, comprising a Poisson-like equation for the wake potential and a nonlinear Schr\"{o}dinger equation governing the spatiotemporal evolution of the thermal matter wave envelope, whose dispersion coefficient is proportional to the beam thermal emittance. In the strongly nonlocal regime, an Ermakov-Pinney type equation
for the evolution of the beam's cross-section was derived, and the possibilities were discussed of an instability leading to the beam blowup and of a stable self-modulation in the form of sausage-like transverse oscillations. {Fully similar results have been obtained by using a kinetic approach, where the study of the self consistent PWF excitation has been provided by the Vlasov-Poisson-type system of coupled equations
\cite{2014EPJD...68..210F,2014EPJD...68..271F}.
In addition, with the same kinetic approach, the self-modulation instability has been successfully described in recent works
\cite{2016NIMPA.829..426A}.}
In the present paper we derive a hydrodynamic description for the interaction between a long relativistic electron beam with an unmagnetized, overdense plasma. {Similarly} to the thermal wave model, the random component of the transverse motion of the beam particles is modelled by the finite temperature of the fluid, which is taken to be anisotropic. We demonstrate that in the case of a small but finite variation in the direction of the propagation, a nonlinearity associated with the adiabatic thermodynamic evolution of the beam produces the localization of the latter. Such coherent structure is in the state of an unstable equilibrium that, upon a small perturbation, may either collapse or {disperse} when the number of beam particles is, respectively, bigger or smaller than that in the stationary state. However, a fine tuning of the initial beam profile may sufficiently postpone these instabilities until other processes, not included in our model, enter the picture and determine the actual evolution of the beam. {However, for a thermodynamic process that is predominantly 2-D the related nonlinearity vanishes and, on the long timescale, the system is governed by the {nonlinear plasma wake force (given by the gradient of the effective potential)} that is responsible also for the betatron-like oscillations observed in the earlier works \cite{2014EPJD...68..210F,2014EPJD...68..271F,2014JPhCS.482a2014F,2016NIMPA.829..426A}.}
\section{Fluid and Vlasov descriptions}\label{relativisticVlasov}
The system {under study} consists of a quiescent plasma pierced by a relativistic beam of electrons, denoted in the rest of the paper by the subscripts $p$ and $b$, respectively. The beam propagates along the $z$ axis with a relativistic speed $u$, $c-u\ll c$ and it is relatively long, $L_{b\Vert}\gg L_{b\bot}$, where $L_{b\Vert}$ and $L_{b\bot}$ are the characteristic lengths of inhomogeneity of the beam in the directions parallel and perpendicular to its direction of propagation, respectively. Expressing the electric and magnetic field in terms of the electrostatic and vector potentials, $\vec E = -\nabla\phi - \partial\vec A/\partial t$ and $\vec B=-\nabla\times\vec A$ and using the Coulomb gauge $\nabla\cdot\vec A=0$, the Poisson's equation and the Ampere's law take the familiar form
\begin{eqnarray}
&& \label{Poissons}
\nabla^2\phi = \left(e/\varepsilon_0\right)\left(n_b + n_p - n_i\right),
\\
&& \label{Amperes}
\left({\partial^2}/{\partial t^2}-c^2\nabla^2\right)\vec A = -\nabla\;\partial\phi/\partial t-\left({e}/{\varepsilon_0}\right)\left(n_b \vec v_b + n_p \vec v_p\right),
\end{eqnarray}
where $-e$ is the electron charge, $n_b$, $n_p$, and $n_i$ are the densities of the beam and plasma electrons, and of the plasma ions, $\vec v_b$, $\vec v_p$ are the hydrodynamic velocities of the beam and plasma electrons, while the temporal variation is considered as sufficiently rapid so that the plasma ions are immobile.
We restrict our study to regimes in which the temporal variation in the reference frame comoving with the beam is sufficiently slow and we conveniently introduce the variables
\begin{equation}\label{variables}
\vec{r}\,' = \vec{r} - \vec{e}_z \, u \, t, \quad \quad t'= t,\quad \quad \vec v_b\!^\prime = \vec v_b - \vec e_z u,
\end{equation}
which implies ${\partial}/{\partial t} = {\partial}/{\partial t'} - u \; {\partial}/{\partial z'}$. Furthermore, we assume that the electric field is sufficiently weak so that the fluid velocities $\vec v_p$ and $\vec v_b\!^\prime$ are nonrelativistic, i.e. we adopt the following scaling
\begin{equation}\label{nonrelativistic_fluid}
\partial/\partial t'\sim \vec v_b\!^\prime\cdot\nabla \sim \vec v_p\cdot\nabla \sim\epsilon \, u\;\partial/\partial z, \quad {\rm where} \quad \epsilon\ll 1.
\end{equation}
The plasma density $n_p$ and fluid velocity $\vec v_p$ are readily calculated from the appropriate hydrodynamic equations. We consider a regime in which the density of the beam scales as $n_b\sim n_p-n_i \sim n_i \, v_p/c$ and, as a consequence, the Lorentz forces acting on an electron in the beam does not exceed the Coulomb force, $|\vec e_z u \times \vec B_0|\lesssim |\vec E|$, provided $u\sim c$.
We take also that the plasma electrons are nonrelativistic and that the potential $\phi$ is sufficiently small,
\begin{equation}\label{scaling_phi}
{e\phi/m_0 u^2\lesssim \epsilon \, (\partial/\partial z)/\nabla_\bot,}
\end{equation}
$m_0$ being the electron rest mass, which permits us to neglect all convective nonlinearities and also the Lorentz force in the momentum equation for the plasma electrons. We study a regime in which the background plasma is cold, and the continuity and momentum equations for plasma electrons take the simple form
\begin{eqnarray}
&&\label{contin_p}
-u\;{\partial n_p}/{\partial z} + n_{p 0} \nabla\cdot\vec v_p = 0,
\\
&&\label{momentum_p}
-u\;{\partial \vec v_p}/{\partial z} = \left(e/m_0\right)\left(\nabla\phi - u \; \partial\vec A/\partial z\right),
\end{eqnarray}
$n_{p 0}$ being the unperturbed density of plasma electrons. Eliminating $\vec v_p$ and using the Poisson's equation (\ref{Poissons}), we have
\begin{equation}\label{first_p}
\left(\omega_{pp}^2 + u^2\,\partial^2/\partial z^2\right)\nabla^2\phi = \left(e/\varepsilon_0\right)u^2\,\partial^2 n_b/\partial z^2,
\end{equation}
where $\omega_{pp} = \sqrt{n_{p 0} e^2/m_0\varepsilon_0}$ is the plasma frequency of the background plasma. With the same accuracy, from $z$-components of the momentum equation (\ref{momentum_p}) yielding ${v_p}_z=(e/m_0 u)(\phi - u A_z)$, and of the Ampere's law (\ref{Amperes}) we get
\begin{equation}\label{second_p}
\left(\omega_{pp}^2 + u^2\,\partial^2/\partial z^2- c^2\nabla^2\right)\left(\phi-u A_z\right)+ c^2\nabla^2 \phi = \left(e/\varepsilon_0\right)u^2\, n_b.
\end{equation}
Finally, combining Eqs. (\ref{first_p}) and (\ref{second_p}) we obtain
\begin{equation}\label{final_p}
\left(\omega_{pp}^2 + u^2\,\partial^2/\partial z^2\right)\left(\omega_{pp}^2 + u^2\,\partial^2/\partial z^2 - c^2\nabla^2\right) \left(\phi - u A_z\right) = \left(e/\varepsilon_0\right)\left[\omega_{pp}^2 + \left(u^2-c^2\right)\partial^2/\partial z^2\right]u^2 n_b,
\end{equation}
which, for a long beam, $\nabla_\bot\gg\partial/\partial z$, whose velocity is relativistic, $u\approx c$, simplifies to the well known result
\begin{equation}\label{final_p_apr}
\left(\omega_{pp}^2 - c^2\nabla_\bot^2\right) \left(\phi - u A_z\right) = \left(e/\varepsilon_0\right)u^2 n_b,
\end{equation}
The density of the beam, $n_b$, is calculated from the well-known textbook expression for the relativistic Vlasov equation,
\begin{equation}\label{rel_Vlasov}
\frac{\partial f_b}{\partial t} + \vec{\rm v}\cdot\frac{\partial f_b}{\partial\vec{r}} - e\left(\vec{E} + \vec{\rm v}\times\vec{B}\right)\cdot\frac{\partial f_b}{\partial\vec{\rm p}} = 0,
\end{equation}
where $f_b(t,\vec{r},\vec{\rm p})$ is the distribution function of the beam electrons, while $\vec{\rm v}$ and $\vec{\rm p}$ are the electrons' velocity and momentum, that are mutually related as
\begin{equation}\label{vel_momen}
\vec{\rm v} = \frac{\vec{\rm p}}{m_0\sqrt{1 + {\rm p}^2/\left(m_0^2 c^2\right)}} \quad \Leftrightarrow \quad
\vec{\rm p} = \frac{m_0\vec{\rm v}}{\sqrt{1 - {\rm v}^2/c^2}}.
\end{equation}
However, solving the full Vlasov equation can be very tedious in a general case. In this paper, we do not account for the kinetic effects, most notably the wave breaking of the Langmuir wave associated with the plasma wake and the resonant particle effects (such as the trapping and acceleration of resonant electrons), and we solve instead the appropriate moments of the Vlasov equation (\ref{rel_Vlasov}). Multiplying the Vlasov equation (\ref{rel_Vlasov}) by $1$, $\vec{\rm p}$, and ${\rm v}_i\, {\rm p}_j$, respectively, and integrating for the entire momentum space, we obtain the relativistic fluid equations of continuity and momentum, and the equation for the $(i,j)$ component of the pressure tensor
\begin{eqnarray}
&&\label{rel_cont}
\partial n_b/\partial t + \nabla\cdot\left(n_b \vec v_b\right)=0
\\
&&\label{rel_moment}
\left(\partial/\partial t + \vec v_b\cdot\nabla\right)\vec p_b = -e\left(\vec E + \vec v_b\times \vec B\right)-\left(\nabla\cdot \mathbf{P}\right)/n_b,
\\
&&\label{rel_pres_tens}
\partial P_{k,l}/\partial t + \nabla\cdot\left(P_{k,l}\,\vec v_b\right) + P_{j,k}\;\partial {v_b}_l/\partial x_j + P_{j,l}\;\partial {v_b}_k/\partial x_j = 0,
\end{eqnarray}
where we have used the standard notation from tensor algebra $\nabla\cdot\mathbf{P} = \vec e_\alpha \, \partial P_{\alpha, \beta}/\partial x_\beta$. In the equations (\ref{rel_pres_tens}) for the stress tensor in a collisionless plasma, we have neglected both the thermal convection and the off-diagonal terms of $\mathbf{P}$, i.e. we have $\mathbf{P} = P_\bot(\mathbf{I} - \vec{e}_z \vec{e}_z) + P_\Vert\,\vec{e}_z \vec{e}_z$ where $\mathbf{I}$ is a unit tensor, viz. $I_{\alpha, \beta} = \delta_{\alpha, \beta}$ and $\delta_{\alpha, \beta}$ is the Kronecker delta. We rewrite these equations using the variables defined in Eq. (\ref{variables}) and introduce also $\vec v_b\!^\prime = \vec v_b - u\,\vec e_z$. When the fluid velocity $\vec v_b\!^\prime$ is nonrelativistic, in the momentum equation (\ref{rel_moment}) we may set
\begin{equation}\label{effective_E}
\vec E + \vec v_b\times \vec B \; \approx \; \vec E + u \,\vec e_z \times \vec B \; \approx \; - \nabla\left(\phi - u A_z\right),
\end{equation}
and our fluid equations{, rewritten through the variables defined in Eq. (\ref{variables}),} readily obtain the form
\begin{eqnarray}
&&\label{rel_cont_1}
\partial n_b/\partial t' + \nabla\cdot\left(n_b \vec v_b\!^\prime\right)=0
\\
&&\label{rel_moment_1}
\left(\partial/\partial t' + \vec v_b\!^\prime\cdot\nabla\right)\vec p_b = e\nabla\left(\phi - u A_z\right) - \left(\nabla_\bot P_\bot + \vec e_z\;\partial P_\Vert/\partial z \right)/n_b,
\\
&&\label{rel_pres_tens_1}
\partial P_\bot/\partial t' + \nabla\cdot\left(P_\bot\,\vec v_b\!^\prime\right) + P_\bot\nabla_\bot\cdot\vec v_b\!^\prime = 0,
\\
&&\label{rel_pres_tens_2}
\partial P_\Vert/\partial t' + \nabla\cdot\left(P_\Vert\,\vec v_b\!^\prime\right) + 2 P_\Vert \; {\partial v'_b}_z/\partial z = 0.
\end{eqnarray}
The pressures $P_\bot$ and $P_\Vert$ will be calculated using the Grad's approximation, expressed through the equilibrium solution of the Vlasov equation and using a limited number of its moments. As the equilibrium solution we use the J\"{u}ttner distribution with a particle drift, which is one of the several relativistic versions of the Maxwellian distribution function that can be found in the literature. In the case of an anisotropic electron temperature, it has the form \cite{2013PhPl...20d4501N}:
\begin{equation}\label{Juttner}
f_0\left(\vec{\rm p}_\bot, {\rm p}_z\right) = C \, \exp\left[-\alpha_\bot \gamma_u\left(\gamma - \beta_u \, {\rm p}_z/mc\right)-\left(\alpha_\Vert-\alpha_\bot\right)\sqrt{1 + \gamma_u^2\left({\rm p}_z/m_0 c - \beta_u\gamma\right)^2}\right],
\end{equation}
where
\begin{equation}\label{relativistic_notation}
\alpha_\bot = m_0 c^2/T_{0\bot}, \quad \alpha_\Vert = m_0 c^2/T_{0\Vert}, \quad \gamma = \sqrt{1+(\vec{\rm p}_\bot^{\,2} + {\rm p}_z^2)/m_0^2 c^2}, \quad \gamma_u = 1/\sqrt{1-\beta_u^2}, \quad \beta_u = u/c,
\end{equation}
and $C$ is determined from the normalization, $\int_{-\infty}^\infty d^3\vec{\rm p} \; f_0 = n_0$. $T_{0\bot}$ and $T_{0\Vert}$ are the perpendicular and parallel temperatures in the comoving frame \cite{1994tdcp.book.....R}.
For nonrelativistic temperatures $\alpha_{\bot, \Vert}\gg 1$, the J\" uttner distribution (\ref{Juttner}) reduces to a shifted Maxwellian, with appropriate relativistic corrections to the electron mass and temperature
\begin{equation}\label{shifted_Maxwellian}
f_0\left(\vec{\rm p}_\bot, {\rm p}_z\right) = \frac{n_0}{\left(2\pi m_0\right)^\frac{3}{2} T_{0\Vert}^\frac{1}{2} \, T_{0\bot} \,\gamma_u}\; \exp\left[-\frac{\vec{\rm p}_\bot^{\,2}}{2 m_0 T_{0\bot}} - \frac{\left({\rm p}_z - m_0 u \, \gamma_u\right)^2}{2 m_0 T_{0\Vert} \gamma_u^2}\right].
\end{equation}
A reliable approximation for the distribution function in the presence of finite potentials $\phi$ and $A_z$, referred to as the Grad's moment approximation, is obtained when in the equilibrium distribution (\ref{shifted_Maxwellian}) we make the substitutions $n_0\to n_b$, $\vec{\rm p}\to\vec{\rm p}-\vec p_b$, and $T_{0\bot,\Vert}\to T_{\bot,\Vert}$.
Perpendicular and parallel pressures $P_\bot$ and $P_\Vert$ are then readily calculated as the appropriate integrals of such distribution function, viz.
\begin{equation}\label{approx_pressures}
P_\bot = n_b T_\bot/\gamma_u \quad {\rm and} \quad P_{\Vert} = n_b T_\Vert \gamma_u.
\end{equation}
From these expressions for the perpendicular and perpendicular pressures we note that $P_{\Vert}/P_{\bot} \sim \gamma_u^2 \; T_{\Vert}/T_{\bot}$. This may be a very big number and permits us to estimate the relevant scalings of the beam dynamics in the following way:
\\
In the regime $n_b \, e (\phi-u A_z)\sim P_\bot\ll P_\Vert$, the parallel convection is negligible, viz. ${v_b}'_z\partial/\partial z \ll \vec v_b\!^\prime\cdot \nabla_\bot$, when
\begin{equation}\label{sc1}
\partial/\partial z \ll \left(T_\bot/T_\Vert\right)^\frac{1}{2}\gamma_u^{-1} \, \nabla_\bot
\end{equation}
Eqs. (\ref{rel_pres_tens_1}) and (\ref{rel_pres_tens_2}) for the pressures $P_\bot$ and $P_\Vert$, remain coupled even for a weak $z$-dependence, Eq. (\ref{sc1}), provided $P_\Vert \, \partial{v_b}'_z/\partial z \gtrsim P_\bot \nabla_\bot\cdot\vec v_b\!^\prime $, which corresponds to
\begin{equation}\label{sc2}
\partial/\partial z \gtrsim \left(T_\bot/T_\Vert\right)\gamma_u^{-2} \, \nabla_\bot
\end{equation}
Conversely, Eq. (\ref{rel_pres_tens_2}) for $P_\Vert$ is decoupled and can be discarded when $P_\Vert \, \partial{v_b}_z/\partial z \ll P_\bot \nabla_\bot\cdot \vec v_b\!^\prime $, which corresponds to
\begin{equation}\label{sc3}
\partial/\partial z \ll \left(T_\bot/T_\Vert\right)\gamma_u^{-2} \, \nabla_\bot.
\end{equation}
Finally, using $P_\Vert = \gamma_u^2 \, P_\bot(T_\Vert/T_\bot)$ with $T_\Vert/T_\bot \approx {\rm constant}$, and making an appropriate combination of Eqs. (\ref{rel_pres_tens_1}), (\ref{rel_pres_tens_2}), and (\ref{rel_cont_1}), we obtain
\begin{equation}\label{adiabatic_process}
\left(\partial/\partial t' + \vec v_b\!^\prime\cdot\nabla\right)\left(n_b^{-\kappa} \, P_\bot\right) = 0,
\end{equation}
which is, essentially, the equation of state for an adiabatic thermodynamic process. Here, from the scalings (\ref{sc2}) and (\ref{sc3}), we note that we have $\kappa = 5/3$ for a not too long beam (\ref{sc2}), and $\kappa = 2$ for an extremely long beam (\ref{sc3}).
\section{Numerical results}
Now, under the scalings (\ref{sc1}) and (\ref{sc2}), or (\ref{sc3}), we can rewrite our basic equations (\ref{final_p}) and (\ref{rel_cont_1})-(\ref{rel_pres_tens_2}) as
\begin{eqnarray}
&& \label{final_p_apr_F}
\left[\left(c^2/\omega_{pp}^2\right)\nabla_\bot^2 - 1\right] U_w = -\left(m_0 u^2 /n_{p0}\right)\,n_b,
\\
&&\label{rel_cont_F}
\partial n_b/\partial t' + \nabla_\bot\cdot\left(n_b \, \vec {v_b}^\prime_\bot\right)=0,
\\
&&\label{rel_moment_1_F}
m_0 \gamma_u \left(\partial/\partial t' + \vec {v_b}^\prime_\bot\cdot\nabla_\bot\right)\vec {v_b}^\prime_\bot = \nabla_\bot\left(U_w - \tau \, n_b^{\kappa-1}\right),
\end{eqnarray}
where $U_w = e(\phi-u A_z)$, $\tau = (T_{0\bot}/n_{b 0}^{\kappa-1})[\kappa/(\kappa-1)]$, and $\kappa$ takes the values $5/3$ and $2$ under the scalings (\ref{sc2}) and (\ref{sc3}), respectively. Using scaled variables $t''=t'/{\rm T}$, $\vec{v_b}^{\prime\prime}_\bot = \vec {v_b}^\prime_\bot/{\rm V}$, $\vec{r}\;^{\prime\prime}\!\!\!\!\!_\bot = \vec{r}_\bot/{\rm R}$, $U_w''=U_w/{\rm W}$, and $n_b''=n_b/{\rm N}$, where
\begin{equation}\label{normalizations}
{\rm N} = n_{bc}\left(\frac{n_{p0}}{n_{bc}}\;\frac{T_{bc}}{m_0 u^2}\;\frac{\kappa}{\kappa-1}\right)^\frac{1}{2-\kappa},\quad
{\rm W} = m_0 u^2\; \frac{\rm N}{n_{p0}}, \quad
{\rm V} = u \, \sqrt{\frac{\rm N}{\gamma_u n_{p0}}}, \quad
{\rm T} = \frac{c}{\omega_{pp} u}\,\sqrt{\frac{\gamma_u n_{p0}}{\rm N}}, \quad
{\rm R} = \frac{c}{\omega_{pp}},
\end{equation}
where $n_{bc}$ and $T_{bc}$ are the beam density and temperature in the centre of the structure, respectively, our equations (\ref{final_p_apr_F})-(\ref{rel_moment_1_F}) are rewritten in a dimensionless form an with no physical parameters involved, viz.
\begin{eqnarray}
&& \label{final_p_apr_F1}
\left(\nabla_\bot^2 - 1\right) U_w = -n_b,
\\
&&\label{rel_cont_F1}
\partial n_b/\partial t + \nabla_\bot\cdot\left(n_b \vec {\;v_b}_\bot\right)=0,
\\
&&\label{rel_moment_1_F1}
\left(\partial/\partial t + \!\vec {\;v_b}_\bot\cdot\nabla_\bot\right)\!\vec {\;v_b}_\bot = \nabla_\bot\left(U_w - n_b^{\kappa-1}\right),
\end{eqnarray}
where for simplicity, the notations $''$ are omitted hereafter. It is worth noting that in the derivation of the Poisson's-like equation (\ref{final_p_apr}) or (\ref{final_p_apr_F1}), which accounts for the plasma response to the beam density $n_b$, we have neglected small terms of the order $\partial/\partial t'$ and $\vec v_p\cdot\nabla$ (the latter are assumed to be of the same order as the convective terms $\vec v_b\!^\prime\cdot\nabla$ in the hydrodynamic equations for the beam). Thus, Eqs. (\ref{final_p_apr_F1})-(\ref{rel_moment_1_F1}) describe essentially an adiabatic evolution of the potential $U_w$ in which the plasma instantaneously responds to the slow changes of the beam density $n_b$.
A stationary solution (denoted with the subscript "$0$"), with $\partial/\partial t = \vec {v_b}_\bot = 0$ is readily obtained, substituting $n_{b 0} = U_{w 0}^\frac{1}{\kappa-1}$ into the Poisson's-like equation (\ref{final_p_apr_F1}), yielding an equation from which all physical parameters have been scaled out, viz.
\begin{equation}\label{hole-like}
\left(\nabla_\bot^2 - 1\right)U_{w 0} + U_{w 0}^\frac{1}{\kappa-1}=0.
\end{equation}
When the beam is extremely long, i.e. in the strictly 2D case (\ref{sc3}) with $\kappa=2$, the above Eq. (\ref{hole-like}) is linear {and can not possess a localized solution that is well-behaved at $r=0$. An appropriate localization in such strictly 2D regime is possible only in the presence of a finite fluid velocity, viz. $\vec {v_b}_\bot\ne 0$, when the convective nonlinearity associated with the curlfree component of $\vec {v_b}_\bot$ produces a {nonlinear plasma wake force} on the slow time scale. The latter drives the modulational instability which, eventually, saturates into a coherent soliton state described in Refs. \cite{2014EPJD...68..210F,2014EPJD...68..271F,2014JPhCS.482a2014F,2016NIMPA.829..426A}.}
Conversely, in the presence of a finite dependence along $z$, (\ref{sc2}), our Eq. (\ref{hole-like}) contains a nonlinear term $U_{w 0}^{3/2}$ which produces the localization of the solution. Physically, the localization comes from the 3-D adiabatic process described by the appropriate equation of state (\ref{adiabatic_process}).
The cylindrically symmetric solution of Eq. (\ref{hole-like}) is displayed in Fig. \ref{hole1}.
\begin{figure}[htb]
\centering
\includegraphics[width=70mm]{Cylindrical_Uw_nb.eps}
\caption{Cylindrically symmetric stationary structure, found as the solution of Eq. (\ref{hole-like}) for $U_{w0}$ (red) and the corresponding $n_{b0}$ (black), in a three-dimensional case with $\kappa = 5/3$.}
\label{hole1}
\end{figure}
The stability of the adiabatic evolution of such solution with respect to small perturbations can be assessed if we linearize Eqs. (\ref{final_p_apr_F1})-(\ref{rel_moment_1_F1}) around the stationary solution, setting $n_b=n_{b 0} + \epsilon n_{b1}$, $\!\vec {\; v_b}_\bot = \epsilon \!\!\! \vec {\;\; v_{b1}}_\bot$, and $U_w = U_{w 0} + \epsilon U_{w 1}$, when to the first order in the (small) bookkeeping parameter $\epsilon$ we obtain
\begin{eqnarray}
&&\label{veza_U1_n1}
n_{b1} = -\left(\nabla_\bot^2 - 1\right) \, U_{w1},
\quad {\rm where} \quad
\nabla_\bot^2 = \frac{\partial^2}{\partial r^2} + \frac{1}{r}\frac{\partial}{\partial r}
\\
&&\label{perturbed}
-\omega^2 n_{b1} +\frac{1}{r}\frac{\partial}{\partial r}\left\{n_{b0}\, r \, \frac{\partial}{\partial r}\left[U_{w1}-\left(\kappa-1\right)n_{b0}^{\kappa - 2} n_{b1}\right]\right\} = 0.
\end{eqnarray}
Here we have applied the Fourier transformation in time and assumed a cylindrically symmetric perturbation. Multiplying Eq. (\ref{perturbed}) by $r \, [U_{w1}^*-(\kappa-1)\, n_{b0}^{\kappa - 2} n_{b1}^*]$, where $^*$ denotes the complex conjugate, doing some simple manipulations and taking that all quantities vanish at $r\to\infty$, we arrive at
\begin{equation}\label{omega_estimate}
\omega^2 = -\frac{\int_0^\infty{r\, dr \, n_{b0}\left|\left(\partial/\partial r\right)\left[U_{w1}-\left(\kappa-1\right)n_{b0}^{\kappa-2}n_{b1}\right]\right|^2}}
{\int_0^\infty{r\, dr \, \left[\left|U_{w1}\right|^2 + \left|\partial U_{w1}/\partial r\right|^2-\left(\kappa-1\right)n_{b0}^{\kappa-2}\left|n_{b1}\right|^2\right]}},
\end{equation}
Obviously, the right-hand-side of Eq. (\ref{omega_estimate}) is a real quantity and its numerator is positive definite, which implies that small perturbations
of the stationary solution are either purely growing/damped ($\omega^2 <0$) or purely oscillating functions of time ($\omega^2 >0$), depending on the sign of the {denominator}. Using the following simple estimate, we can indicate that both oscillating and growing/damped solutions may exist. If we estimate the stationary beam profile by a rectangular shape, $n_{b0}(r)\approx n_{bm}\, h(r_m-r)$ with $n_{bm}\approx 4$ and $r_m\approx 2.5$, we can rewrite Eq. (\ref{perturbed}) simply as
\begin{eqnarray}
&&\nonumber
\left(\nabla_\bot^2+k_+^2\right)\left(\nabla_\bot^2+k_-^2\right)\, U_{w1} = 0, \quad {\rm when} \quad r<r_m
\\
&&\label{rectangular_nb0}
\left(\nabla_\bot^2-1\right)\, U_{w1} = 0, \quad {\rm when} \quad r>r_m,
\end{eqnarray}
where
\begin{equation}\label{wavenumbers}
k_\pm^2 = \frac{1}{2\left(\kappa-1\right)n_{b m}^{\kappa-1}}\left\{\omega^2+n_{b m}-\left(\kappa-1\right)n_{b m}^{\kappa-1}\pm
\sqrt{\left[\omega^2+n_{b m}-\left(\kappa-1\right)n_{b m}^{\kappa-1}\right]^2 + 4 \, \omega^2\left(\kappa-1\right)n_{b m}^{\kappa-1}}\right\}.
\end{equation}
A small perturbation is now readily written as $U_{w1} = a_+ J_0(k_+ r) + a_- J_0(k_-r)$ for $r<r_m$ and $U_{w1} = b\, K_0(r)$ for $r>r_m$, where $a_\pm$ and $b$ are constants of integrations. From the requirements that $U_{w1}$, $\partial U_{w1}/\partial r$, and $\nabla_\bot^2 U_{w1}$ are continuous functions at the edge of the beam, $r=r_m$ (i.e. that the Coulomb and pressure forces on the electron fluid are finite and that there are no surface charges at the edge of the beam) and eliminating the constants of integration $a_\pm$ and $b$ we readily obtain the dispersion relation for the frequency of the linear mode $D(\omega^2)=0$, where
\begin{equation}\label{disprel}
D\left(\omega^2\right) = k_-\left(1+k_+^2\right)\frac{J_1\left(k_- r_m\right)}{J_0\left(k_-r_m\right)} +
k_+\left(1+k_-^2\right)\frac{J_1\left(k_+ r_m\right)}{J_0\left(k_+ r_m\right)} -
\left(k_-^2 - k_+^2\right)\frac{K_1\left(r_m\right)}{K_0\left(r_m\right)}.
\end{equation}
\begin{figure}[htb]
\includegraphics[width=70mm]{dispersion_function.eps}
\caption{Real (black) and imaginary (red) parts of the approximative dispersion function $D(\omega^2)$, defined in Eq. (\ref{disprel}). The equation $D(\omega^2)=0$ reveals the existence of two unstable modes, with $\omega^2<0$, and a sequence of oscillation modes with $\omega^2>0$.} \label{disp_func}
\end{figure}
Dispersion function $D(\omega^2)$ is displayed in Fig. \ref{disp_func}. We can readily see that there exist two unstable modes featuring $\omega^2<0$, with the more unstable one corresponding to the branch point of the characteristic wavenumbers $k_\pm$, see Eq. (\ref{wavenumbers}). Besides these, owing to the periodicity of the Bessel functions $J_0$ and $J_1$, there is also a sequence of oscillating solutions with purely real frequencies, i.e. with $\omega^2>0$. A numerical solution of the dispersion relation $D(\omega^2) = 0$, using Eq. (\ref{wavenumbers}) with $n_{bm}= 4$ and $r_m= 2.5$ yields $\omega^2 = -9, \; -0.633, \; 1.820, \; 8.415, \; 18.49, \; 31.82, \; 48.36, \;...$ . Using the normalizations defined in Eq. (\ref{normalizations}) and $\kappa = 3/2$, we find that the effective width of the stationary state of the beam is $r_m \sim 2.5 \; c/\omega_{pp}$, while the characteristic frequencies and growthrates of its linear perturbations scale (in physical units) as $\omega \sim \omega_{pp} (n_{p0}/\gamma_u\, n_{b0})^\frac{1}{2}(T_{b0}/m_0 c^2)\sim \omega_{pp}/\gamma_u^2$.
We have also solved numerically the full system of equations (\ref{final_p_apr_F1})-(\ref{rel_moment_1_F1}), see Figs. \ref{DG} and \ref{DG2}. {The solution appears to be spreading when the beam contains less electrons than the stationary state shown in Fig. \ref{hole1}, and collapsing when it contains more electron than the stationary state. Conversely, when we adopted an initial condition that contained the same number of electrons as the stationary state but with a different spatial profile, the solution remained stable for a considerably longer time, and even performed a limited number of oscillations (one or two) before the final collapse or spreading took place.} Such behavior comes from the fact that, with our somewhat arbitrary choice of the initial beam profile, besides the stable oscillating linear modes we have inevitably excited also the linearly unstable modes that eventually prevailed in our numerical example. However, the growthrate scales as $\sim\omega_{pp}/\gamma_u^2$ and in an experiment we can finely tune the initial beam profile so that such blowup or dispersion of the beam (i.e. the selffocussing or defocussing) is sufficiently delayed so that other physical effects enter the picture, including the two-stream instability, longitudinal (paraxial) variation, {and the nonlinear plasma wake force that can be responsible for the creation of different coherent nonlinear states such as 2D solitons.}
{It is important to note that the maximum value (in physical units) of the stationary solution (\ref{hole-like}), displayed in Fig. \ref{hole1}, is given by $n_{bc}\approx 4 N$. Using the normalizations Eq. (\ref{normalizations}), this readily yields the condition for the stationary state in the form $T_{bc}/m_0 c^2 = C\; n_{bc}/n_{p0}$ with $C\approx 0.63$. This expression coincides with that found earlier in the Vlasov description, using a macroscopic theory and the virial theorem, see Eq. (35) in \cite{2014EPJD...68..210F}, where the same functional dependence was obtained with the value $C\approx 1/2$. Such small discrepancy probably comes from the deviation of our solution (\ref{hole-like}) from the exact Gaussian, that was used in \cite{2014EPJD...68..210F} as the initial condition.}
\begin{figure}[htb]
\includegraphics[width=120mm]{Damp-Grow.eps}
\caption{The solutions of the full system of nonlinear equations (\ref{final_p_apr_F1})-(\ref{rel_moment_1_F1}). The initial conditions are adopted as $n_b(r, t=0) = n_{b0}(\alpha r)$ and $U_w(r, t=0) = U_{w0}(\alpha r)$, with $\alpha = 1.1$ (left) and $\alpha = 0.95$ (right).} \label{DG}
\end{figure}
\begin{figure}[htb]
\includegraphics[width=120mm]{Damp-Grow_2.eps}
\caption{Same as Fig. \ref{DG}, except that the initial conditions are adopted so as to contain the same number of electrons as a stationary beam, viz. $n_b(r, t=0) = \alpha^2 n_{b0}(\alpha r)$ and $U_w(r, t=0) = \alpha^{2(\kappa-1)} U_{w0}(\alpha r)$, with $\alpha = 1.05$ (left) and $\alpha = 0.9$ (right).} \label{DG2}
\end{figure}
\section{Conclusions}
In this paper we presented a self-consistent nonlinear hydrodynamic theory of the propagation of a long and thin relativistic electron beam through an unmagnetized and overdense plasma, in a configuration that is typical for the {plasma} wakefield acceleration scheme. The random component of the trajectories of the beam particles and of their velocities was modelled by an effective anisotropic temperature. It was demonstrated that in the presence of a finite velocity spread in the parallel direction and for not too long a beam, the beam dynamics could be approximated as a fully 3-D adiabatic expansion/compression. The resulting nonlinearity provided the localization of the beam in the transverse direction and produced a coherent stationary state, {even when the {effects of the nonlinear plasma wake force are} small or absent}. The linear analysis revealed that a small perturbation of such coherent state could be unstable, resulting either in the defocussing or focussing of the beam when the number of the beam particles was initially larger or smaller than that of the stationary state. Numerical calculations demonstrated that the lifetime of the beam could be significantly extended by the appropriate fine tuning, so that the transverse oscillations predicted earlier within the thermal wave model, could be observed. {Conversely, in a system that is predominantly two dimensional, this kind of thermodynamic nonlinearity vanishes and on the long timescale the system is governed by {the plasma wake force} responsible also for the betatron-like oscillations observed in the earlier works \cite{2014EPJD...68..210F,2014EPJD...68..271F,2014JPhCS.482a2014F,2016NIMPA.829..426A}.}
\begin{acknowledgements} This work was supported in part (D.J. and M.B.) by the MPNTR 171006 and NPRP 8-028-1-001 grants. D.J. acknowledges financial support from the INFN's fondo FAI and the hospitality of the Dipartimento di Fisica "Ettore Pancini", Universita di Napoli "Federico II", Italy.
\end{acknowledgements}
| {
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} | 1,107 |
Caratteristiche tecniche
Gioca come trequartista, ma può essere impiegato anche in attacco, come seconda punta.
Carriera
Club
Comincia a giocare al Poissy. Nel 2002 passa al Rouen. Nel 2003 si trasferisce al Stade Brest. Nel 2004 viene acquistato dal Poissy. Nel 2007 si trasferisce al Paris FC. Nel 2008 passa al Stade Lavalloise. Nel 2010 viene acquistato dal . Nel 2011 passa al Beauvais. Nel 2012 si trasferisce al Mont-de-Marsan. Nel 2013 si accasa allo Chambly Thelle. Nel 2015 viene acquistato dal Poissy, in cui torna per la terza volta in carriera.
Nazionale
Nato in Francia, ma di origini guadalupensi, sceglie di vestire la maglia della Nazionale guadalupense. Il debutto arriva il 6 gennaio 2007, nell'amichevole Martinica-Guadalupa (3-0). Ha messo a segno la sua prima rete con la maglia della Nazionale il 16 gennaio 2007, in Guadalupa-Guyana (3-4), in cui ha siglato la rete del definitivo 3-4, trasformando un calcio di rigore. Ha partecipato, con la Nazionale, alla Gold Cup 2007. Ha collezionato in totale, con la maglia della Nazionale, 13 presenze e due reti.
Altri progetti
Collegamenti esterni
Calciatori della Nazionale guadalupense | {
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} | 1,688 |
Q: httpservice result format I use flex and php to retrieve information from my sql database. What i have been doing is formatting the result from the database query into xml and then putting the result into an arraycollection. It all works fine except one of my database fields contains a lot of text that website users enter and as xml doesn't like tags like <> so I leave it un formatted but I would like to format it. Would I be better using flashvars or text than xml? Or is there another way?
A: You should have your database code that produces the XML escape special XML characters (e.g. <, > and &) with the standard entities (<, > and &) and stick with XML.
A: Or wrap text within CDATA.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,664 |
\section{Introduction\label{sec:introduction}}
In the fields of science and engineering, a common task is the
calculation of periodic solutions of nonlinear ordinary differential
equations. In our study, we will focus on ODEs of arbitrary
dimension involving generic, \ie possibly strong and non-smooth
nonlinear functions. In particular, we will address systems that can
comprise distinct states so that the nonlinear functions are only
piecewise defined. In mechanical engineering, such nonlinearities
arise \eg in structural systems with contact joints, where stick,
slip and lift-off are often considered as locally distinct states
\zo{john1989}. In electrical engineering, examples for such systems
are electrical circuits, where \eg transistors, rectifiers and
switches induce distinct system states. A rheological example are
superelastic shape memory alloys where the phases and phase
transformations between \eg martensite and austenite phase can be
regarded as distinct states \zo{schm2004}. Of course, many other
examples can be found in various fields of science and
engineering.\\
In order to find periodic solutions to such problems, analytical
approaches are often not applicable and computational methods have
to be employed. Besides the family of time integration methods, so
called frequency domain methods are commonly used due to their often
superior computational efficiency. The basic idea of frequency
domain methods is to choose a truncated Fourier ansatz for the
dynamic variables, thereby exploiting the periodic nature of the
solution. This class of methods gives rise to nonlinear algebraic
systems of equations. Depending on whether the solution is sought in
the frequency domain or in a collocated time domain, and whether the
residual is formulated in the frequency or time domain, the methods
can be grouped into (Multi-)Harmonic Balance Method
\zo{urab1965,nayf1979}, Trigonometric Collocation Method
\zo{sall2011} and Time Spectral Method \zo{gopi2005}. Among these
methods, the Multi- or High-order Harmonic Balance Method (HBM) is
probably the most commonly applied method.\\
For the HBM, it is generally necessary to compute the spectrum of
the nonlinear function that governs the ODE. This task can generally
be performed by different methods. In the following, we will focus
on those methods that are capable of treating systems with distinct
states.\\
The Alternating-Frequency-Time (AFT) scheme \zo{came1989} is one of
the most commonly applied approaches in this context. The AFT scheme
involves a sampling of the nonlinear function and subsequent
back-transformation into frequency domain. Advantages of this method
are the broad applicability, the comparatively small implementation
effort and the low computational effort for evaluating the residual
function. The latter aspect is particularly true if the (Inverse)
Fast Fourier Transform is used for the transformation between time
and frequency domain. A drawback is that nonlinearities with
distinct states involve special treatment. A sampling of the
nonlinear function is not straight-forward, because the current
state at a specific time instant is not always a priori known. So
called predictor-corrector schemes \zo{guil1998} are frequently
employed to perform the switching between different states for these
systems. In classical AFT schemes, the sampling points are fixed,
and do not need to coincide with the state transition time instants.
This inherently induces discretization errors. Hence, the
sensitivity of the transition time instants with respect to
arbitrary parameters cannot be captured accurately, resulting in
inaccurate derivatives, in particular for higher order
derivatives.\\
More recently, a purely frequency-based formulation was proposed by
\textsc{Cochelin and Vergez} \zo{coch2009}. The authors applied the
Asymptotic Numerical Method to expand the periodic solution into a
power series based on high-order derivatives of the nonlinear
function. In order to obtain these derivatives efficiently, a so
called quadratic recast is performed where the original system of
equations is transformed into a system of only quadratic order. An
advantage of this method is the computationally robust and efficient
continuation of the solution. A drawback is obviously the required
quadratic recast which can be difficult for generic types of
nonlinear functions. Moreover, systems with distinct states need to
be artificially smoothed in order to accomplish a closed-form
quadratic recast. This smoothing procedure induces
inaccuracies compared to the original non-smooth model.\\
In order to avoid the shortcomings of a required recast or the
degenerated accuracy due to sampling, a pure frequency domain
formulation for the original system with distinct states can instead
be used. Such an approach necessarily involves the direct
calculation of the transition time instants between the states. For
high-order HBM, these approaches have only been developed for
special types of nonlinearities so far. For example \z{Petrov and
Ewins}{petr2003b} developed an analytical formulation of the HBM for
piecewise linear friction interface elements in structural dynamical
problems. In this study, the approach in \zo{petr2003b} is extended
to generic systems with an arbitrary number of distinct states, see
\sref{methods_of_analysis}. Analytical formulations can be developed
in case of piecewise polynomial systems, as it will be shown in
\ssref{pp}. Moreover, the formulation facilitates the analytical
calculation of gradients of up to second order as an inexpensive
postprocessing step, see \ssref{sensana}. To demonstrate the
capabilities and the performance of the proposed methodology,
several numerical examples are studied in \sref{numerical_examples}.
Finally, conclusions are drawn in \sref{conclusions}.
\section{Methods of Analysis\label{sec:methods_of_analysis}}
\subsection{Harmonic Balance Method for systems with distinct states\label{sec:hbm}}
Consider a system whose dynamics can be described by a first-order
ordinary differential equation,
\e{\dot{\mm y} = \mm f\left(\mm y,t\right)\fk}{nonlinear_ode}
in which \g{\dot{()}} denotes derivative with respect to time $t$.
It is assumed that the generally nonlinear function \g{\mm f} is
piecewise defined within closed regions of the state space of \g{\mm
y}. These closed regions in state space are denoted \textit{states}
throughout this paper. These states shall not be confused with the
vector \g{\mm y} which is sometimes also referred to as state in
literature since it represents a point in state space.
\fss[thb]{fig1}{Illustration of the dynamics of a system with
distinct states}{0.75}
\\
As time evolves, the system can assume several states, see
\fref{fig1}. A transition between these states is termed state
transition and the corresponding time instant is called state
transition time instant in the following. The system enters a
specific state at the corresponding transition time \g{t^-} and
leaves it at \g{t^+}. Each possible state $k$ consists of a
nonlinear function \g{\mm f_k}, transition conditions \g{g_{kl}}
which roots define a transition hypersurface to state $l$, and
internal variables \g{\mm v_k}:
\ea{\nonumber \text{\underline{Definition of `state $k$'}} && \\
\nonumber \text{Nonlinear function:} & \mm f_k\left(\mm y(t),\mm
v_k,t\right)\fk &\\
\nonumber \text{Transition conditions:} & g_{kl}\left(\mm y(t),\mm
v_k,t\right)\,, & \forall l\in\mathcal L_k\fk\\
\text{Internal variables:} & \mm v_k\left(\mm y(t^-),\mm
f(t^-)\right)\fp}{state}
The set \g{\mathcal L_k} is a set of integers indicating a possible
next state, the system can assume after being in state $k$.\\
It is assumed that the function \g{\mm f} is smooth within a state
and continuous at the state transitions. It should be noted at this
point that the advantages of the proposed method can be particularly
exploited for the case of piecewise polynomial systems, as it will
be shown in \ssref{pp}, although the derivations in the following
are not restricted to these.\\
Internal variables are introduced in the state definition
\erefo{state} to facilitate the treatment of hysteresis effects. In
a hysteretic system, the dynamics do not explicitly depend on the
current value \g{\mm y(t)} but on the time history of \g{\mm y}.
Internal variables can therefore be used to carry this
history-dependent effect over the state hypersurface, which
manifests itself in the dependence of the nonlinear function \g{\mm
f_k} and the transition hypersurface \g{g_{kl}} on \g{\mm v_k}, see
\eref{state}. Note that hysteretic systems will also be addressed in
the numerical examples. For systems without these effects, of
course, the
introduction of internal variables is not necessary.\\
Periodic, steady-state solutions to \eref{nonlinear_ode} are sought
in this study. To this end, the High-order Harmonic Balance Method
can be applied \zo{nayf1979}. Hence, a Fourier series truncated to
harmonic order \g{H} represents the ansatz for the dynamic variables
\g{\mm y(t)},
\e{\mm y(t) \approx \suml{n=-H}{H}{\mm Y_n \ee^{\ii n\Omega
t}}}{fourier_ansatz}
Herein, \g{\Omega} is the fundamental angular frequency of the
response and \g{\ii = \sqrt{-1}} is the imaginary unit. The Fourier
coefficients \g{\mm Y_n} are symmetric, \g{\mm Y_{-n} =
\overline{\mm Y}_{n}}, where \g{\overline{()}} denotes complex
conjugate, since \g{\mm y(t)} is a real-valued function in time.
Substitution of \eref{fourier_ansatz} into the differential equation
\erefo{nonlinear_ode} and Fourier-Galerkin projection \zo{urab1965}
gives rise to a nonlinear algebraic system of equations in the
unknowns \g{{\mm Y}_{n}} and \g{\tau_j^-,\tau_j^+},
\ea{\text{solve} & \ii n\Omega\mm Y_n - \mm F_n\left(\mm Y_{-H},
\cdots, \mm Y_{H}\right) = \mm 0\,,\quad
n=-H,\cdots,H\fk\label{eqm_hbm}\\
\text{with} & \mm F_n = \frac{1}{2\pi}\intl{(2\pi)}{}\mm f(\mm
y,\tau)\ee^{-\ii
n\tau}\dd\tau=\frac{1}{2\pi}\suml{j=1}{J}\intl{\tau_j^-}{\tau_j^+}\mm
f(\mm y,\tau)\ee^{-\ii n\tau}\dd\tau\fk \label{fnl_hbm}\\
\text{subject to} &
\tau_J^+=\tau_1^-+2\pi\,,\quad\tau_j^+=\tau_{j+1}^-\,\,\,\,
\forall\,\, j=1\cdots J\fp\label{pstates_hbm}}{hbm}
For convenience, the normalized time \g{\tau=\Omega t} has been
introduced. During one period of oscillation, the system assumes a
total number of $J$ states. It should be emphasized that neither the
set of states nor the state transition time instants
\g{\tau_j^-,\tau_j^+} are a priori known. As indicated in the
constraint \erefn{pstates_hbm}, the \g{\tau_j^-,\tau_j^+} are
periodic and continuous to cover an entire time period, as a
consequence of the periodic ansatz. In this study, it is proposed to
directly compute the periodic set of transition time instants
\g{\tau_j^-,\tau_j^+}, which is developed in the following
subsection.\\
Once the transition time instants are known for given \g{\mm Y_n},
the integrals in \erefn{fnl_hbm} can be evaluated to formulate the
residual in \erefn{eqm_hbm}. Owing to the piecewise definition of
the function \g{\mm f}, it is convenient to split up the integral in
\erefn{fnl_hbm} into $J$ summands, where each of the summands is an
integral with the transition time instants as integral limits.
\subsection{Periodic set of state transition time instants\label{sec:statetrans}}
\fss[h!]{fig2}{Algorithm for the calculation of the periodic set of
state transition time instants}{0.75}
In \fref{fig2}, an algorithm is summarized that is capable of
finding a periodic set of state transition time instants for an
arbitrary system with or without distinct states. Starting from an
initial time \g{\tau_1^-}, state $k$ and according initial function
value \g{\mm f(\tau_1^-)}, the next states are iteratively computed
until a periodic set of state transitions is found. It is therefore
assumed that a periodic set of state transitions exists and the
algorithm is attracted to it. During the numerical studies, no case
was observed where this assumption was disproved.\\
After evaluating the internal variables, the next roots \g{\tau_l}
of all possible state transition conditions \g{g_{kl}} are computed
and the minimum is taken. In the special case when there is no next
state, the system remains in this state for all times and the
algorithm terminates. Note that this includes the special class of
systems with only a single state.\\
If a next state exists, the current state is stored for subsequent
evaluation of the Fourier coefficients. If the state transitions are
periodic - according to a specified tolerance $\varepsilon$ - the
algorithm can terminate, otherwise $j$ is incremented and the loop
is repeated.
\subsection{Computation and continuation of the solution}
In general, the solution to
\x{Eqs.~(\ref{eqm_hbm})-(\ref{pstates_hbm})} cannot be obtained in
closed form and an iterative numerical procedure has to be employed
instead. In this study, a Newton-Raphson method combined with a
predictor-corrector continuation scheme was used \zo{seyd1994}. The
numerical performance of the solution procedure was greatly enhanced
by providing analytically calculated gradients of the residual, as
derived in the following subsection.
\subsection{Analytical calculation of gradients and sensitivities of
the solution \label{sec:sensana}} Gradients of the residual are
often required in a numerical solution procedure for the algebraic
system of equations in \erefn{eqm_hbm}. Moreover, higher-order
derivatives at the solution point can be used to expand the solution
in a Taylor series. An approximate solution thus becomes available
in the vicinity of the current solution point in parameter space
without the need for re-computation. The Taylor expansion with
respect to the unknown variables can be employed as a predictor in a
numerical continuation procedure. Taylor expansions with respect to
system parameters are particularly interesting for
parametric studies, uncertainty analysis and optimization.\\
In this study, the analytical calculation of gradients of first and
second order is presented. We focus on the Fourier coefficients of
the nonlinear function \g{\mm f} since the sensitivities of the
other term in \erefn{eqm_hbm} is straight-forward. The first and
second order sensitivities of \g{\mm F_n} read
\ea{\frac{\partial\mm F_n}{\partial\psi} =
\suml{j=1}{J}\intl{\tau_j^-}{\tau_j^+}\frac{\partial \mm
f}{\partial\psi}\ee^{-\ii n\tau}\dd\tau + \mm
f(\tau_j^+)\frac{\partial\tau_j^+}{\partial\psi} - \mm
f(\tau_j^-)\frac{\partial\tau_j^-}{\partial\psi}=
\suml{j=1}{J}\intl{\tau_j^-}{\tau_j^+}\frac{\partial \mm
f}{\partial\psi}\ee^{-\ii n\tau}\dd\tau\fk\label{eq:testbild}\\
\frac{\partial^2\mm F_n}{\partial\phi\partial\psi} =
\suml{j=1}{J}\intl{\tau_j^-}{\tau_j^+}\frac{\partial^2 \mm
f}{\partial\phi\partial\psi}\ee^{-\ii n\tau}\dd\tau +
\frac{\partial\mm
f(\tau_j^+)}{\partial\psi}\frac{\partial\tau_j^+}{\partial\phi} -
\frac{\partial\mm
f(\tau_j^-)}{\partial\psi}\frac{\partial\tau_j^-}{\partial\phi}\fp\quad}{fnl_sensana}
Herein, \g{\phi,\psi} are arbitrary scalar variables such as the
components of the Fourier coefficients \g{\mm Y_n}, the frequency
\g{\Omega} or any system parameter. The Leibniz integral rule was
applied to derive \erefs{testbild}{fnl_sensana} since the integral
limits might and often do depend on the parameters. Note that the
last two summands in the first-order sensitivity cancel each other
out in the sum over one period due to the assumed continuity of
\g{\mm f} and
the periodicity condition in \erefn{pstates_hbm}.\\
The calculation of the derivative of the function \g{\mm f} within a
state is typically straight-forward. In contrast, the sensitivities
of a transition time instant \g{\tau_j} is more complex. It has to
be calculated by implicit differentiation of the active transition
condition \g{g(\tau_j)=0}. The resulting first- and second-order
sensitivities of the transition time instants read
\ea{\nonumber \frac{\partial\tau_j}{\partial\psi} &=&
\left[\frac{\partial g}{\partial\tau}\right]^{-1}\frac{\partial g}{\partial\psi}\fk\\
\frac{\partial^2\tau_j}{\partial\phi\partial\psi} &=&
\left[\frac{\partial
g}{\partial\tau}\right]^{-1}\left[\frac{\partial^2
g}{\partial\phi\partial\psi} + \frac{\partial
g}{\partial\phi\partial}\frac{\partial\tau_j}{\partial\psi} +
\frac{\partial
g}{\partial\psi\partial}\frac{\partial\tau_j}{\partial\phi}
\frac{\partial^2
g}{\partial\tau^2}\frac{\partial\tau_j}{\partial\phi}
\frac{\partial\tau_j}{\partial\psi}\right]\fp}{tau_sensana}
In \eref{tau_sensana}, all functions are evaluated at the transition
time instant \g{\tau_j}. It should be remarked that the time
derivative of the transition condition \g{\frac{\partial
g}{\partial\tau}} is nonzero at a regular zero crossing so that the
inverse in \eref{tau_sensana} is well-defined. Note that only
first-order derivatives of the state transition time instants
\g{\tau_j} are directly included in \eref{fnl_sensana}. However,
second-order derivatives may be required for the calculation of the
sensitivities of the internal variables \g{\mm
v_k\left(\mathbf{y}\left(\tau_j^-\right),
\mathbf{f}\left(\tau_j^-\right)\right)}, see definition
\erefo{state}.
\subsection{Application to piecewise polynomial systems\label{sec:pp}}
All previous developments are valid for the class of piecewise
smooth systems. In the sequel of this study, we will focus on the
large subclass of piecewise polynomial systems. For this class, all
functions \g{\mm f_k, g_{kl}, \mm v_k} are polynomials in the
components of \g{\mm y}, which makes the efficient formulation of
the previously derived expressions particularly cheap. In order to
solve \x{Eqs.~(\ref{eqm_hbm})-(\ref{pstates_hbm})} the basic
operations (a) \textit{add/subtract}, (b) \textit{multiply}, (c)
\textit{integrate} and (d) \textit{calculate roots} are required to
find the periodic set of state transitions and to carry out the
integration indicated in
\erefn{fnl_hbm}. These operations can be directly performed in Fourier space.\\
The \textit{multiplication} of two scalar functions
\g{a(\tau),b(\tau)} with associated Fourier coefficients \g{\mm
A=[A_{-H},\cdots,A_H],\mm B=[B_{-H},\cdots,B_H]} can be expressed as
a convolution in Fourier space,
\e{\mathcal F\lbrace a\cdot b\rbrace = \mm A \ast\mm
B\fp}{fourier_product}
Herein, \g{\mathcal F} indicates the Fourier Transform and \g{\ast}
denotes convolution. Note that powers of a Fourier series can be
calculated by recursive multiplication.\\
The \textit{integration} of a truncated Fourier series in the time
interval \g{\tau^-} to \g{\tau^+} can be expressed as follows (see
\arefo{fourier_integral}):
\e{\intl{\tau^-}{\tau^+} a(\tau)\ee^{-\ii n\tau}\dd\tau =
\left(\tau^+-\tau^-\right) A_n + \suml{m=-H,m\neq
n}{H}\frac{\ee^{\ii\left(m-n\right)\tau^+}-
\ee^{\ii\left(m-n\right)\tau^-}}{\ii\left(m-n\right)}A_m\fp}{fourier_integral}
This equation can be applied to the evaluation of the integrals in
\erefn{fnl_hbm}.\\
There are efficient as well as robust numerical methods for the
\textit{calculation of the roots} of a truncated Fourier series, see
\eg \zo{boyd2006}. Most of these methods simply compute the roots of
the associated complex polynomial in \g{z=\ee^{\ii\tau}}. Such
methods are available in many computational software frameworks like
\matlab.\\
It is important to note that the harmonic order is increased by the
convolution in \eref{fourier_product}, \ie when products or powers
of a Fourier series are generated. It is therefore proposed to
truncate the Fourier series of the nonlinear function \g{\mm F_n} to
the original order $H$ in \erefn{eqm_hbm} so that the resulting
number of equations is equal to the number of unknowns.
\subsection{On the numerical performance and accuracy of the proposed method\label{sec:numperformance}}
In all numerical studies of the piecewise polynomial systems
presented in \sref{numerical_examples}, the computational bottleneck
was observed to be the root finding of the complex polynomials
involved in the calculation of the state transition time instants.
State-of-the art polynomial root finding algorithms are based on the
computation of the eigenvalues of a so called companion matrix, for
which the computational complexity increases approximately with the
number of harmonics cubed \g{\mathcal O(H^3)}. The interested reader
is referred to \zo{boyd2006} for a detailed analysis of the
computational cost for this operation.\\
In contrast to the root finding operation, carrying out time-domain
integration, differentiation and multiplication by means of
summation and matrix multiplication according to the derived
closed-form expressions in \ssref{pp} is comparatively efficient.
This has some noteworthy implications for the analytical
calculations of the gradients: The evaluation of first and second
order derivatives represents an efficient post-processing step,
since their calculation only involves comparatively cheap vector and
matrix multiplications, as indicated in \ssref{sensana}.\\
It should be remarked that the accuracy of the proposed method,
particularly regarding the gradients, relies on the direct
calculation of the state transitions. The conventional AFT scheme is
characterized by an inherent discretization error. This causes a
severe limitation for the achievable accuracy. In this context, it
is interesting to note that in a piecewise linear system, the
second-order derivatives essentially result from the sensitivities
of the transition time instants, which can be easily verified from
\eref{tau_sensana}. As these sensitivities are not captured by the
AFT scheme, the second-order sensitivities would be identical to
zero in this case. This emphasizes the superiority of the proposed
method with respect to the AFT scheme regarding accurate sensitivity
analysis.
\section{Numerical examples\label{sec:numerical_examples}}
We have implemented the methodology proposed in
\sref{methods_of_analysis} in a computational framework in the
\matlab software environment. We used an object-oriented software
architecture to exploit operator overloading capabilities. For
example, we defined a Fourier series class that implements the
required operations add/subtract, multiply, integrate and compute
roots. Moreover, we developed and used an Automatic Differentiation
class similar to the one described in \zo{fort2006,neid2010} to
carry out the analytical sensitivity analysis up to second order. A
database of state formulations was created that
includes the nonlinearities presented in this section.\\
The numerical examples in this study comprise structural dynamical
systems. Application of the proposed methodology to fields other
than structural dynamics, \eg electrical networks, is considered
straight-forward but beyond the scope of this study. For structural
dynamical systems, the vectors \g{\mm y,\mm f} can be written as
follows:
\ea{\mm y = \vector{\mm x\\ \mm{\dot x}}\,,\quad \mm f =
\vector{\mm{\dot x}\\ -\mm M\inv\left(\mm D\mm{\dot x}+\mm K\mm x +
\mm f_{\mathrm e}(t)+\mm{\fnl}(\mm x,\mm{\dot
x})\right)}\fp}{dynsys}
Herein, \g{\mm M,\mm D, \mm K} are structural mass, damping and
stiffness matrices, \g{\mm x} is the vector of generalized
displacements, \g{\mm f_{\mathrm e},\mm{\tilde f}} are generalized
excitation and nonlinear forces.\\
The numerical examples can be categorized in two groups. In the
examples in \ssrefs{2dof_cubic}-\ssrefo{fgl} a 2-Degree-of-freedom
(DOF) system with an attached single nonlinear element \g{\fnl} is
considered, see \fref{fig3}. The example for \ssref{beam} is a
cantilevered beam with contact constraints and will be described
later. It should be noted that the example systems with a small
number of DOFs were considered for clarity reasons. The methodology
proposed in this paper can generally be applied to systems with
arbitrary number of DOFs, including large-scale Finite Element
Models.
\fss[t]{fig3}{2-DOF system with nonlinear element}{1.0}
\\
For the 2-DOF system, the structural matrices and the nonlinear
force vector have the following form:
\ea{\nonumber \mm M = \matrix{cc}{1 & 0\\ 0 & 1}\,,\,\,\mm D=\mm
0\,,\,\, \mm K = \matrix{cc}{2 & -1\\ -1 &
2}\,,\,\,\mm{\fnl}=\vector{\fnl\\ 0}\fp}{2dof}
The corresponding vector of generalized coordinates is \g{\mm x\tra
= \matrix{cc}{\xnl & x_2}}, where \g{\xnl} denotes the nonlinear
DOF.
\subsection{2-DOF system with cubic spring\label{sec:2dof_cubic}}
For a first demonstration of the methodology, a cubic spring
nonlinearity is considered. The nonlinearity can be described by a
single state without transition conditions and no internal
variables. In the notation introduced in \eref{state}, the state
definition reads as listed in \tref{2dof}.
\tab[h]{lc}{ & state 1\\
$\fnl$ & $0.5 \xnl^3$\\
$g$ & (-)\\
$v$ & (-)}{State definition of a system with cubic spring}{2dof}
\\
This example emphasizes once again that purely polynomial, \ie
smooth nonlinearities are a special case of the piecewise polynomial
class treated in this study.
\myf[t]{fig4a}{fig4b} {}{}{.45}{.45}{Frequency-Energy-Plot of a
2-DOF system with cubic spring (~(a) overview, (b) detail~)}
\\
The 2-DOF system is investigated in autonomous configuration. The
proposed method was used for the calculation of the nonlinear normal
modes. Great convergence behavior was ascertained.\\
A thorough study of the nonlinear normal modes of this system can be
found in \z{Kerschen \etal}{kers2009} and shall not be repeated
here. Instead, only the so called Frequency-Energy-Plot (FEP) of the
first nonlinear mode is depicted in \frefs{fig4a}-\frefo{fig4b}.
Throughout this study, amplitude and frequency axes in the figures
are scaled by their values for the linear case. The eigenfrequency
increases with the modal amplitude due to the stiffening effect of
the cubic spring. For large amplitudes, the energy localizes in the
left mass in \fref{fig3}. The system exhibits several internal
resonances in the considered modal amplitude range \zo{kers2009},
causing so called tongues in the FEP, see \fref{fig4b}. Apparently
several harmonics are required to accurately predict the nonlinear
modal interactions.
\subsection{2-DOF system with piecewise polynomial spring\label{sec:snap_through}}
Again, the 2-DOF system is considered, however, the cubic spring is
now replaced by a piecewise polynomial spring. The
force-displacement characteristic is given by the function depicted
in \fref{fig5a}. The nonlinearity was defined by introducing three
states, each with a polynomial force \g{\fnl} as listed in
\tref{snap_through}. As it was shown in \sref{pp}, the High-order
Harmonic Balance residual equations can be formulated analytically
for this class of systems by the new technique proposed in this
paper.
\tab[h]{lccc}{ & state 1 & state 2 & state 3\\
$\fnl$ & $-(\xnl-1)^2+1$ & $-(\xnl-2)$ & $(\xnl-3)^2-1$\\
$g$ & $g_{12}=\xnl-1$ & $g_{21}=g_{12}$, $g_{23}=g_{32}$ & $g_{32}=\xnl-3$\\
$v$ & (-) & (-) & (-)}{State definition of a system with piecewise
polynomial spring}{snap_through}
\myf[b!]{fig5a}{fig5b}{}{}{.45}{.45}{Characteristics of a 2-DOF
system with piecewise polynomial spring (~(a) force-displacement
relationship, (b) forced response function~)}
\\
The central part of the characteristic is linear with negative
slope. The two neighboring states have a quadratic
force-displacement characteristic. Note that the piecewise
polynomial spring is conservative with a unique force-displacement
relationship. Hence, the state formulation does
not require any internal variables.\\
A harmonic force excitation at the linear mass in a frequency range
close to the first eigenfrequency is imposed. The forced response
function was calculating using the proposed method and is
illustrated in \fref{fig5b}. Overhanging branches occur in the
forced response characteristic: For moderate vibration amplitudes,
\ie for small vibrations around the equilibrium point \g{\xnl=0},
the system exhibits softening behavior and the amplitude-frequency
curve is bent to the left. The effective stiffness decreases with
increasing amplitude due to negative slope in the force-displacement
characteristic. For larger vibration amplitudes, the effective
stiffness increases due to the quadratic branches, resulting in a
stiffening behavior and the amplitude-frequency curve is bent to the
right. Apparently, several harmonics have to be considered in the
harmonic expansion to accurately predict the dynamic behavior of the
system.
\subsection{2-DOF system with elastic Coulomb friction element\label{sec:twodoffric}}
An elastic Coulomb friction or Masing element \zo{masi1923} is
attached to the 2-DOF system in \fref{fig3}. Tangential stiffness
$\kt$ and friction force limit $\fc$ characterize this nonlinearity.
The nonlinearity can assume two states: Stick (state 1) and slip
(state 2), see \tref{elastic_coulomb}.
\myf[b!]{fig6a}{fig6b}{}{}{.48}{.42}{Forced response of a system
with elastic Coulomb nonlinearity (~(a) forced response functions
for different values of the normal load, (b) typical time history~)}
\myf[t!]{fig7a}{fig7b}{}{}{.45}{.45}{Resonance amplitude as a
function of the normal load (~(a) variation of the excitation level
\g{\epsilon}, (b) variation of the tangential stiffness \g{\kt}~)}
\tab[h]{lcc}{ & state 1 & state 2\\
$\fnl$ & $\kt(\xnl-v_1)$ & $v_2\fc$\\
$g$ & $g_{12}=\fnl^2-(\fc)^2$ & $g_{21}=\dot\xnl$\\
$v$ & $v_1=\xnl(\tau_j^-)-\frac{\fnl(\tau_j^-)}{\kt}$ &
$v_2=\sgn\fnl(\tau_j^-)$}{State definition of a system with elastic
Coulomb friction element}{elastic_coulomb}
\\
If the elastic friction force reaches its limit value \g{\fc}, \ie
\g{g_{12}=0}, a stick-to-slip transition occurs. If a reversal point
is reached (\g{\dot{\xnl}=g_{21}=0}), a stick phase is initiated.
Internal variables for the elastic Coulomb element are the Coulomb
slider position $v_1$ and the slip direction $v_2$.
\\
Again, a harmonic force excitation is imposed at the linear mass. In
\fref{fig6a}, the forced response function in the vicinity of the
eigenfrequency of the first mode is depicted for different values of
the friction force limit \g{\fc}. A tangential stiffness value of
\g{\kt=0.35} was specified. For large values of \g{\fc}, the Coulomb
element is fully stuck so that the hysteresis degenerates to a line
and no damping effect is introduced by the friction element. For
vanishing values of \g{\fc}, the slider can slip freely so that the
hysteresis is flat and again there is no friction damping effect. In
between these extreme cases, a significant amplitude reduction due
to friction damping can be achieved. Moreover, the resonance
frequency increases as the value of \g{\fc} increases due to the
coupling effect of the friction element. The backbone curve that
connects the maxima of the forced response functions was directly
calculated by applying the strategy described in \zo{krac2012a} to
the methodology proposed in this paper. A typical time history of
both displacement \g{\tilde x} and nonlinear force \g{\tilde f} is
illustrated in \fref{fig6b}. Owing to the moderate value of \g{\kt},
the response remains essentially harmonic. The transitions between
stick and slip state can be well-observed from the time
history of the force in \fref{fig6b}.\\
The suitability of the analytically formulated sensitivities is now
investigated. To this end, the resonance amplitude of the first mode
is depicted as a direct function of \g{\fc} in
\frefs{fig7a}-\frefo{fig7b}. These so called optimization curves are
often used for design purposes, see \eg
\zo{bert1998,petr2006b,krac2012b}. In addition to the nominal
parameter set, the optimization curve is also illustrated for
slightly smaller and larger ($\pm 25\%$) excitation level and
tangential stiffness values. The results were obtained by
second-order Taylor expansion about the reference solution (Taylor
exp.). For comparison, the optimization curves were also computed
directly at the new parameter point (Re-computed). The results agree
well in a wide range of the \g{\fc} value. However, the Taylor
expansion fails in predicting the fully stuck configuration, \ie for
very high \g{\fc} values. As it was also reported in
\zo{brau1993,krac2012a}, it is not possible to accurately predict
the dynamic behavior beyond the point where the order or number of
states change.
\subsection{2-DOF system with superelastic shape memory alloy\label{sec:fgl}}
The hysteresis effect of superelastic shape memory alloys (SMA) can
be employed for damping of mechanical structures. A sophisticated
modeling approach would involve constitutive as well as
thermodynamical aspects, see \eg \zo{bern2003}. This is, however,
regarded as beyond the scope of this study and a simplified
rheological piecewise linear model \zo{schm2004} shall be considered
instead. The associated hysteresis can be described by five distinct
states as illustrated in \fref{fig8} and listed in \tref{fgl}. The
system features a purely elastic state (1). The forward and reverse
transformation between austenite and martensite phase is described
by the states (2) and (4). Beyond a certain displacement, a linear
onset (3) is used to describe the superelastic behavior. Depending
on the displacement evolution in time, an intermediate state (5) can
also be reached. Note that the point symmetry of the hysteresis is
exploited in the state definition in \tref{fgl}.
\fss[b!]{fig8}{Approximated hysteresis of a superelastic shape
memory alloy}{1.0}
\myf[t!]{fig9a}{fig9b}{}{}{.45}{.45}{Modal properties of a 2-DOF
system with superelastic shape memory alloy (~(a) eigenfrequency,
(b) modal damping~)}
\tab[h!]{lccc}{ & state 1 & state 2 & state 3\\
$\fnl$ & $k\xnl$ & $v_2\left(\fst+\fc\right)$ &
$k(\xnl-v_3 a)$\\
$g$ & $g_{12}=\fnl^2-(\fst+\fc)^2$ & $g_{23}=kv_2\xnl-ka-\fst-\fc$,
& $g_{34} = \fnl-v_3(\fst-\fc)$\\
& & $g_{25}=\dot\xnl$ &\\
$v$ & (-) & $v_2=\sgn\,\xnl(\tau_j^-)$ & $v_3=v_2$\\
\hline\\
\hline
& state 4 & state 5\\
$\fnl$ & $v_4\left(\fst-\fc\right)$ & $k(\xnl-v_5)$\\
$g$ & $g_{41} = kv_3\xnl-\fst+\fc$, &
$g_{52}=\fnl^2-(\fst+\fc)^2$,\\
& $g_{45} = \dot\xnl$ & $g_{54}=\fnl^2-(\fst-\fc)^2$\\
$v$ & $v_4=v_2$ & $v_5 =
\xnl(\tau_j^-)-\frac{\fnl(\tau_j^-)}{\kt}$}{State definition of a
system with superelastic shape memory alloy}{fgl}
\\
The SMA-type nonlinearity was also applied to the 2-DOF system in
\fref{fig3}. The Nonlinear Modal Analysis technique proposed in
\zo{laxa2009,krac2013d} was employed in conjunction with the
formulations of the nonlinearities proposed in this study.
Eigenfrequency and the modal damping ratio were computed with
respect to the modal amplitude of the first mode. The results are
depicted in \frefs{fig9a}-\frefo{fig9b}. For small vibration
amplitudes, the system always remains in state 1, \ie the damping
vanishes and the eigenfrequency is constant. For moderate vibration
amplitudes, the phase transformation occurs to a certain extent so
that the damping value increases and the eigenfrequency is reduced
due to the softening effect. For large vibration amplitudes, the
effect of the hysteresis cycles becomes smaller again so that
eigenfrequency and
damping value asymptotically approach their linearized values again.\\
As in the previous example, the sensitivities of the nonlinear
dynamic analysis results have been computed to formulate a
second-order Taylor series in the system parameters. Using the
sensitivity results, the modal properties have been expanded with
respect to the parameter \g{\fc}, \cf \frefs{fig9a}-\frefo{fig9b},
for \g{\pm 25\%} deviation from the nominal value. The results agree
well with the re-computed results.
\subsection{Beam with friction and unilateral contact\label{sec:beam}}
\fss[b]{fig10}{Cantilevered beam with friction and unilateral
contact at its free end}{0.7}
As a final example, a clamped beam with combined friction and
unilateral contact was investigated. The system is depicted in
\fref{fig10}. A finite element code was used to mesh the geometry
and derive the structural matrices of the cantilevered beam for the
initial configuration. The finite element model comprised $10,098$
DOFs. A single node-to-ground contact element was attached to the
free end as depicted in \fref{fig10}. In contrast to the example in
\ssref{twodoffric}, the contact model additionally accounts for the
variation of the normal load and possible lift-off. The contact
nonlinearity can thus assume three distinct states as listed in
\tref{nt_elastic}: Separation (state 1), stick (state 2), slip
(state 3). System parameters are tangential stiffness $\kt$,
friction coefficient \g{\mu}, normal stiffness $\kn$ and normal gap
$g$.
\myf[t]{fig11a}{fig11b}{}{}{.45}{.45}{Comparison of
Alternating-Frequency-Time scheme with proposed method (~(a)
accuracy of function and derivative, (b) computational effort~)}
\tab[h]{lccc}{ & state 1 & state 2 & state 3\\
$\mm \fnl$ &
$\vector{0\\ 0}$ & $\vector{\kn\left(\xnl\nnn+g\right)\\
\kt\left(\xnl\ttt-v_2\right)}$ & $\vector{\kn\left(\xnl\nnn+g\right)\\
v_3\mu\kn\left(\xnl\nnn+g\right)}$ \\
$g$ &
$g_{12}=\xnl\nnn+g\,\text{if}\,\kn{\dot\xnl}\nnn>\kt{\dot\xnl}\ttt$,
& $g_{21}=\xnl\nnn+g$, &
$g_{31} = g_{21}$,\\
&
$g_{13}=\xnl\nnn+g\,\text{if}\,\kn{\dot\xnl}\nnn\leq\kt{\dot\xnl}\ttt$
& $g_{23}=\left(\fnl[2]\right)^2-\left(\mu\fnl[1]\right)^2$ &
$g_{32}=\dot{\fnl}[2]-\kt{\dot\xnl}\ttt$\\
$v$ & (-) & $v_2=\xnl\ttt(\tau_j^-) - \frac{\fnl[2](\tau_j^-)}{\kt}$
& $v_3=\sgn\,\fnl[2](\tau_j^-)$}{State definition of a system with
friction and unilateral contact}{nt_elastic}
\\
For this particular example, a comparison with the conventional AFT
scheme was performed in terms of accuracy and computational effort,
\cf \frefs{fig11a}-\frefo{fig11b}. Only the nonlinear force
calculation is considered for the comparison. Random vectors of
complex displacement amplitudes was generated. A number of $1,000$
random vectors was large enough to obtain convergence of the
performance statistics. Seven harmonics have been considered in the
analysis. In \fref{fig11a}, the mean, minimum and maximum error of
the force and the Jacobian are depicted with respect to the number
of time samples $\ntd$ per period used in the AFT scheme. The
accuracy of the AFT scheme can be increased by increasing the number
of time steps. An larger number of time steps yields a better
accuracy, but also a higher computational effort. The computational
effort $T_{AFT}$ essentially increases linearly with the number of
time steps. The effort quickly exceeds the one required for the
proposed method ($T_{ana}$), \cf \fref{fig11b}. It has to be
remarked that only the nonlinear force is considered in the
performance comparison. It is expected that the resulting error in
the predicted response is less significant than that of the force or
the Jacobian.
\myf[t]{fig12a}{fig12b}{}{}{.48}{.42}{Forced response of a beam with
friction and unilateral contact (~(a) forced response functions for
different values of the normal load, (b) orbits along the backbone
curve~)}
\myf[h!]{fig13b}{fig13a}{}{}{.45}{.45}{Resonance properties for
different orders of the harmonic balance approach (~(a) resonance
amplitude, (b) resonance frequency~)}
\\
A harmonic force excitation was imposed at the center of the free
end in a frequency range near the second bending eigenfrequency. The
forced response for varied normal preload \g{N=-g/\kn} is depicted
in \fref{fig12a} along with the backbone curve. The results are
generally similar to the ones presented in \ssref{twodoffric}.
Again, it can be ascertained that there exists an optimum normal
preload that minimizes the resonance amplitude. For smaller preload
values, the contact node may lift off during one period of
oscillation. This causes a softening effect, leading to overhanging
branches in the forced response characteristic. In \fref{fig12b},
some periodic orbits corresponding to points on the backbone curve
are illustrated in a three-dimensional section through the phase
space. According to expectations, a multiharmonic character can be
ascertained from the response. In particular, the static component
of the displacement is varying with the vibration
amplitude.\\
In \frefs{fig13b}-\frefo{fig13a}, resonance amplitude and frequency
are depicted as a direct function of the normal preload \g{N}.
Several harmonics are required to achieve asymptotic convergence of
the resonance properties. This particularly holds for smaller values
of \g{N} \ie when the oscillation of the normal load and partial
separation gain influence on the dynamics of the system.
\section{Conclusions\label{sec:conclusions}}
A method was proposed that allows for an analytical formulation of
the high-order Harmonic Balance Method for the dynamic analysis of
systems with distinct states. The method can be applied to generic
nonlinearities that can be described by piecewise polynomial
functions and state transition conditions.\\
The methodology not only facilitates the computation of the periodic
solution but also provides accurate sensitivity data of the solution
to arbitrary system parameters that can be used \eg for design
studies. It was shown that the approach can be superior to the
conventional Alternating-Frequency-Time scheme in terms of accuracy
and computational efficiency, in particular if the sensitivities of
the transition time instants between the states are of
interest.\\
The method was applied to several structural dynamical systems with
conservative and dissipative nonlinearities in externally excited
and autonomous configurations. Generally good performance and
robustness of the numerical method were observed.\\
Possible future work includes a comparison of the method to the
Harmonic Balance formulation of the Asymptotic Numerical Method, as
introduced in \zo{coch2009}, and the application to other
engineering fields such as electrical switching networks.
\begin{appendix}
\section{Definite integral of a truncated Fourier series\label{asec:fourier_integral}}
A truncated Fourier series $a(\tau)$ is considered,
\e{a(\tau) = \suml{m=-H}{H}A_m\ee^{\ii m\tau}\fp}{a1}
Substituting this definition into \eref{fourier_integral} yields
\e{\intl{\tau^-}{\tau^+} a(\tau)\ee^{-\ii n\tau}\dd\tau =
\intl{\tau^-}{\tau^+}\suml{m=-H}{H}A_m\ee^{\ii m\tau}\ee^{-\ii
n\tau}\dd\tau = \suml{m=-H}{H}A_m\intl{\tau^-}{\tau^+}\ee^{\ii
(m-n)\tau}\dd\tau\fp}{a2}
The indefinite integral of the integral in the last part of
\eref{a2} can be expressed as
\e{\int\ee^{\ii (m-n)\tau}\dd\tau = \begin{cases}\tau &
m=n\\
\frac{\ee^{\ii(m-n)\tau}}{\ii(m-n)} & m\neq n\end{cases}\fp}{a3}
The case $m=n$ thus has to be treated separately. For convenience,
the sum in \eref{a2} is therefore split up. Substituting \eref{a3}
into \eref{a2} finally gives
\eal{\intl{\tau^-}{\tau^+} a(\tau)\ee^{-\ii n\tau}\dd\tau
&=&\left[A_n\tau + \suml{m=-H,m\neq
n}{H}A_m\frac{\ee^{\ii(m-n)\tau}}{\ii(m-n)}\right]_{\tau^-}^{\tau^+}\\
&=& \left(\tau^+-\tau^-\right) A_n + \suml{m=-H,m\neq
n}{H}\frac{\ee^{\ii\left(m-n\right)\tau^+}-
\ee^{\ii\left(m-n\right)\tau^-}}{\ii\left(m-n\right)}A_m\fp}{a4}
\end{appendix}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,000 |
Filmmaker Who Outed ACORN in Prostitution Sting Arrested
By Dan Riehl 2010-01-26T17:54:23
Update 2/1/10: The report below was written on January 26, based on initial wire service reports of the incident, much, if not all of which came into dispute in the days afterward. As of February 1st, there are no allegations of any bugging plot in the FBI affidavit that was released early last week and a law enforcement official has conceded that the four men were not attempting to wiretap or intercept calls. Furthermore, legal representation for the accused has gone on record stating there were no intentions to bug the phones in the Senator's office. Pajamas Media.com regrets having relied on the erroneous initial wire reports of the incident, and we are happy to present this correction to the record. So as not to completely airbrush our article into the memory hole, the original report follows below.
Conservative activist James O'Keefe has been arrested by the U.S. Marshal's Service for allegedly attempting to bug the offices of U..S. Senator Mary Landrieu. Also arrested were Robert Flanagan, the son of a federal prosecutor, and two others.
O'Keefe and partner Hannah Giles rose to fame for a video sting against the community activist group ACORN. The videos appeared on Andrew Breitbart's Big Government website. According to a report on Twitter, Big Government's editors claim to have had no knowledge of events in Louisiana.
From a report in Hotline on Call:
O'Keefe gained fame and notoriety last year when, dressed as a pimp, he videotaped ACORN employees advising him and a female companion about running a prostitution scheme. The Times-Picayune reported that O'Keefe gave a speech on Jan. 21 to Libertarian Pelican Institute in New Orleans about investigative reporting and the role of new media.
O'Keefe is said to have been recording his alleged accomplices' actions on a cell phone after they entered Landrieu's offices dressed as telephone repairmen.
Flanagan and Basel were "each dressed in blue denim pants, a blue work shirt, a light green fluorescent vest, a tool belt and a construction-style hard hat when they entered the Hale Boggs Federal Building," the release said.
After they entered the building, the two men told a staffer in Landrieu's office they were telephone repairmen, according to the release. They then asked for, and were granted, access to the reception desk's phone system.
O'Keefe, meanwhile, recorded their actions with a cell phone.
(View the four-page arrest affidavit here.)
Landrieu's office is not commenting, citing an ongoing investigation. If convicted, the four men could face fines of up to $250,000 and up to ten years in jail.
https://pjmedia.com/blog/filmmaker-who-outed-acorn-in-prostitution-sting-arrested/ | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 5,703 |
\section{Introduction}
\label{sec:intro}
The detection of strong {\ifmmode{{\rm Ly}\alpha~}\else{Ly$\alpha$~}\fi} emission lines has become an essential
method in extra-galactic astronomy to find distant star-forming
galaxies
\citep{PartridgePeebles,Rhoads00,Gawiser2007,Koehler2007,Ouchi08,Yamada2012,Schenker2012,Finkelstein2013}.
The galaxies detected using this method receive the
name of {\ifmmode{{\rm Ly}\alpha~}\else{Ly$\alpha$~}\fi} emitters (LAEs).
A detailed examination of this galaxy population has diverse
implications for galaxy formation, reionization and the large scale
structure of the Universe.
Attempts to fully exploit the physical information included in the {\ifmmode{{\rm Ly}\alpha~}\else{Ly$\alpha$~}\fi}
line require an understanding of all the physical factors involved in
shaping the line.
Due to the resonant nature of this line, these physical factors
notably include temperature, density and bulk velocity field of the
neutral Hydrogen in the emitting galaxy and its surroundings.
A basic understanding of the quantitative behavior of the {\ifmmode{{\rm Ly}\alpha~}\else{Ly$\alpha$~}\fi} line
has been reached through analytic studies in the case of a static
configurations, such as uniform slabs
\citep[][]{Adams72,Harrington73,Neufeld90} and uniform spheres
\citep{Dijkstra06}.
Analytic studies of configurations including some kind of bulk flow
only include the case of a sphere with a Hubble like expansion flow
\citep{LoebRybicki}.
A more detailed quantitative description of the {\ifmmode{{\rm Ly}\alpha~}\else{Ly$\alpha$~}\fi} line has been
reached through Monte Carlo simulations \citep{Auer68,Avery68,Adams72}.
In the last two decades these studies have become popular due to the
availability of computing power.
Early into the 21st century, the first
studies focused on homogeneous and static media
\citep{Ahn00,Ahn01,Zheng02}.
Later on, the effects of clumpy media \citep{Hansen06} and
expanding/contracting shell/spherical geometries started to be studied
\citep{Ahn03,Verhamme06,Dijkstra06}. For a recent review, we refer the interested reader to \citet{review}.
Similar codes have applied these results to semi-analytic models of
galaxy formation \citep{Orsi12, Garel2012} and results of large
hydrodynamic simulations \citep{CLARA,Forero12,Behrens13}.
Recently, Monte Carlo codes have also been applied to the results of
high resolution hydrodynamic simulations of individual galaxies
\citep{Laursen09,Barnes11,Verhamme12,Yajima12}.
Meanwhile, recent developments have been focused on the systematic
study of clumpy outflows \citep{DijkstraKramer} and anisotropic
velocity configurations \citep{Zheng2013}.
The recent studies of galaxies in hydrodynamic simulations
\citep{Laursen09,Barnes11,Verhamme12,Yajima12} have all shown
systematic variations in the {\ifmmode{{\rm Ly}\alpha~}\else{Ly$\alpha$~}\fi} line with the viewing angle. These
variations are a complex superposition of anisotropic density
configurations (i.e. edge-on vs. face-on view of a galaxy), the
inflows observed by gas cooling and the outflows included in the
supernova feedback process of the simulation. These bulk flows
physically correspond to the circumgalactic and intergalactic medium
(CGM and IGM). These effects are starting to be studied
in simplified configurations that vary the density and wind
characteristics \citep{Zheng2013,Behrens2014}.
However, in all these efforts the effect of rotation,
which is an ubiquitous feature in galaxies, has not been
systematically studied. The processing of the {\ifmmode{{\rm Ly}\alpha~}\else{Ly$\alpha$~}\fi} photons in a
rotating interstellar medium (ISM) must have some kind of impact in
the {\ifmmode{{\rm Ly}\alpha~}\else{Ly$\alpha$~}\fi} line morphology.
Performing that study is the main goal of this paper. We investigate for the
first time the impact of rotation on the morphology of the {\ifmmode{{\rm Ly}\alpha~}\else{Ly$\alpha$~}\fi}
line. We focus on a simplified system: a spherical gas cloud with
homogeneous density and solid body rotation, to study the line
morphology and the escape fraction in the presence of dust. We base
our work on two independent Monte Carlo based radiative transfer codes
presented in \cite{CLARA} and \cite{DijkstraKramer}.
This paper is structured as follows: In \S \ref{sec:implementation} we
present the implementation of bulk rotation into the Monte Carlo
codes, paying special attention to coordinate definitions. We also
present a short review of how the {\ifmmode{{\rm Ly}\alpha~}\else{Ly$\alpha$~}\fi} radiative transfer codes work
and list the different physical parameters in the simulated grid of
models. In \S \ref{sec:results} we present the results of the
simulations, with special detail to quantities that show a
clear evolution as a function of the sphere rotational velocity. In \S
\ref{sec:discussion} we discuss the implications of our results. In
the last section we present our conclusions. The Appendix presents the
derivation of an analytic expression to interpret
the main trends observed in the Monte Carlo simulations.
In this paper we express a photon's frequency in terms of the
dimensionless variable $x\equiv (\nu -\nu_a)/\Delta\nu_{\rm D}$, where
$\nu_{\rm \alpha}=2.46\times 10^{15}$ Hz is the Ly$\alpha$ resonance
frequency, $\Delta\nu_{\rm D} \equiv
\nu_{\alpha}\sqrt{2kT/m_pc^2}\equiv \nu_av_{\rm th}/c $ is the Doppler
broadening of the line which depends on the neutral gas temperature
$T$ or equivalently the thermal velocity
$v_{\rm th}$ of the atoms. We also use the parameter $a$ to define the
relative line width as $a=\Delta\nu_{\alpha}/2\Delta\nu_{\rm D}$,
where $\Delta\nu_{\alpha}$ is the intrinsic linewidth. For the
temperature $T=10^4$K used in our radiative transfer calculations the
thermal velocity is $v_{\rm th}=12.8${\ifmmode{{\mathrm{\,km\ s}^{-1}}}\else{\,km~s$^{-1}$}\fi}.
\section{Models of bulk gas rotation}
\label{sec:implementation}
Describing the kinematics of gas rotation in all generality is a
complex task, specially at high redshift where there is still missing
a thorough observational account of rotation in galaxies beyond
$z>1.0$. Even at low redshifts there is a great
variation in the shape of the rotation curve as observed in HI
emission as a function of the distance to the galaxy center. However
there are two recurrent features. First, in the
central galactic region the velocity increases proportional to the radius,
following a solid rotation behavior. Second, beyond a certain radius
the rotation curve tends to flatten. An ab-initio description of
such realistic rotation curves in simulations depends on having access to
the dynamic evolution of all mass components in the galaxy: stars, gas
and dark matter. Such level of realism is extremely complex to
achieve, specially if one wants to get a systematic description based
on statistics of simulated objects.
Following the tradition of studies of {\ifmmode{{\rm Ly}\alpha~}\else{Ly$\alpha$~}\fi} emitting systems,
we implement a model with simplified geometry. We assume that the gas
is homogeneously distributed in a sphere that rotates as a solid body
with constant angular velocity. This simple model will contain only
one free parameter: the linear velocity at the sphere's surface, $V_{\rm
max}$.
\subsection{Detailed Implementation of Rotation}
In the Monte Carlo code we define a Cartesian coordinate system to
describe the position of each photon. The origin of this system
coincides with the center of the sphere and the rotation axis is defined
to be $z$-axis. With this choice, the components of the gas bulk velocity
field, $\vec{v} = v_{x}\hat{i} + v_{y}\hat{j} + v_{z}\hat{k}$, can be
written as
\begin{equation}
v_{x}=-\frac{y}{R}V_{\rm max}, \label{subeq1}
\end{equation}
\begin{equation}
v_{y}=\frac{x}{R}V_{\rm max}, \label{subeq2}
\end{equation}
\begin{equation}
v_{z}=0, \label{subeq3}
\end{equation}
where $R$ is the radius of the sphere and $V_{\rm max}$ is the linear
velocity at the sphere's surface. The minus/plus sign in the
$x$/$y$-component of the velocity indicates the direction of
rotation. In this case we take the angular velocity in the same
direction as the $\hat{k}$ unit vector. With these definitions we can
write the norm of the angular velocity as $\omega=V_{\rm max}/R$.
For each photon in the simulation we have its initial position inside
the sphere, direction of propagation $\hat{k}_{\rm in}$ and reduced
frequency $x_{\rm in}$.
The photon's propagation stops once they cross the
surface of the sphere. At this point we store the position, the outgoing direction
of propagation $\hat{k}_{\rm out}$ and the reduced frequency $x_{\rm
out}$. We now define the angle $\theta$ by $\cos\theta = \hat{k}_{\rm out}\cdot
\hat{k}\equiv \mu$, it is the angle of the outgoing photons with
respect to the direction of the angular velocity. We use the variable $\mu$ to
study the anisotropy induced by rotation. Fig. \ref{fig:geometry}
shows the geometry of the problem and the important variables.
\begin{figure}
\begin{center}
\includegraphics[width=0.4\textwidth]{f1.pdf}
\end{center}
\caption{Geometry of the gas distribution. The angular velocity vector
is parallel to the unit vector $\hat{k}$. In order to describe the
departures from spherical symmetry we use the polar angle $\theta$
formed by the direction of the outgoing photons with respect to the
$z$-axis. We define define the variable $\mu\equiv\cos\theta$ to
report to present our results. Computing the spectra for photons in
a narrow range of $\mu$ is equivalent to having a line-of-sight
oriented in that direction.
\label{fig:geometry}}
\end{figure}
\begin{figure*}
\begin{center}
\includegraphics[width=0.95\textwidth]{f2.pdf}
\end{center}
\caption{
2D histogram showing the number of photons that escape with frequency
$x$ forming an angle $\theta$ (parametrized as $|\cos\theta|$) with the
rotation axis.
The rotational velocity ($0,100,200,300${\ifmmode{{\mathrm{\,km\ s}^{-1}}}\else{\,km~s$^{-1}$}\fi}) increases from left to
right and the optical depth ($10^5$, $10^6$, $10^7$) from top to
bottom.
The {\ifmmode{{\rm Ly}\alpha~}\else{Ly$\alpha$~}\fi} photons are initialized at the center of the sphere.
Two main results can be read from this figure.
First, the line morphology depends on the viewing angle.
Second, the line can become single peaked for high rotational
velocities.
\label{fig:CentralSpec} }
\end{figure*}
\begin{figure*}
\begin{center}
\includegraphics[width=0.95\textwidth]{f3.pdf}
\end{center}
\caption{Same as Fig. \ref{fig:CentralSpec} for {\ifmmode{{\rm Ly}\alpha~}\else{Ly$\alpha$~}\fi} photons
initialized homogeneously throughout the sphere.
\label{fig:HomSpec}}
\end{figure*}
\begin{figure*}
\begin{center}
\includegraphics[width=0.95\textwidth]{f4.pdf}
\end{center}
\caption{Shape of the {\ifmmode{{\rm Ly}\alpha~}\else{Ly$\alpha$~}\fi} line for different maximum rotational
velocities for a LoS perpendicular to the rotation axis
($|\mu|\sim 0$). The continuous (dashed) line represents the central
(homogeneous) source distributions. The continuous thin line
represents the intrinsic homogeneous spectrum. The panels follow the same
distribution as in Fig.s \ref{fig:CentralSpec} and \ref{fig:HomSpec}.
\label{fig:differentvelocities}}
\end{figure*}
\begin{figure*}
\begin{center}
\includegraphics[width=0.95\textwidth]{f4-2.pdf}
\end{center}
\caption{Shape of the {\ifmmode{{\rm Ly}\alpha~}\else{Ly$\alpha$~}\fi} line for different maximum rotational
velocities for a LoS perpendicular to the rotation axis
($|\mu|\sim 1$). The continuous (dashed) line represents the central
(homogeneous) source distributions. The continuous thin line
represent the intrinsic homogeneous spectrum. The panels follow the same
distribution as in Fig.s \ref{fig:CentralSpec} and \ref{fig:HomSpec}.
\label{fig:differentvelocities2}}
\end{figure*}
\subsection{Brief Description of the Radiative Transfer Codes}
Here we briefly describe the relevant characteristics of the two
radiative transfer codes we have used.
For a detailed description we refer the reader to the original papers
\cite{CLARA} and \cite{DijkstraKramer}.
The codes follow the individual scatterings of {\ifmmode{{\rm Ly}\alpha~}\else{Ly$\alpha$~}\fi} photons as they
travel through a 3D distribution of neutral Hydrogen.
The frequency of the photon (in the laboratory frame) and
its direction of propagation change at every scattering.
This change in frequency is due to the peculiar velocities of the
Hydrogen absorbing and re-emitting the photon.
Once the photons escape the gas distribution we store their direction of
propagation and frequency at their last scattering.
The initialization process for the {\ifmmode{{\rm Ly}\alpha~}\else{Ly$\alpha$~}\fi} photons specifies its
position, frequency and direction of propagation.
We select the initial frequency to be exactly the {\ifmmode{{\rm Ly}\alpha~}\else{Ly$\alpha$~}\fi} rest-frame
frequency in the gas reference frame and the direction of propagation
to be random following an flat probability distribution over the sphere.
It means that for photons emitted from the center of the sphere
$x_{\rm in}=0$, while photons emitted at some radii with a peculiar
velocity $\vec{v}$ have initial values $x_{\rm in}$ depending on its direction of
propagation: $x_{\rm in}=\vec{v}\cdot\hat{k}_{\rm in}/v_{\rm th}$.
We do not include the effect of turbulent velocities in the
initialization.
We neglect this given that the induced perturbation should be on the
close to the thermal velocity, $12.8${\ifmmode{{\mathrm{\,km\ s}^{-1}}}\else{\,km~s$^{-1}$}\fi}, which is one order of
magnitude smaller than the velocity widths ($100$-$500${\ifmmode{{\mathrm{\,km\ s}^{-1}}}\else{\,km~s$^{-1}$}\fi}) in the
static case.
If dust is present, the photon can interact either with a Hydrogen
atom or dust grain.
In the case of a dust interaction the photon can be either absorbed or
scattered.
This probability is encoded in the dust albedo, $A$, which we chose
to be $1/2$.
In order to obtain accurate values for the escape fraction of
photons in the presence of dust, we do not use any accelerating
mechanism in the radiative transfer.
The codes treat the gas as homogeneous in density and temperature.
This implies that the gas is completely defined by its geometry
(i.e. sphere or slab), temperature $T$, Hydrogen optical depth
$\tau_{\rm H}$, dust optical depth $\tau_{\rm a}$ and the bulk
velocity field $\vec{v}$.
\subsection{Grid of Simulated Galaxies}
\label{sec:models}
In the Monte Carlo calculations we follow the propagation of $N_{\gamma}=10^5$
numerical photons through different spherical galaxies.
For each galaxy we vary at least one of the following parameters: the maximum
rotational velocity $V_{\rm max}$, the hydrogen optical depth $\tau_{H}$,
the dust optical depth $\tau_{a}$ and the initial distribution of photons
with respect to the gas.
In total there are $48$ different models combining all the possible
different variations in the input parameters.
Table \ref{table:models} lists the different parameters we used to
generate the models. The results and trends we report are observed in both
Monte Carlo codes.
\begin{table}
\begin{center}
\begin{tabular}{llc}\hline\hline
Physical Parameter (units) & Symbol & Values\\\hline
Velocity ({\ifmmode{{\mathrm{\,km\ s}^{-1}}}\else{\,km~s$^{-1}$}\fi}) & $V_{\rm max}$&$0,\ \ 100,\ 200,\ 300$\\
Hydrogen Optical Depth & $\tau_{H} $ & $10^{5},\ 10^{6},\ 10^{7}$\\
Dust Optical Depth & $\tau_{a}$ & $0$,$1$\\
Photons Distributions & & Central, Homogeneous\\\hline\hline
\end{tabular}
\caption{
Summary of Physical Parameters of our Monte Carlo Simulations.}
\label{table:models}
\end{center}
\end{table}
\section{Results}
\label{sec:results}
The main results of this paper are summarized in Fig.
\ref{fig:CentralSpec} and \ref{fig:HomSpec}.
They show 2D histograms of the escape frequency $x$ and outgoing angle
$\theta$ parametrized by $|\mu|$.
Taking into account only photons photons around a value
of $|\mu|$ gives us the emission detected by an observer located at an
angle $\theta$ with respect to the rotation axis.
We have verified that the solutions are indeed symmetric with respect
to $\mu=0$. We have also verified that the total flux is the same for all $\mu$.
From these figures we can see that the line properties change with
rotational velocity and depend on the viewing angle $\theta$.
In the next subsections we quantify the morphology changes with with
velocity, optical depth and viewing angle.
We characterize the line morphology by its total intensity, the full
width at half maximum, (FWHM) and the location of the peak maxima.
In order to interpret the
morphological changes in the line we also report the median number of
scatter for each {\ifmmode{{\rm Ly}\alpha~}\else{Ly$\alpha$~}\fi} photon in the simulation.
For the models where dust is included we measure the escape fraction
as a function of rotational velocity and viewing angle.
\subsection{Line Morphology}
\label{sec:angles}
The first column in both Fig. \ref{fig:CentralSpec} and
\ref{fig:HomSpec} shows that for the static sphere the line properties
are independent of $|\mu|$, as it is expected due
to the spherical symmetry.
However, for increasing rotational velocities, at a fixed optical
depth, there are clear signs that this symmetry is broken.
If the viewing angle is aligned with the rotation axis, $|\mu|\sim
1$, the {\ifmmode{{\rm Ly}\alpha~}\else{Ly$\alpha$~}\fi} line keeps a double peak with minor
changes in the morphology as the rotational velocity increases.
However, for a line of sight perpendicular to the rotation axis,
$|\mu|\sim 0$, the impact of rotation is larger.
The double peak readily transforms into a single peak.
This is clear in Fig. \ref{fig:differentvelocities} where we
present the different line morphologies for $|\mu|\sim 0$ in the
homogeneous and central configurations.
The panels have the same distribution as Fig. \ref{fig:CentralSpec}
and \ref{fig:HomSpec}.
There are three clear effects on the line morphology as the rotational
velocity increases.
First, the line broadens; second, the double peaks reduce their intensity; and
third, the intensity at the line centre rises.
The last two effects are combined to give the impression that the double
peaks are merged into a single one at high rotational velocities.
\subsection{Integrated Line Intensity}
\label{sec:intlineint}
\begin{figure}
\begin{center}
\includegraphics[width=0.4\textwidth]{f5.pdf}
\end{center}
\caption{Integrated flux distribution as a function of the
viewing angle as parametrized by $\mu$. Continuous (dashed)
correspond to central (homogeneous) source distribution.
The models correspond to an optical depth of $\tau_{\rm
H}=10^5$ and rotational velocities of $100${\ifmmode{{\mathrm{\,km\ s}^{-1}}}\else{\,km~s$^{-1}$}\fi}, $200${\ifmmode{{\mathrm{\,km\ s}^{-1}}}\else{\,km~s$^{-1}$}\fi} and
$300${\ifmmode{{\mathrm{\,km\ s}^{-1}}}\else{\,km~s$^{-1}$}\fi}. The distributions are flat in the range of models probed
in this paper, meaning that the integrated flux for all viewing
angles is the same.
\label{fig:muhisto}}
\end{figure}
We now consider possible variations in the integrated flux with
respect to the viewing angle $\theta$.
To this end we define the normalized flux seen by an observer at an
angle $\mu$ by:
\begin{equation}
F(\mu) = \frac{2\Delta N}{N\Delta \mu},
\end{equation}
where $\mu=\cos\theta$, $N$ is the total number of outgoing photons,
$\Delta N$ is the number of photons in an angular bin $\Delta
\theta$. This definition satisfies the condition
$\int_{-1}^{1}F(\mu)d\mu/2=1$. In the case of perfect spherical
symmetry one expects a flat distribution with $F(\mu)=1$.
Fig. \ref{fig:muhisto} shows the results for a selection of models
with $\tau_{\rm H}=10^{5}$, different rotational velocities and the two
types of source distributions. This shows that $F(\mu)$ is consistent with being flat, apart
from some statistical fluctuations on the order of 2\%.
This is a remarkable result: while the rotation axis defines preferential direction, the
integrated flux is the same for all viewing angles in the range of parameters explored in this paper. This can be understood from the fact that
{\it radiative transfer inside a sphere that undergoes solid-body
rotation proceeds identical as inside a static sphere}: we can draw
a line between any two atoms within the rotating cloud, and their
relative velocity along this line is zero (apart from the relative
velocity as a result of random thermal motion), irrespective of the
rotation velocity of the cloud. This relative velocity is what is
relevant for the radiative transfer\footnote{This point can be further
illustrated by considering the path of individual photons: let a
photon be emitted at line center ($x=0$), in some random direction
${\bf k}$, propagate a distance that corresponds to $\tau_0=1$,
scatter fully coherently (i.e. $x=0$ after scattering in the gas
frame) by 90$^{\circ}$, and again propagate a distance that
corresponds to $\tau_0=1$. The position where the photon scatters
next does {\it not} depend on the rotation of the cloud, nor on
${\bf k}$.}
\subsection{Full Width at Half Maximum}
\label{sec:widthpeak}
\begin{figure*}
\begin{center}
\includegraphics[width=0.95\textwidth]{f6.pdf}
\end{center}
\caption{FWHM for the non-dusty models as a function of the viewing
angle parametrized by $|\cos\theta|$. Continuous (dashed) lines correspond
to central (homogeneous) source distributions. The general trend is
of an decreasing line width as the line of sight becomes parallel to the
rotation axis.
\label{fig:widthvsmu}}
\end{figure*}
\begin{figure*}
\begin{center}
\includegraphics[width=0.95\textwidth]{f7.pdf}
\end{center}
\caption{FWHM for the non-dusty models as a function of
rotational velocity $V_{\rm max}$ for observers located
perpendicular to the rotation axis.
The left panel shows the results in velocity units while the right
panel normalizes the data by the FWHM in the static case.
Continuous (dashed) lines correspond to central (homogeneous)
source distributions.
The straight lines represent the fit to the data using the
expression in Eq. (\ref{eq:fwhm}).
\label{fig:widthsvsvelocity}}
\end{figure*}
We use the full width at half maximum (FWHM) to quantify the line
broadening.
We measure this width from the line intensity histogram by finding the
values of the velocities at half maximum intensity.
We use lineal interpolation between histogram points to get a value
more precise than the bin size used to construct the histogram.
Fig. \ref{fig:widthvsmu} shows the FWHM for all models as a function
of the viewing angle.
The FWHM increases for decreasing values of $\mu$ (movement from the
poles to the equator) and increasing values of $V_{\rm max}$.
In Fig.
\ref{fig:widthsvsvelocity} we fix $|\mu|<0.1$, i.e. viewing angle
perpendicular to the rotation axis, to plot the FWHM as a function of
rotational velocity.
We parametrize the dependency of the line width with $V_{\rm max}$ as
\begin{equation}
{\rm FWHM}^2 = {\rm FWHM}_{0}^2 + V_{\rm max}^2/\lambda^2,
\label{eq:fwhm}
\end{equation}
where FWHM$_{0}$ is the velocity width in the static case and $\lambda$
is a positive scalar to be determined as a fit to the data.
With this test we want to know to what extent the new velocity width can be
expressed as a quadratic sum of the two relevant velocities in the
problem.
All the models fall into a single family of lines in the plane shown
in the right panel of Fig. \ref{fig:widthsvsvelocity}, justifying
the choice of our parametrization.
We fit simultaneously all the points in two separate groups, central
and homogeneous sources.
We find that these values are $\lambda_{\rm c} = 0.83 \pm 0.06$ and
$\lambda_{\rm h}= 0.82\pm 0.05$ respectively.
\subsection{Line Maxima}
\label{sec:maxima}
\begin{figure*}
\begin{center}
\includegraphics[width=0.95\textwidth]{f8.pdf}
\end{center}
\caption{Position of the line maxima as a function of maximum
rotational velocity $V_{\rm max}$. Continuous (dashed) lines
correspond to central (homogeneous) source distributions. A value
of $x_{\rm max}=0$ indicates that line becomes single
peaked. \label{fig:maximumvsvelocity}}
\end{figure*}
We measure the peak maxima position, $x_m$, to quantify the transition from
double into single peak profiles.
In Fig. \ref{fig:maximumvsvelocity} we show the dependence of $x_m$ with
the viewing angle parametrized by $|\cos\theta|$ for different
rotational velocities.
There are two interesting features that deserve attention.
First, for a viewing angle parallel to the rotational axis ($\mu\sim
1.0$) the maxima of all models with the same kind of source
initialization are similar regardless of the rotational velocity.
Second, at a viewing angle perpendicular to the rotation axis ($\mu\sim 0.0$) a
large fraction of models become single peaked.
This feature appears more frequently for homogeneously distributed
sources if all the other parameters are equal.
\subsection{Dusty Clouds: Escape Fraction}
\label{sec:escapefraction}
\begin{table}
\begin{center}
\begin{tabular}{c cccccc}
\hline \hline
Source & $\tau_{H}$ & & $\ V_{\rm max}$& & \\
Distribution& & & ({\ifmmode{{\mathrm{\,km\ s}^{-1}}}\else{\,km~s$^{-1}$}\fi}) & & \\
& & 0 & 100 &200 & 300\\ \hline
Homogeneous & $10^{5}$& 0.263 & 0.263 & 0.263 & 0.263 \\
& $10^{6}$ & 0.291 & 0.292 & 0.293 & 0.293 \\
&$10^{7}$ & 0.228 & 0.228 & 0.228 & 0.228 \\
Central & $10^{5}$ & 0.096 & 0.096 & 0.096 & 0.096 \\
&$10^{6}$ & 0.066 & 0.066 & 0.066 & 0.066 \\
&$10^{7}$ & 0.015 & 0.016 & 0.016 & 0.015 \\
\hline
\end{tabular}
\caption{
Escape fraction values for all dusty models. }
\label{table:escape}
\end{center}
\end{table}
We now estimate the escape fraction $f_{\rm esc}$ for the dusty
models. The main result is that we do not find any significant dependence
with either the viewing angle nor the rotational velocity. This is consistent with our finding in \S~\ref{sec:intlineint}, that radiative transfer inside the cloud does not depend on its rotational velocity. For completeness we list in Table \ref{table:escape} the escape
fraction for all models.
We now put these results in the context of the analytic solution for
the infinite slab\citep{Neufeld90}.
In Neufeld's set-up the analytic solution depends
uniquely on the product $(a\tau_{\rm H})^{1/3}\tau_{A}$ where
$\tau_{A} = (1 - A)\tau_{a}$, valid only in the limit $a\tau_{\rm
H}\gg 1$.
At fixed values of $\tau_{a}$ the escape fraction monotonically
decreases with increasing values of $\tau_{\rm H}$.
This expectation holds for the central sources.
But in the case of homogeneous sources the escape fraction increases
slightly from $\tau_{\rm H}=10^5$ to $\tau_{\rm H}=10^{6}$
The naive interpretation of the analytic solution does not seem to
hold for photons emitted far from the sphere's center.
We suggest that increasing $\tau_{\rm H}$ from $10^{5}$ to $10^{6}$ causes a
transition from the 'optically thick' to the 'extremely optically
thick' regime for a noticeable fraction of the photons in the
homogeneous source distribution.
In the optically thick regime, {\ifmmode{{\rm Ly}\alpha~}\else{Ly$\alpha$~}\fi} photons can escape in
'single flight' which corresponds to a scenario in which the
photon resonantly scatters $10^4-10^5$ times until it is scattered
into the wing of the line ($x\sim 3-4$).
At these frequencies the medium is optically thin, and the photons can
escape efficiently in a single flight.
In contrast, in an extremely optically thick medium {\ifmmode{{\rm Ly}\alpha~}\else{Ly$\alpha$~}\fi}
photons escape in a `single excursion' \citep{Adams72}.
Here, photons that are scattered into the wing of the line escape from
the medium in a sequence of wing scattering events.
In both cases, {\ifmmode{{\rm Ly}\alpha~}\else{Ly$\alpha$~}\fi} photons resonantly scatter $10^4$-$10^5$ times.
Because we keep our clouds the same size, the mean free path of Lya
photons that scatter resonantly is 10 times larger for the case
$\tau_{\rm H}=10^5$ than for $\tau_{\rm H}=10^6$.
If we compute the average distance $D$ travelled by {\ifmmode{{\rm Ly}\alpha~}\else{Ly$\alpha$~}\fi}
photons through a medium of size $R$ as a function of line center
optical depth $\tau_{\rm H}$, then we find that during the transition
from optically thick to extremely optically thick the mean traversed
distance $D$ actually decreases slightly.
This decrease is unique to this transition region, and $D$ generally
increases with $\tau_{\rm H}$ at other values of $\tau_{\rm H}$.
\subsection{Average Number of Scatterings}
\label{sec:scatterings}
\begin{figure*}
\begin{center}
\includegraphics[width=0.45\textwidth]{f12.pdf}
\includegraphics[width=0.45\textwidth]{f13.pdf}
\end{center}
\caption{2D histogram of the logarithm of the average number of scatterings as function of $\mu$ (left) and the maximum rotational velocity $V_{\rm
max}$ (right). The left panel shows the behaviour for $\tau=10^{5}$ and
$V_{max}=300${\ifmmode{{\mathrm{\,km\ s}^{-1}}}\else{\,km~s$^{-1}$}\fi} as a function of $\left|\cos\theta\right|$, the color indicates the number of photons per bin. In the
right panel the continous (dashed) lines represent the results for
the central (homogeneous) model. The independence of $N_{\rm scatt}$ with $\mu$ and $V_{\rm max}$ is
present in all models.
\label{fig:Nscatt} }
\end{figure*}
The number of scatterings affects the escape frequency of a {\ifmmode{{\rm Ly}\alpha~}\else{Ly$\alpha$~}\fi}
photon. Studying this quantity further illustrates the independence of
the integrated flux and the escape fraction on rotational velocity.
In Fig. \ref{fig:Nscatt} we show the average number of scatterings
$\langle N_{\rm scatt}\rangle$ as a function of the cosinus of the
outgoing angle $|\cos\theta|$ and the rotational velocity
$V_{\rm max}$.
From the right panel observe that the number of scatterings and the
outgoing angle are independent.
This plot corresponds to the specific case of the central model with
$\tau=10^5$ and $V_{\rm max}=300${\ifmmode{{\mathrm{\,km\ s}^{-1}}}\else{\,km~s$^{-1}$}\fi}, but we have verified that this
holds for all models.
The right panel of Fig. \ref{fig:Nscatt} shows how the average
number of scatterings is also independent from the rotational
velocity.
The lower number of average scatterings in the homogeneous source
distribution is due to a purely geometrical effect.
Photons emitted close to the surface go through less scatterings
before escaping.
In static configurations it is expected that the optical depth correlates number of
scatterings.
This has been precisely quantified in the case of static infinite
slab.
In that model for centrally emitted sources the average number of
scatterings depends only on the optical depth $\langle N_{\rm
scatt}\rangle=1.612\tau_{\rm H}$ \citep{Adams72,Harrington73}, for
homogeneously distributed sources $\langle N_{\rm
scatt}\rangle=1.16\tau_{\rm H}$ \citep{Harrington73}.
In our case we find that for the central model the number of
scatterings is proportional to the optical depth, with $\langle N_{\rm
scatt}\rangle= (1.50, 1.00, 0.92)\tau_{\rm H}$ for optical depth
values of $\tau_{\rm H} = (10^{5}, 10^{6}, 10^{7})$ respectively.
For the homogeneous sources we find that $\langle N_{\rm
scatt}\rangle= (0.99, 0.59, 0.51)\tau_{\rm H}$.
\section{Discussion}
\label{sec:discussion}
\subsection{Towards an analytical description}
There is a key result of our simulations that allows us to build an
analytical description for the outgoing spectra.
It is the independence of the following three quantities with the rotational
velocity and the viewing angle: integrated flux, average number of
scatterings and escape fraction.
As we explained in \S~\ref{sec:intlineint}, the best way to understand this is that radiative transfer inside a sphere that undergoes solid-body rotation
proceeds identical to that inside a non-rotating sphere. While scattering events off atoms within the rotating cloud impart
Doppler boosts on the Ly$\alpha$ photon, these Doppler boost are only
there in the lab-frame. Therefore, in the frame of the rotating gas cloud all atoms are
stationary with respect to each other and the scattering process
proceeds identical as in the static case (also see \S~\ref{sec:intlineint} for an additional more quantitative explanation).
This result allows us to analytically estimate the spectrum emerging from a rotating cloud:
The spectrum of {Lyman $\alpha$~} photons emerging from a rotating gas cloud is identical as for the static case in a frame that is co-rotating
with the cloud. However, the surface of cloud now moves in the lab-frame.
Each surface-element on the rotating cloud now has a bulk
velocity with respect to a distant observer. In order to compute the
spectrum one can integrate over all the surface elements in the
sphere with their corresponding shift in velocity and an additional
weight by the surface intensity.
Fig~\ref{fig:comparison} shows some examples of analytic versus full MC
spectra using this approach (the implementation details are in the Appendix).
The left panel shows the results for different rotational velocities
in the case of $\tau_{H}=10^7$ and an observer located perpendicular
to the axis of rotation ($i=0$ in the scheme of Fig~\ref{fig:scheme}
in the Appendix). The right panel shows the results for different viewing angles in the
case of $\tau_{H}=10^7$ and a rotational velocity of $V_{\rm
max}=300${\ifmmode{{\mathrm{\,km\ s}^{-1}}}\else{\,km~s$^{-1}$}\fi}.
The two methods clearly give good agreement, though not perfect. In particular, the left panel shows that the MC gives rise to a spectrum that is
slightly more concentrated towards the line centre. As we explain in Appendix~\ref{sec:app}, we do not expect perfect agreement, because this requires an analytic solution for the spectrum of Ly$\alpha$ photons emerging from a static, optically extremely thick cloud {\it as a function of the angle at which they escape from the sphere}. This solution does not exist in the literature. It is possible to get better agreement my modifying the surface brightness profile.
In any case, the analytic calculation closely captures the results
obtained from the full calculations from the MC simulations.
As such, they are extremely useful and provide us with a quick tool to
verify our calculations at the first order level.
\begin{figure*}
\begin{center}
\includegraphics[width=0.49\textwidth]{fig10a.pdf}
\includegraphics[width=0.49\textwidth]{fig10b.pdf}
\end{center}
\caption{
Comparison of the Monte Carlo results against the analytic
solution. The left panel explores the results of different velocities.
The right panel presents the results for two different observers:
paralel and perpendicular to the rotational axis, $\mu=1$ and $\mu=0$
respectively.
\label{fig:comparison} }
\end{figure*}
\subsection{Impact on the interpretation of simulated and
observational data}
We now compare our findings to other computational results and discuss
its possible implications for the interpretation of observational data.
{\bf Escape at Line Center.} Our models have shown that rotation
enhances the flux density at line center (see Fig. \ref{fig:differentvelocities}). It has
recently been proposed that galaxies with Lya spectral lines that
contain flux at line center may be `leaking' ionizing (LyC) photons
\citep{Behrens2014,2014arXiv1404.2958V}. The main reason for this possible
connection is that the escape of ionizing (LyC) photons requires
$N_{\rm HI} < 10^{17} $cm$^{-2}$. The same low column densities facilitate the escape of
{\ifmmode{{\rm Ly}\alpha~}\else{Ly$\alpha$~}\fi} photons at (or close to) line center. Our work suggests that
rotation may provide an alternative explanation.
{\bf Single peaked lines}. The presence of single peaked profiles has
been associated to inflow/outflow dynamics
\citep{Verhamme06,DijkstraKramer}.
Gas bulk rotation can also be considered as a probable origin for that
behaviour, provided that the observed single peak is highly
symmetric.
Similarly, in the case of double peaked lines with a high
level of flux at the line center, rotation also deserves to be
considered in the pool of possible bulk flows responsible for that feature,
specially if the two peaks have similar intensities.
{\bf Systemic velocities}. There are observational measurements for the
velocity shift between the {\ifmmode{{\rm Ly}\alpha~}\else{Ly$\alpha$~}\fi} and other emission lines. In our study
we find that the position of the peak maxima can suddenly change with
rotation and viewing angle. Namely the line can become single peaked
for high rotational velocities and viewing angles perpendicular to the
rotation axis.
{\bf Galaxy simulations with gas rotation}. \cite{Verhamme12} studied {\ifmmode{{\rm Ly}\alpha~}\else{Ly$\alpha$~}\fi}
line emission in two high resolution simulations of individual
galaxies.
The main purpose of their study was to assess the impact of two
different ISM prescriptions.
However, each simulated galaxy had a disc structure with a clear rotation pattern in
the ISM and inflowing gas from the circum-galactic region.
The configuration had an axial symmetry and they reported a strong dependence of both
the escape fraction and the total line intensity as a function of the
$\theta$ angle.
From our study, none of these two quantities has a dependence either
on the inclination angle or the rotational velocity.
We suggest that he effect reported by \cite{Verhamme12} is
consistent with being a consequence of the different hydrogen optical
depth for different viewing angles and not as an effect of the bulk
rotation.
{\bf Zero impact on the {\ifmmode{{\rm Ly}\alpha~}\else{Ly$\alpha$~}\fi} escape fraction}. Study of
high redshift LAEs in numerical simulation often requires the
estimation of the {\ifmmode{{\rm Ly}\alpha~}\else{Ly$\alpha$~}\fi} escape fraction in order to compare their
results against observations
\citep{CLARA,Dayal2012,Forero12,Orsi12,Garel2012}. Most of these
models estimate the escape fraction from the column density of dust and
neutral Hydrogen. The results of our simulation indicate that the
rotational velocity does not induce additional uncertainties in those
estimates.
\section{Conclusions}
\label{sec:conclusions}
In this paper we quantified for the first time in the literature the effects
of gas bulk rotation in the morphology of the {\ifmmode{{\rm Ly}\alpha~}\else{Ly$\alpha$~}\fi} emission line in
star forming galaxies.
Our results are based on the study of an homogeneous sphere
of gas with solid body rotation.
We explore a range of models by varying the rotational speed, hydrogen
optical depth, dust optical depth and initial distribution of {\ifmmode{{\rm Ly}\alpha~}\else{Ly$\alpha$~}\fi}
photons with respect to the gas density.
As a cross-validation, we obtained our results from two independently
developed Monte-Carlo radiative transfer codes.
Two conclusions stand out from our study.
First, rotation clearly impacts the {\ifmmode{{\rm Ly}\alpha~}\else{Ly$\alpha$~}\fi} line morphology; the width and
the relative intensity of the center of the line and its peaks are
affected.
Second, rotation introduces an anisotropy for different viewing
angles.
For viewing angles close to the poles the line is double peaked and it
makes a transition to a single peaked line for high rotational
velocities and viewing angles along the equator.
This trend is clearer for spheres with homogeneously distributed
radiation sources than it is for central sources.
Remarkably, we find three quantities that are invariant with respect
to the viewing angle and the rotational velocity: the integrated flux,
the escape fraction and the average number of scatterings.
These results helped us to construct the outgoing spectra of a
rotating sphere as a superposition of spectra coming from a static
configuration. This description is useful to describe the main
quantitative features of the Monte Carlo simulations.
Quantitatively, the main results of our study are summarized as
follows.
\begin{itemize}
\item In all of our models, rotation induces changes in the line morphology
for different values of the angle between the rotation
axis and the LoS, $\theta$. The changes are such that for
a viewing angle perpendicular to the
rotation axis, and high rotational velocities the line becomes single peaked.
\item The line width increases with rotational
velocity. For a viewing angle perpendicular to the rotation axis
This change approximately follows the functional form ${\rm FWHM}^2
= {\rm FWHM}_{ 0}^2 + (V_{\rm max}/\lambda)^2$, where FWHM$_{0}$
indicates the line
width for the static case and $\lambda$ is a constant. We have
determined this constant to be $\lambda_{\rm c}=0.83 \pm 0.06$ and
$\lambda_{\rm h}=0.82\pm 0.05$ for the central and homogeneous source
distributions, respectively.
\item At fixed rotational velocity the line width decreases as $|\mu|$
increases, i.e. the smallest value of the line width is observed for
a line of sight parallel to the ration axis.
\item The single peaked line emerges at viewing angles $\mu\sim 1$ for
when the rotational velocity is close to than half the FWHM$_0$.
\end{itemize}
Comparing our results with recent observed LAEs we find that
morphological features such as high central line flux, single peak
profiles could be explained by gas bulk rotation present in these
LAEs.
The definitive and clear impact of rotation on the {\ifmmode{{\rm Ly}\alpha~}\else{Ly$\alpha$~}\fi} morphology
suggests that this is an effect that should be taken into account at
the moment of interpreting high resolution spectroscopic data. In
particular it is relevant to consider the joint effect of rotation the
and ubiquitous outflows (Remolina-Gutierrez et al., in prep.)
because rotation can lead to enhanced escape of {\ifmmode{{\rm Ly}\alpha~}\else{Ly$\alpha$~}\fi} at line center, which
has also been associated with escape of ionizing (LyC) photons
\citep{Behrens2014,2014arXiv1404.2958V}
\section*{Acknowledgments}
JNGC acknowledges financial support from Universidad de los
Andes.
JEFR acknowledges financial support from Vicerrectoria de
Investigaciones at Universidad de los Andes through a FAPA grant.
We thank the International Summer School on AstroComputing
2012 organized by the University of California High-Performance
AstroComputing Center (UC-HiPACC) for providing computational
resources where some of the calculations were done.
The data, source code and instructions to
replicate the results of this paper can be found
here {\texttt{https://github.com/jngaravitoc/RotationLyAlpha}}.
Most of our code benefits from the work of the IPython and Matplotlib
communities \citep{IPython,matplotlib}.
We thank the referee for the suggestions that allowed us to greatly
improve and better frame the interpretation of our simulations.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,921 |
Q: sizeof operator evaluates in which stage of compilation in gcc sizeof is a compile time operator. See here.
Compilation has many stages. At which stage is the sizeof operator evaluated?
A: Typically, after the preprocessor runs and produces the preprocessed translation unit (whole header files pasted in the place of #include, #define's substituted all over the place, inactive branches of #ifdef conditionals completely removed, etc.), the compiler runs. Most modern compilers are usually also able to do the preprocessing themselves, but for historic reasons, the C preprocessor (cpp) and the C compiler (cc) are at least conceptually distinct. The output of the former serves as input to the latter.
At consequent stages, it is entirely up to the internal implementation of the compiler what these stages are and what their order is. The most "traditional" pipeline, however, is:
*
*Lexing: separation of tokens from one another;
*Parsing: interpreting the combinations of tokens according to the language grammar and producing a parse tree;
*Producing an Abstract Syntax Tree: the parse tree is taken as input and a more usable, better annotated tree is produced;
*Scope analysis: matching the used identifiers with their respective declarations, emitting errors in case of undeclared identifiers;
*Type checking: checking whether the type of each expression matches the expected type in the particular context. After this stage has passed and no errors have been emitted, the program is considered syntactically and semantically correct, so we can proceed with the next step;
*Code generation and optimisation: possibly at this stage, the compiler would emit, for example, 4 in place of the abstract node that represents sizeof(int). It would also chew up constant expressions like 3 + 4 into 7.
Note that sizeof can be evaluated at runtime in case it is applied to a variable-length array (C99 onwards):
int n;
n = ...;
int vl_arr[n];
sizeof(vl_arr); // could be evaluated at runtime if "n" is not known at compile-time
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 7,443 |
UInt32 BGSBipedObjectForm::AddSlotToMask(UInt32 slot)
{
data.parts |= slot;
return data.parts;
}
UInt32 BGSBipedObjectForm::RemoveSlotFromMask(UInt32 slot)
{
data.parts &= ~slot;
return data.parts;
}
UInt32 TESSpellList::GetSpellCount() const
{
return (unk04) ? unk04->numSpells : 0;
}
SpellItem* TESSpellList::GetNthSpell(UInt32 n) const
{
if (unk04 && n < unk04->numSpells) {
return unk04->spells[n];
}
else return NULL;
}
UInt32 TESSpellList::GetShoutCount() const
{
return (unk04) ? unk04->numShouts : 0;
}
TESShout* TESSpellList::GetNthShout(UInt32 n) const
{
if (unk04 && n < unk04->numShouts) {
return unk04->shouts[n];
}
else return NULL;
}
class ItemCounter
{
UInt32 m_count;
TESForm * m_item;
public:
ItemCounter(TESForm * item) : m_count(0), m_item(item) {}
bool Accept(TESContainer::Entry * entry)
{
if (entry->form == m_item)
m_count += entry->count;
return true;
}
UInt32 Count() const { return m_count; }
};
UInt32 TESContainer::CountItem(TESForm * item) const
{
ItemCounter v(item);
Visit(v);
return v.Count();
}
UInt32 TintMask::ToARGB()
{
return MAKE_COLOR((UInt32)(alpha * 255), color.red, color.green, color.blue);
}
SInt32 PlayerSkills::ResolveAdvanceableSkillId(SInt32 actorValue)
{
if(actorValue - kAdvanceableSkillOffset >= 0 && actorValue < kNumAdvanceableSkills + kAdvanceableSkillOffset)
return actorValue - kAdvanceableSkillOffset;
return -1;
}
float PlayerSkills::GetSkillPoints(BSFixedString actorValue)
{
if(data) {
UInt32 avId = LookupActorValueByName(actorValue.data);
SInt32 skillId = ResolveAdvanceableSkillId(avId);
if(skillId != -1)
return data->levelData[skillId].points;
}
return 0.0;
}
void PlayerSkills::SetSkillPoints(BSFixedString actorValue, float points)
{
if(data) {
UInt32 avId = LookupActorValueByName(actorValue.data);
SInt32 skillId = ResolveAdvanceableSkillId(avId);
if(skillId != -1)
data->levelData[skillId].points = points;
}
}
SInt32 PlayerSkills::GetSkillLegendaryLevel(BSFixedString actorValue)
{
if(data) {
UInt32 avId = LookupActorValueByName(actorValue.data);
SInt32 skillId = ResolveAdvanceableSkillId(avId);
if(skillId != -1)
return data->legendaryLevel[skillId];
}
return -1;
}
void PlayerSkills::SetSkillLegendaryLevel(BSFixedString actorValue, UInt32 level)
{
if(data) {
UInt32 avId = LookupActorValueByName(actorValue.data);
SInt32 skillId = ResolveAdvanceableSkillId(avId);
if(skillId != -1)
data->legendaryLevel[skillId] = level;
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,094 |
Q: How to upload a bunch of files with Python requests module When I using the requests module to upload at least 400 files with one request, I got the Too many open files error on macOS system.
I have tried ulimit -n 20000.
Also checked:
*
*sysctl kern.maxfiles got 98304
*sysctl kern.maxfilesperproc got 49152
However, it didn't work.
Here is my code:
import os
import requests
url = 'http://127.0.0.1:8000/api/upload'
file_path = '/Users/BonJu/Projects/downloads'
file_list = os.listdir(file_path)
files = []
for file in file_list:
try:
source = open(os.path.join(file_path, file), 'rb')
files.append(('file', source))
except Exception as e:
print('File: %s, Error: %s' % (file, e.__str__()))
continue
response = requests.post(url=url, data={'uploader': 'admin'}, files=files)
Terminal result:
File: test_252.docx, Error: [Errno 24] Too many open files: '/Users/BonJu/Projects/downloads/test_252.docx'
File: test_253.docx, Error: [Errno 24] Too many open files: '/Users/BonJu/Projects/downloads/test_253.docx'
File: test_254.docx, Error: [Errno 24] Too many open files: '/Users/BonJu/Projects/downloads/test_254.docx'
...
File: test_418.docx, Error: [Errno 24] Too many open files: '/Users/BonJu/Projects/downloads/test_418.docx'
Because it is an API server which links the sent log files to the issue I need all the files to be sent in one request, otherwise the manager will get several mails and won't be able to address the issues.
Is there any solution for this situation?
Solution
I finally adjusted my API to save a temp file for storing the upload logs and pass a status parameter to control the final output.
my code:
payload = {
'status': 'finish',
'uploader': 'admin'
}
response = requests.post(url=url, data=payload, files=files)
API:
@api_view(['post'])
def upload(request, debug, api_version):
status = request.POST.get('status')
file_list = request.FILES.getlist('file')
if status == 'finish':
# open the temp file and insert the last logs then output
else:
# create/insert the logs and save to a temp file
return Response({'status': status, 'files': file_list})
A: I would suggest make a single tar file from all of them.
or closing the files after reading their content (instead of leaving their file descriptor open until you post the request
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 2,133 |
{"url":"https:\/\/www.lumoslearning.com\/llwp\/resources\/common-core-practice-tests-and-sample-questions\/practice-test.html?cid=1566&lid=34651&qid=296975","text":"# Lumos StepUp High School Algebra 2 - Mathematics Practice test + Workbook Writing Functions in Different Forms to find Critical Information\n\n#### Get Full Access to Lumos StepUp High School Algebra 2 - Mathematics Practice test + Workbook\n\nCurrently, you have limited access to Lumos StepUp High School Algebra 2 - Mathematics Practice test + Workbook. The Full Program includes,\n\nOnline Program\n\n## We can write the equation for Kinetic Energy as $$KE\\left(v\\right)=\\frac{1}{2}mv^2$$. We know that the vertex exists at v = 0. Rewrite the equation assuming an additional velocity of $$3m\/s$$.\n\n### Ratings\n\nRate this Question?\n0\n\n0 Ratings & 0 Reviews\n\n5\n0\n0\n4\n0\n0\n3\n0\n0\n2\n0\n0\n1\n0\n0","date":"2019-10-23 00:08:40","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.5110056400299072, \"perplexity\": 2372.4397137211636}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-43\/segments\/1570987826436.88\/warc\/CC-MAIN-20191022232751-20191023020251-00061.warc.gz\"}"} | null | null |
Good Morning Delmarva
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Maryland officials unveil Smart Pond technology
Deana Harley,
FRUITLAND, Md. – In Maryland, a new technology is changing the way ponds throughout the state sustain and clean themselves. On Tuesday, officials from across the state gathered at one pond in Fruitland that will be one of the first ponds to be equipped with the technology, but officials say they aren't stopping with just local ponds.
"So welcome to the future, this is the future of storm water regulation and management," Ben Grumbles, Secretary of the Maryland Department of the Environment, said on Tuesday.
It's the future in the form of a smart pond.
"Normally this pond would just drain, and it wouldn't do anything else, but water would fall and come in from parking lot and buildings and then it would just fill up and then drain," Dr. Charles Glass, Assistant Secretary for the Maryland Department of Transportation, said.
But with this new technology unveiled, ponds like one in Fruitland will be able to sustain themselves more effectively.
"This technology a little allows it to fill up, and, like a bathtub, hold the water if we want to, or release the water if we want to based on storms coming or storms already being passed," Dr. Glass said.
Once this system is used in storm ponds throughout Maryland, officials say it will be used to clean one of the areas most important assets: the Chesapeake Bay.
"The Chesapeake Bay is one of the most important estuaries in the world really, certainly the United States," Rich Dailey, Senior Director of Environment Safety and Health Compliance for Walmart, said.
But none of this would be possible without a first-of-its-kind public to private partnership between several state agencies, including the Maryland Department of Transportation, and a private corporation: Walmart. And you may be thinking – why is Walmart involved with ponds in Maryland?
"This is really an extension of who we are as a company," Dailey said. "In 2005, our CEO at the time, Lee Scott, committed the company to taking a global leadership position in environmental sustainability."
It's a partnership and an initiative that these officials say will stretch far wider than the Eastern Shore.
"We're also looking at the Great Lakes region and Tampa as our next areas for expansion of this type of program," Dailey said.
Categories: Local News, Maryland
Tags: ben grumbles, chesapeake bay, Dr. Charles Glass, environment, fruitland, maryland, maryland department of the environment, maryland department of transportation, officials, Ponds, Rich Dailey, Smart Pond, storm water, walmart, water
Laurel man arrested, facing felony assault charges
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"redpajama_set_name": "RedPajamaCommonCrawl"
} | 5,114 |
Q: Angular Momentum of a Mass that is Tangent to a Tractrix A trailer follows a tractrix path. The trailer hitchpoint trajectory is a straight path. The trailer has a mass moment of inertia which accumulates angular momentum as it travels from an articulated position towards the asymptote.
I need an expression for hitchpoint travel in terms of (t) versus angular momentum.
Any help would be gratefully received.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 7,662 |
\section[Introduction]{Introduction
Spiking statistics of various neuronal models under a random stimulation has been studied
in the framework of two main approaches. The first one is named in \cite{Stein1967}
as ``Gaussian'', because it describes random stimulation by means of Gaussian noise,
see e.g. \cite{Bryant1976}.
This approach has developed into the well-known diffusion approximation methodology, see
\cite{Capocelli1971}.
The second approach is named in \cite{Stein1967} as ``quantal'', because it takes into account
the discrete nature of the influence any input impulse may have on its target neuron.
The wide area of research\cite{Soltic2010,Yamanishi2012,Rossell2013}
and applications\cite{Nichols2010,Luque2012,Zhang2014}
known as
spiking neural networks. see
\cite{Ghosh-Dastidar2009} for a review, could be considered
as utilizing the quantal approach.
For a recent review of mathematically rigorous results as regards neuronal spiking statistics
in the both approaches see \cite{Sacerdote2013}.
We study here mathematically
rigorously, in the framework of quantal approach, spiking
statistics of inhibitory neuron model belonging to a class of models (see Sec. 2.1, below)
with fast $Cl$-type inhibitory delayed feedback.
This construction is stimulated with Poisson stream of excitatory input impulses.
For this configuration it was proven
in the previous paper \cite{Kravchuk2013}
for a concrete neuronal model --- the binding neuron%
\footnote{Detailed description of the binding neuron model can be found in \cite{Vidybida2014}.\\ See also {\tt https://en.wikipedia.org/wiki/Binding\_neuron}.}
with threshold 2 ---,
that statistics of its ISIs is essentially non-Markov. In this paper, we refine
and extend methods
of \cite{Kravchuk2013} making them applicable to any
inhibitory neuron with fast {\it Cl}-type inhibition satisfying a number of simple and
natural conditions. Under those conditions, we prove rigorously that
ISI statistics of a neuron with delayed fast $Cl$-type inhibitory feedback
stimulated with a Poisson stream of input impulses
cannot be represented as a Markov chain of any
finite order.
A similar statement has been proven recently for a class of excitatory neuronal models \cite{Vidybida2015}.
Here we extensively use the approach of paper \cite{Vidybida2015} modifying it as necessary
to cover the case of the $Cl$-type fast inhibition.
\section{Definitions and assumptions}
\subsection[Neuron without feedback]{Neuron without feedback}\label{neuron}
We expect that a neuron
satisfies the conditions Cond0-Cond2 of \cite{Vidybida2015}. As regards the Cond3, Cond4 of \cite{Vidybida2015}, for the
{\em Cl-}type fast inhibition those conditions should be modified as follows:
\begin{itemize}
\item Cond3: Just after firing, neuron appears in its resting state.
\item Cond4: The output interspike interval (ISI) distribution can be characterized with a
probability density function (pdf) $p^0(t)$, which is {\it continuous} with
\begin{equation}\label{nolj}
p^0(0)=0,
\end{equation}
positive:
\begin{equation}\label{pos}
t>0\Rightarrow p^0(t)>0,
\end{equation}
and bounded:
\begin{equation}\label{bound}
\sup\limits_{t>0} p^0(t)< \infty.
\end{equation}
Also, we impose on the function $p^0(t)$ the following condition: $t<0\Rightarrow p^0(t)=0$
in order to have it defined for all real numbers.
\end{itemize}
The Cond3, above, limits the set of models as compared to \cite{Vidybida2015}. Namely, it claims
that the standard state of \cite[Cond3]{Vidybida2015} be exactly the resting state of neuron. This requirement
is imposed due to the specifics of {\em Cl-}type fast inhibition. Without specifying neuronal model
exactly, one can define the resting state as that achieved if the neuron does not receive any input
for infinitely long time. For our purpose it is important that after receiving inhibitory impulse,
the neuron appears in exactly
the same state as immediately after firing. And the state after receiving
{\em Cl-}type inhibitory impulse can be only the resting state, see Sec. \ref{Bj}, below.
The Cond4, above, seems to be as natural for any neuronal model stimulated with Poisson stream as does
corresponding condition from \cite{Vidybida2015}. Its difference from \cite{Vidybida2015} has a
pure mathematical nature and is suitable for the proof in the case of {\em Cl-}type fast inhibition.
It seems, that this conditions are natural for many threshold-type neuronal models known in the
literature, see \cite{Burkitt,Chacron2003,Jolivet2004,Jolivet2006} and citations therein.
But this still has to be proven by calculating corresponding $p^0(t)$.
At least, all the five conditions are satisfied for the binding neuron model and for the basic
leaky integrate-and-fire
(LIF) model, see \cite{Vidybida2007,Vidybida2014b}, where $p^0(t)$ is calculated exactly for each model, respectively.
\subsection{Feedback line action}\label{line}
We expect that the feedback line satisfies Prop1, Prop2 of \cite{Vidybida2015}.
The Prop3 of \cite{Vidybida2015} should be modified for the {\em Cl-}type fast inhibition
as follows:
\begin{itemize}
\item Prop3: The impulse conveyed to the neuronal input is the fast $Cl$-type inhibitory impulse.
This means that after receiving such an impulse, the neuron appears in its resting state.
This exhausts the action of the inhibitory impulse
in a sense that it has no influence on further neuronal states created by next
excitatory impulses. It as well does not affect neuron
if founds it in its resting state.
\end{itemize}
\subsubsection[Biological justification]{Biological justification\footnote{see also text in Sec. 2.2 Feedback line action of \cite{Vidybida2015}}}\label{Bj}
Neurons, which send inhibitory impulses onto their own body, or dendrites are known in real
nervous system, see \cite{Bekkers1998,Bacci2003,Bacci2004,Smith2002}.
The chief inhibitory neurotransmitter in the nervous system is
Gamma-amino\-bu\-tyric acid (GABA). The GABA can activate several types of receptors, the main of which are
GABA$_a$ and GABA$_b$. If GABA$_a$ receptors are activated, the excitable membrane
becomes permeable for $Cl^-$ ions. If a neuron is partially excited, that is its
membrane is depolarized to an extent, the $Cl^-$ current
cancels this depolarization since the $Cl^-$ reversal potential is close to the resting
potential. For the same reason, the $Cl^-$ current does not appear if the membrane
is at its resting potential. This explains partially where does the Prop3, above, come from.
Another case is with GABA$_b$ receptors activation. This causes $K^+$ ions permeability.
The outward $K^+$ current is able to hyperpolarize membrane even below its resting potential.
The remarkable difference between GABA$_a$ and GABA$_b$ mediated inhibition is rather
different kinetics of the corresponding $Cl^-$ and $K^+$ currents. Namely, according to
\cite{Benardo1994},
the $Cl^-$ current rise time is 1 - 5 ms, and the decay time constant is about 10 - 25 ms.
The $K^+$ current rise time is 10 - 120 ms, and the decay time constant is about
200 - 1600 ms. The $K^+$ current can be even slower, see \cite{Bacci2004,Storm1988a,Storm1990}.
Having in mind this contrast in the speed of $Cl^-$ and $K^+$ transients, we idealize
the $Cl^-$ current kinetics as having infinitesimally short rise time and infinitely
fast decay, both can be achieved with infinitely large $Cl^-$ conductance at the moment of receiving
inhibitory impulse. This additionally explains where does the Prop3, above, come from.
Finally, it should be mentioned that our consideration will be valid also for artificial hardware
neurons, see \cite{Rossell2012,Wang2013},
and abstract neurons used in mathematical studies, provided Cond0-Cond4 and Prop1-Prop3,
above, are satisfied.
\bigskip
The important for us consequence of Prop2 of \cite{Vidybida2015} is that at any moment of time the feedback
line is either empty, or conveys a single impulse. If it does convey an impulse, then its state
can be described with a stochastic variable $s$, $s\in ]0;\Delta]$, which we call further ``time to live''.
The variable $s$ denotes the exact time required by the impulse to reach the output end of the line,
which is the neuron's input
for inhibitory impulses, and to leave the line. Here, $\Delta$ denotes the delay duration in the
feedback line.
\section{Results}
Our purpose here is to prove the following Theorem\footnote{A similar theorem for the {\bf excitatory}
feedback line has been proven in \cite{Vidybida2015}}:
\begin{theorem}
Let a neuronal model satisfies conditions Cond0-Cond4, above.
Expect that the model is extended by introducing a delayed {\bf fast {\it Cl}-type
inhibitory} feedback line,
which satisfies the Prop1-Prop3, above. Then, in the stationary regime,
the output stream of ISIs of the neuron cannot be presented as a Markov chain
of any finite order.
\end{theorem}
\subsection[Proof outline]{Proof outline\footnote{The proof outline is very similar to that of \cite{Vidybida2015}, therefore, it is abridged here. Please, consult with \cite{Vidybida2015}.}}
Let $p^{inh}(t_{n+1}\mid t_n,\dots,t_0)dt_{n+1}$ denotes the conditional probability
to get the duration of $(n+2)$-nd ISI in the interval
$[t_{n+1};t_{n+1}+dt_{n+1}[$ provided
that previous $n+1$ ISIs had duration $t_n,\dots,t_0$, respectively.
From the definition in \cite[Ch.2 §6]{Doob1953},
one can obtain the necessary condition
\begin{multline}
\label{def}
p^{inh}(t_{n+1}\mid t_{n},\ldots,t_{1},t_{0})
= p^{inh}(t_{n+1}\mid t_{n},\ldots,t_{1}),~~
\\
t_i\in\mathbb{R^+},~ i=0,\ldots,n+1,
\end{multline}
required for the stochastic process $\{t_{j}\}$ to be $n$th order Markov chain.
We intend to prove that the relation (\ref{def}) does not hold for any $n$. For this purpose
we calculate exact expression for $p^{inh}(t_{n+1}\mid t_n,\dots,t_0)$ as
\begin{equation}\label{defcond}
p^{inh}(t_{n+1}\mid t_n,\dots,t_0)=
\frac{p^{inh}(t_{n+1}, t_n,\dots,t_0)}{p^{inh}(t_n,\dots,t_0)}
\end{equation}
from which it will be clearly seen that the $t_0$-dependence in $p^{inh}(t_{n+1}\mid t_n,\dots,t_0)$
cannot be eliminated whatever large the $n$ is.
In the Eq. (\ref{defcond}), expression $p^{inh}(t_n,\dots,t_1)$ denotes the
joint probability density function of ISIs duration of neuron with the fast {\em Cl-}type inhibitory delayed feedback.
Let us introduce the conditional joint probability density
$p^{inh}(t_{n+1},\dots,t_0\mid s)$, which denotes the conditional probability density to get
$n+2$ consecutive ISIs $\{t_{n+1},\dots,t_0\}$ provided that at the beginning of the first ISI ($t_0$)
the time to live of impulse in the feedback line was equal to $s$.
This conditional probability can be
used to calculate required joint pdfs as follows
\begin{equation}
\label{pdfs}
p^{inh}(t_{n+1},\dots,t_0)=
\int\limits_0^\Delta p^{inh}(t_{n+1},\dots,t_0\mid s)f^{inh}(s)\,ds,
\end{equation}
where
$f^{inh}(s)$ is the stationary pdf which describes distribution of times to live
at the beginning of any ISI in the stationary regime.
In what follows we analyze the structure of functions
$f^{inh}(s)$ and $p^{inh}(t_{n+1},$ $\dots,t_0\mid s)$. It appears that $f^{inh}(s)$ has a singular
component $a\delta(s-\Delta)$ with $a>0$, and $p^{inh}(t_{n+1},\dots,t_0\mid s)$
has jump discontinuities at definite hyper-planes in the $(n+3)$-dimensional
space of its variables $(t_{n+1},\dots,t_0,s)$.
After integration in (\ref{pdfs}), some of those discontinuities will
survive in the $(n+2)$-dimensional space of variables $(t_{n+1},\dots,t_0)$,
and one of those survived has its position depending on $t_0$.
The $t_0$-dependent jump discontinuity
will as well survive in the $p^{inh}(t_{n+1}\mid t_n,\dots,t_0)$ for any $n$,
provided that $t_{n},\dots,t_0$ satisfy the following condition:
\begin{equation}\label{Domain}
\sum\limits_{i=0}^n t_i<\Delta,
\end{equation}
where $\Delta>0$ is the full delay time in the feedback line.
Taking into account that the equation in the necessary condition (\ref{def}) must hold
for any set of $t_{n+1},\dots,t_0$, we conclude that (\ref{def}) cannot be satisfied for
any $n$.
\subsection{The proof}
\subsubsection{Structure of functions $p^{inh}(t_{n+1},\dots,t_0 \mid s)$}
Specifics of the feedback line action together with condition (\ref{Domain}) results
in a very simple structure of $p^{inh}(t_{n+1},\dots,t_0\mid s)$ at different parts of the
integration domain in (\ref{pdfs}). Those parts are defined as follows:
\begin{multline}\nonumber
D_k=\{s\mid \sum\limits_{i=0}^{k-1}t_i<s \le \sum\limits_{i=0}^{k}t_i\},\,
k=0,\dots,n,
\\
D_{n+1}=\{s\mid \sum\limits_{i=0}^{n}t_i <s \le \Delta\}\,.
\end{multline}
As regards the structure itself, the following representation can be derived
similarly as it was done in \cite{Vidybida2015}:
\begin{multline}\label{struk}
p^{inh}(t_{n+1},\dots,t_0 \mid s)=
\\=
p^{inh}(t_{n+1},\dots,t_{k+1} \mid \Delta)\,
p^{inh}\left(t_k \mid s-\sum\limits_{i=0}^{k-1} t_i\right)
\prod\limits_{i=0}^{k-1}p^0(t_i),
\\
s\in D_k,\quad k=0,\dots,n,
\end{multline}
\begin{equation}
\label{strun+1}
p^{inh}(t_{n+1},\dots,t_0 \mid s)=
p^{inh}\left(t_{n+1} \mid s-\sum\limits_{i=0}^{n} t_i\right)
\prod\limits_{i=0}^{n}p^0(t_i),\quad
s\in D_{n+1}.
\end{equation}
\begin{equation}
\label{strukf}
p^{inh}(t_{n+1},\dots,t_{k+1} \mid \Delta)=
\\
p^{inh}\left(t_{n+1} \mid \Delta-\sum\limits_{i=k+1}^{n} t_i\right)
\prod\limits_{i=k+1}^{n}p^0(t_i).
\end{equation}
Here $p^{inh}(t\mid s)$ denotes the conditional pdf to get ISI of duration $t$ if
at its beginning, time to live of impulse in the feedback line is $s$.
Representation of $p^{inh}(t_{n+1},\dots,t_0 \mid s)$ by means of $p^0(t)$ and $p^{inh}(t\mid s)$
found here is
similar to that found in \cite{Vidybida2015} for the excitatory case.
But the structure of function $p^{inh}(t \mid s)$, used in that representation, is different.
\subsubsection{Structure of function $p^{inh}(t \mid s)$}\label{Spts}
Expect that at the beginning of an ISI, there is an impulse in the
feedback line with time to live $s$. Then the probability that this ISI
will have its duration $t<s$ does not depend on the feedback
line presence. Therefore,
\begin{equation}\label{pinit}
t<s\, \Rightarrow\, p^{inh}(t\mid s) = p^0(t).
\end{equation}
In the opposite situation, receiving of an ISI duration greater than $s$
happens if (i) the neuron does not fire during interval $]0;s[$
and (ii) the neuron starts at its resting state (Prop3, above) at the moment $s$
and fires at $t>s$. Realizations of events (i) and (ii) depend on
disjoint segments of the input Poisson stream.
Therefore, (i) and (ii) are statistically independent. The probability of (i)
is as follows:
\begin{equation}\label{P0}
\mathbf{P}^0(s) = 1 - \int\limits_0^{s} p^0(t)dt.
\end{equation}
The probability of (ii) is $p^0(t-s)$. This gives
\begin{equation}\label{pisnit}
t>s\, \Rightarrow\, p^{inh}(t\mid s) = \mathbf{P}^0(s)p^0(t-s).
\end{equation}
It can be concluded from (\ref{pinit}) and (\ref{pisnit}) that
$$
\lim\limits_{t\uparrow s} p(t\mid s) = p^0(s)\quad
\text{and}\quad
\lim\limits_{t\downarrow s} p(t\mid s) = 0.
$$
Now, taking into account (\ref{nolj}) and (\ref{pos}) from Cond4, above, we conclude that the function
$p^{inh}(t\mid s)$
considered as a function of two variables $(t,s)$, $t\ge0$, $s\in\,]0;\Delta]$ has a jump
discontinuity along the straight line $t=s$. The magnitude of this jump is
$p^0(s)$, and it is strictly
positive for positive $t$. Concrete values of $p^{inh}(t\mid s)$ along the line $t=s$ does not matter and can be chosen
arbitrarily.
Finally, for $p^{inh}(t\mid s)$ we have%
\footnote{Compare this with \cite[Eq. (11)]{Vidybida2013}, where $p^{inh}(t\mid s)$ is
calculated exactly for the binding neuron model.}
\begin{equation}\label{pts}
p^{inh}(t\mid s) = \chi(s-t) p^0(t) + \mathbf{P}^0(s)p^0(t-s),
\end{equation}
where $\chi(s)$ is the Heaviside step function.
\subsubsection{Structure of probability density function $f^{inh}(s)$}
It appeared that the structure of $f^{inh}(s)$ is exactly the same as it was found
in \cite{Vidybida2015} for the excitatory case. This structure is as follows%
\footnote{The proof can be found in \cite{Vidybida2015}.}$^,$%
\footnote{Compare this with \cite[Eqs. (14)-(16)]{Vidybida2008a}, where $f(s)$ is calculated exactly for the binding neuron model.}
\begin{equation}\label{ff}
f^{inh}(s)=g(s) + a\delta(s-\Delta),
\end{equation}
where $a>0$ and $g(s)$ is bounded continuous function vanishing out of interval $]0;\Delta[$.
\subsubsection{Form of $p^{inh}(t_{n+1},\dots,t_0)$ and $p^{inh}(t_{n},\dots,t_0)$ after integration in (\ref{pdfs})}
Let $D=\bigcup\limits_{k=0}^n D_k$.
At $D$, representations (\ref{struk}) and (\ref{strukf}) are valid. Also at $D$,
$f^{inh}(s)$ reduces to $g(s)$.
Therefore,
\begin{multline}\label{intD}
\int\limits_D p^{inh}(t_{n+1},\dots,t_0\mid s) f^{inh}(s)\, ds =
\\=
\sum\limits_{k=0}^np^{inh}\left(t_{n+1}\mid \Delta -\sum\limits_{i=k+1}^n t_i\right)\times
\\\times
\prod\limits_{\vbox{\footnotesize\hbox{$i=0$}\hbox{$i\ne k$}}}^np^0(t_i)
\int\limits_{D_k}p^{inh}\left(t_k\mid s-\sum\limits_{j=0}^{k-1} t_j\right) g(s)ds.
\end{multline}
The first factor (with fixed $k$, $0\le k\le n$) in the r.h.s. of Eq. (\ref{intD}) is as follows:
$$
p^{inh}\left(t_{n+1}\mid \Delta -\sum\limits_{i=k+1}^n t_i\right).
$$
Due to Eq. (\ref{pts}), this factor does have a jump discontinuity along the hyperplane
$
\sum\limits_{i=k+1}^{n+1}t_i=\Delta
$
in the space of variables $(t_0,\dots,t_{n+1})$. Notice, that the position of this hyperplane
does not depend on $t_0$ for any $k\in\{0,\dots,n\}$.
The second factor in the r.h.s. of Eq. (\ref{intD}) is as follows:
$
\prod\limits_{\vbox{\footnotesize\hbox{$i=0$}\hbox{$i\ne k$}}}^np^0(t_i),
$
and it is continuous.
The third factor in the r.h.s. of Eq. (\ref{intD}) can be transformed as follows:
\begin{multline}\label{ThirdFactor}
\int\limits_{D_k}p^{inh}\left(t_k\mid s-\sum\limits_{j=0}^{k-1} t_j\right) g(s)ds=
\int\limits_{\sum\limits_{j=0}^{k-1} t_j}^{\sum\limits_{j=0}^{k} t_j}
p^{inh}\left(t_k\mid s-\sum\limits_{j=0}^{k-1} t_j\right) g(s)ds=
\\
=\int\limits_0^{t_k}p^{inh}(t_k\mid s) g\left(s+\sum\limits_{j=0}^{k-1} t_j\right)ds=
\int\limits_0^{t_k}
\mathbf{P}^0(s)
p^0(t_k- s)
g\left(s+\sum\limits_{j=0}^{k-1} t_j\right)ds.
\end{multline}
The last expression is continuous with respect to variables $(t_0,\dots,t_{n+1})$.
Therefore, one can conclude that expression (\ref{intD}) does not have a jump discontinuity,
which position depends on $t_0$.
Consider now the remaining part of integral in (\ref{pdfs}). With (\ref{strun+1})
taken into account one has:
\begin{multline}\label{intDn+1}
\int\limits_{D_{n+1}} p^{inh}(t_{n+1},\dots,t_0\mid s) f^{inh}(s)\, ds
=
\\=
\prod\limits_{i=0}^n p^0(t_i)
\int\limits_{D_{n+1}}p^{inh}\left(t_{n+1}\mid s-\sum\limits_{i=0}^{n}t_i\right) f^{inh}(s)ds.
\end{multline}
Here, the first factor, $\prod\limits_{i=0}^n p^0(t_i)$ is continuous and strictly positive
for positive $t_i$. The second factor can be transformed as follows:
\begin{multline}\label{2fak}
\int\limits_{D_{n+1}}p^{inh}\left(t_{n+1}\mid s-\sum\limits_{i=0}^{n}t_i\right) f^{inh}(s)ds
=
\\=
\int\limits_{\sum\limits_{i=0}^{n}t_i}^\Delta
p^{inh}\left(t_{n+1}\mid s-\sum\limits_{i=0}^{n}t_i\right) f^{inh}(s)ds=
\\=
\int\limits_0^{\Delta-\sum\limits_{i=0}^{n}t_i}
p^{inh}(t_{n+1}\mid s) f^{inh}\left(s+\sum\limits_{i=0}^{n}t_i\right)ds.
\end{multline}
Now, let us use representations (\ref{pts}) and (\ref{ff}) in order to figure out
which kind of discontinuities does the expression (\ref{2fak}) have.
Due to (\ref{pts}) and (\ref{ff}), expression (\ref{2fak}) will have four terms.
The first one we get by choosing the first term both in (\ref{pts}) and (\ref{ff}):
$$
A_{11}=\int\limits_0^{\Delta-\sum\limits_{i=0}^{n}t_i}
\chi(s-t_{n+1}) p^0(t_{n+1})
g\left(s+\sum\limits_{i=0}^{n}t_i\right)ds.
$$
This term is either equal to zero, if $t_{n+1}>\Delta-\sum\limits_{i=0}^{n}t_i$,
or otherwise transforms into a continuous function of variables $(t_0,\dots,t_{n+1})$. Moreover,
$$\lim\limits_{t_{n+1}\uparrow \Delta-\sum\limits_{i=0}^{n}t_i} A_{11}(t_{n+1})=0.$$
The second one we get by choosing the second term in (\ref{pts}) and the first term in (\ref{ff}):
$$
A_{21}=
\int\limits_0^{\Delta-\sum\limits_{i=0}^{n}t_i}
\mathbf{P}^0(s)
p^0(t_{n+1} - s)
g\left(s+\sum\limits_{i=0}^{n}t_i\right)ds.
$$
This is as well a continuous function of variables $(t_0,\dots,t_{n+1})$.
The third one we get by choosing the first term in (\ref{pts}) and the second term in (\ref{ff}):
\begin{multline}\label{third}
A_{12}=
a\int\limits_0^{\Delta-\sum\limits_{i=0}^{n}t_i}
\chi(s-t_{n+1}) p^0(t_{n+1})
\delta\left(\sum\limits_{i=0}^{n}t_i +s - \Delta\right)\,ds=
\\
=a \chi\left(\Delta-\sum\limits_{i=0}^{n+1}t_i\right)p^0(t_{n+1}).
\end{multline}
This term has a jump discontinuity along the hyperplane
\begin{equation}\label{hyper}
\sum\limits_{i=0}^{n+1}t_i = \Delta\,.
\end{equation}
The forth one we get by choosing the second term in (\ref{pts}) and the second term in (\ref{ff}):
\begin{multline}\nonumber
A_{22}=
a\int\limits_0^{\Delta-\sum\limits_{i=0}^{n}t_i}
\mathbf{P}^0(s)
p^0(t_{n+1} - s)
\delta\left(\sum\limits_{i=0}^{n}t_i +s -\Delta\right)\,ds=
\\=
\mathbf{P}^0\left(\Delta-\sum\limits_{i=0}^{n}t_i\right)
p^0\left(\sum\limits_{i=0}^{n+1}t_i - \Delta\right).
\end{multline}
This is as well a continuous function of variables $(t_0,\dots,t_{n+1})$.
After taking into account the above reasoning, we conclude that the required joint
probability density has the following form
\begin{equation}
\label{pn+1}
p^{inh}(t_{n+1},\dots,t_0)=
p^w(t_{n+1},\dots,t_0)+
a\chi\left(\Delta-\sum\limits_{i=0}^{n+1}t_i\right)
\prod\limits_{j=0}^{n+1} p^0(t_j).
\end{equation}
where function $p^w(t_{n+1},\dots,t_0)$ does not have a jump discontinuity depending on $t_0$, and the second term in (\ref{pn+1}) does have such a
discontinuity along the hyperplane (\ref{hyper}).\bigskip
\noindent
{\it Form of $p^{inh}(t_{n},\dots,t_0)$ after integration}\label{Form}\medskip
\noindent
If (\ref{Domain}) is satisfied, then we have similarly to (\ref{struk}), (\ref{strun+1})
\begin{multline}\nonumber
p^{inh}(t_{n},\dots,t_0 \mid s)=
\\=p^{inh}(t_{n},\dots,t_{k+1} \mid \Delta)\,
p^{inh}\left(t_k \mid s-\sum\limits_{i=0}^{k-1} t_i\right)
\prod\limits_{i=0}^{k-1}p^0(t_i),
\\
s\in D_k,\quad k=0,\dots,n-1,
\end{multline}
$$
p^{inh}(t_{n},\dots,t_0 \mid s)=
p^{inh}\left(t_n \mid s-\sum\limits_{i=0}^{n-1} t_i\right)
\prod\limits_{i=0}^{n-1}p^0(t_i),\, s\in D_n.
$$
Again due to (\ref{Domain}), and in analogy with (\ref{strukf}) we have instead
of the last two equations the following one:
\begin{multline}
\label{strn}
p^{inh}(t_{n},\dots,t_0 \mid s)=
p^{inh}\left(t_k \mid s-\sum\limits_{i=0}^{k-1} t_i\right)
\prod\limits_{\vbox{\footnotesize\hbox{$i=0$}\hbox{$i\ne k$}}}^{n}p^0(t_i),\quad
\\
s\in D_k,\,\, k=0,\dots,n.
\end{multline}
It is clear that expression similar to (\ref{strun+1}) turns here into the following
\begin{equation}\label{strn+1}
p^{inh}(t_{n},\dots,t_0 \mid s)=
\prod\limits_{i=0}^{n}p^0(t_i),\quad s\in D_{n+1}.
\end{equation}
Now, due to (\ref{strn}), (\ref{strn+1}) we have
\begin{multline}\label{pn}
p^{inh}(t_{n},\dots,t_0)=\int\limits_0^\Delta p^{inh}(t_{n},\dots,t_0 \mid s)f^{inh}(s)ds=
\\=
\sum\limits_{k=0}^n
\prod\limits_{\vbox{\footnotesize\hbox{$i=0$}\hbox{$i\ne k$}}}^{n}p^0(t_i)
\int\limits_{D_k}
p^{inh}\left(t_k \mid s-\sum\limits_{i=0}^{k-1} t_i\right)
g(s) ds+
\prod\limits_{i=0}^{n}p^0(t_i)\int\limits_{D_{n+1}}f^{inh}(s) ds.
\end{multline}
From calculations similar to those made in Eq. (\ref{ThirdFactor}) it can be concluded
that $p^{inh}(t_{n},\dots,t_0)$ is continuous at the domain defined by (\ref{Domain}).
\subsubsection{$t_0$-dependence cannot be eliminated in $p^{inh}(t_{n+1} \mid t_n,\dots,t_0)$}
Now, with representations (\ref{pn+1}) for $p^{inh}(t_{n+1},\dots,t_0)$ and
(\ref{pn}) for $p^{inh}(t_{n},\dots,t_0)$ we can pose a question about the form
of $p^{inh}(t_{n+1}\mid t_n,\dots, t_0)$. The latter can be found as defined in (\ref{defcond}).
First of all notice that due to (\ref{pn}) and Cond4,
$p^{inh}(t_{n},\dots,t_0)$ is strictly positive for positive ISIs.
This allows us to use it as denominator in the definition (\ref{defcond}).
Second, it can be further concluded from (\ref{pn}) and Cond4, that $p^{inh}(t_{n},\dots,t_0)$ is
bounded.
The latter together with continuity of $p^{inh}(t_{n},\dots,t_0)$
means that any discontinuity of jump type present in the $p^{inh}(t_{n+1},\dots,t_0)$
appears as well in the $p^{inh}(t_{n+1}\mid t_n,\dots, t_0)$. It follows from the above
and from Eq. (\ref{pn+1})
that the conditional pdf $p^{inh}(t_{n+1}\mid t_n,\dots, t_0)$ can be represented in the
following form:
\begin{multline}\label{firepr}
p^{inh}(t_{n+1}\mid t_n,\dots, t_0) =
p^w(t_{n+1}\mid t_n,\dots, t_0)+
\\+
Z(t_{n+1},\dots,t_0)\chi\left(\Delta-\sum\limits_{i=0}^{n+1}t_i\right),
\end{multline}
where $p^w(t_{n+1}\mid t_n,\dots, t_0)$ does not have any jump type discontinuity
which position
depends on $t_0$, and $Z(t_{n+1},\dots,t_0)$ is strictly positive function:
$$
Z(t_{n+1},\dots,t_0)=
\frac{a\prod\limits_{i=0}^{n+1}p^0(t_i)}{p(t_{n},\dots,t_0)}.
$$
The representation (\ref{firepr}) thus proves that for any $n$,
conditional pdf $p^{inh}(t_{n+1}\mid t_n,\dots, t_0)$ does depend on $t_0$
(the second term in (\ref{firepr}))
and this dependence cannot be eliminated.
$\Box$
\section{Conclusions and Discussion}
Conclusions made in \cite{Vidybida2015} for the class of {\bf excitatory} neuronal models
are as well valid for the studied here case of the class of {\bf fast {\em Cl-}type inhibitory}
neuronal models. Therefore, we do not replicate the text here in order to minimize overlap with
the previous paper.
At the same time, it is worth mentioning that the question of to how extent the Poisson
hypothesis is acceptable in neuroscience has been already discussed, see \cite{Averbeck2009}.
The experimentally observed presence of memory in the ISIs output of real neurons has been
reported many times, see \cite{Lowen1992,Shinomoto1999,Ratnam2000,Nawrot2007,Maimon2009}.
Also several theoretical models of how the memory could appear are offered,
see \cite{Chacron2003,Rospars1993a,Lnsk1999,Benedetto2013,Kass2005,Avila-Akerberg2011}.
One more basic reason for non-Markov statistics in a neuronal network is revealed
by B. Cessac in \cite{Cessac2011}, where the discrete time model network of the standard
leaky integrate-and-fire neurons is considered. In this model, the interneuronal communication
lines have zero delays and external input is deterministic, but synaptic weights are subjected to
uncorrelated random fluctuations. For this model, it is established in \cite{Cessac2011}
that the stochastic process, which output is the set of firing states of all neurons the network
is composed of, will be non-Markovian as well. What could be the reason
of memory appearance in the stochastic process of neuronal firing states
if there are no delays in this model?
The effect may be explained as follows. Any LIF neuron in the network considered loses memory
about its previous states at firing moment. But neurons, which do not fire at that same moment, still
retain their memory about their previous states. Therefore, the whole network retains some information
about its previous state after firing of any individual neuron (provided all neurons do not fire synchronously)
the network is composed of.
The instantaneous firing state of the network specifies which neurons are in
the firing state at a given
moment $t$, but says nothing about state of neurons, which are quiescent at that moment.
Knowledge of
earlier firing states of the network can improve our predicting ability as regards
states of the quiescent neurons
at the moment $t$, and finally, as regards the network firing state at the next moment,
$t+1$. This brings about a memory into the network's stochastic activity.
On the other hand, individual neurons in such a network can still have a Poisson-like activity,
if not considered together, see example in \cite{Izhikevich2003}.
\bibliographystyle{ws-ijns}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 5,277 |
layout: post
title: Data Science Hackathon
author: Lewis Gavin
comments: true
tags:
- datascience
- hadoop
- R
---
To spread the recent data science techniques used in certain projects at work, we decided to host our first Hackathon! The idea was to take a day off from the usual day job so everyone had the opportunity to learn something new and have some fun, in a safe environment.
The theme of the Hackathon was around Text Mining, as this is an interesting area that requires no background in analytics, stats or maths to get involved.
## The Setup
People signed up in pairs and were provided with overview sessions to attend in weeks prior to the event. These sessions went over the basics of Text Mining. Hints were provided around the dataset and techniques but the actual "exam question" wasn't released until the day.
A Slack channel was set up to share information and updates with all involved and was used to provide support throughout the day.
There were no constraints on the technology that could be used, however there was a pre-packaged virtual machine to help containing Python and RStudio - plus a Cloudera cluster with Spark, Hive and Impala.
## The Dataset and Exam question
The dataset provided was the [Amazon Fine Food Reviews dataset from Kaggle](https://www.kaggle.com/snap/amazon-fine-food-reviews). This dataset contained around half a million rows of reviews on food items sold on Amazon. This particular dataset was chosen due to the large amounts of free text available - perfect for Text Mining! Not only that, but the context was food based, meaning it was a neutral dataset - easy for anyone to understand.
By providing half a million rows text of data, this gave the contestants the opportunity to see how long calculations can take on large datasets - proving the importance of sampling when performing exploratory analysis.
The exam question for the day was purposefully kept broad - "Find some insights in the data using Text Mining Techniques". This gave people the opportunity to discover and play as they pleased, encouraging learning of a whole bunch of different techniques. There were some starter questions to help get the ball rolling including: *What were the most common phrases (bi/tri-grams)?* and *what sentiment can be derived from reviews on certain products?*
Contestants were asked to produce a "one pager" outlining their Hypotheses, Method, Visualizations & Insights and What they learnt. These would be shared to each other at the end of the day and voted on by all participants to produce 2 winners.
## Outcomes
Naturally, due to the nature of Text Mining and the lack of simple visuals to present textual findings - nearly every team produced a **wordcloud**. However most teams backed up their wordclouds with other interesting findings.
Most teams used R and produced Document-Term Frequency matrices to do the bulk of their analysis however most teams found (Due to RStudio only being able to run on a single core) that they were unable to process the whole dataset. A small number of teams used Spark and due to it being able to process in parallel and in memory, could perform their analysis on the whole data set. The findings were only slightly different but demonstrated limitations of certain tools.
Some teams chose to visualise their findings using R plots, others exported their data and used Tableau. One team looked at modeling the relationships between review submitters and products using Neo4J. Their hope was to be able to demonstrate common products between individuals to aid with recommendations.
Overall, the day was a huge success and greatly received. People really enjoyed the opportunity to do something a little different and there wasn't a single team that could say they didn't learn something new. Hackathons are a great way to get people interacting collaboratively and really help introduce new topics in an interesting way. I hope to be a part of more in the future!
| {
"redpajama_set_name": "RedPajamaGithub"
} | 1,199 |
{"url":"http:\/\/lambda-the-ultimate.org\/node\/846","text":"## Automatic type inference via partial evaluation\n\nAutomatic type inference via partial evaluation. Aaron Tomb, Cormac Flanagan. PPDP\u00e2\u20ac\u212205.\n\nType checking and type inference are fundamentally similar problems. However, the algorithms for performing the two operations, on the same type system, often differ significantly. The type checker is typically a straightforward encoding of the original type rules. For many systems, type inference is performed using a two-phase, constraint-based algorithm.We present an approach that, given the original type rules written as clauses in a logic programming language, automatically generates an efficient, two-phase, constraint-based type inference algorithm. Our approach works by partially evaluating the type checking rules with respect to the target program to yield a set of constraints suitable for input to an external constraint solver. This approach avoids the need to manually develop and verify a separate type inference algorithm, and is ideal for experimentation with and rapid prototyping of novel type systems.\n\nAlso somewhat relevant to the discussions here about type checking as abstract interpretation.\n\n## Comment viewing options\n\n### On first reading, I don't thi\n\nOn first reading, I don't think the particular technique they describe would continue to work when the type language has nontrivial binding structure. (By \"nontrivial\", I mean \"you need to do alpha-conversion when substituting type variables\" -- system F, the polymorphic lambda calculus, is an example. ML-style polymorphism isn't, though.) To apply this to languages like System F you'd need higher-order unification, which is known to be undecidable. You could probably get a partial algorithm if you used this technique with lambda-Prolog or Twelf, though.\n\nPersonally, I think that when you are designing a type system for a language, as opposed to making a custom type system for a program analysis, you should specify the modes of the typechecking rules (a la bidirectional type checking). Among other things, this lets you declaratively specify where type annotations are and aren't needed.\n\n### I did almost the same thing (\n\nI did almost the same thing (without the partial evaluation stuff) a couple of months ago for a advanced programming language course. I implemented an inferencer for the simple typed lambda calculus and the polymorphically typed lambda calculus. Both work briljantly and are very readable.\n\nTo apply this to languages like System F you'd need higher-order unification, which is known to be undecidable. You could probably get a partial algorithm\n\nAccording to cardelli the accepted algorithm for polymorphic lambda calculus is indeed partial. This, however, is not a big problem since the terms seldom arise in pratice. See Cardelli - Types for a nce discussion of the system and the algorithm.\n\nHere is the code for the type inference for the simple typed lambda calculus. Surprisingly it not only checks types, but infers them and can even calculate environments in which terms have certain types..\n\nIt can be called as follows:\n\n infer( [[]], lambda( x:nat, x), X). \n\nlambda terms are represented as the lambda( , ) atom and application by apply( , )\n\nThe first parameter given is the environment ( empty list of lists in this case.). Ascription is done using the \":\" operator. The query should result in the answer (nat -> nat).\n\nps.\nHope i don't break any post etiquette with the length f this post.\n\n:- op( 101, yfx, : ),\nop( 400, yfx, ->).\n\n%%%%%%%%%%%%%%%%%\n% Main F1 rules %\n%%%%%%%%%%%%%%%%%\n\n% basic types\nbasic( K ) :-\nmember( K, [nat, bool] ).\n\n% Env []\ninfer( [[]], valid).\n\n% Env x\ninfer( [[X,A]|Rest], valid ) :-\ninfer( Rest, A),\nnot( member( [X,A], Rest)).\n\n% Type Arrow\ninfer( Env, A -> B ) :-\ninfer( Env, A),\ninfer( Env, B).\n\n% Type Const\ninfer( Env, K ) :-\nbasic( K ),\ninfer( Env, valid).\n\n% Type Bool\ninfer( Env, bool) :-\ninfer( Env, valid).\n\n% Val True\ninfer( Env, true, bool) :-\ninfer( Env, valid).\n\n% Val False\ninfer( Env, false, bool) :-\ninfer( Env, valid).\n\n% Val x\ninfer( [[X,A]|Rest], X, A ) :-\ninfer( [[X,A]|Rest], valid).\n\n% Val Fun\ninfer( Env, lambda(X : A, M), A -> B ) :-\ninfer( [[X,A]|Env], M, B).\n\n% Val Appl\ninfer( Env, apply(M, N), B) :-\nbasic( A ),\ninfer( Env, M, A -> B),\ninfer( Env, N, A).","date":"2019-08-19 14:27:51","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.6705448031425476, \"perplexity\": 4605.917550359139}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-35\/segments\/1566027314752.21\/warc\/CC-MAIN-20190819134354-20190819160354-00388.warc.gz\"}"} | null | null |
In the immortal words of someone who stopped to assist as I was pushing a dead Meganne off a roundabout onto the verge "What's the problem? I'm a mechanic, I've got a computer!"
EV's? Nope, no spanners, not allowed to use'm, elf'n'safety doncha know! Looking at a whole page of stuff in the Sealey trade catalogue, essential for any aspiring EV technician, all you're allowed to do is place it on a rubber mat, cordon it off & display appropriate warning signs.
Hot brake could be caused by the bearing if there's a bit of play in it as it can allow the disc (along with the wheel) to tilt slightly, so it will tend to rub against one of the pads.
Had this happen on my first Nova. Stuffed bearing replaced and the previously hot enough to melt wheel trims brake ran perfectly cool.
In fairness, it was a car for on the farm only...and I was 12 - and the car was free - Wouldn't let anything actually on the road get that bad!
I had a thought today that I could swap the hub off the 307 onto the C4, since the 307 had a new wheel bearing and ball joint only about 30k ago. What are the odds that it's the same part...???
Is the car that is leaving going to the scrap man ? Are you going to try to fit the faulty hub back to the scrapper car or just hand it over as an extra bonus piece ?
Had a decent run out in the new C4 at the weekend, up to Tyndrum in the highlands. It's really very quiet - and with lots of low down pull, and very long gears it's easy to find it's going much faster than you think! 6th is geared out to only 2000 rpm at 70 mph, yet it will pull 6th from below 50, even though the engine is still tight. Computer was showing 55mpg there are back, which is not bad considering I wasn't really hanging back. Ride is smoother than the old one, but still on the stiff side one up! Steering is a little dead at straight ahead, but it seems to hang on round the bends. It's got daft 'mood' lighting in the overhead console in the front - I found it very distracting, so had to work out how to turn it off! Headlights are OK, would benefit from some high output bulbs - not a patch on the MK1s mind you, they are the best halogens out of the box I have ever had.
It sounds good. Did they put that 8v engine in the later C5?
Turns out the C4 doesn't need a new wheelbearing - it was just the disc heating up and making a rumbling / vibrating noise just like a wheel bearing! MPG has also recovered from 38 to 45 since getting a new caliper fitted!
Fixed the stuck front seat in the old C4 today - turned out that the adjuster bar had been pushed down so far that it was stuck under the plastic side trims on the seat, so it couldn't be pulled up to disengage the ratchet on the runner. I imagine that one or other of the small feet has stood on it.... Took quite a lot of percy-whatsit to get it back on the right side of the trim!!
It's only a NiMh affair as well probably (although there is a replacement, so maybe it's a LiIon).
That's that ruled out then!
3008 with Grip Control a bit few and far between however, so will need to keep a watch out. Perhaps just a 'normal' one and a set of chains will do!
Left field entry is a 4WD Dacia Duster.... I am strangely attracted to it; but it does mean abandoning a lifetime of PSA cars. Not sure I am ready for that.... It is mostly French (Renault) under the skin though. And it's cheeeep for what it is! | {
"redpajama_set_name": "RedPajamaC4"
} | 1,076 |
\section{Introduction}
In this article, we develop regularity theory of weak solutions for the problem
\begin{equation}\label{maineqn}
-\Delta_p u+\mathcal{L}(u)=0\text{ in }\Omega,
\quad 1<p<\infty,
\end{equation}
where $\Omega$ is a bounded domain in $\mathbb{R}^n$. The local $p$-Laplace operator is defined by
\begin{equation}\label{plap}
\Delta_p u=\text{div}(|\nabla u|^{p-2}\nabla u),
\end{equation}
and $\mathcal{L}$ is the nonlocal $p$-Laplace operator given by
\begin{equation}\label{fracplap}
\mathcal{L}(u)(x)=\text{P.V.}\int_{\mathbb{R}^n}|u(x)-u(y)|^{p-2}(u(x)-u(y))K(x,y)\,dy,
\end{equation}
where P.V. denotes the principal value. Here $K$ is a symmetric kernel in $x$ and $y$ such that
\begin{equation}\label{kernel}
\frac{\Lambda^{-1}}{|x-y|^{n+ps}}\leq K(x,y)\leq \frac{\Lambda}{|x-y|^{n+ps}},
\end{equation}
for some $\Lambda\geq 1$ and $0<s<1$.
Note that the $p$-Laplace operator $\Delta_p$ reduces to the classical Laplace operator $\Delta$ for $p=2$.
When $K(x,y)=|x-y|^{-(n+ps)}$, the operator $\mathcal{L}$ is the fractional $p$-Laplace operator $(-\Delta_p)^{s}$, which further reduces to the fractional Laplacian $(-\Delta)^s$ for $p=2$.
A prototype problem of type \eqref{maineqn} is
\begin{equation}\label{maineqnprot}
-\Delta_p u+(-\Delta_p)^s u=0,
\quad
1<p<\infty,
\quad
0<s<1.
\end{equation}
Before describing our contribution, let us discuss some of the known results.
In the local case, the regularity theory for the $p$-Laplace equation $\Delta_p u=0$ has been studied extensively, for example, see Lindqvist \cite{Lind}, Mal\'{y}-Ziemer \cite{Maly} and references therein.
For the nonlocal $p$-Laplace equation
\begin{equation}\label{fracp}
(-\Delta_p)^s u=0,
\end{equation}
a scale invariant Harnack inequality holds for globally nonnegative solutions. However, such inequality fails when the solution changes sign as shown by Kassmann \cite{KassmanHarnack} in the case $p=2$, see also Dipierro-Savin-Valdinoci \cite{Valjems}. These results have been extended to $1<p<\infty$ by Di Castro-Kuusi-Palatucci \cite{KuusiHarnack}.
In addition, a weak Harnack inequality for supersolutions of \eqref{fracp} has been discussed in \cite{KuusiHarnack}.
They introduced a nonlocal tail term to compensate for the sign change in the Harnack estimates.
Di Castro-Kuusi-Palatucci \cite{Kuusilocal} has studied local boundedness estimate along with H\"older continuity of solutions for \eqref{fracp}. See also Brasco-Lindgren-Scikorra \cite{Brascolind} for higher regularity results. For lower semicontinuity results of supersolutions, we refer to Korvenp\"{a}\"{a}-Kuusi-Lindgren \cite{lscfrac}.
For the mixed local and nonlocal case with $p=2$, i.e.
\begin{equation}\label{maineqnclas}
-\Delta u+(-\Delta)^s u=0,
\end{equation}
Foondun \cite{Fo} has proved Harnack inequality and local H\"older continuity for nonnegative solutions.
Barlow-Bass-Chen-Kassmann \cite{BBCK} has obtained a Harnack inequality for the parabolic problem related to \eqref{maineqnclas}.
Chen-Kumagai \cite{CK} has proved Harnack inequality and local H\"older continuity for the parabolic problem of \eqref{maineqnclas}.
Such a parabolic Harnack estimate has been used to prove elliptic Harnack inequality for \eqref{maineqnclas} by Chen-Kim-Song-Vondra\v{c}ek in \cite{CKSV}.
For more regularity results related to \eqref{maineqnclas}, we refer to Athreya-Ramachandran \cite{AR}, Chen-Kim-Song \cite{CKSheatest} and Chen-Kim-Song-Vondra\v{c}ek \cite{CKSVGreenest}.
The arguments in these articles combine probability and analysis.
Moreover, the Harnack inequality is proved only for globally nonnegative solutions.
Recently, an interior Sobolev regularity, a strong maximum principle and a symmetry property among many other qualitative properties of solutions to \eqref{maineqnclas} has been studied by Biagi-Dipierro-Valdinoci-Vecchi \cite{BSVV2, BSVV1}, Dipierro-Proietti Lippi-Valdinoci \cite{DPV20, DPV21} and Dipierro-Ros-Oton-Serra-Valdinoci \cite{DRXJV20}.
There also exist regularity results for a nonhomogeneous analogue of \eqref{maineqnclas}. More precisely, Athreya-Ramachandran \cite{AR} has proved Harnack inequality by probabilistic and analytic methods and authors in \cite{BSVV2, DPV20} has obtained boundedness, interior as well as boundary regularity results by analytic techniques.
Biagi-Dipierro-Valdinoci-Vecchi \cite{En21} has obtained interior regularity results for a nonhomogeneous version of \eqref{maineqnprot}.
We establish the following regularity results for weak solutions (Definition \ref{subsupsolution}) of \eqref{maineqn} with $1<p<\infty$ and $0<s<1$.
\begin{itemize}
\item Local boundedness of weak subsolutions (Theorem \ref{thm1}). The argument is based on an energy estimate (Lemma \ref{energyest}), the Sobolev inequality and an iteration technique (Lemma \ref{iteration}).
\item Local H\"older continuity of weak solutions (Theorem \ref{Holder}). Local H\"older continuity is not a direct consequence of the Harnack inequality in the nonlocal case, see \cite{BBCK, Kassweakharnack}.
We follow the approach of Di Castro-Kuusi-Palatucci \cite{Kuusilocal} in which the local boundedness estimate and the logarithmic energy estimate (Lemma \ref{loglemma}) play an important role.
\item Harnack inequality (Theorem \ref{thm2}) for weak solutions and weak Harnack inequality for weak supersolutions (Theorem \ref{thm3}). The expansion of positivity (Lemma \ref{DGLemma}), the local boundedness result and a tail estimate (Lemma \ref{tailest}) are crucial here.
\item Lower and upper semicontinuity of weak supersolutions and subsolutions, respectively (Theorem \ref{lscthm1} and Corollary \ref{uscthm}). This result is an adaptation to the mixed local and nonlocal case of a measure theoretic approach (Lemma \ref{lscthm}) in Liao \cite{Liao}. We refer to Banerjee-Garain-Kinnunen \cite{BGK} for an adaptation of this approach to a class of doubly nonlinear parabolic nonlocal problems.
\end{itemize}
In contrast to the techniques from probability and analysis introduced in \cite{AR, BBCK, CKSheatest, CKSVGreenest, CKSV, CK, Fo}, our approach is purely analytic and based on the De Giorgi-Nash-Moser theory.
To the best of our knowledge, all of our main results are new for $p\ne2$. Moreover, some of our main results (Theorem \ref{thm1}, Theorem \ref{thm3}, Theorem \ref{lscthm1} and Corollary \ref{uscthm}) seem to be new even for $p=2$.
Furthermore, our approach applies to sign changing solutions.
In this respect, our Harnack estimate (Theorem \ref{thm2}) also extends the result of Chen-Kim-Song-Vondra\v{c}ek \cite{CKSV} and Foondun \cite{Fo} to sign changing solutions.
We introduce a tail term (Definition \ref{def.tail}), that differs from the one discussed in \cite{KuusiHarnack}, and a tail estimate (Lemma \ref{tailest})
that capture both local and nonlocal features of \eqref{maineqn}.
Technical novelties include an adaptation of the expansion of positivity technique (Lemma \ref{DGLemma}) for the mixed problem.
This article is organized as follows. In Section 2, we discuss some definitions and preliminary results.
Necessary energy estimates are proved in Section 3. In Sections 4 and 5, we establish the local boundedness and H\"older continuity results.
In Sections 6 and 7, we obtain a tail estimate and the expansion of positivity property.
In Section 8, we prove Harnack and weak Harnack estimates. Finally, in Section 9, we establish the lower and upper semicontinuity results.
\section{Preliminaries}
In this section, we present some known results for fractional Sobolev spaces, see Di Nezza-Palatucci-Valdinoci \cite{Hitchhiker'sguide} for more details.
\begin{Definition}
Let $1<p<\infty$ and $0<s<1$. Assume that $\Omega$ is a domain in $\mathbb R^n$.
The fractional Sobolev space $W^{s,p}(\Omega)$ is defined by
$$
W^{s,p}(\Omega)=\bigg\{u\in L^p(\Omega):\int_{\Omega}\int_{\Omega}\frac{|u(x)-u(y)|^p}{|x-y|^{n+ps}}\,dx\,dy<\infty\bigg\}
$$
and endowed with the norm
$$
\|u\|_{W^{s,p}(\Omega)}=\bigg(\int_{\Omega}|u(x)|^p\,dx+\int_{\Omega}\int_{\Omega}\frac{|u(x)-u(y)|^p}{|x-y|^{n+ps}}\,dx\,dy\bigg)^\frac{1}{p}.
$$
The fractional Sobolev space with zero boundary value $W_{0}^{s,p}(\Omega)$ consists of functions $u\in W^{s,p}(\mathbb{R}^n)$ with
$u=0$ on $\mathbb{R}^n\setminus\Omega$.
\end{Definition}
Both $W^{s,p}(\Omega)$ and $W_{0}^{s,p}(\Omega)$ are reflexive Banach spaces, see \cite{Hitchhiker'sguide}.
The space $W^{s,p}_{\loc}(\Omega)$ is defined by requiring that a function belongs to $W^{s,p}(\Omega')$ for every $\Omega'\Subset\Omega$,
where $\Omega'\Subset\Omega$ denotes that $\overline{\Omega'}$ is a compact subset of $\Omega$.
Throughout, we write $c$ or $C$ to denote a positive constant which may vary from line to line or even in the same line.
The dependencies on parameters are written in the parentheses.
The next result asserts that the standard Sobolev space is continuously embedded in the fractional Sobolev space, see \cite[Proposition 2.2]{Hitchhiker'sguide}.
The argument applies a smoothness property of $\Omega$ so that we can extend functions from $W^{1,p}(\Omega)$ to $W^{1,p}({I\!\!R}^n)$ and that the extension operator is bounded.
\begin{Lemma}\label{locnon}
Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^n$, $1<p<\infty$ and $0<s<1$.
There exists a positive constant $C=C(n,p,s)$ such that
$
||u||_{W^{s,p}(\Omega)}\leq C||u||_{W^{1,p}(\Omega)}
$
for every $u\in W^{1,p}(\Omega)$.
\end{Lemma}
The following result for the fractional Sobolev spaces with zero boundary value follows from \cite[Lemma 2.1]{Silvaarxiv}.
The main difference compared to Lemma \ref{locnon} is that the result holds for any bounded domain,
since for the Sobolev spaces with zero boundary value, we always have zero extension to the complement.
\begin{Lemma}\label{locnon1}
Let $\Omega$ be a bounded domain in $\mathbb{R}^n$, $1<p<\infty$ and $0<s<1$.
There exists a positive constant $C=C(n,p,s,\Omega)$ such that
\[
\int_{\mathbb{R}^n}\int_{\mathbb{R}^n}\frac{|u(x)-u(y)|^p}{|x-y|^{n+ps}}\,dx\, dy
\leq C\int_{\Omega}|\nabla u(x)|^p\,dx
\]
for every $u\in W_0^{1,p}(\Omega)$.
Here we consider the zero extension of $u$ to the complement of $\Omega$.
\end{Lemma}
The following version of the Gagliardo-Nirenberg-Sobolev inequality will be useful for us,
see \cite[Corollary 1.57]{Maly}.
\begin{Lemma}\label{c.omega_sobo}
Let $1<p<\infty$, $\Omega$ be an open set in ${I\!\!R}^n$ with $\lvert\Omega\rvert<\infty$ and
\begin{equation}\label{kappa}
\kappa=
\begin{cases}
\frac{n}{n-p},&\text{if}\quad 1<p<n,\\
2,&\text{if}\quad p\geq n.
\end{cases}
\end{equation}
There exists a positive constant $C=C(n,p)$ such that
\begin{equation}\label{e.friedrich}
\biggl(\int_\Omega \lvert u(x)\rvert^{\kappa p}\,dx\biggr)^{\frac{1}{\kappa p}}
\le C \lvert\Omega\rvert^{\frac{1}{n}-\frac{1}{p}+\frac{1}{\kappa p}} \biggl(\int_\Omega \lvert \nabla u(x)\rvert^{p}\,dx\biggr)^{\frac{1}{p}}
\end{equation}
for every $u\in W_0^{1,p}(\Omega)$.
\end{Lemma}
Next, we define the notion of weak solution to \eqref{maineqn}.
\begin{Definition}\label{subsupsolution}
A function $u\in L^{\infty}(\mathbb{R}^n)$ is a weak subsolution of \eqref{maineqn} if $u\in W_{\loc}^{1,p}(\Omega)$ and for every $\Omega'\Subset\Omega$ and nonnegative test functions $\phi\in W_{0}^{1,p}(\Omega')$, we have
\begin{equation}\label{weaksubsupsoln}
\begin{gathered}
\int_{\Omega'}|\nabla u|^{p-2}\nabla u\cdot\nabla\phi\,dx+\int_{\mathbb{R}^n}\int_{\mathbb{R}^n}\mathcal{A}(u(x,y)){(\phi(x)-\phi(y))}\,d\mu\leq0,
\end{gathered}
\end{equation}
where
\[
\mathcal{A}(u(x,y))=|u(x)-u(y)|^{p-2}(u(x)-u(y))
\quad\text{and}\quad
d\mu=K(x,y)\,dx\,dy.
\]
Analogously, a function $u$ is a weak supersolution of \eqref{maineqn} if the integral in \eqref{weaksubsupsoln} is nonnegative for every nonnegative test functions $\phi\in W_{0}^{1,p}(\Omega')$.
A function $u$ is a weak solution of \eqref{maineqn} if the equality holds in \eqref{weaksubsupsoln} for every $\phi\in W_{0}^{1,p}(\Omega')$ without a sign restriction.
\end{Definition}
\begin{Remark}\label{rkreg}
The boundedness assumption, together with Lemma \ref{locnon} and Lemma \ref{locnon1}, ensures that Definition \ref{subsupsolution} is well stated and
the tail that will be defined in \eqref{loctail} is finite.
Under the assumption that the tail in \eqref{loctail} is bounded, our main results Theorem \ref{thm1}, Theorem \ref{Holder}, Theorem \ref{thm2} and Theorem \ref{thm3} hold true without the a priori boundedness assumption on the function.
In such a case, the local boundedness follows from Theorem \ref{thm1}.
\end{Remark}
It follows directly from Definition \ref{subsupsolution} that $u$ is a weak subsolution of \eqref{maineqn} if and only if $-u$ is a weak supersolution of \eqref{maineqn}.
Moreover, for any $c\in\mathbb{R}$, $u+c$ is a weak solution of \eqref{maineqn} if and only if $u$ is a weak solution of \eqref{maineqn}.
We discuss some further structural properties of weak solutions below.
We denote the positive and negative parts of $a\in{I\!\!R}$ by $a_+=\max\{a,0\}$ and $a_-=\max\{-a,0\}$, respectively. Also, the barred integral sign denotes the corresponding integral average.
\begin{Lemma}\label{Solsubsup}
A function $u$ is a weak solution of \eqref{maineqn} if and only if $u$ is a weak subsolution and a weak supersolution of \eqref{maineqn}.
\end{Lemma}
\begin{proof}
It follows immediately from Definition \ref{subsupsolution}, that a weak solution $u$ of \eqref{maineqn} is a weak subsolution and a weak supersolution of \eqref{maineqn}.
Conversely, assume that $u$ is both weak subsolution and weak supersolution of \eqref{maineqn}.
Let $\Omega'\Subset\Omega$ and $\phi\in W_0^{1,p}(\Omega')$. Then $\phi_{+}$ and $\phi_{-}$ belong to $W_0^{1,p}(\Omega')$. Since $u$ is a weak subsolution, we have
\begin{equation}\label{Sub}
\int_{\Omega'}|\nabla u|^{p-2}\nabla u\cdot\nabla\phi_{+}\,dx+\int_{\mathbb{R}^n}\int_{\mathbb{R}^n}\mathcal{A}(u(x,y))(\phi_+(x)-\phi_{+}(y))\,d\mu\leq 0.
\end{equation}
Analogously, since $u$ is a weak supersolution, we have
\begin{equation}\label{Sup}
\int_{\Omega'}|\nabla u|^{p-2}\nabla u\cdot\nabla\phi_{-}\,dx+\int_{\mathbb{R}^n}\int_{\mathbb{R}^n}\mathcal{A}(u(x,y))(\phi_{-}(x)-\phi_{-}(y))\,d\mu\geq 0.
\end{equation}
Subtracting \eqref{Sub} and \eqref{Sup} and using $\phi=\phi_{+}-\phi_{-}$, we obtain
\begin{equation*}
\int_{\Omega'}|\nabla u|^{p-2}\nabla u\cdot\nabla\phi\,dx+\int_{\mathbb{R}^n}\int_{\mathbb{R}^n}\mathcal{A}(u(x,y))(\phi(x)-\phi(y))\,d\mu\leq 0.
\end{equation*}
The reverse inequality holds by replacing $\phi$ with $-\phi$. Hence, $u$ is a weak solution of \eqref{maineqn}.
\end{proof}
Next, we show that the property of being a weak subsolution is preserved under taking the positive part. Then, it follows immediately that $u_-$ is a weak subsolution of \eqref{maineqn}, whenever $u$ is a weak supersolution of \eqref{maineqn}.
\begin{Lemma}\label{cutoffsub}
Assume that $u$ is a weak subsolution of \eqref{maineqn}.
Then $u_+$ is a weak subsolution of \eqref{maineqn}.
\end{Lemma}
\begin{proof}
Consider functions $u_k=\min\{ku_{+},1\}$, $k=1,2,\dots$.
Then $(u_k)_{k=1}^\infty$ is an increasing sequence of functions in $W^{1,p}_{\loc}(\Omega)$ and $0\leq u_k\leq 1$ for every $k\in\mathbb{N}$.
Let $\phi\in C_c^{\infty}(\Omega')$ be a nonnegative function. By choosing $u_k\phi\in W_0^{1,p}(\Omega')$ as a test function in \eqref{weaksubsupsoln}, we obtain
\begin{equation}\label{cutofftest}
0\geq\int_{\Omega'}|\nabla u|^{p-2}\nabla u\cdot\nabla(u_k\phi)\,dx+\int_{\mathbb{R}^n}\int_{\mathbb{R}^n}\mathcal{A}(u(x,y))(u_k(x)\phi(x)-u_k(y)\phi(y))\,d\mu
=I_1+I_2.
\end{equation}
\textbf{Estimate of $I_1$:} We observe that
\begin{equation}\label{cutoffI1}
I_1=\int_{\Omega'}|\nabla u|^{p-2}\nabla u\cdot\nabla(u_k\phi)\,dx=k\int_{\Omega'\cap\{0<u<\frac{1}{k}\}}\phi|\nabla u|^p\,dx+\int_{\Omega'}u_k|\nabla u|^{p-2}\nabla u\cdot\nabla\phi\,dx.
\end{equation}
\textbf{Estimate of $I_2$:} Let $x,y\in\mathbb{R}^n$. First, we consider the case when $u(x)>u(y)$.\\
If $u_k(x)=0$, then $u_k(y)=0$. Hence, we have
\begin{equation}\label{cutoffI2case1}
(u(x)-u(y))^{p-1}(u_k(x)\phi(x)-u_k(y)\phi(y))=0.
\end{equation}
If $u_k(y)>0$, then $u(y)=u_+(y)$.
Under the assumption $u(x)>u(y)$, it follows that $u(x)=u_{+}(x)$ and $u_k(x)>u_{k}(y)$.
This implies that
\begin{equation}\label{cutoffI2case2}
\begin{split}
(u(x)-u(y))^{p-1}(u_k(x)\phi(x)-u_k(y)\phi(y))
&=(u_+(x)-u_{+}(y))^{p-1}(u_k(x)\phi(x)-u_k(y)\phi(y))\\
&\geq (u_+(x)-u_{+}(y))^{p-1}u_k(x)(\phi(x)-\phi(y)).
\end{split}
\end{equation}
If $u_k(y)=0$ and $u_k(x)>0$, then $u(x)>0\geq u(y)$ and hence
\begin{equation}\label{cutoffI2case3}
\begin{split}
(u(x)-u(y))^{p-1}(u_k(x)\phi(x)-u_k(y)\phi(y))
&=(u(x)-u(y))^{p-1}u_k(x)\phi(x)\\
&\geq(u_+(x)-u_{+}(y))^{p-1}u_k(x)\phi(x)\\
&\geq(u_+(x)-u_{+}(y))^{p-1}u_k(x)(\phi(x)-\phi(y)).
\end{split}
\end{equation}
Therefore, from \eqref{cutoffI2case1}, \eqref{cutoffI2case2} and \eqref{cutoffI2case3} we have
\begin{equation}\label{cutoffI2casexgeqy}
\mathcal{A}(u(x,y))(u_k(x)\phi(x)-u_k(y)\phi(y))
\geq(u_+(x)-u_+(y))^{p-1}u_k(x)(\phi(x)-\phi(y)).
\end{equation}
When $u(x)=u(y)$, the estimate \eqref{cutoffI2casexgeqy} hods true. In case of $u(x)<u(y)$, by interchanging the roles of $x$ and $y$ in the above estimates, we arrive at
\begin{equation}\label{cutoffI2casexleqy}
\mathcal{A}(u(x,y))(u_k(x)\phi(x)-u_k(y)\phi(y))
\geq(u_+(y)-u_+(x))^{p-1}u_k(y)(\phi(y)-\phi(x)).
\end{equation}
Combining the estimates \eqref{cutoffI1}, \eqref{cutoffI2casexgeqy} and \eqref{cutoffI2casexleqy} in \eqref{cutofftest} and letting $k\to\infty$, along with an application of the Lebesgue dominated convergence theorem, we obtain
\begin{equation}\label{cutoffest}
\int_{\Omega'}|\nabla u_+|^{p-2}\nabla u_+\cdot\nabla\phi\,dx+\int_{\mathbb{R}^n}\int_{\mathbb{R}^n}\mathcal{A}(u_+(x,y))(\phi(x)-\phi(y))\,d\mu\leq 0.
\end{equation}
By a density argument \eqref{cutoffest} holds for every $\phi\in W_0^{1,p}(\Omega')$.
This shows that $u_+$ is a weak subsolution of \eqref{maineqn}.
\end{proof}
\section{Energy estimates}
The following energy estimate will be crucial for us.
We denote an open ball with center $x_0\in{I\!\!R}^n$ and radius $r>0$ by $B_r(x_0)$.
\begin{Lemma}\label{energyest}
Let $u$ be a weak subsolution of \eqref{maineqn} and denote $w=(u-k)_{+}$ with $k\in\mathbb{R}$.
There exists a positive constant $C=C(p,\Lambda)$ such that
\begin{equation}\label{energyesteqn}
\begin{split}
&\int_{B_r(x_0)}\psi^p|\nabla w|^p\,dx+\int_{B_r(x_0)}\int_{B_r(x_0)}|w(x)\psi(x)-w(y)\psi(y)|^p \,d\mu\\
&\leq C\bigg(\int_{B_r(x_0)}w^p |\nabla\psi|^p\,dx+\int_{B_r(x_0)}\int_{B_r(x_0)}{\max\{w(x),w(y)\}^p|\psi(x)-\psi(y)|^p}\,d\mu\\
&\qquad+\mathop{\rm {ess\,sup}}\limits_{x\in\supp\psi}\int_{{\mathbb{R}^n\setminus B_r(x_0)}}{\frac{w(y)^{p-1}}{|x-y|^{n+ps}}}\,dy
\cdot\int_{B_r(x_0)}w\psi^p\,dx\bigg),
\end{split}
\end{equation}
whenever $B_r(x_0)\subset\Omega$ and $\psi\in C_c^{\infty}(B_r{(x_0)})$ is a nonnegative function.
If $u$ is a weak supersolution of \eqref{maineqn}, the estimate in \eqref{energyesteqn} holds with $w=(u-k)_{-}$.
\end{Lemma}
\begin{proof}
Let $u$ be a weak subsolution of \eqref{maineqn}.
For $w=(u-k)_+$, by choosing $\phi=w\psi^p$ as a test function in \eqref{weaksubsupsoln}, we obtain
\begin{equation}\label{energytest}
\begin{split}
0&\geq\int_{B_r(x_0)}|\nabla u|^{p-2}\nabla u\cdot\nabla(w\psi^p)\,dx+\int_{\mathbb{R}^n}\int_{\mathbb{R}^n}\mathcal{A}(u(x,y))(w(x)\psi(x)^p-w(y)\psi(y)^p)\,d\mu\\
&=I+J.
\end{split}
\end{equation}
Proceeding as in the proof of \cite[Page 14, Proposition 3.1]{Verenacontinuity}, for some constants $c=c(p)>0$ and $C=C(p)>0$, we have
\begin{equation}\label{energyIest}
\begin{split}
I&=\int_{B_r(x_0)}|\nabla u|^{p-2}\nabla u\cdot\nabla(w\psi^p)\,dx\\
&\geq c\int_{B_r(x_0)}\psi^p|\nabla w|^p\,dx-C\int_{B_r(x_0)}w^p|\nabla\psi|^p\,dx.
\end{split}
\end{equation}
Moreover, from the lines of the proof of \cite[Pages 1285--1287, Theorem 1.4]{Kuusilocal}, for some constants $c=c(p,\Lambda)>0$ and $C=C(p,\Lambda)>0$, we have
\begin{equation}\label{energyJest}
\begin{split}
J&=\int_{\mathbb{R}^n}\int_{\mathbb{R}^n}\mathcal{A}(u(x,y))(w(x)\psi(x)^p-w(y)\psi(y)^p)\,d\mu\\
&\geq c\int_{B_r(x_0)}\int_{B_r(x_0)}|w(x)\psi(x)-w(y)\psi(y)|^p\,d\mu\\
&\qquad-C\int_{B_r(x_0)}\int_{B_r(x_0)}\max\{w(x),w(y)\}^p|\psi(x)-\psi(y)|^p\,d\mu\\
&\qquad-C\mathop{\rm {ess\,sup}}\limits_{x\in\supp\psi}\int_{{\mathbb{R}^n\setminus B_r(x_0)}}{\frac{w(y)^{p-1}}{|x-y|^{n+ps}}}\,dy
\cdot\int_{B_r(x_0)}w\psi^p\,dx.
\end{split}
\end{equation}
By applying \eqref{energyIest} and \eqref{energyJest} in \eqref{energytest}, we obtain \eqref{energyesteqn}.
In the case of a weak supersolution, the estimate in \eqref{energyesteqn} follows by applying the obtained result to $-u$ .
\end{proof}
Next we define a tail which appears in estimates throughout the article.
\begin{Definition}\label{def.tail}
Let $u$ be a weak subsolution or a weak supersolution of \eqref{maineqn} as in Definition \ref{subsupsolution}.
The tail of $u$ with respect to a ball $B_r(x_0)$ is defined by
\begin{equation}\label{loctail}
\Tail(u;x_0,r)=\bigg(r^{p}\int_{\mathbb{R}^n\setminus B_r(x_0)}\frac{|u(y)|^{p-1}}{|y-x_0|^{n+ps}}\,dy\bigg)^\frac{1}{p-1}.
\end{equation}
\end{Definition}
We prove an energy estimate which will be crucial to obtain a reverse H\"older inequality for weak supersolutions of \eqref{maineqn}.
\begin{Lemma}\label{energyforrev}
Let $q\in(1,p)$ and $d>0$.
Assume that $u$ is a weak supersolution of \eqref{maineqn} such that $u\geq 0$ in $B_R(x_0)\subset\Omega$ and denote by $w=(u+d)^\frac{p-q}{p}$.
There exists a positive constant $c=c(p,\Lambda)$ such that
\begin{equation}\label{energyrevest1}
\begin{split}
&\int_{B_r(x_0)}\psi^p|\nabla w|^p\,dx
\leq c\bigg(\frac{(p-q)^p}{(q-1)^\frac{p}{p-1}}\int_{B_r(x_0)}w^p|\nabla\psi|^p\,dx\\
&+\frac{(p-q)^p}{(q-1)^{p}}\int_{B_r(x_0)}\int_{B_r(x_0)}\max\{w(x),w(y)\}^p|\psi(x)-\psi(y)|^p\,d\mu\\
&+\frac{(p-q)^p}{(q-1)}\bigg(\mathop{\rm {ess\,sup}}\limits_{z\in\supp\psi}\int_{\mathbb{R}^n\setminus B_r(x_0)}K(z,y)\,dy
+d^{1-p}R^{-p}\Tail(u_{-};x_0,R)^{p-1}\bigg)\int_{B_r(x_0)}w^p\psi^p\,dx\bigg),
\end{split}
\end{equation}
whenever $B_r(x_0)\subset B_{\frac{3R}{4}}(x_0)$ and $\psi\in C_c^{\infty}(B_r(x_0))$ is a nonnegative function.
Here $\Tail(\cdot)$ is defined in \eqref{loctail}.
\end{Lemma}
\begin{proof}
Let $d>0$, $v=u+d$ and $q\in[1+\epsilon,p-\epsilon]$ for $\epsilon>0$ small enough.
Then $v$ is a weak supersolution of \eqref{maineqn}.
By choosing $\phi=v^{1-q}\psi^p$ as a test function in \eqref{weaksubsupsoln}, we obtain
\begin{equation}\label{revestfinal}
\begin{split}
0&\leq \int_{B_r(x_0)}|\nabla v|^{p-2}\nabla v\cdot\nabla( v^{1-q}\psi^p)\,dx\\
&\qquad+\int_{B_r(x_0)}\int_{B_r(x_0)}\mathcal{A}( v(x,y))( v(x)^{1-q}\psi(x)^p- v(y)^{1-q}\psi(y)^p)\,d\mu\\
&\qquad+2\int_{\mathbb{R}^n\setminus B_r(x_0)}\int_{B_r(x_0)}\mathcal{A}( v(x,y)) v(x)^{1-q}\psi(x)^p\,d\mu\\
&=I_1+I_2+2I_3.
\end{split}
\end{equation}
\textbf{Estimate of $I_1$:} We observe that
\begin{equation}\label{revI1est}
\begin{split}
I_1&=\int_{B_r(x_0)}|\nabla v|^{p-2}\nabla v\cdot\nabla( v^{1-q}\psi^p)\,dx\\
&\leq(1-q)\int_{B_r(x_0)} v^{-q}|\nabla v|^{p}\psi^p\,dx+p\int_{B_r(x_0)} v^{1-q}|\nabla\psi||\nabla v|^{p-1}\psi^{p-1}\,dx\\
&=(1-q)J_1+J_2,
\end{split}
\end{equation}
where
$$
J_1=\int_{B_r(x_0)} v^{-q}|\nabla v|^{p}\psi^p\,dx
$$
and
$$
J_2=p\int_{B_r(x_0)} v^{1-q}|\nabla\psi||\nabla v|^{p-1}\psi^{p-1}\,dx.
$$
\textbf{Estimate of $J_2$:} By Young's inequality, we obtain
\begin{equation}\label{revJ2est}
J_2=p\int_{B_r(x_0)} v^{1-q}|\nabla\psi||\nabla v|^{p-1}\psi^{p-1}\,dx\leq\frac{q-1}{2}J_1+\frac{c(p)}{(q-1)^\frac{1}{p-1}}\int_{B_r(x_0)}|\nabla\psi|^p v^{p-q}\,dx.
\end{equation}
By applying \eqref{revJ2est} in \eqref{revI1est}, for some constant $c=c(p)>0$, we have
\begin{equation}\label{revI1estfinal}
\begin{split}
I_1&\leq\frac{1-q}{2}\int_{B_r(x_0)} v^{-q}|\nabla v|^{p}\psi^p\,dx+\frac{c}{(q-1)^\frac{1}{p-1}}\int_{B_r(x_0)}|\nabla\psi|^p v^{p-q}\,dx\\
&=-\frac{q-1}{2}\Big(\frac{p}{p-q}\Big)^p\int_{B_r(x_0)}\big|\nabla( v^\frac{p-q}{p})\big|^p\psi^p\,dx
+\frac{c}{(q-1)^\frac{1}{p-1}}\int_{B_r(x_0)}|\nabla\psi|^p v^{p-q}\,dx.
\end{split}
\end{equation}
\textbf{Estimate of $I_2$:} Following the lines of the proof of \cite[Pages 1830--1833, Lemma 5.1]{KuusiHarnack} for $w= v^\frac{p-q}{p}$, with some positive constants $c(p,q)$ and $c(p)$, we obtain
\begin{equation}\label{revI2est}
\begin{split}
I_2&=\int_{B_r(x_0)}\int_{B_r(x_0)}\mathcal{A}( v(x,y))( v(x)^{1-q}\psi(x)^p- v(y)^{1-q}\psi(y)^p)\,d\mu\\
&\leq -c(p,q)\int_{B_r(x_0)}\int_{B_r(x_0)}|w(x)-w(y)|^p\,\psi(y)^{p}\,d\mu\\
&\qquad+\frac{c(p)}{(q-1)^{p-1}}\int_{B_r(x_0)}\int_{B_r(x_0)}\max\{w(x),w(y)\}^p|\psi(x)-\psi(y)|^p\,d\mu.
\end{split}
\end{equation}
\\
\textbf{Estimate of $I_3$:} Following the lines of the proof of \cite[Page 1830, Lemma 5.1]{KuusiHarnack} for $w= v^\frac{p-q}{p}$, we obtain
\begin{equation}\label{revI34est}
\begin{split}
I_3&=2\int_{\mathbb{R}^n\setminus B_r(x_0)}\int_{B_r(x_0)}\mathcal{A}( v(x,y)) v(x)^{1-q}\psi(x)^p\,d\mu\\
&\leq c\bigg(\mathop{\rm {ess\,sup}}\limits_{z\in\supp\psi}\int_{\mathbb{R}^n\setminus B_r(x_0)}K(z,y)\,dy\\
&\qquad\qquad+d^{1-p}\int_{\mathbb{R}^n\setminus B_R(x_0)}(u(y))_{-}^{p-1}|y-x_0|^{-n-ps}\,dy\bigg)
\int_{B_r(x_0)}w^p\psi^p\,dx,
\end{split}
\end{equation}
with $c=c(p,\Lambda)>0$. By applying \eqref{revI1estfinal}, \eqref{revI2est} and \eqref{revI34est} in \eqref{revestfinal}, we obtain \eqref{energyrevest1}.
\end{proof}
Next, we obtain a logarithmic energy estimate.
\begin{Lemma}\label{loglemma}
Assume that $u$ is a weak supersolution of \eqref{maineqn} such that $u\geq 0$ in $B_R(x_0)\subset\Omega$.
There exists a positive constant $c=c(n,p,s,\Lambda)$ such that
\begin{equation}\label{logest}
\begin{split}
&\int_{B_r(x_0)}|\nabla\log(u+d)|^p\,dx+\int_{B_r(x_0)}\int_{B_r(x_0)}\bigg|\log\bigg(\frac{u(x)+d}{u(y)+d}\bigg)\bigg|^p\,d\mu\\
&\qquad\leq cr^n\big(r^{-p}+r^{-ps}+d^{1-p}R^{-p}\Tail(u_-;x_0,R)^{p-1}\big),
\end{split}
\end{equation}
whenever $B_r(x_0)\subset B_\frac{R}{2}(x_0)$ and $d>0$.
Here $\Tail(\cdot)$ is given by \eqref{loctail}.
\end{Lemma}
\begin{proof}
Let $\psi\in C_c^{\infty}(B_\frac{3r}{2}(x_0))$ be such that $0\leq\psi\leq 1$ in $B_\frac{3r}{2}(x_0)$, $\psi= 1$ in $B_r(x_0)$,
and $|\nabla\psi|\leq\frac{8}{r}$ in $B_\frac{3r}{2}(x_0)$.
By choosing $\phi=(u+d)^{1-p}\psi^p$ as a test function in \eqref{weaksubsupsoln}, we obtain
\begin{equation}\label{logesttest}
\begin{split}
0&\leq\int_{B_{2r}(x_0)}\int_{B_{2r}(x_0)}\mathcal{A}(u(x,y))((u(x)+d)^{1-p}\psi(x)^p-(u(y)+d)^{1-p}\psi(y)^p)\,d\mu\\
&\qquad+2\int_{\mathbb{R}^n\setminus {B_{2r}}(x_0)}\int_{B_{2r}(x_0)}\mathcal{A}(u(x,y))(u(x)+d)^{1-p}\psi(x)^p\,d\mu\\
&\qquad+\int_{B_{2r}(x_0)}|\nabla u|^{p-2}\nabla u\cdot\nabla((u+d)^{1-p}\psi^p)\,dx\\
&=I_1+I_2+I_3.
\end{split}
\end{equation}
\textbf{Estimate of $I_1$:} Following the lines of the proof of \cite[Pages 1288--1289, Lemma 1.3]{Kuusilocal} and using the properties of $\psi$, for some positive constant $c=c(n,p,s,\Lambda)$, we obtain
\begin{equation}\label{logI1}
\begin{split}
I_1&=\int_{B_{2r}(x_0)}\int_{B_{2r}(x_0)}\mathcal{A}(u(x,y))((u(x)+d)^{1-p}\psi(x)^p-(u(y)+d)^{1-p}\psi(y)^p)\,d\mu\\
&\leq-\frac{1}{c}\int_{B_{2r}(x_0)}\int_{B_{2r}(x_0)}K(x,y)\bigg|\log\bigg(\frac{u(x)+d}{u(y)+d}\bigg)\bigg|^p\psi(y)^p\,dx\, dy+cr^{n-ps}.
\end{split}
\end{equation}
\textbf{Estimate of $I_2$:} Following the lines of the proof of \cite[Page 1290, Lemma 1.3]{Kuusilocal} and using the properties of $\psi$, for some positive constant $c=c(n,p,s,\Lambda)$, we obtain
\begin{equation}\label{logI2}
\begin{split}
I_2&=2\int_{\mathbb{R}^n\setminus {B_{2r}}(x_0)}\int_{B_{2r}(x_0)}\mathcal{A}(u(x,y))(u(x)+d)^{1-p}\psi(x)^p\,d\mu\\
&\leq c d^{1-p}r^n R^{-p}\Tail(u_-;x_0,R)^{p-1}+c r^{n-ps}.
\end{split}
\end{equation}
\textbf{Estimate of $I_3$:}
Arguing similarly as in the proof of \cite[Pages 717-718, Lemma 3.4]{Kin-Kuusi} and using the properties of $\psi$, for some positive constant $c=c(p)$, we have
\begin{equation}\label{logI3}
I_3=\int_{B_{2r}(x_0)}|\nabla u|^{p-2}\nabla u\cdot\nabla((u+d)^{1-p}\psi^p)\,dx\leq-c\int_{B_r(x_0)}|\nabla\log(u+d)|^p\,dx+c r^{n-p}.
\end{equation}
Hence using \eqref{logI1}, \eqref{logI2} and \eqref{logI3} in \eqref{logesttest} along with the fact that $\psi\equiv 1$ in $B_r(x_0)$, the estimate \eqref{logest} follows.
\end{proof}
As a consequence of Lemma \ref{loglemma}, we have the following result.
\begin{Corollary}\label{cor}
Assume that $u$ is a weak solution of \eqref{maineqn} such that $u\geq 0$ in $B_R(x_0)\subset\Omega$. Let $a,d>0$, $b>1$ and denote
$$
v=\min\bigg\{\bigg(\log\bigg(\frac{a+d}{u+d}\bigg)\bigg)_+,\log b\bigg\}.
$$
There exists a positive constant $c=c(n,p,s,\Lambda)$ such that
\begin{equation}\label{corest}
\fint_{B_r(x_0)}|v-(v)_{B_r(x_0)}|^p\,dx\leq c\Big(1+d^{1-p}\Big(\frac{r}{R}\Big)^p\Tail(u_-;x_0,R)^{p-1}\Big),
\end{equation}
whenever $B_r(x_0)\subset B_\frac{R}{2}(x_0)$ with $r\in(0,1]$.
Here $(v)_{B_r(x_0)}=\fint_{B_r(x_0)}v\,dx$ and $\Tail(\cdot)$ is given by \eqref{loctail}.
\end{Corollary}
\begin{proof}
By the Poincar\'e inequality {from \cite[Theorem 2]{Evans}}, for a constant $c=c(n,p)>0$, we have
\begin{equation}\label{Sobapp}
\fint_{B_r(x_0)}|v-(v)_{B_r(x_0)}|^p\,dx\leq c r^{p-n}\int_{B_r(x_0)}|\nabla v|^p\,dx.
\end{equation}
Now since $v$ is a truncation of the sum of a constant and $\log(u+d)$, we have
\begin{equation}\label{trunprp}
\int_{B_r(x_0)}|\nabla v|^p\,dx\leq \int_{B_r(x_0)}|\nabla\log(u+d)|^p\,dx.
\end{equation}
The estimate in \eqref{corest} follows by employing \eqref{logest} in \eqref{trunprp} along with \eqref{Sobapp} and the fact that $r\in(0,1]$.
\end{proof}
\section{Local boundedness}
We apply the following real analysis lemma. For the proof of Lemma \ref{iteration}, see \cite[Lemma 4.1]{Dibe}.
\begin{Lemma}\label{iteration}
Let $(Y_j)_{j=0}^{\infty}$ be a sequence of positive real numbers such that
$Y_0\leq c_{0}^{-\frac{1}{\beta}}b^{-\frac{1}{\beta^2}}$ and $Y_{j+1}\leq c_0 b^{j} Y_j^{1+\beta}$,
$j=0,1,2,\dots$, for some constants $c_0,b>1$ and $\beta>0$. Then $\lim_{j\to\infty}\,Y_j=0$.
\end{Lemma}
Our first main result shows that weak subsolutions of \eqref{maineqn} are locally bounded.
This result comes with a useful estimate.
\begin{Theorem}\label{thm1}(\textbf{Local boundedness}).
Let $u$ be a weak subsolution of \eqref{maineqn}.
There exists a positive constant $c=c(n,p,s,\Lambda)$, such that
\begin{equation}\label{locbd}
\mathop{\rm {ess\,sup}}\limits_{B_{\frac{r}{2}}(x_0)}\,u
\leq \delta \Tail\big(u_{+};x_0,\tfrac{r}{2}\big)+c\delta^{-\frac{(p-1)\kappa}{p(\kappa-1)}}\bigg(\fint_{B_r(x_0)}u_{+}^p\,dx\bigg)^\frac{1}{p},
\end{equation}
whenever $B_r(x_0)\subset\Omega$ with $r\in(0,1]$ and $\delta\in(0,1]$.
Here $\kappa$ and $\Tail(\cdot)$ are given by \eqref{kappa} and \eqref{loctail}, respectively.
\end{Theorem}
\begin{proof}
Let $B_r(x_0)\subset\Omega$ with $r\in(0,1]$. For $j=0,1,2,\dots$, we denote
$r_j=\frac{r}{2}(1+2^{-j})$, $\bar{r}_j=\frac{r_j+r_{j+1}}{2}$,
$B_j=B_{r_j}(x_0)$ and $\bar{B}_{j}=B_{\bar{r}_j}(x_0)$.
Let $(\psi_j)_{j=0}^{\infty}\subset C_c^{\infty}(\bar{B}_j)$ be a sequence of cutoff functions such that
$0\leq\psi_j\leq 1$ in $\bar{B}_{j}$, $\psi_j=1$ in $B_{j+1}$ and $|\nabla\psi_j|\leq\frac{2^{j+3}}{r}$ for every $j=0,1,2,\dots$.
For $j=0,1,2,\dots$ and $k,\bar{k}\geq 0$, we denote
$k_j=k+(1-2^{-j})\bar{k}$,
$\bar{k}_j=\frac{k_j+k_{j+1}}{2}$,
$w_j=(u-k_j)_{+}$ and $\bar{w}_j=(u-\bar{k}_j)_{+}$.
Then there exists a constant $c=c(n,p)>0$ such that
\begin{equation}\label{locest}
\begin{split}
\Big(\frac{\bar{k}}{2^{j+2}}\Big)^\frac{p(\kappa-1)}{\kappa}
\bigg(\fint_{B_{j+1}}w_{j+1}^p\,dx\bigg)^\frac{1}{\kappa}&=(k_{j+1}-\bar{k}_j)^\frac{p(\kappa-1)}{\kappa}
\bigg(\fint_{B_{j+1}}w_{j+1}^p\,dx\bigg)^\frac{1}{\kappa}\\
&\leq c\bigg(\fint_{\bar{B}_j}|\bar{w}_j\psi_j|^{p\kappa}\,dx\bigg)^\frac{1}{\kappa},
\end{split}
\end{equation}
where $\kappa$ is given by \eqref{kappa}.
By the Sobolev inequality in \eqref{e.friedrich}, with $c=c(n,p,s)>0$, we obtain
\begin{equation}\label{locbdSob1}
\begin{split}
\bigg(\fint_{\bar{B}_j}|\bar{w}_j\psi_j|^{p\kappa}\,dx\bigg)^\frac{1}{\kappa}
&\leq c r^{p-n}\int_{B_j}|\nabla(\bar{w}_j\psi_j)|^p\,dx\\
&\leq c r^{p-n}\bigg(\int_{B_j}\bar{w}_j^p |\nabla\psi_j|^p\,dx+\int_{B_j}\psi_j^{p}|\nabla\bar{w}_j|^p\,dx\bigg)=I_1+I_2.
\end{split}
\end{equation}
\textbf{Estimate of $I_1$:} Using the properties of $\psi_j$, for some $c=c(n,p,s)>0$, we have
\begin{equation}\label{locestI1}
I_1=cr^{p-n}\int_{B_j}\bar{w}_j^p |\nabla\psi_j|^p\,dx
\leq c2^{jp}\fint_{B_j} w_j^{p}\,dx.
\end{equation}
\textbf{Estimate of $I_2$:} By Lemma \ref{energyest}, with $c=c(n,p,s)$ and $C=(n,p,s,\Lambda)$ positive, we obtain
\begin{equation}\label{locestI2}
\begin{split}
I_2&=cr^{p-n}\int_{B_j}\psi_j^{p}|\nabla\bar{w}_j|^p\,dx\\
&\leq Cr^{p-n}\bigg(\int_{B_j}\bar{w}_j^p|\nabla\psi_j|^p\,dx
+\int_{B_j}\int_{B_j}\max\{\bar{w}_j(x),\bar{w}_j(y)\}^p|\psi_j(x)-\psi_j(y)|^p\,d\mu\\
&\qquad+\int_{B_j}\bar{w}_j(y)\psi_j(y)^p\,dy
\cdot\mathop{\rm {ess\,sup}}\limits_{y\in\supp\psi_j}\int_{\mathbb{R}^n\setminus B_j}{\bar{w}_j(x)}^{p-1}K(x,y)\,dx\bigg)\\
&=J_1+J_2+J_3.
\end{split}
\end{equation}
\textbf{Estimates of $J_1$ and $J_2$:} To estimate $J_1$, we use the estimate of $I_1$ in \eqref{locestI1} above and to estimate $J_2$, proceeding similarly as in the proof of the estimate $(4.5)$ in \cite[Page 1292]{Kuusilocal} and again using the properties of $\psi_j$, for every $r\in(0,1]$, we obtain
\begin{equation}\label{locestJ12}
J_i\leq c(n,p,s,\Lambda)2^{jp}\fint_{B_j}w_j^p\,dx,\quad j=1,2.
\end{equation}
\textbf{Estimate of $J_3$:} We observe that $w_j^p\geq(\bar{k}_j-k_j)^{p-1}\bar{w}_j$.
For any $\delta\in(0,1]$, we have
\begin{equation}\label{locestJ3}
\begin{split}
J_3&=
C(n,p,s,\Lambda)r^{p-n}\int_{B_j}\bar{w}_j(y)\psi_j(y)^p\,dy
\cdot\mathop{\rm {ess\,sup}}\limits_{y\in\supp\psi_j}\int_{\mathbb{R}^n\setminus B_j}{\bar{w}_j(x)^{p-1}}K(x,y)\,dx\\
&\leq c2^{j(n+ps)}r^p\fint_{B_j}\frac{{w}_j(y)^p}{(\bar{k}_j-k_j)^{p-1}}\,dy\int_{\mathbb{R}^n\setminus B_j}\frac{w_j(x)^{p-1}}{|x-x_0|^{n+ps}}\,dx\\
&\leq c\frac{2^{j(n+ps+p-1)}}{\bar{k}^{p-1}}\Tail(w_0;x_0,\tfrac{r}{2})^{p-1}\fint_{B_j}w_j(y)^p\,dy\\
&\leq c2^{j(n+ps+p-1)}\delta^{1-p}\fint_{B_j}w_j(y)^p\,dy,
\end{split}
\end{equation}
with $c=c(n,p,s,\Lambda)>0$, whenever $\bar{k}\geq\delta\Tail(w_0;x_0,\tfrac{r}{2})$.
Here we used the fact that
$$
\frac{|x-x_0|}{|x-y|}\leq\frac{|x-y|+|y-x_0|}{|x-y|}\leq 1+\frac{\bar{r}_j}{r_j-\bar{r}_j}\leq 2^{j+4},
$$
which holds for $x\in\mathbb{R}^n\setminus B_j$ and $y\in\supp\psi_j=\bar{B}_j$.
By applying \eqref{locestJ12} and \eqref{locestJ3} in \eqref{locestI2}, we obtain
\begin{equation}\label{locestI2final}
I_2\leq c(n,p,s,\Lambda)2^{j(n+ps+p-1)}\delta^{1-p}\fint_{B_j}w_j^p\,dx
\end{equation}
for every $\delta\in(0,1]$.
Inserting \eqref{locestI1} and \eqref{locestI2final} into \eqref{locbdSob1} we have
\begin{equation}\label{locbdSob12}
\begin{split}
\bigg(\fint_{\bar{B}_j}|\bar{w}_j\psi_j|^{p\kappa}\,dx\bigg)^\frac{1}{\kappa}&\leq c(n,p,s,\Lambda)2^{j(n+ps+p-1)}\delta^{1-p}\fint_{B_j}w_j^p\,dx.
\end{split}
\end{equation}
Setting
$$
Y_j=\bigg(\fint_{B_{j}}w_{j}^p\,dx\bigg)^\frac{1}{p},
$$
and
$$
\bar{k}=\delta\Tail(w_0;x_0,\tfrac{r}{2})+c_0^{\frac{1}{\beta}}b^{\frac{1}{\beta^2}}\bigg(\fint_{B_r(x_0)} w_{0}^p\,dx\bigg)^\frac{1}{p},
$$
where
$$
c_0=c(n,p,s,\Lambda)\delta^{\frac{(1-p)\kappa}{p}},
\quad
b=2^{(\frac{n+ps+p-1}{p}+\frac{\kappa-1}{\kappa})\kappa}
\quad\text{and}\quad
\beta=\kappa-1.
$$
From \eqref{locest} and \eqref{locbdSob12} we obtain
\begin{equation}\label{lociteration}
\frac{Y_{j+1}}{\bar{k}}\leq c(n,p,s,\Lambda)2^{j(\frac{n+ps+p-1}{p}+\frac{\kappa-1}{\kappa})\kappa}\delta^{\frac{(1-p)\kappa}{p}}\Big(\frac{Y_j}{\bar{k}}\Big)^{\kappa}.
\end{equation}
Moreover, by the definition of $\bar{k}$ above we have
$$
\frac{Y_0}{\bar{k}}\leq c_0^{-\frac{1}{\beta}}b^{-\frac{1}{\beta^2}}.
$$
Thus from Lemma \ref{iteration}, we obtain $Y_j\to0$ as $j\to\infty$.
This implies that
$$
\mathop{\rm {ess\,sup}}\limits_{B_{\frac{r}{2}}(x_0)}\,u\leq k+\bar{k},
$$
which gives \eqref{locbd} by choosing $k=0$.
\end{proof}
\section{Oscillation estimates}
The following local H\"older continuity result for weak solutions of \eqref{maineqn} follows from Lemma \ref{osclemma} below.
\begin{Theorem}\label{Holder}(\textbf{H\"older continuity})
Let $u$ be a weak solution of \eqref{maineqn}. Then $u$ is locally H\"older continuous in $\Omega$.
Moreover, there exist constants $\alpha\in(0,\frac{p}{p-1})$ and $c=c(n,p,s,\Lambda)$, such that
\begin{equation}\label{Holderest}
\osc_{B_{\rho}(x_0)}\,u=\mathop{\rm {ess\,sup}}\limits_{B_{\rho}(x_0)}\,u-\mathop{\rm {ess\,inf}}\limits_{B_{\rho}(x_0)}\,u
\leq c\Big(\frac{\rho}{r}\Big)^\alpha\bigg(\Tail(u;x_0,r)+\bigg(\fint_{B_{2r}(x_0)}|u|^p\,dx\bigg)^\frac{1}{p}\bigg),
\end{equation}
whenever $B_{2r}(x_0)\subset\Omega$ with $r\in(0,1]$ and $\rho\in(0,r]$.
Here $\Tail(\cdot)$ is given by \eqref{loctail}.
\end{Theorem}
We prove the next result by arguing similarly as in the proof of \cite[Lemma 5.1]{Kuusilocal}.
\begin{Lemma}\label{osclemma}
Let $u$ be a weak solution of \eqref{maineqn} and $0<r<\frac{R}{2}$ for some $R$ such that $B_{R}(x_0)\subset\Omega$ with $r\in(0,1]$. For $\eta\in(0,\frac{1}{4}]$, we set
$r_j=\eta^j\frac{r}{2}$ and $B_{j}=B_{r_{j}}(x_0)$ for $j=0,1,2,\dots$.
Denote
\begin{equation}\label{omm}
\frac{1}{2}\omega(r_0)=\Tail\big(u;x_0,\tfrac{r}{2}\big)+c\bigg(\fint_{B_r(x_0)}|u|^p\,dx\bigg)^\frac{1}{p},
\end{equation}
where $\Tail(\cdot)$ is given by \eqref{loctail}, $c=c(n,p,s,\Lambda)$ is the constant in \eqref{locbd} and let
\begin{equation}\label{om}
\omega(r_j)=\Big(\frac{r_j}{r_0}\Big)^{\alpha}\omega(r_0),
\quad j=1,2,\dots,
\end{equation}
for some $\alpha\in(0,\frac{p}{p-1})$.
Then
\begin{equation}\label{oscest}
\osc_{B_j}u\leq\omega(r_j),
\quad j=0,1,2,\dots.
\end{equation}
\end{Lemma}
\begin{proof}
Lemma \ref{Solsubsup} and Lemma \ref{cutoffsub} imply that $u_+$ and $(-u)_+$ are weak subsolutions of \eqref{maineqn}.
By applying Theorem \ref{thm1} with $u_+$ and $(-u)_+$, we observe that \eqref{oscest} holds true for $j=0$.
Suppose \eqref{oscest} holds for every $i=0,\ldots,j$ for some $j\in\{0,1,2,\dots\}$. To obtain \eqref{oscest}, by induction, it is enough to deduce \eqref{oscest} for $i=j+1$. We prove it in two steps below. In Step $1$, we obtain the estimate \eqref{s1} below related to $u_j$, where $u_j$ will be defined in \eqref{uj}. In Step $2$, we use the estimate \eqref{s1} along with an iteration argument to conclude the proof of \eqref{oscest}.
We observe that either
\begin{equation}\label{p1}
\frac{\big|B_{2 r_{j+1}}(x_0)\cap\big\{u\geq\mathop{\rm {ess\,inf}}\limits_{B_j}+\frac{\omega(r_j)}{2}\big\}\big|}{|B_{2 r_{j+1}}(x_0)|}\geq\frac{1}{2}
\end{equation}
or
\begin{equation}\label{p2}
\frac{\big|B_{2 r_{j+1}}(x_0)\cap\big\{u\leq\mathop{\rm {ess\,inf}}\limits_{B_j}+\frac{\omega(r_j)}{2}\big\}\big|}{|B_{2 r_{j+1}}(x_0)|}\geq\frac{1}{2}
\end{equation}
holds. Let
\begin{equation}\label{uj}
u_j=
\begin{cases}
u-\mathop{\rm {ess\,inf}}\limits_{B_j}u,\quad \text{if }\eqref{p1} \text{ holds},\\
\omega(r_j)-(u-\mathop{\rm {ess\,inf}}\limits_{B_j}u),\quad\text{if }\eqref{p2} \text{ holds}.
\end{cases}
\end{equation}
Then $u_j$ is a weak solution of \eqref{maineqn}. Also, in both cases of \eqref{p1} and \eqref{p2}, $u_j\geq 0$ in $B_j$ and
\begin{equation}\label{p12}
\frac{\big|B_{2 r_{j+1}}(x_0)\cap\big\{u_j\geq\frac{\omega(r_j)}{2}\big\}\big|}{|B_{2 r_{j+1}}(x_0)|}\geq\frac{1}{2}.
\end{equation}
\textbf{Step $1$:} We claim that
\begin{equation}\label{s1}
\frac{|B_{2 r_{j+1}}(x_0)\cap\{u_j\leq 2\varepsilon\omega(r_j)\}|}{|B_{2 r_{j+1}}(x_0)|}\leq\frac{\hat{C}}{\log(\frac{1}{\eta})},
\end{equation}
where $\varepsilon=\eta^{\frac{p}{p-1}-\alpha}$ for some positive constant $\hat{C}$ depending only on $n,p,s,\Lambda$ and the difference between $\frac{p}{p-1}$ and $\alpha$ via the definition of $\varepsilon$. To this end, we will apply the logarithmic estimate from Corollary \ref{cor}, where a tail quantity appears. We set
\begin{equation}\label{mu}
\mu=\log\bigg(\frac{\frac{\omega(r_j)}{2}+\varepsilon\omega(r_j)}{3\varepsilon\omega(r_j)}\bigg)
=\log\bigg(\frac{\frac{1}{2}+\varepsilon}{3\varepsilon}\bigg)\approx\log\Big(\frac{1}{\varepsilon}\Big)
\end{equation}
and define
\begin{equation}\label{theta}
\Theta=\min\bigg\{\bigg(\log\bigg(\frac{\frac{\omega(r_j)}{2}+\varepsilon \omega(r_j)}{u_j+\varepsilon \omega(r_j)}\bigg)\bigg)_+,\mu\bigg\}.
\end{equation}
By \eqref{p12} we have
\begin{equation}\label{lset}
\begin{split}
\mu&=\frac{1}{|B_{2 r_{j+1}}(x_0)\cap\{u_j\geq\frac{\omega(r_j)}{2}\}|}\int_{B_{2 r_{j+1}}(x_0)\cap\{u_j\geq\frac{\omega(r_j)}{2}\}}\mu\,dx\\
&=\frac{1}{|B_{2 r_{j+1}}(x_0)\cap\{u_j\geq\frac{\omega(r_j)}{2}\}|}\int_{B_{2 r_{j+1}}(x_0)\cap\{\Theta=0\}}\mu\,dx\\
&\leq\frac{2}{|B_{2 r_{j+1}}(x_0)|}\int_{B_{2 r_{j+1}}(x_0)}(\mu-\Theta)\,dx=2(\mu-(\Theta)_{B_{2 r_{j+1}}(x_0)}),
\end{split}
\end{equation}
where $(\Theta)_{B_{2 r_{j+1}}(x_0)}=\fint_{B_{2 r_{j+1}}(x_0)}\Theta\,dx$.
Integrating \eqref{lset} over the set $\{B_{2 r_{j+1}}(x_0)\cap \Theta=\mu\}$ we get
\begin{equation}\label{lsetj}
\begin{split}
\frac{|B_{2 r_{j+1}}(x_0)\cap\{\Theta=\mu\}|}{|B_{2 r_{j+1}}(x_0)|}\mu&\leq\frac{2}{|B_{2 r_{j+1}}(x_0)|}\int_{B_{2 r_{j+1}}(x_0)}|\Theta-(\Theta)_{B_{2 r_{j+1}}(x_0)}|\,dx.
\end{split}
\end{equation}
Applying Corollary \ref{cor} with $a=\frac{\omega(r_j)}{2}$, $d=\varepsilon\omega(r_j)$ and $b=e^\mu$ for some constant $c=c(n,p,s,\Lambda)$ we obtain
\begin{equation}\label{c1app}
\begin{split}
\fint_{B_{2 r_{j+1}}(x_0)}|\Theta-(\Theta)_{B_{2 r_{j+1}}(x_0)}|^p\,dx
\leq c\bigg((\varepsilon\omega(r_j))^{1-p}\Big(\frac{r_{j+1}}{r_j}\Big)^p\mathrm{Tail}(u_j;x_0,r_j)^{p-1}+1\bigg).
\end{split}
\end{equation}
Noting that $\eta\in(0,\frac{1}{4}]$, $\alpha\in(0,\frac{p}{p-1})$ along with $r\in(0,1]$ and following the lines of the proof of the estimate $(5.6)$ in \cite[Pages 1294--1295]{Kuusilocal}, we obtain
\begin{equation}\label{tailest1}
\mathrm{Tail}(u_j;x_0,r_j)^{p-1}\leq c\eta^{-\alpha(p-1)}\omega(r_j)^{p-1}
\end{equation}
for some positive constant $c$ depending only on $n,p,s$ and the difference between $\frac{p}{p-1}$ and $\alpha$, but independent of $\eta$. Therefore, using the estimate \eqref{tailest1} in \eqref{c1app} we obtain
\begin{equation}\label{c1app12}
\begin{split}
\fint_{B_{2 r_{j+1}}(x_0)}|\Theta-(\Theta)_{B_{2 r_{j+1}}(x_0)}|\,dx\leq C
\end{split}
\end{equation}
for some positive constant $C$ depending only on $n,p,s,\Lambda$ and the difference between $\frac{p}{p-1}$ and $\alpha$.
The estimate \eqref{s1} follows by employing \eqref{c1app12} in \eqref{lsetj}.\\
\textbf{Step $2$:} Now we use an iteration argument to obtain \eqref{oscest} for $i=j+1$. To this end, for every $i=0,1,2,\dots$, let
$\rho_i=(1+2^{-i})r_{j+1}$, $\hat{\rho}_i=\frac{\rho_i +\rho_{i+1}}{2}$, $B^i=B_{\rho_i}(x_0)$ and $\hat{B}^i=B_{\hat{\rho}_i}(x_0)$.
Recalling that $\varepsilon=\eta^{\frac{p}{p-1}-\alpha}$, we denote
$k_i=(1+2^{-i})\varepsilon\omega(r_j)$ and
$$
A^i=B^i\cap\{u_j\leq k_i\},\quad i=0,1,2,\dots.
$$
Let $w_i=(k_i -u_j)_+$ and $(\psi_i)_{i=0}^{\infty}\subset C_c^{\infty}(\hat{B}^i)$ be such that $0\leq \psi_i\leq 1$ in $\hat{B}^i$, $\psi_i =1$ in $B^{i+1}$ and $|\nabla\psi_i|\leq\frac{c 2^i}{\rho_i}$ in $\hat{B}^i$, with $c=c(n,p)>0$. By applying the Sobolev inequality in \eqref{e.friedrich}, for $\kappa$ as defined in \eqref{kappa}, we obtain a constant $c=c(n,p,s)>0$ such that
\begin{equation}\label{Poincareless1}
\begin{split}
(k_i-k_{i+1})^p\bigg(\frac{|A^{i+1}|}{|B^{i+1}|}\bigg)^\frac{1}{\kappa}&\leq\bigg(\fint_{B^{i+1}}w_i^{\kappa p}\,dx\bigg)^\frac{1}{\kappa}\leq c \bigg(\fint_{B^{i}}w_i^{\kappa p}\psi_i^{\kappa p}\,dx\bigg)^\frac{1}{\kappa}\\
&\leq c r_{j+1}^p\fint_{B^i}|\nabla(w_i \psi_i)|^p\,dx\leq cr_{j+1}^p(I+J),
\end{split}
\end{equation}
where
$$
I=\fint_{B^i}w_i^{p}|\nabla\psi_i|^p\,dx
\quad\text{and}\quad
J=\fint_{B^i}|\nabla w_i|^{p}\psi_{i}^p\,dx.
$$
\textbf{Estimate of $I$:} Since $u_j\geq 0$ in $B_{j}$, we have $w_i\leq k_i\leq 2\varepsilon\omega(r_j)$ in $B^i$. Thus, using the properties of $\psi_i$ above, for some constant $c=c(n,p)>0$, we have
\begin{equation}\label{Jite}
\begin{split}
I&=\fint_{B^i}w_i^{p}|\nabla\psi_i|^p\,dx\leq cr_{j+1}^{-p}(\varepsilon\omega(r_j))^p 2^{ip}\frac{|A^i|}{|B^i|}.
\end{split}
\end{equation}
\textbf{Estimate of $J$:} By Lemma \ref{energyest}, we obtain a constant $C=C(p,\Lambda)$ such that
\begin{equation}\label{tite}
\begin{split}
\int_{B^i}|\nabla w_i|^{p}\psi_{i}^p\,dx&\leq C(J_1+J_2+J_3),
\end{split}
\end{equation}
where
$$
J_1=\int_{B^i}w_{i}^p |\nabla\psi_i|^p\,dx,\quad J_2=\int_{B^i}\int_{B^i}{\max\{w_i(x),w_i(y)\}^p|\psi_i(x)-\psi_i(y)|^p}\,d\mu
$$
and
$$
J_3=\mathop{\rm {ess\,sup}}\limits_{x\in\hat{B}^i}\int_{{\mathbb{R}^n\setminus B^i}}{\frac{w_i(y)^{p-1}}{|x-y|^{n+ps}}}\,dy
\cdot\int_{B^i}w_i \psi_{i}^p\,dx.
$$
From \eqref{Jite} we have
\begin{equation}\label{j1}
J_1\leq cr_{j+1}^{-p}(\varepsilon\omega(r_j))^p 2^{ip}|A^i|,
\end{equation}
with $c=c(n,p)>0$. For $x\in\hat{B^i}$ and $y\in{I\!\!R}^n\setminus B^{i}$, we have
\begin{equation}\label{nee1}
\begin{split}
\frac{1}{|y-x|}
&=\frac{1}{|y-x_0|}\frac{|y-x_0|}{|y-x|}
\leq\frac{1}{|y-x_0|}\Big(1+\frac{|x-x_0|}{|y-x|}\Big)\\
&\leq\frac{1}{|y-x_0|}\Big(1+\frac{\hat{\rho}_i}{\rho_i -\hat{\rho}_i}\Big)
\leq\frac{2^{i+4}}{|y-x_0|}.
\end{split}
\end{equation}
By applying \eqref{nee1}, \eqref{tailest1}, the properties of $\psi_i$, $r\in(0,1]$ and proceeding along the lines of the proof of the estimates $(5.12)$ and $(5.15)$ in \cite[Page 1297]{Kuusilocal}, we obtain
\begin{equation}\label{jmm}
J_{m}\leq Cr_{j+1}^{-p}(\varepsilon\omega(r_j))^p 2^{i(n+p)}|A^i|,
\quad m=2,3,
\end{equation}
for some positive constant $C$ depending on $n,p,s,\Lambda$ and the difference between $\frac{p}{p-1}$ and $\alpha$. Using \eqref{j1} and \eqref{jmm} in \eqref{tite}, we obtain
\begin{equation}\label{Iite}
\begin{split}
J&=\fint_{B^i}|\nabla w_i|^{p}\psi_{i}^p\,dx\leq Cr_{j+1}^{-p}(\varepsilon\omega(r_j))^p 2^{i(n+p)}\frac{|A^i|}{|B^i|},
\end{split}
\end{equation}
for some positive constant $C$ depending only on $n,p,s,\Lambda$ and the difference between $\frac{p}{p-1}$ and $\alpha$.
Let
\[
Y_i=\frac{|A^i|}{|B^i|},
\quad i=0,1,2,\dots.
\]
Noting that $k_i- k_{i+1}=2^{-i-1}\varepsilon \omega(r_j)$ and applying \eqref{Jite} and \eqref{Iite} in \eqref{Poincareless1}, we get
\[
Y_{i+1}\leq C 2^{i(2p+n)\kappa}Y_{i}^{\kappa},
\]
for some constant $C$ depending only on $n,p,s,\Lambda$ and the difference between $\frac{p}{p-1}$ and $\alpha$. From Step $1$, by \eqref{s1}, we have
$$
Y_0\leq\frac{\hat{C}}{\log(\frac{1}{\eta})},
$$
for some positive constant $\hat{C}$ depending only on $n,p,s,\Lambda$ and the difference between $\frac{p}{p-1}$ and $\alpha$. Let
$$
c_0=C,\quad b=2^{(2p+n)\kappa},\quad \beta=\kappa-1
\quad\text{and}\quad \eta_1=c_{0}^{-\frac{1}{\beta}}b^{-\frac{1}{\beta^2}}.
$$
By choosing $\eta=\frac{1}{2} \min\big\{\frac{1}{4},e^{-\frac{\hat{C}}{\eta_1}}\big\}$ we have $Y_0\leq\eta_1$.
Thus by Lemma \ref{iteration} we deduce that $\lim_{i\to\infty}Y_i=0$ and therefore,
$u_j\geq\varepsilon\omega(r_j)\text{ in }B_{j+1}$.
Using the definition of $u_j$ from \eqref{uj}, we obtain
\begin{equation}\label{oscj1}
\text{osc}_{B_{j+1}}\,u\leq (1-\varepsilon)\omega(r_j)=(1-\varepsilon)\eta^{-\alpha}\omega(r_{j+1})\leq\omega(r_{j+1}),
\end{equation}
where we have chosen $\alpha\in(0,\frac{p}{p-1})$ (depending on $n,p,s,\Lambda$) small enough such that
\[
\eta^\alpha\geq 1-\varepsilon=1-\eta^{\frac{p}{p-1}-\alpha}.
\]
Thus \eqref{oscj1} proves the induction estimate \eqref{oscest} for $i=j+1$. Hence the result follows.
\end{proof}
\section{Tail estimate}
The following tail estimate will be useful for us.
\begin{Lemma}\label{Tail}
Let $u$ be a weak solution of \eqref{maineqn} such that $u\geq 0$ in $B_R(x_0)\subset\Omega$.
There exists a positive constant $c=c(n,p,s,\Lambda)$ such that
\begin{equation}\label{tailest}
\Tail(u_{+};x_0,r)\leq c\mathop{\rm {ess\,sup}}\limits_{B_r(x_0)}\,u+c\Big(\frac{r}{R}\Big)^\frac{p}{p-1}\Tail(u_{-};x_0,R),
\end{equation}
whenever $0<r<R$ with $r\in(0,1]$.
Here $\Tail(\cdot)$ is given by \eqref{loctail}.
\end{Lemma}
\begin{proof}
Let $M=\mathop{\rm {ess\,sup}}\limits_{B_r(x_0)}\,u$ and $\psi\in C_c^{\infty}(B_r(x_0))$ be a cutoff function such that
$0\leq\psi\leq 1$ in $B_r(x_0)$, $\psi=1$ in $B_{\frac{r}{2}}(x_0)$ and $|\nabla\psi|\leq\frac{8}{r}$ in $B_r(x_0)$.
By letting $w=u-2M$ and choosing $\phi=w\psi^p$ as a test function in \eqref{weaksubsupsoln} we obtain
\begin{equation}\label{tailtest}
\begin{split}
0&=\int_{B_r(x_0)}|\nabla u|^{p-2}\nabla u\cdot\nabla(w\psi^p)\,dx\\
&\qquad+\int_{B_r(x_0)}\int_{B_r(x_0)}\mathcal{A}(u(x,y))(w(x)\psi(x)^p-w(y)\psi(y)^p)\,d\mu\\
&\qquad+2\int_{B_r(x_0)}\int_{\mathbb{R}^n\setminus B_r(x_0)}\mathcal{A}(u(x,y))w(x)\psi(x)^p\,d\mu\\
&=I_1+I_2+I_3.
\end{split}
\end{equation}
\textbf{Estimate of $I_1$:} By Young's inequality, the estimate
\begin{equation*}
\begin{split}
|\nabla w|^{p-2}\nabla w\cdot\nabla(w\psi^p)
&=|\nabla w|^p \psi^p+p\psi^{p-1}w|\nabla w|^{p-2}\nabla w\cdot\nabla\psi\\
&\geq\frac{1}{2}|\nabla w|^p\psi^p-c(p)|w|^p|\nabla\psi|^p
-c(p)M^p|\nabla\psi|^p,
\end{split}
\end{equation*}
holds in $B_r(x_0)$.
By the properties of $\psi$, we have
\begin{equation}\label{tailestI1}
I_1=\int_{B_r(x_0)}|\nabla u|^{p-2}\nabla u\cdot\nabla(w\psi^p)\,dx\geq -c(p)M^p r^{-p}|B_r(x_0)|.
\end{equation}
\textbf{Estimate of $I_2$ and $I_3$:}
Proceeding along the lines of the proof of the estimates $(4.11)$ and $(4.9)$ in \cite[Pages 1827--1828]{KuusiHarnack} and using the fact that $r\in(0,1]$,
we obtain a constant $c=c(n,p,s,\Lambda)>0$ such that
\begin{equation}\label{tailestI2}
\begin{split}
I_2&=\int_{B_r(x_0)}\int_{B_r(x_0)}\mathcal{A}(u(x,y))(w(x)\psi(x)^p-w(y)\psi(y)^p)\,d\mu\geq -cM^p r^{-p}|B_r(x_0)|,
\end{split}
\end{equation}
and
\begin{equation}\label{tailestI3}
\begin{split}
I_3&=2\int_{B_r(x_0)}\int_{\mathbb{R}^n\setminus B_r(x_0)}\mathcal{A}(u(x,y))w(x)\psi(x)^p\,d\mu
\geq cM r^{-p}\Tail(u_{+};x_0,r)^{p-1}|B_r(x_0)|\\
&\qquad-cMR^{-p}\Tail(u_{-};x_0,R)^{p-1}|B_r(x_0)|-cM^p r^{-p}|B_r(x_0)|.
\end{split}
\end{equation}
The estimate in \eqref{tailest} follows by applying \eqref{tailestI1}, \eqref{tailestI2} and \eqref{tailestI3} in \eqref{tailtest}.
\end{proof}
\section{Expansion of positivity}
The following lemma shows that the expansion of positivity technique applies to mixed problems.
\begin{Lemma}\label{DGLemma}
Let $u$ be a weak supersolution of \eqref{maineqn} such that $u\geq 0$ in $B_R(x_0)\subset\Omega$.
Assume $k\geq 0$ and there exists $\tau\in(0,1]$ such that
\begin{equation}\label{expangiven}
\big|B_r(x_0)\cap\{u\geq k\}\big|\geq \tau|B_r(x_0)|,
\end{equation}
for some $r\in(0,1]$ with $0<r<\frac{R}{16}$. There exists a constant $\delta=\delta(n,p,s,\Lambda,\tau)\in(0,\frac{1}{4})$ such that
\begin{equation}\label{expan}
\mathop{\rm {ess\,inf}}\limits_{B_{4r}(x_0)}\,u\geq\delta k-\Big(\frac{r}{R}\Big)^\frac{p}{p-1}\Tail(u_{-};x_0,R),
\end{equation}
where $\Tail(\cdot)$ is given by \eqref{loctail}.
\end{Lemma}
\begin{proof}
We prove the lemma in two steps.\\
\textbf{Step 1.} Under the assumption in \eqref{expangiven}, we claim that there exists a positive constant $c_1=c(n,p,s,\Lambda)$ such that
\begin{equation}\label{expanstep1}
\Big|B_{6r}(x_0)\cap \Big\{u\leq 2\delta k-\frac{1}{2}\Big(\frac{r}{R}\Big)^\frac{p}{p-1}\Tail(u_{-};x_0,R)-\epsilon\Big\}\Big|
\leq \frac{c_1}{\tau\log\frac{1}{2\delta}}|B_{6r}(x_0)|
\end{equation}
for every $\delta\in(0,\frac{1}{4})$ and for every $\epsilon>0$.
Let $\epsilon>0$ and $\psi\in C_c^{\infty}(B_{7r}(x_0))$ be a cutoff function such that
$0\leq\psi\leq 1$ in $B_{7r}(x_0)$, $\psi=1$ in $B_{6r}(x_0)$ and $|\nabla\psi|\leq\frac{8}{r}$ in $B_{7r}(x_0)$.
We denote $w=u+t_{\epsilon}$, where
$$
t_{\epsilon}=\frac{1}{2}\Big(\frac{r}{R}\Big)^\frac{p}{p-1}\Tail(u_{-};x_0,R)+\epsilon.
$$
Since $w$ is a weak supersolution of \eqref{maineqn}, we can choose $\phi=w^{1-p}\psi^p$ as a test function in \eqref{weaksubsupsoln} to obtain
\begin{equation}\label{DGLtsteqn1}
\begin{split}
0&\leq\int_{B_{8r}(x_0)}|\nabla w|^{p-2}\nabla w\cdot\nabla( w^{1-p}\psi^p)\,dx\\
&\qquad+\int_{B_{8r}(x_0)}\int_{B_{8r}(x_0)}\mathcal{A}( w(x,y))( w(x)^{1-p}\psi(x)^p- w(y)^{1-p}\psi(y)^p)\,d\mu\\
&\qquad+2\int_{\mathbb{R}^n\setminus B_{8r}(x_0)}\int_{B_{8r}(x_0)}\mathcal{A}( w(x,y)) w(x)^{1-p}\psi(x)^p\,d\mu\\
&=I_1+I_2+I_3.
\end{split}
\end{equation}
\textbf{Estimate of $I_1$:} Proceeding similarly as in the proof of \cite[Pages 717--718, Lemma 3.4]{Kin-Kuusi} and using the properties of $\psi$, we obtain a constant $c=c(p)>0$ such that
\begin{equation}\label{EstI1}
\begin{split}
I_1&=\int_{B_{8r}(x_0)}|\nabla w|^{p-2}\nabla w\cdot\nabla( w^{1-p}\psi^p)\,dx\leq -c\int_{B_{6r}(x_0)}|\nabla\log w|^p\,dx+cr^{n-p}.
\end{split}
\end{equation}
\textbf{Estimate of $I_2$:} Arguing as in the proof of the estimate of $I_1$ in \cite[page 1817]{KuusiHarnack} and using the fact that $r\in(0,1]$, we
obtain a constant $c=c(n,p,s,\Lambda)>0$ such that
\begin{equation}\label{EstI2}
\begin{split}
I_2&=\int_{B_{8r}(x_0)}\int_{B_{8r}(x_0)}\mathcal{A}( w(x,y))( w(x)^{1-p}\psi(x)^p- w(y)^{1-p}\psi(y)^p)\,d\mu\\
&\leq -\frac{1}{c}\int_{B_{6r}(x_0)}\int_{B_{6r}(x_0)}\bigg|\log\Big(\frac{ w(x)}{ w(y)}\Big)\bigg|^p\,d\mu+c r^{n-p}.
\end{split}
\end{equation}
\textbf{Estimate of $I_3$:} Here we follow the proof of the estimate of $I_2$ in \cite[Pages 1817--1818]{KuusiHarnack}. To this end, we write
\begin{equation}\label{estI3l}
\begin{split}
I_3&=2\int_{\mathbb{R}^n\setminus B_{8r}(x_0)}\int_{B_{8r}(x_0)}\mathcal{A}( w(x,y)) w(x)^{1-p}\psi(x)^p\,d\mu=2(I_3^{1}+I_3^{2}),
\end{split}
\end{equation}
where
$$
I_3^{1}=\int_{\mathbb{R}^n\setminus B_{8r}(x_0)\cap \{w(y)<0\}}\int_{B_{8r}(x_0)}\mathcal{A}( w(x,y)) w(x)^{1-p}\psi(x)^p\,d\mu
$$
and
$$
I_3^{2}=\int_{\mathbb{R}^n\setminus B_{8r}(x_0)\cap \{w(y)\geq 0\}}\int_{B_{8r}(x_0)}\mathcal{A}( w(x,y)) w(x)^{1-p}\psi(x)^p\,d\mu.
$$
\textbf{Estimate of $I_3^{1}$:} Using the definitions of $w$ and $t_{\epsilon}$, the assumption on the kernel $K$ and the fact that support of $\psi$ is inside $B_{7r}(x_0)$, we get
\begin{equation}\label{l11}
\begin{split}
I_3^{1}&\leq c r^n\int_{\mathbb{R}^n\setminus B_{8r}(x_0)}\Big(1+\frac{(w(y))_-}{t_{\epsilon}}\Big)^{p-1}|y-x_0|^{-n-ps}\,dy\\
&\leq cr^{n-ps}+cr^{n}t_{\epsilon}^{1-p}R^{-p}\mathrm{Tail}(u_-;x_0,R)^{p-1}\leq cr^{n-p},
\end{split}
\end{equation}
with $c=c(n,p,s,\Lambda)$. Here we also used the hypothesis that $u\geq 0$ in $B_R(x_0)$ and $r\in(0,1]$.\\
\textbf{Estimate of $I_3^{2}$:} Let $x\in B_{8r}(x_0)$. Suppose $y\in \mathbb{R}^n\setminus B_{8r}(x_0)$ such that $w(y)\geq 0$. If $w(x)-w(y)\leq 0$, then $\mathcal{A}(w(x,y))\leq 0$. If $w(x)-w(y)\geq 0$, then $\mathcal{A}(w(x,y))\leq w(x)^{p-1}$. Therefore, again using the assumption on the kernel $K$ and the fact that $\psi$ is supported in $B_{7r}(x_0)$ along with $r\in(0,1]$, we get
\begin{equation}\label{l12}
I_3^{2}\leq c\int_{\mathbb{R}^n\setminus B_{8r}(x_0)}\int_{B_{7r}(x_0)}|y-x_0|^{-n-ps}\,dx\,dy\leq cr^{n-p},
\end{equation}
with $c=c(n,p,s,\Lambda)$. By applying \eqref{l11} and \eqref{l12} in \eqref{estI3l}, we have
\begin{equation}\label{estI3}
I_3\leq cr^{n-p},
\end{equation}
with $c=c(n,p,s,\Lambda)$.\\
By using \eqref{EstI1}, \eqref{EstI2} and \eqref{estI3} in \eqref{DGLtsteqn1}, we obtain
\begin{equation}\label{estI123}
\begin{split}
\int_{B_{6r}(x_0)}|\nabla\log w|^p\,dx
+\int_{B_{6r}(x_0)}\int_{B_{6r}(x_0)}\bigg|\log\Big(\frac{ w(x)}{ w(y)}\Big)\bigg|^p\,d\mu\leq cr^{n-p},
\end{split}
\end{equation}
for some constant $c=c(n,p,s,\Lambda)$. For $\delta\in(0,\frac{1}{4})$, we denote
$$
v=\biggl(\min\biggl\{\log\frac{1}{2\delta},\log\frac{k+t_{\epsilon}}{ w}\biggr\}\biggr)_{+}.
$$
By \eqref{estI123}, we have
\begin{equation}\label{pcase}
\int_{B_{6r}(x_0)}|\nabla v|^p\,dx
\leq\int_{B_{6r}(x_0)}|\nabla\log w|^p\,dx\leq c r^{n-p}.
\end{equation}
From \eqref{pcase}, by H\"older's inequality and Poincar\'e inequality (see \cite[Theorem 2]{Evans}), we obtain
\begin{equation}\label{pcase1}
\int_{B_{6r}(x_0)}|v-(v)_{B_{6r}(x_0)}|\,dx
\leq c r^{1+\frac{n}{p'}}\bigg(\int_{B_{6r}(x_0)}|\nabla v|^p\,dx\bigg)^\frac{1}{p}\leq c|B_{6r}(x_0)|,
\end{equation}
where $p'=\frac{p}{p-1}$ and $(v)_{B_{6r}(x_0)}=\fint_{B_{6r}(x_0)}v\,dx$. We observe that
$\{v=0\}=\{ w\geq k+t_{\epsilon}\}=\{u\geq k\}$.
By the assumption \eqref{expangiven}, it follows that
\begin{equation}\label{useofgiven}
|B_{6r}(x_0)\cap\{v=0\}|\geq\frac{\tau}{6^n}|B_{6r}(x_0)|.
\end{equation}
Following the proof of \cite[Page 1819, Lemma 3.1]{KuusiHarnack} and using \eqref{useofgiven}, we obtain
\begin{equation}\label{pcase2}
\begin{split}
\log\,\frac{1}{2\delta}
&=\frac{1}{|B_{6r}(x_0)\cap\{v=0\}|}\int_{B_{6r}(x_0)\cap\{v=0\}}\Big(\log\frac{1}{2\delta}-v(x)\Big)\,dx\\
&\leq\frac{6^n}{\tau}\Big(\log\frac{1}{2\delta}-(v)_{B_{6r}}\Big).
\end{split}
\end{equation}
Now integrating \eqref{pcase2} over the set $B_{6r}(x_0)\cap\{v=\log\frac{1}{2\delta}\}$ and using \eqref{pcase1}, we obtain a constant $c_1=c_1(n,p,s,\Lambda)$ such that
$$
\Big|\Big\{v=\log\frac{1}{2\delta}\Big\}\cap B_{6r}(x_0)\Big|\log\frac{1}{2\delta}\leq\frac{6^n}{\tau}\int_{B_{6r}(x_0)}|v-(v)_{B_{6r}(x_0)}|\,dx
\leq\frac{c_1}{\tau}|B_{6r}(x_0)|.
$$
Hence, for any $\delta\in(0,\frac{1}{4})$, we have
$$
\big|B_{6r}(x_0)\cap\{ w\leq 2\delta(k+t_{\epsilon})\}\big|\leq\frac{c_1}{\tau}\frac{1}{\log\frac{1}{2\delta}}|B_{6r}(x_0)|.
$$
This implies \eqref{expanstep1}.\\
\textbf{Step 2.} We claim that, for every $\epsilon>0$, there exists a constant $\delta=\delta(n,p,s,\Lambda,\tau)\in(0,\frac{1}{4})$ such that
\begin{equation}\label{expanaux}
\mathop{\rm {ess\,inf}}\limits_{B_{4r}(x_0)}\,u\geq\delta k-\Big(\frac{r}{R}\Big)^\frac{p}{p-1}\Tail(u_{-};x_0,R)-2\epsilon.
\end{equation}
As a consequence of \eqref{expanaux}, the property \eqref{expan} follows.
To prove \eqref{expanaux}, without loss of generality, we may assume that
\begin{equation}\label{assume}
\delta k\geq\Big(\frac{r}{R}\Big)^\frac{p}{p-1}\Tail(u_{-};x_0,R)+2\epsilon.
\end{equation}
Otherwise \eqref{expanaux} holds tue, since $u\geq 0$ in $B_R(x_0)$.
Let $\rho\in[r,6r]$ and $\psi\in C_c^{\infty}(B_{\rho}(x_0))$ be a cutoff function such that $0\leq\psi\leq 1$ in $B_{\rho}(x_0)$.
For any $l\in(\delta k,2\delta k)$, from Lemma \ref{energyest} and the proof of \cite[Pages 1820--1821, Lemma 3.2]{KuusiHarnack} for $w=(l-u)_{+}$,
for some constant $c=c(n,p,s,\Lambda)$, we obtain
\begin{equation}\label{energyapp}
\begin{split}
&\int_{B_{\rho}(x_0)}\psi^p|\nabla\,w|^p\,dx
+\int_{B_{\rho}(x_0)}\int_{B_{\rho}(x_0)}|w(x)\psi(x)-w(y)\psi(y)|^p\,d\mu\\
&\leq c\int_{B_{\rho}(x_0)}w^p\,|\nabla\psi|^p\,dx+c\int_{B_{\rho}(x_0)}\int_{B_{\rho}(x_0)}\max\{w(x),w(y)\}^p|\psi(x)-\psi(y)|^p\,d\mu\\
&\qquad+cl\mathop{\rm {ess\,sup}}\limits_{x\in\supp\psi}\int_{{\mathbb{R}^n}\setminus B_{\rho}(x_0)}\big(l+(u(y))_{-}\big)^{p-1}\,K(x,y)\,dy
\cdot|B_{\rho}(x_0)\cap\{u<l\}|\\
&=J_1+J_2+J_3.
\end{split}
\end{equation}
We apply Lemma \ref{iteration} to conclude the proof. For $j=0,1,2,\dots$, we denote
\begin{equation}\label{parameter}
l=k_j=\delta k+2^{-j-1}\delta k,
\quad
\rho=\rho_j=4r+2^{1-j}r,
\quad
\hat{\rho_j}=\frac{\rho_j+\rho_{j+1}}{2}.
\end{equation}
Then $l\in(\delta k,2\delta k)$, $\rho_j,\hat{\rho_j}\in(4r,6r)$ and
$$
k_j-k_{j+1}=2^{-j-2}\delta k\geq 2^{-j-3}k_j
$$
for every $j=0,1,2,\dots$.
Set $B_j=B_{\rho_j}(x_0),\,\hat{B}_j=B_{\hat{\rho}_j}(x_0)$ and we observe that
$$
w_j=(k_j-u)_{+}\geq 2^{-j-3} k_j\chi_{\{u<k_{j+1}\}}.
$$
Let $(\psi_j)_{j=0}^{\infty}\subset C_c^{\infty}(\hat{B}_j)$ be a sequence of cutoff functions such that
$0\leq\psi_j\leq 1$ in $\hat{B}_j$, $\psi_j=1$ in $B_{j+1}$ and $|\nabla\psi_j|\leq\frac{2^{j+3}}{r}$.
We choose $\psi=\psi_j$, $w=w_j$ in \eqref{energyapp}. By the properties of $\psi_j$, we obtain
\begin{equation}\label{I1jest}
J_1=\int_{B_j}w_j^p|\nabla\psi_j|^p\,dx\leq{c(p)2^{jp}}k_j^{p}r^{-p}|B_j\cap\{u<k_j\}|.
\end{equation}
Now proceeding along the lines of the the proof of \cite[Page 1822, Lemma 3.2]{KuusiHarnack}, for any $r\in(0,1]$, we get
\begin{equation}\label{I2jest}
\begin{split}
J_2&=\int_{B_j}\int_{B_j}\max\{w_j(x),w_j(y)\}^p|\psi_j(x)-\psi_j(y)|^p\,d\mu\\
&\leq c(n,p,s,\Lambda)2^{jp} k_j^{p}r^{-p}|B_j\cap\{u<k_j\}|.
\end{split}
\end{equation}
To estimate $J_3$, we follow the proof of \cite[Page 1823, Lemma 3.2]{KuusiHarnack}. To this end, we observe that, for any $x\in\mathrm{supp}\,\psi_j\subset\hat{B}_j$ and $y\in\mathbb{R}^n\setminus B_j$, we have
\begin{equation}\label{ne}
\frac{|y-x_0|}{|y-x|}=\frac{|y-x+x-x_0|}{|y-x|}\leq 1+\frac{|x-x_0|}{|y-x|}\leq 1+\frac{\hat{\rho}_j}{\rho_j -\hat{\rho}_j}=2^{j+4}.
\end{equation}
Using \eqref{ne} and the properties of the kernel $K$, we have
\begin{equation}\label{ne1}
\begin{split}
&\mathop{\rm {ess\,sup}}\limits_{x\in\supp\psi_j}\int_{\mathbb{R}^n\setminus B_j}\big(k_j+(u(y))_{-}\big)^{p-1}\,K(x,y)\,dy\\
&\qquad\leq c2^{j(n+ps)}\int_{\mathbb{R}^n\setminus B_j}\big(k_j+(u(y))_{-}\big)^{p-1}|y-x_0|^{-n-ps}\,dy\\
&\qquad\leq c2^{j(n+ps)}\bigg(k_j^{p-1}r^{-ps}+\int_{\mathbb{R}^n\setminus B_R(x_0)}(u(y))_{-}^{p-1}|y-x_0|^{-n-ps}\,dy\bigg)\\
&\qquad= c2^{j(n+ps)}\Big(k_j^{p-1}r^{-p}+r^{-p}\big(\frac{r}{R}\big)^p \mathrm{Tail}(u_-;x_0,R)^{p-1}\Big)\\
&\qquad\leq c2^{j(n+ps)}k_j^{p-1}r^{-p},
\end{split}
\end{equation}
with $c=c(n,p,s,\Lambda)$.
Here we also used the fact that $r\in(0,1]$ along with \eqref{assume}, $\delta k<k_j$ and the fact that $u\geq 0$ in $B_R(x_0)$.
Therefore, from \eqref{ne1}, we obtain
\begin{equation}\label{I3jest}
\begin{split}
J_3&=ck_j\mathop{\rm {ess\,sup}}\limits_{x\in\supp\psi_j}\int_{\mathbb{R}^n\setminus B_j}\big(k_j+(u(y))_{-}\big)^{p-1}\,K(x,y)\,dy
\cdot|B_j\cap\{u<k_j\}|\\
&\leq c2^{j(n+ps)}k_j^{p}r^{-p}|B_j\cap\{u<k_j\}|,
\end{split}
\end{equation}
with $c=c(n,p,s,\Lambda)$.
By using \eqref{I1jest}, \eqref{I2jest} and \eqref{I3jest} in \eqref{energyapp}, we obtain
\begin{equation}\label{J123est}
\int_{B_j}\psi_j^p|\nabla w_j|^p\,dx\leq c 2^{j(n+ps+p)}k_j^{p}r^{-p}|B_j\cap\{u<k_j\}|,
\end{equation}
with $c=c(n,p,s,\Lambda)$.
By applying the Sobolev inequality in \eqref{e.friedrich} along with \eqref{I1jest} and \eqref{J123est}, for $\kappa$ defined in \eqref{kappa}, we obtain a constant $c=c(n,p,s,\Lambda)$ such that
\begin{equation}\label{Poincareless}
\begin{split}
(k_j-k_{j+1})^p&\bigg(\frac{|B_{j+1}\cap\{u<k_{j+1}\}|}{|B_{j+1}|}\bigg)^\frac{1}{\kappa}
\leq\bigg(\fint_{B_{j+1}}w_j^{\kappa p}\psi_j^{\kappa p}\,dx\bigg)^\frac{1}{\kappa}\leq c \bigg(\fint_{B_{j}}w_j^{\kappa p}\psi_j^{\kappa p}\,dx\bigg)^\frac{1}{\kappa}\\
&\leq c r^{p}\fint_{B_j}|\nabla(w_j \psi_j)|^p\,dx\leq c 2^{j(n+ps+p)} k_j^{p}\frac{|B_j\cap\{u<k_j\}|}{|B_j|}.
\end{split}
\end{equation}
Let
$$
Y_j=\frac{|B_j\cap\{u<k_j\}|}{|B_j|},
\quad j=0,1,2,\dots.
$$
From \eqref{Poincareless} we have
\begin{equation}\label{Iteapp}
Y_{j+1}\leq c_2\, 2^{j(n+2p+ps)\kappa} Y_j^{\kappa},
\quad j=0,1,2,\dots,
\end{equation}
for some constant $c_2=c_2(n,p,s,\Lambda)$.
We choose
$c_0=c_2$, $b=2^{(n+2p+ps)\kappa}>1$ and $\beta=\kappa-1>0$
in Lemma \ref{iteration}. By \eqref{assume}, we have
$$
k_0=\frac{3}{2}\delta k\leq 2\delta k-\frac{1}{2}\Big(\frac{r}{R}\Big)^\frac{p}{p-1}\Tail(u_{-};x_0,R)-\epsilon.
$$
By \eqref{expanstep1} we have
\begin{equation}\label{iteini}
Y_0\leq
\frac{\big|B_{6r}(x_0)\cap\big\{u\leq 2\delta k-\big(\frac{r}{R}\big)^\frac{p}{p-1}\Tail(u_{-};x_0,R)-\epsilon\big\}\big|}
{|B_{6r}(x_0)|}
\leq\frac{c_1}{\tau\log\frac{1}{2\delta}}
\end{equation}
for some constant $c_1=c_1(n,p,s,\Lambda)$ and for every $\delta\in(0,\frac{1}{4})$.
Using \eqref{iteini} we choose $\delta=\delta(n,p,s,\Lambda,\tau)\in(0,\frac{1}{4})$ such that
$$
0<\delta=\frac{1}{4}\exp\bigg(-\frac{c_1 c_0^\frac{1}{\beta} b^\frac{1}{\beta^2}}{\tau}\bigg)<\frac{1}{4},
$$
so that the estimate $Y_0\leq c_0^{-\frac{1}{\beta}}b^{-\frac{1}{\beta^2}}$ holds.
By Lemma \ref{iteration} we conclude that $Y_j\to0$ as $j\to\infty$.
Therefore, we have
\[
\mathop{\rm {ess\,inf}}\limits_{B_{4r}(x_0)}\,u\geq \delta k,
\]
which gives \eqref{expanaux} and so \eqref{expan} holds.
\end{proof}
\section{Harnack inequalities}
Proceeding similarly as in the proof of \cite[Lemma 4.1]{KuusiHarnack}, along with an application of Lemma \ref{DGLemma}, we obtain the following preliminary version of
the weak Harnack inequality, compared to Theorem \ref{thm3}.
\begin{Lemma}\label{WeakHarnacklemma}
Let $u$ be a weak supersolution of \eqref{maineqn} such that $u\geq 0$ in $B_R(x_0)\subset\Omega$.
There exist constants $\eta=\eta(n,p,s,\Lambda)\in(0,1)$ and $c=c(n,p,s,\Lambda)\geq 1$ such that
\begin{equation}\label{wk}
\bigg(\fint_{B_r(x_0)}u^{\eta}\,dx\bigg)^\frac{1}{\eta}
\leq c\mathop{\rm {ess\,inf}}\limits_{B_r(x_0)}\,u+c\Big(\frac{r}{R}\Big)^\frac{p}{p-1}\Tail(u_{-};x_0,R),
\end{equation}
whenever $B_r(x_0)\subset B_R(x_0)$ with $r\in(0,1]$.
Here $\Tail(\cdot)$ is defined in \eqref{loctail}.
\end{Lemma}
The following Harnack inequality follows with a similar argument as in the proof of \cite[Theorem 1.1]{KuusiHarnack}.
For convenience of the reader, we give a proof here in the mixed case. To this end, the following iteration lemma from \cite[Lemma 1.1]{GGacta} will be useful for us.
\begin{Lemma}\label{iteration1}
Let $0\leq T_0\leq t\leq T_1$ and assume that $f:[T_1,T_2]\to[0,\infty)$ is a nonnegative bounded function. Suppose that for $T_0\leq t<s\leq T_1$, we have
\begin{equation}\label{itt}
f(t)\leq A(s-t)^{-\alpha} +B +\theta f(s),
\end{equation}
where $A,B,\alpha,\theta$ are nonegative constants and $\theta<1$.
Then there exists a constant $c=c(\alpha,\theta)$ such that for every $\rho,R$ and $T_0\leq\rho<R\leq T_1$, we have
\begin{equation}\label{itt1}
f(\rho)\leq c(A(R-\rho)^{-\alpha}+B).
\end{equation}
\end{Lemma}
\begin{Theorem}\label{thm2}(\textbf{Harnack inequality}).
Let $u$ be a weak solution of \eqref{maineqn} such that $u\geq 0$ in $B_R(x_0)\subset\Omega$.
There exists a positive constant $c=c(n,p,s,\Lambda)$ such that
\begin{equation}\label{Harnackest}
\mathop{\rm {ess\,sup}}\limits_{B_{\frac{r}{2}}(x_0)}\,u\leq c\mathop{\rm {ess\,inf}}\limits_{B_r(x_0)}\,u+c\Big(\frac{r}{R}\Big)^\frac{p}{p-1}\Tail(u_{-};x_0,R),
\end{equation}
whenever $B_r(x_0)\subset B_\frac{R}{2}(x_0)$ and $r\in(0,1]$.
Here $\Tail(\cdot)$ is given by \eqref{loctail}.
\end{Theorem}
\begin{proof}
Let $0<\rho<r$. Then by Lemma \ref{Solsubsup} and Theorem \ref{thm1}, for every $\delta\in(0,1]$, there exists a positive constant $c=c(n,p,s,\Lambda)$ such that
\begin{equation}\label{h1eqn}
\mathop{\rm {ess\,sup}}\limits_{B_{\frac{\rho}{2}}(x_0)}u\leq \delta\mathrm{Tail}(u_{+};x_0,\frac{\rho}{2})+c\delta^{-\gamma}\bigg(\fint_{B_{\rho}(x_0)}u_{+}^p\,dx\bigg)^\frac{1}{p},
\end{equation}
where $\gamma=\frac{(p-1)\kappa}{p(\kappa-1)}$ and $\kappa$ as in \eqref{kappa}.
By Lemma \ref{Tail} and \eqref{h1eqn}, we obtain
\begin{equation}\label{h2eqn}
\mathop{\rm {ess\,sup}}\limits_{B_{\frac{\rho}{2}}(x_0)}u
\leq c\delta\Big(\mathop{\rm {ess\,sup}}\limits_{B_{\rho}(x_0)}u+\Big(\frac{\rho}{R}\Big)^\frac{p}{p-1}\mathrm{Tail}(u_{-};x_0,R)\Big)+c\delta^{-\gamma}\bigg(\fint_{B_{\rho}(x_0)}u_{+}^p\,dx\bigg)^\frac{1}{p},
\end{equation}
for some constant $c=c(n,p,s,\Lambda)$. Let $\frac{1}{2}\leq\sigma^{'}<\sigma\leq 1$ and $\rho=(\sigma-\sigma^{'})r$.
Using a covering argument, it follows that
\begin{equation}\label{h2eqn1}
\begin{split}
\mathop{\rm {ess\,sup}}\limits_{B_{\sigma^{'}r}(x_0)}u
&\leq c\frac{\delta^{-\gamma}}{(\sigma-\sigma^{'})^{\frac{n}{p}}}\bigg(\fint_{B_{\sigma r}}u^p\,dx\bigg)^\frac{1}{p}
+c\delta\mathop{\rm {ess\,sup}}\limits_{B_{\sigma r}(x_0)}\,u+c\delta\Big(\frac{r}{R}\Big)^\frac{p}{p-1}\mathrm{Tail}(u_{-};x_0,R)\\
&\leq c\frac{\delta^{-\gamma}}{(\sigma-\sigma^{'})^{\frac{n}{p}}}(\mathop{\rm {ess\,sup}}\limits_{B_{\sigma r}(x_0)}\,u)^\frac{p-t}{p}
\bigg(\fint_{B_{\sigma r}(x_0)}u^t\,dx\bigg)^\frac{1}{p}+c\delta\mathop{\rm {ess\,sup}}\limits_{B_{\sigma r}(x_0)}\,u\\
&\qquad+c\delta\Big(\frac{r}{R}\Big)^\frac{p}{p-1}\mathrm{Tail}(u_{-};x_0,R)
\end{split}
\end{equation}
for every $t\in(0,p)$ with a constant $c=c(n,p,s\Lambda)$.
Young's inequality with exponents $\frac{p}{t}$ and $\frac{p}{p-t}$ and choosing $\delta=\frac{1}{4c}$ in \eqref{h2eqn1} implies that
\begin{equation}\label{h2eqn2}
\mathop{\rm {ess\,sup}}\limits_{B_{\sigma^{'}r}(x_0)}u\leq \frac{1}{2}\mathop{\rm {ess\,sup}}\limits_{B_{\sigma r}(x_0)}u
+\frac{c}{(\sigma-\sigma^{'})^\frac{n}{t}}\bigg(\fint_{B_{r}(x_0)}u^t\,dx\bigg)^\frac{1}{t}+c\Big(\frac{r}{R}\Big)^\frac{p}{p-1}\mathrm{Tail}(u_{-};x_0,R)
\end{equation}
for every $t\in(0,p)$ with a constant $c=c(n,p,s,t,\Lambda)$.
Using Lemma \ref{iteration1} in \eqref{h2eqn2}, we have
\begin{equation}\label{h3eqn}
\begin{split}
\mathop{\rm {ess\,sup}}\limits_{B_{\frac{r}{2}}(x_0)}\,u
&\leq c\bigg(\fint_{B_{r}(x_0)}u^t\,dx\bigg)^\frac{1}{t}+c\Big(\frac{r}{R}\Big)^\frac{p}{p-1}\mathrm{Tail}(u_{-};x_0,R),
\end{split}
\end{equation}
for every $t\in(0,p)$ with a positive constant $c=c(n,p,s,t,\Lambda)$. Combining the above estimate \eqref{h3eqn} with Lemma \ref{WeakHarnacklemma} and choosing $t=\eta\in(0,1)$, the result follows.
\end{proof}
We have the following weak Harnack inequality for supersolutions of \eqref{maineqn}.
\begin{Theorem}\label{thm3}(\textbf{Weak Harnack inequality}).
Let $u$ be a weak supersolution of \eqref{maineqn} such that $u\geq 0$ in $B_R(x_0)\subset\Omega$.
There exists a positive constant $c=c(n,p,s,\Lambda)$ such that
\begin{equation}\label{wkest}
\bigg(\fint_{B_{\frac{r}{2}}(x_0)}u^{l}\,dx\bigg)^\frac{1}{l}
\leq c\mathop{\rm {ess\,inf}}\limits_{B_{r}(x_0)}\,u+c\Big(\frac{r}{R}\Big)^\frac{p}{p-1}\Tail(u_{-};x_0,R),
\end{equation}
whenever $B_r(x_0)\subset B_{\frac{R}{2}}(x_0)$, $r\in(0,1]$ and $0<l<\kappa (p-1)$.
Here $\kappa$ and $\Tail(\cdot)$ are given by \eqref{kappa} and \eqref{loctail}, respectively.
\end{Theorem}
\begin{proof}
We prove the result for $1<p<n$. For $p\geq n$, the result follows in a similar way. Let $r\in(0,1]$, $\frac{1}{2}<\tau'<\tau\leq\frac{3}{4}$ and we choose $\psi\in C_c^{\infty}(B_{\tau r}(x_0))$ such that $0\leq\psi\leq 1\text{ in }B_{\tau r}(x_0)$, $\psi=1$ in $B_{\tau' r}(x_0)$ and $|\nabla\psi|\leq\frac{4}{(\tau-\tau')r}$.
For $d>0$ and $q\in(1,p)$, we set
\[
v=u+d
\quad\text{and}\quad
w=(u+d)^\frac{p-q}{p}.
\]
Noting $r\in(0,1]$, the property of $\psi$ above and the proof of the estimate $(5.11)$ in \cite[Page 1834]{KuusiHarnack}, there exists a constant $c=c(n,p,s,\Lambda)$ such that
\begin{equation}\label{revestlocal}
I_1=\int_{B_r(x_0)}w^p|\nabla\psi|^p\,dx\leq\frac{c r^{-p}}{(\tau-\tau')^p}\int_{B_{\tau r}(x_0)}w^p\,dx,
\end{equation}
\begin{equation}\label{revestmax}
I_2=\int_{B_r(x_0)}\int_{B_r(x_0)}\max\{w(x),w(y)\}^p|\psi(x)-\psi(y)|^p\,d\mu
\leq\frac{c r^{-p}}{(\tau-\tau')^p}\int_{B_{\tau r}(x_0)}w^p\,dx.
\end{equation}
Assume that $\Tail(u_{-};x_0,R)$ is positive. Then for any $\epsilon>0$ and $r\in(0,1]$ choosing
$$
d=\frac{1}{2}\Big(\frac{r}{R}\Big)^\frac{p}{p-1}\,\Tail(u_{-};x_0,R)+\epsilon>0,
$$
and noting that
\begin{equation}\label{sup}
\mathop{\rm {ess\,sup}}\limits_{z\in\supp\psi}\int_{\mathbb{R}^n\setminus B_r(x_0)}K(z,y)\,dy\leq c(n,p,s,\Lambda)r^{-p},
\end{equation}
we obtain
\begin{equation}\label{revestnonlocal}
\begin{split}
I_3&=\bigg(\mathop{\rm {ess\,sup}}\limits_{z\in\supp\psi}\int_{\mathbb{R}^n\setminus B_r(x_0)}K(z,y)\,dy+d^{1-p}R^{-p}\Tail(u_{-};x_0,R)^{p-1}\bigg)
\int_{B_r(x_0)}w^p\psi^p\,dx\\
&\leq\frac{c(n,p,s,\Lambda)r^{-p}}{(\tau-\tau')^p}\int_{B_{\tau r}(x_0)}w^p\,dx.
\end{split}
\end{equation}
If $\Tail(u_{-};x_0,R)=0$, we can choose an arbitrary $d=\epsilon>0$ and again using \eqref{sup} the estimate in \eqref{revestnonlocal} follows.
Now using Sobolev inequality in \eqref{e.friedrich} and the fact that $\psi\equiv 1$ in $B_{\tau' r}$, $r\in(0,1]$, along with Lemma \ref{energyforrev} and the estimates \eqref{revestlocal}, \eqref{revestmax} and \eqref{revestnonlocal}, we have for $p^*=\frac{np}{n-p}$ with $1<p<n$,
\begin{equation}\label{revSobloc}
\begin{split}
\bigg(\fint_{B_{\tau' r}(x_0)} v^\frac{n(p-q)}{n-p}\,dx\bigg)^\frac{p}{p^*}
&=\bigg(\fint_{B_{\tau' r}(x_0)}w^{p^*}\,dx\bigg)^\frac{p}{p^*}
\leq\bigg(\fint_{B_{\tau r}(x_0)}|w\psi|^{p^*}\,dx\bigg)^\frac{p}{p^*}\\
&\leq(\tau r)^{p-n}\int_{B_{\tau r}(x_0)}|\nabla\,(w\psi)|^p\,dx
\leq\frac{c}{(\tau-\tau')^p}\fint_{B_{\tau r}(x_0)}w^p\,dx,
\end{split}
\end{equation}
with $c=c(n,p,s,q,\Lambda)$. Using $q\in(1,p)$ and the Moser iteration technique as in \cite[Theorem 8.18]{GT2001} and \cite[Theorem 1.2]{Tru2}, we get
\begin{equation}\label{rev1}
\bigg(\fint_{B_{\frac{r}{2}}(x_0)} v^l\,dx\bigg)^\frac{1}{l}
\leq c\bigg(\fint_{B_\frac{3r}{4}(x_0)} v^{l'}\,dx\bigg)^\frac{1}{l'},
\quad 0<l'<l<\frac{n(p-1)}{n-p}.
\end{equation}
Let $\eta\in(0,1)$ be given by Lemma \ref{WeakHarnacklemma} and then choosing $l'=\eta\in(0,1)$ and observing that
$$
\bigg(\fint_{B_{\frac{r}{2}}(x_0)}{u}^l\,dx\bigg)^\frac{1}{l}
\leq\bigg(\fint_{B_{\frac{r}{2}}(x_0)} v^l\,dx\bigg)^\frac{1}{l},
$$
we obtain from \eqref{rev1}
\begin{equation}\label{weakHarnackfinalest}
\bigg(\fint_{B_{\frac{r}{2}}(x_0)}u^l\,dx\bigg)^\frac{1}{l}
\leq c\mathop{\rm {ess\,inf}}\limits_{B_r(x_0)} v+c\Big(\frac{r}{R}\Big)^\frac{p}{p-1}\Tail(u_{-};x_0,R),
\end{equation}
for all $0<l<\frac{n(p-1)}{n-p}$.
For any $\epsilon>0$, choosing
$$
d=\frac{1}{2}\Big(\frac{r}{R}\Big)^\frac{p}{p-1}\Tail(u_{-};x_0,R)+\epsilon,
$$
in \eqref{weakHarnackfinalest} and letting $\epsilon\to 0$, the result follows.
\end{proof}
\section{Semicontinuity}
Before stating our results on pointwise behavior, we discuss a result from Liao \cite{Liao}.
Let $u$ be a measurable function that is locally essentially bounded below in $\Omega$.
Let $\rho\in(0,1]$ be such that $B_\rho(y)\subset\Omega$. Assume that $a,c\in(0,1)$, $M>0$ and
$\mu_-\leq\mathop{\rm {ess\,inf}}\limits_{B_\rho(y)}u$.
Following \cite{Liao}, we say that $u$ satisfies the property $(\mathcal{D})$, if there exists a constant $\tau\in(0,1)$ depending on $a,M,\mu_-$ and other data
(may depend on the partial differential equation and will be made precise in Lemma \ref{DGL}), but independent of $\rho$, such that
$$
|\{u\leq\mu_-+M\}\cap B_\rho(y)|\leq\tau|B_\rho(y)|,
$$
implies that $u\geq\mu_-+aM$ almost everywhere in $B_{c\rho}(y)$.
Moreover, for $u\in L^1_{\loc}(\Omega)$, we denote the set of Lebesgue points of $u$ by
$$
\mathcal{F}=\bigg\{x\in\Omega:|u(x)|<\infty,\,\lim_{r\to 0}\fint_{B_r(x)}|u(x)-u(y)|\,dy=0\bigg\}.
$$
Note that, by the Lebesgue differentiation theorem, $|\mathcal{F}|=|\Omega|$.
The following result follows from \cite[Theorem 2.1]{Liao}.
\begin{Lemma}\label{lscthm}
Let $u$ be a measurable function that is locally integrable and locally essentially bounded below in $\Omega$. Assume that $u$ satisfies the property $(\mathcal{D})$.
Then $u(x)=u_*(x)$ for every $x\in\mathcal{F}$, where
\[
u_*(x)=\lim_{r\to 0}\mathop{\rm {ess\,inf}}\limits_{y\in B_r(x)}u(y).
\]
In particular, $u_*$ is a lower semicontinuous representative of $u$ in $\Omega$.
\end{Lemma}
Since $u$ is assumed to be locally essentially bounded below, the lower semicontinuous regularization $u_*(x)$ is well defined at every point $x\in\Omega$.
Our final regularity results stated are consequences of Lemma \ref{lscthm} and Lemma \ref{DGL} below.
\begin{Theorem}\label{lscthm1}(\textbf{Lower semicontinuity}).
Let $u$ be a weak supersolution of \eqref{maineqn}.
Then
\[
u(x)=u_*(x)=\lim_{r\to 0}\mathop{\rm {ess\,inf}}\limits_{y\in B_r(x)}u(y)
\]
for every $x\in\mathcal{F}$.
In particular, $u_*$ is a lower semicontinuous representative of $u$ in $\Omega$.
\end{Theorem}
As a Corollary Theorem \ref{lscthm1}, we have the following result.
\begin{Corollary}\label{uscthm}(\textbf{Upper semicontinuity}).
Let $u$ be a weak subsolution of \eqref{maineqn}.
Then
\[
u(x)=u^*(x)=\lim_{r\to 0}\mathop{\rm {ess\,sup}}\limits_{y\in B_r(x)}u(y)
\]
for every $x\in\mathcal{F}$.
In particular, $u^*$ is an upper semicontinuous representative of $u$ in $\Omega$.
\end{Corollary}
We prove a De Giorgi type lemma for weak supersolutions of \eqref{maineqn}.
\begin{Lemma}\label{DGL}
Let $u$ be a weak supersolution of \eqref{maineqn}.
Let $M>0$, $a\in(0,1)$, $B_r(x_0)\subset\Omega$ with $r\in(0,1]$ and
$\mu_-\leq\mathop{\rm {ess\,inf}}\limits_{B_r(x_0)}\,u,\,\lambda_-\leq\mathop{\rm {ess\,inf}}\limits_{{I\!\!R}^n}\,u$.
There exists a constant $\tau=\tau(n,p,s,\Lambda,a,M,\mu_-,\lambda_-)\in(0,1)$ such that if
$$
|\{u\leq\mu_-+M\}\cap B_r(x_0)|\leq\tau|B_r(x_0)|,
$$
then $u\geq\mu_-+aM$ almost everywhere in $B_{\frac{3r}{4}(x_0)}$.
\end{Lemma}
\begin{proof}
Without loss of generality, we may assume that $x_0=0$. For $j=0,1,2,\dots$, we denote
\begin{equation}\label{parameter1}
\begin{split}
k_j&=\mu_{-}+aM+\frac{(1-a)M}{2^j},
\quad
\bar{k}_j=\frac{k_j+k_{j+1}}{2},\\
\quad
r_j&=\frac{3r}{4}+\frac{r}{2^{j+2}},
\quad\bar{r}_j=\frac{r_j+r_{j+1}}{2},
\end{split}
\end{equation}
$B_j=B_{r_j}(0)$, $\bar{B}_j=B_{\bar{r}_j}(0)$, $w_j=(k_j-u)_{+}$ and $\bar{w}_j=(\bar{k}_j-u)_{+}$.
We observe that $B_{j+1}\subset\bar{B}_j\subset B_j$, $\bar{k}_j<k_j$ and hence $\bar{w}_j\leq w_j$.
Let $(\psi_{j})_{j=0}^{\infty}\subset C_{c}^{\infty}(\bar{B}_j)$ be a sequence of cutoff functions satisfying $0 \leq \psi_{j} \leq 1$ in $B_j$, $\psi_{j}= 1$ in $B_{j+1}$,
$|\nabla\psi_j|\leq\frac{2^{j+3}}{r}$.
By applying Lemma \ref{energyest} to $w_j$, we obtain
\begin{equation}\label{energyapp2}
\int_{B_j}|\nabla w_j|^p \psi_j^p\,dx\leq
C(n,p,s,\Lambda)(I_1+I_2+I_3),
\end{equation}
where
\[
I_1=\int_{B_{j}}\int_{B_{j}}{\max\{w_j(x),w_j(y)\}^p|\psi_{j}(x)-\psi_{j}(y)|^p}\,d\mu,
\quad I_2=\int_{B_j}w_j^p|\nabla\psi_j|^p\,dx
\]
and
\[
I_3=\mathop{\rm {ess\,sup}}\limits_{x\in\supp\psi_j}\int_{{\mathbb{R}^n\setminus B_{j}}}{\frac{w_j(y)^{p-1}}{|x-y|^{n+ps}}}\,dy
\cdot\int_{B_{j}}w_j\psi_j^p\,dx.
\]
Since $u\geq \lambda_-$ in ${I\!\!R}^n$, noting the definition of $k_j$ from above, we have
\begin{equation}\label{nelsc}
w_j=(k_j-u)_+\leq (M+\mu_--\lambda_-)_+=L\text{ in }{I\!\!R}^n.
\end{equation}
Let $A_j=\{u<k_j\}\cap B_j$. We estimate the terms $I_j$, for $j=1,2,3$, separately.\\
\textbf{Estimate of $I_1$:} Using $w_j\leq L$ from \eqref{nelsc} along with $\frac{r}{2}<r_j<r$, $r\in(0,1]$ and the properties of $\psi_j$, we obtain
\begin{equation}\label{estI1}
I_1=\int_{B_{j}}\int_{B_{j}}{\max\{w_j(x),w_j(y)\}^p|\psi_{j}(x)-\psi_{j}(y)|^p}\,d\mu
\leq C\frac{2^{jp}}{r^{p}}L^p|A_j|,
\end{equation}
with $C=C(n,p,s,\Lambda)$.\\
\textbf{Estimate of $I_2$:} Using the properties of $\psi_j$ and the fact that $w_j\leq L$ from \eqref{nelsc}, we have
\begin{equation}\label{estI2}
I_2=\int_{B_{j}}w_j^p|\nabla \psi_j|^p\,dx\\
\leq C\frac{2^{jp}}{r^{p}}L^p|A_j|,
\end{equation}
with $C=C(n,p,s,\Lambda)$.\\
\textbf{Estimate of $I_3$:} For every $x\in\supp\psi_{j}$ and every $y\in\mathbb{R}^n\setminus B_j$, we observe that
\begin{equation}\label{I2}
\frac{1}{|x-y|}=\frac{1}{|y|}\frac{|x-(x-y)|}{|x-y|}\leq\frac{1}{|y|}(1+2^{j+5})\leq\frac{2^{j+6}}{|y|}.
\end{equation}
Then using $r_j>\frac{r}{2}$, $w_j\leq L$, $0\leq\psi_j \leq 1$, we obtain
\begin{equation}\label{estI3final}
I_3=\mathop{\rm {ess\,sup}}\limits_{x\in\supp\psi_j}\int_{{\mathbb{R}^n\setminus B_{j}}}{\frac{w_j(y)^{p-1}}{|x-y|^{n+ps}}}\,dy
\cdot\int_{B_{j}}w_j\psi_j^p\,dx
\leq C(n,p,s)\frac{2^{j(n+ps)}}{r^{p}}L^p|A_j|,
\end{equation}
for every $r\in(0,1]$.
By using \eqref{estI1}, \eqref{estI2} and \eqref{estI3final} in \eqref{energyapp2}, we have
\begin{equation}\label{energyapp4}
\begin{split}
\int_{B_j}|\nabla w_j|^p \psi_j^p\,dx\leq C\frac{2^{j(n+ps+p)}}{r^{p}}L^p|A_j|,
\end{split}
\end{equation}
with $C=C(n,p,s,\Lambda)$.
Noting that $B_{j+1}\subset\bar{B}_j\subset B_j$, $\bar{w}_j\leq w_j$ and using the Sobolev inequality in \eqref{e.friedrich}, we obtain
\begin{equation}\label{dist}
\begin{split}
\frac{(1-a)M}{2^{j+2}}|A_{j+1}|&=\int_{A_{j+1}}(\bar{k}_j-k_{j+1})\,dx
\leq\int_{B_{j+1}}\bar{w}_j\,dx\\
&\leq\int_{B_{j+1}}w_j\,dx
\leq|A_j|^{1-\frac{1}{p\kappa}}\Big(\int_{B_j}w_j^{p\kappa}\psi_j^{p\kappa}\,dx\Big)^\frac{1}{p\kappa}\\
&\leq C r^{1+\frac{n}{p\kappa}-\frac{n}{p}}|A_j|^{1-\frac{1}{p\kappa}}\Big(\int_{B_j}|\nabla(w_j\psi_j)|^p\,dx\Big)^\frac{1}{p},
\end{split}
\end{equation}
with $C=C(n,p,s)$ and $\kappa$ as given in \eqref{kappa}.
By using \eqref{energyapp4} together with the fact that $w_j\leq L$ and the properties of $\psi_j$ in \eqref{dist}, we get
\begin{equation}\label{dist8}
\begin{split}
\frac{(1-a)M}{2^{j+2}}|A_{j+1}|\leq C(n,p,s,\Lambda) L r^{\frac{n}{p}(\frac{1}{\kappa}-1)}2^\frac{j(n+ps+p)}{p} |A_j|^{1-\frac{1}{p\kappa}+\frac{1}{p}}.
\end{split}
\end{equation}
Hence, we obtain from \eqref{dist8} that
\begin{equation}\label{dist7}
\begin{split}
|A_{j+1}|&\leq \frac{C(n,p,s,\Lambda)L r^{\frac{n}{p}(\frac{1}{\kappa}-1)}}{(1-a)M}2^{j(2+\frac{n+ps}{p})}|A_j|^{1-\frac{1}{p\kappa}+\frac{1}{p}}.
\end{split}
\end{equation}
By dividing both sides of \eqref{dist7} with $|B_{j+1}|$ and noting that $|B_j|<2^{n}|B_{j+1}|$ together by $r_j<r$, we obtain
\begin{equation}\label{dist4}
\begin{split}
Y_{j+1}&\leq\frac{C(n,p,s,\Lambda)L}{(1-a)M}2^{j(2+\frac{n+ps}{p})}Y_j^{1+\frac{1}{p}(1-\frac{1}{\kappa})},
\end{split}
\end{equation}
where we denoted $Y_j=\frac{|A_j|}{|B_j|}$, $j=0,1,2,\dots$. By choosing
\begin{align*}
c_0&=\frac{C(n,p,s,\Lambda)L}{(1-a)M},
\quad
b=2^{(2+\frac{n+ps}{p})},
\quad
\beta=\frac{1}{p}\Big(1-\frac{1}{\kappa}\Big),
\quad
\tau=c_0^{-\frac{1}{\beta}}b^{-\frac{1}{\beta^2}}\in(0,1)
\end{align*}
in Lemma \ref{iteration} gives $Y_j\to0$ as $j\to\infty$, if $Y_0\leq\tau$.
This implies that $u\geq\mu_{-}+aM$ almost everywhere in $B_{\frac{3r}{4}}(0)$.
\end{proof}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 6,113 |
Q: Handling TCP Packet Fragmentation Well it seems very ambiguous to me how Packet Fragmentation occurs as it never happens with my Local tests and i don't know what to do in order to handle a fragmented packet before notifying my application that i actually received some info.
Here is the way i receive from a connected Client Socket
var asynchronousState = (AsynchronousState) ar.AsyncState; // AsynchronousState is an entity that Holds each connected socket's Buffer, IPEndPoint and so on...
try {
Socket socket = asynchronousState.Socket;
int length = socket.EndReceive(ar);
if (0 != length) {
if (null != ClientReceive) {
var bytes = new byte[length];
Array.Copy(asynchronousState.Buffer, bytes, length);
ClientReceive(asynchronousState, bytes);
Array.Clear(asynchronousState.Buffer, 0, asynchronousState.Buffer.Length);
}
if (socket.Connected) {
socket.BeginReceive(asynchronousState.Buffer, 0, asynchronousState.Buffer.Length, SocketFlags.None, HandleAsyncReceive, asynchronousState);
return;
}
}
DisposeSocket(asynchronousState);
}
catch (SocketException exception) {
if (exception.SocketErrorCode != SocketError.Disconnecting &&
exception.SocketErrorCode != SocketError.NotConnected &&
exception.SocketErrorCode != SocketError.ConnectionReset &&
exception.SocketErrorCode != SocketError.ConnectionAborted &&
exception.SocketErrorCode != SocketError.Shutdown) {
Console.WriteLine(exception);
Core.Logger.Log(exception);
}
DisposeSocket(asynchronousState);
}
How i construct my Packets
public class ExchangeMessage : PacketStructure //Packet Structure is a custom builder similar to `BinaryWrite/Reader` : This returns a byte[] or builds from a byte[]
{
public int Length;
public int Type;
public byte[] PublicKey
{
get { return ReadArray(140, 4); }
set { WriteArray(value, 4); }
}
public ExchangeMessage(byte[] receivedPacket) : base(receivedPacket) {}
public ExchangeMessage(int length, int type) : base(length, type)
{
Length = length;
Type = type;
}
}
Now that i invoke the ClientReceive event handler whenever i receive something expecting that the application will be able to deserialize or rebuild my Packet back. What if the packet was fragmented ?
I am very confused as by researching the topic i find a lot of different opinions among them was something i am skeptical about that says If you are building your Packet by yourself, it will not be fragmented!
A: The Sockets API won't deliver you packet fragments, only complete packets. The TCP/IP stack will buffer received fragments until it has a complete packet, and if some fragments are lost the whole packet will be discarded and have to be retransmitted in its entirety.
This is one reason that Path MTU detection improves performance -- it prevents mid-channel fragmentation and the corresponding increase in packet error rates. Sending smaller packets will result in just as many errors, but entire lost packets can be dealt with via selective-ACK which is much more efficient than discarding received fragments of a partial packet.
If you used a lower-level networking API you might be able to see packet fragments.
Note that TCP packets do not correspond 1:1 to send() calls. The Nagle algorithm can combine multiple writes into a single packet, and a single write which is larger than the path MTU will generate multiple packets. I think this is what Bob was alluding to in his comment.
Fragmentation, packet loss, and retransmission is all handled inside TCP/IP. Your application doesn't need to worry about it. Your application should treat a TCP socket as a stream of bytes.
The bytes you put in come out in the same order. How long it takes and how many come out at once is not guaranteed and is beyond your control.
Since you want to treat your data as having structure, not mere bytes, you will have to add that structure yourself. Length prefixes and record separators are both popular ways of doing this.
For example, HTTP uses record separators for the command/response and metadata. The HTTP command and all headers are separated by \r\n. The disadvantage is that if separators appear in the data they need to be escaped. This example is borrowed from wikipedia.
HTTP/1.1 200 OK
Date: Mon, 23 May 2005 22:38:34 GMT
Server: Apache/1.3.3.7 (Unix) (Red-Hat/Linux)
Last-Modified: Wed, 08 Jan 2003 23:11:55 GMT
ETag: "3f80f-1b6-3e1cb03b"
Content-Type: text/html; charset=UTF-8
Content-Length: 131
Accept-Ranges: bytes
Connection: close
<html>
<head>
<title>An Example Page</title>
</head>
<body>
Hello World, this is a very simple HTML document.
</body>
</html>
Length-prefixing is also used. The Content-Length header gives the length of the payload. This allows any byte to appear in the payload.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 9,299 |
\section{Introduction}\label{s.intro}
In this paper we deal with the problem of finding
useful criteria for the strong convergence of a bounded operator semigroup $T$
on a Banach space $E$, with a special view on the asymptotics of
certain semigroups of positive operators on Banach lattices.
This problem is
not at all new, but
we refrain from even trying to give a list of
relevant literature at this point. (However, cf.{} Section
\ref{s.classical} below.) Rather,
let us stress some features that distinguish our approach from most
others.
Classically, asymptotics of operator semigroups focusses on {\em
strongly continuous} one-parameter semigroups $T=(T_t)_{t\in
[0,\infty)}$. However, there are important instances
of operator semigroups which lack strong continuity, e.g.,
the heat semigroup on the space of
bounded continuous functions on $\mathbb{R}$ or on the space of finite measures over
$\mathbb{R}$ or, in an abstract context, dual semigroups of
$C_0$-semigroups on non-reflexive spaces. (See the recent paper \cite{KunzeNL}
for a more involved concrete example.) Hence, there is a need
for results on asymptotics beyond $C_0$-semigroups.
Secondly, besides the ``continuous time'' case just mentioned, there
is an even more fundamental interest in ``discrete time'', i.e., in the asymptotics of the powers
$T^n$ of a single operator $T$. From a systematic point of view, it is
desirable to try to cover both cases at the same time as far as
possible. This is the reason why we
consider general semigroup representations $(T_s)_{s\in S}$---where
$(S,+)$ is an Abelian semigroup with zero element $0$---%
as bounded operators on a Banach space
without any further topological assumptions (see Section \ref{s.rep}).
It may not come as a surprise that non-trivial results about
asymptotics can be obtained---even in such a general setting---by employing
the so-called Jacobs--deLeeuw--Glicks\-berg (JdLG) theory for compact (Abelian)
semitopological semigroups. In fact, the role
of the JdLG-theory for asymptotics is well-established. Usually,
it is applied to the semigroup
\[ \calT := \mathrm{cl}\{ T_s \suchthat s\in S\}
\]
(closure in the strong or weak operator topology) and hence rests on
a ``global'' compactness requirement for the whole semigroup. This
is appropriate for the abovementioned ``classical'' cases
(powers of a single operator, $C_0$-semigroups)
because there the strong compactness of $\calT$ is necessary for the
convergence of $T$. However, typical examples
of non-continuous shift semigroups (left shift on $\Ell{\infty}(0,1)$
or $\mathrm{c}_{\mathrm{0}}(\mathbb{R}_+)$) show that such semigroups may converge strongly to
$0$ without being relatively strongly compact. (The left shift
semigroup on $\mathrm{c}_{\mathrm{0}}(\mathbb{R}_0)$ is not even eventually relatively strongly compact.)
In order to cover also these more general situations, we introduce the set
\[ \calT_\infty := \bigcap_{t\in S} \mathrm{cl}\{ T_{s{+}t} \suchthat s\in S\}
\]
which we call the {\emdf semigroup at infinity}. It turns out that
$\calT_\infty$ is a good replacement for $\calT$ {\em if}
$\calT_\infty$ is strongly compact and not empty. In particular, its
minimal projection, $P_\infty$, satisfies
\[ P_\infty x = 0 \quad \Leftrightarrow\quad \lim_{s\in S} T_s x = 0 \qquad (x\in
E)
\]
(see Theorem \ref{rep.t.Pinfty}).
Not surprisingly, in the mentioned ``classical''
cases the condition that $\calT_\infty$ be strongly compact and not empty
is actually equivalent to relative strong
compactness of the original semigroup (cf. Remark \ref{rep.r.sac}),
and hence the use of $\calT_\infty$ is then---in
some sense---unnecessary. In a general appoach going beyond
these classical cases, however, it is our means of choice.
\medskip
In a next step (Section \ref{s.abs}), we describe a convenient
set-up that warrants the crucial property (i.e.: $\calT_\infty$ is
non-empty and compact). To this end,
the new notion of {\em quasi-compactness
relative to a subspace} is introduced. This condition generalizes the
traditional notion of quasi-compactness of a semigroup and plays a
central role in our first main result (Theorem \ref{abs.t.main}).
\medskip
After these completely general considerations, and from then on until
the end of the paper, we confine our attention to
{\em positive} operator semigroups on Banach {\em lattices}.
Our second main result,
Theorem \ref{psg.t.main}, appears to be a mere
instantiation of Theorem \ref{abs.t.main} in such a setting.
However, the theorem gains its
significance from the fact that the
required quasi-compactness condition is intimately linked to
the well-known property of {\em AM-compactness} of positive
operators which, in turn, occurs frequently when dealing with
``concrete'' positive semigroups arising in evolution equations and
stochastics (see Appendix \ref{app.int}).
\medskip
The main thrust of Theorem \ref{psg.t.main} is that it reduces
the study
of the asymptotic properties of certain positive semigroups
to the following special case: $E$ is an atomic
Banach lattice and $T$ embeds into a strongly compact group
of positive invertible operators thereon. Hence, in the subsequent Section \ref{s.grp}
we analyze this situation thoroughly and establish a structure theorem
which is reminiscent of the Peter--Weyl theorem
and its consequences for Banach space representations of compact
groups (Theorem \ref{grp.t.structure}).
\medskip
Putting the pieces together, in Section \ref{s.conv} we formulate
several consequences regarding the asymptotic (and
spectral-theoretic) properties of a positive semigroup $T =
(T_s)_{s\in S}$ satisfying the
conditions of Theorem \ref{psg.t.main}.
Particularly important here is the fact that
we can identify (in Theorem \ref{conv.t.main1})
two intrinsic properties of the semigroup $S$ that imply
the convergence of $T$: an algebraic one (essential divisibility of
$S$) and a topological one (under the condition that
$S$ is topological and $T$ is
continuous). Interestingly enough, both are applicable in the case $S=
\mathbb{R}_+$, but none of them in the case $S= \mathbb{N}_0$.
\medskip
Finally, in Section \ref{s.classical} we review the pre-history of the
problem and show how the results obtained so far by other people
relate to (or are covered by) our findings. Interestingly, in this
pre-history the spectral--theoretic results (``triviality of the
point spectrum'') have taken a much more prominent role than the
asymptotic results. We end the paper with a new and unifying result in
this direction (Theorem \ref{clas.t.gluha}).
\subsection*{Relevance and Relation to the Work of Others}
The present work can be understood as a continuation and
further development of the recent paper
\cite{GerlachConvPOS} by M.~Gerlach and the first
author. Many
ideas in the present paper can already be found in \cite{GerlachConvPOS}, like that
one can go beyond strong continuity by combining the JdLG-theory with the
concept of AM-compactness; or that under AM-compactness conditions
a purely algebraic property
(divisibility) of the underlying semigroup suffices to guarantee the
strong convergence of the representation.
However, we surpass our reference in many
respects:
\begin{aufziii}
\item A general Banach space principle (Theorem \ref{abs.t.main}) is
established and identified as the theoretical core which
underlies the results of \cite{GerlachConvPOS}. This principle, which is based
on the new notion of ``quasi-compactness relative to a subspace''
and on our systematic study of the ``semigroup at infinity'' (Theorem \ref{rep.t.Pinfty}),
has potential applications in the asymptotic theory of semigroups
without any positivity assumptions.
\item As a consequence of 1), the
two main results from \cite{GerlachConvPOS}, Theorem 3.7 and
Theorem 3.11, are now unified. Moreover, our results hold without
requiring the semigroup to have a quasi-interior fixed point.
\item A general structure theorem for representations
of compact groups on atomic Banach lattices (Theorem
\ref{grp.t.structure}) is established. This result is auxiliary to---but
actually completely independent of---our principal enterprise, the asymptotics of operator semigroups.
From a different viewpoint, it is a contribution to the theory of
positive group representations as promoted by Marcel de Jeu (Leiden)
and his collaborators.
\item A new spectral-theoretic result (Theorem \ref{clas.t.gluha})
about the properties of unimodular eigenvalues is establied.
\end{aufziii}
\medskip
To understand the relevance of the results obtained in this paper, one best looks into the
pre-history of its predecessor \cite{GerlachConvPOS}. We decided
to place such a historical narrative {\em after} our systematic
considerations, in Section \ref{s.classical}.
That gives us the possibility to then refer freely to the results proven
before, and to explain in detail their relation to the results obtained
earlier by other people.
\subsection*{Notation and Terminology}
We use the letters $E,F, \dots$ generically to denote Banach spaces or
Banach lattices over the scalar field $\mathbb{K} \in \{ \mathbb{R} , \mathbb{C}\}$.
The space of bounded linear operators is denoted by
$\BL(E;F)$, and $\BL(E)$ if $E= F$; the space of compact operators
is $\CO(E;F)$, and $\CO(E)$ if $E= F$.
Frequently, we shall endow
$\BL(E;F)$ with the strong operator
topology (sot). To indicate this we use terms like ``sot-closed'',
``sot-compact'' or speak of ``strongly closed'' or ``strongly compact''
sets etc. A similar convention applies when the weak operator
topology (wot) is considered. Whereas for a set $A\subseteq E$ the set
$\mathrm{cl}_{\sigma}(A)$ is the closure of $A$ in the weak ($= \sigma(E;E')$)
topology on $E$, the sot-closure and the wot-closure of a set
$M \subseteq \BL(E;F)$
are denoted by
\[ \mathrm{cl}_{\sot}(M)\quad \text{and}\quad \mathrm{cl}_{\wot}(M),
\]
respectively. We shall frequently use the following auxiliary
result, see \cite[Corollary~A.5]{Engel2000}.
\begin{lem}\label{intro.l.sotcp}
Let $E$ be a Banach space. Then a bounded subset $M \subseteq \BL(E)$ is
relatively strongly (weakly) compact if and only if the orbit
\[ Mx := \{ Tx \suchthat T \in M\}
\]
is relatively (weakly) compact for all $x$ from a dense subset of $E$.
\end{lem}
The set $\BL(E)$
is a semigroup with respect to operator multiplication. Operator
multiplication
is sot- and wot-separately continuous, and it is sot-simultaneously
continuous on norm-bounded sets.
For the definition of a semigroup
as well as for some elementary definitions and results from
algebraic semigroup theory, see Appendix \ref{app.sgp}.
We shall freely use standard results and notation
from the theory of Banach
lattices, with \cite{Schaefer1974} and \cite{Meyer-Nieberg1991}
being our main references. If $E$ is a Banach lattice the set $E_+:=\{
x\in E \suchthat x\ge 0\}$ is its cone of positive elements. In some
proofs we confine tacitly to real Banach lattices, but there should be no
difficulty to extend the arguments to the complex case.
\section{Representations of Abelian Semigroups}\label{s.rep}
Throughout the article,
$S$ is an Abelian semigroup (written additively) containing a neutral
element $0$.
Observe that for each
$s\in S$ the set
\[ s{+}S := \{ s+r \suchthat r \in S \} \subseteq S
\]
is a subsemigroup of $S$. We turn
$S$ into a directed set by letting
\[ s \le t \quad \stackrel{\text{\upshape\tiny def.}}{\Iff} \quad t \in s{+}S \quad \Leftrightarrow\quad
t{+}S \subseteq s{+}S.
\]
For limits of nets $(x_s)_{s\in S}$ with respect to this direction,
the notation $\lim_{s\in S} x_s$ is used. Note that $0 \le s$ for all $s\in S$.
\begin{ex}
Observe that in the cases $S= \mathbb{Z}_+$ and $S= \mathbb{R}_+$ the so-defined
direction and the associated notion of limit coincides
with the usual one.
\end{ex}
\medskip
\noindent
A {\emdf representation} of $S$ on a Banach space $E$ is any mapping
$T: S \to \BL(E)$ satisfying
\[ T(0) = {\mathop{\mathrm{I\mathstrut}}} \quad \text{and}\quad T(s+t) = T(s)T(t) \qquad (t,s\in S).
\]
In place of $T(s)$ we also use index notation $T_s$, and
often call $T= (T_s)_{s\in S}$ an {\emdf operator semigroup} (over
$S$ on $E$). The {\emdf fixed space} of the representation $T$ is
\[ \fix(T) := \bigcap_{s\in S} \ker(T_s - {\mathop{\mathrm{I\mathstrut}}}) =
\{ x\in E \suchthat T_s x = x \,\,\text{for all $s\in S$}\}.
\]
An operator semigroup $(T_s)_{s\in S}$ is {\emdf bounded} if
\[ M_T := \sup_{s\in S} \norm{T_s} < \infty.
\]
Boundedness has the following useful consequence.
\begin{lem}\label{rep.l.ct0}
Let $T= (T_s)_{s\in S}$ be a bounded operator semigroup on the Banach
space $E$. Then for each vector $x\in E$ the following assertions
are equivalent:
\begin{aufzii}
\item $0 \in \cls{\{T_s x\suchthat s\in S\}}$.
\item $\lim_{s\in S} T_s x = 0$.
\end{aufzii}
\end{lem}
\begin{proof}
Suppose that (i) holds
and $\veps > 0$. Then there is $s\in S$ such
that $\norm{T_{s}x} \le \veps$. But then
\[ \norm{T_t x} \le \veps \,M_T \qquad \text{for all $t \in s{+}S$.}
\]
It follows that $\lim_{s\in S} T_s x = 0$, i.e.,
(ii). The converse is trivial.
\end{proof}
Given an operator semigroup $T= (T_s)_{s\in S}$ on a Banach space $E$,
a subset $A$ of $E$ is called {\emdf $T$-invariant} if
$T_s(A) \subseteq A$ for all $s\in S$. A closed, $T$-invariant
subspace $F$ of $E$ gives rise to a {\emdf subrepresentation}
by restricting the operators $T_s$ to $F$.
Such a subrepresentation
is called {\emdf finite-dimensional ($d$-dimensional)} ($d\in \mathbb{N}$)
if $F$ is finite-dimensional ($d$-dimensional).
A one-dimensional subrepresentation is given by
a scalar representation $\lambda: S \to \mathbb{K}$ and a non-zero vector
$u\in E$ such that
\[ T_s u = \lambda_s u\qquad \text{ for all $s\in S$.}
\]
The corresponding mapping $\lambda$ is then called an {\emdf
eigenvalue} of $T$, and $u$ is called a corresponding
{\emdf eigenvector}. Obviously,
\[ \text{$\lambda$ is constant}\quad \,\Leftrightarrow\,\quad \lambda = \car
\quad\,\Leftrightarrow\,\quad u \in \fix(T).
\]
An eigenvalue $\lambda =
(\lambda_s)_{s \in S}$ is called {\emdf unimodular} if $\abs{\lambda_s} = 1$
for each $s \in S$. (So the constant eigenvalue $\car$ is unimodular.)
An unimodular eigenvalue $\lambda$ of $T$ is called a {\emdf torsion eigenvalue}
if there is $m \in \mathbb{N}$ such that $\lambda_s^m = 1$ for all $s\in S$.
\medskip
If $E$ is a Banach lattice, a semigroup $(T_s)_{s\in S}$ on $E$ is
called {\emdf positive} if the positive cone $E_+$ is
$T$-invariant, i.e., if each operator $T_s$ is positive.
And a positive semigroup
is called {\emdf irreducible} or said to {\emdf act
irreducibly} on $E$ if $\{0\}$ and $E$ are the only
$T$-invariant closed ideals of $E$. (Recall
that a subspace $J$ of $E$ is an ideal if
it satisfies:\quad
$x\in E, \, y \in J, \,\,\abs{x} \le \abs{y} \quad \dann \quad x\in
J.$)
\subsection*{The Semigroup at Infinity}
Given an operator semigroup $T = (T_s)_{s\in S}$ we write
\[ T_S := \{ T_s \suchthat s\in S\} \,\,\subseteq \,\, \BL(E)
\]
for its range, which is a subsemigroup of $\BL(E)$. And we abbreviate
\[ \calT := \mathrm{cl}_\sot \{T_s \suchthat s\in S\} \quad \text{and}\quad
\calT_s := \mathrm{cl}_\sot \{T_t \suchthat t \ge s\}\quad (s\in S),
\]
and call
\[ \calT_\infty := \bigcap_{s\in S} \calT_s =
\bigcap_{s\in S} \mathrm{cl}_\sot \{T_t \suchthat t \ge s\}
\]
the associated {\emdf semigroup at infinity}. In effect,
$\calT_\infty$ is the set of sot-cluster points of the net $(T_s)_{s\in
S}$.
Note that $\calT_\infty$ is multiplicative and even satifies
\[ \calT \cdot \calT_\infty \subseteq \calT_\infty.
\]
But it may be empty (in which case it
is, according to our definition in Appendix \ref{app.sgp},
not a semigroup\footnote{We apologize for this little
abuse of terminology.}.)
\subsection*{The JdLG-Splitting Theory}
One of the principal methods to prove strong convergence of a bounded
semigroup is to employ the splitting theory of Jacobs, de Leeuw and
Glicksberg as detailed, e.g., in \cite[Chapter~16]{Eisner2015}.
Usually, this theory is applied to the semigroup
$\calT$ or to its wot-counterpart $\mathrm{cl}_\wot\{ T_s \suchthat
s\in S\}$. In contrast, we shall apply it to $\calT_\infty$. If
$\calT_\infty$ is a strongly compact semigroup, the JdLG-theory
tells that it contains a unique minimal idempotent, which we denote
by $P_\infty$. (Minimality means that $P_\infty \cdot \calT_\infty$ is
a minimal ideal in $\calT_\infty$.) The range of $P_\infty$ is denoted
here by
\[ E_\infty := \ran(P_\infty).
\]
Observe that $Q T_s = T_sQ$ for each $s\in S$ and each $Q\in
\calT_\infty$.
In particular, $E_\infty$ is $\calT$-invariant.
\begin{thm}\label{rep.t.Pinfty}\label{rep.t.spec}
Let $T = (T_s)_{s\in S}$
be a bounded operator semigroup on the Banach
space $E$ such that the associated semigroup at infinity,
$\calT_\infty$, is strongly compact and non-empty.
Then the following additional assertions hold:
\begin{aufzi}
\item $\calT P_\infty = \calT_\infty P_\infty$.
\item $T$ is relatively strongly compact on $E_\infty$, i.e.,
\[ \calG := \mathrm{cl}_\sot\{ T_s\res{E_\infty} \suchthat s\in
S\}\subseteq \BL(E_\infty)
\]
is a strongly compact group of invertible operators on $E_\infty$.
Moreover,
\[ \calG = \calT\res{E_\infty} := \{ Q \res{E_\infty} \suchthat Q \in \calT\}.
\]
\item For each $x \in E$ the following statements are equivalent:
\begin{aufzii}
\item $P_\infty x= 0$.
\item $0 \in \mathrm{cl}_\sigma\{T_s x \suchthat s\in S\}$.
\item $\lim_{s\in S} T_s x = 0$.
\item $Rx = 0$ for some/all $R\in \calT_\infty$.
\end{aufzii}
\item If $(\lambda_s)_{s\in S}$ is a unimodular eigenvalue of $T$ with
eigenvector $0 \neq x\in E$, then $x\in E_\infty$ and there is a unique eigenvalue
$\mu = (\mu_Q)_{Q\in \calG}$ of $\calG$ such that $\lambda_s =
\mu_{T_s}$ for all $s\in S$.
\item If $\mu = (\mu_Q)_{Q\in \calG}$ is an eigenvalue of $\calG$ on
$E_\infty$, then $\lambda_s := \mu_{T_s}$\, ($s\in S$) is an
unimodular eigenvalue of $T$.
\end{aufzi}
(We suppose $\mathbb{K} = \mathbb{C}$ for assertions {\rm d)} and {\rm e)}.)
\end{thm}
\begin{proof}
a)\ Since $\calT \calT_\infty \subseteq \calT_\infty \calT$, we have
$\calT P_\infty = \calT P_\infty P_\infty \subseteq \calT_\infty
P_\infty \subseteq \calT P_\infty$.
\smallskip\noindent
b)\ By a) we have $\calT\res{E_\infty} = \calT_\infty \res{E_\infty}$, and
the latter is a strongly compact group of invertible operators on $E_\infty$
by the JdLG-theory. Since restriction is a sot-continuous operator from $\BL(E)$ to
$\BL(E_\infty; E)$, $\calT\res{E_\infty} \subseteq \calG$. The
converse inclusion follows from $\calG P_\infty \subseteq \calT$,
which is true because $P_\infty \in \calT$.
\smallskip\noindent
c)\ If $Rx = 0$ for {\em all} $R\in \calT_\infty$, then clearly
(i) holds, and (i) implies that $Rx=0$ for {\em some} $R\in
\calT_\infty$. On the other hand, this latter statement obviously
implies $0 \in \cls{\{T_s x\suchthat s\in S\}}$, which is equivalent
to $\lim_{s\in S} T_s x = 0$, i.e., (iii).
If, in turn, (iii) holds and $\veps > 0$ is fixed, then there is
$s\in S$ such that $\{ T_tx \suchthat t \ge s \} \subseteq
\Ball[0,\veps]$. Hence, also $\calT_\infty x \subseteq
\Ball[0,\veps]$. As $\veps > 0$ was arbitrary, $\calT_\infty x =
\{ 0\}$, i.e., $Rx= 0$ for all $R\in \calT_\infty$.
Finally, (iii) obviously implies (ii). Conversely,
starting from (ii) we apply $P_\infty$ to obtain
\[ 0 \in \mathrm{cl}_\sigma\{ T_sP_\infty x \suchthat s\in S\}.
\]
However, by b) the set $\{ T_sP_\infty x \suchthat s\in S\}$
is relatively strongly compact and hence its weak and its strong
closures must coincide. This yields
\[ 0 \in \cls{\{ T_sP_\infty x \suchthat s\in S\}},
\]
which implies $P_\infty x = P_\infty(P_\infty x) = 0$ by what
we have already shown.
\smallskip\noindent
d)\ Let $0\neq x\in E$ be an eigenvector for the unimodular eigenvalue
$(\lambda_s)_{s\in S}$ of $T$. Define $y := x - P_\infty x$. Then
$P_\infty y = 0$ and hence $T_s y \to 0$. On the other hand, since
$P_\infty$ commutes with every $T_s$, $T_sy = \lambda_s y$ for all
$s\in S$. As $\abs{\lambda_s} = 1$, it follows that $y = 0$ and hence
$x\in E_\infty$. The remaining statement now follows easily since $\mathbb{C}
x$ is $T$-invariant and $T_S\res{E_\infty}$ is dense in $\calG$.
\smallskip\noindent
e)\ is obvious.
\end{proof}
As a corollary we obtain the following
characterization of the strong convergence of a semigroup.
\begin{cor}
For a bounded operator semigroup $T = (T_s)_{s\in S}$
on a Banach space $E$ the following assertions are equivalent:
\begin{aufzii}
\item $T$ is strongly convergent;
\item $\calT_\infty$ is a singleton;
\item $\calT_\infty$ is non-empty and strongly compact and acts as
the identity on $E_\infty$;
\item $\calT_\infty$ is non-empty and strongly compact and $T$
acts as the identity on $E_\infty$.
\end{aufzii}
In this case:\quad $\lim_{s\in S} T_s = P_\infty$.
\end{cor}
\begin{proof}
(i)$\dann$(ii): If $T$ is strongly convergent with $P := \lim_{s\in S} T_s$ being its limit, then
$\calT_\infty= \{P\}$ is a singleton.
\smallskip\noindent
(ii)$\dann$(iii): If $\calT_\infty=\{P\}$ is a singleton, then it is
clearly non-empty and strongly compact. It follows that $P =
P_\infty$, and hence $\calT_\infty$ acts as the identity on
$E_\infty$.
\smallskip\noindent
(iii)$\dann$(iv): Suppose that $\calT_\infty$ is non-empty and
strongly compact and acts as the identity on $E_\infty$. Let
$Q\in \calT_\infty$. Then, by the equivalence
(i)$\,\Leftrightarrow\,$(iv) in Theorem \ref{rep.t.Pinfty}.c),
$Q({\mathop{\mathrm{I\mathstrut}}} - P_\infty) = 0$ and hence $Q = QP_\infty = P_\infty$. So it
follows that $\calT_\infty= \{P_\infty\}$. Moreover,
since $T_s \calT_\infty \subseteq \calT_\infty$ for each $s\in
S$, we obtain $T_sP_\infty = P_\infty$ and hence $T_s = {\mathop{\mathrm{I\mathstrut}}}$ on
$E_\infty$ for all $s\in S$.
\smallskip\noindent
(iv)$\dann$(i): Suppose that (iv) holds. Then
\[ \lim_{s\in S} T_s = \lim_{s\in S} (T_sP_\infty + T_s({\mathop{\mathrm{I\mathstrut}}} -
P_\infty)) = P_\infty + \lim_{s\in S} T_s({\mathop{\mathrm{I\mathstrut}}} - P_\infty) = P_\infty
\]
strongly, by the equivalence (i)$\,\Leftrightarrow\,$(iii) of Theorem \ref{rep.t.Pinfty}.c).
\end{proof}
Theorem \ref{rep.t.Pinfty} and its corollary yield
the following strategy to prove strong convergence of an
operator semigroup:
\begin{aufziii}
\item Show that $\calT_\infty$ is non-empty and strongly compact.
\item Show that $\calT_\infty$ (or, equivalently, $T$) acts as the identity on $E_\infty:= \ran(P_\infty)$.
(For this one may employ the additional information
that $\calT$ acts on $E_\infty$ as a compact group.)
\end{aufziii}
\begin{rem}\label{rep.r.sap}
Suppose that $T= (T_s)_{s\in S}$ is a strongly relatively compact operator
semigroup on $E$ with minimal idempotent $P \in \calT$.
Then, of course, $\calT_\infty$ is non-empty and strongly
compact, and hence a closed ideal of
$\calT$. It follows from the minimality of $P$ in $\calT$ and
$P_\infty$ in $\calT_\infty$ that
\[ P \calT \subseteq P_\infty \calT \subseteq P_\infty \calT_\infty
\subseteq P \calT_\infty \subseteq P \calT.
\]
Hence $P\calT = P_\infty \calT_\infty$, which implies that $P =
P_\infty$. So, in the case that $T$ is relatively strongly compact,
passing to the semigroup at infinity yields the same
JdLG-decomposition of $E$ as working with $\calT$.
\end{rem}
We end this section with a technical, but useful characterization
of the property that $\calT_\infty$
is non-empty and compact.
\begin{prop} \label{rep.p.sac}
For a bounded operator semigroup $T = (T_s)_{s\in S}$
on a Banach space $E$ the following assertions are equivalent:
\begin{aufzii}
\item $\calT_\infty$ is non-empty and strongly compact.
\item Every subnet of $(T_s)_{s \in S}$ has a strongly convergent
subnet.
\item Every universal subnet of $(T_s)_{s \in S}$ is strongly convergent.
\item For each $x \in E$ every subnet of $(T_s x)_{s \in S}$ has a
convergent subnet.
\item For each $x \in E$ every universal subnet of $(T_s x)_{s \in
S}$ converges.
\end{aufzii}
If $S$ contains a cofinal sequence, then the above
assertions are also equivalent to:
\begin{aufzii} \setcounter{aufzii}{5}
\item For every $x \in E$ and every cofinal sequence $(s_n)_{n \in \bbN} \subseteq
S$, the sequence $(S_{s_n}x)_{n \in \bbN}$ has a convergent subsequence.
\end{aufzii}
\end{prop}
\begin{proof}
(i)$\dann$(iv): Suppose that (i) holds and let $x \in E$. The net
$(T_s ({\mathop{\mathrm{I\mathstrut}}} - P_\infty)x)_{s \in S}$ converges to $0$ according to Theorem~\ref{rep.t.spec}.c).
On the other hand, the net $(T_s P_\infty x)_{s \in S}$ is contained
in the compact set $\calT_\infty P_\infty x$ due to
Theorem~\ref{rep.t.spec} a), so each of its subnets has a convergent
subnet. This shows (iv).
\smallskip\noindent
(iv)$\dann$(v): This follows since a universal net with
a convergent subnet must converge.
\smallskip\noindent
(v)$\dann$(iii): Let
$(T_{s_\alpha})_{\alpha \in I}$ be a universal subnet
of $(T_s)_{s \in S}$. Then for each $x \in E$,
the net $(T_{s_\alpha}x)_{\alpha \in I}$ is universal and hence, by
(v), convergent. Thus,
$(T_{s_\alpha})_{\alpha \in I}$ is strongly convergent.
\smallskip\noindent
(iii)$\dann$(ii)$\dann$(i) and (iv)$
\dann$(vi) all follow from
Theorem \ref{uni.t.cluster}.
\smallskip\noindent
(vi)$\dann$(i): Suppose that $S$ admits a cofinal sequence.
Then by Theorem \ref{uni.t.cluster} for each $x\in E$ the
set $C_x := \bigcap_{t\in S} \mathrm{cl}\{T_s x\suchthat s\ge t\}$ is
non-empty and compact. Since $\calT_\infty x \subseteq C_x$,
it follows that $\calT_\infty$ is strongly compact.
In order
to see that $\calT_\infty$ is not empty, fix a
cofinal sequence $(s_n)_n$. By (vi) and since $E$ is metrizable,
it follows that for each $x\in E$ the set $\{ T_{s_n}x \suchthat
n \in \mathbb{N}\}$ is relatively compact. Hence, $\{ T_{s_n} \suchthat
n \in \mathbb{N}\}$ is relatively strongly compact. It follows that
the sequence $(T_{s_n})_n$ has a cluster point, which is a member
of $\calT_\infty$ since $(s_n)_n$ is cofinal.
\end{proof}
\begin{rems}\label{rep.r.sac}%
\begin{aufziii}
\item
Assertion~(vi) in Proposition~\ref{rep.p.sac} is called {\emdf strong
asymptotic compactness} in \cite[p.\,2636]{Emelyanov2001}.
\item Proposition~\ref{rep.p.sac} has an interesting
consequence
in the ``classical'' cases where $S = \mathbb{N}_0$ or $S= \mathbb{R}_+$ and
$T$ is strongly continuous (cf. the Introduction). Namely, in these cases
one can actually dispense with
the semigroup at infinity, because
$\calT_\infty$ is strongly compact and non-empty {\em if and only if}
$\calT$ is strongly compact.
\end{aufziii}
\end{rems}
In the next section we shall present another situation in which
$\calT_\infty$ is non-empty and strongly compact.
\section{The Abstract Main Result}\label{s.abs}
Suppose that
$E$ and $F$ are Banach spaces such that $F$ is densely embedded
in $E$:
\[ F\, \stackrel{d}{\hookrightarrow}\, E.
\]
Reference to this embedding is usually suppressed and $F$
is simply regarded as a subspace of $E$. We take the
freedom to consider an operator on $E$ also as an operator from $F$ to
$E$. (This amounts to view $\BL(E) \subseteq \BL(F;E)$ via the
restriction mapping.)
A semigroup $(T_s)_{s\in S}$ on $E$ is called {\emdf
$F$-to-$E$ quasi-compact}, or {\emdf quasi-compact relatively to $F$},
if for each $\veps > 0$ there is $s\in S$ and
a compact operator $K: F \to E$ such that
\[
\norm{T_s - K}_{\BL(F;E)} < \veps.
\]
Note that we do not require that $K$ can be
extended to a bounded operator on $E$. In effect,
the condition of being $F$-to-$E$ quasi-compact can be expressed as
\[ \dist( \{T_s\suchthat s\in S\},
\CO(F;E)) = 0,
\]
where ``$\dist$'' refers to the distance induced
by the norm on $\BL(F;E)$.
\begin{thm}\label{abs.t.main}
Let $E$ and $F$ be Banach spaces,
with $F$ being densely embedded into $E$, and let $(T_s)_{s\in S}$ be
a bounded operator semigroup on $E$ which restricts
to a bounded operator semigroup on $F$ and
is $F$-to-$E$ quasi-compact. Then the following assertions hold:
\begin{aufzi}
\item $\calT_\infty$ is a strongly compact and non-empty.
\item Each element of $\calT_\infty$ is compact as an operator from
$F$ to $E$.
\item $\calT$ acts on $E_\infty$ as a sot-compact group of invertible
operators.
\item For $x\in E$ the following assertions are equivalent:
\begin{aufzii}
\item $\lim_{s\in S} T_s x= 0$;
\item $x \in \ker P_\infty$;
\item $0 \in \mathrm{cl}_\sigma\{ T_sx \suchthat s\in S\}$.
\end{aufzii}
\end{aufzi}
\end{thm}
\begin{proof}
a) and b)\
By passing to an equivalent norm on $F$ we may suppose that
each $T_s$, $s\in S$, is a contraction on $F$. Let
$\Ball_F$ and $\Ball_E$ denote the closed unit balls of $E$ and $F$,
respectively.
Let $\veps > 0$ and choose $s\in S$ and $K \in \CO(F;E)$ such that
$\norm{T_s- K}_{\BL(F;E)}\le \veps$. Then
\[ T_{t{+}s}(\Ball_F) = T_s T_t(\Ball_F) \subseteq
T_s(\Ball_F) \subseteq K(\Ball_F) + \veps \Ball_E
\]
for each $t \in S$, and therefore
\begin{equation}\label{AM.e.main}
\calT_{s}(\Ball_F) \subseteq \cls{K(\Ball_F)} +
\veps \Ball_E.
\end{equation}
Now, let $(T_{s_\alpha})_{\alpha}$ be any universal subnet of
$(T_s)_{s\in S}$ (Lemma \ref{uni.l.kelley}) and let $x\in \Ball_F$. Then
the net $(T_{s_\alpha}x)_\alpha$ is a universal net in $E$. Moreover,
\eqref{AM.e.main}
shows that for each $\veps> 0$ this net has a tail
contained in the $\veps$-neighborhood of some compact set. Hence,
by Lemma \ref{uni.l.cauchy},
it is a Cauchy net and thus convergent in $E$.
Since $F$ is dense in $E$ and $T$ is bounded,
$(T_{s_\alpha}x)_\alpha$ converges for {\em every} $x\in E$. In other
words, $(T_{s_\alpha})_\alpha$ is strongly convergent. As its
limit must be a member of $\calT_\infty$, it follows that
$\calT_\infty \neq \emptyset$.
It also follows from \eqref{AM.e.main} that
$\calT_\infty (\Ball_F) \subseteq \cls{K(\Ball_F)} + \veps \Ball_E$.
As $\cls{K(\Ball_F)}$ is compact, it admits a finite $\veps$-mesh.
Hence, $\calT_\infty (\Ball_F)$ admits a finite $2\veps$-mesh. Since
this works for each $\veps > 0$, $\calT_\infty (\Ball_F)$ is relatively
compact in $E$.
In particular, it follows that $\calT_\infty \subseteq \CO(F;E)$
and that for each $x\in F$
the orbit $\calT_\infty x$ is relatively compact in $E$.
Since $T$ is bounded on $E$ and $F$ is dense in $E$,
$\calT_\infty$ is relatively strongly compact (Lemma \ref{intro.l.sotcp}).
But $\calT_\infty$ is strongly closed, so it is strongly compact as claimed.
\smallskip\noindent
c)\ and d) follow from a) by Theorem \ref{rep.t.spec}.
\end{proof}
\begin{rem}
Theorem \ref{abs.t.main} seems to be new even for $C_0$-semigroups. In that case, by Remark
\ref{rep.r.sac}.b), it follows a posteriori that the $C_0$-semigroup is
relatively compact.
\end{rem}
In the next sections we shall see that our set-up from above
has a quite natural instantiation in the
context of semigroups of
positive operators on Banach lattices with a quasi-interior point.
\section{Positive Semigroups and AM-Compactness}\label{s.psg}\label{s.AM}
From now on, we consider {\em positive} semigroups $T=(T_s)_{s\in S}$
on Banach lattices $E$. The role of $F$ in our abstract setting
from above will be taken by the {\em principal ideal}
\[ E_y := \{ x\in E \suchthat \text{there is $c \ge 0$ such that
$\abs{x}\le cy$}\}
\]
for some $y \in E_+$, endowed with the natural
AM-norm
\[ \norm{x}_y := \inf\{
c \ge 0 \suchthat \abs{x}\le cy\}.
\]
Since we need that $F= E_y$ is dense in $E$, we have to require
that $y$ is {\em a quasi-interior point} in $E$.
As we further need $E_y$-to-$E$ quasi-compactness,
it is natural to ask which
operators on $E$ restrict to compact operators from $E_y$ to
$E$. It turns out that these are precisely the {\em AM-compact}
operators,
i.e., those that map order intervals of $E$ to
relatively compact subsets of $E$, see Lemma \ref{int.t.AM}.
There are a couple of useful theorems that help to identify
AM-compact operators. For example, operators
between $\Ell{p}$-spaces induced by positive integral kernel functions
and positive operators that ``factor through $\Ell{\infty}$-spaces''
are AM-compact. (Proofs of these well-known facts are presented in
Appendix \ref{app.int}, see Theorems \ref{int.t.AM} and \ref{int.t.fact-Linfty}.)
\subsection*{The Range of a Positive Projection}
When we apply Theorem \ref{abs.t.main} to a semigroup of positive operators,
the resulting projection $P_\infty$ will be positive, too.
The following
is a useful information about its range.
\begin{lemdef}\label{psg.l.ranP}
Let $E$ be a Banach lattice and let $P$ be a positive projection
on $E$. Define
\[ \norm{x}_P := \norm{P\abs{x}}\qquad (x\in \ran(P)).
\]
Then the following assertions hold:
\begin{aufzi}
\item $\norm{\cdot}_P$ is an equivalent norm on $\ran(P)$.
\item The space $\ran(P)$ is a Banach lattice with respect to the
order induced by $E$ and the norm $\norm{\cdot}_P$. Its modulus
is given by
\[ \abs{x}_P := P\abs{x} \qquad (x\in \ran(P)).
\]
\end{aufzi}
{\rm The Banach lattice $\ran(P)$ endowed with the
norm $\norm{\cdot}_P$ as in a) and b) is
denoted by $[\ran(P)]$.}
\begin{aufzi}\setcounter{aufzi}{2}
\item If $y\in E_+$ then $P(E_y) \subseteq [\ran(P)]_{Py}$.
In particular, if $y\in E_+$
is a quasi-interior point of $E$ then
$Py$ is a quasi-interior point of $[\ran(P)]$.
\end{aufzi}
\end{lemdef}
\begin{proof}
This is essentially \cite[Proposition~III.11.5]{Schaefer1974}.
\end{proof}
In order to obtain further insight into the
relation of the closed ideals in $[\ran(P)]$ and in $E$,
we define for any Banach lattice $E$ the mapping
\[ \Phi(J) := \mathrm{cl}\bigl\{ x\in E \suchthat
\abs{x}\le y\,\,\text{for some $y\in J_+$}\bigr\}\qquad (J \subseteq E).
\]
If $J_+ = J \cap E_+$ is a cone, then $\Phi(J) = \Phi(J_+)$ is
the smallest closed ideal in $E$ containing $J_+$.
\begin{thm}\label{psg.t.ideals}
Let $E$ be a Banach lattice and $P$ a positive projection on $E$, and
let $\Phi$ be defined as above. Then the following assertions hold:
\begin{aufzi}
\item If $I$ is a closed $P$-invariant ideal in $E$ then $P(I) = I \cap
\ran(P)$ is a closed ideal in $[\ran(P)]$.
\item If $J$ is a closed ideal in $[\ran(P)]$ then $\Phi(J)$ is
$P$-invariant and the smallest closed ideal in $E$ containing $J$.
Moreover,
\[ J = P(\Phi(J)) = \Phi(J) \cap \ran(P).
\]
\end{aufzi}
\end{thm}
\begin{proof}
a)\ Let $I \subseteq E$ be a closed $P$-invariant
ideal and let $J := I \cap \ran(P)$. Then $J$ is a closed subspace
of $\ran(P)$. And if $x\in \ran(P)$ and $y \in J$ with $\abs{x}_P \le
\abs{y}_P$, it follows that
\[ \abs{x} = \abs{Px}\le P\abs{x} = \abs{x}_P \le \abs{y}_P = P\abs{y}
\in I
\]
by $P$-invariance. Hence, $x\in J$ and therefore $J$ is an ideal in
$[\ran(P)]$. Moreover, again by $P$-invariance,
\[ J = PJ = P(I \cap \ran(P)) \subseteq P(I) \subseteq
I \cap \ran(P),
\]
and hence $I \cap \ran(P) = J = P(I)$.
\smallskip
\noindent
b)\ Let $J$ be any closed ideal of $[\ran(P)]$. Then $\Phi(J)$
is the smallest closed ideal in $E$ containing $J$. (In fact, if
$x\in J$ then $\abs{x} = \abs{Px}\le P \abs{x} = \abs{x}_P \in J_+$,
and hence $J\subseteq \Phi(J)$.) It is also $P$-invariant, for
if $\abs{x} \le y \in J_+$ then $\abs{Px} \le P\abs{x} \le Py = y$.
This also shows that $P(\Phi(J)) \subseteq J$, and since $J \subseteq
\Phi(J)$, it follows that $P(\Phi(J)) = J$.
\end{proof}
\subsection*{The Main Result for Positive Semigroups}
We are now prepared for our second main theorem.
\begin{thm}\label{psg.t.main}
Let $T= (T_s)_{s\in S}$ be a bounded and positive
operator semigroup on a Banach lattice $E$
with a quasi-interior point $y \in E_+$. Suppose, in addition,
that $T$ is $E_y$-to-$E$ quasi-compact and
restricts to a bounded semigroup on $E_y$. Then the following
assertions hold:
\begin{aufzi}
\item $\calT_\infty$ is
strongly compact and non-empty and consists of AM-compact operators.
\item $[\ran(P_\infty)]$ is an atomic Banach lattice with order continuous
norm and quasi-interior point $P_\infty y$.
\item The semigroup $\calT = \mathrm{cl}_\sot\{T_s\suchthat s\in
S\}$ acts on $[\ran(P_\infty)]$ as
a compact topological group of positive, invertible operators.
\item If $(T_s)_{s\in S}$ acts irreducibly on $E$, then $\calT$ acts
irreducibly on $[\ran(P_\infty)]$.
\end{aufzi}
\end{thm}
\begin{proof}
a)\ follows from Theorem \ref{abs.t.main} and Lemma \ref{AM.l.AM-char}.
\smallskip\noindent
b)\ Each order interval $J$ of $E_\infty = [\ran(P_\infty)]$ is of the form
$J = J' \cap E_\infty$, where $J'$ is an order interval of $E$.
Since $P_\infty$ is AM-compact but restricts to the identity on
$E_\infty$, it follows that $J = P_\infty(J) \subseteq P_\infty(J')
$ is relatively compact. By \cite[Theorem~6.1]{Wnuk1999}, this implies that
$E_\infty$ as a Banach lattice is atomic and has order continuous
norm.
\smallskip\noindent
c)\ This follows again from Theorem \ref{abs.t.main}.
\smallskip\noindent
d)\ Suppose that $J\neq \{0\}$ is a closed $\calT$-invariant ideal in
$[\ran(P_\infty)]$. Then the set
\[ \{ x\in E \suchthat \abs{x}\le y \,\, \text{for some $y \in J_+$}\}
\]
is $T$-invariant, and hence $\Phi(J)$ is a closed $T$-invariant ideal
in $E$ containing $J$. By irreducibility, $\Phi(J) = E$, and hence
$\ran(P_\infty) = P(E) = P(\Phi(J)) = J$ by Theorem \ref{psg.t.ideals}.
\end{proof}
\begin{rems} \label{psg.r.main}
\begin{aufziii}
\item The assumption that $T$ restricts to a bounded semigroup on $E_y$ is
for instance satisfied if $y$ is a {\emdf sub-fixed point} of $T$,
i.e., if $T_t y \leq y$ for all $t \in S$.
\item Certainly, if $T_s$ is AM-compact for some $s \in S$ then
$T$ is $E_y$-to-$E$ quasi-compact.
\end{aufziii}
\end{rems}
We conclude this section with
the following result, essentially proved by Gerlach and Glück
in \cite[Lemma~3.12]{GerlachConvPOS}. It shows that
in certain situations
it suffices to require merely that $T_s$ {\em
dominates} a non-trivial AM-compact operator for some $s\in S$.
\begin{lem} \label{classical.l.dom}
Let $T = (T_s)_{s \in S}$ be a bounded, positive and irreducible
semigroup on a Banach lattice $E$ with
order continuous norm and a quasi-interior sub-fixed point $y$ of $T$.
Suppose that there are $s \in S$ and an AM-compact operator
$K\neq 0$ on $E$ with $0 \le K \le T_{s}$. Then $T$ is $E_y$-to-$E$
quasi-compact and restricts to a bounded semigroup on $E_y$. In
particular,
Theorem \ref{psg.t.main} is applicable.
\end{lem}
\section{Compact Groups of Positive Operators on Atomic Banach Lattices}
\label{s.grp}
In view of our general strategy, Theorem~\ref{psg.t.main} suggests
to look for criteria implying that a positive group representation on
an atomic Banach lattice with order-continuous norm is trivial. To
this end, we first summarize some known results about atomic Banach
lattices.
\subsection*{Atomic Banach Lattices}
Recall that an {\emdf atom} in a Banach lattice is any element
$0 \neq a\in E$ such that its generated principal ideal, $E_a$, is
one-dimensional: $E_a = \mathbb{K} \cdot a$. We denote by
\[
A = A_E := \{ a\in E_+ \suchthat \text{$a$ atom},\, \norm{a} =1\}
\]
the set of positive atoms of norm one.
For distinct $a,\:b\in A$ one has
\[ \abs{a- b} = \abs{a + b} = a + b \ge a
\]
and hence $\norm{a-b} \ge \norm{a} = 1$. This shows that $A$ is a discrete set
with respect to the norm topology.
A Banach lattice $E$ is
called {\emdf atomic}, if $E$ is the smallest band in $E$ that
contains all atoms. In other words,
\[ A^d : = \{ x\in E \suchthat \abs{x} \wedge a = 0 \,\,\text{for all
$a\in A$}\} = \{0\}.
\]
For each $a\in A$ the one-dimensional subspace $E_a = \mathbb{K} a$ is a
projection band, with corresponding band projection $P_A$ given by
\[ P_a x := \sup [0,x]\cap \mathbb{R} a = \sup\{ t\in\mathbb{R}_+ \suchthat ta \le x\}
\cdot a \qquad (x\in E_+).
\]
(See, e.g. \cite[Thm. 26.4]{Luxemburg1971} and cf.{}
\cite[Prop. 1.2.11]{Meyer-Nieberg1991}.) The next result is a
consequence of \cite[p.143, Ex. 7]{Schaefer1974}
and \cite[Thm. 1.2.10]{Meyer-Nieberg1991}.
For the convenience of the reader, we give a proof.
\begin{thm}\label{grp.t.atomic-base}
Let $E$ be a Banach lattice and let $A$ be its set of positive
normalized atoms. Then for each finite subset
$F\subseteq A$ the space
\[ \spann(F) = \bigoplus_{a\in F} \mathbb{K} a
\]
is a projection band with band projection\quad $\displaystyle P_F = \sum_{a\in F} P_a$.
\quad Suppose, in addition, that $E$ is atomic. Then
\begin{equation}\label{grp.eq.atomic}
{\mathop{\mathrm{I\mathstrut}}}_E = \sum_{a\in A} P_a
\end{equation}
as a strongly order convergent series. Each band $B$ in $E$ is generated
(as a band) by $A \cap B$.
\end{thm}
\begin{rem}
There are different notions of ``order convergence''
in the literature, see \cite{Abramovich2005}. We employ the definition
found in \cite[Definition~1.1.9 i)]{Meyer-Nieberg1991}. For the case
of \eqref{grp.eq.atomic} this simply means
\begin{equation}\label{grp.eq.atomic-full}
x = \sup_F \sum_{a\in F} P_a x \qquad \text{for all $x\in E_+$},
\end{equation}
where the supremum is taken over all finite subsets of $A$.
\end{rem}
\begin{proof}[Proof of Theorem~\ref{grp.t.atomic-base}]
Fix a finite set $F \subseteq A$. If $a, b \in F$ with $a \neq b$,
then $a \wedge b = 0$ and hence $P_a P_b = 0$. It follows that
\[ P_F := \sum_{a\in F} P_a
\]
is a projection (and $F$ is a linearly independent set).
Again by the pairwise disjointness of the elements of $F$,
\[ \sum_{a\in F} P_a x = \bigvee_{a\in F} P_a x \le x \qquad (x\in E_+).
\]
This shows that $0 \le P_F \le {\mathop{\mathrm{I\mathstrut}}}$, and hence $P_F$ is a band
projection \cite[Lemma 1.2.8]{Meyer-Nieberg1991}. Since, obviously,
$\ran(P_F) = \spann(F)$, the first assertion is proved.
In order to prove \eqref{grp.eq.atomic-full}
fix $x\in E_+$ and let $y\in E_+$ be such that $y \ge P_Fx$ for all
finite $F\subseteq A$. Then $y\ge P_a x$ for each $a\in A$ and hence
\[ 0 \le x - (x\wedge y) \le x - P_a x \perp a
\]
If $E$ is atomic, it follows that $x = x\wedge y$, i.e., $x\le y$.
This yields \eqref{grp.eq.atomic-full}.
Finally, let $B\subseteq E$ be any band and let $0 \le x\in
B$. Then for each $a\in A$, $P_a x \in B$ (since $0 \le P_a x \le x$
and $B$ is an ideal). Hence, either $P_ax = 0$ or $a \in B$.
It follows from \eqref{grp.eq.atomic} that
$B$ is generated by $A\cap B$.
\end{proof}
With this information at hand we now turn to
the representation theory.
\subsection*{A Structure Theorem}
Let $G\subseteq \BL(E)$ be a group of positive, invertible operators
on $E$. (In particular, $G$ consists of lattice homomorphisms.)
Then for each $g\in G$ and $a\in A$ the element $g \cdot a \in
E$ must be an atom again. In effect
\begin{equation}\label{grp.eq.vphi_g}
\vphi_g(a) := \norm{g\cdot a}^{-1} (g\cdot a) \in A.
\end{equation}
It is easy to see that
\begin{equation}\label{grp.eq.vphi}
\vphi:
G \to \mathrm{Sym}(A),\qquad g\mapsto \vphi_g
\end{equation}
is a group homomorphism from $G$ to the group of all bijections on
$A$. The corresponding action
\[ G \times A \to A, \qquad (g,a) \mapsto \vphi_g(a)
\]
is called the {\emdf induced action} of $G$ on $A$.
For each $a\in A$ the orbit mapping
\[
G \to A,\qquad g \mapsto \vphi_g(a)
\]
of the induced action
is continuous (with respect to the strong operator topology on $G$).
If, in addition, $G$ is strongly compact, then each orbit
\[ \vphi_G(a) = \{ \vphi_g(a) \suchthat g\in G\}
\]
is finite (since $A$ is discrete).
We denote by $A/G$ the set of all these orbits. Then
$A/G$ is a partition of $A$ into finite subsets.
\begin{lem}\label{grp.l.fix}
In the described situation, suppose that $G$ is compact.
Then for $a\in A$ and $g\in G$:
$g \cdot a = a \,\,\,\Leftrightarrow\,\,\, \vphi_g(a) = a$.
Furthermore: $g = {\mathop{\mathrm{I\mathstrut}}}_E \,\,\,\Leftrightarrow\, \,\,\vphi_g = \mathop{\mathrm{id}}\nolimits_A$.
\end{lem}
\begin{proof}
Fix $a \in A$ and $g\in G$. If $g \cdot a = a$ then $\vphi_g(a)=
a$, since $\norm{a} = 1$. Conversely, suppose that $\vphi_g(a) = a$.
Then $g^n \cdot a = \norm{g \cdot a}^n a$ for all $n\in \mathbb{Z}$. By
compactness, $\norm{g \cdot a} = 1$, and hence $g\cdot a = a$ as claimed.
\smallskip\noindent
Suppose that $\vphi_g(a)= a$ for all $a\in A$. Then, as we have just
seen, $g \cdot a = a$ for all $a\in A$. So $g$ leaves all atoms
fixed. Since $g$ acts a lattice isomorphism and hence is order
continuous, it follows from Theorem \ref{grp.t.atomic-base} that $g = {\mathop{\mathrm{I\mathstrut}}}_E$.
\end{proof}
We can now prove a theorem that is reminiscent of the Peter--Weyl
structure theorem and its applications to Banach space
representations of compact groups.
\begin{thm}[Structure Theorem]\label{grp.t.structure}
Let $E \neq \{0\}$ be an atomic Banach lattice and let
$A$ be its set of positive normalized atoms. Let
$G\subseteq \BL(E)$ be a strongly compact group of positive invertible
operators on $E$, and let
$A/G$ be the set of orbits of elements
of $A$ under the induced action of $G$ on $A$. Then the following assertions hold:
\begin{aufzi}
\item For each orbit $F\in A/G$ the band $\spann(F)$ is
$G$-invariant, the corresponding band projection $P_F$ is
$G$-intertwining, and $G$ acts irreducibly on $\spann(F)$.
\item If $B\neq\{0\}$ is a $G$-invariant band in $E$ on which
$G$ acts irreducibly, then $B = \spann(F)$ for some $F \in A/G$.
\item $I = \sum_{F\in A/G} P_F$ as a strongly order-convergent
series.
\item In the case $\mathbb{K} = \mathbb{C}$, each eigenvalue of $G$ on $E$ is
torsion.
\item If $G$ acts irreducibly on $E$, then $\dim(E)< \infty$ and
$G$ has only finitely many eigenvalues.
\end{aufzi}
\end{thm}
\begin{proof}
a)\ It is obvious that $\spann(F)$ is $G$-invariant and $G$ acts
irreducibly on it. Since $G$ consists of
lattice automorphisms, also $\spann(F)^d$ is $G$-invariant, and hence
$P_F$ is $G$-intertwining.
\smallskip\noindent
b)\ Let $B\neq\{0\}$ be any $G$-invariant band in $E$. Then $B$ is generated
(as a band) by $A \cap B$. By $G$-invariance,
$A \cap B$ is a union of $G$-orbits (for the induced action), i.e., a
union of elements of $A/G$.
Hence, if $G$ acts irreducibly on $B$, $A\cap B$ must coincide with
precisely one $G$-orbit of $A$, i.e., $A \cap B \in A/G$.
\smallskip\noindent
c)\ follows from Theorem \ref{grp.t.atomic-base} since $A/G$ is partition of
$A$.
\smallskip\noindent
d)\ Let $\lambda : G \to \mathbb{C}$ be an eigenvalue of $G$ on $E$ and $0
\neq x\in E$ a corresponding eigenvector. Then $\lambda$ is a
continuous homomorphism, and since $G$ is compact, $\lambda$ is
unimodular. By c), one must have $y := P_F x \neq 0$ for some $F\in A/G$,
and by a), $y$ is also an eigenvector corresponding to $\lambda$.
Let $g \in G$ and $n := \abs{F}$, the length of the (induced) $G$-orbit
$F$. Then $g^{n!}$ acts (induced) on $F$ as the identity. Hence, by
Lemma \ref{grp.l.fix}, $g^{n!}$ acts (orginally) as the identity on
$\spann(F)$. This yields
\[ y = g^{n!}y = \lambda_g^{n!}y
\]
and hence $\lambda_g^{n!} = 1$. As $g\in G$ was arbitrary, the eigenvalue $\lambda$
is torsion.
\smallskip\noindent
e)\ If $G$ acts irreducibly on $E$, then b) tells that
$E$ is finite dimensional. As
eigenvectors belonging to different eigenvalues have to be linearly
independent, there can only finitely many eigenvalues, as claimed.
\end{proof}
\begin{rem}
For the special case of Banach sequence spaces, Theorem \ref{grp.t.structure}
has been first proved by de Jeu and Wortel in \cite[Theorem 5.7]{Jeu2014}.
\end{rem}
We now shall list several criteria
for the group $G$ in Theorem~\ref{grp.t.structure}
to be trivial. In Section~\ref{s.conv} we will translate those criteria
into sufficient conditions for the strong convergence of positive operator semigroups.
A positive linear operator $T$ on a Banach lattice $E$ is called
{\emdf strongly positive} if $Tf$ is a quasi-interior point for every
non-zero positive vector $f \in E$.
\begin{cor} \label{grp.c.misc}
Let $G$ be a strongly compact group of positive invertible operators
on an atomic Banach lattice $E\neq \{0\}$.
Then each one of the following
assertions implies that $G = \{{\mathop{\mathrm{I\mathstrut}}}_E\}$:
\begin{aufziii}
\item $G$ is divisible (cf.~Appendix~\ref{app.sgp}).
\item $G$ has no clopen subgroups different from $G$.
\item $G$ contains a strongly positive operator.
\item Every finite-dimensional $G$-invariant band of $E$ on which $G$ acts irreducibly
has dimension $\le 1$.
\item $\mathbb{K} = \mathbb{C}$ and $G$ is Abelian and does not have any non-constant
torsion eigenvalues.
\end{aufziii}
\end{cor}
\begin{proof}
By Lemma \ref{grp.l.fix} it it suffices to prove
in each of the mentioned cases that the group homomorphism
$\vphi$, defined in \eqref{grp.eq.vphi}, is trivial. We fix $a\in A$
and abbreviate $F := \vphi_G(a)$.
\smallskip\noindent
1)\ $G$ acts transitively on $F$, which is a finite set.
By a standard result from group theory, each homomorphism from a
divisible group into a finite one must be trivial (see
\cite[Lemma~2.3 and Proposition~2.4]{GerlachConvPOS} for a proof).
Hence $F = \{a\}$.
\smallskip\noindent
2) The set $H := \{g \in G \suchthat \vphi_g(a) = a\}$
is a clopen subgroup of $G$ (since $A$ is discrete), so $H = G$.
\smallskip\noindent
3) Suppose that $g\in G$ is strongly positive. Then
$\vphi_g(a)$ is a quasi-interior point and an atom, hence $\dom(E) =1$.
In particular, $F = \{a\}$.
\smallskip\noindent
4)\ By Theorem \ref{grp.t.structure}, $\spann(F)$ is a
finite-dimensional $G$-invariant
band of $E$ on which $G$ acts irreducibly. Hence $1 \le \abs{F} =
\dim(\spann(F))\le 1$, by assumption. If follows that $F= \{a\}$.
\smallskip\noindent
5)\ Let $m := \abs{F}$. We may
consider $G$ as a compact Abelian group of $m \times m$-matrices
acting on $\spann(F) \cong \mathbb{C}^m$. Since $G$ is commutative,
it is simultaneously diagonalizable. Each diagonal entry in a
simultaneous diagonalization is an eigenvalue of $G$.
By Theorem \ref{grp.t.structure} such an eigenvalue is torsion, and hence, by
asumption, trivial. This means that $G$ acts as the identity on $\spann(F)$,
which implies that $F = \{a\}$.
\end{proof}
\section{Convergence of Positive Semigroups} \label{s.conv}
We shall now combine Theorem~\ref{psg.t.main} with the findings of the
previous section to obtain general results about strong convergence
of positive operator semigroups. In all results of this section we shall
take the following
hypotheses (those of Theorem~\ref{psg.t.main}) as a starting point:
\begin{itemize}
\item $E$ is a Banach lattice;
\item $T= (T_s)_{s\in S}$ is a positive and bounded operator semigroup
on $E$;
\item $T$ restricts to a bounded semigroup on $E_y$ and
is $E_y$-to-$E$ quasi-compact for some quasi-interior point $y\in E_+$.
\end{itemize}
Let us call these our {\emdf standard assumptions} for the remainder
of this paper. The standard assumptions warrant that Theorem \ref{psg.t.main}
is applicable, and we freely make use of this fact in the following.
\subsection*{Spectral-Theoretic Consequences}
We first draw some spectral-theoretic conclusions.
\begin{thm}\label{conv.t.spec}
Suppose that an operator semigroup $(T_s)_{s\in S}$ on
a complex Banach lattice $E$ satisfies the standard assumptions.
Then the following assertions hold:
\begin{aufzi}
\item Each unimodular eigenvalue of $T$ is torsion.
\item If $T$ is irreducible, then it has only finitely many unimodular
eigenvalues.
\item $T$ is strongly convergent if and only if $T$ has no
non-constant torsion eigenvalue.
\end{aufzi}
\end{thm}
\begin{proof}
a)\ By Theorem \ref{rep.t.spec},
each unimodular eigenvalue of $T$ is the restriction
of an eigenvalue of
\[ \calG := \calT\res{E_\infty},
\]
where $E_\infty = [\ran(P_\infty)]$. Since the latter space is atomic and $\calG$ is
compact, Theorem \ref{grp.t.structure} yields that this eigenvalue is torsion.
\smallskip\noindent
b)\ If $T$ acts irreducibly on $E$ then, by Theorem \ref{psg.t.main},
$\calG$ acts irreducibly on $[\ran(P_\infty)]$. Hence, $\calG$ has
only finitely many unimodular eigenvalues by Theorem \ref{grp.t.structure}.
By Theorem \ref{rep.t.spec}, each eigenvalue of $T$ is the restriction of
an eigenvalue of $\calG$. This proves the claim.
\smallskip\noindent
c)\ The ``if''-part follows from Corollary \ref{grp.c.misc}. For the
``only if''-part we suppose that $T_s \to P$ is strongly
convergent. Then $P = P_\infty$ and $\calG =
\{{\mathop{\mathrm{I\mathstrut}}}_{E_\infty}\}$. Since each unimodular eigenvalue of $T$ is the
restriction of an eigenvalue of $\calG$ (Theorem \ref{rep.t.spec}),
$T$ has no non-constant unimodular eigenvalues.
\end{proof}
\subsection*{Sufficient Conditions for Convergence}
Apart from the spectral characterization of the previous theorem,
Corollary \ref{grp.c.misc} yields the following sufficient conditions
for the convergence of a positive semigroup.
\begin{thm}\label{conv.t.main1}
Suppose that an operator semigroup $(T_s)_{s\in S}$ on
a Banach lattice $E$ satisfies the standard assumptions.
In addition, let at least one of the following conditions
be satisfied:
\begin{aufziii}
\item $S$ is essentially divisible (e.g.: $S$ is divisible or generates
a divisible group; cf.~Appendix~\ref{app.sgp});
\item $S$ carries a topology such that $T$ is
strongly continuous and the
only clopen subsemigroup of $S$ containing $0$ is $S$ itself (e.g.: $S$ is connected).
\item $T_s$ is strongly positive for some $s\in S$.
\end{aufziii}
Then $T$ is strongly convergent.
\end{thm}
\begin{proof}
Theorem~\ref{psg.t.main} is applicable, so it suffices to consider the case
that $E\neq \{0\}$ is atomic and has order-continuous norm and that $\calT$ is a compact group of positive invertible operators on $E$. We must
show that $\calT$ acts trivially on $E$.
\smallskip\noindent
1)\ By Corollary \ref{grp.c.misc} it suffices to show that $\calT$ is
divisible. But this follows from a straightforward compactness
argument.
\smallskip\noindent
2)\ Let $\calH$ be any clopen subgroup of $\calT$. Then
$H := \{ s\in S \suchthat T_s \in \calH\}$ is a clopen subsemigroup of
$S$. (Note that $H \neq \emptyset$ since $\calH \neq \emptyset$ is open and
$T_S$ is dense in $\calT$.) By hypothesis, $H = S$, so
$T_S \subseteq \calH$. Since $\calH$ is closed and $T_S$ is dense in
$\calT$, it follows that $\calH= \calT$. Hence, $\calT= \{ {\mathop{\mathrm{I\mathstrut}}}_E\}$
by Corollary \ref{grp.c.misc}.
\smallskip\noindent
3)\ If $T_s$ is strongly positive, then $\calT$ contains a strongly
positive operator, and we conclude with the help of
Corollary \ref{grp.c.misc}.
\end{proof}
\begin{rems}
\begin{aufziii}
\item Condition 1) is satisfied, in particular, if
$S =\mathbb{R}_+$ (divisible semigroup), but also
if $S= \{0\} \cup [1, \infty)$ (not divisible, but generating a
divisible group). Note that in the latter case,
the semigroup direction is just a subordering of the natural one,
but does not coincide with it. Nevertheless, the associated
notions of ``limit'' do coincide.
\item Condition 1) is also satisfied when $S = [0, \infty)$, endowed
with the semigroup operation $(a,b) \mapsto a \vee b =
\max\{a,b\}$. The semigroup direction coincides with the natural ordering.
This semigroup is neither divisible nor
does it generate a divisible group (it is not even cancellative).
However, it is essentially divisible.
On the other hand, this example is a little artificial, as each
element of $S$ is an idempotent, and hence a representation
$T= (T_s)_s$ is just a family of projections with decreasing ranges
as $s$ increases. For such semigroups, the question of
convergence can often be treated by other methods.
\item The semigroup of {\emdf positive dyadic rationals} is
\[ D_+ := \{0\} \cup \bigl\{ \tfrac{k}{2^n} \suchthat k, n \in \mathbb{N}_0
\bigr\}.
\]
The semigroup direction on $D_+$ coincides with the usual ordering.
It is easy to see that $D_+$ is not essentially divisible.
If we endow $D_+$ with its natural topology, $D_+$ is not
connected. However, $D$ is the only clopen subsemigroup of $D_+$
containing $0$.
(Actually, apart from $D_+$ itself there is no other
{\em open} subsemigroup of $D_+$ containing $0$.)
Hence, from Theorem \ref{conv.t.main1}
it follows that each strongly continuous representation
of $D_+$ that satisfies the standard hypotheses is strongly convergent.
Without strong continuity, however, this can fail.
In fact, let $D = D_+ - D_+$ denote the group generated by $D_+$ in the
real numbers. Then $D$ has a subgroup of index $3$ (namely $3D$), so by the
same construction as in the proof of \cite[Theorem~2.5]{GerlachConvPOS}
we can find a positive and bounded representation of
$(D,+)$ on the Banach lattice $\bbR^3$. The restriction of this representation
to $(D_+,+)$ satisfies the standard conditions, but it does not converge.
\item With Theorem \ref{conv.t.main1}, Condition 3), we
generalize a result of Gerlach, cf.{} \cite[Theorem~4.3]{Gerlach2013}.
\end{aufziii}
\end{rems}
\medskip
\subsection*{Lattice Subrepresentations}
A closed linear subspace $F$ of Banach lattice $E$ is called a
{\emdf lattice subspace}, if it is a Banach lattice
with the order induced by $E$ but with respect to an equivalent norm.
A lattice subspace need not be a sublattice.
(By Theorem \ref{psg.l.ranP}, the range of a positive projection is always a
lattice subspace.)
Given a representation $T= (T_s)_{s\in S}$ on a Banach lattice $E$,
each $T$-invariant lattice subspace gives rise to a
{\emdf lattice subrepresentation}. So the lattice subrepresentations
are those subrepresentations where the underlying space is
a lattice subspace.
\begin{thm} \label{conv.t.latsub}
Suppose that an operator semigroup $T= (T_s)_{s\in S}$ on
a Banach lattice $E$ satisfies the standard assumptions.
In addition, suppose that each finite-dimensional
lattice subrepresentation of $T$ is at most one-dimensional.
Then $T$ is strongly convergent.
\end{thm}
\begin{proof}
As in the proof of Theorem \ref{conv.t.main1}
it suffices to consider the case
that $E\neq \{0\}$ is atomic and that
$\calT$ is a compact group of positive invertible operators on $E$.
It then follows that each finite-dimensional
$T$-invariant band of $E$ is at most one-dimensional.
Corollary~\ref{grp.c.misc} is applicable and yields the claim.
\end{proof}
\section{Conclusion: Some Classical Theorems Revisited} \label{s.classical}
In this section we start with a little historical survey and end with
demonstrating how our approach leads to far-reaching generalizations
of the ``classical'' results.
\subsection*{Historical Note}
In 1982, Günther Greiner in the influential paper \cite{Greiner1982a}
proved the following result as ``Corollary~3.11'':
\begin{thm}[Greiner 1982]\label{clas.t.greiner}
Let $T = (T_s)_{s\ge 0}$ be a positive contraction $C_0$-semigroup on
a space $E = \Ell{p}(\prX)$, $1\le p < \infty$, with the following
properties:
\begin{aufziii}
\item There is a strictly positive $T$-fixed vector;
\item For some $s_0 > 0$ the operator $T_{s_0}$ is a kernel operator.
\end{aufziii}
Then $\lim_{s\to \infty} T_s x$ exists for each $x\in E$.
\end{thm}
For the proof, Greiner employed what has become known as ``Greiner's
$0/2$-law'' (see \cite[Theorem~3.7]{Greiner1982a} and also \cite{Greiner1982})
and a result
of Axmann from \cite{Axmann1980}. Both results have involved proofs and make
use of the lattice structure on the regular operators on Banach
lattices with order-continuous norm.
The relevance of Greiner's theorem
derives from the fact that the assumptions can be frequently verified
for semigroups arising in partial differential equations or in
stochastics.
\medskip
For a long time, Greiner's theorem stood somehow isolated within the
asymp\-totic theory of (positive) semigroups.
The ``revival'' of Greiner's theorem as a theoretical result began
with a paper of Davies \cite{Davies2005} from 2005. Davies showed that
the peripheral point spectrum of the generator $A$ of a $C_0$-semigroup
$T$ of positive contractions on a space $E = \Ell{p}(\prX)$, $1\le p <
\infty$, has to be trivial in
the following cases: (1) $\prX$ is countable with the counting measure
and (2) $\prX$ is locally compact and second countable and $T$ has the
Feller property (i.e., each $T_s$ for $s > 0$ maps $E$ into the space of
continuous functions). Case (1) was subsequently generalized by
Keicher in \cite{Keicher2006} to bounded and positive $C_0$-semigroups on
atomic Banach lattices with order-continuous norm, and by Wolff
\cite{Wolff2008} to more general atomic Banach lattices.
Shortly after, Arendt in \cite{Arendt2008} generalized Davies' results
towards the following theorem.
\begin{thm}[Greiner 1982/Arendt 2008]\label{clas.t.arendt}
Let $A$ be the generator of a positive contraction $C_0$-semigroup $T
= (T_s)_{s\ge 0}$ on a space $E = \Ell{p}(\prX)$, $1\le p <\infty$.
Suppose that for some $s_0 > 0$ the operator $T_{s_0}$ is a kernel operator.
Then $\spec_{\mathrm{p}}(A) \cap \ui \mathbb{R} \subseteq \{0\}$.
\end{thm}
This result is ``Theorem 3.1'' in Arendt's paper
\cite{Arendt2008}. Interestingly, as observed by Gerlach in
\cite{Gerlach2013}, it already appears in Greiner's 1982 paper, namely
in the first paragraph of his proof of Theorem \ref{clas.t.greiner}
(i.e., his ``Corollary~3.11''). We
will thus call Theorem \ref{clas.t.arendt} the {\em Greiner--Arendt theorem}.
Arendt points out that Theorem \ref{clas.t.arendt} implies Davies' result:
in case (1) every positive operator is a kernel operator, whereas in case (2) the Feller
property implies that each $T_s$ for $s > 0$ is a kernel
operator. (This follows from Bukhvalov's characterization of kernel
operators, cf.~\cite[Corollary 2.4]{Arendt2008}.)
Let us briefly sketch Arendt's proof of Theorem \ref{clas.t.arendt}:
If $f\in E$ is an eigenvector of $A$ for the eigenvalue $\lambda \in \ui \mathbb{R}$,
then $T_s\abs{f} \ge \abs{f}$
for all $s\ge 0$. Since each $T_s$ is a contraction and the norm on
$\Ell{p}$ is strictly monotone, it follows that $\abs{f}$ is a fixed
point. By restricting to the set $\set{ \abs{f} > 0}$ one
can assume that $\abs{f}$ is strictly positive. Next, from the
weak compactness of the order interval $[0, \abs{f}]$ it follows that the semigroup
is weakly relatively compact. Then the JdLG-theory enters
the scene and reduces to problem to an atomic Banach lattice with
order continuous norm. Finally, Keicher's analysis from \cite{Keicher2006} shows that the
dynamics there must be trivial, and hence $\lambda=0$.
Arendt's paper is remarkable in several respects. First of all,
his proof of Theorem \ref{clas.t.arendt} employs the JdLG-theory which is
central also to the more recent work of Gerlach
and Glück, and to the present paper. Secondly, Arendt recalls
Greiner's Theorem \ref{clas.t.greiner} and gives a
proof (building,
as Greiner did, on Theorem \ref{clas.t.arendt})
under the additional assumption that the semigroup is
irreducible. (This proof appears
to be the first complete one in English language, cf. \cite[Remark
4.3]{Arendt2008}.)
Thirdly, Arendt promotes Greiner's result by illustrating its use with
several concrete examples.
Most remarkable of all, however, is what is {\em not} written in
\cite{Arendt2008}: namely that
Greiner's Theorem \ref{clas.t.greiner} almost
directly implies the Greiner--Arendt Theorem \ref{clas.t.arendt}. Indeed, one starts
exactly as in Arendt's proof until one has found the quasi-interior fixed point
$\abs{f}$; then Greiner's theorem tells that $\lim_{s\to \infty}
T_sf$ exists, and hence $\lambda = 0$ follows.
\medskip
In the following years the topic was taken up by M.~Gerlach and
J.~Glück. Gerlach \cite{Gerlach2013} discussed Greiner's approach in a general
Banach lattice setting and extended it to semigroups that merely dominate a kernel operator; he also noted
that the dominated kernel operator can be replaced by a compact
operator. In their quest to find a unifying framework, and stimulated
by ``Corollary~3.8'' in Keicher's paper \cite{Keicher2006},
Gerlach and Glück in \cite{Gerlach2017,GerlachConvPOS} finally
identified ``AM-compactness'' as the right property generalizing the
different cases. Alongside a unification, this led also to a major
simplification, since AM-compactness is much more
easily shown directly than by passing through the concept of a kernel
operator. (E.g., it follows directly from Theorem \ref{int.t.fact-Linfty} that a
Feller operator as considered by Davies is AM-compact.)
Finally, Gerlach and Glück realized that strong continuity of the
semigroup can be dispensed with, since arguments
requiring time regularity can be replaced by purely
algebraic ones. This led to proofs for most of the above-mentioned results
for semigroups without any time regularity.
\medskip
Somewhat independently from the above development, Pich\'or and
Rudnicki proved convergence results for Markov semigroups which merely dominate a
non-trivial kernel operator \cite[Theorems~1 and~2]{Pichor2000}. Their
results are closely related to (and earlier than) the results of Gerlach \cite{Gerlach2013},
but their approach is different, focusing on $L^1$-spaces and employing methods from stochastics.
Later on, in \cite{Pichor2016, Pichor2018a},
these authors adapted their original results to various situations
involving semigroups
on $L^1$-spaces, with numerous applications in mathematical biology.
\begin{rem}
As noted above,
the implication ``Theorem \ref{clas.t.greiner}$\dann$Theorem
\ref{clas.t.arendt}'' is almost immediate. Now,
in hindsight, it becomes clear that Arendt in \cite{Arendt2008} was
also very close to proving the converse implication
``Theorem \ref{clas.t.arendt}$\dann$Theorem \ref{clas.t.greiner}''.
Indeed, the weak compactness of order intervals in $\Ell{p}$-spaces
for $1\le p < \infty$ implies that on such spaces a positive bounded
semigroup $T=(T_s)_{s\ge 0}$
with a quasi-interior fixed point is relatively weakly compact. Hence,
the ``triviality of the peripheral point spectrum'' asserted by
Theorem \ref{clas.t.arendt} implies that $T$ acts trivially
on the ``reversible'' part of the corresponding JdLG-decomposition.
One can then infer strong convergence of $T$ if one knows
that $T$ is not just relatively weakly, but even relatively {\em strongly}
compact. And the latter holds, in fact,
since kernel operators are AM-compact; but in this context this
was noted only later by Gerlach and Glück.
\end{rem}
\subsection*{Old Theorems in a New Light}
Let us now review some of the abovementioned results in the light of
our actual findings. First of all, consider the following result,
which is merely an instantiation of Theorem~\ref{conv.t.main1} a), to
the (divisible!) semigroup $\mathbb{R}_+$.
\begin{thm}\label{clas.t.greiner-new}
Let $T = (T_s)_{s\ge 0}$ be a positive and bounded (but not necessarily strongly
continuous) semigroup on a Banach lattice $E$
with the following
properties:
\begin{aufziii}
\item There is a quasi-interior point $y\in E_+$ and $c > 0$ such
that $T_s y \le c y$ for all $s\ge 0$.
\item For some $s_0 > 0$ the operator $T_{s_0}$ is AM-compact.
\end{aufziii}
Then $\lim_{s\to \infty} T_s x$ exists for each $x\in E$.
\end{thm}
Theorem \ref{clas.t.greiner-new} is a slight strengthening of Theorem~4.5 from
\cite{GerlachConvPOS}, where the quasi-interior point $y$ is required
to be $T$-fixed. It implies
Greiner's Theorem \ref{clas.t.greiner} as a special case:
simply note that on $E:= \Ell{p}(\prX)$ a strictly positive function
is quasi-interior and that a kernel operator is AM-compact (Theorem \ref{int.t.AM}).
\medskip
In the next result, the requirement that one of the semigroup
operators is AM-compact is relaxed towards a mere domination property, however
on the expenses of strengthening other hypotheses.
\begin{thm}\label{clas.t.pichor-new}
Let $T = (T_s)_{s\ge 0}$ be a positive, bounded and irreducible (but not necessarily strongly
continuous) semigroup on a Banach lattice $E$ with order continuous norm.
Suppose that the conditions are satisfied:
\begin{aufziii}
\item There is a quasi-interior point $y\in E_+$ such
that $T_s y \le y$ for all $s\ge 0$.
\item For some $s_0 > 0$ there is an AM-compact operator $K\neq 0$ with $0 \le K \le T_{s_0}$.
\end{aufziii}
Then $\lim_{s\to \infty} T_s x$ exists for each $x\in E$.
\end{thm}
\begin{proof}
By Lemma~\ref{classical.l.dom}, $T$ satisfies the standard assumptions
(see Section~\ref{s.conv}). Hence, as $\mathbb{R}_+$ is a divisible semigroup,
the assertions follow from Theorem~\ref{conv.t.main1} a).
\end{proof}
Theorem \ref{clas.t.pichor-new} is a generalization of the
abovementioned results \cite[Theorems~1 and~2]{Pichor2000} of Pich\'or
und Rudnicki for stochastic $C_0$-semigroups on $L^1$-spaces.
For $C_0$-semigroups on Banach lattices, the theorem is due to Gerlach \cite[Theorem~4.2]{Gerlach2013}.
We note that irreducibility of the semigroup can be replaced by other assumptions
ensuring that the AM-compact operator $K$ is ``sufficiently large'' when compared
with the semigroup. A very general result of this type was proved by
Gerlach and Gl\"uck in \cite[Theorem~3.11]{GerlachConvPOS}.
\bigskip
Let us finally return to the spectral-theoretic results (by
Davies, Keicher, Wolff and Greiner--Arendt) discussed above. In this direction, we
establish the following general theorem.
\begin{thm}\label{clas.t.gluha}
Let $T = (T_s)_{s \in S}$ be a bounded and positive
semigroup on a Banach lattice $E$. Suppose that for some
$s\in S$ the operator $T_s$ is AM-compact.
Then the following assertions hold:
\begin{aufzi}
\item Each unimodular eigenvalue is torsion.
\item If $T$ is irreducible then there are only finitely many unimodular
eigenvalues.
\item Suppose that {\rm 1)} $S$ is essentially divisible or {\rm 2)}
$T$ is strongly continuous
with respect to some topology on $S$ such that the only clopen
subsemigroup of $S$ containing $0$ is $S$ itself.
Then the only possible unimodular eigenvalue of $T$ is the constant one.
\end{aufzi}
\end{thm}
\begin{proof}
We combine the classical ideas from Scheffold \cite{Scheffold1971}
as employed by Keicher in \cite[Theorem
3.1]{Keicher2006} with the theory developed in this paper.
\smallskip\noindent
a)\ Let $\lambda = (\lambda_s)_{s\in S}$ be a unimodular eigenvalue of
$T$, and let $0\neq z \in E$ be a corresponding eigenvector.
Abbreviate $y := \abs{z} \in E_+$. Then
\[ 0 \neq y = \abs{z} = \abs{\lambda_s z} = \abs{T_sz}\le T_s \abs{z}
= T_s y\qquad
(s\in S).
\]
It follows that the net $(T_sy)_{s\in S}$ is increasing.
\smallskip\noindent
One can find a positive
linear functional $\vphi \in E'_+$ such that $\vphi(y) > 0$. Define
\[ J := \{ x\in E \suchthat \lim_{s\in S} \vphi( T_s\abs{x}) = 0\}.
\]
It is routine to check that $J$ is a closed and $T$-invariant ideal.
Since $\vphi(T_sy) \ge \vphi(y) > 0$ for all $s\in S$, we have $y \notin J$.
Moreover, $T_ty -y \in J$ since
\[ \vphi(T_s\abs{T_t y - y}) = \vphi T_s (T_t y - y) =
\vphi(T_{t{+}s}y)
- \vphi(T_s y) \qquad (s\in S)
\]
and $(\vphi(T_sx))_{s\in S}$ is increasing and bounded.
\smallskip\noindent
Since $J$ is a closed $T$-invariant ideal, the quotient space
$E_1 := E/J$ naturally carries the structure of a Banach lattice,
and the representation $T$ on $E$ induces a representation $\hat{T}$
on $E_1$ by
\[ \hat{T}_s(x{+}J) := T_s x + J \qquad (s\in S,\, x\in E).
\]
Let $\hat{z} := z+J$ and $\hat{y} := y + J$ be the equivalence classes of $z$ and $y$ in $E_1= E/J$,
respectively. Since the canonical surjection is a lattice
homomorphism, $\hat{y} = \abs{\hat{z}}$ in $E_1$. Since $y \notin J$,
$\hat{z} \neq 0$. It follows that $\hat{z}$ is an eigenvector
of $\hat{T}$ for the eigenvalue $\lambda$.
\smallskip\noindent
Since $T_s y - y\in J$ for each $s\in S$,
the point $\hat{y}$ is $T$-fixed for the induced semigroup on
$E_1$. Moreover, by the hypothesis and Theorem \ref{AM.t.factor}, for some $s\in S$
the operator $\hat{T}_s$ is AM-compact. Hence, when we restrict to the
closed ideal $E_2 := \cls{F_{\hat{y}}}$ generated by $\hat{y}$ in
$E_1$,
we find that the semigroup $\hat{T}$ restricted to $E_2$ satisfies the
standard assumptions. Theorem \ref{conv.t.spec} then yields that
that $\lambda$ must be torsion.
\smallskip\noindent
c)\ We start again as in the proof of a). By Theorem
\ref{conv.t.main1}, either condition 1) and 2) implies that
$\hat{T}$ on $E_2$ is convergent. Then, by Theorem \ref{conv.t.spec}, we conclude that
$\lambda$ is constant.
\end{proof}
Theorem \ref{clas.t.gluha} generalizes the Greiner--Arendt Theorem
\ref{clas.t.arendt}: simply specialize $S = \mathbb{R}_+$ and note that kernel
operators are AM-compact (Theoren \ref{int.t.AM}). A fortiori, it
generalizes Davies' results from \cite{Davies2005}. However, it also
implies Keicher's result \cite[Theorem ~3.1]{Keicher2006} (but not
Wolff's), as on an atomic Banach lattice with order-continuous norm all
order intervals are relatively compact, and hence all bounded operators are
AM-compact. Finally, Theorem \ref{clas.t.gluha} also generalizes
\cite[Theorem~4.19]{GerlachConvPOS}.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,532 |
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Page Copyright
Première partie
Deuxième partie
Troisième partie
Remerciements
Biographie
À mes parents
Ceux qui connaissent le jour de Brahma
qui dure mille âges
et sa nuit, qui ne prend fin qu'après mille âges
ceux-là connaissent le jour et la nuit.
Et la foule des êtres,
indéfiniment ramenée à l'existence,
se dissout à la tombée de la nuit
et renaît au lever du jour.
Et toutes les créatures sont en moi
comme dans un grand vent
sans cesse en mouvement dans l'espace.
Je suis l'être et le non-être,
l'immortalité et la mort [...]
# Première partie
1
Lorsqu'il était petit, il avait vu mourir un arbre-Gomphal.
*
Ils trouvent Lian à l'aube du jour de l'an, sur l'esplanade près du port, au bord de la Mer. Nu, inconscient, ensanglanté. Il a dû se blesser aux grillages interdisant l'accès aux quais. Il a de la chance, c'est Jill Estérazy qui le découvre, avec Jaques Hueng, alors qu'ils font leur habituel périple avant la foule, elle à grandes poussées rythmées sur les roues de son fauteuil roulant, lui à côté, au petit trot, avec le chien. Ils l'emportent en hâte à la commune toute proche sur les genoux de Jill, étrange madone. Quand il ouvre les yeux, et qu'ils lui demandent comment il s'appelle, ils entendent mal son murmure, ils comprennent : " Liam ". Et comme il est blond, avec des yeux gris bleu trop pâles dans sa face brune, et qu'ensuite il semblera comprendre leur anglam, ils lui font faire des faux papiers au nom de Liam Shaunessy.
*
Lorsqu'il était petit, il avait vu mourir un arbre-Gomphal.
L'arbre a toujours été là devant la maison, en plein soleil, avec les boules duveteuses de ses rejetons autour de lui, au ras du sol, comme une couronne. Lian a vu naître le dernier : d'abord une branche étirée jusqu'au sol, déformant la masse sphérique des filaments à la luminescence maintenant atténuée. Puis la tête de l'arbre-Gomphal retrouve sa luminosité habituelle, sa belle ligne courbe. L'arceau, d'abord souple et doux au toucher, est devenu aussi dur et rugueux que les autres : on peut s'y appuyer, s'y suspendre, tourner autour en s'y accrochant d'une main. Au ras du sol, là où l'arceau plonge en terre, une minuscule pousse verte pointe bientôt. Devient, en une journée, une brindille. Trois jours plus tard, un brouillard blanchâtre se matérialise à son sommet : une huitième réplique miniature de l'arbre-Gomphal, mais comme les autres inachevée, sans vie ; quand on effleure leurs filaments, même pas de vague picotement – il ne faut jamais toucher ceux de l'arbre-Gomphal, Lian l'a appris à ses dépens.
L'arbre a ses petits régulièrement, un par année : huit arceaux, huit petits Gomphali, huit années. Lian est dans sa troisième saison, mais il sait déjà qu'herbes, feuilles et arbres vivent un autre temps que lui, un temps qui revient sans cesse sur son chemin, alors que les êtres humains, et lui, Lian, qui en est un, continuent tout droit, sans jamais revenir – mais pour aller où ?
Et où placer l'arbre-Gomphal, dans ce commencement d'ordre qu'il a trouvé au monde ? L'écorce du tronc (qu'on a le droit de toucher, elle) est souple et douce comme une peau et, en collant l'oreille tout contre, on peut sentir comme une vibration... Mais les filaments enchevêtrés de sa chevelure, lorsque Père-Nathénèk les fait observer de près à Lian, ressemblent bien à de l'herbe ou à des feuilles, très minces, longs et étroits, un bruit de soie quand le vent les frotte les uns contre les autres. Le nom même, " arbre-Gomphal "... Lian a l'habitude de dire " Père-Nathénèk " et " Mère-Laraï " quand il n'utilise pas les termes plus enfantins " béra ", " ati ". Et l'arbre est une personne : il pense, il sent, il se souvient. Il parle, même – c'est ce que disent les histoires de Laraï. Alors, il s'appelle " Gomphal ", même si ce n'est pas une personne de la même façon que Nathénèk ou Laraï. Comme eux il a des enfants, comme eux il traverse les saisons sans vraiment changer, comme eux il a toujours été et sera toujours là. Comme eux, surtout, il est unique : il n'y a pas d'autre arbre-Gomphal aux alentours ; les rejetons ne comptent pas : si petits, si inertes quand on les touche...
Mais voilà que les arceaux de l'arbre-Gomphal sont cassés ! Ils se dressent de travers dans le ciel, tout noirs ; la brise les agite comme du papier, il y en a des morceaux par terre, secs et creux. Et, là où s'étaient timidement blottis les rejetons de l'arbre, se gonflent huit sphères chatoyantes, deux fois plus grosses que la veille.
L'arbre lui-même a rétréci. La peau de son tronc s'écaille en lamelles, qui brillent d'une lueur sourde sur le sol, la boule ternie de ses filaments semble si pesante...
Nathénèk dit : « L'arbre-Gomphal va mourir. »
Mourir, cela arrive parfois aux gens. L'autre père de Lian, Dougall, son père-par-le-sang, est " mort " ainsi quand Lian était tout petit. Au début, il pensait que c'était seulement une façon de dire que Père-Dougall n'était plus là. Comme Micahmee, la sœur de Père-Nathénèk. Un jour, Micahmee a quitté le plateau pour aller vivre dans l'Ouest ; elle écrit, des fois. Mais on n'a jamais dit qu'elle était " morte ". Dougall, c'est différent. Mère-Laraï a dit " il est mort ", ses yeux se sont assombris d'une façon particulière. Cette absence-là est triste, pas du tout comme celle de Micahmee. Il y a deux façons de ne pas être là, alors : on " part " ou bien on " meurt ". Quand on meurt, c'est plus triste.
Pourtant, on dit aussi des plantes et des animaux qu'ils " meurent ", et ni Laraï ni Nathénèk ne semblent en avoir beaucoup de peine. Parce qu'animaux et plantes dorment, en réalité, invisibles dans la terre, pour se réveiller plus tard et s'étirer vers le soleil. Mais les gens qui meurent, d'une certaine façon, c'est pareil, Nathénèk l'a bien expliqué à Lian : on les emporte dans l'île d'Ëlmadziulan, après avoir fait pousser une coque de cristal bleu autour de leur corps, et on les installe sur l'île. Puis, quand vient le temps, on les retire du cristal, on les plante en terre, et avec eux on plante des graines de tingalyai, l'arbre de vie ; l'arbre pousse en puisant la substance du mort, et quand il est devenu assez grand pour avoir des graines, on les récolte et on les envoie à tout le monde, et on les plante dans les cours intérieures, dans les rues et dans les villes et sur les digues, partout, pour avoir de l'eau.
« Les tingalyai vivent très, très longtemps, au moins deux mille saisons, a continué Nathénèk. Et ainsi les morts restent avec nous. Et quand l'arbre a fini sa vie, nous en faisons des objets, des meubles, des statuettes, et le souvenir dure ainsi pendant encore des générations. » Fasciné, Lian a regardé les trois tingalyai qui s'abreuvent au lac et alimentent le bassin et les canalisations de la maison. « Est-ce que Père-Dougall est ici, alors ?
— Non, a dit Nathénèk avec un soupir. Il est ailleurs. » Et comme ses yeux s'étaient assombris aussi, comme ceux de Mère-Laraï, Lian n'a pas insisté.
Pour l'arbre-Gomphal, Nathénèk a dit " mourir " et Lian demande tout de suite : « Comme Père-Dougall ? » L'ombre familière passe dans les yeux de Nathénèk, qui dit avec lenteur : « Oui, comme ton père-Dougall. »
Avec un mélange d'angoisse et de curiosité, Lian observe les progrès de la mort chez l'arbre-Gomphal.
D'abord la rupture des arceaux, la vie soudaine des petits : leur boule est animée de scintillements lents, leur tronc mince s'est assoupli. Puis la sphère du gros arbre s'éteint complètement : à la place de la blancheur frémissante irisée par le soleil, une masse triste et terne. Ensuite elle change de forme. Ou plutôt elle cesse d'avoir une forme. Les filaments fusionnent en une pellicule opaque ; il en pend de grands morceaux sur les branches soudain devenues visibles, comme du linge mouillé.
Les herbes, la mousse, les petites plantes qui vivaient dans la lumière atténuée de l'arbre commencent à se faner. Deviennent noires, se recroquevillent, un grand cercle au pied de l'arbre, comme de la terre brûlée.
Pendant deux jours, rien. Les petits Gomphali vibrent et scintillent, leur père se tient immobile et silencieux au milieu de leur cercle enfin animé. Le troisième jour, alors que le soleil atteint le milieu du ciel, les grandes feuilles opaques drapées sur les branches redeviennent translucides. L'arbre frissonne, une vague se propage dans sa blancheur ternie, une neige de particules légères se met à tomber. Elles disparaissent en l'air avant d'atteindre le sol. Il ne reste bientôt plus que les branches, des traits noirs qui dessinent encore une sphère si on les regarde de loin, mais on peut voir le ciel au travers.
Lian se tient devant Laraï, et il demande, la nuque appuyée contre le ventre de sa mère : « Il va mourir, maintenant, ati ?
— Il a vécu très, très longtemps, et il a eu beaucoup d'enfants.
— Mais il va mourir ! Pourquoi ?
— Pour que ses enfants puissent vivre », dit Nathénèk derrière eux, un peu trop vite, un peu trop fort, et Lian sent se raidir les muscles du ventre de Laraï. Il renverse la tête pour regarder sa mère ; à l'envers, c'est difficile de déchiffrer l'expression d'un visage ; Laraï hoche simplement la tête : « Oui. » Mais, comme dans l'arbre-Gomphal, Lian a vu passer en elle une vague non de lumière mais d'obscurité.
Cette nuit-là, il y a un grand craquement devant la maison, comme un cri. De la fenêtre de sa chambre, Lian regarde, terrifié. Le squelette noir oscille sous les lunes avec ses doigts mutilés, un mouvement lent d'abord, qui s'amplifie, et enfin, réduites en poussière toutes les racines qui l'attachaient à la terre, l'arbre-Gomphal s'incline, et s'écroule.
Quelques jours plus tard, deux des petits ont rompu le cercle. Et bientôt un, deux, trois, tous les autres commencent à s'éloigner aussi de l'arbre abattu. Lian contemple le sol nu, les trous profonds où s'étaient trouvées leurs racines. Nathénèk retourne l'un des petits arbres et montre à Lian les racines à présent pseudopodes qui se tordent lentement.
« Remets-le, béra, remets-le par terre ! » crie Lian, horrifié et apitoyé. Avec comme un scintillement plein de colère, la petite boule duveteuse se remet à avancer. Ils bougent le jour et ils se nourrissent la nuit, dit Nathénèk.
« Mais où est-ce qu'ils vont ?
— Vers le Nord. Ils remontent au pays des Gomphali, près du Leïtltellu. Ça leur prendra des dizaines et des dizaines de saisons. Ensuite, ils redescendront vers le Sud, là où il fait plus chaud, et ils se chercheront une place au soleil pour s'arrêter et avoir leurs propres enfants.
— Et après, ils mourront ? »
Nathénèk lui ébouriffe les cheveux : « Ce sera dans très, très longtemps, quand ta mère-Laraï, toi et moi, nous serons partis depuis des années.
— Quand nous serons morts ? » dit Lian, épouvanté, le dernier cri de l'arbre dans les oreilles.
Nathénèk fronce les sourcils. Il s'accroupit pour mettre sa tête au même niveau que celle de Lian, il lui prend les bras et il les serre, fort : « Écoute-moi bien » – il a l'air un peu fâché – « les gens ne meurent pas. Ils s'en vont. Ils partent avec la Mer et ils vivent pour toujours. C'est là que nous irons, Mère-Laraï, toi et moi...
— Et Père-Dougall ? »
Nathénèk se relève ; son visage à contre-soleil est un masque d'ombre dans lequel Lian ne voit rien : « Quelquefois, très rarement, il arrive qu'on ne puisse pas rejoindre la Mer. C'est ce qui est arrivé à ton père-Dougall. Mais si tu es un bon garçon, tu n'auras pas à t'en faire, tu rejoindras la Mer quand ton temps sera venu. »
Il part de sa démarche chaloupée et Lian reste là avec le petit Gomphal qui rampe lentement, presque imperceptiblement, vers le Nord. Est-ce que Nathénèk veut dire que Père-Dougall n'a pas été bon ? Qu'il est mort à cause de cela ? C'est quoi, la Mer ? Comment sait-on que le temps est venu ?
*
Il ne reprend pas conscience avant trois jours. Jack ou d'autres changent ses pansements, le nourrissent, lui donnent à boire, le baignent. Jill ne le quitte pas une minute. Quand il se réveille et qu'il la voit, il est simplement surpris. La chambre ne l'étonne pas trop : c'est le genre de décor qu'il se rappelle, avec les fenêtres en losanges, aux encadrures sculptées, les murs où alternent bandes de pierre rouge et bandes de pierre dorée. Le lit est plus haut, les meubles ont des formes inhabituelles, le modèle de la lampe de chevet diffère de ceux qu'il connaît, mais ce n'est pas important, sans doute des particularités locales, il n'est jamais allé sur la côte ouest jusqu'alors.
Ensuite, il commence à comprendre. Parce qu'il ne saisit pas très bien l'accent de Jill, d'abord, ne l'a jamais entendu auparavant – et qu'elle répond à sa question : « Mais c'est l'accent de Bird-City. » Ensuite parce que d'autres membres de la commune font leur apparition, et que ce sont tous de purs Virginiens : il sait bien qu'il ne se trouvait pas dans le sud-est du continent ni dans l'île de Krillialtaoz quand la Mer est partie.
Quand la Mer est partie.
Ils s'inquiètent alors en voyant son expression, lui disent de ne pas avoir peur, que personne ne lui fera de mal, qu'ils sont tous des amis. C'est là qu'ils lui demandent son nom et entendent " Liam ".
2
Un matin, vers la fin de la première lunaison de Hékeltéñu, Laraï et Nathénèk commencent à préparer les bagages, et Lian comprend qu'il sera du voyage. Il n'a jamais quitté la maison. Il ne sait trop s'il doit s'en réjouir ou s'en inquiéter, mais il est plutôt soulagé. La semaine précédente, dans son lit, le soir, il entendait les voix de ses parents, qui s'efforçaient de rester feutrées. Laraï ne voulait pas partir, Nathénèk voulait que Lian les accompagne et répétait : « Il ne va pas rester ici toute sa vie ! »
C'est un très long voyage ; Lian somnole souvent, bercé par le rythme hypnotique des sabots des deux aski attelés à la carriole. Un jour, il s'est encore endormi, mais quand il se réveille on est sorti des montagnes, la carriole roule sur une route de dalles rouges et polies, à travers des collines aux boisés aimables, bien différents des grands arbres sauvages auxquels il est habitué. On arrive bientôt à une rivière – il n'a jamais vu autant d'eau courante. On charge la carriole et les aski sur un petit bateau à aubes muni d'une cheminée d'où s'échappent des panaches de fumée blanche. Dans un halètement pressé, le bateau quitte le quai, s'engage dans le courant, et les rives défilent à toute allure sous les yeux écarquillés de Lian.
Ils restent à l'écart sur le bateau, mangent entre eux, ne parlent pas aux matelots ni au capitaine, un grand et gros homme à la peau très foncée, au crâne couvert de petites nattes noires cruellement serrées, et que Lian regarde de loin, un peu effrayé. Il n'a jamais vu personne d'autre que son père et sa mère. Il préfère regarder les arbres, puis les collines qui ondulent de chaque côté des rives.
Après la rivière, c'est la savane, à perte de vue, une étendue presque plane, bien plus grande que le plateau. Les grandes herbes en sont déjà à moitié couchées, toutes bleuies par le soleil, il y pointe seulement de rares arbustes rabougris – mais, parfois, la boule blanche d'un Gomphal s'y arrondit, majestueuse. Malheureusement, la plaine devient vite aussi monotone que la montagne et la rivière. On s'arrête bien dans une " auberge " ou un " relais ", de temps en temps, mais très brièvement, pour acheter de quoi manger ; on n'y couche que lorsqu'il pleut, le reste du temps on dort sous les étoiles. Dans les auberges et les relais, on ne va pas au " dortoir " avec tout le monde, on prend une chambre, et c'est là qu'on mange ; Lian ne sait s'il en est satisfait ou déçu ; mais c'est fascinant, tous ces gens qui ne sont ni Laraï ni Nathénèk, tous différents, et les enfants, surtout, qui courent partout – Lian aimerait bien courir avec eux, mais Laraï a été très claire : il ne doit jamais s'éloigner seul de la chambre. D'un autre côté, parfois, il y a des gens qui les regardent d'un drôle d'air, ses parents et lui, quand ils arrivent dans une auberge. Pas vraiment méchant, mais surpris, ou compatissant, ou ennuyé. En réalité, Lian s'en rend compte, c'est surtout lui qu'on regarde – ou qu'on s'efforce de ne pas regarder, ce qui est encore plus bizarre. Quand il demande pourquoi à Laraï, elle répond : « Parce qu'ils ne te connaissent pas », et il doit s'en contenter, car Nathénèk ne dit rien. Est-ce que tous les gens se connaissent, alors, dans les auberges ?
Laraï et Nathénèk ne parlent à personne, pourtant. Aux premières heures de l'aube, on repart, et le voyage recommence, dans le cliquetis régulier des sabots sur les dalles polies. Une fois, Lian aperçoit au loin un troupeau de tovik qui filent la corne haute, crinière et queue emportées par la course. Il voudrait les voir revenir, mais la plaine infinie les a avalés.
Il essaierait bien de poser des questions, mais il comprend très bien que ses parents n'en ont pas envie. Ils échangent entre eux des paroles brèves ; quelquefois Nathénèk se met à chanter, mais la voix de Laraï se joint rarement à la sienne, et il finit par se taire.
Et enfin, enfin, le paysage change à nouveau, la plaine s'étage en collines de plus en plus élevées, et même parfois rocailleuses, d'où souffle un vent plus chaud. « Les collines près de la Mer », dit Nathénèk avec un soupir de contentement. L'herbe y est plus jaune, il y a de vrais arbres, et de plus en plus souvent des maisons, d'abord isolées, puis groupées en hameaux. Sur la route, maintenant, on rattrape d'autres carrioles et de gros chariots de toutes sortes, remplis de gens aux habits gaiement colorés, cinq, six par chariot, parfois plus. Plus légère, la carriole de Nathénèk les double et on échange des saluts polis avec leurs passagers. Tout le monde a l'air très joyeux.
À la nuit, le vingt-cinquième jour (Lian sait déjà compter sur ses doigts : cela fait cinq mains qu'ils sont partis), ils arrivent au sommet de la dernière rangée de collines, la plus haute. En contrebas s'étend une plaine obscure, car les lunes ne sont pas encore levées. Disposées à intervalles réguliers au flanc de la longue colline, des moitiés de ronds bleutés brillent dans la pénombre.
« C'est ça, la Mer ? C'est là qu'on va ?
— Non, dit Nathénèk. Plus au Nord, au lieu du rassemblement. »
Les moitiés de ronds bleus sont de grosses pierres arrondies presque aussi hautes que la carriole, et la route les suit. Bientôt des taches de lumières sourdes, au loin, deviennent des tentes rondes, carrées, en triangle, dressées en groupe ici et là, avec des feux, des carrioles et des chariots, des aski dételés qui paissent dans les allées, et même quelques tovik qui les dominent de la tête et de l'encolure, avec des rubans tressés dans leur crinière. Lian a un peu mal au cœur ; les auberges, ce n'était rien, il n'a jamais vu tant de monde à la fois.
Laraï choisit une place à l'écart, la tente est bientôt dressée, le feu allumé, le repas en train de cuire sur les braises. Le ventre plein, Lian se sent mieux. Il y a de la musique quelque part au centre du camp, mais une main le rattrape au vol. « Reste là, Lian ! » Pourquoi Laraï est-elle fâchée ? Il proteste : « Mais, ati, la musique... »
Le visage de Laraï semble se défaire ; elle s'agenouille près de lui : « Nous irons ensemble plus tard, Lian. Tu ne dois pas y aller tout seul. Promets-moi de rester avec Nathénèk pour l'instant. »
Elle n'est pas fâchée, elle a peur ! Étonné, inquiet, Lian promet. Elle s'en va, revient bientôt avec des beignets sucrés en forme de spirale dont Lian se bourre, ravi. Après, il a tellement sommeil qu'il oublie la musique.
Un bruit de voix assourdies le réveille ; on parle dehors à mi-voix ; c'est toujours la nuit ; l'ouverture de la tente découpe un morceau de ciel étrangement violet. « Il le faut », dit une voix inconnue. Une ombre apparaît dans l'entrée. Père-Nathénèk. Il vient secouer doucement Lian : « Viens, Lian, viens voir la Mer. » Dehors, deux autres silhouettes, celle de Mère-Laraï et une autre, un homme, moins grand qu'elle, moins grand que presque tout le monde. Dans le ciel, les trois petites lunes ont disparu et la grosse lune n'est plus pleine : un ovale noir est en train de flotter lentement au travers, et cela fait comme un œil.
Un grand silence règne à présent sur le campement, et pourtant, tout le monde marche vers le bas de la colline, vers la ligne des pierres phosphorescentes. Lian a essayé de prendre la main de sa mère, mais Laraï semble distraite et sa main reste inerte dans celle de Lian ; quand il la lâche, pour voir, elle ne le retient pas. Mais c'est la nuit, la lumière de la lune est trop étrange, il y a trop de monde autour d'eux : il reste près de Laraï. Au bout d'un moment, une autre main enveloppe la sienne ; il croit que c'est son père-Nathénèk, mais c'est le petit homme inconnu. Ils se regardent un moment tout en avançant. L'homme n'est pas très vieux, il a des cheveux sombres qui lui descendent sur les sourcils ; son visage est un peu bizarre, Lian ne saurait dire pourquoi. Il ne sourit pas vraiment, mais il a l'air gentil. Comme ni Laraï ni Nathénèk ne disent rien, Lian accepte sa compagnie.
Tout à coup, il ne sait comment, il se retrouve avec l'inconnu en avant de la foule ; devant eux, sous la lumière violette, la plaine obscure est immobile et déserte au-delà des pierres bleutées. Derrière eux, les bruissements se taisent peu à peu : la foule a cessé d'avancer. Lian se sent soudain très vulnérable, comme si cette présence invisible le poussait malgré lui en avant, mais il ne veut pas dépasser la ligne des pierres. Il n'ose se retourner pour voir où est Laraï.
L'inconnu ne bouge pas. Personne ne bouge. Le silence devient intolérable. Et puis soudain, d'une seule voix, la foule sans visage se met à chanter. Lian tourne la tête alors, vite, n'aperçoit ni sa mère ni son père mais une forêt de bras levés vers le ciel, et il s'agrippe plus fort à la main du petit homme en regardant de nouveau devant lui.
Le chant semble durer éternellement. Il n'en comprend pas les paroles, il n'est même pas sûr qu'il y ait des paroles : c'est comme le ruisseau, la nuit, à côté de la maison, quand il ne dort pas, s'il faisait juste un peu plus attention il pourrait reconnaître une voix qui lui parlerait. Parfois le chant est sur le point de s'éteindre, presque inaudible, puis il reprend de plus belle, des phrases longues et basses d'abord sur lesquelles roulent ensuite des motifs de plus en plus courts, de plus en plus aigus. Ensuite, le tonnerre des voix graves vient peu à peu noyer les voix hautes, et le chant s'inverse encore, un flux et un reflux régulier, comme un bercement. Lian sent ses yeux se fermer. S'il lâchait la main de l'inconnu, il flotterait dans l'espace violet et il resterait là, balancé entre la terre et le ciel, pour toujours...
Le chant s'arrête brusquement, presque brutalement, au sommet d'une phrase haute, et Lian tressaille comme s'il avait trébuché. La lune est toute violette, avec le rond noir dedans. Tout le monde attend de nouveau, une énorme bulle de silence qui se gonfle derrière lui... Et soudain, loin devant, là où le ciel rejoint la plaine, un trait brillant apparaît, une nappe, non, une vague, non, un mur de lumière bleue, un éclair qui se précipite sur eux ! Lian fait un pas en arrière, mais la main de l'inconnu le retient. Il ferme les yeux.
Un grand cri retentit derrière lui, poussé par des centaines de poitrines, un énorme cri de joie, qui lui fait rouvrir les yeux en tremblant.
La chose terrifiante qui l'instant d'avant allait tout engloutir lèche le pourtour des pierres, étrangement scintillante. Bleue. D'un bleu comme Lian n'en a jamais vu, vivant, frémissant, ourlé contre la roche, comme si la masse agitée de lents frémissements était plus solide que liquide... Et il y a cet éclat scintillant qui flotte au-dessus, une brume qui se perd dans le ciel, impalpable, magique. Fasciné, oubliant tout le reste, Lian s'exclame : « Oh, la lumière ! » Il n'a plus peur. Il s'arrache à l'étreinte du petit homme, en trois pas il est au bord de la chose lumineuse et bleue, et il y plonge ses mains.
À travers sa stupeur, alors, il entend le cri sourd de la foule derrière lui. Il se retourne, atterré : il a dû faire quelque chose de mal ! Le bleu palpite au creux de ses mains encore réunies en coupe... Tout le monde le regarde avec une expression horrifiée ou incrédule. Mais pas le petit homme, qui a l'air très triste. Lian écarte les doigts, le bleu toujours impalpable glisse, coule, tombe en scintillant dans l'herbe, se divise autour de ses pieds nus et retourne se fondre dans la lumière.
Mais il n'a rien senti, rien touché.
Conscient du murmure qui agite maintenant la foule, plein d'incertitude et d'effroi, il éclate en sanglots convulsifs.
*
On le traite comme tout le monde à la commune – compte tenu des circonstances de sa découverte, et du fait qu'il ne parle presque pas, au début. Il flotte dans un étonnement rêveur bientôt mêlé d'une hésitante gratitude. Si on trouve son physique un peu curieux, on ne le dit pas. N'ont-ils pas remarqué ses pupilles plutôt ovales, la forme trop carrée de ses ongles, sa paupière nictitante, l'opercule qui lui ferme à volonté l'oreille externe, sa quasi-absence de pilosité corporelle ? Apparemment pas, ou alors ils ont décidé de ne pas lui poser de questions sur le sujet. À certains échanges, il comprend cependant qu'on n'a pas appelé de médecin : il n'a pas subi d'examen poussé, ses blessures n'étaient pas assez graves pour cela. Il est seulement " en état de choc ". On semble tenir pour acquis qu'il a été attaqué par des agresseurs inconnus individuellement mais dont l'existence collective semble aller de soi. Plus tard, il trouvera cela un peu curieux : on n'est pas vraiment des opposants, mais on est apparemment prêt à recueillir et à cacher sous de faux papiers une victime supposée de la " police fédérale ".
Personne ne fait de commentaire non plus sur ses différences invisibles, le comportement général semble impliquer qu'on ne les remarque pas non plus. C'est alors qu'une certaine gratitude point en lui. C'était vrai, alors, personne n'est capable de le repérer, ici ? Ici, il est normal ?
3
Lian ne se rappelle pas vraiment le voyage de retour, qu'il a tendance à confondre avec l'aller – mêmes paysages, mêmes auberges, mêmes silences. Mais il sait qu'on a dû lui expliquer ce qu'il pouvait comprendre de la Mer, à trois saisons. Comment, annoncée par les aïlmâdzi, elle est apparue la première fois sur les rivages et dans les ports de Tyranaël préparés depuis longtemps pour sa venue, car elle est bien plus haute que les plus hautes marées des océans d'eau salée. Comment elle a recouvert Aritu et Paalu, les deux continents de l'Ouest et de l'Est évacués par leurs habitants, n'en laissant que les plus hautes montagnes sous forme d'îles. Et comment elle a disparu deux saisons plus tard lors de l'éclipse de soleil, ainsi que l'avaient vu les aïlmâdzi, laissant derrière elle les océans intacts et des étendues de terre dépourvues de toute végétation mais où peu à peu ont repoussé plantes et arbres. Et deux saisons plus tard, de nouveau avec l'éclipse de la lune, et toujours comme l'avaient vu les aïlmâdzi, elle est revenue.
« C'est quoi les aïlmâdzi ? » se décide alors à demander Lian, puisqu'on ne lui explique toujours pas. Il ne se rappelle plus qui a répondu, Nathénèk ou Laraï, seulement que la réponse a tardé à venir – « Des gens qui ont le don de voir dans les nombreuses maisons de Hananai » – et qu'on lui a raconté alors l'histoire de la création, adaptée à son âge : Hananai habite plusieurs maisons à la fois, et dans beaucoup de maisons il y a les mêmes pièces, avec les mêmes meubles, mais juste un peu différents, et elle y accueille les mêmes gens, juste un peu différents ; et là l'histoire devient drôle, quand on lui fait imaginer tous les Nathénèk et toutes les Laraï et tous les Lian différents qui peuvent habiter les nombreuses demeures – un Lian rond comme une balle, ou en forme de cube avec seulement des pieds... L'histoire de la Mer perd de son intérêt ensuite. Elle était presque terminée de toute façon, et ce sont les fins que Lian trouve souvent les plus ennuyeuses : le Retour et le Départ de la Mer rythment le passage de l'année pour les Ranao ; elle transforme en glorieuse lumière le corps de ceux qu'elle a jugés dignes de la rejoindre et de mêler au sien leur esprit. Elle n'a pas dissous le corps de Lian lorsqu'il l'a touchée, certes, mais c'est seulement parce que son temps n'est pas encore venu. Il doit être bon et sage, et un jour, lui aussi, il ira rejoindre la Mer.
Il y aurait eu bien d'autres questions à poser, sans doute moins périlleuses – d'où vient la Mer ? Comment ? De quoi est-elle faite ? Est-ce une personne ? Mais Lian demande, sans savoir pourquoi : « Père-Dougall n'a pas été bon, lui, c'est pour ça qu'il n'a pas pu rejoindre la Mer ? », et il perd alors tout désir de poser d'autres questions : Laraï serre les lèvres en regardant au loin, et Nathénèk finit par répondre : « Non, mais quelquefois, c'est ainsi », sur un ton que Lian commence à connaître : plus-tard-quand-tu-seras-grand.
Et il s'occupe à devenir grand ; il oublie le voyage, et la Mer trop lointaine. Il a bien trop d'autres choses à apprendre. Car il grandit : les Gomphali lui arrivent maintenant presque aux genoux, et les arbustes s'écartent toujours davantage de la maison, plus vite à mesure qu'ils grandissent eux aussi. Il faut marcher plus loin à chaque lunaison pour les voir, petites silhouettes obstinées que leur trajectoire finira bien par amener aux premières pentes pierreuses de la montagne. Plus haut, escaladeront-ils le roc nu, les pentes de neige et de glace ? Quelques-uns seulement, dit Nathénèk ; et quelques-uns seulement sentiront un jour dans le lointain l'immensité orangée du Leïtltellu, le grand lac chaud du Nord. Les autres seront emportés dans les tempêtes, glisseront dans les précipices ; leurs filaments trop faibles cesseront de leur fournir l'énergie nécessaire, leurs racines gèleront et les affameront... Mais à quoi aura servi le sacrifice de leur père-Gomphal, alors ?
Si tous les petits Gomphali rejoignent le Nord, explique Nathénèk avec patience, ils seront trop nombreux, ils épuiseront le sol du territoire où ils se rassemblent pour la fécondation, et ils ne pourront pas redescendre procréer dans le Sud. L'impitoyable barrage des montagnes, maintenant et plus tard dans le Nord, ne laisse passer que les plus gros, les plus forts, ceux qui sont capables de creuser la glace assez profond pour s'ancrer dans les tempêtes, ceux dont les filaments peuvent capter la moindre parcelle de lumière à travers le brouillard le plus épais. Ceux-là gagnent le droit de vivre, et de descendre vers la chaleur féconde des eaux du Leïtltellu.
La calme conviction de Nathénèk est contagieuse, et Lian ne trouve rien à reprendre à ces explications : la loi qui fait disparaître certains Gomphali pour que l'espèce entière puisse survivre est assez claire ; c'est ainsi – une expression qui revient souvent dans la conversation, avec Nathénèk. Pourtant, une obscure protestation monte en Lian chaque fois qu'il observe l'obstination patiente des Gomphali. Un jour, il en choisit un, le plus petit, le dernier parti, et chaque matin, pendant une semaine, il pose de grosses pierres devant l'arbuste ; pour éviter l'obstacle, le Gomphal choisira sûrement le moindre effort : il infléchira sa route et, à force de l'infléchir, il finira par se diriger vers la passe qui traverse les montagnes en toute sécurité, au sud-ouest. Il pourra toujours repartir ensuite vers le Nord, n'est-ce pas ?
Mais chaque soir, quand Lian vient vérifier les progrès de son protégé, il trouve une trace bien nette autour des pierres, et le Gomphal toujours plus près de la montagne fatale. « Pourquoi il ne veut pas que je l'aide ? » demande Lian à Nathénèk, frustré. L'arbre ne se détourne que pour éviter les surfaces liquides très larges et très profondes, lui explique son père ; même l'ombre ne l'arrête pas, en plein jour, car il peut sentir la lumière. Il doit aller vers le Nord à cette période de sa vie, c'est inscrit dans ses gènes, il ne peut faire autrement et ce n'est pas gentil de l'en empêcher. « Il y a des choses dans le monde qu'on ne peut pas changer, Lian. C'est comme... le soleil qui se lève à l'est. On n'y peut rien. C'est ainsi. »
*
Au début, il dort beaucoup. Mais à mesure que les jours passent et deviennent des semaines, il finit quand même par se rendre compte peu à peu que tous les gens de la commune sont au moins des sensitifs : la façon dont ils se parlent, leur comportement à l'égard les uns des autres, tout cela lui semble familier. Il se l'explique pourtant aisément, sans inquiétude : un simple écho rémanent de la mutation disparue chez eux. Mais ne voient-ils pas qu'ils ne le voient pas ? Et si oui, pourquoi ne disent-ils rien ? Par politesse ? Ils ne se privent pas de lui poser des questions sur ses antécédents, au début – il s'installe dans sa feinte amnésie après que Jill Estérazy a déclaré : « Il ne se souvient de rien, le pauvre, c'est normal, laissez-le tranquille ! » Une fois qu'il est capable de se lever et commence à se déplacer dans la commune, toujours accompagné de Jill qui l'a pris sous son aile, une certaine perplexité menace de faire surface : ils ne perçoivent bel et bien pas sa différence. Ils le voient quand il arrive dans leur dos, ils savent que c'est lui derrière la porte avec Jill (mais peut-être après tout entendent-ils dans les deux cas le bruit du fauteuil roulant, bien qu'il soit fort silencieux). Ils devinent ce qu'il désire, ce qu'il ressent – même si quelquefois, surtout au début, ils se trompent complètement.
Il trouve encore : la Mer. La Mer lui a laissé une sorte de... coloration protectrice. Pourquoi pas ? Et quelle autre explication possible ? Les jours passent, cela ne s'efface pas avec le temps, au contraire, il demeure caméléon – avec quelques éclipses, mais rares, et personne ne semble s'en inquiéter. Il accepte la situation telle quelle, sans l'approfondir, flottant toujours dans le même étonnement lointain et sans angoisse. L'important, c'est que, pour les autres, il est comme tout le monde.
4
Lian accompagne désormais ses parents dans leurs rondes en forêt. Parfois tous les trois ensemble, quelquefois seul avec Nathénèk, mais plus souvent avec Laraï. Il apprend vite à faire silence pour ne pas troubler les véritables habitants de la forêt, qu'on voit rarement dans la pénombre bourdonnante traversée de lumière et d'insectes, mais qu'on entend toujours, cliquetis, aboiements brefs, ronflements, souffles rauques, le grincement rythmé du nätzinllad, l'oiseau qui ne vole pas et marche au ras du sol, les trilles obstinés de la mastanlid à mi-hauteur et, loin dans le ciel presque ininterrompu de feuillage, le bavardage intermittent des natlànkaï tandis que, invisibles et vifs, ils collectent les graines qu'ils cacheront et oublieront plus tard.
Laraï sait le nom de toutes les plantes, de tous les animaux, de tous les insectes. Dans la forêt, Laraï connaît tout. Elle ne se perd jamais. Elle a des points de repère – des petites figurines de terre cuite, de pierre ou de bois, peintes de couleurs vives, toutes différentes, accrochées ici à une branche basse, là nichées dans un creux de rocher. « C'est ce que faisaient les Hébao d'autrefois, les premiers gardiens de la forêt : chacune et chacun gardait et cultivait un territoire bien précis. C'était une faute très grave de déplacer des marques qui ne vous appartenaient pas. »
Toutes différentes, les figurines, tailles et couleurs, postures aussi, mais toujours la même créature, mi-animale mi-humaine : debout, assise, couchée, tel bras levé ou plié, ou telle jambe, tournés comme la tête dans une direction ou une autre. Toutes petites, pas plus grandes que la main de Lian, mais avec des détails amoureusement fignolés : autour de la tête et jusqu'au ras des yeux, un halo de cheveux très finement bouclés d'où sortent deux petites protubérances, un peu comme des cornes, mais terminées par une sorte de petite boule aplatie ; le corps est mince et sinueux, presque sans hanches et sans épaules, avec un torse bombé, des jambes musclées et des bras nerveux découverts par une courte tunique sans manches décorée de minuscules poissons. C'est Liani-Alinoth. Elle a créé les êtres humains par inadvertance. Elle dormait, les jambes dans l'eau, et voilà qu'une petite bête aquatique plus audacieuse que les autres lui grimpe dessus, à l'air libre. Elle commence à suffoquer, la pauvre, en plein soleil. Ses cris réveillent Liani-Alinoth – et Laraï raconte toujours exactement de la même façon cette partie de l'histoire, sans rien expliquer, laissant toujours Lian perplexe : « Liani-Alinoth hésite entre la colère, parce que la petite bête est sortie de son élément, l'admiration, parce que la petite bête est sortie de son élément, et la pitié – parce que la petite bête est sortie de son élément. » Mais en fin de compte, Liani-Alinoth souffle sur la bestiole pour lui permettre de respirer dans l'air, et fait pousser un arbre de vie dans sa main pour la protéger du soleil. Et ainsi Liani-Alinoth devient la Gardienne des eaux et des forêts.
Et Laraï conclut avec un sourire grave : « C'est ta gardienne aussi, Lian, je t'ai donné son nom : " le lieu de la vie ". »
Lian est flatté, mais un peu perplexe : « Je croyais que c'était Hananai qui avait créé les êtres humains. »
Laraï se met à rire : « Oh, elle les a créés ! Mais Hananai est la divinité qui va masquée, Lian, et elle possède autant de noms que de demeures. Pour les anciens Hébao, elle s'appelait Liani-Alinoth. »
Laraï connaît toutes les histoires. Souvent, quand elle montre à Lian quelles baies il faut manger, où creuser pour trouver les bonnes racines, à quel cri on reconnaît quel animal, de quelles couleurs se méfier chez les champignons, les fleurs ou les serpents, elle lui raconte comment les anciens Hébao l'ont appris pour la première fois. Quand il se promène dans la forêt, de plus en plus souvent sans accompagnateurs à mesure qu'il grandit, il entend bruire les histoires autant que les feuillages et les animaux. Il n'est jamais seul : l'esprit des anciens Hébao l'accompagne.
Et puis, un jour, il apprend que ses parents et lui ne vivent pas seuls non plus sur le plateau de Landaïeïtan. Il remarque dans les lianes basses des figurines qui ne sont pas celles de Laraï. Il reste frappé de stupeur et son père vient presque buter contre lui. « Quoi, Lian ? » D'un doigt tremblant, Lian désigne la marque. « Ah, trop loin », dit simplement Nathénèk sans se troubler, « nous sommes chez Gundiel ». Et il rebrousse chemin. Une fois à la maison, il lui montre pour la première fois une grande carte du Landaïeïtan – ce morceau de papier coloré, c'est le plateau, leur plateau ? Le territoire dont Laraï est chargée s'étend au nord-ouest, autour du Leïtnialen en forme de croissant – le lac Perle, à cause de la couleur gris-rose de ses eaux. Leur maison, invisible sur la carte, quel scandale, est située au bord du lac, là où il s'incurve. Ce n'est pas le seul du Landaïeïtan : toute son étendue en est parsemée, en général des lacs assez petits, souvent allongés, ou en forme de larme. Et d'autres maisons, d'autres fermes de gardiens, sont également réparties sur tout le plateau. Celle des Olaïlliu est la plus proche, à environ vingt-cinq langhi plus à l'est – l'est, c'est à droite de la carte.
« Tu finiras bien par rencontrer leurs enfants dans tes promenades, remarque Nathénèk avec nonchalance, ils vont souvent se baigner au bassin Tarli. » Et il désigne de l'ongle, sur la carte, à l'extrême limite du territoire de Laraï, un petit rond bleu bien trop petit pour être un lac – sans prêter attention aux sourcils froncés de Laraï. Lian a bien vu, lui, même si l'idée qu'il existe non seulement d'autres adultes mais d'autres enfants sur le plateau lui fait écarquiller les yeux. « Est-ce qu'ils ont le droit ? »
Nathénèk éclate de rire en roulant la carte : « Bien sûr ! Et de se promener dans notre territoire comme nous dans le leur. Tout le monde, dans la réserve du Landaïeïtan. La terre n'appartient à personne. Nous sommes des gardiens, ici, Lian. Nous observons et répertorions les animaux et les plantes, nous cultivons parfois celles-ci dans nos petits jardins ou dans la forêt pour savoir comment elles poussent, nous les étudions au laboratoire pour déterminer comment elles pourraient aider tout le monde sur Atyrkelsaõ, mais nous sommes des gardiens. »
Lian a cinq ou six saisons à cette époque-là, il accompagne depuis un bon moment ses parents dans la forêt et il commence à savoir ce qu'ils font sur le plateau. Ce n'est pas pour cela qu'il sursaute : « Atyrkelsaõ ? »
Il entend Laraï faire " tss " entre ses dents. Pourquoi est-elle agacée ?
« Le monde où nous vivons, Lian, dit Nathénèk avec bonhomie. Tu te rappelles quand même bien que nous vivons sur une planète ? »
Oui, oui, une des perles dans le collier dans le tiroir dans la commode dans la chambre dans la maison de Hananai, comme le dit la chanson ; une grosse boule ronde qui tourne autour du soleil en même temps que la lune – elle-même une planète, puisqu'elle a des lunes. « Elle ne s'appelle pas Tyranaël ? Toutes les histoires de Laraï se passent sur Tyranaël... »
Nathénèk ne répond pas tout de suite ; il regarde Laraï. Avec un soupir, celle-ci s'avance dans la pièce et vient s'asseoir à la table : « Eh bien, ça dépend, Lian. »
Lian, qui sent venir une histoire, s'appuie aux genoux de sa mère. Elle lui caresse un bras. « Il y a très longtemps, mille saisons au moins, nous vivions sur une planète que nous appelions Tyranaël. Un jour, les aïlmâdzi ont vu un grand danger, et nous sommes partis pour l'éviter... » Elle hésite un moment, puis reprend comme si elle avait voulu dire autre chose mais avait changé d'avis : «... dans une autre demeure de Hananai, sur une planète exactement pareille. Comme deux graines de fofolod dans la même gousse. Les mêmes rivières, les mêmes montagnes, les mêmes lacs, tout au même endroit... mais sans les villes et les ports et les ponts et les routes, rien de ce que nous avions bâti sur Tyranaël. Nous avons dû reconstruire. »
Elle se tait, les yeux perdus au loin. Lian est abasourdi. Comment fait-on pour passer d'une demeure de Hananai à une autre ? Mais il n'a pas le temps de poser la question, Laraï reprend : « Certains d'entre nous... pensaient qu'il ne fallait pas tout recommencer comme avant, et on les a appelés darnao. D'autres, les dânao, voulaient reconstruire un certain nombre de choses de la même façon, pour que nous ne perdions pas la mémoire. Nous avons beaucoup discuté et nous avons convenu de nous partager le travail. Les dânao s'occuperaient de ce qui serait pareil, et les darnao de ce qui serait différent. Par exemple, sur le Landaïeïtan de Tyranaël, il y avait beaucoup de mines et de carrières partout. Mais ici, sur Atyrkelsaõ, certaines plantes et certains animaux n'existent nulle part ailleurs que sur le plateau. Les darnao ont décidé d'en faire une grande réserve, et les dânao ont ouvert ailleurs les mines et les carrières dont nous avions besoin. »
Elle se tait encore. Lian remarque : « Ça se ressemble, dânao, darnao...
— Bien sûr, sourit Nathénèk approbateur. Les enfants du souvenir, les enfants du devenir : deux faces de la même médaille. »
Laraï hoche la tête en souriant aussi, finalement. « Et c'est pour cela que nous, les darnao, nous préférons appeler cette planète-ci " Atyrkelsaõ " plutôt que " Tyranaël ".
— L'autre côté de l'éternité, dit Nathénèk avec gravité en regardant Laraï.
— L'autre côté de l'éternité », acquiesce Laraï à mi-voix en le regardant aussi.
Quelque chose passe entre eux, comme souvent, et Lian devine avec plaisir qu'ils ne sont plus fâchés l'un après l'autre – même s'il ne sait toujours pas pourquoi ils étaient fâchés.
Ce soir-là, Lian rêve de grosses perles qui font la ronde autour du soleil, toutes pareilles, et elles poussent des tiges et deviennent des Gomphali, sans jamais cesser de tourner ni d'être des perles, et lui est suspendu dans les branches – il fait bien attention de ne pas se faire piquer par les filaments. Dans le rêve, il doit sauter d'une perle à l'autre, mais il a très peur de tomber dans le vide, alors il se répète le mouvement dans sa tête, mais il ne saute jamais.
*
Il y a beaucoup de monde à la commune, une cinquantaine de personnes – il n'a pas vécu avec autant de monde depuis plus d'une saison ; des foules dans des rues ou sur des bateaux, ce n'est pas pareil. Ici, on le connaît par son nom – " Liam " –, on lui parle, on s'attend à ce qu'il réponde. À ce qu'il partage les repas pris en commun. Bientôt, à ce qu'il les prépare avec les autres quand c'est son tour. À ce qu'il travaille comme tout le monde. On ne lui demande pas de se trouver du travail à l'extérieur, heureusement – un autre fait acquis dont il ne comprend pas vraiment la raison d'abord, mais qu'il accepte encore avec gratitude. Il devra seulement travailler dans la commune. Que sait-il faire ? Il propose avec prudence : « Jardiner. Travailler le bois. Faire de l'aquarelle, jouer du piano. » Les deux premières offres suscitent une approbation sans réserve. On le lâche bientôt sur la terrasse qui couronne la maison, entièrement reconvertie en jardin potager et en verger. Il lui faut un moment pour s'adapter aux plantes locales, il n'en connaît vraiment ni culture ni usage, ce ne sont que des noms pour lui – heureusement, il peut aller consulter en catimini les manuels dans la bibliothèque de Jill Estérazy.
Jill est la trésorière de la commune, et son « institutrice » : elle apprend aux tout jeunes enfants à lire, à écrire et à compter en attendant qu'ils aillent " à la grande école ". La blonde Suzane, sa nièce, s'occupe des enfants un peu plus âgés en alternance avec Andreï Bingham, un petit homme jovial dans la quarantaine, à la barbe et aux cheveux poivre et sel frisés en auréoles jumelles, autour de son menton et autour de son crâne. Il y a une vingtaine d'enfants en tout à la commune, petits et " moyens " mêlés, et une demi-douzaine d'adolescents qu'on voit seulement les matins et les soirs, avant leur départ pour l'école du quartier, après leur retour.
À la fin de la première semaine – une semaine de vacances pour les adultes et pour les enfants – la commune se vide tous les jours en alternance, matinée, après-midi, des deux tiers de ses adultes qui vont travailler à l'extérieur. Ne restent que Jill, Suzane et Andreï, et " les grands-mères et les grands-pères ", comme les appellent les enfants, un groupe d'une demi-douzaine de vieilles personnes (dont les grands-parents d'Andreï), qui font presque tout ensemble, une sorte de club à l'intérieur de la commune. On les aime et on les respecte, avec une note souvent indulgente, ou un peu agacée quand ils radotent aux réunions.
Le fonctionnement de la commune est assez simple : on prend les décisions à la majorité, et après des discussions animées. Les enfants sont apparemment les enfants de tout le monde (comme d'ailleurs " les grands-pères et les grands-mères "), et il faut à Liam plusieurs semaines pour déterminer avec certitude qui sont les parents de qui – il ne se presse pas, c'est comme un jeu, sans importance. On met aussi l'argent en commun, on se partage les tâches, et gare à qui essaie d'en faire moins et de recevoir plus que sa part. Il y a des séances de réflexion, toutes les fins de semaine, où chacun évalue sa performance, reçoit les commentaires des autres, le cas échéant des points de démérite. Pas de points de mérite, même si l'on félicite et encourage chaleureusement enfants et adolescents lorsqu'ils ont bien travaillé. Les adultes, eux, cela va de soi ; ce sont leurs fléchissements qu'on souligne.
On s'habille de façon simple, voire monotone. Deux modèles de robes, de tuniques, de combinaisons et de pantalons, une palette de couleurs limitée pour les hommes, blanc, brun, gris, noir et bleu, un peu plus riante pour les femmes (on a droit aux autres couleurs primaires, rouge et jaune, et au vert). Les cheveux sont portés courts ou aux épaules ; si on frise, c'est naturellement, comme Suzane ou Jill. On se rattrape sur les accessoires, foulards, ceintures, bijoux de pacotille, mais avec discrétion – l'ostentation est l'un des manquements qu'on ne rate pas aux séances de réflexion.
Mais on mange correctement, bien que sans excès ; les enfants jouent et rient sans contrainte, une fois qu'ils ont fait leurs devoirs et aidé aux tâches collectives. On est en bonne santé – pas de dodus à la commune : on fait de l'exercice en jardinant, on va travailler à pied à l'extérieur partout ou on le peut et, tous les matins, après le petit-déjeuner, on se livre à des séances communes d'assouplissement et de méditation, même Jill dans son fauteuil roulant, même les grands-pères et les grands-mères. C'est une vie simple et saine ; on a les uns pour les autres une affection discrète mais réelle, compte tenu des inévitables frictions. Qui, plusieurs fois, en viennent aux cris, aux empoignades et aux pleurs – entre enfants comme entre adultes – et doivent se régler en séance de groupe. Mais après un examen de conscience poussé, et grâce aux commentaires et conseils des uns et des autres, les problèmes semblent se résoudre sans séquelles.
Lian enregistre tout cela avec le même intérêt flottant. Ces gens, cette demeure, sa présence parmi eux sont si improbables que parfois il en serait presque amusé.
5
Un jour, distrait par le chant d'un oiseau inconnu qu'il essaie de mémoriser pour le transcrire dans son carnet – à défaut du portrait de l'oiseau, qui se déplace tout le temps –, Lian pénètre sans le remarquer dans une clairière. La surface n'en est pas plane mais bombée, et la végétation n'est pas tout à fait la même que dans la forêt environnante. Les oreilles et l'esprit fixés sur le chanteur invisible, Lian gravit le monticule central en écartant distraitement buissons et arbustes. Et trébuche au sortir du dernier buisson, pris au dépourvu par le large espace circulaire qui s'ouvre devant lui, par le miroitement et l'odeur de l'eau, par sa couleur aussi, un vert profond où se détachent le vert presque blanc des lentilles d'eau et le vert amande des roseaux. Il s'accroupit, étonné. Un trou d'eau ? À en juger par les plantes, l'odeur et les petites taches rouges et noires des ueldânani affairées qu'il voit nager ici et là près des bords, ce n'est pas de l'eau stagnante ; il peut même déceler à la surface le léger friselis d'un courant, car il n'y a pas de vent en cette fin d'après-midi.
Intrigué, il décide de faire le tour – bien trente lani de diamètre. Il pose un pied prudent dans l'eau, tout près du bord, est surpris de sentir comme la pente est raide et dure sous la pellicule de mousse glissante, et se résout à circuler à l'extérieur en s'accrochant quand c'est nécessaire aux branchages, car il a parfois à peine la place de poser les pieds. Ayant déterminé d'où vient le courant, il redescend du monticule et va explorer la clairière dans cette direction, qui serait l'amont d'un hypothétique ruisseau. Rien.
Perplexe, il revient au trou d'eau, à l'étang, il ne sait comment l'appeler... et soudain il se rappelle les petits ronds bleus sur la carte montrée par Nathénèk. Bassin. C'est un bassin ! Il s'assied, les jambes dans l'eau – presque chaude en surface, elle se rafraîchit vite, il doit y en avoir profond. Mais ce n'est pas... naturel, un bassin, n'est-ce pas ? Il glisse une main le long de la surface moussue si étrangement dure, gratte de l'ongle, rencontre vite de la pierre. Y met les deux mains, gratte avec plus d'énergie. Quand l'eau s'éclaircit, il a dégagé un morceau de surface rocheuse légèrement incurvée, écarlate, de l'ultellaod sans aucun doute, mais pas lisse – elle est marquée d'innombrables petites dépressions régulières. Plane, pourtant, la surface, égale. Sculptée, c'est le mot qui lui vient à l'esprit, même s'il le sait inexact : Laraï lui a montré comment travailler un morceau de pierre avec pour tout outil un autre morceau de pierre.
Il continue à gratter la mousse sur le pourtour du bassin, au ras de l'eau, et tombe enfin sur un joint où la pierre légèrement arrondie s'ajuste, sans un interstice, à une autre pierre également travaillée. Pas de ciment. Il suit le joint vers le fond jusqu'à avoir de l'eau à l'épaule, mais sans en sentir la fin. Il faut en avoir le cœur net : il se débarrasse de son sac à dos, se déshabille, prend son souffle et plonge, yeux ouverts sous sa paupière transparente. Difficile de voir quand même dans l'eau trop riche, mais il tâtonne le long du joint et peut le sentir s'incurver en s'enfonçant. La lumière de la surface est à au moins trois lani au-dessus de lui quand il touche un autre joint perpendiculaire et le suit sur un lani ou deux sans en rencontrer d'autre. À bout de souffle, stupéfait, il remonte et crève la surface. D'où viennent ces pierres énormes ? Il n'y a pas d'ultellaod par ici, il faut aller au sud-ouest du Leïtnialen pour en trouver les premières veines. Et qui les a travaillées ainsi, à la main, qui a créé ces bassins ? Les gardiens du plateau, aux temps lointains où les Ranao se sont installés dans leur nouvelle demeure ? Mais pourquoi ?
Au moins, cette petite baignade l'aura rafraîchi. Il se hisse sur le rebord, se glisse entre les buissons et se retourne sur le dos, un bras sur les yeux, pour se laisser sécher. Ses parents lui recommandent toujours de ne pas exposer trop longtemps sa peau nue au soleil, mais sous ces grandes feuilles dentelées à l'odeur musquée, il est assez protégé. Il laisse les bruits de la clairière envahir sa conscience, avec ceux de la forêt, une aura plus lointaine. L'oiseau capricieux chante-t-il encore ? Oui. Une mélodie longue et complexe, où se détachent des motifs récurrents, d'abord brefs et aigus, puis longs et plus graves... Lian chantonne à mi-voix en essayant de suivre l'oiseau puis, engourdi par la chaleur, il chantonne seulement dans sa tête. Même cela, au bout d'un moment, c'est trop, et il se contente d'être, saturé de chaleur humide, de sons et de parfums.
Un bruit d'éclaboussures le tire brusquement de sa somnolence et il se dresse sur son séant. Dans la pénombre, il se demande un instant, égaré, si c'est la nuit, mais non : de gros nuages ont envahi le ciel, le soleil a disparu, le vent s'est levé, le temps a fraîchi, il va pleuvoir. Il a dû dormir plusieurs heures, malgré tout, c'est le milieu de l'après-midi, il a faim ; et il va devoir penser à rentrer.
De nouveau le bruit d'éclaboussures. Lian observe la surface obscure d'où s'élèvent de légères vapeurs, comme des petites langues chaudes. Des ondulations viennent lécher le rebord près de lui. Il essaie de se rappeler le bruit, un peu inquiet. Un gros animal. Une malangai ? C'est ce qu'il y a de plus gros dans cette partie du plateau, d'après Laraï ; elles vivent près de l'eau où elles aiment s'amuser, et se sauvent si on les dérange ; mais il n'en a pas vu de traces aux alentours de toute façon... Un karaïker ? Lian frissonne, mais il sait qu'il joue à se faire peur : si un tigre géant était un habitué du bassin, il en aurait encore plus remarqué les traces aux alentours – ce sont les premières que ses parents lui ont appris à reconnaître, même si les quelques karaï du plateau vivent plus à l'ouest, dans la partie la plus sauvage de la réserve. Oui, mais un karaïker fou, un mangeur d'humains ?
Qui se baignerait bien gentiment au lieu d'avoir avalé Lian tout cru pendant qu'il dormait ? Lian aime bien se raconter des histoires, mais il y a des limites ; il hausse les épaules en observant encore un moment la surface de plus en plus indistincte sous son voile de vapeur, tandis que la lumière continue à décliner et que résonnent les premiers grondements lointains de l'orage. Allons, il ferait mieux de retourner dans la forêt et de se fabriquer un abri en prévision de l'averse.
Il se lève, s'habille en hâte, assure son sac sur ses épaules et se détourne après un dernier coup d'œil au bassin.
Et dans cette fraction de seconde, il voit, sous la vapeur blanchâtre, près du bord, à ses pieds. Deux grands yeux jaunes et ronds au ras de l'eau, sous une toison hirsute, brillante de gouttelettes. Y avait-il deux petites protubérances, comme des cornes molles et mouvantes, à travers les poils – les cheveux ? Il n'est pas sûr. Il gardera l'impression d'un corps sombre et sinueux juste sous la surface, sans épaules et sans hanches, deux longs bras étendus, une grande queue – ou bien étaient-ce des jambes jointes ?
Lian fait volte-face et dévale le monticule, le cœur dans la gorge, il n'arrête pas de courir quand les premières gouttes de pluie, lourdes et pressées, lui fouettent le visage. Il s'engouffre dans la forêt, saute par-dessus les troncs abattus, fonce à travers les rideaux de lianes, et finalement trébuche et tombe. Il se redresse à quatre pattes, terrifié : le poursuit-on ? Impossible de rien entendre dans le déluge. Au bout d'un moment, comme nul monstre ne surgit, il se blottit sous une large branche, honteux et dégoulinant, pour attendre la fin de l'averse.
Laraï et Nathénèk le contemplent d'un air étrange, cette nuit-là, à la maison, quand il leur raconte. Un mélange de stupeur et... d'envie ? « Ça pourrait être une naaludan, remarque Nathénèk.
— Ça n'a pas de cornes », dit aussitôt Lian qui a parfois aperçu les timides mais mélodieux nageurs blancs dans le lac, et Laraï renchérit : « Une naaludan ne serait pas allée vers lui. »
Il a vu une moatrani. « Nous les appelons ainsi, moatranao, nous ignorons quel nom elles se donnaient. C'étaient les premières... les ancêtres. Les autres ancêtres, car il y a très, très longtemps, des créatures identiques ont existé aussi sur Tyranaël, et nous sommes leurs descendants. Mais ici, sur Atyrkelsaõ... »
Laraï cherche ses mots, ce qui ne lui est pas habituel, et Nathénèk enchaîne : «... elles ne se sont pas développées comme sur Tyranaël. Il y a des centaines de milliers d'années, elles ont cessé de changer. Et elles sont devenues de plus en plus rares, parce que ce qui ne change pas est condamné à disparaître, c'est ainsi.
— Certains pensent qu'il n'y en a plus du tout sur Atyrkelsaõ, reprend Laraï, mais on en a vu parfois, dans des endroits sauvages.
— Ou bien on a voulu en voir, s'obstine Nathénèk.
— Lian ne connaissait même pas leur existence ! »
Lian réfléchit, les sourcils froncés, inquiet de leur désaccord. Cette créature lui fait surtout penser aux figurines de Laraï, à Liani-Alinoth, mais il n'ose pas le dire. « Est-ce que ce sont... des gens ?
— Oui, Lian », dit Laraï d'une voix un peu triste, et Nathénèk ne la contredit pas. « Du moins l'ont-elles été, il y a très, très longtemps, et qui peut dire ce qui leur en reste si elles existent encore ? Ce sont eux qui ont construit les bassins. »
Le lendemain, elle emmène Lian dans une autre clairière, au nord du lac, plus près de la montagne. Le relief en a été bouleversé par un ancien tremblement de terre et le bassin est à sec depuis longtemps. Les énormes pierres sont en partie disjointes, certaines sont même fêlées, mais on en distingue encore très bien l'agencement. Et Laraï montre à Lian l'arrivée des conduits souterrains qui amenaient l'eau depuis un torrent disparu.
Les torrents peuvent disparaître ? Laraï sourit : « Il y a une centaine de milliers d'années, Lian, près de cinq cent mille saisons. Le plateau n'était pas comme aujourd'hui. Le continent était différent. Tout change, tu sais, les plaines deviennent des montagnes et les montagnes des plaines, les rivières s'assèchent, les mers s'ouvrent et se referment... Tout se transforme. Pas à la même vitesse, voilà tout. »
Elle s'accroupit, caresse la pierre usée du conduit et murmure, pensive : « Et les moatranao ont construit ceci, des dizaines de bassins semblables dans tout le sud-est, il y a une centaine de milliers d'années... Elles ont taillé et transporté les pierres, elles ont construit les canalisations, parfois sur des langhi de long, elles ont édifié les monticules, ils ont façonné les bassins.
— Mais pourquoi ?
— Tu te rappelles l'histoire de Liani-Alinoth ? Comme nous, à l'origine, les moatranao étaient des créatures aquatiques. Nous pensons qu'elles ont eu plus longtemps que nous besoin d'eau pour se reproduire – mais pas n'importe quelle eau. De l'eau calme mais non stagnante, pas trop d'espace, une température et un ensoleillement particuliers, et même certains types d'animaux et de végétation dans l'eau et aux alentours, qui sait ? Et quand elles ne les ont plus trouvés à l'état naturel, ou avec difficulté, elles les ont recréés elles-mêmes. Ou encore plus tôt dans leur évolution, à mon avis, par choix et non par besoin. Parce qu'elles étaient devenues... – elle sourit à Lian –... des gens. Et que les gens, où qu'ils soient, tendent à changer le monde autour d'eux, parce qu'ils changent eux-mêmes. »
Lian contemple le bassin effondré en écartant de son front un nuage d'insectes grésillants. « Elles n'ont jamais rien construit d'autre ?
— Non. Avant de découvrir les bassins, nous étions persuadés qu'il n'y avait jamais eu de conscience sur Atyrkelsaõ. Que c'était une maison vide, sans passé... » La voix de Laraï se perd dans un murmure, elle ne parle plus vraiment à Lian. « Une maison sans fantômes.
— Mais si Atyrkelsaõ est exactement comme Tyranaël, " deux graines de fofolod dans la même gousse ", pourquoi il y avait des Ranao sur l'une et des moatranao sur l'autre ?
— Ah mais, Lian, les graines de fofolod ne sont jamais exactement semblables dans leur gousse. Les demeures de Hananai non plus. C'est comme les gouttes de pluie. Il y a des différences entre elles, que nos yeux ne peuvent pas toujours voir, mais qui existent.
— Il pourrait aussi y avoir une demeure, alors, où les Ranao et les moatranao existeraient en même temps, et où les moatranao seraient encore vraiment des gens ?
— Oui », finit par dire Laraï, avec une drôle de voix dure. « Mais je ne sais pas s'ils existeraient très longtemps ensemble. »
Et Lian garde pour lui le " pourquoi ? " qui lui est inévitablement monté aux lèvres. Désormais, cependant, quand il se promène dans la forêt, s'il n'est jamais seul, ce n'est plus à cause de l'esprit des anciens Hébao – ils n'ont jamais connu cette forêt-là, ce ne sont que des souvenirs dans les histoires de Laraï –, mais l'esprit des moatranao l'accompagne, qui ont été des gens et qui avaient sûrement plein d'histoires aussi, dont personne ne saura jamais rien.
*
La commune est située dans le quartier 4 de la zone Ouest de Bird-City (on dit " Bird " ; on dit aussi " Orlemur Ouest " en parlant du quartier.) Mais Liam ne sort pas. Pas prudent, même avec ses faux papiers, il vaut mieux pas, dit Jill. Ce n'est pas comme s'il voulait sortir, non plus. L'espace de la maison lui suffit, le toit-terrasse, sa chambre, la cuisine, la salle à manger, la cour centrale, le grand bassin ombragé de son arbre-à-eau. Une bulle confortable. Il ne va pas la crever en essayant d'en sortir. D'ailleurs, Jill a raison, il courrait le risque de se faire repérer ; il se le dit par acquit de conscience : il a peine à croire à l'existence de la ville au-delà de la commune. Il pourrait la voir, pourtant : il lui suffirait de s'approcher des parapets de la terrasse et de regarder la rue en contrebas, dont il entend pourtant la rumeur. Mais il ne le fait pas.
Il est bien. Entre son travail au jardin, à la cuisine, à l'atelier et les longues parties d'échecs avec Jill ou Andreï, il ne s'ennuie pas. Par acquit de conscience, encore, il a parcouru en diagonale les livres de la bibliothèque ; il ne tient pas à connaître le monde qui entoure la commune ; il est certain d'en savoir tout ce qui compte. Ce sont surtout des manuels techniques, d'ailleurs, jardinage, médecine, bricolage, quelques romans... rien de très intéressant. Il glane des détails, pour remplir les cadres qu'il possédait déjà. On écrit Année, ici, on écrit Mois – a-t-on besoin de se rappeler qu'on vient d'ailleurs, d'un monde où le temps se mesurait en d'autres années et d'autres mois ? – mais, bizarrement, pas de majuscule à saison, bien qu'on écrive, et pense donc, toutes les saisons avec des majuscules ; il n'y a que douze Mois, les jours ont des noms différents, ils durent trente-cinq heures, tout le monde fait la méridienne ; mais c'est l'Hiver aussi à Bird-City, même si l'Année commence avec le retour de la Mer...
Il est pourtant bien obligé d'entendre les discussions des travailleurs de l'extérieur, à la table des repas, et quelques autres informations vont rejoindre celles qui dorment dans son paysage intérieur, curiosités épinglées comme des papillons. La ville semble fonctionner sur le même modèle que la commune, le reste de la région aussi, sans doute – mais aucun des membres de la commune n'a jamais quitté la région, à peine la ville : on ne sort pas sans passeport de la province, on ne circule pas sans laissez-passer temporaire d'une région à l'autre de la province, et en ville il faut avoir en tout temps son idicarte, sa carte d'identification, sur soi. Bird semble, en gros, une association de communes. Des séances de réflexion ont lieu à l'échelle du quartier, puis de la zone, et on y examine avec la même gravité attentive la performance des uns et des autres. On a des quotas à respecter – une partie des fruits et des légumes du jardin reviennent au quartier sous forme de conserves, une portion du salaire des travailleurs de la commune revient à la zone. On reçoit des points de démérite, sinon. Et des remontrances, par écrit, du conseil de quartier ou du conseil de zone. Le Conseil de Ville (le seul qui a droit à des majuscules) fait effectuer par ses fonctionnaires des inspections-surprises : pour les enfants et leur niveau d'instruction, pour l'hygiène (la cuisine, les toilettes et les douches étant toujours impeccables, personne ne s'en soucie beaucoup) et – Liam ne peut se le décrire autrement – pour le degré de satisfaction général des membres de la commune. Tout le monde le prend avec philosophie : c'est comme la pluie et le soleil. Lui, il flotte avec équanimité au-dessus, à côté, en dehors de tout cela. Il se promène de sa chambre à la salle à manger, en passant par la terrasse et la cuisine, il travaille, il mange, il écoute, il joue aux échecs, il dort, les semaines s'ajoutent aux semaines. La routine est délicieuse, délicieux le calme de la grande maison le matin et l'après-midi, l'ombre tiède des arbres de la terrasse, la terre moite et fertile entre ses doigts, les plantes de mieux en mieux apprivoisées... Il s'est trouvé une place. Elle lui semble évanescente parfois, comme un rêve, parfois il se dit qu'il va se réveiller, qu'êtres et choses, ici, ne sont que des fantômes, mais il écarte vite ces percées d'angoisse, les oublie. Il s'est trouvé une place. Ici, il est normal. Il vit, et il mourra, comme tout le monde. C'est tout ce qui compte.
6
Un jour, bien sûr, au bassin Tarli, Lian rencontre les enfants des voisins. Cela se fait simplement, un après-midi d'automne. Il a pris son dana-dana et il est passé par la piste, comme les autres fois : ça rallonge, mais c'est plus praticable pour des roues – il a sept saisons, il est capable de pédaler sur une demi-douzaine de langhi aller puis retour dans une journée. Lorsqu'il y arrive, ce jour-là, il entend enfin des rires, des cris, des bruits de plongeon ; il y a tellement pensé que la curiosité est bien plus forte que la crainte, et il les observe, Maïli, Miniaz, Tarmel, il connaît leurs noms, Nathénèk les lui a dits. Miniaz aux nattes rousses et Maïli aux courtes boucles brunes sont des alnaldilim, enfants d'un même père, en l'occurrence Naritias ; Tarmel aux cheveux noirs est l'enfant de l'autre époux de Gundiel, Kavnid. Lian connaît même le nom de la mère de Gundiel, qui vit avec eux, Olaïli, et de sa sœur cadette, Shalaï.
Et eux le connaissent aussi, même s'ils ne l'ont jamais vu – Laraï ou Nathénèk se rendent aux réunions de travail et aux assemblées générales du plateau, même s'ils n'y emmènent jamais Lian. Il croyait que c'était normal. Il apprend qu'il est différent.
Son corps est un peu différent, il le savait, il a vu ses parents nus, mais il pensait qu'en grandissant il deviendrait comme Nathénèk – tout change, n'est-ce pas ? C'est ce que Laraï lui avait dit quand il avait remarqué la différence entre son pénis et celui de son père : « Nathénèk est un adulte et toi un petit garçon. » Mais Lian regarde Tarmel quand le petit sort de l'eau – il regarde le pénis de Tarmel, qui se secoue pour s'égoutter et se déploie ensuite au soleil, et c'est comme celui de Nathénèk, en plus petit, une sorte de courte trompe cylindrique et musclée qui, une fois sèche et réchauffée, se recourbe de nouveau vers l'arrière et devient presque invisible, à l'abri maintenant dans le léger repli de chair, le long du pubis ; elle ne pend pas tout droit et pointu au bas de son ventre, toujours visible, comme le pénis de Lian.
Pourtant Tarmel n'est pas un adulte, il a seulement quatre saisons ! Lian, désemparé, contemple son propre sexe avec un début d'inquiétude.
Tarmel, qui a suivi la direction de son regard, redéploie son pénis et le fait tourner de droite à gauche : « Et ça ne sert pas pour faire pipi, non plus », dit-il d'un ton définitif.
Lian n'est pas pareil. Ils examinent ensemble leurs différences, tous également curieux. Le brun de sa peau est plus clair, ses cheveux presque blancs – blonds, c'est le soleil qui les décolore, « blonds », concède Maïli. Sous la paupière nictitante transparente, ses yeux sont gris-bleu, une couleur inconnue des Ranao, et la paupière plus ovale que verticale. Et il a un nombril. Il croyait aussi que le petit nœud changerait avec l'âge, se résorberait : Nathénèk n'en a pas ; sur son ventre comme sur celui de Laraï, à peu près à la place du nombril, une petite bande verticale de peau plissée, un peu plus claire, mais c'est tout. Les petits Olaïlliu ont la même bande verticale, et pas de nombril non plus.
Miniaz effleure le ventre de Lian et dit : « Moi, je trouve ça mignon. » Elle trempe une main dans l'eau et, sur les planches déjà sèches du ponton, elle dessine rapidement un petit rond avec deux taches pour les yeux et une tache pour la bouche, puis un autre rond plus grand, avec aussi trois points en triangle ; Maïli ajoute des traits : bras, jambes et un sexe masculin. C'est un jeu, maintenant. Lian dessine les filles : leurs seins ne sont pas encore formés, mais les petites aréoles sont apparentes, avec les semis de taches symétriques en dessous, comme Laraï, là où les ancêtres femelles des Ranao avaient une double rangée de tétines. Il ajoute les nattes de Miniaz, pour faire plus ressemblant.
« Eh, c'est quoi, ça, des cornes ? » proteste le modèle.
Une impulsion fait dire à Lian : « Non, tu n'es pas une moatrani. »
Et il en est bien content, car Maïli, avec un reniflement dédaigneux, lui sert sa réplique suivante sur un plateau : « Personne n'en a jamais vu, des moatranao ! »
La discussion se détourne alors résolument du physique comparé de Lian et des petits Olaïlliu.
Plus tard dans la matinée, alors qu'ils attendent la carriole qui les a amenés là, Tarmel propose de jouer à cache-cache. Puis il se mord les lèvres tandis que Miniaz dit, juste un peu trop vite : « On n'a pas le temps.
— C'est quoi, ce jeu-là ? » demande Lian.
Il y a un petit silence, puis Maïli soupire : « Celui qui s'y colle compte jusqu'à cent sans regarder, et tous les autres vont se cacher. Il faut les retrouver le temps de compter jusqu'à cent. Le premier à avoir été trouvé s'y colle au tour suivant. »
Un autre petit silence. « On peut y jouer », dit Miniaz à Maïli.
« Ça serait juste se cacher », rétorque celle-ci.
Lian les regarde l'une après l'autre, agacé : de quoi parlent-elles ? « Pourquoi ? » proteste-t-il.
Maïli semble hésiter, puis hausse les épaules. Elle se couche par terre, la tête dans les bras. « Cachez-vous, sans vous cacher – pas toi, Lian. Je compte seulement jusqu'à cinquante. »
Déconcerté, il regarde Miniaz et Tarmel se dissimuler, sans un bruit, parfaitement invisibles, dans les hautes herbes et dans un buisson touffu d'arpelai. Quand Maïli se relève, elle ne se déplace même pas pour chercher. Elle tend un doigt : « Tarmel. » Elle en fait autant pour Miniaz. Pas une hésitation. Elle ne peut pas les avoir vus, mais elle sait exactement où ils se trouvent.
« On n'a pas besoin de se voir avec les yeux, nous autres. Si on ne se cache pas, si on ne s'empêche pas de voir à l'intérieur, on se trouve tout de suite. C'est pour ça que ça s'appelle " cache-cache " : il faut se cacher des deux façons. »
Et Miniaz enchaîne, presque d'un ton d'excuse : « Si tu t'y collais, on aurait juste à se cacher, puisque tu ne peux pas nous voir à l'intérieur. Ça prendrait des heures, ça ne serait pas très amusant... Nous, si on ferme les yeux » – et elle le fait – « c'est comme si tu n'étais pas là du tout. On ne te voit pas à l'intérieur. On ne te trouverait jamais, tu gagnerais tout le temps. Ça ne serait pas juste.
— Voilà Shalaï ! » dit Maïli. Elle semble soulagée. Pas de carriole en vue, pourtant. Mais les autres enfants rangent leurs affaires et s'élancent vers la piste qui passe près de la clairière ; Miniaz se retourne pour crier : « Viens nous voir à la maison quand tu veux, Lian ! » Quelques minutes plus tard, une carriole sort en effet de la forêt. Quoi, Shalaï aussi, ils la voient à distance ?
Ils ne lui ont pas dit qu'il est un halatnim. Cela n'aurait pas été poli, et ils avaient déjà dépassé la limite.
Quand il rentre ce soir-là, il ne sait pas trop comment s'y prendre, avec ses parents. Il range ses affaires, va se laver, leur montre ses carnets où, comme d'habitude, il a consigné ce qu'il a vu de nouveau ou d'insolite. Puis, comme s'il s'agissait simplement d'une autre variété de créatures exotiques – et c'est bien le cas, d'une certaine façon – il conclut : « Et j'ai rencontré les Olaïlliu au bassin Tarli. »
Il voit le raidissement de Laraï, la lenteur avec laquelle Nathénèk finit de couper sa tranche de pain. Puis son père demande : « Vous vous êtes bien amusés ? »
Et Lian s'entend dire : « Oui, mais je ne peux pas jouer à cache-cache avec eux. »
Ils lui expliquent, alors. Les gens qui ont des pouvoirs et ceux qui n'en ont pas. « C'est ainsi. » Ce n'est pas Laraï qui dit " halatnim ", c'est Nathénèk. Laraï parle de mutation ; Lian s'illumine : voilà pourquoi il n'est pas tout à fait pareil, bien sûr, comme les kâstelladaï qui sont des versions différentes des kâstulladaï, grises au lieu d'être noires, et avec des pattes plus courtes et un bec plus fin. Laraï dit « Oui, les halatnim sont une variante des Ranao. » Nathénèk a l'air de vouloir parler mais elle répète, et ce n'est pas pour Lian : « Les halatnim sont des Ranao », et Nathénèk finit par hocher la tête en silence.
Ensuite, Laraï raconte à Lian l'histoire d'Oghim. Pas en entier. Juste l'ombre perdue du petit prince, la quête pour la retrouver, les pouvoirs acquis en chemin : comment, premier tzinan, Oghim s'envole pour échapper au karaïker, comment, premier keyrsan, il déplace la Fleur de Palang sans la toucher pour ne pas en être foudroyé, et comment enfin, premier danvéràn, il parle à l'homme fou sans passer par ses oreilles sourdes. Pas à haute voix, précise Nathénèk, à l'intérieur. « Comme Miniaz, Maïli et Tarmel se voient ? » Non, eux, ce sont simplement des lâdzani, ils sentent les émotions des gens, à plus ou moins grande distance. Beaucoup de Ranao sont ainsi. Quelques-uns, plutôt rares, ont tous les pouvoirs, comme Oghim ; on les appelle des hékel – il y en a une sur le plateau, Kéryan ; c'est chez elle qu'on fait les assemblées et les réunions. D'autres Ranao, assez rares aussi, présentent tel ou tel pouvoir, mais l'intensité en est si limitée que cela ne sert à rien.
Lian ouvre de grands yeux : « Vous en avez, vous ? »
Nathénèk se met à rire : « Mon don a des hauts et des bas – surtout des bas ! Je peux planer à environ un demi-pied du sol. N'importe quoi de plus haut, je bute dedans. Et sur un ou deux lani, pas plus : après ça, fini. Ne me demande pas une démonstration, je ne l'ai pas fait depuis des années et ça prend de l'entraînement, comme n'importe quel effort musculaire. Ta mère peut faire bouger de petits objets.
— Ce n'est pas important, intervient Laraï avec force. Comme ce n'est pas important d'être une lâdzan ou une danvéràn. Il y a bien des façons différentes de savoir ce que les gens pensent ou ressentent. Tu vois, Lian, quand Oghim arrive dans l'île des Ékelli et qu'ils lui offrent le choix entre son ombre et ses pouvoirs, il leur dit " Reprenez vos pouvoirs, Ô Ékelli. La puissance de l'amour est aussi grande, et elle ne me séparera pas de mes frères humains "...
— C'est quoi, les Ékelli ?
— Dans l'ancien temps, on croyait que c'étaient des envoyés de Hananai, dit Nathénèk.
— Des gardiens, comme les dzarlit ? » Liani-Alinoth est une dzarlit, créée par Hananai pour garder les rivières et les forêts, sauf qu'elle ne le savait pas encore quand elle a sauvé la petite bête aquatique.
Laraï dit " oui " avec un peu d'impatience et reprend : « Ce qui est important, Lian, c'est comment nous nous traitons les uns les autres. Nous n'avons pas besoin d'être des lâdzani ou des danvérani pour t'aimer, ni toi pour nous aimer, n'est-ce pas ? »
Une vraie question, posée sans sourire, et Lian répond avec gravité : « Non. » Et c'est la vérité, mais en même temps il se demande pourquoi les Ékelli ont quand même laissé leurs pouvoirs à Oghim. Laraï l'embrasse, Nathénèk lui caresse les cheveux, ils reprennent leur repas. Au cours de la conversation, Lian leur dit qu'on l'a invité à se rendre chez les Olaïlliu quand il le désire.
« Pourquoi pas ? » dit Nathénèk en souriant. Entre lui et Laraï passe un de ces silences où Lian a appris à reconnaître un désaccord, mais sa mère se contente de hocher la tête en répétant « Pourquoi pas ? », et il n'en demande pas davantage.
*
On est à la fin de Février – l'Hiver se termine, déjà ? – Tony Vogel et Marla Lopez, qui travaillent dans une usine quelque part à la périphérie de la ville, protestent contre l'augmentation des quotas sur leurs chaînes de montage. Un des grands-pères remarque de sa voix un peu chevrotante : « Ils commencent la guerre drôlement tôt cette Année !
— Mais non », dit quelqu'un d'autre, un jeune, Liam ne se rappelle pas qui, « l'entrepôt de l'Arsenal a brûlé, ils doivent reconstituer les stocks d'uniformes. »
La conversation continue après cela, mais Liam ne l'entend pas, la poitrine comme traversée d'une aiguille chauffée à blanc. La guerre ! La guerre ? Puis, peu à peu, le ton et la teneur de la conversation se fraient un chemin dans sa conscience pétrifiée. Personne ne semble inquiet ni choqué. La guerre. C'est... comme un phénomène naturel récurrent, banal. La guerre, ou la campagne, contre les rebelles (le plus souvent les Rèbs, rarement les Sécessionnistes). On s'en passerait bien, mais les Rèbs ne veulent pas se calmer, ils essaient toujours de prendre Dalloway, il faut bien défendre la place. Et tenir le front (ou la frontière, ou la ligne Ewald). C'est une question de principe, d'honneur, de bon sens (selon qui parle). Pas de quoi s'énerver. Si ton numéro sort, tu y vas. C'est comme ça. Quand ? On hausse les sourcils, puis on se rappelle que Liam est amnésique, on explique avec patience : après le départ de la Mer, et jusqu'à son retour. Depuis quand ? Depuis... On se consulte du regard, on consulte le grand-père, qui consulte un autre grand-père... Longtemps. Ça a commencé vers 128, par là. « Ah oui, intervient un troisième grand-père, mais il y a eu l'insurrection, avant, en 104. Ça n'a jamais vraiment arrêté après.
— Janos, proteste une grand-mère, la guerre n'a vraiment commencé qu'après 126.
— Et qu'est-ce que tu crois que c'était, l'insurrection ? Une partie de campagne ? Deux Années, ça a duré ! Un million et demi de morts ! Le pays ravagé ! Terrible, je te dis, terrible ! Quand on a vu ça...
— Oh, Grand-père, dit un des adolescents narquois, vous n'étiez même pas né...
— J'ai fait la campagne de '26, mon garçon, on est allés à Dalloway en train, et je te dis, il y avait encore des ruines partout ! Terrible, terrible... »
7
Également située au bord d'un lac – plus petit, le lac Aleïdan –, la maison des Olaïlliu est bien plus vaste que celle de Lian ; ils l'habitent plus nombreux, aussi. Lian fait pêle-mêle connaissance de leur mère, de leurs pères, de leur grand-mère et de leur tante, du banker qui a choisi de vivre avec eux, Tiuli, un grand noir et feu de la variété des montagnes. Au début, il se sent mal à l'aise parmi tous ces adultes inconnus, il craint un peu que ce ne soit comme dans les auberges, lors de ce fameux voyage jusqu'à la côte. Mais on est très aimable. Le mot halatnim ne sera jamais prononcé devant lui, on ne commentera ni les habitudes solitaires de sa famille, ni le fait qu'il soit enfant unique ; on sait la valeur de la politesse, chez les Olaïlliu. On discute plutôt gravement de son expérience de la forêt, de sa rencontre avec la peut-être moatrani, des découvertes récentes de Gundiel, des expériences de Kavnid, biologiste résident d'Aleïdan comme Nathénèk l'est de Leïtnialen. On lui fait visiter le laboratoire, lui laisse observer des cultures au microscope. Lian est plutôt flatté de toute cette attention. Ensuite, Shalaï lui ouvre ses albums remplis d'aquarelles de fleurs, de papillons et d'oiseaux. Il est tout retourné : lui aussi, il veut utiliser de la couleur dans ses carnets d'expédition, pourquoi Laraï ne lui a-t-elle pas appris à en utiliser ? « C'est trop compliqué en forêt, remarque Shalaï. Il fait trop humide. Il vaut encore mieux noter ce qu'on voit, et peindre quand on est rentré. » Lian admet qu'elle a raison, mais il est bien décidé à apprendre l'aquarelle, et Shalaï s'offre à lui donner des leçons. Il faudra revenir.
Après le repas, Gundiel et Kavnid font de la musique, à quatre mains, sur un instrument que Lian trouve d'abord assez comique, habitué qu'il est aux flûtes de Nathénèk : une espèce de grosse boîte de bois verni plate et triangulaire posée sur trois pieds, avec à l'avant, sur toute la largeur, un étroit plateau rectangulaire muni des touches alternées d'ivoire et d'ultiéki violet : vraiment pas le genre d'instrument qu'on peut emporter avec soi dans la forêt, ce " piano " ! Mais quand la mélodie s'élève, il écoute les notes rondes et vibrantes, transporté, et il n'a plus envie de sourire.
Ils sont en train de chanter tous en chœur une ballade que Lian est bien content de connaître lorsqu'une nouvelle visiteuse arrive, une grande femme maigre dans la cinquantaine, yeux noirs attentifs, courts cheveux de jais, peau d'un bronze doré. « Kéryan Bataliz Dialadnu », se présente-t-elle à Lian, qui touche les mains tendues en donnant son propre nom, intérieurement aux aguets : voilà donc la hékel Kéryan ! Elle semble ne rien avoir de bien particulier, pourtant. Elle grignote avec appétit les biscuits offerts, tout en sirotant du thé, chante avec les autres le cas échéant – elle a une belle voix grave un peu rauque. Entre deux chansons, elle interroge Lian à mi-voix sur sa rencontre avec la moatrani. Puis elle se désintéresse de lui.
Pendant le reste de l'après-midi, les autres enfants lui font visiter la maison, les jardins, les cabanes qu'ils sont en train de construire dans la forêt proche, une chacun. Ils scient, ils clouent, ils rabotent ; ensuite, pour se reposer, ils se racontent des histoires. En attendant le dernier repas de la journée, Shalaï donne à Lian sa première leçon d'aquarelle. Quand il va se coucher, le soir, avec Tarmel, il a du mal à s'endormir. Trop de monde, trop de choses nouvelles – mais ce n'était pas désagréable du tout, tous ces gens ensemble, cette chaleur, ces rires...
« J'aimerais bien avoir des frères et des sœurs », conclut Lian de retour chez lui le lendemain, lors de la collation de mi-matinée, pour essayer de résumer son expérience. Le silence qui suit lui fait lever les yeux. Laraï continue à manger. Au bout d'un moment, elle pose sa cuillère, s'essuie les lèvres, boit une gorgée d'eau. Ses yeux ne regardent pas Lian. Ni Nathénèk. « Je ne peux pas créer d'autres enfants, Lian », dit-elle d'une voix calme et un peu froide.
Lian n'ose pas demander pourquoi.
*
Liam se fait raconter l'histoire de l'insurrection par les grands-pères, qui ne demandent que ça. L'Insurrection, avec une majuscule, en 104, au départ de la Mer. Dans toutes les grandes villes en même temps, Bird, Cristobal, Nouvelle-Venise, Morgorod, Tihuanco. Les usines qui sautent, les voies de chemin de fer, les ponts. Les rebelles retranchés dans les villes, ensuite, attendant le grand soulèvement général de la population, qui n'a pas lieu. Et finalement chassés par les troupes fédérales. Le début de leur longue retraite vers leurs places fortes jusqu'alors secrètes, dans les montagnes et les hauts plateaux de Nouvelle-Dalécarlie au nord-ouest, dans les McKelloghs à l'est, dans les monts Alcubierre au sud-ouest, et surtout en Licornia, à l'abri des montagnes Rouges.
« Mais qu'est-ce qu'ils voulaient ?
— Renverser le gouvernement, prendre le pouvoir, quoi d'autre ? » grognent à plusieurs voix les grands-pères. Revenir en arrière, annuler tous les progrès accomplis. Des idéologues, des fanatiques, surtout à l'époque. Ils se sont un peu calmés, à force de crever de faim derrière leurs montagnes. Enfin, la population, parce que les dirigeants... Ce sont eux qui font attaquer la Ligne, qui poussent autour de Dalloway. « Si ça ne tenait qu'à moi, je ferais sauter Dalloway, moi, ça calmerait tout le monde ! »
Dalloway, l'ancien spatioport, se trouve au-dessus de l'influence de la Mer, sur un haut plateau réaménagé, à plus de deux mille mètres d'altitude. Les principaux complexes d'ordinateurs se trouvent toujours là, dont dépendent les satellites météo, les satellites de communication et le centre de coordination pour la défense rapprochée de Virginia. Un point stratégique essentiel, Dalloway. Un saillant d'environ cinquante kilomètres de large tenu par les Fédéraux, débordant du mauvais côté de la Ligne, une épine dans le pied des rebelles. Pourtant, lors des négociations de paix, on était presque arrivé à une entente : territoire neutre, également accessible à tous. Mais non, ils voulaient Dalloway pour eux tout seuls, les maudits Rèbs !
« Dieu sait pourtant qu'Ewald s'est mis en quatre pour que les négociations aboutissent, le pauvre. C'est terrible, quand même. S'il y avait eu quelqu'un de bonne foi aussi en face, tout ça aurait été terminé... »
8
Ce matin d'automne-là, Lian va vérifier, comme chaque jour, où en est rendu son petit Gomphal. Il a dix saisons, à présent, mais l'arbuste lui arrive toujours aux genoux : ils grandissent ensemble. Le Gomphal a parcouru environ cinq cents lani depuis le début de son voyage vers le nord ; il arrive dans une zone rocailleuse, l'embouchure du ruisseau qui se déverse dans le lac. Le soleil n'a pas encore dépassé les montagnes. Lian contemple l'arbuste aux filaments encore endormis dans l'ombre. Comment traversera-t-il ? Faudrait-il lui construire un pont, pour lui faciliter la tâche ? De temps en temps, par acquit de conscience, Lian place devant lui plusieurs grosses pierres pour lui barrer la route, mais le petit Gomphal les a toujours contournées pour reprendre son chemin vers le nord.
Soudain exaspéré sans bien savoir pourquoi, Lian va chercher d'autres pierres et se met à les entasser autour de l'arbuste, un enclos circulaire qui lui arrive aux genoux.
Le Gomphal frémit dès que le soleil jaillit des crêtes de l'est. Et alors, hein, que va-t-il faire, maintenant ?
La lumière atteint le sommet de la boule duveteuse, les filaments scintillent, une racine-pseudopode tâte la première pierre...
Une voix retentit derrière Lian, et il sursaute : c'est une voix d'homme, inconnue, qui a posé une question – dans une langue inconnue. Il se retourne. Et voit un étranger. Un petit homme étrange. Pourquoi, étrange ? Il ne sait pas très bien. L'inconnu porte des vêtements dépourvus de marques d'identification, ordinaires pour la saison, sauf peut-être le pantalon avec une pièce de cuir à l'intérieur des jambes, mais ce n'est pas cela. Il répète sa question, et Lian reconnaît tout de même son prénom à la fin de la phrase – cet homme le connaît ? – mais le reste lui est toujours aussi impénétrable. Ce n'est sûrement pas du setlâd ! Il continue à dévisager le petit homme : mince, à peu près l'âge de Nathénèk, une tignasse noire en désordre... La lumière du soleil sculpte un visage aux traits vaguement familiers – est-ce possible ? Mais non, ce n'est pas de là que vient cette sensation bizarre ; quelque chose ne va pas dans ce visage. Les sourcils sont trop fins, trop écartés, le trop grand nez trop droit, les lèvres trop minces... Et la peau, brunie, mais une nuance bizarrement claire. Quelque chose ne va pas avec les yeux, non plus.
« C'est toi qui as mis les pierres, Lian ? » répète l'inconnu, enfin compréhensible, mais avec une intonation curieuse. Reproche, réprimande ? Non, une sorte de... lassitude. Qu'est-ce que ça peut lui faire, si on a mis des pierres ? Déconcerté, prêt à être agacé, Lian hoche la tête en marmonnant " Oui ", et se retourne machinalement vers le Gomphal.
Avec lenteur, avec difficulté, presque à la verticale, le petit arbre est en train de se hisser sur la paroi de sa prison, en direction du nord.
Lian hausse les épaules, cueille le Gomphal presque arrivé au sommet de son escalade, en faisant attention de ne pas en toucher les filaments ; il traverse le ruisseau, pieds nus dans l'eau froide, l'arbuste à bout de bras. Après une cinquantaine de lani, il s'arrête, le pose dans l'herbe, en regarde les pseudopodes tâter le sol puis reprendre leur lente reptation vers le nord. L'étranger n'a pas bougé et lui adresse un regard paisible quand il revient vers lui : « Je m'appelle Odatan », dit-il sans rien offrir d'autre, comme c'est le droit des voyageurs.
Lian retient un sourire : drôle de nom, un étranger qui s'appelle " l'étranger ". « Lian Dougallad Laraïnu », répond-il en tendant les mains, car on lui a appris à être poli, même dans son isolement.
L'étranger lui serre les mains un instant, une expression curieuse passe sur son visage, mais il ne dit rien et ils se mettent en route côte à côte vers la maison. Lian observe le petit homme à la dérobée, à la fois mal à l'aise et curieux : l'impression de familiarité se fait plus intense, comme s'il avait déjà marché ainsi près de cet inconnu, mais quand, où ? La caravane de nuages qui passait sur le soleil s'éloigne enfin, et Lian comprend soudainement ce qui le frappait dans les yeux de l'étranger, à la pupille maintenant étrécie dans la lumière : ils sont très clairs, trop clairs – gris-bleu, comme les siens.
Et comme si cette découverte avait soudain tiré le souvenir de son sommeil, il se rappelle, oui, il a déjà rencontré cet inconnu : sur la côte, au bord de la Mer !
Il s'immobilise. Le petit homme en fait autant, paisible, le dévisage en silence.
« Vous êtes... un halatnim ? » s'entend demander Lian, au mépris cette fois de toute politesse, mais il n'a pu s'en empêcher.
Et à sa grande surprise, l'étranger répond : « Non. » Sans paraître fâché, avec même une sorte de petit sourire triste. Comme il n'ajoute rien et se remet en marche, Lian n'ose insister et lui emboîte le pas.
En s'approchant de la maison, il entend la voix de sa mère qui crie : « C'est toi qui l'as fait venir ! »
Laraï et Nathénèk se trouvent dans la cour intérieure, près du bassin où boit un grand tovker zébré noir et roux. Ils ne les ont pas vus arriver. Laraï est assise à côté de Nathénèk, agrippée des deux mains au rebord du bassin et elle répète, les dents serrées : « C'est toi, c'est toi !
— Mais non, par la Mer ! » explose soudain Nathénèk en frappant l'eau du bassin d'une grande gifle qui les éclabousse tous deux et fait broncher le tovker. « Ce n'est pas moi et je le regrette !
— Ce n'est pas Nathénèk », dit l'étranger de sa voix calme.
Laraï et Nathénèk se retournent en sursautant. Lian, horriblement embarrassé, se glisse vers le tovker et fait mine de le contempler. L'animal tourne la tête vers lui avec un bref son guttural, comme surpris ; c'est un jeune : sur son front, la corne incurvée n'est pas très longue ; la barre de poils sombres plus épais, comme des sourcils jointifs au-dessus des grands yeux bruns au regard liquide, lui donne une expression à la fois sévère et perplexe, comme à tous ses congénères.
« Lian, reprend le petit homme avec naturel, voudrais-tu indiquer à Nagal-Rasser où il pourrait paître ? »
Le tovker se détourne du bassin et quitte la cour, tandis que Lian trotte pour se maintenir à sa hauteur, à la fois reconnaissant et irrité : on l'éloigne.
Une fois dans la prairie où broutent les quelques aski de la maison, Lian s'assied dans le creux de son rocher favori, qui devient vraiment trop étroit pour son confort. Il regarde les aski accueillir l'étranger noir et roux ; comme le tovker n'a pas encore atteint sa pleine croissance, il ne les domine que d'une demi-encolure ; assemblés en rond autour de lui, les aski s'approchent tour à tour en sifflant doucement, le museau levé, et les naseaux frémissants se touchent.
Avec agacement, Lian se rend compte qu'il essaie de ne pas penser à ce qui se passe à la maison entre les adultes. Il se trémousse pour mieux s'installer dans le creux du rocher, ramène ses genoux sous son menton et les entoure de ses bras. Pourquoi aurait-il peur, de toute façon ? C'est plus déroutant qu'inquiétant. L'inconnu a toutes les caractéristiques d'un halatnim, pourtant, les yeux, la peau – encore plus claire que celle de Lian, en fait ! En quelle langue parlait-il ? Et il voyage avec un tovker... Lian contemple la haute silhouette zébrée tandis que l'animal broute paisiblement tout près de lui. Un tovker a choisi cet Odatan pour compagnon ! C'est comme dans les anciennes légendes de Laraï. Il n'y plus de rois ni de princes depuis bien longtemps, mais les tovik se choisissent encore des compagnons parmi les humains – les tovik de Tyranaël, car il n'y en avait pas sur Atyrkelsaõ lorsque les Ranao y sont arrivés, ils les ont amenés avec eux, ceux qui ont voulu venir. Laraï ne lui a jamais expliqué comment se faisait le choix, simplement qu'il était rare, et précieux, et un signe de grande valeur. Odatan est un homme de grande valeur, alors. Qu'est-ce qu'un homme de grande valeur ressemblant à un halatnim vient faire à Leïtnialen ?
Encore une autre variété de halatnim, peut-être. Est-ce qu'on le voit, cet Odatan, quand on ne le regarde pas ?
Lian se mord les lèvres à la pensée qui vient de lui traverser l'esprit. Puisqu'on ne le voit pas, lui, si on ne sait pas qu'il est là, il pourrait se glisser jusqu'à la maison et...
Il se renfonce dans son rocher, honteux, et en même temps un peu étonné. Il n'a jamais eu cette idée auparavant, ni chez les Olaïlliu ni avec ses parents. Il lui arrive toujours de surprendre quelqu'un, bien sûr, et maintenant qu'il sait pourquoi, s'il le faut, il s'en excuse ; on ne lui en tient de toute façon jamais rigueur. Mais le faire délibérément, ce serait... impoli. Et puis, si on lui a dit de s'éloigner, c'est parce qu'on ne veut pas qu'il soit là. On a des choses à se dire qu'il ne doit pas entendre. Qui ne le concernent pas.
Ou bien qui le concernent, justement ? Mais alors, pourquoi ne doit-il pas les entendre ? L'inquiétude renaît, s'alourdit, devient insupportable. Et finalement, Lian quitte son rocher. Il se glisse dans le passage voûté qui donne sur la cour intérieure, où le visiteur se trouve toujours avec ses parents, en train de dire d'une voix patiente : « Il faudra bien le lui apprendre un jour. » Et se retourne vers le buisson d'atlevet rose derrière lequel s'est dissimulé Lian : « Et il ne voudra peut-être pas toujours rester sur le plateau, n'est-ce pas, Lian ? »
Le feu aux joues, et stupéfait, Lian se redresse. Entend derrière lui un petit reniflement. À l'entrée du passage, le tovker renifle à nouveau, comme amusé, secoue sa crinière et fait volte-face pour retourner vers la prairie.
« Viens, Lian », dit Nathénèk. Il ne semble pas vraiment fâché.
« Pourquoi voudrait-il quitter le plateau ? » demande Laraï entre ses dents serrées. Elle, elle est fâchée.
« Aimerais-tu aller ailleurs sur Atyrkelsaõ, Lian ? » dit Odatan.
Lian lance un regard à Nathénèk, qui incline la tête. Il s'approche en examinant la question, vite affolé. Aller ailleurs ? Quitter Leïtnialen ? Quitter ses parents ?
« Pas maintenant, mais plus tard », précise le petit homme.
L'affolement diminue un peu. « Quand ?
— Dans quatre ou cinq saisons. Pour ton eïldaràn.
— L'eïldaràn ne le concerne pas ! » lance Laraï d'une voix basse et intense.
— Si ce que dit Odatan est vrai, il doit la subir quand même, c'est plus sûr », réplique Nathénèk.
Eh, ont-ils déjà oublié qu'il est là ? Lian fait un pas en avant : « C'est quoi, l'eïldaràn ? »
Ils se tournent tous vers lui.
« L'eïldaràn, dit le petit homme d'une voix posée, c'est l'ouverture des portes. Celles qui peuvent être ouvertes. Les gens qui possèdent des dons ne le savent pas toujours. Quelquefois, la porte s'ouvre toute seule. Quelquefois, il faut l'aider. Toi, rien n'entre, rien ne sort, et tu es un naïstaos. Mais c'est peut-être que ta porte est si bien fermée qu'elle en est invisible. L'eïldaràn pourrait nous le dire. »
Lian apprend ainsi le nom de ceux qui ne peuvent pas rejoindre la Mer, naïstoï : les séparés, les enfermés. Il n'est pas le seul. Beaucoup de halatnim sont ainsi, mais aussi parfois des Ranao. Pas tous enfermés de la même façon que lui, cependant, et certains sortent tout seuls, ou avec de l'aide. Son père-Dougall était un naïstaos aussi. Mais pas vraiment comme lui. Toi, dit Odatan avec l'expression de quelqu'un qui ne veut pas avoir l'air triste, tu es spécial.
Lian comprend. Mais surtout, à l'expression de ses parents, il devine pour la première fois que leurs silences n'étaient pas simplement remplis d'histoires qu'il entendrait plus tard, quand il serait grand. Il y avait dans ces silences des choses qu'on ne voulait pas lui dire du tout, jamais. Qu'on lui cachait. Il les regarde tour à tour, Nathénèk, Laraï, il voit l'effort qu'ils font pour ne pas détourner les yeux.
Est-ce que ce sont des mensonges, quand on ne vous dit pas ce qu'on devrait vous dire ?
Quand Lian pensait à Tyranaël, jusque-là, le monde d'où étaient venus les Ranao, c'était comme à une légende : à travers des légendes. Les explications de Laraï étaient restées très floues sur la façon dont les Ranao étaient arrivés sur Atyrkelsaõ. " Le Jour où la Mer était couverte de Vaisseaux " – cent millions de navires, disait l'histoire. Nathénèk avait aussitôt rectifié : cela ne s'était pas réellement passé ainsi. « La Mer ne nous a pas amenés tous ensemble en une seule fois. Nous étions bien plus de cent millions, de toute façon ! Il a fallu près de cent vingt saisons, et pendant tout ce temps les gens sont venus petit à petit, à mesure qu'on installait ce qu'il fallait pour les accueillir.
— Mais il y a eu le jour du grand passage, en 17 LH, avait insisté Laraï. Des centaines de milliers de navires sur la Mer.
— Après quoi il a continué d'en arriver jusqu'en 30 LH, et même encore quelques retardataires après », s'était obstiné Nathénèk.
Lian ne voyait pas la raison de ce débat et s'en était désintéressé. « LH », les lettres qui suivaient les années dans toutes les dates inscrites dans les carnets, voulaient dire " Lita Hdiaël ", " après le grand voyage " ; peu lui importait si ce voyage-là n'en avait pas été un de cent millions de vaisseaux et avait seulement constitué le début, relativement modeste, de l'installation de ses ancêtres sur Atyrkelsaõ. Il préférait contempler le Leïtnialen quand le lac se perdait dans la brume, et l'imaginer comme la Mer, soudain couvert de voiles multicolores à perte de vue. Mais l'idée avait fait son chemin souterrain, jusqu'au jour où il avait soudain compris que, lorsque la Mer quittait Atyrkelsaõ, elle retournait sur Tyranaël. L'image des graines de fofolod dans leur gousse avait pris une dimension nouvelle : c'était plutôt... comme ces graines jumelles qu'on trouve parfois, soudées par un côté, sauf que là c'était plutôt comme une sorte de... conduit, et c'était la Mer, le conduit ! « Est-ce qu'on pourrait revenir avec la Mer, revenir sur Tyranaël ? » avait-il demandé à Nathénèk, tout excité. Mais son père avait secoué la tête, gravement : « Non, Lian. Seulement dans un sens, le passage. »
Et Laraï avait ajouté, comme si elle était un peu fâchée : « Et il n'y a plus rien pour nous de l'autre côté. »
Mais après l'arrivée d'Odatan, tout change, comme au jeu des mille bornes où, lorsqu'on commence à retourner les cartes une à une, le jeu se modifie au fur et à mesure, en cascade, prenant avec chaque carte des configurations nouvelles. Tyranaël existe dans une autre demeure de Hananai et non, comme Lian avait pris l'habitude d'y penser, dans une autre chambre de la même maison que lui, loin, derrière une porte fermée. Et elle n'est plus " Tyranaël " que dans la mémoire des Ranao. L'autre côté de la Mer a d'autres habitants, qui lui ont donné un autre nom dans leur propre langage, Virginia, qui l'ont transformée, qui ignorent tout d'Atyrkelsaõ, pour qui les Ranao sont des fantômes et pas même des légendes. Les Terriens, les Virginiens, comme le dit plutôt Odatan : d'autres êtres, différents des Ranao, mais humains. Comme les moatranao ? Non, pas du tout : ils sont venus à travers l'espace, dans des vaisseaux qui n'ont pas de voiles, d'un monde complètement différent né d'un autre soleil très lointain. Ce sont des gens, en tout cas. Et on est parti de Tyranaël bien avant qu'ils n'y arrivent ? En quoi étaient-ils un danger ?
Le visiteur ne dit rien ; il regarde Nathénèk et Laraï. Nathénèk et Laraï se regardent. « Que fait-on, Lian, quand les pitlànkraï essaiment ? » demande enfin Laraï, d'un ton distant.
De l'autre côté du lac, deux saisons plus tôt, il a suivi avec elle, sur plusieurs langhi, la large zone dévastée, et il a vu les carcasses parfaitement nettoyées des petits animaux qui ne se sont pas écartés assez vite. Il avait demandé avec effroi : « Qu'est-ce qu'on ferait si elles venaient de notre côté ? »
Lian regarde sa mère, puis son père. Quand les minuscules et féroces insectes sont en marche, on se gare. Mais les Terriens n'étaient pas, ne sont pas, des pitlànkraï, sûrement ! On n'aurait pas pu s'arranger avec eux ?
Laraï dit : « Non.
— C'est ainsi, soupire Nathénèk.
— Chacun de son côté », ajoute Odatan après un petit silence.
Lian finit par acquiescer – ce sont les adultes, ils doivent savoir, n'est-ce pas ? Et puis, c'est de l'histoire ancienne, on ne peut plus rien y changer. Les Ranao sur Atyrkelsaõ, les Virginiens sur Virginia...
Lian se raidit brusquement. Chacun de son côté. Mais la Mer... Il regarde les trois adultes tour à tour, un peu inquiet : « Est-ce qu'ils pourraient venir sur Atyrkelsaõ, les Virginiens, avec la Mer ?
— Depuis environ deux cents saisons, quelques-uns réussissent à passer avec elle, dans un seul sens aussi », dit le visiteur, qui l'observe avec attention. « C'est... un don, comme pour les aïlmâdzi, les danvérani ou les keyrsani, mais très rare. Les halatnim sont des hybrides, les enfants que les passeurs ont créés avec les Ranao. »
Le regard abasourdi de Lian passe de Laraï à Nathénèk. Il balbutie enfin : « Mon père... mon père-Dougall était un passeur ?
— Un halatnim. »
Lian se sent les jambes un peu molles. Il va s'asseoir sur le rebord du bassin, près de Laraï. Il espère un moment qu'elle va lui parler ou le toucher, mais elle ne bouge pas. Elle fixe ses mains croisées sur ses genoux et elle ne dit rien. Nathénèk non plus.
Pas comme les moatranao, les Virginiens. Nés d'un monde complètement différent. Pas du tout la même espèce, alors ?
« Mais comment peuvent-ils avoir des enfants avec les Ranao ? »
Laraï dit, d'une voix sarcastique et dure que Lian ne lui connaît pas : « Avec difficulté. »
Elle se lève et quitte la cour. Nathénèk esquisse un mouvement pour la rattraper, mais Odatan l'arrête, se tourne vers Lian : « On les a modifiés un peu pour que ce soit possible. » Puis, un peu plus bas, comme pour lui-même : « Ils ne sont pas si différents des Ranao. »
*
Thomas Ewald, c'est le Président de Virginia à l'époque de l'Insurrection. Il a perdu ses deux fils et sa fille pendant les combats autour de Morgorod – un horrible accident, une erreur de calcul, ils ont été écrasés sous un bombardement effectué par les Fédéraux. Et après bien des examens de conscience, il décide que la guerre civile a assez duré, Thomas Ewald. Il tend la main, il propose une trêve – non sans avoir eu bien du mal à convaincre son propre gouvernement. On offre aux rebelles une Année de désengagement total, pendant lequel ils pourront se rendre, eux et leurs sympathisants, en Licornia. Laissez-passer pour tous ceux qui le veulent, pas de questions, pas de contrôles. Et les rebelles acceptent – pardi ! Une dizaine de millions de personnes se retrouvent à la fin de l'Année dans le sud-est. Et là, en profitant lâchement du retour de la Mer qui neutralise en grande partie l'avantage technologique des Fédéraux, ils frappent. Dans les grandes villes, encore, attentats, sabotages : ils ont laissé derrière eux une cinquième colonne d'agents décidés à tout. Ils attaquent Dalloway, où avaient lieu les pourparlers de paix. Ils échouent. La chasse aux clandestins commence partout sur le continent. On établit la ligne Ewald, tout le long des montagnes Rouges et jusqu'à la côte est, un glacis protecteur d'une quinzaine de kilomètres de large, hérissé à intervalles de forts, de casemates, semé de mines et de senseurs, où rien ne peut bouger sans être repéré. On s'installe dans ses positions de part et d'autre de la Ligne, une trêve armée qui dure seize Années. Dans le reste du continent, la vie reprend tant bien que mal, on panse les blessures, on relève les ruines, on veut croire que c'est fini. Plus de soixante saisons s'écoulent : deux générations n'ont jamais connu la guerre. Et en 126, les rebelles attaquent de nouveau Dalloway.
Depuis, de temps à autre, le ciel s'allume dans les montagnes Rouges. Les rebelles essaient ici ou là de traverser la ligne Ewald, les Fédéraux les repoussent. La plupart du temps, pas grand-chose : escarmouches, tirs symboliques. C'est surtout autour de Dalloway. Pas question de laisser se relâcher la vigilance, cependant – chaque fois, les Rèbs ont essayé d'en profiter. C'est pourquoi l'effort de guerre s'intensifie dans les deux Mois qui précèdent le départ de la Mer : il faut équiper les troupes fraîches qu'on enverra effectuer leur tour sur le front. Les conscrits sont tirés au sort dans la classe des dix-huit/vingt saisons. C'est un très lourd fardeau sur l'économie de maintenir ces dix millions d'hommes sur la Ligne tout le temps. On n'aime vraiment pas les Rèbs à la commune – si on n'aime pas trop non plus certains aspects du gouvernement fédéral.
Quand Liam demande des précisions sur l'idéologie des rebelles, on fronce le nez : capitalisme sauvage, pollution effrénée, individualisme immoral... Liam ne comprend pas tout très bien, mais ni le mot " mutants " ni le mot " pouvoirs " ne sont prononcés une seule fois, c'est l'essentiel. Bon, il y a la guerre. Mais entre normaux. L'histoire telle qu'on la lui a racontée, il peut la comprendre.
La bulle a éclaté, pourtant. Il ne peut plus s'imaginer qu'il rêve. Il est réveillé. Il n'arrivera pas à se rendormir.
9
Dans l'après-midi, le visiteur s'en va. Laraï n'a pas reparu. Devant la maison, le tovker recourbe une patte antérieure pour permettre à son cavalier de monter en s'aidant du sabot comme d'une marche mouvante. Une fois sur la selle, le petit homme passe ses jambes dans les cuissières obliques et enroule une de ses mains dans la crinière du tovker. Jamais de bride ni de mors, bien sûr, pour un Libre Compagnon.
« Bon chemin, dit Nathénèk en levant une main.
— Bon chemin, répond le petit homme. Veux-tu monter sur Nagal-Rasser, Lian, et faire un petit bout de route avec moi ? »
Lian a à peine le temps de se tourner vers Nathénèk qu'il se sent attrapé sous les bras par-derrière et hissé d'un seul mouvement vers Odatan, qui l'aide à s'installer sur l'avant de la selle. « Assieds-toi en tailleur, c'est le plus confortable pour quelqu'un de ta taille. Je te servirai de dossier – et mon chapeau de parasol ! Accroche-toi. » Lian attrape une touffe de crinière. Le tovker ne bronche pas. Nathénèk les contemple avec une expression indéchiffrable, la tête rejetée en arrière, les yeux plissés sous le soleil.
Le tovker se met en route, au pas, sans un signal de son cavalier. Ils suivent la berge du lac en direction du sud-ouest. Ce n'est pas du tout comme sur un asker ; on est plus haut, d'abord, plus haut même que dans la carriole : le monde paraît différent ainsi, un peu comme depuis un arbre, mais un arbre animé. Et puis, assis en tailleur sur cet énorme dos, c'est difficile de se croire un cavalier ; on a plutôt l'impression... d'être sur un bateau !
« Nagal-Rasser fait toujours ce que vous voulez ? » Aussitôt la question posée, Lian la trouve idiote : le tovker a choisi Odatan !
« La plupart du temps. Et quand il ne veut pas... on en discute. » La voix du petit homme a une inflexion amusée.
Dans les histoires de Laraï, les tovik sont toujours des animaux intelligents, et qui parlent. Mais Lian a dix saisons, il sait faire la différence entre les animaux des histoires et les véritables animaux : Tiuli, par exemple, le banker des Olaïlliu, ne lui a jamais adressé la parole ! Il se met à rire : « Et comment vous faites ?
— Nous nous parlons en esprit, dit l'autre après une petite pause. Les tovik sont comme des danvérani. La mutation qui a affecté les Ranao, à l'aube des temps, sur Tyranaël, les a également touchés. Plusieurs autres animaux aussi. »
Lian se retourne pour voir s'il est sérieux. « Les banki, les lladao ? » Les histoires de Laraï seraient donc vraies ?
« Plutôt des lâdzani, eux.
— Les Gomphali ? » Dans l'histoire de Lileïniloo, c'est le Gomphal qui alerte la Hébaë de la présence du Gardien du Vent. Lian n'y croyait plus, pas davantage qu'au reste, mais...
« Oui, mais parce que ce sont des créatures de la Mer. »
Lian abasourdi s'adosse à nouveau contre la poitrine du petit homme. Au bout d'un moment, il murmure : « Tout le monde, alors, sauf les halatnim.
— Non, les halatnim aussi. Au début, les premiers passeurs qui sont venus avec la Mer étaient tous des danvérani. Mais ils n'ont jamais été très nombreux. Le don s'est perdu chez leurs enfants à mesure que les générations se sont succédé. Un peu comme... du sirop dilué dans trop d'eau. Ceux de la sixième génération, ta génération, Lian, sont maintenant souvent de simples lâdzani. »
Lian se retourne encore, avec un soudain espoir : « Moi, on ne me voit pas si on ne me regarde pas, mais vous saviez que je me cachais dans le buisson !
— Non. Mais Nagal-Rasser t'avait suivi, et je t'ai vu dans son esprit », dit Odatan ; il esquisse un petit sourire. « Nagal-Rasser me sert parfois d'yeux et d'oreilles... »
Lian digère l'information, déçu, un peu vexé. « Vous lui aviez demandé de me surveiller ?
— Non. Mais je te l'ai dit : Nagal-Rasser fait aussi ce qu'il veut. »
Le tovker secoue un peu la tête avec un petit sifflement ronflant, comme s'il avait compris leur échange – et Lian réalise qu'il a bel et bien compris, qu'il les écoute par l'intermédiaire d'Odatan. Du coup, il redevient muet, tandis qu'ils s'engagent dans l'ombre des tingai bordant cette partie du lac. Mais il a des questions à poser, il les posera et ce n'est pas ce... ce bateau sur pattes qui va l'en empêcher !
« Mon père-Dougall, c'était un lâdzani ?
— Un petit danvéràn. Il entendait et voyait un peu, mais pas très loin ni très clair. »
Au bout d'un moment, Lian dit tout bas : « Et moi je ne suis rien du tout, même pas un lâdzan. »
Les bras d'Odatan se resserrent autour de lui : « Ne dis pas ça, Lian ! Crois-tu que ce soit si plaisant d'entendre ce que les autres pensent, sans pouvoir le contrôler ? Ton père-Dougall était ainsi. Ça l'a rendu très malheureux. Il aurait voulu être un vrai danvéràn. Mais tes petits amis Olaïlliu, ce ne sont pas des danvérani et ils s'en passent très bien, non ? Et ça ne les dérange même pas de ne pas te voir comme eux se voient les uns les autres. Ce sont tes amis quand même, n'est-ce pas ?
— Oui..., admet Lian.
— Tu connais les plantes et les animaux de la forêt encore mieux qu'eux. Tu es capable de vivre tout seul sur le plateau. Tu sais faire de l'aquarelle, sapristi ! »
Lian réfléchit : « Je vais apprendre à jouer du piano, aussi, décide-t-il soudain.
— Très bonne idée. »
Au bout d'un moment, Lian reprend : « Mais l'eïldaràn, alors, Mère-Laraï a raison, ça ne sert sûrement à rien, je n'ai pas besoin d'aller à Hleïtzer... »
Le petit homme se dégage soudain des cuissières, prend Lian éberlué à bras-le-corps et glisse avec lui à bas du tovker. Il traverse la piste sans le lâcher, l'assied sur un tronc de bois flotté au bord de la plage. Son visage est grave, mais résolu : « Écoute-moi bien, Lian. Les dons... sont plutôt comme des ruisseaux : ils s'enfoncent dans la terre et on croit qu'ils ont disparu, mais parfois ils sont simplement en train de se transformer. Tu te rappelles, quand la Mer est arrivée ? »
Pris au dépourvu par le changement de sujet, Lian hoche la tête.
« Tu as vu sa lumière. La lumière de la Mer. Les vrais naïstoï ne la voient pas. Ils ne voient que du brouillard. Il en naît quelques-uns comme toi depuis deux générations. Une nouvelle variété de naïstoï. C'est ce qui me fait penser que tu as peut-être une porte à ouvrir. Que le don s'exprime d'une autre façon chez toi. »
— Je deviendrais... un danvéràn ? » souffle Lian incrédule.
Le petit homme se mord les lèvres : « Peut-être. Peut-être pas. Ce n'est encore jamais arrivé ici. » Sa voix se raffermit : « Mais si on n'essaie pas, on ne saura pas. Et puis quelquefois, surtout après la quinzième saison, chez les halatnim... les portes s'ouvrent toutes seules. Et si cela arrive... » – le regard clair s'assombrit – « Il vaudrait mieux que ce soit à Hleïtzer, où il y aurait du monde pour t'aider, qu'ici sur le plateau.
— Pourquoi ?
L'autre dévisage Lian un moment comme s'il cherchait une explication, soupire en haussant un peu les épaules : « Imagine que tu aies toujours vécu dans le noir. Et que tout d'un coup tu voies le soleil. Tu serais aveuglé, tu aurais mal, tu aurais peur. Non ? »
Lian acquiesce, en se demandant où l'autre veut en venir.
« Mais » – et le petit homme croise ses mains devant Lian, paumes tournées vers lui, à la hauteur de ses yeux – « s'il y a quelqu'un pour te protéger du soleil, pour t'expliquer ce qui se passe, pour te montrer comment faire pour vivre dans la lumière, c'est mieux pour toi, non ? » Il décroise les mains pour conclure : « Et comme les gens qui ont très peur et très mal peuvent être très dangereux, c'est mieux aussi pour ceux qui les entourent. »
Lian réfléchit un moment : « Mais il y a des gens sur le plateau. Il y a mes parents, à Leïtnialen. »
L'autre soupire de nouveau : « Ils ne pourraient pas t'aider. Seuls des hékel le pourraient. »
Lian ramasse des cailloux et les jette un à un dans le lac à ses pieds. « C'est Kéryan qui vous a fait venir ? »
Odatan ramasse un caillou aussi, bien plat, et l'envoie rebondir en ricochets. « Non, Lian. » Il le regarde de biais, les yeux plissés. « Je venais te voir. Je te connais depuis que tu es né, et même avant. Ce n'est pas la première fois que nous nous rencontrons, le sais-tu ? »
Lian murmure : « Oui. » Dans le silence, un souvenir de nuit violette passe entre eux.
Une autre série de ricochets. Lian se prend au jeu, trouve un caillou plus approprié, le lance. Trois rebonds. Il peut faire mieux. Il se met à chercher pour de bon autour de lui.
Après un bref concours, que Lian gagne d'une courte tête, le petit homme se lève. « Je dois partir, Lian. »
Lian le contemple, la main en visière devant les yeux parce qu'il est face au soleil. « Vous reviendrez me chercher, pour l'eïldaràn ? »
Le visage de l'autre est perdu dans l'ombre de son chapeau, mais Lian entend sa voix sourire quand il dit : « D'accord. »
En revenant à la maison, après avoir regardé le petit homme disparaître dans la forêt, Lian trouve Nathénèk assis à la table ronde installée sous l'arbre qu'ils ont planté en remplacement du Gomphal. Il épluche des tubercules de sarsinit, en en grignotant un de temps en temps. Lian hésite, infléchit son chemin et vient s'asseoir près de lui sur le banc circulaire. Nathénèk lui adresse un rapide coup d'œil, esquisse un sourire, reprend son travail. Lian contemple le visage familier de son père – son autre père, mais c'est Nathénèk qui lui a chanté les premières chansons, qui lui a appris à rouler en dana-dana, à pêcher à la sagaie... C'est Nathénèk, son vrai père, il a toujours su que cela n'a rien à voir avec le sang ; il le voit bien chez les Olaïlliu, où Kavnid s'entend mieux avec Miniaz qu'avec son fils Tarmel.
« Odatan dit... que je deviendrais peut-être un danvéràn, si je vais à Hleïtzer. »
Nathénèk hoche la tête sans cesser de gratter les tubercules : « Ça te plairait ? »
Lian réfléchit, hausse les épaules : « Tant qu'à faire, je préférerais voler, comme toi. »
Nathénèk sourit sans rien dire.
Lian retourne entre ses doigts une fleur du kaïringa tombée sur la table, puis reprend : « On me plantera dans l'île d'Ëlmadziulan, alors, quand je serai mort. Je deviendrai un tingalyai.
— Oui », dit Nathénèk d'une voix paisible, et Lian considère donc l'éventualité sans trop de crainte non plus. Vivre deux mille années au bord d'un beau bassin et donner l'eau et la vie à tous, ce n'est pas si épouvantable. Et puis, il fera plein de graines.
Et il a dix saisons, il a bien le temps.
Il demande quand même : « Pourquoi vous ne m'avez jamais rien dit ? » Et, en entendant sa propre voix, il comprend que ce n'est pas un reproche mais une véritable curiosité.
Les mains de Nathénèk s'immobilisent un instant, reprennent leur tâche. « Pas " rien ", dit-il enfin. Et tu as commencé à en apprendre pas mal avec tes petits amis Olaïlliu.
— Ati ne voulait pas tellement que je les rencontre », remarque Lian, pour qui quantité d'événements passés viennent en un éclair de prendre un sens nouveau. Il ne dit pas " toi, oui ", mais ce n'est pas nécessaire.
Nathénèk pèle un autre tubercule en silence. Son bon visage rond a une expression inhabituelle, triste et presque dure tout à coup. « Ta mère-Laraï... a été très malheureuse autrefois. » Il semble chercher ses mots, reprend avec un soupir : « Avec ton père-Dougall. Mais pas seulement à cause de lui. Il était très malheureux aussi. Elle pensait... que tu le serais moins en sachant moins de choses. »
Il jette un coup d'œil à Lian : « Es-tu plus heureux maintenant ? »
C'est une véritable question aussi, et Lian l'examine comme elle le mérite. Il conclut : « Je ne suis pas plus malheureux. » Il réfléchit encore un instant et rectifie : « Je ne suis pas malheureux. »
Ce n'est pas tout à fait vrai. Il y a dans son cœur une étrange petite place silencieuse, et il ne veut pas trop savoir ce qui se trouve là et qui se tait. Mais ça ne fait pas mal, alors il répète : « Je ne suis pas malheureux. »
Nathénèk l'attire contre lui, l'embrasse, et pendant un moment, serré contre la poitrine de son père, Lian se dit que rien, absolument rien ne peut lui faire du mal, ni tout ce qu'il ignore, ni tout ce qu'il apprendra, plus tard, quand il sera grand, quand il ira à Hleïtzer.
Laraï n'évoquera jamais la visite d'Odatan. Mais deux semaines environ après son passage, alors que Lian se trouve chez les Olaïlliu pour ses leçons d'aquarelle et maintenant de piano, il voit arriver Kéryan Bataliz Dialadnu, qui s'ébroue dans l'entrée – les pluies sont un peu en avance. Il n'y a personne pour l'accueillir, tout le monde travaille à cette heure-ci dans la maison, les ateliers ou le laboratoire, mais cela ne semble pas la déranger. Elle se rend dans la cuisine où Naritias prépare la collation de mi-journée, et revient s'asseoir dans le divan bas ; elle boit son thé en écoutant les gammes plus ou moins laborieuses de Lian. Lorsqu'il a fini et que Gundiel le libère, Kéryan se lève et s'avance vers lui. Après avoir échangé un sourire avec la hékel, Gundiel quitte la pièce.
« Lian, ta mère-Laraï m'a demandé de t'enseigner la langue de ton père, si tu le désires. Le désires-tu ? »
Lian, pris au dépourvu, contemple le visage maigre et grave qui se penche vers lui. « La langue de mon père ?
— La langue des passeurs, et des halatnim qui veulent la parler. La langue qu'on parle de l'autre côté. »
Lian fait mine de ranger les feuilles de musique pour se donner le temps de réfléchir. « Comment c'est ? »
La hékel prononce une longue phrase incompréhensible, presque sans accentuation ni ligne mélodique, et rythmée de pauses bizarres, mais pas désagréable.
« Ça veut dire quoi ?
— Qu'on a retrouvé l'éternité, et que c'est la Mer partie avec le soleil. »
Lian hausse les sourcils malgré lui et Kéryan esquisse un petit sourire : « De la poésie.
— Est-ce que c'est difficile ?
— Moins que le setlâd. »
Un bon argument. Mais Lian, à qui Nathénèk essaie d'apprendre à avoir l'esprit pratique, demande encore : « Est-ce que j'en ai besoin ?
— Tu en auras sans doute besoin si tu vas à Hleïtzer », réplique posément la hékel.
Outre l'aquarelle et le piano, Lian apprendra désormais le virginien chez les Olaïlliu. Une langue plus facile en effet que le setlâd : les mots ne changent pas de sens selon leur accentuation, par exemple, s'ils changent – quelques-uns seulement, les homonymes – selon leur graphie (et alors il faut deviner le sens au contexte, mais c'est bien plus simple) ; et on ne peut pas les utiliser aussi créativement qu'avec le setlâd, où l'on peut faire glisser des mots les uns dans les autres à volonté pour en créer de nouveaux, ou en inventer de toutes pièces au fil du besoin ou de la fantaisie à partir des racines connues de tous. Mais le vocabulaire virginien n'est pas si simple que cela, et la syntaxe non plus ; de nombreux mots existent en six ou sept versions parfois assez différentes – des variations locales, dit Kéryan, mais apparemment tout le monde doit les connaître. Même chose pour la syntaxe : il y a des règles générales de fonctionnement, mais à l'intérieur quantité d'exceptions, également des " variations locales ". Finalement, le principe en est un peu celui du setlâd : plusieurs langues voisines ou distinctes qui ont fini par s'accommoder les unes des autres sans disparaître. Une des caractéristiques les plus déroutantes pour Lian, cependant, c'est la rigidité et la pauvreté de l'attribution des genres. Féminin, masculin, et ce " neutre " aux formes si souvent masculines qui n'équivaut absolument pas au registre setlâd englobant du " han'maï " – d'ailleurs intraduisible en virginien : " sous le regard de Hananai " ne veut rien dire si on ignore la nature multiple de la Divinité.
Mais une langue, c'est comme une forêt, et Lian l'explore avec autant de curiosité et de fascination qu'il a appris à connaître celle du Landaïeïtan. Au début, quand il rentre à Leïtnialen, il essaie de partager ses découvertes. La réaction de Laraï l'en décourage vite, si Nathénèk tolère un peu mieux ses comptes-rendus – mais Nathénèk ne parle pas le virginien. De fait, Lian finit par comprendre qu'il ferait mieux de ne jamais parler cette langue en leur présence. Il a le droit d'étudier, et même d'apporter des livres à la maison, mais c'est tout. Il n'ose demander à sa mère si elle parle la langue virginienne, ou si elle la parlait, au moins, avec son père-Dougall ; tout d'un coup, cela ne lui semble plus aussi évident que lorsque Kéryan s'est offerte à la lui apprendre.
10
Un jour, au début d'Avril, un des grands-pères meurt. C'est Stefan, le grand-père d'Andreï, l'un des plus vieux. Il vient de se lever de la table du souper, et il tombe, il s'écroule, tout d'un bloc, sa canne rebondit sur les dalles avec un claquement sec. On se précipite, en premier Paula Eklosz, l'infirmière de la commune, mais on sait déjà qu'il est mort. Elle lui touche le cou, le poignet, par acquit de conscience. Secoue la tête d'un air navré et passe la main sur les yeux fixes. La grand-mère d'Andreï s'est affaissée dans les bras des autres grands-mères. On l'escorte dans sa chambre.
« Aide-moi, Liam », dit Paula à Liam qui se trouve être le plus proche d'elle parce qu'il s'est précipité comme les autres, quoique sans comprendre. Il saisit le vieil homme sous les bras, ils le portent dans sa chambre à côté de celle de la grand-mère qu'on peut entendre sangloter par la porte restée ouverte, à travers les murmures apitoyés des autres. Ils l'étendent sur le lit. Avec douceur, Paula lisse les cheveux blancs en désordre, remet les vêtements en place, croise les mains noueuses du vieil homme sur sa poitrine. Doit prendre conscience de l'immobilité pétrifiée de Liam, car elle relève la tête, le regarde et dit avec compassion : « Il avait cent trois ans, Liam. Il a bien vécu. Et c'est une belle mort, sans douleur. »
Cette nuit-là, très tard, Liam se glisse dans la chambre du mort. Plus personne, on a mis fin à minuit à la veillée funèbre. On a pourtant laissé allumée la lampe à gaz, en guise d'ultime veilleuse – ou bien craint-on que le mort ne se réveille dans le noir ? Liam augmente la flamme, s'assied sur le bord du lit. Le vieil homme semble-t-il plus petit ? On dirait qu'il dort.
Liam reste là sans pensée, opaque, vide. Au bout d'un moment, il entend la porte s'ouvrir. Des pas. Il se lève, machinalement. C'est Suzane, en robe de nuit, boucles blondes toutes ébouriffées, les yeux rougis. Ce grand-père-là était son préféré. Ils se regardent longuement, que comprend-elle ? Elle dit « Oh, Liam ! », le prend dans ses bras avec des sanglots muets. Il la serre contre lui, il se rend bien compte qu'elle en a besoin, même si lui est sans larmes.
Au bout d'un long moment, elle s'écarte un peu, s'essuie les yeux en murmurant : « Il est tard, viens, il faut aller se coucher. » Ils quittent la chambre du mort, main dans la main. Quand ils arrivent à la chambre de Suzane, elle ne lui lâche pas la main, alors il entre avec elle. Comme il hésite, incertain de ce qu'il doit faire, elle dit d'une voix enrouée : « Reste un peu avec moi ? » Elle se couche dans le lit ouvert. Il s'assied, caresse ses cheveux lumineux comme de l'or dans la lueur de la lampe de chevet. Après un moment, elle se pousse, tapote le matelas près d'elle en murmurant « Viens là, va ». Il obéit, s'étend près d'elle sur le lit étroit. Elle éteint la lampe, se blottit contre lui. Il sent ses mains sur lui, sur sa poitrine, sur son cou. Il comprend, elle a besoin d'être rassurée. Avec une douceur désolée, il lui caresse les épaules, les cheveux. Maintenant, il a envie de pleurer, mais il ne sait pas vraiment pourquoi. Après un moment, elle se blottit de nouveau contre lui sans bouger, et il écoute son souffle devenir régulier tandis que les premières lueurs de l'aube se glissent à travers les rideaux.
*
La première fois qu'ils ont été jusqu'au bout, Miniaz s'est appuyée sur un coude pour contempler le corps nu de Lian, suivant d'un doigt les contours de ses pectoraux pour en agacer de l'ongle les aréoles sensibles, descendant jusqu'au nombril, puis jusqu'au sexe redevenu tout timide, encore mouillé d'elle et de lui. Elle a dit : « Ça fait bizarre », avec un sourire complice, et Lian ne s'est pas senti blessé : il trouvait ça plutôt bizarre aussi, surtout après avoir vu Tarmel leur démontrer de ce que pouvait un pénis rani en érection ; en l'occurrence, ce qu'il lui enviait le plus en cet instant, c'était le contrôle de l'éjaculation ; il croyait pourtant avoir fait des progrès en ce sens, mais il a été pris par surprise. Comme Miniaz a ajouté, en frottant d'un air coquin une joue contre son épaule : « Mais c'était plutôt agréable », la conversation a pris très vite un autre tour, plus satisfaisant pour tout le monde, cette fois. Plus tard seulement, curieux, il lui a demandé : « Bizarre comment ? », et elle a froncé le nez : « Toucher quelqu'un ainsi sans le toucher. Mais c'est plutôt... intéressant. Plus mystérieux. On ne sait pas tout de suite ce qui fait plaisir ou pas... » Elle lui a fait un clin d'œil : « Et quand ça marche, alors, ça vaut vraiment la peine ! »
Avec Maïli, c'est différent. Moins complice. Et puis elle est devenue fertile très tôt, il a fallu limiter un peu les explorations. Tarmel, c'est complètement autre chose encore, très expérimental quelquefois – mais en général paisible et silencieux, parce que toucher quelqu'un sans le toucher plonge Tarmel dans l'extase : il pourrait passer des heures à caresser Lian et à en être caressé. C'est tout de même Miniaz que Lian préfère. Mais elle a une saison de plus que lui et va partir bientôt pour son eïldaràn – tous les adolescents un peu doués d'Atyrkelsaõ, les hékellin, participent au rituel, et Miniaz est une petite danvéràn. Elle ne le lui a jamais vraiment dit, on ne parle jamais directement de ce qui touche aux pouvoirs chez les Olaïlliu, du moins pas en présence de Lian, mais il le sait.
« Quand tu reviendras de Hleïtzer, le mois prochain, tu voudras encore de moi ? plaisante-t-il à demi.
— Oh, je ne reviendrai pas tout de suite », dit Miniaz, distraite par les lèvres et les mains de Lian sur elle. Comme elle se tait, il relève la tête, la regarde. Il a appris à reconnaître cette expression : elle regrette ce qu'elle vient de dire. Intrigué, un peu inquiet, il la presse. Elle dit : « C'est juste une épreuve. Un rite de passage, si tu veux. On le subit entre quinze et vingt saisons environ, ça dépend. Dans les îles. Quand on est prêt, mais toujours après... l'eïldaràn. Kéryan a dit que j'étais prête. » Elle bat des paupières et ajoute : « Les gens comme toi » – on ne dit jamais " naïstoï " chez les Olaïlliu, et " halatnim " seulement par rarissime étourderie – « en sont dispensés. »
Ensuite, c'est elle qui s'emploie à le distraire, et il se laisse faire. Depuis la visite d'Odatan, il peut apprécier la politesse des Olaïlliu, la façon gracieuse dont ils l'ont toujours traité, enfants et adultes, sans effort vraiment apparent une fois passée la période d'adaptation. Ce doit être pourtant difficile, pour des gens habitués à toujours savoir au moins ce que les autres ressentent – les disputes ou les discussions de Miniaz, Maïli et Tarmel entre eux sont assez différentes de celles qu'ils ont avec lui. Ils se touchent souvent, par exemple – avec lui, ils essaient de contrôler ce réflexe ; la proximité physique intensifie les perceptions des lâdzani.
La petite place silencieuse s'élargit dans son cœur.
Il a presque quinze saisons et il est en train de mémoriser à voix haute un bref poème pour la leçon suivante de Kéryan, qui portera sur les registres archaïques du virginien ; il l'a choisi lui-même dans un des livres qu'elle lui a prêtés. C'est un poème très bref, en mode dit " latinam ", et dans un ancien registre dit " franca ", une sorte de dialogue dont on ne connaît pas les interlocuteurs, plutôt un fragment de conversation surpris au passage ; mais il en aime l'arbitraire étrangeté, et surtout l'envolée finale, qu'il a reconnue avec un certain amusement lorsqu'il l'a lue tout haut.
« Elle est retrouvée. – Quoi ? L'Éternité. »
Dans son dos, soudain, la voix altérée de Laraï : « C'est la mer allée avec le soleil. »
Il se retourne. Sa mère se tient sur le seuil, les lèvres blanches, les mains serrées en poings au creux de la poitrine comme si elle avait reçu un coup. « Qui... qui t'a donné ce poème ? » murmure-t-elle entre ses dents serrées, les yeux étincelants.
Lian balbutie : « Le livre... Kéryan me l'a prêté. J'ai choisi le poème. Il y en a plein... » Et comme Laraï se tient toujours rigide dans l'embrasure, il ajoute en hâte : « Je peux en apprendre un autre si tu veux ! »
Laraï passe une main dans ses cheveux en disant « Non, non, ça va... ». Elle se redresse avec lenteur, fait quelques pas hésitants dans la chambre, effleurant au passage les habits drapés sur le dossier de la chaise, le premier tiroir à demi tiré de la commode. Lian ne lui a jamais vu cet air égaré. Elle se laisse tomber dans l'autre chaise, le dévisage un moment avec une tendresse désespérée qui l'affole : qu'y a-t-il, qu'a-t-il fait ? Mais elle lui caresse la joue : « Ce n'est rien, Liani, ce n'est rien. » En prenant une grande inspiration, elle croise les bras, s'appuie au dossier de la chaise et parcourt la pièce du regard. « Commence à être petite, cette chambre », murmure-t-elle pour elle-même. Lian, qui avait plutôt craint un commentaire sur le désordre, se détend un peu. Le silence retombe, mais n'est plus aussi inquiétant.
« Ton père-Dougall aimait beaucoup ce poème, dit enfin Laraï à mi-voix. C'est lui qui me l'a appris. J'ai craint... » Elle se mord les lèvres en haussant une épaule et poursuit, sur un ton différent, plus délibéré : «... j'ai pensé que c'était un tour de Kéryan. Les hékel aiment bien faire ce genre de choses. Vous prendre par surprise. » Son visage s'est durci et elle regarde au loin, les sourcils froncés. C'est vrai que Kéryan a une façon un peu curieuse de parler, quelquefois, ou plutôt de ne rien dire, comme les " points d'orgue " au piano, et dans le silence s'engouffrent parfois une phrase musicale imaginaire mais surprenante, des idées qu'on ignorait avoir... Avec un petit tressaillement intérieur, surprise, colère, Lian voit soudain dans une autre lumière le comportement de Kéryan ; aurait-il choisi ce poème, si bizarrement musical fût-il, s'il n'y avait reconnu la première phrase virginienne qu'il a entendu énoncer ?
Laraï feuillette le livre d'une main distraite, puis elle le repousse et s'adosse de nouveau dans la chaise avec un soupir.
« Comment il était... » demande impulsivement Lian, pour conclure plus bas, soudain conscient de ce qu'il fait, mais il a commencé, il doit finir : «... Père-Dougall ? »
Laraï hoche la tête, dit aussitôt. « Un homme en colère. » Puis, comme un remords : « Un homme qui ne savait pas être heureux. Mais quand il oubliait d'être malheureux, c'était bien. »
Lian pose ses bras sur la table et son menton sur ses bras, pour être plus près d'elle – il est bien trop grand maintenant pour grimper sur ses genoux. Elle lui sourit et lui caresse une main.
« Pourquoi il était malheureux ?
— Ah. » Un profond soupir. Laraï se carre dans sa chaise. « Son ancêtre, qui est venu de Virginia avec la Mer, était un danvéràn très puissant. Le fils de cet homme l'était moins. Sa petite-fille, la grand-mère de Dougall, l'était moins encore. Comme sa fille, la mère de Dougall. » Elle parle avec une lenteur délibérée, choisissant ses mots avec soin. « Et lui, Dougall, encore moins qu'elle.
— Les pouvoirs se sont perdus », acquiesce Lian pour lui montrer qu'il comprend.
Elle hoche la tête. « Être des danvérani comptait beaucoup pour les premiers halatnim. Voir le don se perdre les a blessés à chaque génération. Certains plus que d'autres. Quand tu es venu, Dougall a pensé... »
Elle se penche de nouveau vers Lian, lui serre les bras : « Il était heureux d'avoir un enfant, Lian, comprends-le bien. Il avait de l'amour pour toi, et pour moi. Mais... » Son visage se contracte. « C'est plus difficile d'aimer les autres quand on ne s'aime pas soi-même. Et il pensait... que tu n'aurais pas dû être comme tu étais, et que c'était sa faute. Stupide, mais c'est ce qu'il a pensé... »
Après une pause, elle murmure : « Ils ne sont pas comme nous, ils ne savent pas ce qui est important. Peut-être... Les pouvoirs existent parmi nous depuis des milliers de saisons. Mais pour eux, de l'autre côté, depuis bien moins longtemps, quatre cents ou cinq cents saisons tout au plus. Ils n'ont peut-être pas encore eu bien le temps de comprendre. »
Lian essaie d'évoquer les quelques souvenirs flous qu'il a de Dougall. Des disputes violentes entendues de loin dans un demi-sommeil angoissé, de grandes mains brusques... N'en a-t-il pas d'autres ? Cette silhouette à contre-jour, ces bras qui le font sauter en l'air, était-ce Dougall ou Nathénèk ? Non, Dougall, une odeur particulière se dégageait de ses vêtements, à la fois douce et âcre, qui n'a jamais été celle de Nathénèk ; Lian serait incapable de la décrire, mais s'il la sentait de nouveau il est sûr qu'il la reconnaîtrait.
« Est-ce qu'on vivait à Hleïtzer, tous les trois ? »
Laraï sort de sa méditation avec un petit sursaut : « Oui. C'est là que tu es né. Mais je ne voulais pas... y rester. Nous sommes venus sur le Landaïeïtan. »
Lian regarde autour de lui, incrédule : « Ici ?
— Non ! Non... plus à l'est. Mais ton père-Dougall n'aimait pas trop la forêt. Et finalement, il est parti. » L'ombre familière passe sur le visage de Laraï. « Il est mort.
— Il est mort comment ?
— Un accident. » Laraï répète, un peu plus bas, un peu plus dur : « Un accident.
— Et tu as rencontré Père-Nathénèk après », dit Lian, pour changer le sujet.
Le visage de Laraï s'éclaire d'un léger sourire un peu mélancolique : « Non, en fait, je l'ai rencontré pendant que Dougall était encore là. Après sa mort, nous avons travaillé ensemble un temps. Et puis nous avons décidé de vivre ensemble. C'est sa maison, ici. Il y habitait avec sa sœur. »
Elle plonge dans une profonde rêverie. Lian espère que ce sont des souvenirs heureux. Mais l'expression anxieuse reparaît dans ses yeux, ses traits se contractent. « C'est Nathénèk qui a voulu, pour la Mer. Que tu la voies, peut-être. Qu'elle te voie, sûrement. Je ne pensais pas... »
Elle se mord violemment les lèvres.
« Quoi ? » s'écrie Lian, affolé de nouveau.
Et alors, comme si elle se déchirait à l'intérieur : « Oh, Lian, nous ne serons jamais ensemble dans la Mer ! Tu ne peux pas... Tu ne peux pas la rejoindre ! Odatan nous l'avait dit, pourtant, que tu devais être... un naïstaos, mais je ne voulais pas le croire. Quand je t'ai vu... avec la Mer dans les mains, j'ai cru que j'allais mourir. »
Lian balbutie : « La Mer ? »
Laraï caresse convulsivement les mains, les bras, les épaules de Lian, tout en se forçant à prendre de grands respirs : « Peut-être... pas. C'est pour ça... l'eïldaràn. Odatan a peut-être raison. Il y a peut-être une chance... que tu ne sois pas vraiment un naïstaos. Que tu voies le Signe de la Mer, qu'elle te libère. Oh, qu'elle te libère, Lian, pour elle ! Ça m'est bien égal si tu n'es pas un hékel, ni un danvéràn, ni rien, mais que tu puisses rejoindre sa lumière, que nous soyons ensemble tous les trois un jour dans sa lumière ! »
Le cœur battant à tout rompre, Lian se rappelle la nuit violette de l'éclipse, le cri sourd de la foule derrière lui, l'absence déroutante de sensations quand il a touché la scintillante surface bleue. Et le visage triste et calme de l'inconnu – d'Odatan.
Plus tard, il s'interrogera sur le rôle et la nature d'Odatan – mais Nathénèk ne pourra pas lui en dire grand-chose : un Virginien, un passeur, un danvéràn. Plus tard. Pour le moment, en regardant Laraï lutter contre les larmes, il se sent tout creux, il a peur de bouger, peur de se casser. Il ne comprend pas tout à fait pourquoi sa mère a si mal – que Laraï est profondément croyante, que rejoindre la Mer n'est pas pour elle cette histoire merveilleuse et vague qu'on lui a racontée à lui quand il était petit, il y a longtemps, mais une réalité vivante, une certitude, un espoir de tous les instants. Plus tard, il saura. Maintenant, il comprend seulement que sa mère a mal. À cause de lui. À cause de ce qu'il est. Et il a mal aussi comme si c'était de sa faute – même si, stupide, ce n'est pas sa faute. Et, pour la première fois, quand Nathénèk atterré dira " C'est ainsi ", la formule magique ne parviendra pas vraiment à étouffer la révolte de Lian : « Mais pourquoi ? » Il sait désormais ce que recèle la petite tache de silence, en lui. Il ne pourra plus l'oublier.
La troisième semaine d'Atéhatéñu, le onzième mois de l'année, à la toute fin de l'automne, alors que Lian a quinze saisons et demie, Odatan revient à Leïtnialen. Il salue Lian en virginien. Lian lui répond en setlâd. Le visiteur ne fait pas de commentaires. Ils savaient qu'il arrivait, les bagages sont prêts depuis la veille au soir. Pas de grandes démonstrations. Nathénèk embrasse Lian, Lian étreint Laraï. Il n'a pas d'autres adieux à faire. Les Olaïlliu ne savent pas qu'il partait, Kéryan s'en chargera bien. Le Gomphal, il est allé le voir la veille au soir, à l'heure où l'ombre l'immobilise.
Et ils s'en vont.
*
Les habitants de la commune ne parlent presque jamais de la Mer, y pensent à peine. Ils vivent pourtant à seulement un demi-kilomètre des quais, à l'extrême limite est du quartier, au bord du canal qui encercle l'ancien port. Jill et Jaques vont toujours y faire leurs promenades matinales, le long de la grande esplanade herbeuse parsemée d'arbres et de buissons qui longe le canal et ouvre sur les grilles interdisant l'accès direct aux quais. Mais pour eux, ce n'est pas " le bord de la Mer " ni " le port ", simplement l'esplanade Carghill. Même l'océan existe à peine pour eux – pas de Cristobal-sur-l'Océan sur l'autre côte, huit cents kilomètres à l'ouest, où l'on s'installerait après le départ de la Mer : tout le territoire intermédiaire est un simple blanc sur la carte. Liam n'a même pas à voir le scintillement bleuté de la Mer au-dessus des édifices déserts du port ancien : la fenêtre de sa chambre donne sur la ville et, sur la terrasse, une haie impénétrable d'arbres-trolls masque la vue au nord-ouest.
L'absence de la Mer dans la vie de la commune ne dérange nullement Liam, au contraire. Il la remarque, tout de même, distraitement, une autre curiosité. On dit aux tout petits enfants : « Si tu n'es pas sage, on t'emmène à la Mer. » C'est pratiquement la seule mention qui en est faite dans leur vie quotidienne. Ils apprennent ensuite l'histoire de Virginia – la colonisation, la première expédition décimée par la Mer, les naufragés éparpillés sur tout le continent, chaque groupe persuadé, jusqu'au retour de la seconde expédition, d'être les seuls survivants... Et si la première lutte victorieuse contre la Terre est également liée à un retour de la Mer, la dernière et définitive guerre contre les Terriens agresseurs s'est déclenchée lors d'un de ses départs. Et les rebelles aussi réactivent la guerre lorsque la Mer s'en va. On pourrait être heureux de son retour – il marque la fin de la campagne, le début de deux saisons paisibles – mais on en est simplement soulagé, avec un peu d'agacement parfois : si la Mer n'existait pas, tout ça serait réglé depuis longtemps, et la vie serait bien plus simple.
Les Anciens, c'est un peu différent. Il reste assez de fresques, des mosaïques dans les cours intérieures – les édifices eux-mêmes, la ville. (Toutes les villes, les routes, les canaux... Mais Liam s'arrête toujours au bord de cette dérive.) On n'évoque pas les Anciens à la commune – on vit chez eux, on ne les voit plus à force de les voir. Même pas des fantômes. Leur énigmatique disparition ne suscite plus l'ombre d'une curiosité, depuis le temps. Leur absence si présente fait partie du décor au même titre que tout le reste, les arbres, les pierres écarlates ou dorées, les oiseaux bleus et blancs qui tournent au-dessus de la ville pour se percher sur les rares tourelles encore debout sur les terrasses. Tout sauf la Mer a perdu son nom, ici, pour en trouver un autre. Anciens, arbres-rois, arbres-à-eau, racalous, oiseaux-de-clocher. Liam ne les appelle jamais autrement, comme il n'est plus jamais " Lian " pour lui-même. Une nouvelle vie lui a été donnée, Liam en est reconnaissant, il ne veut pas la questionner. Les autres, dont c'est la vie normale, ne le font pas : il trouve facile de les imiter. Il s'intègre aux histoires qu'ils se racontent, d'autres histoires. Sur la Mer, sur les Anciens, sur les Terriens.
Cette histoire-là s'est vu ajouter récemment une suite fascinante, mais nul ne semble fasciné à la commune, et Liam n'a donc pas envie de l'être : des Terriens arrivent. Enfermés dans Lagrange, un astéroïde qu'ils ont décroché de leur lointain système solaire, ils voyagent depuis des Années dans l'espace. Depuis environ vingt saisons ils se promènent dans le système de Virginia. Ce sont eux, cette oblongue tache lumineuse qu'on voit briller la nuit, parfois, quand le ciel est dégagé. « Z'ont rien de mieux à faire que de déranger tout le monde avec leurs moteurs à ions ? » grommelle un des grands-pères, astronome amateur convaincu. « Bien des Terriens, ça ! »
Ils arrivent, ils vont passer, ils vont s'en aller. Sans s'arrêter. Le système d'Altaïr leur sert simplement à freiner et à modifier leur trajectoire, un jeu compliqué de ricochets entre ses planètes. C'est ce qu'ils ont dit quand on a pris contact avec eux. Silence total depuis. Les voyageurs se soucient aussi peu de Virginia que les Virginiens d'eux. On n'en est même pas vexé. Une vieille histoire, les Terriens, la Terre, dont les notes en bas de page n'intéressent plus personne. Liam, après une brève curiosité vite distraite, se contente de regarder parfois la lumière dans le ciel, la nuit, quand il va se coucher. C'est joli.
11
Ils se dirigent vers la passe Arpaïel, au sud-ouest, le chemin le plus court pour contourner les Ltellaïllia si l'on désire se rendre à Hleïtzer depuis cette région du plateau. Les premières heures du voyage sont extrêmement silencieuses. Lian s'en accommode fort bien. Il préfère contempler autour de lui sa forêt qu'il va quitter pour longtemps, au moins une saison puisque l'eïldaràn ne peut avoir lieu qu'au retour et au départ de la Mer. Et ensuite il reviendra sur le Landaïeïtan, quel que soit le résultat. Mais il préfère ne pas trop penser aux résultats possibles de l'eïldaràn ; il a appris à ne pas s'appesantir sur cette incertitude-là. Il préfère écouter de toutes ses oreilles les sons familiers, sentir sur sa peau l'humidité chaude et douce qui diffuse en une légère brume toujours mystérieuse la lumière verte filtrée par les hautes frondaisons, humer les parfums, laisser formes et couleurs emplir son regard, sans vouloir les prendre ni même les comprendre, pour simplement être dans la forêt, être la forêt.
À peine une heure après la collation de mi-matinée, Odatan tire une couverture de son sac et la déroule à l'ombre d'un jeune tingai : il doit faire une petite sieste, dit-il en virginien. Ils repartiront vers la quinzième heure.
Trois heures de perdues ? « Vous vous fatiguez vite », remarque Lian en setlâd, un peu narquois.
L'autre s'installe sur sa couverture, réplique en virginien : « Ce n'est pas ça. Je ne suis pas un hybride de sixième génération, moi, Lian. Sur Virginia, tout le monde fait la méridienne. Les Terriens ont évolué sur une autre planète, où la journée est bien moins longue qu'ici, et si nous n'arrêtons pas en milieu de journée, notre organisme tend à se dérégler. »
Lian ignorait l'existence de la méridienne, s'il connaît les mesures virginiennes du temps – il a appris, en découvrant les mots nouveaux, " jour ", " heure ", " minute ", " seconde ", que s'ils ont des équivalents setlâd, leurs durées ne sont pas les mêmes ; la journée rani compte vingt-huit heures, la journée virginienne trente-cinq.
Mais Odatan vient surtout de lui rappeler qu'il est un pur Virginien, passé assez récemment avec la Mer. Lian examine l'autre, qui a fermé les yeux et croisé les mains sur sa poitrine (bizarrement : seulement les dernières phalanges, les doigts pliés et entrelacés) : ce visage étroit et anguleux aux yeux trop rapprochés, ce grand nez aquilin, l'arc effilé des sourcils, les lèvres trop minces, c'est ça, un Virginien. Et le brun clair de la peau, et la petite taille – à quinze saisons et demie, Lian est presque aussi grand que lui – et les ongles roses et pointus, et les poils sur les mains, sur les bras, sûrement partout. Lian se prend à détailler le dormeur avec une animosité qui l'étonne un peu – il y a pourtant là certains traits qu'il a l'habitude de voir lorsqu'il se rencontre lui-même dans un miroir ; il n'a pas de poils sur le corps, certes, juste le duvet presque imperceptible d'un vrai Rani, mais son nez, sa bouche, ses sourcils à lui ne sont pas très ranao. Ni la couleur de sa peau. Ni ses yeux.
L'autre a les yeux fermés, mais il ne dort pas. Il dit : « Désolé, Lian, c'est ainsi. Tu peux faire un peu de yoga en attendant. Ou lire. Il y a quelques livres dans ma sacoche. J'ai le dernier Malani Opted.
— C'est qui, Malani Opted ?
— Une halatnim qui écrit des histoires passionnantes. »
Lian fait « pfff » et commence plutôt à se dérouiller les articulations dans les exercices préparatoires à la satlàn – qui n'est pas vraiment du " yoga ", malgré ce qu'en dit le dictionnaire virginien de Kéryan. La méditation en mouvement a toujours un effet apaisant sur lui. Odatan redressé sur un coude l'observe un moment et remarque, toujours en virginien : « Tu vas un peu vite, dans la troisième phase. »
Lian pense très fort : Fais ta sieste, Virginien, t'ai-je demandé ton avis ?
Comme s'il avait entendu, mais Lian sait bien que non, le petit homme se recouche les mains croisées sur la poitrine et ferme les yeux.
Quand ils sont repartis, et qu'à un moment Odatan lui adresse de nouveau la parole en virginien, Lian agacé demande : « C'est quoi, un test ? » – en virginien, car la notion d'épreuve, ou même celle d'essai, est différente en setlâd : ce ne sont pas des situations où l'un des partenaires prétend en savoir plus que l'autre. Le petit homme lui jette un regard en biais et répond, en setlâd : « Des essais. Je veux m'assurer que tu comprends bien le virginien. Tu en auras besoin à Hleïtzer, avec tes futurs compagnons. Il y en a parmi eux qui ne parlent pas du tout le setlâd, ou très mal. Quelques communautés halatnim ont choisi de ne parler que leur langue. C'est leur droit.
— Des premières générations ?
— Plutôt le contraire, murmure Odatan.
— Et vous êtes là depuis combien de temps, vous ? »
Ils sont descendus de leurs montures pour se dégourdir les jambes ; le petit homme continue à marcher en silence, vraiment petit auprès de Nagal-Rasser.
« Longtemps », dit-il enfin.
Lian lui lance un regard sceptique que l'autre ne voit pas car il regarde droit devant lui. Si Odatan a la quarantaine, c'est bien le maximum. Il connaissait Laraï et Dougall bien avant sa naissance, s'il faut en croire Nathénèk. Mais l'inflexion descendante indique, en setlâd, qu'un sujet est clos, et Lian change plutôt d'angle : « Et vous travaillez au Haëkelliaõ de Hleïtzer. Vous êtes un hékel, alors.
— Pas vraiment au Haëkelliaõ. Et je ne suis pas un hékel. Juste... un bon danvéràn passé avec la Mer. »
Du coup, Lian oublie sa stratégie. Il imagine une petite barque perdue dans l'immensité bleue, la voile gonflée par le Vent du Retour, l'abordage – quelque part, dans l'inconnu, puisque les Virginiens ignorent tout d'Atyrkelsaõ. « Pourquoi êtes-vous passé ? murmure-t-il avec un soudain respect vaguement craintif.
— Par hasard. Je suis tombé dans la Mer. »
Lian s'immobilise, tirant brusquement sur la bride des deux aski, qui renâclent ; Odatan fait encore un pas puis se retourne vers lui tandis que Nagal-Rasser s'immobilise à son tour avec une majestueuse lenteur.
« Tombé ? »
L'autre hoche la tête, très calme : « Les passeurs n'ont pas besoin de barque. »
Lian murmure enfin : « Vous êtes... un naïstaos ?
— Oui. Tous les passeurs. »
Lian se sent les jambes un peu molles et s'appuie contre le flanc de Métyi, son asker de monte, qui émet un sifflement doux et perplexe. Dans son souvenir, des petits ruisselets de lumière bleue, délicatement ourlés, s'égouttent de ses doigts pour retourner se fondre dans la masse scintillante de la Mer.
« Vous vouliez savoir... la Mer, quand j'étais petit... »
Le visage de l'autre prend une expression horrifiée : « Oh, Lian, non ! On ne fait pas ce genre d'expérience avec la Mer ! Et je ne savais pas. Lorsque tu as été conçu, on a simplement cru Laraï malade, au début, parce qu'on ne percevait absolument pas le bébé dans son ventre. On m'en a parlé, j'ai reconnu le symptôme. Mais... » Il se mord les lèvres, répète, les traits contractés : « Non, Lian. Quand tu as touché la Mer, alors seulement j'ai compris ce que tu pouvais être. »
Lian murmure. « Un naïstaos.
— Un naïstaos qui voit la lumière de la Mer, Lian. »
Lian ne dit rien. Odatan pousse un petit soupir. « Je voyais la lumière aussi, de l'autre côté, avant de devenir un danvéràn. Et j'étais comme toi, complètement enfermé derrière une porte invisible. Tous les passeurs. L'eïldaràn pourrait être pour vous ce que le passage avec ou sur la Mer est pour nous. »
Lian hausse les épaules : « Je deviendrais un danvéràn.
— Peut-être.
— Mais je resterais un naïstaos. La Mer me rejetterait toujours. »
L'autre se donne sur la cuisse une claque exaspérée qui fait broncher Métyi. « La Mer ne rejette pas les naïstoï, ça n'a rien à voir, être bon, être méchant, ce sont des superstitions, ça, Lian ! »
Lian n'écoute pas. D'une voix qui s'enroue, il répète, de plus en plus fort pour couvrir la voix du petit homme : « Mais je ne pourrais pas rejoindre la Mer. Qu'est-ce que ça peut me faire d'être un danvéràn si je ne peux pas rejoindre la Mer ! ? »
À l'aveuglette, il s'agrippe à la crinière de Métyi, se hisse sur la selle, enfonce ses genoux dans les flancs rebondis. Avec un petit sifflement de protestation, l'asker part au grand trot sous les futaies.
*
Le septième jour, vers le milieu de la matinée, alors qu'ils se sont enfin engagés dans la passe Arpaïel, Nagal-Rasser s'immobilise, Odatan saute à terre, s'approche d'un bloc de granit au bord du torrent et fait signe à Lian, qui obtempère. En dégringolant le long de la montagne en des temps immémoriaux, le roc a éclaté, et une longue face plane de pierre gris-rose est à nu, miroitant au soleil – et striée de fines marques verticales entrecroisées.
« Karaïker », dit Odatan.
Lian n'a jamais vu des rochers marqués ainsi ; en forêt, les karaï laissent des traces sur le sol et sur les arbres – mais entamer du granit ? Toutefois, le territoire de Laraï et même le plateau sont loin, les voyageurs ont déjà passé deux crêtes de montagnes : Lian se trouve désormais en territoire inconnu et doit se fier davantage à son compagnon. Il se dresse sur la pointe des pieds, pose une main sur les stries les plus proches, qui la débordent largement de toute part. Un adulte. Par réflexe, il regarde autour de lui, tend l'oreille – le karaïker, prédateur dominant, n'est pas particulièrement discret lorsqu'il n'est pas en chasse. Mais ils n'ont pas vu de troupeaux d'aski sauvages dans les environs, et ce ne sont pas des marques récentes, car Métyi aurait manifesté son inquiétude – elle a un odorat bien plus développé que celui des humains. Néanmoins, comme son compagnon, Lian sort le fusil de son étui, y glisse un chargeur de fléchettes – avec les précautions d'usage : l'enduit qui paralyse un karaïker est instantanément mortel pour les humains. Tout en retirant ses gants, il remarque quand même : « Les karaï ne vivent pas aussi haut dans les montagnes d'habitude, non ? »
Odatan hoche la tête : « Celui-ci doit être un chasseur d'agraï. »
Lian se met à rire. « Une légende ! » Les karaï et les agraï ne peuvent être rivaux : leurs territoires ne se recoupent pas, les karaï étant bien plus à l'aise dans les forêts et dans les zones médianes relativement boisées des montagnes, les agraï ayant besoin de terrain découvert, le plus escarpé possible, pour reprendre leur vol après l'atterrissage – et ils respirent fort mal, de surcroît, en dessous d'une certaine altitude. Même leurs proies de prédilection sont différentes – les aski sauvages pour les karaï, et les maëliki, blancs acrobates des précipices, pour les agraï.
« Toutes les légendes ne sont pas mensongères », dit Odatan, en passant la courroie de son fusil sur son épaule. « Karaï et agraï sont des lâdzani aussi, à leur manière. Ceux de Tyranaël, entendons-nous bien, pas ceux d'Atyrkelsaõ. Comme les banki et les lladao : seulement ceux que les Ranao ont amenés avec eux. » Le petit homme murmure, pensif : « La mutation n'a jamais eu lieu, ici. Et les moatranao ne sont jamais devenus des Ranao. »
Lian digère sa stupeur en silence. Les deux graines de fofolod ne sont pas si semblables, après tout.
Nagal-Rasser se remet en route et ils le suivent – la masse et la corne aiguë du tovker sont leur première ligne de défense en cas d'attaque frontale inopinée. Métyi ferme la marche : la brise est favorable, son odorat les préviendra si un karaïker les suit de trop près. Le large torrent bruit à leur droite, la pente incurvée de la vallée est trop bien dégagée sur leur gauche pour une approche furtive. Au bout d'une demi-heure sans incident, alors que Lian commence à se détendre, un son étrangement humain tombe du ciel pour se répercuter entre les parois rocailleuses de la vallée, comme un cri de douleur impossiblement prolongé, aigu, lointain et rauque. Odatan tend un bras, doigt pointé : très haut à l'ouest, du côté où les montagnes s'étagent au-dessus de la passe, plane un large trait noir faussement immobile.
Lian, plus fasciné qu'inquiet, contemple son premier agraïllad. Avec un cri pareil, pas étonnant que le rapace géant ait nourri l'imagination des conteurs. « Et lui, alors, c'est un chasseur de karaï ? » demande-t-il en plaisantant à son compagnon.
Odatan dit « Peut-être », sans sourire. Croit-il vraiment que l'oiseau va les attaquer, dans cette vallée où les courants ascendants sont insuffisants pour lui permettre de remonter et alors que – à part peut-être les deux aski – ils ne ressemblent en rien à ses proies habituelles ? L'autre continue à marcher, les sourcils froncés.
Au bout d'une heure, il doit décider que ses craintes ne sont pas fondées : il remonte en selle. Lian en fait autant. Le petit homme se met soudain à chantonner par à-coups, après un moment, à voix basse, comme sans en avoir conscience ; Lian finit par reconnaître certains vers, à défaut de la mélodie : c'est la légende des karaï et des agraï.
Kithal et Paguid, les Gardiens des montagnes, accidentellement créés sœur et frère jumeaux par Hananai, sont en conflit. Hananai, lassée de leurs déprédations, les transforme en félins et en rapaces géants pour les jeter sur les montagnes en leur disant " Régnez, maintenant ! ", – mais, dans cette version, sans remords : elle ne leur donne pas le don de parole pour la dernière nuit de l'hiver. Avec les petits morceaux restants, elle façonne la première femme et le premier homme, cependant, comme dans la version connue de Lian. Leurs descendants vivent d'abord dans les plaines de Paalu, le continent de l'est et, parfois, sans qu'ils en comprennent la raison, une irrésistible nostalgie les saisit lorsqu'ils regardent du côté de l'ouest et de l'immense barrière de montagnes où trône le Hanultellan aux épaules de nuages. Alors, ils abandonnent tout et vont s'y installer. Là, ils chassent karaï et agraï, qui eux aussi gardent un souvenir confus de leur origine et se haïssent mortellement, mais en pure perte, puisque les forêts du karaïker sont impraticables pour l'agraïllad et inversement les sommets escarpés de celui-ci pour le karaïker.
L'histoire d'Odatan dévie à partir de là. Les deux humains où dort la substance des Gardiens jumeaux sont frères issus du même père, et non sœur et frère – Kithulai s'appelle Kalalu, et Paguyn Merrim. Et au lieu de se compléter dans le chant et la danse, comme Kithulai la créatrice de chants et Paguyn aux pieds légers, Kalalu est un éleveur et Merrim un chasseur. Ils collaborent au début – Merrim capture des aski sauvages que Kalalu apprivoise, dresse, et apprend à traire et à tondre. Mais leur harmonie ne tarde pas à se rompre, là aussi. Merrim, affamé au retour d'une chasse infructueuse, tue par accident le plus bel asker de Kalalu ; Kalalu, outragé de voir ce crime rester impuni par le village, jette le cadavre ensanglanté dans le grand chaudron où l'on conserve tout le lait du troupeau communal ; les villageois furieux de voir leur lait ainsi souillé chassent les deux frères, chacun de son côté, Merrim dans la forêt, Kalalu dans la montagne. Kalalu doit devenir chasseur pour survivre – et Merrim apprendre à se nourrir aussi de racines et de noix.
Odatan cesse de chantonner. Lian, qui commençait à se prendre à l'histoire, demande : « Et ensuite ? »
Un petit muscle tressaute dans la joue de son compagnon. « Kalalu se fait tuer dans la montagne par un agraïllad, Merrim rencontre le fantôme de son frère dans la forêt et, fatalement distrait, se fait tuer par le karaïker qu'il était en train de traquer. Les Gardiens des montagnes, enfin libérés, retournent auprès de Hananai qui attribue les précipices à Paguid et les pics à Kithal. Fin. »
Lian est déçu, presque choqué. La légende que Laraï lui a apprise est bien plus satisfaisante. « Oui », dit Odatan avec un drôle de petit sourire, « plus civilisée. »
Et quand Lian lui demande d'où vient sa version à lui, il répond : « Des Krilliadni. » Puis en virginien, comme si c'était une traduction, un ton plus bas : « Des chasseurs. »
Mais Krilliadni ne veut pas dire " chasseurs " en setlâd. Simplement " ceux du froid ", d'après l'étymologie. Le terme désigne en réalité les habitants des îles d'Aalpaalu et d'Aalaritu, parties montagneuses des continents Est et Ouest seules laissées à découvert par la Mer lorsqu'elle se trouve là, et seules habitées lorsqu'elle ne l'est pas. Un peu étrange, à vrai dire, dans la mesure où la majeure partie de ces îles se trouve dans des zones tempérées ; les plus au sud d'Aalpaalu, Krillialtaoz et Krilliadi, correspondent cependant aux sommets les plus élevés d'Atyrkelsaõ, dont le fameux Hanultellan aux neiges éternelles, qui atteint près de sept mille lani d'altitude – la montagne autrefois sacrée des Paalani. Alors, " Krilliadni ", et le nom a dû se généraliser à partir de là.
« Pourquoi " chasseurs " ?
— Des chasseurs chassent », répond Odatan à mi-voix, toujours en virginien. Puis, de nouveau en setlâd : « On ne t'a jamais parlé des grandes chasses ?
— Pas vraiment. Un rituel de passage, après l'eïldaràn. Ça ne concerne pas les naïstoï, apparemment, ils en sont écartés. Quel rapport avec les Krilliadni ?
— Les naïstoï en sont dispensés, Lian », intervient l'autre d'un ton un peu las.
C'était l'euphémisme employé aussi par Miniaz, Lian ne relève pas : « Quel rapport avec les Krilliadni ?
— Les Krilliadni participent aux grandes chasses », dit Odatan ; perché comme il est maintenant sur Nagal-Rasser, presque un demi-lani plus haut que Lian, son visage est indéchiffrable ; mais sa voix est neutre, trop. « On va dans leurs îles pour l'épreuve. Une saison. Il faut survivre. C'est dangereux. »
Il n'élabore pas davantage et Lian n'a pas envie de le pousser, soudain un peu inquiet pour Miniaz, même si elle s'est beaucoup améliorée dans sa connaissance de la survie en forêt depuis leur enfance. Ils continuent à avancer en silence, Odatan lointain sur Nagal-Rasser, Lian morose sur Métyi. L'autre le surprend en murmurant : « Mais n'importe qui peut aller vivre chez les Krilliadni en dehors des grandes chasses, à n'importe quel moment de sa vie. C'est un choix.
— Vous y êtes allé, vous ? » demande Lian, narquois : depuis qu'ils voyagent ensemble, il a eu le temps de se rendre compte que le petit homme aime bien son confort.
« Oui, dit Odatan, le prenant au dépourvu. Plusieurs années. » Il observe Lian du haut de Nagal-Rasser, un peu ironique. « La vie y est beaucoup plus dure que sur le continent, mais elle semble parfois plus... simple. » Son sourire s'efface. « Une illusion, bien entendu. »
*
Liam aime les enfants de la commune, surtout les plus jeunes. Quelquefois, quand il a fini de travailler dans le jardin ou à l'atelier, il vient s'asseoir sous l'arcade de la cour intérieure près des salles où Jill, Suzane et Andreï font la classe, et il se laisse couler dans le temps au rythme hypnotique de l'eau de la fontaine, dans le bassin, et avec l'autre musique aussi : voix chantonnantes, murmures, rires, questions. Quand les petits sortent en courant, plusieurs s'arrêtent souvent près de lui pour lui raconter avec enthousiasme ce qu'ils ont appris ou poursuivre à son bénéfice une querelle dont il comprend qu'il doit être l'arbitre. Les petits l'aiment bien aussi. Il ne sait pas pourquoi. Peut-être parce qu'il parle toujours d'une voix égale, est toujours prêt à les écouter, toujours avec gravité. Ou parce qu'ils n'ont pas grandi avec lui, qu'il vient d'ailleurs – de l'extérieur de la commune, c'est-à-dire, on n'a jamais précisé d'où, et pour cause ; il se trouve investi pour eux d'une aura de fascinant exotisme.
Après la collation qui suit la méridienne, et le soir, après le souper, petits et moyens se rassemblent parfois autour de Jill, Suzane et Andreï, pour des histoires. Quand Liam est de cuisine, il va les rejoindre après avoir fini de ranger. Pas tellement pour entendre les histoires, il les connaît vite par cœur, mais pour regarder les visages des enfants, étonnés, effrayés, ravis. Pour eux, il n'y a pas de différence. Tout est réel. Tout est vrai.
Un soir, quand vient le temps de la dernière histoire, les enfants font preuve de beaucoup d'habile mauvaise foi. Jill et les deux autres déclarent forfait après s'être fait dire à plusieurs reprises : « Mais celle-là, on la connaît ! », et Jill lance, en plaisantant à demi : « Demandez à Liam, il doit en savoir d'autres ! » Les enfants se retournent vers lui. Pris de court, il proteste, mais certains des petits viennent s'agripper à ses genoux, implorants, « Oh oui, Liam, raconte-nous une histoire ! » Vaincu, il improvise. Croit improviser. « Il était une fois deux enfants qui vivaient au pied d'une très haute montagne, tout seuls dans la forêt. Un garçon et une fille. La fille s'appelait Katrina... » (Katrin la brunette s'épanouit, les autres rient en se poussant du coude)... « et le garçon Patriki. » (Nouveaux rires. Patrik bombe le torse.) « Ils se débrouillaient très bien dans la forêt. Patriki chassait avec son arc et ses flèches, et Katrina capturait et dressait des cabals sauvages.
— Elle ne peut pas, dit un des garçons plus âgés. C'est trop grand pour des enfants, un cabal sauvage.
— Si, si, elle peut ! protestent d'autres, surtout Katrin.
— C'étaient de plus petits cabals, en ce temps-là, tranche Liam. Il y a très longtemps. Ils n'avaient pas encore grandi. Mais on pouvait utiliser leur poil laineux pour faire des habits, et on pouvait les traire pour avoir du lait. C'est ce que Katrina avait découvert, et elle le faisait très, très bien. » Là, selon l'habitude de Jill et des autres, il renvoie l'histoire aux enfants, leur demandant comment Katrina s'y prend pour tondre et traire les cabals et quels animaux Patriki chasse dans la forêt – les hypothèses fusent, raisonnables ou délibérément absurdes. Les animaux de la forêt inspirent davantage les enfants, cependant : « Des souris ! Des éléphants ! Des dragons ! Des tigres ! Des cro-cro-diles !... »
Il est temps de reprendre le fil de l'histoire.
« Justement, oui, des tigres, il y en avait dans la forêt, des tigres géants, rayés blanc et bleu. Mais Patriki ne les chassait pas. Il savait que ses flèches ne les toucheraient jamais, parce que c'étaient des tigres magiques.
— Ils étaient invisibles ! propose une petite.
— Non, on ne voyait que les rayures bleues ! contre une autre.
— On ne voyait que les rayures bleues », accepte Liam, se lançant aussitôt dans la description de la première rencontre entre Patriki et ces rayures bleues qui flottent mystérieusement à travers la forêt. Puis il reprend : « Et les tigres n'étaient pas les seuls animaux magiques, en ce temps-là. Il y avait aussi, très haut dans la montagne, des aigles géants, rayés bleu et blanc, et ils étaient magiques parce qu'ils parlaient. Les tigres aussi parlaient. Mais ils ne parlaient jamais aux aigles, parce qu'ils ne pouvaient pas s'entendre. »
Il s'arrête alors, pétrifié, en prenant enfin conscience de ce qu'il est en train de raconter. Les enfants se trémoussent, impatients. Katrin demande : « Pourquoi ils ne s'entendaient pas ?
– Les aigles étaient très haut en l'air, et les tigres tout en bas par terre, réplique un des plus vieux, d'un air supérieur. Ils ne pouvaient pas crier assez fort. »
Liam contemple les visages attentifs tournés vers lui, à la fois désemparé et attendri. Ils ne comprennent pas, bien sûr. Ils comprennent autre chose. Et pourquoi pas ?
Il continue avec le fil qu'on lui a tendu, revient en arrière pour raconter l'origine des tigres et des aigles magiques : en ce temps-là, une très puissante magicienne s'occupait du soleil, des étoiles et des lunes, et des rivières, des forêts et des montagnes ; comme elle ne pouvait pas tout faire elle-même, elle décide de créer des gardiens. Et en créant le Gardien des montagnes, elle a le hoquet et voilà que des jumeaux lui sortent de la bouche. Et le Gardien s'appelle... Ariki, et la Gardienne Arina, et la magicienne... est juste la Magicienne – mais questions et suggestions se font de moins en moins fréquentes à mesure que l'histoire se déploie, les bouches restent entrouvertes, les yeux s'écarquillent. Frère et sœur, repentants après leur terrible querelle, partent à la recherche l'un de l'autre, quittant leur exil respectif, Patriki la forêt, Katrina la montagne, et chacun rencontre l'animal-totem de l'autre, qui s'est hasardé lui aussi en dehors de son territoire habituel et s'est blessé ; comme c'est la dernière nuit de l'Hiver, ils leur parlent au lieu de les achever. Katrina ramène l'aigle vers les hauteurs de la montagne sur un travois, Patriki le tigre vers les profondeurs de la forêt, ils se rencontrent à mi-chemin, se tombent dans les bras, se racontent leurs aventures, et le tigre et l'aigle en viennent à se parler aussi, ce qu'ils n'ont jamais tenté jusqu'alors la dernière nuit de l'Hiver malgré le don que la Magicienne leur avait fait de la parole.
« J'aimerais tant pouvoir courir dans la neige blanche des sommets, dit enfin le tigre d'une voix rêveuse. Comment est-ce, là-haut, mon frère l'aigle ?
— Les volcans rougeoient au milieu des glaciers, comme un amour qui ne finit jamais. Quelquefois, quand les chemins d'air m'emportent vers le soleil, je vois des étoiles en plein jour et le ciel est si bleu qu'il en est presque noir. Mais j'aimerais tant planer dans l'ombre de la forêt... Comment est-ce, en bas, mon frère le tigre ?
— La terre est noire et profonde, et les arbres sont les racines du ciel. Parfois, quand je cours à travers la nuit, j'entends le cœur qui bat sous l'écorce du monde, comme un amour qui ne finit jamais... »
Et Katrina et Patriki, réconciliés, retournent vivre ensemble pour toujours dans leur village. Et les Gardiens querelleurs Ariki et Arina, enfin libérés de leur sortilège, reviennent auprès de la Magicienne, qui leur demande alors de quoi ils désirent être désormais les Gardiens.
« Des montagnes, dit Ariki sans hésiter, là où elles s'enracinent dans les profondeurs de la terre, là où le sang du monde coule encore.
— Des montagnes, dit Arina sans hésiter, là où elles touchent le ciel, là où elles s'ouvrent au soleil en profonds précipices. »
La Magicienne hocha la tête : « Les montagnes, dit-elle, là où le ciel et la terre s'aiment d'un amour qui ne finit jamais... » Et elle sourit à Arina comme à Ariki : « C'est bien. Qu'il en soit ainsi jusqu'à la fin des temps. »
Il y a un silence perplexe quand il se tait. Jill dit en souriant : « Il est tard, ce coup-ci, on va se coucher. On dit merci à Oncle Liam pour la belle histoire. » Un chœur désordonné de voix enfantines le remercie et les petits partent sous la houlette d'Andreï, les arguments faisant rage pour et contre le choix final des Gardiens. Suzane s'étire avec un sourire ravi : « Je ne savais pas que tu étais un conteur d'histoires ! En as-tu d'autres comme ça ? »
Depuis la nuit qu'ils ont passée si chastement ensemble, elle recherche sa compagnie, l'embrasse sur la joue pour lui dire bonjour et bonsoir, lui touche la main ou le bras quand elle lui parle. Jill remarque, avec un amusement indulgent : « Il peut sûrement en inventer d'autres pour nous dépanner, hein, Liam ? »
Hésitant, inquiet, tenté, il contemple le sourire de Suzane. Et finalement, il dit : « Je suppose que oui. »
12
Au sortir de la passe, sur environ deux cents langhi, la route descend en paresseux lacets le long d'une série de plateaux. « La route des carrières et des mines », a dit Odatan, une véritable route dallée d'ultellaod écarlate, de plus en plus large à mesure qu'on descend vers les collines et la plaine d'Eïtyrhondal, perdues dans une brume laiteuse en contrebas. Il doit y avoir des villages invisibles dans la forêt, car des pistes fréquentées débouchent ici et là sur la route, avec des gens affairés, à pied ou sur des aski, circulant dans les deux directions. Puis, à partir de l'avant-dernier plateau, les pistes deviennent des routes secondaires pavées, et la circulation se fait plus dense, essentiellement de gros charrois à vapeur chargés de billes de bois, de lingots de minerai ou de pierres brutes qui descendent vers la plaine ou remontent à vide, mais aussi des carrioles légères attelées d'aski, de lents chariots tirés par des pâlukaï au front têtu, des cabriolets, des cavaliers et encore, toujours, des marcheurs, femmes, hommes ou enfants, toujours chargés ou non de fardeaux divers. Il y a aussi des charrois communaux, également à vapeur, bien identifiés par leurs couleurs vives et la liste de leurs destinations peinte le long de leurs côtés. À ce moment-là, la route s'est divisée en deux voies, une qui monte, une qui descend, et la circulation s'organise de gauche à droite sur chaque voie, dans un aimable désordre, plus lente à gauche, plus rapide à droite.
Lian et son compagnon voyagent sur le côté gauche, bien que, comme plusieurs cavaliers, ils piquent parfois un trot sur la bande herbeuse séparant les deux voies. Un tovker pourvu d'un cavalier est un spectacle assez rare, de toute évidence (pendant tout leur voyage au sortir des montagnes, Lian n'en verra qu'un seul autre), et les regards les suivent longtemps ; il voudrait penser que c'est la seule raison pour laquelle les piétons se retournent sur leur passage ou ralentissent parfois sur l'autre voie.
S'il y a des relais et des auberges sur les routes attenantes, parfois signalés par des panneaux, Odatan ne s'y arrête pourtant pas. Pendant trois jours, ils continuent à dormir à la belle étoile, en épuisant les provisions qui leur restent.
Ils ont quitté le Leïtnialen depuis vingt-quatre jours quand ils arrivent en vue de la plaine. Il fait nettement plus sec et l'air est riche et un peu lourd pour Lian, habitué à respirer en altitude ; il sourit à l'idée qui lui traverse soudain l'esprit : après avoir été karaïker, le voilà en passe de devenir agraïllad !
La route se suspend à des falaises abruptes au-dessus des collines, taillée à vif dans le roc, où des veines écarlates d'ultellaod miroitent en alternance avec des épaulements de pierre couleur crème, assez friable : en plusieurs endroits, des grillages recouvrent la face de la falaise pour retenir les éboulis. À un tournant, les voyageurs découvrent que leur voie est obstruée par un long charroi renversé en travers. Deux carrioles se sont déjà arrêtées. L'accident vient d'arriver : des sacs de sciure, de copeaux et de bûchettes sont répandus un peu partout, on s'affaire à soigner les deux passagers qu'on a étendus dans l'herbe. Le charroi a apparemment perdu une roue en pleine descente – et il devait aller un peu trop vite, aussi ; mais le conducteur a réussi à le détourner vers la falaise, qu'il est allé percuter avec violence ; la cabine est à moitié écrasée. Constatant qu'on n'a pas besoin d'eux pour les blessés – trois personnes s'en occupent avec une compétence évidente –, Lian et Odatan vont dégager la route, ou du moins un passage, en ramassant les sacs éparpillés et les débris du véhicule.
Tout se passe très vite. Ils viennent de saisir à deux un lourd sac de bûchettes tombé entre la falaise et le charroi renversé quand il y a comme un craquement sourd juste au-dessus d'eux, un grondement qui s'amplifie. Lian lève la tête, le temps de percevoir un mouvement dans l'énorme veine d'ultellaod qui les surplombe, de lâcher le sac en même temps que son compagnon, de fermer les yeux.
De les rouvrir avec stupeur, parce qu'il n'est pas enseveli sous des tonnes de roc, et de sauter à l'écart pour se protéger, quand même, tandis que des petits morceaux d'ultellaod pleuvent autour de lui et explosent en éclats coupants sur la route. Les deux blocs détachés de la falaise, dont l'un est presque aussi gros que la remorque du charroi, sont apparemment passés à l'horizontale au-dessus d'eux pour aller se poser entre les deux voies. Lian reste pétrifié, tandis que les derniers cailloux ricochent sur la route.
Odatan se tient le bras en grimaçant, du sang coule entre ses doigts. Lian se précipite vers lui, l'autre secoue la tête, « Ce n'est rien », mais la coupure est profonde – les éclats d'ultellaod sont tranchants comme des rasoirs. Lian improvise un tourniquet avec le lacet qui ferme son encolure de chemise, tire son compagnon vers le petit groupe des sauveteurs encore agenouillés près des deux blessés. Une femme assez âgée à la peau bronzée se lève, les regarde approcher avec un calme étrange. Elle incline la tête : « Bon chemin, Odatan.
— Grâce à toi, Kilasi », réplique Odatan d'une voix un peu enrouée. Lian l'aide à s'asseoir dans l'herbe, la vieille femme s'agenouille près de lui et examine la plaie tandis que Lian court chercher la trousse d'urgence dans les fontes de sa selle.
Kilasi nettoie la blessure, applique du désinfectant et commence à enrouler avec soin un bandage. Mais il faut des sutures ! La vieille femme réplique : « Ce ne sera pas nécessaire. » Quand Lian insiste, scandalisé, Odatan intervient : « Elle sait ce qu'elle fait, Lian. » Et sa voix habituellement paisible est si rude que Lian, pris au dépourvu, se tait. Kilasi se relève, aide Odatan à en faire autant. Elle le dépasse d'une bonne tête.
D'autres véhicules se sont arrêtés sur l'autre voie aussi. Il faudra détourner la circulation autour du plus gros bloc dont les extrémités débordent sur la voie montante comme la voie descendante, mais tout le monde est indemne. Lian contemple le spectacle, abasourdi. Puis, lorsque le charroi renversé s'ébranle en grinçant et vient se ranger lentement le long de la falaise pour dégager le passage, sans que personne y touche, il comprend enfin.
Kilasi s'est affaissée, les yeux fermés, contre Odatan ; il la soutient de son bras indemne. « Pour le reste, souffle-t-elle, ils se débrouilleront. » Odatan lui tapote l'épaule d'un air compatissant en disant « Mais oui ! ». Sous le long manteau de voyage de la vieille femme, la tunique bleue n'est plus simplement une tache de couleur, ainsi que Lian l'a distraitement perçue la première fois. C'est la tunique d'une hékel. Avec, à l'ourlet et au col, la bande noire des keyrsani.
Après qu'elle s'est remise, Odatan fait les présentations. « Kilasi Mattiéïl Lunmiu. Lian Dougallad Laraïnu. » Elle tend les mains, Lian les prend ; les yeux noirs le dévisagent un moment avec intérêt dans leur entrelacs de rides bienveillantes, se détournent avec une politesse sans affectation. « Je vais emmener les blessés au relais Dnaõzer. Lian peut prendre Nagal-Rasser. Il y a une place dans mon cabriolet, tu viens avec moi, ça vaudra quand même mieux. »
Odatan émet un petit « tsss » agacé, mais il se laisse faire, sans doute parce que Lian manifeste beaucoup d'enthousiasme à l'idée de monter le tovker. Le petit homme s'assied à l'avant du cabriolet avec la hékel tandis qu'on installe les blessés sur les banquettes à l'arrière. Les deux aski sont attachés aux montants de la capote. Et Lian se hisse sur le dos du tovker, qui l'accueille avec son habituelle équanimité et part au trot dans l'herbe à côté du cabriolet dès qu'il a passé les jambes dans les cuissières. Brièvement déséquilibré, Lian enroule ses mains dans la crinière soyeuse, cherche un instant le rythme, assez différent de celui de Métyi et, quand il l'a trouvé, il se laisse emporter, un grand sourire aux lèvres. Si on le regarde, désormais, se dit-il avec une satisfaction maligne, on aura vraiment de quoi s'étonner : un halatnim, et un naïstaos, monté sur un tovker !
Le relais Dnaõzer : un grand morceau de forêt transformé en parc, avec à l'entrée une vaste prairie où paissent des aski et des pâlukaï ; de l'autre côté de la route des véhicules s'alignent un peu au hasard dans une aire dégagée couverte de gravillons. Une demi-douzaine de femmes et d'hommes en tuniques bleues attendent Kilasi ; on décharge les blessés sans exiger de la hékel des explications que, de toute évidence, on détient déjà, et l'on s'occupe de dételer et de ranger son cabriolet. Puis on leur demande avec sollicitude s'ils désirent emprunter l'un des dana-dana alignés près de l'entrée – il y en a une autre variété, les masdana, à trois roues, des espèces de mini-cabriolets propulsés par les pédaliers situés aux places arrière et sous le siège du conducteur assis à l'avant au guidon. « Non », dit Kilasi. « Toujours pas, même un masdana ? » fait Odatan. « Non-non-non ! » réplique Kilasi ; ils échangent un sourire amusé. « Mais si vous pouvez faire porter nos bagages au relais... », suggère Kilasi redevenue sérieuse aux gardiens de l'entrée. On lui assure que ce sera fait.
Ils s'engagent dans une allée bordée d'arbres moussus et de grands buissons aux feuilles vernies dont Lian ignore le nom. Elle serpente entre prairies herbeuses et bosquets ponctués çà et là de plaques de fleurs, de buissons et de pièces d'eau miroitantes ; d'autres sentiers s'y embranchent, plus étroits. Un peu partout, à moitié dissimulées par l'herbe ou les feuillages, se dressent de petites stèles de pierre dorées, incrustées de filaments argentés de sirid. Lian, curieux, s'accroupit pour en examiner une bien visible au bord du chemin. « Ne la déterre pas, Lian, dit Kilasi, c'est une offrande. »
Le relais n'est pas simplement une auberge, mais un musée, et un sanctuaire. « Sur Tyranaël, avant les guerres de l'Unification, se dressait ici une grande ville, la capitale des Paalani, Dnaõzer. C'est de là qu'est partie la toute dernière guerre, et non loin de là qu'elle s'est achevée, dans la plaine, à Hanat-Naïkaõ, avec la mort de l'Envoyé divin, Ktulhudar. De l'Autre Côté, on avait conservé les ruines de la ville. Ici, sur Atyrkelsaõ, les reconstituer n'aurait eu aucun sens. On s'est donc rangé à l'avis des darnao et, à l'emplacement exact de la ville, ils ont construit le sanctuaire et dessiné le parc. Les visiteurs d'ascendance paalao, mais d'autres aussi, plantent ces stèles en souvenir de l'Envoyé divin – et de Tyranaël. »
Encore une histoire qu'il ne connaissait pas ; Lian se dit, avec une certaine ironie, que ce n'est sûrement pas la dernière. Kilasi et Odatan vont-ils se relayer pour éduquer le petit sauvage sorti de sa forêt ? Mais non, Kilasi a passé un bras sous le bras valide d'Odatan et ils continuent leur marche dans un silence amical. Après avoir parcouru un demi-langhi au moins, ils arrivent au relais – Lian en avait deviné la proximité aux rires et aux appels qui résonnaient parfois à travers les arbres. Le complexe circulaire, qui fait au moins cent vingt lani de diamètre, se dresse dans une vaste clairière, sur une élévation de terrain qui évoque aussitôt pour Lian les monticules des moatranao. Mais le bassin est un anneau relativement étroit, une dizaine de lani ; on le traverse par un pont pour arriver dans l'anneau suivant, d'une vingtaine de lani, pelouses, fleurs, buissons bas, plantes vertes, et les tingalyai qui alimentent le complexe en eau, régulièrement espacés et d'un âge vénérable à en juger par leur taille et le renflement de leur tronc-réservoir. Une arcade couverte à colonnade enserre le corps principal du bâtiment rond et modulaire, dont les cinq segments sont recouverts de toits légèrement bombés faits de briques de verre hexagonales, épaisses et translucides. Au point le plus haut de l'édifice, une autre coupole, avec en son centre une explosion de feuillage cuivré – un jardin intérieur ? Sous l'arcade, on distingue des tables où l'on doit se restaurer à l'heure des repas, des fauteuils et des bancs où pour l'instant des gens se reposent ou bavardent, lisent, écrivent, jouent à des jeux divers, tout en surveillant les jeunes qui nagent et plongent dans le bassin, en compagnie d'une troupe de banki de petite taille, la variété daru des plaines.
Dans le grand hall, il y a du monde mais pas trop, une vingtaine de personnes venues comme eux s'enregistrer et dont certaines attendent leur tour sur une banquette, ou en examinant des objets placés dans de nombreuses vitrines le long des murs : coupes, vases, bijoux, tapisseries. D'autres font la file au comptoir d'accueil. Après s'être enquis d'un dortoir où il reste de la place pour un groupe de quatre, l'homme qui se trouve devant Lian et Odatan va pour s'éloigner, et bute dans Lian. Son visage aux méplats accusés de Paalao passe presque instantanément de la surprise à l'irritation.
« Jamais assez de place pour vous », grommelle-t-il, d'une voix qui veut être entendue.
Lian devine plus qu'il ne voit les regards se fixer sur eux pour se détourner presque aussitôt. « Veuillez m'excuser », balbutie-t-il machinalement en reculant d'un pas pour laisser davantage d'espace ; il lui semblait pourtant s'être tenu à une distance suffisante. Mais l'autre ne s'éloigne pas ; il le fixe avec une curiosité hostile, dit encore sans discrétion : « Et au relais Dnaõzer, en plus... »
Personne ne s'est jamais comporté ainsi avec Lian. Il est complètement désemparé. Le cœur lui débat, il a mal à la poitrine comme s'il avait trop couru. A-t-il peur ? Qu'est-ce qu'il a ?
« Pourquoi pas au relais Dnaõzer ? » demande Odatan, paisible.
L'autre se tourne vers lui, le dévisage un instant en silence ; ses traits se crispent, il se mord les lèvres : « Parce que la véritable Dnaõzer est perdue à jamais pour mon peuple », laisse-t-il enfin échapper d'une voix qui s'étouffe d'émotion. « Par votre faute. » Il tend une main tremblante vers Lian. « Et vous venez semer vos graines néfastes sur Tyranaël, maintenant. Mais Hananai vous voit, et vous juge. Regardez ce qui attend votre descendance, Virginien. Naïstoï vous êtes, et naïstoï vous redevenez. La justice de Hananai.
— Sais-tu à qui tu parles ? » s'écrie Kilasi – elle s'étrangle presque.
Odatan lui pose une main sur le bras. « Il parle à un Virginien. Et il a le droit de le faire, n'est-ce pas ? Plus que le droit, en ces lieux, le devoir. Ktulhudar serait satisfait, Paalao, lui qui a dit à ton peuple " vous serez grands dans la paix, et non dans la guerre ", lui qui a sacrifié sa vie humaine pour guérir les hommes de leur poison. Tu as laissé sortir le venin qui te blesse, te sens-tu mieux à présent ? »
L'homme reste un instant pétrifié, puis s'écarte avec une exclamation inarticulée et quitte le hall presque en courant. Odatan se tourne vers les autres personnes présentes – un grand silence règne, tout le monde les regarde, maintenant. « En reconnaissez-vous aussi le goût ? Vous tente-t-il ? »
La plupart secouent négativement la tête ; quelques-uns baissent les yeux. Puis une grande femme à la peau dorée de Tyrnaë dit d'une voix claire : « Je le reconnais, mais il ne me tente pas. Cet enfant n'est pas responsable des volontés de la Mer. Pas plus que toi ni tous les tiens, passeur. »
Un murmure d'assentiment court dans le hall. Odatan hoche la tête.
Et c'est terminé. La tension s'évapore, chacun revient à ses occupations. Lian, encore sous le choc, les jambes molles, regarde stupidement son compagnon qui se retourne vers le hékel du comptoir d'accueil et demande une chambre pour deux. Le jeune homme le contemple, les yeux écarquillés, se reprend avec un sursaut, balbutie : « Oui, oui, dans la section nord, 221 », et lui tend deux clés. Sans lui demander son nom ni celui de Lian pour son registre. Odatan donne l'une des clés à Lian, empoche l'autre, tandis que le jeune hékel pose une autre clé devant Kilasi en murmurant : « Le 237 ». Mais elle, il inscrit son nom dans le registre, même s'il ne le lui a pas demandé. « Vos bagages sont en route, ajoute-t-il.
— Merci.
— Allons au sanctuaire, Lian », dit Odatan d'une voix étrange, distante.
— Tu devrais te reposer, béra, murmure la vieille hékel.
— C'est plutôt toi qui le devrais. »
Elle secoue la tête. Ils se regardent un instant en silence, puis Odatan se dirige vers l'un des couloirs qui s'ouvrent dans le hall, à la droite du comptoir. Ils le suivent. Lian est encore sous le coup du " béra " adressé par la vieille femme à Odatan, qui a bien quarante saisons de moins qu'elle. Mais comme ils entrent presque tout de suite dans le sanctuaire, il n'a pas le temps de poser de question.
*
Liam raconte des histoires aux enfants de la commune. La grande histoire de la création du monde, et des humains que la Magicienne doit se reprendre à trois fois pour créer. La drôle d'histoire des Atiolai, la première race humaine, qui croient que les montagnes de l'est pondent le soleil à neuf tous les matins, pour que les montagnes de l'ouest l'avalent tous les soirs. La triste histoire de Merrim et Kalalu, les amants à jamais séparés par l'ouverture de la Passe de la Hache. La joyeuse histoire de Pit-Pit au chapeau vert, et comment le tout petit Gardien des petites choses gagne son pari avec les autres Gardiens en jetant simplement un tout petit caillou sur la Terre, et le tout petit caillou tombe sur la tête du cheval du vilain Prince, et le Prince tombe de son cheval et perd la mémoire et devient gentil, et il rencontre la Princesse, et il l'aime même quand il a retrouvé la mémoire, et à la fin leurs royaumes ennemis ne se font plus la guerre – tout ça pour un tout petit caillou.
« Pourquoi il a un chapeau, Pit-Pit ?
— Parce qu'il a demandé aux autres Gardiens un de leurs cheveux s'il gagne le pari, et il se fabrique un chapeau magique avec tous ces cheveux magiques, pour se protéger de la colère des autres Gardiens fâchés d'avoir perdu leur pari.
— Mais pourquoi il est vert, le chapeau de Pit-Pit ?
— Parce qu'il est vert. »
Des histoires.
13
Le sanctuaire de Dnaõzer, on y pénètre par l'est et on en fait le tour complet d'abord par l'extérieur en direction de l'ouest, en suivant le large corridor qui l'encercle. Le mur de gauche est entièrement blanc, comme le sol du corridor, recouverts tous deux d'un enduit épais qui étouffe les sons. Des fresques vernies recouvrent le mur de droite, exactement recopiées de celles qui ornent le puits funéraire de Ktulhudar sur Tyranaël. D'abord les Paalani en paix malgré eux, vaincus par le Tyrnaë Markhal, troisième du nom, aidé par les Envoyés divins. Atéhoni, la fille aveugle et muette du puissant roi Atsulad, et sa prophétie, " Ô mon père, laisse-moi vivre et mêler notre sang à celui de l'étranger tyrnaë, car de notre union naîtra un dieu, et le nom des Atsuladi deviendra immortel. " La naissance miraculeuse de l'enfant, au cours d'un orage qui lui donne son nom, Ktulhudar, Fils de l'Éclair. Et, lorsqu'il entre dans l'âge d'homme, à sa seizième saison, les trois autres miracles : l'Épée des Atsuladi sort seule de son fourreau pour se donner à lui, la lourde Selle ancestrale ne pèse pas plus qu'un fétu entre ses mains, et un tovker franchit les portes ouvertes sans l'aide d'une main, pour venir lui rendre obéissance, " noir comme la nuit d'avant Bélek, mais sa corne était blanche ".
Ensuite la guerre, la guerre, la guerre, et Lian contemple les fresques avec un sentiment de nausée. Il sait ce qu'est la guerre, certaines des histoires de Laraï l'évoquaient – mais les guerres des Hébao, escarmouches, embuscades, combats singuliers, quelques dizaines de combattants rivalisant d'adresse, d'astuce et de courage dans la forêt. Les batailles rangées, les soldats par milliers, les massacres qui durent des jours, la guerre qui dure des années, cela dépasse son entendement. Il contemple les fresques pourtant, et il est obligé de voir, de savoir. Les artistes de l'antiquité – cette histoire remonte à plus de huit mille saisons – avaient un sens atroce du détail. Lian saturé ne voit bientôt plus que du sang, du sang, du sang. Et partout Ktulhudar triomphant, à l'armure toujours sans tache, à l'épée jamais rougie.
Jusqu'au sac d'Almïundaz, la ville ouverte des Hébao. Et là, dans la nuit et les flammes, Ktulhudar couvert de sang pour la première fois de sa vie passe au cou de la Hébaë Eylaï le lourd collier d'or que seule sa main peut ouvrir une fois qu'il s'est refermé sur un cou d'esclave.
On est revenu au point de départ. On pousse la très lourde porte de bois sombre, dépourvue d'ornement – et il faut y mettre les deux mains d'abord, et toute sa force, mais ensuite elle s'ouvre aisément, comme emportée par sa propre masse. On peut alors entrer au plus secret du sanctuaire. On en suivra les contours dans le sens inverse, d'ouest en est. Comme dans le corridor, les briques de verre de la coupole y atténuent la lumière du jour, mais il y fait encore plus sombre. Trois salles distinctes s'y ouvrent, comme les trois chambres d'un cœur rani, séparées par une paroi en biseau qui ne se rend pas au centre du sanctuaire. Et au centre se dresse un arbre, un tingalyai, le plus grand que Lian ait jamais vu. Il a été planté là il y a près de neuf cents saisons, et autour de lui on a édifié le sanctuaire. La partie inférieure de son énorme tronc renflé se perd dans l'obscurité d'une chambre souterraine, un puits creusé à même le roc et dont on n'a pas adouci les arêtes vives. Les branches du sommet s'étendent comme cent bras et jaillissent vers le ciel par la coupole ouverte, filtrant le soleil en reflets cuivrés.
Des oiseaux chantent dans l'arbre. Des dizaines, peut-être des centaines d'oiseaux de toutes sortes. On ne les entendait pas de l'extérieur, et on comprend alors à quel point les parois doivent être épaisses, qui séparent les deux faces du sanctuaire. Leurs chants résonnent entre les parois de la salle, puis se taisent quand la porte se referme avec un choc sourd. Alors, dans le silence, on peut entendre la voix de l'eau souterraine qui court, invisible, au pied de l'arbre de vie.
Et l'histoire de Ktulhudar reprend. Comme lors du premier périple extérieur, la voix d'Odatan et celle de Kilasi psalmodient en alternance la légende ancienne. Comment Ktulhudar édicte ses nouvelles lois de la guerre, plus humaines, irritant contre lui le riche Argad et le rusé Balinduz, chefs des Coalisés. Comment il libère la Hébaë et comment elle referme elle-même autour de son cou le collier d'or. Comment il l'épouse, malgré l'opposition de ses alliés. Et de nouveau du sang : celui de l'assassin auquel échappe l'épouse du Prince. Ensuite, dans une blancheur aveuglante, la visite de Ktulhudar désespéré aux autres Envoyés, ses sœurs et frères : " Rendez-la immortelle, et je mettrai fin à la guerre. " Les voici tous deux, le Prince et son épouse, vêtus de leurs plus riches atours, devant les Envoyés invisibles au cœur de leur forteresse immaculée. Mais le Prince retourne seul en son palais de Dnaõzer, " et lorsque Balinduz lui demanda où était la Princesse, son visage devint si terrible que Balinduz s'évanouit à ses pieds ".
Ensuite, c'est l'adieu de Ktulhudar aux siens, et la prophétie. " Je suis l'Épée et la Blessure, l'Arbre et le Bûcheron. Mon séjour parmi vous s'achève. Voici venu le temps de la paix. Trop de haine habitait encore le cœur des hommes, et de peur qu'elle n'empoisonne la terre à jamais, je suis venu moissonner la dernière moisson du mal. Vous serez grands un jour, Paalani, non par les armes et dans la guerre, mais dans la paix. Demain nous livrerons la dernière bataille, et je vous quitterai. Mais rappelez-vous que mon esprit sera toujours avec vous. "
Et maintenant, la grande bataille, la dernière bataille, la bataille de Hanat-Naïkaõ, presque aux portes de Dnaõzer, dans la plaine où l'armée de Markhal, avec celles de ses alliés aritnai et hébao, a repoussé l'armée des Coalisés. Du sang, du sang, le dernier sang. Et Ktulhudar au visage d'orage, immobile au milieu de la mêlée sur son tovker noir. Soudain, autour de lui, ses gardes disposés en quinconce, les fidèles Kraïkerdaru, semblent s'endormir les yeux grands ouverts. Le Prince brandit son épée et lance un cri sauvage, dans une langue inconnue de tous. Sa noire monture se cabre, et une flèche de feu vient frapper Ktulhudar en pleine poitrine.
« Il se fit un grand silence alors sur le champ de bataille, car la lumière du soleil s'était obscurcie, murmure Kilasi. Une voix surnaturelle résonna, et chacun l'entendit dans son cœur comme si elle s'adressait à lui seul, et la voix disait " PAIX ! " Les armes tombèrent, les soldats s'étreignirent, et l'empereur Markhal, troisième du nom, s'en vint chercher le corps de Ktulhudar au milieu des rangs paalani. On le porta jusqu'à son camp sur le long bouclier des hommes d'Aritu, tandis que Hébao et Tyranao chantaient leurs chants funèbres et que les Paalani en pleurs faisaient gronder le tonnerre de leurs tambours dans la nuit. »
Ils sont revenus près de la lourde porte, à l'est, du côté où le soleil se lève. Il se lève aussi dans la dernière image de la fresque. Odatan enchaîne, d'une voix sourde où résonne une curieuse amertume : « Au matin, les Envoyés divins emportèrent le corps mortel de leur frère sur Ékellulan, leur île du Nord où il reposerait auprès de son épouse. Et leur lumière brillait désormais d'un feu pur, car il les avait enfin guéris de la folie de l'immortalité. »
Le silence revient, assez longtemps pour que quelques oiseaux se mettent à chanter dans les ramures de l'arbre de vie. Lorsque Kilasi reprend la parole, Lian ne sait si elle s'adresse à lui ou à Odatan : « Mais Ktulhudar, pour expier le terrible prix de cette guérison, décida de marcher pour toujours sur la terre des humains. Et des lointains enfants de ses enfants naquit Oghim, le prince sans ombre. » Puis, posément : « Et tu es là, petit halatnim, parce qu'Oghim, le premier aïlmâdzi, a vu le premier passeur qui viendrait de Tyranaël. »
Laraï n'avait pas dit qu'Oghim était le premier aïlmâdzi, ni qu'il avait vu des Virginiens... Mais Lian y songe à peine ; il entend à peine Odatan murmurer « Je suis fatigué, je vais faire ma méridienne », et la lourde porte se refermer sur lui. Il est trop perdu dans l'histoire de Ktulhudar, dans les formes et les couleurs de la fresque que le miroitement de la lumière à travers les branches du tingalyai anime de mouvements magiques. Immobile auprès de Kilasi, comme en transe, il écoute le chant des oiseaux se mêler peu à peu, triomphant, au bruissement de l'eau souterraine.
À pas lents, ils quittent enfin le sanctuaire, dont ils ont été les seuls visiteurs pendant tout ce temps. Ils ne repassent pas par le hall, s'engagent au sortir du corridor principal dans un lacis de couloirs déserts qui les mènent à la section nord du relais. Lian se sent la tête légère, un fourmillement nerveux dans tout le corps. Les images des fresques lui tournent dans la tête. Si ancienne, l'histoire, et pourtant, la voix des deux conteurs était empreinte de chagrin et de pitié comme si elle s'était déroulée la veille... Lian sait qu'il est un trop bon auditeur – pendant toute son enfance, Nathénèk l'a gentiment taquiné pour sa crédulité, veillant à remettre en perspective les histoires de Laraï. Mais si les tovik sont des sortes de danvérani, et les banki des sensitifs comme les humains – et même les karaï et les agraï – quelle perspective pour cette histoire-ci ?
« Qu'est-ce qui est vrai, dans l'histoire de Ktulhudar ? » demande Lian en levant la tête vers la vieille hékel.
Elle le regarde de biais avec un petit sourire. « Tout. » Il va protester mais elle poursuit – et son sourire n'est pas dénué de tristesse : « Tout ce que tu décides de croire, petit halatnim. C'est nous, en dernier ressort, qui décidons de la vérité.
— Mais Ktulhudar, il a réellement existé ? » insiste Lian avec une légère impatience.
Le sourire de la vieille femme se fait plus franchement amusé : « Tu m'as demandé ce qui était vrai dans cette histoire. Nous ne parlons pas de la réalité. Seule Hananai connaît et le réel et le vrai. Nous autres humains devons nous contenter d'imaginer le réel et d'en chercher la vérité. »
Lian réfléchit un moment, les sourcils froncés. « Le soleil se lève à l'est, c'est la réalité et c'est vrai. Je ne l'imagine pas. » Le visage de Nathénèk passe soudain dans sa mémoire. « C'est ainsi. Je ne peux pas le changer en décidant que ce n'est pas vrai.
— Le changer, non. Mais te changer, oui, en décidant de croire que ce n'est pas vrai. »
Lian éclate d'un rire scandalisé : « En devenant fou, vous voulez dire ! »
Kilasi s'est immobilisée. Son visage est soudain grave : « Oui, Lian, en devenant fou. » Puis elle reprend : « Mais écoute. Il était une fois un village où les gens croyaient que les montagnes de l'est accouchaient à neuf du soleil tous les matins pour en nourrir tous les soirs les montagnes de l'ouest. Un jour s'en vint une vieille voyageuse qui avait vu le soleil se lever et se coucher bien des fois sur l'océan. Elle en avait déduit à force que la terre était ronde, et tournait autour du soleil. Les villageois, saisis de compassion, décidèrent de la guérir de sa folie. Ils la privèrent de sommeil, de nourriture, d'eau et de lumière. Et lorsqu'elle sortit de son cachot, elle les remercia : elle savait maintenant que les montagnes de l'est accouchaient d'un soleil neuf tous les matins, et que celles de l'ouest s'en nourrissaient tous les soirs. Elle n'était plus folle.
— Mais le soleil se levait quand même toujours à l'est », marmonne enfin Lian, buté.
Kilasi se met à rire. « Oh, Lian, je ne dis pas que la réalité n'existe pas, ni même que nous ne pouvons pas en connaître des fragments. Pourquoi sinon serais-je une guérisseuse ? » Elle redevient grave. « Ktulhudar a réellement existé, comme tous les empereurs nommés Markhal, et la Hébaë Eylaï et bien d'autres. Mais ce qu'ils étaient, et ce qu'ils ont fait, c'est le sens qui en importe pour nous, maintenant comme déjà en leur temps. Lorsque Dalgyan, l'historien tyrnaë qui s'était volontairement rendu à Ktulhudar, a commencé d'écrire son histoire, Les Chroniques du Royaume du Milieu, il voulait garder trace des événements pendant qu'ils se déroulaient, mais il voulait surtout les comprendre, parce que l'existence et les actes de Ktulhudar niaient tout ce qu'il avait été habitué à croire, et qu'il ne voulait pas devenir fou. Il est des parts de la réalité que notre esprit ne peut appréhender aisément – n'appréhendera peut-être jamais – et nous essayons de les approcher, de les apprivoiser, en discernant non pas ce qu'elles signifient en elles-mêmes, sous le regard de Hananai, mais ce qu'elles signifient pour nous.
— Leur vérité, non leur réalité », traduit Lian à mi-voix. Et, après une pause : « Quelle était la vérité de Ktulhudar, alors ?
— Pour ceux qui l'ont suivi, et pour nous aujourd'hui, c'est qu'il n'y a plus jamais eu de guerres entre les Ranao », dit Kilasi avec un sourire approbateur. Et elle ajoute, un peu plus bas : « Et que nous avons préféré abandonner Tyranaël plutôt que de voir renaître la guerre. »
Lian s'immobilise, le cœur brusquement serré. « Les Virginiens... font la guerre ? »
Kilasi le dévisage. « Ils nous l'auraient faite », dit-elle enfin, avec une curieuse hésitation. Puis, plus assurée : « Et ils se la font entre eux, depuis très longtemps. Ou ils la faisaient lorsque les derniers passeurs nous ont donné des nouvelles de l'autre côté. Il y a bien longtemps de cela, cependant. Hananai sait ce qui a pu arriver depuis. »
Lian se hâte pour rattraper la vieille femme, qui s'est remise en marche ; ils se trouvent maintenant dans la section nord, et elle vérifie les numéros des chambres. « Les derniers passeurs ? Il n'en passe plus ? »
Elle lui lance un regard de côté : « Pas depuis quatre-vingts saisons.
— Quatre-vingts... Mais... attendez... »
La hékel s'est immobilisée. Il la dévisage en essayant de donner forme à sa stupeur. « Odatan... est un passeur... »
Kilasi hoche la tête.
« Il a à peine quarante saisons !
La hékel hésite un bref instant : « Il est plus vieux qu'il n'en a l'air.
— C'est... le dernier passeur ? » Voilà pourquoi les gens le regardaient de ce drôle d'air, alors, dans le hall !
Kilasi semble prendre une décision : « Seul Odatan peut te parler de lui-même, s'il le juge bon, Lian. Je ne le peux pas. »
Elle a en fait utilisé la formule consacrée, " la discrétion m'oblige au silence " : elle le pourrait, mais a déterminé qu'elle ne le doit pas. Le réflexe de politesse joue aussitôt, et Lian n'insiste pas. Ils se remettent en marche. Au bout d'un moment, Lian se rend compte qu'un fragment de phrase lui tourne dans la tête : " la folie de l'immortalité ". La folie des Envoyés divins. Et de Ktulhudar, alors, puisqu'il en était un aussi. En quoi était-ce une folie ? Comment la terrible guerre déclenchée par Ktulhudar a-t-elle pu en guérir les Envoyés ? Mais pas Ktulhudar lui-même, apparemment, puisqu'il a décidé de " marcher pour toujours sur la terre des humains ". Sous sa forme divine ? humaine ? Cela n'a pas de sens. Quelle était la vérité de Ktulhudar, pour Ktulhudar ?
*
« Tu devrais les écrire, tes histoires », dit Suzane à Liam, le jour où il taille avec elle leurs initiales dans le tronc de l'arbre-roi, sur la terrasse ; elle lui a expliqué que c'est une coutume des amoureux : les arbres-rois durent près de trois cents saisons ; un large anneau de cœurs et de lettres entoure en effet la circonférence de l'arbre, certains si anciens que l'écorce s'est presque refermée sur eux. Une impulsion lui a fait raconter alors à Suzane une variante de la création des humains, où pour protéger l'imprudente créature aquatique, la désormais omniprésente Magicienne fait pousser dans sa main l'arbre salvateur – qui n'est pas un arbre-à-eau dans cette version archaïque, mais un arbre-roi.
« Pourquoi les écrire ?
— Pour que d'autres en profitent, voyons ! Et puis les histoires pour enfants sont toujours en demande dans les communes. Ce serait un appoint intéressant pour nous si tu les publiais. »
Avec tous ces arguments, Liam se laisse convaincre. Il se fait une liste des histoires qui lui reviennent à l'esprit, choisit les plus aisément transformables. Et au début, c'est un plaisir inouï, qu'il n'essaie pas d'examiner. Simplement, il se rappelle et il transpose ; quelquefois, le déguisement est trop bizarre, il rit tout bas, et si Suzane est là, elle vient déposer un baiser sur le sommet de son crâne – elle ne l'a jamais entendu rire auparavant.
Il écrit. Toutes les histoires qu'il a racontées aux enfants, et le conte de Lilinilou et de Patirivi le Gardien du Vent, et d'autres aventures de Pit-Pit au chapeau vert, et comment la Magicienne a créé le premier arbre-à-eau, et la grande quête de Mata et de Gata, partis à la recherche de la Fleur magique volée par le Sorcier Écorché.
Des histoires.
14
Ils quittent le relais après le repas, vers la seizième heure, sans avoir revu Kilasi ; Odatan dit au revoir à Nagal-Rasser, qui repart de son côté ; tous deux montés sur les aski, qui feront le voyage avec eux jusqu'à Hleïtzer, Lian et le petit homme s'en vont prendre le bateau à vapeur qui monte d'écluse en écluse jusqu'au plateau d'Ellnitzer, à l'extrême pointe sud du Grand Lac, un peu plus de mille langhi par les canaux. Le vapeur transporte plus de fret que de passagers et ne se traîne guère à plus de seize langhi à l'heure, une allure qui semble au début presque vertigineuse pour Lian accoudé au bastingage. On s'arrête deux fois par jour, afin de décharger et charger du cargo, débarquer et embarquer des passagers, et refaire le plein de gaz. Le petit homme tombe gilet et chemise au premier arrêt pour mettre la main à la pâte. « Eh, qu'est-ce que vous faites ! s'interpose Lian, vous ne devriez pas... » Puis il se rend compte avec un temps de retard que l'autre ne porte plus son bandage au bras.
La blessure ne saigne pas, ne suinte même pas. Elle est, en fait, déjà cicatrisée – une mince et longue ligne rougeâtre bien nette sur la peau brune.
Odatan suit le regard de Lian, semble un instant se figer puis, d'un ton comme résigné : « Je guéris vite. » Il attrape une grosse caisse : « Un coup de main, Lian ? »
Lian obtempère, abasourdi. Le petit homme tient l'arrière de la caisse et dirige les opérations. Une fois revenu sur le pont, Lian retrouve sa voix : « Kilasi a dit... que vous êtes plus vieux que vous n'en avez l'air.
— Kilasi, hein ? fait Odatan entre ses dents. Elle a toujours eu tendance à trop en dire. »
Lian hésite, se lance : « Vous êtes le dernier passeur. »
Odatan assure sa prise sur les poignées d'une autre caisse. « Non. »
Lian médusé le contemple. « Il n'en passe plus depuis...
— Quatre-vingts saisons », achève l'autre à sa place. Puis avec un petit sourire sans joie : « Je suis plus vieux que je n'en ai l'air. »
Il assure sa prise sur les poignées de la caisse, Lian tourne le dos et en fait autant de l'autre côté, à tâtons. Pas d'inflexion descendante, pourtant, dans la dernière phrase du petit homme. Mais au diable la politesse de mise entre Ranao ! Odatan est un Virginien et Lian un halatnim, non ?
« C'est parce que vous guérissez vite ? lance Lian par-dessus son épaule. Une autre mutation ? »
La réponse tarde un peu à venir : « Oui.
— Il y en a d'autres comme vous ici ? »
Silence. Ils descendent la passerelle dans le brouhaha des quais. « Pas que je sache.
— De l'Autre Côté ?
Ils déposent la caisse près de la première : « Pas que je sache. »
La voix est un peu enrouée. Lian se retourne, contemple le visage brun trop pâle, les yeux clairs sous la tignasse sombre. Il s'entend dire, comme on proteste : « Mais vous n'êtes pas immortel », ridicule, mais c'est dit, il ne peut pas revenir dessus.
« J'espère bien que non », dit le petit homme avec un sourire en biais, comme s'il le mettait au défi de poursuivre.
Du coup, Lian demande : « Vous connaissez Kilasi depuis longtemps ?
— Depuis sa naissance.
Lian avale sa salive, mais s'entête. « Elle vous a appelé " béra "...
— Une figure de style.
— Vous ne vieillissez pas ?
— Très lentement.
— Mais vous avez quel âge ? » s'exclame Lian, vaincu.
L'humeur joueuse de l'autre s'est déjà effacée. Il s'essuie le front d'un geste distrait. Il est moins poilu que Lian ne l'aurait cru, les bras, les jambes, la poitrine un peu. « Je suis vieux », murmure-t-il, comme pour lui-même. Puis, avec une ombre de sourire : « Mais la jeunesse, la vieillesse, ce sont des choses relatives, n'est-ce pas ? Te rappelles-tu comme tu étais quand tu avais trois ou quatre saisons ? »
Lian fait une grimace. Le petit homme hoche la tête. « Eh bien, moi, quand je me rappelle mes quinze saisons... » Son regard se fait lointain et il termine d'une voix assourdie : «... je me sens vieux aussi. »
Il se redresse : « Mais pas au point de ne pouvoir décharger encore quelques caisses. Et toi, te sens-tu trop jeune ? »
Ils retournent sur le pont du caboteur.
*
Et puis le plaisir que Liam éprouve à déguiser les histoires atteint un sommet, et se retourne de l'autre côté en douleur. Parce qu'il les relit, ces histoires. Parce qu'à force de les relire, de se relire, il perçoit soudain autrement les modifications, les glissements, ce qu'il a dû inventer pour remplir les trous là où il a dû sauter par-dessus un fragment original intransposable. À travers les trous, à travers les métamorphoses, à travers les mensonges, il se rappelle ce qu'il sait, et qu'il ne pourra jamais dire. De plus en plus, sous les ratures, il y a les autres noms, les autres mots qui poussent. Tingalyai, Matal Ughataï, Iptit, Askorch, Pian-Dzaïri... Quelquefois, Liam reste figé de longues minutes, le cœur battant douloureusement, comme ébloui : il voit double.
Une nuit, il ne peut pas dormir, il se lève sans faire de bruit pour ne pas réveiller Suzane et il va écrire, d'un seul jet, comme en transe, le conte des Collines de la Mer. La Mer s'était ramassée, comme un chat fait le gros dos, pendant qu'on construisait les digues qui en protégeraient l'intérieur des terres d'Atyrkelsaõ ; et l'on voyait son moutonnement bleu à l'horizon, comme de hautes collines, comme des plateaux, comme des montagnes. Et une hardie navigatrice nommée Ramaïel décida un jour d'y aller naviguer. Une fois arrivée dans les Collines de la Mer, autour et sur les Collines de la Mer, car son navire pouvait escalader les pentes bleues, elle découvrit tout un monde, des plantes et des animaux et d'autres créatures étranges. Et elle en invita plusieurs à bord de son vaisseau, émerveillée, pour les montrer à son retour. En cours de route, les créatures riaient et devisaient avec Ramaïel et ses marins, mais quand le bateau arriva au rivage, elles se défirent, telle l'eau d'une fontaine lorsque le jet en est coupé, pour retourner en ruisselets bleus jusqu'à leur source, car c'étaient des créatures de la Mer : elle s'était divertie à les créer comme elle l'avait fait au temps jadis lorsqu'elle était arrivée pour la première fois sur Tyranaël, mais elle ne désirait pas les voir vivre sur Atyrkelsaõ.
Et il contemple la page, les yeux brûlants, ce n'est pas un conte, il sait que ce n'est pas un conte, la Mer s'est vraiment retirée loin de tous les rivages du continent, pour ne pas envahir l'intérieur des terres pendant qu'on construisait les digues de l'Autre Côté, les digues d'Atyrkelsaõ, mais il ne peut pas écrire cette histoire-là, il ne peut pas la raconter, même pas en mentant, rien sur la Mer, pas ici, il est de l'Autre Côté, du mauvais côté ! Il rature d'abord " Atyrkelsaõ ", puis " Tyranaël ", puis avec une énergie furieuse, la poitrine broyée dans un étau, il rature tout à grands traits, si fort qu'il passe au travers du papier.
Le grattement de son stylo réveille Suzane, ou peut-être n'a-t-il pu retenir un gémissement, il entend la voix endormie de la jeune fille qui dit son nom, « Liam ? » Il a envie de lui crier « Lian, je m'appelle Lian ! », mais lui dit n'importe quoi d'une voix contrainte, « Dors, tout va bien », et ce n'est pas vrai, il ne savait pas qu'il avait ouvert une porte, non, pas une porte, une digue, la digue a sauté, et les souvenirs se précipitent, il est débordé, il coule, il se noie.
15
Ils arrivent à Hleïtzer vers la vingt et unième heure. Mille deux cent cinquante langhi depuis Ellnitzer, trente et unième jour de voyage depuis le Leïtnialen ; ils ont passé l'équateur dans le premier tiers du voyage sur le lac : on a sonné la corne de brume, le cuisinier a distribué des biscuits spéciaux en forme de sablier, et l'on est resté tard dans la nuit sur le pont à faire de la musique et à raconter des histoires de fantômes ; Lian n'a pas vu le rapport, mais il a écouté les histoires avec intérêt. Il essaie de se faire à l'idée que c'est l'été de l'hémisphère nord. À cette altitude et à huit cents langhi seulement au nord de l'équateur, à vrai dire, les différences ne sont pas énormes. Le port est impressionnant, plus de trois langhi de large, rempli de bateaux de toutes tailles, péniches, vapeurs, voiliers de plaisance, barques et jonques chargées à ras bord de produits hétéroclites. La ville étage ses anneaux concentriques – canaux, zones vertes, quartiers habités – sur une série de plateaux en escaliers, jusqu'à la montagne au pied de laquelle Odatan désigne un immense édifice rouge et doré entouré de verdure, dans l'axe de la voie principale qui monte à l'assaut des plateaux et se divise en deux pour le contourner : « Le Haëkelliaõ. »
Le petit homme est resté très silencieux tandis qu'ils abordaient, les yeux fixés sur le panorama de la cité. Il ne dit rien non plus tandis qu'ils gravissent les plateaux pour se rendre à leur destination. On leur a encore proposé dana-dana et masdana dans le port, mais Odatan a décliné avec un petit sourire : « Trop de côtes ! » ; il y en a en assez grand nombre dans les rues, pourtant. Mais Lian se concentre sur l'endroit où son asker va poser les pattes, sur le dos de son guide dans ses habits de voyage un peu fripés, juste en avant de lui, pour ne pas se laisser engloutir ; la foule est relativement clairsemée pourtant en ce début de soirée, et l'on y parle, comme partout en public, à mi-voix ; mais cette rumeur même, ces mouvements colorés, ces odeurs inhabituelles... Lian se surprend à répéter intérieurement quelques litanies de satlàn.
Le grand calme du Haëkelliaõ est un soulagement : presque tout le monde y est en train de dîner. Une hékel en courte tunique bleue les salue et les débarrasse de leurs aski à l'entrée. À travers la vaste cour carrée presque déserte, où alternent pelouses d'un jaune éclatant et aires dallées d'écarlate autour de l'immense bassin ovale ombragé de plusieurs tingalyai, deux silhouettes s'avancent vers eux, une fille rousse et dorée qui évoque aussitôt Miniaz pour Lian, un grand garçon mince et brun aux yeux noirs.
« Kyrin et Argelos vont s'occuper de toi, Lian, dit Odatan. Tout ira bien. » Et il amorce un mouvement pour faire demi-tour.
Lian ne peut s'empêcher de s'exclamer : « Vous ne restez pas là ? »
Le petit homme a un sourire presque embarrassé : « Je n'habite pas au Haëkelliaõ quand je viens à Hleïtzer. Je vous rendrai visite de temps à autre. Tu n'as pas besoin de moi. Tout ira bien, tu verras. » Et il disparaît d'un pas pressé sous la grande voûte d'entrée.
Ils emmènent Lian dans sa chambre, proche des leurs, y dînent avec lui en lui expliquant à deux voix les us et coutumes du Haëkelliaõ, en lui parlant d'eux, en l'amenant à parler de lui, de son voyage, de ses parents, du plateau – mais pas d'Odatan, et une obscure réticence l'empêche d'aborder lui-même le sujet. Au cours de la conversation, il apprend qu'un autre naïstaos – ils énoncent le mot sans affectation, c'est rafraîchissant – réside dans une chambre voisine. Tomas Liéouélyn. Le seul autre au Haëkelliaõ. Lian est surpris, mais pas par le nom étrange, ni même par l'absence du garçon. Odatan lui avait dit qu'ils n'étaient pas nombreux, mais... « Il y en a dans les autres Haëkelliani, à Markhalion, Ansaalion, Krisaõ et Hébuzer. Pas plus d'une dizaine en tout, à peu près un pour environ trois mille halatnim », dit Kyrin. Et Argelos remarque, pensif : « C'est fascinant de vivre en un temps où se développent de nouveaux dons. »
Kyrin Melamnaz Alianrad, Argelos Karghad Tandiu. Dans leur seizième saison comme Lian, ils vivent au Haëkelliaõ depuis trois mois déjà, en prévision aussi de l'eïldaràn, mais sans grande incertitude : tous deux danvérani déclarés, ils attendent simplement de savoir s'ils seront hékel, et s'y préparent. Les ancêtres hébao de Kyrin vivaient sur le Landaïeïtan, de l'Autre Côté, raconte Lian dans la première lettre qu'il enverra à ses parents, la semaine suivante. Argelos descend de Paalani installés au nord du Hleïtan au moment de la Conquête. Kyrin est très décidée, très pratique. Argelos ne dit pas grand-chose, mais toujours quand il faut. Beau aussi, à sa façon. Peut-être parce qu'il est si doux, si paisible. Tout en angles, pourtant, le visage taillé plutôt raide comme tous les Paalani, avec des yeux très noirs, comme des puits, mais étincelants.
Des deux, il ne sait lequel l'attire le plus. Le souvenir de Miniaz le pousse vers Kyrin et ses rondeurs accueillantes, mais la paix lumineuse d'Argelos est tout aussi puissante. Thomas, c'est très différent. Lian a du mal à en parler à ses parents. En leur écrivant, il se rend compte qu'il ignore ce qu'ils savent, si ses découvertes au Haëkelliaõ et dans Hleïtzer en sont pour eux – si elles font partie de ce qu'ils lui ont caché – et sa plume devient maladroite.
Thomas vient le voir, ce premier soir, quand il est seul ; il a dû guetter le départ des deux autres. Il entre, les sourcils froncés, dit d'un ton abrupt : « Tu es Lian Flaherty », en virginien, et c'est à peine une question.
Il est de la même taille que Lian, et roux – mais un roux comme Lian n'en a jamais vu, pas rouge comme Miniaz : orange, comme certaines variétés de sarsinit. Et ses cheveux semblent avoir déteint sur ses sourcils et sur toute sa peau, très claire, constellée de petites taches orange pâle aussi. Jusqu'à ses yeux, qui sont d'un riche marron roux. Lian le dévisage, fasciné. Les sourcils, le nez, les lèvres, sont bien plus virginiens que ranao.
Lian se reprend, un peu embarrassé de son indiscrétion, rectifie en virginien également : « Je suis Lian Dougallad Laraïnu.
— Oui, dit l'autre avec impatience, mais ton père s'appelait Dougall Flaherty. Je suis Tomas Liéouélyn. »
C'est ainsi que Lian l'entend, il apprendra plus tard que ça s'écrit " Thomas Llewellyn " ; Thomas est très jaloux de l'orthographe virginienne de ses noms. Pour l'instant, le garçon aux cheveux orange (comme Lian se le décrira souvent au début) fait des yeux le tour de la pièce, voit les restes du repas de Lian, dit : « Alors, ils t'ont fait le numéro d'introduction ? » Les inflexions, autant que le vocabulaire, indiquent en virginien de l'ironie, un doute jeté sur la sincérité de Kyrin et d'Argelos. Lian, un peu choqué, réplique avec calme en setlâd : « Ils m'ont accueilli. » Puis, se rappelant soudain ce qu'Odatan lui a dit des halatnim qui refusent de parler la langue rani, il traduit en virginien : « Ils m'ont accueilli comme un des leurs, et je n'ai aucune raison de douter de leur gentillesse. »
L'autre hésite et reprend en setlâd, comme embarrassé : « Non, je voulais dire... Ils ont fait pareil pour moi. Ce sont eux qui vont s'occuper de nous. On travaillera ensemble, tous les quatre. » Lian comprend alors que Thomas n'est pas hostile, mais aussi incertain que lui : il l'exprime autrement, voilà tout. Il lui sourit : « Ça devrait être intéressant », dit-il en virginien – et dans le bon registre, moitié sérieux moitié plaisant, car Thomas lui sourit en retour.
Thomas a été élevé par sa famille virginienne. Il y a des purs Virginiens, sur Atyrkelsaõ, qui ne sont pas des passeurs. On les appelle des Keldarao. Ils ne sont pas très nombreux, environ six mille, concentrés surtout dans le sud-est du continent et dans les îles d'Aalpaalu. « Chez les Chasseurs ? » Non, à l'écart. Au début, Lian ne peut que spéculer sur leurs raisons de ne vouloir se croiser qu'entre eux ; un peu plus tard, quand Thomas l'aura emmené chez le vieux Raül pour assister aux réunions du " club ", il comprendra mieux. La mère de Thomas était malgré tout une halatnim, de troisième génération, son père par le sang un pur Keldaran ; ce n'est pas la situation de Lian, mais il n'osera quand même pas en parler à ses parents dans sa première lettre. Les parents de Thomas sont morts dans un accident quand il avait six saisons, une terrible tragédie. Il avait vécu jusqu'alors avec eux parmi la famille halatnim de sa mère ; sa famille virginienne, où l'on est danvéràn de père en fils (en fille aussi, sans doute, mais Thomas ne le dit pas), a demandé sa garde ; la famille de sa mère la lui a accordée. Thomas n'en raconte pas plus, mais Lian voit à travers sa désinvolture, malgré la neutralité des phrases virginiennes – à cause d'elle. Thomas le halatnim, le naïstaos, l'invisible, Thomas le rejeté.
Et si Lian ne raconte pas à ses parents l'origine de la blessure, comment pourrait-il ensuite leur faire comprendre son indulgence envers Thomas – la solitude du garçon aux cheveux orange, son besoin inarticulé d'affection, sa gentillesse maladroite, désarmante, entre deux sautes d'ironie et de colère ; et même, oui, sa violence. Intense, Thomas, imprévisible – dangereux. Et sans expliquer tout cela, comment Lian pourrait-il décrire à ses parents l'attrait que Thomas exerce sur lui ? Il se l'explique mal à lui-même, au reste, après le doux Tarmel. C'est peut-être surtout l'exotisme, cette crinière rousse, cette peau semée de soleil... Thomas n'accepte d'être touché par Lian que dans les jeux, cependant, ou dans des bagarres pour rire qu'il suscite toujours le premier. Il n'admet en lui-même ni tendresse ni désir, même si parfois sa première éducation rani refait surface et s'il se laisse aller un peu, du moins dans le ton de voix plus caressant, dans des regards, dans des sourires. Lian, qui jouit bientôt de la compagnie de Kyrin, sinon de celle d'Argelos, pense " pauvre Thomas " et, en redoublant de gentillesse discrète, il respecte le choix de l'autre, sans comprendre que ce n'est pas un choix. Et Thomas l'aime, avec férocité. Mais Lian ne le sait pas. Thomas non plus.
Et Lian ne parle pas non plus en détail de Thomas dans sa première lettre à ses parents, et plus jamais après, parce que le garçon aux cheveux orange l'emmène bientôt dans des lieux où ils n'aimeraient sans doute pas le voir. Chez Raül Costa, où se rassemble, deux fois par semaine, le Club Virginien de Hleïtzer. Une vingtaine d'hommes et de femmes, des halatnim de quatrième génération, où les traits virginiens sont plus nets que chez Lian. On l'examine avec une certaine suspicion, malgré ses cheveux blonds et ses yeux pâles, jusqu'à ce que Thomas ait expliqué qu'il est un naïstaos comme lui et le présente comme « Lian Flaherty ». À la grande stupeur de Lian, les visages se détendent, le vieux Raül remarque avec un sourire grave, « un descendant du premier capitaine ! », référence qui échappe totalement à Lian, et on l'admet dans les délibérations du Club.
Ce sont plutôt des conférences, des exposés, des discussions sur un thème, avec parfois des conférenciers invités. On parle uniquement en virginien, bien entendu. Et exclusivement de Virginia. À commencer, pour le bénéfice de Lian, par l'histoire héroïque de la colonisation – il apprend alors que, par son père, il est peut-être un descendant lointain du capitaine de la première expédition terrienne sur Virginia. De la Terre, le monde d'où sont pourtant venus les Virginiens, on ne dit presque rien : un ancien ennemi, vaincu après de terribles sacrifices, écrasé, disparu, bon débarras. On s'étend plutôt tout à loisir sur les grandes réalisations de la technologie virginienne en réponse au " défi unique de Virginia ", l'usage universel du gaz, la vitesse et la puissance de milliers d'ingénieuses machines mues par l'électricité quand la Mer n'est pas là, et même quand elle est là, au-dessus des mille mètres fatidiques où s'exerce son influence neutralisante.
Lian a entrevu tout cela dans le dictionnaire de Kéryan et au détour de certains textes qu'elle a utilisés pour lui apprendre le virginien : ce qu'il entend surtout, au début, ce sont des histoires. Exotiques aussi, fascinantes, parfois inquiétantes (quand il essaie par exemple d'imaginer Hleïtzer-de-l'autre-côté – Morgorod – remplie d'automobiles, ou encore sa zone industrielle). Mais il entend aussi autre chose. Selon les membres du Club, et même s'ils ne le disent pas en clair, les Ranao sont une culture en stagnation, incapable de changer, condamnée à court ou à moyen terme. Lian se rappelle Laraï, Nathénèk, et les Olaïlliu : l'existence des darnao, les enfants du devenir et du changement, ne contredit-elle pas cette assertion ? On écarte son objection d'un revers de main : les darnao sont aussi conservateurs que les dânao, ils veulent garder Atyrkelsaõ intacte ! Et ils ont tous fait l'union sacrée pour rejeter les améliorations techniques suggérées par les passeurs et leurs descendants. Oh, les idées nouvelles, les Ranao les acceptent assez volontiers : pour les collectionner, comme si c'étaient des papillons ! Comme si c'étaient... des curiosités. Sans vouloir les mettre à l'épreuve, sans jamais rien en tirer de concret.
— Il y avait un piano, là où j'ai appris le virginien », insiste Lian, pour une fois qu'il peut contribuer à la conversation.
Lartigues émet un reniflement dédaigneux : « Oh, oui, bien sûr, un peu de musique, de poésie, ce qui ne risque pas de déranger l'ordre établi. Mais ça ne les intéresse pas vraiment. Une autre technologie, une autre science à côté d'eux, et ça ne les intéresse pas ! Aucune véritable curiosité, aucun sens pratique !
— Leurs moteurs à vapeur, ils en font tout un plat, s'exclame Lee Ségura, le trésorier du Club, mais enfin, ça a été inventé bien avant le moteur à gaz, c'est de la préhistoire, le moteur à vapeur ! Et qu'est-ce que ça fait si les moteurs à gaz naturel sont un peu polluants ? On se déplace plus vite en automobile, c'est drôlement plus pratique ! »
Costa renchérit : « Ah, ça, ce n'est pas comme leurs charrois – " charrois " ! Pourraient pas les appeler " camions " ou " autobus ", oh non ! Bien caractéristique, ça, " charroi " ! Bon, un moteur à gaz, ça pétarade un peu, mais suffit d'un bon silencieux pour régler le problème ! C'est moderne, ça va vite, c'est puissant, qu'est-ce qu'il leur faut de plus ? Pourquoi ils s'accrochent à leurs vieux trucs primitifs ?
— C'est comme pour l'électricité », reprend Lartigues – visiblement, le numéro leur est familier. « Ça ne serait quand même pas sorcier de s'équiper d'un barrage ou deux, ou d'une centrale thermique, histoire d'avoir l'électricité quand la Mer n'est pas là, hein ? Mais non, le progrès, connais pas. " À quoi ça sert ? ", toujours la même rengaine. " Est-ce qu'on en a besoin ? " À quoi ça sert, l'électricité ? À faire fonctionner les frigos. À quoi ça sert, les frigos ? À conserver les aliments durant la saison chaude. Et là : " On a déjà les chambres froides souterraines, alors à quoi ça sert ? " À croire qu'il n'y a jamais rien à améliorer chez eux, que tout est parfait ici !
— Et c'est pas comme s'ils nous laissaient développer nos propres affaires, reprend le vieux Costa, non, monsieur. Pas question d'installer une ligne de chemin de fer dans le sud-est ! Où est notre liberté, là-dedans ? Hein ? Où sont nos droits ? »
Lian écoute l'échange sans oser intervenir avec les " oui, mais " qui se pressent sur ses lèvres : d'après ce qu'il en a vu, les moteurs à vapeur ranao ont un rendement égal, ils sont moins compliqués à produire et à entretenir, le biogaz qui les alimente ne nécessite pas de zone industrielle, par exemple ! Et les Ranao savent ce qu'est l'électricité : ils ont choisi, après l'arrivée de la Mer, de ne plus l'utiliser du tout, ce qui lui semble également rationnel. Et les chemins de fer, ils en ont eu, on en voit dans les fresques datant de l'Unification, mais ils ont finalement décidé aussi que non, ils n'en voulaient plus.
Les membres du Club ont quand même un peu raison ; Lian comprend le choix rani des transports communaux, mais si Odatan avait eu une automobile, à gaz ou pas, le voyage aurait duré moins longtemps, au moins à partir de la passe Arpaïel ; il serait parti plus tard du plateau, et surtout il n'aurait pas dû subir les regards de tous ces gens, en route – pas de la véritable curiosité, comme le dit Lartigues, mais au mieux de la pitié, une forme comme une autre de rejet. Et l'agressivité du Paalao, au relais Dnaõzer.
Entre deux exposés sur les merveilles technologiques ou culturelles de Virginia, on décrit l'excellence et la justice de son système politique, la fameuse démocratie. Lian ne connaît pas assez bien le système rani pour saisir ce qu'ils ne disent pas ainsi – il sait simplement que, sur le plateau, la hékel Kéryan est chargée de transmettre à l'extérieur les besoins et les désirs des gardiens de la réserve, et réciproquement ; il ne voit pas très bien la différence – excepté sans doute le fait que Kéryan n'ait pas été élue mais simplement choisie par les gardiens du plateau ; et le mode de consultation et de communication, puisque c'est une danvéràn. Mais il sent une histoire en creux, le fil d'une critique implicite des Ranao, encore.
« S'ils sont si mécontents du continent, pourquoi ne vont-ils pas vivre chez les Chasseurs ? » finit-il pourtant par dire à Thomas, après une de ces séances de récriminations.
« Ils sont vieux, dit Thomas avec dédain. Ils aiment leur confort. C'est plus facile de parler. » Son regard se durcit. « Mais chez les Chasseurs, on ne parle pas pour ne rien dire. Rani, halatnim, Virginien, peu importe où on est né, ce qu'on est. Ce qui compte, c'est ce qu'on fait, pas les pouvoirs. Des hommes libres, les Chasseurs. Des hommes fiers.
— Argelos dit... – Lian regrette d'avoir évoqué Argelos, Thomas a déjà haussé les épaules, mais il s'obstine –... que ce sont des gens qui n'arrivent pas à trouver leur place, à faire leur paix avec ce qui doit être...
— Bien sûr, la ligne du parti ! » s'exclame Thomas avec une ironie furieuse puis, voyant que Lian ne comprend pas : « Le discours officiel. Ce que ça veut vraiment dire, c'est que les Chasseurs dérangent l'ordre établi. Ils critiquent, ils questionnent, ils refusent, ils ont le courage de leurs opinions, ils n'hésitent pas à prendre leurs affaires en main... Pourquoi crois-tu qu'il y a tellement de halatnim là-bas ? On s'arrange pour qu'ils s'exilent, en espérant qu'il leur arrivera un accident. La vie est dure, dans les îles. Mais eux, les vertueux Ranao, ils n'ont porté la main sur personne, n'est-ce pas, ils peuvent continuer à dire " Il ne faut pas tuer " ! » Il hausse de nouveau les épaules ; « Qu'est-ce qu'il te dirait d'autre, Argelos, c'est un futur hékel ! »
Et en toute honnêteté, Lian est obligé de reconnaître qu'on ne lui a jamais rien dit qui contredise formellement cette interprétation. L'idée le frappe soudain : ce que dit Thomas des Chasseurs, ce que les membres du Club disent sans le dire des Ranao, c'est une interprétation : une autre façon de voir l'histoire. Leur vérité à eux. Et comme il devine d'où parle Thomas, quelle peine secrète nourrit les fureurs du garçon aux cheveux orange, comme l'humiliation secrète des halatnim descendants de passeurs, maintenant presque dépourvus de pouvoirs et de surcroît parfois naïstoï, il fait taire ses objections.
Un jour, l'une des membres du Club, Ferdina Marquis, arrive très irritée à une réunion. Elle est depuis longtemps en dispute avec sa voisine, une Rani : les arbres de la terrasse de celle-ci empiètent sur sa propre terrasse, et le hékel consulté a déclaré que ces arbres n'avaient pas à être coupés, ni même élagués.
« " On ne se dispute pas pour des arbres ", il a dit, avec cet air qu'ils ont, vous savez, d'en savoir toujours plus que vous ! Si ces arbres me font de l'ombre et que je veux du soleil, c'est quand même bien mon droit ! Ça ne les tuerait pas, ces arbres, d'être élagués ! Ça ne la tuerait pas, cette vieille chouette de Kitlaz ! Mais non, il a fallu qu'elle mette un maudit hékel dans le circuit ! Comme si on avait besoin de trente-six mille pouvoirs pour régler un pur problème technique ! Cette maison est à moi, c'est ma terrasse, et je ne veux pas de son ombre à elle sur ma terrasse à moi, c'est quand même simple ! »
Lian, un peu éberlué par l'usage répété du possessif, voit ses compagnons hocher gravement la tête. « Ça n'arriverait pas s'il y avait des lois claires et précises, au lieu de ces interminables médiations, grommelle Lartigues.
— Ça n'arriverait surtout pas s'ils ne croyaient pas les hékel indispensables ! lance Ferdina. Ah, ils l'ont, la combine, les hékel, on ne peut pas dire. Eux et leurs fameux pouvoirs. Les danvérani, je vous dis, c'est les pires !
— Tout serait plus simple s'il n'y avait pas de pouvoirs, voilà tout, lance Thomas, les sourcils froncés. Comme chez les Chasseurs. Et comme de l'Autre Côté. »
Tout le monde acquiesce avec conviction, et Lian ouvre de grands yeux : « Les passeurs viennent de l'Autre Côté ! proteste-t-il.
— Ils venaient, dit Costa, l'index dressé.
— Ils ne viennent plus », renchérit Lartigues ; et, déjà prêt à s'irriter : « Tu n'étais pas au courant ? Ils ne te l'ont même pas dit ?
— Si, mais...
— Il n'en vient plus parce qu'il n'y en a plus, explique Thomas avec une certaine satisfaction – il aime combler les lacunes de Lian. « Il n'y a plus de mutants de l'Autre Côté. Les normaux ont gagné la guerre. »
C'est la première fois que Lian entend le terme " mutants " utilisé au Club, où l'on ne parle que très rarement des pouvoirs. Mais surtout, les normaux. Voilà un terme que Lian n'avait jamais entendu utiliser ainsi. L'implication est curieuse : les mutants ne sont donc pas normaux ? Lian a toujours employé " mutants " de façon descriptive, comme il dirait " chaud ", ou " vert " : un élément naturel, relevant du " c'est ainsi ". En couple avec " normaux ", cependant, le terme prend une autre résonance.
Et les normaux ont gagné la guerre. Contre les mutants. Kéryan l'avait bien dit, au relais, que les Virginiens se faisaient la guerre entre eux...
« Ils ne lui ont rien dit ! » conclut Lartigues, se méprenant sur son silence et prenant les autres à témoin de cette nouvelle duplicité des Ranao.
Lian proteste par acquit de conscience : « Ils m'ont dit qu'il y avait une guerre, mais qu'on ne savait pas où elle en est maintenant que les passeurs ne passent plus.
— Pardi ! s'exclame Costa.
— Les passeurs ne passent plus parce qu'il n'y a plus de passeurs, répète Thomas avec une sombre allégresse. Plus de mutants de l'Autre Côté. Ils ont fini par disparaître. Juste des gens comme nous, maintenant. Normaux. »
Lian observe ses compagnons à la dérobée : ils hochent tous la tête. " Normaux " ? D'après ce qu'Odatan lui a laissé entendre des halatnim de dernière génération, ce sont en général au moins des petits lâdzani. Il n'a pas posé la question, cependant. Des mutants, en tout cas, si limitées soient leurs capacités. Et Thomas, le pauvre Thomas, naïstaos fils et petit-fils de danvéràn et si farouchement anti-pouvoirs, est un mutant aussi – même si sa mutation fait de lui comme de Lian, ma foi, c'est vrai, un " normal " au sens où ils l'entendent : totalement dépourvu de pouvoirs. Est-ce donc la raison pour laquelle on les a acceptés dans le Club avec tant d'enthousiasme ? En quoi est-ce si souhaitable, de ne pas avoir ce que possède tout le monde ?
Ah mais, si personne ne l'a... Et d'après ce qu'ils disent, de Virginia, tout le monde y serait ainsi, sans pouvoirs. Lian songe soudain à la brève histoire contée par Kéryan, les villageois qui croyaient voir le soleil pondu à neuf par les montagnes tous les matins et la voyageuse dissidente guérie par eux de sa " folie "... Son sourire involontaire s'efface : les membres du Club Virginien, et Thomas, ne sont pas fous, bien sûr. Mais si ce qu'ils disent à propos des passeurs et de la guerre est vrai, ils sont simplement... du mauvais côté des montagnes. Pour toujours.
Et la pensée le traverse, inattendue, douloureuse, avant qu'il ait pu la réprimer : et moi aussi, si l'eïldaràn ne donne rien.
*
Lian ne touche plus aux contes. C'est Suzane, déconcertée, qui prend l'initiative de les taper à la machine et d'envoyer le manuscrit au bureau culturel du Conseil de Ville. Lui, une nostalgie déchirante l'envahit maintenant, à tout moment, irrésistible, physique. Il regarde une mosaïque, une fresque, un arbre, et il reste cloué sur place, aveuglé, sourd à tout ce qui l'entoure.
Une lettre leur arrive bientôt, des félicitations et la permission de publier. Quatre semaines plus tard, Liam reçoit les épreuves. Il s'est un peu calmé, ou bien le fait pour ses histoires d'être imprimées leur confère à nouveau un degré tolérable d'irréalité. Il corrige les épreuves. Les renvoie. Se fait rappeler par retour du courrier que le manuscrit n'a toujours pas de titre. Il se mord les lèvres pour retenir le premier qui lui est venu à l'esprit, Contes de Tyranaël, reste muet, les yeux perdus dans le vague, incapable d'en trouver un autre. Suzane claque des doigts devant son visage, agacée : « Où es-tu, Liam, il lui faut un titre, à ce recueil, reviens parmi nous, veux-tu ? »
Il murmure « Je suis ailleurs », d'une voix si altérée qu'elle comprend seulement le dernier mot, mais s'illumine : « Légendes d'ailleurs ! Ah, oui, parfait ! »
Dans le courant de Mai, Légendes d'ailleurs est publié par le Pic, les Presses Intercommunales de Bird-City, dans une jolie édition cartonnée de format poche ; on a choisi de représenter sur la couverture Katrina et Patriki avec le tigre et l'aigle – trop de monde pour cet espace restreint, et les deux animaux sont bien raides et proportionnellement trop petits, mais peu importe, tout le monde s'accorde à trouver le livre magnifique, on organise une fête à la commune pour sa sortie officielle. Dans les semaines suivantes, on trouve quelques commentaires dans les bulletins des communes de Bird qui l'ont acheté, souvent élogieux sur le style mais parfois plus réservés sur le contenu : trop bizarre, ou trop compliqué, ou pas assez réaliste.
Suzane se hérisse à la place de Lian, s'agace de son absence de réaction. Le livre, dans l'ensemble et malgré les réserves de certains adultes, est un succès auprès des plus jeunes, on s'en rend compte lorsqu'on reçoit les premiers chèques du Pic, au début de Juin. Suzane voudrait bien que Lian en écrive un autre – pas d'abord pour l'argent supplémentaire qui va dans les coffres de la commune, mais parce qu'elle aime ses histoires, elle le lui répète, elle est fière de lui. Il donne des réponses évasives, ou répond à côté, ou ne répond pas. Suzane essaie de comprendre pourquoi il n'est pas plus enthousiaste, amène même le sujet lors d'une séance de réflexion après avoir constaté sa réticence à en discuter avec elle. « Ce n'est pas comme un quota de conserves », finit par dire Jill, volant à la rescousse de Lian qui ne dit rien et regarde ses mains, lointain, abattu. « On ne peut pas le forcer à écrire s'il n'en a plus envie.
— Mais pourquoi n'en a-t-il plus envie ? » proteste Suzane.
Lian devine que les regards se sont tournés vers lui, hausse les épaules et ment : « Je n'ai plus d'idées d'histoires. »
Ce n'est pas vraiment un mensonge. Ce qu'il voudrait raconter, ce ne sont pas des histoires, et il ne le peut pas.
Est-ce que c'est un mensonge, quand on ne peut pas dire ce qu'on veut dire ?
Cette nuit-là, quand Suzane le rejoint chez lui et devient tendre, peut-être pour essayer encore de le convaincre ensuite, ou pour lui faire comprendre qu'elle a accepté la situation – il ne sait, comment saurait-il ? – il n'arrive pas à lui répondre comme elle le désire. Il regarde sa blondeur rousse, il la caresse avec une tendresse désespérée, il se laisse caresser, mais il pense à Miniaz, à Kyrin, et avec elles, dans l'élan irrésistible des souvenirs, s'en viennent tous les autres, ceux qu'il se rappelle et ceux qu'il voudrait ne pas se rappeler. Suzane est déçue, inquiète, blessée. Un moment, elle essaie de parler, de le faire parler, mais que pourrait-il lui dire ? Il étouffe. Elle reste un instant assise au bord du lit, soupire, « Mieux vaut dormir, alors », et retourne dans sa chambre.
Lian reste couché sur le dos, les yeux ouverts dans la pénombre, tandis que la sueur sèche sur sa peau en picotant. Laraï, Nathénèk, Odatan, Dougall. Thomas. Imprononçables à jamais, les paroles qui n'ont pas été prononcées. Il ne savait pas. Il sait maintenant. Trop tard. Perdu, perdu pour toujours, l'autre côté.
Au bout d'un moment il se lève, il va à tâtons au petit bureau, allume la lampe. Sur une page blanche, il écrit, d'un trait : Lorsqu'il était petit, il avait vu mourir un arbre-Gomphal. Et il ne peut pas aller plus loin. Il déchire la feuille en menus morceaux, il la jette dans la corbeille, une neige de particules légères, et il reste assis là, mains à l'abandon sur la table, les yeux fixés sur la lueur un peu vacillante du gaz.
16
Au bout de la quatrième semaine de son séjour au Haëkelliaõ, Lian est amoureux, terriblement amoureux de Kyrin. Il ne sait pas comment c'est venu. Au début, la première fois qu'ils ont fait l'amour ensemble, il a bien compris qu'elle était autant poussée par la curiosité que par une affection réelle, mais sans passion, et il n'en a pas été dérangé. Il était prêt à une relation détendue, amicale, complice, comme avec Miniaz. Et puis, elle a refusé de poursuivre. Elle ne le désirait pas, elle l'a dit clairement, Lian n'aurait même pas pensé à insister. Mais quelque chose est arrivé, il ne sait quoi, il ne s'est pas rendu compte : peu à peu la présence de Kyrin, son contact, même simplement amical, lui sont devenus aussi nécessaires que de respirer.
Peut-être parce que Kyrin est toujours avec Argelos. Ils ne sont pas amants, pourtant – Kyrin dit " pas encore ", d'une manière qui plonge Lian dans des abîmes de perplexité chagrine. Mais ils sont ensemble : unis par le lien invisible, incompréhensible, des danvérani. Ce n'est ni dans leurs paroles ni dans leurs silences – devant Lian et Thomas, ils échangent toujours à haute voix. C'est dans la tête de Lian, dans ce qu'il imagine de leur intimité – il n'a osé leur demander ce qui se passe réellement. Et en vérité, il ne sait qui il envie le plus, Argelos d'être avec Kyrin, ou Kyrin avec Argelos.
Il ne comprend pas très bien au début ce qui se passe entre lui et Argelos. Il l'attribue d'abord, avec espoir, à une attirance réciproque. Il voit bien comme Argelos lui parle, le regarde, s'occupe de lui plus que de Thomas. Un après-midi, dans un des jardins intérieurs où ils se sont réfugiés pour échapper à la chaleur, ils se font sécher côte à côte après s'être baignés dans le petit étang. Kyrin n'est pas là, Thomas est en ville ; Lian contemple le dos d'Argelos qui s'est couché sur le ventre ; des gouttelettes brillantes cascadent le long de ses flancs, d'autres brillent au creux de ses reins. D'un doigt un peu timide, Lian les chasse. Argelos tourne la tête vers lui. Ils se regardent un moment sans rien dire, puis Lian voit le visage d'Argelos s'assombrir ; le garçon brun se redresse sur un coude avec une expression consternée en soufflant : « Oh, Lian... Non. Ce n'est pas ça. »
Lian reste muet, désemparé, atterré aussi d'avoir si mal compris. À la fin, il ne peut s'empêcher de protester à mi-voix : « Pourquoi ? »
Argelos secoue la tête d'un air navré et finit par dire : « C'est ainsi. »
L'expression familière allume en Lian un bref éclair de révolte, mais insister serait impensable. Il passe la main dans l'herbe rase et fine à côté de lui – un bien pâle substitut pour la peau d'Argelos – et murmure : « Y a-t-il quelque chose au monde qui ne soit pas ainsi ? »
Argelos se met à rire : « Oh, Lian, bien sûr !
— On ne peut pas faire un pas ici sans buter sur des " c'est ainsi " ! La Mer, les dons... »
Argelos est redevenu sérieux ; il s'assied, les bras autour des genoux, les sourcils un peu froncés : « C'est vrai, murmure-t-il. Il y a si longtemps que nous vivons avec que nous ne les voyons plus très bien, ou alors uniquement par rapport à nous. » Il sourit, non sans ironie : « On ne peut pas vivre chaque instant de chaque jour avec le mystère, on exploserait ! C'est pour cela que nous prêtons tant de masques à Hananai et qu'elle les accepte : elle sait que sans eux, constamment face à sa lumière sans déguisement, nous deviendrions fous. »
Il contemple Lian avec une intense fascination, et Lian se sent tressaillir d'espoir, malgré tout. Il ne peut pas le regarder ainsi et ne pas...
« Quand les passeurs ont commencé à arriver... », reprend Argelos. Il se détourne et soupire : « Nous nous sommes vus par leurs yeux, ceux d'entre nous qui étaient en contact avec eux, et pour beaucoup le choc a été presque intolérable. Pas du tout comme lorsque les aïlmâdzi avaient vu avec les Étrangers, avant qu'ils n'arrivent sur Tyranaël. Pour les aïlmâdzi, oui, un choc, mais ils sont habitués à voir par d'innombrables yeux, ils connaissent les nombreuses demeures de Hananai. La plupart se sont remis. Et pour tous, et même les hékel qui les avaient aidés à transcrire leurs visions dans les plaques mémorielles, c'étaient des histoires, étranges, parfois effrayantes, mais ni plus ni moins que toutes les autres visions des aïlmâdzi. Un mystère, certes, mais un mystère habituel. La plupart des Ranao ne touchent jamais de plaques mémorielles, les visions font partie... de ce qui est ainsi. On élabore parfois des histoires à partir de ces visions, certaines entrent petit à petit dans notre mémoire, quand nous pouvons les apprivoiser, nous les approprier, d'autres restent à jamais énigmatiques. La plupart des gens s'interrogent rarement sur elles – s'interroge-t-on sur l'air qu'on respire ? »
Argelos est terriblement beau ainsi, mince visage ardent, yeux étincelants, lèvres entrouvertes sur ses dents humides. Lian détourne les yeux, accablé. Il commence à comprendre confusément d'où vient son erreur, la nature de la fascination qu'il exerce sur Argelos. Il conclut à mi-voix : « Mais avec les passeurs, c'était différent. Ils étaient là... directement, sans intermédiaires.
— Oui, soupire Argelos. Et dans la demeure même de Hananai que la Mer nous avait offerte. Ensuite, bien sûr, nous avons replacé dans une perspective différente les étonnements des passeurs, leurs craintes, leurs colères, en comprenant mieux d'où ils venaient, avec quels yeux ils nous voyaient, et voyaient la Mer, et les dons et tout le reste. L'orage s'est apaisé, l'étang est redevenu tranquille... » Il se redresse en se frottant les bras pour en écarter les insectes, son visage se détend, ses yeux brillent d'une lueur plus douce : « Oh, nous n'étions plus comme avant. Notre paysage avait changé, s'était élargi. Nous nous sommes employés à l'explorer. À l'accepter. On ne peut pas faire que les passeurs ne soient pas passés. »
Lian hausse les épaules sans pouvoir déguiser son amertume : « C'est ainsi. »
Argelos a de nouveau tourné la tête vers lui et le regarde, mais Lian ne veut pas se voir dans ses yeux. Halatnim. Naïstaos.
L'autre se doute-t-il de ce qu'il ressent ? Il effleure le bras de Lian. « Les passeurs constituent pour certains d'entre nous un mystère profond, Lian. Presque... effrayant. » Sa voix est triste, mais ferme. « Pas leur don – c'est comme les autres dons, et la Mer, et tout le reste, il fait partie des univers de Hananai et obéit à ses lois. Mais leur existence même... Tu comprends, ils voyageaient entre les demeures de Hananai avec la Mer, non pas comme les aïlmâdzi, en esprit, mais dans leur chair – en elle mais sans être absorbés par elle ! »
Lian lui jette un coup d'œil à la dérobée : Argelos regarde au loin, passionné, fervent, intolérable de beauté. « Ils étaient plus proches d'elle, d'une certaine façon, que ne l'ont jamais été tous ceux qui l'ont rejointe, ou le Communicateur même, qui parle avec son Esprit. Que l'envers de ce don, son autre face, ait été l'impossibilité pour les passeurs de se joindre à la lumière de la Mer... Qu'il se soit, apparemment, perdu, et que sa seule survivance soit justement l'impossibilité de la rejoindre dans leur chair... » Le garçon secoue la tête et conclut à mi-voix : « Les dons font partie pour nous de ce qui est, nous savons d'où ils viennent, nous en avons étudié le fonctionnement, nous les servons de notre mieux. Mais nous ne comprenons toujours pas leur nature profonde, ni la direction de leurs métamorphoses. Et ce don-là... est encore plus mystérieux que celui des aïlmâdzi, pour nous. »
Et moi, a soudain envie de lui crier Lian, est-ce que j'existe en dehors de mon absence de don ? Il se retient, bien sûr. Il ne veut pas se fâcher avec Argelos – Argelos ne se fâcherait pas, il serait simplement consterné, peiné, et que peut-il y faire ? C'est ainsi.
« C'est un peu comme pour les Ékelli », murmure encore Argelos, qui a continué à tourner autour de ses propres questions. « Qu'ils aient pu franchir les portes d'une des demeures mais n'aient jamais pu en repartir, même avec la Mer... », il se tourne vers Lian avec un sourire à la fois ironique et navré : « C'est la limite de ce que je peux supporter comme mystère ! »
Lian fait un effort. Si c'est ce qu'Argelos peut et veut lui donner, la seule façon dont ils peuvent être ensemble, il ne va pas le refuser, n'est-ce pas ? « En quoi est-ce un mystère ? La Mer est à sens unique pour tout le monde, les Ékelli, les Ranao, les passeurs. »
Et il va ajouter, en se forçant à l'humour, " C'est ainsi, non ? ", quand Argelos se retourne vers lui avec vivacité : « Ah, mais pourquoi, puisqu'elle, elle passe, et puisque les aïlmâdzi vont partout avec elle, au moins en esprit ? Lorsque l'Esprit de la Mer nous a proposé de nous installer ici, les baïstoï dans la Mer nous ont simplement dit que ce serait à sens unique. Que même passer dans un seul sens serait difficile – comme naviguer sur la Mer juste avant son départ ou après son retour, mais en infiniment plus pénible encore. Aucune explication. Certains ont compris que c'était à cause de nos limitations, ou des lois de l'univers – que c'était ainsi. D'autres ont pensé que la Mer voulait nous protéger des Étrangers. Et d'autres que c'était la volonté de la Mer pour des raisons qui nous dépassent et qu'elle n'a pas choisi de nous communiquer. Nous avons dû nous en accommoder, comme de la pluie ou du beau temps. C'est devenu ainsi. Mais pendant toutes les années où les premiers ont tout installé pour ceux qui suivraient, et pendant toutes les années où nous avons effectué le passage, les débats n'ont jamais cessé. Tout le monde n'est pas passé. Beaucoup ont choisi de rejoindre la Mer. Certains ne sont venus ici que très tard... et quelques-uns ont même choisi de rester sur Tyranaël. Et jusqu'à ce que nous ayons la preuve que les Étrangers étaient vraiment arrivés, le débat a continué.
— Jusqu'au premier passeur.
— Oh non, bien avant ! Les premiers Étrangers que la Mer a absorbés, au tout début... L'Esprit de la Mer les a protégés tant bien que mal, la plupart étaient devenus fous. Oh, il lui a fallu très, très longtemps pour les guérir et arriver à établir un contact avec eux. »
Malgré le chagrin et la frustration qui lui brûlent la poitrine, Lian ne peut s'empêcher de commencer à être curieux. On ne lui a jamais parlé de ces détails – ses parents ne le voulaient pas, les Olaïlliu le croyaient sans doute au courant ou avaient décidé que l'ensemble du sujet était trop périlleux ; et le Club Virginien... a sa propre ligne du parti. « Vous auriez pu les contacter, non, ceux qui n'étaient pas dans la Mer ? Envoyer des messages ? »
Argelos secoue la tête avec gravité : « La Mer a toujours refusé tous les messages, y compris au temps où nous étions en train de nous installer sur Atyrkelsaõ et où la majorité de notre peuple se trouvait encore sur Tyranaël. C'était très clair depuis le début : un choix sans retour, pour qui que ce soit. »
Il se balance un peu d'avant en arrière, le regard lointain : « Non, le débat a surtout fait rage alors parce que la toute première vision de la première aïlmâdzi à avoir vu avec eux, Eïlai Liannon Klaïdaru, s'était réalisée. Les aïlmâdzi ne savent pas quelle demeure ils visitent : elle peut ressembler à la leur, mais le détail qui l'en sépare est parfois si infime qu'ils ne le remarqueront jamais. » Sa voix a pris une intonation différente, presque anxieuse : « Après Eïlai, tous les aïlmâdzi se sont mis à voir avec les Étrangers, les mêmes visions terribles, guerre, massacres, ravages...
— Mais si on ne sait pas où les visions sont vraies, pourquoi les avoir crues ? s'étonne Lian.
— Parce qu'avant la première arrivée de la Mer, ça avait été pareil : tous les aïlmâdzi l'avaient vue apparaître, et c'est pour cela que nous avions décidé de tout préparer pour l'accueillir. Nous ne pouvions courir ce risque. »
Argelos reste un moment silencieux puis reprend tout bas : « Et elle est arrivée. Mais pas comme les aïlmâdzi l'avaient vu : nous n'avons pas été engloutis. Alors on s'est demandé... Était-ce parce qu'ils avaient vu que ce n'était pas arrivé ? Avaient-ils simplement vu une autre demeure où de toute éternité nous avions été engloutis, et vivions-nous dans la demeure où, de toute éternité, nous avions décidé de nous préparer à cause de leur vision – et n'avions pas été engloutis ? Avions-nous changé de demeure sans le savoir dès l'instant où nous avions décidé de nous préparer ? » Sa voix se fait plus grave encore : « Avons-nous, grâce aux aïlmâdzi, le pouvoir... de changer l'avenir ? Ou bien voient-ils ce qui, de toute éternité, doit être, et n'avons-nous de liberté que celle de faire ce qui doit être ? »
Lian contemple Argelos, accablé ; il se rend bien compte que le garçon est bouleversé, qu'il vient de lui confier une de ses préoccupations les plus profondes – mais tout ce qu'il peut penser en cet instant, c'est : et voilà ce qu'il voit chaque fois qu'il me regarde ! ?
Argelos n'en a pas conscience ; il est trop pris par sa propre angoisse. « On se l'est demandé depuis que les aïlmâdzi existent, tout ça, soupire-t-il enfin, et on n'a jamais trouvé de réponse satisfaisante. Mais la plupart du temps, c'était très... théorique : les visions concernent en général des incidents assez mineurs, parfois importants pour ceux qui les vivent, mais impossibles à vérifier, même rétrospectivement – qui sait quels détails diffèrent, ce qui nous a échappé ? Mais avec les visions de la Mer... et celle des Étrangers... Une fois, bon. Deux fois... D'autant que le premier aïlmâdzi, Oghim, avait vu un Étranger, on pouvait le comprendre à présent, le reconnaître dans sa vision. Une vision si brève... À peine si on pouvait y deviner la Mer – c'est seulement après l'arrivée de la Mer qu'on a compris le décor de la vision d'Oghim. » Argelos est presque amusé, tout d'un coup : « Imagine, on avait voulu croire qu'il s'agissait d'une créature de la Mer, cet homme qui n'était pas un Rani, puisqu'il en sortait ! »
Lian n'imagine rien du tout ; il répète en écho, abasourdi : « Une créature de la Mer ?
— Oh, dit Argelos étonné, tu ne sais pas ? » Peut-être trouve-t-il bienvenue cette occasion de changer un peu de sujet : il raconte longuement à Lian comment, au temps de sa première venue sur Tyranaël, la Mer s'est amusée à créer toute sorte de créatures étranges qu'on pouvait apercevoir, parfois, sur le rivage. Et comment elle l'a fait encore, sur Atyrkelsaõ, pendant qu'on construisait les digues. Lian le dévisage, mais non, Argelos est sérieux. Les Gomphali sont vraiment des créations de la Mer, comme l'a dit Odatan.
Dans le silence qui suit, Argelos revient à ce qui le travaille : « Oghim avait vu l'un des Étrangers d'Eïlai sortir intact de la Mer, murmure-t-il. Eïlai les avait vus absorbés. » Il se tourne de nouveau vers Lian, le regard étincelant : « Imagines-tu le tourbillon de questions que cela soulevait, que cela soulève encore ? Avec le temps, après le Grand Passage, on a un peu oublié. On avait assez à faire de s'accommoder de la nouvelle demeure, de toutes ses différences, de toutes ses ressemblances. La vision d'Oghim est redevenue ce qu'elle avait toujours été pour nous, plus importante pour ce qu'elle signifiait – l'existence future des aïlmâdzi et la nature de leur don – que par son contenu même.
— Jusqu'au jour où le premier passeur est arrivé », dit Lian, avec une sorte d'obscure satisfaction, comme on gratte une plaie.
Argelos hoche la tête : « Une troisième vision peut-être réalisée, dit-il d'une voix un peu altérée. Pour beaucoup d'entre nous, c'était trop. Oghim avait vu cet Étranger, et à cause de cela nous l'avions reconnu dans la vision d'Eïlai, et à cause de cela nous avions décidé d'accepter l'offre de l'Esprit de la Mer et de venir ici... Et parce que nous étions ici nous avions pu accueillir le premier passeur. Tout ce que nous avions fait jusque-là n'était-il donc destiné qu'à réaliser la vision d'Oghim ? Qu'en était-il de notre liberté sous le regard de Hananai ? Vous avez un mot, les Virginiens : " prédestination ". Nous n'en avions pas. Vous dites " c'est écrit ", mais ce n'est pas la même chose que " c'est ainsi ". Nous reconnaissons par là la demeure de Hananai où nous nous trouvons vivre, ses lois, et notre existence à l'intérieur de ces lois. Mais nous savons, depuis que les aïlmâdzi existent, qu'ailleurs ça a été, c'est, ce sera autrement. C'est pour cela qu'existe le mode han'maï, pour pouvoir parler des possibles. Et quelquefois... c'est effrayant. Nous vivons aujourd'hui sur une Tyranaël où notre race ne s'est jamais développée, Lian ! Il y a une Tyranaël où nous avons été engloutis par la Mer, une autre où la venue des Étrangers a causé notre disparition et la leur. Imagine ce que nous avons pensé, lorsque les premiers passeurs ont commencé à arriver ! »
Lian l'imagine fort bien, pour le coup : « Les visions des premiers aïlmâdzi pouvaient se réaliser sur Atyrkelsaõ et non sur Tyranaël.
— Aucun d'eux n'avait jamais vu d'Étrangers sur Atyrkelsaõ indiscutablement reconnaissable comme telle, mais ça ne voulait rien dire, n'est-ce pas ? Il y a tant de ressemblances entre nos deux demeures...
— On a quand même dû se calmer quand on a vu que les passeurs étaient rares, remarque Lian un peu narquois. Et encore plus quand ils ont cessé de passer. »
Argelos l'observe un moment, puis dit avec une légère tristesse : « Non. Quand on a réussi à leur permettre de créer des enfants avec nous, Lian. »
Lian se détourne, à la fois blessé et irrité. Argelos a fait exprès de lui rappeler sa nature ! Mais le garçon brun se penche soudain vers lui, prend son visage entre ses mains et l'oblige à le regarder : « Tu es des nôtres, Lian », dit-il avec une tendresse que Lian ne peut pas ignorer. « Ça n'a absolument rien à voir. Je t'aime. Mais juste pas comme ça. Je n'y peux rien, et toi non plus. »
Lian a accepté – avec mélancolie : c'est ainsi, Argelos lui restera physiquement inaccessible. L'affection inaltérable que lui manifeste le mince garçon aux yeux ardents, sa confiance, lui sont des compensations suffisantes, il se l'affirme, il le croit. Mais c'est comme si le refus de Kyrin, après celui d'Argelos, avait exaspéré en lui des émotions indistinctes qui se croisent, glissent les unes dans les autres, s'inversent, et au bout du compte, il ne sait trop comment, il se retrouve amoureux de Kyrin. Terriblement. Au point de penser à l'eïldaràn avec une terreur et un espoir qu'il n'a jamais éprouvés jusque-là, qui le font trembler. Qui le font pleurer, une nuit, dans les bras de Kyrin. Qui lui font dire, d'une voix brisée : « Est-ce que tu m'aimerais, Kyrin, si je n'étais pas un naïstaos ? » et il l'a dit en setlâd, bien sûr, les termes et l'inflexion qui se traduiraient en virginien, littéralement, par " m'aimerais-tu dans nos enfants ? ", par " serais-tu la mère de nos enfants ? "
Et Kyrin, atterrée mais honnête, et toujours directe, répond : « Mais je t'aime, Lian ! » – elle dit comme Argelos : " je t'aime en toi-même ", ce qu'on dit aux amis – « mais je ne pourrais t'aimer de cette façon. Tu es un halatnim. Tu peux créer des enfants avec d'autres halatnim, mais pas avec moi. Et je désire créer des enfants, Lian. »
Il n'a entendu que le début et la fin de sa phrase. Il murmure, accablé : « Les enfants d'Argelos.
— D'Argelos ou d'un autre, Lian, mais des enfants ranao. »
Il reste un instant pétrifié. Kyrin, c'est Kyrin qui dit ça ? Ils ont raison, alors, au Club, quand ils disent qu'en réalité tous les Ranao sont racistes, que c'est ce que cache leur politesse ? Pourquoi l'a-t-elle accepté dans son lit, alors, la première fois, par... par perversité ? Il se lève, et crache : « Je ne te contaminerai pas plus longtemps de ma présence ! »
Mais la poigne solide de Kyrin le rattrape, et il ne peut pas s'en libérer – à âge égal, elle est aussi grande et forte que lui. « Que fais-tu, Lian ? »
Il bégaye, presque aveuglé de rage douloureuse : « Tu ne veux pas souiller ton sang avec le mien, je comprends très bien ! »
Elle le rassied sur la banquette d'une traction brusque. « Mais qu'est-ce que tu racontes ? s'écrie-t-elle d'un ton horrifié. Ce n'est pas ça ! Je ne veux pas subir les modifications génétiques nécessaires, c'est tout. Si je le faisais, je ne pourrais plus créer d'enfants ensuite avec des hommes de mon peuple, ou alors ce serait tellement dangereux... »
Elle le dévisage d'un air presque suppliant : « Je ne pourrais peut-être plus créer d'enfants avec Argelos, Lian. Ou avec n'importe quel autre Rani que j'aimerais de cette façon. C'est peut-être égoïste de ma part, mais je ne veux pas courir ce risque...
— Quel risque ? ! Les Ranao et les Virginiens sont compatibles ! »
Kyrin reste silencieuse un moment, les yeux agrandis, puis elle murmure : « Tu ne sais pas ? »
Et après la dénégation furieuse et déjà secrètement terrifiée de Lian, elle lui prend les mains, un geste bien inutile entre eux puisqu'il ne sent rien que ses mains, et elle explique, avec gentillesse, avec compassion, mais toujours directe et sans circonlocutions. Les Ranao sont d'origine amphibienne plus récente que les Virginiens. Les mères allaitent leurs enfants, mais ceux-ci se développent en elles sans contact biochimique, dans un œuf souple qui se résorbe après crevaison de la paroi et naissance de l'enfant. Il n'y a donc pas de cordon ombilical, pas d'ombilic sur le ventre des enfants ranao, seulement la cicatrice froncée indiquant la résorption du sac vitellin.
Lian pose machinalement la main sur son ventre, sent son nombril à travers le tissu léger de sa tunique, reste pétrifié.
Une femme virginienne peut créer sans grand problème des enfants avec un père rani, poursuit Kyrin de sa voix douce, impitoyable, une fois effectuées chez elle, avant la conception, les modifications génétiques nécessaires. Mais c'est différent pour une femme rani qui désire avoir l'enfant d'un père virginien, ou même halatnim. Les embryons métis ont besoin de l'apport immunitaire de la mère, leur œuf n'est pas étanche, la mère développe un tissu spongieux qui vient irriguer le sac vitellin. Ce tissu se dégrade, mais le sac vitellin ne peut ni le résorber ni se résorber ; il se trouve expulsé à la naissance, avec l'enfant, qui est donc pourvu de l'équivalent d'un nombril. Mais les modifications génétiques sont irréversibles. L'enfant conçu par une mère rani avec un père rani grandit bien dans son œuf étanche, mais le même tissu spongieux vient encercler celui-ci, causant de graves problèmes de développement et, en dernier ressort, la mort de l'enfant – et parfois de la mère – si l'on ne se livre pas à une intervention chirurgicale délicate et dangereuse, à laquelle de toute façon peu de nouveau-nés survivent.
Lian regarde sans le voir le visage apitoyé de Kyrin. Il se rappelle. Nathénèk et Laraï nus avec lui dans le Leïtnialen, la première fois qu'il a remarqué leurs différences ; Miniaz sur le ponton au milieu du bassin Tarli, lui effleurant le nombril ; le pénis mobile et musclé de Tarmel dans sa main... Lian a le vertige, tandis que se recompose à la vitesse de l'éclair le paysage de sa petite enfance, de son enfance, de toute sa vie jusqu'à présent. L'expression de Laraï, à la table du repas familial, quand elle a dit : « Je ne peux pas créer d'autres enfants, Lian ». C'était elle qu'il regardait, comme toujours, il ne regardait pas Nathénèk, mais quelle expression, alors, sur le visage de Nathénèk ? Quelle histoire, entre eux ? Quelle histoire, en vérité, entre Laraï et Dougall ? Entre Dougall et Nathénèk ? Pourquoi Dougall est-il parti, réellement ? Comment est-il mort, réellement ?
Pourquoi ne lui ont-ils jamais rien dit ?
Mais il peut se rappeler, il peut déchiffrer, maintenant, ses souvenirs. La tension, légère mais toujours présente, entre Laraï et Nathénèk. Ils dormaient dans la même chambre, pourtant. Il ose à peine imaginer, comprendre. La tendresse, entre eux, toujours un peu triste, parfois un peu forcée : les baisers, les caresses au passage, comme des rituels. Comme des exorcismes. Les longues randonnées en forêt, le travail incessant. Mais pourquoi ? Il y a des façons de... Il les a apprises lui-même avec Maïli, sûrement Laraï et Nathénèk les connaissaient aussi ! Leur image lui échappe, se brouille sous les sens nouveaux qui viennent de lui apparaître. Le calme bonhomme de Nathénèk, ses silences devant les sautes d'humeur de Laraï. Comme il la débarrassait parfois de Lian, quand elle était ainsi. Et ses regards sur Lian. Lian les avait crus pensifs et aimants ; ils l'étaient sans doute, mais tristes aussi – nostalgiques.
Une compassion horrifiée lui coupe le souffle à mesure qu'il imagine, que l'histoire se ramifie. Et toujours, les chemins viennent se recroiser au même endroit : lui, l'enfant, l'unique enfant. L'enfant qui a déçu Dougall, qui a changé Laraï pour toujours, qui a frustré Nathénèk.
Lian s'enfuit de la chambre de Kyrin, sans un mot – il ne peut parler, il suffoque. Il court sans rien voir dans des corridors, dégringole et escalade des escaliers, traverse des cours intérieures, des jardins obscurs, des esplanades, se retrouve enfin dans la grande cour centrale, sous le ciel étoilé. Il halète ou sanglote, il ne sait. Il se laisse tomber sur le rebord du bassin, dans l'herbe au pied d'un des tingalyai, hors d'haleine. Il a dû croiser du monde dans sa course aveugle, et Kyrin a dû alerter Argelos, car au bout d'un moment, une mince silhouette se détache de la pénombre pour venir s'asseoir tout près de lui. Argelos ne dit rien. Lian écoute la nuit, le froissement de l'eau dans le bassin, le ramage des oiseaux nocturnes dans les jardins du Haëkelliaõ et dans les tingalyai. Son propre souffle qui s'apaise. Argelos, si proche. Leurs bras s'effleurent à chaque respir. Et tout à coup c'en est trop de nouveau pour Lian, il étreint Argelos, se serre contre lui en tremblant. Et les bras d'Argelos se referment sur lui, la main d'Argelos lui caresse les cheveux, le murmure d'Argelos passe sur son front, un souffle chaud, doux, triste : « C'est ainsi, Liani, c'est ainsi... »
Pendant les semaines qui précèdent encore l'eïldaràn, Lian n'écrira pas d'autres lettres à ses parents. Il essaie une ou deux fois, reste paralysé pendant une éternité devant la feuille blanche. Ensuite, il n'essaie même plus.
C'était toujours Nathénèk qui répondait, de toute façon.
*
À la toute fin de la première semaine de Juillet, Jill vient trouver Lian à l'atelier. L'air soucieux, elle lui tend une lettre, sans rien dire.
L'enveloppe porte un simple cachet rouge qu'il ne reconnaît pas, sans autre indication – le cachet du Conseil de Ville est un cercle formé d'une ronde d'oiseaux-de-clochers stylisés, celui du conseil de zone un triangle, également bleu, et le conseil de quartier n'a pas de sigle. Mais la lettre est bien adressée à Liam Shaunessy, Commune Carghill 1, 1, avenue Trenton Ouest, Z0Q4.
C'est une convocation pour le lendemain 15 juillet, à 9h30, à la Base Quintin, 18336, rue Otchkis, Z0Q2.
Il a été conscrit. Recruté. Dans l'armée. Demain à midi, la Mer s'en va. Demain commence la nouvelle saison de guerre.
Un grand silence se fait en lui. Il dit enfin, d'une voix hésitante : « Je n'irai pas, c'est tout.
— C'est ton avis de conscription, Liam ! Ton numéro est sorti. Si tu ne te rends pas à la caserne, la police militaire viendra te chercher comme déserteur ! »
Comme toujours lorsqu'elle réfléchit furieusement, elle donne des petits coups de paume sur les roues de sa chaise, comme si elle n'attendait qu'un signal pour se propulser en avant ou en arrière. Elle le dévisage, sourcils foncés, d'un air bizarrement calculateur : « Tu ne te rappelles toujours rien, ce qu'ils t'ont fait, pourquoi ils te l'ont fait ? »
Lian répond, avec une totale sincérité : « Non. » Il ne lui demande pas ce qu'elle croit qu'on lui a fait ni pourquoi. Il ne l'a pas demandé en deux saisons, il ne va pas commencer maintenant. Ce n'est pas pertinent, de toute façon : quoi qu'elle pense, elle est dans l'erreur.
Elle hausse enfin les épaules, le visage durci, semble prendre une décision. « On n'y peut rien. S'ils n'ont pas bougé jusqu'à présent... J'ai fait ce que j'ai pu. »
Il se sent bizarrement détaché, à présent. Il y a une saison pour la guerre, une saison pour la paix. Une saison pour l'hiver, une pour l'été. On n'y peut rien. Ici, c'est ainsi : « Qu'est-ce que je fais, alors ?
— Tu vas à la Caserne Quintin demain. J'irai avec toi, il faut bien. Après ça... Tu as eu une grave dépression nerveuse il y a deux saisons – c'est ce qu'il y a dans ton dossier. Ça aidera peut-être. De toute façon, joue les idiots. Pas trop, mais... Réponds de travers quand on te pose des questions, trompe-toi... »
Il se dit, avec un vague amusement, qu'il ne lui sera sans doute pas très difficile de se tromper. Un peu surpris quand même, il se dit aussi qu'il devrait être terrifié. Mais après deux saisons à la commune et dans le quartier sans avoir jamais été inquiété, il a peine à penser qu'on va soudain le repérer. Ici, il est normal. Il fera comme tout le monde.
Le lendemain à midi, quand la Mer s'en va, Lian ne la voit pas : il est dans sa chambre, à faire sa valise. Ce n'est pas comme s'il voudrait voir la Mer partir, non plus.
17
Quand Lian et Thomas arrivent chez Raül Costa ce jour-là, une semaine avant l'eïldaràn, ils trouvent le Club en effervescence. On a une visiteuse de marque. Une femme d'une quarantaine de saisons, grande, mince et musclée, avec des cheveux coupés très courts, un casque noir qui épouse son crâne oblong. Et la peau d'une nuance que Lian n'a jamais vue, ni le bistre foncé de certains Paalani de la côte est, ni le cuivre parfois intense des Hébao, non, un brun si profond qu'il en paraît presque noir. Lartigues la présente : Kalaï Malawi. Elle arrive des îles de l'est. Il ne le dit pas, mais tout le monde a des yeux pour voir, c'est une pure Virginienne, une Keldaran.
La femme incline la tête pour un bref salut, puis parcourt l'assistance des yeux – dans cette face impassible, Lian fasciné ne voit d'abord que ce mouvement, à cause du contraste des pupilles noires avec le blanc de l'œil ; puis les détails deviennent perceptibles ; si la face n'était aussi étroite, le nez assez plat et la bouche aux lèvres pulpeuses seraient curieusement assez ranao. Mais c'est surtout la couleur de la peau qui fascine Lian ; il croyait tous les purs Virginiens comme Odatan, petits et pâles, et mesure sa naïveté : il existe de nombreuses variantes physiques parmi eux, bien sûr, comme parmi les Ranao. D'ailleurs, quand la femme prend la parole, Lian met un moment à la comprendre : elle utilise surtout le mode afran du virginien ; il ne l'a jamais pratiqué au Club où l'on emploie surtout le latinam dans son registre 'spanic.
Kalaï Malawi leur parle des communautés de Keldarao installées au sud-est de Krillialtaoz, dans l'archipel d'Aalapaalu. Elle leur décrit leurs réalisations, leurs progrès, leurs projets. Puis elle en arrive au véritable sujet de sa visite : les communautés ont besoin de leur aide, de l'aide de tous les Virginiens fidèles. L'hiver a été particulièrement inclément, la saison des cultures a commencé très tard, les réserves sont basses, mieux vaut prévenir que guérir, tous les dons sont acceptés. Elle parle avec une certaine raideur : il lui en coûte sans doute de demander, en particulier à des halatnim, si « fidèles Virginiens » soient-ils. Est-ce pour cela qu'on l'écoute avec cet air de satisfaction secrète ?
Après un temps de réflexion, les propositions fusent : le fils de Raül Costa travaille dans une minoterie et pourrait négocier l'achat de grain, un cousin de Lartigues se mettra en rapport avec sa belle-famille à Hébuzer pour rassembler des fonds parmi les Clubs Virginiens de l'ouest... Et plusieurs sortent des carnets pour faire des chèques.
L'argent existe chez les Ranao, parallèlement au troc. Mais Lian et Thomas, en tant que candidats à l'eïldaràn, sont pris en charge par le Haëkelliaõ ; comme tous les autres candidats, ils se rendent utiles en participant aux travaux d'entretien et à la vie de tous les jours. On leur donne chaque semaine un carnet de notes d'échange à valeurs fixées, contre lesquelles ils peuvent s'ils le désirent troquer de menus items en ville. Ils ne sortent guère du Haëkelliaõ, sinon pour se rendre au Club, et leur carnet est presque plein, mais ce n'est pas du tout la même chose que ces " chèques " de la " Banque Virginienne ", Thomas le lui explique à mi-voix ; lui, il peut s'en tirer aisément : il contactera sa famille à Paaltaïr, sur la côte est.
Lian est embarrassé de ne pouvoir contribuer, mais surtout très surpris : si les Keldarao craignent d'être dans le besoin, pourquoi ne le signalent-ils pas aux hékel ?
Un silence soudain s'abat sur la salle, et Lian devine son impair avant d'en comprendre la cause exacte. « Nous sommes économiquement indépendants des Ranao », dit enfin Kalaï Malawi, encore plus roide, en cherchant des yeux son interlocuteur.
« Lian vient du Landaïeïtan », s'empresse de dire Thomas ; Lian retient une mimique agacée : Thomas utilise toujours ce détail lorsqu'il veut défendre son ignorance ; il croit bien faire, mais c'est agaçant de le voir décrire par implication les gardiens de la réserve comme des ignorants. Si Lian ne sait pas tout ce qu'il devrait savoir, c'est la faute de ses parents, pas celle des gardiens en général.
Kalaï Malawi dévisage Thomas, puis Lian, avec une brève stupeur, puis recouvre son impassibilité initiale. Vient-elle seulement de comprendre ce qu'ils sont ? Est-elle choquée de trouver des naïstoï au Club ? Moins étonnant que d'y trouver une danvéràn, se dit Lian morose – Malawi en est certainement une, comme beaucoup de Keldarao ; Thomas et lui n'ont aucun pouvoir, eux, au moins !
Lartigues, un peu gêné, les présente : « Thomas Lléwelyn, Lian Flaherty.
— Flaherty ? » dit Kalaï Malawi avec une petite grimace vite réprimée. « Le fils de Dougall Flaherty ?
Le cœur de Lian manque un battement : « Vous connaissiez mon père ? »
Cette idée semble bien déplaisante à Kalaï Malawi, dont le rictus s'accentue : « Pas... personnellement. Mais j'étais déjà au Comité Directeur à l'époque de cet incident malencontreux. Il n'aurait pas dû aller chez les Chasseurs.
— Un bon choix, dit Thomas, agressif.
— Une décision regrettable, réplique Malawi offensée. Le meurtre avait eu lieu en territoire keldaran. Nos tribunaux à nous auraient très bien réglé l'affaire, et sans toute cette publicité négative. »
Une fois revenu au Haëkelliaõ, Lian lance au premier hékel rencontré : « Je dois parler à Odatan. » Et il retourne dans sa chambre d'où il chasse Thomas, puis Argelos venu aux nouvelles, et où il marche de long en large pendant près d'une heure, incapable de s'asseoir, ou de penser.
Quand le petit homme se présente enfin, Lian ne lui laisse même pas ouvrir la bouche : « Vous m'avez menti ! Mon père... » Il ne peut en dire davantage, c'est comme si ce simple mot lui avait enfin coupé les jambes, il se laisse tomber sur le lit.
Odatan reste un moment immobile, puis se dirige à pas lents vers la chaise du bureau, la tire, s'y assied. « Personne ne t'a menti. Dougall était mort pour ta mère, comme pour ton père-Nathénèk, du jour où il est parti dans les îles.
— Elle m'a dit qu'il était mort dans un accident ! » s'écrie Lian d'une voix qui se déchire.
Odatan bat des paupières, mais ne détourne pas les yeux. « Il a tué un homme par accident. Quand on va chez les Chasseurs pour ce genre de raison, on est mort pour ceux qui restent. On n'a jamais eu de ses nouvelles. Les Chasseurs de l'Est ne sont pas renommés pour leur pacifisme. Pour autant qu'on le sache, il est bel et bien mort. Laraï a pensé que tu n'avais pas besoin de savoir. »
Lian donne un violent coup de poing dans le mur, ne sent même pas la douleur, hurle : « Qui décide de ce que j'ai besoin de savoir, de quel droit ? »
Odatan s'est figé. Puis il s'affaisse un peu dans sa chaise. Il se passe les mains sur la figure et reste silencieux, coudes sur les cuisses, le menton sur ses mains croisées. Les yeux clairs sont fixés sur Lian, mais ne le voient pas. « Curieux », murmure-t-il enfin comme pour lui-même, « nous voulons éviter les erreurs de nos prédécesseurs, et nous en commettons d'autres, et ça revient au même... » Il se redresse avec un soupir : « Nous faisons ce que nous croyons pouvoir ou devoir faire, Lian. Parfois, nous nous trompons. »
Lian regarde fixement le visage navré du petit homme, ce visage à la mensongère jeunesse, et il dit, comme on frappe : « Ça ne sert à rien, alors, d'être vieux ? »
Un terrible éclair de souffrance passe dans le regard pâle. Au bout d'un moment, Odatan se lève, et s'en va.
Et bientôt, c'est l'eïldaràn. Lian en a tellement peur, s'est tellement raidi contre sa peur, qu'il ne sent plus rien. Il dort d'un profond sommeil pendant la nuit du retour, ne va pas regarder l'éclipse de lune, ne participe pas aux réjouissances. Il passe comme anesthésié à travers le début de la matinée du lendemain, dans sa chambre, sans manger, comme on le leur a prescrit, en ne buvant un peu d'eau qu'une fois toutes les trois heures. Il devrait faire la satlàn, mais à quoi bon ? À un moment donné, on frappe timidement à la porte – « C'est Thomas » – il le laisse entrer. Le garçon aux cheveux orange ne dit rien, va s'asseoir dans l'embrasure de la fenêtre, les bras autour des genoux. Ils attendent ensemble le moment où l'on viendra les chercher à leur tour.
Après la douzième heure, avec Argelos et Kyrin, et parmi une centaine d'autres candidats, on les conduit sur la place du Haëkelliaõ, du côté nord. Un cordon de hékel en tunique bleue scintillante les sépare de la foule silencieuse des parents, des amis. Lian ne jette pas un seul regard de ce côté. Ni Nathénèk ni Laraï ne sont là, il l'a exigé – il ne sait pas s'ils seraient venus, de toute façon, ne veut pas le savoir. Après sa dernière lettre, il leur a renvoyé les leurs sans les ouvrir. Il n'y en a eu que deux. Ont-ils compris d'eux-mêmes ou Odatan s'en est-il chargé, avec Kéryan ? Peu importe, il ne veut pas le savoir non plus. Retournera-t-il jamais sur le plateau ? Il n'arrive pas à l'imaginer, à se voir devant Laraï, devant Nathénèk. De toute façon, le futur n'existe pas. Le temps s'arrête à l'eïldaràn.
La façade nord du Haëkelliaõ est une paroi un peu oblique sans terrasses intermédiaires, très lisse malgré les réseaux argentés du sirid qui y courent en fins filaments presque invisibles et constituée uniquement de pierre dorée au contraire des autres façades où elle alterne avec l'ultellaod. " Voir le Signe de la Mer. " Comment pourrait-il soudain y apparaître, le fameux Signe de la Mer ? Lian jette un regard en biais à Thomas, qui contemple la façade, les lèvres agitées par intermittence d'un murmure silencieux : comme on le leur a prescrit, il suit les méandres intérieurs de la satlàn.
Une main vient saisir la main de Lian, qui sursaute. Argelos. Les yeux noirs, intenses, presque implorants. Lian respire profondément. Une idée vagabonde vient le distraire, mantra, on dit mantra en virginien, mais il la repousse. Il se calme, il se concentre, il plonge. Cela n'a jamais été aussi difficile, mais la présence d'Argelos l'aide, sa force paisible. Il commence à flotter. Bientôt les doigts d'Argelos toujours entrelacés aux siens ne sont plus qu'un fil ténu, à l'extrême limite de sa conscience ; quand Argelos le lâche, il s'en rend à peine compte, ce n'est pas très important.
« Lian ! »
Quelqu'un le secoue. Une rumeur autour de lui. Il ouvre les yeux sur le visage consterné d'Odatan. Il a fermé les yeux ? Il devait regarder la façade, pour voir le Signe de la Mer ! Thomas n'est pas là. Le cordon des hékel s'est rompu. Des parents étreignent leurs enfants ; certains s'éloignent déjà avec eux. Quelques candidats pleurent ; d'autres rient, exultants ; d'autres sont immobiles, les yeux encore fixes, avec une expression d'indicible stupeur.
À quelques pas de lui, Argelos et Kyrin, face à face, extatiques.
Lian les contemple sans comprendre, en comprenant trop bien, balbutie : « C'est fini ? C'est fini ? Mais je n'ai pas regardé ! J'avais les yeux fermés !
— Tu aurais vu le Signe même avec les yeux fermés, Lian », murmure le petit homme abattu. Il prend Lian par les épaules, le retourne vers lui : « Mais ce n'est pas forcément fini. On peut essayer d'autres méthodes. Avec de l'aëllud...
— De l'aëllud », répète Lian en écho, sans savoir ce qu'il dit.
« Une drogue. Ça peut aider. Ça avait donné quelques résultats... »
Lian recule d'un pas en s'arrachant à l'étreinte du petit homme, murmure : « Non... » Puis, plus violemment : « Mais non ! » Il évite les mains tendues pour le retenir, recule encore, bouscule il ne sait qui, se retourne et marche, lentement d'abord, puis plus vite, puis court, droit devant.
Il ne va pas très loin, le parc du Haëkelliaõ se referme sur lui, sa lumière verte trouée de soleil, ses oiseaux jacassants, ses parfums familiers, déchirants. Il s'appuie au tronc d'un kaïringa, l'enlace, haletant, la joue contre l'écorce odorante. Il voudrait pleurer, il ne peut pas. Après un moment, il se laisse glisser contre le tronc, recroquevillé entre les racines, les yeux fermés. Après un autre moment, il entend des pas sur la mousse. Il n'ouvre pas les yeux, il ne veut même pas deviner qui est venu le rejoindre, il se veut invisible, absent, mort.
« On n'en a rien à foutre », murmure enfin la voix butée, désespérée, de Thomas – c'est seulement Thomas, bien sûr que c'est Thomas. Il répète : « On n'en a rien à foutre, ça ne nous concerne pas, on ferait mieux d'aller chez les Chasseurs, c'est tout des manigances de hékel, tout ça !
— Non, murmure Lian, non. » Argelos et Kyrin, leur visage illuminé. Ensemble. Danvérani, hékel. Ensemble.
« Si, dit furieusement Thomas, si ! Le Signe de la Mer, c'est juste les hékel qui leur mettent ça dans la tête, ils le savent d'avance, pour les futurs hékel, c'est juste de la frime, la Mer ne fait rien du tout !
— Je l'ai touchée », dit Lian, les yeux toujours fermés. Il se sent curieusement détaché. « Quand j'étais petit. Au retour de la Mer. Je l'ai prise dans mes mains. Ils m'ont dit que j'irais la rejoindre quand je serais grand. Comme tout le monde. Mais pas nous. Tout le monde, mais pas nous. Les naïstoï. Les enfermés, les séparés. Tous ceux que j'aime, ensemble, et je ne les rejoindrai jamais. Ma mère, mon père, Argelos, Kyrin. Jamais. Nous ne nous retrouverons jamais dans la Mer.
— Mais c'est pas vrai ! s'écrie Thomas d'une voix étranglée. Elle absorbe seulement la matière, la Mer, rien d'autre ! Les esprits, la lumière, c'est leurs histoires de hékel ! La Mer, c'est juste... un dépotoir à populations superflues, voilà, une grande poubelle ! »
Lian ouvre les yeux et contemple Thomas, Thomas qui essaie de ne pas pleurer, mais des larmes roulent sur ses joues, une à une. Thomas croit-il vraiment ce qu'il dit ? Mais il croit ce qu'il a besoin de croire. C'est son droit. Et quelle importance, la réalité de la Mer ? C'est sa vérité qui compte. Sa vérité : le désespoir de Laraï. De Nathénèk, de Dougall. De Thomas. Il a raison, Thomas : cela ne les concerne pas, toutes ces histoires de hékel. De Ranao. Ils ne sont pas, n'ont jamais été, ne seront jamais des Ranao. Ni des Virginiens. Halatnim, hybrides, entre-deux, des sans-pouvoirs, voilà leur réalité, leur vérité. Et tant mieux, oui, tant mieux, à quoi servent-ils donc, ces fameux pouvoirs ? Toute cette souffrance. Ceux qui ont des pouvoirs, ceux qui n'en ont pas, et toutes ces vies brisées – Dougall, Laraï, Nathénèk. Thomas, qui pleure maintenant pour de bon en hoquetant « Allons-nous-en, allons-nous-en ! », et Lian, qui le serre contre lui, et caresse les cheveux orange en murmurant : « Oui, oui. »
Quand ils rentrent au Haëkelliaõ et que Lian se rend dans sa chambre pour préparer ses affaires, il y trouve Odatan. Qui ne dit rien. Lian lui lance avec défi : « Je vais chez les Chasseurs avec Thomas ! » Il est prêt à tout faire si on essaie de les en empêcher.
Le petit homme reste un instant comme pétrifié, puis incline la tête. « Vous êtes libres, et aussi de vous tromper », murmure-t-il d'une voix enrouée. « Mais pense à ceci : pour approcher la vérité, il faut d'abord que ce soit elle qu'on cherche. »
Lian hausse les épaules avec violence.
Après un long silence, Odatan dit encore : « Ton père est allé au village de Lïu-Ganzer, sur Krillialtaoz. » Puis il tourne les talons et quitte la chambre.
# Deuxième partie
18
La Base Quintin se trouve dans la ville nouvelle, au-delà du grand canal de ceinture marquant la limite de la ville des Anciens. Pas de zones concentriques dans la ville nouvelle ; les bâtiments y sont des grandes boîtes hétéroclites à l'air fragile avec leurs grandes façades apparemment de verre coloré, alignées les unes à côté des autres le long de rues et d'avenues à angles droits – plantées d'arbres-rois et d'arbres-à-eau, tout de même. Lian n'aurait pu y aller tout seul : ils prennent deux ferries et trois bus pour se rendre à la Base – à gaz, toujours, le départ de la Mer n'a pas changé grand-chose aux habitudes de la ville bien que l'électricité soit maintenant disponible ; la circulation est assez dense, mais il y a bien davantage de bicyclettes et de gazobus que d'automobiles particulières, et beaucoup de piétons aussi sur les trottoirs. Personne ne remarque Lian. Jill reste silencieuse et préoccupée pendant tout le trajet.
Ce n'est pourtant pas la première fois qu'il sort de la commune. Vers le début du Printemps, Suzane l'a convaincu d'aller se promener avec elle sur l'esplanade, les soirs, après le souper ; sans doute avait-elle envie, elle aussi, de se montrer au bras d'un garçon, comme les autres filles – Jill les accompagnait, comme s'ils avaient eu besoin d'un chaperon... On n'a jamais remarqué Lian non plus lors de ces promenades. Ils ne sont pas allés ailleurs que sur l'esplanade, cependant – après deux saisons dans le quartier 4, Lian n'en connaît toujours que la géographie abstraite des cartes de Bird-City affichées dans le bureau de Jill. Et il n'a jamais pris de ferries ni de bus – pas très différents d'un caboteur ou d'un charroi communal, à vrai dire, sinon, le gazobus, par le châssis de matériaux composites, le nombre des passagers et les portes à air comprimé.
Sur le pont des ferries, accoudé aux fenêtres grandes ouvertes des bus, Lian essaie de ne pas trop regarder, mais il voit quand même. Les cicatrices de la guerre sont encore présentes. Dans la vieille ville, surtout, là où des édifices des Anciens ont été détruits, on a reconstruit à leur place des bâtiments modernes ; le contraste est surprenant, presque choquant. C'est moins net dans la ville nouvelle : impossible de dire si tel ou tel terrain vague est récent ou non. Mais sur au moins une place on a délibérément laissé des ruines au milieu d'un petit parc : un monument aux morts de l'Insurrection. Il n'y a pas d'arbres, juste des pelouses et des buissons de prunelliers roses.
Jill abandonne Lian bien avant les portes de la Base, à une centaine de mètres. Elle lui désigne l'entrée, marmonne « Bonne chance », fait faire demi-tour à son fauteuil roulant et s'éloigne. De nombreux jeunes gens sont descendus du bus en même temps qu'eux, quelques-uns sont venus en voiture, beaucoup en bicyclette ou à pied, presque tous aussi avec des membres de leur famille. Après avoir regardé Jill disparaître dans la foule, Lian tourne les talons, franchit en même temps que d'autres garçons et filles les grilles ouvertes. Depuis qu'il a quitté la commune, deux heures et demie plus tôt, il s'efforce de se laisser traverser par tout ce qui l'entoure comme s'il était un simple réceptacle vide, sans que les sensations deviennent des perceptions, les émotions des sentiments. Sans pensée, sans volonté, absent. Ailleurs.
*
Dougall Flaherty n'est pas à Lïu-Ganzer ; il y a vécu une année, se trouvait assurément depuis plus longtemps dans les îles de l'Est car il connaissait bien les coutumes des Chasseurs ; il est reparti depuis deux saisons sur le continent. Nemgorot, le guérisseur rani du village, ne peut en dire davantage.
L'homme ne correspond pas à l'idée que Lian s'était faite d'un Chasseur : ni très grand ni très fort, avec des mains fines, et presque aimable. Il leur offre l'hospitalité pour les premiers jours ; s'ils veulent rester à Lïu-Ganzer, il leur faudra bâtir leur propre demeure, ce n'est pas la place qui manque ; il les aidera. Comme la plupart des villages d'Aalpaalu, Lïu-Ganzer ne compte qu'une centaine de Chasseurs, femmes et enfants compris, dont le tiers à peine sont des halatnim – guère d'enfants, à vrai dire. Lian est d'abord un peu surpris ; mais pourquoi pas des femmes et des enfants ? Quelques-uns doivent venir avec leur famille. Certains en créent peut-être de nouvelles. Et des femmes doivent bien pouvoir se révolter aussi, quelquefois.
Lian est allé chez les Chasseurs pour devenir un Chasseur lui-même autant et plus que pour trouver son père, n'est-ce pas ? Il ne comptait pas vraiment le trouver, en vérité ; ils ne vont pas repartir. Ils construisent donc leur cabane avec l'aide de Nemgorot. Après quoi Lian explore le village. C'est vite fait : une vingtaine de bâtisses de pierre et de bois autour d'une place de terre battue. La plupart des autres Chasseurs préfèrent vivre à l'écart. Il y a quelques maigres jardins. La seule véritable ressource est la chasse ; cueillette et pêche viennent en appoint. Certains villages plus au sud-est, peu nombreux, entretiennent des petits champs ; on échange des denrées avec eux. Dans cette partie de Krillialtaoz, au pied du Hanultellan, la terre est rare, et ingrate : la vraie reine, c'est la pierre sous toutes ses formes, de la montagne au caillou. Heureusement, c'est l'été. Les épaules majestueuses du Hanultellan portent leur manteau de neiges éternelles, comme les autres pics moins élevés de la chaîne des Hanultellarn, de part et d'autre de Diât Strag (que Thomas s'entête à appeler par son nom virginien, le défilé de la Hache), tout proche du village ; mais le reste du haut plateau est couvert de la végétation rase et obstinée qui marque l'été dans cette partie des îles.
La vie au village est simple, austère. Les adultes sont généralement taciturnes. Seuls les enfants animent la place de leurs rires et de leurs cris. Des hommes fiers, des hommes libres, disait Thomas. Pourquoi ces hommes n'en semblent-ils pas plus heureux ? Les femmes non plus, d'ailleurs. Ils doivent l'être, pourtant, puisqu'ils restent... Car enfin, on peut partir, on peut cesser d'être un Chasseur : Dougall est revenu sur le continent – Odatan aussi, mais Lian refuse de songer à Odatan. Pourquoi Dougall est-il parti, après avoir vécu si longtemps dans les îles ? Et si on peut quitter les îles, qu'en est-il des théories de Thomas ?
Peut-être Thomas dirait-il simplement que les îles ont brisé Dougall, et qu'il est retourné par faiblesse ou lâcheté vivre sur le continent aux conditions des Ranao. Mais Thomas est devenu très taciturne. Bien que l'environnement soit fort différent de celui du Landaïeïtan, Lian s'y adapte vite, plus vite que son compagnon, qui a toujours vécu dans des villes et à qui il faut presque tout apprendre, la chasse, la pêche, les plantes utiles – et il n'a pas eu le temps de s'endurcir physiquement en route vers les îles : le bateau de la Mer les a débarqués directement au port de Trélinaõ, sur la côte, seulement quelques jours de voyage ensuite. Mais si la réalité des îles ne correspond pas à l'histoire qu'il a imaginée, le garçon aux cheveux orange n'en dit rien ; il essaie d'être plus Chasseur que les Chasseurs et ne parle plus que par monosyllabes.
*
Après un examen physique expéditif, à la suite duquel on le déclare bon pour le service, des tests écrits également rudimentaires dont il ignore les résultats, et une entrevue plus rapide encore avec les deux agents de la section politique, Lian se retrouve dans le baraquement qui lui a été assigné. On l'a simplement mesuré, pesé, on a vérifié sa vision et l'acuité de son ouïe, excellentes, qu'il n'a pas les pieds plats, qu'il est droitier – il n'a pas dit qu'il est ambidextre – l'état de ses dents – nul n'a besoin de savoir qu'elles repoussent au fur et à mesure. Une brève angoisse quand on lui a piqué un doigt à l'aide d'une petite aiguille, afin d'évaluer la compatibilité de son sang avec le sérum universel, mais on a simplement laissé tomber la goutte rouge sur une languette de papier absorbant, qui n'a pas changé de couleur : il est compatible ; c'est alors qu'on l'a expédié chez le barbier après l'avoir déclaré apte. Après quoi on lui a octroyé son équipement, et on l'a expédié avec d'autres de la Compagnie B, 44e Régiment d'infanterie légère, dans le baraquement 12 – lit 9.
Lian prend la pochette de plastique où se trouvent sa montre, son idicarte et les divers menus objets qu'il avait sur lui avant l'examen physique... La renverse sur le lit, le cœur soudain étreint d'angoisse, vérifie deux fois. Étale ses habits, secoue sa chemise, fouille avec une angoisse croissante les poches de son pantalon, examine une fois de plus le contenu de la pochette de plastique. Il avait réussi à rester presque absent jusque-là, calme, détaché, mais c'est comme si tout ce vide offrait simplement plus de place à l'accablement qui l'envahit.
« Perdu quelque chose ? » dit le garçon en train de s'installer à sa gauche, un grand maigre aux cheveux noirs tout bouclés.
« Ma bague », murmure Lian d'une voix altérée.
L'autre continue à ranger ses affaires dans le coffre situé au pied de son lit. « Important ?
— Tout ce qui me reste... » Cette seule pensée l'étouffe, il se tait.
« Une fille ? » dit l'autre.
L'intonation n'était qu'à demi ironique, c'est la première personne qui lui adresse la parole en tant que personne et non en tant que numéro depuis le début de la journée. Lian répond, à mi-voix : « De ma famille.
— Orphelin ? »
L'intonation est plutôt compatissante, cette fois. Du coup, Lian se sent la gorge nouée, hoche seulement la tête.
« Pas moi, marmonne l'autre comme pour lui-même. La vie est mal faite. »
Lian reste un instant déconcerté, puis dit simplement « Quelquefois ». Il s'oblige à bouger. Après avoir passé sa montre à son poignet, remis le reste moins l'idicarte dans la pochette, il jette un coup d'œil au garçon qui continue à ranger méthodiquement ses affaires. La vingtaine, peau très brune comme presque tout le monde ici, grand et carré d'épaules mais trop mince pour sa taille, un visage tout en méplats, joues creuses et pommettes saillantes, yeux noirs, orbites profondes, grande bouche au dessin sarcastique.
« Elle était comment, ta bague ? » reprend le garçon, les sourcils un peu froncés.
« En or, avec un petit cabochon de rubis poli. »
L'autre fait « mmm ». Il ferme son coffre, se redresse, dévisage Lian à son tour : « Grayson James, dit-il sans tendre la main. On dit Gray. Ça a l'air qu'on en a pour un moment ensemble.
— Liam Shaunessy », dit Lian ; il a failli dire " Lian ", s'est repris à temps.
Autour d'eux, les rangées se remplissent, le brouhaha monte, des portes d'armoires claquent. Avec un soupir, Liam commence à ranger son équipement, l'uniforme de rechange et l'uniforme de parade dans l'armoire, le reste dans le coffre.
« De toute façon, ils vont tout nous faire refaire », remarque Grayson James.
— Comment ça ?
— Il y a la manière ordinaire, et la manière militaire. » Il donne un petit coup de pied dans le coffre de Lian : « Pas réglementaire, comme ça. Le mien non plus. Personne. Vont tout flanquer par terre et nous faire recommencer. »
Lian considère son coffre à moitié plein, désemparé. « Ce n'est pas la peine de ranger maintenant, alors...
— Oh si ! Sinon, crac, punition.
— Mais ça ne sera pas rangé comme il faut.
— Ça ne fait rien. »
Et devant la mine déconfite de Lian, l'autre se met à rire : « N'essaie pas de comprendre ! On est chez les militaires, ici. Finis de ranger, va. »
Lian s'exécute en demandant. « Tu as déjà été conscrit ?
— Non, mais on m'a raconté ! Pas toi ? » Le jeune homme s'est assis en tailleur sur son lit, mains croisées entre les genoux.
« Non, murmure Lian.
— C'est simple : il faut faire ce qu'on nous dit et essayer de penser le moins possible. Surtout pendant la période d'entraînement. »
Avec une ombre d'ironie, Lian se dit que ça ne devrait pas lui être trop difficile.
Au bout d'une demi-heure, alors que les recrues commencent à s'ennuyer et à chahuter pour de bon, le sergent-instructeur Maja Turner fait son apparition, une femme d'une cinquantaine d'années qui se tient très droite, musculature nerveuse, voix trompeusement calme qui peut atteindre un volume et une intensité surprenants. Un silence parfait règne bientôt dans le baraquement 9, alors qu'ils se tiennent tous au garde-à-vous. Une espèce de jeu stupide, se dit Lian en regardant droit devant lui comme les autres, tout le monde le sait, mais personne n'ose le dire.
L'instructrice nomme un chef de section, un garçon à la peau presque noire nommé Ritchie Abdul. Ensuite, elle leur fait vider coffres et casiers, un par un, et recommencer le rangement à zéro. Lian oublie le conseil de son voisin et sourit. Il se fait donner dix pompes. Ne sait pas ce que l'instructrice veut dire, se fait demander s'il a un problème. Quelques recrues ne peuvent retenir un gloussement. Dix pompes chacun. Lian en prend cinq de plus. Maintenant, il a compris. Il s'exécute.
Quand vient le temps de la méridienne, Lian fourbu aurait presque sommeil. Il écoute les autres dormeurs – certains ronflent déjà. Il pense avec une stupeur incrédule qu'il est un soldat maintenant. Qu'il va vivre pendant des semaines, des Mois, avec tous ces inconnus – et ces inconnues : les baraquements sont séparés, mais il y a autant de filles que de garçons parmi les recrues et tout le monde sera entraîné ensemble. Et il ne se trouve pas là parce qu'il l'a choisi mais parce que le hasard en a décidé pour lui. Ensuite, il essaie de se dire que les inconnus ne le resteront sans doute pas longtemps ; et qu'eux non plus n'ont pas choisi d'être là. Peu importe, l'absurdité profonde de la matinée écoulée le laisse éberlué. Les règlements, les règles, les ordres. Ce qu'on ne peut pas faire. Ce qu'on est obligé de faire. Les grades, les insignes, les formes adéquates de salut. La tête lui tourne un peu.
L'après-midi est consacré à des exercices de marche en formation, répétitifs, intensément ennuyeux, mais il se rend compte qu'il n'a en effet pas grand mal à ne pas penser. Après quoi on les envoie à la collation puis aux douches en leur ordonnant de revêtir leur uniforme de parade, et on les présente au commandant de la Base, toutes les recrues réunies, dix-huit cents garçons et filles au garde-à-vous sous la pluie chaude, pendant plus d'une heure. C'est d'une telle étrangeté, ces rangées d'uniformes où l'on n'arrive plus à distinguer des individus, que Lian dégoulinant écoute à peine le discours du colonel Diaz, quand celui-ci daigne enfin monter sur le podium. La voix fulminante amplifiée par les haut-parleurs leur garantit, longuement, qu'ils sont la pire classe de recrues qu'on ait jamais vu de sa vie mais qu'ils vont devenir des soldats, des vrais soldats, les meilleurs des soldats. Ensuite, on les fait repartir pour un tour de parade.
Lian passe de nouveau aux douches après la parade, dans les tout derniers ; personne n'a encore fait de remarque sur son corps pratiquement dépourvu de pilosité, mais une obscure prudence le pousse à limiter les risques d'exposition. Quand il revient à sa rangée en se passant la main dans les cheveux – quelle sensation bizarre, cette courte brosse rêche – Grayson James assis sur son lit en slip, le torse nu et encore humide, le regarde arriver avec un sourire en biais ; avant même que Lian se soit assis sur son lit, il lui tend ses deux poings fermés. « Devine quelle main. »
Lian reste un moment ébahi, puis, comme l'autre insiste, désigne la main droite du garçon. Qui s'ouvre sur la bague au cabochon de rubis.
Lian souffle : « Où l'as-tu trouvée ?
— Oh », dit Grayson, nonchalant mais visiblement très content de lui, « il y a quelquefois des erreurs... administratives, lors du recrutement... Du personnel aux doigts collants. Mais on peut toujours s'arranger. »
Il faut un moment à Lian pour comprendre qu'on a essayé de lui voler sa bague. Il la passe à son médius avec une gratitude stupéfaite. « J'ai une grande dette envers toi », murmure-t-il avec gravité.
L'autre, l'air un peu surpris, finit par dire d'un ton plaisant : « Oh, je me ferai payer, compte là-dessus ! »
Ensuite, comme ils ont une quinzaine de minutes avant l'extinction des feux, ils parlent un peu. Lian préférerait s'abstenir, il se doute que n'importe quelle conversation normale peut receler des dizaines de pièges pour lui, mais il ne va pas refuser, surtout maintenant. Grayson demande : « Tu viens d'où, Liam Shaunessy, tu fais quoi ?
— Quartier 4. Jardinier.
— Biologiste ?
Lian se rappelle une réflexion de Jill : « Pas de diplômes, mais oui, si on veut. Et toi ?
— Cristobal, mais j'étudie ici, alors... Quartier 1, l'université Œniken. Littérature et mathématiques. » Et, avec une emphase bouffonne : « Les arts et les sciences ! » Le sourire s'efface. « Ça passe le temps...
— En attendant quoi ? » demande Lian, curieux du changement de ton.
Le garçon fait une petite grimace faussement ironique : « Mes parents me destinent à la politique.
— Et toi ? »
Grayson hausse les épaules en silence. Au bout d'un moment, Lian conclut à mi-voix : « Tu penses que tu n'as pas le choix.
— Difficile d'aller contre la tradition familiale », dit l'autre ; il ne prétend plus à la désinvolture. Puis le sourire sarcastique revient : « Pourquoi, tu penses qu'on a toujours un choix ? »
Lian reste un instant figé. « On peut être beaucoup de choses. Il s'agit de poser les bonnes questions », murmure-t-il enfin, la gorge un peu serrée, en revoyant le visage grave d'Odatan. « Personne ne peut choisir qui tu es, sinon toi-même. »
« Ha ! dit Grayson, va dire ça à Turner et aux autres instructeurs !
— Ils décident ce que nous faisons, pas qui nous sommes.
— Et nous ne sommes pas ce que nous faisons ? » rétorque Grayson.
Lian baisse la tête, soudain accablé de souvenirs. « Pour les autres, pas forcément pour nous », admet-il d'une voix qui s'éraille.
« Mmm, dit l'autre, toujours sarcastique. Tu veux dire que si je finis par faire ce que mes parents veulent me voir faire, ce sera par lâcheté. »
Lian, navré du malentendu, s'apprête à protester, mais l'autre lève déjà une main : « Non, ce n'est pas ce que tu veux dire, d'accord. Juste moi : je me trouve un peu trop... conciliant avec eux. Mais d'un autre côté, tu n'as pas tort, je suis peut-être simplement de mauvaise foi, inconscient. Peut-être que ça me tente, en réalité, la politique. » Il fronce les sourcils, poursuit à mi-voix. « Ce serait tellement plus facile. Descendre sa pente. Il y a des héritages qui viennent de loin. Trop de poids accumulé. » Un petit rire bas, empreint d'amertume. « C'est dans les gènes. »
Lian tressaille, se penche vers l'autre : « Mais nous ne sommes pas que nos gènes, Gray ! »
Le garçon le dévisage, comme déconcerté, finit par dire : « Je suppose que non... », avec une note interrogative. Puis le sourire ironique revient : « En tout cas, ne pas oublier : pour l'armée, on est ce qu'on fait et on fait ce qu'on vous dit ! »
Et justement le sergent Turner vient ordonner l'extinction des feux. Grayson adresse un clin d'œil à Lian, qui esquisse un sourire en retour. Les lumières s'éteignent.
19
Lian et Thomas s'intègrent vite au village, sans doute parce qu'il n'y a rien où s'intégrer véritablement ; on les traite avec la totale indifférence qui est, Lian finit par le comprendre, la politesse des îles. Ou bien les habitants du village ne sont même pas des lâdzani, ou bien peu leur importe ce que vous êtes, cette partie-là au moins de l'histoire de Thomas serait vraie, même s'il est des Chasseurs qu'ils ne voient jamais, ceux qui vivent à l'écart.
La seule exception est Nemgorot, qui parle un peu plus, qui sourit parfois. La règle de discrétion, en usage ici plus encore que sur le continent, interdit d'en demander plus qu'il ne veut bien en dire, mais il semble posséder des connaissances étendues, pas seulement en médecine. Et pourtant, c'est un Chasseur lui aussi. Avec quoi n'a-t-il pu faire sa paix ? Il a souvent les sourcils froncés, comme s'il luttait contre une migraine tenace... Il n'est pas que le guérisseur du village : on vient lui présenter des sujets de querelle, il s'occupe des enfants, du partage de la pêche et de la chasse... « C'est peut-être le chef », dit Thomas, sceptique, un jour de rare loquacité. Nemgorot se met à rire lorsque Lian lui transmet ce commentaire. « Je n'ai pas d'autorité. Je suis seulement... utile. Ils me tolèrent justement parce que je ne suis pas un chef. » Le terme virginien sonne bizarrement dans la phrase en setlâd, et le sourire de Nemgorot prend une tonalité différente, plus sombre : « À cause de ma faiblesse. Parce que je suis moins qu'un Chasseur. » Mais comme il termine sa phrase par une inflexion descendante, Lian n'insiste pas.
Pourtant, après trois semaines, Lian se rend compte qu'ils sont en observation. Un nommé Siridelln vient leur rendre visite, accompagné d'un nommé Kamlaz. Kamlaz est un pur rani, pas Siridelln – un nom inhabituel, mais c'est celui qu'il s'est choisi lorsqu'il a rejoint les Chasseurs. " Métal sans fin ", sans doute approprié : le halatnim semble aussi dur, froid et indestructible que du sirid. Lian n'avait pas songé à se choisir un autre nom, et une vague prudence le retient d'évoquer son père ; il dit simplement « Lian ». « Darkalla », dit Thomas, qui y a pensé, lui. Lian le regarde à la dérobée, un peu inquiet : " Frère sombre " ? Mais les deux autres hochent la tête avec une réticente approbation, même si le nom est un mélange bizarre de virginien et de setlâd. Ils leur offrent de venir chasser avec eux. Thomas accepte en dissimulant mal son plaisir et Lian, un peu surpris, accepte aussi. On leur donne rendez-vous pour le lendemain à l'aube.
Il se rend vite compte qu'on les met à l'épreuve – qu'on leur fait passer un test. Siridelln et Kamlaz ne sont pas seuls ; deux autres couples de Chasseurs, dont trois halatnim qu'ils n'ont jamais vus, se relaient pour courir et traquer avec eux. Pour les évaluer. Heureusement, en trois semaines, Lian a retrouvé ses jambes et son souffle ; c'est plus difficile pour Thomas, qui n'en a jamais eu – et il n'a pas encore tout à fait fini de s'adapter à l'altitude, on est plus haut ici qu'à Hleïtzer ; mais il s'accroche avec une obstination farouche, et Lian traîne plus lentement qu'il ne le pourrait pour rester au contact ; si Thomas lui en est reconnaissant, il ne le manifeste pas.
Ils passent une journée entière à arpenter ainsi le haut plateau, escaladent même quelques pentes pour traquer des maëliki. Ils réussissent à en capturer chacun un, comme on le leur a ordonné, les redescendent sur leur dos et s'arrêtent près d'un torrent qui s'encaisse entre des parois sonores. On leur demande alors de tuer et de dépouiller les animaux. Thomas égorge sans broncher le premier maëlik, l'écorche avec un peu plus de maladresse, étant encore novice ; Lian se raidit quand vient son tour – sur le Landaïeïtan, on traque pour observer, non pour manger, et on ne tue que quelques rares spécimens afin de les étudier ; les seuls animaux qu'il ait jamais chassés ainsi, ce sont des poissons. Mais chez les Chasseurs, on fait comme les Chasseurs. Ils ont leurs propres lois, qui ne sont pas celles du continent, et ces maëliki nourriront plusieurs familles. C'est d'ailleurs de toute évidence encore un test, car lorsqu'il se relève, moins ensanglanté que Thomas, Siridelln lui tape sur l'épaule avec une mimique approbatrice.
Ils rentrent fourbus au village ce soir-là, mais Thomas rayonne de satisfaction. Les jours suivants, on vient encore les chercher, pour les soumettre à d'autres épreuves d'adresse ou d'endurance que Lian passe haut la main, Thomas à force d'acharnement. Après une semaine, Lian comprend l'obstination de son compagnon : on les juge enfin dignes d'être initiés dans le culte du Karaïker.
On les emmène une nuit dans une caverne éclairée de torches fumeuses, au centre de laquelle repose sur une pierre plate l'énorme crâne poli d'un félin géant. Une demi-douzaine d'hommes, et une femme, nus, excepté un pagne de peau, zébré comme leur visage et tout leur corps de suie et d'ocre rouge. On fume du maalt, une herbe au parfum douceâtre, à la fumée âcre qui irrite la gorge – Lian en reconnaît subitement l'odeur : Dougall ! Il a l'impression d'être un peu ivre, ensuite. On les peint aussi de noir et de rouge. On chante en s'accompagnant de tambours – une longue histoire confuse où il croit reconnaître un écho de Paguyn et Kithulai. Chacun s'entaille une main et laisse sa marque sur la poitrine nue de Lian et de Thomas, qui en font autant à leur tour. Ensuite on boit, et pour le coup Thomas fait preuve de plus d'endurance que Lian, qui se réveille tard le lendemain dans leur cabane avec un mal de tête à fendre les montagnes.
Comme Thomas est déjà reparti chasser, Lian vaque seul au jardin. Il voit passer Nemgorot, lui dit qu'il a été intronisé avec Thomas dans le culte du Karaïker : il trouve tout cela plutôt amusant. « J'ai entendu les tambours », dit Nemgorot, sans sourire.
« C'est quoi, ce culte, au juste ? Thomas semble au courant, mais je n'en ai jamais entendu parler.
— Un culte très ancien », répond simplement le guérisseur, avec une inflexion descendante. Il s'éloigne, et Lian se remet à bêcher, perplexe. Quand Thomas revient de sa chasse, et qu'il l'interroge, il se rend compte que son compagnon ne savait pas grand-chose ; c'est juste une rumeur qui courait sur les Chasseurs, elle est vraie, il en est heureux. Ils sont de vrais Chasseurs, de vrais hommes libres, maintenant que la confrérie du Karaïker les a adoptés. Lian retient un sourire indulgent : Thomas s'est trouvé une place.
*
Vers le milieu de la deuxième semaine d'entraînement, Grayson, qui reçoit du courrier, lui, sort un livre d'une grosse enveloppe matelassée. Lian, une fois de plus déçu dans son attente d'un petit mot de Suzane ou de Jill, reconnaît avec stupéfaction et embarras un exemplaire de Légendes d'ailleurs.
« Hmm, dit Grayson avec un faux sérieux. Liam Shaunessy. Ce nom me dit quelque chose... »
Lian hésite, mais il ne va pas nier. Grayson feuillette le livre : « Tu ne m'avais pas dit que tu écrivais.
— Juste des histoires pour enfants. »
Grayson relève la tête avec un large sourire, et Lian comprend avec retard que l'autre est aussi embarrassé que lui, mais pas pour la même raison : « Eh bien, je suis un enfant comblé. » Il sort un stylo-feutre d'une de ses poches : « Tu me le dédicaces ? »
Lian prend le livre, constate qu'il a été lu – et sans doute plusieurs fois. Déconcerté, touché malgré lui, il prend le stylo et écrit d'un trait : " À Grayson, l'enfant comblé ".
L'autre lit la dédicace, sourit. « Mais ce n'est pas juste pour les enfants.
— Non, concède Lian.
— Tu vas écrire autre chose ? »
Lian ne peut retenir un petit tressaillement chagriné. Il a emporté un gros carnet, mais n'y a encore rien écrit. Il a le sentiment qu'il n'y écrira jamais rien. Grayson l'observe avec une sorte de timidité sûrement bien inhabituelle chez lui.
« Je ne sais pas, murmure enfin Lian. J'aimerais écrire... » Il cherche un moment, s'entend déclarer – et c'est une révélation pour lui : «... l'histoire de mes parents.
— Eh bien, pourquoi pas ? »
Encore stupéfait du brusque sentiment de certitude qui l'a envahi, Lian écoute la question résonner en lui. Pourquoi pas ? Il contemple le visage curieux et innocent de Grayson ; il lui doit bien d'être aussi honnête que lui : « J'aurais peur que ce ne soit pas assez vrai.
— Une biographie, c'est une biographie. Si tu connais bien les faits...
— Oh, ce serait un roman. »
Grayson reprend sans se démonter : « Pourquoi, tu n'as pas assez d'informations factuelles sur eux ? »
Lian hésite, les yeux au loin, essayant de formuler ce qu'il découvre lui-même en cet instant : « Trop, pas assez. On se raconte toujours des histoires sur ses parents. Les faits... ne sont jamais que nos souvenirs ou ceux des autres, une autre histoire. »
Et il a soudain envie de fermer les yeux, comme s'il pouvait ne plus voir ainsi le visage de Nathénèk, de Laraï, d'Odatan.
« Il y a quand même bien des façons de vérifier les faits, s'obstine Grayson qui ne peut pas comprendre. Des documents, des témoins... »
Lian secoue la tête : « Pas de documents. Et les témoins sont... inaccessibles.
— Eh bien, alors, si tu écris un roman, tu n'as pas à te préoccuper de la vérité des faits, tu inventes ce qui t'arrange, ce qui convient à ton histoire. »
Lian ne peut s'empêcher de sourire un peu tristement : « Ah, mais il y a toujours au moins deux versions des faits. Quand les choses sont simples. Et puis, il n'y a pas que la vérité des faits. Il faut que les mensonges soient vrais aussi. »
Cette fois, Grayson est dérouté : « De vrais mensonges ?
— Des mensonges vrais. Leur vérité pour moi. Et je ne suis pas sûr de pouvoir l'écrire. Pas encore. Trop près. Trop loin. » Il hausse les épaules en signe d'impuissance : « Pas la bonne distance.
— Tu pourrais sûrement écrire d'autres contes, en tout cas », dit Grayson en feuilletant de nouveau le livre.
Lian sourit, avec tristesse ; c'est ce qu'il n'a pas réussi à expliquer à Suzane : « Non, je ne pourrais pas », murmure-t-il, sans espérer être compris là non plus. Mais Grayson le dévisage un moment avec intensité puis dit en hochant la tête : « Oui. Encore des contes, maintenant, ce seraient seulement des mensonges, n'est-ce pas ? »
20
Six semaines ont passé depuis le retour de la Mer. Lian en prend conscience avec stupeur, un matin : se peut-il que l'équivalent d'une saison seulement se soit écoulé depuis qu'il a quitté le Landaïeïtan ? Il s'est efforcé de ne pas penser à ses parents depuis qu'il est à Lïu-Ganzer. À ce qu'ils ont dû ressentir lorsqu'ils ont cessé de recevoir ses lettres. À ce qu'ils ont dû penser lorsqu'ils ont appris – sûrement – qu'il est allé rejoindre les Chasseurs. Il croyait les avoir oubliés, voilà que des myriades d'images surgissent soudain dans sa mémoire, claires et précises comme s'il était parti la veille. Un regret soudain le saisit... mais pour quoi ? Pour le Landaïeïtan où on l'a nourri de faussetés ? Pour le Haëkelliaõ de Hleïtzer, d'où l'ont fait fuir les quelques vérités enfin découvertes ? Rien à regretter – sûrement pas l'ignorance ; il préfère savoir ce qu'il sait et être ici.
Il le doit : il le paie assez cher. Une pensée trop dérangeante, qu'il oublie aussitôt.
C'est la dernière semaine du mois des Chasseurs. Curieux, tout de même, qu'un mois leur soit dédié, et ce depuis des temps extrêmement lointains, alors qu'ils sont, dans les termes de Thomas, des révoltés – et donc des rejets de la société rani. Ils vivent encore plus indépendants que les Keldarao – un Chasseur change de nom s'il le désire, disparaît dans les îles, le continent n'a plus aucun droit sur lui. Si l'on envoie sur leur territoire les jeunes Ranao respectueux des règles édictées par les hékel, c'est assurément pour leur faire prendre conscience de ce qui les attend s'ils les enfreignent.
Un matin, c'est à peine l'aube, le village dort encore, Lian se réveille en sursaut : on lui secoue le bras. Il reconnaît Siridelln et Kamlaz dans la pénombre.
« L'heure de la chasse, alla », dit Siridelln.
Il l'appelle " frère " : ce sont les affaires du culte. Avec un soupir, et parce que Thomas est déjà tout équipé, Lian se lève et s'équipe à son tour. Mais ils lui désignent la machette qui lui sert à se frayer un chemin dans le sous-bois, en contrebas du village, quand il va à la pêche. Ils ne portent eux-mêmes ni arc ni flèches ni lance ; seulement de larges coutelas et de gros sacs passés en bandoulière. Avec un haussement d'épaules, Lian raccroche arc et carquois près de la porte. Encore une épreuve. Il pensait en avoir terminé avec ça.
« On y va ? » dit-il avec désinvolture.
Siridelln hoche la tête, impassible, et s'éloigne au petit trot, aussitôt suivi par Kamlaz. Lian et Thomas leur emboîtent le pas.
La matinée s'allonge, se termine. Ils ont pris la piste en contrebas du village, près du lac, la suivent vers la vallée. Elle est difficile à lire : trois animaux, de taille moyenne. Pas d'odeur spécifique, pas de laissées. L'herbe est foulée, la terre brouillée par une averse pendant la nuit, et Lian n'arrive pas à identifier les traces. Il est presque certain de n'en avoir jamais suivi de telles, cependant, mais il n'a pas l'intention de poser aux autres des questions compromettantes. Drôle de gibier, en tout cas, qui suit sans jamais s'arrêter un chemin presque rectiligne. L'après-midi s'étire sans rencontre. L'allure est curieusement lente, pour une traque ; on s'arrête même pour manger.
La nuit tombe. Deux des petites lunes éclairent les traces, toujours aussi visibles, toujours aussi illisibles. Ils ne dorment donc pas, ces animaux ? Et ils mangent quoi ? Rien n'indique qu'ils se soient nourris depuis le matin. Ils ne mangent pas, ils n'excrètent pas : logique. Mais Lian croit de plus en plus à une sorte de plaisanterie. Un peu longue, mais ce n'est pas comme s'ils avaient vraiment autre chose à faire.
Siridelln s'arrête enfin sur un petit promontoire au-dessus d'un amas de roches à moitié recouvert de buissons. Kamlaz disparaît dans la pente à gauche de Lian. Quand il revient, il désigne l'amas de rochers.
Ils descendent du promontoire en silence, contournent le monticule rocailleux avec précaution. Une large fissure, mal dissimulée par des branchages : une caverne. Kamlaz a dégainé son coutelas ; sur un hochement de tête de Siridelln, il se dirige vers l'entrée obscure. Va-t-il attaquer ces animaux seul ? Ils sont sûrement dangereux, si on doit les surprendre ainsi au repos... Voilà en quoi consiste l'épreuve, alors : les surprendre.
Kamlaz revient presque aussitôt, un éclat de dents blanches sous les lunes, du sang sur son coutelas. Lian n'a rien entendu. Siridelln tire une corde et un assez large bandeau de peau de sa sacoche, disparaît à son tour dans la caverne. Lian tend l'oreille : silence. Plus une cérémonie rituelle qu'une épreuve, se dit-il, vaguement amusé : le premier doit tuer sans réveiller les autres animaux, le second doit capturer... et le troisième ? En soumettre un à mains nues ? Ils sont quatre, de toute façon : que fait le quatrième ?
Siridelln reparaît et leur fait signe de venir. Lian se faufile derrière Kamlaz et Thomas par l'entrée étroite de la caverne. Quand ses yeux se sont habitués à la pénombre, il voit le gibier.
Une masse immobile, ficelée comme un paquet. Une autre un peu plus loin, dans une flaque de sang qui s'élargit. Une troisième un peu à l'écart. Sous une couverture. Près d'un feu presque éteint.
Siridelln se penche déjà sur la troisième silhouette, la secoue... Elle se redresse en sursaut, sans crier. Essaie de bondir, mais la main de Siridelln s'est refermée sur sa gorge tandis que de l'autre il lui retourne violemment les bras dans le dos. Kamlaz la ligote avec sa corde.
Siridelln remet du bois sur le feu, qui bondit en pétillant. La silhouette garrottée est celle d'une femme, non, une très jeune fille. Siridelln se tourne vers Thomas avec un sourire féroce : « Tu as besoin d'une femme, petit frère. En voilà une. »
Une pierre grosse comme un poing s'envole brusquement du sol pour venir lui frapper l'épaule. Avec un rugissement sourd, il brandit son coutelas devant le visage de la jeune fille : « Recommence et tu meurs tout de suite ! »
Il se tourne vers Lian, à présent, en désignant l'autre masse ficelée, un garçon d'environ dix-huit saisons aux yeux exorbités au-dessus du bâillon de peau qui lui écrase la figure : « Il est à toi. Tue-le. »
Lian bondit sur Siridelln sans même penser à dégainer sa machette.
Après, il doit les regarder torturer le garçon et se servir de la fille avant de la tuer à son tour. Siridelln, Kamlaz. Thomas ne fait rien. Quand ils ont voulu le tuer, lui, Thomas a hurlé « Non ! ». Ils ont ricané, mais ils ont renoncé. Ils ont ordonné à Thomas de le ligoter et de le bâillonner et, après une hésitation, Thomas a obéi, les yeux détournés.
Encore après, quand ils ont joué aussi un peu avec Thomas, ils détachent Lian : « Tu n'es pas digne d'être un frère du karaïker. Va rejoindre Nemgorot et les autres lâches. Et que ta chasse ne croise jamais la nôtre », dit Siridelln. Il lui crache dessus, Kamlaz en fait autant, et ils entraînent Thomas couvert de sang, les yeux fous. Lian essaie de ramper depuis le fond de la caverne, mais ils lui ont cassé des côtes et l'éclair rouge de la douleur le foudroie. Le feu s'est éteint. La lumière des lunes disparaît. C'est l'heure noire d'avant l'aube. Ensuite, le soleil monte, illumine la caverne. Lian ferme les yeux, pour ne pas voir. Bientôt les mouches bourdonnent. Et sur lui aussi. Il dérive entre évanouissements et hébétude. Une autre nuit. Un autre matin. Il a tellement soif. Un bruit à l'entrée de la caverne. Ils sont revenus l'achever. Non. Nemgorot. Et un autre homme du village. Ils le retournent sur le dos. Il s'évanouit.
*
L'entraînement commence. La Base Quintin se trouve à l'extrême limite de la ville nouvelle ; ses terrains s'étendent sur une cinquantaine de kilomètres carrés, épais boisés, étangs, petite rivière, marais, champs découverts, collines, zones rocailleuses, et même des escarpements rocheux hauts de plusieurs dizaines de mètres. Lian comprend vite que le paysage a été remanié de façon délibérée et pourquoi, tandis qu'avec les autres il marche, trotte, court, en portant son sac plein de sable et de roches. D'abord sur les chemins, ensuite en dehors. Il s'est rouillé à la commune malgré les exercices et le jardinage, mais au bout d'une semaine il commence à retrouver son endurance de la forêt. Certains autres ne sont pas aussi chanceux, Grayson, en particulier. Lian traîne à l'arrière du peloton pour rester avec lui et Chris, une des filles, – petite et mince, et dont les charges sont calculées en fonction du poids et de la taille comme pour tout le monde, mais qui, comme Grayson, n'a apparemment jamais fourni de réel effort physique de toute sa vie.
Quand on passe à l'entraînement avec le véritable matériel, Turner leur dit : « Débrouillez-vous, mais il faut amener tout ça du point A au point B. » C'est plus simple : on se répartit les charges selon les capacités. Lian, qui a convaincu les autres de fonctionner ainsi, se retrouve chef de section pendant une semaine, un peu embarrassé. C'est lui qui doit diriger les séances d'examen, tous les trois jours, en présence du commissaire politique attaché au bataillon, le capitaine Lars Trent – un petit homme mince et droit d'une cinquantaine de saisons, voix toujours douce, traits toujours impassibles, et qui ne dit jamais grand-chose : sa simple présence suffit à assurer le sérieux général. Grayson l'appelle « notre conscience de poche » ou, plus souvent, « Jiminy », une référence qui échappe à Lian et à la plupart des autres ; il doit leur raconter l'histoire de Pinocchio. Tout le monde s'écroule de rire. Les autres membres de la section ne sont pas plus respectueux qu'il ne le faut à l'égard de Trent – hors de sa présence.
On a commencé l'entraînement au tir sur cible. Tant qu'on s'est contenté de monter et de démonter les armes, Lian a trouvé l'exercice plutôt intéressant, comme un casse-tête. Et au début, malgré le bruit, c'est intéressant aussi d'utiliser ces machines qu'il connaît maintenant par cœur, et dont il apprécie l'ingénieuse complexité. Ce ne sont pas des arcs et des flèches, mais le principe est le même. Il se concentre, il contrôle son souffle comme la satlàn lui a appris à le faire et il tire sur les lointaines cibles rondes : c'est une simple épreuve d'adresse. Après une courte période d'adaptation, il voit ses cibles lui revenir avec de beaux tirs groupés ; l'instructeur est très content de lui, le cite en exemple aux autres.
Un jour, les cibles qui se dressent au fond du champ de tir ne sont plus des cercles rouge et blanc. Ce sont des silhouettes en tenue de camouflage. Lian se fige. L'instructeur de tir le remarque vite : « Alors, Shaunessy, on prend des vacances ? »
Lian ferme les yeux. C'est juste du carton, du bois et de la peinture, se répète-t-il en essayant de contrôler le battement de son cœur. Il se mord les lèvres, vise, tire. Manque, à plusieurs reprises. « Dis donc, Shaunessy, tu es devenu daltonien ou quoi ? » dit l'instructeur déconcerté et mécontent. Juste du carton, juste du carton. Lian se concentre, vise bas, les jambes. Touché. « Quand même ! » dit l'instructeur, en passant au suivant.
Lian s'affaisse dans son créneau, le front contre la crosse de son arme.
Une fois que l'instructeur s'est assez éloigné, Grayson lui souffle : « Qu'est-ce qu'il y a ? »
Lian avale sa salive ; il a la nausée. Il réussit à dire, en s'y reprenant à deux fois : « On va tuer... des gens.
— Qu'est-ce que tu croyais qu'on allait faire ? » s'exclame Grayson à mi-voix, éberlué. Puis, devant le silence accablé de Lian, son regard plein de détresse, il finit par murmurer : « Tu sais, quand on y sera, là-bas, tu pourras tirer à côté, personne ne s'en rendra compte. »
Au bout d'un moment, comme l'instructeur revient vers eux, Lian recommence à tirer, les dents serrées. Il vise avec encore plus de soin, les jambes, les bras. Il ne tirera pas à côté. Il ne tuera personne par accident. Il ne tuera personne.
Dans le combat à mains nues, la satlàn vient de nouveau à son aide – ce n'était pas seulement une danse, aux temps archaïques, mais une danse guerrière. Il apprend le reste. Une pression appliquée pendant tel laps de temps rend inconscient, un coup à tel ou tel endroit paralyse. La même pression peut tuer, ailleurs le coup est fatal. Peu importe, il apprend, mais il n'utilisera pas.
Il y a les exercices de combat rapproché, à l'arme blanche, d'abord avec des sacs de sable, puis des mannequins. Lian réussit encore à compartimenter – c'est du sable et de la toile, juste du sable et de la toile. Et même pour le combat corps à corps, avec des partenaires : on arrête toujours avant le coup mortel. C'est ce qu'il fera. Grayson, en le voyant livide quand même, la première fois, lui dit que les combats rapprochés sont extrêmement rares, sur le front. Il hoche la tête en silence, sombrement résolu à se préparer à toutes les éventualités.
Quand on les place en situation de combat simulé, avec les mitrailleuses lourdes et les fusils automatiques qui tirent à blanc, les explosions, la fumée, il ne craque pas. Pendant les embuscades dans les bois, section contre section, avec les fusils électroniques et les senseurs qui allument leur petite lumière rouge pour vous dire que vous êtes mort, il ne craque pas. Il a trouvé comment voir, comment comprendre où il se trouve. Il est ce qu'il choisit d'être, personne ne peut en décider à sa place. Ceci est son entraînement aux Grandes Chasses, et il ne tuera pas.
Deux, trois, bientôt quatre semaines. Tout n'est pas aussi dur. Lian aime les exercices de nuit, être lâché n'importe où sur le champ de manœuvre, et nager, escalader, courir pour retourner à la Base, dans un temps limité, seul ou avec la section au complet. Il y est très doué – sa vision de nuit l'y aide autant que son habitude ancienne de la forêt, à vrai dire, mais il les passe sous silence. Les séances d'instruction technique, il aime aussi ; c'est fascinant, quelquefois, de voir ce que toutes ces machines peuvent faire. Toutes ne sont pas destinées à tuer. Et, de toute façon, il ne les utilisera pas ainsi.
Quatre, cinq, bientôt six semaines. Les sections prennent leur forme définitive, l'entraînement de base se termine. Lian est plus fort, plus rapide, plus habile. Presque calme, aussi. Il est sûr de savoir ce qu'il fait, ce qu'il fera.
Et puis un jour, au baraquement, Grayson discute avec deux autres gars et fait un commentaire sarcastique de trop ; le ton monte, Lian s'interpose, un des gars lui saute dessus. Lian se retrouve à genoux sur lui, la main levée pour le coup qui lui écrasera la carotide. Se redresse d'un bond, les oreilles bourdonnantes, va vomir dans les toilettes où il reste à genoux, tête basse, incapable de se relever.
La voix de Grayson résonne entre les carreaux de céramique derrière lui : « Liam ? »
Il appuie sur le bouton de la chasse d'eau, s'essuie la bouche et se relève tant bien que mal pour sortir du cubicule en chancelant. Il ne peut pas regarder Grayson. Il va se rincer la bouche à un lavabo, tombe face à face avec lui-même dans le miroir. Se détourne, tout raide, comme si ses articulations s'étaient grippées. « Je ne peux pas », s'entend-il murmurer d'une voix rouillée aussi. « Je dois partir d'ici.
— Mais qu'est-ce que tu as ? » demande Grayson, irrité et inquiet.
Lian le regarde avec désespoir. Même lui, il ne comprend pas ! Comment lui expliquer ? Lian voudrait s'arracher la chair des os, se couper les mains, les jambes, la tête, réduire en pièces ce corps qui l'a trahi : il disciplinait son esprit, et son corps a court-circuité son esprit, acquis des réflexes qui n'ont pas besoin de lui pour se déclencher !
« Gray, j'ai failli tuer Fred ! Je ne me suis pas arrêté parce que je le voulais mais juste... parce que je me suis rendu compte à temps ! Il faut que je m'en aille ! Je suis mort si je reste ! »
Grayson change d'expression. Il s'approche enfin, tout près, mais sans toucher Lian. « Tu ne peux pas déserter, Liam, c'est la rééducation assurée ! Écoute, on va bientôt commencer les entraînements spécialisés, demande à être médic. »
Lian reste un moment en suspens, balbutie : « Médic ? Je peux... demander ?
— C'est pas mal tard pour te déclarer objecteur de conscience, mais ça peut sûrement s'arranger. »
Lian s'appuie au lavabo, les jambes molles. Demander. Poser les bonnes questions. Imbécile, il n'a jamais pensé à demander !
Ils lui font rencontrer le psychologue, à qui il explique très simplement ce qui s'est passé. Il n'a rien à avouer, rien à inventer, juste la vérité nécessaire : comment il avait cru qu'il pourrait, comment il a compris qu'il se trompait. Le psychologue n'insiste pas, d'ailleurs ; pas plus que le capitaine Trent, qui écoute Lian en hochant la tête. On ne lui fera pas faire ce qu'il ne veut ou ne peut pas faire ; on a besoin de soldats efficaces ; tout le monde a sa place, il s'agit simplement de trouver laquelle. Les instructeurs sont déçus, ils voyaient très bien Lian comme éclaireur, mais ils se font apparemment une raison. Au début d'Août, quand les spécialisations commencent, il va suivre les cours et les entraînements des infirmiers. Avec Grayson, qui lui fait la surprise. « Je n'allais tout de même pas te laisser t'amuser tout seul ! » lui souffle-t-il en entrant le premier jour dans la salle d'instruction et en observant sa stupeur avec un évident plaisir.
Ils apprennent toutes les façons dont un corps humain est vulnérable, encore, et ils en ignoraient la moitié. Au début, c'est pire que l'entraînement au combat pour Lian – et infiniment pire pour Grayson, qui s'évanouit lors de la première opération à laquelle ils assistent pendant leur séjour à l'hôpital de la Base. Mais Lian serre les dents et reste jusqu'au bout. Il regarde les silhouettes en blouse verte qui s'affairent avec leurs étincelantes armes de vie, le corps inerte, ouvert, sanglant, mais vivant et qui vivra, et il sait qu'il a trouvé sa place. Mieux que ne pas tuer. Guérir.
21
Lian ne dit rien pendant que Nemgorot soigne les entailles de son visage, de sa poitrine et de son dos, quand il panse ses côtes cassées. Il ne dit rien quand Siridelln et Kamlaz passent devant sa cabane en riant très fort. Il ne dit rien non plus quand Thomas vient le voir. Il ne le regarde même pas. Au bout d'un moment, Thomas se lève et sort sans rien dire non plus.
Une semaine passe, deux, trois. Lian ne parle pas. Il recommence à sortir pour jardiner un peu, puis aller à la pêche, puis poser des collets. Les entailles se sont cicatrisées, les croûtes sont tombées ; ses côtes ne lui font plus mal quand il respire à fond. Mais il ne parle pas.
Un jour, alors qu'il est en train de nettoyer des poissons sous l'auvent de la cabane, Nemgorot traverse la pluie qui détrempe la place pour venir s'asseoir près de lui.
« C'est pour ça aussi que Dougall est parti », dit-il à mi-voix, sans le regarder, comme s'il se parlait à lui-même.
Lian continue à écailler son poisson.
« Tous les Krilliadni ne sont pas ainsi, reprend Nemgorot.
— Il ne faut pas tuer », martèle Lian d'une voix rauque, déshabituée de parler, et qui fait tressaillir le guérisseur.
« La loi... est différente ici, murmure enfin Nemgorot. Lian, as-tu jamais pensé à ce qu'elle signifie sur le continent ? »
Lian hausse les épaules.
« Elle signifie " il n'est pas bon de mourir ". Non, laisse-moi continuer ! Quelle est la fin de la vie, sur le continent, son but et son terme ?
— Rejoindre la Mer », dit enfin Lian entre ses dents serrées.
« Mourir, dit Nemgorot avec douceur. Ce qui termine la vie, ici comme dans le reste de l'univers, c'est la mort, Lian. Ceux qui ne rejoignent pas la Mer, quels qu'ils soient, où qu'ils soient, meurent. Et seule Hananai sait ce qu'il advient alors d'eux. »
Lian laisse tomber le poisson écaillé dans la bassine, en prend un autre.
« La Mer n'a pas toujours été là, reprend le guérisseur. Des centaines de milliers de générations sont tombées en poussière avant qu'elle n'apparaisse pour la première fois, et elles ont toutes connu la mort comme un destin inévitable. La Mer est là depuis longtemps – elle ne le sera peut-être pas toujours. La mort, elle, dansera encore avec la vie quand la Mer aura disparu. La mort, Lian, et la souffrance. Rien ne peut les faire disparaître. Croire qu'elles n'existent pas parce qu'on ne les rencontre pas souvent sur le continent...
— Parce qu'elles sont dans les îles ! » Les écailles giclent sous la pluie.
« Parce que tout les combat sur le continent. Sur Tyranaël, sur Atyrkelsaõ, depuis des milliers d'années. Mais personne ne doit les oublier. Oublier que la vie est fragile, l'humanité difficile. Et chaque adolescent, quand son temps est venu, doit aller dans les îles pour mériter sa vie, et l'éternité de la Mer, en risquant la violence et la mort et le mal. »
Lian n'écaille plus son poisson : il le déchiquette, méthodiquement.
« On les entraîne, poursuit Nemgorot d'une voix rauque mais obstinée. Certains écoutent mal. D'autres sont présomptueux, ou maladroits, ou imprudents. Ils ne prennent pas soin de leur équipement, ils ne regardent pas où ils vont ni ce qu'ils font... Il y a des accidents. Certains meurent sous les pattes d'un karaïker. Et certains... aux mains de Krilliadni. Mais la majorité revient sur le continent.
— Et ceux qui ne reviennent pas ? gronde Lian. Ces trois-là n'avaient pas droit à la Mer, comme tous les autres ?
— La Mer n'est pas un droit, Lian. L'éternité de la Mer n'est pas un droit. C'est une possibilité. Sur le continent, il y a des accidents, des maladies incurables. On tue, même, parfois, sur le continent. Et la Mer... La Mer se gagne, et difficilement. L'illumination qui permet de la rejoindre, l'achèvement, certains n'y parviennent jamais. Et certains... » La voix de Nemgorot se brouille : «... certains refusent la Mer. »
Le couteau de Lian s'immobilise.
« La Mer est un choix, Lian, le nôtre avant d'être le sien. »
Lian a jeté par terre les restes méconnaissables du poisson et le couteau que la pluie lave à grosses gouttes pressées ; son cœur bat douloureusement dans sa poitrine. « Et les naïstoï, ils ont le choix ? »
Nemgorot soupire : « Lian, certaines choses sont. Nous ne pouvons faire que la terre ne soit pas ronde, que la nuit ne suive le jour et la mort la vie. Nous n'y sommes pour rien non plus si la Mer est là, si l'éternité est possible pour certains et pas pour d'autres, si certains se voient octroyer des dons et d'autres pas. C'est ainsi.
— Et c'est bien parce que c'est ainsi ? » gronde Lian.
Le silence de Nemgorot est plus long. « Je n'ai pas dit que c'est bien ou mal, Lian, murmure enfin le guérisseur. Seulement que c'est ainsi.
— Et Siridelln et Kamlaz, et les trois qui n'auront plus jamais le choix, c'est ainsi ! ? »
Nemgorot détourne les yeux. « D'une certaine façon, oui », dit-il enfin d'une voix brisée.
Lian se lève en renversant son tabouret : « Non, non et non !
— Que leur feras-tu ? » s'écrie Nemgorot avec une passion soudaine. « Les tueras-tu à leur tour, pour les punir ? Et toi alors, qui te tuera ?
— Combien en ont-ils tués ? » rétorque Lian, les yeux brûlants.
Nemgorot s'affaisse sur lui-même : « Trop. Mais...
— Quand un karaïker tue trop de Krilliadni, quand il prend le goût du sang, est-ce qu'on ne le chasse pas sans honneur, avec des trappes à pieux ? »
Il y a une sorte de désespoir, maintenant, dans le regard du guérisseur : « Lian, dit-il enfin, ni Siridelln ni Kamlaz ne sont des karaï.
— Ils sont pires : ils savent qu'ils tuent pour le plaisir !
— Tu le dis toi-même, réplique Nemgorot d'une voix sourde. Pire que des bêtes : des humains. » Il se redresse soudain, prend une grande inspiration en se passant les mains sur la figure : « Écoute-moi, Lian. N'as-tu pas essayé de tuer Siridelln ?
— Pour défendre les autres !
— Mais lorsqu'ils t'ont laissé vivant dans la caverne, ne les aurais-tu pas tués, si tu l'avais pu ? »
Lian remet son tabouret sur ses pieds, se rassied lentement. « Et toi ? » finit-il par dire d'un ton buté.
Nemgorot esquisse un sourire las : « Nos races sont un peu différentes, jeune halatnim. Nous sommes beaucoup plus anciens que vous. Votre violence est plus neuve. Pourtant, même nous, nous n'avons pu faire disparaître la nôtre.
— On peut faire disparaître Siridelln et Kamlaz !
— Oui, en étant aussi violent. C'est ce qui arrivera un jour : ils rencontreront plus fort qu'eux et périront selon leur propre loi.
— Aux mains d'un autre Siridelln ! »
Nemgorot soupire : « Pas forcément. » Il médite un moment. « Thomas t'a sauvé la vie, dit-il avec lenteur. Deux fois. Il est venu me prévenir. Il n'avait rien fait, il était juste resté là. Aurais-tu voulu le tuer, lui aussi ? »
Lian croise les bras pour réprimer le tremblement qui l'a saisi. Quand il est sûr de pouvoir parler, il dit : « Il n'a rien fait. Il est juste resté là. Il est déjà mort. »
Le silence se prolonge.
« Il est mort », dit enfin Nemgorot d'une voix presque inaudible. « Il s'est coupé la gorge il y a trois semaines. »
Lian regarde la pluie tomber. Il ne sent rien, il ne pense rien, il ne bouge pas. Il ne pourrait pas.
« Lian », reprend enfin Nemgorot après un très long moment, comme s'il avait pris une décision, « aurais-tu puni les hasallim s'ils s'étaient défendus avec succès, s'ils avaient tué Siridelln et Kamlaz ? »
Lian cherche sa voix, murmure : « Non.
— Ils se seraient punis eux-mêmes ! » Le guérisseur s'est levé, penché vers Lian, sa voix tremble. « Ils se seraient punis parce que leur négligence, en pays de Krilliadni, aurait induit autrui en tentation pour son malheur ! Crois-tu qu'ils ignorent ce qui les attend ici ? On leur recommande d'éviter les villages, de s'assurer qu'on ne les suit pas, de toujours avoir un veilleur... Ces trois-là ne l'ont pas fait, ou mal. S'ils s'en étaient sortis vivants en utilisant leurs dons, et les autres morts, ils auraient été horrifiés, ils se seraient imposé une pénitence, peut-être à vie ! C'étaient des aspirants hékel ! »
Sa voix se casse brusquement.
« Toi aussi, dit soudain Lian. Tu es un hékel. Tu n'as jamais été un Krilliadni. »
Nemgorot s'est rassis : « Oui, je suis un Krilliadni, dit-il enfin d'une voix trop calme. J'ai renoncé à la Mer. Et non, je n'ai jamais été un Krilliadni comme Siridelln ou Kamlaz. »
Il se lève, fait deux pas dans la pluie. Quand il se retourne, son visage ruisselant est indéchiffrable.
« La dernière fois que nous avons entendu parler de ton père, Lian, il se trouvait dans le Nord, à Ansaalion. »
22
Six jours après son départ de Lïu-Ganzer, à l'un des arrêts du caboteur sur le canal qui relie la pointe nord du grand lac à la Hleïtsaõ, Lian aperçoit une silhouette familière parmi les nouveaux passagers qui montent sur le pont. Il reste à la place qu'il s'est choisie.
Il porte toujours les habits de peau qu'il avait troqués contre les siens au village – ses autres vêtements ne lui faisaient plus, il avait beaucoup grandi en un mois. Rien ne signale vraiment qui il est ni d'où il vient hormis peut-être les cicatrices de son visage tailladé. Pourtant, une fois sur le continent, il a constaté qu'on lui offrait vivre et couvert – dans des limites raisonnables, mais sans question. C'est la coutume pour ceux qui reviennent des îles. « Tu es le bienvenu parmi nous », lui dit-on quand il remercie. Et puis, sur le bateau de la Mer qui traversait le détroit, et en remontant la rivière jusqu'au Hleïtan, il s'est rendu compte aussi qu'il ne s'attirait aucun regard curieux ou irrité ; on lui souriait même, parfois. On voit en lui un Chasseur et un halatnim, de retour sur le continent : cela suffit à tout le monde. Toutes ses différences, celles qui se voient et les autres, se trouvent comme désarmées : il ne dérange personne puisqu'il porte bien haut son étiquette. Au début, il était furieux qu'on le prenne pour un Chasseur : il a essayé de dissuader ses hôtes, il était prêt à travailler pour son écot. Ils ont été surpris, alors, et même un peu inquiets. Mais il n'a pas voulu discuter plus avant : si toutes ces histoires de Ranao tournent finalement à son avantage, pourquoi ne pas en profiter ? Quelle importance ? Cela ne le concerne pas. Et il a trop hâte de se rendre à Ansaalion.
Il ne bouge pas lorsqu'Odatan vient s'accouder près de lui au bastingage, à côté de la grande roue à aubes de bâbord. À quoi bon ? Un instant, très brièvement, il s'est imaginé en train de prendre le petit homme à bras le corps et de le jeter par-dessus bord, mais si Odatan se trouve là, ce n'est sûrement pas pour se laisser décourager par une démonstration aussi grossière. Il suffit de ne pas lui adresser la parole, de ne pas lui répondre. Le petit homme ne dit rien. Ils assistent côte à côte au chargement des dernières caisses, regardent la grande roue se mettre lentement en branle après le coup de sifflet annonçant le départ. Lian, comme à son habitude, se perd dans le mouvement de la roue, le bruissement de l'eau cascadante, hypnotique, un mantra qu'il n'a pas besoin de répéter lui-même. Quand Odatan s'éloigne, c'est à peine s'il le remarque.
Le canal traverse en ligne presque droite la haute plaine d'Utyrkreïtao, entre les deux rivières : trois jours de voyage avec les arrêts. Il en reste deux. Ce n'est pas un très grand bateau et, même si Lian vit sur le pont, il doit aller chercher à manger au réfectoire comme tout le monde. Il aperçoit Odatan à plusieurs reprises pendant ces deux jours. La première fois qu'ils se croisent dans la coursive, l'autre ne fait pas mine de ne pas le voir, ne détourne pas les yeux, incline brièvement la tête, passe. Les autres fois, Lian met un point d'honneur à soutenir aussi son regard – sans inclinaison de tête, bien entendu.
Quand ils arrivent à la Hleïtsaõ, Lian a un bref espoir, mais voit le petit homme monter après lui sur le premier des caboteurs qui doivent l'emmener jusqu'à Ansaalion. Il ne va pas attendre le prochain et retarder son voyage, n'est-ce pas ? Ni au deuxième, ni au troisième transfert. Le quatrième caboteur est un vraiment petit bateau, pour passagers seulement. Quand ils se croisent, inévitablement, dans la coursive du réfectoire, ils essaient de s'éviter mais partent tous deux en même temps à droite, puis à gauche, face à face chaque fois. Odatan s'immobilise et s'efface contre la paroi, avec une ombre de sourire. Lian passe et remonte sur le pont avec son plateau. Il se sent soudain ridicule.
Des joueurs qu'il a déjà remarqués sur le bateau précédent sont encore rassemblés autour de leur grande carte du continent. Odatan s'arrête à son tour pour les observer. Lian demande à mi-voix – et ils sont les seuls badauds : « À quoi jouent-ils ? » Le petit homme répond, très naturellement, en setlâd : « Odhabaï », pour traduire ensuite : « Le jeu de la perfection. » Lian ferait bien mine encore un moment de parler à la cantonade, mais il abandonne cet ultime enfantillage, même s'il continue en virginien : « C'est quoi ? »
La carte, il l'a reconnue, est celle du continent principal, Hébu, sans la Mer. On joue avec un dé à six chiffres et six couleurs, et deux jeux de cartes ; l'un est fort mince – les vingt-huit métiers traditionnels, auxquels correspondent des pions aux couleurs du dé, un par joueur, et des figurines représentant chaque métier ; l'autre est beaucoup plus épais : les cartes renvoient à des villes, des villages, parfois des bois, des collines, des étangs ou des points arbitraires sur le continent – du moins arbitraires pour Lian : de toute évidence, ils ne le sont pas pour les joueurs. Il y a présentement cinq joueurs, mais ils doivent toujours être au moins deux. Les cartes attribuent à chacun un point de départ et un métier, le dé un chiffre et la couleur correspondante. Jaune pour les danvérani, vert pour les tzinao, noir pour les keyrsani, roux fauve pour les aïlmâdzi et rouge pour les kvaazim. Et bleu pour les hékel, qui ont tous les dons.
Mais une autre catégorie n'est pas attribuée par le dé : les krilliadni, en blanc. Apparemment, hékel et krilliadni sont deux joueurs essentiels. Quand il y a moins de sept joueurs, et même lorsqu'il n'y en a que deux, on tire le dé selon des règles compliquées pour déterminer qui sera le hékel et qui le krilliadni.
Le but du jeu consiste à être parfait, " elnaos ", à finir le premier en ayant déposé tous ses pions sur la carte après avoir exercé chacun des vingt-huit métiers. On roule le dé, chacun joue quand sa couleur sort, on s'échange cartes de lieux et de métiers, et le krilliadni joue toujours après le hékel.
Le fonctionnement du jeu paraît simple lorsqu'Odatan le décrit ainsi, mais Lian se rend vite compte, en suivant la partie en cours, qu'il doit y avoir quantité d'autres règles dont le petit homme n'a pas fait mention. On ne roule pas souvent le dé, pions et figurines se déplacent bien lentement, chaque coup suscite entre les joueurs d'interminables discussions. On y aligne des informations factuelles (souvent très détaillées) sur métiers ou lieux, des citations – parfois fort longues – dont Lian ne peut dire s'il s'agit de poésie, de philosophie, de textes légaux, ou des trois à la fois. À d'autres moments, deux joueurs se mettent à discuter entre eux à mi-voix, et leur ton indique un échange serré de questions et de réponses ; les autres écoutent avec attention. Et quelquefois personne ne fait rien, et tout le monde contemple longuement la carte d'un air méditatif.
Le krilliadni, en tout cas, bénéficie d'un statut particulier : chaque fois qu'il joue, il peut demander à n'importe quel joueur de prendre son métier, tandis que lui-même peut prendre n'importe quel autre métier et avancer ses pions trois tours d'affilée. Inversement, le hékel est au service de tous les joueurs, y compris le krilliadni, peut être détourné de son chemin n'importe quand et, contrairement aux autres joueurs, n'utilise jamais pour lui-même aucun des pouvoirs dont il est pourvu.
« Jolie propagande », remarque Lian, toujours en virginien, après les explications d'Odatan. « Pauvre hékel toujours en train de se sacrifier pour le bien de tous ! »
Il a un peu trop élevé la voix et le joueur qui joue le hékel lève la tête d'un air surpris, voit qui a parlé, se concentre de nouveau sur le jeu.
« As-tu déjà oublié Nemgorot ? » murmure Odatan.
Pris de court par tout ce qu'implique la question, Lian réplique enfin, un peu au hasard : « Nemgorot s'est puni. »
Odatan soupire : « Non. Nemgorot n'a pas voulu jouir de ce que tous ne peuvent recevoir.
— Et vous ? » contre-attaque Lian.
Un petit muscle tressaille dans la joue d'Odatan. « Je ne suis pas un hékel », répond-il – à côté, mais Lian ne se soucie plus d'être poli et précise : « Pourquoi êtes-vous allé dans les îles ? »
Le petit homme tourne la tête vers lui, le regarde un moment avec une calme patience, jusqu'à ce que Lian se sente presque rougir, mais du diable s'il détournera les yeux. Odatan dit enfin : « Parce que je ne savais pas où était ma place, et que la paix se refusait à moi. »
Lian hausse presque les épaules. La ligne du parti.
« Crois ce que tu choisis de croire », murmure Odatan avec une résignation triste.
« Très bien. Expliquez-moi en quoi ce jeu n'est pas de la propagande. »
Odatan lui prend le bras et l'entraîne à l'écart des joueurs, qui se sont tous arrêtés, les bras croisés, et les regardent de l'air ostentatoirement patient de qui est dérangé. Ils s'asseyent sur des caisses, à l'arrière, toujours à l'abri de l'auvent de toile tendu sur le pont. Le petit homme contemple les planches du pont à ses pieds, méditatif. « D'abord, la propagande des uns est la vérité des autres, Lian. Ensuite... je suis un peu mal placé pour te parler du jeu. Je n'y ai jamais joué moi-même. Quant à la recherche de la perfection, de l'achèvement... Je suis un naïstaos, pas un elnaos, je ne rejoindrai jamais la Mer, je ne peux guère en parler non plus. Mais je peux te parler de la vie. »
Il relève la tête, esquisse un sourire en regardant Lian qui mange posément, assis en face de lui : « C'était le nom du jeu, autrefois, " Le jeu de la vie ". Tu n'as pas eu l'occasion de le remarquer sur le Landaïeïtan, ni tellement après, mais les Ranao voyagent beaucoup, pendant presque toute leur existence. Quand ils sont jeunes, ils restent quelques saisons ici, puis là, vont encore ailleurs, changent chaque fois de métier, parfois de nom. Même ceux qui restent longtemps au même endroit, dix, vingt, trente saisons, qui ont fondé une famille, finissent par aller ailleurs une fois les enfants grandis, et parfois leurs enfants partent avec eux. Parfois au hasard. Le plus souvent selon les mêmes règles qui régissent le jeu, et qui ne sont écrites nulle part. Je les étudie depuis que je suis ici, en observant des joueurs, et je commence seulement à comprendre certaines progressions. La culture rani est complexe, très ancienne. Quatre cultures, et tous les hybrides qu'elles ont produits au cours des siècles, les Aritnai avec les Hébao, les Paalani avec les Hebao et les Aritnai, les Tyranao avec tout le monde... Plus de sept mille cinq cents années d'histoire documentée, Lian, plus de trente mille saisons, peux-tu imaginer cela ? L'histoire documentée des humains, sur la Terre, n'avait guère plus de cinq mille saisons quand ils ont quitté leur planète. »
Et le jeu est un aide-mémoire autant qu'un guide, pour les Ranao. Ils ne voient pas seulement de l'espace lorsqu'ils regardent la carte et s'y déplacent. Ils voient aussi du temps. Non seulement leur histoire, mais la façon dont leurs sociétés se sont transformées, et pourquoi.
« Les dons, dans leur forme actuelle, n'ont été intégrés qu'assez récemment au jeu, six cents années, peut-être. Ils existent depuis... oh, deux mille années. La Mer est arrivée il y a un peu plus de mille années. Sans doute un jour fera-t-elle partie aussi du jeu. Pour l'instant, on a seulement supprimé les deux autres continents de la carte, et on a changé le nom du jeu. »
Lian finit posément son bol de légumes. « Et les Chasseurs, on les a intégrés quand ?
— En même temps que les hékel. C'est l'addition la plus récente, quatre cent cinquante années. Ils existaient depuis trois fois plus longtemps. »
Mille huit cents saisons. « C'est vrai que le progrès n'est pas exactement foudroyant chez les Ranao, remarque Lian, sarcastique.
— Le gain des uns est la perte des autres, dit Odatan.
— Et la manipulation des uns le hasard des autres. » Lian, qui n'est plus amusé du tout, pose son plateau sur le pont. « Vous êtes ici par hasard, bien entendu.
— Oui et non. J'étais en chemin vers Ansaalion. J'ai pensé que j'aimerais faire route avec toi, et je t'ai attendu.
— Comment savez-vous que je vais à Ansaalion ?
— Ton père s'y trouve.
— Et vous l'avez appris à Nemgorot.
— Non. Je viens de l'apprendre. Il doit le tenir d'une autre source. »
Mais quelle importance, après tout ? « Et si je ne désire pas votre compagnie ? »
Le petit homme ouvre un peu les bras, englobant le reste du bateau : « Il y a de la place. »
Lian ramasse son plateau, se lève. « Vous allez faire quoi, à Ansaalion ? » demande-t-il pour avoir le dernier mot, car il ne croit pas obtenir de réponse.
« M'occuper d'une petite naïstaos », dit Odatan d'une voix patiente.
Il serait prêt à poursuivre la conversation ? Pas de chance, Lian n'a vraiment plus rien à lui dire.
*
La troisième semaine d'Août, ils quittent la Base pour Morgorod, où ils compléteront leurs spécialisations. À la gare, Lian jette un coup d'œil curieux à la locomotive ; l'aspect en est un peu différent, mais le principe exactement le même que celui des engins ranao vus autrefois dans les fresques. On a mis le bataillon dans un train spécial, pas de civils à bord. Sur les quais bondés, parents et amis se pressent entre les uniformes bruns et les gros sacs ; on s'étreint, on fait semblant de sourire, on essaie de ne pas pleurer. Lian est monté tout de suite dans le compartiment assigné à leur section, en tête du train, avec Grayson qui n'a pas d'adieux à faire non plus ; il est venu étudier à Bird-City contre le gré de sa famille de Cristobal, il n'a formé aucune relation durable à l'Université Œniken, c'est tout ce qu'il a jamais consenti à dire : il est aussi laconique sur son passé que Lian, quoique pour des raisons évidemment différentes ; il ne lui a jamais demandé d'où lui venaient ses cicatrices, par exemple ; ils s'accordent à se ménager réciproquement dans leurs silences.
Et le train part, quatre mille kilomètres de voyage vers l'Est, vers Morgorod. Une demi-journée à travers les Plaines Dorées, puis, après avoir franchi la Dandelion à Bellac, où l'on accroche les wagons d'un autre bataillon, le reste de la journée jusqu'au port militaire d'Anspach, sur le Dolgomor ; là, on transfère les wagons dans d'énormes ferries pour la traversée du lac, qui les amènera directement à Morgorod.
À Hleïtzer. À travers la plaine d'Atéhonal jusqu'à Hanatsaõ, la ville double sur la grande rivière, à travers les collines et le plateau d'Utyrkreïtao, jusqu'à l'immense étendue verte et grise du Hleïtan dans le lointain. Après la première heure de voyage où l'on traverse la zone industrielle de Bird-City, une fois dans la campagne où la présence virginienne s'efface presque, Lian doit bientôt regarder ailleurs ou fermer les yeux : les boisés, les champs, les villages entrevus, le double ruban écarlate de la route qui s'approche et s'écarte entre les arbres, tout ressemble trop à ses souvenirs. Les seuls détails qui ne concordent pas – et il les suit longuement du regard, au début – ce sont les espèces de pylônes métalliques qui se dressent ici et là, d'une dizaine de mètres de haut, aux flancs incurvés, avec une petite boule étincelante fichée au sommet ; ils brillent pourtant du caractéristique éclat argenté du sirid, et Lian sait que les Virginiens n'utilisent pas cet alliage. Il n'ose poser de question directe, lance à la cantonade, « Ils font drôles, ces pylônes », et Chris enchaîne : « Je me suis toujours demandé ce qui fait briller la boule, moi.
— Réaction chimique à l'atmosphère, non ? lance Delgado.
— C'est ce qu'ils disent ! » fait Abdul, qui doute toujours de toutes les versions officielles.
« Artefact des Anciens, abracadabra, mystère et boule de lumière », conclut Grayson avec un sourire en biais.
Lian ne regarde plus les pylônes.
Lorsqu'ils arrivent à Bellac, dont la moitié ouest s'arrondit toujours au bord de la rivière, pratiquement intacte, c'est comme un soulagement de passer sur le large pont blanc de trois kilomètres, en solide béton virginien, qui relie les deux rives, de voir comme l'autre moitié jumelle de la cité, sur la rive est, a été saccagée par l'insurrection et la guerre et presque entièrement reconstruite sur le modèle à angles droits. Le port d'Anspach est également un répit bienvenu, tout en lignes sévères et fonctionnelles, faisceau de voies ferrées, longs quais gris, alignements d'entrepôts, de réservoirs, de baraquements ; il n'y avait rien sur le lac à cet endroit, de l'Autre Côté.
La gueule caverneuse et sonore des ferries engouffre leur wagon ; comme le train a été doublé à Bellac et qu'on commence par vider les wagons de queue, c'est seulement après une bonne heure de voyage que Lian et Grayson peuvent monter à leur tour sur le premier pont pour s'accouder au bastingage. Heureusement, il fait nuit maintenant, le lac est juste une vaste étendue obscure sous le ciel couvert, on voit seulement les lumières du ferry qui est parti en premier, quelque part en avant, sur la gauche. Et il est bientôt l'heure de l'extinction des feux.
Mais ils arrivent à Morgorod en plein milieu de la journée du lendemain. La ville se déplie peu à peu sous le soleil, le port au premier plan, encore assez étranger avec ses troupeaux de réservoirs blancs et gris et surtout les silhouettes dégingandées de ses grues, bizarres animaux aux longs cous et aux pattes de métal ajouré sur le ciel laiteux. L'étagement des plateaux, ensuite, les épaules asymétriques de la montagne à l'arrière-plan... Les autres recrues excitées se hèlent d'un pont à l'autre, se désignant tel ou tel détail du doigt ; c'est la véritable capitale de Virginia, la haute ville se trouve au-dessus de l'influence de la Mer quand elle est là, on y a l'électricité pendant toute l'Année et, parmi les slogans lumineux de ses enseignes géantes, malgré les ravages de l'Insurrection, elle dresse toujours fièrement ses tours, ses pyramides, ses forteresses aux façades miroitant dans la lumière.
Lian contemple les deux rubans de l'avenue principale qui monte tout droit à travers les plateaux en se divisant autour du vaste édifice écarlate et doré du Capitole, au pied de la montagne : le parc est dévoré par des tours plus hautes que les murailles obliques du Haëkelliaõ. Lian sait pourquoi Odatan était si silencieux, lorsqu'ils sont arrivés à Hleïtzer : la double vision est presque insoutenable.
23
Lian a établi une trêve tacite avec Odatan ; ils ne s'évitent pas, se parlent peu, de sujets anodins – Lian en virginien, Odatan en setlâd. Ils jouent aux cartes, ou aux échecs sur le petit échiquier de voyage d'Odatan, s'attirant les regards curieux de quelques passagers. Certains sont intéressés, quelques-uns demandent à apprendre les règles, d'autres les connaissent déjà. Ce jour-là, à l'un des arrêts, tandis qu'on embarque et débarque, une passagère qui se rend comme eux à Ansaalion a remplacé Lian devant Odatan, une petite Hébaë grassouillette d'âge moyen nommée Gamrit. Elle renverse son roi pour la deuxième fois en soupirant : « Ce jeu est définitivement trop agressif pour nous. »
Ce n'est pas la première fois qu'on fait ce genre de commentaire autour de l'échiquier. Lian ne peut s'empêcher de murmurer : « Hypocrites !
— Non, Lian », dit Odatan en replaçant les pièces dans leur boîte. « Ils sont tous allés dans les îles.
— Ils n'y ont pas appris grand-chose !
— À poser des questions », dit Gamrit en virginien.
Lian se fige ; il ne lui était pas venu à l'esprit qu'on pouvait le comprendre : sauf à Hleïtzer, et au Haëkelliaõ, il n'a presque jamais rencontré de Ranao bilingues. Il dévisage la petite femme, surpris : intervenir comme elle vient de le faire n'est pas habituel non plus – mais les Hébao sont notoirement directs. Il répète – en passant au setlâd, une politesse moqueuse : « Des questions, dites-vous. »
La femme répond en setlâd à son tour, sans broncher : « Certaines ont des réponses, et les grandes chasses nous obligent à les trouver. Qu'est-ce que la loi, comment elle vit en nous et par nous, comment nous sommes libres de la choisir, libres de la changer, mais non libres des conséquences de nos choix. D'autres n'ont pas de réponse, et elles nous obligent à nous en souvenir. Pourquoi pouvons-nous tuer ? Pourquoi pouvons-nous mourir ? Pourquoi la mort quand il y a l'éternité de la Mer ? Les grandes chasses nous donnent ces questions, en les inscrivant parfois dans notre chair, et nous les emportons ensuite avec nous, pour toujours. »
Elle suit des yeux, avec gravité, les cicatrices encore visibles sur le visage de Lian : « Tu as quitté les îles. Et tu es le bienvenu parmi nous. »
Lian reste muet. Puis il se met à rire, avec une fureur stupéfaite. La femme se trouble alors, se lève, une main tendue en geste d'excuse, mais il ne la touche pas. Elle reste un moment immobile, navrée, puis salue Odatan en silence et s'éloigne.
Dans le dernier tiers de son cours, alors qu'elle se rapproche de sa source dans le Leïtltellu, la grande rivière se fait moins large et plus tumultueuse, encaissée entre des collines de plus en plus abruptes qui deviennent bientôt des falaises de granit veinées d'écarlate. Le dernier bateau, dans la matinée du septième jour de voyage, s'arrête à une dizaine de langhi des chutes Ohmalad, au pied du plateau de Hmaladukreïtao, et tous les passagers pour Ansaalion se rendent jusqu'au canal qui double la rivière désormais infranchissable ; d'écluse en écluse, trois autres jours de voyage les mèneront à leur destination.
Sur le dernier caboteur, au sortir de la passerelle, Odatan examine son billet pour s'assurer du numéro de sa cabine ; placé par le hasard à deux pas de lui, Lian s'apprête à explorer le pont pour se trouver une place où dormir. Il entend quelqu'un s'exclamer : « Oghimiu ! », lève machinalement les yeux. Un Paalao aux cheveux tout blancs se précipite vers eux. Vers Odatan, devant lequel le vieil homme tombe à genoux, extatique, les mains levées, en répétant « Oghimiu, bénis-moi, Odatan Oghimiu ! »
Odatan reste un instant pétrifié, puis il prend les mains du vieillard, essaie de le relever en murmurant « Non, non... ». Mais le vieux s'accroche à lui en répétant « Bénis-moi, bénis-moi, Oghimiu, et je serai parfait ! »
Lian apitoyé s'approche, déprend les mains du vieil homme de la veste d'Odatan et le met debout : « Venez, grand-père, ce n'est pas celui que vous croyez, venez à l'ombre. »
Le vieil homme le dévisage d'un air incrédule : « Ne sais-tu pas en quelle compagnie tu voyages, jeune halatnim ? s'écrie-t-il d'une voix fervente. C'est Matieu Odatan Oghimiu, le premier passeur, l'enfant de la prophétie ! Il porte l'éclair sur son front, le fils d'Oghim ! » Le vieillard s'arrache à l'étreinte de Lian, se précipite de nouveau aux pieds d'Odatan.
Lian prend alors conscience de la curieuse immobilité des autres passagers autour d'eux. Du silence, qui a gagné le quai. Des regards fixés sur Odatan, qui ont peine à se détourner.
Le petit homme, les traits contractés, pose une main sur la tête du vieillard.
« Nous sommes bénis, bénis de ta présence ! » s'exclame celui-ci d'une voix qui se brise en sanglots. « Oh, ma vie est parfaite maintenant que j'ai pu te revoir et te toucher. Merci, Oghimiu, merci ! »
Un homme et une femme se détachent enfin de la foule, viennent prendre le vieux Paalao chacun par une main et l'entraînent à l'écart en lui murmurant des paroles apaisantes ; cette fois, il se laisse faire. Odatan n'a pas bougé. Tête basse, les bras ballants, sans regarder personne, il déclare d'une voix distante : « Je dois me reposer. » Il ramasse son sac et se dirige vers la porte menant aux cabines. Mouvements et bruits reprennent sur le pont, par saccades, tandis qu'on s'écarte de son chemin en feignant de poursuivre le sien.
Lian médusé bondit avec un temps de retard, rattrape la porte qui se rabat et dégringole l'escalier pour rejoindre Odatan dans la coursive. L'autre ne s'arrête pas, ne le regarde pas. Son visage paraît taillé dans la pierre, et la phrase à la fois moqueuse et scandalisée que Lian allait lui adresser meurt sur ses lèvres. Odatan ouvre la porte de sa cabine, ne la referme pas. Lian entre derrière lui. Le petit homme place son sac sur la couchette, en sort posément ses affaires de toilette, des habits propres, un livre. Lian cherche une chaise et s'y laisse tomber, saisi d'un vertige. Toutes les questions semblent dérisoires face à l'énormité devinée. Des éclairs d'images, des fragments de dialogues passés se bousculent dans sa mémoire. Odatan à Dnaõzer, Odatan au bord du Leïtnialen – et, la première rencontre, Odatan au bord de la Mer. Finalement, la seule question qu'il se trouve capable de poser, celle qui n'est pas totalement absurde pour lui, c'est : « Mes parents, ils savaient ?
— Oui », dit Odatan, dans un souffle rauque. Il se redresse. « Et toi, tu n'avais pas besoin de le savoir ! »
Il fixe sur Lian un regard étincelant, comme s'il le mettait au défi de répliquer, mais Lian n'a rien à dire ; que pourrait-il dire ?
« Je ne l'ai jamais caché. Je n'ai jamais voulu... de secret, de mensonge », reprend le petit homme de la même voix obstinée, déchirée. « Mais je n'allais pas non plus tout le temps le crier sur les toits ! Tu as bien vu ! Au début, ils voulaient faire de moi je ne sais quel être surnaturel ! Oh, pas tous. Il y en avait qui étaient terrifiés. La vision d'Oghim, la prophétie réalisée. Croire est une chose. Avoir la preuve... Mais la preuve de quoi ? Ce n'est pas comme si c'était la première fois que des visions se réalisaient peut-être, la Mer est arrivée, les Virginiens sont arrivés sur Tyranaël ! Et ce n'est même pas sûr, que je sois le Matieu d'Oghim. Les détails... ça ne concorde pas exactement. Mais c'était assez, c'était trop, comme une pierre dans un étang et les cercles n'en finissaient pas d'aller tout bouleverser, encore, oh, il y en a eu des nouveaux Krilliadni à cause de moi... »
Le petit homme se passe les mains sur la figure comme s'il voulait s'arracher la peau, ou s'obliger à se taire. Il respire à plusieurs reprises entre ses doigts, profondément. Contemple Lian foudroyé. Secoue la tête en se laissant tomber sur la couchette.
« Et juste quand je commençais à trouver une paix, dans les îles, je me suis rendu compte... que je ne pouvais pas être tué. » Sa voix est plus mesurée, sans inflexion. « Je n'avais pas beaucoup changé physiquement depuis que j'étais passé, non plus, mais je n'y avais pas vraiment prêté attention. Trop occupé. » Une ombre d'ironie : « Du coup, je suis retourné sur le continent. J'avais quitté ma famille, mes enfants, mes amis, mais je ne voulais plus manquer un instant de leurs vies, maintenant que je savais comment j'allais les perdre. »
La phrase s'est achevée dans un murmure. Lian ne bouge pas. Il a presque peur de respirer.
« Et là, d'autres passeurs ont commencé à arriver. Il fallait les aider. Pendant un temps, ça m'a aidé aussi. Eux, ils ne me voyaient pas tout à fait de la même façon. » Un nouvel éclair d'ironie, plus cruel : « Ça n'a pas duré. Au fil des générations, eux aussi ont cru de moi ce qu'ils avaient besoin de croire. Un fou. Un menteur. Les autres me voyaient bien en étalon reproducteur – mes précieux gènes intacts de danvéràn et de Dieu sait quoi encore, pour retarder la dérive génétique des halatnim. Les Keldarao, en particulier, m'auraient bien gardé pour eux. Pas de chance, je me reproduis avec qui je le désire, quand je me reproduis. Un traître, alors. Ah, l'hystérie, quand les premiers têtes-de-pierre ont recommencé à naître parmi eux ! »
Le terme virginien au milieu du flot de setlâd fait sursauter Lian. Le petit homme doit s'en rendre compte car il précise : « Le nom qu'on donnait aux gens comme toi de l'Autre Côté. Aux gens comme moi. Pas tous des passeurs, remarque. Ils ont eu cet espoir, au début, ici, avec les premiers naïstoï. Ils en ont envoyé un sur la Mer. Elle l'a ramené sur la rive avant de partir, bien sûr, comme elle l'a toujours fait de tout ce qu'on a essayé de faire passer depuis Atyrkelsaõ. Ils n'avaient pas voulu me croire. L'homme est devenu fou de désespoir. »
La tête basse, le petit homme caresse d'un air absent la couverture du livre posé près de lui. Après un long silence que Lian ne saurait comment interrompre, il reprend d'une voix lente : « On dit qu'on ne peut pas mentir à tout le monde tout le temps. Mais on ne peut pas non plus dire la vérité à tout le monde tout le temps. Le plus dur, finalement, c'est d'être entre les deux, de ne jamais savoir quand dire quoi, et à qui. »
Il relève la tête, son visage a une expression presque implorante : « Je ne veux rien de toi, Lian, dit-il d'une voix qui se brise. On a essayé... de me forcer à ouvrir mes portes, quand j'avais ton âge. De l'Autre Côté. Je ne te forcerais jamais. Personne. Je pensais que l'Esprit de la Mer et les hékel pourraient... Mais je me trompais. Et c'est fini. Tu as choisi d'essayer, tu as choisi de ne pas continuer, ça s'arrête là. Je veux seulement faire un bout de chemin avec toi, comme je te l'ai dit. Pas de motif secret, pas de machination...
— Choisi ? » souffle Lian avec un temps de retard.
« On ne t'a pas forcé, Lian. Tu pouvais dire non n'importe quand.
— Même si je ne le savais pas ? »
Une grimace douloureuse brouille les traits du petit homme. « Il fallait demander. »
Lian dit, mais avec moins de violence qu'il n'en avait eu l'intention. « Comme si j'avais pu !
— On peut toujours, Lian. Poser des questions.
— Ce n'est pas poli ! »
Odatan soupire : « Mais si on veut poser une question quand même, malgré les règles de politesse, c'est qu'elle est importante, n'est-ce pas ? Qu'on n'est ni irrité ni honteux de son ignorance, et qu'on n'a pas peur d'apprendre. » Il regarde dans le vide, accablé. Murmure au bout d'un moment : « J'aurais dû continuer à respecter le choix de Laraï. J'étais tellement aveuglé par mes certitudes... Non, ça ne sert à rien d'être vieux. » Il dévisage longuement Lian. « Un jour, je te demanderai de me pardonner, quand tu pourras m'entendre. J'aurais dû te laisser sur le plateau.
— Non », dit Lian.
Il répète, étonné : « Non. » Il ne sait trop comment expliquer, s'expliquer à lui-même cette soudaine certitude. Il n'aurait pas pu rester sur le plateau. Trop de secrets, trop de mensonges, trop de questions, leur poids accumulé aurait fini par crever l'œuf protecteur, et il serait parti. Il ne sait pas où, peut-être pour les îles, mais pas forcément. Ou pas tout de suite. Il dit tout haut : « Je serais parti. » Ensuite, incertain de ce qu'il ressent, il se lève. Son regard croise celui d'Odatan. Le premier passeur, toujours vivant, et tout d'un coup – un paradoxe que Lian ne peut appréhender, seulement reconnaître – c'est bien plus horrible d'être vivant que d'être mort. Odatan, Matieu Odatan Oghimiu, le passeur de la prophétie d'Oghim. Mais c'est toujours le même petit homme à la tignasse emmêlée, aux yeux pâles, au regard douloureux.
« Vous devriez vous reposer, s'entend-il dire d'une voix sans tranchant. C'est l'heure de votre méridienne. »
24
Ansaalion est la plus ancienne cité du continent – sur Tyranaël. Sur Atyrkelsaõ, elle a moins de mille saisons, mais c'est assez pour lui assurer une patine respectable. Ils arrivent vers la vingtième heure, c'est le plein été, le soleil rase l'horizon. Depuis qu'il a quitté la zone tropicale, Lian a découvert les splendeurs des couchers de soleil. Une mort lente, mais belle. En plissant les yeux et en rêvant un peu, on ne distingue plus la limite entre le ciel et l'eau. Le soleil perd sa blancheur éblouissante, devient jaune, orange, puis d'un rouge plus sombre, alors que le ciel alentour coule du bleu laiteux du jour en un bleu azur, puis indigo, avant de virer lentement au violet. Un halo rose et vieil or se dessine pour diffuser bientôt en une tache pourpre autour du soleil qui glisse rapidement sous l'horizon. Quand le temps est clair, au moment ultime où la dernière goutte de sang sombre disparaît, un bref embrasement vert turquoise semble jaillir du point de feu rouge et vient étinceler – oh, si brièvement – à la surface du Lac... Ensuite, quand la lueur rémanente du soleil s'efface, en été du moins, les animalcules phosphorescents montent des profondeurs et l'on peut rêver encore qu'on est accoudé, divinité pensive, au bord d'un ciel étoilé. La ville elle-même, avec le réseau de ses rues liquides, tremble dans la nuit comme un être vivant, la projection à l'échelle humaine d'une des minuscules créatures translucides qui habitent ses eaux.
Bâtie au bord d'une mer d'eau douce, Ansaalion est enracinée dans les multiples îles parsemant l'embouchure de la Hleïtsaõ – les îles originelles et les îles artificielles – inextricablement mêlée au lac d'où vient toute subsistance, la chair rosée de l'atéhan, les fruits oblongs de l'anahi qui flotte dans les anses tranquilles en épais tapis cuivré, le bois et les baies des grands tingai penchés sur les eaux tièdes qui baignent leurs racines... Protégés par les neiges et les brumes éternelles des hauts plateaux montagneux qui encerclent la fosse d'effondrement du lac, les Tyranao n'ont connu pendant des éternités que l'étendue flamboyante du Leïtltellu ; aux temps anciens, avant d'avoir compris les masques de Hananai, ils vénéraient le soleil et les lunes, mais loin derrière les divinités mystérieuses de l'eau. Sur les murs des édifices, aux proues des bateaux et des barques, dans le langage quotidien, ce sont encore elles qui apparaissent ici et là, pieusement conservées et reconstituées avec tout le reste. La Mer elle-même semble avoir moins droit de cité à Ansaalion que cette myriade de créatures surnaturelles, dzarlit ou divinités plus anciennes encore : Héhlilu, Anamaliõn, Podzer, et le tout petit, le tout malicieux, Iptit au Chapeau vert.
Lian arpente la cité pendant plusieurs jours à la recherche de Dougall, le plus souvent en barque, quelquefois à pied, dans la fragrance à la fois suave et acidulée des kaïringa qui débordent des terrasses en longs arceaux verdoyants piquetés de fleurs, alternant avec les senteurs mêlées des plats exotiques. Des enfants plongent et nagent ici et là en s'éclaboussant, petites grenouilles nues à la peau dorée. Les reflets des palais et des anciens temples ondulent avec majesté dans les grands canaux, et sur les murs de mosaïque le miroitement incessant de l'eau ; rumeur paisible sur les petits canaux de traverse, appels, plaisanteries, brefs sifflements avertisseurs échangés d'une barque à l'autre, réverbérés par les parois de pierre vers le ciel laiteux. Lian est tombé amoureux d'Ansaalion, au premier regard, et même sa quête infructueuse de Dougall ne parvient pas à assombrir ces quelques jours pendant lesquels, parfois, il se sentirait presque libre.
Plus d'une semaine après son départ des îles, les suppositions des Ranao continuent à lui procurer vivre et couvert. « Et si un Chasseur essaie de continuer ainsi pendant tout le reste de sa vie ? » a-t-il demandé, sarcastique, à Odatan, tandis qu'ils parcouraient les derniers langhi du canal et qu'il s'interrogeait sur les limites de la bonne volonté rani à son égard. Le petit homme a haussé un sourcil : « Cela n'arrive généralement pas. Quand on quitte les îles, c'est pour de bon. Tu n'as pas l'intention de continuer ainsi toute ta vie, n'est-ce pas ? »
Leurs chemins se sont séparés au débarcadère, après une fin de voyage un peu embarrassée où chacun à bord s'efforçait de respecter les règles de politesse et veillait à ce que chacun les respectât ; ils n'ont pas revu le vieux Paalao, en tout cas. Entre Lian et Odatan, pendant ces deux derniers jours, beaucoup de silences, mais à la tonalité presque paisible, comme si l'énormité de la révélation avait brûlé d'un coup toute l'animosité de Lian à l'égard du petit homme. Quand ils se quittent, Odatan lui donne les informations dont il dispose à propos de Dougall. Lian murmure, un peu accablé : « Je ne sais même pas à quoi il ressemble. » Le petit homme répond avec une curieuse mélancolie : « À toi. »
Un bref retour de curiosité pousse Lian : « Vous le connaissez bien ? »
Odatan le dévisage un moment avec une sorte d'angoisse puis, comme on se jette à l'eau : « C'est un de mes petits-enfants, Lian. »
Lian croit d'abord qu'il parle de façon symbolique. Ensuite, il est moins stupéfait qu'il ne l'aurait cru ; d'une certaine façon, cela explique beaucoup de choses. Ce qui est inexplicable, c'est qu'Odatan persiste à vouloir des enfants. « Je suis... votre arrière-petit-fils, alors. »
Odatan murmure d'un air un peu contrit : « J'avais plusieurs raisons de m'occuper de toi, tu vois.
— Et de Dougall ? »
Le regard bleu s'assombrit. « Je suis un père, quand j'ai des enfants. Mais pas un grand-père, ni la suite. J'y ai renoncé. C'est... trop difficile. Pour tout le monde. Dougall... Ce qui l'a blessé, c'était de savoir qui j'étais, ce que j'étais, et...
— De comparer », achève Lian assombri.
Odatan hoche la tête en silence. Lian médite un instant. « Kilasi ? »
Encore un hochement de tête, une ombre de sourire affectueux. « Une de mes arrière-petites-filles. Elle ne compare pas.
— C'est une hékel, rétorque Lian.
— Mais elle n'a pas pour don principal d'être une danvéràn », réplique à son tour le petit homme sans se troubler ; puis, un ton plus bas : « Et c'est une naïstaos. »
Lian reste un instant pétrifié. « On peut... on peut être hékel et...
— On peut être beaucoup de choses, Lian, dit Odatan avec douceur. Il s'agit de poser les bonnes questions. »
Ensuite, il le serre dans ses bras avec beaucoup de naturel, en lui disant : « Rappelle-toi que tu es libre, Lian. Personne ne peut choisir qui tu es, seulement toi. »
*
La Base Potemkine se trouve au sud-est de Morgorod, dans la zone ultramoderne reconstruite après la guerre civile. Elle ressemble, en un peu plus petit, à la Base Quintin. Lian retrouve avec gratitude la routine militaire un peu relâchée pendant le bref voyage. Les entraînements et les cours reprennent, surtout de l'instruction technique maintenant, avec une note d'excitation et de nervosité sous-jacente pour les recrues : la ligne Ewald est à moins de quinze cents kilomètres, le prochain voyage sera pour s'y rendre, cette fois-ci, c'est du sérieux. Mais ils plaisantent quand même entre eux : « C'est l'Année de la comète ! » – c'est ainsi qu'ils appellent la tache lumineuse de l'astéroïde-vaisseau qui continue, indifférent aux affaires des planétaires, à se déplacer lentement dans le ciel. Dans des histoires virginiennes – héritées de la Terre, mais les Virginiens ne les voient plus ainsi depuis longtemps –, les comètes sont des signes. Ils ont choisi d'y voir le signe que la guerre finira bientôt. Ils n'y croient pas vraiment, bien sûr.
Lian continue à apprendre avec obstination comment réparer des corps brisés. Il étudie le matin, l'après-midi, le soir. Il écrit dans son carnet, aussi – tout ce qu'il se rappelle des plantes médicinales utilisées de l'Autre Côté et que la pharmacopée virginienne ignore souvent, bien que l'existence et les propriétés en soient souvent connues depuis le début de la colonisation, il l'a bien constaté : plusieurs poussaient même sur la terrasse de la commune. Quand il a quartier libre, il se promène sur les terrains de la Base pour trouver celles qu'il peut préparer lui-même avec l'équipement disponible. Discrètement, mais Grayson l'aide – il a des ressources apparemment infinies de débrouillardise, et une façon bien à lui de régler les problèmes de sous-officiers trop curieux ou de magasiniers trop avares. « Ne me pose pas de questions », dit-il toujours d'un air bouffon de conspirateur. Lian n'y songe même pas. Un soir, alors que Grayson a réussi à lui procurer le demi-litre de lait de chaux dont il a besoin pour traiter sa récolte de houx bleu, ou sinon la décoction de baies serait un poison foudroyant, Lian ne peut pourtant s'empêcher de remarquer : « Ça va faire mal, dis donc, quand tu vas collecter mes dettes... » Gray le dévisage avec l'habituel sourire sarcastique qui ne trompe plus Lian depuis longtemps : « Je serai gentil. »
25
Lian dévide le fil de Dougall à travers la ville : il a travaillé comme débardeur au port ; il a été cuisinier dans une des hostelleries du Canal Aïrmôn ; il a aidé à la reconstruction du Grand Théâtre, après l'incendie du mois précédent...
Le fil s'arrête là. Y a-t-il un Club Virginien à Ansaalion ? Oui, mais on n'y a pas entendu parler d'un Dougall Flaherty – bien que le nom soit vaguement familier, remarque la vieille femme qui l'accueille à l'adresse du Club. Elle a un peu froncé le nez en le voyant entrer, il ne sait si ce sont ses habits de Chasseur, sa " tête-de-pierre " ou la nature de son enquête, mais elle semble soulagée de le voir partir. Et maintenant, quoi ? Il en est réduit à interroger les gens aux alentours des endroits où a travaillé Dougall.
Il se trouve depuis près de six jours à Ansaalion et il commence à se décourager quand une batelière vient le trouver dans le petit parc où il a élu domicile. Elle a appris qu'il cherchait un halatnim aux yeux bleus et aux cheveux blancs – Lian rectifie machinalement, « blonds ». Elle en a vu un s'embarquer il y a environ trois semaines de cela pour Mérèn-Ilïu. Le cœur de Lian se met à battre : la piste se réchauffe ! Il s'embarque derechef pour Mérèn-Ilïu. Sur le bateau, l'idée le traverse : c'est aussi de là qu'Oghim s'embarque pour l'île des Ékelli, dans la légende. Et tout d'un coup, avec un sentiment pénible de dislocation, il se rappelle que la légende n'en est pas une, ou pas vraiment, que la vision du Rêveur s'est réalisée, qu'Odatan... Puis il se raidit. Réel ou vrai, tout cela ne le concerne pas. Sa vérité à lui, c'est qu'il est venu chercher son père.
Le bateau est un lent caboteur qui dessert tous les petits villages blottis sur les rives. Pêcheurs et bûcherons, dans cette partie de la région : la riche mosaïque des terrasses amoureusement entretenues autour d'Ansaalion a vite disparu le long de la côte rocailleuse. Pas ici que viennent se marier les arbres-Gomphal, mais le long de la rive sud du Leïtltellu, là où les hauts plateaux consentent une ébauche de plaine ; Lian n'en a pas aperçu beaucoup pendant son voyage sur le canal.
Pas de Dougall deux jours après, à Mérèn-Ilïu, une assez grosse bourgade installée au bord d'une anse étroite et profonde, ombragée de hauts résineux dont il ne connaît pas le nom et dont alternent curieusement branches vert sombre et branches blanchâtres. On se rappelle le passage du halatnim ; il a travaillé avec les pêcheurs pendant quelques jours pour payer son écot. Il est parti presque tout de suite pour Ëlmadziulan ; peut-être s'y trouve-t-il encore, à moins qu'il ne soit reparti vers l'ouest depuis l'île, car on ne l'a pas revu à Mérèn-Ilïu. On désigne à Lian la masse brumeuse de l'île toute proche, qui barre l'horizon de l'anse. Un bateau s'y rend tous les deux jours. Demain, à l'aube.
En s'installant dans le lit qu'on a mis à sa disposition à l'auberge – « Tu es le bienvenu parmi nous », encore, et avec une conviction plus accentuée que partout ailleurs – Lian a encore dans l'oreille le nom de l'île tel que le prononcent les habitants du village, avec le glissement entre " ëlmadzi " et " aïlmâdzi ". En virginien, ça donnerait quelque chose comme " l'île du rêveur éternel ". Des darnao sont passés par là, qui ont changé le nom d'Ékellulan. Ktulhudar ne s'y retrouverait pas – Oghim non plus. Lian hausse les épaules dans le noir. On est sur Atyrkelsaõ, ici ; ni Oghim ni son ancêtre n'ont jamais posé le pied dans cette île-ci. Cette île-ci, personne n'y rêve, personne n'y dort ; c'est seulement l'île des Morts.
*
Le rythme des entraînements et des exercices en grandeur réelle ne change pas ; les nouvelles recrues, qui se considèrent maintenant comme des soldats chevronnés, piaffent presque d'impatience. Depuis bientôt deux Mois, on n'a connu que l'univers réglementé des Bases – avec l'exception toute relative du voyage depuis Bird. On sait qu'à la fin d'Août, avant de partir pour la Ligne, on aura quartier libre pendant cinq jours – un vrai quartier libre, avec permission d'aller en ville. Ils se racontent ce qu'ils feront – envirosims, vrais concerts, holovids, restaurants, mais surtout les bars et les boîtes ; la vie de Morgorod n'est de toute évidence pas aussi austère que celle de Bird-City. Lian trouve leurs fantasmes monotones, mais il les comprend : amours et amitiés se sont noués et dénoués pendant ces deux Mois – tant que les relations entre recrues n'interfèrent pas avec le service et qu'on est discret, elles sont légales. Mais après tout ce temps, on se connaît trop, on a envie de sang neuf – comme le dit Delgado au cours d'une conversation sur le sujet, une expression qui choque un peu Lian.
Il a été très clair au début, surtout avec Chris qui a souffert d'un cas aigu de culte du héros lorsqu'il l'a aidée lors de l'entraînement de base. Après cela, les quatre autres filles de la section ne lui ont causé aucun problème ; Vitale a eu une brève relation enjouée avec Grayson et Nan a flirté avec Delgado, mais comme Martine et Térésa elles se sont vite cherché de la compagnie ailleurs ; c'est la même chose dans les autres sections – « On pratique l'exogamie, a remarqué Grayson avec ironie, ça développe la cohésion entre les différentes armes. » Quant aux garçons, il n'y a pas eu d'ouvertures, ou bien Lian n'a pas su déchiffrer les éventuels signaux : l'armée sanctionne l'ostentation et le désordre dans les relations amoureuses, non leur nature, mais de toute évidence ceux et celles qui préfèrent la compagnie de leur propre sexe sont quand même plus discrets que les autres ; ces comportements ont dû être acceptés assez tard dans l'évolution de la société virginienne, et plus tard encore dans son armée, a déduit Lian de ses observations. Ce n'est pas comme s'il était tenté, de toute façon, et par qui que ce soit. La trêve qu'il a établie avec lui-même est encore bien trop fragile.
26
Le lendemain est un jour d'aube claire à Mérèn-Ilïu ; le soleil levant diffuse en halo un mince anneau diaphane autour de l'autre cercle, où les rayons lumineux dessinent la grande croix ; aux points d'intersection de la croix et de l'anneau principal clignotent les petits arcs-en-ciel évanescents. Lian, qui s'est perdu dans leur contemplation, arrive juste à temps pour le départ du bateau à vapeur, on embarque les derniers passagers. Ils sont plus nombreux qu'il ne l'aurait cru, une quarantaine. Pendant le court laps de temps que dure le voyage, ils restent appuyés au bastingage ou arpentent le pont par groupes de deux ou trois, l'air méditatif. Il y a au moins trois hékel, dont un assez âgé à la tunique bleue et rouge de kvaazim.
Pas de Barrière autour de cette île-ci, qui n'est pas l'île d'Oghim et n'a jamais été celle des Ékelli. Des plages de sable rosé bordées des silhouettes familières des tingai et des kaïringa, bientôt remplacées par ces résineux zébrés, que les gens du coin appellent " ûtelgànai ". Ensuite, un petit port où dansent une dizaine de barques de pêche, un quai, quelques entrepôts. Deux charrois communaux attendent au débarcadère, moteur au ralenti, émettant des petites bouffées de vapeur blanche. En voyant Lian se présenter dans le premier véhicule, le conducteur hoche la tête, murmure « Bienvenue ». Lian va s'asseoir tout au fond sans répondre et s'accoude dans le châssis métallique à claire-voie. Le charroi est bientôt plein ; une des hékel, une femme d'une cinquantaine de saisons, tyrnaë mâtinée d'aritnao, s'est installée sans façon près de Lian avec son gros sac. Le charroi démarre. Sans doute ce qu'il y a de plus rapide sur Atyrkelsaõ, après les bateaux de la Mer. Le vent de la course fait claquer par intermittence la toile épaisse tendue entre les ridelles métalliques, qui protège les passagers du soleil estival. C'est la première fois que Lian emprunte un charroi communal, la nouveauté est divertissante un temps. Puis il se perd dans le glissement des arbres sombres de chaque côté de la route, un mantra d'une autre sorte.
Cent vingt-cinq langhi, le trajet jusqu'au centre de l'île prend plus de deux heures. La voisine de Lian ne lui adresse pas la parole, mais personne ne parle à personne. Il y a seulement le martèlement étouffé du moteur, le claquement de la toile, parfois une bouffée de parfums balsamiques. Lian aurait pensé qu'on apercevrait les champs des morts depuis la route mais non, la barrière des ûtelgànai s'avère impénétrable. De toute façon, il n'est pas venu visiter un champ des morts. Au complexe central, il trouvera Dougall, ou on lui dira où le trouver. C'est tout ce qui lui importe.
La route de dalles écarlates ondule avec les collines, puis redescend vers la cuvette centrale. On aperçoit enfin le Haëkelliaõ de l'île, un grand édifice pentagonal à trois étages de taille décroissante, bordés de verdure, aux parois où alternent bandes écarlates et bandes dorées. Une esplanade l'entoure, elle-même encerclée par un bassin en anneau sur lequel se penchent des tingalyai, puis, encore, par les pins omniprésents dont les zébrures presque horizontales accentuent l'effet d'emprisonnement. Le charroi franchit un petit pont, tourne à gauche et s'immobilise sur l'esplanade devant l'entrée principale, suivi du second charroi à moitié plein, qui déverse comme lui ses passagers au soleil. Il est encore tôt, à peine la sixième heure, mais il fait déjà chaud ; une odeur inattendue d'écurie arrive avec le vent. Lian suit les visiteurs dans l'édifice.
Une femme assez âgée, en qui il reconnaît avec surprise une halatnim, accueille les visiteurs. La plupart n'en ont pas besoin et disparaissent dans des couloirs à l'entrée ; Lian comprend qu'il s'agissait de travailleurs du Haëkelliaõ. À la dizaine d'autres vrais visiteurs, la femme montre comment se servir des registres indiquant le secteur où se trouvent ceux qu'ils cherchent ; ils pourront ensuite emprunter les aski ou les carrioles du Haëkelliaõ ; elle leur donne une petite carte indiquant les aires de repos où ils pourront camper. Lian les regarde partir, stupéfait. Camper, dans les champs des morts ?
La femme se tourne vers lui – il sait qu'elle l'a déjà remarqué, avec ses habits de peau. Elle lui adresse un sourire un peu hésitant : « Que puis-je pour toi ? »
Il lui explique le but de sa visite et le visage ridé prend une expression désolée : « Dougall a travaillé avec nous pendant presque deux semaines, mais il est retourné à Ansaalion. » La femme soupire en jetant un coup d'œil circulaire sur la salle, dont les gradins sont couverts de plaques de sirid comme toutes les salles du Haëkelliaõ – les plaques d'Ékellulan, amenées de Tyranaël lors du Grand Passage : « Tenir compagnie aux morts n'est pas chose aisée. Et il avait choisi de travailler aux plantations. Certains de nos bénévoles durent moins longtemps. »
Lian contemple aussi les plaques argentées, aux voix muettes pour lui. Des images, des émotions, des mots, les souvenirs mensongers de survivants eux-mêmes morts depuis longtemps. C'est ce qui reste de toutes ces vies. Et de lui, que restera-t-il ? Il fait volte-face, sans bien savoir où il va, entend la voix de la halatnim derrière lui, hésitante : « Le charroi ne repart que cet après-midi. Veux-tu que je te fasse visiter le Haëkelliaõ ? »
Il dit en hâte « Non, merci », puis après réflexion se retourne et dévisage la femme. « Vous êtes une naïstaos ? »
Elle tressaille un peu, mais répond avec calme : « Oui.
— Vous travaillez ici depuis longtemps ?
— Une vingtaine de saisons.
— Parce que vous êtes une naïstaos. »
Elle incline la tête de côté, réfléchit. « Au début, oui. Je voulais m'endurcir.
— Et vous vous êtes endurcie. »
Elle sourit avec indulgence : « Non. J'ai trouvé ma paix. »
Il sort, cette fois, fait quelques pas rapides, ébloui par les rebonds de la lumière sur les dalles. Puis il ralentit : il n'a nulle part où aller. La dernière carriole de visiteurs disparaît dans l'une des grandes allées en étoile autour du Haëkelliaõ. Il est seul en compagnie du vent et des hauts ûtelgànai qui ploient en murmurant leurs branches alternées d'ombre et de lumière. Mais comme il a commencé d'avancer, il continue, franchit le pont qui s'offre à lui et se retrouve dans l'allée extérieure du premier cercle des morts, les plus anciens. Parmi les arbres et les buissons courent en ondulant les petits murets bas de pierre dorée où sont serties les plaques-souvenirs des morts. Et les grandes statues de cristal bleuté, silencieuses, miroitant au soleil, toutes semblables. À l'intérieur, tous différents, les cadavres préservés, aux yeux ouverts : grands corps, corps ronds, corps fermes ou ridés, bruns, dorés ou bistres, des femmes, des hommes, des enfants même – nus, vulnérables, inaccessibles.
Il suit l'allée circulaire en marchant très vite et en regardant droit devant lui, mais le cercle est un cercle et, devant lui, il y a toujours d'autres statues translucides. Il remarque tout de même des silhouettes animées : ici un groupe de plusieurs personnes chargent des statues sur des carrioles basses ; ailleurs, deux hommes bêchent ; à un moment donné, il entend un roulement et un bruit de sabots derrière lui et une autre carriole passe, tirée par un asker ; dans la carriole sont étendus trois longs paquets étroits recouverts d'un tissu bleu. Des morts dont un kvaazim a réduit en poudre l'enveloppe de cristal, et qu'on s'en va planter avec leur graine de tingalyai sur le ventre.
Lian fait encore quelques pas, mais simplement parce qu'il a acquis un certain élan. Et puis il voit une statue qui n'est pas comme les autres. Le cristal est resté brut au lieu d'être taillé pour imiter la forme humaine. On l'a fichée en terre debout comme les autres cependant ; ses faces biseautées, légèrement asymétriques, lui confèrent une profondeur plus bleue. Un petit homme maigre et brun y repose ; cheveux blancs coupés en brosse touffue, étroit visage au nez en bec de rapace, lèvres minces, paupières fortement bridées, et tout le reste : un pur Virginien. Un Keldaran, dans la partie la plus ancienne du champ des morts ? Lian s'agenouille, fouille dans l'herbe, mais sans trouver de plaque gravée d'un nom.
« Wang Shandaar », dit une voix féminine derrière lui. Il sursaute et se retourne. C'est sa voisine du charroi, la hékel, tenant les bras d'une brouette remplie d'instruments aratoires. « Je t'ai surpris, pardonne-moi », ajoute-t-elle.
Lian dit machinalement « Ce n'est rien », puis il répète : « Wang Shandaar ?
La hékel hoche la tête en contemplant le bloc de cristal avec respect. « Le premier Virginien venu de son plein gré avec la Mer, environ cinq cents saisons après le Grand Passage. »
Lian reste un instant interdit : « Je croyais qu'Odatan...
— Shandaar n'était pas un passeur. Et ce n'était pas le premier des siens à franchir la Mer. Une trentaine de saisons plus tôt, un bateau et son équipage étaient passés. Par accident, ils n'auraient pas dû se trouver sur la Mer lors de son départ. Tous morts, hélas. La Mer était bien trop puissante pour eux, alors : ils n'étaient pas prêts. » La hékel reprend un peu plus bas, après une pause : « Oh, comme on en a débattu alors sur tout Atyrkelsaõ ! Quelle effrayante merveille de voir encore se réaliser peut-être des visions des aïlmâdzi ! »
Elle médite un instant, les yeux fixés sur le bloc bleuté. « Et finalement, on a décidé de les renvoyer de l'Autre Côté, ces malheureux passeurs malgré eux, sans rien indiquer de notre présence. La Communicatrice de l'époque a demandé à l'Esprit de la Mer, et la Mer a accepté de faire une exception. Mais en examinant ceux que nous avions ainsi renvoyés, Shandaar avait imaginé un moyen d'atténuer le choc du passage, et finalement, il l'a mis en œuvre. Seul. Et la Mer l'a poussé dans le port de Hébuzer. Il a survécu trois jours à son passage. Pendant trois jours, parmi nous, je crois qu'il a été heureux. C'était un homme plein de curiosité. Et on a décidé de le garder ici, pour toujours, sans le replanter. Pour ne jamais oublier. Parce qu'il avait donné sa vie pour nous rejoindre, parce que, à sa façon, et le premier des siens, il croyait en la Mer. »
Lian déconcerté ne regarde plus le bloc de cristal, mais la hékel : « Vous en parlez comme si c'était hier...
— Ceux qui l'ont recueilli l'ont raconté pour des plaques, qui se trouvent ici. Si tu veux, je te les dirai. »
Il secoue la tête en marmonnant une vague politesse et reprend son chemin sans but. Des anciens passeurs, des visions peut-être réalisées, des plaques aux voix éternellement muettes pour lui – rien de tout cela ne le concerne. Il y a seulement les statues de cristal bleuté qui lui répètent son destin, rangées après rangées après rangées de cadavres. Et parmi ces statues, quelque part, loin à la périphérie des champs de morts, il y a un Thomas roux à la gorge tranchée, et le corps déchiré d'une aspirante hékel dont il ne connaît pas le nom et...
Il presse le pas, plus vite, encore plus vite. Traverse le pont qui mène à la route empruntée plus tôt, continue tout droit entre les rangées funèbres d'ûtelgànai. Au bout d'un moment il ralentit, adoptant sans en avoir conscience son ancien rythme de marche en forêt, qu'il pouvait soutenir pendant des heures. Quand il s'arrête, l'estomac noué d'une crampe, l'heure de la collation de mi-matinée doit être passée. Il s'assied, fouille dans son sac à la recherche des noix toujours conservées en cas de besoin. Puis il repart à la même allure, en évitant soigneusement de penser. Quelque part au début de l'après-midi, le charroi s'arrête à sa hauteur, presque vide, le conducteur lui fait signe de monter. Une fois au port, il convainc un des travailleurs des entrepôts de lui laisser emprunter sa barque, et il retourne à Mérèn-Ilïu à force de rames. Vers la fin de l'après-midi, il est de nouveau dans le vapeur qui va vers le sud. Le lendemain à la même heure, il est de retour à Ansaalion.
Au débarcadère, un homme s'avance vers lui. Il n'est pas très grand ; assez maigre, il flotte dans des vêtements de charpentier délavés par le soleil. Il porte un grand chapeau de cuir dont l'ombre lui coupe le visage en deux, dérobant la couleur de ses yeux et soulignant la ligne amère de sa bouche. Il s'immobilise devant Lian qui en fait autant, le cœur battant. Comme l'homme reste silencieux, Lian déclare, incertain, en virginien : « Je suis Lian Flaherty.
— Tu devrais te trouver d'autres habits », dit enfin l'autre en setlâd. Puis, d'une voix qui s'enroue un peu : « On m'a dit que tu me cherchais. »
27
Le matin où commence la période de quartier libre, Lian reste à la Base ; il n'est pas certain de vouloir aller en ville ; mais Grayson revient le chercher après la méridienne, et il insiste tellement, charmeur, que Lian se laisse convaincre – par amitié, par acquit de conscience, pour se mettre à l'épreuve. Vers le milieu de l'après-midi, il a compris que c'était une erreur. Visiter le Capitole et y voir le Haëkelliaõ à chaque détour de couloir, dans chaque fresque, s'avère trop pour lui. En attendant que les autres aient fini leur tour guidé, il va s'asseoir dans la cour centrale. Il se perd délibérément dans l'élan incessant des fontaines, contemple l'énorme arbre-à-eau qui ombrage le bassin, se demande fugitivement d'où provenait sa graine, évite aussitôt cette pensée, comme les caliches du bassin viennent examiner ses doigts trempés dans l'eau mais se détournent en éclairs rouges et noirs dès qu'ils bougent. Il a envie de rentrer à la Base, mais une obstination agacée le fait rester : il ne va pas céder aussi facilement ! Et puis, Grayson serait trop déçu.
À la fin de la soirée, après avoir mangé dans un restaurant plein d'autres recrues et essayé plusieurs bars, ce qui reste de la section se rend dans une des boîtes à soldats de la haute ville – les filles et un des garçons ont retrouvé leurs flammes du moment et décidé de faire cavaliers seuls. On accède à la salle par un long escalier ; elle s'étend en contrebas, immense. Le bruit est assourdissant, on ne perçoit de la musique qu'un rythme de base, lourd et insistant, qui vous résonne dans le ventre. Des lumières multicolores fulgurent dans toutes les directions, des danseuses et des danseurs s'exhibent sur de petites scènes rondes, presque aussi dénudés que les clients qui se pressent sur les pistes de danse. Même avec l'air conditionné, il fait chaud ; les peaux sont moites, les parfums insistants, les yeux fiévreux, les mains fureteuses. Lian reste un instant confondu au sommet de l'escalier, tandis que leurs trois compagnons dégringolent vers la salle avec des ululements joyeux presque inaudibles dans le tumulte. Rien de ce qu'il a vécu à la commune de Bird, encore moins à la Base, ne lui a laissé imaginer ce genre d'endroit.
Une fille rousse vient se coller contre lui sans rien dire, une autre enlace Grayson en silence aussi, mais le langage de leurs corps ne laisse place à aucune équivoque. « Le repos du guerrier ! » hurle Grayson à l'oreille de Lian d'un air égrillard. Lian a envie de dire « Quel repos ? », mais il sait qu'il ne s'entendra pas, et Grayson non plus, s'il ne hurle pas – et il ne veut pas hurler. Il secoue la tête et fait demi-tour.
Le silence de la rue est presque aussi douloureux que le choc sonique précédent. Il fait sombre. Morgorod est toujours en black-out, la nuit, pendant la saison de campagne. Lian se laisse tomber sur le rebord d'un des caissons où l'on enferme les arbres, dans la haute ville, et se force à prendre de grandes inspirations pour contrôler le tremblement qui l'a saisi. Quel plaisir peut-on trouver à cette... oblitération par le bruit ? Puis il se sent soudain glacé, malgré l'air tiède et moite de la nuit tropicale. Oblitération. Oubli. Ils combattent le mal par le mal. Rituel propitiatoire. Le rythme lancinant de la musique et les lumières stroboscopiques pour le futur tonnerre et les futurs éclairs du front. Le repos du guerrier.
Au bout d'une durée imprécise, la porte s'ouvre et se referme dans son dos, une lointaine bouffée de bruit sauvage, puis de nouveau le silence. À peine surpris, il entend la voix sarcastique de Grayson : « C'est pas ton truc, hein ?
— Non.
— Quoi, tu n'aimes pas les filles ?
— Pas comme ça. »
Gray vient s'asseoir près de lui. Ses habits sont tout froissés, ses cheveux noirs en désordre.
« Oh la la, quel puritain ! Tu as bien eu des petites amies, quand même ?
— Pas en ce moment.
— Oooh, dit Grayson avec une compassion bouffonne, une belle histoire d'amour qui a foiré ! »
Ce genre de réflexion ne lui ressemble pas ; il doit être ivre. Lian hausse les épaules, se lève. « Je rentre. »
Grayson le rattrape au bout de quelques enjambées. « À pied ?
— Seulement douze kilomètres. On a fait bien plus que ça en une nuit.
— Tu ne sais pas t'amuser, Lian. Pas bon pour la santé, ça !
— Tant mieux si tu t'es amusé, Gray. »
Au bout d'un moment, Grayson marmonne : « Eh, je suis resté dix minutes, ça me prend juste un peu plus de temps pour m'amuser. »
Ils avancent en silence, du même pas – ils sont de la même taille.
« Tu n'avais pas quelqu'un, dans ta commune ? » Grayson semble dégrisé, tout à coup.
« Suzane ? Non. Pas vraiment.
— Qu'est-ce qui n'a pas marché ? »
Lian hésite, mais la réponse est si évidente : « J'étais trop ailleurs.
— Ah, les femmes n'aiment pas ça, c'est vrai. » Puis, plus bas, avec une sorte de ressentiment : « C'est drôle, moi, je préfère les gens qui ne sont pas trop là. »
Après quelques centaines de mètres, ils aperçoivent un taxi en maraude, lui font de grands signes. Le chauffeur consent à s'arrêter, malgré leurs uniformes – tous les transports sont gratuits pour les militaires – s'illumine quand Grayson agite son idicarte : « On paie les braves gens, nous. » Ils roulent bientôt dans l'un des interminables boulevards qui traversent la ville nouvelle et conduisent à la Base. Grayson regarde par la fenêtre sans rien dire. Lian, un peu embarrassé, murmure : « Tu n'avais pas besoin de rentrer... »
L'autre hausse un peu les épaules : « J'étais fatigué. »
Comme on arrive près de la Base, Grayson s'anime soudain, le nez collé à la vitre : « Hé, la comète ! » Lian se penche, suit le doigt tendu : le ciel est découvert, pour une fois, et Lagrange est en effet bien visible, comme un lointain sabre de lumière avec la traînée blanche qui le précède et se perd entre les étoiles ; depuis la ville, on ne le voyait pas, il était dissimulé par la montagne.
Une fois rentrés à la Base, ils se débarrassent de leurs vêtements de sortie tout fripés et Grayson va chercher son petit télescope – où il se l'est procuré, c'est l'habituel mystère. En maillot de corps et short dans la chaleur humide, ils grimpent sur le toit plat du baraquement, Lian étale la couverture qu'il a récupérée au passage sur son lit. Le black-out étant en vigueur à la Base aussi, la pollution lumineuse est nulle et la " comète " brille dans le ciel de toute sa splendeur faussement immobile. C'est une de ses passes les plus rapprochées de Virginia, Lagrange ne sera plus jamais aussi clairement visible. Avec le télescope, on distingue encore mieux les détails du jet d'ions, non un trait rectiligne comme on le croit à l'œil nu mais un brouillard qui se dissipe en nappes et volutes translucides, masquant légèrement les constellations proches. Lian est saisi d'une sorte de vertige : cette grosse tache de lumière est en fait une masse de glace et de rocs filant à travers l'espace à des vitesses inouïes, emportant quarante mille êtres humains dans son ventre, et ces nuages ténus couvrent en réalité des centaines de millions de kilomètres.
« Tu imagines », murmure Grayson dont les pensées ont bien entendu suivi une pente tout autre, « si une comète de cette taille venait s'écraser sur Virginia ? Hop, fini les humains. »
La perspective ne semble pas l'atterrer outre mesure. Il lui tend le télescope.
« C'est un astéroïde, remarque Lian.
— Astéroïde, comète ! La main de Dieu. Qu'est-ce que c'est, nos petits jeux stupides, nos petites guerres, en face de ça ?
— La main de Dieu ? » dit Lian, plutôt amusé maintenant. Gray va-t-il partir dans une de ses grandes méditations métaphysiques ? Mais il le préfère ainsi.
« Dieu, ou le hasard, même chose, question de point de vue. Nous regardons dans nos télescopes et nous nous sentons importants parce que nous voyons des choses immensément lointaines, et en face, il y a Dieu, ou l'univers, même chose, question de point de vue, mais eux ils nous voient par le petit bout de leur lorgnette, et nous sommes très, très petits. Tellement petits que même ton petit gardien des petites choses ne voudrait pas s'occuper de nous.
— Mais quelle humilité, Gray », sourit Lian en se couchant sur le dos pour observer plus confortablement le ciel.
L'autre hausse les épaules sans répondre, et Lian comprend soudain qu'il est sérieux. « Il n'y a rien de trop petit pour le petit gardien des petites choses », remarque-t-il avec douceur – et c'est à lui de perdre son sourire : Laraï le lui disait toujours, quand il omettait dans un dessin tel ou tel détail d'une plante ou d'un animal en arguant que c'était trop petit pour être important.
Il cherche le trait lumineux, le trouve, l'observe en silence en essayant d'imaginer qu'il le voit bouger.
Grayson dit brusquement, d'une voix basse et intense : « Qu'est-ce que je fais, Lian ? Qu'est-ce que je veux ? »
Lian se redresse, déconcerté. Il a beau distinguer aisément Grayson dans l'obscurité avec sa vision de nuit, il ne peut comprendre d'où vient cette soudaine angoisse. Il lui retourne ses questions avec une douceur prudente : « Je ne sais pas, Gray. Que fais-tu ? Que veux-tu ? »
Un long silence. « Je mens, murmure enfin la voix torturée. Je me mens, je te mens. Tellement... de mensonges. Et pourquoi, vraiment ? Je ne sais plus. Ce n'est plus pareil. Tu as tout changé. »
Lian contemple le visage obstinément détourné, les boucles emmêlées qui retombent sur le front têtu, il a envie de sourire, il a la gorge serrée. Il tend une main, effleure le bras nu de Grayson, qui tressaille et tourne vers lui un regard aveugle, terrifié, implorant. Et c'est Argelos, et c'est Thomas, Lian voit maintenant ce qui l'a ainsi attiré en son compagnon depuis le premier jour, il sourit, un peu triste, mais en même temps c'est ainsi que ce doit être.
Grayson reste pétrifié un instant, puis gémit, désespéré : « Tu ne comprends pas, tu ne comprends pas...
— Je comprends quelque chose, dit Lian, étrangement pacifié. Et toi ? »
Il caresse du bout de l'index les lèvres entrouvertes. Grayson le regarde toujours sans vraiment le voir dans le noir, éperdu. Puis il balbutie : « Je n'ai... je n'ai jamais...
— Veux-tu ? »
Grayson est incapable de parler, maintenant. Il hoche simplement la tête. Il tremble. Lian lui caresse la joue – un peu rêche, une barbe d'un jour – l'épaule, si douce par contraste. « C'est très simple, Gray. Juste toi et moi. »
Et il l'embrasse.
28
Le dernier festival de l'été commence à Ansaalion, le festival du gaad, en ce premier jour de Gaadtéñu. L'épi rouge, symbole des anciens Aritnai, apparaît partout. La ville en profite pour faire toilette, on récure édifices et demeures, on repeint à neuf les emblèmes des barques et des bateaux et la coutume veut que, en ce temps de l'année, on se procure de nouveaux habits. Des odeurs nouvelles s'entrecroisent au-dessus des canaux, celle du gadrundel, le gros gâteau rempli de graines croquantes et de fruits confits auquel chaque famille donne une forme particulière, dictée par ses traditions, et qu'on fait cuire dans les fours de quartiers parce qu'il est trop gros pour les fours familiaux, celle des pâtés d'atéhan aux mille recettes différentes, celle des kilnis, les minuscules beignets rouges qu'on mange à toute heure sur les canaux et les places... Dans le port, les visiteurs affluent de toute la province. Pendant les quatre jours du festival, la circulation augmente sur les canaux ; les barges d'approvisionnement et les bateaux communaux ont la priorité ; ensuite le flot diminue peu à peu et on laisse de nouveau place aux bateaux individuels.
Des flots de musique s'échappent d'un peu partout : le festival est l'occasion de joutes serrées entre les Tyranao et leurs cousins les Hébao dont musique et chant sont sans doute, avec l'affabulation sans frein, les principaux talents. On se déguise, aussi, en souvenir du subterfuge de la maligne Héhlilu pour soutirer aux puissances souterraines l'épi de gaad qu'elles conservaient jalousement. On peut être qui on le désire pendant le festival du gaad, empruntant à celle-ci ou celui-là son métier, son caractère et même son visage – pour lui présenter un portrait parfois sans indulgence, quitte à se rencontrer soi-même ensuite au détour d'un canal ou d'une rue, également caricaturé. On en profite aussi pour sortir accessoires et costumes anciens conservés avec soin dans les coffres familiaux, ainsi bien sûr que toutes les légendes et traditions qui peuvent s'y rattacher.
Dougall emmène Lian chez un tailleur, c'est la première chose qu'ils font ensemble. Il troque des modifications intérieures extensives de l'atelier de l'artisan contre un ensemble complet, deux chemises, pantalons longs et courts, gilet pour toutes occasions ; Lian refuse toutefois de se séparer de sa longue veste de peau, qu'il a faite lui-même, solide, imperméable, et pleine de poches utiles. Ensuite, en attendant que les habits retouchés soient prêts, ils vont s'asseoir à la terrasse d'une taverne.
Après un long silence, Dougall demande : « Pourquoi maintenant ? »
Lian envisage plusieurs réponses, aucune satisfaisante, choisit : « Je ne savais pas que tu étais vivant. »
Un éclair de stupeur douloureuse passe dans le regard bleu, masqué aussitôt par une ébauche de sourire sarcastique : « Elle ne t'a rien dit.
— Ils ne m'ont jamais rien dit ! Ils m'ont toujours menti !
Dougall le contemple un instant, un sourcil levé : « Jamais ? Toujours ? »
Lian hausse les épaules, buté : « Trop souvent.
— Ils voulaient te protéger, murmure enfin Dougall.
— De la vérité ? !
— La vérité...
— De la réalité, alors ! » s'exclame Lian exaspéré. Dougall ne va pas les défendre ? « Tu aurais fait la même chose ? »
Dougall regarde au loin. « Je n'aurais pas été un bon père. Il valait mieux que tu grandisses avec Nathénèk.
— Et maintenant ? lance Lian.
— Maintenant, tu vas entrer dans ta dix-septième saison, et tu apprends à poser des questions. » L'ironie s'efface. Le regard bleu revient sur Lian, suit les cicatrices de ses joues, se détourne encore. « Tu es allé chez les Krilliadni », murmure Dougall d'une voix sourde. Son visage à lui ne porte pas de cicatrices.
« Et alors ? » dit Lian au bout d'un moment furieux, terrifié. « Je te cherchais ! Et leurs questions, on n'en a rien à faire, ce sont des questions de Ranao !
— Ces questions-là n'ont pas de race, Lian ! » Dougall baisse la voix, les traits contractés. « Et tant qu'on a peur de son ignorance, on ne peut approcher la vérité. »
Il ne va pas se mettre à parler comme un hékel, non plus ! Lian fait tourner sa chope de bière sur la table, exaspéré, les yeux fixés sur les bulles qui n'en finissent pas de mourir à la surface.
Un choc dans le petit canal devant eux, un cri bref, l'éclaboussure d'un plongeon. Deux barques viennent de se heurter. Le pilote de la barque noire et verte jaillit de l'eau en se propulsant d'un seul mouvement à bord de la barque blanche et rouge, commence à protester, ruisselant. L'autre batelier, d'abord apaisant, s'énerve à son tour et répond avec vivacité. Sans un regard pour leurs occupants, les autres bateliers contournent les deux barques qui dérivent lentement au milieu du canal.
Mais la rumeur des voix qui flottaient sur le canal se tait à cet endroit. Une poche de silence s'élargit peu à peu autour des deux barques. À la terrasse de la taverne, tout le monde s'est immobilisé, les conversations se sont interrompues. Lian se rend compte qu'il est le seul à observer l'altercation, avec Dougall.
Sur un éclat de voix plus haut que les précédents, l'homme de la barque verte et noire lève soudain la main et gifle l'autre. L'autre batelier se fige, lèvres blanches sur ses dents serrées. Après un instant de parfaite immobilité, il prend une profonde inspiration, recule d'un pas dans sa barque qui tangue et s'incline, les deux mains offertes, dans le salut du plus profond respect.
L'autre homme semble se réveiller en sursaut ; il regarde autour de lui comme s'il avait soudain pris conscience du silence, puis saisit les mains tendues en s'inclinant à son tour. Comme par enchantement, les voix du canal s'élèvent à nouveau autour d'eux. Le pilote de la barque blanche et rouge godille rapidement pour rattraper l'autre barque, le second batelier saute à bord de son embarcation et reprend en sens inverse son trajet interrompu.
Lian laisse échapper un reniflement sarcastique. « Ces gens sont si polis ! »
Dougall ne dit rien, les mains croisées autour de sa chope de verre. « Sais-tu combien d'habitants il y a à Ansaalion, Lian ? dit-il soudain. Près de trois millions. Et la population totale d'Atyrkelsaõ ? Huit cent cinquante millions environ. Et il y en a eu jusqu'à un milliard et plus, après le Grand Passage. Sur Tyranaël, au moment de l'arrivée de la Mer, quand toute la population d'Aritu et de Paalu a été déplacée, près de trois milliards sur ce seul continent. Crois-tu qu'ils auraient pu continuer à exister sans politesse ? Leurs cultures ont survécu à des chocs dont un seul aurait détruit la nôtre. Ils sortaient de terribles guerres quand la mutation... Connais-tu le jeu de la perfection ?
— Un peu, dit Lian en haussant les épaules, surpris du changement de sujet.
— Le hékel ne gagne jamais, l'as-tu remarqué ? S'il gagne, il a perdu.
— C'est un jeu ! Ce sont les hékel qui exercent le pouvoir réel ! »
Dougall repose sa chope avec un claquement sec : « Mais non ! Ils facilitent à tous l'exercice du leur. Et ils en paient le prix. S'ils atteignent l'illumination, s'ils sont en mesure de rejoindre la Mer mais qu'on a encore besoin d'eux, ils restent. Il y a peut-être autant de hékel que de Krilliadni à l'île des morts ! Ils ont fait ce choix à mesure que les dons se développaient. La communauté. Parce qu'ils l'avaient déjà choisie avant les dons, de façon répétée, depuis très longtemps. Ce n'est pas forcément le bon choix, ni le seul. On en a fait d'autres, de l'Autre Côté par exemple... Mais c'est celui qui leur a permis, à eux, de survivre. C'est ainsi. »
Lian le dévisage avec une fureur incrédule. « C'est ainsi ? répète-t-il d'une voix rauque. Et il faut aller mourir dans les îles ? »
Dougall se mord les lèvres : « Le bien du plus grand nombre...
— Mourir pour le bien-être d'une communauté dont on ne fait pas partie ?
— Nous faisons partie de la communauté rani, Lian ! proteste Dougall alarmé.
— Alors pourquoi dis-tu " ils " en parlant d'eux ? »
Silence. Dougall dit enfin, le regard assombri : « Parce que ma mère, et mon grand-père... » Sa voix s'éteint, reprend. « Mais les enfants de tes enfants seront des Ranao à part entière, si tu y veilles.
— Nous sommes des naïstoï ! » gronde Lian, les poings serrés.
Dougall se fige. Un muscle tressaute dans sa mâchoire. « Ils en ont eu aussi. Presque plus maintenant. Cette mutation-là n'a pas duré de l'Autre Côté non plus : il n'y a plus de passeurs. Ici, pour nous, ce n'est qu'une résurgence passagère, j'en suis certain. Les enfants de tes... »
Lian frappe des deux mains sur la table. « S'il y a des enfants de mes enfants ici, moi, je serai mort ! C'est moi qui suis ici maintenant, pas les enfants de mes enfants ! »
Dougall se raidit : « Peut-être devrais-tu être ailleurs, alors ! » dit-il entre ses dents serrées.
« J'y suis allé, comme toi ! »
Soudain conscient du silence qui les entoure, Lian adresse aux Ranao les plus proches un regard furieux ; les yeux restent détournés ; le silence persiste.
Au bout d'un moment, d'une voix dépourvue d'inflexion, Dougall dit : « Tes habits doivent être prêts, Lian. Viens. »
Chez le tailleur, une fois Lian vêtu à neuf de pied en cap, Dougall semble se détendre un peu. Il lisse le col de la chemise, tire sur le gilet pour mieux l'ajuster, passe enfin une main maladroite dans les mèches blondes de Lian en marmonnant «... coupe de cheveux ». Lian bat des paupières à ce contact, la gorge subitement serrée. Mais Dougall fait un pas en arrière, l'examine une dernière fois d'un œil critique et lance par-dessus son épaule au tailleur et à ses apprentis : « Beau travail. »
Puis il retire de son annulaire gauche une bague qu'il tend à Lian : « Ça complétera. » L'or massif enserre sans griffes un cabochon rond et poli, rouge, la nuance du rubis et non de l'ultellaod.
« Je t'échangerais aussi bien les habits du petit contre cette bague, remarque le tailleur en s'approchant, intéressé. J'ajouterais même un manteau d'hiver.
— C'est un bijou de famille, dit Dougall.
— Dans ce cas, soupire l'autre, il n'a pas de prix pour toi. Tant pis. Viens demain, je te montrerai les modifications que j'envisage pour l'atelier. »
Une fois dehors, Lian dit à Dougall : « Je t'aiderai. »
Dougall ajuste son chapeau de cuir sur sa tête, regard de nouveau invisible dans l'ombre : « Je ne suis pas un artisan, je ne forme pas d'apprentis », dit-il enfin avec une réticence évidente.
« Je sais comment travailler le bois ! proteste Lian. Sûrement assez pour t'aider, en tout cas ! »
Dougall reste un moment silencieux puis concède : « D'accord. Sept jours. Mais ensuite, je pars pour l'ouest, et tu ne viendras pas avec moi. Tu me cherchais, tu m'as trouvé, je suis heureux de t'avoir rencontré, Lian. Ça s'arrête là. Je ne suis que ton père-par-le-sang. Juste... un homme, comme n'importe quel homme rencontré en chemin. Ce que tu dois savoir, je ne peux pas te l'apprendre. Personne. »
Et avant que Lian ait pu protester, il tourne les talons en disant : « Viens, il est tard, j'ai faim. »
Ils se rendent sur la place du Haëkelliaõ, où est installée la foire et où l'on peut manger pour rien aux étals de dégustation. Il y a foule, mais plus pour assister aux spectacles qui se déroulent sur les tréteaux ou au hasard des improvisations un peu partout. Avec leurs plateaux et une cruche de vin d'arpelai, ils s'asseyent comme plusieurs badauds sur le rebord d'une des fontaines, non loin d'une scène où des masques improvisent sur des motifs traditionnels. Lian va reporter les plateaux – la cruche n'est pas encore vide – et revient près de Dougall qui est en train d'allumer une pipe, tandis qu'une joute musicale a remplacé les dialogues comiques : des jeunes filles tournent et sautent au son aigu des pilpai et des tikhouti, un jongleur lance des boules multicolores vers le ciel tout en exécutant les premières figures de la maalsaïtlàn, la danse archaïque à l'origine de la satlàn. Alors qu'il s'approche de Dougall, l'odeur à la fois douce et âcre de la fumée le fait tressaillir, il la reconnaît, doublement : c'est celle du maalt.
Un éclair dans sa mémoire, la caverne aux ombres dansantes, les frères du karaïker et leurs balafres noires et rouges, la grande pipe qu'on se passait de main en main... Il s'assied, confusément angoissé, tend la main vers la cruche.
Légère, la cruche. Vide. Il n'en a pas bu une goutte. Il lève les yeux : Dougall le regarde fixement. Son visage semble se brouiller, se défaire, il ouvre la bouche, mais au même moment les cloches de la ville et les cornes de bateaux se mettent à résonner, c'est la vingt et unième heure, le commencement officiel du festival.
Dougall se lève un peu lourdement, sans dire ce qu'il allait dire. Lian en fait autant, décontenancé. Autour d'eux, les réjouissances reprennent de plus belle. Soudain une bande de personnages déguisés séparent Lian de Dougall ; ils portent sur la tête des masques énormes, plus grotesques qu'effrayants, et ils entraînent Lian dans leur farandole. Il n'a pas encore retrouvé ses esprits que le groupe de masques est déjà plus loin. L'un d'eux, aux vêtements noirs et rouges et dont le masque de karaïker richement ouvragé s'orne de crocs argentés, arrive derrière Dougall, se ramasse et lui saute sur le dos avec un rugissement bien imité. Ils roulent tous les deux sur le sol.
Dougall se relève, il a perdu son chapeau, ses cheveux lui retombent dans les yeux. L'autre se met à danser autour de lui en lançant des petits coups de pattes rapides, le touchant à l'épaule, à l'estomac, au flanc...
Dougall reste immobile, comme pétrifié. Ses yeux étincelants comme du verre suivent les mouvements du masque. Et tout d'un coup, avec un cri inarticulé, il saisit l'un des bras tendus, effectue un mouvement rapide de torsion et l'autre se retrouve sur le dos. La main de Dougall accroupi sur lui s'élève et retombe, une fois. Le masque s'immobilise. Dougall aussi.
Les autres avaient cessé de danser et encourageaient leur compagnon avec des rires. Certains font maintenant mine de se prosterner devant Dougall en psalmodiant l'ancien titre des rois paalani, « Kar Karaïlan ! Kar Karaïlan ! » – " tueur de karaï, tueur de karaï ". D'autres, dont un adolescent à masque de tovker blanc, feignent de donner des coups de pied au karaï mort, « Tu as trouvé ton maître, Arundaz, lève-toi et marche, maintenant, Arundaz ! »
Ils se taisent et s'immobilisent les uns après les autres. La silhouette rouge et noire ne bouge toujours pas. Dougall non plus. Toute la scène se détache pour Lian avec une précision irréelle dans les lumières tremblantes des lampions et des grandes torchères. Enfin, le masque au tovker blanc retire sa tête pour découvrir le visage anxieux d'un très jeune adolescent. « Arundaz ? » Puis, d'une voix plus aiguë : « Ati ? » Il s'agenouille près du corps inerte, en face de Dougall toujours pétrifié. Avec une hâte fébrile, il détache le masque aux crocs d'argent. En dessous, le visage d'une jeune femme aux yeux fixes, qu'il effleure d'une main hésitante.
Le silence s'étend de proche en proche comme une marée sourde et pourtant vibrante, mis en relief de façon aiguë par les rires et la musique qui s'élèvent toujours aux confins de la place.
Dougall se relève avec des mouvements lents, comme s'il nageait. Lian voudrait courir vers lui, le secouer, le faire revenir à lui, mais il est incapable de bouger. La foule s'écarte, pourtant, en face d'eux. Devant trois hékel en tunique bleue scintillante. L'un d'eux prend avec douceur le bras de Dougall. Entre les deux autres, le corps inerte se soulève sans qu'ils le touchent, et ils s'éloignent dans l'allée que la foule ouvre pour eux. L'adolescent au masque de tovker s'est relevé et regarde dans leur direction avec des yeux d'aveugle, même quand la foule s'est refermée ; il tient toujours le masque de karaïker ; ses compagnons l'entourent et l'entraînent en silence. Comme à un signal, tout le monde se détourne et s'écarte de l'endroit où se trouvent le hékel et Dougall ; les bruits de la fête reprennent peu à peu, avec une note tremblante, angoissée.
Le hékel a lâché le bras de Dougall. Ses lèvres ne remuent pas mais, comme s'il avait parlé, le visage de Dougall perd son effrayante fixité. Sa maigre silhouette semble se recroqueviller, s'affaisser. Il ne regarde pas du côté de Lian, il suit le hékel et disparaît bientôt dans la foule.
Lian revient à lui avec un sursaut, court derrière eux, Trop de monde sur son chemin. Sans le regarder, sans méchanceté, comme par inadvertance, on fait obstruction à ses efforts.
« C'est fini, Lian, laisse-le, c'est fini », dit une voix brisée derrière lui. Il se retourne, Odatan, c'est Odatan, bien sûr, et Lian s'arrache à la main qui s'est posée sur son bras, en criant : « Où vont-ils ? Que vont-ils lui faire ? Laissez-moi passer ! »
Les yeux dilatés du petit homme semblent noirs dans la lumière bondissante des torchères. « C'est la deuxième fois qu'il tue, il est naikraël maintenant. Ils l'emmènent au Haëkelliaõ. Laisse-le. »
Lian continue à se débattre, les oreilles bourdonnantes. " Naikraël " : " négation, vide, absence éternels ". Naikraël ? Il se fraie un chemin vers le centre de la place et cette fois on le laisse passer. Il arrive trop tard, Dougall a disparu derrière une masse de hékel en tuniques scintillantes, qui occupent toute la voûte d'entrée, tête baissée, mains offertes à hauteur de poitrine. Lian pense un moment se jeter en courant contre eux, les bousculer, entrer de force, mais ils sont trop nombreux. Trop immobiles. Ils se laisseront trop frapper.
Les bruits de la fête résonnent, des silhouettes vont et viennent autour de lui. Il se sent lourd, lourd et froid, prêt à s'enfoncer dans les profondeurs de la terre. Il a vaguement conscience à ses côtés de la présence des masques silencieux. Il se laisse tomber plus qu'il ne s'assied sur les dalles, devant la première rangée de hékel. Il ferme les yeux.
Après un espace de temps incolore, c'est le silence qui les lui fait rouvrir. Les hékel se sont écartés. Une silhouette vêtue d'une tunique brune s'avance entre eux. Lian ne reconnaît pas tout de suite Dougall : il semble plus grand, plus maigre, son visage est très calme. Ses yeux... une pellicule vitreuse, blanchâtre, les recouvre. Il tient une longue canne avec laquelle il tâte le sol à petits coups.
Lian se relève à genoux, tend la main vers lui : « Béra... »
Le visage pacifié frémit à peine. Dougall continue d'avancer à pas hésitants à travers la place, où chacun, sans le regarder, s'écarte de lui. Deux mains appuient sur les épaules de Lian, quelqu'un murmure à son oreille : « Il a choisi l'aveuglement, il n'existe plus, on ne doit plus le voir ni lui parler, le toucher ni l'aider, ne le sais-tu donc pas ? » Il se débat pour se libérer, se retourne vers un inconnu qui fait un pas en arrière et s'incline d'un air navré, les deux mains offertes à hauteur de poitrine.
Avec un cri de rage, Lian se détourne, court derrière Dougall. La silhouette brune est presque arrivée au bord du canal de ceinture, elle y arrive, elle se penche en tâtonnant vers les anneaux d'amarrage. Lian crie, en virginien : « Dougall ! Père ! »
Les mains de Dougall s'affairent avec maladresse à tirer une barque vers le quai, à défaire le nœud qui la tient captive... Lian le contemple, pétrifié. Lentement, en s'y reprenant à plusieurs fois, Dougall monte dans la barque, repousse de sa canne le bord du quai. La barque glisse entre les autres, semble hésiter un moment puis s'éloigne peu à peu, prise dans le courant qui la poussera de canal en canal vers le sud jusqu'à la Hleïtsaõ, dans les rapides, puis dans les chutes.
Le mois des Moissons passe, le mois des Fermiers, le mois des Chemins. L'été a viré à l'automne. Lian redescend lentement vers le sud. Au gré des routes, selon le travail qui s'offre ici ou là, ricochant de l'ouest à l'est au nord et au sud, toujours au sud, tel un joueur inepte mais obstiné d'Odhabaï. Il aide aux moissons puis à la cueillette des fruits sur les rives de la Toïtsaõ, recommence plus loin, d'autres moissons, d'autres fruits, peu importe. Il a perdu sa veste, il ne sait plus où. Ses habits tombent en loques. On lui en donne d'autres, il les prend. Il est batelier pendant trois lunaisons, charroyeur pendant une autre, vagabond dans l'intervalle. Il dort n'importe où, il y a toujours un champ pour l'accueillir, puis, lorsque la pluie commence à étoiler la poussière de l'été sur les routes, quelque part le long de la Hleïtsaõ, il y a des auberges, des fermes, des hangars, des arbres. Il mange ce qu'on place devant lui, ne répond pas aux questions détournées, ne voit pas les regards apitoyés qui l'accompagnent. Il ne sait pas où il va, seulement que c'est vers le sud. En chemin, il croise des Gomphali. Ils ont laissé leurs frères mourir dans les montagnes, ils vont faire leurs petits quelque part au soleil, et disparaître. Quelle importance ? Quelle importance ?
Un jour, à la traversée d'une rivière, loin à l'intérieur des Plaines Bleues, il rencontre Argelos. Il attend le bac, un jeune homme vêtu de bleu scintillant surgit près de lui et c'est Argelos. Il n'a pas beaucoup changé. Il est hékel maintenant. Il va à Markhalion. Lian veut-il venir avec lui à Markhalion ?
Markhalion se trouve bien au nord-ouest de la route de Lian, mais pourquoi ne pas retourner en arrière, pourquoi ne pas aller à Markhalion, il n'est jamais allé à Markhalion.
À Markhalion, par un incompréhensible télescopage du temps, c'est le début de la nouvelle année, les fêtes du départ de la Mer, odeurs, couleurs, musiques. Plus d'une saison s'est écoulée. Mais quelle importance ? La Mer brille bleu au-delà des quais de pierre polie. À midi, elle disparaîtra. De ce côté-ci du continent, il y a une éclipse de soleil pour accompagner son départ. Les elnoï, les Parfaits qui vont se joindre à elle, marchent en habits bleus dans les rues, l'air serein. Lian mange d'autres gâteaux, voit d'autres danses. Perd Argelos dans la foule, n'essaie pas de le retrouver. Devant le Haëkelliaõ, le matin du départ, alors que l'éclipse, lentement, commence à dévorer le soleil, des vagues d'adolescents se succèdent, plusieurs centaines, des filles, des garçons, avec des hékel. Ici et là dans leurs rangs, certains tombent dans une sorte de ravissement, les yeux tournés vers la façade du Haëkelliaõ. Ensuite, quelques-uns pleurent. Des adultes viennent embrasser les adolescents, en larmes aussi parfois. La Mer a désigné les futurs hékel.
Lian marche au hasard le long des quais. Au hasard, il s'arrête. Il se trouve dans les faubourgs, presque de l'autre côté de la baie. À l'ouest, Markhalion s'allume de torchères dans la fausse nuit de l'éclipse à moitié complète. Le ciel tourne au mauve. Les Parfaits doivent être rassemblés dans le grand amphithéâtre du port, prêts à monter dans les petites barques que le vent poussera au large, et d'où ils se jetteront dans la Mer. Quelque part le long de la Hleïtsaõ, y a-t-il une petite barque vide qui suit le courant en tournoyant ?
Ce quai est désert, trop loin du cœur de la fête. La masse bleue de la Mer semble presque immobile, légèrement ourlée là où elle touche la pierre, en bas des marches. Des irisations presque imperceptibles courent dans sa brume scintillante. Lian se déshabille. Il descend l'escalier jusqu'au bord. Au bout d'un moment, il fait un pas de plus. Il y a encore des marches sous la surface. La Mer n'est ni chaude ni froide, ni solide ni liquide. Elle n'existe pas pour son corps, son corps à lui n'existe pas pour elle, et pourtant elle est là. Il continue à descendre les marches. La masse bleue lui arrive à mi-cuisses à présent. Il se penche, laisse baigner ses mains dans cette absence visible, bouge un peu les doigts. Il ne sent rien. Les bateaux flottent sur la Mer, tout flotte, qui n'est pas absorbé par la Mer. Il s'allonge, se retrouve étendu sur le dos. La Mer le porte. Il se met à faire de lents mouvements de bras. Il n'a pas l'impression d'avancer, mais quand il jette un coup d'œil derrière lui, le quai s'est éloigné. Il continue. Des pensées paresseuses l'accompagnent. Qu'arrivera-t-il s'il s'éloigne assez du bord, au-dessus de l'abîme que la Mer recouvre ? Le repoussera-t-elle vers le quai avant de disparaître ? Partira-t-elle en le laissant s'écraser sur la terre noire à nouveau découverte, mille lani plus bas ? Ou se retrouvera-t-il sur le quai après le Départ, incapable de dire comment il y est revenu, une coque vide, incohérent, comme cet homme qu'il a vu un jour et dont on disait que la Mer l'avait refusé ?
Un soudain vertige l'envahit. Il ferme les yeux.
# Troisième partie
29
Alicia tombe. Elle tombe vers la planète au fond du puits. Depuis que la navette est entrée dans l'atmosphère, la tour de contrôle de Saint-Exupéry l'a prise en charge et elle n'a rien à faire qu'à essayer de ne pas penser qu'elle tombe. Elle sourit : elle sait bien qu'elle ne tombe pas – elle vole, ce qui est déjà assez bizarre en soi. Elle a pourtant aligné assez d'heures de sim depuis des années, mais elle a décidé de se refuser l'échappatoire trop facile du " c'est juste comme dans les sims ", elle ne veut pas se laisser dériver dans la légère auto-hypnose où elle pourrait se convaincre que sensations et perceptions sont exactement identiques à celles auxquelles elle s'est habituée. Elles ne le sont pas. Ceci est la réalité. Le point culminant de vingt-deux ans d'entraînement. Le but de toute sa vie.
Elle peut se permettre un sourire sarcastique : personne pour la voir. Tous les liens audio et visio avec Lagrange sont coupés, silence total de Saint-Exupéry aussi, ils la pilotent en automatique. Les derniers moments de sursis. Et ensuite, encore trois semaines à vivre en bocal, comme si la demi-quarantaine à bord de Lagrange n'avait pas suffi. Sans compter le temps écoulé à bord de la navette depuis qu'elle a quitté la station. Non, ils veulent encore trois semaines, pour être sûrs. Leurs semaines à eux, quatorze jours chacune. Ne devrait pas s'appeler " quarantaine " quand ça dure plus de soixante jours !
Mais ça aurait pu être pire. Au début des négociations, ils parlaient de la renvoyer en orbite dans la navette pour toute leur période d'observation. Quelle paranoïa ridicule. Que feront-ils quand leur propre émissaire reviendra sur Virginia ? Sera-t-il jamais assez " décontaminé " pour eux ? Bon, leur fameuse Peste terrienne a fait quatre-vingts millions de morts, mais c'était il y a près de quatre siècles. D'ailleurs, " terrienne ", " terrienne ", il n'y en a jamais eu la moindre manifestation sur Terre à l'époque, pas plus qu'ailleurs dans le système solaire. Phobies de planétaires. Phobies de Virginiens. Ce n'est pas le seul problème auquel elle aura affaire une fois rendue à destination.
Allons, un peu d'honnêteté. Bien sûr que quatre siècles, c'est plus que suffisant pour voir se développer de part et d'autre des virus sans danger pour les locaux, mais mortels pour les non-immunisés. Et on ne se fera pas faute de l'immuniser, elle, c'est sûr ! Déjà dans Lagrange... les dernières semaines n'étaient pas particulièrement agréables... Ennui mortel, en plus, à revoir pour la millionième fois les mêmes bandes, à refaire les mêmes exercices de sim, à vérifier pour la millionième fois qu'elle est prête, prête, prête pour la mission de sa vie.
Et ce n'est pas comme si tu n'avais pas de petites phobies non plus, eh, Alice qui tombe au fond du puits gravitationnel ? Pas vraiment une phobie : un souvenir de phobie, à la rigueur. Son corps est parfaitement conditionné, pas de nausée, pas de panique, sa terreur bien lagrangienne des surfaces convexes est depuis bien longtemps maîtrisée, mais c'est tout dans la tête, pour peu qu'on veuille se laisser aller, la réalité n'a rien à voir. Et elle peut bien admettre qu'elle joue à se faire des sensations. Elle sait qu'elle ne tombe pas, qu'elle vole en une rase trajectoire à travers l'atmosphère de Virginia et qu'elle se posera en douceur sur la piste d'atterrissage de la base militaire Saint-Exupéry – dans un coin très à l'écart de la piste, mais bien en sécurité sur le plancher des vaches, comme ils disent.
En même temps, pourtant, dans un recoin de sa cervelle, la petite Alicia est encore là, et ses cauchemars enfantins, l'atmosphère de Virginia comme une soupe de plus en plus épaisse, de plus en plus étouffante, et elle, lourde, lourde, qui coule en suffoquant, et elle essayait de s'imaginer dans un scaphandre pour lutter contre la terreur, elle descendrait, mais dans un scaphandre, et ils lui avaient retiré même ce maigre réconfort, non, elle n'aurait pas de scaphandre, elle se tiendrait toute nue à la surface – enfin, dans ses vêtements, mais sans aucune protection au fond, tout au fond du puits et quand elle lèverait la tête, elle ne verrait pas la concavité normale de l'horizon, comme dans les cavernes de Lagrange, avec les champs ou la forêt, les canaux ou le lac au-dessus d'elle, s'enroulant à l'intérieur du cylindre du monde : il y aurait seulement du ciel, à perte de vue. Oh, quelle crise d'hystérie elle avait piqué la première fois qu'ils l'avaient plongée dans le sim ! C'était pourtant sous un ciel relativement plein, où le regard pouvait s'accrocher dans la distance à de grands beaux strato-cumulus – mais encore bien trop vide, et bien trop convexe pour elle la ligne d'horizon quand, affolée, elle avait essayé de regarder ailleurs ; c'était la mer, en plus ; ils auraient au moins pu lui mettre des montagnes ! Elle avait vomi partout. Marti n'avait pas été content. Sa faute aussi, il n'avait pas voulu choisir l'approche progressive recommandée par les psy. Trois ans, elle avait seulement trois ans ! Il savait très bien quel effet ça aurait sur elle. Il l'avait fait exprès, comme tout le reste. Il a toujours voulu la voir échouer.
Un bref retour de rage douloureuse lui fait serrer les dents, puis elle s'oblige à se détendre. Elle est passée au travers de tous les obstacles que son père a mis sur son chemin, elle est ici, maintenant. Inutile de se faire le coup du pénible souvenir d'enfance. C'est bel et bien la mission de sa vie, on l'y a entraînée toute sa vie... non, elle s'y est entraînée : elle l'a faite sienne, et elle la réussira. Et Marti sera bien déçu. Rien que d'imaginer sa colère impuissante – et qui devra rester invisible – ça vaut la peine. Et puis, soyons quand même moins mesquine, un peu de tenue, elle est en voie d'entrer tout debout dans l'Histoire. Le Premier Contact Depuis La Séparation. La Grande Réunion des Frères Ennemis. Le Retour de la Terre sur Virginia. Ah, non, pas ça, elle peut le penser en ces termes, tout le monde le fait plus ou moins dans Lagrange, mais surtout ne jamais le dire devant des Virginiens. Terrible faux pas diplomatique, à proscrire absolument, quelle horreur, la Terre ne " revient " pas sur Virginia ! Et c'est vrai, en plus. Lagrange n'est pas la Terre. Lagrange 5 a levé l'ancre il y a bien longtemps, rompu ses amarres, fui la Terre en perdition. On n'y reviendra jamais, sur la Terre, et elle ne reviendra jamais nulle part. Dieu sait où en est la Terre maintenant. Lagrange n'a jamais essayé de le savoir, du moins pas officiellement. On préfère fantasmer, bien sûr, un désert sans vie ou une planète péniblement vivante mais sans humains, qu'on a eu raison d'abandonner à son triste sort. En réalité, et malgré toutes les fulminations des Néo-Jugementalistes, il y survit des poches de civilisation chaotiques et cahotantes, plus ou moins retournées à des archaïsmes divers... Quelque chose comme Virginia, par exemple.
Ah, non, encore un horrifique faux pas, discipline ton mauvais esprit, Alicia ! Et puis, la Mer est absente de Virginia depuis quatre de leurs Mois, là-bas, en bas, on est en pleine période moderne. Ou du moins le gouvernement et les militaires. Les fameux rebelles sécessionnistes aussi, sans doute, même s'ils n'ont jamais essayé de contacter Lagrange – ou pas pu : le gouvernement fédéral y a sûrement veillé. Et résolument modernes, les zones de la planète qui se trouvent au-dessus de deux mille mètres d'altitude, modernes depuis toujours, ou enfin depuis le début de la colonisation. La version virginienne de la modernité, c'est-à-dire qui n'a plus grand-chose à voir avec celle de Lagrange. Peut-on dire " une modernité archaïque " ? Mais, au moins, l'électricité.
Elle soupire. Avec tous les sims et les bandes visio et les tonnes de documents et les sessions avec les historiens, les sociologues, les psychologues, elle a l'impression de connaître cette damnée planète comme si elle l'avait faite ! « Illusion dangereuse », répétait le vieux Krantz, index levé. Il avait raison, bien sûr. Elle est tout à fait prête à la méfiance. Elle regrette seulement un peu de ne pas se sentir plus curieuse. « Tu ne vas pas là pour faire du tourisme », lui a sèchement rappelé Marti le jour où, fascinée, elle devait avoir douze ans, elle a commis l'erreur de dire lors d'une session d'instruction : « Tous ces mystères, les Anciens, la Mer... » La tête de Marti ! Ce n'étaient pas des mystères, l'énigme des Anciens était résolue depuis longtemps – ils s'étaient très certainement jetés avec armes et bagages dans la Mer au cours de grands auto-génocides religieux ; quant à la Mer elle-même, elle ne serait pas là lorsque la trajectoire de Lagrange l'amènerait à destination. La jeune Alicia avait remarqué une fois de plus alors, mais en le gardant pour elle, que ni Marti ni personne n'avait d'interprétation expéditive pour se débarrasser de la Mer, mystère ou énigme...
Ce n'est pas comme si elle voulait faire du tourisme, non plus : trop d'informations sont venues laminer sans répit ce qu'elle a pu éprouver au début de réelle curiosité. Mais non, Père, je ne vais pas faire du tourisme ni fraterniser avec les indigènes. Je vais chercher le Secret Perdu de la Propulsion Greshe, afin que les vaillants Lagrangiens puissent continuer à des vitesses moins escargotes leur voyage sans fin à travers la galaxie, et la galaxie suivante, et tout l'Univers pendant que vous y êtes. Et, surtout, ne pas vous laisser déranger par l'idée qu'une partie des vaillants Lagrangiens préféreraient peut-être s'arrêter une fois pour toutes, et pourquoi pas sur Virginia, hein ?
Mais cette idée-là est trop hérétique, même pour le noyau central de l'hérétique faction stoppiste. À vrai dire, Alicia ne la soutient, avec un amusement pervers, que pour voir les vieux birbes rappeler une énième fois, très rationnels, le handicap majeur de la planète, cette division entre zones modernisables et zones archaïques, à cause de la Mer. Pas question pour eux d'exprimer le motif profond de leur refus, de leur réflexe – bien sûr, trop incorrect : " Oui, mais il y a des Virginiens sur Virginia. " Au mieux, les moins hypocrites parlent de " problèmes d'adaptation avec les indigènes ". Heureusement qu'on peut noyauter aux franges du noyau. Être l'hérétique de quelqu'un, même des hérétiques. Cher Eduardo. Elle lui a quand même appris à dire " oui, mais ", à force. Oh, la tête des dignes camarades, ce jour-là, quand il a dit, l'innocent : « Oui, mais nous serions si peu nombreux que nous nous intégrerions vite à leur population. » L'assimilation, horreur. Personne n'a rien dit – ainsi confrontés avec leur réflexe raciste, que pouvaient-ils dire ? – et on est passé à un autre sujet. Comment l'aimes-tu, ton fils stoppiste, Marti ? Et si tu savais à quel point il l'est...
Elle hausse un peu les épaules, agacée de la tournure qu'ont prise ses réflexions – mal à l'aise. C'est pourtant une de ses victoires les plus délicieuses sur leur père, avoir réussi à convertir Eduardo. Pour l'instant du moins : Marti doit penser que cela lui passera, comme il croit que cela lui a passé à elle ; et il a peut-être raison, le gamin n'a que vingt ans (eh, gamine, tu en as vingt-cinq ! Oui, mais. Ses vingt dernières années à elle ont compté double.) Elle y prendrait sans doute davantage plaisir si Eduardo était conscient du véritable enjeu.
Conscient d'être un pion dans la partie qu'elle mène contre leur père ? Sûrement pas ! Le petit en aurait sans doute de la peine. Elle n'a pas du tout envie de lui faire de la peine. Elle l'aime bien, finalement. Et si elle l'a manipulé, c'est pour faire pièce aux manipulations des autres, Marti en tête. Et puis, elle ne l'a pas vraiment manipulé, au contraire, elle l'a déconditionné. Elle a élargi ses horizons. Elle lui a montré qu'il avait d'autres options. Plutôt ironique, en fin de compte. Il aurait bien aimé aller tout de suite sur Virginia, lui, je lui en ai tellement raconté... Mais ce n'est pas lui qui est en ligne pour devenir le futur Capitaine, hein, Marti ? C'est moi. C'est moi l'aînée, la préfabriquée. C'est moi qui dois faire mes preuves. Et je les ferai, je te le garantis.
*
Dans l'avion qui les emmène à Dalloway, un gros transport de personnel plein à craquer qui s'arrache avec difficulté à la piste de la Base Potemkine, Lian est au début aussi paralysé que ses compagnons ; il a vu comme eux des avions atterrir et décoller à la Base, et il a fait comme eux des entraînements avec hélijets et autoplaneurs ; mais jamais aussi haut ; il n'a jamais été malade lors de ces exercices, il était trop concentré sur l'exercice lui-même, le bruit avait un effet comme hypnotique – et puis, on s'élevait rarement au-dessus de quinze cents mètres. Une fois, à Hleïtzer, il était monté dans un ballon, avec Thomas, mais ça n'a rien à voir non plus. Ici, on grimpe, on grimpe, ça n'en finit pas, la surface du monde danse, s'incline, tourne, tombe et se perd dans la distance. Plusieurs recrues se mettent à vomir. Lian fait jouer ses maxillaires pour se déboucher les oreilles et avale sa salive, les dents serrées. Le ronronnement des moteurs est trop lointain, il faut sans cesse reconstruire la certitude du mouvement trop régulier. Après l'excitation angoissée de la montée en puissance pour s'arracher à la piste, après l'éclosion au soleil à l'altitude de croisière, loin au-dessus de la couverture de nuages, on a l'impression de faire du sur-place, c'est magique – et en même temps, chaque fois que Lian regarde les ailes qu'on peut voir ployer un peu, il ne sait s'il a envie de rire ou de hurler.
Environ à la moitié du trajet, la couverture de nuages s'effiloche puis disparaît. On est trop haut maintenant même pour le vertige : les vagues des collines, puis des plateaux qui se pressent au pied des montagnes Rouges n'offrent qu'un relief incertain, il suffit de tricher un peu avec ses perceptions pour y voir une simple carte un peu froissée. La ligne Ewald traverse sur toute sa longueur le haut plateau de Dalloway pour se perdre dans la distance au nord-est et au sud-ouest. On en voit clairement depuis le ciel la zone interdite, une longue balafre rectiligne alternativement grisâtre ou rougeâtre là où défoliants et bulldozers ont mis à nu du granit ou de la paragathe. Lorsque l'avion effectue le grand tournant qui le fait remonter vers le nord-est et sa destination, un long lac gris terne glisse sous l'aile, en plein milieu de la zone désertifiée, puis disparaît. Sur les cartes d'état-major, c'est le lac Boomerang – la forme en est caractéristique. Lian ferme les yeux pour ne pas voir son Leïtnialen.
30
La gigantesque mer intérieure du bouclier septentrional glisse vers l'ouest – le " lac Mandarine ", joli nom, mais pourquoi pas " mer ", techniquement plus exact ? Ça sonnerait moins bien, c'est vrai, l'amère Mandarine... Et surtout, il n'y a sur Virginia d'autre mer que la Mer, ils ont dû le décider au début de la colonisation et la Commission de toponymie virginienne s'est débrouillée avec. Après tout, c'est de l'eau douce, comme les deux autres " lacs " ; alors, d'accord, lac Mandarine. Trop haut pour le voir de cette couleur, juste une vaste surface jaunâtre et triangulaire, presque étranglée en son milieu sur une île minuscule dans la distance – l'île d'Aguay entourée de sa fameuse et infranchissable barrière de radiations naturelles. À courte portée, les radiations, heureusement, sinon voilà une zone idéale de peuplement qui aurait été perdue pour la postérité : sources chaudes, eaux poissonneuses, rives hyper-fertiles, climat tempéré en plein milieu de la froidure nordique...
Et dans une région accidentée plutôt inhospitalière aussi, le nord-ouest de la cordillère des McKelloghs qui étire ses dents pointues du nord au sud tout le long du continent. Alicia se permet un bref coup d'œil vers l'écran où s'arrondit la convexité bien visible de la planète à cette altitude, se perdant dans une brume blanchâtre – et au-dessus, le bleu laiteux du ciel ; qui se fonce, devient noir, et loin, loin dans le noir, il y a Lagrange, invisible à présent, mais qui achève ses orbites de yoyo entre les planètes du système d'Altaïr pour se rendre à destination. La navette a pénétré trop loin dans l'atmosphère à présent, il faut un effort trop délibéré pour imaginer la profondeur du ciel, un léger vertige menace : plutôt se laisser croire qu'il n'y a rien au-delà de cette cloche impalpable.
La navette amorce le grand virage qui va lui faire suivre le plateau des Deux-Rivières, le long de la chaîne ; la cime majestueuse du Catalin, le plus haut sommet de la planète à près de neuf mille mètres, glisse à son tour dans les écrans avec ses voisins plus modestes mais néanmoins couronnés de nuages, ou bien ce sont des écharpes de neige soufflées par les vents en altitude. Alicia ne peut s'empêcher de les contempler, l'estomac un peu noué. Il n'y a pas de montagnes dans Lagrange, encore moins d'hiver. Juste des collines avec de la rocaille artificielle pour les amateurs de varappe et de la pluie plus ou moins programmée. Mais ici, en cette deuxième moitié de ce qui est le long Octobre de Virginia, à cette latitude et à ces altitudes, l'hiver est déjà commencé – ou n'a jamais fini en ce qui concerne glaciers et neiges éternelles.
Le but du voyage est proche, à présent. Il fait beau sur cette région des McKelloghs. Exactement comme dans les sims, le relief bien dégagé – la couverture de neige précoce en moins, qui en accentue les détails. Il faut en reconstituer le code paradoxal : les champs en plus foncé, les forêts et les bois plus clairs. L'aspect tridimensionnel est tout de même mieux rendu. (Alicia hausse les épaules : évidemment, idiote, ce n'est pas un sim ! Mais les bandes visio en tridi ne font pas le même effet non plus. Tout a l'air plus petit, en tridi...) Les élévations de terrain se froncent comme les plis d'une épaisse couverture, on distingue les tortillements des moindres petits cours d'eau, entre l'élan des deux immenses rivières qui découpent le plateau longeant les McKelloghs, le large flot plus ou moins boueux de la Dandelion, les rives escarpées de la Holodbolchoï aux blancs rapides.
La navette est en trajectoire d'approche basse à présent, difficile de prétendre qu'on regarde une carte, c'est vraiment la surface, et pas du tout comme lorsqu'on fait de l'aéroglisseur dans Lagrange. Pas tellement la planéité, toute relative dans cette région quand même assez accidentée, mais... le désordre, Alicia n'a pas d'autre mot pour se décrire son impression, comme si tous les fractals de tous les programmes de sim n'arrivaient pas vraiment à égaler ce chaos organique. Heureusement, repérées avec une sorte de soulagement, des traces humaines : villes, routes, barrages, carrières, grandes cicatrices blanches des passages de lignes à haute tension, enfin la civilisation. Les civilisations. Assez facile, à cette altitude, de distinguer la superposition des humains sur les indigènes disparus : les grandes taches uniformes des cultures intensives, en longs rubans parallèles, ne s'accordent pas avec le relief du terrain comme le font ailleurs les terrasses ou les petits champs repris des anciens occupants de la planète ; et surtout les villes : le plan circulaire des Anciens, encerclé par les damiers monotones des villes nouvelles entre les lésions des zones commerciales et industrielles.
Et, dans la montagne rabotée, les longues pistes grises et noires de Saint-Exupéry, rubans de tarmac entrecroisés, hangars, entrepôts, radars, baraquements, complexe central et tour de contrôle. Finies les vacances, Alicia. Au travail.
Et en fin de compte, elle se retrouve en scaphandre au fond du puits : le temps de passer de la navette au véhicule isolé puis dans le tube flexible qui la conduit à ses appartements, et elle la garde jusque-là, sa combinaison spatiale, même si elle aurait pu la retirer une fois dans le sas du tube ; elle veut faire preuve de bonne volonté. Le comité de réception, au pied de la plateforme mobile de la navette, est en combinaison isolante aussi. On se serre les gants avec cérémonie en s'adressant par l'intermédiaire des interphones les paroles historiques requises et dûment répétées. Après quoi, pendant que les techniciens également en combinaison grouillent autour de la navette pour l'arroser d'agents décontaminants avant de l'emmener s'abreuver au lac proche, Alicia s'engouffre dans son cordon ombilical inversé, toute seule, et les autres se rendent dans la salle de contrôle d'où l'on conduira le reste des introductions sur un mode sagement virtuel. Avant son départ de la station, elle a fait la connaissance par l'intermédiaire des écrans d'Alan Grumberg, son homologue virginien, mais elle ne le rencontrera pas : il se trouve déjà dans la navette avec son propre scaphandre isolant, fin prêt à partir pour de bon lui aussi après toute une vie de simulations. Un gars laconique d'une trentaine d'années, format bulldog. On veut espérer qu'il a été correctement entraîné pour sa mission diplomatique : ils ont quand même disposé de tout le temps nécessaire après le premier contact. Ce qui ne vaut évidemment pas quatre cents ans d'obsession maniaque, mais on fait avec ce qu'on a. Non, elle exagère, quand même : on ne savait pas vraiment où on allait quand on a désamarré la station, au moment des Troubles ; ça n'a été voté qu'après une cinquantaine d'années, quand ils en ont eu assez de ramper à travers l'espace. Et le projet Greshe a seulement démarré quand Marti a résolu d'en faire le projet de sa vie – et de la vie de quelques autres. On ne peut pas dire qu'il n'a pas de suite dans les idées, il avait quoi, treize, quatorze ans ?
Les appartements sont de dimensions plus que respectables pour qui est habitué aux locaux exigus de Lagrange, et pourvus de tout le confort moderne virginien. La console de communication, entre autres – elle ferait saliver les connaisseurs rétro-techno de Lagrange – est assez fonctionnelle. Et de toute façon, Alicia a appris à se servir de toutes ces antiquités sans prix, comme elle a appris le virginien. Pauvres linguistes, ils se sont bien amusés pendant un temps à extrapoler la dérive du dialecte depuis la rupture totale de la colonie avec la Terre, mais on y a vite mis fin : ce serait inutile, on capterait les émissions radio assez tôt – s'il y en avait, évidemment – pour effectuer la mise à jour bien avant la naissance de l'émissaire prévu. L'évolution vraisemblable de la société virginienne, scénarios, options, ça, c'était important pour La Mission, sociologues et psychologues ont toujours été abondamment subventionnés. La langue... les détails peuvent toujours être rectifiés sur place sans trop d'obstruction des indigènes ; la non-communication peut même constituer un outil utile – on a le temps de réfléchir et d'observer pour vérifier l'exactitude des scénarios socio-politiques, par exemple, pour lesquels les indigènes sont sûrement moins enclins à fournir de l'aide. Les techniciens ont proposé, Marti Coralàn a disposé, l'assemblée a avalisé, fin de la discussion.
Et de fait, les indigènes ne font pas d'obstruction. Ils n'ont voulu aucune communication directe de données de Virginia à Lagrange et réciproquement, pour diverses raisons de sécurité (ils craignent sûrement davantage les contaminations informatiques que l'espionnage des rebelles et croient sans doute autant que les Lagrangiens aux vertus de la non-information. Tous aussi maniaques les uns que les autres !). Leur seul caprice – avec l'échange d'ambassadeurs solitaires, et ma foi, on n'avait pas non plus envie sur Lagrange d'envoyer trop de monde se faire spirituellement contaminer, hein ? Et puis, c'était une façon de soutenir la thèse officielle, qui est même vraie : non, non, nous voulons seulement retrouver les plans de la propulsion Greshe, pas question de débarquer même pour du tourisme sur (pouah !) le plancher des vaches ; et avec ou sans les plans de la propulsion Greshe, nous repartirons ensuite vers de nouvelles aventures et vous ne nous reverrez jamais. Les planétaires gardiens des vaches ont choisi de ne point être offensés par le contenu subliminal de ces déclarations : mais c'est parce que leur phobie réciproque est la meilleure garante de la bonne foi entre Lagrangiens et Virginiens. On se rassure comme on peut. Et si la théorie officielle se voit contredite un peu par la suite des événements, eh bien, l'univers n'est pas une horloge.
Comme convenu, Alicia trouve dans sa console tout un tas de cadeaux, données historico, socio et le reste – sans aucun doute péniblement officielles, mais on ne va pas se plaindre, celles qu'elle apporte le sont tout autant ; elle aura en tout cas de quoi s'occuper pendant les trois semaines d'observation.
Après quelques échanges utilitaires avec ses gardiens, pardon, ses hôtes, sur le bon usage de ses appartements et la routine à venir, Alicia déclare qu'elle va maintenant se reposer, on lui souhaite une bonne méridienne et on coupe le contact. Elle ne doute pas qu'on ne continue à la suivre en audio, peut-être même en visio, et si c'est le cas elle souhaite bien du plaisir aux voyeurs, mais en réalité elle s'en moque : elle y est préparée depuis toujours ; elle pourrait aisément repérer et neutraliser les engins espions, mais on a décidé qu'elle serait parfaitement transparente, question de confiance. Seulement en cas d'extrême urgence, le contact avec Lagrange par l'intermédiaire du transmetteur dont les éléments inoffensifs et méconnaissables sont disséminés parmi ses affaires de toilette ; ses hôtes, ses gardiens, n'ont nul besoin d'en connaître l'existence. En cas de vie ou de mort, exclusivement. Oui, Capitaine. Mais j'ai le transmetteur, et Eduardo a un récepteur aussi, que dites-vous de ça, Capitaine ? En cas de vie et de mort – ou si elle trouve les données convoitées, et alors les hérétiques des hérétiques se mettront en branle.
Elle se déshabille en bâillant. Elle est morte de fatigue, maintenant que l'adrénaline retombe : elle vit à l'heure de Virginia depuis sa demi-quarantaine dans Lagrange. Le plus étrange, sûrement, cette ultime séparation alors qu'elle était encore là, ces deux semaines dans sa bulle à l'intérieur de la bulle de Lagrange. Pour les autres, bien sûr : les contrôleurs, et Eduardo. Pour elle... pas vraiment. Depuis sa toute petite enfance elle travaille à se rendre étrangère. Elle a appris à vivre dans un autre monde – non, sur une planète –, une autre société. Elle en a toujours trop su pour être à l'aise dans Lagrange. Les autres enfants se sont chargés de lui rabattre le caquet les quelques fois où elle a essayé de jouer les importantes avec tout ce qu'elle apprenait et dont ils n'avaient pas la moindre idée – et quand Marti a appris ces indiscrétions, quelle fusillade ! Elle a tout dû garder pour elle – jusqu'à la rencontre avec Eduardo, en cachette de tout le monde. Ensuite, bien sûr, quand elle a eu une dizaine d'années, on a appris en grandes pompes officielles quelle était son auguste destinée. Personne n'a été très impressionné, à vrai dire. Elle était déjà à part, avec sa double vision secrète et ses " oui, mais " ; au mieux – dans les meilleurs des cas – on a compris pourquoi. Sinon, on a continué à ne pas beaucoup la fréquenter : trop dérangeante. Oh, les psychologues ont toujours veillé à la rectitude de son allégeance lagrangienne, au cas où. Elle leur a toujours dit ce qu'ils désiraient entendre. Elle ne se fait pas d'illusions, cependant : une fois parmi les Virginiens, elle se trouvera simplement de l'autre côté de sa double vision. La seule chose qui compte, c'est qu'elle sache où elle se trouve réellement, elle : ailleurs, peu importe, là où elle aura choisi d'être. Les psychologues croient ce qu'ils ont besoin de croire, comme Marti et le Conseil, tout le monde. Et elle, elle ne croit qu'en elle-même. Elle sait qui elle est. Eux, ils n'ont pas besoin de le savoir.
Alors qu'elle se frotte avec délice sous la douche, l'idée la frappe qu'elle est de nouveau dans une bulle pour plus de quarante jours. Sauf qu'à l'extérieur de la bulle, cette fois, c'est une planète entière, pas une autre bulle mais une grosse, très grosse boule – et du plein, pas du creux. Elle quitte le cubicule rempli de vapeur, vite amusée de son malaise : elle ne va pas devenir claustrophobe, maintenant ?
Elle dort mal. Elle rêve, le vieux rêve. La station a explosé, ils flottent tous dans le grand silence du vide, mais ils flottent ensemble. Elle, elle est toute seule, très loin, et elle essaie de nager pour les rattraper, si elle les rattrape elle ne mourra pas tout de suite, mais ils s'éloignent sans entendre ses appels, bien sûr, dans le vide on ne vous entend pas crier. Sauf si on est dans un scaphandre, mais elle n'est pas en scaphandre. Elle a laissé sa combinaison à l'entrée de la chambre et ils ont dû l'emporter pour la décontaminer...
Elle se réveille complètement, enfin, arrachée au sommeil par ses tentatives confuses de rationaliser le cauchemar, ou capable de le rationaliser parce qu'elle était déjà réveillée, peu importe, elle est réveillée. Et la désorientation frappe quand même, un très bref instant – les dimensions de la pièce, les couleurs, l'odeur, les textures, les formes, l'intensité de la lumière qui point lorsqu'elle se lève – pas besoin de claquer des doigts ou de parler, quand même. Elle respire plusieurs fois, à fond, en espérant qu'elle n'a pas crié tout haut – qu'en déduiraient les observateurs ? Elle a dormi comme une bûche, la méridienne est finie depuis longtemps. Elle a faim. C'est le moment de vérifier le bon fonctionnement du matériel ad hoc.
La nourriture est sans surprise : rien d'exotique ; les experts ont sûrement mis au point un programme d'adaptation bien gradué. Alicia replace le plateau vide dans la poubelle à succion, bâille avec ostentation, non, non, vous n'êtes pas en train de m'observer, et va s'installer à la console pour en continuer l'exploration en circuit fermé. Mais un malaise rémanent s'obstine, le rêve ne veut pas s'éteindre. Son plus ancien cauchemar. Le premier dont elle se souvienne – la soupe suffocante de Virginia n'est venue qu'après. Elle sait exactement de quand il date. Cinq ans. Quand elle a découvert qu'elle n'était pas unique.
Alicia le Fantôme. Le nom qu'elle s'était donné à l'insu des autres, son secret à elle toute seule. Les entraînements étaient commencés depuis deux ans ; une fois les premières horreurs passées, elle s'était convaincue qu'elle en était très fière – elle l'était : l'attention de son père, de sa mère, de ses autres dresseurs, tous ces mystères partagés avec les grands seulement, ce qu'elle savait et que les autres devaient ignorer... Elle n'avait pas vraiment d'amis, elle préférait jouer toute seule : elle s'entraînait pour sa mission – on ne lui avait pas dit exactement de quoi il s'agissait, farfouiller dans des banques de données archaïques, ou sinon elle aurait eu le fantasme moins facile. Non, elle se rendrait sur une planète, un jour, parmi des planétaires, des gens qui n'aimaient pas les Lagrangiens, et elle devrait alors accomplir sa mission – héroïque et dangereuse, c'est ce qu'elle avait conclu en écoutant l'envers des discours adultes ; malgré les réassurances de Virginia, on se préparait encore pour tous les scénarios, à cette époque-là, y compris des planétaires soudain hostiles et des recherches à effectuer malgré eux. Elle devait donc apprendre à devenir une héroïne, forte, endurante, habile, pleine de ressources, et surtout capable de se faufiler partout sans se faire repérer. Du coup, elle avait pris l'habitude de déambuler sous la peau de Lagrange : conduits d'aération, gymkhana de tubulures, postes de contrôle souterrains, innombrables machines entretenant l'existence du monde. L'illusion du monde, car, elle l'avait vite compris, ce qu'elle considérait comme le vrai monde était un mensonge, et ces profondeurs étaient l'envers d'un décor. Le vrai monde n'était pas là, il se trouvait encore plus loin à l'extérieur de la bulle concave de Lagrange, et il n'était pas gentil, il était même carrément méchant, le vrai monde. Mortel. Le vide de l'espace autour de ces couches successives d'entrailles mécaniques, de terre, d'air, de plantes, d'animaux et d'humains, autour de la dure enveloppe de ferro-nickel et sa pellicule d'autres machines, les yeux, les oreilles de Lagrange, et les énormes jambes qui pédalaient si fort pour ralentir Lagrange.
La station, c'est à cette époque-là aussi qu'elle avait appris ce terme, en même temps que l'histoire du monde. Juste un astéroïde en forme de patate, évidé et transformé pour être ancré dans un coin du réseau invisible de forces qui maintenaient en équilibre le Soleil, la Terre et la Lune – encore de nouveaux termes à apprendre tandis que le petit point de Lagrange, qui s'appelait alors O'Neil, se détachait lentement puis de plus en plus vite du système solaire terrien reproduit en animation et fausses couleurs sur l'écran (" pour atteindre cinq pour cent de la vitesse luminique ", disait la voix onctueuse de la commentatrice). Autrefois, les gens n'avaient pas vécu à l'intérieur du monde, mais à l'extérieur, à la surface d'un autre monde, beaucoup, beaucoup plus gros, mais ce n'était pas très impressionnant sur l'écran ; le sim, un peu plus tard, oui, horriblement, mais cette première expérience avait été une pure découverte, d'abord déroutante – quoi, le monde n'avait pas toujours été le monde ? – puis inquiétante – le monde pouvait ne pas être le monde ? – mais enfin plutôt séduisante : le monde pouvait être autre chose que ce qu'il était. Et puis, la progression était intellectuellement satisfaisante : on vivait maintenant à l'intérieur du monde, lui-même à l'intérieur de l'astéroïde, lui-même en mouvement dans le grand ventre noir de l'espace... Sauf que ce ventre n'était pas très accueillant, au contraire. Comme tous les autres enfants de Lagrange, cependant, elle avait appris à ne pas y penser très souvent. Mais, au contraire des autres enfants, elle savait qu'elle quitterait un jour Lagrange pour se rendre sur un autre monde. Sur une planète dont la surface ne serait pas du tout dangereuse de la même façon que celle de Lagrange. Où c'était l'intérieur, en réalité, qui serait dangereux, on suffoquait si on descendait trop profond – c'est là que, l'idée de la pression atmosphérique se mélangeant avec celle de la pesanteur, elle avait développé pour se rassurer son histoire du scaphandre, aussitôt tuée dans l'œuf par ses instructeurs.
Mais sur cet autre monde, avec ou sans scaphandre, elle accomplirait sa mission, La Mission qu'elle et elle seule pouvait accomplir, pour laquelle elle s'entraînait déjà plusieurs heures par jour, et même quand ses instructeurs la laissaient tranquille, car alors elle devenait Alicia le Fantôme, circulant comme une ombre à l'envers du décor, et même dans le décor – un fantôme doit aussi travailler à se glisser un peu partout sans se faire remarquer là où il y aurait du monde pour le voir, n'est-ce pas ?
Et un jour, alors qu'elle rampe en silence à l'envers d'un plafond en essayant de reconstituer mentalement le plan des bureaux de son complexe d'habitation, pour se situer, elle entend des voix familières, son père, sa mère. Elle s'immobilise en se mordant les lèvres avec un plaisir coupable : elle est là et ils ne le savent pas ! Ils parlent de leurs obscures affaires d'adultes, et elle va repartir en redoublant de prudence muette quand sa mère dit : « Mais Alicia est l'aînée. » Plus que son prénom, plus même que le dernier mot, c'est d'abord l'intonation qui arrête Alicia : une résignation agacée.
« Règlement stupide, grommelle son père. L'arbitraire érigé en loi. Pourquoi le premier enfant du Capitaine, peu importe le sexe, devrait-il hériter de la fonction ?
— Parce que l'enfant a été prévu pour, indépendamment du sexe, Marti, dit la mère d'Alicia, ironique. L'alliance harmonieuse du hasard et de la nécessité.
— Oh, je t'en prie, Greer, ne me ressors pas ces vieilles rengaines !
— Ce n'est pas comme si j'avais eu le choix non plus, Marti, dit la mère d'Alicia d'une voix sèche. J'ai servi de ventre, c'est tout. Techniquement c'est ta fille, pas la mienne. Et je ne vois pas pourquoi tu fais tant d'histoires. Elle a encore le temps d'échouer, Dieu sait, et à ce moment-là tu pourras toujours essayer de pousser Eduardo. Quoique, légalement, c'est toute une pente à remonter, je te souhaite bien du plaisir. »
Et à ce moment-là, Alicia, qui a confusément saisi qu'elle ne devrait pas entendre cette conversation, ou du moins ne pas la comprendre, essaie de continuer à ramper à l'intérieur du plafond, mais elle ne fait pas aussi attention qu'elle le devrait : elle pose les mains à un endroit qui n'est pas assez solide – elle l'aurait bien vu si elle n'avait pas pensé à autre chose, essayé de ne pas penser à autre chose – et le plafond cède dans un grand craquement poussiéreux et elle tombe aux pieds de ses parents médusés, la tête la première.
Légère commotion cérébrale, une semaine à l'hôpital, et personne n'a jamais évoqué l'incident, personne ne lui a même jamais demandé ce qu'elle faisait dans le plafond. Quelquefois, lorsqu'elle y pensait, elle se demandait si elle n'avait pas rêvé. Mais elle avait déjà assez exploré l'envers du monde pour savoir que tout a un envers, et que même les rêves peuvent avoir un ventre caché de réalité. Elle a retourné celui-là sur le dos, et elle a vu. En allant fouiller dans les banques de données interdites, mais elle n'a pas demandé la permission non plus : le dossier personnel de son père, de sa mère, le sien propre. Et celui d'Eduardo, le petit frère de deux ans et demi dont elle avait ignoré jusque-là l'existence. Pas vraiment un frère, une moitié de frère, ils ont le même père, c'est tout. Greer non plus n'est pas sa mère à lui. Marti Coralàn a une autre famille, sa vraie famille ; il ne vit plus avec Greer – ils se sont séparés au moment de la naissance d'Alicia. C'est juste pour les besoins de la cause, de La Mission, pour le bénéfice d'Alicia qu'ils prétendent. Quand Alicia dort, Marti et Greer ne dorment pas ensemble. Il est parti chez lui, elle est partie chez elle, et c'est de là qu'ils viennent le matin ; c'est là aussi qu'ils se trouvent lorsqu'ils partent plusieurs jours pour leur " travail ", et que les instructeurs s'occupent seuls d'Alicia. Marti ne voulait pas mélanger son vrai fils avec sa fille de hasard, sa fille forcée.
Oh, qu'il a été furieux quand il les a trouvés ensemble ! Trois mois, quand même, elle a réussi à garder le secret avec Eduardo pendant trois mois... Et ensuite, Marti ne pouvait plus s'opposer aux exigences des psychologues menés par le vieux Krantz qui l'avait à l'œil : les dégâts potentiels étaient déjà assez importants, on avait assez sacrifié au caprice de Marti, le contrôle des dommages devait commencer tout de suite : la petite continuerait à voir son frère. Et finalement, elle est allée habiter chez Marti et son autre femme, qu'elle a appris avec obéissance à appeler Maman. Beaucoup, beaucoup plus tard, elle a réussi à décrypter le code des donneurs, à la Banque, et elle a appris qui était sa mère génétique – morte depuis deux générations. Mais ce n'était plus très important alors, juste une question de principe. À ce moment-là, elle s'était habituée à se considérer comme la fille de personne. Alicia le Fantôme.
Mais c'est alors qu'elle a commencé à faire ce rêve, après l'hôpital. Le vrai sujet du rêve, ce qui la réveille en sanglots encore quelquefois, malgré elle, ces larmes qu'elle essuie avec rage, ce n'est pas à cause du vide mortel où culbutent les morceaux de la station éclatée, ce sont les trois silhouettes qui s'éloignent en flottant, Marti, et Greer ou l'autre Maman, mais ensemble, avec entre eux le tout petit scaphandre d'Eduardo qu'ils tiennent chacun par une main.
31
Lian et ses compagnons n'iront jamais dans le complexe central de Dalloway, les tours de contrôle et les bâtiments administratifs de l'ancien spatioport reconverti, ne les voient que brièvement alors que, accompagnés d'avions de chasse aux allures de moustiques, ils font leur approche au-dessus du grand lac rond jouxtant hangars et zones d'entretien pour aller se poser à l'extrémité des pistes d'atterrissage. Leurs baraquements se trouvent près de l'hôpital, au nord ; on les y conduit dans une flotte d'antiques gazobus bleu et noir tout ferraillants. Ils n'y resteront pas longtemps, quelques jours, le temps d'être instruits dans ce qui sera désormais leur tâche principale : patrouiller la Zone II, entre le défilé où passent route et voie ferrée, au nord, et un semis de trois petits lacs, à l'est (" Los Tres Niños ", dit la carte en mode 'spanic archaïque). Le saillant de Dalloway est comme une ligne Ewald en miniature sur trois côtés, mais même du côté du lac on a installé le même genre de défenses pour ne pas être pris à revers. La zone d'interdiction totale s'étend autour du complexe lui-même : sur cinq kilomètres de large, pas une herbe, pas un buisson, un sol râclé à l'os et truffé de senseurs et de mines à pression ; si une bestiole un peu trop grosse vient à se perdre dans ce désert, elle déclenche des feux d'artifice, mais il y a longtemps que les animaux du plateau ont appris leur leçon. La Zone II est moins féroce : les champs de mines y sont limités aux couloirs d'infiltration probables, les senseurs plus espacés, et on a fait grâce à la végétation, sauf autour des points d'appui, des ensembles de bunkers situés sur des élévations de terrain et entourés d'un glacis d'herbe rase comme celui où la Section 3 du 2e Peloton de la Compagnie Écho du 101e Régiment d'infanterie légère va prendre ses quartiers pour le reste de la saison, au bord du plus grand des Trois Enfants.
Les premières semaines de Septembre sont tranquilles. La Section 3 apprend à connaître son territoire. Au début, ils sont nerveux : c'est la partie de la Ligne la plus proche de l'ennemi. En patrouille, Lian est aux aguets et examine comme les autres chaque détail du paysage où peut se dissimuler un rebelle, arbres, lits encaissés des petits torrents, ces hauts buissons fleuris, cet amoncellement de rocaille, ces deux troncs abattus soudés par les draperies de mousse ; quelquefois, il déraperait presque – le rythme particulier d'un chant d'oiseau dans le lointain, une bouffée de parfum, un jeu de la lumière à travers les feuilles, ou juste l'allure familière de la marche en forêt quand on doit faire silence ; mais la plupart du temps son attention est celle de Liam Shaunessy, et le plateau n'est que celui de Dalloway, avec les noms du plateau de Dalloway fermement inscrits dans l'esprit de Liam Shaunessy comme sur la carte.
Ils font de la surveillance, mais surtout, finalement, de la maintenance ; au début, toutes les alertes qui résonnent dans les bunkers les font bondir, dents serrées, à la recherche de leurs armes, mais l'attitude nonchalante des techniciens préposés aux instruments les convainc bientôt de se détendre. Des senseurs claquent, grignotés par la faune ou simplement par l'usure normale, des mines sautent sur un animal malchanceux, à cause d'un arbre qui tombe, dans un petit glissement de terrain – une fois, lors d'un orage, à cause d'un éclair mal placé, un champ entier explose de proche en proche, un vrai tir de barrage qui les réveille en sursaut, le cœur dans la gorge, cette fois ça y est, les Rèbs attaquent... Mais non, les sondes automatiques de reconnaissance ne voient rien, fausse alerte, encore. Grayson lui adresse un clin d'œil : « Plutôt calme pour la guerre, l'Année de la comète ! » La radio transmet des rapports d'escarmouches et de points chauds un peu partout le long de la Ligne et au sud-ouest de Dalloway, mais dans le secteur nord de la Zone II, on s'ennuierait presque.
Ils passent quand même trois jours à reconstituer le champ de mines dévasté, qui borde le plus petit des Trois Enfants ; malgré la proximité de la frontière, et sans doute à cause de la tension suscitée par le travail, dangereux malgré tout, ils sont particulièrement exubérants, se baignent et jouent au water-polo dans le lac, les soirs ; la dernière nuit, alors qu'à la fin de l'après-midi Ritchie surpris par un envol de papillons géants a failli se faire sauter avec l'engin qu'il manipulait, Grayson et Lian se glissent hors de la tente qu'ils partagent avec Delgado et Stuyvesant, s'amusent à échapper à la vigilance des trois sentinelles et se rendent au bord du lac pour y nager nus en silence ; ensuite, ils font l'amour. Lian n'est plus aussi surpris de l'urgence de son compagnon, comme il l'a été au début de leur séjour sur le front. Un réflexe profond unit érotisme et mort chez Grayson, chez tous les autres aussi sans doute – un tropisme que Lian n'éprouve pas avec la même intensité, mais dont il a appris à reconnaître les signes, maintenant ; ce sont les soupirs de Ritchie et de Vitale, dans la tente voisine, qui l'ont réveillé, et Grayson aussi.
Et puis, vers la fin d'Octobre, alors qu'ils sont bien installés dans leur routine – alors que Lian se surprend de plus en plus souvent à déraper vers le Landaïéïtan –, les senseurs se mettent à tomber en panne à une fréquence accélérée, des mines sautent presque tous les jours. Lorsqu'ils se rendent sur place, cependant, aucune trace de rebelles, et la reconnaissance aérienne ne décèle pas d'activité dans leur secteur. Ils réparent, remplacent, multiplient les patrouilles. Rien n'y fait. Les senseurs continuent à claquer, les mines à exploser. Et on ne trouve même pas de débris de bestioles alentour.
Après environ neuf jours de ce régime, et alors que la Section revient au bunker encore bredouille, les techniciens soupirent, plus agacés qu'inquiets : « Allez, c'est les gremlins. Ils ont décidé de venir jouer dans notre coin. On va y avoir droit pour le reste de la saison. »
Des petites unités de rebelles, de la taille d'une Section, dix, quinze soldats maximum, qui se livrent à ces opérations de harcèlement continu. Connaissent le plateau comme leur poche, n'entrent jamais au contact, de vrais courants d'air, on ne les reconnaît qu'aux fréquences de pannes du matériel, c'est pour ça qu'on les appelle " gremlins ". « On aurait vu leurs traces ! » proteste Nan vexée. Non, justement, quand on repère leur piste, c'est là qu'il faut se méfier : ils vous conduisent là où vous les suivez.
Le jour où ils tombent sur des marques récentes, Nan, qui est le chef de section, décide de passer outre à l'avertissement des techniciens. Ils suivent les méandres capricieux de la piste pendant toute une après-midi, dans les parties les plus inhospitalières de leur secteur, de plus en plus proches de leur gibier, apparemment, mais sans jamais réussir à l'entendre ou à le voir en direct. Et en fin de compte, alors qu'ils font une pause, fourbus et exaspérés, pour décider s'ils rentrent au poste ou continuent pendant la nuit avec leur équipement de vision nocturne, une avalanche de fruits pourris leur dégringole dessus de toutes parts, des cailloux, des mottes de terre boueuse. Ils réduisent les arbres environnants en charpie, mais quand le vacarme des armes automatiques s'est tu et qu'ils vont examiner les lieux dans le silence de mort qui est tombé sur la forêt, rien. Et Lian ne peut longtemps faire l'hypothèse d'une troupe de chachiens, qui ont parfois ce type de comportement lorsqu'on empiète sur leur territoire, car le tir de barrage en aurait bien tué au moins quelques-uns, et il n'y a pas une trace de sang dans les échardes de bois et la bouillie de feuilles.
Les techniciens haussent les épaules, mi-apitoyés mi-narquois : « C'est les gremlins, les gars. On vous avait prévenus.
— Mais ils auraient pu nous tuer n'importe quand, alors, murmure Lian abasourdi.
— Oh, ne pensez pas qu'ils ne le feront jamais ! » dit le plus âgé des techniciens, un grand sec du nom de Passeron. « Ils vous endorment, et après, crac, ils vous descendent. C'est des vicieux, les gremlins. Jamais se relâcher, avec eux ! »
Ils redoublent de vigilance dans leurs patrouilles, triplent les sentinelles lorsqu'ils vont s'occuper des senseurs et des mines qui continuent à sauter n'importe quand – autant de mains de moins pour le travail, qui dure deux fois plus longtemps ; ils ont à peine fini de réparer ou de remplacer ici, un message du bunker les prévient que ça a sauté là ; quelquefois ils restent sept, huit jours d'affilée loin du poste, dans la chaleur humide et sans répit, à manger des rations et à dormir entassés dans leurs minuscules tentes de campagne sans pouvoir vraiment se laver ou mettre des habits secs. Leur humeur s'en ressent. Le jour où on leur tire dessus, c'est presque un soulagement, ils répliquent avec un enthousiasme disproportionné. Une fois la fumée retombée, cependant, ils se rendent compte que personne n'a été touché parmi eux. Pas de trace des gremlins, bien entendu, sur la petite crête d'où sont partis les tirs rebelles. Lian examine depuis la hauteur le ruisseau en contrebas, près duquel ils ont été cloués pendant une demi-heure. Pas beaucoup de couvert. Ou bien les Rèbs tirent comme des pieds, ou bien ils les ont délibérément manqués. Il penche pour la seconde hypothèse.
« Jusqu'au jour où ils ne nous manqueront pas, murmure Grayson.
— Si j'en vois un, en tout cas, grommelle Delgado, je ne le raterai pas, moi, je vous le dis. »
Deuxième semaine de Novembre. Troisième. Les gremlins sévissent toujours. Lian a de plus en plus de mal à prendre la situation au sérieux, malgré les avertissements moroses de Passeron et des autres techniciens – « Et si c'était juste une préparation psychologique à une attaque massive, hein ? Ils nous endorment et crac ! » Il n'y a pas eu de véritable bataille rangée sur la Ligne depuis des décennies, pas même à Dalloway qui est pourtant le point d'étincelle. Quand une action implique cent cinquante hommes de part et d'autre, on parle de bataille ! Tout regroupement, tout mouvement massif de troupes se ferait repérer par les patrouilles aériennes et les sondes ; même une infiltration de densité moyenne se ferait repérer. Et les gremlins n'ont jamais même blessé des membres de la Section 3 – sinon dans leur amour-propre, comme lorsqu'ils leur ont volé une caisse de senseurs au nez et à la barbe des sentinelles, qui n'y ont vu que du feu et pourtant jurent ne jamais s'être endormies.
« C'est quoi, cette guerre-là ? » proteste Ritchie Abdul lors d'une des séances de réflexion avec le capitaine Trent qui fait son tour bihebdomadaire des postes. Comme les autres membres de la Section, le radio est passé de la surprise à la colère puis à une autre sorte d'étonnement, plus fondamental, plus proche du malaise. Lian garde le silence, mais n'en pense pas moins. Une autre sorte de guerre, Ritchie. La guerre comme se la faisaient les Hébao, autrefois ; un jeu, grave, mais sans victimes humaines ; tous les rebelles ne sont peut-être pas aussi assoiffés de sang fédéré qu'on nous l'a toujours dit. Quelquefois, il se prend à imaginer qui sont ces soldats inconnus, de l'autre côté : des jeunes, comme eux, des vétérans ? S'amusent-ils autant à leurs tours que lui, pour sa part, commence à le faire en essayant de les prévoir et de les déjouer ? Si par extraordinaire, par erreur, ils se rencontraient au détour d'un chemin, que feraient-ils ?
« Les réflexes prendraient le dessus », dit Grayson en haussant les épaules.
Lian murmure : « Ça se domine, des réflexes, Gray », et l'autre émet un petit « tsk » agacé : « Pas forcément au contact, et pas quand il y a surprise. Tout le monde n'est pas comme toi, saint Lian. Ne te fais pas trop d'illusions, OK ? Je ne tiens pas à te ramasser en petits morceaux, ça fait désordre. »
Comme Lian sait que l'inquiétude de son compagnon est réelle, malgré le ton sarcastique, il ne relève pas.
Dernière semaine de Novembre. Tout d'un coup, l'état-major s'énerve. Quelqu'un, en haut lieu, a décidé qu'il était temps de mettre fin aux activités des gremlins du secteur nord de la Zone II. Pour le moral des troupes, peut-être, ou bien un général a envie de faire des exercices grandeur nature sur le terrain. Dès que les gremlins du secteur nord se manifestent à nouveau – ils font maintenant sauter les senseurs selon un rythme bien particulier, comme s'ils avaient décidé de signer leurs attaques – on fait donner les hélicoptères, trois pelotons sautent dans le secteur nord, le bouclent et commencent à resserrer la nasse. La Section 3 est au premier rang des opérations – ils se sont précipités dans le secteur dès qu'ils ont eu confirmation de la signature de panne. Lian est persuadé que tout ce brouhaha ne servira à rien, que les gremlins passeront à travers les mailles du filet. C'est avec un scepticisme total qu'il se dirige, avec le reste de sa Section, vers la petite élévation de terrain où le QG déclare que se sont retranchés les rebelles : comment le sauraient-ils, hein ? Les infrarouges ne donnent jamais rien, avec les Rèbs.
Des tirs nourris les accueillent, et cette fois-ci on ne joue plus : Vitale pousse un cri presque incrédule « Médic ! ». Lian bondit pour l'aider à se traîner à l'abri. Une blessure à la cuisse, pas très grave, la balle a raté de loin l'artère fémorale ; Lian désinfecte, recolle, panse, administre la drogue qui annulera l'effet de choc, et Vitale reprend son fusil pour se mettre à mitrailler avec une conviction nouvelle l'amas de rocher où sont bel et bien coincés, il faut le croire, les rebelles.
« Contact confirmé au point N23-47 ! » crie Ritchie dans son micro pour couvrir le bruit de la fusillade. La section prend position tout autour du petit monticule, par groupes de deux ; les rebelles sont techniquement encerclés. Ne reste plus qu'à attendre les renforts ; les plus proches, la 3e Section du 1er Peloton et deux Sections du 3e, sont à environ un kilomètre et demi au sud et à l'ouest de leur position.
Lian va rejoindre Grayson derrière le rocher où il s'est accroupi. « On voit leur vraie couleur, maintenant, hein, à tes gremlins ! » dit l'autre, les sourcils froncés malgré son effort pour rester sarcastique. Lian, angoissé, écoute les tirs qui alternent entre coups isolés et roulements crépitants. Les trois Pelotons ont apporté leurs armes lourdes ; une fois qu'ils auront fait leur jonction, ils réduiront les rebelles en poussière. En petits morceaux. En bouillie.
Lian se faufile vers Nan. « On pourrait leur proposer de se rendre, non ? » Elle le regarde d'un air surpris : « On nous a dit d'attendre le 3e Peloton, avec le commandant. » Puis ses yeux se plissent : « Ah, dis donc, ça ferait plutôt bien dans le décor s'ils se rendaient à nous, eh ? » Profitant d'une accalmie, elle crie de toutes ses forces : « Vous êtes encerclés et les renforts arrivent, avec de l'armement lourd. Rendez-vous maintenant, ça évitera les dégâts ! »
Les tirs reprennent de plus belle et des éclats de pierre ricochent autour de Nan et de Lian. « C'est pas l'enthousiasme... », marmonne la jeune fille en répliquant d'un bref staccato rageur.
Lian se laisse glisser assis, les bras autour des genoux, incapable de penser. Au bout d'un moment, les tirs se font plus sporadiques, pourtant. S'arrêtent. Il se redresse avec un espoir renouvelé...
Et voit, debout sur le monticule, une silhouette en uniforme vert. Sans arme, les bras levés, longs cheveux dans le vent – c'est une fille, au sommet de l'amas rocheux.
Pas debout. Au-dessus. Au-dessus des rochers. En pleine ascension.
« Ils font une sortie ! » s'écrie la voix de Grayson, et au même moment trois ou quatre silhouettes surgissent, dévalent du monticule en tiraillant au hasard – mais personne ne réplique, ils regardent tous, bouche bée, la fille qui flotte au-dessus des rochers.
Grayson pousse un rugissement inarticulé, Nan et Ritchie semblent se réveiller près de Lian, se retournent, à demi couchés et se mettent à tirer dans le dos des fuyards. Deux des silhouettes vertes s'écroulent, les autres zigzaguent et disparaissent entre les buissons et les arbres qui explosent sous le feu roulant des automatiques.
La fille est retombée, Lian peut voir une partie de son uniforme vert, le torse, les bras à l'abandon sur le rocher, les cheveux qui lui voilent le visage. Elle ne bouge pas.
Les tirs crépitent encore un moment de l'autre côté du monticule, puis s'éteignent. La radio de Ritchie émet des bruits furieux, qui deviennent des questions. Ritchie, puis Nan répondent. Lian voit bouger la silhouette verte recroquevillée à trois mètres de lui, il s'élance, s'agenouille. C'est un petit gars blond d'une vingtaine de saisons, l'épaule et le bras déchiquetés, on voit les tendons et les esquilles d'os ; le sang lui a éclaboussé la figure, sa poitrine se soulève convulsivement. Lian murmure il ne sait quoi, des paroles apaisantes, tout en essayant d'appliquer un pansement à pression d'une main et de fouiller de l'autre dans son sac. Le rebelle se débat, les yeux fous. Lian crie « Gray ! Viens m'aider, j'en ai un vivant ! », le bras en travers de la poitrine du garçon pour l'immobiliser.
Le rebelle cesse brusquement de bouger, Lian se redresse : bon, le choc l'a rattrapé, il va se tenir tranquille. Mais l'autre le regarde, les yeux écarquillés. Regarde Grayson qui vient s'agenouiller près de Lian. Puis encore Lian, toujours avec cette expression d'horreur paniquée. Et soudain, le garçon repousse brusquement Lian, qui va bousculer Grayson ; ils tombent, emmêlés. Le rebelle a roulé vers son fusil mitrailleur, se met le canon dans la bouche et appuie sur la détente. La rafale lui fait exploser le crâne.
32
Vers le milieu de la deuxième semaine de Novembre, Alicia sort enfin de sa bulle. Malgré tous ses entraînements, et les conditionnements supplémentaires qu'elle s'est imposés elle-même, ce n'est pas une minute trop tôt. La solitude n'était pas totale, bien sûr. On lui a régulièrement communiqué des condensés de nouvelles expédiées depuis Lagrange, l'arrivée de la navette, la réception de Grumberg, les petits messages d'Eduardo, tout comme on a envoyé à Lagrange des capsules de son arrivée et de sa quarantaine, et ses messages du genre " hello, je suis bien arrivée, on s'occupe très bien de moi, tout se déroule comme prévu " – anodins, comme ceux d'Eduardo ; si les militaires réussissent à les décoder, chapeau. Mais ils n'y parviendront pas, pas quand le texte de référence est un livre inventé et appris par cœur par l'émetteur et le récepteur, et cérémonieusement détruit ensuite.
Elle s'ennuie. Elle s'ennuie mortellement. Elle a visionné encore des centaines, peut-être des milliers de bandes, plus récentes que celles de la station, d'accord, mais après combien de documentaires de propagande sera-t-elle jugée suffisamment dédouanée ? La production agricole des Plaines Dorées, les secteurs miniers des McKelloghs, les villes nouvelles et leurs zones industrielles, la pêche sur le lac Mandarine – ça, elle connaissait déjà, de réputation, et il faut reconnaître que le caviar et les filets de poisson-poison, frais ou fumés, quand ils en ont inclus dans le menu, étaient absolument succulents ; elle y développerait assez facilement une accoutumance, il faudra persuader Lagrange d'en stocker.
On lui a également fourni la version virginienne d'envirosims, des tridis et même des films bidimensionnels – sans doute empruntés à des musées – ainsi que des livres. Assez intéressant, plus d'un point de vue sociologique qu'artistique à vrai dire, et encore : une bonne partie des allusions lui échappent encore, malgré les entraînements, normal. Non, son esprit est relativement bien diverti, c'est son corps qui proteste d'être enfermé ! Heureusement qu'une salle de gymnastique bien équipée est incluse dans ses appartements – sans fanfreluches technologiques superflues, mais pas moins efficace pour autant. Alicia y passe des heures. C'est la natation qui lui manque le plus – mais même s'il y avait une piscine, elle n'a pas emporté de maillot de bain : il n'est pas prévu qu'elle aura du temps à perdre une fois au travail. La verdure lui manque aussi, et l'air frais – ou du moins les seuls qu'elle connaisse, leurs équivalents lagrangiens reconstitués. Mais ça ne doit pas être tellement différent sur Virginia, n'est-ce pas ?
Enfin, enfin, le sas de sa chambre s'ouvre, et elle sort. Dans le complexe militaire de Saint-Exupéry, d'abord, dont on lui fait visiter les zones permises. Elle rencontre en direct les gens auxquels elle a parlé et qu'elle a vus dans les écrans de sa console depuis trois semaines. Le général Le Plantier, commandant de la Base, ses subalternes, officiers, ingénieurs et techniciens, leurs épouses, leurs époux, et la dizaine de représentants officiels du gouvernement – elle a eu le temps d'apprendre tous les noms. Sauf pour les assistants interchangeables de la conseillère Jenifer Ashton, qui se renouvellent souvent semble-t-il, elle s'est fait des listes, elle ne commet pas d'erreur. Ce n'est pas comme si elle avait hâte de rencontrer tous ces gens en chair et en os, à vrai dire (plutôt en muscles et en os : ces Virginiens sont terriblement en forme) : ils semblent aussi raides et guindés en direct qu'en virtuel. Elle renonce vite à serrer les mains qu'on lui tend avec une hésitation parfois perceptible. Et elle apprend vite à élargir les limites de son espace personnel pour respecter le leur – simple habitude de planétaires, parce qu'ils disposent de davantage de place, ou essaient-ils plus ou moins inconsciemment de la tenir le plus à distance possible ? Peu importe. Elle les imite. À la guerre comme à la guerre, n'est-ce pas ?
Inutile d'examiner les banques de données de la Base : celle-ci est de construction bien trop récente ; après l'Insurrection, il a bien fallu trouver un endroit sécuritaire autre que Dalloway pour effectuer les envois de satellites et autres activités du même type ; on a transféré tout ce qu'on a pu à Saint-Exupéry, mais rien que le nécessaire, et on sait exactement ce qu'on y a. Ce qu'elle recherche a bien plus de chances de se trouver dans d'obscurs dossiers maintes fois encryptés, compressés et oubliés, comme leurs programmes de compression et d'encryptage, sous des couches séculaires de poussière électronique. Les machines originelles sont encore là aussi, malgré toutes les modifications ultérieures : on va faire de l'archéologie informatique. Alicia est bien préparée : l'équipement de Lagrange date à peu près de l'époque où les liens ont été rompus entre la Terre et Virginia. Si les Virginiens n'ont pas réussi à trouver le trésor, depuis le temps qu'ils doivent le chercher (un autre espoir déçu de Marti : Alicia ne s'est pas vu couper la mission sous le pied), c'est sûrement faute d'avoir continué à développer le savoir-faire nécessaire. Elle a bon espoir. Elle trouvera. Peut-être dans les banques d'une des Tours de communication, dans une des cinq grandes villes ; plus vraisemblablement à Dalloway. Elle aurait bien commencé par là, mais on en a décidé autrement en hauts lieux. Comme les fouilles à Dalloway risquent d'être bien plus longues que partout ailleurs, on ira en dernier, à la fin de Novembre : la Mer reparaît à la fin de Décembre, les opérations militaires s'interrompent alors, on aura la paix ensuite pour la majeure partie des recherches. Alicia a acquiescé sans rien dire ; elle espère bien qu'elle n'aura pas besoin de plus d'un de leurs Mois pour trouver les données !
Pas question d'accéder aux ordinateurs de Dalloway à partir de Saint-Ex, bien entendu, ni d'ailleurs de n'importe quelle Tour de communication : ils fonctionnent en circuit fermé, trop dangereux sinon, avec la proximité des rebelles. Quant aux Tourcoms, eh bien, oui, elles sont reliées entre elles, mais il y a eu tellement de remises à jour des machines comme des logiciels, il se peut que des dossiers traînent un peu partout, isolés, inaccessibles à distance. Il faut voir sur place. Elle comprend très bien : les Virginiens veulent promener l'ambassadrice lagrangienne sur le continent, pour prouver hors de tout doute à leurs lointains visiteurs qu'ils ont davantage à échanger contre leurs bonnes grâces technologiques que la seule, et hypothétique, propulsion Greshe. C'est prévu, Alicia se conformera au scénario. Ce n'est pas comme si elle avait son mot à dire : la procédure a été déterminée avant même sa naissance.
On la promène donc d'abord dans la Base, laboratoires, ateliers, centres de contrôle, hangars d'entretien. On en est visiblement très fier, elle en est visiblement très impressionnée, tout le monde est bien content.
Ce soir-là, dîner de gala chez Le Plantier, servi par des ordonnances en uniforme blanc. Intime : seulement une vingtaine de personnes, la moitié des civils – Jenifer Ashton, la déléguée du gouvernement, et ses assistants en combinaisons ou costumes bruns ou gris, à peine plus colorés qu'elle. On échange de graves et ennuyeuses considérations sur la visite et sur l'industrie spatiale virginienne – très limitée, mais à laquelle l'arrivée des visiteurs a donné un regain de vie (on évite avec tact d'évoquer la remise à neuf concomitante d'une partie des installations de défense planétaire, bien entendu). On essaie de pousser Alicia à boire, et à parler, ce qu'elle évite avec une égale agilité en dissimulant sa surprise : elle aurait cru les Virginiens plus habiles. Mais pour des négociateurs aussi prudents et rusés derrière les écrans, ils sont curieusement ineptes au face à face. Au bout de trois heures de ce régime, elle est presque tentée de boire pour alléger son ennui. La tension et la fatigue de la journée aidant, elle se sent flotter de plus en plus loin des conversations et des rires discrets, à la fois incrédule et vaguement scandalisée. Cette ridicule comédie, et elle en a peut-être pour des Mois ! ? Et pour quoi ? Pour faire la nique à Marti Coralàn ? Absurde – et soudain curieusement pénible d'être ainsi isolée de ces gens alors qu'elle se trouve enfin parmi eux. Les simulations ne l'avaient pas préparée à cela. On a traversé des milliards de kilomètres, des générations sont nées et ont disparu, les deux branches de l'humanité terrestre sont bel et bien réunies après des siècles de séparation, et que fait-on, maintenant ? On ment, tout le monde ment, tout le monde sait bien que tout le monde ment...
Elle essaie de se calmer, un peu inquiète. Petite crise d'adaptation quand même, maintenant qu'elle est en direct, elle savait que c'était possible. Et non, question mensonges, elle espère bien qu'ils ignorent au moins avoir affaire à Alicia Coralàn, la Fille du Capitaine – pas question de laisser savoir aux planétaires quelle otage potentielle ils détiendront là, a décrété Lagrange, et Marti Coralàn a dû s'incliner. Quand elle l'a appris, elle a tout de suite choisi elle-même son nom d'emprunt : Alice, pour le puits, et Cortès, pour sa mère génétique. Ils n'ont pas émis de commentaires, du moins pas en sa présence.
Or, en regardant les autres dîneurs, Alice Cortès a quand même envie de faire quelque chose d'inattendu, crier, sauter sur la table, dire la vérité, une vérité, n'importe laquelle. Mais Alicia se rend compte qu'il n'y en a aucune qui ne soit potentiellement compromettante. Elle se contente de dire, un peu penaude : « Je me demande comment les livres d'histoire raconteront tout ceci.
— Les vôtres ou les nôtres ? » remarque un des assistants de Jenifer Ashton, un nouveau, elle n'a pas retenu son nom. L'intonation est surprenante dans son sarcasme discret, presque complice. Alicia observe mieux l'homme qui vient de parler : un de ceux à qui la combinaison gris passe-partout va mieux qu'aux autres, de type vaguement latino comme elle, peau très brune, cheveux gris argent – une coquetterie, sûrement, même pour des Virginiens –, la quarantaine mince, longue face anguleuse, trop d'expressivité pour être un agent de la sécurité comme Armitadj, Brandelles et Pyrce, les trois autres membres de base de l'équipe d'Ashton avec l'assistant Delplace. Il ne la regarde pas, mange avec application son dessert, yeux noirs à demi voilés par les lourdes paupières un peu bridées. Seule l'intonation était ironique, rien n'en transparaît sur le visage incliné. Et les autres, autour, ne tiennent pas compte de son commentaire tout en s'empressant de fournir les leurs – vraiment un subalterne, alors, Alicia n'est même pas sûre qu'on le lui ait présenté, ni qu'il ait pris la parole depuis le début du dîner ; on ne la lui a sûrement pas adressée non plus. Pas si subalterne qu'il n'ait été invité, quand même. Un peu vieux pour un apprenti assistant. Mais la bureaucratie virginienne est un buisson si éternellement foisonnant...
On fait assaut de considérations philosophiques auxquelles elle doit se prêter, en dissimulant son ennui – le commentaire de l'anonyme, au bout de la table, était plus proche de ce qu'elle voulait dire, et ne pouvait pas dire. Mais il ne se mêle pas à la conversation.
Après le repas, on prend café et liqueur sous la véranda couverte. La vue est à couper le souffle sur les hauts plateaux, à l'ouest, et le ciel étoilé. Heureusement, on ne voit pas Lagrange, dissimulée par les montagnes à cette heure, ou Alicia n'y couperait pas d'autres tentatives plus ou moins maladroites de lui tirer des vers du nez. Elle s'est assise le plus à l'écart possible, – on l'a laissée faire – et hume son cognac indigène plus qu'elle ne le sirote mais cela lui donne une contenance et la dispense de parler comme de se voir adresser la parole. L'anonyme vient tout de même se camper devant la vitre près d'elle, les mains dans le dos – peut-être par politesse, pour qu'elle ne reste pas trop ostensiblement seule – et sans rien dire. Un double égard qui mériterait récompense. Si elle lui parle, ça fera peut-être avancer sa carrière, à ce brave homme.
« Vous ne croyez pas en l'objectivité historique, alors ? » C'est un sujet qu'on a réussi à ne pas aborder pendant l'échange de propos précédent, mais Alicia ne sait pas trop comment engager autrement la conversation.
L'autre se retourne à demi. « Pourquoi, vous y croyez, vous ? » dit-il enfin, comme après mûre réflexion, comme si la question méritait d'être considérée avec sérieux.
Elle ne peut s'empêcher de rire : « Y a-t-il encore quelqu'un pour y croire ? »
Il sourit, mais sans ironie. « Certains mensonges sont peut-être plus vrais que la vérité. »
Elle le dévisage, un peu surprise mais surtout amusée : « Mmmm. Et c'est quoi votre mensonge, à vous autres ? »
Cette fois, le sourire est sarcastique : « Ah, mais si nous vous le disons, quel mérite ? »
Elle rit encore : « J'ai autre chose à faire de toute façon ! Tant que vous restez cohérents...
— Mais c'est là tout le problème, n'est-ce pas ? Ne pas se perdre entre les vrais mensonges et les mensonges vrais...
— Tout le monde vit dans les mensonges », remarque-t-elle, un peu méfiante tout à coup, par retour de réflexe même si c'est sans doute absurde. « On ne les appelle simplement pas ainsi. Mythes, légendes, religions, idéologies. Des histoires. Chaque société, chaque culture a ses propres histoires. Elles sont toutes utiles.
— Mais pas toutes vraies. »
Elle hausse les épaules : « Vraies par rapport à quoi ?
— Pas toutes bonnes, en tout cas. L'histoire que Virginia s'est racontée sur la Terre a abouti à des massacres. Ça a mal tourné. »
Alicia est vraiment surprise, cette fois. C'est un des sujets absolument tabous. « Il aurait suffi de s'en raconter une autre, une bonne, dit-elle avec une prudence renouvelée, pour que ça tourne autrement.
— Peut-être. On ne le saura jamais, n'est-ce pas ?
L'intonation est curieusement attristée. Alicia fronce les sourcils : « C'est à nous de changer l'histoire, non ? »
Le visage de son interlocuteur s'illumine d'un sourire d'abord incertain puis plus franc : « Oui. Oui, c'est à nous. »
Ils se regardent un moment, lui avec un plaisir incompréhensible, et Alicia déroutée – de quoi sont-ils en train de parler, exactement ? Ce type n'est pas aussi coincé que les autres, en tout cas. Rafraîchissant. Et elle ne sait même pas comment il s'appelle. Elle se lève, fait jouer les muscles de ses épaules avec un soupir. Le stratagème qu'elle a mis au point au-cas-où ne lui a pas encore servi jusque-là, mais c'est le moment ou jamais : « Excusez-moi, mais comment épelez-vous votre nom, exactement ? » L'orthographe virginienne des noms propres a souvent des caprices inattendus.
« A-n-d-e-r-s-o-n », dit-il aussitôt, et précise ensuite que son prénom s'écrit Graëme, et non Graham. Le diagnostic d'Alicia semble se confirmer : politicien, qui a les bons réflexes. Tant qu'à se faire remarquer, qu'il n'y ait pas erreur sur la personne, n'est-ce pas ?
Ou bien il veut seulement être aimable, se dit-elle quand même aussi, un peu honteuse, avec l'étrangère parachutée de si loin.
« Je suis arrivé aujourd'hui, offre-t-il encore, mais je me rends à Nouvelle-Venise avec la délégation. »
Il semble impliquer qu'il n'en fera pas partie une fois rendu à destination. Un peu déçue, Alicia demande : « Vous connaissez bien la ville ?
— Mais vous serez très occupée... »
Elle se trompe exprès : « Pas vingt-quatre... trente-cinq heures sur trente-cinq, j'espère ! C'est une cité particulièrement pittoresque, d'après ce que j'en sais. »
Holà, Alicia ! Pas de tourisme, pas de fraternisation, tu te rappelles ? Mais ce sont leurs règles à eux, là-haut, pas les siennes. Ou du moins c'est elle qui décide de leur application. Et puis, ce type semble plus ouvert que les autres, on pourra peut-être en apprendre quelques petites choses intéressantes, qui sait ?
Anderson l'observe avec une discrète surprise. Quoi, s'imagine-t-il qu'elle est en train de flirter ? La société virginienne est pourtant assez égalitaire pour que ce genre d'a priori n'y soit pas un réflexe – ce n'est pas comme dans Lagrange ; il y a presque autant de femmes que d'hommes parmi les soldats de la base, leurs officiers, les ingénieurs et les techniciens civils.
Elle dévisage son interlocuteur, la tête un peu rejetée en arrière – vraiment grands, ces Virginiens ! Pas laid, dans le genre intellectuel – peu courant chez les politiciens... Es-tu en train de flirter, Alicia ? Quelle idée absurde ! Et justement, non dépourvue d'un charme pervers... Ils auraient une attaque, là-haut. Mais Marti serait trop content.
« Si vous pouvez vous libérer, je serais très heureux de vous accompagner », dit Anderson, avec juste un peu plus d'enthousiasme que ne le nécessiterait la politesse.
33
Un mort dans la Section 3, Bob Mayer. Trois blessés, outre Vitale. Lian soigne, stabilise, l'hélijet de l'équipe médévac emporte. Ensuite, ils ramassent les cadavres vêtus de vert avec les membres de la 3e Section du 2e Peloton, qui a fait sa jonction. Les autres sont retournés à leur fort quand ils ont appris le résultat de la bataille, apparemment. Lian ne pense pas, il ne ressent rien, son corps a pris les commandes. Huit rebelles tués, les autres ont réussi à s'échapper. La fille, sur le monticule, ne porte aucune blessure apparente. « Encore une suicidée », marmonne le soldat inconnu en la faisant rouler dans le sac noir. « Mais ils sont dingues, ces Rèbs ! » Lian entend vaguement la remarque à travers le brouillard, regarde mieux le visage de la fille. Lèvres cyanosées, joues marbrées. Il prend la main molle : le bout des doigts est bleui aussi. La machine à diagnostic intérieure se met en route, automatiquement : attaque cardiaque massive. Suicidée ? Le soldat referme le sac, en empoigne la tête. « Eh, on y va ? » Lian prend l'extrémité du sac et ils redescendent l'aligner auprès des autres.
Ils retournent au bunker en hélijet. Lian regarde à ses pieds parce qu'il se trouve avoir la tête baissée, habité uniquement par le sifflement des moteurs, le battement du rotor. Les survivants de la Section 3 ne disent rien. L'excitation du combat est retombée, ils sont plutôt épuisés, et même un peu hébétés. Finalement Delgado remarque, d'une voix comme hésitante : « Ils ont bien failli nous avoir avec leur truc, hein ? La fille qui se rend...
— Salope de Rèb, dit Nokuda. Bob serait peut-être encore là si je l'avais descendue plus tôt. »
Lian tressaille, répète : « Descendue ? » L'intonation incrédule de sa voix les tourne vers lui. « Personne ne l'a descendue. »
Nokuda le dévisage en fronçant les sourcils : « Eh, je sais quand même bien ce que j'ai fait !
— Elle n'était pas blessée. Elle a eu une crise cardiaque.
— Ça, tu peux le dire : une balle en plein cœur ! » aboie Nokuda avec un rire bref.
Les autres hochent sombrement la tête.
« Personne ne lui a tiré dessus », murmure Lian. Il a l'impression de discuter dans un de ces rêves où l'on s'obstine, mais sans conviction. « Elle flottait en l'air et les autres ont déboulé et vous leur avez tiré dessus, mais pas sur elle.
— Qu'est-ce que tu racontes, elle flottait en l'air ? dit Ritchie en rigolant. Elle s'est levée, les bras en l'air, comme si elle se rendait.
— Elle flottait. Je l'ai vue. Vous l'avez tous vue. C'est pour ça que personne n'a tiré, au début. »
Les autres le dévisagent avec un mélange de perplexité et d'amusement inquiet. « Liam, dit Grayson avec patience, elle faisait semblant de se rendre, Nokuda a compris que c'était une ruse et il l'a tirée. »
Lian se retourne vers lui ; de vagues volutes de scandale agitent son brouillard, à présent. « Tu as crié " ils font une sortie ", et la fille est tombée, et ensuite tout le monde s'est mis à tirer sur ceux qui se sauvaient. Avant, elle flottait en l'air. Ses pieds ne touchaient pas le sol. »
Grayson lui passe un bras autour des épaules d'un air navré : « Tu as mal vu, Liam, dit-il avec douceur. Elle s'est levée tout d'un coup, alors, l'effet de surprise...
— Ouais, dit Ritchie, ça m'a fait le même effet, un moment, Liam. Comme si elle flottait. Elle était drôlement grande, ça doit être pour ça, aussi. »
Les autres acquiescent. Eux aussi, ça leur a fait ça. Mais c'était la taille de la fille, oui, ses bras levés, la surprise. Il y a une note hésitante dans leur voix. Ils ne sont pas tout à fait sûrs, ils sont prêts à admettre ce qu'il a cru voir, ou du moins à essayer de se l'expliquer avec lui... Lian ne sait plus trop, du coup. Peut-être qu'il s'est trompé ? La surprise, le soulagement de voir peut-être les rebelles se rendre, cette grande fille aux cheveux qui flottaient dans le vent...
Mais non ! Non ! Il en est presque désolé, ce n'est pas qu'il désire croire ce qu'il a vu, mais il sait qu'il l'a vu. Il secoue la tête sans rien dire, les autres haussent les épaules ou froncent les sourcils, déconcertés. Grayson le dévisage, visiblement inquiet pour lui. « On est tous fatigués, Liam, dit-il enfin. On va se reposer et on en reparlera. »
On ne se repose pas tout de suite : le capitaine Trent est là pour féliciter la section ; il y aura des médailles à la clé. Il les encourage à lui raconter le combat. Quand on en est au moment crucial, où la rebelle fait mine de se rendre, Delgado remarque en riant : « Et là, Liam a été tellement impressionné qu'il a cru la voir flotter en l'air ! » Trent se tourne vers Lian d'un air discrètement soucieux : « Ça va mieux, maintenant, Shaunessy ? »
Lian regarde les visages souriants de ses compagnons, hésite, finit par secouer la tête en marmonnant : « Je l'ai vue. » Les autres se mettent à rire, mal à l'aise, ou lèvent les bras au ciel. Trent ne sourit pas. Il regarde Lian comme s'il ne le voyait pas vraiment. Puis il semble sortir de sa méditation avec un petit sursaut, murmure : « Le premier vrai combat, Shaunessy, ça fait toujours un choc. »
Tandis que les autres continuent à parler avec Trent, Lian s'éloigne – ils ne le remarquent pas, même Grayson – et il va prendre sa douche. Ensuite, il reste étendu sur son lit, en slip ; l'air conditionné a des ratés dans leur bunker, ce sera une nuit sans. Il voudrait que Grayson soit là. Pour que Gray lui répète encore une fois que non, il n'a pas vu ce qu'il a vu, et si Gray lui-même ne le croit pas, c'est bien qu'il se trompe, n'est-ce pas ? Il n'a pas vu ce qu'il a vu. Il n'a pas vu une tzinan parmi les rebelles. Il n'a pas vu une tzinan se sacrifier pour que ses compagnons rebelles puissent s'échapper. Il n'a pas vu une tzinan rebelle morte d'une absence de balle en plein cœur.
Et le mot est une clé, tzinan, et il tourne dans la serrure et la boîte s'ouvre. Il y a toujours des mutants sur Virginia. Les autres, de l'Autre Côté, Costa, Lartigues, tous les membres du Club, Thomas... Ils se trompaient. Ils se trompaient : se mentaient. Croyaient ce qu'ils avaient besoin de croire, et il voulait le croire aussi, il n'en a jamais parlé avec Argelos ou les hékel du Temple, il n'a jamais demandé, il n'a jamais posé de questions à Odatan, il préférait cette histoire-là, il voulait la croire, et elle a été vraie tant qu'il l'a crue – le soleil était un œuf, les montagnes le pondaient le matin et les autres montagnes l'avalaient le soir, mais comment pourrait-il le croire maintenant, comment a-t-il pu le croire, comment a-t-il pu être assez fou pour... ?
Qui est fou ? Tout le monde croit les mutants disparus, ici. Tous les membres de la Section ont réussi à s'expliquer ce qu'ils ont vu afin de ne pas l'avoir vu. Qui est-il pour dire qu'il a raison et qu'ils se trompent ?
Il se redresse soudain, le cœur battant. Ou pour dire qu'ils se trompent totalement ? Ils le croient peut-être parce que... parce que c'est vrai, presque vrai. Parce que les mutations sont peut-être simplement rares, très rares, extrêmement rares, maintenant. Et seulement chez les rebelles, si cela se trouve ! Combien sont-ils, maintenant, en Licornia ? Dix, neuf millions, moins ? Un réservoir génétique limité, en tout cas, et depuis des dizaines d'Années. Pas étonnant si les mutations ont mis plus de temps à disparaître de chez eux, si elles reparaissent parfois, un accident, une rareté, une curiosité. Et bien sûr, ils s'en serviraient. Comme cet après-midi, pour la valeur de choc, et ça a failli marcher, en plus ! Et la pauvre fille est bien morte d'une crise cardiaque, parce qu'elle a trop demandé à son trop faible pouvoir. En face, les Fédéraux, quand ça arrive, on ne peut pas y croire, évidemment, on n'y pense même plus, ou on préfère ne pas y penser : les mutations ont disparu depuis si longtemps, et elles ont fait tant de mal !
Lian reste là, à demi appuyé sur un coude, hors d'haleine comme s'il venait d'échapper... à quoi ? Il ne sait pas. Ou bien il sait : à la folie. Grayson arrive sur ces entrefaites. Lian lui dit presque... mais se tait, attristé – encore une des confidences qu'il ne peut faire à Gray. Est-ce que ce sont des mensonges, quand on ne vous dit pas ce que vous ne pouvez pas entendre ? L'autre se déshabille avec des gestes brusques, se rend dans les douches sans un mot. Il n'a pas l'air content. Parce que Lian s'est entêté ? Parce qu'il a dû le défendre devant les autres et qu'il n'aime pas attirer l'attention sur eux deux ? Ou parce qu'il est dérangé, peut-être, dans l'histoire qu'il se raconte, qu'il a bien le droit de se raconter – lui aussi il a vu la fille se rendre, seulement se rendre.
Quand Grayson revient, Lian se lève et va à sa rencontre. Comme il n'y a personne d'autre qu'eux dans le dortoir, il le prend par le cou, appuie son front contre le sien. Grayson reste les bras ballants. « Liam, dit-il d'une voix tendue, il faut arrêter avec cette histoire de rebelle qui vole, là. Si jamais Trent décide que tu es en train de craquer... »
Lian ébauche un sourire : « Ne t'en fais pas, je suis calmé. C'était juste une illusion d'optique. »
Grayson reste un instant figé, puis le prend par les épaules pour le tenir à bout de bras en le dévisageant avec intensité : « Tu es sûr ? Vraiment sûr ? »
Lian incline la tête. Ce n'est pas un mensonge, n'est-ce pas, quand on vous dit ce que vous avez besoin d'entendre ? Grayson s'illumine ; il serre Lian contre lui. Après quelques instants, son sexe se tend sous la serviette-éponge ; ses mains sur Lian deviennent plus précises, le dos, le creux des reins – Gray aime la peau de Lian, au grain si lisse, il le lui a souvent dit, ça l'excite incroyablement. Lian murmure : « Les autres vont arriver.
— Oh, ils en ont bien encore pour une heure avec Trent », dit Grayson, la voix rauque. Ils s'étendent l'un près de l'autre. Grayson tremble. Il jouit presque tout de suite, aux premières caresses. Et il se met à sangloter, sans larmes, un bras sur les yeux. Lian se redresse sur un coude, consterné de son inconscience, de son égoïsme : pour Gray aussi, c'était le premier combat, les premiers morts. Il le prend dans ses bras avec une tendresse désolée, caresse les cheveux encore mouillés. L'autre s'accroche à lui comme s'il se noyait. « Qu'est-ce que je peux faire, Liam ? murmure-t-il, qu'est-ce que je peux faire ? »
Lian lui embrasse le front, apaisant : « Rien, Gray. C'est comme ça. »
34
Depuis le petit aéroport où atterrit leur avion, au nord des McKelloghs, Alicia et la délégation se rendent en convoi automobile discret jusqu'à Nouvelle-Venise. Très discret, le convoi, mais c'est depuis longtemps entendu de part et d'autre, il n'y aura ni réception officielle, ni parades ni bains de foule – pas après avoir fait croire si longtemps à la population que Lagrange ne s'arrêterait pas. Et puis, on est en guerre, ce serait un trop grand risque de publiciser la présence de l'ambassadrice lagrangienne.
La guerre est un peu loin, à vrai dire, dans l'autre moitié du continent, mais d'après ce qu'Alicia sait des rebelles, et même en tenant compte de l'inévitable propagande fédérale, elle préfère ne pas attirer leur attention. Elle est en train de succomber à la contamination locale : leur nom officiel est " Sécessionnistes ", mais tout le monde les appelle " rebelles ", et elle commence à en faire autant. Ma foi, à la guerre comme à la guerre, encore, même si ce n'est vraiment pas la sienne. La route traverse une section de territoire qui a été l'une des places fortes rebelles pendant la guerre d'attrition, après l'Insurrection ; on lançait de là des raids sur la région du lac et Nouvelle-Venise. Même après tout ce temps, il reste des traces de batailles sans merci, des morceaux entiers de forêt dévastée qui n'ont pas repoussé, ou mal, des chaos de rochers noircis ou à demi fondus, des cratères vitrifiés. Avec une vague sensation de nausée, Alicia contemple le paysage bouleversé. Certes, les forces fédérales n'y sont pas pour rien, mais comment peut-on en arriver là, comment les rebelles ont-ils pu déclencher une guerre civile aussi impitoyable, aussi meurtrière – aussi longue ? Un tel entêtement... Et des deux côtés, en plus. Alors que la solution d'Ewald était évidente depuis le début, les laisser tous aller dans le sud-est et faire sécession. Sauf que ce n'était pas vraiment leur choix, aux rebelles, ils ne l'ont accepté que sous la contrainte au moment où la victoire se décidait contre eux, et ils ne s'y sont jamais résignés. Et maintenant, ils sont tous là en train de se tirailler dessus à la frontière, toutes les deux saisons. Des campagnes saisonnières. Quelle barbare absurdité.
Mais ce n'est pas comme si tout le monde pouvait désamarrer une station Lagrange et partir au petit bonheur dans le cosmos quand la situation devient trop intenable. En laissant les autres crever comme bon leur semble sur le plancher, avec les vaches.
Le reflet d'Alicia fronce les sourcils dans la vitre fumée du compartiment arrière où elle se trouve avec Jenifer Ashton et trois uniformes – Anderson est dans la voiture de queue. D'où lui vient cette idée idiote ? Elle n'a jamais avalé la thèse du trauma souterrain qui, selon certains psychologues hérétiques de Lagrange, grignote tout un chacun dans la station – on a quitté le système solaire depuis près de quatre siècles, pour l'amour du ciel, personne n'a une mémoire aussi longue ! Elle a vu les documents, comme tout le monde : les catastrophes climatiques sans répit, de plus en plus généralisées, la panique, les émeutes, les épidémies, les guerres-éclairs des dictateurs désespérés d'acquérir des territoires moins menacés – et la certitude générale, horrible, paralysante : ce n'était pas la première fois mais cette fois-ci il n'y aurait pas de salut, parce que le point d'équilibre avait été dépassé, parce que tous les freins avaient sauté, parce que le bolide roulait à une vitesse accélérée vers le fond de l'abîme, parce que le reste de la Confédération solaire était dans un état trop précaire lui-même pour ne pas se laver les mains de la Terre – et parce que la voie des étoiles était fermée à jamais, sans la propulsion Greshe. Et alors, que pouvaient-ils faire d'autre, les premiers Lagrangiens, que de s'emparer de la seule barque de sauvetage et s'enfuir ? Ils n'avaient pas de boule de cristal pour leur dire que cent cinquante ans plus tard un nouvel état d'équilibre aurait été atteint malgré tout, plusieurs tours de spirale plus bas, et que la Terre serait encore vivante, et même une partie de l'humanité. Ils étaient de bonne foi quand ils croyaient devoir être les seuls survivants.
Mais de moins bonne foi, hein, Alicia, quand ils ont alors capté le message, " Revenez, on a besoin de vous ! " et qu'ils l'ont enterré au plus profond de leurs banques de données. Elle en a eu, du mal, pour le récupérer. Curieux, quand même : enterré, mais pas effacé...
Peu importe : elle n'a pas éprouvé le besoin d'en parler. Elle avait satisfait sa curiosité, c'était suffisant. Ni alors ni jamais elle n'a éprouvé la moindre crise de conscience, pas sur des choix faits par des gens morts depuis presque trois cents ans ! C'est comme ce paysage dévasté, ces reliques d'affrontements fratricides. Absurde, ou navrant, mais en dernier ressort, cela ne la concerne pas.
Avec les ingénieurs et les techniciens du lieu, Alicia retourne pendant sept jours les entrailles des banques de données à la Tourcom de Nouvelle-Venise. Sans grande conviction, mais c'est le scénario. Elle dort sur place, elle mange sur place. Pendant les pauses, elle contemple le panorama, le long haut plateau bleuté du massif des Brouillards, au sud-ouest, ou l'horizon un peu trop infini du lac, mais surtout la ville en contrebas, avec son dessin de mandala. Vraiment mandarine, le lac sous le soleil, et la ville presque irréelle à force d'être une ville des Anciens. Depuis l'altitude de sept cents mètres où s'ouvre le bouquet des installations de la Tour, aucune présence virginienne n'est décelable. Elle est d'ailleurs minimale : il n'y avait pas assez de place pour construire une ville nouvelle, et de toute façon cette région a été un fief des Vieux-Colons dès le début de la colonisation, des gens qui avaient décidé de se faire indigènes pour de bon et préféraient conserver les précieuses terrasses de culture plutôt que d'y agrandir la ville. Seulement trois millions d'habitants, Nouvelle-Venise, pas plus que du temps des Anciens. Et pratiquement pas un poil d'électricité, même quand la Mer est absente comme en ce moment. Quelquefois, lorsqu'Alicia se rappelle qu'il y a des générations de gens, sur cette planète, qui n'ont jamais utilisé un intercom, un ordi ou un bain sonique, jamais joué dans un sim, jamais même vu un film, bi ou tridi, elle éprouve malgré tous ses entraînements une sensation d'étrangeté totale, d'autant plus déboussolante que ces gens, s'ils se trouvaient dans Lagrange, n'attireraient pas un regard. Plus foncés, c'est tout, mais il y a des maniaques du bronzage parmi les Lagrangiens, et quelques atavismes.
Le septième jour, dans l'après-midi, elle décide qu'elle s'est assez conformée au scénario. Elle a trouvé des tas de trucs bizarres, et même quelques petites choses utiles – pour les Virginiens ; elle les leur abandonne avec magnanimité – mais de propulsion Greshe, point. Elle déclare forfait. Passons à la Tour suivante, je vous prie. On exprime des regrets, on la remercie de ses trouvailles, et Jenifer Ashton vient la cueillir au pied de la Tour, avec son entourage. Parmi lequel, discret et à l'écart, se tient Anderson.
« Nous allons vous emmener à l'hôtel » dit Ashton pleine de sollicitude après avoir manifesté à son tour ses regrets, « vous pourrez vous reposer.
— Je préférerais me détendre en faisant autre chose, déclare Alicia. Visiter la ville, par exemple. Serait-ce possible, discrètement ? Avec une ou deux personnes, pas plus ? »
Jenifer Ashton fait oh, et fait ah, ce n'était pas prévu, elle ne connaît pas du tout la ville, il va falloir trouver un guide sûr et mettre un itinéraire sur pied...
« Écoutez, puisque ce n'était pas prévu, personne ne pourrait savoir que je me promène, n'est-ce pas ? Discrétion, je veux bien, mais je ne vais quand même pas rester enfermée tout le temps comme un poisson dans un bocal ! » proteste Alicia, carrément prête à feindre un caprice s'il faut en venir là.
« Nouvelle-Venise a été tirée au sort comme première ville, les risques sont en effet extrêmement minimes... », intervient Delplace, dûment inquiet de l'incident diplomatique qu'il a pour rôle de sentir poindre et d'éviter. « Andy et Gregor peuvent suivre en protection rapprochée. Et Anderson est qualifié. Il connaît très bien la ville, je crois ? »
L'intéressé hoche la tête, impassible : « Je me ferais un plaisir d'accompagner Serra Cortès. »
Alicia révise son évaluation d'Anderson, un peu déçue. " Qualifié ", hein ? Pour la protéger au cas où ? Pas un fonctionnaire subalterne, alors, un agent aussi secret que les autres. Mais pourquoi pas ? Ce sera sûrement une promenade intéressante quand même, dans une perspective différente, voilà tout.
Nouvelle-Venise mérite son nom. La Venise terrienne désormais engloutie n'a sans doute jamais été aussi propre, cependant. Le système d'égouts des Anciens était impeccable, et l'est toujours tant qu'on l'entretient, comme c'est visiblement – et d'après l'absence d'odeurs intempestives – le cas. L'écologie autorégulatrice des canaux est restée d'une remarquable efficacité aussi, sans doute parce que les habitants de Nouvelle-Venise ont choisi dès le début de ne pas la soumettre à des pressions trop fortes : les algues et les herbes des fonds, tout comme les caliches, omniprésents et voraces petits poissons rouges et noirs, ne sont pas conçus pour combattre une pollution normale. Ou enfin, terrienne, rectifie intérieurement Alicia, avec ironie. Mais le biogaz utilisé par les descendants des Vieux-Colons pour alimenter leurs bateaux et leurs barques ne laisse pas de résidus. De toute façon, la ville entière semble d'une propreté méticuleuse, canaux grands et petits, places, rues, ruelles. Dans le Nord, on n'a apparemment jamais tout à fait oublié la leçon des premières Catastrophes sur Terre – au contraire des rebelles, et des Terriens.
Et on n'a pratiquement rien modifié dans cette ville. Tout juste si on décèle ici ou là, habilement déguisées, les tubulures qui amènent dans certains édifices l'autre variété de gaz, celle qui est distribuée sur tout le continent à partir des énormes gisements du Dolgomor. Nouvelle-Venise et sa région sont toujours restées à part, encore plus archaïques que tout le reste de Virginia, comme plus à l'ouest la Nouvelle-Dalécarlie. Ce ne sera pas pareil à Cristobal, Bird ou Tihuanco – encore moins à Morgorod, dont la haute ville semble sortie tout droit d'un cauchemar futuriste du début du XXIIe siècle !
Anderson ne dit rien de tout cela ; c'est Alicia qui le pense avec obstination, au début, pour résister à l'attrait pernicieux de la cité. Et au choc, parce que c'est son premier vrai contact, sans intermédiaire, avec la réalité de Virginia. Mais elle résiste, par réflexe. Ce n'est pas comme si elle n'en avait pas fait cent fois l'expérience dans les sims, cette ville. Plutôt mille fois, c'étaient ses sims préférés jusqu'à ce que Marti s'en rende compte et avertisse les psychologues. D'accord, les sims ne lui rendent pas tout à fait justice : il manquait les parfums, les fleurs le long des quais, sur les branches des arbres-rois retombant des terrasses, et jusqu'à l'odeur de l'eau, bizarrement acidulée là où elle recouvre les évents volcaniques sous-marins qui tiédissent le lac et son climat. Mais Alicia connaît en réalité Nouvelle-Venise aussi bien qu'Anderson, elle aurait pu s'y promener toute seule. Ou enfin, presque, ses souvenirs des sims remontent quand même à une dizaine d'années.
Et puis, ce ne serait sûrement pas aussi fascinant. Agent secret, peut-être, mais Anderson en sait bien plus que les sims sur Nouvelle-Venise. Et quand il ne sait pas, il invente. À un moment donné, leur barque passe près d'une longue fresque – parfaitement conservée, comme toutes les autres, sous son épais vernis d'origine – dépeignant sur plusieurs édifices de suite toute une histoire qu'il lui conte par le menu. C'est d'abord une grande scène de chasse en forêt assez incongrue dans ce royaume ancien de pêcheurs. Le personnage vêtu de riches vêtements, là, monté sur son cabal, c'est le prince Khaliad, un enfant gâté qui ne voulait pas devenir roi. Au-dessus, dans les nuages, toutes ces créatures bizarres sont des divinités en train d'observer les absurdités des humains. Et là, debout sur ce minuscule nuage, c'est Iptit, le dernier créé, le petit dieu des petites choses, en train de lancer le caillou doré qui va changer la destinée de Khaliad, et celle d'Arani, cette jeune personne vivant dans la forêt, ici, avec son chachien. Grande, rousse et bien en chair, elle collecte des herbes tandis que Tamal le chachien est suspendu par la queue dans les branches au-dessus d'elle, en train de se balancer. Et ici, Khaliad se réveille, il a perdu la mémoire après être tombé de sa monture effrayée par le choc du petit caillou, et il est recueilli par une joyeuse bande de saltimbanques. Voici sa rencontre avec Arani, sur la place du marché où, en chassant Tamal qui lui a volé son caillou-talisman, il renverse l'étal de la jeune guérisseuse...
Alicia, surprise et amusée, finit par arrêter Anderson : « Je ne savais pas que vous étiez un conteur. Avez-vous inventé des histoires sur toutes les fresques de Nouvelle-Venise ? »
Il ne répond pas tout de suite – elle commence à s'habituer à ses pauses méditatives : « Nous en avons tous inventés sur les Anciens, depuis le début de la colonisation. Tellement de vides à remplir... Alors, des poètes, des écrivains, des musiciens...
— Ô Reine, amante, sœur... », fredonne Alicia en souriant, les premiers vers de l'oratorio d'Œniken, le grand compositeur virginien – et le seul, en ce qui la concerne, mais elle espère que la discussion ne portera pas là-dessus.
Anderson poursuit : «... visage dressé sur le sable des heures, pour quel marin perdu regardes-tu la Mer ? » Il sourit aussi ; il a une plaisante voix de baryton, un peu voilée.
« Je croyais qu'on ne parlait pas trop des Anciens », remarque Alicia, un peu malicieuse.
« On n'en parle pas, mais on en rêve. Tous les enfants de Virginia en rêvent. Et ils deviennent grands...
— Les musiciens, les écrivains, les poètes.
— Ceux qui n'oublient pas leurs rêves. »
Il regarde au loin, avec une légère tristesse. « Mais vous n'êtes pas un musicien », reprend Alicia, un peu déconcertée.
— Ni un écrivain ni un poète, dit-il après une petite pause. Je vous raconte seulement ce que j'ai lu. »
La barque continue d'avancer au rythme régulier de son moteur, tandis qu'il la guide entre bateaux et pontons. On est en Hiver, et c'est le Nord, mais la température est douce même si le ciel est couvert. Alicia regarde défiler le reste de la fresque – elle peut se raconter sa propre histoire maintenant : Khaliad tombe amoureux d'Arani, il lui offre la moitié de son caillou-talisman... et il s'en va ? « Que s'est-il passé ? Arani pleure...
— Ah, dit Anderson avec gravité, Khaliad a retrouvé la mémoire, mais du même coup il a oublié ce qu'il a fait depuis qu'il l'a perdue, et sa vie avec Arani. Il retourne chez son père et devient un bon prince, au lieu de l'enfant gâté qu'il était. »
Devant les portes du palais royal – plus petit et d'architecture différente, ce n'est pas l'édifice abritant le gouvernement provincial, bien qu'il occupe comme lui la place centrale bien reconnaissable malgré la perspective tordue – s'étire maintenant une longue file de femmes jeunes et vieilles, portant toutes un caillou doré, et elles passent devant Khaliad, qui a l'air bien affligé.
« Je sais ! dit Alicia. Il a l'autre moitié, et il va retrouver Arani de cette façon.
— Et ensemble, ils mettent fin à la guerre entre leurs deux peuples – le peuple d'Arani avait été conquis par celui de Khaliad. De surcroît, Khaliad a découvert grâce au caillou les propriétés de la pierre dorée, la pseudo-pyrite, qui restitue au contact des êtres vivants l'énergie emmagasinée. Et à partir de ce temps-là, on commence à l'utiliser systématiquement dans la construction.
— Bien trouvé ! » s'exclame Alicia en riant.
Anderson hoche la tête, tout en continuant à guider la barque. Au bout d'un moment, il murmure, pensif : « Petites causes, grands effets. Nous racontons les histoires, les histoires nous racontent, qui sait ? Il finit toujours par se passer quelque chose.
— Quelque chose d'inattendu, j'espère ! » remarque Alicia en pensant au scénario où elle joue son rôle, surprise d'entendre le vague ressentiment qui perce dans sa voix. Il la regarde. Pendant un instant, elle se plaît à imaginer qu'il donne un grand coup de barre, lance le moteur à plein régime et vire dans un canal perpendiculaire, au grand dam des deux suiveurs dans leur propre barque. Mais bien sûr il se contente de sourire – un peu comme le soir du dîner, avec un plaisir surpris, hésitant, chaleureux : « Oui, espérons. Iptit a toujours son mot à dire. »
On repart de Nouvelle-Venise pour retourner au petit aéroport au pied du Catalin, d'où on traversera en biais la moitié du continent pour aller atterrir dans un autre petit aéroport du massif de la Tête, également situé à plus de deux mille mètres ; de là, cent cinquante kilomètres en voiture jusqu'à Cristobal, où se trouve la Tourcom suivante. Encore heureux que la Mer soit absente, sinon ç'aurait été quoi, chemin de fer à vapeur et carriole ?
Anderson n'est pas du voyage.
La Tourcom de Cristobal est comme toutes les autres – c'étaient des modules préfabriqués, livrés presque tout montés au début de la colonisation, en même temps que les colons en frigo. Elle aussi a été placée au point le plus haut de la ville, au sud-est, dans la ville nouvelle – seulement sept cent cinquante mètres, la Tour, c'est toujours ça d'économisé grâce au relief environnant, celle de Bird fait le kilomètre, elle. En entrant dans la petite pièce spartiate qui va constituer ses quartiers pendant toute la durée de son séjour, au sommet de la Tour, Alicia retient un soupir. On lui avait offert de loger au Palais du Gouvernement provincial, mais la perspective des allées et venues l'a rebutée : elle préfère être sur place et davantage maîtresse de son temps ; elle regrette pourtant presque d'avoir refusé l'offre de ses hôtes – même si de toute évidence eux aussi la préfèrent tout le temps au même endroit, ça simplifie la surveillance.
Elle se met au travail avec une autre fournée d'ingénieurs et de techniciens – on est apparemment jaloux de son territoire, d'une province à l'autre –, et se rend compte très vite qu'elle en a pour un moment. Les modifications des machines sont labyrinthiques, les banques de données monstrueuses, et les vieilles archives s'empilent et s'emboîtent, fossilisées, dans caves et greniers électroniques. Nouvelle-Venise a beau être la capitale provinciale du Nord, le Nord lui-même n'est pas un des grands centres économiques et politiques de Virginia. Cristobal, au contraire... Bird-City, l'ancienne capitale, ce sera encore pire. Et ne parlons même pas de Morgorod !
Après sept jours, elle a un bref espoir : des plans ! Mais c'est seulement une antique description de moteurs ioniques, comme ceux de Lagrange, en moins sophistiqué. Elle se passe les mains sur la figure, tandis que les techniciens se lèvent en silence, visiblement dégoûtés, pour se dégourdir les jambes. C'est le milieu de l'après-midi, juste après la méridienne, et avec la retombée de l'adrénaline elle a envie de tout envoyer promener. Une demi-semaine, et elle a peut-être grignoté la moitié des archives ! Et ce n'est pas comme si les spécialistes locaux lui étaient d'un grand secours non plus. À croire qu'ils se traînent les pieds. Quoi, c'est une question d'honneur, si nous n'avons pas trouvé, elle ne trouvera pas ? Mais quels imbéciles !
« Vous allez vous brûler si vous continuez comme ça. »
Elle ne se retourne pas tout de suite, avec plus d'exaspération que de plaisir. La cavalerie à la rescousse, hein ? Il faut la distraire, la pauvre petite ?
« Vous avez encore des histoires à me raconter ?
— Une escapade », sourit la voix d'Anderson.
Elle fait pivoter son fauteuil, le dévisage : « Pas quand c'est permis. »
Il écarte un peu les mains d'un air d'excuse, sans se départir de son calme souriant : « Mieux que rien. »
Elle finit par dire : « Pourquoi pas ? » en haussant les épaules, et il lui demande aussitôt, la prenant au dépourvu : « Vous avez emporté un maillot de bain ? »
Ils traversent la ville en voiture – celle d'Anderson, un gazillac ordinaire de milieu de gamme ; les anges gardiens sont invisibles derrière, mais après le premier coup d'œil machinal Alicia se fait un devoir de ne plus vérifier. D'ailleurs, pas d'itinéraire compliqué : on va chercher le grand canal de ceinture de la ville ancienne, et la rue qui le double, on le suit tout du long et on sort de la ville pour longer la digue vers le nord-ouest. Alicia commence à se douter de leur destination, mais elle a décidé de ne rien demander. Anderson n'explique rien non plus et s'il n'est pas anxieux de remplir le silence, elle ne l'est pas davantage.
Ils s'arrêtent devant une grande villa ancienne, au creux de la courbe en S dessinée par la Digue. Anderson en ouvre la grande porte d'entrée avec une bonne vieille clé, s'efface devant Alicia. « La maison de famille. On ne s'en sert jamais en cette saison. Il doit bien y avoir des maillots qui vous feront. »
C'est une demeure ancienne classique à un seul étage, bâtie autour du traditionnel bassin, avec terrasse luxuriante sur le toit, tourelle à oiseaux et arbres-à-eau, qui n'a apparemment pas été reconditionnée pour sacrifier à la modernité virginienne : ni électricité ni gaz. Il fait sombre quand ils entrent dans la salle de séjour. Anderson se déchausse et enlève ses chaussettes, reste un moment pieds nus sur les dalles. Une sourde marée lumineuse s'arrondit bientôt autour de lui, repoussant la pénombre fraîche des volets clos. « Maillot de bain », lui rappelle Alicia, agacée. Croit-il qu'elle va pousser des cris de surprise émerveillée ?
« Maillot de bain. Attendez-moi là. » Ses pas s'éloignent. Alicia s'est laissée engloutir par un divan et regarde la lueur de la pierre s'éteindre. Et puis, quand même, elle se penche et pose la main sur une dalle, le temps de la voir s'illuminer à son contact. Se redresse quand elle entend Anderson revenir.
Après avoir trouvé un maillot adéquat, ils remontent en voiture et se dirigent, comme elle s'y attendait, vers le parc de la Tête. Anderson conduit en silence, détendu, un coude à la portière, cheveux gris emmêlés par le vent.
« Ce n'est pas comme si je n'y étais jamais allée », remarque enfin Alicia, toujours agacée.
Il lui adresse un regard en biais, avec un grand sourire sarcastique : « Mais pas avec moi. Et le parc a un peu changé depuis vos sims, quand même. »
Elle se tasse dans son siège, morose, tandis que la campagne défile. Qu'est-ce qu'elle fait là ? Elle devrait être à la Tour en train de travailler. Chaque minute qui s'écoule est une minute de plus qu'elle devra passer sur cette misérable planète, avec ces planétaires stupides !
De fait, le parc n'est pas exactement celui qu'elle a visité. Des sentiers différents. Des œuvres d'art inconnues. Davantage d'arbres et de buissons. Ou bien ils ont l'air plus nombreux parce qu'elle sait que ses pieds foulent vraiment l'herbe, parce que ses narines hument les senteurs mêlées des fleurs et des feuilles. Elle a l'impression d'être pour la première fois dehors ; la visite de Nouvelle-Venise ne compte pas, même avec les jardins et les parcs : c'était une ville. Mais ici, pas un seul bâtiment. Rien que la nature sauvage. Ou enfin presque : ce parc est totalement artificiel, somme toute, comme la falaise-digue à la pointe de laquelle il s'étend ; les Anciens l'ont aménagé de toutes pièces, pas un seul repli de terrain qui n'ait été voulu, comme l'agencement des bosquets et des prairies, les ruisseaux, les petits lacs : tout du faux, comme Lagrange.
Alicia, tu es de mauvaise foi. Du faux qui date de centaines de siècles, est-ce encore du faux ? Et puis les plantes ne sont pas " fausses ". À part les arbres-à-eau, et encore, il n'y en a sans doute pas une seule d'origine dans ce parc, elles ont toutes été remplacées au cours du cycle naturel des saisons, sous un vrai soleil, plantées dans de la vraie terre. Pas d'entrailles mécaniques sous la peau verte des prairies. Non, ce n'est pas le même genre de fausseté que dans Lagrange. Un de ces mensonges vrais dont parlait Anderson à Nouvelle-Venise, alors – devenu vrai avec le temps, comme tant de mensonges.
Presque personne dans les allées, dans les prairies. C'est tout de même à plus de trente kilomètres de la ville, on est en semaine, les adultes travaillent, à cette heure-ci, à Cristobal. Et ceux qui ne travaillent pas restent apparemment chez eux, même s'il fait un temps splendide pour la saison, il faut le reconnaître : pas trop chaud, pas trop frais, et un ciel d'un bleu tendre, presque dégagé de sa sempiternelle couverture de fins nuages en haute altitude.
Serviettes sur l'épaule, et tenant un panier de pique-nique qu'il a rempli à la villa, Anderson s'arrête au bord d'un des petits lacs parsemant le parc. Alicia examine l'eau d'un œil sceptique. De la vraie eau sauvage. Avec plein de bestioles sauvages dedans. Elle sait qu'elle ne risque rien, et le lac de Lagrange ressemble à s'y méprendre à ce lac, en plus grand, mais elle hésite un peu à se lancer. Il n'y a pas trop de plantes aquatiques, heureusement, et elles se trouvent toutes de l'autre côté, bambous, joncs, nénuphars, lentilles d'eau en voile vert et ondulant le long de la rive.
Anderson désigne le lac d'un ample geste du bras : « Pour vous toute seule », dit-il d'un ton encourageant.
Alicia se déshabille en marmonnant : « Quoi, vous avez fait chasser tout le monde du parc ? » Puis, sans attendre la réponse qui ne vient pas, un peu honteuse de son obstination dans la mauvaise humeur, elle s'élance sur le ponton et plonge. Au bout d'une trentaine de mètres, elle se retourne sans pouvoir retenir un rire exultant, continue à crawler sur le dos plus lentement, en savourant la douceur de l'eau, son goût étrange – pas désagréable, juste différent. Anderson s'est assis en tailleur sur la berge. « Vous ne venez pas ? »
Il répète : « Pour vous toute seule.
— Je vous invite, allez ! »
Il se déshabille et plonge à son tour.
Une fois séchés et rhabillés, ils se dirigent vers la Tête en suivant l'un des cheveux de pierre qui grimpent et convergent sur le promontoire. Alicia n'a pas vu la Tête en arrivant, l'avion est allé se poser au sud-ouest du massif et n'a jamais suivi la côte, mais elle s'en est approchée cent fois dans les sims depuis l'océan, pour la voir grandir dans le lointain : une tache bleutée d'abord – dans les sims de crépuscule, les plus spectaculaires, quand l'enduit luminescent de la sculpture géante se met à briller – puis un visage de plus en plus immense, légèrement tourné vers le ciel, une femme, une Ancienne, la reine, l'amante et la sœur d'Œniken, ou plus vraisemblablement une déesse, peut-être en attente, peut-être en prière, peu importe, gigantesque tête de proue de la falaise gigantesque. Treize cents mètres presque à pic sur la plaine en contrebas quand la Mer est absente ; Alicia y a fait de la varappe plus d'une fois, remontant par la voie la plus difficile, celle qui aboutit sur la tiare dont est coiffée la Tête en passant par la courbe de la joue et l'oreille ; si l'enduit lumineux n'est pas lisse, heureusement, la pierre est travaillée avec minutie, il n'y a pas beaucoup de bonnes prises. Elle aurait bien aimé faire l'escalade pour de vrai, mais ce serait inimaginable, bien entendu. Une petite baignade dans un lac, c'est tout ce qu'on lui permet.
Alicia jette un coup d'œil en biais à Anderson. Ils n'ont pas échangé trois phrases depuis leur arrivée au parc. Est-il fâché d'avoir à la cornaquer encore ? Il n'a pas le choix, il est en service commandé, lui aussi. Alicia continue à l'observer, il ne semble pas s'en rendre compte et continue à marcher comme si de rien n'était. Elle essaie de maîtriser l'impatience qui point de nouveau.
« Pourquoi êtes-vous ici avec moi ? »
Si elle croyait le prendre au dépourvu, c'est raté. « Pour vous distraire, j'espère, répond-il aussitôt sans broncher.
— Ils espèrent. »
Il la regarde enfin, en haussant les sourcils : « Si je rends votre séjour plus agréable et vous dispose plus favorablement à notre égard, ça ne fera sûrement pas de mal. Mais je n'ai pas d'ordre de mission en poche, si c'est ce que vous voulez savoir.
— Qu'est-ce que vous faites ici ? Et à Nouvelle-Venise ?
Il esquisse un sourire : « Quelqu'un m'a offert une faveur non sollicitée, dit-il enfin. Vous avez dû remarquer le taux élevé de renouvellement dans la délégation ? On profite de votre présence pour... alimenter les résumés. Distribuer des points, se gagner des faveurs, payer des dettes... Politicailleries habituelles. J'allais à Nouvelle-Venise, on m'a fait faire un crochet par Saint-Ex. Ça a pris une tournure un peu imprévue. On m'a demandé de recommencer ici, j'ai accepté. Je pouvais refuser.
— Ce n'est pas mal vu de fraterniser avec les horribles Lagrangiens ?
Il réplique, du tac au tac : « Ce n'est pas mal vu de fraterniser avec les horribles Virginiens ? »
Alicia sourit plus largement, prête à rester dans le mode sarcastique, mais il s'est immobilisé et la fixe, très sérieux, presque avec reproche. Elle laisse son sourire s'effacer, ne détourne pas les yeux. « Non. Je ne suis pas comme ça.
— Eh bien, moi non plus ! Nous ne sommes pas nos étiquettes ! »
Elle l'observe, surprise de ce soudain éclat. Il prend une grande respiration, ébauche un sourire : « Et je suis curieux.
— On vous en donne le droit ?
— On le prend, non ? »
Elle ne peut s'empêcher d'éclater de rire, incrédule : « Vous ne me ferez jamais croire que vous êtes un politicien ! »
Il se met à rire aussi, vraiment amusé : « J'espère bien que non. Mais vous avez consulté mon dossier. »
Elle s'efforce de rester impassible. Fils d'une famille d'importants de père en fils, mais assistant secrétaire d'un assistant secrétaire, des transferts presque toujours horizontaux, un graph de carrière quasiment plat. Elle avait parfaitement le droit de consulter son dossier officiel, rien de secret là-dedans, routine, elle n'est pas allée fouiller plus profond : elle ne va pas alimenter la paranoïa de ses hôtes en leur révélant ses réelles capacités de pirate informatique. Mais ils savent l'un et l'autre que cela se fabrique, des dossiers. Elle le regarde bien en face, d'un air entendu : « Un dossier. »
Il lui rend son regard, soudain très sérieux : « Mon emploi m'importe peu. Je suis dans la politique par paresse ou par atavisme. Une planque comme une autre. » Et, un ton plus bas : « Nous ne sommes pas nos étiquettes.
— Aucune ambition ? » commente-t-elle avec le scepticisme amusé qui s'impose, mais plus par réflexe que par conviction.
Il fronce les sourcils et réplique : « Et vous ? » d'un ton soudain mordant.
Elle bat des paupières, prise au dépourvu : « Trouver la propulsion Greshe. »
Il la dévisage d'un air sévère, presque comme un juge, et elle se rappelle brusquement qu'il a au moins vingt ans de plus qu'elle : « Que vous la trouviez ou non, vous allez retourner là-haut et repartir. Vous vous y êtes préparée toute votre vie. Vingt-cinq ans. Et après ? »
Alicia reste prise dans le feu des yeux noirs, exigeants, incapable d'échapper à l'écho de la question, la question qu'elle a si souvent essayé de ne pas se poser. Et finalement, elle dit : « Je serai libre. Libérée », mais elle entend bien son intonation, elle sait que c'est plutôt un souhait, un espoir, un vœu pieux. Elle n'a même pas encore décidé si elle se ferait débarquer ou non avec les stoppistes – si tout se passe comme prévu, bien sûr : il reste encore à retrouver cette maudite propulsion Greshe.
Anderson fait : « Ah », avec une tristesse mélancolique qui la laisse déroutée et sans possibilité de réplique. Ils se remettent à marcher. Alicia a enfoncé ses poings dans les poches de sa combinaison, oppressée. Qu'est-ce qu'elle a ? Pour qui il se prend, ce bonhomme ? Elle marmonne obstinément, pour se l'entendre dire : « Je veux juste faire ce que j'ai à faire, et après, être tranquille.
— Moi aussi, murmure Anderson. Nous sommes plus que ce que nous faisons. »
Elle lui jette un regard à la dérobée : un petit muscle saute dans sa joue.
« Qu'est-ce qui est important pour vous, alors ? » demande-t-elle impulsivement, sincère.
Il tourne la tête vers elle. « En ce moment ? » dit-il enfin. Et, un ton plus bas, d'une voix soudain un peu éraillée : « Vous. »
Elle imagine plusieurs répliques, les écarte toutes, reste avec sa protestation : « Mais pourquoi ? » Ça ne peut pas être... Elle n'est même pas vraiment jolie, juste... moyenne. Et volontaire, et intelligente, et elle sait dire " oui, mais " – et sur Lagrange, ça ne paie pas.
Il s'immobilise : « Pourquoi ! ? » s'exclame-t-il, incrédule. Puis il semble se reprendre. « Parce que vous venez... de tellement loin... de tellement longtemps... » Il avale sa salive, les traits soudain contractés. « Et nous sommes là à essayer de nous parler, juste vous et moi, à essayer de nous connaître, à tâtonner... »
Il la dévisage d'un air presque furieux, elle ébauche un mouvement de retrait, déconcertée. Il tend la main aussitôt, touche son bras nu en murmurant : « Non... », et elle reste paralysée sous le regard intense, comme implorant, des yeux noirs rivés aux siens. Anderson prend sa main, et elle se laisse faire. Il l'attire vers lui, lui caresse avec lenteur la joue, les lèvres, comme fasciné. Elle se laisse faire encore, fascinée de sa fascination. Il se penche, il va l'embrasser, il l'embrasse. Ses lèvres sont sèches et chaudes, juste ses lèvres, il lui a même lâché la main, mais elle ne recule pas. Il la regarde, les yeux agrandis, avec une sorte de sourire tremblant. Puis il lui prend la main et sans un mot, ils reprennent leur chemin, vers la Tête, et la grotte du motif central du bandeau de pierre, où ils s'étendent sur la nappe du pique-nique pour faire l'amour.
Pendant qu'il la caresse, interminablement, avec une délicatesse infinie, attentive, délicieuse, elle se dit " pourquoi pas ", elle se dit " je ne le reverrai jamais ", elle se dit " ce sera mon secret ". Et puis elle ne se dit plus rien parce que les caresses deviennent plus précises, et elle s'enhardit à son tour, ne plus penser, juste être là, ne rien rater. Elle est surprise par le plaisir, et même lorsqu'il la pénètre, il n'y a aucune douleur, c'est terriblement excitant, pour un peu elle jouirait encore. Ensuite, il recommence à la caresser, toujours avec la même lenteur hypnotique, elle doit l'arrêter parce que c'est trop, maintenant, de l'autre côté de la passion.
Il lui embrasse le bout des doigts. A-t-il l'air lointain ou simplement rêveur ?
« À quoi pensez-vous, après ? »
Il a l'air surpris, elle se rend compte qu'elle l'a vouvoyé, mais elle ne va pas se reprendre. Il semble méditer un instant : « À comment c'était.
— Et c'était comment ?
Il dit « Éblouissant » d'une voix altérée, en la regardant presque avec angoisse et, déroutée, elle le croirait presque. Ensuite il demande : « Et vous ? »
Elle hésite – elle pense aux sims défendus. Mais techniquement... Et elle aussi, c'est la vérité quand elle dit, sans détourner les yeux : « La première fois. Pas vraiment... le plaisir. Mais avec quelqu'un. »
Il la dévisage avec intensité, lui caresse le front, la joue, les lèvres, la prend dans ses bras et murmure dans ses cheveux quelque chose comme " Merci ".
35
La deuxième semaine de Décembre, le temps pour les blessés de reprendre leur place dans la Section, et pour tout le monde de s'habituer à la remplaçante de Bob Mayer, Trance Huitzinga, une fille d'une efficacité méticuleuse et qui ressemble à une souris, un capitaine des Opérations spéciales vient leur rendre visite. Ils ont fait leurs preuves en patrouille et au combat, on a une mission à leur confier : un campement rebelle a été repéré non loin de la Ligne, au nord-est, dans le secteur de Solmedad. C'est la première fois depuis longtemps que l'ennemi s'installe aussi près de la vallée où passent route et chemin de fer. Pas question de lui laisser croire qu'il pourrait tenter quoi que ce soit de sérieux. La mission de la Section 3 : se rendre en territoire rebelle, détruire le camp. Des autoplaneurs les déposeront à proximité, reviendront les prendre quand ils auront terminé : insertion rapide, précision chirurgicale, on entre, on sort.
Ils étudient les cartes et préparent leur matériel. Ils étaient plutôt taciturnes depuis la confrontation finale avec les gremlins, même l'exubérant Abdul, et même si Lian n'a jamais évoqué de nouveau ce qu'il avait vu lors de l'incident. Mais cette mission, et la confiance qu'on leur manifeste, semble les avoir sortis de leur morosité. Ils partent à l'aube, alors que la brume se traîne encore le ventre sur les montagnes, accrochée dans les hauts arbres fantomatiques. Les deux autoplaneurs passent au ras des cimes, dans un silence presque parfait, franchissent la Ligne sans incident et se posent dans une petite vallée rocailleuse. On débarque le matériel, on vérifie une dernière fois le code d'appel – on observera un silence radio total jusqu'à l'extraction – et les autoplaneurs redécollent et disparaissaient, étranges poissons plats dans la mer de brume. Nan prend l'arrière avec Lian et Grayson, Nokuda prend la pointe.
Après trois heures de marche dans la forêt, ils arrivent à la crête qui, selon leur carte, doit se trouver en face et un peu au-dessus de l'endroit où est établi le campement ennemi, à environ huit cents mètres. Pas de sentinelle avancée. Les rebelles se trouvent dans leur territoire, certes, mais si près de la Ligne... Un peu surpris, ils prennent leurs positions, s'aplatissent entre les buissons, examinent les lieux à la jumelle. Grayson passe les siennes à Lian, déconcerté : « On est au bon endroit ? » Nan est déjà en train de vérifier la carte. « Oui », dit-elle, les sourcils froncés. « Un piège ? » murmure Ritchie.
On ne voit rien, du moins pas d'où ils se tiennent. Les détecteurs de mouvement n'indiquent rien d'humain non plus.
Ils envoient des éclaireurs. Après une demi-heure de tension croissante, ils repèrent juste en face d'eux les pulsions lumineuses codées qui leur servent à communiquer entre eux à distance : ils peuvent y aller.
Il y a eu un campement sur le petit plateau de Solmedad, assez important, et pendant assez longtemps ; on voit encore les trous des poteaux de tentes, des marques de feux, l'herbe foulée dans les endroits passants, et quelques détritus, étonnamment rares. Les rebelles ont nettoyé avec soin avant de quitter la place – ce qui ne concorde guère avec les barbares environnementaux décrits à Lian par les grands-pères, à la commune. Mais ils ont quitté la place. Depuis au moins une semaine. Et n'y ont même pas laissé de pièges, comme le prouve l'examen minutieux du site.
Les membres de la Section 3 sont perplexes, et déçus. Toute cette prudence, toute cette adrénaline gaspillées pour rien ! Ils vérifient encore une fois la carte, non, ils ne se sont pas trompés. Ils explorent la jungle environnante : pas de traces de passages récents.
« C'était un test ? proteste Ritchie scandalisé.
— De quand elles dataient, leurs infos ? » marmonne Huitzinga, encore plus déçue qu'eux tous, car c'était sa première mission.
« Voulaient voir si on a toujours la forme, après l'autre jour », dit Delgado avec une intonation interrogative, mais sans conviction. Lui aussi, il est irrité.
Ils font une pause pour manger, reprennent leur matériel et repartent en sens inverse. Il fait plein jour maintenant. L'humidité est exténuante, les insectes exaspérants. Lian marche en état d'alerte flottante, trop soulagé pour être étonné du résultat de leur mission. Test, erreur, il s'en moque. Pas de sang. Un répit.
Ils arrivent dans une zone relativement dégagée, un grand champ ondoyant d'herbes et de fougères où la brise roule des vagues d'un jaune bleuâtre et d'où émergent çà et là des arbustes aux grasses feuilles vernies doublées d'épines. Herbes et fougères poussent dru, si haut qu'on doit traverser le champ comme un étang, de la végétation jusqu'à mi-poitrine, armes tenues à bras levé.
« Est-ce qu'il y a des serpents, là-dedans ? » murmure Huitzinga. C'est sa phobie, dont l'entraînement n'a pas réussi à la débarrasser ; ils s'en sont rendu compte très vite et ne lui laissent rien passer. Ritchie, qui marche derrière elle, dit d'un ton faussement rassurant : « Seulement des non venimeux », et Huitzinga se crispe, jetant autour d'elle des regards paniqués. Grayson renifle derrière Lian, amusé. « Il n'y en a pas, Trance, dit Lian en haussant un peu les épaules, les serpents aussi aiment pouvoir se déplacer avec un minimum d'obstacles. »
Et brusquement, à leur gauche, une demi-douzaine de silhouettes vertes se dressent dans les fougères, des coups secs résonnent. Huitzinga saute sur le côté comme si on lui avait donné une gigantesque claque. Ritchie lève brusquement les bras, son fusil mitrailleur virevolte et plonge dans les fougères, et le radio part à la renverse vers Lian, qui tend les bras par réflexe, mais quelqu'un lui saute sur le dos et l'aplatit par terre, il ne voit plus rien. Encore quelques coups secs, entrecoupés de deux uniques et brèves rafales de fusil mitrailleur, plus loin à l'avant de la colonne. Lian a le nez dans les tiges écrasées des fougères, Grayson l'étrangle presque pour l'immobiliser. Un gémissement devant eux à leur gauche, Huitzinga, Ritchie ? Lian veut dire leur nom, mais il n'a pas articulé une syllabe que la main dure de Grayson vient le bâillonner. Encore des coups secs, espacés. Des tirs de semi-automatiques. Détachés, méthodiques, on achève les blessés. Les froissements réguliers dans les fougères sont les enjambées des tueurs qui se rapprochent.
Soudain, Grayson se propulse d'un seul élan par-dessus le corps de Lian jusqu'à celui de Ritchie, tâtonne, se retourne, à demi couché, avec le revolver du radio qu'il pointe à deux mains, haletant, « Pardonne-moi, c'est la seule façon », et tire.
L'explosion, c'est tout ce que se rappellera Lian.
36
Alicia ne revoit pas Anderson pendant le reste de son séjour à Cristobal, et elle ne demande pas de ses nouvelles. À Bird-City non plus, où elle passe une autre semaine entière, plus cinq jours – encore sans rien trouver. Le séjour dans l'ancienne capitale est d'un ennui mortel, même si Alicia a accepté de séjourner au Présidium, le Palais du Gouvernement (pour se rendre compte qu'on ne lui faisait effectivement cette proposition que dans l'espoir de la lui voir refuser ; tant pis pour eux). Mais elle n'a pas envie d'aller se promener, comme le lui propose avec insistance l'un des nouveaux assistants d'Ashton ; elle reste dans sa chambre et essaie d'imaginer de nouvelles stratégies pour venir à bout des programmes rétifs. Bird est une ville plutôt laide, bien plus marquée que Cristobal par l'Insurrection et la guerre subséquente, avec des poussées incongrues d'architecture " moderne " en plein milieu des édifices anciens. Tout en s'acharnant sur les codes et les langages de programmation antiques, Alicia n'essaie pas de nier qu'elle pense à Anderson – normal, les hormones en folie, l'exotisme, l'interdit... Elle ne le reverra plus, il a poussé la fraternisation un peu trop loin, il doit être puni, le pauvre homme – ou bien une fois lui a suffi, Alicia, l'exotisme a des limites pour tout le monde, mais elle refuse de se laisser déranger même par cette idée, elle ne le reverra plus, tout est sous contrôle.
À Tihuanco ensuite, sur le Lac Doré – le seul des trois à presque mériter ce nom de lac, seulement huit cent cinquante kilomètres de longueur – la ville pourrait être intéressante ; les combats ont eu lieu dans la ville nouvelle, sur la rive du lac, et non dans la cité ancienne elle-même qui élève intacte sur sa montagne en pain de sucre son architecture unique de terrasses circulaires reliées par des rampes et des escaliers, comme la tour de Babel des anciennes gravures. Peut-être que les drogues adaptant son métabolisme à tous ces voyages commencent à s'emmêler les pieds (elle sait bien que non : Cristobal, Bird et Tihuanco ont à peine une heure de décalage), mais Alicia est morose tout du long. Le lac n'est pas doré, la température est pourrie, il pleut presque tout le temps, les responsables locaux sont particulièrement guindés, et quand, par pur esprit de contradiction, elle demande à aller visiter la ville, on lui dit " Non " presque sans fioritures, comme si Delplace en avait déjà assez de son rôle de paratonnerre diplomatique.
Elle partirait pour Morgorod d'une humeur massacrante si, sur sa console au Palais gouvernemental où on l'a encore logée sans lui demander son avis cette fois, elle ne trouvait la veille du départ une image rebondissant paresseusement d'un coin à l'autre de l'écran. Une des illustrations originales d'Alice au pays des merveilles, mais à la place de " Drink Me ", sur la bouteille, on a écrit " Read Me ". Elle pose un doigt sur l'image, sans résultat, pense à toucher l'étiquette de la bouteille, idiote, la voit s'ouvrir : « À bientôt. » Pas de signature, et elle ne va pas faire de recherches qui pourraient être compromettantes pour tout le monde. Mais elle s'embarque d'un cœur plus léger dans l'avion militaire pour Morgorod. Juste de la curiosité, bien sûr. Elle a décidé qu'elle en avait le droit.
À Morgorod, dans la haute ville qui est d'un rétro encore plus délirant maintenant que dans les sims, la Tourcom n'est qu'une tour parmi d'autres, car cette partie de la ville, sur le dernier plateau avant la montagne, est située à plus de deux mille mètres : on se demande pourquoi Morgorod n'a pas été dès le début la capitale de Virginia ; mais c'est seulement depuis l'Insurrection, comme par défi aux rebelles tout proches derrière leur redoute des montagnes Rouges... La Tour se trouve à un demi-kilomètre du Capitole, le siège du gouvernement fédéral ; comme les gouvernements provinciaux dans leurs capitales respectives, on a installé celui-ci dans le vaste complexe que les Anciens édifiaient au centre de leurs cités – palais ou temple ou n'importe quoi, mais une petite ville dans la ville : il y travaille au moins cinq mille personnes dont la moitié vivent là en permanence, et il reste quand même assez de place pour assurer à Alicia un séjour tranquille à l'abri des indiscrets. Chaque matin, une limousine anonyme l'emmène à la Tour, la ramène le soir – elle a transigé avec elle-même : elle fait la méridienne sur place. Mais elle a décidé de prendre désormais son temps. Lagrange est encore en train de négocier son approche au point L4 du système ; la troisième semaine de Novembre n'est qu'au tiers écoulée ; ce n'est pas comme si un compte à rebours tictaquait quelque part.
Elle n'essaie pas de trouver Anderson. Il vient à la Tour, comme elle l'espérait, le surlendemain de son arrivée. Elle le voit apparaître à la porte de la salle où elle travaille avec les techniciens laconiques de Morgorod, et s'entretenir avec un des ingénieurs venu à sa rencontre ; elle continue son travail sans broncher. C'est la fin de la journée, excellent minutage – après le énième terabyte de données météo conservées Dieu sait pourquoi, elle n'a pas l'intention de faire des heures supplémentaires aujourd'hui. Il vient à sa station, regarde avec elle en silence l'écran dérouler menu après menu de dossiers aux titres cryptiques, flasher quand quelque chose s'accroche dans les filtres, puis reprendre, fausse alerte. Finalement il murmure : « Fascinant. »
Elle entend bien qu'il plaisante, souffle en retour : « Ça vaut tous les mantras.
— Vous n'avez pas besoin de vous détendre, alors. »
Cette fois, elle lève la tête vers lui et l'enveloppe d'un seul regard, surprise du choc sourd d'excitation qui lui vibre jusqu'au bout des doigts ; cheveux d'argent ébouriffés, face anguleuse aux rides un peu lasses, yeux à demi voilés, grande bouche faussement sérieuse, elle ne se le rappelait pas si... intéressant.
« J'ai besoin de me réveiller, plutôt.
— Ça peut s'arranger. »
Il la ramène au Capitole, où elle se change, puis l'entraîne dans une véritable escapade cette fois : dans la basse ville, déjà en proie à son fameux brouillard hivernal et où cloches et clochettes de repérage font retentir leurs incessants et mélodieux carillons. Ils se tiennent par la taille, « Surtout, ne me lâchez pas », a dit Anderson, tandis que, après avoir débarqué du ferry, ils passent de rues en canaux en ruelles. Ou bien, sans clochettes aux chevilles, ils ont semé leurs anges gardiens depuis longtemps, ou bien ceux-ci suivent leur trajet bien tranquilles depuis leur voiture parce qu'Anderson porte un traceur. Peu importe, c'est amusant quand même de jouer aux gendarmes et aux voleurs dans le brouillard en évitant les passants pourvus de clochettes, eux – rares à cette heure tardive, il est vrai, les passants : on se couche tôt à Morgorod pendant la saison de campagne, avec le blackout. Et oui, bien sûr, les sims, mais c'est vrai qu'elle ne l'a jamais fait avec Graëme et que ce n'est pas pareil.
Ils arrivent enfin à leur destination, un restaurant chic sur le vieux port, dont les fenêtres sont soigneusement masquées et l'intérieur éclairé autant par la lueur de la pierre dorée que par celle des lampes à gaz. Pour les dîneurs, il est tard, et la salle est pratiquement déserte. Alicia se laisse tomber dans la chaise que lui tire aimablement le maître d'hôtel dans une alcôve bien abritée – hors d'haleine, ravie. « Comment avez-vous fait...
— J'ai travaillé assez longtemps à Morgorod pour avoir eu le temps d'apprendre à me repérer dans la basse ville. Et j'ai une excellente mémoire auditive.
— Non, je voulais dire, comment avez-vous fait pour venir me retrouver ? Je vous croyais puni pour fraternisation excessive. »
Il paraît un instant désarçonné de son erreur d'interprétation, puis se met à rire : « Oh, non ! Personne n'est au courant. Et j'ai été bien sage, j'ai fait tout ce que j'avais à faire.
— Personne n'est au courant ?
— Et ne le sera, si nous sommes raisonnables. »
À la façon dont il la regarde, dont il se retient de la toucher, elle se demande, un peu ivre sans avoir rien bu, lequel des deux devra l'être le plus. Anderson poursuit d'un ton amusé : « Vous aviez l'air de si mauvaise humeur à Bird qu'on a décidé en haut lieu de me recruter, puisque vous aviez semblé apprécier ma compagnie. Je me suis fait prier. J'ai tenu à terminer les projets en cours avant d'être mis en vacances.
— À Bird ? proteste Alicia, à la fois amusée et scandalisée. Je n'étais pas de mauvaise humeur à Bird ! »
Il sourit : « On vous a crue de mauvaise humeur et on m'a contacté sans que j'aie rien à faire, c'est ce qui compte.
— Et on ne se doute vraiment de rien », remarque Alicia, masquant son embarras de scepticisme ironique.
Il demande, très grave : « Y a-t-il quoi que ce soit à soupçonner ? »
Elle se défend encore : « Qu'en pensez-vous ? »
Il la dévisage un long moment sans répondre, les yeux soudain voilés. « Peu m'importe ce qu'ils peuvent penser », murmure-t-il d'une voix basse, presque féroce.
En un éclair, elle décide définitivement que ça lui est égal aussi, ce qu'ils peuvent penser là-haut. Tant que tout cela n'entrave pas sa mission, après tout, ils n'ont rien à dire. Et ils n'ont pas à le savoir, non plus – elle n'en a évidemment pas parlé dans ses petits messages bihebdomadaires, et Virginia n'envoie certainement pas à Lagrange des rapports détaillés sur elle dans son dos. Le seul message qui intéresse Lagrange, c'est celui où elle annoncera avoir découvert ce qu'elle recherche. Elle hoche la tête : « Moi aussi. »
Il se tend vers elle, se force de justesse à ne pas lui toucher la main, prend à la place son verre d'eau qu'il fait tourner sans y boire. « Ce n'est pas... pensable pour eux », dit-il enfin, les sourcils un peu froncés. Il lève les yeux et la regarde bien en face. Elle acquiesce avec gravité : « Contre nature, hein ? Les miens aussi, je pense. Oh, pas consciemment, mais... »
Il a un sourire triste : « Oui. »
Elle le dévisage un moment, un peu perdue. Que sont-ils en train de se dire, encore ? Saisie d'un accès de remords, d'honnêteté, de crainte, elle dit : « Je repartirai, vous savez. » Elle ne le savait pas, elle n'en est toujours pas sûre, mais c'est le scénario, en cet instant sa seule certitude.
Et il dit encore, plus bas, les yeux détournés : « Oui. »
37
Lian reprend conscience à l'hôpital de Dalloway, au cours de la troisième semaine de Décembre. Il ne sait pas la date, bien entendu, seulement que du temps s'est écoulé. Quand il bouge, avec un gémissement inarticulé, il y a un brusque mouvement blanc près de lui, un visage emplit tout son horizon, une femme, peau brun foncé, cheveux crépus, regard étincelant de plaisir, grand sourire : « Eh bien, vous voilà revenu parmi nous ? » Le visage disparaît, on fait le tour du lit, on manipule quelque chose, des instruments sans doute, il y a des bips sonores, puis le visage souriant se penche à nouveau de l'autre côté. « Tout ira bien, maintenant. » Lian referme les yeux, étourdi.
Plus tard, ils lui disent qu'il est tombé dans le coma après son arrivée à l'hôpital, pendant une semaine ; sa blessure à la tête n'était pas si grave, la balle a ricoché en lui fêlant simplement le crâne, mais ils ont craint des complications imprévues. Les premiers examens indiquent pourtant qu'il devrait se remettre sans séquelles majeures. Il n'éprouve pas de tressaillement d'inquiétude à l'idée qu'il s'est trouvé pendant au moins quatorze jours inconscient entre les mains de médecins qui ont pu l'examiner sous toutes les coutures : l'idée ne l'effleure même pas.
On le traite avec beaucoup d'égard. Il a une chambre individuelle, avec une infirmière pour lui tout seul. Des gens en uniforme lui expliquent plus tard : il est un héros. Seul survivant de la bataille de Solmedad avec les rebelles, il en a neutralisé une section entière à lui tout seul. Il ne réagit pas. Ces énoncés lui parviennent à travers un vide atone, détachés les uns des autres, et de lui. Il dort, se nourrit sans le savoir par perfusion, urine sans le savoir grâce au cathéter, on le lave, il dort. Entre-temps, rien. L'infirmière lui parle, il entend – l'idée d'une réponse ne l'effleure pas non plus : il flotte dans un univers d'où la causalité est absente ; les paroles qu'on lui adresse sont des objets aussi arbitraires et indéniables que le pied du lit qu'il peut voir lorsqu'il a les yeux ouverts, la table de nuit qu'il peut voir lorsqu'il tourne la tête à droite, les instruments de surveillance à gauche, le torse et la tête de l'infirmière, ses mains.
Un jour cependant, et c'est un événement aussi arbitraire que tous les autres mais plus rien n'est pareil ensuite, il se réveille pour de bon. L'infirmière lui a dit quelque chose, il n'a pas répondu, elle lui caresse les cheveux avec un petit soupir – on lui a retiré son gros pansement depuis un moment déjà, des mèches ont repoussé. Il regarde la main passer au-dessus de son visage, plusieurs fois, et soudain, indépendamment de sa volonté, sa main à lui se lève pour prendre le poignet de la jeune femme. Il en observe la paume, la retourne – la peau est un peu plus sombre sur le dessus – il écarte les doigts un par un. Sa peau à lui est beaucoup plus claire, et maintenant il regarde sa propre main. Et il s'entend dire, « Seul survivant ? » comme si pendant toutes ces semaines, depuis qu'il a entendu cette phrase, son cerveau avait travaillé à son insu à élaborer cette seule question.
On vient le voir, les docteurs, et un homme mince à la voix douce, d'une cinquantaine de saisons, qu'il connaît mais dont il n'arrive pas à retrouver le nom, jusqu'à ce que les autres l'appellent " Captaine Trent ". C'est Trent qui raconte à nouveau l'histoire, d'abord presque mot pour mot : seul survivant de la bataille de Solmedad, Lian a neutralisé l'ennemi à lui tout seul. Pour cet acte héroïque, on lui a décerné l'Étoile du Congrès, la plus haute distinction militaire virginienne. Il est un héros de guerre.
Lian contemple ses mains alternativement ouvertes et refermées sur le drap devant lui. Il dit « Neutralisé ? » Il reconnaît l'euphémisme habituel, mais il a l'impression qu'il devrait pouvoir le connecter à ses mains, et il n'y arrive pas.
« Oui, dit Trent. Tu ne te rappelles pas ?
— Amnésie résiduelle, marmonne un des docteurs. Ça passera sans doute. »
Trent vient s'asseoir sur le bord du lit, dévisage Lian avec attention. « Qu'est-ce que tu te rappelles ? demande-t-il avec douceur.
— Non », dit Lian, en réponse à la première question de Trent : son cerveau fonctionne encore au ralenti. Mais la causalité s'y est suffisamment rétablie pour qu'il commence à éprouver un étonnement diffus, et quand la deuxième question vient s'ajouter à la première, leur soudaine synergie doit éveiller des réflexes qu'il ignorait avoir, car il dit : « Pas grand-chose. »
Trent lui raconte alors l'histoire reconstituée de la bataille : l'équipe d'évacuation a trouvé les cadavres des rebelles éparpillés dans le champ de fougères ; d'après les traces qui s'entrecroisaient, ils ont évidemment pourchassé Lian après avoir tué les autres membres de la patrouille, et il leur a échappé, jouant avec eux à un jeu mortel de cache-cache pour les neutraliser un par un. On l'a trouvé à côté du cadavre de Ritchie Abdul, son fusil mitrailleur encore en main.
Lian dit avec lenteur – c'est une longue phrase, celle-là : « Je n'avais pas d'arme.
— Tu avais pris celle du radio. »
Lian regarde ses mains. Au bout d'un moment, il demande : « Tous morts ? »
Trent comprend d'abord de travers : « Tous les sept », puis, devant l'expression patiente de Lian, il rectifie : « Oui, toute ta Section.
— Grayson, réussit à dire Lian, mais il veut être sûr.
— On ne sait pas, dit Trent en baissant la tête. On n'a pas retrouvé assez de... Officiellement, il est disparu au combat. Il semble avoir sauté sur une grenade. » Sa voix s'effiloche dans le silence ; ensuite il répète, avec une douceur apitoyée : « Tu ne te rappelles rien ? »
Lian ferme les yeux. « Non », dit-il, et il sait qui ment, et pourquoi. « Rien. »
Trent reste un instant immobile, puis il se lève avec un soupir, tapote l'épaule de Lian : « La guerre est finie pour toi, en tout cas, mon garçon. » Il sort en entraînant les médecins.
Lian passe les trois dernières semaines de Décembre en convalescence à l'hôpital de Dalloway. Ses mains, ses bras sont sans force ; au début il est incapable de tenir quelque chose, peut à peine s'asseoir seul dans son lit, encore moins se hisser hors du fauteuil où on l'installe. On lui fait de la rééducation par principe : aucun trauma physique, aucun trouble endocrinien ne justifient ces symptômes. On espère davantage des séances avec les psychologues, mais on est déçu : le patient refuse de parler de sujets autres qu'anodins, refuse de répondre aux questions, refuse les autres approches indirectes – interprétation de taches, visionnement d'images, associations de mots, psychodrame... À la fin de la première semaine, son handicap physique s'atténue, il redevient au moins capable de s'occuper seul de lui-même ; il est par ailleurs toujours cohérent, calme et poli, et c'est tout de même un héros certifié : on ne s'acharne pas trop. Il peut maintenant s'habiller, se déplacer, tenir une fourchette, un verre plein, une assiette, même si ses mains sont toujours incapables d'exercer une pression supérieure à deux kilogrammes. Vers le milieu de la deuxième semaine, il demande des pinceaux et de l'aquarelle. Sa coordination est redevenue excellente. Les psychologues l'observent avec espoir. Mais il peint uniquement des plantes, fleurs, herbes, buissons, arbres. De mémoire, car il ne va jamais s'installer pour travailler dans le petit parc de l'hôpital, s'il y effectue les obligatoires promenades journalières ; les reproductions sont néanmoins d'une stupéfiante exactitude. Les psychologues estiment que c'est bon signe, même s'ils sont toujours inquiets à propos de l'état de sa mémoire : non seulement il ne se rappelle pas le combat de Solmedad, mais son amnésie semble avoir fait tache d'huile sur les semaines précédentes : il n'a du premier affrontement victorieux de sa Section avec les rebelles dans la Zone II que des souvenirs flous et lacunaires. On a essayé des drogues psychoactives et mnémotropiques, sans grand résultat, mais comme il semble par ailleurs parfaitement fonctionnel – excepté sa myopathie partiellement sous contrôle – et que la saison de campagne s'achève, il est démobilisé deux jours avant le retour de la Mer et mis dans le dernier avion en partance pour Morgorod.
De là, il repart presque aussitôt pour l'ouest et Bird-City, dans un train bondé de soldats démobilisés comme lui. Il a droit à un traitement de faveur, cependant : un compartiment-couchette pour lui tout seul, en tête du train. À quel point est-il spécial pour ceux qui ont décidé de le traiter ainsi ? Il se le demande, distraitement, sans y attacher beaucoup d'importance. On l'a relâché, c'est tout ce qu'il veut savoir. Personne n'a jamais fait aucune remarque sur ses particularités physiques ou autres à l'hôpital de Dalloway, et on l'a toujours traité par ailleurs exactement comme les autres patients : sa coloration protectrice tient toujours le coup. Quand il doit se déplacer dans le train pour se rendre aux toilettes et au restaurant, si on le remarque, c'est avec un respect admiratif, à cause de ses médailles. Pas l'Étoile du Congrès, elle lui sera décernée en grande pompe la semaine suivante, lui a-t-on dit, lors de la parade qui clôturera la campagne de l'An 148 à Bird-City. Il n'a pas l'intention de s'y rendre. Après sa première expédition au wagon-restaurant, il tire les rideaux du compartiment et enfile ses vêtements civils. La guerre est finie. La guerre est finie.
38
Alicia passe à Morgorod presque deux semaines, qui lui semblent moins longues que tout le temps écoulé depuis son arrivée à Saint-Exupéry. Elle travaille d'arrache-pied le jour, fait sagement sa méridienne dans un local ad hoc de la Tour, puis trois heures de conditionnement physique, et retourne travailler parfois assez tard dans la soirée – mais pas les jours où Anderson vient la chercher pour une autre escapade-surprise. C'est quelquefois très officiel, comme l'après-midi où il l'emmène visiter, horreur, la zone férocement industrielle de Morgorod. Et certes, il y a là des usines " ultramodernes ", et oui, on a raison d'être fier de l'ingéniosité typiquement virginienne mise en œuvre dans la partie de la zone qui se trouve en dessous de l'influence de la Mer, mais défend-il vraiment le principe de ce travail répétitif et ennuyeux, les dégâts écologiques malgré tous les efforts, la triomphante affirmation rétro du XXIIe ou du XXIIIe siècle dans la zone avec et, dans la zone sans, cette version baroque du XIXe siècle d'avant l'électricité, lourd aussi sur l'environnement ?
« Mais justement, si vous nous aidez... »
Il s'arrête en plein élan, pose un index sur ses lèvres puis sur celles d'Alicia : « Non, murmure-t-il, non. »
Pour faire preuve de bonne volonté à son tour, elle remarque après un petit silence : « Mais la situation s'est énormément améliorée depuis une centaine de saisons, tout de même...
Il lui adresse un sourire reconnaissant : « La participation collective s'est améliorée. Les comités d'usines collaborent au contrôle de la production et travaillent avec les comités de citoyens, à tous les niveaux. On a dû faire des choix difficiles, c'est ce qui a mené à l'Insurrection, au départ. Nous avons fait le choix de la collectivité. Il faut le refaire sans cesse. » Il soupire, le regard lointain. « Le système a des ratés, bien sûr, il n'y en a pas de parfait, mais dans l'ensemble, compte tenu des circonstances – et nous n'avons pas vraiment décidé des circonstances, de cette guerre qui nous pompe chaque Année pendant deux saisons – les gens sont heureux.
— Pourquoi n'y mettez-vous pas fin, à cette guerre, une fois pour toutes ?
Il se redresse dans le lit, choqué : « Une solution finale, Alice ? »
Atterrée du malentendu – comment peut-il croire cela d'elle ? – elle proteste : « Mais non, je voulais dire, faites la paix ! »
Il se mord les lèvres, dit enfin : « Il faut être deux pour ça. Les Rèbs se sont calmés, sûrement, mais... Et de notre côté... Disons que le degré de confiance implicite n'est pas assez élevé pour que nous décidions de retirer unilatéralement nos billes. Et pourtant, sur le plan économique...
— Nous pourrions servir d'intermédiaire », dit impulsivement Alicia.
Mais, avec un sourire mélancolique, il répète : « Non », un doigt sur ses lèvres, en l'attirant vers lui. Elle répond à ses caresses, essaie de se perdre dans sa peau comme il semble pouvoir le faire si aisément dans la sienne ; elle sait bien qu'il a raison, que lorsqu'ils sont ensemble, il ne doit y avoir personne d'autre avec eux, aucun autre temps non plus que le présent. Mais pendant un instant elle hésite entre plaisir et douleur, tant il y a d'urgence angoissée dans leur étreinte, et après, elle a envie de pleurer.
Vers la fin de la cinquième semaine de Novembre, il la surprend : il lui fait un cadeau. Un collier d'or émaillé des Anciens, assez lourd, une copie évidemment, mais d'âge vénérable. Alicia est moins frappée par le geste ou la valeur certainement considérable de l'objet que par sa nature ouvertement symbolique. Une fois le collier refermé sur son cou, lui dit Graëme avec le grand sourire en biais qu'il a lorsqu'il est sincère et embarrassé de l'être, elle ne pourra pas l'ouvrir. Elle se met à rire pour rester dans le mode léger : « Vraiment ? » Le collier se referme avec un petit cliquetis discret. Elle tâtonne à l'aveuglette, ne sent aucune trace de fermeture. Elle le retourne vers sa poitrine, amusée, mais c'est plutôt une collerette : il ne s'écarte pas assez du cou pour qu'on puisse en distinguer le mécanisme. Examiné dans un miroir non plus : impossible à dire avec les lignes complexes de l'émaillage bleu. « Très bien, je me rends. »
Il passe les mains dans son cou, un déclic, le collier défait glisse vers ses seins nus ; elle le rattrape, mais il le lui prend avant qu'elle ait pu l'examiner. « Le veux-tu ? »
Protester que c'est un cadeau trop coûteux serait simplement une échappatoire, indigne d'eux. Elle sait ce qu'elle doit dire, même si ça fait mal, bien plus mal qu'avant, de se le rappeler, de le lui rappeler – elle est de moins en moins sûre, mais ils n'en ont plus jamais parlé : « Tu me montreras comment on le défait, quand je partirai ? »
Elle voit la vague d'ombre qui passe sur son visage, la façon dont il respire, par petits coups, comme s'il avait mal : « Oui, dit-il très bas, sans la quitter des yeux, mais j'espère que tu voudras le garder. »
Elle balbutie, « Bien sûr ! », elle lui prend le collier des mains, se le passe autour du cou et en presse les extrémités l'une contre l'autre, au hasard, mais il y a un déclic et elle peut sentir reposer contre sa peau la lourdeur du métal. Ce jour-là, ils font éperdument l'amour, et elle a autant de mal à laisser Graëme partir, ensuite, que lui à s'en aller.
Cela devient un rituel entre eux : elle met le collier, elle le garde jusqu'à ce qu'il vienne la chercher à la Tour, et il le défait quand ils se retrouvent. C'est incroyablement érotique ce poids tiède contre sa peau, sous ses vêtements, à l'insu de tous. Parce qu'ils savent tous deux le poids réel de ce collier, ils plaisantent. Elle plaisante : « Je reprends le collier », en se préparant à aller travailler, un matin qu'il est encore avec elle – une de leurs très rares imprudences. Il la regarde avec incrédulité, puis éclate de rire en même temps qu'elle, un rire libérateur, et déchiré.
Ils ne font pas l'amour chaque fois qu'ils se voient. Souvent, simplement, parfois en se promenant, ils parlent, de tout et de rien, en évitant avec une politesse tendre et triste les sujets dangereux. Elle sait qu'elle ne peut vraiment se confier sans se compromettre et s'abstient donc de trop l'interroger sur lui. De loin, avec prudence, elle a compris qu'il n'est pas marié, qu'il a eu des liaisons ; elle sent des blessures, n'insiste pas, se sent soudain très jeune et très ignorante avec son existence tout entière gaspillée à apprendre la vie sur une planète où elle ne restera sans doute pas vivre. Lagrange, ou Virginia, c'est paradoxalement moins périlleux – tant qu'on reste dans le mode neutre de l'information. Une fois, la conversation aboutit au mariage et aux arrangements familiaux des Lagrangiens. À demi endormie par les lentes caresses de Graëme, Alicia a réagi moins vite qu'elle ne l'aurait fait ordinairement, et même là, elle hésite à changer de sujet, consciente à la fois de l'écueil mais aussi de l'obscure motivation qui peut pousser Graëme à nager aux alentours. Il prétend à la simple curiosité objective, bien sûr : Lagrange n'a pas fourni beaucoup de données sur ce point. Elle décide de répondre dans le même registre.
« On donne sperme et ovules à seize ans à la banque génétique : il y a des emplois dangereux, les accidents sont toujours possibles, la population est trop limitée pour risquer de perdre quelque potentiel que ce soit. Par ailleurs, le premier enfant est « aléatoire » : un seul des parents est le parent donneur. Et porteur, le cas échéant, si c'est la mère ; mais le tout est réglé par contrat préalable. L'autre parent est anonyme et choisi au hasard dans la banque, afin d'assurer le brassage génétique. Choisi au hasard, sauf pour les détenteurs de postes héréditaires : on sélectionne pour les traits souhaitables, les prédispositions – et non, pas pour le sexe. C'est un pari calculé, bien entendu, on voit à l'usage. C'est seulement s'il y a " incapacité démontrée ", ultérieurement, qu'on n'hérite pas du poste. »
Et elle n'a pas dû réussir à demeurer dans le strict registre informationnel, car Graëme murmure, après l'avoir dévisagée avec attention : « Tu es une première enfant. »
Elle hausse un peu les épaules : il a eu entre les mains le dossier d'Alice Cortès : « Oui. »
Il sourit en secouant un peu la tête : « Comme c'est bizarre, ces postes héréditaires. Le poste d'ambassadeur ? »
Il plaisante à demi, mais Alicia tressaille intérieurement. « Non, on l'a créé pour moi. On m'a créée pour lui. Qu'est-ce que ça peut te faire ? ! » Elle a senti que sa voix lui échappait encore, mais il est trop tard pour revenir en arrière. Graëme ne sourit plus. Il la dévisage avec une soudaine intensité triste : « Je ne peux pas deviner ce qui te blesse, Alice », murmure-t-il.
Elle essaie de revenir au mode léger : « Oh, juste un cas classique de rivalité entre frère et sœur. Mon père voulait un garçon.
— On est encore comme ça chez vous ? Je croyais... Lagrange n'est pas une société égalitaire de ce point de vue ?
— En droit oui, mais pas forcément dans les têtes. Ça a toujours varié d'une culture à l'autre, sur Terre. Il se trouve qu'une forte proportion des premiers Lagrangiens venaient de cultures ayant un sens profond de... la différence. Alors, la plus vieille différence du monde, ça s'efface mal. »
Il lui caresse le bras avec sollicitude, songeur. Elle ne s'est jamais sentie aussi seule avec lui. Elle a envie de crier.
« Et ton frère ? »
Elle hésite, mais puisqu'Alice Cortès a un frère aussi, elle peut presque dire la vérité – louée soit la petite cervelle machiavélique de Marti, pour une fois, même s'il a exigé cette donnée dans le dossier sous prétexte d'empêcher les lapsus dangereux, une façon détournée de mettre en doute sa compétence : « C'est leur véritable enfant. Oh, un garçon charmant. Sérieux. Doué. De toute façon, les enfants essaient toujours de se conformer aux histoires que leurs parents leur ont inventées, n'est-ce pas ? »
La main de Graëme s'est immobilisée ; Alicia lève les yeux : il la regarde, mais il ne la voit pas. « On croit qu'on le doit, murmure-t-il, jusqu'à ce qu'il soit trop tard. Et puis les parents meurent, les histoires changent. On survit. » Le brouillard se dissipe, le regard noir l'enveloppe d'une caresse désolée : « Il n'est peut-être pas encore trop tard pour toi, Alice. »
Elle s'oublie, elle écarte son chagrin pour poser sa main sur la main de Graëme et dire avec douceur : « Quand a-t-il été trop tard pour toi ? »
Il tressaille un peu, comme surpris, essaie de sourire : « Très tôt », et puis il se laisse aller dans l'oreiller, un bras sur les yeux.
Et comme le silence se prolonge, comme elle ne peut pas le supporter, Alicia choisit soudain de donner ce qu'elle peut donner, elle trahit son propre silence : « La fonction d'ambassadeur n'existait pas », murmure-t-elle, très vite, pour ne plus pouvoir se reprendre. « Mais celle de capitaine, oui, et elle est héréditaire – sauf incapacité démontrée. »
Graëme reste muet un moment. Il a laissé retomber son bras et il la contemple, le visage absolument dénué d'émotion. Puis il penche un peu la tête, elle peut pratiquement voir les rouages s'enclencher... mais il lui arrive d'un côté où elle ne l'attendait pas si tôt, sur un ton un peu amusé, un peu incrédule, mais fondamentalement sérieux : « Et tu veux vraiment devenir capitaine ? »
Le raccourci laisse Alicia presque molle de soulagement, de gratitude. Elle se raidit, regarde Graëme droit dans les yeux : « Je ne sais pas. Mais je ne veux pas échouer. »
Ils se dévisagent, puis Graëme affirme, avec un sourire presque féroce : « Tu n'échoueras pas.
— Merci du vote de confiance », murmure-t-elle ; elle voudrait en être aussi certaine.
Il l'attire à lui, et elle se laisse aller contre la poitrine si étonnamment lisse – il utilise des dépilatoires, une autre coquetterie, l'inverse de ses cheveux argentés. Après une pause, un peu inquiète quand même, elle demande, les lèvres sur la veine qui bat sourdement au cou de Graëme : « Tu vas le leur dire ?
La réponse est lente à venir, et Alicia se redresse, un peu alarmée ; mais il demande en retour, très sérieux : « Tu veux que je leur dise ? »
S'il pense que ça pourrait lui servir à quelque chose, à elle, elle ne voit vraiment pas comment ; mais elle ne veut pas lui demander quelle machination compliquée vient de lui traverser l'esprit : « C'est à toi que je l'ai dit. »
Il hoche la tête sans sourire non plus, lui caresse la joue d'un revers de doigt : « Toi et moi, alors. » Et il la serre contre lui.
39
Lian arrive à Bird le lendemain de son départ de Morgorod, après la méridienne. Des foules de parents et d'amis venues accueillir les soldats se pressent sur les quais, il y a même un orchestre. Personne ne l'attend, personne ne le remarque : il est en civil. Il bourre ses habits militaires dans une consigne de la gare et laisse la clé sur la porte. En sortant de la salle de consigne, son gros sac en bandoulière, il entre presque en collision avec un petit vieux croulant sous plusieurs sacs bruns bourrés à craquer. Il l'aide à les ramasser et à les ranger dans un casier, sans un mot ; le petit vieux lui serre les mains avec effusion, ah mon garçon, il devrait y en avoir davantage des comme vous. Lian n'écoute pas, se détourne dès que le vieux lui a lâché les mains, sort de la gare.
Il ne peut pas dire que rien n'a changé à Bird-City : il ne connaît pas Bird-City. Mais une fois quittées la gare et ses retrouvailles bruyantes et émues, la ville ressemble à ce qu'il en a vu lorsqu'il en est parti, deux saisons, deux siècles plus tôt. Les passants affairés sur les trottoirs, les vendeurs de sandwichs et leurs tricycles, les hordes de bicyclettes, les gazobus bleu et noir dans les avenues et les rues, le sifflet des ferries haletants aux débarcadères. La température est plaisante, tiède ; le vent souffle sans discontinuer du nord-ouest – le vent de la Mer, qui annonce son retour, mais pour les citoyens de Bird c'est simplement le vent du Nouvel An, deux semaines de répit bienvenu dans la chaleur équatoriale de l'Hiver. La guerre n'existe pas pour tous ces gens. Ils ont raison. Elle est finie.
Il a pris une carte de la ville à la gare – il ne se fie pas à sa mémoire pour retrouver son chemin. En examinant le plan, il se dit que Jill avait choisi un périple bien compliqué pour se rendre avec lui à la Base Quintin depuis la commune : il suffisait de prendre le ferry du canal Carghill, puis l'autobus 207, qui suit tout du long l'avenue Otchkis. Peut-être Jill voulait-elle lui faire visiter la ville, pour une fois qu'il s'y déplaçait... Depuis la gare, c'est plus compliqué. Elle se trouve au sud-est, aux confins de la ville nouvelle, au pied de la Tourcom. Gigantesque, la Tour. Lian l'avait aperçue dans le lointain depuis la terrasse de la commune, bien sûr, mais sa ressemblance avec les tourelles à oiseaux des Anciens lui en avait masqué les dimensions véritables – dans son idée, c'était simplement une tourelle beaucoup plus haute que toutes les autres. Extrêmement plus haute, il peut le constater à présent. Un peu plus d'un kilomètre de haut, afin de pouvoir assurer toute l'Année l'électricité à ses services essentiels : banques de données, communications, météo... Heureusement, elle est doublée d'un matériau qui prend la couleur du ciel environnant, et sa mince tige aux flancs incurvés n'abrite que les ascenseurs menant au bouquet du sommet : sa masse n'est pas aussi oppressante qu'elle pourrait l'être, même quand on se trouve comme Lian à son pied, la tête renversée en arrière, à en suivre l'élan gracieux et interminable vers le ciel.
Saisi de vertige, il s'arrache à sa contemplation. Il n'est pas là pour faire du tourisme. Il n'est là pour rien, en réalité, mais il y est. Et le seul endroit où il peut se rendre, c'est la commune. Le silence de Jill, de Suzane ? Il refuse d'être inquiet. Peut-être préféraient-elles ne pas lui écrire parce que c'était trop dur, elles avaient peur pour lui, elles ne voulaient pas le démoraliser... Et puis, des lettres se perdent, les postes commettent des erreurs – une de ses dernières lettres, à Morgorod, lui a été retournée " destinataire inconnu " ! On l'a recueilli alors qu'on ne savait pas même d'où il venait, on l'accueillera, bien sûr, maintenant qu'il revient de la Ligne.
Il vérifie une dernière fois son trajet : trois bus pour aller rejoindre le Grand Canal Trenton Est, qui suit la digue le long d'Orlemur-Est, un premier ferry, et le second dans le petit canal Carghill, qui contourne le port ancien et s'arrête juste devant la commune. Lian aurait préféré des ferries tout du long, mais il n'y a pas de canaux dans la ville nouvelle.
Dans le deuxième bus, entre deux arrêts, un homme barbu en combinaison de travail tachée de graisse se lève brusquement à l'avant et commence à distribuer des feuilles volantes, mal imprimées sur du papier grossier. La plupart des passagers les refusent d'un air ennuyé ou en soupirant un « Ah non, pas encore ! » exaspéré ; d'autres les prennent mais les plient sans les lire et les jettent dans la corbeille à papier au moment de descendre à leur arrêt. Certains les chiffonnent même avec une certaine violence – sans les jeter par terre, bien entendu. Lian s'est assis tout au fond. Il secoue la tête sans regarder l'homme quand il lui tend une feuille sous le nez – il entraperçoit SOLIDARITÉ en grosses lettres grasses au-dessus d'un texte serré. « Travailles-tu, camarade ? » dit l'homme d'une voix revêche.
Lian lève les yeux : l'autre a l'air moins irrité que las et découragé. « Non », dit Lian vaguement apitoyé. Et, mais seulement parce qu'il pense ainsi couper court à la discussion : « Je reviens de la Ligne. »
L'autre lui agite la feuille devant la figure : « Prends ça, alors, et penses-y. Penses-y maintenant plutôt que dans quelques semaines, quand tu redeviendras comme tout le monde. »
Lian hausse les épaules, déjà distrait, et prend la feuille. L'autre se détourne, appuie sur la sonnette d'arrêt, descend d'un pas lourd. Par acquit de conscience, Lian parcourt la feuille en diagonale : c'est un appel à une manifestation pour la fin de l'après-midi sur l'avenue de la Victoire. " Pour une véritable participation ! " " De vrais comités d'usine ! " " Contrôle des quotas ! " " Liberté d'association ! " Le texte est hérissé de points d'exclamation, et Lian vite lassé repose la feuille sur le siège vide près de lui. Rien de tout cela ne le concerne. Il ne redeviendra pas comme tout le monde : il n'a jamais été comme tout le monde. Et il se déplace dans une bulle temporelle restreinte, où le futur se limite pour lui au trajet du bus, du prochain bus, à peine à celui du premier ferry. La commune, c'est encore plus flou.
À bord du dernier ferry, ce sont de très jeunes gens, garçons et filles, qui distribuent les feuilles, d'autres feuilles, sur du papier orange criard, et ils font l'article en même temps, ils vous interpellent, ils vous prennent par le bras, ils veulent discuter. C'est pour la même chose, une grande manifestation conjointe, étudiants-ouvriers, même combat, toute la population est appelée à manifester devant le Présidium, il faut exiger la limitation des quotas, la levée des mesures d'exception ! « Mais qu'est-ce que vous vous énervez, les jeunes, la campagne est finie ! » disent les quelques passagers qui acceptent de discuter, plus pour se distraire pendant le trajet que par conviction, sans doute. « Les mesures seront levées demain. Et les quotas vont changer aussi. Vous n'êtes pas censés être encore en classe, à cette heure-ci ? C'est demain, les vacances.
— Et la prochaine campagne, et toutes les autres ! ? s'exclame une des adolescentes avec passion. Il n'y a aucune raison de maintenir les mesures de guerre, même pendant les saisons de campagne. À quand remonte la dernière action rebelle à l'arrière de la Ligne, hein ? Le dernier sabotage ? Pas depuis que je suis née ! Les Sécessionnistes sont en Licornia, et ils n'en sortiront jamais. Le couvre-feu, le black-out, les contrôles, ça n'a plus aucun sens ici, ou n'importe où ailleurs sur le continent ! On devrait essayer de faire la paix !
— Pfff, écoute-moi ça », dit une femme d'une cinquantaine de saisons, avec une indulgence souriante. « C'est jeune et ça croit que le monde commence avec eux.
— Ça leur passera avant que ça me reprenne », glousse son compagnon, la soixantaine gaillarde. « Tu te rappelles, la manif' de 141 ? »
L'adolescente exaspérée apostrophe Lian : « Tu dois bien comprendre, toi ! ? »
Lian hausse les sourcils et se détourne, vaguement étonné : pourquoi comprendrait-il, que devrait-il comprendre ? C'est cette malheureuse fille qui ne comprend pas, ces adolescents, les passagers sur le ferry, dans les gazobus, partout dans la ville, tout le monde. Ils ne comprennent pas. Ils ne savent pas qu'ils s'agitent en vain, qu'ils vivent dans une histoire qu'on a fabriquée pour eux. Et lui, il est coincé là aussi : il sait, mais ça ne lui sert à rien non plus.
*
À la toute fin de la dernière semaine de Novembre, Alicia subit une énième inoculation : les rebelles ont apparemment mis au point un nouveau virus, on ne prend pas de risques avec des gens qui vont se trouver en première ligne. L'épisode laisse à Alicia une impression bizarre – pas l'inoculation, qui provoque une simple fièvre bénigne, quand elle se réveille après la méridienne c'est déjà passé – mais le rappel soudain qu'elle va se rendre à Dalloway. " En première ligne. " Dans ce qui est, somme toute, un peu l'équivalent d'une forteresse assiégée.
Après trois derniers jours de travail à la Tourcom de Morgorod – on fait les choses consciencieusement ou on ne les fait pas – Alicia se retrouve, avec la délégation réduite à Ashton et Delplace, et Graëme, à la Base Potemkine d'où ils vont s'envoler vers Dalloway. Voir tous ces uniformes affairés est sans doute une préparation adéquate à ce qui s'en vient, se dit-elle en essayant de rester impassible, mais elle ne peut se déprendre d'un certain malaise. Qui s'accentue lorsqu'elle voit par le hublot les escorteurs de l'escadrille de chasse. Elle a toujours eu du mal à prendre cette guerre au sérieux ; c'est un conflit d'intensité très basse, une routine, des escarmouches, à la rigueur un bras de fer militaire où personne n'est capable de l'emporter parce que – d'après les spécialistes de Lagrange, et l'attitude des officiels virginiens l'a confirmé à Alicia – les Fédéraux ne veulent pas réellement l'emporter et les rebelles n'en ont pas les moyens. Mais l'idée qu'il serait néanmoins techniquement possible pour ceux-ci de descendre l'avion en vol d'un missile bien placé... Ça deviendrait presque réel, tout d'un coup.
La ligne Ewald est bien réelle aussi, et Alicia la regarde se déplier dans le hublot avec une fascination horrifiée. La dévastation totale impliquée par cette ligne blanchâtre qui s'étire au milieu des montagnes... Le roc et la terre nue, et, autour des forts et des casemates, une armée de machines meurtrières prêtes à pulvériser tout ce qui bouge. C'est d'une absurdité si totale qu'elle en a presque la nausée. Graëme assis près d'elle devait l'observer, il se penche : « Ça va ? » demande-t-il avec sollicitude.
Elle désigne le hublot du menton : « C'est un peu... accablant. »
Il soupire : « Oui. Mais il y a des situations qui viennent de trop loin, elles sont pratiquement impossibles à transformer, ou alors... avec des solutions radicales, et nous y avons renoncé, heureusement. Ensuite, tout ce qui reste, c'est... le triage. Sauver ce qui peut l'être. Vous connaissez ça, sur Lagrange. » Il esquisse un sourire en rectifiant : « dans Lagrange. » Il se trompe toujours.
Elle répond à son sourire et ferme les yeux. Elle préfère quand même ne pas voir les montagnes martyrisées.
Dalloway a été modifié depuis l'Insurrection ; on a dynamité la route et la voie de chemin de fer qui reliaient l'ancien spatioport à Léonovgrad maintenant en territoire rebelle ; les nouvelles voies d'accès passent plus au nord et arrivent directement, par un large souterrain, au complexe central. La défense du périmètre extérieur est si dense qu'un moustique n'y passerait pas. Sur trois côtés, et même sur le lac pourtant techniquement en territoire fédéral, c'est la zone de protection totale, bourrée de senseurs, de mines et d'armes automatiques, quadrillée par d'incessantes patrouilles ; pas une herbe ne pousse dans ce carré de quatorze kilomètres de côté. La zone suivante est un peu plus vaste et pas mal plus verte, malgré des taches glabres autour des bunkers servant de points d'appui, mais sur dix kilomètres de large tout autour du périmètre rapproché, c'est encore le royaume des senseurs, des mines et des patrouilles. Après, vers l'est, c'est de nouveau la forêt tropicale, le territoire ennemi ; les installations des rebelles sont quant à elles parfaitement invisibles.
On loge la délégation dans l'aile sud du complexe central. Au sous-sol, bien entendu, plus sécuritaire. On attribue d'ailleurs à Alicia un capitaine Bertran – pas de prénom – qui ne doit pas la lâcher d'une semelle, un grand type brun plat et carré d'une quarantaine de saisons, aux cheveux rasés de si près qu'il semble chauve, et auprès de qui les agents habituels des services secrets semblent presque des chiots joueurs. Il salue avec raideur, sans un mot, absolument impassible. « Il va rester tout le temps avec moi ? » dit ensuite Alicia à Graëme, assez horrifiée.
« Vois-le comme... toutes les personnes responsables du service aux chambres dans un hôtel rassemblées en une seule », dit Graëme. Un petit sourire entendu : « Et non, pas tout le temps. » Il redevient grave : « Tu as autre chose à faire que d'apprendre le bon usage de Dalloway. Il te servira de guide partout où tu voudras aller. Et puis, s'il arrive n'importe quoi...
— Dalloway doit être l'endroit le mieux défendu de toute la planète ! proteste Alicia.
— L'endroit le plus attaqué aussi, réplique Graëme. Surtout cette Année, semble-t-il. » Il médite un instant, l'air sombre. « On m'a trouvé du travail pour justifier ma présence, dans la section politique. Je ne pourrai pas être là aussi souvent que je le voudrais. Ça me rassure de savoir Bertran avec toi, Licia. »
Le diminutif l'attendrit – Graëme ne l'utilise que lorsqu'ils sont seuls, bien sûr, et dans les moments de grande tendresse ; il la prend par les épaules : « Ce que tu cherches est sûrement ici », ajoute-t-il avec un sourire un peu triste. « C'est tout ce à quoi tu dois penser maintenant, n'est-ce pas, à le trouver ? »
40
Une fois descendu du dernier ferry, Lian trouve l'esplanade Carghill inchangée, elle, tandis qu'il en suit les chemins de traverse le long des parterres toujours fleuris pour se rendre jusqu'à la commune. Guère de monde dehors : les trajets sont lents sur les canaux, et c'est déjà l'heure de la collation d'après-midi. Personne ne prête attention à Lian parmi les quelques passants. Il ne s'arrête pas au bord de l'esplanade pour contempler les deux étages de la façade rouge et dorée de la commune, de l'autre côté de la rue, il s'engage tout de suite sous la voûte d'entrée.
Une silhouette féminine traverse la cour intérieure d'un pas pressé, en diagonale, sans doute en retard pour la collation. Une femme, la quarantaine, qu'il ne connaît pas, mais c'est normal, ce doit être une nouvelle. Elle infléchit son chemin vers lui quand il la hèle après avoir posé sa valise : « Oui, que puis-je faire pour vous ?
— Pourriez-vous prévenir Jill ? Je suis Liam Shaunessy. »
La femme fronce un peu les sourcils : « Jill ?
— Jill Estérazy », dit Lian, un peu déconcerté : il n'y a qu'une Jill à la commune.
La femme semble encore plus perplexe : « Nous n'avons pas de Jill Estérazy... Vous êtes sûr que vous êtes au bon endroit ? »
Lian reste un instant muet. Dit encore : « Jill Estérazy ! » comme si la simple répétition devait déclencher les souvenirs de la femme. Puis, comme elle secoue la tête : « Commune 1, 1 avenue Trenton Ouest ! »
La femme continue à secouer la tête d'un air à la fois dérouté et un peu agacé : « Non, je suis désolée, il n'y a pas de Jill Estérazy ici.
— Jaques Hueng », dit Lian avec une irritation croissante où point l'angoisse. « Je veux voir Jaques ! » Ont-ils décidé de l'ignorer ? De le proscrire ? Mais pourquoi ?
« Pas de Jaques Hueng non plus. Vous devez vraiment faire erreur.
— Écoutez », dit-il entre ses dents serrées, prêt à tout pour se faire entendre, « je reviens de la Ligne, j'ai habité ici deux saisons, je sais quand même bien de quoi je parle ! Je veux voir Jaques ! »
La femme se mord les lèvres, plus embarrassée qu'agacée à présent : « Je suis vraiment désolée, mais je ne connais pas les gens dont vous parlez. » Et plus bas, presque compatissante : « Je suis née ici, vous savez. »
Il la regarde fixement pendant un moment, incapable d'articuler un mot. Son cœur bat à tout rompre, il est obligé d'ouvrir la bouche pour respirer à petites goulées oppressées. « Je veux... » Sa voix est inaudible, il se reprend : « Je veux voir les autres membres de la commune. »
La femme écarte les mains d'un air impuissant et, sans l'attendre, il traverse la cour au pas de charge pour se rendre dans la salle commune. La femme s'élance derrière lui avec un temps de retard : « Eh ! »
On ne le voit pas entrer, les conversations continuent, les rires, la circulation des assiettes et des pichets. Il s'avance entre les tables, sous le battement feutré des grands ventilateurs, et le silence se fait au fur et à mesure tandis que des visages surpris et interrogateurs se tournent vers lui.
Tous inconnus.
« Quelqu'un connaît une Jill Estérazy ou un Jaques Hueng, ici ? » demande la voix résignée de la femme, à l'entrée. Un murmure de négations déconcertées passe dans la salle. Lian se retourne, hébété ; la femme a croisé les bras et le regarde en secouant un peu la tête avec commisération. Il dit faiblement : «... pas possible », et elle vient le prendre par la main, avec douceur, pour le reconduire dans la cour.
« Je suis désolée, murmure-t-elle, vraiment désolée. Vous devriez aller à l'hôpital Willamette, ils pourraient peut-être vous aider. Je peux vous accompagner. »
Il dit : « Non, non », il s'arrache à la main bienveillante, franchit de nouveau la voûte, ne s'arrête pas quand la femme lui crie « Eh, votre sac ! », court après lui en essayant de lui mettre la bandoulière sur l'épaule, en vain. Il suit le trottoir d'un pas mécanique. Se retrouve devant l'édifice de la commune voisine. Entre. Personne dans la cour. Au bruit des voix, il devine la salle à manger, s'y rend, voit d'autres visages inconnus se lever vers lui mais il ne les connaissait pas vraiment, les gens de l'autre commune. Il balbutie : « Jill ? Jill Estérazy ? Suzane Estérazy ? Jaques Hueng, Paula Éklosz ? »
Un homme se lève, les sourcils froncés : « Qui cherches-tu, mon garçon ? Ce n'est pas ici.
— Mais vous les connaissez ! Ils habitent à côté, dans la commune 1 ! Jill, Suzane... »
Le visage de l'homme a pris une expression plus méfiante : « Non, on ne les connaît pas. Il n'y a jamais eu personne de ce nom dans le coin. Tu dois te tromper, mon gars. »
Un autre homme plus jeune se lève à une table, ils s'en viennent ensemble vers Lian, qui recule d'un pas en balbutiant, encore : « Je reviens de la Ligne. J'habitais là. À côté. Ils habitaient là. »
La mention de la Ligne a fait ralentir les deux hommes, qui échangent un regard. « Oh, mon garçon, tu devrais retourner à Willamette... soupire le plus âgé. Viens, on va t'accompagner. »
Lian recule vers la cour : « Je ne suis pas fou !
— Bien sûr que non, mon gars, juste... fatigué. Ça peut arriver, d'être fatigué, c'est normal. Surtout quand on revient de la Ligne. Tu as bien gagné le droit de te reposer, maintenant. Viens, on va aller avec toi... »
Lian hurle « Non ! » et bondit dans la cour. La femme de la commune 1 s'y trouve avec le sac, l'air désemparé. Les deux autres ont suivi Lian, le plus jeune amorçant un mouvement tournant qui l'amène devant la voûte d'entrée. Le plus vieux se rapproche en murmurant des paroles apaisantes. Lian se rend compte qu'il a pris une posture de défense, malgré ses mains affaiblies. Horrifié, il laisse ses bras retomber. Recule jusqu'au bassin et s'assied sur le rebord, en répétant : « Je ne suis pas fou. »
Les deux hommes se sont immobilisés, d'autres membres de la commune aussi, qui étaient sortis de la salle à manger. Le plus vieux s'approche lentement, mains ouvertes bien en évidence, un peu tendues. Lian ne bouge pas. L'homme s'assied à son tour au bord du bassin, non loin de lui, sans le regarder. « Je me rappelle, quand j'ai fait mon service. Pas toujours drôle, la Ligne. Tu étais dans quelle zone ? »
Lian marmonne « deux », tout en suivant des yeux un adolescent qui a sauté sur une des bicyclettes rangées dans le râtelier près de l'entrée et disparaît sous la voûte.
« Ah, dit l'autre, oui. Les patrouilles, et les gremlins, hein ? Près de Dalloway ? »
Lian se lève et l'homme le considère d'un œil alarmé. « Je vais très bien, dit-il d'une voix qu'il pense calme. Je ne vais pas vous déranger plus longtemps. »
Quelques hommes esquissent un mouvement pour s'interposer, mais il les regarde sans rien dire, et finalement ils s'écartent. La femme lui tend encore son sac au passage. Il l'ignore.
Il se retrouve dehors. La lumière a baissé, une petite averse se prépare. Il marche droit devant lui sur le trottoir. La voiture de police le rattrape après quelques centaines de mètres. On est très poli, surtout après avoir vérifié son identité militaire. Il n'a jamais habité à la commune, cependant ; il a une autre adresse, quelque part dans le quartier 1. On lui redonne son sac, on lui propose de contacter la Tour pour lui, et on le fait malgré ses protestations. Pas de Jill ni de Suzane Estérazy à Bird-City, mais il y a d'autres Estérazy, veut-il les contacter ? Il dit, « non », il dit, « je dois me reposer ». On lui offre de le reconduire en voiture. Il dit, « je préfère marcher un peu, ça me fera du bien ». On le laisse finalement partir.
Il fait demi-tour quand il est certain que la voiture de police s'est éloignée, laisse glisser son sac de son épaule et revient à la commune, sa commune. Il fait le tour dans la ruelle, passe par l'escalier extérieur, escalade la porte de la terrasse. Va droit à l'arbre-roi qui se dresse à son emplacement habituel, comme tous les autres arbres de la terrasse, mais si les plantes et les arbres et les oiseaux le reconnaissent, ils ne peuvent pas le lui dire, ils ne parlent pas, ici. Il cherche, l'anneau des amoureux s'est beaucoup élargi sur le tronc noueux. Mais là, comme il se les rappelle, dans le double cercle, les initiales entrelacées, S, L. Ses jambes se dérobent sous lui, il s'agenouille, le front contre l'écorce, les bras autour de l'arbre. Il n'est pas fou. Il n'est pas fou. C'est pire.
Il se retrouve dans la rue sans savoir comment. L'averse a chassé les rares passants des trottoirs luisants. La rue est vide, le quartier, la ville pourraient aussi bien être vides aussi : il est seul. Il n'est pas fou. Tout le monde l'est sauf lui et ça revient au même. Il a bien vu, tous ces gens ne mentaient pas, ils sont certains, ils ont vécu dans cette commune toute leur vie, et ils savent qu'il n'y a jamais eu là de Jill, de Suzane, de Jaques – ni de Liam Shaunessy. Ils sont tranquilles. Persuadés. Ont été persuadés, comme Delgado, et Ritchie et Nan et tous les autres – comme Ritchie, et Nan et Delgado et tous les autres auraient été persuadés, bien tranquilles, bien en sécurité, s'il n'avait pas été là, lui et ses yeux rétifs, son esprit rétif, inaccessible, impossible à manipuler ! Mais non, il était là, comme un caillou dans leur chaussure, et chaque fois qu'ils le regardaient, une petite pointe de doute les traversait malgré tout, juste un peu, juste trop pour Trent et tous les autres Trent de la Ligne, et c'est pour ça, pour ça qu'on les a massacrés. Éliminés. Neutralisés. Plus de doute, plus d'incertitude, plus de fille rebelle qui vole au-dessus des rochers, plus d'histoire à raconter, peut-être, en plaisantant, une fois revenus chez eux.
Et maintenant, la commune. Quand, la commune ? Après son enrôlement, quand Jill et Suzane ne répondaient plus à ses lettres ? Plus tard, " destinataire inconnu " ? Ils l'ont aidé, ils lui ont procuré des faux papiers, ils l'ont peut-être pris pour un rebelle, c'étaient des sympathisants. Ou bien Jill, seulement Jill – ses paroles elliptiques le jour de la conscription, le trajet tortueux qu'ils ont suivi pour se rendre à la Base : elle avait peur, oh, comme elle devait regretter de l'avoir hébergé, mais il était trop tard. Seulement Jill, alors, mais c'est pareil. Toute la commune, disparue. Non : la commune n'a jamais existé. Seulement dans ses souvenirs à lui, qui n'est pas fou. Et lui, on le laisse courir depuis le début, mais pourquoi ? Pour qu'il pousse d'autres sympathisants à se dénoncer ? Quel rôle joue-t-il sans le savoir dans l'histoire secrète que se racontent les véritables tireurs de ficelles, les mutants, les télépathes, oh, ils sont partout, ils sont toujours là, ils n'ont jamais cessé de l'être, comment a-t-il pu être aussi aveugle, et maintenant toute la commune a disparu, cinquante personnes, effacées de l'existence, Suzane, Jill, Jaques, Paula, les grands-pères et les grands-mères, Katrina, Patrik, oh, les petits, non, oh, non !
« Lian ! »
Quelqu'un le secoue. On le tient par les bras et on le secoue. On le tient par les bras, on le secoue et on l'appelle " Lian ". La stupeur le paralyse, et on ne s'écroule pas, assommé par une manchette, à ses pieds. On est un petit vieillard aux cheveux blancs coupés en brosse drue, aux yeux très pâles dans sa maigre face brune, vaguement familière. On répète « Lian ? » d'un air soulagé quand on réussit à accrocher son regard. « Ne restons pas là. » Une voiture est arrêtée près du trottoir, un vieux cabriolet gazillac d'un modèle antique. Comme on voit qu'il ne bouge pas, s'est planté les deux pieds sur les dalles et ne se laissera pas entraîner, on ajoute d'un ton pressant : « Je m'appelle Dutch Grangier. Je suis un ami de Matieu. Je te cherche depuis tellement longtemps, Lian, deux saisons ! Mais invisible comme tu es, et couvert... Et l'autre se protège tellement bien aussi, là... Seulement quatre semaines que je t'ai repéré. Mais tout va bien maintenant, tu es en sécurité. Viens, Lian, il finira bien par comprendre, il faut partir tout de suite ! »
C'est à peine si Lian a entendu ce qui a suivi le nom de Matieu. Il n'a pas la force de résister à la main sèche qui le pousse dans la voiture, claque la portière. Le petit vieillard démarre en trombe. Lian réussit enfin à balbutier : « Matieu ?
— Des Années que je ne communiquais plus avec lui, trop frustrant, tu comprends, quelques dizaines de secondes aux passages de la Mer et puis plus rien... », dit le vieux sans le regarder, le cou tendu pour mieux voir au-dessus du volant un peu trop haut pour lui. « Lui m'a contacté lors du dernier passage, il y a deux saisons. Je t'ai cherché dans toute la ville pendant des semaines ! L'aiguille et la botte de foin. Pas évident, dans une ville de six millions d'habitants, surtout quand l'aiguille est déguisée en foin ! Je n'avais pas pensé que quelqu'un pouvait te couvrir. M'occupe plus de tout ça depuis des Années, est-ce que je sais, moi, toutes les taupes qu'il y a à Bird-City ? »
Lian murmure : « Les taupes ? »
Le petit vieux lui jette un rapide regard en biais, prend un virage sur les chapeaux de roues. « Excuse-moi, je suis un peu énervé, ça m'a pris plus longtemps que prévu pour passer à travers la couverture. » Un petit rire sec, pas amusé du tout : « Ces téleps de dernière génération commencent à être presque aussi forts que moi, incroyable mais vrai. Mais là, je t'ai, tu peux être tranquille, il ne te retrouvera pas. Ah, zut, sens interdit, d'où il sort, celui-là, l'était pas là la dernière fois ! »
La voiture fait une embardée et repart tout droit, tourne dans la rue suivante qui longe un canal. « Jill », dit le petit vieux d'une voix plus calme ; il a dû prendre conscience de l'hébétude de Lian, il parle un peu plus lentement : « Jill était une agente des rebelles, une clandestine. À Bird depuis au moins huit Années. Très, très profondément infiltrée. Jamais été utilisée. S'était un peu relâchée, sans doute. Et pas moyen de prévenir ou de poser des questions, les communications ne marchent que dans un seul sens avec les taupes, question de sécurité. Il n'y a plus de bloqués chez les Gris – ah, les Fédéraux – seulement chez les rebelles, et même là, rares, maintenant, mais... Son réflexe a été de te protéger, elle n'a pas vraiment réfléchi. Heureusement. Et quand elle a réfléchi, il était trop tard, elle était obligée de continuer à te couvrir. Encore heureux qu'elle n'ait pas été du genre enclin aux solutions expéditives... Surtout quand ton numéro est sorti à la conscription ! Mais je comprends son raisonnement. Les Fédéraux n'avaient rien fait depuis deux saisons, c'était un risque à courir, ta disparition à ce moment-là aurait attiré une attention tout aussi dangereuse... »
Lian essaie de dire « Jill », sa voix se bloque dans sa gorge et il reprend : « Jill n'a jamais existé. »
Le petit vieux freine à un feu rouge, de justesse. « Jill pensait qu'elle était compromise, une fois que tu étais au service ils se rendraient compte, c'était fichu. Elle n'avait pas compté avec les manigances de la petite superstar, là... Même moi, ça m'a pris des Mois pour comprendre qu'il t'avait trouvé, qu'il te couvrait. Mais quelle idée aussi d'écrire ce bouquin, Lian ! En tout cas, Jill a quitté Bird par les voies clandestines habituelles. Quand elle a enfin pu parler de toi à quelqu'un, il était trop tard. Les Fèds avaient déjà escamoté tout le monde. » Le vieux reprend plus bas, d'une voix sombre : « Hyperréaction, comme d'habitude. Ne devaient même pas savoir de quoi il s'agissait vraiment, mais par principe, quand les parents se sont rendu compte que le petit avait pris des initiatives discutables... On nettoie derrière, comme d'habitude. Oh, il a dû être furieux quand il l'a appris. Et ensuite il a dû se dire que ça te ferait un autre choc, ma foi, que tu débloquerais peut-être après ça. Rien compris ! Mais comment le pourrait-il ? Il ne sait même pas qu'il réinvente la roue, le pauvre gamin... Stupéfiant, qu'ils l'aient laissé s'amuser aussi longtemps. Ah, mais bien sûr, ce n'est pas n'importe qui. Va quand même passer un mauvais quart d'heure maintenant qu'il t'a perdu. Doit se demander pourquoi ta bague n'émet plus ! »
Le petit vieux émet un gloussement sarcastique, tout en passant une intersection à l'orange. Il a moins de chance à l'intersection suivante, ralentit en grommelant.
Lian ouvre la portière et zigzague dans le boulevard entre les voitures qui klaxonnent frénétiquement. Il saute sur le trottoir, continue à courir. Il ne veut plus entendre les explications, les vérités, les folies du vieil homme – c'est de sa faute, c'est de sa faute, ils sont tous morts, non, non, ils sont tous morts et lui il est vivant, et c'est de sa faute !
41
Alicia trouve, à la fin de la dernière semaine de Novembre. Elle est en train de faire des heures supplémentaires, Graëme n'étant pas disponible ce soir-là, pris dans une de ses sessions très fermées du bureau politique de Dalloway ; les autres techniciens sont allés se coucher. Et c'est très simple, trop. L'un des utilitaires d'archivage qu'elle a apporté de Lagrange est le bon, les archives compressées s'ouvrent sur les dessins techniques du Mercure, et des données sur les moteurs Greshe, y compris un court document hérissé de formules expliquant succinctement mais avec précision la théorie de leur fonctionnement. Le premier choc passé, joie, soulagement, Alicia éprouve cependant un curieux malaise. Ce n'est pas parce qu'elle n'a personne à qui apprendre la nouvelle, Graëme étant absent – de toute façon le premier à le savoir devait être Eduardo ; et ce n'est pas même parce qu'elle va maintenant changer de scénario : elle n'y songe pas encore vraiment. Mais c'était si... facile. Elle essaie d'ironiser : un retour de flamme de La Mission Héroïque et Dangereuse, Alicia ? Mais elle sait depuis trop longtemps que c'était une question de patience mécanique, si les informations se trouvaient bien sur Virginia : arriver à la bonne archive par éliminations successives et utiliser les bons programmes. Elle en a éliminé beaucoup, les jours et les semaines passées en sont les pénibles témoins ; et cette archive était encore perdue dans un tas d'autres vieux chiffons qu'il a fallu trier un par un, interminablement, de dix façons différentes. Oui, mais. Avec tout le temps dont les Virginiens ont disposé pour chercher avant elle, ils n'ont jamais trouvé ? Ils ont dû mettre des équipes là-dessus. Et ils ont dû commencer par Dalloway, eux. D'accord, ils n'avaient pas ces programmes-là, mais leurs informaticiens sont compétents, surtout pour l'équipement dont ils disposent dans les Tours et à Dalloway. Et ils n'ont pas été fichus d'extrapoler un seul bon algorithme pour les archives qu'ils n'arrivaient pas à ouvrir ?
Du coup, par habitude, parce que Graëme n'est pas là et qu'elle n'a bien sûr pas sommeil, Alicia examine l'envers de sa découverte. Et décèle aux alentours des ombres, des empreintes suspectes, des traces de manipulation récente – elle utilise ses véritables talents de pirate, là, elle veut en avoir le cœur net ; elle a conclu depuis longtemps que ses stations ne sont pas sous surveillance, et puis ce n'est pas comme si elle était en train d'essayer de pirater les banques de Dalloway présentement en utilisation. Or l'envers du décor ne ment pas : on a placé cette information là. Pour qu'elle la trouve, quoi d'autre ? On l'a menée en bateau, comme Lagrange – depuis quand, pourquoi ? Elle est plus déconcertée que furieuse, au début : quel intérêt peuvent y avoir les Virginiens ? Leur position aurait été bien plus forte s'ils avaient eu les données en main. Mais non, la véritable question qui fait mal, tout de suite, c'est : Graëme, Graëme est-il au courant ?
Elle dissimule toute trace du fait qu'elle a copié la précieuse archive après l'avoir ouverte, éteint sa station, récupère le fidèle chien de garde Bertran dans la salle attenante et retourne à ses appartements. Il est tard, la soirée est bien entamée. Graëme ne devrait plus en avoir pour très longtemps. Prendre une douche, se calmer, réfléchir surtout, réfléchir. Eduardo et les autres peuvent attendre encore un peu.
Après avoir congédié Bertran, elle arrache sa veste brune et la jette sur le lit – elle a adopté la tenue militaire, à Dalloway : plus discret. Elle s'immobilise en voyant dans le miroir, à moitié dissimulé par son t-shirt blanc, le collier qu'elle a mis le matin, comme d'habitude. Un élan d'exaspération chagrine la jette vers la porte, lui fait rappeler Bertran qui s'éloigne dans le couloir de son allure élastique et résolue. Il revient sur ses pas, l'air à peine interrogateur.
« Pouvez-vous défaire ce truc, Bertran ? »
Il reste un instant immobile, les yeux fixés sur le collier. Qu'est-ce qu'il a ? Face-de-bois est surpris ? Il a intérêt à ne pas faire de commentaires ! Mais il se penche, déclenche le mécanisme du premier coup, et c'est à Alicia d'être surprise et de le cacher. N'importe qui connaît ce genre de collier, alors, sur Virginia ? Si ça se trouve, c'est un cadeau habituel, un cliché, et elle qui... Mortifiée, elle dit sèchement : « Ce sera tout, merci, Capitaine », referme la porte et jette le collier sur la table de nuit.
Elle prend sa douche, longuement, mais cela ne change pas grand-chose. Elle n'arrive toujours pas à trouver d'explication satisfaisante. Les Fédéraux avaient les données en main, c'est la seule certitude. Depuis quand, impossible de le savoir pour l'instant. On a voulu être certain que Lagrange entrerait dans le système et s'amarrerait au point L4 ? Mais de toute façon, s'ils avaient la carte en main, nous serions passés par leurs conditions. Sont-ils si paranos qu'ils croient que nous les aurions attaqués pour obtenir les données sans échange de technologies, et si stupides qu'ils pensent pouvoir nous attaquer pour s'emparer de nos joujoux ? Ils ont peut-être remis partiellement en état leur système de défense planétaire, mais ce ne sont pas quelques missiles à peine furtifs qui viendront à bout de nos défenses anti-débris spatiaux !
Ou alors l'hypothèse vraiment parano – la seule existence de Lagrange est une menace pour les Fédéraux, ils veulent la station à proximité pour la détruire. Mais c'est toujours aussi stupide, les défenses de Lagrange sont toujours aussi efficaces ! Et ce n'est pas leur émissaire solitaire qui peut faire quoi que ce soit là-haut, la Sécurité doit le surveiller avec deux fois plus de prudence que moi ici ! Et puis, autre problème : s'ils sont si paranos, et s'ils ont placé l'info pour que je la trouve, comment peuvent-ils être absolument certains que je ne vais pas feindre de ne pas l'avoir trouvée et repartir avec ? Ils ne savent même pas que j'en suis techniquement capable. Après quoi Lagrange prétendrait que après tout, non, on n'a rien eu, on ne donne rien, on repart, et qu'est-ce que vous pouvez y faire, hein ? Ou bien...
Mais non, il y a des limites, elle ne va pas se laisser entraîner dans cette spirale infernale et soupçonner les Fédéraux d'avoir des talents informatiques cachés, eux aussi. Elle s'en serait rendu compte, depuis le temps. Ou alors on serait déjà venu l'arrêter ou quelque chose. Les boîtes dans les boîtes, c'est bien joli, mais il ne faut pas surestimer la duplicité de l'ennemi. Et qu'est-ce qu'elle raconte, " l'ennemi ! ". L'adversaire, à la rigueur.
Elle se roule en boule sur le lit, furieuse de se sentir aussi misérable. Interroger Graëme. Le doit-elle ? Si Graëme fait partie de... Ah non, elle ne va pas repartir là-dedans, il faut s'arrêter quelque part – elle veut s'arrêter là, elle ne veut pas croire que Graëme... elle ne peut même pas terminer la phrase, et même si c'était le cas, pourquoi, POURQUOI ?
Elle se rhabille et appelle Bertran. Elle lui dit qu'elle doit voir Graëme de toute urgence, toutes affaires cessantes. Elle songe confusément que c'est un test : si Bertran refuse... si elle est consignée dans ses quartiers, si Graëme refuse de la voir... Mais Bertran acquiesce sans rien dire ; il ne contacte même pas le central pour prévenir Graëme. Est-ce suspect ? Graëme s'attend-il à ce qu'elle vienne le trouver ? Oh, tais-toi, Alicia !
Elle se rend à la suite de Bertran dans une partie du complexe où elle n'a jamais mis les pieds. Quelques gardes, au début, et ensuite, apparemment, un saint des saints : seulement des systèmes de sécurité à cartes et empreintes. Mais Bertran détient tous les bons codes – Bertran est bel et bien son passe-partout dans la base, comme le lui avait promis Graëme ; elle n'avait simplement jamais eu l'occasion de l'utiliser ainsi. Une preuve qu'ils ont confiance en elle, et donc en Lagrange, non ? Elle se sent un peu honteuse de sa crise d'hystérie.
Et elle arrive dans un couloir au bout duquel Bertran lui ouvre la porte d'une petite pièce brillamment éclairée, comme une antenne chirurgicale : la propreté, l'odeur, les instruments de contrôle... Elle s'immobilise sur le seuil, prise au dépourvu : elle croyait arriver dans une salle de conférences. Est-ce son arrivée inopinée, tout le monde semble figé sur place. Un garde en uniforme brun, trois autres hommes en blouses blanches, infirmiers ou médecins. Graëme. Et une espèce de table oblique où est attachée une femme nue.
Tout ce monde immobile, comme tétanisé, et l'entrée d'Alicia fait voler le tableau en éclats. Graëme se tourne vers elle, stupéfaction, horreur. Les quatre autres s'effondrent par terre comme des masses, sans un soupir, mais pas Graëme. Les attaches qui retenaient les poignets et les chevilles de la femme sautent et la femme se propulse vers Alicia en criant : « Alicia Coralàn, ils vous mentent ! »
Graëme n'a pas bougé. Et puis il rugit « Non ! », une protestation désespérée, il bondit vers le garde étalé à ses pieds, prend son pistolet et tire. Le visage de la femme se fracasse. Le sang gicle, avec des esquilles d'os et de cervelle, sur Alicia qui s'était instinctivement mise en posture de défense.
Sans savoir ce qui s'est passé dans l'intervalle, elle se retrouve dans des passages déserts avec Graëme qui murmure d'une voix atterrée, « C'est fini, tout va bien », tout en la portant à moitié. Bertran n'est nulle part en vue. Alicia balbutie : « Qu'est-ce que... qu'est-ce qui s'est passé ? » Il continue à l'entraîner et elle se laisse faire, toute molle.
« La rebelle t'a attaquée... » Et, suppliant : « Mais qu'est-ce que tu faisais là, Alicia ! ? »
Elle s'entend dire, de loin : « J'ai trouvé... les documents... la propulsion Greshe. »
Graëme ne ralentit pas, mais son étreinte se resserre autour de sa taille : « Mon Dieu, c'est splendide ! Oh, ma pauvre chérie, et il a fallu que... Est-ce que c'est bien ce que vous vouliez ? »
L'esprit d'Alicia a comme un hoquet, revient au début de leur échange : « Attaquée ?
— Une rebelle, Alicia. On l'a prise en train de s'infiltrer dans le périmètre, elle venait pour toi, pour te tuer.
— Elle a dit... " ils vous mentent "...
— Pour te paralyser, le temps d'être assez près de toi.
— Mais ses menottes... ont sauté... et elle n'a pas... couru, elle...
— Chut, Alicia, mon amour, tu es sous le choc, attends qu'on soit dans ta chambre. »
Il l'aide à se déshabiller, à se nettoyer sous la douche brûlante, le visage, les cheveux, elle en a partout, elle grelotte, la réaction. Il veut appeler un médecin pour lui donner un sédatif, mais elle fait un effort surhumain pour se calmer, elle dit, non, non, je suis très bien, elle refuse avec énergie d'être anesthésiée et finalement Graëme renonce. Elle marche de long en large dans son peignoir, les bras croisés, en essayant de réprimer les frissons qui la secouent par intermittence. Quand il essaie de la prendre dans ses bras en lui disant « Tu devrais t'étendre », elle se dégage avec brusquerie.
Il se laisse tomber sur le bord du lit, consterné. Prend le collier, en ferme et en ouvre le mécanisme : « Tu as réussi à le défaire ? »
Elle lance entre ses dents serrées, sinon elles claquent : « Bertran. »
Il répète à mi-voix, « Bertran », et elle trouverait l'intonation curieuse si elle n'avait pas tant d'efforts à faire pour cesser de trembler.
« Ma pauvre chérie, murmure-t-il avec une compassion navrée, quelle horrible histoire, et justement quand tu touches au but ! »
Elle a trop de mal à parler, avec ses maxillaires contractés, elle va au plus court, en s'interrompant quand ses muscles lui échappent : « Les données étaient là. Pour que je les trouve. Vous les aviez. Tout du long. » Mais ce n'est plus ça, l'important, ce n'est plus ça ! « Elle ne m'a pas. Attaquée. Elle s'était. Arrêtée.
— Elle fonçait sur toi !
— Comment s'est-elle libérée de ses attaches ? » s'écrie Alicia tout d'un trait avec la soudaine énergie du désespoir, qui l'emporte sur ses contractions nerveuses. « Que faisiez-vous à cette femme, que faisais-tu là ? ! »
Graëme est très calme. Il a mis ses mains dans son dos – pour s'empêcher de la toucher ? « Les commandos rebelles sont conditionnés, ils se suicident quand ils sont pris et interrogés. C'est pour ça qu'on les déshabille et qu'on les attache. Elle devait avoir un conditionnement encore plus profond, déclenché par ta vue. Force hystérique, elle a fait sauter ses liens. »
C'est plausible, c'est plausible, n'est-ce pas ? Alicia essaie de se rejouer la séquence des événements, elle ne sait plus très bien, tout le monde était comme paralysé, puis les autres se sont affaissés, pourquoi ? mais pas Graëme, pourquoi ? et après que la femme s'est libérée, Graëme n'a pas agi tout de suite, il y a eu un délai, la surprise, mais vraiment long même pour de la surprise, et la femme aurait largement eu le temps de sauter sur Alicia, mais elle s'était arrêtée, elle ne voulait sûrement pas la tuer.
Elle secoue la tête, brusquement affolée, « Non, non », elle ne sait pas si elle dit non à Graëme ou à ses propres pensées incohérentes. Graëme s'approche, mais elle s'écarte à nouveau, bras pliés, mains raidies, « Ne me touche pas ! » Il se fige, il la regarde comme si c'était lui et non elle qui se noyait, murmure d'une voix éraillée : « Licia, je ne peux pas savoir ce que tu crois avoir vu, mais je t'assure que tu te trompes. J'ignore comment les rebelles ont su que tu étais ici, mais cette femme était un assassin venu pour te tuer.
— Elle savait mon nom ! Qui lui a dit mon nom ?
Il s'écrie : « Je n'en sais rien, je n'en sais rien, je te jure que je ne l'ai dit à personne », et il est si visiblement désespéré...
« Mais tu m'as menti, tu n'es pas un... un petit fonctionnaire, tu étais en train d'interroger cette femme.
— Le médecin était en train de l'interroger, j'étais seulement témoin, ça fait partie du travail de la section politique à Dalloway, la routine, tu crois que ça m'amuse ?
— La routine ? Une rebelle venue m'assassiner ?
— Nous venions seulement de l'apprendre !
Ils restent un moment dressés l'un contre l'autre, et Graëme s'effondre sur le lit : « Oh, Alicia, non, non », et c'est vrai, il ne peut pas faire semblant, pas comme ça, il est vraiment effondré.
« Mais les données, vous avez menti, vous les aviez trouvées. Depuis quand ?
— Une vingtaine de saisons. Mais je voulais que tu les trouves, Alicia, que toi tu les trouves.
— Quoi, moi ! ? »
Il la contemple un moment sans bouger, l'air bouleversé. Puis il se lève, il lui prend les mains malgré sa résistance, d'abord, et quand il parle c'est d'une voix basse, angoissée mais résolue : « Écoute-moi bien, Alicia. Ils avaient un accord secret avec ton père. Tu n'étais pas censée réussir. Tu devais être discréditée avant même de venir à Dalloway. Ils m'ont manipulé – c'était mieux si je ne savais rien, au début, si j'étais sincère. Ils avaient étudié ton profil, le mien, ils savaient que nous étions... compatibles. Si ça n'avait pas marché, ils avaient un deuxième choix, et un troisième... Quand j'ai compris, après Cristobal... Je me suis tenu à l'écart. Et j'ai essayé de me traîner les pieds aussi longtemps que je l'ai pu, quand ils sont revenus me chercher officiellement. Ils ne savaient pas où nous en étions, toi et moi, mais tu n'avais même pas vu l'autre type qu'ils t'avaient flanqué dans les pattes. Heureusement qu'ils m'ont recontacté avant de prendre des mesures radicales du genre drogues et mises en scène. »
Alicia reste muette de stupéfaction, de chagrin, de fureur.
« J'ai choisi, Alicia. Toi et moi. Je les ai convaincus que le plan de ton père était trop gros, que tu ne te laisserais pas faire, que tout cet aspect de la magouille se retournerait contre lui et ultimement contre eux qui y auraient collaboré. Qu'il était puissant dans Lagrange, mais pas tout-puissant, qu'il avait des comptes à rendre. Qu'il valait mieux procéder en douceur. La partie essentielle de son plan, c'est d'envoyer Eduardo réussir là où tu aurais échoué, et somme toute ce serait te discréditer assez, il finirait par s'en contenter. Ils ont accepté, ils ont retiré toute la surveillance rapprochée. À partir de là, je n'avais plus qu'à placer les données à Dalloway pour que tu les trouves. J'ai... des amis. »
Il lui embrasse le bout des doigts, elle se laisse faire, dans le brouillard : « Je ne pouvais pas te le dire, je n'osais pas. J'avais déjà assez de mal à ne pas me trahir, nous trahir – et j'ai l'habitude de toutes ces saloperies depuis presque toujours, moi. Tu es très bien entraînée et intelligente et habile mais... tu n'étais pas préparée pour ça ! Je ne pouvais pas courir ce risque pour nous, ma chérie. Triage, Alicia, j'ai fait pour le mieux, tu comprends, n'est-ce pas, dis-moi que tu comprends ? »
Alicia hoche la tête machinalement, mais il ne comprend pas, lui, intérieurement une autre Alicia hurle et rage, comment a-t-il pu faire ça, comment a-t-il pu la manipuler ainsi, il n'avait pas le droit, pas le droit de choisir pour elle, pas lui, pas lui aussi, l'accumulation des mensonges a atteint le point critique, Marti, Graëme, c'est la même chose, et il ne se rend même pas compte, il ne se rend même pas compte de ce qu'il a fait, elle ne veut même pas essayer de savoir s'il dit la vérité ou pas, quelle partie de ce qu'il dit est vraie, ou ce qu'il croit vrai, elle explose enfin, elle lui crie : « Je ne pourrai jamais plus te croire, tu ne comprends pas, je ne pourrai jamais plus te croire ! »
Il proteste, plaintif : « Mais c'est la vérité, Alicia, je l'ai fait pour toi ! »
Oh, elle voudrait le secouer, elle voudrait le frapper, pour qu'il comprenne, elle se sent déborder d'une telle violence qu'elle en est épouvantée, recule : « Laisse-moi, va-t'en ! »
Il reste un long moment muet, immobile, puis il dit d'un ton distant, blessé : « Je te dis la vérité, Alicia. Tu es sous le choc, mais tu comprendras plus tard, j'en suis sûr. Tu verras que c'était le mieux à faire. Tu devrais prendre un calmant. Tu aurais les idées plus claires. »
Elle se force à dire : « Non, ça va aller. J'ai vraiment besoin d'être seule. Dans ces cas-là... je t'assure, il vaut mieux me laisser seule. Je t'en prie. » Et elle se hait, et elle le hait d'avoir à feindre, à demander, à dépendre.
Elle regarde la porte se refermer sur lui. Pas de clé qui tourne dans la serrure – la serrure est à carte magnétique, ils n'ont qu'à annuler son code s'ils le désirent. Elle jette un coup d'œil autour d'elle, réprime durement une velléité de frisson, se force à prendre de grandes respirations. L'éclat bleu et doré du collier la nargue, sur la table de nuit. Il faut faire quelque chose, n'importe quoi, mais agir, si peu que ce soit. Et elle fait ce qu'elle n'a jamais fait auparavant, s'ils la surveillent, ils sauront que le jeu peut se jouer à deux : elle scanne la chambre.
Rien ? Pas la moindre miette de senseur-espion ? Elle n'en ressent même pas de la satisfaction. Ça ne veut rien dire. Elle est à Dalloway. Tout le maudit complexe est une prison ! Elle s'assied sur le lit, pose le scanneur sur la table de nuit et se passe les mains sur la figure. Quoi, maintenant, quoi faire ?
Et son regard tombe sur la table de nuit, et le scanneur, et le collier près duquel est posé le scanneur, et la petite lumière ambrée qui dénonce. Dans le collier. Un traceur.
Elle s'habille, ramasse les divers petits objets usuels où sont dissimulés les composants du transmetteur, les enveloppe avec soin dans un morceau du rideau de douche déchiré, avec le cube des données sur la propulsion Greshe – et elle redevient Alicia le Fantôme.
42
Lian marche.
Il voit – vert, brun, rouge, doré, brun, noir, gris, blanc, roux, multicolore, des arbres, des murs, des lampadaires à gaz, des affiches, des bicyclettes, des gazillacs, des camions, des gazobus, des ferries, des passants, des promeneuses, des chiens, des tricycles de vendeurs ambulants, des fontaines, des allées, des parcs, des canaux, des escaliers, des ruelles, des rues, des feux de circulation, des trottoirs. Il entend – le grondement proche ou lointain de la circulation, les voix paisibles, rieuses, fâchées, pressantes, distraites, les sonneries aigrelettes des bicyclettes, les roulades musicales des oiseaux dans les parcs, de brefs aboiements de chiens, des sifflets de ferries, des bruissements frais de fontaines. Il sent, il goûte les mille parfums des arbres, des buissons, des fleurs, des herbes dans les parterres, tombant des terrasses, sur les remblais des canaux – acidulés, poivrés, sucrés, violents, délicats, doux, âcres – et tout autour et par-dessus, l'odeur de la ville, plastique et caoutchouc chauds, asphalte suintant, ferraille, poussière. Il goûte aussi son sang, métallique, parce qu'il s'est violemment mordu la lèvre inférieure, il ne sait pas quand, et y passe machinalement sa langue de temps à autre. Sur sa peau, seulement le frottement de ses vêtements et la chaleur tiède et le vent. Il ne touche personne, même s'il s'en va en aveugle : son corps a encore pris les rênes, avec tous ses entraînements – peut-être qu'on l'évite aussi, mais il l'ignore. Il ne veut plus rien savoir, jamais. Il ne pense pas. Il marche.
Soudain, il ne marche pas seul. Il y a du monde autour de lui, devant, derrière. Des femmes, des hommes, jeunes, vieux, en habits de ville ou de travail, au coude à coude, allant du même pas, certains bras dessus bras dessous. Les uns sourient, d'autres froncent gravement les sourcils, d'autres encore semblent anxieux. Ici et là des jeunes gens portent des jeunes filles à califourchon sur leurs épaules. Ils rient. Il y a des enfants. Beaucoup de gens tiennent des bâtons au bout desquels sont accrochés de grands ou de petits panneaux de carton souple où sont écrits des mots, à la main, en majuscules et en couleur. À quelques mètres devant lui, on brandit une longue et large banderole gonflée comme une voile malgré les petits trous qu'on y a percés. Il déchiffre machinalement, en transparence, le mot à l'envers, SOLIDARITÉ. De l'arrière, de l'avant, arrivent des chants scandés, qu'on reprend autour de lui et qui passent, comme des vagues.
Quand s'est-il trouvé pris dans la manifestation ? Il l'ignore. Mais il continue à marcher avec tout le monde le long de l'avenue de la Victoire, en direction du Présidium. On occupe toute la largeur de l'avenue vidée de sa circulation, les deux voies, en épargnant cependant l'étroit terre-plein médian et ses longs parterres fleuris. C'est un peu comme à la parade, mais le rythme change d'un groupe à l'autre, d'une personne à l'autre – et il n'y a pas d'uniformes. Ce n'est pas le même jeu : on marche ensemble, mais pas pour marcher. Pour quoi, alors ? Pour les rêves inscrits sur les bouts de carton, sur les bannières ? Sur les trottoirs, peu de spectateurs ; davantage aux fenêtres ou penchés aux parapets des terrasses. « Avec nous, avec nous ! » scandent les manifestants. Quelques silhouettes jaillissent des entrées pour se joindre à la foule, surtout des jeunes, déclenchant des flots de reproches, d'ordres ou de supplications depuis les terrasses ; parfois on leur court après pour les arrêter, mais la mer des manifestants se ferme sur eux, se reforme en un mur souriant.
Entre les têtes moutonnantes de la foule, Lian entraperçoit maintenant au loin les hautes parois obliques du Haëkelliaõ – du Présidium –, la façade principale tournée vers le nord et la Mer encore absente, toute dorée, mais veinée de verdure sombre là où branches et lianes retombent depuis la dernière terrasse. Les rires se font moins fréquents, les conversations se taisent, les filles sont descendues des épaules des garçons ; on scande les slogans avec plus de force, comme si la colère montait – ou la peur.
« C'est là qu'on va voir ! » lance l'homme qui marche devant Lian, à la cantonade.
« Si c'est le gouvernement du peuple, par le peuple, pour le peuple ! » scande un autre, farouche, à sa gauche.
« Parce que c'est nous, le peuple ! » s'exclame un troisième, un de ceux qui tiennent des porte-voix et veillent aux bords de la foule.
Un chœur d'acquiescements sombres leur répond.
« Ils ne peuvent pas nous tirer dessus, hein ? » dit une voix plus timide, juste à côté de Lian, tandis qu'un bras se glisse sous le sien. « C'est une manif' pacifique, même s'ils l'ont interdite... » Lian a comme un éblouissement : une vingtaine de saisons, trapue, courts cheveux noirs, peau bistre, noirs yeux bridés, la jeune fille ressemble à Nan. Il effleure la main qui s'agrippe à son bras, murmure : « Mais non. » Il ne sait même pas si la fille l'a entendu par-dessus le nouveau slogan qui déferle à travers la foule, mais elle semble rassurée, elle lui sourit.
Puis une autre vague passe dans le long serpent des manifestants : de proche en proche on ralentit, on s'arrête. Lointaine, bien qu'amplifiée par des haut-parleurs, une voix résonne, nette, calme, impérative : « Dispersez-vous et rentrez chez vous. Cette manifestation est illégale. Dispersez-vous et rentrez chez vous. »
Il y a un bref flottement. « Tu parles ! » s'exclame un adolescent. « Et quoi encore ? » grogne une matrone. Un nouveau slogan se déclenche, lancé par le porte-voix d'un des membres du service d'ordre de la manifestation, « Nous-sommes-le-peuple, Nous-sommes-le-peuple ! », qui se répand d'avant en arrière le long des manifestants, repris enfin comme par une seule énorme voix, tandis que la foule se remet en marche avec une lenteur délibérée. « Ils bloquent la place, on s'assoira devant, suivez les directives, on reste calmes, faites passer, on reste calmes », lancent les bénévoles qui circulent aussi le long du cortège. Lian continue à avancer en jetant de temps à autre un coup d'œil dans les rues transversales ; elles sont vides, et bloquées au bout d'une centaine de mètres par des véhicules de police.
« Dispersez-vous et rentrez chez vous », gronde de nouveau la voix amplifiée, plus proche, « cette manifestation est illégale. Dispersez-vous...
— Nous-sommes-le-peuple ! Nous-sommes-le-peuple ! »
On ralentit encore, les rangs se tassent davantage, on s'arrête : l'avant de la manifestation est arrivé devant le barrage qui empêche l'accès à la place de la Victoire. Lian se hausse sur la pointe des pieds, comme plusieurs, pour voir. Ceux qui sont moins coincés dans la foule escaladent les caissons des arbustes, sur le terre-plein central. « Oh la la », lance l'un d'eux, un adolescent, à gauche de Lian – d'une voix qui se veut consternée mais sonne plutôt excitée – « la place est pleine de fourmis !
— Juste la garde, pas l'armée, alors ? » dit quelqu'un, une femme, soulagée.
« Eh, faudrait voir ça qu'ils nous mettent l'armée ! » grommelle un homme d'une soixantaine de saisons. « On n'est pas des rebelles ! »
Lian veut en avoir le cœur net. Il se fraie un chemin jusqu'au terre-plein, escalade à son tour un caisson. À environ deux cents mètres en avant, sur toute la largeur de l'avenue, s'alignent plusieurs rangées immobiles de silhouettes en armures brunes luisantes, casquées, masquées, portant de grands boucliers rectangulaires et des longs bâtons flexibles. Plus loin en arrière, on devine les véhicules blindés qui cordonnent la place elle-même.
Les directives arrivent de l'avant, en même temps que Lian voit les gens s'asseoir par vagues dans l'avenue. L'adolescent guetteur préfère rester juché sur son caisson, comme plusieurs autres et Lian, qui regarde cette mer de têtes brunes, blondes, rousses, grises, ondoyant un peu autour de lui, à perte de vue. Son cœur se serre, il ne sait pourquoi, ce n'est pas de l'angoisse ni du chagrin mais une exaltation bizarre, à la fois sombre et chaleureuse, comme les braises d'un feu dans la nuit, comme lorsqu'il regardait dormir les autres à la belle étoile, pendant ses tours de garde, en patrouille, sur la Ligne. Le chant reprend, « Nous-sommes-le-peuple ! Nous-sommes-le-peuple ! », clair mais un ton plus bas, comme si on fredonnait au bord d'un berceau – qui veut-on charmer, les fourmis casquées, là-bas, immobiles ?
« Pour la dernière fois, dispersez-vous, rentrez chez vous, cette manifestation a été déclarée illégale par l'assemblée de vos représentants élus », clame la voix électrifiée, énorme à présent.
Le slogan revient en vague depuis le fond de l'avenue. Lian se surprend à scander intérieurement : nous-sommes-le-peuple, nous-sommes-le-peuple.
Des éclatements sourds, devant, des sifflements, quelques cris. Des filets blanchâtres commencent à s'élever en tournoyant. Des silhouettes se lèvent et courent ramasser des grenades pour les relancer en direction des gardes. Plusieurs, près de Lian, sortent des mouchoirs et des foulards qu'ils se nouent sur le nez et la bouche ; plusieurs ont de petits aérosols à pompe dont ils arrosent les masques. « Les andouilles ! dit un gamin jubilant, avec le vent, ils vont tout prendre dans la gueule ! » En avant, des manifestants commencent pourtant à se relever en masse pour échapper à la fumée, malgré les exhortations des porte-voix.
Les gardes chargent. D'abord au pas. Puis à la course, suivis par les tireurs de lacrymogènes qui envoient une troisième bordée beaucoup plus profondément dans les rangs de la foule encore assise – on en relance le plus possible, encore partiellement protégé de la fumée par le vent. En tête de la manifestation, sur une dizaine de rangées, presque personne n'a bougé – conviction ou paralysie, peu importe. La garde arrive au contact des assis en plein élan, bâtons levés. Les rangées suivantes s'éparpillent pour échapper à la charge, se font rattraper ; quelques-uns essaient de contourner les gardes pour se rapprocher de la place, mais la plupart s'engouffrent dans les rues avoisinantes pour les trouver barrées et bientôt enfumées à leur tour – et le vent s'y fait bien moins sentir que dans l'avenue. On reflue en toussant vers la seule voie ouverte, où les premiers rangs des gardes ont été suivis par les voitures d'incendie et leurs lances à pleine pression, qui vous fauchent les jambes. On tombe, on dérape sur plusieurs mètres, la peau arrachée, on est violemment plaqué contre des murs, des poteaux, d'autres manifestants qui dégringolent à leur tour.
De nouveaux tirs serrés – plus des gaz, on a dû enfin se rendre compte que le vent n'était pas propice, mais une pluie de petites balles de caoutchouc dur, en tir tendu. Lian saute par réflexe derrière le caisson, accroupi, tandis que les projectiles hachent l'arbuste et les touffes de fleurs. Autour de lui, on court, on crie. On tombe, et un autre réflexe l'envoie vers l'homme qui s'est étalé tout près, pour le tirer à l'abri relatif du caisson. « Merci », halète le type, un des gars du service d'ordre, il porte le brassard bleu. Il se redresse avec une grimace. « En plein dans le dos, les salauds. Viens, on se tire !
— Toutes les rues sont barrées.
— Ça ne fait rien, faut essayer, s'ils me ramassent, je suis cuit. »
Lian se fige, soudain glacé. Le crâne du jeune rebelle explose au ralenti dans sa mémoire.
Des silhouettes casquées passent autour d'eux, apparemment trop concentrées sur les fuyards pour prêter attention à ce qui ne bouge pas. Lian aide l'autre à se relever, à demi accroupi. Ils échangent un regard puis, d'un commun accord, se lancent dans l'avenue.
Lian se retrouve bientôt avec son compagnon parmi des hommes et des femmes qui courent ensemble, par petits groupes de trois ou quatre, une douzaine en tout mais ensemble, c'est clair pour Lian. Ils zigzaguent avec habileté entre les nuages de fumée, les corps étendus et les gardes et bifurquent soudain, les uns après les autres, dans une rue étroite bloquée par deux voitures de police placées en biais. Que croient-ils faire ? Mais ils arrivent aux barrières, les sautent ou les écartent, puis escaladent les voitures. Pas de grenade lacrymogène, aucune réaction. Les cinq ou six policiers embusqués derrière les véhicules restent immobiles tandis qu'on les bouscule pour passer. Lian continue à courir.
Des crépitements secs le tirent de son hébétude : les policiers se sont réveillés aussi. À côté de lui une femme trébuche et s'effondre, deux des hommes en avant aussi, les autres se plaquent contre les murs ou zigzaguent pour échapper aux balles, puis s'enfoncent sous les voûtes des entrées qui s'ouvrent à droite et à gauche. Certaines sont fermées par des grilles. Lian en trouve une ouverte, longe un bassin, gravit quatre à quatre l'escalier d'une terrasse, se rend compte qu'il n'est pas seul, un de ses compagnons l'a suivi. Il défonce la porte de l'escalier extérieur à coups de pied et dégringole dans la ruelle arrière.
Des claquements secs résonnent dans le lointain du côté de l'avenue, des coups de feu, maintenant, de plus en plus nourris. Et ici, autour d'eux, des bruits de course. Lian voit passer des silhouettes au croisement de la ruelle avec une rue transversale, policiers ou autres fuyards, impossible à dire. Il part dans la direction opposée, vers l'ouest, pour s'éloigner le plus possible de l'avenue. L'autre le suit.
Lian prend soudain conscience qu'il fait sombre, que la nuit est tombée. Ils foncent dans un lacis de ruelles obscures et d'allées herbeuses chichement éclairées de petits lampadaires à gaz, encombrées de poubelles, de linge qui sèche, de balancelles et de caisses, déclenchant au passage des hurlements de chiens. Soudain l'autre trébuche et s'effondre en balbutiant : « Attends, attends ! » Lian se laisse tomber près de lui, hors d'haleine. C'est un homme assez jeune, la trentaine, maigre et nerveux, qui a réussi à ne pas perdre ses lunettes pendant la course. Il semble écouter, les yeux fermés. « Sont à cinq rues, dans McEwen. Parallèles. On peut encore. Quelle heure ? »
Pris au dépourvu, Lian consulte sa montre, se rend compte qu'elle ne marche plus : un choc violent en a enfoncé le boîtier. L'autre secoue la tête, toujours haletant : « Ça fait rien. Faut aller au nord. Le port. Rendez-vous. »
Il essaie de se relever, mais ses jambes se dérobent sous lui. Lian le soutient, sent quelque chose de visqueux contre sa paume. Dans l'ombre, on ne voit pas la couleur du sang, mais sa main en est couverte. Il rassied l'autre qui proteste faiblement, examine la blessure. La balle est entrée de bas en haut dans l'épaule en évitant de justesse l'omoplate, a dû ricocher sur la clavicule en la fêlant ; le pectoral gauche, à la sortie, est un dégât de chairs éclatées. En cherchant autour de lui, Lian voit une corde à linge où traînent quelques chiffons ; il improvise un pansement de fortune immédiatement rougi. L'autre a perdu beaucoup de sang, il devrait être en état de choc, mais il parle encore, d'une voix entrecoupée : « Au port. Il faut aller. Au port. »
Lian retire sa chemise, la déchire pour faire un pansement supplémentaire plus solide, ses mains ont retrouvé leur force, il n'y pense même pas. C'est la nuit à présent. Les petites lunes ont commencé à monter à l'est, la pénombre est moins dense. Le bruit lointain de l'émeute se fait de plus en plus sporadique. Les poursuit-on encore ? Sans doute : des chiens aboient, de façon intermittente, dans tout le quartier à l'est de leur position. Aller au port avec le type, il pourrait, il pourrait le porter, il connaît la direction générale, c'est suffisant. Mais si le blessé ne reçoit pas des soins immédiats, ça ne servira à rien de l'emmener jusqu'au port !
L'autre respire à petits coups, en retenant chaque fois un gémissement. Tout d'un coup, d'une voix très claire, comme s'il avait rassemblé ses dernières forces pour parler, il dit : « Va au port, camarade. Laisse-moi. »
Lian reste pétrifié. Puis, avec une rage soudaine, il frappe le mur près du blessé, qui tressaille mais ne le quitte pas des yeux. « Non ! » Il répète « Non ! » et soudain il explose : « Comment... comment pouvez-vous laisser arriver ça ! ?
— On ne peut pas... contrôler... tout le monde... tout le temps. Ni eux... ni nous. Va au port. N'y vont... jamais. Trop peur. Surtout... cette nuit. »
Lian gronde de nouveau : « Non. Je ne vais pas te laisser là !
— Je ne serai... pas là, dit l'autre avec un faible sourire. Va, camarade. Je t'en prie.
— Non », répète Lian, mais tout bas, presque d'un ton implorant. Il regarde la ruelle obscure à la recherche d'il ne sait quoi, il ne sait qui pour l'aider, étreint d'un désespoir brûlant.
Le corps du blessé se convulse près de lui. Il se retourne juste à temps pour voir l'autre s'affaisser. Il le redresse, affolé, incrédule malgré la puanteur qui assaille ses narines – il la connaît bien, pourtant, il l'a assez sentie sur la Ligne, le jour des gremlins. La tête part en arrière sur le cou ramolli, cogne le mur. Derrière les lunettes de travers, les yeux sont révulsés ; la bouche est ouverte ; sur une molaire, en bas, à gauche, dans la salive, on peut voir la pellicule gélatineuse d'une capsule écrasée.
43
Alicia n'essaie même pas la porte. Elle a étudié les plans de Dalloway, du temps où elle devait effectuer une Mission Héroïque et Dangereuse, et à la veille de partir encore, les nouveaux plans, par acquit de conscience. Les données ont un peu changé, mais pas tellement dans le complexe central. L'envers du décor est toujours là, derrière les murs, les plafonds. Elle se faufile. Là où il le faut, une ou deux fois, elle passe dans le dos des gardes. On est en régime de nuit, tout a l'air bien tranquille. Y aura-t-il une alerte si on découvre qu'elle n'est pas dans sa chambre ? Peut-être pas, mais elle s'en moque. Elle accède au rez-de-chaussée, dans une ancienne salle de consigne transformée en réserve. À partir de là, c'est facile : la porte se laisse faire sans déclencher d'alarme. Alicia se retrouve sur le tarmac, trois cents mètres à parcourir avant d'atteindre les hangars, sous les pinceaux des projecteurs, pas de problème, la routine. Les détecteurs de mouvement et tout le reste, c'est à la périphérie plus que dans la base elle-même. Elle se faufile entre les caisses, les véhicules, les bidons, les non-identifiables bâchés. Échappe à deux patrouilles. Longe deux kilomètres de hangars et après, voie presque libre, c'est le lac.
Elle s'est fait rappeler par son souffle court qu'elle se trouve à plus de deux mille mètres et qu'elle n'y est pas encore tout à fait habituée, malgré le séjour à Morgorod. Elle se glisse plus posément, sans éclaboussures, dans l'eau frisquette. Toutes ces jolies défenses rétro-modernes sont plus conçues pour empêcher d'entrer que de sortir, ce qui aide. Le lac, presque parfaitement circulaire, se trouve dans le périmètre de protection totale, mais c'est surtout une ligne de mines et de détecteurs de mouvement au milieu, et la même chose le long de l'extension triangulaire du périmètre autour de la moitié sud du lac. C'est un scénario qu'elle a répété plusieurs fois avec succès : il faut passer le premier barrage, continuer le long de la rive et ensuite ramper dans un aqueduc des Anciens qui alimentait l'une de leurs mines à ciel ouvert du secteur, disparue quand on a raboté le plateau. Il débouche un kilomètre à l'intérieur de la zone moyenne.
À ce moment-là, l'excitation nerveuse d'Alicia est retombée ; elle continue parce qu'elle a commencé et ne peut pas faire demi-tour – ne veut pas. Et elle ne veut pas penser à ce qu'elle fera si l'arrivée de l'aqueduc a été condamnée ; elle avait monté son sim après avoir découvert l'existence de l'aqueduc en comparant des documents datant de la colonisation aux derniers travaux de sécurité effectués à Dalloway momentanément redevenu propriété terrienne, juste avant la Libération. Il y a des chances raisonnables qu'on ait encore oublié ce conduit à moitié sous-marin. Sinon... Mais une chose à la fois. Pas de passé, pas de futur pour le Fantôme, juste l'instant présent, seul réel.
Elle arrive sans embûches à la ligne de mines et de senseurs flottants, prend son souffle et plonge. Une dizaine de mètres, elle peut le faire aisément ; les senseurs sont réglés sur la surface et non sur la profondeur. Ensuite, elle continue à suivre la rive, très lentement, en émergeant à peine pour respirer à intervalles irréguliers : elle pourrait être un poisson venant gober des insectes à la surface, les senseurs ne vont pas s'énerver pour un poisson, et puis elle continue à s'éloigner de la base. Et, finalement, elle arrive là où doit se trouver l'orifice de l'aqueduc. Elle explore un moment sous l'eau, angoissée, mais c'est là, et ça a bel et bien été obstrué, mais un gros animal, peut-être une des loutres géantes qui vivent encore dans le lac, a défoncé la grille il y a longtemps ; on n'a jamais pris la peine de la réparer. En s'arc-boutant, on peut écarter assez le métal pour se faufiler. Environ encore dix mètres sous l'eau, et on émerge.
Ça pue et il fait très noir, dans le conduit de l'aqueduc ; elle ne voit absolument rien après les premiers mètres. Elle patauge sur les mains et les genoux. Et soudain, elle donne la tête la première dans quelque chose de mou et un petit pinceau de lumière s'allume, presque aveuglant. Alors qu'elle s'apprête à défendre chèrement sa vie, elle reconnaît dans les taches d'ombres et de lumières le visage de Bertran. Qui lui fait signe de se taire et vient lui murmurer à l'oreille : « Senseurs à la sortie, directionnels. Suivez-moi. » Il éteint sa lampe, elle l'entend remuer tandis qu'il se retourne et s'éloigne. Incapable de penser, elle le suit.
À la sortie, vers le haut, une autre grille. Dont Bertran désactive les senseurs d'une façon qui ne déclenche apparemment aucune alarme nulle part : les projecteurs continuent à peindre leur lumière violente sur le paysage, danse bien réglée des ombres qui s'allongent, s'étrécissent, disparaissent, reviennent. Ils s'intègrent souplement à cette danse, s'éloignent davantage encore du périmètre de protection totale. Ils se trouvent maintenant dans la zone moyenne, où la végétation leur offre un luxe d'abris possibles. Entre les mines et les bouquets de senseurs, bien entendu, mais Bertran semble en connaître fort bien les emplacements. « Restez très près de moi. » Elle obtempère et se faufile derrière lui de buisson en arbre en rocher, direction sud, au moins trois kilomètres, jusqu'à la limite de la zone plane qu'on a écorchée dans le plateau. Toujours dans le périmètre de protection moyenne, et presque six kilomètres à faire pour en sortir, mais ils seront encore mieux à l'abri dans le relief naturel du plateau. Et après ? Alicia ne veut pas se poser la question, ni s'interroger sur Bertran. Plus tard. Le scénario a pris un tour inattendu, mais c'est toujours le scénario. Il faut sortir de Dalloway pour de bon.
Et puis quelque chose tourne mal alors qu'ils approchent des éboulis menant sur la hauteur. Une patrouille les a-t-elle repérés ? Un tir de barrage se déclenche, éclairs, pulsations de tonnerre, explosions de roches en shrapnels autour d'eux. Ils se mettent à couvert. Une autre explosion, toute proche, un choc violent sur la tête, même pas de douleur, fin de la partie pour Alicia.
Quand elle reprend conscience, elle est perdue dans le temps pendant un bref instant, elle gémit : « Ce n'est pas ma faute, pourquoi vous ne m'avez rien dit ! », elle s'accroche à son père qui la ramasse par terre alors qu'elle vient de tomber à ses pieds depuis l'envers du décor. Elle le regarde, terrifiée, suppliante, et il a changé de visage... Elle se rappelle alors, non, c'est Bertran, elle se redresse, elle se sent un peu floue. Il fait très sombre, mais l'espace respire assez vaste autour d'eux. Le petit faisceau de la lampe-crayon ne va pas très loin, même une fois que ses yeux se sont accoutumés à l'obscurité.
Bertran explique : elle a juste été assommée par un morceau de roc, pas grave. L'important, c'est que l'explosion les a fait tomber dans un des tunnels miniers des Anciens – la région en est truffée, on ne s'est pas donné la peine de tous les repérer, c'est un labyrinthe ; on a simplement bouché les entrées qu'on a trouvées. Mais il connaît un itinéraire, ils pourront quitter le périmètre moyen de cette façon, en marchant une vingtaine de kilomètres à l'intérieur de la montagne. Ensuite ils se trouveront dans une des petites vallées qui découpent les massifs, et de là ils pourront descendre tranquillement vers le sud-ouest, ils seront en dehors de la Ligne – en terrain fédéral, mais c'est la forêt vierge, les risques de détection sont minimes. Il est venu par là. Il a une cache avec de l'équipement, à une soixantaine de kilomètres à l'ouest ; il va falloir marcher.
Elle se lève avec prudence, se tâte la nuque, étonnée de ne pas se sentir plus vaseuse. Une grosse bosse, et c'est très sensible, mais il n'y a même pas de sang. Le petit sac est toujours attaché à sa ceinture, espérons que le paquet ficelé dans le bout de plastique est resté raisonnablement étanche. Après avoir jeté un bref regard au relief indistinct qui est le visage de Bertran dans l'ombre, toujours aussi impassible en vrai agent double qu'en faux capitaine, elle dit simplement : « Allons-y. » Elle ne veut même pas savoir ce qu'il est, un rebelle sûrement, ou même un Fédéral se faisant passer pour un rebelle, elle n'a pas d'énergie à perdre dans ce genre de spéculations improductives. S'éloigner le plus possible de Dalloway, c'est la priorité. Il l'emmène vers l'ouest ? Tant qu'à faire, ça l'arrange : les rebelles dans leur territoire seraient une variable inconnue de trop en la circonstance. Elle s'arrangera pour neutraliser ce type et elle expédiera à Lagrange le code d'extraction immédiate. Elle a en main les données convoitées, la station est maintenant ancrée depuis deux jours au point L4, ils conviendront d'un lieu de récupération, ils enverront la navette avec un moddex, ce sera terminé, et que les Fédéraux essaient de les en empêcher, pour voir !
Ils marchent et ils marchent et ils marchent dans le labyrinthe des tunnels. Elle n'a jamais tenté ce genre d'évasion de Dalloway – l'information dont dispose Bertran était inconnue des auteurs des autres sims, comme d'elle-même – mais c'est un bon plan ; sauf qu'elle n'a jamais eu aussi soif dans un scénario. Et ensuite ils marchent, ils marchent et ils marchent pour arriver à la cache de Bertran. Ils se nourrissent sur la forêt, en attendant – Face-de-bois a des ressources inattendues. Alicia espère surtout que ces denrées exotiques ne vont pas lui flanquer la diarrhée, mais ses craintes sont de courte durée. Elle finit même par prendre goût aux racines de rattèles. Ça goûte la carotte.
Elle croyait s'être habituée à la réalité de Virginia, mais l'escapade au parc de la Tête ou les promenades dans les jardins du Capitole à Morgorod pourraient aussi bien avoir eu lieu dans des sims, ils lui semblent irréels, ces souvenirs de gazon, de sentiers, ces plantes bien sages, ces petits lacs et ces fontaines, tandis que, en sueur, la peau irritée, elle se fraie un chemin avec Bertran à travers les lianes, les ronces, les troncs abattus, les buissons grands comme des arbres, les arbres grands comme des tours, la pénombre humide et verte si épaisse qu'on a parfois l'impression d'être devenu amphibie... Le bruit incessant et chaotique de la faune, ça va, il y en avait dans les sims ; mais la chaleur, l'humidité, les insectes, et les odeurs entêtantes – fleurs, champignons, écorces, pourriture opiniâtre qui nourrit la vie opiniâtre à chaque étage de la forêt... Et il n'y a pas de commande d'arrêt quand on n'en peut plus, il faut continuer, et encore après : c'est Bertran la seule commande d'arrêt, et il a l'air infatigable. Il ménage Alicia, pourtant, elle s'en rend bien compte et met un point d'honneur à ne pas être fatiguée, mais il y a des moments au début où elle tient à peine debout. Toutes ces heures passées à fixer un écran, malgré le conditionnement physique journalier, ça ne vous prépare pas très bien pour la partie Héroïque et Dangereuse de La Mission, hein, Alicia ? Au moins, elle a retrouvé assez d'énergie pour ironiser, c'est bon signe.
Ils arrivent enfin à la cache, dans une partie du plateau où la végétation s'espace un peu. Il y a un petit torrent, Alicia se lave, elle se change, bonheur inouï. Ils font leur premier vrai repas depuis cinq jours. Elle dort douze heures d'affilée.
On continue à marcher ensuite. Face-de-bois ne pouvait pas avoir un tout-terrain dans sa cache ? Trop près, encore trop aisément repérable, bien sûr. Ils suivent une série de larges pistes anciennes, praticables parce qu'encore pratiquées par des camions. Pas en cette saison ; la campagne en a encore pour plus d'un Mois, si près de la Ligne l'interdiction d'exploitation est en vigueur pour les mines, les carrières et les exploitations forestières : on ne rencontre personne. En de nombreux endroits la forêt a eu le temps de reconquérir la piste, il faut dégager, mais on couvre ses bons vingt à vingt-cinq kilomètres par jour. Alicia calcule, horrifiée, qu'on en a au moins pour trois semaines, plus de quarante jours, avant d'atteindre la passe qui quitte le plateau pour se diriger plein ouest vers les collines et la plaine longeant les montagnes Rouges. Et elle qui trouvait archaïques et lents les petits avions empruntés pour sauter d'un bord à l'autre du continent !
Face-de-bois semble capable de marcher pendant des heures sans dire un mot. Alicia essaie d'imiter ce laconisme et de se perdre dans le rythme de la marche, mais elle y a beaucoup de mal : si elle n'y prend garde, elle se retrouve en train de penser à Graëme et à ce qui s'est passé cette nuit-là. À un moment donné, pour empêcher les événements de se rejouer trop complètement dans sa mémoire, elle cède en partie – contre-feu – elle demande : « Je dois vous appeler comment ?
— Bertran.
— Qu'est-ce que vous faisiez là ?
— J'attendais de pouvoir vous emmener. »
Vraisemblable. Une fois les données en main, et l'ambassadrice, ils pouvaient négocier.
« Et comment avez-vous su que j'essaierais de m'échapper, et par où ?
— Psychologie, et tactique élémentaire. »
Elle envisage un bref instant ce que cela implique de surveillance et d'information sur elle, puis s'en détourne résolument : « Et si j'étais restée ?
— Plan B. »
Elle ne va pas demander ce qu'aurait été le plan B. Elle aurait dû continuer à se taire. Elle a presque la nausée. Toutes ces machinations qui se tramaient par-dessus sa tête, sa naïveté, son ignorance, son arrogance de se croire si peu que ce soit aux commandes ; si elle avait une touche pour s'arrêter elle-même, elle l'utiliserait. En l'absence d'une telle touche, elle se secoue pour répartir autrement sa charge sur son dos et presse le pas.
44
Ils prennent Lian le lendemain matin à l'aube, par hasard, un couple de policiers en fourgonnette qui allaient déjeuner et qui jugent bien suspect ce type au torse nu et ensanglanté en train de marcher sur le terre-plein de l'avenue Goulet, dans la ville nouvelle. Il est très loin du port. Il est parti plein sud. Il n'a même pas ralenti quand la lune éclipsée, à minuit, a inondé le ciel de sa luminescence violette pour annoncer le retour de la Mer.
Ils le trouvent encore plus bizarre quand ils sont en contact direct avec lui. Plus de coloration protectrice : il a échappé à tous ses bergers, plus personne, de près ou de loin, pour le faire passer pour normal. Il y pense un instant, distraitement, oublie. Les policiers ne semblent d'ailleurs pas très bien savoir ce qui les dérange ainsi chez lui – en dehors de son torse dépourvu de blessure et de son absence de réaction. Il ne répond pas quand la femme lui demande son nom, mais il a une expression tellement hébétée qu'elle n'insiste pas. Ils cherchent ses papiers. Ses papiers étaient dans sa veste – il a perdu sa veste, il ne se rappelle plus, pendant la manifestation. Les policiers s'illuminent : ah, il était à la manifestation ? Il est rendu bien loin du centre-ville, et toute la nuit est passée, mais voilà qui simplifie les choses. Ils l'embarquent.
Et l'emmènent à la grande prison centrale où l'on est en train de rassembler les manifestants arrêtés la veille au soir. Elle occupe tout un pâté d'édifices au bord du canal encerclant la grande place du Présidium, le long de la voie nord de l'avenue. Ils doivent attendre : il y a un embouteillage de gros fourgons amenant d'autres prisonniers d'autres postes de police, l'avenue autour de la place du Présidium est pratiquement bloquée. Les policiers, philosophes, attendent leur tour. Lian, rencogné seul sur la banquette arrière, menottes aux poings, voit sans vraiment les regarder la place et l'immense édifice du Présidium qui glissent à une allure d'escargot dans la fenêtre tandis que la voiture de police rampe vers sa destination. Ni dans l'avenue ni sur le canal on n'aperçoit trace des événements de la veille. Au milieu de la place quasiment déserte, le Présidium dresse sa façade nord, dorée, aveugle, impassible.
Elle tremble, pourtant, étrangement liquide, comme derrière un voile d'air chaud. Lian cligne des yeux, machinalement : il voit flotter une tache noire, comme s'il avait trop regardé le soleil. Et la tache ne s'efface pas, elle continue à flotter devant lui, dansant sur la façade du Présidium. Pas vraiment noire, d'ailleurs, ou alors par contraste avec l'éclat doré de la pierre et les réseaux étrangement brillants du sirid. Et si exactement circulaire... Il cligne encore des yeux, vaguement curieux à présent : la tache se dédouble. Ou plutôt, non, il y distingue des détails, un autre cercle plus petit, à l'intérieur. Ou une spirale. Peut-être seulement parce que les cercles bougent. Saisi d'un bref vertige, déconcerté, il ferme les yeux en appuyant sa nuque contre le dossier de la banquette.
Les cercles sont toujours là ! Il n'y en a pas deux mais dix, cent, mille, tous parfaitement distincts ! Et son regard sans paupières est attiré comme par un aimant au centre des cercles, là où ils semblent naître du néant, là où point... une lumière ? Immense, brûlante, et le temps ralentit, s'arrête, se contracte...
Explose, un grand jaillissement fixe, et Lian avec lui, retourné comme un gant, mince pellicule en expansion autour de la durée qui gonfle à la vitesse de l'éclair. Et pendant cette éternité, cette seconde, il voit, il entend, il goûte et sent et touche... Quoi, il l'ignore, c'est trop bref, mais il sait qu'en même temps il a été vu, entendu, goûté, senti, enveloppé d'une présence multiforme, immense, terrifiante.
Et l'instant d'après, il n'est plus un avec la surface infinie de la durée infinie, il a retrouvé son centre. Quelque chose a jailli en lui, une volonté – une négation ? – et il flotte à présent dans une bulle très petite, très souple et très imperméable. Il pourrait traverser sans effort cette membrane invisible – mais pourquoi le ferait-il alors qu'il risquerait de toucher à nouveau cette écrasante présence toute proche ? Juste au-delà de la membrane – surprise, curiosité – tourne une unique présence, le vieux petit homme de la gare, Dutch Grangier. Lian reste immobile sur la banquette arrière de la voiture de police tandis que le conducteur, devant, se frotte les tempes en grommelant : « Il va encore durer longtemps, cet embouteillage ? Ça me fout mal au crâne de me traîner comme ça. » Et la femme réplique : « Mais non, Ted, tu as faim, on n'a pas pu prendre le petit-déjeuner, avec le rigolo, là, derrière. Moi aussi, j'ai une sacrée migraine. »
45
Après cinq autres jours de marche, ils arrivent à un camp de forestiers, abandonné et à moitié englouti par la jungle, mais qui sert apparemment aussi de cache aux rebelles infiltrés en territoire fédéral : il y a là un gazillac tout-terrain et des vivres. La durée anticipée du voyage diminue soudain de façon radicale, pour le plus grand soulagement d'Alicia. Il faut toujours dégager la route de temps à autre, et plus largement maintenant, mais on arrive à faire en moyenne trois cents kilomètres par jour, en se réapprovisionnant en gaz dans les caches réparties le long de la route. C'est plus facile de ne pas penser dans les incessants cahots, accrochée au volant quand c'est son tour de conduire, ou lorsqu'on plonge à la machette dans la jungle dévoreuse de piste. Après sa brève discussion avec Bertran, première et dernière erreur, Alicia trouve extraordinairement facile de ne pas récidiver. Aux haltes, elle mange de façon mécanique puis s'endort d'un sommeil de brute.
Après trois jours – pas trop tôt, elle est couverte de bleus – ils arrivent à la passe Toïtovna et rejoignent une véritable route. Brièvement, car celle-ci est encore trop exposée : ils bifurquent presque aussitôt dans un autre réseau de pistes, mais de bien meilleure qualité et pour cause : cette région est plus peuplée. À vrai dire, ce sont de petits villages, parfois des hameaux, parfois quelques bourgades un peu plus grosses, logeant forestiers, mineurs et leurs familles : les compagnies forestières et minières qui exploitent cette partie-là des montagnes Rouges ne ferment pas pendant la saison de campagne, elles – on est à plus de cinq cents kilomètres à l'ouest de la Ligne.
Après la passe, on a quitté la forêt dense et oppressive et l'on descend de plateau en plateau dans un paysage plus dégagé. Alicia souffre pourtant d'une telle surdose de jungle qu'elle conduirait volontiers tout le temps : pendant qu'elle négocie la piste, elle oublierait presque ce qui l'entoure. Quelquefois, si elle se laissait aller, elle aurait la nausée. Elle donnerait n'importe quoi pour ne plus jamais voir un arbre de sa vie.
Vers la fin de la deuxième journée après la passe, son vœu se réalise presque : ils sortent de la forêt pour retrouver la route, devenue à deux voies – mais toujours les dalles jointives des Anciens, la paragathe encore en bon état – qui s'accroche aux flancs rocailleux du dernier haut plateau. Les arbres reculent, s'espacent, disparaissent même parfois pour laisser place à de simples buissons sur le terre-plein entre les deux voies. Enfin un paysage complètement différent ! Sur la route et dans les falaises, c'est la pierre écarlate qui domine – les montagnes Rouges justifient enfin leur nom. Et surtout un gigantesque panorama s'offre en contrebas ; pour un peu Alicia aurait le vertige. L'herbe neuve est jaune en cette saison, le début du printemps dans l'hémisphère sud, mais, la brume de la distance aidant, les Plaines Bleues sont raisonnablement bleuâtres.
Une fois sur la route, la durée recommence à s'écouler à des vitesses un peu plus humaines. On doit bien faire des pointes vertigineuses à cent à l'heure, dans les descentes. Est-ce la relative vitesse, la profondeur du paysage ? Le temps se déplie de nouveau pour Alicia, vers le futur du moins. C'est dit, à la prochaine halte elle utilisera le transmetteur pour se faire sortir de là.
La prochaine halte est une ville ancienne en ruine, rareté presque unique chez les premiers indigènes extraordinairement fidèles à leurs cités à travers le temps. Très respectueux de leurs quelques reliques – la plupart du temps des fragments, d'ailleurs, et non des villes entières – ils les entouraient habituellement de parcs. Ces ruines-ci sont d'une incroyable antiquité, les plus vieilles de la planète, plus de huit mille saisons. Il ne reste que la forteresse centrale, ou le palais, le cœur de la cité en tout cas, sur environ huit cents mètres de large et cinq cents mètres de long. Aucun parc, mais au début de la colonisation on voyait encore que les ruines avaient été entretenues avec soin : on y avait planté une variété d'herbe drue et courte, qui se ressème très longtemps toute seule, et fort inhospitalière à tout ce qui n'est pas elle-même ; des murs branlants avaient été consolidés, voire restaurés, les endroits trop dangereux condamnés... Plus de huit cent cinquante ans après la disparition des Anciens, la nature si infatigablement policée par eux a repris un peu de sa liberté : des arbres ont poussé dans des taches d'herbe affaiblie, des buissons, d'autres herbes, des plantes grimpantes se sont lancés à l'assaut des murs. Et comme l'image qu'en connaît Alicia date elle-même d'au moins quatre cents ans, elle ne reconnaît pas vraiment les lieux ; mais elle sait où elle se trouve.
Ils s'installent à la tombée de la nuit dans l'ancienne auberge toujours intacte, que les sims décrivaient comme l'auberge Vichenska – un grand édifice traditionnel encore abreuvé par son arbre-à-eau, dans le bassin ; variante du modèle habituel, un pylône se dresse aussi dans le bassin à travers les branches, petite boule dûment lumineuse piquée au sommet. Tout a été déménagé bien proprement au moment de la grande migration des rebelles vers le sud-est, plus de cent soixante-dix saisons plus tôt : on pourrait dire que le bâtiment est dans l'état exact où l'ont trouvé les premiers colons terriens... Personne d'autre n'est venu s'y installer : les Virginiens n'ont guère le loisir aujourd'hui, ni le désir, de se livrer au tourisme, surtout dans une région aussi proche de la Ligne. De toute façon, les trouvailles archéologiques faites sur le site au début de la colonisation sont toutes rassemblées depuis longtemps au Musée de Cristobal – armures, tapisseries, mobilier et objets funéraires, et surtout, bien sûr, le fameux sarcophage de cristal bleu, avec la momie intacte de son roi-guerrier. La petite Alicia a souvent contemplé la fresque qui décore les parois du grand puits circulaire au fond duquel on a découvert le sarcophage – en accélérant le sim, car l'énorme dalle de pierre qui bouche le puits et s'y enfonce en tournant met plus de quatre heures pour effectuer la descente complète, de la naissance à la mort du héros. Mais, là encore, Marti a veillé à refroidir sa curiosité.
Ils étendent leurs sacs de couchage dans une grande chambre donnant sur la terrasse. On est encore à plus de mille mètres et les nuits de printemps sont fraîches, surtout avec des fenêtres sans vitres : Bertran installe le brasero à gaz dans la chambre pour suppléer à la chaleur qui sourd des murs et du sol, lumineuse, depuis qu'ils sont entrés et surtout depuis qu'ils se sont déchaussés. Alicia se demande distraitement comment elle va réussir à dormir avec toute cette lumière ; les Anciens disposaient d'écrans pour profiter de la chaleur sans la lumière, mais eux n'en ont pas.
Quand ils ont fini de manger, Alicia étale devant elle les composants du transmetteur, en vérifie le bon état et commence à les monter.
« Il vaudrait mieux pas, dit Bertran. Anderson connaît l'algorithme de dispersion. »
Elle se fige.
« Et de toute façon, continue l'autre de sa voix posée, toutes les communications ont été interrompues avec Virginia depuis le sabotage. »
Alicia répète d'une voix blanche : « Sabotage ?
— On a saboté les ordinateurs de vol. L'insurrection a été arrêtée, mais Lagrange ne pourra pas repartir de L4 avant très, très longtemps, et seulement à condition d'avoir les données sur la propulsion Greshe. »
Elle regarde bouger les lèvres de Bertran, elle entend ce qu'il dit, mais elle ne comprend toujours pas. Elle répète « Insurrection ? » et proteste malgré elle : « Ils ne devaient pas saboter les ordi ! Seulement si... » Puis se force à se taire, mais Bertran conclut, à peine interrogateur : « Si on ne trouvait pas les données sur la propulsion Greshe. »
À quoi bon dissimuler encore ? Elle hoche la tête. Murmure d'une voix blanche : « Quand ?
— Deux jours après notre départ de Dalloway. Mais je ne l'ai appris que tout récemment. Dès votre disparition, Dalloway a prévenu Lagrange : vous avez été enlevée, peut-être tuée, par les rebelles, qui croyaient à tort les données sur la propulsion Greshe déjà en votre possession. Les Fédéraux ont assuré à Lagrange qu'ils étaient eux-mêmes sur une bonne piste et trouveraient bientôt les données. Grumberg a dû prendre contact avec vos amis – on a spéculé très tôt sur l'existence de factions désireuses de débarquer sur Virginia, et c'était son rôle de se trouver des alliés pour immobiliser Lagrange. »
Il ne va pas plus loin, c'est inutile : Alicia disparue, les données bientôt entre les mains des Fédéraux qui les fourniraient à Lagrange, c'était maintenant ou jamais, il fallait coincer la station et négocier le débarquement en échange de la remise en état des ordis de vol. Elle comprend très bien la logique de la situation. Mais Cédric, Moran, Fabienne – Eduardo ? Jamais ils n'auraient... « Pourquoi une insurrection ?
— Un dérapage, une bavure, est-ce que je sais... On a peut-être réagi plus durement que prévu au chantage. Ou Grumberg a poussé à la roue. Vos amis ne devaient pas être très expérimentés. »
Alicia accablée essaie de les imaginer en train de se battre, n'y arrive pas. Se battre pour quoi ? Elle se reprend à deux fois pour demander : « Il y a eu... des morts ? »
Bertran soupire : « Nous n'en savons rien. Les communications ont été interrompues juste après l'insurrection. Il faut supposer que oui. Le sabotage devait sûrement être réversible, n'est-ce pas ? »
C'était si simple. Si on en venait là, ils encryptaient les routines de contrôle des moteurs ioniques et ils échangeaient le code d'encryptage contre le débarquement. Puis elle comprend brusquement où va la question de Bertran, se fige, glacée. Si on a mis fin à l'insurrection, et si Lagrange est toujours coincée, c'est qu'on n'a pas obtenu le code d'encryptage. Moran, c'était elle qui... Si elle avait été prise, elle l'aurait donné, elle n'aurait pas résisté – aucun d'entre eux. Non, oh, non, ils se cachent encore dans les entrailles de Lagrange, l'insurrection est finie mais ils ne les ont pas tous capturés, Moran court encore, Moran ne peut pas être morte !
« Quand les communications seront rétablies », murmure Bertran, pensif, « et elles le seront quand les Fédéraux préviendront Lagrange qu'ils ont retrouvé les données, on découvrira que Grumberg était un agent double des rebelles, et avec la puissance de feu de Lagrange derrière eux, les Fédéraux pourront régler la guerre de façon définitive. »
Alicia a comme un éblouissement, et ensuite, pourquoi fait-il si noir ? Elle se sent vaciller, pose les mains par terre. « Mais ce ne sont pas... les rebelles.
— Non. C'est Graëme Anderson. »
Elle a beau ouvrir la bouche pour respirer, elle n'y arrive pas. Il lui semble qu'elle tombe, elle tend une main pour se rattraper, sent la main de Bertran la saisir, s'y accroche de toutes ses forces en fermant les yeux. Elle serre les dents en avalant sa salive, non, elle ne vomira pas.
Après un moment, la voix de Bertran reprend avec douceur : « C'est sans doute ce que cette femme voulait vous dire, mais elle n'en a pas eu le temps. Elle est allée au plus pressé : vous alerter, n'importe comment. »
Alicia secoue la tête, pas trop, elle a peur de se briser. Elle est complètement défaite, maintenant, elle n'a plus aucune défense contre les images qui se bousculent. Le visage convulsé de la rebelle qui crie...
« Mon nom, balbutie Alicia, elle savait... mon nom.
— Graëme le savait.
— Mais pourquoi... le lui aurait-il dit ?
— Il ne lui a pas dit. »
Les attaches sautent toutes seules, les quatre hommes s'effondrent comme des poupées de chiffon, mais pas Graëme, pas Graëme, et l'élan incroyablement rapide, incroyablement... immobile de la femme vers elle. Tout son corps, immobile, ses jambes, ses pieds. Ses pieds qui ne touchaient pas terre. Elle l'a bien vu. Graëme a profité du choc, ensuite, il a détourné son attention, une distraction, un truc d'illusionniste, mais ce n'était pas ça l'important pour lui – leur histoire à eux, ses mensonges ou non, ses machinations avec la propulsion Greshe – l'important, c'était cette femme qui ne touchait pas terre, et elle l'a bien vue, et il voulait le lui faire oublier et elle aussi elle voulait l'oublier, ça n'avait pas de sens, elle ne pouvait pas avoir vu...
Ils vous mentent. Ils vous mentent. Tandis que la voix de Bertran aligne des mots qu'elle comprendra plus tard, " mutation ", " télépathes ", " têtes-de-pierre ", une pensée paresseuse s'étire dans la tête d'Alicia, de plus en plus lente : ça ne peut pas être un mensonge, ils ne nous ont jamais rien dit. Ensuite, elle ne pense plus.
Quand elle se réveille, elle ne se rappelle rien du reste de la soirée. Elle ne s'est pas évanouie, quand même ? C'est la lumière du jour qui l'a réveillée, plus intense que la lueur de la pierre dorée. Elle se sent bizarrement bien, du moins sur le plan physique. Bertran n'est nulle part, si son sac de couchage est roulé bien proprement en face du sien. Sur le brasero, du café chaud, ou du moins un équivalent virginien. Trois barres de rations nutritives. Alicia est surprise de se sentir affamée, prend une des barres et commence à en peler l'enveloppe protectrice, puis se rappelle qu'elle est comestible et mord à pleines dents. Des oiseaux chantent sur la terrasse. Ciel bleu tendre dans la découpure de la fenêtre. Une belle journée de printemps.
Elle sort sur la terrasse. Les oiseaux se taisent et s'envolent, comme toujours – elle a pu le vérifier au Parc, dans les jardins à Morgorod, dans la forêt : même après quatre cents ans loin de la Terre, les humains de Lagrange portent encore la marque du péché originel pour la vie native de Virginia. Il a fallu une bonne demi-douzaine de générations pour que les stigmates terriens commencent à s'effacer chez les colons. Ce n'était pas ainsi, dans les sims. Licence poétique. Et puis, surtout, nombre des sims qu'elle préférait dataient de bien après la colonisation.
Toujours pas de Bertran. Alicia arrive au parapet qui borde la terrasse, écarte les branches luxuriantes d'un arbre-roi. Vue splendide sur la ville. Une silhouette entre les ruines, dans la grande allée qui aboutit au puits circulaire. Bertran joue les touristes. Ça ne colle pas au personnage. Un Énigmatique Agent Secret Rebelle ne fait pas du tourisme. Mais pourquoi pas ? Nous ne sommes pas nos étiquettes, comme disait quelqu'un. Non, trop tôt, bien trop tôt pour l'ironie, le souvenir la plie presque en deux, un coup de lance chauffée à blanc dans la poitrine, vite, penser à autre chose, Bertran, que fait Bertran dans les ruines ? Il ne bouge pas. A-t-il mis en marche le mécanisme du puits ? Est-il en train de regarder la massive dalle ronde, taillée dans un seul bloc de granit, qui s'enfonce en tournant, tournant, vers la mort du héros ? Peut-être temps pour la petite Alice d'aller voir un autre puits, n'importe quelle distraction, mettre un pied devant l'autre, bien appliqué sur le sol, le pied, et aller rejoindre Bertran.
Il doit l'entendre venir, elle marche si lourdement sur les dalles de l'allée. Mais comme s'ils continuaient une conversation, il dit sans se retourner : « La ville couvrait presque toute cette partie du plateau. Ils l'ont démantelée, vous savez, après la dernière bataille, pour en distribuer les pierres dans toutes les cités et les villages du continent. Ils n'ont laissé que le palais. »
Ce détail ne se trouve nulle part dans la fresque, ni dans les spéculations des archéologues qui ont examiné le site. Encore une de ces histoires que les Virginiens s'inventent pour remplir les trous de leur mémoire d'intrus. Qui l'aurait cru, Face-de-bois est un rêveur ! Alicia hausse les épaules : « Ils avaient tué un dieu, ça valait la peine d'être commémoré.
— Non », dit Bertran – il proteste, comme si elle n'avait pas parlé juste pour dire quelque chose. « Ils commémoraient son sacrifice. La victoire était celle de tous. Celle de la paix.
— Quand un dieu-guerrier meurt, ce n'est sûrement pas une victoire pour son camp. »
Bertran se retourne vers elle. Pour la première fois, dans la lumière sans obstacles, elle remarque ses yeux, d'une étonnante teinte ambrée. Il dit enfin, un ton plus bas – et elle se demande alors avec qui il discute, sûrement pas avec elle : « Mais ce n'est pas l'histoire d'un dieu-guerrier. C'est l'histoire d'un dieu qui aurait voulu pouvoir mourir, qui a aimé une mortelle et qui a été puni pour avoir enfreint les lois de l'univers. »
Elle aboie un rire bref, dont elle n'entend pas le son trop rauque : « C'est un jeu national, inventer des histoires d'amour à vos Anciens ? »
Il la dévisage un moment avec attention. « Aucune histoire d'amour n'est jamais inventée, il y a toujours quelqu'un pour les vivre », dit-il avec cette douceur bizarre qui lui donne envie de fuir – ce n'est pas, ce n'a jamais été le Bertran auquel elle s'est habituée à Dalloway mais elle ne veut pas y penser, pas maintenant. Puis il se redresse avec un soupir : « Prête à partir ? Alors, allons-y. »
46
« Alors, qu'est-ce que vous cherchez comme ça ? » dit en plaisantant le policier au volant du véhicule où se trouve Lian, quand les deux hommes en vêtements civils viennent se pencher à sa fenêtre après avoir examiné un par un chacun des fourgons de prisonniers qui le précèdent. Il sait qui ils sont, bien sûr, et il a bien vu que la circulation est maintenant complètement arrêtée, mais c'est un homme naturellement jovial. Le premier des deux agents montre brièvement un insigne accroché au revers de sa veste et dit, laconique, en désignant Lian du menton : « Lui. »
On ne dit pas " mais " deux fois à des agents fédéraux après le premier « Sécurité nationale ». Les policiers leur remettent Lian, avec un soupir ; ils vont devoir attendre que la circulation débloque pour retourner chez eux, maintenant.
Lian, toujours menotté, traverse la place entre les deux agents. Il fixe les dalles qui défilent à ses pieds : il ne veut pas risquer de voir la façade dorée qui se rapproche. Enfin, une longue voûte sonore les avale. Ils ne s'engagent pas dans la grande cour centrale, bifurquent tout de suite sous l'arcade à colonnade. Autour d'eux passent des visiteurs ou des employés de l'État, affairés, soigneusement indifférents. Lian flotte, calme, lointain, en état de choc. Il lui semble n'avoir jamais perçu avec autant d'acuité, comme si sa bulle protectrice faisait office d'amplificateur : la voix amusée de cette femme, là-haut, accoudée au parapet de la première terrasse, cet oiseau qui déplie une aile sur la branche la plus basse de l'immense arbre-à-eau, au ras du bassin, les reflets miroitants sous lesquels jouent les caliches rouges et noires, la brise acidulée qui a dû passer sur des fleurs de miralilas, le silence massif des deux hommes qui l'encadrent de leur impersonnelle hostilité... C'est son attention qui concentre ces perceptions, il s'en rend compte après un moment, comme si sa simple volonté allait chaque fois découper un morceau de réalité pour l'amener plus près de ses yeux, de ses oreilles, de ses narines – ou comme s'il se trouvait soudain à coïncider, d'une manière intense et immédiate, avec ce qu'il regarde, entend ou goûte. Mais il ne s'étonne pas, ne s'interroge pas. Il est en mode uniquement réceptif.
Il pénètre avec ses gardiens dans un long corridor dallé de mosaïques anciennes ; les murs sont décorés de fresques continues, une congrégation hétéroclite d'animaux terrestres faisant mine de se cacher dans une luxuriante forêt, à gauche, à droite une assemblée non moins improbable d'animaux aquatiques dissimulés dans une forêt sous-marine, habitants de l'eau salée comme de l'eau douce. Le mythe d'Amléandaï, sans doute, quand l'héroïne fondatrice des Hébao s'en va chercher les créatures reprises par Hananai en punition de la première chasse, et qu'elle leur pose des énigmes pour les attirer hors de leurs cachettes, jusqu'à la dernière énigme, qui fait sortir Hananai elle-même, curieuse. Pas grand monde dans ce corridor ; ceux qui croisent Lian et ses gardiens détournent les yeux ; ils ne regardent pas non plus les fresques.
Enfin une porte à double battant, qui se referme sur Lian sans que ses gardiens soient entrés derrière lui. Dans une salle de travail aux murs nus, sans fenêtre, occupée tout entière par une grande table noire en fer à cheval, à l'éclat lustré. Des chaises pivotent vers Lian, tournant vers lui des visages figés, un homme, trois femmes dont une très vieille ; un autre homme au fond, très vieux aussi, dans un plus grand fauteuil ; les autres ont tous environ la cinquantaine, mais ils ont l'air plus âgés, des visages durs, desséchés, terrifiés. Et Grayson qui se lève à demi et retombe dans son siège comme si on lui avait cassé les genoux.
Lian les observe tour à tour avec une sérénité un peu distante, il sait qui ils sont sans savoir comment, ces membres du Conseil restreint du Présidium. Et Grayson, qui le regarde comme s'il était le soleil, aveuglé. Lian attend. Il attend leurs questions. Mais la première n'en est pas une, malgré la tournure interrogative. C'est un refus scandalisé, presque une supplication aussi, chuchotée : « Mais qu'est-ce que c'est que ça ? »
La plus vieille des femmes, la plus proche de Lian ; elle n'a pas de relief, on dirait une feuille de papier chiffonnée ; ses mains remuent sur la table, par saccades réprimées, comme si elle voulait agripper, griffer, déchirer. L'autre femme, près d'elle, hausse les épaules : « Un Rèb, Mélanie. » Celle-là est plus en chair, avec un meilleur contraste, peau très noire dans un ensemble de soie vert mousse, courte paille de cheveux blancs ou d'un blond très clair, impossible à dire sous la lumière artificielle des panneaux à gaz.
« Jamais vu de Rèb comme ça, marmonne la très vieille Mélanie.
— Un faux bloqué », laisse tomber la femme la plus proche de Lian, avec la patience ennuyée de qui explique pour la centième fois. C'est la plus jeune des trois ; elle, elle porte des lunettes complètement noires.
« Plus de bloqués comme ça, s'entête la très vieille Mélanie.
— Ce n'est plus un bloqué, s'exclame Grayson. Vous l'avez bien senti tout à l'heure !
—... sens pas grand-chose maintenant », grommelle le vieillard du fond, éteint par sa veste d'un blanc éclatant.
« On le couvre, et ce n'est pas moi !
— Ah, tu ne vas pas recommencer avec tes histoires à dormir debout », dit sèchement l'homme aux verres teintés. Il se lève, vient se planter devant Lian. « Il se couvre lui-même, c'est tout, une nouvelle variété de protection. C'est un Rèb, qui faisait semblant d'être bloqué et qui s'est trahi. N'est-ce pas, mon garçon ? »
La question est purement rhétorique et Lian ne répondrait pas, même s'il savait de quoi l'autre veut parler.
« Mais bon sang ! s'écrie Grayson avec une rage incrédule, je l'avais bien perdu avant et maintenant on l'a retrouvé, et sans son traceur, non ?
— Je ne sais pas ce que j'ai senti tout à l'heure », remarque le vieux au fond de la salle, obstiné. « Ça ne ressemblait à rien. »
Les autres acquiescent.
« Là n'est pas la question..., reprend la femme aux lunettes noires.
— C'est toute la question ! » proteste Grayson. La femme lui jette un regard fulgurant et il se raidit en se mordant la lèvre inférieure.
« La question, reprend-elle d'une voix dangereuse, c'est : pourquoi les Rèbs ont-ils modifié ainsi son code génétique, et pourquoi nous l'ont-ils envoyé. Une vague caricature d'Ancien !
— Éliminons-le tout de suite », dit la femme aux cheveux blond-blanc, toujours du même ton ennuyé. « Je ne vois pas pourquoi on en fait toute une histoire. Gray s'est bien amusé dans notre dos, on l'a pris sur le fait pour la deuxième fois, é finita la commedia, et revenons aux choses sérieuses.
— Non, reprend la femme aux lunettes noires. Ce n'est quand même pas une coïncidence si la manifestation d'hier a si mal tourné. Je dis qu'ils préparent quelque chose, quelque chose de gros, et que ce garçon est un test.
— La manifestation a mal tourné parce qu'on a de plus en plus de mal à contrôler et qu'il faudrait arrêter d'utiliser ce genre de truc pour crever les abcès, intervient de nouveau Grayson exaspéré. Et il ne peut pas être un rebelle. Je l'ai trouvé par hasard ! S'il n'avait pas publié ce bouquin, je ne l'aurais jamais remarqué ! Il a vécu deux saisons dans cette commune sans jamais rien faire, et ils ne savaient absolument rien, vous avez bien vu ! Et moi j'ai vécu deux saisons avec lui, je sais quand même bien que c'était un bloqué ! Et qu'il ne l'est plus. La question que vous devriez vous poser, c'est ce qu'il est maintenant, ce qu'il pourrait faire pour nous. C'est à peine si on le perçoit, là, même moi...
— Il ne s'agit pas de toi ni de tes puériles manigances, coupe la femme avec une irritation dédaigneuse. Ce n'est pas pour nous dire ce que nous devons faire que tu es là. » Elle se retourne vers Lian et lui adresse un sourire carnassier : « Crois-tu pouvoir nous résister longtemps, petit rebelle ? Aux six meilleurs téleps de toute Virginia ? Si tu parles maintenant, les dégâts seront bien moindres, tu le sais, n'est-ce pas ? »
Lian l'observe, fasciné, sans répondre : parce qu'elle a dissimulé ses yeux, elle se croit inscrutable ; ne sait-elle pas qu'elle a laissé tout le reste de son visage à découvert, à l'abandon, et qu'on peut y lire sa terreur ?
« Non ! » s'exclame Grayson, qui se lève en renversant son fauteuil.
Les autres ne bougent pas, ne le regardent même pas. Ils ne regardent pas Lian non plus, d'ailleurs. Mais sa bulle protectrice résiste souplement à leur assaut. Le leur renvoie en miroir. Ils ne bougent toujours pas, mais leur expression a changé : ils sont sous le choc.
Lian, alors, très posément, avec douceur, leur explique. Qui il est, ce qu'il est, d'où il vient, comment il est venu, et ce qu'il a vu sur la façade dorée de leur Présidium.
Et ils ne l'entendent pas. Ne veulent pas l'entendre ! Ils se ratatinent dans leur fauteuil en croisant les bras, pour s'éloigner davantage, se protéger davantage, et leur stupeur horrifiée ne procède pas de ses révélations mais de la facilité avec laquelle, l'instant d'avant, il a résisté à leurs pouvoirs conjugués. Même Grayson ne le croit pas. Son visage a pris une expression consternée, et il murmure : « Oh, Liam, si ce sont les Rèbs qui t'ont fait ça, s'ils t'ont fait croire ça, nous pouvons t'aider, je te le jure, je peux t'aider ! »
Lui aussi, comme les autres, il a peur, il s'est enfermé derrière une barrière rigide et opaque, lui non plus il ne veut pas reconnaître la vérité.
Soudain agacé par cette obstination dans la folie, Lian traverse sa bulle comme si elle n'avait pas existé – sent, lointaine, la stupeur incrédule du vieux Dutch Grangier qui se trouve dans la cour du Présidium, assis au bord du bassin, faisant mine de nourrir les caliches – et, avec une gravité sévère, il va les toucher, directement : la très vieille Mélanie Roster, la Noire aux cheveux de paille, Toni Elikéra, Tara la femme aux lunettes révélatrices, la mère de Grayson – et le père de Grayson, Brice, derrière ses verres teintés, et le vieux Grégoire Arnussen encore hébété au fond de la salle. Il les regarde, et ils se voient. Tout ce qu'ils ont dit, tout ce qu'ils ont tu, leurs mensonges, leur carapace, volent en éclats. Pour découvrir leurs vieilles, vieilles terreurs, pâles, nues et frissonnantes. Il les regarde – et ils le voient, parce que sa bulle est totalement transparente à présent, ils ne peuvent plus ne pas le voir, nu comme eux, mais sans colère, sans peur, triste et calme. Ils ne peuvent plus ne pas le croire. Ils n'ont plus le choix.
L'idée le traverse, hésitante, qu'il est peut-être cruel. Il se retire, mais ils demeurent pétrifiés, les yeux fixes, et il n'ose les toucher à nouveau pour les sortir de leur catatonie. Il se tourne vers Grayson, qui recule, les mains levées, en balbutiant : « Que... qu'est-ce que... », tandis que son regard affolé passe d'une silhouette immobile à une autre.
Et maintenant qu'ils sont tranquilles tous les deux, Lian le touche aussi, attristé, un peu honteux, avec prudence, presque avec bonté. Il ne le forcera pas. Il se laissera simplement couler en lui. Et le technicien lui tend la bague avec un sourire satisfait : « Particules émettrices dans la colle qui aide à tenir le cabochon. Trois kilomètres de portée, ça devrait suffire ? » Il sourit en retour. Le technicien le regarde soudain d'un air perplexe : « Je peux faire quelque chose pour toi, mon gars ? » « Me suis trompé de porte, excusez-moi. »
Et la colère sourde, le chagrin, « Mes parents me destinent à la politique », et il me regarde, l'innocent, je le piste depuis des semaines, j'ai fait tirer mon propre numéro à la conscription pour être sûr de ne pas le perdre, j'ai dépensé des trésors d'ingéniosité pour dissimuler toutes mes manœuvres à mes parents, ils seraient fiers de moi, si ça se trouve, tant d'efficacité procédurière ! Mais cet étrange garçon blond n'en sait rien, rien – et moi je ne sais rien de ce qu'il pense, de ce qu'il ressent, simplement qu'il a l'air sincère dans sa tristesse quand il dit « Tu penses que tu n'as pas le choix. »
Et la désolation opaque, le vertige de l'alcool, pas assez, l'alcool, la certitude qui tue sous la fausse comète messagère de catastrophe, c'est la catastrophe, je ne peux plus, je ne peux plus, mais je dois continuer, je n'ai pas le choix, et il ne comprend pas, et je ne pourrai jamais lui dire la vérité maintenant, mais oh, touche-moi, Liam, oui, touche-moi, maintenant, pardonne-moi même si tu ne sais rien et si tu comprends de travers, tu as très bien compris, embrasse-moi, oui, juste toi et moi, Liam, juste toi et moi.
Et Oh, cette peau, si incroyablement lisse et douce, ce corps tellement... fermé, on le caresse et on ne sent rien, rien que la peau, la douceur, la tiédeur, les muscles qui tressaillent, les os devinés, et il me caresse, il m'embrasse, seigneur que sa peau est douce, partout, comme une fille, et lui non plus ne sent rien, rien d'autre, pas de bouillie d'émotions, pas d'échos, pas de barrière non plus à garder tout le temps, tout le temps, rien, juste l'autre, complètement autre, quel miracle.
Et Stupeur, incrédulité, fureur, fureur, non, ce n'est pas possible, pas maintenant ! Trent... oh, les salauds, les salauds !... Couche-toi, Liam, couche-toi, ne bouge pas, bon Dieu, ne bouge pas ! Horreur, désespoir, fureur, oh seigneur, ils les ont tous... tais-toi, Liam, tais-toi, tu ne peux rien faire pour eux, on ne peut rien faire, il faut que je fasse quelque chose... Et le revolver de Ritchie, voilà, armé, oui, pas trembler, bouge pas, Liam, bouge pas, pardonne-moi, c'est la seule façon, yeux bleu gris, stupéfaits, bouche ouverte, non, bouge pas, non... mais trop tard, l'explosion, le choc dans les poignets, les oreilles sourdes, horreur, désespoir, non, NON, Liam, NON, fallait pas bouger, je l'ai tué, NON ! Chape de glace, tripes nouées, l'autre est tout près, pas le temps, se lever, bien visible, juste au cas où, mais non bien sûr, savent que je suis là, les salauds, les salauds ! « C'était le dernier, je crois », vas-y, retourne-toi, va vers tes copains, mon salaud, Ritchie, où as-tu mis ton fusil, imbécile ! Et la fureur satisfaite, la nausée de haine comblée quand les trois premières silhouettes vertes s'écroulent, et tu crois que tu vas me tirer dessus, toi ? Tu ne peux même pas bouger, regarde-moi arriver, oui, yeux fous, bouche muette, à bout portant, oui mon salaud, Liam, oh, Liam !
Et des fleurs, il peint des fleurs, qu'est-ce que j'ai fait, plus jamais ensemble, non, non, ce n'est pas possible, je dois aller le trouver, personne ne se rendrait compte, je pourrais lui expliquer, il comprendrait, je suis sûr, je lui ferais comprendre, ce n'est pas ma faute, je n'avais plus le choix, Liam, je ne pouvais plus faire marche arrière, mais j'ai essayé, je te le jure, j'ai essayé de tous les protéger, Trent, pissant dans son froc, la terreur de Dieu, j'étais tellement sûr... Mais il a prévenu mon père, le salaud, il ne s'en tirera pas à si bon compte, je te le jure, je l'aurai au tournant, quand il ne se méfiera pas ! Je ne suis pas n'importe qui, merde ! Et eux aussi, ils vont l'apprendre, ça a assez duré tout ça, le fils à maman à papa, je les emmerde tous royalement, je les conchie, tous ces vieux débris ! Quand tu auras débloqué, je t'expliquerai et tu comprendras, tout, tu comprendras, oh qu'il est maigre, mais comment ils le nourrissent dans cet hôpital, plus que la peau sur les os, sa peau, si douce, sa peau, oh, Liam, pardonne-moi, je n'avais pas le choix, c'était pour toi, tu comprends, pour toi. Je ne pouvais rien dire à personne, Dieu sait ce qu'ils t'auraient fait. Et c'est moi qui t'avais trouvé, pas eux ! On leur montrera, Liam, tu verras, toi et moi, Liam, juste toi et moi.
Grayson a trébuché dans son fauteuil renversé. Il continue à reculer par terre, sur les fesses et sur les mains, vers le coin de la salle. Lian veut s'accroupir pour le rassurer, mais l'autre lève les bras pour se protéger la figure. Lian s'immobilise, la poitrine brûlante. Il a du mal à respirer, il est descendu trop profond, bien trop profond dans la nuit lourde où se débat Grayson, où il s'enfonce. Une petite lumière est encore là, pourtant, il peut la voir qui ricoche affolée entre les parois d'ombre, – mais comment pourrait-il l'arrêter, quelle paix lui offrir, quelle vérité qui soit une paix, il n'en connaît aucune, rien que la blessure, toujours, et la perte, et la mort.
Et alors, parce qu'il n'a rien d'autre à donner, Lian donne à Grayson, pêle-mêle, le visage convulsé de l'homme dont il ne saura jamais le nom, la veille, dans la ruelle, et les visages vivants de leurs compagnons de la Section, parce qu'il ne les a jamais vraiment vus morts mais c'est pareil, et le jeune gremlin inconnu aussi dont le crâne explose en geyser rouge, et les aspirants hékel massacrés il n'a pas pu l'empêcher, et Thomas que son silence a envoyé se trancher la gorge, et le visage de Laraï « Oh, Lian, nous ne serons jamais ensemble dans la Mer ! » et Dougall qu'il n'a pas su retenir au bord de sa violence, et tout au fond, la première angoisse, la silhouette noire du Gomphal écroulé. Un déluge de souffrance, de vies brisées, de questions sans réponses, de réponses qui tuent, et lui pourtant, toujours vivant, il ne sait pas pourquoi mais il est toujours vivant, est-ce que ça veut dire quelque chose, qu'il soit encore vivant, y a-t-il à cette question-là une réponse qui ne tue pas ? Il l'ignore, et tout ce qu'il peut donner, à la fin, la dernière offrande, c'est le petit Gomphal qui escalade son enclos de roches pour remplir peut-être sa destinée, absurde et obstiné, c'est ainsi, camarade, c'est ainsi.
47
Ils descendent à travers les dernières collines, puis obliquent vers le sud et commencent à traverser les Plaines Bleues. Bertran conduit toute la matinée. Après cinq heures de route, Alicia est déjà presque aussi écœurée de l'austérité monotone de la plaine qu'elle l'a été de la monotone luxuriance de la forêt. De l'herbe, de l'herbe, à perte de vue, parfois aussi haute que le tout-terrain, une ondulation jaune verdâtre infinie où surnage ici et là la boule blanche vaguement lumineuse d'un plumetier. Les montagnes sont bientôt trop loin à l'est pour distraire le regard. De temps à autre, – rarement, cette partie des Plaines Bleues est l'un des endroits les plus déserts du continent – l'herbe s'abaisse, il y a des cultures, des villages, on les voit à peine, le mur d'herbe revient encadrer la piste, ou la route, c'est selon ; mais piste ou route, elles sont en bon état, pas beaucoup de cahots non plus pour venir décrocher Alicia de ses pensées. Elle essaie de dormir, elle ne peut pas. Yeux ouverts ou yeux fermés, c'est la même chose : son cerveau en roue libre, zigzaguant comme ivre d'une horreur à l'autre, lui rejouant tout ce qui s'est passé depuis qu'elle a débarqué, mais l'autre version, la vraie version. Graëme. Rien, elle ne peut rien croire de tout ce qu'il lui a dit. Il a toujours su qui elle était. Il a toujours su qu'elle ne repartirait jamais. Un pion de plus dans son jeu. Elle a une conscience aiguë de l'air dans ses cheveux, du frottement de ses vêtements contre ses bras, ses cuisses, ses seins, quand elle touche sa propre peau nue par accident elle tressaille, elle imagine, elle l'imagine, lui, en train de la toucher. " Je suis curieux. " Oui, pervers, contre-nature. Ses caresses interminables, comme on caresse un animal. Un télépathe. Et elle, sourde, aveugle, muette, tête-de-pierre. Comment pouvait-il seulement la toucher ? Elle a envie de hurler, comment a-t-elle pu être aussi stupide... Mais comment aurait-elle pu ne pas l'être ? Le secret le mieux gardé de tout Virginia. L'union sacrée des mutants, Fédéraux et rebelles, tous aussi menteurs les uns que les autres, depuis toujours, depuis qu'ils se mènent cette guerre à éclipses, depuis des siècles, presque depuis le début de la colonisation ! Et maintenant qu'elle le sait, ce n'est pas seulement sa petite histoire à elle qui prend un autre sens, c'est toute l'histoire de Virginia, l'Indépendance, la Libération, les Terriens éternelles marionnettes ignorantes manipulées par les uns, par les autres, par les uns contre les autres, et maintenant aussi. Et maintenant qu'elle sait...
Ils ne la laisseront jamais retourner sur Lagrange.
Elle regarde les immuables murs d'herbe avec un dégoût mêlé d'épouvante, puis une froide ironie vient presque la calmer : ce n'est pas comme si Lagrange allait l'accueillir à bras ouverts, non plus. Il faudrait passer sur le corps de Marti Coralàn. Si les gamins ont parlé... Et puis de nouveau, l'horreur brûlante, Moran ! Non, ne pas penser à Lagrange, rester ici. Sur le plancher des vaches. Où elle est coincée pour toujours, maintenant qu'elle sait le secret des Fédéraux et des rebelles – toi qui aimes connaître l'envers des décors, tu es servie, tu fais maintenant partie de l'élite, des rares élus, hein, Alicia ? Qui ne vont sûrement pas non plus te laisser claironner la vérité sur les toits virginiens.
« Qu'allez-vous faire de moi ?
— Je vous mets à l'abri », dit Bertran comme si cela allait de soi.
« Après.
— Il faut d'abord voir comment la situation va se développer.
— Ils n'ont pas rétabli les communications ?
— Non. »
Ils doivent encore être en train d'essayer de débrouiller les ordinateurs de vol, par pure obstination. Mais ils finiront bien par négocier. Et à ce moment-là... Avantage politique, technologique et militaire garanti une fois pour toutes aux Fédéraux. Sauf si une petite voix vient révéler la vérité. Pas tellement sur leurs machinations, mais sur leur nature. À tous. Étonnant qu'elle soit encore vivante, de fait. Mais ce n'est pas comme si on allait la croire, non plus. Même si elle a la propulsion Greshe. Elle observe à la dérobée le profil calme de son compagnon. Le sait-il ? Elle essaie de se rejouer leur conversation de la veille, elle n'est pas sûre. Si elle communique avec Lagrange, les Fédéraux la repéreront et la reprendront tout de suite. C'est ce qu'il a dit. Mais pourquoi le croirait-elle, lui ? Parce qu'il l'a aidée à sortir de Dalloway ? Il avait l'intention de le faire de toute façon. Parce qu'elle est toujours vivante alors qu'il sait qu'elle sait ? Ils ont besoin d'elle pour négocier éventuellement avec Lagrange. Pas pour sa libération, bien sûr, mais pour la propulsion Greshe. Ah, mais s'il ignore qu'elle l'a ? Et c'est une question de minutage : si les Fédéraux reprennent contact avant elle avec Lagrange, plus de négociations possibles. À la limite, il devrait vouloir qu'elle les contacte maintenant, repérage par les Fédéraux ou pas, un risque à courir. Car elle pourra bien utiliser le transmetteur depuis la Licornia, elle sera chez les rebelles, compromise – à plus d'un titre – non fiable, et détentrice de données inutiles, parce que sûrement déjà en la possession de Lagrange par le biais des Fédéraux.
Elle répète plus durement : « Qu'allez-vous faire de moi ? Si je ne peux pas parler à Lagrange maintenant, je ne vous suis plus d'aucune utilité et j'en sais trop. »
Il lui jette un rapide coup d'œil : « Vous avez vraiment avalé tout ce qu'ils vous ont dit des Sécessionnistes, n'est-ce pas ?
— Personne ne nous a dit que vous étiez tous des télépathes !
— Vraiment pas tous. Mais je comprends votre argument. Pensez-vous cependant que, même s'ils vous croyaient, cela changerait quoi que ce soit à la situation de Lagrange ? Les vôtres sont immobilisés mais ils sont loin, à l'abri de n'importe quelle attaque. Une fois en possession des données, ils peuvent travailler tout seuls dans leur coin à se doter de la propulsion Greshe. Ça leur prendrait plus de temps qu'avec la collaboration des Fédéraux, c'est tout. Non, les capacités particulières des Virginiens n'entreraient pas dans leur équation. Ils pourraient très bien choisir de travailler avec les Fédéraux même en sachant ce qu'ils sont : la télépathie ne fonctionne pas avec les Lagrangiens. Ils vous auraient peut-être envoyée même s'ils l'avaient su ! Ce n'est pas parce qu'il est télépathe que Grumberg a convaincu vos amis. »
Brusquement renvoyée à sa culpabilité, Alicia se raidit, agressive : « Mais combien de Virginiens savent la vérité, du côté des Fédéraux ? »
Bertran a un petit sourire : « De plus en plus. Et tout le monde en Licornia. Pensez-vous qu'ils essaient de le cacher ? Les Fédéraux essaient. Non, Alicia, croyez-moi, on ne va pas vous manger toute crue. »
Elle se rencogne contre la portière, absurdement irritée : il vient de lui dire qu'elle est en sécurité ; mais il vient surtout de lui dire qu'elle ne peut rien faire, n'a plus rien à faire, qu'elle est... superflue, surnuméraire, un dommage collatéral. Que dis-tu de ton Héroïque et Dangereuse Mission maintenant, Alicia ? De tes ridicules petites machinations ? Dangereuses, oui, mais pas pour toi. Le visage de Moran passe dans sa mémoire, leurs visages à tous, leurs yeux brillants, leur excitation, leur confiance. Moran sûrement, mais les autres, y en a-t-il d'autres, des morts, oh, pas Eduardo, pas Eduardo ! Elle se replie sur elle-même en retenant un gémissement.
Elle n'a pas dû le retenir assez bien : « Vous ne pouviez pas savoir », dit Bertran.
Elle se redresse, les bras férocement croisés sur la poitrine, mortifiée : « Ça ne change rien.
— Ça devrait modifier la façon dont vous évaluez vos actions. »
Elle hausse les épaules avec violence. Ses actions ! Quelles actions ? Sa seule action, c'est quand, à cause de sa guerre secrète contre Marti Coralàn, elle a écouté les gamins et s'est persuadée que leur plan idiot avait des chances de fonctionner ! Une fois sur Virginia, elle n'a jamais rien fait qui n'ait été manipulé de bout en bout, que ce soit le scénario de Lagrange ou celui de Graëme.
Inexact, remarque la voix sarcastique en elle : il n'avait sûrement pas prévu que tu irais le surprendre dans son antre.
Parce qu'elle n'était pas censée pouvoir le surprendre. Elle portait le collier, n'est-ce pas, quand elle n'était pas avec lui ? Et à Dalloway elle ne pouvait aller nulle part sans Bertran, non plus.
Bertran. Qui a défait le collier. Qui lui a ouvert les portes interdites.
Elle se tourne vers lui, soudain saisie d'une rage brûlante : « Et c'était quoi, votre scénario à vous ? »
Il comprend vite : « J'attendais que vous ayez les données.
— Vous saviez... » Elle a presque dit " que je les avais ? ", se retient de justesse au cas très, très hypothétique où elle disposerait encore de cette ombre d'avantage – à quelles fins elle n'en a pas la moindre idée, mais elle contrôle ce qu'elle peut. « Vous saviez qu'il interrogeait cette femme ? demande-t-elle plutôt.
— Je savais qu'il restait une survivante. »
Et combien d'autres morts sur ta conscience, Alicia ? Ah mais non, non, elle ne prendra pas ceux-là, elle n'est quand même pas coupable de ce que les rebelles ont choisi de faire !
« Je n'aurais pas pu y aller sans vous. »
Il ne répond pas tout de suite : « Non, dit-il enfin. J'ai pensé... que ce serait plus convaincant de vous faire voir. Me simplifierait le travail ensuite. » Il soupire : « J'aurais dû me douter que les rebelles en enverraient au moins un de la même force que Graëme. »
Quand elle n'est plus aussi suffoquée, elle remarque : « Mais en fin de compte tout a tourné pour le mieux, ça vous a bel et bien simplifié le travail, non ? »
La voiture s'arrête brusquement. « Alicia, dit Bertran d'une voix un peu enrouée, je regrette que ce soit arrivé ainsi. Je voudrais que rien de tout cela n'ait été nécessaire. Mais on ne peut défaire ce qui a été fait. Seulement essayer d'en trouver la juste perspective. Il y a longtemps que j'ai cessé de croire en mon innocence. Peut-être est-il temps pour vous d'en faire autant. »
Elle le dévisage, livide de fureur : « Je n'ai jamais cru être innocente !
— Alors cessez de chercher qui blâmer ! »
Et elle s'entend crier « Je n'ai rien fait ! », mais le regard de Bertran sur elle ne change pas, et elle sait qu'elle ment, elle voudrait pouvoir se dire que c'est la faute de Marti, que tout, depuis le début, a toujours été la faute de Marti, mais elle sait bien que ce n'est pas vrai, que c'est sa faute à elle aussi, ses choix, ses décisions, depuis longtemps, sa faute.
Elle ouvre la portière et saute sur la piste. Se sent stupide tout de suite, elle ne va pas plonger dans l'herbe ! Elle n'a nulle part où aller. La phrase prend soudain toutes ses dimensions, une planète entière et nulle part où aller. Elle entend le moteur du tout-terrain s'arrêter, les pas de Bertran qui contourne le véhicule et arrive derrière elle. Il ne la touche pas, heureusement. Il murmure : « Il ne s'agit pas de blâmer ou de se blâmer, Alicia. Il faut comprendre. Apprendre. Et aller plus loin.
— Nulle part ! » C'est tout ce qu'elle peut dire, elle explose en sanglots. Alors seulement il la prend dans ses bras, quand elle n'a plus la force de lui résister, et il la berce un peu en murmurant : « On arrive toujours quelque part, Alicia. »
Cette nuit-là, elle fait une crise de manque aigu – venue elle ne sait d'où, une souffrance torturante, incompréhensible, odieuse. Graëme. Elle ferme les yeux, elle se mord les lèvres, elle se grifferait pour cesser de penser à lui, mais ce ne sont pas des pensées, c'est son corps qui se rappelle et son corps ne pense pas : il a enregistré Graëme dans chacune de ses cellules et il le lui rejoue. Ses mains, sa bouche, sa langue sur elle, sa peau contre la sienne, son rythme, son souffle, ses gémissements, son odeur, son goût. Des fragments d'images, un regard, un sourire, une échappée d'épaule, la ligne d'une cuisse, jamais au complet, comme s'il la tentait, comme s'il la narguait. Elle se ramasse en boule, d'abord, puis se retourne avec brusquerie sur le dos en écartant les bras, elle ne veut pas se toucher, ce n'est pas supportable, elle chiffonne le sac de couchage dans ses poings, presque tétanisée de rage, de désir. Finalement elle n'y tient plus et sort de la tente à quatre pattes.
Bertran est assis près du feu rougeoyant, il prend toujours le premier tour de garde. Elle se laisse tomber près de lui, en murmurant : « Peux pas dormir. » Il ne dit rien, tisonne un peu le feu qui lance quelques étincelles. La nuit n'est pas sombre. La lueur de la lune, à l'est, diffuse dans le ciel voilé. Pas de point lumineux à chercher dans l'orbe immense du ciel, on ne distingue aucune étoile. Alicia a mal à la poitrine comme si elle avait couru. Elle se balance un peu d'avant en arrière tout en observant Bertran à la dérobée. Puis se laisse aller contre son épaule avec un soupir. Lui passe un bras autour de la poitrine. Il a tourné la tête vers elle. Elle caresse la ligne lisse de sa joue, ses lèvres.
Il lui immobilise la main et dit avec douceur : « Non, Alicia.
— Pourquoi ? Vous n'aimez pas les animaux, vous ? »
Il murmure : « Oh, Alicia ! Ce n'est pas ça du tout. Ce n'était pas ça du tout pour lui. Et vous n'effacerez pas Graëme ainsi.
— Je croyais que ça ne marchait pas avec nous, la télépathie ! »
Il la dévisage avec une consternation angoissée : « Ça ne marche pas. Mais ce n'est pas nécessaire. Ça ne dispense pas d'essayer de comprendre. J'ai vécu autrefois avec quelqu'un à qui je ne pouvais pas tout dire. » Il se tait, reprend un ton plus bas : « Il vous a sûrement confié des secrets qu'il osait à peine s'avouer. »
Un retour de Graëme la plie soudain en deux, " Je ne peux pas savoir ce qui te blesse, Alice. " Elle referme ses mains dans l'herbe, renverse la tête en arrière, convulsée de désespoir furieux et crie à elle ne sait qui, Bertran ou Graëme : « Ça m'est égal ! Plus jamais personne comme ça, plus jamais ! Je préfère crever ! »
Le feu crachote à travers les herbes. La voix de Bertran dit soudain, lointaine, presque amusée : « Les humains disent si facilement jamais. Graëme aussi l'a dit, vous savez. Il y a eu quelqu'un pour Graëme, comme ça, et il a dit " plus jamais ". Et il vous a rencontrée. »
Elle nie, d'une voix qui se brise dans les aigus : « Il ne m'a jamais aimée ! »
Bertran penche un peu la tête de côté en faisant : « Ah... » Il la prend avec fermeté par les épaules et l'attire vers lui avec un sourire triste et indulgent : « Ce serait plus facile ainsi, n'est-ce pas ? Mais ce n'est pas vrai. »
Son dernier éclat l'a comme épuisée. Elle se laisse aller contre lui en fermant les yeux, les bras repliés contre la poitrine. Et elle s'endort.
48
Lian laisse Grayson couché dans le coin, recroquevillé sur lui-même la tête entre les coudes, les genoux sous le menton. On enterrait les morts ainsi, il y a très, très longtemps. Peut-être Grayson ne renaîtra-t-il jamais non plus. Mais il n'y a plus rien à faire pour lui, c'est à lui d'essayer d'être. Flottant dans une compassion navrée, mais détachée, Lian sort de la salle de travail. On essaie de l'arrêter, puis on le laisse passer : il regarde, il trouve son image, il l'efface, on s'écarte, on l'a oublié, on ne l'a jamais vu. Le vieux Grangier est là quelque part à la périphérie de la bulle, silencieux, attentif, mais il ne fait rien. Les gens qui passent dans le couloir aux fresques ne voient pas non plus la transparence tremblante devant laquelle Lian s'arrête, un peu intrigué, parce que la voix de Dutch Grangier lui dit, lointaine : Descends par là. Un escalier aux parois bleutées l'accueille une fois le mur traversé, des marches qui s'enfoncent en tournant dans une luminescence bleutée presque familière. En bas, un espace d'abord obscur, mais Déchausse-toi, dit le vieux. Lian obéit encore, conciliant, et bien sûr, au bout d'un moment, la lueur dorée de la tellaod éclôt autour de sa chaleur vivante et gagne de proche en proche les parois, dessinant confusément des salles, puis des couloirs et encore des salles, toutes désertes.
Dans une salle que rien ne distingue des autres, ouverte sur un couloir tout aussi identique, Grangier dit Attends-moi là, j'arrive. Lian s'immobilise un instant puis il se remet en marche, il ne sait pourquoi. La lente marée de la lumière dorée lui fait signe et il la suit, il descend avec elle, ou bien y a-t-il autre chose qui l'attire dans les profondeurs des souterrains, les profondeurs, où le sang du monde coule encore pour un amour qui ne finit jamais ? Il descend. Les pierres des parois sont plus grosses, d'un travail plus fruste, les plaques de sirid plus nombreuses, plus épaisses et moins bien polies aussi ; quand il les effleure au passage, elles lui renvoient une sensation opaque et massive : il passe à travers les racines du Haëkelliaõ.
Le calme ici est si vaste, si vaste le silence, que Lian doute par moments de sa propre existence. La voix de Grangier ne lui parvient plus, qui avait protesté au début, inquiète, puis irritée puis suppliante. Il descend. Si loin sous l'écorce du monde, la lumière de la tellaod, puisée au soleil, pénètre encore. Elle illumine à présent une petite salle ronde au plafond en coupole, aux parois incurvées. Pas de décoration sinon, dans les parois et jusqu'à l'apex de la coupole, de complexes filigranes de sirid dont la lumière réveille peu à peu l'éclat argenté. Lian suit la vague lumineuse jusqu'au centre de la salle, et là, sans savoir pourquoi, il s'arrête et la laisse continuer sans lui. Il regarde les dalles nues à ses pieds, pensif, étrangement pacifié, les yeux perdus dans la lueur de la tellaod. Cligne des yeux en apercevant un point minuscule d'un éclat différent, presque une couleur. Mais déjà c'est une ligne, qui frise en arabesque, dont chaque volute se déploie en d'autres arabesques, se gonflant peu à peu en un motif d'un bleu intense et pourtant impalpable. Dans le motif, des myriades de points s'allument, qui sont des arabesques en mouvement vers d'autres points, et tous les points sont le centre, et tous les centres forment le même cercle, et les dalles tremblent comme de l'eau, comme la façade du Présidium. Un grand vent se lève dans la tête de Lian, une rumeur, des millions de voix bourdonnantes, mais à distance, comme prudentes, elles tournent autour de lui. Quelques-unes se détachent et viennent à lui, plus claires, plus disciplinées, interrogatives – l'Esprit de la Mer, le cercle toujours renouvelé des baïstoï qui sert de conduit, de porte-parole, d'intermédiaire. Mais Lian les écarte sans même y penser, il va vers la lumière, il reconnaît la lumière qui point derrière eux, à travers eux, tout autour, nulle part ailleurs qu'en lui-même, la présence, immense, attentive, curieuse, intense, de plus en plus intense, trop intense – il lève un bras pour se protéger les yeux, instinctivement, mais sans paupières pour les couvrir ses autres yeux deviennent braises, sa peau fond comme cire, sa chair se carbonise, ses os s'envolent en poudre, il brûle.
49
Deux jours plus tard, au début de la deuxième semaine de Décembre, ils rencontrent le cirque.
Alicia voit d'abord seulement qu'ils doublent une caravane et trois camions poussifs et non identifiés, – ils n'ont pas rencontré un seul véhicule depuis le début de la matinée. Il y a en tête du convoi une petite camionnette d'un bleu poussiéreux qu'ils vont sans doute doubler aussi, mais non, Bertran s'arrête, fait signe, et les autres s'arrêtent aussi. Bertran ne dit pas à Alicia de rester dans le tout-terrain, mais il ne l'invite pas non plus à venir. Dans le doute, elle préfère s'abstenir. Les conducteurs de la camionnette et des trois camions descendent pour venir à la rencontre de Bertran, une douzaine de personnes en tout, des hommes et des femmes, apparemment jeunes et apparemment amicaux.
Après un bref conciliabule, Bertran revient vers le tout-terrain : « On décharge les affaires. On va ranger le tout-terrain dans le dernier camion. »
Alicia obtempère : « On ne va plus en Licornia ?
— Jamais dit qu'on allait en Licornia. Dit que je vous mettais à l'abri. »
Alicia pose le dernier sac au bord de la route, en se forçant à rester impassible ; elle n'est plus très sûre de comprendre la stratégie des rebelles.
« Vous n'avez plus peur que les Fédéraux nous rattrapent ?
— Ils ne vous cherchent plus. Leur thèse officielle est maintenant que vous avez été assassinée. Ils essaient toujours de communiquer avec Lagrange. Et ils cherchent toujours la propulsion Greshe. »
Alicia a un peu plus de mal à ne pas réagir. Bertran n'est pas dupe, car il complète : « J'ai réencrypté les données avant de partir, un autre code, ça va les occuper pendant plusieurs semaines. »
Alicia se mord les lèvres. Il savait que les données étaient là ! Elle demande plutôt : « Ils n'en avaient pas de copies ?
— Ils ont découvert qu'elles ont été effacées, comme l'original. »
Alicia reste un instant interdite, hésitant entre la stupeur et la fureur. Elle se contient, demande d'une voix à peu près calme : « Pourquoi ne les aviez-vous pas prises ?
— Elles n'étaient pas là avant que vous arriviez avec Anderson. Et nous ne pouvons pas contacter Lagrange. Nous avons besoin de vous et de votre transmetteur. » Il fait une petite pause : « Et puis, ç'aurait été dommage de vous être donné tant de mal pour rien. »
Alicia le dévisage avec un scandale incrédule : il se moque d'elle ? Mais elle a déjà admis que tous ses efforts avaient été absurdes, un peu plus un peu moins, elle n'en est plus à ça près. Elle ne demande pas pourquoi il n'a pas simplement effacé aussi les données de Dalloway : les Fédéraux essaieraient avec bien plus d'acharnement de savoir si elle est vivante ou non et de la retrouver s'ils pensaient qu'elle en détient la seule copie. Elle empoigne son sac à dos : « Alors, on fait quoi, maintenant ?
— On voyage avec le cirque, et on attend. »
Elle enregistre sans commentaire que le convoi est un cirque. « On attend quoi ? Qu'ils récupèrent les données ?
— Même s'ils les avaient, ça ne changerait rien : tout le monde est coincé pour le moment. Dalloway appelle, mais ils ne savent même pas si on reçoit leur message. Pas moyen de savoir si c'est volontaire ou non, si le système de communication de Lagrange a été endommagé aussi à la suite de l'insurrection. On attend que Lagrange rétablisse de nouveau le contact. Tant qu'à révéler où vous êtes à Anderson, autant que ce soit en communiquant avec les vôtres quand vous le pourrez réellement.
— Et Lagrange me croira un peu plus si je ne suis pas encore en Licornia.
— Peut-être. Mais, surtout, la Mer revient à la fin de Décembre, dans cinq semaines. Elle égalisera un peu les chances. Espérons qu'ils ne passeront pas trop vite à travers mon encryptage. »
Le cirque, puisque c'est un cirque, ne dispose pas d'un personnel bien nombreux : onze personnes en tout, six gars, cinq filles. Ils savent qui elle est sans que Bertran leur dise plus que son vrai nom, lui serrent tous la main sans réticence, avec enthousiasme même, presque avec respect. Maintenant qu'elle est au courant, Alicia peut mieux apprécier : ces gamins sont de bien meilleurs acteurs que les Fédéraux avec l'intruse tête-de-pierre. Elle enregistre les noms – des prénoms, on n'offre rien d'autre : Karl, Marco, Michelle, Sélim, Antonia, Kat, " pour Katerine ", Tran et Chen, des jumeaux qui se présentent d'une seule voix, puis précisent et se mettent à rire, Djira, Benny, Samanta... Plus grands qu'elle – comme tous les Virginiens, mais elle est habituée – plutôt jolis dans l'ensemble, et surtout minces, musclés, énergiques. Et jeunes. Alicia dissimule sa surprise : deux seulement doivent avoir vingt saisons, Karl et Michelle, et la plus jeune, Samanta, semble en avoir à peine quatorze, ce qui équivaudrait à peu près à seize ans de Lagrange. Difficile de dire qui est le chef – les plus âgés sans doute, mais ce n'est pas évident. Des rebelles infiltrés, en tout cas. Les Fédéraux contrôlent moins bien le continent qu'ils ne le prétendent, remarque Alicia en se dirigeant avec Bertran vers la camionnette bleue. « Ils se concentrent sur les grands centres et la Ligne, précise celui-ci, surtout pendant la saison de campagne. »
Et c'est vrai que cette partie des Plaines Bleues ne rassemble pas un maximum de population. Le cirque semble pourtant ne pas vouloir en rater le moindre petit hameau. Quand Bertran déploie la carte pour lui montrer leur itinéraire, son doigt fait plus de zigzags qu'une fourmi hyperactive. Ils roulent encore pendant deux heures et arrivent à destination à Kamtcha, un presque village – treize édifices anciens disposés en rond autour d'un bouquet d'arbres-à-eau. Ils se garent à proximité, Karl et Michelle descendent de la camionnette pour aller rencontrer les responsables du village. Alicia aurait cru qu'on essaierait de la cacher mais non. Bertran lui tient la portière ouverte : « Venez, il va falloir monter la tente. »
Il veut dire le chapiteau. Un petit, qui prend seulement trois heures à monter, mais c'est vraiment la préhistoire du cirque, Alicia n'a vu de telles installations que dans des documents d'archives. Les rebelles sont-ils si retardataires ? La toile est en synthétique, quand même, comme les câbles, et les deux mâts principaux en fibre de carbone, mais le mode d'emploi est garanti archaïque : on monte les mâts, on plante les piquets, on tire les câbles et on tend la bâche à force de bras. Ensuite, on sort des camions les bancs des gradins et on les assemble autour de la piste, à la main. Ils sont en bois franc, léger et d'une teinte bizarrement violette qui ne doit pas être celle du vernis, mais c'est du vrai bois.
Et tout ce mal pour une soixantaine de personnes à tout casser, enfants compris, qui se rassemblent peu à peu autour de la tente à mesure que s'approche l'heure de la méridienne. Une fois les plates-formes des trapèzes et les autres agrès installés, les forains vont manger un morceau et se reposer dans leur caravane ; on a installé Bertran et Alicia dans la camionnette.
Au début de l'après-midi, Alicia, qui n'a rien d'autre à faire, va regarder la répétition, mais elle s'en lasse vite. Ils sont gracieux, rapides et agiles, mais là aussi c'est l'enfance de l'art : jongleries diverses, monocycle et danse sur une corde raide, pyramide d'équilibristes... Le numéro de trapèze est ordinaire, comme la séance de prestidigitation – même pas un lapin ou une colombe ; qu'est-ce qu'un cirque sans animaux ? Tran et Chen ont un numéro de clowns en miroir assez amusant, si on veut.
Et puis, à vingt-huit heures trente, quand tout le monde a fini de souper, c'est le spectacle. Un peu plus magique tout de même : musique, costumes, paillettes scintillant dans les lumières, la nuit tout autour... Et l'attente des villageois, qui se sont habillés pour l'occasion, les adultes comme les enfants, visages déjà illuminés de plaisir. Ne doivent pas avoir beaucoup de distractions au fond de leur brousse.
Bertran s'initie avec Sélim au programme de la console son et lumière, en face de l'entrée des artistes. Alicia s'est installée dans les coulisses ; on lui a montré quoi faire, elle ne va pas manifester de la mauvaise volonté : elle est capable de passer des accessoires aussi bien que n'importe qui. C'est même d'une absurdité comique, pour changer. Le contraste. Elle regarde les spectateurs s'installer dans les gradins et elle se dit que pas un d'entre eux ne sait que des ex-Terriens tournent au-dessus de leur tête, ou qu'à Dalloway les Fédéraux s'arrachent les cheveux pour désencrypter l'enjeu de toute l'affaire qu'ils ignorent également ; connaissent-ils même le nom de la propulsion Greshe ? Des avenues, des rues et des canaux portent ce nom dans plusieurs villes, mais savent-ils encore de quoi il s'agit ? Se rappellent-ils leurs lointains ancêtres amenés sur Virginia en une fraction du temps qu'il a fallu à Lagrange pour ramper jusqu'ici ?
Mais ils ne savent rien, ces braves gens. Et moins encore l'essentiel, ce que sont les Fédéraux, et ce que sont les fringants jeunes gens en costumes chamarrés en train de virevolter sur la piste. On se demande ce qu'ils font là, d'ailleurs – Alicia pourrait imaginer des couvertures plus discrètes pour des rebelles circulant à l'arrière des lignes ennemies. Ou bien ce cirque n'était-il destiné qu'à la récupérer, elle, depuis le début ? Un peu bizarre quand même. Et qu'auraient-ils fait, alors, si en s'enfuyant de Dalloway elle était partie vers l'est et non vers l'ouest ? Ah, mais Bertran aurait été en train de l'attendre de toute façon. Et puis, " tactique élémentaire ", comme il disait : plus logique de s'enfuir par le lac, moins de périmètre mortel à traverser...
Elle ne peut retenir un sourire ironique : sur la piste, les quatre jongleurs finissent leur numéro ; l'effet de dislocation entre le souvenir de Dalloway et les objets hétéroclites qui volent entre leurs mains est trop appuyé. Ils devaient se retenir pendant la répétition, elle doit s'avouer que c'est assez impressionnant à présent : balles, quilles, assiettes, chapeaux circulent en une navette incessante, au rythme de plus en plus rapide de la musique d'accompagnement, jaillissent dans les airs, retombent presque par miracle dans les mains tendues avec négligence pour les récupérer, repartent... et filent soudain en direction des coulisses, où les accessoiristes les récupèrent – Alicia s'est laissé surprendre. La foule applaudit à tout rompre tandis que les jongleurs saluent puis s'éclipsent en bondissant, remplacés par les équilibristes.
Là encore, Alicia doit admettre que la répétition ne leur rendait pas justice. Est-ce mimétisme involontaire avec l'assistance suspendue à chacun de leurs mouvements, muette, les yeux écarquillés, ou bien a-t-elle envie de retomber elle-même en enfance ? La finale du numéro de danse de Djira sur la corde raide, en particulier, lui laisse le souffle aussi court qu'aux autres spectateurs : double, puis triple saut périlleux, hélice, saut carpé, avec une lenteur étonnante, toujours au bord du décrochement, toujours récupérés avec une impossible grâce. Alicia n'en est pas encore remise que les trapézistes escaladent déjà les agrès, aussi souples et vifs que s'ils étaient à l'horizontale – ils ne grimpent pas, ils coulent vers le haut – et, tandis que la musique change de tonalité et de rythme, ils commencent à se balancer tout en s'enroulant autour de barres et de cordes comme autant de rutilants et souples serpents.
Alicia se sent étreinte d'un vague malaise et pourtant elle est conquise. Elle fait " oh ", et " ah ", et applaudit en même temps que les autres spectateurs, la bouche un peu entrouverte, le cœur dans la gorge. Dix fois elle pense, mon Dieu, ce n'est pas possible, elle va tomber, il va tomber – les trois trapézistes officient à quinze mètres de haut, et sans filet. Mais dix fois les mains trouvent les barres, les chevilles, la taille, les poignets prévus à l'endroit prévu. Avec une nonchalante élégance les corps élastiques s'étirent, se replient, font des vrilles : ils se promènent, ils nagent, ils volent, le vide est leur élément naturel.
Le cœur battant, plus essoufflée qu'eux, Alicia leur tient le rideau lorsqu'ils reviennent dans les coulisses et ne peut s'empêcher de leur crier : « Bravo », comme la foule debout, presque plus incrédule qu'admirative, dans les gradins. Samanta lui adresse un sourire radieux au passage.
Il n'y a pas d'entracte. Tran et Chen viennent cabrioler sur la piste pendant qu'on installe devant les coulisses l'estrade où va avoir lieu le spectacle de magie. Leur numéro aussi est plus amusant lorsqu'ils sont en costume mais, après l'incroyable apothéose des trapézistes, finir par la séance de prestidigitation semble un choix un peu curieux.
Et de fait, même si le rythme est plus rapide qu'à la répétition et si costumes et musique dramatisent les gestes, Alicia ne trouve toujours pas les premiers tours particulièrement impressionnants. Bouquets, plumes, foulards, bougies allumées surgissent avec obéissance des mains de Michelle ou des oreilles du souriant Benny son assistant. On déchire le papier de soie et on le tire reconstitué de la bouteille si ostensiblement vide, après quoi on se verse à boire et on lève son verre à la santé du public. Les tours de cartes suscitent les murmures attendus du public bon public ; le dernier, il faut l'admettre, est spectaculaire : les cartes semblent douées d'une vie propre tandis qu'elles s'étirent impossiblement entre les mains écartées de la magicienne puis s'arrondissent en arc-de-cercle tandis qu'elle les bat – Michelle était l'une des jongleuses, à vrai dire.
Pendant tout le début du numéro, Benny a fait mine d'être toujours un peu maladroit, ou distrait, et Michelle a feint de le tolérer avec de plus en plus d'agacement. Comme il rattrape de justesse – mais sans en rater une, comment fait-il ? – les cartes qu'elle lui envoie d'un revers de main négligent, par brassées, elle se fâche. Elle tape du pied sur la scène, tend vers lui un doigt accusateur, et il décolle, l'air ahuri, pour se retrouver à plat sur le dos à un mètre du sol.
Il se couche sur le côté, accoudé, la joue sur la main, en faisant semblant de prendre son mal en patience, mais Michelle tape encore du pied, dessine un cercle de l'index : Benny se met à tourner, toujours dans la même position détendue, puis feint de protester, gesticule, je veux descendre, mais Michelle inverse simplement le mouvement. Benny se roule en boule d'un air piteux, mais il ne fait que tourner plus vite, d'un bout à l'autre de la scène, comme un toton, toujours suspendu dans le vide.
Les spectateurs quittent la tente à regret après avoir applaudi tous les artistes revenus saluer sur la piste. Ensuite on démonte le système son et lumière et on désassemble tout de suite les gradins ; la tente en dernier, le lendemain quand on repartira, tôt à l'aube. Des conversations s'entrecroisent dehors : la foule ne s'est pas entièrement dispersée, on discute avec animation du spectacle. Quelques personnes offrent un coup de main, les forains acceptent volontiers : plus tôt on aura fini, plus vite on pourra aller manger et se reposer.
Alicia aide à ranger les accessoires. Elle sait maintenant d'où venait le malaise croissant qui a accompagné pour elle la majeure partie du spectacle. Les voltes impossiblement lentes de Djira. Les rattrapages impossibles des trapézistes. Impossibles. Benny non plus ne portait aucun équipement spécial. Ils ne lui ont pas montré la fin de la séance de prestidigitation, lors de la répétition, elle se demande pourquoi, ce n'est pas comme si elle n'était pas au courant, elle. Elle se demande surtout pourquoi ils prennent ce risque stupide. Pour faire la nique aux Fédéraux, ou aux ignorants qu'ils tiennent sous leur coupe ?
Lorsqu'il ne reste plus que le chapiteau, ils se retrouvent dans la caravane pour souper, épuisés mais satisfaits, plusieurs portant encore des traces de maquillages. Ils échangent des commentaires, plaisanteries, félicitations, conseils – à haute voix ; Alicia aurait pensé qu'entre télépathes... Mais Bertran n'a-t-il pas dit que les rebelles n'en étaient pas tous ? Et puis ils sont polis, de toute évidence, ces jeunes gens.
En plein milieu du souper, on frappe à la porte de la caravane. Autour de la table, on échange des sourires dont le sens échappe à Alicia. Samanta se lève et fait entrer deux personnes d'âge moyen, une femme et un homme, à l'air embarrassé mais résolu. Alicia les observe avec un certain amusement : quoi, ils veulent se sauver pour se joindre au cirque ? On les assied à la table, on leur offre à boire, on fait les présentations. Ce sont des éleveurs. Térésa Dillion fabrique du fromage pour le village, Tom Dillion entretient un petit troupeau de cabals. Ils sont nés ici, oui, ils ont des enfants et des petits-enfants, ah oui, les petits ont adoré le spectacle, nous aussi d'ailleurs... Là, ils se taisent, sirotent leur verre sans regarder personne.
« Est-ce que vous êtes des rebelles ? » demande Térésa Dillion de but en blanc, et Alicia manque s'étrangler, mais pas autant que lorsque Michelle répond « Oui » sans broncher.
Les deux visiteurs hochent la tête ; ils n'ont pas l'air plus terrorisé qu'il ne faut ; apparemment, ce n'est pas le but réel de leur visite. « J'ai toujours cru... que c'étaient des bobards, dit le fermier. Mon grand-père me racontait des histoires de l'Insurrection que son père lui avait racontées, on avait vu des rebelles faire des trucs bizarres...
— On se disait que ça devait être de la propagande rebelle, vous savez, continue la femme. Pour faire peur aux troupes, à l'époque. Mais ce que vous avez fait ce soir... Les autres ont beau dire... Le truc de la fin, là, avec le gars qui tourne en l'air, vous ne pouvez quand même pas faire ça avec des miroirs, hein ? Et puis, les trapézistes... Et elle, là... » Elle indique Djira du menton. « Non.
— Et que disent les autres gens du village ? demande Marco.
— Ils trouvent que vous êtes bien entraînés », dit Térésa Dillion avec un petit sourire gêné.
— Nous sommes très bien entraînés », remarque Antonia.
Térésa et Tom Dillion échangent un regard, puis secouent la tête à l'unisson, l'air buté. « Pas à ce point-là, dit la femme. Mais si vous êtes des rebelles, ils ne veulent pas le savoir. Et pour le reste... encore moins. Ils ont la trouille.
— Pas vous », commente Karl.
La femme hausse les épaules : « Un cirque ! » dit-elle, une évidence.
« Et puis, ce n'est pas comme si la sorcellerie existait », ajoute l'homme d'un ton définitif.
La bouteille de vin se soulève toute seule et vient remplir son verre devant lui.
« Si on n'est pas des sorciers, alors, dit Kat, on est quoi ? »
L'homme avale sa salive, prend son verre – en se forçant quand même un peu – boit et repose le verre sur la table. « Je ne sais pas, mais sûrement pas des sorciers. »
Une vague de sourires approbateurs passe autour de la table, et les deux visiteurs se détendent un peu plus. Mais la femme regarde tour à tour les jeunes gens d'un air encore obstiné : « Les histoires qu'on a entendues disaient aussi... » Elle se racle la gorge. «... que les rebelles pouvaient vous forcer à faire des trucs que vous ne vouliez pas faire, parce qu'ils, euh, ils pouvaient entrer dans les esprits. »
Elle hausse un peu une épaule, elle veut seulement en avoir le cœur net, elle espère être rassurée. Mais Karl incline simplement la tête : « Nous pouvons entrer dans les esprits, si on nous y invite. Mais nous ne forçons personne à faire ce que nous voulons. »
Tom Dillion ébauche un sourire : « Ça, c'est une blague, dans les esprits, personne ne peut... »
Tout le monde secoue la tête autour de la table. Sous le feu croisé de tous ces regards graves, les Dillion se recroquevillent un peu. Samanta, qui est la plus proche de la porte, va l'ouvrir.
« Vous pouvez partir quand vous voulez », dit Karl.
L'homme semble bien prêt à le prendre au mot, mais la femme se carre dans sa chaise en croisant les bras. « Montrez-moi. »
Presque aussitôt elle se redresse : « Eh ! » Elle contemple Karl, les yeux écarquillés, ouvre et referme la bouche à plusieurs reprises. Son mari s'est tourné vers elle, inquiet, mais elle dit : « Montrez-lui !
— Voulez-vous qu'on vous montre, Ser Dillion ? » demande Michelle.
Il hésite, mais sa femme hoche la tête, le regard brillant, et il dit « Oui. » Dans la fraction de seconde suivante, son visage prend une expression incrédule. « C'est... c'est moi, ça ?
— C'est comme ça que je vous vois, en tout cas », dit Michelle avec bonne humeur, puis grave à nouveau : « Vous êtes un homme courageux, Ser Dillion. »
Après cela, il y a un grand silence. Térésa Dillion demande « Je pourrais avoir un verre d'eau ? » d'une voix un peu enrouée. On la sert, et tout le monde attend. Y compris Alicia. Elle a voulu chuchoter une question à Bertran, mais il a posé un doigt sur ses lèvres et elle s'est tue, agacée. Ils savent ce qu'ils attendent, eux, de toute évidence.
Térésa Dillion se racle encore une fois la gorge. « Ce n'est pas juste vous, c'est tous les rebelles ?
— Pas tous, mais beaucoup, dit Antonia.
— Et vous demandez toujours... la permission ?
— Nous, oui, dit Sélim. Sauf en cas de vie et de mort. » Et, un ton plus bas, avec tristesse : « C'est quand même la guerre. »
Tom Dillion a froncé les sourcils : « Il y en a qui ne demandent pas la permission ? »
Tous les jeunes gens hochent la tête en silence autour de la table.
« Les Fédéraux, Tom, dit soudain la femme. Je t'ai toujours dit que je n'aimais pas ce type de New Sonora, quand il vient en tournée. Les séances de réflexion avec lui, là, au village... »
Tom Dillion hausse les épaules, mais elle insiste : « Réfléchis cinq minutes, Tom ! Si les rebelles peuvent faire ce genre de choses, tu crois qu'ils seraient encore derrière la Ligne après tout ce temps ? Il doit y en avoir de l'autre bord, chez les Fédéraux, pour leur faire pièce. Réfléchis, Tom ! »
Alicia les observe, fascinée malgré elle. Tom n'a pas trop l'air d'avoir envie de réfléchir davantage ; il commence à prendre conscience de tout ce qu'il vient d'entendre et il arrive à sa limite. Il marmonne : « Infériorité militaire et technologique...
— Ha ! » fait sa femme – la découverte l'excite plutôt, elle. « S'ils peuvent faire voler un bonhomme, tu crois qu'ils ne peuvent pas faire dévier des missiles ? Toutes ces histoires qui courent sur la Ligne, c'est pas juste des histoires !
— On ne peut pas trop... dit Tran.
—... utiliser les dons sur la Ligne, poursuit Chen.
—... les Fédéraux ne veulent pas que ça se sache...
—... et ils suppriment les témoins, conclut Chen.
— Quoi ? » dit Térésa Dillion horrifiée.
Son mari se redresse, alarmé : « Eh, mais on le sait, nous, maintenant ! »
Alicia commencerait presque à s'amuser. Ils ne peuvent pas les dénoncer, après ce qu'on vient de leur dire des témoins. Bien joué. Et ensuite ? On recrute ?
« Il y a une façon simple de vous protéger, dit Michelle. Comme nous nous protégeons nous-mêmes quand c'est nécessaire. Nous plaçons une sorte de... labyrinthe dans notre esprit, et les questions se perdent dedans. Si on insiste trop, ça devient une barrière, et les questions... rebondissent dessus. »
Les deux Dillion digèrent l'information. La femme réagit la première : « Mais vous, ça peut marcher parce que vous êtes des... euh...
— Télépathes, offre Marco. Non, ça marche pour n'importe qui. On peut vous montrer, c'est très simple, et une fois que c'est en place, ça reste. »
Térésa Dillion continue à réfléchir. « Il faut que vous... entriez dans notre esprit pour le faire.
— Oui, dit Karl.
— Quelle garantie... », commence Tom Dillion d'un ton rogue.
« Aucune, dit Michelle avec douceur. Seulement la confiance.
— Nous, on vous a fait confiance, renchérit Samanta. On vous a montré. On n'était pas obligés. »
Et Michelle : « Vous m'avez vue aussi, n'est-ce pas, Ser Dillion, quand je vous ai touché ?
Tom Dillion ne dit rien, mais sa femme murmure : « C'est vrai », en regardant Karl. Elle se redresse : « Je ne vais pas... devenir, euh, télépathe ?
— Pas du tout », dit Karl en souriant.
Elle médite encore un moment, puis prend sa décision : « Allez-y. D'accord. Faites-le.
— Térésa... dit son mari.
— Vous pouvez encore réfléchir, remarque Michelle. Nous ne partons que demain matin après huit heures. Prenez votre temps.
— Tom, tu fais ce que tu veux, mais moi, c'est maintenant. Si je peux être protégée, je ne vais pas rater l'occasion. Et tu sens bien qu'ils disent la vérité.
— Les Fédéraux aussi peuvent avoir l'air sincère !
— Les Fédéraux ne nous ont jamais dit qu'il y avait des gens comme eux !
— Ça ne fait peut-être pas longtemps !
— Oh, Tom, tu es de mauvaise foi. Rappelle-toi les histoires de ton grand-père.
— Pourquoi ils ne nous l'ont pas dit plus tôt, eux ? »
Cette question-là arrête Térésa Dillion en plein élan. Elle se tourne vers les autres : « C'est vrai... pourquoi ? »
Karl hoche la tête : « Nous avions peur aussi. Nous avons eu peur très longtemps. Les Fédéraux ont encore peur.
— Mais pas nous, dit Djira avec un sourire lumineux. Nous, nous n'aurons plus jamais peur. »
Alicia ne peut pas attendre d'être revenue dans la camionnette avec Bertran. « Quel cirque ! s'exclame-t-elle. Quelles foutaises ! C'est ça que vous faites, raconter des histoires aux gens pour les endormir ?
— Pas des foutaises. Avec la barrière-miroir, Térésa Dillion est désormais à l'abri de l'influence des Fédéraux. Tom le sera aussi, s'il finit par se décider à accepter.
— Mais si ces gens ne sont pas des télépathes, rien ne peut...
— Ça prend sur n'importe quel Virginien. Les mutations ont plus ou moins touché tout le monde, rappelez-vous. Les modifications que le contact a apportées à certains fonctionnements neurochimiques du cerveau de Térésa Dillion inhiberont désormais tout autre effort de modification de ce type induit par un contact télépathique. Ce n'est pas transmissible de parents à enfants, cependant. Il faut choisir à chaque fois. »
Alicia s'arrête pour dévisager Bertran ; il est très sérieux. « Vous êtes en train de me dire que la télépathie est de la télékinésie, et que les rebelles ont trouvé un moyen de fabriquer des têtes-de-pierre.
— Pas exactement. C'est un cas un peu particulier, ici. Et la nature de votre barrière est différente, comme celle d'ailleurs des gens qu'on a appelés têtes-de-pierre ici après la Libération. Mais l'effet, à toutes fins pratiques, est similaire.
— Sauf si des rebelles veulent jeter un petit coup d'œil, bien entendu.
Bertran secoue la tête en reprenant son chemin, et elle le rattrape : « Eux non plus ne peuvent pas défaire la barrière-miroir, dit-il avec patience. Seuls les immunisés eux-mêmes peuvent passer au travers s'ils le désirent, mais ils doivent apprendre comment.
— Ah oui, les Rèbs vont se priver eux-mêmes de leur arme la plus puissante !
— Ce n'est pas une arme, Alicia », soupire Bertran.
Ils grimpent dans la camionnette et déroulent leurs sacs de couchage. Il ne fait pas trop chaud, heureusement : on a laissé les portières grandes ouvertes, et la nuit a rafraîchi l'habitacle.
« Dans ce cas, pourquoi pas pendant le spectacle ? Ils se montrent pour ce qu'ils sont, ils expliquent, et hop, barrière-miroir pour tout le monde. »
Le sourire de Bertran est-il amusé ou attristé ? « Ils ont essayé de dire la vérité lors de l'Insurrection, et ça s'est retourné contre eux. Vous dites, je crois, " il n'est pire sourd que celui qui ne veut pas entendre " ? Ou qui ne peut pas. Et puis, ce n'est pas ainsi que procèdent les enfants d'Iptit. »
Alicia a reconnu le nom et elle a tressailli, mais le grand écart est trop déconcertant : « Le petit dieu des petites choses ? Quel rapport ? »
Bertran sourit : « Le gardien du hasard ou du chaos. C'est un mouvement qui s'est développé chez les Sécessionnistes depuis une vingtaine de saisons. Ils croient... en la force accumulée des petits changements. En l'inattendu, aussi. Comme l'arrivée de Lagrange, par exemple, un joker que personne n'attendait dans le jeu... » Il se retourne et fouille dans son sac pour en tirer un livre épais, en format de poche, visiblement lu souvent. « Essayez ça. Vous comprendrez mieux. Peut-être. »
50
Le livre s'intitule L'Autre Rivage, par un nommé Nathan Leray, publié dans une maison d'édition de Licornia nommée Choix, première édition Léonovgrad, Hiver 149, une dizaine ensuite ; aucune information biographique, pas de description du contenu, pas même l'étiquette minimale " roman " ou " biographie " ou " essai ". Alicia commence à le lire dans la camionnette le lendemain matin, doit arrêter parce qu'elle a mal au cœur, reprend à la halte de mi-matinée ; puis ils arrivent dans un autre village et il faut aider à installer le chapiteau. Elle grignote une partie de sa méridienne – elle sait ce qu'elle lit, à présent, une fiction, et ma foi, c'est assez bien tourné, un style élémentaire mais efficace. Est-ce le livre où Graëme avait pris ses histoires d'Anciens ? Elles s'y trouvent, en particulier celle d'Iptit au Chapeau vert, avec Khaliad et Arani. Alicia est tentée de la lire en diagonale, se force à ne pas le faire : si les gamins du cirque se considèrent comme " les enfants d'Iptit ", c'est sûrement important. Mais c'est la même histoire, à quelques détails près. Presque supportable – elle s'attendait à être submergée de souvenirs, mais non. Elle termine pendant l'après-midi, la deuxième partie est nettement plus difficile à avaler que la première, même pour de la fiction, et quand elle part à la recherche de Bertran, son opinion est faite. Si ce livre, comme elle le soupçonne à présent, est à l'origine du mouvement, la bible des gamins du cirque et d'autres comme eux, l'auteur est un escroc qui a réussi, ou un fou qui n'a jamais été enfermé et qui en a trouvé d'autres aussi fous que lui.
« Tout repose sur un jeu de mot, Bertran ! La Mer " s'en va " et donc elle va quelque part, et de là découle tout le reste, y compris les " passeurs ", et les Rêveurs, et les délires sur la nature même de la Mer qui se promène entre deux univers. Mais si la Mer " disparaît ", ça marche déjà moins bien, eh ? Si elle " s'éteint ", c'est encore pire. Non ? »
Bertran est assis sur le rebord de la fontaine dont la vasque occupe le centre du village. Il l'a écoutée sans rien dire, les yeux au loin. Il demande enfin : « Êtes-vous chrétienne ? »
Alicia répond, totalement prise au dépourvu : « Élevée catholique, pour ce que ça a de sens maintenant.
— Eh bien, remarque Bertran d'une voix pensive, si on juge Leray aux œuvres de ses disciples, on peut dire que c'est en effet un escroc ou un fou qui a réussi, comme Jésus. »
Elle est choquée par réflexe. Puis son autre réflexe s'enclenche, oui mais. Bertran n'a pas tort. Les gamins ne forcent personne à connaître la vérité. Ils la montrent à ceux qui veulent voir, qui peuvent voir. Comme les Dillion – deux personnes sur une soixantaine – qui ont été capables de passer à travers les mensonges accumulés, la peur ou l'indifférence... Et ils leur laissent le choix de ce qu'ils ont vu, comme le choix de ce qu'ils feront ensuite. Tom Dillion est venu leur dire au revoir avec sa femme, quand ils ont quitté Kamtcha, mais il a décidé de ne pas se laisser installer la fameuse barrière.
« Je suppose qu'il n'y a pas de cirques à Cristobal ou Morgorod », remarque Alicia, un peu ironique quand même.
« Non, mais le livre circule sous le manteau.
— Puni de mort, si on est pris avec ? »
Bertran secoue un peu la tête, indulgent : « Mais non. Les Fédéraux ont choisi de le traiter comme un mauvais livre, c'est moins dangereux que la répression ouverte. Quand les premiers exemplaires en sont arrivés chez eux, on en a publié quelques critiques littéraires détaillées, des démolitions en règle, juste pas trop féroces, plutôt sarcastiques, et ensuite on l'a oublié en espérant que les gens en feraient autant. Ce qui a été plus ou moins le cas : de la mauvaise fiction... Bien entendu, on confisque tous les exemplaires qu'on peut trouver, et on rééduque, discrètement. Mais il y en a quelques-uns qui passent au travers, des livres, et des lecteurs. »
Alicia fronce les sourcils : « C'est de la fiction.
— Si vous le voulez, dit Bertran.
— Oh, Bertran ! C'est juste une autre manière de montrer en cachant, puisque vous me disiez que la méthode directe n'a pas réussi aux rebelles. Le bon vieux coup des paraboles. On laisse les gens démêler le vrai du faux. »
Les yeux ambrés la contemplent, indéchiffrables. Puis Bertran hoche la tête en soupirant : « Exactement. »
51
Il y a presque deux semaines qu'elle a quitté Dalloway. Un après-midi, alors qu'ils ont fini les répétitions et se détendent un peu en attendant l'heure de la représentation, elle entend l'une des filles du groupe demander à une autre si elle a encore des tampons, et soudain elle se rappelle qu'elle n'a pas eu ses règles depuis... quand ? Un peu trop longtemps. Elle va trouver Michelle, le médecin du groupe : elle a été assez irrégulière depuis qu'elle est arrivée sur Virginia, malgré les drogues utilisées pour l'aider à adapter ses rythmes à ceux de la planète ; les premiers colons ont eu le même genre de problème, elle ne s'inquiète pas, mais à tout hasard... Michelle lui fait une prise de sang, lui demande de l'urine, elle s'exécute et retourne lire dans l'herbe à côté de la camionnette – depuis l'ouvrage de Leray, elle a pris goût à l'archaïsme de la simple lecture, l'ennui aidant, et elle est en train de passer à travers les livres du groupe, en les faisant durer le plus possible.
Quand elle retourne voir Michelle, Bertran est là, elle se demande pourquoi. Et Michelle lui dit qu'elle est enceinte.
Elle proteste tout de suite : « C'est impossible, mes anticonceptionnels longue durée...
—... peuvent se neutraliser », dit Bertran.
Elle se sent glacée, mais bizarrement détachée. Elle dit enfin : « Les immunisations, avant Dalloway.
— Oui. »
Elle le dévisage avec une curiosité distante : « Vous saviez ?
— C'est facile à reconstituer. »
La question suivante devrait être plus difficile, mais elle ne l'est pas ; Alicia se sent tellement calme... « Il sait ?
— On ne vous a pas fait d'analyses à Dalloway. Il y a seulement quoi, trois semaines ? »
Elle se rappelle quand, elle est presque sûre de savoir quand. Ni Michelle ni Bertran ne demandent, et elle en est lointainement reconnaissante, ce qu'elle va faire. Il est bien trop tôt pour choisir. Ou trop tard ?
Ils continuent leur chemin vers le sud, une autre semaine passe. Alicia, flottant encore sur l'onde de choc, regarde le paysage changer – ils se dirigent de nouveau vers les collines et les hauts plateaux des Rouges et la piste suit la rive boisée d'une petite rivière qui se transforme par endroits en torrent – rapides écumeux, chaos de rochers, arbres et buissons suspendus dans la falaise. On retrouve ensuite les méandres paresseux à travers des prairies parfois herbeuses et fleuries, parfois marécageuses, et alors des voiles d'oiseaux s'enlèvent à leur passage. L'Été sous ces latitudes est d'une profusion discrète, mais d'autant plus précieuse. C'est comme une générosité de Virginia, ce changement, cette variété nouvelle des paysages, Alicia se prend à penser pour la première fois sans ironie " c'est beau ". C'est comme si cet enfant dans son ventre était une racine que Virginia avait fait éclore, comme si un atavisme profond répondait maintenant en elle aux dimensions sans limites du ciel et de la terre, comme si Lagrange avait été une couche ancienne de débris alluviaux dans un lac qui se serait étouffé et qu'une crue soudaine, emportant tout, aurait revivifié. Elle ne comprend pas. Elle pense, elle essaie de penser à ce que Graëme lui a fait, causes, conséquences, paramètres de ses propres actions futures, et elle a l'impression que son cerveau refuse tout simplement de les appréhender, grippe, renâcle, cesse de fonctionner. Une fois – ils se trouvent dans le premier camion, Bertran conduit, Karl dort dans la couchette arrière – sa stupeur, son désarroi débordent, elle murmure : « Pourquoi ? ! Je devrais haïr cet enfant, je devrais m'en être déjà débarrassée ! »
Bertran ne dit rien – il doit savoir qu'il n'y a rien à dire, qu'elle seule peut répondre à cette question. Elle s'exclame ensuite, plus accablée que furieuse : « Pourquoi a-t-il fait ça ? Pour m'enchaîner davantage à lui ? Ce n'était pas nécessaire... »
Mais il ne le savait pas forcément. Il n'a jamais rien su, peut-être. Elle se rejoue leurs dialogues, leurs moments ensemble, maintenant qu'elle sait, elle voit tous les malentendus. Il était surpris mais jamais fâché, il riait, il s'excusait, ça l'amusait qu'elle ne devine rien, qu'elle comprenne de travers. Mais lui aussi se trompait – bien plus souvent qu'elle, de fait. Et il riait encore, ça l'amusait encore. Ou bien il faisait semblant tout le temps, est-ce qu'elle sait, en réalité il était exaspéré ! Un enfant, avec elle, avec une tête-de-pierre, pourquoi, pourquoi ?
Elle regarde le paysage défiler à la portière, sans pensées. Bertran prend soudain la parole, et au début, elle ne voit pas le rapport. « Ils sont obligés de contribuer à l'amélioration des lignées. Pas forcément de se marier, mais de se reproduire. Graëme est un surtélépathe, l'un des plus puissants des Fédéraux sinon le plus puissant. Il n'a jamais voulu connaître ses enfants. »
Alicia n'en est plus à une stupeur ni à un chagrin près. Elle demande : « Combien en a-t-il ?
— Trois. » Puis Bertran ajoute, une conclusion : « Il voulait peut-être un véritable enfant. »
Alicia tressaille à l'écho soudain que cette phrase éveille en elle. Elle pourra sans doute en faire quelque chose à un moment donné, mieux l'écouter, mais c'est trop, trop tôt. Elle proteste, blessée : « Vous essayez de l'excuser ! »
Bertran soupire : « Non, j'essaie de comprendre. »
Elle contemple son profil ; il a laissé pousser ses cheveux depuis qu'ils ont quitté Dalloway, ils retombent en petites boucles serrées sur son front, sur sa nuque, il a l'air plus jeune ainsi. Elle demande, confusément agressive : « Vous en avez, vous, des enfants ?
Une très légère pause : « Non.
— Pourquoi pas ? » Elle est surprise : il ferait sûrement un bon père. « Les rebelles sont-ils forcés aussi de contribuer aux lignées ? »
Elle se rend compte, à son léger raidissement, que la question le blesse, mais il est trop tard.
« Non. Mes enfants sont tous morts. »
Elle se fige, puis souffle, consternée : « Oh. Oh, pardonnez-moi. »
Il dit : « Non, non. Il y a longtemps. »
Elle entend bien son intonation, elle lit bien l'expression de sa bouche ; elle murmure avec une douceur coupable : « Mais quand même. »
Il semble longuement méditer, tandis que des émotions complexes se chassent sur son visage. Puis, avec une ébauche de sourire mélancolique, il secoue un peu la tête : « Mais quand même. Je ne regrette rien. »
Alicia se replonge dans la contemplation du paysage.
Une autre semaine passe, elle n'a toujours pas pris de décision, le temps se fait court. Elle devrait être plus angoissée mais, c'est étrange, au lieu d'augmenter, son angoisse diminue. La routine du cirque, peut-être, a sur elle un effet apaisant. La compagnie des enfants – elle pense à eux et elle pense " les enfants " ou " les petits ", et elle a beau ironiser, elle se sent incroyablement vieille avec ses vingt-six ans maintenant – son anniversaire a eu lieu, est déjà loin, sur Lagrange. Si passionnés, si convaincus, les petits, et pourtant une telle paix en eux, une telle innocence... Ils savent qui ils sont, ce qu'ils font, quelle est leur place. Elle les observe au petit-déjeuner, au souper, les repas qui les rassemblent tous – ils prennent les collations et dînent plutôt par groupes sans cesse changeants. Elle devrait se sentir encore plus à l'écart ici que sur Lagrange : elle sait qu'elle est exclue du réseau mental qu'elle ne peut s'empêcher de soupçonner entre eux, même s'ils parlent toujours à voix haute, et même quand elle n'est pas directement présente. Pis encore, elle ne fait pas partie du tissu plus lâche mais plus constant de leurs émotions, celui qui unit plus ou moins tous les Virginiens au contact ou à proximité. Elle devrait se sentir exclue – mais, paradoxalement, elle a l'impression de l'être moins que sur Lagrange. Parce que depuis le début, elle sait, ils savent qu'elle sait, et ce n'est pas si grave ou si important. Pas de mensonges, pas de soupçons, pas d'embarras. Elle ne comprend pas comment il peut en être ainsi – elle a cessé, quelque part au cours de sa deuxième semaine avec eux, de les penser polis et hypocrites : ce préjugé-là ne résiste pas au frottement quotidien. Elle ne comprend pas comment, mais c'est ainsi. Elle n'est pas une étrangère pour eux, on l'a adoptée d'emblée, elle fait partie du groupe – Sélim essaie de lui apprendre à jongler, Benny à faire des tours de cartes. On lui sourit quand les regards se croisent, on lui parle sans qu'elle parle la première, on la consulte, on l'écoute.
Un jour, Samanta vient s'asseoir près d'elle alors qu'elle somnole dans l'herbe après la collation de mi-matinée. Comme l'adolescente ne dit rien, Alicia ne dit rien. Puis Samanta fait un drôle de bruit, ou bien elle bouge d'une drôle de façon : Alicia lui jette un regard à la dérobée, voit qu'elle pleure. Catastrophée, elle lui passe un bras autour des épaules, qu'est-ce qui ne va pas ? Samanta est amoureuse de Benny, Benny n'est pas amoureux d'elle. « Il est tellement désolé, ce n'est pas supportable », gémit la gamine. Elle continue à pleurer dans les bras d'Alicia qui ne sait pas quoi dire. Finalement, elle se redresse et s'essuie la figure en disant d'une petite voix tremblante : « Merci. »
Alicia lui remet des mèches brunes en place derrière les oreilles, mi-amusée mi-chagrine : « Je ne crois pas avoir fait grand-chose.
— Oh si ! » s'exclame la petite avec conviction. Alicia hausse les sourcils. « Tu sais comme c'est, quand on a du chagrin et que quelqu'un essaie de nous aider, ils ont de la peine pour nous, ou bien ça les enrage de ne rien pouvoir faire, et on le sait, et c'est pire ? Nous, c'est vraiment pire, tu comprends, on le sent. Quand on peut, on bloque, bien sûr, mais alors là, c'est vraiment affreux, on se sent encore plus seul, ça devient un cercle vicieux, tu comprends... »
Alicia comprend très bien.
Samanta a un hoquet, renifle et s'essuie encore les yeux avec un sourire embarrassé. « Ça fait drôle de dire ça, c'est sûrement égoïste, mais des fois, on a envie d'avoir de la peine... tout seul. Sans être distrait. C'est... reposant. Et ça passe plus vite. » Elle dévisage Alicia en se mordant un peu une lèvre, incertaine de ne pas commettre un impair : « Tu m'as beaucoup aidée, là, je me sens vraiment mieux.
— Eh bien, à ton service, dit Alicia en souriant pour la rassurer, n'importe quand, n'hésite pas », et la petite s'en va. Alicia contemple sans la voir la page de son livre. Se peut-il que Graëme l'ait trouvée reposante, lui aussi ?
Une autre semaine commence, le délai approche, arrive, est dépassé sans qu'Alicia ait pris une décision, la décision est prise. Elle gardera l'enfant. L'enfant n'est pas responsable. Ce n'est pas l'enfant qu'elle déteste. Elle ne sait plus trop qui elle déteste, elle a un peu le vertige devant ce panorama trop vaste où les responsabilités individuelles tendent à se diluer. Il y a toujours une autre perspective qui s'ouvre derrière, les décors se multiplient, avec leurs envers qui sont la vérité d'un autre décor et finalement c'est aussi grand que la planète, et qui peut dire alors ce qui est décor et ce qui ne l'est pas ? Facile de détester, sur Lagrange : l'espace limité concentrait tout (elle prend soudain conscience qu'elle ne pense plus " dans Lagrange ", depuis quand ?) Mais ici, ce n'est pas pareil. Elle contemple l'horizon des montagnes, les collines où ils sont en train de rouler et qui modifient la perspective des plaines en contrebas. Sauter d'un bout à l'autre du continent en avion lui avait permis de continuer à se mentir, à prétendre que c'était toujours une sorte de scénario limité, bien contrôlé, mais les longues heures dans l'exaspérante monotonie des plaines, pendant les semaines précédentes, ont fini par user aussi cette illusion-là ; pas la pénible randonnée en forêt : jamais d'horizon. Mais ici, on sent qu'on peut toujours aller plus loin. Pas forcément parce qu'on y est prêt, parce qu'on a compris, comme le disait Bertran, mais au moins on peut changer le mal de place.
52
Vers la fin de la septième semaine de Décembre, la nouvelle arrive : les communications sont rétablies avec Lagrange.
La nouvelle arrive en la personne d'un petit vieillard sec mais énergique nommé Simon Fergusson, épais cheveux blancs toujours en désordre au-dessus d'un regard pâle et froid. Il rejoint le cirque dans un petit hameau à l'heure matinale où l'on est en train de démonter la tente. S'il remarque le ventre un peu arrondi d'Alicia, il ne fait pas de commentaires. Il explique, personne ne discute : les Fédéraux ont réussi à récupérer les données encryptées, mais ils ne les ont pas transmises au complet à Lagrange, ils prétendent n'en avoir retrouvé qu'une partie ; ils négocient les premiers transferts de technologie militaire. Temps pour Alice et son transmetteur d'entrer en lice. Fergusson est arrivé avec un ampli linéaire et des antennes directionnelles démontables qui devraient permettre une durée raisonnable de communication avec Lagrange. Après quoi il repartira en quatrième vitesse avec Alicia pour la Licornia, car ils auront sûrement été repérés.
Alicia assemble le transmetteur tandis que les autres, sans un mot, vont finir de ranger bâches et mâts dans les camions. Elle jette un coup d'œil à Fergusson qui finit d'installer son propre matériel. Le laconisme tout militaire du petit homme est sans doute de rigueur, mais elle n'est pas obligée de l'aimer. « Ils ne me croiront pas forcément, vous savez. Ils croient que ce sont les rebelles qui ont poussé les stoppistes à saboter les ordis de vol. Et avec la propagande qu'on leur a fait avaler depuis le début...
— Vous allez leur transmettre les données au complet, remarque le vieil homme sans lever les yeux de ses fils, c'est une preuve.
— Seulement s'ils veulent la voir comme telle. Ils peuvent encore soupçonner je ne sais quelle machination des rebelles. Et ils ne voudront pas automatiquement vous aider pour autant.
— Nous ne voulons pas qu'ils nous aident. Nous voulons qu'ils n'aident pas les Fédéraux. Qu'ils se retirent du jeu.
— C'est dans leur pente », remarque Bertran, qui vient d'arriver après avoir aidé à bourrer les camions. « La faction stoppiste n'a pas exactement le vent dans les voiles en ce moment, et la tendance générale doit être à se retirer complètement des affaires des planétaires. »
Fergusson se redresse, les mains sur les reins, et toise Bertran. « Ah, le fameux capitaine Bertran, notre agent très secret à Dalloway. » Puis, à la cantonade, c'est-à-dire à Alicia, car ils sont seuls tous les trois, il déclare : « L'homme providentiel. C'est lui qui a découvert le projet de sabotage et le fait qu'Anderson jouait dans le dos du Conseil, et c'est lui qui nous a prévenus quand vous étiez rendue à Dalloway. » Il continue à dévisager Bertran qui reste là, paisible, les bras croisés. « Vous auriez pu nous prévenir plus tôt, quand elle était encore à Morgorod.
— L'itinéraire était ultra-secret.
— Mais pas les plans d'Anderson ?
— J'ai eu accès à Grumberg avant son départ, pas à Anderson. J'ai reconstitué à partir de ce que j'ai pu tirer de Grumberg. Mais je ne pouvais pas savoir quel en était le minutage, la faction stoppiste était une hypothèse pour nous comme pour eux, et quant à évaluer quelle influence exacte aurait Alicia dans le tableau... Je ne me doutais pas qu'il voudrait la prendre en main lui-même, c'était une addition tardive au plan. »
Alicia intervient, suffoquée : « Vous saviez qu'ils voulaient saboter... et vous n'avez pas prévenu Lagrange alors que vous étiez à Dalloway ?
— Les communications sont étroitement surveillées, j'aurais risqué de me brûler, et pour quoi ? Les Fédéraux auraient aisément fait passer ça pour de la désinformation émanant d'un agent provocateur. Comme vous le disiez, les Lagrangiens sont prédisposés à penser le pire des rebelles. Quant au reste, encore une fois, je n'ai pu sonder Grumberg qu'à son départ de Dalloway. L'esprit d'un être humain n'est pas un livre ouvert, Alicia, surtout un agent préparé par Anderson. Grumberg ne savait pas qu'il était un agent double des rebelles, par exemple, ce conditionnement-là ne se déclenchait qu'après le sabotage... Et ensuite, difficile de prévoir comment votre relation avec Graëme avait pu modifier son plan. »
Les rouages s'enclenchent dans l'esprit d'Alicia : « Il n'y avait pas d'accord secret au départ avec mon père pour me faire échouer ? »
Bertran semble déconcerté, puis il soupire : « Non, Alicia. Vous deviez trouver les données et faire la livraison, ensuite on sabotait, Lagrange était fermement anti-rebelles, et vous étiez en place pour devenir un capitaine fort bien disposé envers les Fédéraux.
— En tout cas, intervient Fergusson, vous auriez pu encrypter correctement toutes les données, Bertran, et pas seulement la moitié.
— J'étais un peu pressé.
— Vous avez bien eu le temps d'effacer les copies et l'original. »
Bertran sourit : « Ça, ça faisait un moment.
— Vous auriez pu vous arranger pour qu'elle trouve les données plus tôt.
— Et prendre des risques superflus ? dit Bertran en haussant les épaules. Elle devait les trouver de toute façon.
— Vous auriez pu la laisser chercher et nous les donner à nous plus tôt, on aurait tout fait pour contacter Lagrange.
— Et les Fédéraux en auraient fait encore plus pour vous en empêcher, quitte à réchauffer davantage la guerre tiède. C'est ce que vous auriez voulu, des missiles sur Léonovgrad et de l'autre côté des Rouges ? »
Alicia a eu le temps de sortir de son hébétude. Pourquoi le vieux s'acharne-t-il ainsi sur Bertran ? Y a-t-il donc des factions chez les rebelles aussi ? Elle remarque, agacée : « Sans Bertran, je ne serais sans doute pas sortie aussi facilement de Dalloway. »
Fergusson se tourne vers elle avec un petit rictus sarcastique : « Facilement, oui. En échappant entre autres à tous les détecteurs.
— Je savais où ils étaient placés, dit Bertran sans se départir de son calme. Et on a eu de la chance. Vous ne croyez pas en la chance ? »
Cette fois, le vieux ricane carrément : « Iptit.
— Iptit, dit Bertran, très sérieux.
— Iptit a dû être extrêmement occupé cette nuit-là, rétorque Fergusson. Il n'y est pas allé de main morte, avec le tunnel. Les débris l'ont rebouché derrière vous, en plus. Encore utilisable pour nous, mais eux ne l'ont pas trouvé.
— De quoi vous plaignez-vous ? intervient encore Alicia, excédée. Vous avez maintenant une voie d'accès à Dalloway par la porte d'en arrière ! »
Fergusson n'a pas quitté Bertran des yeux ; il marmonne : « Justement...
— J'étais à la bonne place au bon moment, dit Bertran avec une soudaine lassitude. Ça arrive, non ?
— Vous étiez à la bonne place depuis très longtemps. Plus de vingt saisons, si je ne me trompe. Le truc bizarre, c'est qu'il n'y a aucune trace nulle part de votre implantation à Dalloway. »
Bertran se met à rire : « J'espère bien, ou sinon quelqu'un n'aurait pas fait correctement son travail.
— Quelqu'un a trop bien fait son travail, réplique le vieux du tac au tac, nous ne savons même pas qui vous êtes. Josh Bertran, oui, votre couverture est remarquablement bien solide et détaillée, mais personne n'a l'air de savoir exactement qui il y a dessous. Votre lignée, par exemple. Avec une barrière comme la vôtre, et pour avoir réussi à tromper Anderson aussi longtemps, vous ne devez pas être n'importe qui. »
Cette fois Alicia en a assez : « Mais qu'est-ce que c'est que ces histoires, à la fin ? »
Bertran sourit : « Notre ami Simon a de bonnes raisons d'être très méfiant.
— Qu'en savez-vous, de mes raisons ? réplique le vieux.
— Vous êtes un surtélép, ça vient avec le territoire. »
Fergusson le dévisage un moment en silence, les yeux étincelants, puis se tasse un peu sur lui-même. « Vous en êtes un aussi, murmure-t-il, et plus fort que moi.
— Ah bon, c'est ça qui vous dérange ? » lance Alicia, écœurée.
Le vieux a une drôle d'expression, résignation, lassitude. Il dit : « Non. » Il ajoute : « Notre ami Bertran peut s'occuper de vous aider aussi bien moi, je n'en doute pas. Allez-y. Il faut que je mange quelque chose. »
Et il s'éloigne.
Alicia médusée demande : « C'était quoi, ça ? Je vais devoir dire à Lagrange qu'il y a des factions chez les rebelles, en plus ? Ça ne va pas les disposer davantage à écouter.
— Non, dit Bertran, en prenant place devant les appareils du vieux. C'est une histoire entre Simon et moi, purement personnelle. Rien à voir avec les rebelles. Dites-leur ce que vous avez à leur dire. »
Alicia l'observe un moment : « Tout ?
Bertran lui rend son regard, impassible : « Tout ce que vous voulez leur dire. »
Et Alicia leur dit tout. Elle s'identifie, et quand ils sont persuadés que c'est bien elle, elle commence par leur expédier les données sur la propulsion Greshe. Ensuite, elle leur raconte le plus succinctement possible ce qui s'est passé à Dalloway, la véritable nature des forces qui s'affrontent sur la planète – en faisant l'impasse sur ses relations avec Graëme. Elle n'est pas sûre qu'on la croit en ce qui concerne la mutation, mais elle s'en moque. Ils croient ce qu'ils veulent croire, ils feront ce qu'ils veulent avec. La réaction est claire, d'ailleurs, elle le devine vite : comme on en avait l'intention au départ, on ne se mêlera pas de toutes ces sombres histoires de planétaires, et on se débrouillera entièrement sans eux.
On n'a pas encore dit un mot du sabotage, de l'insurrection stoppiste. Elle pourrait revenir sur Lagrange. En héroïne, même, puisqu'elle a échappé aux Fédéraux et récupéré les données. On le lui propose – Marti est obligé de le lui proposer : on peut l'extraire comme prévu par le plan d'urgence, si elle est capable d'échapper encore aux Fédéraux pendant quelques jours. Elle s'entend dire " Non ", se rend compte seulement alors qu'elle a pris la décision de ne pas revenir sur Lagrange, et pas seulement parce qu'elle ne le peut pas, mais parce qu'elle n'en a pas envie. Elle se soupçonne aussitôt : es-tu bien sûre, Alicia, que ce n'est pas la crainte d'être mise en accusation et emprisonnée pour ton rôle dans la conspiration stoppiste ?
Elle lance un coup d'œil à Bertran : « On peut continuer ? » Il hoche la tête. Elle pose des questions brèves sur l'insurrection. Si on est au courant du rôle qu'elle a joué, on ne lui assénera pas des accusations véhémentes, elle s'en doute bien : on la laissera revenir et là, fin de la partie. Mais au cours des échanges, il appert que personne n'a parlé – seulement Grumberg, et il a présenté les jeunes stoppistes comme uniquement manipulés par lui, par les rebelles, sans jamais mentionner Alicia. Moran a été tuée par accident quand la police de Lagrange a pris d'assaut la cache des gamins, il y a eu plusieurs blessés chez les autres conspirateurs. On en a été tellement horrifié – et tellement horrifié surtout d'apprendre que Moran était la seule à connaître les codes d'encryptage – qu'on a complètement laissé tomber le reste : la priorité était à la remise en état des ordis. On a interrogé les prisonniers de façon rudimentaire, ces jeunes imbéciles crédules – et ils n'ont pas offert plus d'information qu'on en demandait. Ils ont protégé Alicia.
Elle dit à Marti qu'elle est la seule véritable responsable.
Pas pour lui asséner son triomphe en pleine face – ni sa fille ni son fils ne seront capitaine, le poste va sortir de la famille, s'il savait comme elle s'en moque maintenant ! Mais parce qu'elle ne veut pas revenir en arrière. Ça marche : il met fin de manière abrupte à la communication.
À ce moment-là, elle se rend compte qu'elle pleure, et que les petits du cirque se sont rassemblés autour d'elle. Sans un mot, Samanta vient l'embrasser, et les autres, chacun son tour. Bertran l'étreint en dernier. Même le vieux Fergusson est là, qui lui tapote l'épaule.
Et maintenant, on s'en va. Dans le tout-terrain, le minimum vital, pas question de s'arrêter pour dormir, seulement pour se réapprovisionner en gaz et en eau, on se relaiera au volant. Alicia, Fergusson et Bertran. Seulement ? Fergusson se méprend, déclare avec un petit sourire sarcastique : « Ce sera plus que suffisant.
— Mais eux ? » dit Alicia en désignant les autres.
« Nous allons partir de notre côté, dit Sélim.
— Nous nous débrouillerons très bien, ajoute Karl. S'il le faut, on s'évaporera dans le paysage. Avec Sélim et Benny pour faire la couverture, on a l'habitude. »
À vrai dire, on ne sait pas exactement ce que vont faire les Fédéraux : tous les plans de tout le monde sont à l'eau, maintenant, à cause d'Alicia – et à cause de Graëme Anderson. Vont-ils essayer par principe de la recapturer ? Ils vont peut-être placer le blâme un plus près de chez eux, remarque Fergusson. Le Conseil du Présidium était de bonne foi : ils voulaient vraiment échanger contre de la technologie les plans de la propulsion Greshe, dans les meilleures conditions pour eux – en s'assurant d'un maximum de bonne volonté de la part de Lagrange. Et que Lagrange reparte, bon débarras. C'est Graëme qui a tout manigancé pour naufrager Lagrange au point L4.
« Mais pourquoi ? » dit Alicia en grimpant dans le véhicule à côté de Bertran.
Fergusson s'installe à l'arrière : « Il voulait une coopération plus active de Lagrange, je pense. Il serait entré dans l'Histoire comme l'homme qui aurait mis fin à la guerre dans le sud-est...
— Mais son scénario n'a pas tourné comme prévu, dit Bertran. Et ce qu'il fera s'il réussit à se sortir de cette histoire-là... Le Conseil ne se laissera sûrement pas convaincre aussi aisément que les autres fois. Il y a des limites à ce qu'ils peuvent tolérer, même d'un surtélépathe essentiel aux lignées, même d'un membre du Conseil.
— Comme s'ils pouvaient le contrôler », murmure Fergusson, qui semble avoir oublié pour un instant sa querelle avec Bertran. Puis, avec un soupir : « On ne sait pas ce qu'ils feront, les uns ni les autres, mais on ne va pas attendre pour voir. »
53
On roule. Ils ont deux mille cinq cents kilomètres à parcourir en se tortillant par des petites pistes entre les caches de réapprovisionnement, au moins une journée entière, trente-cinq heures de route. Alicia surveille le ciel, au début ; Bertran le remarque : « Ils n'enverront pas d'avion. Ou seulement pour débarquer du monde. »
Fergusson glousse à l'arrière : « Ils nous veulent vivants, d'une part, et d'autre part, ils ne peuvent pas risquer un avion. Ils ne savent pas qui nous pouvons avoir, comme mesures antiaériennes. »
Ils vérifient constamment, du moins Alicia le suppose, mais à ce qu'ils disent personne ne semble les suivre. Personne ne suit le cirque non plus, quand Fergusson se permet un bref contact pour vérifier. Après quelques heures, elle ne demande plus. Peut-être les Fédéraux ont-ils décidé de ne plus perdre de temps et d'énergie pour elle ? Elle n'est plus persona grata sur Lagrange, elle ne leur sert plus à rien. Ce sont des pragmatiques. C'est seulement dans les mélodrames que les malfaiteurs frustrés dans leurs machinations essaient à tout prix de se venger.
« Ne sous-estimez pas Graëme, dit Bertran.
— Ou ne le surestimez pas, renchérit Fergusson. Si quelqu'un vient, ce sera lui. »
Alicia ne comprend pas très bien les sous-entendus – Bertran et le vieux trouvent moyen d'être d'accord en se contredisant. Elle entend surtout que Graëme viendrait en personne. Elle souffle : « Graëme ?
— Vous ne risquez rien », lui dit Bertran avec une certitude absolue. Et pour une fois, Fergusson hoche la tête sans réticence.
Ils traversent la pénéplaine qui sépare les derniers contreforts des Rouges de ceux des monts Shœlzer et arrivent en vue du Fjord Blanc, le lendemain très tôt à l'aube. La Mer est revenue dans la nuit, c'est pour cela qu'ils arrivent déjà à la côte : le fjord est presque deux fois plus large quand elle est là.
Pour la première fois de sa vie, Alicia voit vraiment la Mer. Ou plutôt elle ne la voit pas, ce qui confirme pour elle l'authenticité de la chose. C'est la limite des sims, la Mer : toujours reconstituée sans le brouillard qui la recouvre, puisqu'on ne peut pas le photographier – et on n'allait pas pousser l'authenticité jusqu'à simuler ce brouillard qui cache tout... Ici, Alicia ne voit que le brouillard, ou du moins ce qu'on a décidé il y a fort longtemps de désigner par ce terme. Les descriptions virginiennes orales ou écrites parlent d'une sorte de scintillement, parfois même d'une luminescence diffuse, mais ce n'est vraiment pas le cas : juste un mur blanchâtre, immobile, opaque – un vrai mur, qui commence impossiblement raide au bord de la côte et s'élève droit dans le ciel, pour se perdre quand même peu à peu dans la lumière du jour.
Ils roulent vers ce mur immuable. Si loin au sud, c'est pratiquement un désert, et les pistes ne sont pas en très bon état. Si personne ne les suit, ils pourraient aller moins vite. Fergusson, dont c'est le tour au volant, fait « mmmm », mais ne ralentit pas. Bertran dort à l'arrière – Alicia ne sait pas comment il fait. Elle est épuisée, elle n'a pas vraiment fermé l'œil depuis la veille au soir et à chaque cahot trop prononcé elle porte les mains à son ventre par réflexe, agacée chaque fois qu'elle s'en rend compte, mais c'est plus fort qu'elle. Elle a connu un début de grossesse remarquablement libre d'inconvénients – à vrai dire, elle était occupée à courir dans les champs de mines puis à se battre avec la forêt vierge, elle avait autre chose à faire que de vomir tous les matins. Ce serait quand même trop bête s'il arrivait quelque chose maintenant qu'elle a décidé de garder cet enfant.
Il y a à peine une heure qu'ils longent le mur de brouillard vers l'est quand brusquement Fergusson freine et coupe le moteur.
« Graëme », dit Bertran d'une voix sans inflexion.
« C'est seulement maintenant que vous le repérez ? remarque Fergusson avec une surprise appuyée. Ça fait une heure qu'il joue les phares tous azimuts.
— Je ne voulais pas le dire avant vous, au cas où vous ne l'auriez pas vu », réplique Bertran, aimable.
Alicia les dévisage tour à tour, incrédule : « Mais vous n'avez pas bientôt fini ? Combien sont-ils ? »
Ils se retournent vers elle du même mouvement : « Seulement Graëme, dit Bertran. N'est-ce pas, Simon ? »
Fergusson acquiesce d'un air revêche. « Ses hommes sont arrêtés à plus d'un kilomètre. Une trentaine. Ils bouclent le coin.
— Ses hommes ! ?
— Nous ne voulions pas vous inquiéter inutilement, dit Bertran. Il pourrait en avoir deux fois plus, ça ne change strictement rien. »
Alicia reste médusée. Le silence se prolonge. « Qu'est-ce qu'on fait ? » s'exclame-t-elle enfin, partagée entre l'exaspération et un début de panique.
« Il veut vous voir, dit Fergusson.
— Pas question !
— Vous êtes sûre ? dit Bertran.
— Vous ne risquez rien », reprend le vieux. Alicia n'est pas si sûre : il a l'air épuisé.
« Je n'ai rien à lui dire.
— Il veut vous parler.
— Pour me dire quoi ? !
— Vous ne voulez vraiment pas le savoir ? » murmure encore Bertran. Sa voix est empreinte d'une curieuse tristesse. Alicia se perd un moment dans le regard ambré fixé sur elle. Qu'est-ce qu'il veut, une dernière rencontre pour bien conclure l'histoire d'amour ? Il n'y en a pas eu, d'histoire d'amour, juste un énorme mensonge.
« Il est seul, Alicia », dit Fergusson à son tour. Quoi, ils sont de mèche ? Mais le vieil homme poursuit : « Il ne nous laissera pas passer sans vous parler.
— Deux contre un ! »
Fergusson jette un bref regard à Bertran : « Peut-être, mais je préférerais éviter un affrontement... déplaisant, et Bertran aussi, sûrement. » Les yeux pâles la dévisagent un moment puis, comme s'il prenait une décision, d'un ton plus net, le vieil homme déclare : « Et je crois que vous en avez besoin, Alicia. »
Si elle disait non maintenant, ce serait uniquement pour les contrarier ; elle n'en est plus là. Elle dit « très bien », et Fergusson remet le tout-terrain en marche.
Encore trois tournants et, à une cinquantaine de mètres, ils aperçoivent une silhouette assise sur un rocher au bord de la piste.
Alicia descend du tout-terrain. Elle a les jambes engourdies, ils roulent sans discontinuer depuis cinq heures, depuis la dernière cache d'approvisionnement. La silhouette s'est levée et attend. Alicia se retourne, indécise, vers ses deux compagnons. Deux contre un. Et Graëme est venu seul ? Elle se remet en marche.
Il n'a pas changé. Plus maigre, les cheveux plus ternes, peut-être. Vêtu de noir, manipulation, encore, " je porte le deuil de notre amour ". Tu peux le porter, le deuil ! Elle voudrait tourner les talons mais il n'en est plus question maintenant, au contraire, elle se force même à s'approcher presque à portée de bras. Il ne bouge pas. Il la regarde. Elle ne va pas baisser les yeux. Le silence devient insupportable, mais elle n'a rien à lui dire, rien ! Lui aussi, il doit bien savoir que cette rencontre est inutile ! Et pourtant, il est venu. Que pourrait-il lui dire ?
Ce qu'il ne lui a jamais dit, bien sûr : « Je t'aime, Alicia. »
D'une voix presque neutre. Une information, objective. Pour que ce soit clair entre eux. Au moins ça.
Et elle le croit. Elle ne l'aurait cru nulle part ailleurs, mais ici, elle le croit. Il ne serait pas venu là se mettre à la merci de ses ennemis. Il ne serait pas venu là tout seul.
Il ne la quitte pas des yeux. Comme si c'était à elle de parler, de finir.
Et elle s'entend dire, d'une voix moins dure qu'elle ne l'aurait cru : « J'attends un enfant. Pourquoi ? »
Il a fait un brusque pas vers elle, le visage illuminé de joie incrédule. S'immobilise en jetant un bref coup d'œil à sa gauche : « Je ne la toucherai pas. »
Elle regarde aussi : les deux autres sont descendus du tout-terrain et l'ont rattrapée, à sa droite, en longeant le mur de brouillard. À portée de voix, mais peu importe. Elle répète : « Pourquoi ? »
Graëme la contemple avec une avidité désespérée : « Oh, Alicia... » Puis il semble réfléchir un moment, yeux baissés, mains à l'abandon. Dit soudain à mi-voix : « Pour ça. Parce que tu es obligée de demander. Parce que tu ne sais pas ce que je pense, parce que tu dois deviner ce que je ressens. Et moi aussi avec toi, Alicia, moi aussi ! Nous sommes... à égalité. Tu ne peux pas savoir... » Il ferme les yeux dans un soudain élan de passion douloureuse : « Tu ne peux pas savoir à quel point je hais ce que je suis, ce que nous sommes, les Fédéraux, les rebelles, tous des... monstres ! Ça ne devrait pas être ainsi. Cette mutation n'aurait jamais dû exister ! » Il se tend vers elle, les poings serrés dans l'effort qu'il fait pour ne pas la toucher : « Et elle peut disparaître, Alicia. Notre enfant à nous, il sera normal. Normal. Il a au moins cent jours, et je ne le sens pas du tout en toi ! Ça veut dire que j'ai raison, vos gènes sont dominants, les gènes de Lagrange, et maintenant que vous êtes là, si vous vous croisez avec nous, la mutation disparaîtra, Alicia, comprends-tu ? La mutation disparaîtra ! »
Elle le contemple, paralysée par une soudaine épouvante. Ces yeux étincelants, ce sourire triomphant, cette certitude. Il ne peut pas croire... Il ne peut pas avoir fait tout ça pour...
« Après quelques générations, la mutation commencera à se déclencher chez eux aussi, dit Bertran d'une voix un peu altérée. C'est ce qui est arrivé après la Libération aux derniers colons terriens, c'est ce qui est toujours arrivé. »
Graëme se tourne vivement vers lui : « Ce n'est pas pour ça que ça arrivera toujours ! On pourra étudier en détail ce qui se passe. On trouvera un moyen de neutraliser la mutation !
— Quarante mille cobayes, Graëme ? » laisse tomber Fergusson avec lassitude
« Ils comprendront ! Ils se porteront volontaires, quand on leur expliquera ! » Graëme regarde à nouveau Alicia avec ferveur : « Tu comprends, toi, Alicia, n'est-ce pas ? Il faut mettre fin à cette mutation. C'est à cause d'elle, tous ces mensonges qui tuent. C'est elle qui nous empoisonne, depuis le début. C'est elle qui nous a dressés les uns contre les autres. Il faut que ça cesse. On ne va pas se massacrer jusqu'à la fin des temps ! »
Alicia recule d'un pas, malgré elle. Elle s'est presque sentie penser un instant qu'il n'a pas tort ! Mais elle sait bien que non, les mutations ne sont pas responsables de tout ce qu'on a commis en leur nom depuis le début. Elle évoque, presque comme on prie, Samanta et Karl, et Djira, Antonia, Michelle, Benny, tous, les enfants d'Iptit qui n'ont jamais tué personne, jamais menti à personne, qui risquent leur vie chaque jour pour rendre le choix possible à autrui. Elle n'en peut plus, elle se détourne pour s'enfuir vers les deux autres – oui, elle va s'enfuir, mais c'est intolérable tout d'un coup.
Graëme lui prend le bras pour l'arrêter – tressaille comme s'il avait pris une décharge électrique mais murmure quand même d'une voix rauque, le visage contracté : « Non, Alicia, je t'en prie. Tu dois venir avec moi, maintenant, notre enfant a besoin de nous. »
Elle le dévisage, presque plus consternée qu'incrédule : comment peut-il penser une seconde... : « Je suis avec eux, Graëme. Je vais en Licornia avec eux. »
Il se redresse, soudain presque indulgent : « Et comment ? Vous êtes coincés là. Mes hommes vous encerclent. Je suis prêt à attendre aussi longtemps qu'il le faudra. Mais avec l'enfant, c'est un peu différent. Il faut être raisonnable, Alicia. »
Elle secoue la tête : « Par la Mer, Graëme. Nous partons par la Mer. »
Il reste un moment immobile, puis son visage prend une expression navrée : « Oh, Licia. Vous auriez essayé de traverser par les montagnes en contournant le Fjord. Ils t'ont raconté leurs histoires et tu les as crus ? Ils t'ont fait lire ce tissu d'âneries, je parie ? Ne me dis pas que tu l'as cru ? Pas toi, Licia, tu es au-dessus de ça ! » Il la contemple avec une tendresse presque amusée : « On ne peut pas voyager sur la Mer, elle rend fou, c'est ce qui lui est arrivé, à ce pauvre rebelle, il est devenu dingue, et il a écrit ces idioties. Ils savent très bien que ce sont des divagations, mais ils les utilisent pour recruter des malheureux qu'ils envoient se faire massacrer après, comme cette pauvre femme à Dalloway. On ne peut pas voyager sur la Mer, Licia, et si on la touche, on meurt, tu sais bien que c'est vrai, c'est vrai depuis le début de la colonisation, et ça, ce sera toujours vrai ! »
Elle le dévisage, éperdue. Elle entend ses propres arguments, ce sont les arguments qu'elle a opposés à Bertran, et pourquoi en doute-t-elle maintenant ? Parce que c'est Graëme qui les défend ?
Fergusson s'éloigne déjà le long du brouillard. « Venez, Alicia », dit Bertran en lui prenant la main.
Elle se détourne pour le suivre, incertaine, hébétée de chagrin.
« Alicia ! » proteste Graëme en s'élançant vers elle. Il s'immobilise de nouveau, lutte en vacillant pour faire encore un pas en avant, puis renonce. « Qu'allez-vous lui faire ? s'écrie-t-il avec désespoir. Ne les écoute pas, Alicia ! Ils sont capables de te tuer pour perpétuer leurs mensonges ! Viens avec moi, je t'en supplie ! » Et il a peur, il a vraiment peur pour elle, elle peut l'entendre dans sa voix.
Fergusson s'arrête. Puis il va s'agenouiller devant le mur de brouillard. Ses mains disparaissent dans la masse opaque, en ressortent festonnées de bleu, un bleu comme Alicia n'en a jamais vu, intense et pourtant impalpable, plus une lumière qu'une couleur. Des petits filaments s'étirent et retombent, des gouttelettes, qui glissent de nouveau vers le brouillard en serpentant à travers les cailloux, mais dans les mains de Fergusson réunies en coupe, alors qu'il revient vers eux, il y a cet éclat bleu, la Mer. Il s'arrête – la tête un peu rejetée en arrière, il est si petit en face de Graëme – et, avec lenteur, il laisse couler la Mer à ses pieds.
Ensuite, il prend l'autre main d'Alicia et, avec Bertran, ils marchent tous trois le long de la Mer vers le bateau qui les attend.
54
Ils traversent le Fjord Blanc en diagonale vers le sud-est, plus de mille kilomètres pour se rendre au cap Termaine, l'extrême pointe de la Licornia : la côte est du fjord devient inaccessible lorsque la Mer est là, toute en falaises abruptes et crêtes aiguës alignées comme des jeux de cartes sous leur épaisse et rase toison de plantes grasses et d'épineux.
Après une rapide collation, Alicia s'est accoudée au bastingage de poupe entre Fergusson et Bertran. Les trois marins sont invisibles dans les entrailles du bateau, sauf la femme de barre, debout dans son petit cockpit transparent. Alicia se dit qu'elle devrait aller dormir, mais elle est trop énervée, anxieuse aussi. Elle connaît par cœur l'histoire de l'Entre-deux, le bateau perdu lors de la malheureuse tentative d'exploration de l'archipel ouest, au début de la colonisation ; et celle des deux équipages décimés lors de la traversée expérimentale du Grand Golfe, après la première Indépendance. Fergusson et Bertran la protégeront si sa propre immunité naturelle ne tient pas le coup assez longtemps, elle sait qu'elle ne risque rien, mais on ne change pas d'histoires aussi facilement.
Curieusement, le brouillard n'est pas aussi épais lorsqu'on se trouve à l'intérieur : on voit presque d'un bout à l'autre du petit vapeur, et la Mer en dessous, ourlée le long de la coque telle du mercure bleu ; les hélices semblent y laisser une trace plus sombre qui s'efface vite. Tout flotte sur la Mer, elle n'offre pratiquement pas de résistance, à cette vitesse ils arriveront vers minuit.
Alicia contemple la Mer, consciente du globe de brouillard qui semble se déplacer en même temps que le bateau ; elle pense aux mains de Fergusson dégoulinant de bleu, elle a le vertige devant cette nouvelle perspective, trop vaste cette fois, suffocante. Elle murmure enfin, avec une sorte d'effroi : « Tout est vrai, alors, dans le livre ?
Bertran pousse un petit soupir : « C'est certainement la vérité de Nathan Leray.
— Oui, c'est vrai », dit Fergusson plus durement.
Elle le dévisage un moment, ce vieux petit homme tanné par le soleil, cet inconnu avec ses yeux incolores et sa tignasse blanche. Elle le revoit devant Graëme – et le visage pétrifié de Graëme. Elle n'arrive pas à penser clairement : « Vous êtes... un passeur, alors ?
Fergusson semble hésiter : « Non. Certains d'entre nous... » Sa voix ralentit, se perd comme s'il était las d'avoir à expliquer ou ne savait comment le faire.
Bertran reprend à sa place : « Certains d'entre nous sont simplement réfractaires à la Mer. »
Alicia se retourne vers lui, bouche bée : « Vous aussi ?
— Oui. »
Et comme il regarde Fergusson, elle regarde aussi le vieil homme : il a l'air médusé. Alicia lui demande, incertaine : « Vous avez essayé... de passer ?
Fergusson se reprend : « Non, dit-il d'un ton sec.
— Mais alors comment savez-vous...
Il a un drôle de sourire tordu : « Je sais. »
Alicia se tourne vers Bertran : « Et vous ?
Le sourire de Bertran a une tonalité différente, presque amusée : « Je sais que je ne suis pas un passeur non plus. »
Alicia le regarde en clignant des yeux – elle devrait mettre un chapeau, elle va se faire prendre : le soleil ne se soucie pas du brouillard qui n'existe pas pour lui, et tape. « Il n'y en a plus, des passeurs », remarque-t-elle, à moitié interrogative quand même.
« Oh si, potentiellement », dit Bertran en faisant une petite moue. « Mais à un moment donné on s'est rendu compte qu'il suffisait aux bloqués de se trouver sur la Mer juste avant son départ, ou après son retour : l'induction a lieu tout aussi bien sans passer. Sauf que, une fois qu'elle a eu lieu et qu'ils sont devenus des télépathes, la Mer devient dangereuse pour eux. Non seulement ils ne peuvent plus passer mais les séquelles neurologiques sont parfois mortelles. Après les premières tentatives, elle les rejette maintenant, tout simplement. Entre-temps, ils ont fini par apprendre que, de l'autre côté, leurs descendants perdaient toutes leurs capacités et redevenaient même des bloqués. Ça les a pas mal... refroidis. Et finalement, – il jette un petit coup d'œil à Fergusson – ils se sont laissé persuader de consacrer leurs dons... à la révolution. Enfin », rectifie-t-il avec une intonation un peu ironique, « au travail qui devait être fait de ce côté-ci, puisque ce sont presque tous de puissants téléps. »
Fergusson murmure, avec moins d'animosité que ne l'aurait cru Alicia, puisque de toute évidence Bertran et lui poursuivent leur affrontement larvé : « Ils sont plus utiles ici, non ? »
Bertran dit : « Sans doute. »
Alicia écarte leurs sous-entendus indéchiffrables ; cette querelle ne la concerne pas, lui semble surprenante, décevante même – politicailleries. D'un autre côté, ce serait presque rafraîchissant. Rassurant. Ces deux hommes peuvent bien être capables de plonger dans la Mer sans en être affectés, ce sont après tout des êtres humains...
Et puis non, elle a beau essayer, le contraste est trop grand entre leurs petites mesquineries et les énormités dont ils discutent, les véritables dimensions de ce nouveau paysage la dépassent, elle n'arrive pas à s'y installer. Elle proteste à mi-voix en songeant de nouveau au livre de Leray : « Tout vrai ? Des passeurs, des Rêveurs, des... des immortels ? » Elle entend sa propre incrédulité, elle sent qu'elle hausse les épaules.
« Ce serait plus facile, n'est-ce pas, si ce n'était pas vrai, dit Fergusson avec une pointe d'amertume.
— On peut toujours choisir ce qu'on croit, remarque Bertran.
— Choisir ? » Elle éclate d'un rire scandalisé : « Comment ?
— Oh, on se débrouille toujours, murmure Fergusson. C'est une question de survie. »
Tiens, ils sont d'accord, maintenant ? Alicia se tourne vers Bertran, indécise : « Qu'est-ce que vous croyez, vous ? »
Un bref rire muet fait tressauter les épaules de Bertran : « Je crois ce que je peux. Pour le reste, je cherche.
— Vous apprenez », dit Alicia avec un sourire hésitant ; elle se rappelle leur conversation dans les ruines de Dnaõzer.
Fergusson marmonne : « Et vous avez beaucoup de choses à apprendre ?
— Oh », dit Bertran, et son sourire est redevenu mélancolique, « j'ignore énormément de choses, et il y en a plus encore que je ne comprends pas. » Puis, de nouveau ironique : « Pas vous ? »
Ils se toisent, un échange incompréhensible ; Fergusson hausse les épaules et se détourne. Un peu agacée, Alicia regarde la Mer et bascule de nouveau dans une fugue de stupeur incrédule. Elle est en train de voyager sur la Mer ! En désespoir de cause, elle essaie de s'accrocher à quelque chose de solide, de familier – tactique, stratégie, plans – et demande, « Qu'est-ce qui va se passer maintenant ? », mais en réalité elle demande ce qui va se passer pour elle, comment elle réussira à vivre dans ce nouveau décor qui n'arrête pas de se ramifier dans toutes les directions, comme un fractal en folie ; elle se dit, à la fois ironique et catastrophée, qu'elle en sait définitivement trop pour son propre bien.
Comme elle l'espérait, cependant, Bertran enchaîne sur des préoccupations plus immédiates : « Eh bien, on est revenu à peu près à la situation précédente, les Fédéraux dans leur coin, les rebelles dans le leur, et à l'autre bout du triangle, hors-jeu, on le souhaite, Lagrange. De ce côté-ci, le livre continue à circuler, les enfants d'Iptit aussi. De l'autre côté... qui sait ? Pas de passeurs, en tout cas. Peut-être qu'ils avaient besoin des passeurs de ce côté-ci, de l'influx de sang nouveau, pour ainsi dire. La mutation évolue de façon si capricieuse... Et maintenant que le contact a été délibérément interrompu...
— Il peut être rétabli n'importe quand, sans doute, dit Fergusson comme à regret. Mais je ne suis pas sûr de l'accueil qui serait fait ici à des passeurs venus de l'autre côté.
— S'il y en avait, ce serait plus difficile à nier, non ? remarque Alicia. Le livre de Leray... c'est un peu court. »
Fergusson la dévisage et se met à rire tout bas.
« Quoi ? dit-elle.
— Vous pensez toujours que c'est une fiction.
— Une biographie fictive, réplique-t-elle en haussant les épaules, butée. Il y a eu des contacts entre les deux côtés, de toute évidence, au moment des passages de la Mer, des informations ont été échangées. On a choisi ce moyen de les diffuser sur Virginia, avec tout le reste. »
Le vieil homme semble toujours amusé : « Vous admettez que des passeurs sont partis d'ici, mais pas que l'inverse ait eu lieu, dit-il en secouant la tête.
— Et vous ? » rétorque Alicia, agacée.
Bertran vient à la rescousse – de qui exactement ? « On croit ce qu'on peut. Vous en discuterez avec l'auteur. Nous passerons chez lui. »
Fergusson s'est raidi : « Ce n'était pas prévu.
— Vous y voyez un inconvénient ? » demande Bertran, très calme.
Le silence se prolonge. Puis, comme s'il venait de prendre une décision, le vieil homme murmure : « Non. »
Bertran hoche la tête, puis se penche avec une sollicitude inquiète vers Alicia, qui s'accroche brusquement au bastingage car la tête lui tourne un peu : « Ça va ? »
Elle dit, « Oui » – que dire d'autre ? – « C'est juste la fatigue.
— Vous devriez aller vous reposer », dit Fergusson, avec une gentillesse inhabituelle.
Elle décide qu'il a raison.
55
Ils abordent vers minuit au cap Termaine, où ils montent les tentes et s'endorment, épuisés. Le lendemain matin arrive une camionnette d'un vieux modèle, mais entretenue avec amour, et fort confortable. Ils remontent plein nord le long du cap. Il fait un temps radieux, sous un ciel presque bleu. Le paysage est austère, à cette latitude – montagnes râpées à l'ouest, à l'est les collines de plus en plus abruptes qui s'élèvent vers le plateau de la Licornia proprement dite, entre les deux, une steppe herbeuse et ses rares arbustes. Ils roulent sur une route des Anciens, à une seule voie, mais dallée d'imperturbable paragathe : les Anciens – Alicia se force en vain à penser " les Ranao " – s'étaient répartis dans tout le continent bien plus également que ne l'ont fait leurs successeurs. Les nouveaux habitants du sud-est en ont fait autant. Les replis des collines dévoilent de temps à autre des villages de nouveau vivants, entourés de soigneuses cultures en terrasses, de canaux d'irrigation, de quelques boisés, de petits enclos où des cabals paissent, fraternellement, avec des vaches et des chevaux. Une voie de chemin de fer, inattendue, ondule entre montagne et plateau.
Après deux heures de voyage, Alicia aperçoit un troupeau de licornes accroché à l'une des collines entre lesquelles passe la route. Une vingtaine d'adultes, une dizaine de petits, sans doute un clan, car les teintes et les motifs de leurs robes se complètent harmonieusement. Elles ne bougent pas quand la camionnette longe leur prairie, lèvent à peine la tête. Alicia les contemple longuement, le nez sur la vitre, retombée sans honte en enfance. C'est le cadeau ultime de Virginia, la confirmation ultime. Quand elle était petite, elle n'avait jamais cru à la réalité des sims, au contraire des autres enfants – elle en savait déjà trop sur l'envers des décors. Elle pensait toujours confusément que si elle le désirait assez fort elle parviendrait à traverser les sims et à se retrouver de l'autre côté, là où ils étaient vrais. En voyant les licornes, pour la première fois depuis qu'elle se trouve sur Virginia, elle a le sentiment que c'est arrivé. Elle est arrivée.
Pour se rendre chez Nathan Leray, il faut longer le plateau vers le nord puis le traverser plein est : la présence de la Mer rend la côte impraticable là aussi. Ils arrivent dans la fin de l'après-midi à Esperanza, un petit village au bord de la Mer : une vingtaine de demeures anciennes autour d'une place où se dressent les omniprésents arbres-à-eau, vraiment adaptés à tous les climats ; derrière le village, un pylône est bizarrement planté de guingois, aux trois quarts enfoui dans la pente rocailleuse, avec sa sphère qui brille entre les coulées de paragathe.
On vient à leur rencontre quand la camionnette s'arrête devant l'entrée voûtée de la troisième demeure. Quelques enfants, quatre femmes, dont une dans un fauteuil roulant, une demi-douzaine d'hommes d'âges divers. L'un d'eux est plus grand, maigre, un peu voûté – plus vieux aussi, ses cheveux drus sont tout blancs. Il s'avance plus vite que les autres mais, à mesure qu'il s'approche, il ralentit, pour s'arrêter enfin devant Fergusson qu'il dévisage avec intensité. Il murmure enfin : « Dutch ? Je te croyais...
— Mort », complète le vieil homme. La tête renversée en arrière, il contemple l'autre avec une expression à la fois lasse et soulagée : « C'est une très longue histoire. Je te la raconterai demain, tu veux ? Maintenant, j'aimerais... me reposer. »
Il s'éloigne vers l'entrée, tandis que l'homme le suit du regard, comme foudroyé.
Alicia hésite, puis, comme Bertran ne dit rien, elle demande : « Nathan... Nathan Leray ? »
L'homme se retourne vers elle. Il a des yeux très pâles sous les mèches blanches – non, pas blanches, c'est un de ces blonds qui blanchissent au soleil ; pas aussi vieux qu'il en a l'air, le milieu de la quarantaine tout au plus. Étranges, ces yeux, pas vraiment la couleur – gris-bleu très atténué, comme ceux de Fergusson – mais la pupille, plutôt ovale.
L'homme a un sourire un peu embarrassé et Alicia se sent presque rougir d'être surprise à l'examiner ainsi : « Ici, je m'appelle seulement Lian Flaherty », dit-il. Il lui tend les mains : « Vous devez être Alicia Coralàn. »
Cette nuit-là, Bertran s'en va sans rien dire à personne. Il a laissé un cadeau sur la table de la salle commune, enveloppé dans un grand carré de soie bariolé, souvenir du cirque, avec un morceau de papier portant le nom de Simon Fergusson au-dessus de sa propre signature. Ils le trouvent en descendant prendre le petit-déjeuner, le lendemain matin. Le vieil homme déplie le tissu, dévoilant une sphère de métal argenté, grosse comme deux poings et incrustée de trois cabochons de paragathe rose assez rapprochés, qui lui servent de trépied.
Fergusson se laisse tomber sur la chaise la plus proche.
« Ça va ? » demande la femme au fauteuil roulant, inquiète – Jillian, elle s'appelle Jillian. Alicia regarde le vieil homme prendre la sphère et en couvrir les cabochons de sa main droite. Elle les voit s'illuminer, entend quelques exclamations étouffées de surprise parmi les personnes présentes. Reconnaît sa propre incompréhension sur le visage de Flaherty qui est entré en dernier. Fergusson a replacé la sphère sur le carré de soie et la contemple, comme paralysé. Les cabochons sont de nouveau éteints.
Flaherty s'approche, pose les deux mains sur la table en face de lui, se racle la gorge : « Dutch ? »
Le vieil homme tressaille, lève vers l'autre un regard d'abord aveugle, puis son expression hébétée se dissipe, remplacée par une résignation faiblement teintée d'ironie : « Une très longue histoire », murmure-t-il en hochant la tête.
56
Lian descend dans le chemin qui traverse les prairies suspendues en escalier derrière la maison. Les humains font la méridienne, l'heure et demie qui en reste, mais pas les tapoches : elles se dressent à son approche et se battent le poitrail en sifflant. Il siffle en retour. Elles l'observent un moment, puis retournent rassurées dans leur terrier. Il doit y en avoir en train de nidifier. Lian sourit : c'est l'Hiver, mais à cette latitude, si près de l'équateur, il y a toujours quelque chose pour naître et fleurir, mourir ou porter fruit.
Tout le monde est allé se coucher, parce qu'on a sauté la majeure partie de la méridienne pour assister au départ de la Mer emportant les premiers passeurs virginiens depuis cent dix saisons. Lian n'arrive pas à dormir, comme s'il était encore pris dans la turbulence, l'appel d'air de cette porte enfin rouverte sur l'autre rivage qu'il ne reverra pas. Il se refuse la mélancolie facile du " plus jamais ", continue à descendre de prairie en prairie par le chemin rocailleux, arrive à la dernière, maintenant suspendue sur l'absence de la Mer, et s'installe à côté du vieil arbre-Gomphal qui en occupe le centre. Il n'a pas encore commencé à faire ses enfants, cet arbre, mais il est ancien. On a voulu le symbole pour ce départ de nouveaux émissaires vers l'Autre Côté : ils sont partis du même endroit que le premier passeur. C'est l'arbre de Mathieu, la prairie de Mathieu, tout en haut de la pente se dresse la demeure d'où Mathieu Janvier est sorti, il y a si longtemps, sans savoir qu'il allait traverser la Mer et devenir Matieu Odatan Oghimiu. Les Bordes et même leurs descendants l'ont quittée depuis longtemps ; d'autres familles s'y sont installées, et maintenant une commune d'une quarantaine de personnes ; mais la maison est pleine à craquer, avec la célébration dont on a accompagné le départ ; les deux premières prairies, en haut, sont couvertes de tentes ; voitures et carrioles s'amassent sur le terre-plein devant la maison.
Lian a supporté avec patience les cérémonies – ils ne lui ont pas demandé de faire un discours, heureusement – mais vers la fin, la tête lui tournait. Trop de gens. Il en a presque complètement perdu l'habitude à Esperanza, dans leur quasi-désert du bout du monde. Mais il ne pouvait pas ne pas venir en réponse à l'invitation, n'est-ce pas ?
Il s'installe avec un soupir, les bras autour des genoux, contemple le paysage en contrebas, la terre nue des autres prairies suspendues, puis les collines toutes nues aussi qui ondulent vers l'océan invisible au loin. D'ici quelques semaines, la vie aura explosé de nouveau sur le territoire abandonné par la Mer, les herbes auront poussé, insectes et bestioles seront sortis de leur sommeil souterrain, puis les oiseaux reparaîtront par vagues successives et enfin, vers la fin du mois de Juillet, des licornes commenceront par petits groupes leur lente migration vers leurs terres ancestrales, le long de l'isthme de Shandaar, par le défilé de la Hache à travers la barrière des monts Barth et jusque dans les plaines centrales du continent Est. De Paalu, redevenu Aalpaalu de l'autre côté, avec le retour de la Mer. Là-bas, bientôt, les bateaux des hasallim commenceront à voguer vers les îles et leurs Krilliadni, pour les Grandes Chasses.
Lian se redresse soudain, heureux de la diversion, malgré tout : il a senti le contact léger, curieux, voit presque en même temps la licorne apparue entre les arbres dans le chemin tracé par ses congénères, au bord de la Mer absente. Il lui rend son salut, avec un soupir résigné. Son don à éclipses s'est remis à fonctionner tout d'un coup. Encore heureux que ce soit maintenant et pas tout à l'heure, pendant les cérémonies, ou quand la Mer est partie. La dernière fois que ça lui est arrivé, après un retour, pendant la période où la Mer était encore active, il a eu des migraines absolument épouvantables.
Un don qui a des hauts et des bas. Il sourit au souvenir de Nathénèk. Il n'a pas toujours été aussi philosophe. Se réveiller dans la voiture de Dutch, en route vers le sud, et se rendre compte qu'il ne percevait absolument plus rien... Il était encore sous le choc, heureusement, le choc de sa rencontre avec la Mer. Mais avec le temps, il a tout de même fini par accepter sa cécité comme allant de soi – un juste échange, cette éternité fulgurante de savoir contre son don calciné, tout le reste de sa vie à l'explorer, à essayer de comprendre. Revenir dans l'univers du temps, étirer ce point unique, sans dimension, dans l'espace de la causalité, de la séquentialité, des mots...
Oh, il a dû réécrire des dizaines de fois le chapitre où il relatait son contact avec la Mer. Il a gardé les premières versions, les plus proches de l'illumination, non qu'elles soient nécessairement les plus fidèles – il a abandonné cette illusion depuis longtemps – mais pour leurs enchaînements elliptiques, fiévreux, leur vocabulaire approximatif où se fixait et se perdait du même mouvement le souvenir. Pour lui-même, pour se souvenir du souvenir. Pour les autres, il a fini par renoncer, honteux mais résigné : il a recréé un analogue lointain, un reflet, une simulation vraisemblable, acceptable. À peine acceptable – il sourit ; il se rappelle encore mot pour mot la critique parue dans El Pueblo, le grand quotidien de Cristobal : " [...] imitation servile mais maladroite de techniques archaïques, pseudo-flot de conscience à la ferveur apprêtée, suspension infantile de la ponctuation, l'auteur n'a de toute évidence pas compris que, si ce qui se conçoit bien ne s'énonce pas toujours aisément, ce qui se conçoit mal n'a vraiment rien à gagner d'une incohérence doublement redondante ". Le " doublement redondante " l'avait plié en deux de rire, mais en même temps il devait admettre que la période aurait moins bien ronflé sans l'adverbe.
Non, il rit maintenant, mais il ne riait pas de si bon cœur à ce moment-là. C'était important, à ce moment-là, de dire, d'être entendu. Oh, il a essayé la description objective ; il voulait tout inclure, le moindre détail de ce qu'il avait connu avec la Mer, et dans l'ordre, depuis sa naissance dans un univers lointain jusqu'au moment où Oghim l'avait touchée, bien loin de chez elle déjà, éveillant sa curiosité... Mais chaque tentative chronologique a abouti à un échec. L'instantané de la révélation première était un labyrinthe sans cesse jaillissant de lui-même, un infini jeu de miroirs. La naissance même de la Mer se multipliait dans d'innombrables univers où elle s'était vue naître, où elle s'était vue s'éveiller à l'intelligence, où elle avait ensuite maîtrisé le déclic qui la projetait dans un autre univers, et un autre, et un autre, à volonté. Et finalement la voix même de la Mer – pas une voix, il l'avait simplifié ainsi pour pouvoir l'écrire, il y pensait désormais ainsi – éveillait à chaque instant l'écho de toutes ses autres voix, dans tous les autres univers où elle avait laissé des doubles en se divisant pour s'enfanter. La Mer accueillie par les anciens Ranao était repartie depuis longtemps, laissant sa fille, sa sœur, son autre elle-même, satisfaire une curiosité née ailleurs, dans un autre univers immensément lointain, tout proche. Et en même temps elle était là, comme tous les autres, toujours en contact, bien sûr : y a-t-il une partie de l'esprit qui ne soit en contact avec tous les autres, à tout moment ?
Il se rappelle, à un moment il avait pensé Hananai ? avec une terreur sacrée. Et il l'avait entendue, il les avait entendues, rire, dans tous les univers. C'était seulement la Mer, et elle savait rire. Du coup, il n'était pas devenu fou.
Pas fou, mais aveugle, mais sourd, mais muet. Trois jours de coma, puis le réveil dans la voiture et tout le voyage jusqu'au fin fond du sud, et la traversée du Fjord. Pas un mot. Il n'avait pas dit un mot pendant des semaines. Il ne pouvait pas. Il ne voulait pas. S'il parlait, il basculerait de nouveau et pour toujours dans le monde mortel, la révélation se ternirait s'il laissait le temps la toucher, et le temps, c'étaient les mots, la mort, c'étaient les mots.
Puis, pendant quelques secondes, la lumière est revenue. Le don est revenu. Il les a vus, tous ceux qui se trouvaient avec lui dans la maison au bord de la Mer, Dutch, Martel, Sélina, Jhotto, Jill, Sirban...
Jill ?
La lumière s'est éteinte à cet instant, mais il a répété « Jill ? » et il est retombé dans l'univers du temps. Il aurait préféré que ce fût pour une raison plus spectaculaire, ou plus signifiante, mais non, c'était seulement Jill – stupeur, joie, culpabilité, colère, rancune, sentiments transitoires des humains transitoires, pour lui rappeler qui il était, d'où il venait, et qu'il ne pouvait demeurer seul à jamais dans l'illumination de la Mer. Il a même fini par discerner là un dessin satisfaisant comme un dessein : cette femme l'avait trouvé au bord de la Mer, il renaissait une nouvelle fois à Virginia en la retrouvant, la boucle était bouclée. Qu'elle ne fût ni sa mère, ni sa sœur ni son amante, seulement une femme qu'il ne connaissait en fin de compte pas du tout, même s'il avait vécu deux saisons en sa compagnie, cela diminuait un peu la portée du symbole, mais on fait avec ce qu'on a. Il essayait d'écrire sa biographie, et non – ou du moins pas délibérément – une fiction.
Il se met à rire tout seul, un rire qui se perd dans un bâillement. Il s'étire, les mains derrière la tête, recule pour s'adosser plus confortablement au tronc du Gomphal et appuie sa nuque contre le tronc. Il le perçoit – la lumière est restée avec lui, ce sera un jour avec, peut-être. L'arbre est une présence diffuse, une curiosité très lente – ils n'existent pas dans la même durée que les humains, comme si la Mer, en leur donnant vie, avait voulu créer une image bizarrement inversée d'elle-même : des êtres qui se déplacent dans l'espace et le temps, mais dans un seul univers, sur un seul continent, pendant des siècles, et dont les pensées coulent moins vite que la sève.
Lian se laisse glisser dans l'état d'apesanteur qui précède le sommeil, laissant ses propres pensées flotter au hasard et redevenir des sensations tout juste au bord du langage, prêtes à couler sous le seuil, images, sons, les visages des cinq passeurs, surtout la jeune fille, Ivania, radieuse ; la foule entonnant le chant de la Mer, le chant lui-même, et voilà qu'il se mêle à l'autre, celui de sa première rencontre avec la Mer, les images virent et deviennent celles de son enfance, où se précisent soudain le visage rond de Nathénèk, la démarche vive de Laraï, a-t-il bien fait de remettre un exemplaire de son livre aux passeurs, est-il encore temps de dire, à travers les Années ? Voudront-ils entendre ? Pourront-ils entendre ?
Il a recommencé à penser – la rive du sommeil s'éloigne. Au même moment, il sent quelque chose bouger sous sa main, tressaille. Avec un soupir, il ouvre les yeux.
À côté de lui, les racines-pseudopodes de l'arbre-Gomphal se sont emparées d'un petit caillou, qu'elles poussent avec lenteur sur la terre rocailleuse, entre deux taches de fine herbe bleuâtre.
Lian s'écarte, un peu surpris. Après plusieurs minutes, plusieurs cailloux, quelque chose se dessine, une ligne vaguement courbe. Lian observe, fasciné. La lumière du don s'est éteinte de nouveau, il ne perçoit plus l'arbre-Gomphal, mais les racines-pseudopodes continuent leur manège. Il y a maintenant un demi-cercle de cailloux entre les deux touffes d'herbe.
Lian sourit, à la fois ému et incrédule. L'arbre a-t-il pu cueillir cette image en lui, tout à l'heure ? Le souvenir de ses efforts obstinés pour sauver un petit Gomphal qui n'en avait cure ? En l'enfermant pour l'obliger à se sauver. Quelle logique étrange. Mais il la comprend toujours, il la comprend mieux que jamais. Il regarde le ciel laiteux de l'après-midi. Quelque part, loin, très loin au-delà des nuages, dans le noir de l'espace, il y a des gens qui se sont enfermés pour se sauver, et combien de temps leur faudra-t-il pour comprendre qu'ils peuvent sortir eux-mêmes de leur prison ? Mais ce sont les plus difficiles à quitter, n'est-ce pas, les cages qu'on chérit parce qu'on les a édifiées soi-même. Il sait. Il se rappelle les siennes.
Hop, la lumière s'est rallumée, il peut sentir l'intention patiente du Gomphal. Ah, non, pas un jour en pointillé ! Il se redresse, un peu chagrin, entend au même moment un rire de bébé et la voix alarmée d'Alicia : « Maria, attends-moi, Maria ! »
Il se retourne, voit la petite qui titube dans le chemin entre les rochers, les bras tendus pour garder l'équilibre. Il se lève, inquiet à son tour, mais les petites jambes en caoutchouc réussissent à rester tendues, et l'enfant arrive dans l'herbe de la prairie – le changement de texture la fait trébucher, pour le coup, et elle se retrouve assise sur son derrière avec de grands yeux étonnés. Alicia l'attrape en riant et la secoue : « Friponne-friponne ! ». Puis, la fourrant sans cérémonie sous son bras, appuyée sur sa hanche – elle ne peut plus la porter autrement – elle rejoint Lian devant l'arbre-Gomphal.
« La méridienne est déjà finie ?
— Tu n'as pas dormi, hein ? accuse-t-elle en souriant. Tu vas être frais ce soir, au banquet. »
Il pousse un gémissement d'agonie à moitié feint seulement. Il avait oublié le banquet.
Alicia dépose la petite dans l'herbe, où l'enfant choisit un mode d'exploration plus sécuritaire, à quatre pattes. Puis la jeune femme se redresse, les mains sur les reins. Lian effleure d'un doigt son ventre protubérant sous le mince voile de coton. « Ça va ? Tu as dormi, toi ?
Alicia dit : « Un peu. Maria s'est réveillée avant moi. »
Il l'aide à s'asseoir, ce qu'elle accepte avec un petit " tsk " agacé, comme toujours, mais il ne peut s'en empêcher. Elle est à peine aux deux tiers de sa grossesse, mais elle est si petite, et si ronde... Ils vont avoir des jumeaux, il n'arrive pas à se faire à l'idée, c'est magique ! Alicia trouve sans doute la chose un peu moins merveilleuse, surtout par cette chaleur. Elle se laisse aller avec un soupir contre le tronc de l'arbre, enlève son chapeau et s'évente. « Je serai contente quand ce sera fini », dit-elle, et il ne sait pas si elle parle de la grossesse ou du banquet, mais il acquiesce en lui embrassant la main. Elle le regarde un moment, les paupières mi-closes. « Ça va, toi ? » demande-t-elle avec tendresse. Il sait ce qu'elle veut dire. Il répond « Oui », et c'est vrai, la légère mélancolie de tout à l'heure s'est dissipée.
Ils regardent la petite qui leur fait face, accroupie sur ses jambes potelées pour examiner avec gravité une tige d'herbe dentelée par une chenille. Elle a les sourcils et les yeux de Graëme, c'est certain, les paupières, la profondeur noire du regard. Peut-être la bouche. Pour le reste, non, ce sera Alicia. Les pensées de Lian prennent toujours la même pente, dans un premier temps, quand il observe l'enfant ; il n'essaie pas de les en empêcher. C'est assez loin pour lui, maintenant, assez longtemps, il voit tout autrement – il aime à penser " plus clairement ", mais il ne laisse jamais cette hypothèse devenir tout à fait une certitude. Un jour, ce sera sans doute pareil pour Alicia.
Elle le surprend. Elle murmure : « Il est toujours là quelque part... » Elle tourne la tête vers lui : « Crois-tu qu'il viendrait, un jour ? Ici ? »
Pas d'angoisse dans sa voix, juste une question, pensive. Lian regarde les racines-pseudopodes de l'arbre-Gomphal, près de lui, qui continuent à rouler le dernier petit caillou qui fermera le cercle. Graëme a survécu tellement de fois, à tellement d'histoires. Peut-être qu'il accepterait de vivre, à la fin. Les risques de la vérité.
« L'accueillerais-tu ? » demande-t-il à mi-voix.
Alicia réfléchit, le front plissé, puis répond, toujours honnête : « Maintenant, non. Mais plus tard... je ne sais pas. S'il était capable de venir ici... Je ne sais pas. » Et maintenant elle est un peu anxieuse, ses yeux cherchent ceux de Lian – elle n'est pas sûre de ce qu'elle est en train de dire, de ce qu'il est en train d'entendre, elle demande, un ton plus bas, hésitante : « Et toi ? »
Lian réfléchit à son tour. Il prend le caillou qui fermait le cercle, le jette, en fait autant d'un autre à côté. Et finit par dire aussi « Je ne sais pas », mais il sourit, ce n'est pas de l'inquiétude, pas même de l'incertitude, seulement la sereine affirmation des possibles, han'maï, sous le regard de Hananai.
La petite a ramassé le caillou. Elle le retourne entre ses doigts, le considère un moment, les sourcils froncés, puis le met dans sa bouche. Lian bondit avant Alicia, mais déjà la petite a décidé que la sensation ne lui convenait pas et lui tend le caillou luisant de salive. Il la prend dans ses bras en riant et se rassied sous l'arbre. Elle essaie de se dégager : elle a vu bouger les racines-pseudopodes, elle veut aller voir. Lian la laisse faire. L'enfant s'accroupit près des racines, fascinée. Près d'eux, avec lenteur, avec délibération, le Gomphal finit de démanteler le cercle.
## Remerciements
Le récit qui commence avec ce volume est mon premier rêve de science-fiction qui se soit transformé en une histoire, le premier que j'aie écrit – et réécrit, et réécrit... En trente ans, il a subi bien des métamorphoses en même temps que moi. Mais certaines de ces métamorphoses lui sont venues plus spécifiquement de rencontres, et je désire remercier ici ces visiteuses et ces visiteurs après lesquels le paysage se réorganisait autrement.
Dans l'ordre d'apparition : René Ferron-Wherlin, Jean-Joël Vonarburg, François Duban, Bertrand Méheust, Aliocha Kondratiev, Danielle Martinigol, Bruno Chaton, Maximilien Milner, René Beaulieu, Serge Mailloux, Gérard Klein (pour les licornes), Daniel Sernine, Jean-Claude Dunyach, Wildy Petoud, Joël Champetier, Jean-François Moreau, Yves Meynard, Jean Pettigrew, Sylvie Bérard, Denis Rivard.
Et enfin, et surtout, le dernier visiteur, la source des ultimes métamorphoses – les plus essentielles – Norman Molhant, écosystématicien et encyclopédie extraordinaire. Plongeant avec abnégation dans mon paysage au détriment du sien, il m'a donné l'occasion d'éprouver ce rare plaisir, que seule la science-fiction sait m'offrir, de voir mes fantasmes et mes rêves correspondre parfois à ceux de l'univers. Sans lui, cette histoire n'aurait jamais été ce qu'elle devait être. Si elle ne l'est pas, j'en suis seule responsable.
## Biographie
Élisabeth Vonarburg est une des figures les plus marquantes de la science-fiction québécoise. Elle est reconnue tant dans la francophonie que dans l'ensemble du monde anglo-saxon (plus d'une vingtaine de prix majeurs ont couronné ses ouvrages, de Paris à New York en passant par Chicoutimi) et la parution de ses ouvrages est toujours considérée comme un événement. Outre l'écriture de fiction, Élisabeth Vonarburg pratique la traduction (La Tapisserie de Fionavar, de Guy Gavriel Kay, Le Jeu de la passion, de Sean Stewart...), s'adonne à la critique (notamment dans la revue Solaris) et à la théorie (Comment écrire des histoires, La Lignée). Depuis 1973, Élisabeth Vonarburg a fait de la ville de Chicoutimi son port d'attache. En 1998, le Conseil du Statut de la Femme lui décernait le prix « Femmes et littérature » et, en 2007, elle recevait le Prix à la création artistique du Conseil des Arts et des Lettres du Québec, région Saguenay–Lac-Saint-Jean.
De la même auteure chez Alire :
Le Silence de la Cité. Roman.
Beauport, Alire, 1998.
Chroniques du Pays des Mères. Roman.
Beauport, Alire, Romans 027, 1999.
Les Voyageurs malgré eux. Roman.
Lévis : Alire, Romans 124, 2009.
La Maison au bord de la mer. Nouvelles.
Beauport : Alire, Nouvelles 037, 2000.
Le Jeu des coquilles de nautilus. Nouvelles.
Lévis : Alire, Nouvelles 070, 2003.
Sang de pierre. Nouvelles.
Lévis : Alire, Nouvelles 128, 2009.
Tyranaël :
1. Les Rêves de la Mer. Roman.
Beauport : Alire, Romans 003, 1996.
2. Le Jeu de la Perfection. Roman.
Beauport : Alire, Romans 004, 1996.
3. Mon frère l'ombre. Roman.
Beauport : Alire, Romans 005, 1997.
4. L'Autre Rivage. Roman.
Beauport : Alire, Romans 010, 1997.
5. La Mer allée avec le soleil. Roman.
Beauport : Alire, Romans 012, 1997.
Reine de Mémoire :
1. La Maison d'Oubli. Roman.
Lévis : Alire, Romans 085, 2005.
2. Le Dragon de Feu. Roman.
Lévis : Alire, Romans 090, 2005.
3. Le Dragon fou. Roman.
Lévis : Alire, Romans 095, 2006.
4. La Princesse de Vengeance. Roman.
Lévis : Alire, Romans 100, 2006.
5. La Maison d'Équité. Roman.
Lévis : Alire, Romans 101, 2007.
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"redpajama_set_name": "RedPajamaBook"
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Q: Sending an image by http with node.js I'm trying to send an image by HTTP with node.js.
I can send HTML, text but, not images.
Here's a sample of what I'm trying to do:
request(options, function(error, response, body) {
http.createServer(function(req, res) {
res.statusCode = response.statusCode;
res.writeHead(res.statusCode, { 'Content-Type': response.headers['content-type'] });
res.write(body);
res.end();
}
});
and all I get is:
The same happens when I try an icon...
If I wget the image I get the right content I guess, but it doesn't show up in the image viewer either... I don't know how to fix this.
Any idea?
Thanks!
A: Make sure body is a buffer and not an encoded representation of the image (unless it's encoded in such a way that common browsers can decode it if you send out an appropriate Content-Encoding header, in which case you should (you guessed it!) send out the appropriate header.
In particular, don't try to read the image with fs.* methods where you asked for the responses to be UTF-8 and then try to send it "raw".
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Q: Error using Animation package R Tried to run the following code in R:
ani.options(
convert = shQuote("C:/Program Files/ImageMagick-6.9.0-Q16/convert.exe")
)
saveGIF(
{
for (i in 1:10) plot(runif(10), ylim = 0:1)
},
movie.name = "test.gif",
interval = 0.2,
ani.width = 300,
ani.height = 300,
outdir = getwd()
)
Getting the error message:
Executing:
"\"C:/Program Files/ImageMagick-6.9.0-Q16/convert.exe\"" -loop 0 -delay 20 Rplot1.png Rplot2.png Rplot3.png
Rplot4.png Rplot5.png Rplot6.png Rplot7.png Rplot8.png Rplot9.png Rplot10.png "test.gif"
The filename, directory name, or volume label syntax is incorrect.
an error occurred in the conversion... see Notes in ?im.convert
[1] FALSE
Warning messages:
1: running command 'C:\Windows\system32\cmd.exe /c "\"C:/Program Files/ImageMagick-6.9.0-Q16/convert.exe\"" --version' had status 1
2: running command 'C:\Windows\system32\cmd.exe /c "\"C:/Program Files/ImageMagick-6.9.0-Q16/convert.exe\"" -loop 0 -delay 20 Rplot1.png Rplot2.png Rplot3.png Rplot4.png Rplot5.png Rplot6.png Rplot7.png Rplot8.png Rplot9.png Rplot10.png "test.gif"' had status 1
3: In cmd.fun(convert) :
'"\"C:/Program Files/ImageMagick-6.9.0-Q16/convert.exe\"" -loop 0 -delay 20 Rplot1.png Rplot2.png Rplot3.png Rplot4.png Rplot5.png Rplot6.png Rplot7.png Rplot8.png Rplot9.png Rplot10.png "test.gif"' execution failed with error code 1
4: running command '"\"C:/Program Files/ImageMagick-6.9.0-Q16/convert.exe\"" -loop 0 -delay 20 Rplot1.png Rplot2.png Rplot3.png Rplot4.png Rplot5.png Rplot6.png Rplot7.png Rplot8.png Rplot9.png Rplot10.png "test.gif"' had status 127
Could someone please point on my mistake.
A: I had the same issue, and it seems it is due to the path to convert.
If I type:
ani.options(convert = shQuote('C:/Program Files/ImageMagick-6.8.7-Q16/convert.exe'))
And then check the path
ani.options("convert")
[1] "\"C:/Program Files/ImageMagick-6.8.7-Q16/convert.exe\""
However, if I do:
ani.options(convert = 'C:/Program Files/ImageMagick-6.8.7-Q16/convert.exe')
ani.options("convert")
[1] "C:/Program Files/ImageMagick-6.8.7-Q16/convert.exe"
It is the correct path and saveGIF works.... weird things...
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"redpajama_set_name": "RedPajamaStackExchange"
} | 4,245 |
{"url":"https:\/\/gateoverflow.in\/4173\/gate2012-cy-ga-7","text":"6.8k views\n\n$A$ and $B$ are friends. They decide to meet between 1:00 pm and 2:00 pm on a given day. There is a condition that whoever arrives first will not wait for the other for more than $15$ minutes. The probability that they will meet on that day is\n\n1. $1\/4$\n2. $1\/16$\n3. $7\/16$\n4. $9\/16$\n\nedited | 6.8k views\n\nMeeting occurs if the first person arrives between $1:00$ and $1:45$ and the second person arrives in the next $15$ minutes or if both the persons arrive between $1:45$ and $2:00.$\n\nCase 1:\n\n\u2022 $45\/60$ are favourable cases and hence probability of first person arriving between $1:00$ and $1:45$ is $3\/4.$\n\u2022 Probability of second person arriving in the next $15$ minutes $= 15\/60 = 1\/4$\n\u2022 So, probability of one person arriving between $1:00$ and $1:45$ and meeting the other $= 3\/4 \\times 1\/4 \\times 2 = 3\/8$ $(2$ for choosing the first arriving friend$)$\n\nCase 2:\n\n\u2022 Both friends must arrive between $1:45$ and $2:00.$ Probability $= 1\/4 \\times 1\/4 = 1\/16.$\n\nSo, probability of a meet $= 3\/8 + 1\/16 = 7\/16$\n\nCorrect Answer: $C$\n\nby Veteran (422k points)\nedited\n0\nwhy not in second case u did 2C1?\n+2\nFor the second case both have to arrive after 1:45- we do not have a choice here. In the first case, we have an order- first arriving can be one among 2.\n0\n\n@\u00a0sir,\n\nBut in 1:45 to 2:00\n\nIf A comes first then B Or B comes first then A\n\ndoen't\u00a0this count to two diff cases ??\n\nFor such questions which are also known as probability based on areas , u can find easily using grid of 2 dimension as shown below\n\n 1\/2*F F 1\/2 * F F 1\/2 * F 1\/2*F F 1\/2 * F F 1\/2 * F\n\nThe favorable area for the given problem is shown above...\n\nSo no of favourable cells \u00a0= \u00a04 \u00a0+ \u00a06 * 1\/2 \u00a0 \u00a0 = \u00a0 \u00a07\n\nTherefore probability that they meet \u00a0= \u00a0No of favourable cells\/ No of total cells\n\n= \u00a0 7\u00a0\/ 16\n\nHence C) should be the correct option.\n\nby Veteran (101k points)\n0\nBut you ignore the case when A reaches first..\n0\nFor arrival of B we are considering vertical axis.\n0\nyes. But consider the first 'F' from origin. Actually, it won't be F, even if B comes after A. But you consider only B coming first.\n0\n0\n+1\nCorrected..:)\n+5\nHi habib could you explain it bit more.\n\nWe are unable to understand what those cells in table are representing ..?\nprobability that one person meet on that day = 15\/60 = 1\/4\nProb(failing to meet) = 3\/4\nprob(failing to meet by both the persons) = 3\/4 * 3\/4 = 9\/16\nProb( meet on that day by both the persons) = 1-9\/16 = 7\/16\nby Active (1.4k points)\n\nCheck this\n\nby Active (1.8k points)\n+1\n+1\nBest one :)\n+1 vote\nMark this formula for this kind of any question.\n\nx=minutes one could not wait for.\n\nn=60\/x.\n\nprobability they meet= (2n-1)\/n^2\n\nhere x=60\/15 =4.\n\nprobabi;ity=8-1\/16 =7\/16\nby (193 points)\n7\/16\nby (27 points)\nWe can derive one formula also from graph For(n*n)\n\n(n^2-(n-1)^2) \/n^2\n\nfrom given ques n=4\n\n(16-9)\/16\n\nby Active (2.1k points)\n0\nCould you explain bit more please.","date":"2019-11-14 13:30:22","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.659717321395874, \"perplexity\": 2185.6559062411297}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-47\/segments\/1573496668525.62\/warc\/CC-MAIN-20191114131434-20191114155434-00357.warc.gz\"}"} | null | null |
Watching TV With the Red Chinese DVD
Watching TV With the Red Chinese
Quirky blend of comedy and drama, set in 1980, follows the exploits of three Chinese exchange students (Leonardo Nam, James Chen, Keong Sim) living in New York City. All three are at first anxious to immerse themselves in the rich culture of a big American metropolis, but a mugging leads one of them to become more and more disenchanted with his new surroundings. Ryan O'Nan, Gillian Jacobs, Constance Wu co-star. 105 min. Widescreen; Soundtrack: English.
Title: Watching TV With the Red Chinese
Starring: Gillian Jacobs
Director: Shimon Dotan
Studio: MVD Visual
Shimon Dotan | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 2,423 |
namespace visualization_msgs
{
class InteractiveMarker : public ros::Msg
{
public:
std_msgs::Header header;
geometry_msgs::Pose pose;
const char* name;
const char* description;
float scale;
uint8_t menu_entries_length;
visualization_msgs::MenuEntry st_menu_entries;
visualization_msgs::MenuEntry * menu_entries;
uint8_t controls_length;
visualization_msgs::InteractiveMarkerControl st_controls;
visualization_msgs::InteractiveMarkerControl * controls;
virtual int serialize(unsigned char *outbuffer) const
{
int offset = 0;
offset += this->header.serialize(outbuffer + offset);
offset += this->pose.serialize(outbuffer + offset);
uint32_t length_name = strlen(this->name);
memcpy(outbuffer + offset, &length_name, sizeof(uint32_t));
offset += 4;
memcpy(outbuffer + offset, this->name, length_name);
offset += length_name;
uint32_t length_description = strlen(this->description);
memcpy(outbuffer + offset, &length_description, sizeof(uint32_t));
offset += 4;
memcpy(outbuffer + offset, this->description, length_description);
offset += length_description;
union {
float real;
uint32_t base;
} u_scale;
u_scale.real = this->scale;
*(outbuffer + offset + 0) = (u_scale.base >> (8 * 0)) & 0xFF;
*(outbuffer + offset + 1) = (u_scale.base >> (8 * 1)) & 0xFF;
*(outbuffer + offset + 2) = (u_scale.base >> (8 * 2)) & 0xFF;
*(outbuffer + offset + 3) = (u_scale.base >> (8 * 3)) & 0xFF;
offset += sizeof(this->scale);
*(outbuffer + offset++) = menu_entries_length;
*(outbuffer + offset++) = 0;
*(outbuffer + offset++) = 0;
*(outbuffer + offset++) = 0;
for( uint8_t i = 0; i < menu_entries_length; i++){
offset += this->menu_entries[i].serialize(outbuffer + offset);
}
*(outbuffer + offset++) = controls_length;
*(outbuffer + offset++) = 0;
*(outbuffer + offset++) = 0;
*(outbuffer + offset++) = 0;
for( uint8_t i = 0; i < controls_length; i++){
offset += this->controls[i].serialize(outbuffer + offset);
}
return offset;
}
virtual int deserialize(unsigned char *inbuffer)
{
int offset = 0;
offset += this->header.deserialize(inbuffer + offset);
offset += this->pose.deserialize(inbuffer + offset);
uint32_t length_name;
memcpy(&length_name, (inbuffer + offset), sizeof(uint32_t));
offset += 4;
for(unsigned int k= offset; k< offset+length_name; ++k){
inbuffer[k-1]=inbuffer[k];
}
inbuffer[offset+length_name-1]=0;
this->name = (char *)(inbuffer + offset-1);
offset += length_name;
uint32_t length_description;
memcpy(&length_description, (inbuffer + offset), sizeof(uint32_t));
offset += 4;
for(unsigned int k= offset; k< offset+length_description; ++k){
inbuffer[k-1]=inbuffer[k];
}
inbuffer[offset+length_description-1]=0;
this->description = (char *)(inbuffer + offset-1);
offset += length_description;
union {
float real;
uint32_t base;
} u_scale;
u_scale.base = 0;
u_scale.base |= ((uint32_t) (*(inbuffer + offset + 0))) << (8 * 0);
u_scale.base |= ((uint32_t) (*(inbuffer + offset + 1))) << (8 * 1);
u_scale.base |= ((uint32_t) (*(inbuffer + offset + 2))) << (8 * 2);
u_scale.base |= ((uint32_t) (*(inbuffer + offset + 3))) << (8 * 3);
this->scale = u_scale.real;
offset += sizeof(this->scale);
uint8_t menu_entries_lengthT = *(inbuffer + offset++);
if(menu_entries_lengthT > menu_entries_length)
this->menu_entries = (visualization_msgs::MenuEntry*)realloc(this->menu_entries, menu_entries_lengthT * sizeof(visualization_msgs::MenuEntry));
offset += 3;
menu_entries_length = menu_entries_lengthT;
for( uint8_t i = 0; i < menu_entries_length; i++){
offset += this->st_menu_entries.deserialize(inbuffer + offset);
memcpy( &(this->menu_entries[i]), &(this->st_menu_entries), sizeof(visualization_msgs::MenuEntry));
}
uint8_t controls_lengthT = *(inbuffer + offset++);
if(controls_lengthT > controls_length)
this->controls = (visualization_msgs::InteractiveMarkerControl*)realloc(this->controls, controls_lengthT * sizeof(visualization_msgs::InteractiveMarkerControl));
offset += 3;
controls_length = controls_lengthT;
for( uint8_t i = 0; i < controls_length; i++){
offset += this->st_controls.deserialize(inbuffer + offset);
memcpy( &(this->controls[i]), &(this->st_controls), sizeof(visualization_msgs::InteractiveMarkerControl));
}
return offset;
}
const char * getType(){ return "visualization_msgs/InteractiveMarker"; };
const char * getMD5(){ return "311bd5f6cd6a20820ac0ba84315f4e22"; };
};
}
#endif | {
"redpajama_set_name": "RedPajamaGithub"
} | 8,168 |
At home I eat very simple clean food, I love having friends over for dinner and I have a soft spot for fish and seafood.
If you believe this content violates or infringes your rights, Tom Green may not know. You could solve this problem by contacting him. | {
"redpajama_set_name": "RedPajamaC4"
} | 872 |
\section{Introduction}
\label{sec:introduction}
High-mass star formation (SF) sites (hereafter HMSFSs) are the beacons
by which we probe a large part of the physics of external galaxies.
They generally represent the most important tracers of the properties
of their host galaxies in terms of star formation rate (SFR) and
general activity. The most frequent tracers of the star-formation
activity generally use the fact that the abundant UV light coming from
the hot, young stars is absorbed in the vicinity and reradiated in the
form of line or continuum emission. This is true, for example, for
H$\alpha$ \citep[e.g.][]{1998ARA&A..36..189K}, the aromatic emission
bands in the mid-IR \citep[e.g.][]{2007ApJ...666..870C} or the IR
continuum due to solid-state materials \citep[dust,
e.g.][]{1986ApJ...303L..41S}. These tracers work relatively well and
are used to characterise nearby star forming regions and star forming
galaxies out to large redshifts, although interesting discrepancies
have been noted for dwarf galaxies at low SFR
\citep[see][]{2009ApJ...706..599L}. One of the main assumptions that
enters into the quantitative interpretation of these data is the
geometry of the material surrounding the newly formed stars, in
particular, where the UV light is being reprocessed. For example, if
the UV photons escape from the ionised medium this may boost the
aromatic feature strengths and strongly influence the lines
originating from the surrounding photo-dissociation regions (PDRs).
There are indications that this geometry in external galaxies may
qualitatively and quantitatively differ from that observed in the
Milky Way (MW)
\citep[e.g.][]{2006A&A...446..877M,2009A&A...508..645G}. One simple
effect may be that at different metallicities the surrounding medium
is more or less opaque and therefore the UV photons have a different
mean free path. More complex scenarios are also discussed in the
literature. For example, clumpiness of the molecular cloud may lead to
small molecular cores surrounded by large PDRs. {\it Herschel} with
its unprecedented wavelength coverage and angular resolution at
submillimeter (submm) wavelengths provides a unique opportunity to
probe the cold interstellar medium (ISM) and sample the effects of the
environment on the resulting SF tracers. In particular, it is well
suited to trace the distribution of matter around HMSFSs and to map
the way the UV radiation permeates and heats the surroundings. Here we
present a study of two HMSFSs in the Large Magellanic Cloud (LMC)
based on data taken in the HERITAGE program \citep[PI. Meixner,
see][]{meixner_special_issue}.
\section{Data treatment and modeling}
\label{sec:data-treatm-model}
We construct spectral energy distributions (SEDs) of the environment
of two distinct HMSFSs (\object{LHA 120-N 44} and \object{LHA 120-N
63} \citep{1956ApJS....2..315H}, hereafter N44 and N63,
respectively) in the strip of the LMC that was mapped during the
science demonstration phase. The two regions were chosen because they
are the brightest and most isolated regions observed. N44 is the
brightest H~{\sc ii}~complex in the observed strip. Massive star
formation has been active in this region; it contains three OB
associations LH47, 48, and 49 with ages $\gtrsim$ 10 Myr at the
central super-bubble and $\lesssim$ 5 Myr at the super-bubble rims and
surrounding dense H~{\sc ii}~regions, as well as a large number of
massive young stellar objects (YSOs) with ages $\lesssim$ 1 Myr
\citep{1970AJ.....75..171L,1995ApJ...452..210O,2009ApJ...695..511C}.
N44 is also the brightest source at all of the SPIRE wavelengths (250,
350, and 500~$\mu$m) in the strip \citep{meixner_special_issue}. By
contrast, N63 is a simple, roughly round H~{\sc ii}~region. It
contains one OB association LH83 with an age $<$ 5 Myr and a number of
massive YSOs \citep{1996ApJS..102...57B,2008ApJ...678..200C}. At SPIRE
wavelengths, N63 is in a relatively isolated environment. Thus, we use
N44 as an example of a prominent H~{\sc ii}~regions seen in more
distant galaxies and N63 as a comparison whose simple structure makes
it more straightforward to relate dust properties with physical
conditions of the ISM.
The data we use are: {\it 2MASS} J, H, K$_s$
\citep{2006AJ....131.1163S}, {\it Spitzer} IRAC1, 2, 3, 4
\citep{2004ApJS..154...10F}, MIPS 24, 70 and 160 $\mu$m
\citep{2004ApJS..154...25R} and {\it Herschel}
\citep{pilbratt_special_issue} SPIRE 250, 350 and 500 $\mu$m
\citep{griffin_special_issue}. See \citet{meixner_special_issue} for a
description of the SPIRE data treatment. We do not use the PACS
information since the data we have until now does not allow us to
extract extended source fluxes with sufficient accuracy. We have
extracted maps of 40$^{\prime}$$\times$40$^{\prime}$ centered on each
H~{\sc ii} region. We have convolved these data to a spatial
resolution of 38$^{\prime\prime}$ set by the MIPS 160~$\mu$m/SPIRE
500~$\mu$m data. The 2MASS data have been convolved with the beam of
MIPS 160~$\mu$m, the IRAC1$-$4, MIPS 24, 70~$\mu$m have been convolved
using custom-made kernels \citep{2008ApJ...682..336G}. The SPIRE data
have all been convolved to the SPIRE 500~$\mu$m resolution assuming
Gaussian beam profiles with FWHM of 18.1, 25.2 and
36.9$^{\prime\prime}$ for SPIRE 250, 350 and 500, respectively. We
have resampled the convolved images to the pixel scheme of the MIPS
160~$\mu$m image using the IDL/astrolib routine {\it hastrom}.
The center of the HMSFS is determined, by fitting a 2D Gaussian
profile, as the brightest source in the total IR (TIR) image. The TIR
image is obtained by simple integration from 8 to 500~$\mu$m. The
reasoning behind using this definition of the center is that this
location (on a size scale of tens of parsecs ) probably hosts the most
active site of embedded star formation. We extract flux densities (in
Jy) for annuli around the given center (see Fig.~\ref{figMaps}). We
use the following radii for the sizes of the annuli: 35, 47, 78, 125,
200, 330 and 530$^{\prime\prime}$, corresponding to linear sizes of 8,
12, 19, 30, 48, 80 and 128~pc assuming a distance to the LMC of 50~kpc
\citep[e.g.][]{2008AJ....135..112S}. The annuli were chosen to be
larger than the apparent size of the HMSFS so as to be able to also
study the regime in which the emission becomes dominated by the
general LMC. Examples of the extracted SEDs for each region in
different annuli are shown in Fig.~\ref{figResults}a,b. There is a
clear and systematic trend for the far-IR to peak at longer wavelength
for the outer annuli. In the case of N44 this trend is to a large part
offset by the broadness of the far-IR peak. This is an indication that
the emission arises from a broad temperature distribution. We also
note that we detect polycyclic aromatic hydrocarbon (PAH) emission (at
8~$\mu$m) in all SEDs. Since we are, to first order, interested in the
``shape'' of the IR SED as a function of distance, we create relative
flux density maps by dividing the maps at each wavelength by the TIR
map in order to measure the scatter. The scatter inside each annulus
of the relative intensity map was used to estimate the uncertainty
(1$\sigma$) on the flux densities. Note, that this method for
determining the variance causes a large scatter in the near-IR pixels
which are dominated by stellar light. This is the reason for the large
error-bars at these wavelengths in Fig.~\ref{figResults}a,b.
We use a phenomenological dust model, which is described in detail in
\citet{2009A&A...508..645G}, to interpret the observed SEDs. This
model consists of a quantity of dust, with a realistic composition
(PAHs, silicate and graphite) and grain-size distribution (complex
molecules, very small (VSG) and big grains) illuminated by a radiation
field with a range of intensities. The main parameters that we aim to
constrain are: {\it 1)} total dust mass, {\it 2)} the range of
illumination intensities, and {\it 3)} the relative composition, in
particular the mass fraction of PAHs. The radiation field distribution
is represented by a power-law function that describes which fraction
of the matter is exposed to what radiation field
\citep{2002ApJ...576..159D}:
\begin{eqnarray}
&&M_\mathrm{dust} \propto U^{-\alpha}: U_\mathrm{min} < U < U_\mathrm{max}
\label{eqn:dale} \\
&&<{\mathrm U}> =
\frac{1}{M_\mathrm{dust}}
\int_{U_\mathrm{min}}^{U_\mathrm{max}} U
\frac{{\mathrm d} M_\mathrm{dust}(U)}{{\mathrm d} U} {\mathrm d} U,
\end{eqnarray}
where $U$ is the intensity of the incident radiation field the dust is
subjected to ($U$=1 for the diffuse MW environment), $U_\mathrm{min}$
and $U_\mathrm{max}$ the minimum and maximum value of $U$ and $<$U$>$
is the mass-weighted averaged radiation field intensity.
\begin{figure*}
\sidecaption
\includegraphics[width=7cm]{f14628c}
\includegraphics[width=7cm]{f14628d}
\caption{Summary of main results. SEDs of the two SF regions we
study (left). The black symbols show the integrated photometry of
three different annuli, lines are the best fit models and the
coloured triangles represent the synthetic photometry in the
corresponding filters. The derived dust column density (panel~{\bf
c}), mean radiation field energy density (panel~{\bf d}), and
the PAH mass fraction (panel~{\bf e}) as a function of annulus
size are shown on the right. The error-bars on the parameters in
Panels {\bf c,d,e} have been derived by propagating the variance
on the photometry using a Monte-Carlo method (see
Sec.~\ref{sec:data-treatm-model}). }
\label{figResults}
\end{figure*}
The starting point for the modeling is to adopt dust properties that
fit the MW IR emission well \citep[][distribution
BARE-GR-S]{2004ApJS..152..211Z}. The observed 24~$\mu$m fluxes in the
diffuse ISM force us to use a dust size distribution that is different
from the MW values. The best fits are obtained by reducing the mass
fraction of VSGs to total dust by 50 per cent. We use the MW
interstellar radiation field as the shape of the radiation field and
do not vary this. The derived dust masses are robust against the
choice of interstellar radiation field and the mass fraction of VSGs.
However, the relative mass fractions of the various dust constituents
may depend on these choices.
In comparing the dust mass tracers with the gas tracers
\citep{meixner_special_issue,gordon_special_issue,duval_special_issue}
some issues have been raised about the applicability of this
composition to the LMC, in particular the graphitic component. We have
verified that the derived mass and radiation field profiles (the shape
as a function of annulus) are not sensitive to the choice of the
carbon bearing grains. However, the absolute values of the derived
parameters depend on this choice. Uncertainties on the derived
parameters were estimated using a Monte-Carlo evaluation. The fitting
routine was repeated 300 times with the observational constraints
varied randomly, according to their standard deviations and new best
fit parameters are determined. The ensemble of best fit parameters is
used to calculate the error-bars on each parameter (see
Fig.~\ref{figResults}).
\section{Results}
\label{sec:results}
Fig.~\ref{figResults} {\bf a,b} show representative SEDs and the best
fit models. The models fit the data very well over the entire
wavelength range, for all annuli. Most parameters are well constrained
with a distribution of best fit parameters which is roughly symmetric
around the central value. One exception to this is the $<$U$>$ in the
two outer annuli of N44. We find a distribution around the mean which
is heavily skewed to low values of $<$U$>$. We do not detect a
systematic submm excess, i.e. the whole wavelength range up to
500~$\mu$m is well fit by the standard model. The model and the
500~$\mu$m surface brightness agree on average within 3\% with a very
small dispersion. The model is in accordance with the findings of
\citet{gordon_special_issue}, who show that the evidence in the LMC
for a submm excess is weak and if present it is confined to the more
diffuse and fainter environments.
The SED fits allow us to study the derived properties as a function of
annulus (projected distance). We show in Fig.~\ref{figResults}{\bf
c,d,e} as a function of radius, the dust column density
($\Sigma_\mathrm{dust}$), average radiation field intensity ($<$U$>$)
and the mass fraction of PAHs normalised to the MW value. Both sources
show a column density profile that decreases outwards, although the
column density in the central region of N63 is not much higher than in
the outer annuli. N63 causes an increase of a factor 2$-$3 of column
density. The column density towards the central regions of N44 is
$\sim$10 times higher than its environment.
The width of the IR SED, which is well determined using the {\it
SPIRE} data, requires a significant dust mass at lower temperatures.
This is reflected in the low values of $<$U$>$ in panel {\bf d}.
$<$U$>$ exhibits only a small range for the entire sample of SEDs.
$<$U$>$ decreases as a function of distance for N63. N44 does not
exhibit any significant trend and the data are consistent with a
constant $<$U$>$ over a distance scale from 10$-$100~pc. Note that
$<$U$>$ is weighted by dust mass and so it readily traces the
radiation field as seen by the coldest dust along the line of sight.
We calculate the second moment of the radiation field distribution
($\Delta U$), again weighted by dust mass in order to
quantify the range of radiation field that the matter is exposed to.
{\it All} SEDs require a wide range of $U$. For N44 $\Delta U$ is
roughly constant at the value of 30 meaning that the entire region is
typified by a radiation field intensity ranging from 1 to 30. N63
shows an outward decreasing $\Delta U$ from $\sim$50 in the inner
annulus to 2 in the outer. Thus the radiation field in N63 spans 5-55
($<$U$>$-$<$U$>$+$\Delta U$) on the inside and 1-3 in the
outer annuli. This implies that the outer annuli in N63 closely
resemble a diffuse environment.
The fraction of mass contained in PAHs increases with increasing
radius out to about 50 per cent of the MW value. We find a significant
depletion of the PAHs towards the central regions over $\sim$20 and
40~pc, for N44 and N63 respectively. Note that the smaller size of the
depleted region is consistent with the fact that N63 is less prominent
compared to its surroundings (see below). Interestingly, the radius of
the depleted environment corresponds well to the area occupied by the
ionising stars in Fig.~\ref{figMaps}. The fraction of the ionised PAHs
is not well constrained but we have checked that there is no
systematic effect of the fitting procedure that causes the observed
trend in Fig.~\ref{figResults}{\bf e}.
\begin{figure}
\centering
\includegraphics[width=6cm]{f14628e}
\caption{Derived geometrical parameters of the two regions. We show
the dust column density and the fitted profile of a thick shell,
with a power-law density profile (blue line) and a slab (red). The
dashed model corresponds to the case when we fix the exponent of
the power-law to 2 (see text).}
\label{geometry}
\end{figure}
We construct a simple geometrical model to try to constrain the size
and density of the matter in the HMSFSs, i.e. to deproject the
observed column densities. We represent the molecular cloud around the
H~{\sc ii}~region as a geometrically thick shell. To this we add a
slab that represents the disk of the LMC, i.e the diffuse extended
emission. We assume a constant dust column density for the slab and a
power-law profile for the shell
($\rho_\mathrm{dust}=\rho_\mathrm{dust,in}(r/R_\mathrm{in})^{-\gamma}$).
The dust mass profile and the best fit decomposition into these two
components and the corresponding parameters are shown in
Fig.~\ref{geometry}. We have propagated the uncertainties on the dust
column density in each annulus (see Fig.~\ref{geometry}). The
uncertainties are significant. We find in case N63, for which the
over-density is weak compared to the slab (i.e. the disk of the LMC),
that the geometrical parameters are not very well constrained. In
particular, it is hard to constrain the $\gamma$ parameter. The best
fit model has $\gamma=1.2$ which seems flat. If we fix $\gamma$ at 2,
the value expected for a cloud in hydrostatic equilibrium, we find
that the distance where the slab component starts to dominate, i.e.
the intersection of the blue and red lines in Fig.~\ref{geometry},
reduces from $\sim$80 to $\sim$30~pc.
\section{Discussion \& Conclusion}
\label{sec:disc--concl}
Figs.~\ref{figResults} and \ref{geometry} show the power of {\it
Herschel} to determine the matter distribution around HMSFSs. We
investigate the effect the new SPIRE constraints on the derived
parameters and their uncertainties by also fitting the SEDs without
using these SPIRE data. We find that the column density of dust is
often very discrepant (by more than an order of magnitude) from the
values derived using the SPIRE data. The derived $\Sigma_{dust}$
profile for N44 (like Fig.~\ref{figResults} top panel) is virtually
constant at a value of 0.1~M$_{\sun}$\,arcsec$^{-2}$ due to the fact
that the cooler dust, is not well traced by the MIPS data. For N63 the
model over-predicts the derived dust masses for the annuli between 20
to 60 pc by a large factor ($\sim$5). In these cases the model tends
to find very small values of $U_\mathrm{min}$. This artifact could
perhaps be circumvented, when modelling those kind of regions, in
cases where submm constraints are missing, by limiting the allowed
range of $U_\mathrm{min}$ in Eq.~(\ref{eqn:dale}). The difficulty of
constraining the radiation field parameters without the SPIRE bands is
also reflected in the derived uncertainties. In particular the value
of $U_\mathrm{min}$ is ill-constrained which results in large,
asymmetric error-bars on the derived column densities.
N44 and N63 exhibit strikingly different behaviour in the radiation
intensity profile (Fig.~\ref{figResults}{\bf d}.) The lack of a
systematic decrease of $<$U$>$ around N44 indicates that we are
observing dust with a wide range of temperatures along each
line-of-sight. The inner annuli in N44 are affected by the superbubble
to the NE of the OB association, where high values of $<$U$>$ are
expected. The low values of $<$U$>$ for such a luminous SF region may
reflect clumpiness. The profile is clearly incompatible with a
centrally illuminated optically thin irradiation profile. It is clear
from Fig.~\ref{figMaps} that the studied regions are not very
spherically symmetric (azimuthally smooth). In particular, N44
harbours several clusters and the peak of the X-ray emission is
located in a cavity, $\sim$ 20$-$30~pc away from the TIR peak (see
Fig.~\ref{figMaps}). Measuring the azimuthally averaged properties
smears out some of the characteristics. This smearing could have been
the cause for the lack of trend seen in the average $U$ as seen by the
dust (Fig.~\ref{figResults}). We have verified that this small range
of $<$U$>$ is not simply an artifact of this averaging or a wrong
choice of center of the annuli by studying the parameters we derive
pixel by pixel in the maps which makes no assumptions about the
geometry. Indeed the highest $<$U$>$ in N44 is found close to the
center we chose. Except for the very center all other values, with
their scatter, are within the range as depicted in
Fig.~\ref{figResults}. We conclude that the choice of center does not
dominate the lack of trend of N44 in the average radiation field. Thus
this lack of trend reflects the true broad range of irradiation
conditions along all lines of sight in N44 , which is an indication of
the inhomogeneity of the ISM around N44. A simple dust model shows a
deficit in PAHs toward the centers of these two regions. We find no
evidence for a submm excess. We have used the observed dust column
densities surrounding N44 and N63 to derive a 3-D model for these
regions for the first time.
\begin{acknowledgements}
We acknowledge financial support from the NASA Herschel Science
Center, JPL contracts \# 1381522 \& 1381650. We thank the
contributions and support from the European Space Agency (ESA), the
PACS and SPIRE teams, the Herschel Science Center and the NASA
Herschel Science Center (esp. A. Barbar and K. Xu) and the PACS and
SPIRE instrument control centers, without which none of this work
would be possible. We thank the referee, Glenn White, for comments
that have improved the paper.
\end{acknowledgements}
\bibliographystyle{aa}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,475 |
package net.runelite.client.plugins.kourendlibrary;
import com.google.inject.Guice;
import com.google.inject.Inject;
import com.google.inject.testing.fieldbinder.BoundFieldModule;
import net.runelite.api.coords.WorldPoint;
import static org.junit.Assert.assertEquals;
import org.junit.Before;
import org.junit.Test;
public class LibraryTest
{
@Inject
private Library library;
@Before
public void before()
{
Guice.createInjector(BoundFieldModule.of(this)).injectMembers(this);
}
@Test
public void testVarlamoreEnvoyFindingProcess()
{
library.mark(new WorldPoint(1610, 3799, 1), null);
library.mark(new WorldPoint(1608, 3799, 1), null);
library.mark(new WorldPoint(1615, 3799, 1), null);
library.mark(new WorldPoint(1616, 3799, 1), null);
library.mark(new WorldPoint(1621, 3799, 1), null);
library.mark(new WorldPoint(1624, 3796, 1), null);
library.mark(new WorldPoint(1624, 3792, 1), null);
library.mark(new WorldPoint(1624, 3791, 1), null);
library.mark(new WorldPoint(1623, 3789, 1), null);
assertEquals(SolvedState.NO_DATA, library.getState());
library.mark(new WorldPoint(1621, 3789, 1), Book.WINTERTODT_PARABLE);
assertEquals(SolvedState.INCOMPLETE, library.getState());
library.mark(new WorldPoint(1618, 3799, 2), null);
library.mark(new WorldPoint(1613, 3792, 2), null);
library.mark(new WorldPoint(1618, 3790, 2), Book.TRANSPORTATION_INCANTATIONS);
library.mark(new WorldPoint(1609, 3816, 2), Book.RICKTORS_DIARY_7);
assertEquals(SolvedState.COMPLETE, library.getState());
// The Varlamore Envoy book can be found in this bookcase, but should not cause a state reset if not found
library.mark(new WorldPoint(1622, 3816, 2), null);
assertEquals(SolvedState.COMPLETE, library.getState());
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 131 |
## Praise for Murder of Angels
"I love a book like this that happily blends genres, highlighting the best from each, but delivering them in new configurations.... In Murder of Angels, the darkness is poetic, the fantasy is gritty, and the real-world sections are rooted in deep and true emotions. Lyrical and earthy, Murder of Angels is that rare book that gets everything right."
—Charles de Lint
"[Kiernan's] punk-rock prose and the brutally realistic portrayal of addiction and mental illness make Angels fly."
—Entertainment Weekly (A-)
"Kiernan's best book to date, joining her always-prodigious gift for language with a wrenching, compelling story."
—Locus
"Kiernan can write like a banshee.... [She] paints her pages in feverish, chiaroscuro shades. A bridge to the beyond, built out of exquisite dread."
—Kirkus Reviews
## Praise for Low Red Moon
"The story is fast-paced, emotionally wrenching, and thoroughly captivating.... Kiernan only grows in versatility, and readers should continue to expect great things from her."
—Locus
"Low Red Moon fully unleashes the hounds of horror, and the read is eerie and breathtaking.... The familiar caveat 'not for the faint of heart' is appropriate here—the novel is one of sustained dread punctuated by explosions of unmitigated terror."
—Irish Literary Review
"Effective evocations of the supernatural...a memorable expansion of the author's unique fictional universe."
—Publishers Weekly
## Praise for Threshold
"Threshold is a bonfire proclaiming Caitlín R. Kiernan's elevated position in the annals of contemporary literature. It is an exceptional novel you mustn't miss. Highly recommended."
—Cemetery Dance
"A distinctively modern tale that invokes cosmic terrors redolent of past masters H. P. Lovecraft and Algernon Blackwood...a finale that veers unexpectedly from a seemingly inevitable display of supernatural fireworks to a subtly disarming denouement only underscores the intelligence behind this carefully crafted tale of awe-inspired nightmare."
—Publishers Weekly
"[Caitlín R. Kiernan is] the most singular voice to enter the genre since Neil Gaiman popped up in graphic novels and Stephen King made movies live inside books.... If you haven't sampled her work yet, you haven't really been reading the future of horror and dark fantasy, only its past."
—SF Site
"Kiernan's prose is tough and characterized by nightmarish description. Her brand of horror is subtle, the kind that is hidden in the earth's ancient strata and never stays where it can be clearly seen."
—Booklist
"Threshold confirms Kiernan's reputation as one of dark fiction's premier stylists. Her poetic descriptions ring true and evoke a sense of cosmic dread to rival Lovecraft. Her writing envelops the reader in a fog concealing barely glimpsed horrors that frighten all the more for being just out of sight."
—Gauntlet
## Praise for Silk
Winner of the International Horror Guild Award for Best First Novel Finalist for the Bram Stoker Award for Best First Novel Nominated for the British Fantasy Award
"Caitlín R. Kiernan draws her strength from the most honorable of sources, a passion for the act of writing. Her tightly focused, unsparing, entranced gaze finds significance and beauty in the landscape it surveys."
—Peter Straub
"A remarkable novel...deeply, wonderfully, magnificently nasty."
—Neil Gaiman
"A daring vision and an extraordinary achievement.... Caitlín R. Kiernan is an original."
—Clive Barker
"Caitlín R. Kiernan writes like a Gothic cathedral on fire.... [Her] work is populated with the physically freaky, mentally unstable, sexually marginalized characters who have caused so much consternation in conventional circles—but Caitlín R. Kiernan is headed in an entirely different direction. Her unfolding of strange events evokes not horror, but a far larger sense of awe."
—Poppy Z. Brite
"[Kiernan] has what it takes to excite me as a reader.... Think of Poppy Z. Brite with slightly more accessible prose and characters who aren't quite so outré.... I just loved this book and can't wait to see what she writes next."
—Charles de Lint
"An observational coming-of-age novel that astutely and empathetically provides connection between characters and readers...a skillfully constructed Southern gothic of profound creepiness...an incremental triumph of texture and layering, harkening back to an earlier tradition of supernatural fiction, an era when storytelling took as much time as it needed to accrue the maximum effect.... Hers is a dark and mellifluous voice to which we should listen."
—Locus
"Kiernan's writing is meaty, atmospheric, and evocative; her prose is well crafted and terrifically engaging.... Silk is a strong first showing, and Kiernan should have a bright future ahead of her."
—Fangoria
"An engrossing and exquisitely lyrical novel...that could conceivably transfuse characters and settings with a Poppy Z. Brite novel."
—Hellnotes
"A novel with an uncommonly rich texture...should establish [Kiernan] as an important writer of the future. This novel transcends the goth genre."
—Necrofile
"Spun as beautifully as the many webs within.... You absolutely must read it."
—Carpe Noctem
"A masterful story by an extraordinary new voice in literature...on her way to becoming an incredibly well-known—and well-respected—talent. Silk is simply the extraordinary beginning of an incredible journey, both for Kiernan and her readers."
—Alabama Forum
"Kiernan is uniquely herself, but even if you miss the endorsement by Neil Gaiman, you cannot fail to see the kindred spirit that flows through their writing. I feel no risk in voicing the opinion that if you enjoy one, you will relish the other."
—SF Site
## DAUGHTER OF HOUNDS
## CAITLÍN R. KIERNAN
ROC
Published by New American Library, a division of
Penguin Group (USA) Inc., 375 Hudson Street,
New York, New York 10014, USA
Penguin Group (Canada), 90 Eglinton Avenue East, Suite 700, Toronto,
Ontario M4P 2Y3, Canada (a division of Pearson Penguin Canada Inc.)
Penguin Books Ltd., 80 Strand, London WC2R 0RL, England
Penguin Ireland, 25 St. Stephen's Green, Dublin 2, Ireland (a division of Penguin Books Ltd.)
Penguin Group (Australia), 250 Camberwell Road, Camberwell, Victoria 3124,
Australia (a division of Pearson Australia Group Pty. Ltd.)
Penguin Books India Pvt. Ltd., 11 Community Centre, Panchsheel Park,
New Delhi-110 017, India
Penguin Group (NZ), cnr Airborne and Rosedale Roads, Albany,
Auckland 1310, New Zealand (a division of Pearson New Zealand Ltd.)
Penguin Books (South Africa) (Pty.) Ltd., 24 Sturdee Avenue,
Rosebank, Johannesburg 2196, South Africa
Penguin Books Ltd., Registered Offices:
80 Strand, London WC2R 0RL, England
Copyright © Caitlín R. Kiernan, 2007
All rights reserved
REGISTERED TRADEMARK—MARCA REGISTRADA
LIBRARY OF CONGRESS CATALOGING-IN-PUBLICATION DATA:
Kiernan, Caitlín R.
Daughter of hounds / Caitlín R. Kiernan.
p. cm.
ISBN: 978-1-1012-1274-5
I. Title.
PS3561.I358D38 2007
813'.54—dc22 2006018475
Without limiting the rights under copyright reserved above, no part of this publication may be reproduced, stored in or introduced into a retrieval system, or transmitted, in any form, or by any means (electronic, mechanical, photocopying, recording, or otherwise), without the prior written permission of both the copyright owner and the above publisher of this book.
PUBLISHER'S NOTE
This is a work of fiction. Names, characters, places, and incidents either are the product of the author's imagination or are used fictitiously, and any resemblance to actual persons, living or dead, business establishments, events, or locales is entirely coincidental.
The publisher does not have any control over and does not assume any responsibility for author or third-party Web sites or their content.
The scanning, uploading, and distribution of this book via the Internet or via any other means without the permission of the publisher is illegal and punishable by law. Please purchase only authorized electronic editions, and do not participate in or encourage electronic piracy of copyrighted materials. Your support of the author's rights is appreciated.
For my agent, Merrilee Heifetz, for whom I will someday
write a pleasant book...
...and for Poppy and Chris and the city of New Orleans.
May the days ahead be kinder.
In memory of Elizabeth Tillman Aldridge.
(1970–1995)
## AUTHOR'S NOTE
The attentive reader will note several obvious debts that this book owes to the works of Howard Phillips Lovecraft (1890–1937), as well as to Sir Richard F. Burton (1821–1890) and his classic English translation of Antoine Galland's French rendering of the original Arabic Alf Layla (The Thousand and One Nights, c. 850 CE). Considering the medieval legend that anyone foolish enough to read the whole of The Thousand and One Nights would go mad, it seems not unlikely that it may have served, in part, as the inspiration for Lovecraft's Arabic Necronomicon.
Never once has a novel come easily to me, but I think it's safe to say that I've never written one during a time of such personal and emotional turmoil as I endured during the two long years spent writing Daughter of Hounds. My grateful thanks to everyone who has made these difficult times a little less so, but especially to Kathryn (my own heart's Book of Shadows), Jim Shimkus and Jennifer Lee, Byron White, Jada Walker and Katharine Stewart, Anita W. Nicker (my girl in Barcelona), my mother, William Schafer and Subterranean Press, Nar'eth ni'glecti Mericale (figment, familiar, reluctant goddess, and five-time alter ego), John Morgan (this book's first editor), and Liz Scheier (its second) for all their enthusiasm, insight, and encouragement, the intrepid Travis Burton (who got me into the old train tunnel beneath College Hill), and to my agent, Merrilee Heifetz. A special thank you to Dr. Richard B. and Carol Pollnac for their generous aid and hospitality during the summer of 2004, while I was scouring Rhode Island and Massachusetts for the black keys with which to unlock this story, as well as to the staff of the Robert W. Woodruff Library and the William L. Matheson Reading Room (Emory University) and the Providence Athenaeum. And also to the readers who've stuck with me all these years and, particularly, to everyone who's been brave enough to subscribe to Sirenia Digest. Shine on, all of you.
This novel was written on a Macintosh iBook.
All look and likeness caught from earth,
All accident of kin and birth,
Had passed away. There was no trace
Of aught on that illumined face,
Upraised beneath the rifted stone
But of one spirit all her own—
She, she herself, and only she,
Shone through her body visibly.
—SAMUEL TAYLOR COLERIDGE
I lost a world the other day.
Has anybody found?
You'll know it by the row of stars
Around its forehead bound.
—EMILY DICKINSON
## Contents
PROLOGUE
ONE EMMIE
TWO SOLDIER
THREE NEW YORK
FOUR WOONSOCKET
FIVE ANGELL STREET
SIX SHADOW AND FLAME
SEVEN STAR
EIGHT INTERSECTIONS
NINE THE BAILIFF
TEN THE YELLOW HOUSE
EPILOGUE APRIL
ABOUT THE AUTHOR
I PARALLEL LIVES
II LOST GIRLS
## PROLOGUE
I
The ghoul lady takes out her white linen handkerchief and uses one corner to dab at her watering left eye. It's an old wound, a relic of her spent and reckless youth, but it still bothers her sometimes, especially when the weather Above is wet. And today the weather Above is very wet, all of Providence caught up in the final, rainy, death-rattle sighs of something that was a hurricane only a few days before. She sits on the wooden stool that's been provided for her and blinks and gazes down her long muzzle at the dozens of faces staring impatiently back at her from the candlelight and shadows trapped beneath the Old North Burial Ground. The restless assembly of her wards, ghoul pups and human changelings seated together on the damp earth, wriggle about and whisper among themselves. She clicks her teeth together once, a sound that might draw blood, and they grow a little quieter. She wishes again that she were back in the warmth of her own dry burrow, deep beneath the basement of the old yellow house on Benefit Street, the familiar weight of College Hill pressing down around her, protecting her ancient, aching bones and her bad eye from this damned inclement weather.
"Myself, I would have postponed this outing," she says, and not for the first time that night, "but Master Shardlace feels most emphatically that schedules are made to be kept, so here we are, one and all."
Up front, one of the changelings sneezes.
"Likely as not," says Madam Terpsichore, addressing the child directly, "we shall all catch our deaths this evening. But let us not falter an instant in our dedication. At least the program shall not be disrupted." And with that, she shifts her poppy-colored eyes towards the spot where Master Shardlace, lately of the Mystic and Stonington Village warrens, is crouched, half-hidden by the dangling roots of a sycamore tree. He flinches at her glance, and that gives her some small measure of satisfaction. "Wipe your nose," she barks at the child who sneezed, and it does so.
"The question at hand," Madam Terpsichore continues, "that most urgent matter of history and propriety and etiquette which has brought us forth from the succor and haven of our dens, which has brought—nay, dragged—us each and every one out into this tempest..." And she pauses here to spare another acid glance for Master Shardlace and his roots. He pretends not to notice. "The question," she says, "is, indeed, a grave thing."
A few of the students snicker at the pun while Madam Terpsichore dabs at her eye again. One careless moment more than a century ago and she still bears this scar, the ugly mark of a lost instant's indecision, an insult that she would have done well to let pass. Then tonight her eye would not be throbbing and watering as though it envied the storm Above.
"A wonder we are not all drowned," she says dramatically and shakes her head.
"The lesson," Master Shardlace growls softly from his hiding place, prompting her, risking another glare or something more substantial. "If we could only proceed, we would sooner find ourselves home and snug again."
"Oh, most assuredly," Terpsichore hisses between her long incisors and eyeteeth, and he looks quickly down at the ground between his splayed feet and retreats deeper into the tangled veil of sycamore roots. She wonders, for the hundredth or so time, exactly what he might have done to deserve his exile and, more important, why ever Master Danaüs chose to give him safe haven in Providence. And, more important still, what she must have done to so displease the dark gods that she deserves to be weighted with such an officious waste of hide and bone and sinew. Her bad eye weeps, and she wipes the tears away.
"Yes," she sighs. "The lesson at hand." And the ghoul draws a deep breath, filling her lungs with air that smells and tastes and knows of the subtle complexities of mere human death, the turning of great stone wheels upon the infinite axis of time, the sugar-sweet reek of loss and forgetfulness and regret, slow rot and embalming and scurrying black beetles. Above, the storm reminds her that summer has finally given way to autumn, the orangebrowngolden season of harvest, of reaping, of closing doors and grinning pumpkins, and if her kind ever had a season in this world, it would be autumn. She makes a tight fist and squeezes until her claws draw claret droplets of blood; then Madam Terpsichore opens her left hand and holds it out for all to see.
"We play so desperately at being fearsome things," she says, and her sooty lips curl back in an expression that is not nearly so kind as a smile, but still something more charitable than a snarl or a grimace. One of the changelings coughs then, the same girl who sneezed a few seconds before, a pretty, ginger-haired girl who has chosen for herself the name of Sparrow Spooner, a name she borrowed from a tombstone, as has always been the custom of the stolen ones, the Children of the Cuckoo.
"Take strength, child," Madam Terpsichore tells Sparrow Spooner, and the ghul offers her bleeding hand to the girl. "Warm yourself against the cold and the wet and what's to come."
Sparrow Spooner hesitates, glancing anxiously from Madam Terpsichore to the faces of the other students. She can see that some of them are jealous of her, and some are frightened for her, and some are hardly paying any mind at all. A pup named Consequence rolls his yellow eyes, and a boy who hasn't yet taken a name sticks out his tongue at her. She turns back to the ghoul, not pretending that she has a choice, and crawls on her hands and knees until she's kneeling in front of Madam Terpsichore's stool.
"We need the world to think us monsters," the ghoul says to her, "and so monsters we become."
The girl leans forward and begins to lick at the blood oozing from her mistress' leathery, mottled palm.
"We must, all of us, keep apart the night from the day, the world Above from the world Below, the shadows from the sun, and we must keep them apart at any cost," Madam Terpsichore says, watching the others as she gently strokes the child's head with her free hand, her razor claws teasing at Sparrow Spooner's matted ginger hair. "Even if we should find our death of cold in the effort."
"There has been a breach," Master Shardlace grumbles from the safety of his place among the sycamore's dangling, dirt-clod roots. "A trespass has occurred, and we are all—"
"I am coming to that," Madam Terpsichore barks back at him, and he mutters to himself and grows silent again.
Sparrow Spooner stops cleaning her mistress' bleeding left hand and gazes up at Madam Terpsichore. The changeling's lips and chin and the tip of her nose are smeared with sticky crimson, and she absently wipes her mouth on the sleeve of her dingy dress.
"I know you, child. You've come a long, long way, through the Trial of Fire and the Trial of Blades. Next Full Hunger Moon, you're up to face the Trial of Serpents, and, if you survive, you'll win your Confirmation."
The changeling only nods her head, not so dull or frightened that she doesn't understand that the time for words has long since come and gone. The ghoul's blood is bitter and salty on her tongue and burns her throat going down to her belly. But it warms her, too, pushing back some of the chill that's worked its way into her soul.
"Do you know the story of Esmeribetheda and the three gray witches?" Madam Terpsichore asks the changeling, and Sparrow Spooner nods her head again. Of course she knows the story, has known it since she was very young, one of the seventy-four "Parables of Division" recorded in the Red Book of Riyadh and taught to the Children of the Cuckoo before they are even old enough to read the words for themselves.
"Then you remember the crime of Esmeribetheda, don't you?" Madam Terpsichore asks Sparrow Spooner.
"Yes, ma'am," the girl replies and wipes her mouth again. The blood on her face has begun to dry, turning the color of rust.
"Then will you tell us, please?" and she motions towards the students. "Perhaps some of the others have forgotten." For a moment, the chamber beneath the cemetery comes alive with nervous chatter and tittering laughter at Sparrow's predicament. But Madam Terpsichore narrows her eyes and silences them with a look.
"She...Esmeribetheda became curious, and she wanted—"
"Stand up, please," Madam Terpsichore says, interrupting her. "Stand up and face the class, not me. I already know the story."
"So do they," the changeling complains and earns a scowl and another click of the teeth from her mistress. She apologizes for her impudence and gets to her feet, brushing some of the mud from her dress and bare legs, then turns to face the others.
"Esmeribetheda became curious, and she wanted to know how the children of men and women lived, what it was like to have a mother and father. She wanted to know what she'd lost when the Hounds of Cain had stolen her from her crib."
"And what did she do to learn these things?" Madam Terpsichore asks.
"She was sought out by three human witches, Arabian necromancers determined to locate a route to the world Below that they might learn its secrets and gain greater power in their arts. In the desert, at an altar beneath a dead tree that had once served as a temple to the goddess Han-Uzzai, Al-Uzza, youngest daughter of Allah, she was met by a blue-eyed crow. In truth, though, the crow was one of the witches who had disguised herself, and it promised that Esmeribetheda would be reunited with her parents if she'd show the necromancers a doorway and lead them down to the Hall of..." And Sparrow Spooner stops talking and looks over her shoulder at Madam Terpsichore.
"What's wrong, dear?" the ghoul asks her. "Have you forgotten what comes next?"
"No, ma'am," the girl replies. "But they know the story. They know all of it."
"Yes, but we never, ever suffer from hearing a good tale retold, do we? Especially when it's a story with so much to teach us, so much we should take pains to remember."
Sparrow Spooner licks at her dry lips, tasting the ghoul's blood again. The warmth it left in her stomach has already begun to fade, replaced with something hard and cold that twists and turns like a winding ball of pink worms, something much colder than the late November night.
"Continue, please," Madam Terpsichore says.
"Well, Esmeribetheda was shown images of the life she might have lived. She saw herself in her mother's arms. She saw her brothers and sisters. She saw herself growing into a young woman and marrying a handsome man who gave her children of her own, children she could keep. The witches promised her she could have all this back, all that might have been, if she'd show them the road down to the hounds. She agreed that she would, and the crow flew away to tell the other witches."
"She agreed to show them the way?" Madam Terpsichore asks. "Even though she knew perfectly well that it was forbidden for her to reveal those paths to mortal men?"
"Yes," Sparrow Spooner replies, promising herself that whatever's going to happen, she won't cry. She doesn't want the others to see her cry. "She was a very foolish and ungrateful girl. She'd never been able to accept the life she'd been given. On a moonless night, Esmeribetheda led the witches across the sands to a warren doorway. But the hounds knew, and they were waiting for her."
The nameless boy who'd stuck his tongue out at her earlier was now pretending to hang himself, tugging at an imaginary noose before his head lolled to one side in a pantomime of strangulation. The ghoul named Consequence snickered, but Madam Terpsichore seemed not to notice them.
"And what happened next?" she asks Sparrow Spooner.
"The three witches were killed there on the spot and their corpses carried down into the tunnels. Esmeribetheda was led back through the dunes to the dead tree in the desert, and the ghouls hanged her there, and then they set the tree on fire."
"Yes," Madam Terpsichore says, speaking now so softly that only the changeling can hear. "They did. Would you call that justice, child?"
Sparrow glances over at the rootsy place where Master Shardlace is hiding, as though he might decide to help, when she knows damned well that he won't, that she's been brought here tonight instead of some other, later night at his insistence.
"Was it justice?" Madam Terpsichore asks again, and now she rises from her place on the stool, standing up straight so that she looms over the girl and her head almost scrapes against the low roof of the chamber.
"Esmeribetheda just...she only wanted to go home...she only wanted to get back the life that had been taken away from her."
"I know the story, child," the ghoul sighs, almost whispering, and presses her muzzle gently against Sparrow Spooner's cheek. "I have asked you a question."
"She wanted to go home," the changeling says. "That's all. She wanted to go home."
"Your life will be spared," Madam Terpsichore says, not unkindly, her wet nose nuzzling the girl's face, her eyes on the other students. "But there must be a punishment, you understand that?"
"Yes, ma'am," the changeling girl says, her legs gone suddenly so weak that she's afraid she might fall. Her mistress' breath, hot as a summer day, smells of lifeless, broken things that have lain a long time beneath the soil.
"She should die," Master Shardlace growls.
"No, she will live," Madam Terpsichore tells him, "but she will always remember this night and the folly of her actions. She will learn, tonight, that desire is only another demon that would happily see her strung from the branches of a burning tree."
"And what of the witch?" demands Master Shardlace.
"The witch will die, just as the three died in the story of poor, misguided Esmeribetheda." And Madame Terpsichore grips Sparrow Spooner by the back of the head and forces the girl down onto her knees. From the shadows, there comes the rough sound of stone grating against stone, stone ground against metal, and then a sudden gust of fresh night air threatens to extinguish the candles. All the changelings and ghoul pups turn to see the open door leading up to the cemetery and the world Above and to behold the face of the one who has led Sparrow astray from the path set for her by the Cuckoo.
"You be strong, child," Madame Terpsichore tells Sparrow Spooner, and the girl shuts her eyes.
II
The old hearse, a 1948 Caddy slick and long and blacker than the stormy New England night, subtle as a fucking heart attack, rolls unchallenged through the wild Massachusetts night. In the passenger seat, Soldier drifts between her uneasy dreams and the nagging edges of wakefulness, dozing and waking and dozing again to the metronome rhythm of the windshield wipers. The radio's set to a classic-rock station out of Boston, and she's already told that asshole Sheldon that she'll break his goddamn fingers if he so much as touches the dial. He can listen to that indie-rock college shit on his own dime, not when she's trying to catch a couple hours' shut-eye before a job.
After Providence and their brief meeting with the Bailiff and one of his boys at the Dunkin' Donuts on Thayer Street, the hearse left the city and followed I-95 north all the way up to and across the New Hampshire state line, finally doubling back at Hampton Beach, because that's the way the Bailiff had told them to do it. Just like always, everything worked out ahead of time to the letter and in accordance with the Bailiff's precise instructions, the plans he'd cobbled together from star charts and newspaper astrologers and the obscure intersections of geometry and geography, nothing Soldier even pretended to understand. She listened when he talked and did what she was told.
Past the Hamptons, then on to Salisbury and Newburyport, U.S. 1 traded for State 1A, past sleeping houses and fishing boats tied up secure against the storm, across the bridge spanning the brackish confluence of the Merrimack River and Newburyport Bay. Other bridges over other lesser waters, over railroad tracks, Rowley to Ipswich, and when Sheldon jabs her in the arm and tells her to wake up, Soldier tells him to fuck off. But she opens her eyes anyway, squinting out at the dark streetlights and the darker windows of the houses along High Street and the raindrops hitting the windshield. Eric and the Animals are coming through the Caddy's speakers, "White Houses," and at least that's one thing about the night that's all right by her.
"We're there?" she croaks, her mouth dry as ashes, and reaches for the pint bottle of George Dickel she stashed beneath the seat before leaving Rhode Island. "Why the fuck is it so dark?"
"Not quite there, but close enough," Sheldon replies. "Time to rise and shine, Sleeping Beauty."
"Fuck you."
"Babe, if I thought there was time—"
"Why is it so dark?" she asks him again. "What's up with the streetlights?"
"Power's out. The storm, I expect."
"Jesus, I need a goddamn drink," Soldier says, changing the subject, and her hand has to grope about for only a moment before it closes around the neck of the bottle lying on the floorboard behind the heels of her army-surplus boots.
"The Bailiff wants you sober for this one," Sheldon says and glances anxiously at Soldier as she unscrews the cap.
"The Bailiff doesn't pull the fucking trigger, now, does he? Why don't you shut up and watch the road?"
Old Hill Burying Ground rises up on their left, countless listing rows of slate and granite markers lined up like a dutiful army of stone soldiers gathered together beneath the swaying boughs of oaks and hemlocks, an army of the dead standing guard since sometime in 1634. And Soldier remembers this place, the delivery they made there a year or so back, one of her first rides with Sheldon, and they left a heavy leather satchel sitting outside one of the vine-covered mausoleums. She never found out what was inside the satchel, never asked because she never wanted to know. It isn't her job to know.
She takes a drink of the whiskey, and if it doesn't quite drive away the fog in her head, it's a halfway decent start.
"What time is it?" she asks, and Sheldon shrugs.
"You got a watch, lady," he says. "You tell me."
Instead, she takes another swallow of George Dickel, rubs at her eyes, and watches the night slipping by outside the hearse. She realizes that she's sweating and unzips her shabby bomber jacket, a WWII antique she took off a corpse a couple of years back. The fleece lining is nappy and moth-eaten and worn straight through in a few spots.
"It's almost three thirty," Sheldon sighs, checking his wristwatch when it's clear Soldier isn't going to check for herself. "We made pretty good time, all things considered."
"Yeah? All things considered, looks to me like we're cutting this pretty goddamned close," she replies, tightening the cap on the whiskey bottle. "If we miss Bittern—"
"—then I suppose we're fucked, good and harsh. But we're not gonna miss him. Ain't no way that card game's gonna break up until dawn, right? No way, lady, especially not with this blow. Hell, he's probably into Jameson for ten or twelve Gs by now, easy. Ain't no way he's gonna walk with that many Franklins on the line."
"Look, man, all I'm saying is we're cutting it close. It would have been nice if we'd had a little more notice, and that's all I'm saying."
Sheldon Vale slows for a traffic light that isn't working, then steers the hearse off High Street onto North Main. On the radio, Eric Burdon's been replaced by the Beatles' "Continuing Story of Bungalow Bill," and he reaches for the knob.
"Don't touch it," Soldier says.
"Oh, come on. I've been listening to this crap since Providence, and I fucking hate the Beatles."
"Why don't you just worry about getting us to Bittern and forget about what's on the radio," Soldier tells him and returns the bottle to its spot beneath the seat.
"I should have let you sleep."
"Yeah," she says, "you should have let me sleep."
"He gives us as much notice as he can," Sheldon says, and it takes Soldier a second or two to figure out what he means, to remember what she said about the Bailiff.
"You think so? You think that's how it is?"
"Where's the percentage in doing any different?"
"You really think it's all that simple?"
Sheldon snorts and turns left onto State 133 and crosses the swollen, muddy Ipswich River.
"What I think is I was driving this route, running for the Bailiff, when you were still shitting your diapers, and maybe old Terpsichore and Danaüs got their plans all laid out for you, all right, but you don't know even half as much about operations as you like to let on."
Soldier laughs, then goes back to staring out the window. "That was a mouthful, Sheldon. Were you rehearsing that little speech all the way up from Providence?"
Sheldon frowns and wipes condensation off the inside of the windshield with his bare hand.
"You know that's gonna streak," she says. "And you know how the Bailiff feels about hand prints and streaky windshields."
"Yeah, well, I can't fucking see."
Soldier shrugs and folds down the passenger-side sun visor. There's a little mirror mounted there, and she stares for a moment at her reflection, stares at the disheveled woman staring back at her—the puffy, dark half circles beneath her bloodshot eyes, half circles that may as well be bruises, her unkempt, mouse-colored hair that needed a good cutting two or three months ago. There's an angry red welt bisecting the bridge of her nose that'll probably leave a scar, but that's what she gets for picking a fight with one of the ghouls. She sticks her tongue out at herself, then folds the visor up again.
"You look like shit," Sheldon Vale says, "in case you need a second opinion."
"You're a damned helpful cunt, Shelly."
"Shit," he hisses, glancing at the rearview. "I think I missed the turnoff."
"Yep," Soldier says, pointing at a green street sign. "That's the fucking Argilla right there. You missed it. Guess that'll teach you to keep your eyes on where you're going, instead of letting yourself get distracted by my pretty face."
Sheldon curses himself and Jesus and a few of the nameless gods, slows down and turns around in a church parking lot, slinging mud and gravel, and then the hearse's wheels are back on blacktop, rolling along with the rubber-against-wet-asphalt sound that's always reminded Soldier of frying meat. Soon they're on the other side of the river again, retracing the way they've just come, left turns become rights, and there's the cemetery once more.
"What's on your mind, old man?" Soldier asks, because he might be an asshole, and he might have shitty taste in music, but Sheldon Vale can usually be counted on to get you where you're going without a lot of jiggery-pokery and switchbacks.
"You think they're gonna kill that kid?" he asks her and turns off the highway onto a road leading away towards the salt marshes and the sea.
"Don't you think she's kind of got it coming?" Soldier asks him back, and then she has to stop herself from reaching for the bottle again. "I mean, she knew the fucking rules. This isn't some first-year squeaker. She's one moon away from Confirmation. She should have known better."
"She's a kid," Sheldon says, as if maybe Soldier hasn't quite entirely understood that part, and he slows down to check a road sign by the glow of the headlights. "Town Farm Road," he says, reading it aloud. "Man, just once I wish someone else would pull this route."
"Kids screw up," Soldier says. "Kids screw up all the time, just like the rest of us. Kids screw up, and it gets them killed, just exactly like the rest of us."
"So you think they're gonna do her?"
"No, I didn't say that. But this is some pretty serious shit, Shelly. If we're real damn lucky, it's not so serious that we can't put it to rest by visiting Mr. Ass-for-brains Joey Bittern and—"
"She's just a kid," Sheldon says again.
"Some rules, nobody gets to break," Soldier says, watching the half-glimpsed houses and marshy fields and the trees that seem to appear out of nowhere rush past the hearse and vanish in the night behind them. "Some rules you don't even bend. And this isn't anything you don't already fucking know."
The Beatles make way for Jefferson Airplane, and Sheldon looks at the radio in disgust, but doesn't reach for the knob.
"Grace Slick is a fat cow," he says.
"Not in 1967 she wasn't."
Sheldon mutters something under his breath and stares straight ahead at the rain-slick road, the yellow dividing line, the stingy bits of the night revealed in the headlights. And Soldier's starting to wish she'd asked for another driver, beginning to wonder if Sheldon's up to this run.
"Someone does something like this," she says, "I don't care if it's just some kid or one of us, Madam Terpsichore or the goddamn Bailiff himself—"
"You've made your point," Sheldon tells her, and then he turns the wheel as the road carries them deeper into the marshes leading away to the Eagle Hill River and the Atlantic.
"You just don't mess around with shit like that," Soldier says, knowing it's time to shut the fuck up about the kid and let him drive, time to start thinking about the shotgun in the back and exactly what she's going to say to Joey Bittern when they reach the old honky-tonk at the end of Town Farm Road.
"I just don't think it's right," Sheldon Vale mumbles so softly that she barely catches the words over the radio and the storm and the whir of the tires on the road.
"Whole lot of crazy shit ain't right," she replies, then begins singing along with "Don't You Want Somebody to Love?" while Sheldon drives the hearse, and Soldier tries hard not to think about whatever is or isn't happening to Sparrow Spooner back in Providence.
III
In the night Above, the dying storm hammers cold-rain nails and gusting wind down upon the quaintly ancient and indifferently modern rooftops of sleeping Providence, filling metal spouts and concrete gutters far past overflowing, choking sewers, drumming at windowpanes, and wearing away sidewalks and slate shingles and gravestones bit by infinitesimal bit. Demons never die quietly, and a week ago the storm was a proper demon, sweeping through the Caribbean after her long ocean crossing from Africa, a category five when she finally came ashore at San Juan before moving on to Santo Domingo and then Cuba and Florida. But now she's grown very old, as her kind measures age, and these are her death throes. So she holds tightly to this night, hanging on with the desperate fury of any dying thing, any dying thing that might once have thought itself invincible.
But there's still magick left in her, the wild, incalculable sorcery of all storms, great and small. She wields invisible gale-force tendrils that whip and weave themselves through the groaning limbs and branches of an enormous oak tree that has stood watch at the Old North Burial Ground since a forgotten late-summer day in 1764 when it was planted here by Mercy Tillinghast, widow of Colonel Peter Mawney, planted here that something strong would stand to mark his life and her grief, something to shade the family plot from the wide Rhode Island sky. But now two hundred and forty-six summers have come and gone, and the tree, like the storm, is an old, old creature. The wind does not so much kill it as help it to die, helping it to finally lie down as its roots at last pull free of the soggy ground. It falls across the graves of Colonel Mawney and his wife and their children and grandchildren, shattering headstones and driving broken limbs deep into the earth.
Eighty-five miles north and east of the cemetery, Sheldon Vale comes to the end of Town Farm Road and kills the Caddy's engine, and the changeling named Soldier stares eagerly, anxiously out across the night-and rain-shrouded marshes.
And back in Providence, in the world Below, deep within the labyrinth of catacombs and chambers excavated by the ghouls all those long millennia before the first white men came here, like this storm, from lands far across the sea—down there, the sound of the tree falling is the rumbling grandfather of all thunderclaps. The children assembled before Madam Terpsichore and Master Shardlace (who have been joined now by Master Danaüs and Madam Mnemosyne) look up towards the ceiling of the chamber, and more than a few of them cry out in surprise or fear or a little of both. Madam Terpsichore raises her scarred and shaggy head, waiting to see if the layers of earth overhead will hold. It would be almost ironic, she thinks, if the evening's proceedings were to be interrupted by so simple, so random a thing as a falling tree. They might die here, all of them, ghul and pup and changeling brat, victims of a universe with less than no concern for their petty laws and taboos and ideas of justice.
"Be still, all of you!" she growls, perking her ears, flaring her nostrils, listening as the rumbling, root-cracking, wood-splintering cacophony quickly subsides, as dust and loosened bits of dirt and stone sift down and settle over her wards. She counts her heartbeats, waiting for the collapse, for a crushing, smothering burial they would all deserve, venturing so near the surface on a night like this. But apparently they're not meant to die tonight, and the ceiling of the chamber holds, braced up by strong pine timbers and brickwork arches. Terpsichore whispers a hurried, grateful prayer and crosses her heart three times.
"Luck favors fools," she sneers and bares her teeth for Master Shardlace, who has finally come slinking out from behind his sycamore roots to grovel before Danaüs and Mnemosyne.
"This is a very serious matter," Master Danaüs grunts, and brushes dust from his fur.
"So is my life, dear sirrah," Madam Terpsichore replies. "So are the lives of my students—"
"A matter to be settled swiftly," Danaüs continues, as though he has not heard her or doesn't care, "before further and greater damage is done. Damage we can't undo."
"Yes, yes," Shardlace simpers. "That's exactly what I was telling her only a moment ago. That's what I was saying, precisely."
Terpsichore turns and snaps her jaws at Shardlace, half wishing for a fight, for a fair and incontestable chance at his bare throat. But Shardlace only whines and withdraws once again.
"Surely," she mutters. "Then let's have ourselves done with it." And Madam Terpsichore yanks away the burlap hood covering the head and shoulders of the mortal man who has been led through the stormy night, across the wide cemetery, and down into the chamber by the pair of changeling couriers who found him hiding at the center of a circle of salt and hen's blood, mumbling the Twenty-Third Psalm, inside his tiny apartment at 7 Thomas Street. The man is bald, and his eyes are the color of clay. When he sees her face, the bald, brown-eyed man doesn't scream, but his lips tremble, and he shuts his eyes tight. Master Danaüs stands on his right and Madam Mnemosyne on his left, and the two changelings who brought him here are waiting near the rear of the chamber, should he try to run.
"This is him?" Madam Terpsichore asks, and she barks out a dry laugh. "This pathetic...this man is the fearsome magician who has so inconvenienced me and mine? Is this a joke, Danaüs? If so, it's a poor jest, indeed. He's no more than a drowned rat, near as I can tell."
The man's clothes, a white T-shirt and blue jeans, are soaked through and streaked with mud, and rainwater drips from his long salt-and-pepper beard and off the end of his nose. Sparrow Spooner, still kneeling at her mistress' feet, glances at the bald man and sees that he's lost his left tennis shoe. It's hard to tell what color his sock might be for all the mud.
"Sparrow, is this him?" Madam Terpsichore asks, staring down at the changeling girl. "Is this cringing, sodden rat standing here before me the man with whom you have spoken of our secrets?"
"He said—"
"We'll get to that, child. I simply asked you if this is the man."
"Yes," Sparrow Spooner replies, "that's him." And she immediately goes back to looking at the man's muddy feet.
Madam Terpsichore nods once, then tells the man to open his eyes. When he doesn't, she leans forward and whispers in his ear, "Would you prefer, rat, that we open them for you? It would be a pleasure, and thereafter there would be no danger of my having to ever ask you again."
"I've done nothing wrong," the man says unconvincingly, and Madam Terpsichore grins and licks her lips.
"Open your eyes, witch."
The bald man breathes in deeply, exhales the musty, mushroom-scented air, and then opens his clay-colored eyes. He stares into the devouring black-hole pupils of the creature that will be his judge and, most likely, his executioner before this stormy night has finished. He struggles against his terror and panic to fashion a still place somewhere deep inside himself, a mental sanctuary that he fills with an arbitrary litany of mathematics and hermetic symbols, a familiar, impromptu order to stand against the suffocating chaos of his fear.
"Is this not what you've worked for, what you wished for, Mr. Higginson?" Madam Terpsichore asks the man, leaning so close that her stiff gray whiskers brush painfully against the tip of his nose, and he flinches. "Isn't this why you've seduced our poor darling Sparrow into betraying us all?"
"She came to me," he replies, trying too hard to sound courageous, trying harder still not to shut his eyes again.
"Yes, rat, she did. She came to you; she allowed herself to be lured to you, and for that, I promise, she will be punished. But you laid the trap, Mr. Higginson."
"I...I know...what you are," the bald man stutters. "I'd seen so much already. The child, she was...was only a means of further affirmation."
"You, sirrah rat, know nothing!" Madam Terpsichore snarls and places the tip of one clawed index finger lightly against his left temple, pressing only hard enough to draw a thin trickle of blood. "Even now, when you believe that all your careful, exhaustive researches, your divinations and discoveries and unholy dreams have at last been justified, that you have been rewarded with some half-imagined truth; even now you still know nothing, nothing whatsoever."
"I know that you're real," he says, the words slipping recklessly from his lips. "Yes, I know something...I know about Iscariot and Narcissa Snow, about William Wellcome, and, by the gods, I know a few things about Richard Pickman, too."
"You will not dare repeat those names, Mr. Higginson," Madam Mnemosyne admonishes. "Those names are not spoken here, not ever."
"I've even seen one of Pickman's paintings," Elgin Higginson continues undaunted, because he can't imagine he has anything left to lose. Beads of sweat and rainwater roll down his brow and sting his eyes. "Did you really think you'd found them all? That you'd destroyed them all?"
"Enough," Master Danaüs barks and then shoves the bald man so hard that he almost loses his balance and falls. "The child should offer her testimony now. There's nothing more to be gained by listening to this man." Mesdames Terpsichore and Mnemosyne nod and murmur their agreement.
"Yes," Madam Terpsichore says. "Sparrow Spooner, who has passed unscathed through her first two trials, who was chosen by the Cuckoo even before her birth and has been among us for all her short life, will speak now, as is her right and duty." And the ghoul looks down at the child kneeling beside her.
"What else do you want me to say?" the girl asks and glances nervously up at the bald man.
"Only what you have already told me and Master Shardlace—how this man, by duplicity and deceitful magicks, drew you forth from the warrens, from whence you were forbidden to leave before your Confirmation, and how he then tricked you into divulging secrets entrusted to you by the Cuckoo."
"But I've already told you," Sparrow says, still looking up at the bald man, at the rivulet of blood creeping down the left side of his face. "That's him. He's the one."
"Yes, dear, but you now must tell us in his presence."
"I don't see why," the changeling mutters and turns away. All of them are watching her, waiting—all the other students, her ghul masters and mistresses, and the odd bald man from Above who let her walk with him beneath the sun. "There's nothing that I haven't already said. Nothing important."
Madam Terpsichore exchanges a quick glance with Master Danaüs and then drops down onto her skinny haunches, her aged knees and anklebones cracking loudly from the sudden exertion. She places a hand beneath the child's chin and gently turns Sparrow Spooner's head until she's gazing into the girl's eyes again.
"Do not forget that your own fate will be determined this night," Madam Terpsichore tells the changeling. "The decision has already been made to spare your life, but a penance may yet still be uttered that will make you regret our mercy."
When Sparrow Spooner opens her mouth to reply, the ghoul slaps her so hard that her ears ring, and she bites her tongue. Madam Terpsichore's claws leave five parallel gashes in her pale flesh, ragged crimson wounds that will become scars that the changeling will carry all her life, however short or long that might prove to be.
"Now, do as you've been bidden," Madam Terpsichore tells her, and when the girl begins to sob, the ghoul raises her hand as if to strike her again. "Don't you dare cry, Sparrow Spooner. I shall not abide that human weakness in my sight."
"Leave her alone," the bald man says, and he takes a step towards Sparrow Spooner. "If you want to hurt someone, hurt me."
"Don't worry, rat," Madam Mnemosyne replies, pulling him back, showing him almost all of her yellow teeth at once. "We shall do that, momentarily. But there is an order to these things. Do not be impatient for your undoing. It's coming."
"We're waiting, child," Master Danaüs says irritably.
His eyes are like fire, Sparrow Spooner thinks. His eyes are like spoonfuls of fire.
And then she wipes her nose and does as she's been told, because she's seen enough in the eight years since the Cuckoo delivered her to the warrens dug deep beneath College Hill and Federal Hill, beneath Swan Point and the Old North Burial Ground and St. Francis Cemetery, to know that the ghouls hate the contrivances of human speech too much to waste their breath on idle threats. She faces the bald man named Mr. Higginson, the man who bought her frozen lemonade and taught her seven new words for green. The man she showed a particular mausoleum on the west bank of the Seekonk River. The man she drew a crude map for and answered questions she knew she shouldn't. The man she daydreamed might one day become her father.
"I'm sorry," she says to him, and then Sparrow Spooner tells the ghouls everything they want to hear.
IV
At the windy nub end of Town Farm Road, where the asphalt has turned to gravel and mud and potholes, Sheldon Vale and Soldier sit in the hearse, listening to the rain falling hard against the roof of the Caddy. Sheldon lights a Marlboro Red and passes it to Soldier, and she takes a deep drag and keeps her eyes on the place where the uneven edge of the road vanishes into tall brown marsh grass and cattails.
"So, you really think they're gonna kill her?" Sheldon asks.
"Fuck, will you shut up about the kid? Right now, the kid ain't our problem."
"What if they know we're coming? What if—"
"Sheldon, shut up a second and listen to me," and then she takes another pull off the Marlboro and holds the smoke until her head begins to buzz, then exhales through her nostrils. "They don't know jack shit, okay? I scried this one myself last night. I scried it fucking twice, okay, and they don't know bupkes," Soldier says, silently daring Sheldon to make a crack about her less than stellar magickal abilities.
"All the obfuscation lines are in place," she continues, "and they're working like gangbusters. Trust me. Those assholes are sitting in that dive playing fucking New York stud, drinking beer, talking trash, and they don't know we're coming."
Sheldon lights another cigarette for himself and sighs. "You know I get the shakes," he says. "That's all. Give me a minute and they'll pass. Give me a minute and I'll be right as rain."
"I swear, Sheldon, it's beyond me how you ever made it through the trials."
"It wasn't a pretty sight, I'll tell you that," he says, and laughs, then stares out the driver-side window, looking past his reflection at the chain-link fence and NO TRESPASSING signs where the Ipswich town dump begins.
"You ought to cut the lights," Soldier says and taps her Marlboro against the rim of the ashtray. "You're gonna kill the battery, and I'm in no mood to have to walk all the way back to town through this shit."
Sheldon shrugs and switches off the headlights. The rainy night swallows them at once, and Soldier gazes out at the marsh again, waiting for her eyes to adjust to the darkness.
"I've never gotten the shakes," she says. "Not even once. The Bailiff, he told me I would, but it never happened."
"Yeah? That's 'cause you're such a coldhearted little bit of jam."
"Fuck you, Shelly. It just never happened, that's all," and then Soldier considers asking him to turn the radio back on, but she doesn't. There's no more time for music. No more time for anything but the job. Time to be quiet, to focus, time to get her act together and find the frequency of this night. That's how she always thinks of it, the frequency of the night, like turning the dial past static and white noise and all the crap you don't want to hear, turning the dial until you find just the right station.
"I still can't believe Bittern thought he could get away with this," Sheldon says. "I mean, I know he's not the brightest bulb in the pack, but Jesus..."
"What difference does it make? Ten minutes from now he's a corpse. Who gives a sick? The whys and wherefores aren't our concern."
"When he got tight with those assholes down in Jersey, I told the Bailiff then it'd come to this, sooner or later. A little creep like Bittern gets it in his head he's gonna play mobster with the big boys and—"
"Stop it," Soldier growls and snaps her fingers a few inches from his face; Sheldon shuts up and goes back to staring at the dump. "Hell, you go on like that long enough," she says, "and I will get the shakes." She decides she's had enough of the Marlboro and stubs it out in the ashtray.
"You got your goggles, right?" Sheldon asks her.
"I can see just fine without them. I'm going in first, but you better be right there to cover my ass; you understand me?"
"Yeah," he replies, crushing out his own cigarette. "I think I can handle it."
Soldier reaches beneath the seat and retrieves the bottle of Dickel. "Not a word," she says before Sheldon can remind her what the Bailiff told him about keeping her sober. Then she has one last drink before it starts, before the shitstorm barreling her way, one to keep her warm, and she tucks the bottle out of sight again.
"Let's get this over with," she says and pulls the black Beretta 9mm from the shoulder holster inside her jacket. Soldier pops the clip and runs her thumb quickly across the shells, counting off all eight Browning cartridges one by one.
"Man, I hope you're right about the lines," Sheldon says, reaching for the twelve-gauge Ithaca and the box of shells behind his seat.
"Tell me the last time I was wrong," Soldier replies and slides the clip back into the Beretta, then pulls the slide, chambering the first round. She flips off the safety, presses the gun to her lips, and shuts her eyes, whispering a short prayer to Mother Hydra.
"And no one walks out of there alive," Sheldon says, and one of the shotgun shells slips from his fingers and rolls away into the darkness between his legs.
"That's what the man said. No one walks. You've got the gasoline?"
"Five gallons. That ought to do the job."
"Yeah," Soldier says, returning her pistol to its holster. "That ought to do the job. Try to get a few of those shells in the gun, okay?"
"Yeah," Sheldon replies and almost drops another one.
"And hurry the hell up." She reaches for the other shotgun, the massive Mag-10 Roadblocker she picked up after some asshole Thelemite wearing Kevlar body armor and seven different deflection sigils had come at her with a goddamn Sears Craftsman chain saw a few months back.
"That's a scary fucking gun," Sheldon says. "You expecting something in there besides Mr. Bittern and his cardsharp buddies?"
"You just watch my ass, Shelly. Let me worry about the big scary monsters, okay?"
Sheldon finishes loading his shotgun and smiles a nervous smile at her. "It's a deal, lady. I'll look at your ass, and you'll kill all the bad guys."
"Yeah, you crack me up," Soldier says and opens her door. The rain is cold, colder than the November night, and she knows she'll be drenched long before they reach the door of Quaker Jameson's roadhouse.
"Do you see it?" Sheldon shouts at her across the roof of the hearse, shouting to be heard above the storm. "I don't see it anywhere." She glances over her shoulder at him, and he's fiddling about with the resolution lever on his goggles.
"You just fucking follow me," she says. "Forget about those things."
The wind wraps itself about her, giving the rain teeth, whipping at her clothes and the exposed skin of her face as though this is something personal, like maybe the storm has decided it has a vendetta against her. Soldier grits her teeth and squints into the stinging rain. She can see the path leading away through the marsh grass to a rickety, crooked catwalk and, beyond that, the low, sagging roof of Jameson's place. There's the faintest blue-white glow surrounding it all, a gentle, constant pulse of pale alabaster light, the last little bit of the glamour that she can't see through. Perhaps, she thinks, if the cheapskate son of a bitch would hire a halfway decent witch, then she might need goggles, too. But she doubts it.
"I still don't fucking see it," Sheldon mutters, but this time she ignores him. She's found her frequency, the clear channel she needs to see this whole mess through, and she knows that he'll follow her, that he'll be where she needs him to be when she needs him there. The big shotgun is heavy and solid and comforting in her hands, and that's something else she knows she can rely on.
Soldier can't hear the gravel crunching beneath her boots, not over the wind, but she can feel it. She steps across a deep puddle and into the tall grass, and the gravel is immediately replaced by mud. Even through the storm, she can smell the marsh and the backwaters of Paine Creek, that musky, fishy odor that's not so very different from the smell of sex. And all of it makes her stronger, the gun and the storm and the marshes, the knowledge of the flooded creek flowing to the river that flows, in turn, down to Plum Island Sound. Now that she has the frequency, now that she's so hard in fucking tune, there's no such thing as distraction, only these sensations and her concentration growing so sharp that it almost hurts. Ten more steps carry her from the mud and out onto the catwalk; she can feel the weathered old boards squeak and flex as she moves quickly towards the roadhouse.
"Hell, that place was stuck together back before the Revolution," the Bailiff said once, some other night when she'd had some other business out at Quaker Jameson's. "Started out as a whorehouse for the Masonic and Rosicrucian types, some place they could dip their wicks in whatever struck their fancies without drawing too much attention, if you get my drift. Mr. Benjamin Franklin himself had a few good pokes in that place."
Thirty-three more steps, and Soldier reaches the wide landing where the catwalk meets the front porch. The light from electric lamps leaks out through the milky antique windowpanes, and this is probably the only place in all of Essex County with electricity tonight, she thinks, wondering if it's an enchantment or just a backup generator. She steals a quick look over her shoulder, then, just to be sure, and there's Sheldon Vale in his ridiculous goggles. He motions towards the roadhouse with the barrel of his own shotgun, and Soldier gives him the thumbs-up. Sheldon nods his head, and she reaches for the door handle, something ornate cast in copper gone cancerous with verdigris, black and green shades of corrosion, and in the last instant before she touches it, the handle sparks. A single fleeting arc of sun-colored light that bridges the space between her palm and the handle, and she curses under her breath and jerks her hand back.
"Fuck this," she hisses and grips the handle, squeezing hard against the protective charm, collapsing it into something so small and ineffectual it couldn't even fry an ant. She feels the magick coursing through her and bleeding off into the night, losing itself in the turmoil of the storm. And then the door swings open wide, bathing her in yellow incandescence, and Soldier pumps the shotgun once as she steps across the threshold and points it at the first thing that moves.
A skinny, redheaded boy in a Sex Pistols T-shirt stares back at her from the other end of the Roadblocker. He drops the serving tray he's carrying, and five or six mugs and a couple of shot glasses shatter when it hits the floor, spraying beer and whiskey at her feet.
"On your fucking knees, faggot," she snaps at him, and a second later the boy's spread out on the roadhouse floor with all that beer and broken glass. Soldier swings the shotgun around and takes aim at the table where Joey Bittern and Quaker Jameson and two others have all put down their playing cards and drinks and are watching her. One of them's a morning gaunt, perched there on a bar stool like something that would happen if a stork knocked up an orangutan, and the other, seated across the table from the gaunt, is an orchid-skinned demon smoking a cigar and looking about twice as pissed off as anything she's ever had the misfortune to come across.
"You told us the door was locked," the demon says and turns away from her, glaring at Jameson from beneath its scaly beetled brows. "You also told us we'd hear her coming."
No one said anything about demons, Soldier thinks, channeling the thought the way that Madam Melpomene taught her years ago, sending it straight back at Sheldon, to the spot by the doorway where she hopes like hell he's still standing. I'm pretty goddamn sure the Bailiff didn't say dick about any fucking demons.
There's no reply, nothing rolling back from Sheldon's mind to hers, and Joey Bittern grins like a cartoon wolf and lights a cigarette. He's a big man, half Portuguese, half Narragansett Indian, one hundred percent asshole, and he blows a smoke ring at the rafters supporting the high ceiling.
"You're fifteen minutes late, Soldier," he says and pretends to check a nonexistent wristwatch. "What held you up, Mr. Vale? You miss that turn again?"
"Put your fucking hands down on the fucking table," Soldier says. "All of you. Now!"
"See, I'm thinking this game's a little steep for your tastes, sweetmeats," Bittern says and blows another smoke ring at the ceiling. "I'm thinking you talk like a big dog, but when the chips are down—if you'll excuse the pun—you got nothing to bring to the table but talk."
"That's a goddamn big motherfucking shotgun," Quaker Jameson says nervously.
"You said we'd hear the bitch coming," the demon says again, and then it picks its cards up off the table. "I like to pissed myself."
"I said put your hands on the table—"
"This is about the little Spooner girl, right?" Joey Bittern asks her. "Am I correct in assuming that those mangy curs down on Benefit Street have sent you out here to slap my paddies for pointing our friend Mr. Higginson in the right direction? That's all I did you know, point him—"
"Talk to me, Shelly," Soldier says, tightening her finger on the trigger. "Tell me that you're fucking back there somewhere."
"Oh, I'm back here, Soldier girl," Sheldon replies, "but I think you better be quiet and listen to the man."
And there it is, presto-change-o, abrafuckingcadabra, suddenly plain as daylight, and there's no time left to think about how fucking stupid she's been, how she should have seen it coming, how the Bailiff should have seen it coming a mile away. She's in the frequency, and she pulls the trigger, but the Mag-10 answers with a hollow, harmless click.
"Very thoughtful, Mr. Vale," Bittern says. "It's good to see you're the dependable sort. An eye for details and all that."
The morning gaunt makes a gurgling sound that's probably meant to be laughter. The air between her and the card table has begun to shimmer and twist back upon itself, like heat rising from blacktop on a summer's day, and so she knows that the four at the table have been removed from the reach of bullets, anyway.
"Shelly," Soldier says, speaking as calmly as she can still manage, "do you even begin to have any idea how much shit you're in?"
"You just let me worry about that," he tells her.
"Well said, Mr. Vale," Joey Bittern mumbles around the filter of his cigarette. "You're a man of few words, but you choose them well."
"Did you learn that from a fucking fortune cookie?" Soldier asks, her eyes moving quickly from Bittern to the orchid-colored demon, from the demon to Jameson to the gurgling night gaunt. "Give him a chance, he'll talk your goddamn ear off. Ain't that right, Shelly?"
"How about we cut the fuckin' la-di-da chitchat," the demon scowls, "kill the changeling bitch, and get back to the game? Or maybe, Joey, you think all this drama's gonna make me forget how far in the hole you are?"
"See, Soldier? Some people just aren't capable of savoring the moment," Bittern laughs and shakes his head. "They always have to be rushing things. Some people"—and he glares at the demon—"they just don't quite appreciate the sheer, astounding elegance of deceit done right."
"Whatever they decide to do with you, Shelly," Soldier says, reaching into her jacket for the 9mm, "I just hope it's going to hurt for a long, long time."
And then she hears Sheldon Vale take a step forward, the soles of his boots scuffing across the floorboards, her senses jacked up so goddamn high and tight that she even hears his index finger squeeze the Ithaca's trigger, hears the hammer fall, and then there's thunder—the night cracking open to show the storm precisely how it's done, to teach it something about tempest and destruction. There's no more than a foot between the muzzle of the shotgun and Soldier, and it tears through her easy as a hot knife through butter...
. . and she's stumbling, falling towards the demon's shimmering bulletproof barrier, driven forward by the force of the blast...
...and there's a wet spray of blood and bone and mangled entrails moving out before her, the lead shot opening her like a butcher's prize sow, like a bouquet of bloodflowers, like Noah's goddamn forty-day flood...
And seconds (tick)
have become (tock)
entire minutes.
Soldier shuts her eyes because she doesn't want to have to die looking at her own guts and stinking Joey Bittern's ugly, fucking grin. He's killed me, she thinks, and wonders if all that crazy shit Madam Terpsichore and Master Danaüs taught her about Mother Hydra and Father Kraken and the lives waiting for her out past death is anything more than pretty storybook tales designed to
make...
this...
easier.
I'll know in a moment, she thinks, and then all the world is pain.
And
then
"And no one walks out of there alive," Sheldon says, and one of the shotgun shells slips from his fingers and rolls away into the darkness between his legs.
"That's what the man said. No one walks. You've got the gasoline?"
"Five gallons. That ought to do the job."
"Yeah," Soldier says, beginning to remember things that haven't happened yet, an instant of déjà vu so strong that it's nauseating, so strong it almost seems to knock the breath from her lungs. She grips the Beretta, her hand trembling just the slightest bit, and she doesn't return it to its holster. "Yeah," she says again. "That ought to do the job. Try to get a few of those shells in the gun, will you?"
"Yeah," Sheldon says and almost drops another one.
"Christ, you're a clumsy asshole, Shelly," she whispers and swallows, her throat dry as August dust, and right now she'd give almost anything for whatever's left in the bottle beneath the seat. Sheldon slips an orange shell into the Ithaca 37 Classic and looks up at her.
"What's wrong?" he asks.
"What?"
"You look like you just saw a fucking ghost," he replies and takes another shell from the box. "I told you to stay sober. The Bailiff told you to stay sober."
"What the hell did they offer you?" she asks, and when he looks up again, Soldier raises the 9mm and puts two in his skull, right between the eyes. Bang, bang. Easy as falling off a log, exactly what the doctor ordered. Sheldon Vale slumps back against the driver-side door of the old hearse, and his whole body shudders once and is still.
"You sold me out, you bastard," she whispers. "You fucking sold me out."
She sets the pistol on the dash and reaches for the bottle of whiskey beneath the seat and sits drinking it, listening to the rain and watching the dim alabaster glow wreathing Quaker Jameson's roadhouse. The air trapped inside the hearse smells like cordite and blood, cigarettes and whiskey. When the pint's empty, Soldier lays the bottle down next to her and reaches for the Mag-10 tucked behind her seat.
"Five gallons," she says. "Yeah. That ought to be plenty."
V
Sparrow Spooner opens her eyes again when the man from Above begins to scream and pray to the Catholic god he abandoned more than half a lifetime ago. She isn't sure how long she's been huddled there, alone in this muddy, dark corner of the chamber, naked and shivering beyond the candlelight. Madam Mnemosyne laid her down here when they'd finished taking two fingers from her left hand, and she told Sparrow not to move a muscle and to keep her eyes tightly shut and her mouth shut tighter still, unless she wanted worse than she'd already gotten.
"When we are done, she must remain useful to us," Madam Terpsichore said to the other ghouls, before the cutting began. "She still has potential, this one, despite the gravity of her recent indiscretions." And Sparrow understood that Madam Terpsichore was the only thing left standing between her and the full wrath of the hounds. They would have seen her dead—Danaüs and Mnemosyne and that mongrel bastard Shardlace who'd sniffed out her forbidden liaisons with Mr. Higginson. They would have made a terrible example of her, something for the rest of the Children of the Cuckoo to see. Would have carved her up like a body taken for the dissection slab and then hung all the divided bits of her throughout the warren, her hands and feet and innards strung on baling wire and dried grapevines and left to rot, left for the other changelings to see again and again and again.
Master Shardlace wanted them to take her tongue, that she might never utter another of their secrets. Madam Mnemosyne suggested that a binding geas would be more appropriate and far more effective and would not so diminish her value as a courier and messenger. But, in the end, Master Danaüs left the decision to Terpsichore, and she asked only for two fingers from Sparrow Spooner's left hand, the pinkie and ring finger. They used one of the ceremonial knives, carved from greenish soapstone and graven with images of the writhing, many-eyed daughters of Mother Hydra. Madam Terpsichore cauterized the stumps with a pinch of belladonna and a few words of musical, alien language, some arcane tongue that Sparrow has yet to learn. The ghoul threaded the severed fingers onto plastic fishing line and hung them about Sparrow's throat.
"You will wear that until I say otherwise," Madam Terpsichore told her, and then Madam Mnemosyne carried her to the spot where she now sits with only the earthworms and tiny white mushrooms for company, listening to the man named Higginson scream.
Sparrow Spooner opens her eyes, even though no one's told her that she could, and she squints past the gloom and the ache in her hand that has begun to move slowly, steadily up her entire arm. Madam Terpsichore has spread wide her long, clawed fingers, jabbing them towards the night sky hidden by the ceiling of the chamber, above the fallen tree and the Old North Burial Ground, above Providence and the storm, and she's calling out the names of gods and darker things. As for Mr. Higginson, Sparrow sees that he's been given to the ghoul pups and changelings, all of whom have left the places where they were sitting and are clambering over his thrashing body. It isn't often that they get fresh meat, fresh and living meat, and it doesn't get any fresher than this; so this is one more part of her punishment, that she will not be permitted to take part in this rarest of feasts, even though it's her doing that the man's down here.
Madam Terpsichore asks almighty Father Kraken to let this business end here, tonight, with no further repercussions. Sparrow knows that particular prayer, the snipping of a dangling thread, the tying off of a knot against any maleficent consequence. She begins to whisper it along with the ghul, then stops, too distracted by the man's screams and the vicious feeding noises of her peers. Two of the pups are wrestling over a slippery, fat length of intestine; Sparrow Spooner's empty stomach growls, and she licks her lips hungrily.
Madam Mnemosyne and Master Danaüs bark and yap their encouragement to the frenzied, blood-spattered pack, and Master Shardlace laughs and mocks the screams of the dying man. Madam Terpsichore squats next to Mr. Higginson, brushing aside two or three of her students, and she snatches the man's soul as it exits his right ear like a greasy puff of smoke. So, Sparrow thinks, he'll be dead soon, and I won't ever get a father. I won't even get so much as a single mouthful of him.
And then she feels something wash over her, something quick and dizzying and bittersweet that she's never felt before, something that she might almost have mistaken for simple déjà vu (which, she has been taught, is only the memory of the world flowing backwards through her from the end of time), if not for the way Madam Terpsichore turns and looks at her—a questioning, startled look from her red eyes to ask, You felt that, too, didn't you, child? The others seem oblivious, but last semester Sparrow Spooner ranked first in her Discord and Continuity class. She has often felt very faint things before, subtle or fleeting magicks and cosmic shudders that almost everyone else has missed. Things like the summoning that Mr. Higginson cast over the warren below the yellow house on Benefit Street, drawing her out into the day.
Be still, Madam Terpsichore's eyes say. Be still and be silent. We will speak of this later. And then she looks away again, squeezing the captured soul until there's nothing left in her palm but a dull green, pea-soup-colored stone, which she swallows at once.
One of the older changelings, a tall, blond-headed boy named Nehemiah Sweet, bites through the man's carotid artery, and Mr. Higginson is finally gone, gone forever, body and soul and everything in between. And then Sparrow Spooner closes her eyes again, shuts them tight as she can, just like Madam Mnemosyne said she should; she tries hard not to hear the delicious sounds of bone snapping and blood spilling and teeth ripping through sinew and gristle, tries to hear the storm still raging overhead instead. She doesn't let herself think about what she might or might not have felt, whatever it was or wasn't that Madam Terpsichore might, or might not, have felt as well. The pain from her missing fingers helps to distract her, the pain and the falling rain Above and the thunder, and before the others have finished with Mr. Higginson's carcass, she's asleep.
VI
When Soldier is finally done with the twenty-three incantations that have forever sealed the doors and windows of Quaker Jameson's roadhouse, the measured, angry stream of syllables to secure every loose board and ill-fitted sheet of corrugated aluminum—and maybe she's not much of a magus, but, by the Mother and Father, she knows how to plug a hole—she begins emptying the five gasoline cans. The demon and the morning gaunt have already gone, of course. This was never their fight, just something they came to see, a little mortal entertainment to interrupt the monotony of their long, long lives. The demon went down, vanishing into some deep, secret cavern or melding seamlessly with the weathered metamorphic strata below the marsh. The gaunt went straight up, a glittering, firework ghost climbing into the low clouds and disappearing with a distinct crack. Soldier knew better than to try to stop either of them.
By the time the last of the gallon cans is almost empty, Jameson and Joey Bittern have both given up trying to escape and have started making her promises, offers of power and money and sex and other things that she either doesn't need or has all she wants. They press their lips to the windowpanes and shout for her to see reason and stop. They make threats and curse the hounds and every changeling ever stolen from its crib or stroller or the arms of its mother. She listens for a little while, standing there in the rain, and then she turns and walks back to the muddy edge of the road, trailing the last of the gasoline behind her.
"I'm thinking this game's a little steep for your tastes, sweetmeats," she whispers and lights one of the highway flares from the Caddy's glove compartment. It paints the night with a sizzling ruby glow, and for a moment Soldier holds it at arm's length, staring into the heat and brightness, the dazzling inferno cocktail of potassium nitrate, magnesium, strontium, and sulfur.
"I'm thinking you talk like a big dog," she says, and imagines she hears the roar of Sheldon's shotgun again, imagines the pain as it tears her apart and the smirk on Joey Bittern's face. "But, when the chips are down, you got nothing to bring to the table but talk." And then she tosses the flare away, towards the tall grass and the catwalk, and Soldier watches as the flames rush hungrily back towards the roadhouse.
VII
Upstairs in the big gray house on Angell Street, Emmie Silvey lies in her bed, listening to the rain drumming against the roof and the bedroom window. Her father is sitting on the edge of the bed. He's been reading to her from one of her books, Moon Mouse, but he keeps pausing to listen to the storm. This last time there was a sudden cracking sound from the sky just as Arthur the mouse asked his mother how far away the moon is, a sound that might have been thunder, and he stopped reading and stared at the black, rain-slicked window.
"Deacon, it's only thunder," Emmie says. "What does Arthur's mother tell him?"
"Baby, how many times in the last three years have I read you this book?" he asks her, but his voice sounds sleepy and far away. He's been drinking tonight. She can smell it. She can always smell it.
"I don't know," she replies, because she doesn't. Sadie bought her the book before going away to New York, gave it to her for her fifth birthday. Inside the cover, Sadie wrote, Happy fifth birthday, my beloved moonchild. And she explained to Emmie that she'd been born on Halloween night, just as the full moon was rising in the sky.
"Deacon," Emmie says again. "What happens next?"
"Oh," her father says, turning away from the window and smiling sleepily down at her. "You know, I could read you another book tonight. How about some Dr. Seuss? Or Where the Wild Things Are?"
"No, I want to hear how far away the moon is."
Her father sighs and looks at the book lying open in his lap. "Very far," he tells her. "It was farther than the meadow. Farther even than the farmer's cornfield. Farther than the wheatfields."
"Golden wheat," Emmie says, correcting him. "Farther than the fields of golden wheat. I can tell you're only pretending to read."
"Yeah," he says and smiles at her. "Kiddo, you know this old book by heart. You don't need me to read it to you anymore."
"I need to hear the way you say it."
"Is that so? I think you're just being ornery."
"What's ornery?"
"What you're being," he replies.
"That's another of your Alabama words, isn't it?"
"Not really," Deacon says. "Lots of people who never lived in Alabama say ornery." And then he continues reading, and Emmie goes back to listening to his voice. There's a little bit of a slur from the beer, but it's still something comforting against the rain and wind and lightning. Arthur the mouse crosses the meadow and the cornfield and the golden wheatfield and finally climbs an old fire escape up to the moon, which is made of yellow cheese, just like his mother said that it might be. He shows her the picture, the big wheel of Swiss on someone's kitchen table, Arthur the mouse slipping in through an open window.
"It should be white," she says, as she almost always does, and her father nods and agrees with her, as he almost always does.
"Yes, but this is poetic license," he says.
"What's that?"
"Permission to lie," he tells her and goes back to reading. Arthur the mouse runs home to his mother, and he tells her how he nibbled at the moon.
"He would have broken his teeth on the rocks," Emmie says and rolls over on her left side to face the window.
"Yes, I imagine he would've," her father says.
"When is Sadie coming home?" she asks, and he sighs again. Whenever he sighs, the beer smell grows stronger, and she wrinkles her nose.
"Sadie lives in New York now, Emmie. You know that."
"We could go see her. We could ride the train."
"Maybe when the weather's warmer," her father says, and starts reading again. Arthur is alarmed at the half-moon, amazed that he managed to nibble so much of it away.
"We could go see the dinosaurs," she says, "when we go to see Sadie."
"We will. I promise. We'll go to see the dinosaurs. Do you want me to finish this?"
"I know how it ends," she says, wishing the rain would stop so she could see the moon again, to be sure it was still up there, so she could fall asleep without the sound of the storm in her ears. "He didn't eat it all. He only nibbled."
"I told you that you know this story by heart," her father says. "I told you that you were being ornery."
"You know everything," Emmie says and wiggles farther beneath the warm flannel sheets and blankets and the quilt that Sadie bought when she went to Tennessee. The quilt is white with autumn-colored maple leaves stitched one to the other.
"I don't know everything."
"You know magic," Emmie says.
"No, baby, I just know a few tricks. I've told you, it ain't nothing but tricks."
Emmie nods her head, though she isn't sure that she understands the difference. "I want to hear some music," she says. "I want to hear Doris Day."
"It's too late for music," her father says. "You have to get up for school—"
"It's not too late for music. Music isn't like stories. You don't have to look at the pictures."
"If I play Doris Day, will you go to sleep?" he asks, and she nods again.
"But you have to stay with me until I'm asleep," she tells him.
"It's only rain," her father assures her. "It can't hurt you, you know?"
She watches the rain streaking the glass a moment, pelting the windowpane like it wants inside, and she thinks maybe Deacon doesn't know everything, after all.
"'You Are My Sunshine,'" she says.
"Yes, well," her father says, "it's very sweet of you to say so."
"No, silly," Emmie says and turns her head to frown at him. "That's the song I want to hear. I want you to play 'You Are My Sunshine.'"
"And you promise you'll go to sleep if I do?"
"If you promise to stay with me until I do."
"Then it's a deal," Deacon Silvey says and stands up, wobbling a little, and crosses the room to the shelf where Emmie keeps all her CDs. It takes him only a moment to find the disc with "You Are My Sunshine" on it, and that's something else that Sadie gave Emmie. She has three Doris Day CDs, and they were all gifts from Sadie. He hates Doris Day, and sometimes he thinks this whole thing, his daughter's fascination with Doris Day, is a plot to drive him the rest of the way insane. He presses a button on Emmie's pink and white Hello Kitty stereo, and the lid pops obediently open to accept the CD.
"When's it gonna stop?" Emmie asks.
"What? You mean the rain?"
And she doesn't answer right away; while he waits, Deacon forwards the CD to track five, sets it on repeat, and presses the button marked PLAY. In a moment the room is filled with the shrill din of accordions and the steady thump thump thump of a double bass.
The other night, dear, as I lay sleeping...
"Yeah," Emmie says. "When's the rain gonna stop?"
"Soon," he replies. "It'll stop tomorrow. By this time tomorrow, it'll have blown all the way out to sea."
"That's good," she says, and Deacon goes back to sit on the bed with her.
"It's only rain," she says.
"Nothing but," he assures her and leans over to kiss her forehead. In the lamplight, her yellow eyes seem almost golden. He brushes a few strands of ash blonde hair from her face.
"It doesn't scare me," she tells him, sounding as though she really means it.
"That's 'cause you're a big girl now, Emmie. Big girls don't have to be afraid of storms anymore."
And he sits with her, listening to the rain and thunder, listening through the rain for anything else that might be out there in the night, the things he's spent the past eight years listening and waiting for. The polka music fills the room, and he stays with Emmie until she's asleep. Then he switches off the lamp. The night-light will be enough until morning; Emmie's never been afraid of the dark.
You are my sunshine, my only sunshine.
You make me happy when skies are gray...
Deacon Silvey switches the stereo off on his way out of the room, pausing a moment to be sure that Emmie doesn't wake up. When she doesn't, he leaves her and pulls the door shut behind him.
## I
## Parallel Lives
One need not be a chamber to be haunted,
One need not be a house....
—EMILY DICKINSON
## ONE
## Emmie
Emmie and Deacon sit together on one of the long antique benches lined up neatly inside the old Kingston Station, daughter and father waiting impatiently with all the other people headed south to New Haven or New York or wherever it is they're all bound at half past eleven on a cold Saturday morning in February. Old Kingston Station instead of Providence Station because Deacon says he likes the long drive, and, besides, he has a friend in West Kingston he hasn't seen for a while. He's sipping at a can of Diet Pepsi from a vending machine near one of the windows, and Emmie is silently wishing that the train would hurry up. She loves taking the train to see Sadie, even when she has to ride alone, even when her father is too busy with his shop to go with her. She pretends she's Eva Marie Saint in North by Northwest, or she pretends that all the sights rushing by outside the windows of the Amtrak are places that she's never seen before, exotic, far-off places from the books she's read, or she just sits listening to the reassuring drone of the wheels on the tracks and pretends nothing at all.
"Did you pack your toothbrush?" her father asks, and Emmie nods her head, even though she didn't pack her toothbrush because she has another one at Sadie's. "Did you think to pack clean underwear?" he asks.
"Yes," she says, wishing he wouldn't ask her questions about her underwear in public.
"More than one pair?"
"Yes, Deacon. More than one pair."
There's a round mirror mounted halfway up the wall directly in front of them, the sort of mirror she's seen in convenience stores so the clerks will know if anyone's shoplifting, stuck up there between a wide bay window and the door with a glowing red-orange exit sign mounted above it. Emmie watches herself and her father, the two of them caught in that distorted, fish-eyed reflection. She nods her head, and the girl in the mirror nods her head, too.
"Why are you nodding your head?" Deacon asks, and Emmie shrugs. The girl in the mirror shrugs right back.
"I just felt like it."
"Oh," Deacon says and sips his Pepsi. "I thought I might have missed something."
"No, you didn't miss anything," she tells him.
In the mirror, her father is wearing the gray wool sweater she gave him for his last birthday, the sweater and a pair of corduroys, and she can see the puffy reddish half circles beneath his green eyes, the half circles he gets when he's been drinking too much and sleeping too little. The half circles that mean he's spending too much time remembering things he shouldn't. Her own eyes are almost the same color as sunflowers, and they glint faintly in the morning light. She's the only kid in her school with yellow eyes, the only person she's ever seen anywhere with yellow eyes. Sadie says it means that her soul is golden, and she'll live to see more sunny days than rainy nights, but Deacon says that's bullshit and nonsense, that it doesn't mean anything at all that her eyes are yellow. He says lots of people have yellow eyes. The fat black tomcat that lives next door with a skinny old man named Mr. Bloom has eyes almost the same color as hers.
"Don't you talk to strange people," her father says, staring down at his soda can.
"How do I know if they're strange people?" she asks, even though she understands what he means. "What if they're just pretending to be normal and fool me?"
"Don't talk to people you don't know."
"Even the steward?"
"No, Emmie, not even the steward."
"But what if she's strange?" she asks, still watching their reflection in the round mirror on the wall.
"The damn train's late," Deacon says, though Emmie's pretty sure it isn't. She knows it means he doesn't want to talk about strange people anymore.
"You're gonna miss me," she says. Deacon shrugs his broad shoulders and takes another swallow of Diet Pepsi, then wipes his lips on the back of his left hand.
"Yes, you will, too," she tells his reflection. That other girl's lips move in unison with hers, the girl with the same yellow eyes and shoulder-length black hair and the same new pink-and-white, zebra-striped fur coat as Emmie. Maybe it's not a mirror at all, she thinks. Maybe it's a window. That's an old thought, one that she's never even told Deacon about, that mirrors might really be windows, that there might be some other world with some other her, and every now and then the two of them just happen to pass by the same windows and see each other and play this mocking game.
"You got the money I gave you?"
"Yeah," Emmie replies, suddenly bored with the mirror and looking down at her shoelaces instead. "I got it."
"And your cell phone's charged?"
"Mostly. It's charged enough."
"When you get back," her father says, setting the empty can down on the station floor between his feet, "we'll bake some of those snickerdoodle things you like. I'll take a day off and we'll see a movie or something like that. Maybe we'll drive out to the Beavertail Lighthouse and watch the waves. What do you think?"
"Sure. I guess so," Emmie sighs, not bothering with the least bit of enthusiasm because she's never thought watching the waves out at Beavertail Point is even half as much fun as her father seems to think it is, and she kicks at her backpack with the toes of her boots. "I think the train's late," she says.
"No, it's not," her father tells her, but he checks the clock on the wall anyway. "It's not quite time yet." Then he tugs at the zipper of Emmie's coat and smiles. "How's the new coat working out for you?" he asks. "Does Santa Claus have good taste or what?"
"I know all about Santa Claus," she says, and a small brown bird lands on the window ledge, flutters its wings, and then pecks at the sill.
"Do you now?" Deacon asks, and he cocks one eyebrow and rubs at his chin, rubbing at the rough and mostly gray stubble growing there because he hasn't shaved for a few days. She knows that he knows she doesn't believe in Santa Claus anymore, and that he isn't really surprised, but she doesn't say so. She smiles for him and kicks her backpack again. "Well, hell," he says. "I guess you also know all about getting too old for presents and toys and such, then, right?"
"No. I'm still a kid," she replies. "And kids get toys on Christmas, whether they believe in Santa Claus or not. It's a rule."
"A rule? Now, I don't know about that."
And then the train's pulling into the station, right on time, give or take a minute here or there, and the woman behind the ticket counter is talking loudly over the PA, reciting destinations and train numbers. Some of the people in the waiting room have started getting to their feet, shuffling their bags about, talking among themselves the way people talk whenever something's about to happen. And others, the ones who were already standing, are moving out onto the platform, out into the freezing Rhode Island morning. Emmie looks at the window again and realizes that all the noise has frightened the bird away.
"I don't have to go," Emmie says, turning away from the window and towards her father. "I could stay with you."
"What's wrong now?" Deacon asks her, the lines on his forehead wrinkling so she knows he's worried or confused or both. "You've got your ticket, Emmie. And Sadie's expecting you. All I've heard out of you for the last two weeks is how much you want to go to New York and see Sadie."
"But I could stay," she says, glancing back at the window, the place where the bird had been. "I could stay if you wanted me to."
"Are you sick?" he asks and presses a palm to her forehead.
"No. I'm not sick, Deacon. I'm fine."
"Then you're being silly," he says and picks her backpack up off the station floor, lifting it in a very decisive, end-of-discussion way. "I've got a lot of work to do this week. You'd have to spend half your winter break hanging around the shop with Jack."
"I like the shop," she says so softly she's almost whispering, wishing the bird had picked a different windowsill to land on.
"You'd be bored out of your skull, and you'd drive him crazy in the bargain."
"I said I like the shop, Deacon."
"You like New York, too. And that's where you're going. I know how much you've been looking forward to this. You and Sadie both."
"Yeah," Emmie sighs, admitting defeat, reluctantly admitting that Deacon's right, but also starting to relax a little. Seeing the bird there she felt confused, trying to clearly recollect something that she'd spent a lot of time trying hard to forget, and then, remembering, she felt angry and guilty and scared. She wanted to go back home with Deacon and lock herself in her room, wanted to forget about trains and New York City. But now the small brown bird is gone, and it seems to have taken all those bad feelings with it, and she can look forward to the trip again. Now she feels silly, just like her father said, and she smiles up at him and points at the door to the platform.
"I got a train to catch," she says, just like someone in a movie would say.
"You sure?" Deacon asks uncertainly. "If you really don't want to go—"
"I'm okay. I was worried about you, that's all."
"Yeah? Well, don't you do that. I'll be cool as a moose, kiddo. I always am."
"Okay," she says and smiles for him again. Emmie resists a last glance at the windowsill and lets Deacon lead her out into the cold, out onto the platform with all the other people.
She was almost seven when the thing with the bird happened, the thing that there was no way to take back and, she'd discovered, apparently no way for her to forget about, either, no matter how hard she tried. Deacon said that's how it always worked, that the harder you try to not remember something, the harder it is not to remember. "Just try not thinking of a white elephant sometime," he would say, like she was supposed to know what he was talking about. It hadn't been a white elephant. She'd never even seen a white elephant and thought he might have made them up. It had been a starling, a starling on a Saturday afternoon in October, the scraggly remains of the autumn leaves clinging stubbornly to the trees up and down Angell Street, all their bright colors turned dark and dull by the cold. Listening to the way they rustled against one another in the wind, Emmie imagined that's what beetles would sound like, if beetles ever learned to talk.
The big rhododendron bushes growing around the front of their house were still green, though the ferns were turning shades of yellow and brown and dying back for the winter. She was playing on the porch while her father watched television, and at first she mistook the sounds the bird was making for the dry, rustling noises of the dead leaves. She had brought all her dinosaurs, the ones that Sadie had sent her, down from upstairs, and she'd lined them up on the porch rail—the Styracosaurus and Triceratops, the Parasaurolophus and spiny Edmontonia, the Tyrannosaurus rex and the Apatosaurus rearing up, enormous on its hind legs. She wasn't sure exactly what the dinosaurs were about to do, because she always liked it to be a surprise, whatever happened next, but she knew that there would be trouble. There was almost always trouble when she let the Tyrannosaurus too near the plant-eating dinosaurs.
And then Emmie noticed the starling, lying in the grass not far from the bottommost step. It was watching her nervously with its tiny black-bead eyes, and its breast feathers shimmered in the sunlight like oil on the road after a thunderstorm. Right off, she could tell that there was something wrong with it, the way the bird was holding its left wing crookedly, the way it was just lying there, watching her.
"Fly away, bird," she said, but it didn't. She tried to ignore it and go back to playing with her dinosaurs, moving the Triceratops in between the Tyrannosaurus and the other herbivores. The meat-eater was huge and his mouth was filled with teeth like knives, but his belly was soft, and he was afraid of the Triceratops' long horns. The frustrated Tyrannosaurus hissed, and the Triceratops let out a throaty bellow to say it wasn't going away, and if the Tyrannosaurus wanted something to eat, it would have to look elsewhere.
Down in the yard, the starling ruffled its feathers and hopped clumsily once or twice, then lay still again. Emmie looked over her shoulder at the front door and the living room window. She couldn't see Deacon and thought that he might have gone to the bathroom or to get something from the kitchen. He'll be back in a second, she told herself, and tried to concentrate on keeping the Tyrannosaurus from sneaking past the Triceratops.
The starling ruffled its dark and speckled feathers again, flapped at the air with its right wing, and made a hoarse sort of whistling whooee sound.
"Shut up, bird," Emmie said. "Shut up and fly away." The Triceratops lunged at the Tyrannosaurus, but her fingers slipped, because the starling was distracting her, and she knocked the horned dinosaur off the railing and into the ferns and rhododendron leaves below. "Shit," she hissed, hissing like the hungry Tyrannosaurus, cursing the way Deacon does when he can't find the remote control or whenever a lightbulb blows.
Whooee, the starling whistled.
"That's your fault," she scolded the bird, then looked again to see if her father had come back. He hadn't, and now she'd have to go down the stone porch steps and dig about in the bushes for the escaped dinosaur. Now she'd have to walk past the starling.
Whooee, it whistled, and then it made an anxious clicking sound and fluttered its right wing.
"If you were a smart bird," Emmie said, starting down the steps, "you wouldn't fly into windows. If you were a smart bird you wouldn't make so much noise." As she got closer, the bird tried to hop away, but kept tumbling over in the grass and weeds. It was easy to catch the starling, and Emmie knelt in the shadow of the house and stroked its back with an index finger. The bird stared up at her with its black eyes and trembled in her hand. She could tell that its left wing was broken, and probably one of its legs, too, and there was blood and bird shit matted in its feathers. It was warm, and she could feel its heart beating, beating so fast it should burst.
I ought to tell Deacon about you, she thought. Deacon would know what to do with a hurt bird. But the words in her head felt more like something she'd thought a long, long time ago than something that she was thinking right then. She'd forgotten all about the lost Triceratops; it was getting hard to think about anything but the way the bird's heart was thumping against her palm, the fear in its glistening eyes, the way it trembled more when she touched it than when she didn't.
Gently she began to lift the wing that looked as if it might be broken, the left wing hanging limp and seemingly useless, but suddenly the starling stabbed at her hand with its sharp yellow beak. Emmie cried out, though she was more startled than hurt. Where the bird had pecked her, near the base of her thumb, there was blood, a single crimson drop welling out of her, growing larger and larger, and soon it would be running down her wrist and dripping onto the grass. She pictured the horns of the Triceratops piercing the leathery flesh of the Tyrannosaurus' belly, gouging wounds to drive it away and maybe even kill it, because, she thought, if you kill something, it can never try to hurt you again.
Deacon would know how to help you, bird.
The bird's heart was beating so fast now that Emmie couldn't tell where one heartbeat ended and another began. They had blurred together into one seamless sensation traveling out of the injured starling and into her. She felt a little dizzy, a little sick to her stomach, but she felt something else, too, something that felt good. Something that made her heart beat faster, and all at once the rustling of the dead leaves and the faded colors of the trees and the smell of the autumn day—everything—grew so perfectly clear, it was as though she'd never heard or seen or smelled anything in her life, nothing before that moment but the dim shadows of things and never the things themselves.
The front door creaked open, and she heard her father's footsteps on the porch, and Emmie almost turned to see, almost turned to hold the bird and her bleeding hand out for him to fix, but the starling pecked her again, harder than before, drawing still more blood.
And that's when she closed her fingers tight around its body and squeezed. Her heart was racing, beating almost as fast as the bird's had beaten, and the yard and Angell Street and the sky beyond the trees, all of it was so very loud, so vivid, so perfectly defined, that there was no room left inside her for the things that Deacon was saying. They were only another part of the whole, only splinters that would distract and ruin if she let them, splinters that would draw her back to that place where there was nothing but shadows waiting for her. Emmie squeezed even harder, breathless now at the ease with which the starling's bones snapped in her hand, at the contrast between cool, dry feathers and the warmth and wet leaking out of the bird.
Somewhere Deacon was telling her to drop it, drop it, drop it now over and over again. And then his hands were on her shoulders, pulling her up off her knees, shaking her so hard that her teeth clacked together and she bit her lip. The blood tasted almost as sweet and rich as chocolate syrup.
And then it was over—just like that—and Angell Street was only a street again, and there was nothing at all remarkable in the sound of passing cars or a leaf blower or the wind whispering through the maple tree outside her house.
"Drop it!" Deacon shouted again, and she did.
She wanted to tell him that she was sorry, because she was, because the starling was lying dead at her feet, as broken and empty as anything she'd ever seen, and she'd done that. She needed him to understand that she'd only wanted to help the bird, that its wing was broken, and it couldn't fly, and she'd wanted to find Deacon so that they could fix it. But her father was too busy cursing and dragging her up the stairs to listen, too busy to hear what she couldn't quite remember how to say. He dragged her past the dinosaurs, across the porch, and into the house. She almost tripped going over the threshold, but he caught her. He slammed the front door, and the lock clicked like breaking bones between his strong fingers.
He's going to kill me, she thought, certain that's just what she deserved for murdering a poor crippled starling, an eye for an eye, like she'd once heard someone say on TV, but then they were moving again, all the way down the hall to the bathroom. Deacon made her sit on the toilet seat while he ran hot water in the sink and found a bar of Ivory soap and a washcloth and a bottle of peroxide, and by the time he turned back to her she was crying.
He told her that he was sorry, held her and said that he hadn't meant to scare her, that he was frightened, too, and didn't understand. He was afraid, he said, and she needed to explain what had happened. But Emmie was sobbing too hard to talk, and after a little while Deacon stopped trying to make her. Instead, he used the soap and washcloth and hot water to clean all the blood and bird shit and feathers off her sticky hands, then cleaned out the places where the starling's beak had torn her skin, dabbed at the cuts with the peroxide, and that made her cry even harder. He put two Band-Aids on her hand, one across the other, hiding her wounds.
Later, after Emmie had stopped crying and he'd scrubbed her face and helped her change her clothes, after he'd brought her dinosaurs inside (except for the Triceratops, which had still been hiding in the bushes) and put the dead bird inside a green garbage bag, there was a trip to the emergency room and a tetanus shot.
"Birds and people can get a lot of the same diseases," he told her while they were waiting to see a doctor. "You should always leave them alone, especially the sick ones." And he did one of his magic tricks for her, the one where he pulls a nickel out of Emmie's ear and then puts it in his mouth and spits out a quarter.
That night she lay in bed, her arm aching from the tetanus shot, the Band-Aid cross gone and her hand wrapped up in gauze like an Egyptian mummy's, and she listened to Deacon talking on the phone downstairs. She knew that Sadie was on the other end, even though she couldn't make out many of the words. The words didn't matter. She could tell from the tone of his voice. Emmie lay there, listening, waiting for sleep and not wanting it to come, afraid the dead starling might return when her eyes were shut. It would get out of the green garbage bag and sit on her windowsill, pecking at the glass, watching her while she slept. It would call out, Whooee, whooee, so all the other starlings in Providence would come and do the same. If it could, it would change her dreams and make them into nightmares instead. It would haunt her, so she wouldn't ever be able to forget what she'd done. She listened to her father on the phone and waited for the birds. And the last thing Emmie heard before finally falling asleep was Deacon crying, and part of her wished that the starlings would come, that their hard yellow beaks would shatter the windowpane, and the birds would carry her aloft and away somewhere she'd never have to hear that sound again.
It's only been a short while since the train left Kingston when Emmie Silvey first notices the woman watching her, the woman seated on the other side of the aisle and one row forward. Emmie's been staring out the window, saving the book in her backpack for later on, when she gets bored with Connecticut. They've just passed the Mystic station, a handful of white sailboats bobbing listlessly in the little harbor, a few fishing boats moored at weathered piers, the wide gull-littered sky stretched out like a painting above the water and the shore. She's sitting on the landward side of the car and was glancing across the aisle for a better view when she caught the woman looking back at her. Not merely in her direction, and certainly not at anything behind her, but directly at her. When the woman sees that Emmie's caught on, she smiles and doesn't look away. The woman has curly brown hair tied back in a long ponytail, and she's wearing a black leather blazer with a gray turtleneck sweater underneath. Emmie's close enough to see that the woman's eyes are a pale hazel-brown; she doesn't return the smile, but goes back to looking out the window, wishing that there were someone sitting in the empty seat next to her. A big man with a red beard and wind-chapped cheeks, perhaps, or an old woman who might once have been a schoolteacher or a librarian, someone like that in the vulnerable space between Emmie and the brown-eyed woman.
She's not really looking at me, Emmie tells herself, though she knows it isn't the truth. She was just looking around the train, that's all. And then Emmie tries to make a counting game from the pattern printed on the back of the seat in front of her, the upholstery pattern that reminds her somewhat of a piece of hard candy or Sadie's rugs from Argentina. Broad bands of maroon on either side, then a riot of vertical stripes straight down the center, narrow and even narrower bands of purple and greenish blue, gray-blue, and rusty orange. She counts nineteen or twenty bands, though it's hard to be certain exactly how many there are, since some of the colors grade together, bleeding imperceptibly from one to the next. The blues and purples make up the outermost bands, with the orangey browns located at the center. But the whole time Emmie's really thinking about the brown-eyed woman, not the upholstery pattern, and, after only five minutes or so, she steals another quick glance in her direction. The woman is still watching and smiles a second time.
Emmie turns quickly away, not exactly frightened, not yet, because there are always the stewards and lots of other people crowded onto the train, but the woman's definitely making her nervous now. Stop, she thinks. Don't look at me again, pretending her mind can reach out all the way to the woman's seat, can reach inside her head and make her want to do anything but stare at people she doesn't know and has no business spying on. Then the train slows down and shudders once or twice, rolling up to the station platform at New London; some people get off and others get on, and Emmie sits with her fingers crossed, hoping that the brown-eyed woman is going only this far, or, at the very least, that someone boarding will take the empty seat next to her. She watches the busy platform and waits, but no one sits down beside her, and when the train is moving again, she risks another peek and sees that the woman is still sitting right there, so maybe she intends to ride all the way down to Manhattan. Emmie hauls her backpack up into the empty seat and tries to concentrate on the world rushing by outside the windows—beds of granite and slate carved to make way for the railroad; the twisted, arthritic fingers of bare limbs against the sky; brick buildings boarded up and scarred with graffiti; abandoned warehouses and gas stations; a tiny, neglected graveyard sheltered by enormous evergreens. At the crossings, automobiles wait impatiently for the train to pass so that they can go on about their business. Everything is just exactly as it ought to be, all the familiar sights and sounds of the trip, but she remembers her cell phone in the front pocket of the pack and thinks about taking it out and calling Deacon to tell him about the woman. And then she thinks about calling just to hear the sound of his voice.
"Would you mind if I sat here?" someone asks her, and "No, I don't mind," Emmie says eagerly before she even looks to see who it is, too relieved that the seat won't be empty anymore to care. But then she turns her head, turns to move her backpack from the seat, and she sees that it's the hazel-eyed woman. She's much taller than Emmie expected her to be, and there's a tattoo on the back of her left hand—two intersecting triangles, one red, one black, forming a six-pointed star, and there's something that looks like a T at the very center. For a moment Emmie doesn't move, trying desperately to figure out a way to tell the woman that she's changed her mind, that she wants to be alone, or the seat's taken, and her father will be back from the restroom in just a minute or two. But the woman's smiling, and her eyes are so bright that they almost make Emmie squint, and she moves the pack from the seat to her lap.
"Are you going to New York?" the woman asks, sliding easily into the seat. Her voice sounds a little hoarse, like maybe she's been crying or sick or she's only half-awake. "That's where I'm headed, and I've always hated not having anyone to talk to on the train." The woman holds out her right hand, the one without the tattoo, for Emmie to shake. "My name's Saben," she says. "Saben White. And you're...?"
"You have a tattoo," Emmie says, not shaking the woman's hand.
"Oh, yeah. That," Saben White replies and looks down at the design worked into her skin. "I've had that for a long time. I forget about it."
"What's it supposed to be?" Emmie asks, hugging her backpack closer and wondering where Deacon is now, and if he's thinking about her, if he's worrying about her.
"Well," the woman says, frowning thoughtfully, examining her tattoo like maybe she's never looked at it closely before. "That's what's called the Seal of Solomon. It's called other things, too, but—"
"Who's Solomon?" Emmie asks, interrupting and not caring if it's rude, because it's also rude to stare at people you don't know and then sit down where you're not wanted.
"He was a king, the Israelite king who built a great temple in Jerusalem where the Ark of the Covenant would be safe. To the Hebrews, he's known as chacham mi'kol ha'adam—the wisest of all men."
"Are you Jewish?" Emmie asks.
"Not at all," Saben White replies.
"Then why do you have that tattoo on your hand?"
"Like I said, it's called other things, too. It means other things to other people besides the Jews. And it keeps me safe."
"From what?" Emmie asks, squinting skeptically at the woman.
"Oh, lots of stuff. All the things someone like me needs protecting from. You're a very curious little girl; you know that?"
"My father said I shouldn't talk to you."
"Did he? That's odd. I don't think your father even knows me. I mean, I don't know him."
"That's the point," Emmie replies, beginning to get annoyed at the way the woman seems to be talking in circles, the way she's only acting like she doesn't know what Emmie's trying to say. "I don't know you. He doesn't know you, either."
"Oh, yeah," the woman says and taps at her forehead with the ring finger of her right hand. "It's that old 'don't talk to strangers' routine. Okay—"
"So I think you should go back to your seat," Emmie says.
"Is that what I should do?"
Emmie sighs and looks out the window again. The train is crossing a bridge, but she isn't sure which one it is or which river it's ferrying them over. The woman distracted her, and now she's lost track of bridges and rivers and just about everything else. This one's wide enough to be the Connecticut River, but that would mean that they're almost to Old Saybrook, and she's not sure they could have gone that far already. It would also mean she's somehow missed crossing the Niantic, and she never misses the rivers, one of the ways she's learned to measure the train's progress so she knows how much farther until New York and Sadie, how much distance has accumulated between her and Deacon. She does the math in her head—it's fifteen minutes from Kingston to Westerly, ten minutes from Westerly to Mystic, then fourteen minutes from Mystic to—
"You're right," the woman says, and Emmie gets confused and stops counting stations. "I don't know your dad, not personally, and he doesn't know me, but I do know who he is. He owns the old bookshop on Thayer Street, right? I've been in there a couple of times."
"Yeah, but lots of people know he owns the shop. That doesn't mean you're not a stranger. Kids who talk to strangers go missing. It happens all the time, every day."
"His name is Deacon Silvey," the woman continues, "and right now you're on your way to New York to visit your stepmom, Sadie Jasper. I've read all her books, by the way. So, see, you might not know me, but I'm really not such a stranger, either."
Emmie stares silently at the woman for a moment, wishing she could be sure which river is underneath her, feeling lost and sorry that she didn't lie and tell Deacon that she was feeling sick. He'd have taken her back home if she had, and she'd be safe in her own bed by now, and none of this would be happening.
"You know my name, don't you?" she asks the woman, who nods her head yes and picks a bit of fuzz off her gray sweater.
"I do," she admits. "And there's a good reason for that, but I probably shouldn't tell you what it is. Not just yet, anyway. I shouldn't tell you too much all at once. I don't want to frighten you."
"Too late," Emmie says, and now she's sure that it's the Connecticut River out there, and they'll be stopping at the Old Saybrook station in just a few more minutes. "You already have."
"I knew that I might, and I'm sorry about that. I promise, I didn't want to upset you or your father."
"What do you want?" Emmie asks, keeping her eyes on the window now, the bridge ending as the river gives way to land again.
"Just to talk to you, that's all. I swear. I'm not going to hurt you. I won't even touch you. You mean a lot to me, Emmie."
"Why is that, Saben? You don't even know me, so how could I mean a lot to you? That doesn't make any sense."
"Jesus," the woman says, "but you're a precocious one, aren't you?"
"I know what that means," Emmie says, because she does, and knows that sometimes adults pick words that they think kids won't understand.
"Of course you do. You're a smart little girl, Emmie. I know all about that, too, about the school and all." And now Saben White's voice is starting to sound hoarser than when she first sat down, and Emmie almost asks her what's wrong, if she's sick, then remembers that she has no business talking to the woman in the first place.
"I don't talk to strangers," she insists.
"Kids go missing every day," the woman says and nods her head again. "You shouldn't talk to strangers. Especially not to strangers on trains. They're probably the worst kind of all."
"Deacon never told me that," Emmie says, glancing back to the hazel-eyed woman.
"I expect there's all sorts of things he hasn't told you. And things he never will."
"What's that supposed to mean?" Emmie asks.
"It means that I need to remember why I have this thing," she says and points at the Seal of Solomon tattooed on her hand. "Do you know what the sacred number is, Emmie? Has anyone ever told you?" Emmie shakes her head; she's noticed that a man sitting across the aisle from them, an old Asian man wearing bifocals and a yellow scarf, is listening to what Saben White is saying.
I could ask him to help, she thinks. I could tell him to make her leave me alone.
"See these two triangles?" the woman asks her. "The one pointing up, the black one, stands for fire and masculine energy. Like your father. And the red one, that's for water and for feminine energy, like your mother. Or Sadie. Water and fire, so that's two of the elements, and if you look at the way the base of each triangle bisects the other just beneath its apex—that's what you call the top of a triangle—"
"I know that," Emmie says, still watching the old man, wishing that he'd say something, anything at all. It's rude to stare, and he should tell her to stop doing it.
"Okay," Saben White says hoarsely, "when you look at how the base of each triangle bisects the other just beneath its apex, you find the symbols for earth and air, which gives you all four of the elements—fire, water, earth, and air. And a triangle has three sides—"
"That's why it's called a 'triangle,'" Emmie replies, and the old man in the yellow scarf blinks at her.
"Right, Emmie. Now, what do you get when you add three to four?"
"You get seven."
"And seven is the sacred number," the woman says, and her eyes are shining like she's about to start crying. "So, this star here really has seven points."
Emmie looks from the useless old man back to the tattoo, counting off the points one by one. "No," she tells Saben White. "There are only six."
"Only six that you can see," the woman explains, and now the train has reached Old Saybrook, and people are beginning to move about. The man with the yellow scarf has started reading a newspaper, and Emmie gives up on him. "But there are still seven points there. Three and four, but the seventh, it's invisible."
"I don't think I believe in invisible things," Emmie says. "I don't think invisible things are real."
"You'll learn otherwise. Sooner or later you'll learn that invisible things can be a lot more real than the things you see. And sometimes that's why they're invisible."
"That's stupid," Emmie mutters, and turns to the window again, gazing out at Old Saybrook Station. "Now you're not even making any sense."
"I think I'd better get off here," the woman says. "I think it'd be best if I got off here."
"Yeah," Emmie agrees, relieved because the woman wasn't a kidnapper or a sex pervert after all, just some crazy person who can't count and doesn't know how to mind her own business. "It's probably best. I have a book in here I want to read." And she pats her backpack.
"You be careful, Emmie," the woman says. "And listen, I know this isn't going to make much sense, either, but you need to stay away from horses."
"What?"
"Just while you're in New York City this time, stay away from horses, okay? I think it's very important."
I think you're a nut job, Emmie almost says, never mind if it's mean to talk to crazy people that way, and she opens her mouth to say something else, but Saben White is already up and moving, pushing her way between the other passengers. In only a moment, Emmie's completely lost sight of her, so she turns back to the window and waits for the woman to appear on the platform, but she never does. A few minutes more and the train pulls out, rolling on towards New Haven and Bridgeport and Penn Station. Emmie forgets all about the book in her backpack, and the cell phone, too, and just watches the world through the train window as the morning turns quickly to afternoon, trying to remember everything the woman said about the Seal of Solomon, about four and three making the sacred number, in case she decides to tell Sadie or Deacon about it later. She might, and she might not. Sometimes she has no idea what the right thing to do is, and maybe it would only make them worry.
There are no clouds in the sky, but Emmie wishes it would snow, just the same.
Three years ago, not long before Sadie and Deacon finally stopped fighting, and Sadie left Providence and didn't come back, Emmie woke late one night to find her stepmother sitting alone at the foot of her bed. Emmie lay very still and quiet, unsure whether Sadie had realized she was awake, afraid that this was something important, something terrible, and as soon as Sadie knew she wasn't asleep any longer she'd begin talking, which would be the start of the terrible, important thing. Sadie was sitting with her back to the night-light, so Emmie couldn't see her face, even though she could see better in the dark than most people. Sadie's face was only a shadow, framed by her thick black hair, and Emmie tried hard to pretend that she'd come in to make sure everything was all right or because she'd forgotten to kiss Emmie good night (though she hadn't). Maybe she'd heard a noise outside, a cat rummaging about in the garbage cans or the wind brushing around the eaves of the old house, and thought that Emmie must have heard it too and might be frightened. Maybe that was all it was. She waited impatiently to find out, listening to one of her favorite Patty Duke songs set on repeat in her CD player—"The World Is Watching Us"—one of the songs she most liked to sleep to, turned down low enough so it wouldn't annoy Deacon, who hated Patty Duke just as much as Doris Day.
"Hey there, pumpkin," Sadie said very softly, and Emmie thought that she could almost see her stepmother's words, like wisps of red velvet in the darkness. "I'm sorry I woke you up."
"That's okay," Emmie replied. "It wasn't a very interesting dream."
"I hate those."
"This one wasn't even interesting enough to hate."
"Well, those are even worse," Sadie said, and then she didn't say anything else for a while, and Emmie lay there and waited, because she didn't know what else to do. Finally Sadie turned towards her, and Emmie could see her face illuminated in the glow of the night-light. Sadie's eyes sparkled like wet blue jewels. Emmie often wanted eyes the same color as Sadie's eyes; no one thought girls with yellow eyes were pretty.
"I need to ask you something, Emmie. I know that I can, because you're a very smart kid. Hell, I've never met a kid as smart as you."
And so it was something important and terrible. Emmie shut her eyes again, shut them tight, pretending this was just another part of the very uninteresting dream that Sadie had interrupted. Sometimes she'd rather she weren't so smart, or that Deacon and Sadie had never learned she was anything but a normal kid who couldn't be told the things that only grown-ups were supposed to know. Just another normal, unremarkable girl with Sadie's pale blue eyes.
"I might be leaving soon," Sadie said. "I might be better off living in New York. It might be better for my writing."
Emmie opened her eyes again. The Patty Duke song ended and immediately started over.
"What about Deacon?" she asked.
"I think Deacon might be better off with me living in New York, too. That's one of the reasons I'm considering it. I didn't want you to find out later and think I was keeping it a secret."
"Does Deacon think he'd be better off with you living in New York?" Emmie asked, though she was pretty sure that Sadie hadn't yet said anything to Deacon about leaving, which meant she was keeping it a secret.
"We still need to talk about it. I wanted to talk to you first."
"Don't you think that's kind of backwards?"
And then Sadie just stared at her for a moment, the same way her second-grade teacher had stared at her when Emmie had memorized the entire multiplication table in one afternoon, the same way that her father had stared at her when he discovered that she knew how to play the piano even though she'd never had lessons. Emmie didn't like that stare. It had always made her feel like something in a jar.
"I don't know," Sadie said at last and looked away again, the shadows hiding her face once more.
"He loves you," Emmie said. "He'd do anything for you. He told me that he would."
"He won't stop drinking for me. He won't do it for you, either."
And Emmie didn't argue, because she knew that was the truth, and there was hardly ever any point in arguing with the truth.
"You're going to divorce him?"
"That's not what I said," Sadie replied. "I just don't think it's good that we live together anymore."
Maybe this is the dream, Emmie thought. Maybe this is the dream, and it's being awake that isn't very interesting. She wished that could be how it was, that it could all be as simple as her getting turned around inside her head, and in the morning she'd wake up and tell Sadie, and Sadie would tell her what a very silly dream it had been.
"I won't leave him," Emmie said. "If that's what you came to ask me, I won't."
Sadie took a deep, hitching breath and wiped at her face with the front of her T-shirt. "Yeah," she said. "That's what I thought you'd say."
"Then why'd you have to wake me up, if you already knew?" Emmie asked, feeling suddenly more angry than sad or confused, and not caring if Sadie knew.
"I needed to be sure; that's why."
"You can't make me leave him either, Sadie. If you did, he would..." And she paused, not sure if she wanted to say what she was thinking, afraid that saying it out loud might be enough to make it so. But holding the words back was like being sick and trying not to vomit, and they came anyway. "He would die, if we both left him. He wouldn't have anything if we were both gone."
Sadie was crying now, and she shook her head. "I would never do that, Emmie. I'd never force you to come with me."
"But you want me to. You think that I should."
"I love you, sweetie, and I want what's best for you, that's all. But I'd never make you come, not if you want to stay."
"He's my father," Emmie said and sat up, pushing back the covers. "He's my father, and this is my house."
"It's my house, too," Sadie said, almost as though it was something she'd forgotten and was trying to remember, and she wiped her nose again. "Jesus, I'm sorry, Emmie. I promised myself that I wasn't going to cry."
"I think that was probably a dumb promise," Emmie said. "I don't see the point in making yourself dumb promises you know you can't keep."
"I can't stand the thought of being so far away from you," Sadie said, almost whispering now.
"Then don't leave us," Emmie replied, hearing the bite in her voice, the recrimination, and feeling only the smallest bit guilty about it. "Nobody's making you go. No one said you should."
Sadie took another deep breath and held it for what seemed like a very long time before Emmie finally heard her exhale. "This is hard, and you're not making it any easier."
"I'm not supposed to, Sadie. I'm just a kid, and you're my mother. I don't care what's best for your writing, not if it means that you're leaving me and Deacon alone."
Sadie turned back towards Emmie, and now her blue eyes were even brighter than before, bright with tears and anger and frustration and things that Emmie hadn't yet learned words for. Emmie flinched, as though her stepmother meant to strike her, as though something had been thrown. She almost called out for her father, but then the Patty Duke song ended again, and in the moment of silence before it started over, her panic dissipated, fading away like the red, used-up air that had spilled out of Sadie's nostrils.
"My writing pays the bills, little girl," she said. "My writing keeps your father in beer and whiskey and..." And then she held her good right hand up almost like a crossing guard, a gesture that Emmie understood at once—stop, don't go any farther, don't say another word, shut up—though she wasn't sure whether it was intended for her or for Sadie.
"Christ," her stepmother muttered, biting at her lower lip and shaking her head, and then she looked away from Emmie, staring down at the rumpled comforter instead. "I don't believe I just said that. I'm sorry, Emmie. That sounded like something my mother would have said."
"Maybe that's important," Emmie told her and scooted a few inches closer to her stepmother, close enough to be hugged if that's what Sadie needed to do next.
"Maybe so," Sadie agreed and then didn't say anything else for a couple of minutes, long enough that "The World Is Watching Us" ended again and started again. "But I'm still sorry. You should be asleep."
"I have school tomorrow."
"Yeah, you do. I shouldn't have awakened you. I shouldn't have brought this up in the middle of the night."
"I won't say anything to Deacon," Emmie said, hoping that Sadie wouldn't either. Hoping that by morning Sadie would have decided moving to New York was a ridiculous idea, that leaving them wouldn't help her writing at all.
"Thanks, pumpkin."
"Just don't give up on us yet," Emmie said, and Sadie did hug her then, held her tight and promised that she'd never give up on Emmie, no matter what, which wasn't particularly reassuring, since it still left the possibility that she might give up on Deacon and leave them anyway.
And later, after Sadie had left the room and there were only the outside noises and Patty Duke and all the strange creaks and thumps that an old house makes at night when it thinks no one is listening, Emmie told herself a story to make her sleepy again. It was a new story, mostly, not one she'd made up earlier or heard or read in a book. Sadie was in the story, but she was much younger, hardly more than a girl, and she hunted sea monsters with a pirate ship stolen from the Barbary Coast. Her bad hand had been replaced with a shiny mechanical contraption of cogs and gears with a timepiece set conveniently in the palm, and she searched the far horizon with a long brass telescope that had been given to her by the court astronomer of one sultan or another. The stolen pirate ship was called the Harbinger, because it was a word that Emmie loved and used whenever she felt using it was appropriate.
"Then turn not pale, beloved snail," the Sadie in the story howled boldly up at the fearsome Atlantic sky, "but come and join the dance!" (And Emmie had reluctantly stolen that part from Alice's Adventures in Wonderland, because she was too tired to think of anything better.)
She kept meaning to put Deacon in the story somewhere, because, as Sadie had taught her, telling stories was a sort of magick, a powerful sort of magick, and maybe doing so would help keep Sadie in Providence with them. But then the Harbinger suddenly ran aground on an island that was really the shell of an enormous man-eating turtle named Archelon, and by the time the giant turtle had been vanquished and the ship was on its way again, racing across the frigid Labrador Sea in search of seals and yetis and polar bears, she was falling asleep, sliding easily from the story into her dreams.
There were no pirate ships in her dream, or sea monsters, and Sadie didn't have a robotic hand, but her father was there, and her mother, too—the tall, plain woman who'd given birth to her, whom Emmie had only ever seen in photographs, because Chance Silvey had died the same day Emmie was born. She met her mother in dreams, sometimes, and sometimes they talked, and this time she told Emmie not to worry, that Deacon was a whole lot stronger than she thought. "He's slain dragons," her mother said, and smiled, and Emmie smiled, too, knowing it must be true, and the dream rose and fell like a French brigantine lost on the vast and briny deep, taking her places, and then on to other places, towing her soul slowly towards the dawn.
Sadie and Emmie take a cab from Pennsylvania Station, because Sadie said that she wasn't in the mood for the subway, and that was fine with Emmie. It was always much better to be aboveground, with the noise and movement of the city pressing in all around her, than stuffed into some dingy subway car hurtling headlong through the oily dark with no idea what might be going on overhead. The driver is a black woman with a thick Caribbean accent, and she asks Emmie how old she is before starting the meter. Then Sadie's cell phone rings, and she promises Emmie that it will take her only a second, just a minute, that it's someone from her agent's office, so she has to take the call. But Emmie knows that it always takes more than a second or a minute, and that's fine, too. There are far too many wonderful things to see, and if she talks to Sadie she might miss something. She'll have plenty of time to talk to Sadie later. So, the driver drives, and Sadie talks to the phone, and Emmie sits with her nose and lips against the cold taxi window and watches the city as the woman with the Caribbean accent navigates the crowded, steaming roads.
"Well, he would say that, wouldn't he," Sadie laughs, but not the sort of laugh that means she thinks something's funny, and then she digs about in her purse until she finds a yellow pack of Juicy Fruit gum. She gives a stick to Emmie and takes one for herself. "No, I've got my step-daughter down for a few days. He'll just have to wait until next week. They already said that I had until March, didn't they?"
The cab turns south onto Broadway, and Sadie laughs again and says good-bye to the person on the other end of the phone call. "I'm turning this damned thing off," she says, pressing a button on the cell phone and dropping it into her purse, then zipping the purse shut. "Now it's just me and you, pumpkin."
"We should get some pizza," Emmie says hopefully, reluctantly turning away from the smudgy cab window, from all those people and the other automobiles, the trash blowing along the sidewalks and the stunted little trees huddled in the lee of the tall buildings, the storefronts and restaurants. They've just passed a big yellow sign promising Ray's Famous Pizza, and Emmie wonders if the driver would go back if Sadie paid her extra.
"Pizza?" Sadie asks and wrinkles her nose. "You sure you want pizza? I thought we could get some sushi after we dropped your stuff at the apartment."
"I'd rather have pizza," Emmie replies. "Sausage pizza with lots of green olives. We can have sushi later, I promise. We'll have sushi tomorrow."
"I know you never get it, that's all."
Emmie smiles and starts folding the strip of silver foil from her piece of Juicy Fruit, working it nimbly between the tips of her fingers. "I don't eat raw fish," she says, lowering her voice, imitating her father. "You can get goddamned tapeworms eating raw fish. You want a tapeworm, Emma Jean Silvey? You want a tapeworm long as your arm?"
And then they're both laughing so hard that Emmie drops her bit of foil and has to scrounge around on the floorboard behind the driver to find it again.
"You're going to get filthy down there," Sadie tells her. "You'll probably get a tapeworm long as your arm."
"Nope. I found it," Emmie says and holds up the foil so Sadie can see. Then the taxi hits a bump, and she almost drops it again. "Tapeworms hate pizza, by the way."
"Fine, you want pizza, then we'll get you some pizza." And Sadie sees that Emmie's almost finished folding the strip of foil into a tiny bird. A couple of months ago she sent Emmie a book on origami. Last time she spoke with Deacon, he complained that Emmie had folded a twenty-dollar bill into a kangaroo.
"Can I have that one?" Sadie asks, and Emmie nods her head.
"Sure," she says and gives the foil bird to Sadie.
"You're getting good at this."
"Yeah. I found more books in the library," Emmie tells her. "It's easy, once you get the hang of it. I'm teaching some kids at school."
"You are? How's school been working out these days?"
"Better," Emmie says, and then she catches the driver watching her in the rearview mirror.
"Just better?"
"Boring," Emmie adds, keeping her eyes on the driver's reflection. "Math is boring, history is even worse, and I hate playing goddamn volleyball."
"Everyone hates volleyball," Sadie says, still inspecting the precise angles of the origami bird. "That's why they make you do it. What about English? What are you reading?"
Emmie wishes that the driver would stop staring at her, wishes that Sadie would notice and make the woman stop. It's just my eyes, Emmie thinks. She's noticed my eyes, and she's never seen a girl with yellow eyes before. And that makes her think about the woman on the train again, Saben White and the star tattooed on the back of her hand. Seven points, even though Emmie could only see six. She starts to tell Sadie that the driver's staring at her, but then the woman with the Caribbean accent looks away, her eyes darting back to the road in front of them, back to the Manhattan traffic.
"Is something wrong?" Sadie asks, and Emmie almost tells her about the driver, and about Saben White as well. She knows that's exactly what her stepmother would want her to do, what Deacon would want her to do. Deacon almost got in a fight once with a guy on Thayer Street who was staring at Emmie's eyes, not just staring but pointing and making jokes about her. She remembers her father grabbing the man by the collar of his shirt and telling him that if he didn't apologize he'd be picking his teeth up off the ground. The man apologized, though Emmie had secretly wished that he wouldn't, had wanted to see Deacon hurt him for embarrassing her. Everyone on the street had stopped whatever they'd been doing and looked at them, looking at her, all those curious, peering green and brown and blue eyes brushing against her like dirty hands.
If I tell her, things will only be worse, she thinks, and so Emmie keeps it to herself for now and tells Sadie about her English class, instead, about Gulliver and To Kill a Mockingbird and a long poem by Robert Frost. Sadie gets her to recite part of the poem, the part that she remembers, and in a few more minutes the taxi is turning off Broadway, and the ride's almost over. When they reach St. Mark's Place, Sadie leans forward and tells the woman that this will do, this is fine. Emmie gets out and stands on the curb while Sadie pays the driver, stands staring up at the wintry slice of afternoon sky above the rooftops, and Emmie imagines that her eyes are that same color, like broken shards of china. There's an airplane up there, a black dot trailing white smoke, and as long as she's pretending, she makes it a spaceship on its way back to a planet where everyone has eyes like hers.
## TWO
## Soldier
"I'm asking you nicely—you need to stop staring at that damned watch of yours," Odd Willie Lothrop grumbles, plucking an unlit Winston from his thin lips. "You're making me jumpy."
"She's late," Soldier says for the seventh or eighth time since Saben failed to show for their two-o'clock. That was almost thirty-five minutes ago now, and the Bailiff's already called twice to see what the hell's going on, why the three of them aren't on their way from Cranston to Woonsocket. She's made Odd Willie take both the calls because she doesn't like talking to the Bailiff when he's pissed off—nobody does—but Soldier knows the next one will be for her, for her personally and no one else, and she'll have to come up with some sort of halfway credible excuse to cover Saben White's ass—again. This will make twice in one month, and Soldier doesn't know why the bitch hasn't already been handed her walking papers, why someone hasn't put a couple in her skull and left her floating facedown in a marsh somewhere.
"Jesus God, I fucking hate Woonsocket," Soldier says and takes another sip from her third cup of coffee, which has started getting cold and isn't doing anything at all for her hangover. "I don't know why this one couldn't have gone to Kennedy. He's sure as hell got a lot more experience up there than I do." She stares out the tinted, fly-specked plate-glass window of the Dunkin' Donuts at the squalor scattered up and down the gray asphalt strip of Warwick Avenue. Soldier only ever comes this far south when the Bailiff says that she has to, when he consults his tea leaves or his crystal ball, or whatever it is he uses to make up his mind, and picks a meeting place down here. Then she has no choice but to leave Providence and make the crossing into Kent County over the poisonous, filthy waters of the narrow Pawtuxet River.
"Kennedy's a nebbish," Willie says.
"And I'm up shit creek," Soldier replies and checks her watch again; Odd Willie shakes his head and puts the Winston back between his lips.
"Maybe she got stuck in traffic," he mumbles around the filter, and Soldier wants to hit him in the face. She takes another sip of coffee instead, something almost as bitter as her mood, and thinks about stepping outside for a smoke.
"You always been this high-strung?" Odd Willie asks. "I mean, were you this way before the wingding with Sheldon and Joey Bittern?"
Soldier sets down her cup and stares at Odd Willie across the tabletop; he knows better than to start talking shit about Sheldon Vale, especially when she's sober and her butt's on the line. His shiny coal-black hair is slicked back into a crooked sort of duck's ass, his dewy sage-colored eyes peering out at her from beneath the place his eyebrows would be if he didn't shave them off once a week. He has a silver tooth right up front, and his cheekbones are so high and hollow, his face so pinched and drawn in on itself, Willie could easily score some extra cash renting himself out on Halloween. But pretty hasn't ever been very high on the Bailiff's list of desirable qualities in his underlings.
"Is it true," Soldier asks, "that people started calling you Odd Willie 'cause someone caught you fucking a dead cat?"
He laughs and makes a gun with his thumb and forefinger, then jabs it against his right temple. "Hell, I guess you done told me," he mutters around his cigarette. "Bang, bang."
"Hey, I'm just saying, you hear shit, that's all."
"And you heard that people started calling me Odd Willie 'cause I fucked a dead cat?"
"Something like that," she replies, looking out the window again. "At least, I think it was a cat. Might have been a dead raccoon."
"So, you're sitting there telling me you ain't never fucked nothing dead?" And he takes out the pink plastic comb he carries with him everywhere and runs it through his hair a few times.
Soldier nods her head. "I suppose," she says, "this is when you lose all respect for me, right? Alas, I have been forever diminished in your sight."
"Hey, baby, it ain't nothing can't be fixed."
"Don't do me any favors," she says, sick of Odd Willie and sick of the bright daylight making her headache worse, just fucking sick, and she reconsiders punching him in the face.
Odd Willie Lothrop shrugs, chews on the end of his cigarette, and puts the pink comb away. "Fine," he says. "Whatever. I already knew you were a tight ass. So, what you got against Woonsocket, anyway?"
"You're fucking kidding me, right?" But Odd Willie just shakes his head and grins a big, stupid grin for her, flashing his silver incisor. Soldier rolls her eyes. "Well, it's nothing that couldn't be fixed with napalm and bulldozers," she tells him, "maybe a few well-placed exorcisms here and there."
"I see," Odd Willie says and stops grinning. "Well, this'll be my first time up that way, so what do I know?"
And that's news to Soldier, the sort of news she wishes someone would let her in on before it's too late to make other, less suicidal arrangements. But this is how it's been since that night at Quaker Jameson's roadhouse. No one's ever come right out and said that any of it was her fault, almost walking into a trap that way, not being absolutely shit-sure what was what before she set out for Ipswich. And no one's ever blamed her for killing Sheldon, either, but there have been questions, questions she hasn't been very good at answering. So she keeps drawing these crap runs with the likes of Odd Willie Lothrop, and she gets stuck making excuses for Saben White because there's no way she's gonna blow it again, and whenever the Bailiff so much as dreams she's been at the bottle, it all gets a little bit worse.
"So, when do you tell me what this gig's about, anyway?" Willie asks. "I've never been much for drawing it out, savoring the razor's edge of suspense and all that happy crap."
"When we're on the road," Soldier says and checks her watch. "When we're moving, and when we're not sitting in fucking Dunkin' Donuts."
"You're kind of paranoid, too," he says and starts picking at the eviscerated remains of a Boston creme–filled doughnut lying on the napkin in front of him. The sight of it makes Soldier's stomach lurch, so she goes back to watching squalid Warwick Avenue for some sign of Saben, waiting for the phone to ring again.
"If you say so," she tells him.
"No, man, I'm serious," Odd Willie replies, giving up on the doughnut again. "People overhear some weird shit and ninety-nine times out of a hundred they're not gonna believe a single word of it. See, it just sails right on over their heads." He demonstrates by slicing the air above his greasy black hair with his left hand. "Someone hears us yakking about work, the kind of work we do, and you might as well be telling them the pope's a goddamn yeti. All this paranoid cloak-and-dagger, secret-handshake shit, you ask me, it's a total waste of resources."
"Like I said, if you say so. But I still don't talk shop in fucking Dunkin' Donuts."
"Maybe you'd be a little less ap-pre-hen-sive over at a Tim Hortons or a goddamn Krispy Kreme," Odd Willie snorts, wringing apprehensive until every syllable is bled bone-dry.
"Are you stoned?"
"Now, now," he says and grins, flashing that silver eyetooth for her, and he taps the windowpane hard with his knuckles. "Glass houses, Soldier. You always gotta remember about those glass houses."
"I'm sober," she tells him and reaches for the lukewarm coffee. "I haven't had a drink—"
"—since you passed out last night," Odd Willie says. "You're so hungover your hair hurts."
"I'm just fine," she lies, and Odd Willie nods his head and holds up what's left of the doughnut, just a few inches from her nose.
"Is that a fact? Wanna bite?"
And then her cell phone starts ringing again, and he drops the doughnut and licks his fingers, sits back in the booth, and points at the phone.
"Your turn," he says.
"One day we're gonna dance," she whispers, picking up the phone, speaking softly enough that there's no chance anyone but Odd Willie will hear her. "Just me and you and something sharp."
"So, you can talk dirty," he laughs, smug and dumb and entirely pleased with himself, laughs loudly and arches the prickly flesh where his eyebrows should be. "It's a date, girlo. Just you and me and a couple of pigstickers, but right now, you better answer that thing." And he points at the phone.
But when she looks, the number coming up on the display isn't one of the Bailiff's after all; it's Saben White's, and Soldier takes a deep breath, feeling something faintly like relief, and tosses the phone to Odd Willie. "You tell her she's got five minutes," and Soldier holds up all the fingers on her right hand for emphasis. "Just five fucking minutes. Then she's gonna have lots worse things to worry about than explaining this shit to the Bailiff."
Willie Lothrop nods, the humor in his face not quite draining completely away, and answers the phone.
"Yo, little miss slowpoke. You know you're screwed, right?" he chuckles, then waits while Saben White says whatever it is she has to say for herself.
"Five minutes," Soldier tells him again. "Not a goddamn second more." There's a fresh bottle of Dickel waiting for her out in the parking lot, wrapped in brown paper and tucked under the front seat of the old black Dodge Intrepid she's been driving the last three weeks. She can taste the whiskey, sweet and strong, burning her throat and her belly, easing the pain behind her eyes and the storm in her gut.
"No motherfucking way," Willie says and smacks himself in the forehead. An old woman sitting at another booth turns and stares at them. She has faintly lavender hair and a very large mole on her chin; Soldier stares back at her until she mumbles something under her breath and looks away. "That's fucked-up," Willie giggles, the way he does when he's nervous or scared.
"Just hand me the phone," Soldier says impatiently and holds out her hand.
"Rocky Point," Odd Willie says to Saben White, ignoring Soldier. "Right, yeah. Don't you worry; we'll find you. You just keep your ass put," and then he hangs up.
"What the hell was that?" Soldier demands, and Odd Willie smacks himself in the head again, even harder than before.
"We got trouble," he says, and before she can ask him what sort of trouble, the old woman with the lavender hair turns back around and glares at them. "What the fuck is your problem, lady?" Odd Willie asks her, and the old woman narrows her eyes like an angry cat and doesn't say a word.
"Are you going to tell me—" Soldier begins, but Willie interrupts her.
"I swear to God, Soldier, if that old bat keeps staring at me like that, I'm gonna do her right here, in front of God and everyone."
"You're very unpleasant," the old woman croaks at him, and Soldier realizes that it's not a mole on her chin, but a dab of chocolate icing. "This used to be a good neighborhood."
"Yeah?" Odd Willie asks. "And just when the hell was that?"
"Let it go," Soldier says, her heart suddenly beating too fast, her palms gone cold and sweaty because she knows this is something bad, that it has to be something wicked bad to have Odd Willie losing his shit in public over a nosy old woman. "What did Saben say to you? Where the hell is she?"
"She's out at Rocky Point," he replies, glaring furiously back at the old woman. "She says she's got a fucking body in the trunk," and he raises his voice and yells, "Yo, Grandma, did you hear that over there? I said, she's got a fucking body in the trunk of her fucking car!"
The old woman shakes her head. "You Mafia trash," she sneers. "This used to be a decent neighborhood."
"I'm telling you," Odd Willie growls, "the bitch better stop looking at me."
And then Soldier sees one of the Hispanic girls behind the counter reaching for the telephone, and she grabs Odd Willie by one arm and hauls him up out of the booth. The old woman points a crooked finger at them, jabbing at the air like she means to poke it full of holes.
"That's right," she says. "You get out of here. Both of you, go on. I bet you got people to kill. I bet you got drugs to sell."
"Suck my dick, you shriveled old cunt."
"Wop trash," the old woman snaps back at him, then waves her finger about dramatically, like an incompetent magician finishing up a particularly difficult trick, before returning to her doughnuts.
"We're leaving," Soldier says, speaking to Odd Willie and anyone else who's listening. "We're leaving right now."
"She's calling the cops," he says and motions at the girl behind the counter with the antenna of the cell phone. "You better tell her to stop, or I'll have to shoot her. Hell, I'll shoot everyone in this dump."
And it could go that way, Soldier thinks, remembering all the crazy shit she's heard about Willie Lothrop. It could go that way, for sure, and maybe the Bailiff would be there to clean up the mess, and maybe he wouldn't. Maybe they'd both die waiting to find out, or maybe they'd die a little farther along, but it'd be their asses, either way.
She snatches her cell phone from Willie's hand and leans close, speaking directly into his right ear. His hair smells like a medicine cabinet. "Listen, you psycho shit," she hisses. "We're walking out that door, and you're not doing anything unless it's something I've fucking told you to do."
And Odd Willie nods once, just once, but it's enough that she lets herself start believing that maybe Saben White really is her biggest problem, and maybe she's not going to die in a Dunkin' Donuts on Warwick fucking Avenue.
"It's a priest," Willie says and giggles again, still watching the girl behind the counter.
"What? What's a priest?"
"The body locked in Saben's trunk. She killed herself a fucking priest."
And Soldier takes him roughly by the arm then, and leads him out into the February cold, out into the unwelcoming sunlight, her hangover and Woonsocket and the new bottle of bourbon all forgotten. Odd Willie gets into the Dodge as soon as she unlocks the doors; he takes a 9mm from the holster on his ankle and checks the clip. "I'm cool," he says. "I just can't stand people staring at me, that's all. It messes with my head." Soldier doesn't reply, doesn't say another word to him, doesn't even dare look over her shoulder until there are a couple of miles between the two of them and whatever is or isn't happening on Warwick Avenue.
"You're going the wrong way," he says, which is true, and Soldier turns around in the parking lot of a Shell station and heads for the old amusement park at Rocky Point.
Soldier first met the Bailiff when she was only five years old. So much of her childhood is only a vague, uncertain blur, but that first sight of the Bailiff, that's clear as moonlight through clean windows and un-cloudy skies, as Sheldon Vale used to say. One night, she'd been told to put on her good calico dress, cornflower blue with tiny rosebuds, and then allowed to leave the tunnels and go into the basement of the yellow house on Benefit Street, up the creaky basement stairs into the house itself, and she found the Bailiff in a musty room filled with books and curio cabinets and an enormous globe cast in bronze, the whole world borne on the shoulders of a kneeling giant. When she'd opened the library door, the Bailiff stopped reading the book lying open on the writing desk in front of him and stared at her.
"Well now, child, exactly what would you be after?" he asked, and Soldier almost slammed the door, almost turned and ran back through the house, all the way back down to the safety of the tunnels. Instead, she squeezed the crystal doorknob and stared at the fat man, his smooth bald head and great gray beard, the biggest man she'd ever seen, and she imagined that he could hold the world on his shoulders, too, if it ever came to that. He was wearing a shabby navy blue seersucker suit and white tennis shoes, and he held a china teacup in his left hand, his pinkie finger extended like a small, plump sausage.
"Well, child? Are you dumb? Are you deaf? Do you have a name yet, or shall I just call you whatever strikes my fancy?"
"Soldier," she said and took one cautious step into the room.
"Soldier," the Bailiff replied, the way the word slipped from his lips making her think that perhaps he'd never heard it before and wasn't sure what it meant. He set his teacup down on a stack of books. "What kind of name is that for a pretty little girl such as yourself?"
"The name I chose," she told him.
"Then who am I to be asking questions? It's a fine name, Soldier. And which boneyard did you steal it from?"
She shrugged and eased the door shut behind her.
"Don't you even know?"
"Are you God?" she asked, setting his question aside for later, then glanced at the globe again. The painted continents were drifting across its surface, colliding with one another, pushing up new mountain ranges, tearing deep rifts for new seas to fill.
"Which one?" the Bailiff asked and sat up a little straighter in his chair.
She thought a minute, not having expected that particular question, and then she said, "The god of men. The church god. The one who let his son die, so that the people he'd made wouldn't go to hell." It was something that she'd overheard some of the older children and pups talking about one day when they should have been asleep, and though none of it had made any sense to her, she thought that it might be the right answer to the fat man's question. Or at least the sort of answer that would make her look like she knew about such things.
"Ah," the Bailiff said and rubbed thoughtfully at his beard. "And why would you ever think a thing like that, little Soldier, that I could be the god of man?"
"You look like him," she said, still watching the silent dance of continents and oceans over the circumference of the enormous globe.
"Do I?"
"Yes," Soldier replied. One of the pups had said that the god of man lived in a great house in the sky, where he was held prisoner by all the creatures that he'd created, and that he had a beard.
"Looks can be deceiving," he told her. "Our eyes are great liars and even worse judges of divinity."
"That's okay," Soldier sighed, finally turning away from the globe and back towards the Bailiff. "I don't think you're anyone's god. Not really."
"Well, I'm disposed to suspect you're correct about that," he told her and picked up his teacup again. "And don't you think I'm not grateful for that one small mercy, either."
"Are you the Cuckoo?" she asked.
"Why are you standing all the way over there?" he replied, watching Soldier over the rim of his cup. "Whatever I am, I don't usually eat little girls. Especially not one so full of questions."
Soldier looked back at the closed door, hesitating a moment more. It wasn't too late to leave, and she wasn't sure she was supposed to be in this room with this man. Maybe he was one of the house's secrets, and Madam Terpsichore had warned them all how dangerous secrets could be.
"Questions invariably give me indigestion," the Bailiff said and slurped his tea.
"Well, are you?" Soldier asked again, taking another two or three steps nearer the Bailiff. "Are you it? Are you the Cuckoo?"
"Not likely," he replied, sniffing at the contents of his teacup. "Have you ever seen a cuckoo? Nasty little things. Members of the family Cuculidae. Never met one yet was worth so much as a plug nickel and the time of day."
"I was...I was looking for something," she said, which Soldier thought was probably true. "They said I could come upstairs. They said—"
"They say an awful lot of things, don't they?"
"They do," she admitted reluctantly, hoping no one else was around to hear, trying to remember whatever it was she was supposed to be doing in the yellow house.
"The hounds are indeed damned garrulous beasts, and they do dearly love the sound of their own flapping tongues," the Bailiff muttered, half to himself, and held out the teacup to Soldier, who was now standing close enough to take it from him, if she wanted it. "Oh, it's a smidgen bitter, I warn you," he said, staring down into the cup and then back to Soldier. "Too much fenugreek in the mix, I suspect."
She took the cup from him and saw there was a little liquid left inside. It was very dark and smelled almost like turpentine.
"One thing's certain, though. We all got our burden of questions, little Soldier," the Bailiff told her and took a handkerchief from his breast pocket, white linen embroidered with tiny blue flowers, and he wiped at his wide forehead. "They follow us into this world, and damn it, they follow us out." Then he noticed that she wasn't drinking and frowned. "Don't you like tea?"
"Is that what this is?"
At that, the Bailiff laughed out loud and dabbed the handkerchief at the corners of his brilliant emerald eyes. "Well, that's what they tell me," he said. "But there's never as much truth in the telling as in the tasting, I've always heard. Have yourself a sip, and tell me what you would have it called if not tea."
Soldier, who had once taken a dare to eat a large spider that one of the pups had found beneath her bunk and then been sick for a week, sniffed the tea again and wrinkled her nose. "It smells like alley apples," she said.
"Oh, it's not quite as bad as that. I drink it almost every day."
"Is that why your hair fell out?" Soldier asked, sloshing the dark liquid about in the china cup. The cup had primroses painted on it, and there was a chip and a crack and the handle looked as though it had broken off and been glued back on again.
"Well," the Bailiff said, folding his handkerchief and returning it to the pocket of his seersucker suit. "They told me you had a mouth on you, child, and I daresay there's a wit to match."
And then she tasted the tea, because it seemed inevitable that she would, because this man might be the Cuckoo, despite what he'd said, or he might be the god of man and churches or a secret that shouldn't have been found out. Whichever, it was better not to make him angry, she thought, better if he didn't go complaining to the pale, silver-eyed people who kept the house that she'd interrupted his reading and then refused his hospitality.
It wasn't tea, and it didn't taste like turpentine. It tasted like something much, much worse.
"Such a face," the Bailiff said and shook his head. Soldier swallowed, gagged, coughed without covering her mouth, and then started to set the empty cup down on the edge of the writing desk, but, "No," the Bailiff told her. "That's not the last of it."
"Yes, it is," she said and wiped the back of one hand across her lips, as though she might wipe the taste away. "I drank it all. I wish I hadn't, but I did."
"Yes, you drank it," the Bailiff agreed, "but that doesn't mean the cup's quite empty."
"Oh," Soldier said and looked into the cup again. The Bailiff was right. The bottom was covered with a sort of sludge. It reminded her of the mud that got all over everything whenever it rained too much and the tunnels began to flood. It reminded her of the time she cut her thumb on a scalpel she hadn't had permission to touch. It reminded her of the empty skins snakes leave behind.
"They see something in you, dear girl," the Bailiff said, and she was dimly aware of the old, dry wood of his chair creaking as he shifted his bulk from one side to another. "It frightens them. It makes them wonder if it means their doom, what you are, what you may yet become, or if perhaps you can become a more perfect servant of the Cuckoo."
"I don't know," Soldier whispered, and her voice seemed to have shrunk down to a sound no larger than the scritching of insect legs against bare dirt walls.
"Nor do I," replied the Bailiff. "But we must endeavor to find out, don't you think?"
"I'm going to be sick," she told him and realized that the stuff in the bottom of the cup had become ashes or had been ashes all along.
"It would be better if you didn't. We'd only have to start this all over again some other time."
So she took a deep breath and struggled to think of anything at all but the wriggling sensation the dark liquid had left in her throat, the roiling, almost painful fullness much farther down inside. "I don't want to look at this anymore," she said and set the cup on the desk before he could protest, set it next to the book he'd been reading when she came in. The brittle paper was the color of ancient, rotting teeth, and she didn't recognize any of the words because they were all in Latin, and the ghouls held back Latin until the seventh year.
"Not to worry, child," the Bailiff said and ran the tip of one finger around the rim of the cup. "You've seen the start of it. The rest can wait." His skin against the porcelain made a high, clear ringing, and hearing it made Soldier's stomach feel better.
"They need their prophecies and portents," he said. "It keeps them on an even keel, if you know what I mean."
"That's pretty," Soldier said, meaning the ringing his finger was drawing from the cup. "I didn't see any bells in there," and she thought perhaps he was hiding them in his hand somewhere. A hand that big might hide almost anything at all.
"Do you recollect your mum?" the Bailiff asked. "Do you sometimes see her face?"
Soldier frowned. It was a stupid question, and whoever he was, he ought to at least have the sense to know that. Anyone who could read Latin ought to know that much. "No," she said. "The Cuckoo took me when I was a baby, just like all the others."
"Of course it did. But do you recall her face?"
"Our mothers are gone, gone, gone," she replied, repeating one of the first lessons the ghul had taught her. She didn't want to be talking, and she silently asked Mother Hydra to tell the fat man to shut up so she could listen to the music from the teacup. "Our mothers are gone from us forever."
"That's what they told you?" he asked.
"Yes. When the Cuckoo took me from her, he took all my memories of her, too. It made me..." And she paused, looking for exactly the words that Madam Mnemosyne had used. "It made us hollow, so we could be filled with other things."
"Did it now? Sure, that's what they say, all right, and you're a smart girl what knows all her lessons."
Soldier realized that the music had stopped, but she felt much better, just the same.
"You do what you're told," the Bailiff says, "and don't think I'm telling you to do nothing otherwise. You answer to the hounds, and you always will. That's the way of it, little Soldier, for better or ill. But you should know this, too," and now he lowered his voice, and his eyes seemed to flicker and grow dim. That frightened Soldier, those glistening emeralds bruising until they were almost brown, though she wasn't sure why it frightened her, and she took a step back from him, but then the Bailiff put a hand on her shoulder so she couldn't run away.
"They don't know even half what they pretend to know," he said, speaking in a hushed, conspiratorial tone. "And they've got no problem with making up the difference. They're something terrible, something more than this old world, just the way they tell it, but they ain't gods, no more than you or me. Don't you ever forget that, girl, no matter what they might say to the contrary."
"I'm sleepy," she said, because she was. Suddenly Soldier was very sleepy, and she wondered if she'd ever be able to find her way back downstairs before her dreams caught up with her.
"That's just the tea. The nap will do you good."
"Will I see you again?" she asked.
"You will," the Bailiff said, "and often. They want me to try to tell them if you're what they've been waiting for all this time, what they've been praying would never track them down."
"Am I?" she asked groggily.
"I guess we'll have to see about that, won't we?" he replied and laughed again. And then there was no use trying to stay awake even one minute longer, sleep grown so thick and warm and welcoming about her. He rose from his chair and took her in his strong arms, and the fat man who said he wasn't the god of men or the Cuckoo carried Soldier back through the yellow house and down the stairs to the tunnels far Below.
Near the dead end of Rocky Point Avenue, the ruins of the abandoned amusement park sprout like something cancerous, some gaudy, carnival-colored malignancy, grown wild and shattered from the banks of Narragansett Bay. There were rides and attractions and restaurants here as far back as the end of the nineteenth century, more than a hundred years of Rhode Islanders stuffing themselves with deep-fried clam cakes and doughboys, then getting sick on the carousels and Ferris wheels and roller coasters, until a bankruptcy finally shut the place down in 'ninety-six. Since then, the relentless vandalism of the seasons—snow and frost, rain and salt wind and summer heat—and the more random desecrations of teenagers have taken their toll. Now the park sits empty and forgotten by almost everyone but land developers and kids looking for someplace they shouldn't be. A few deer and the occasional coyote stroll the Midway, and rats nest in the boarded-up ice-cream stands. Something else for Soldier to hate about Warwick, another place she'd never go if not going there were up to her.
Saben White's rusty old Chevy Impala is sitting directly in front of the park's front gates—six vertical concrete columns painted hideous shades of blue and red and yellow, neons faded now almost to pastels, and there's a sagging cross span with giant red-and-white plywood letters and the gaps where other letters are missing to gaily spell out ROCK PO NT. When they pull up, Saben's standing near the rear of the car, smoking a cigarette.
For the last half mile, Odd Willie's been singing what he swears is an old radio and TV jingle for the park—Come with your family, come with your friends. Rocky Point is open, 'cuz it's summertime again! When he sees Saben, his voice rises to a strained, ear-rending falsetto, Frankie Valli on crack, and Soldier tells him to shut the hell up.
He giggles and lights another Winston. "Hey, I'm thinking maybe it's not a real priest," he says, talking through a cloud of smoke. "Maybe it was an impostor, and we're in the clear, safe as houses, cool as Eskimo poop."
"Oh, it'll be a real priest," Soldier replies, because she's been stuck with Saben just long enough to know that she never fucks anything up halfway, that with her it's all or nothing. "You just stay sharp and don't go freaking out on me again, you understand?"
"Roger-dodger, Captain Kangaroo," Odd Willie snorts and takes another long drag off his cigarette.
"You are such a fucking asshole," she says, pulling in close beside the Chevy, not certain if she means Willie Lothrop or Saben White but pretty damned sure it doesn't make any difference. Soldier shifts into park and kills the engine, and Willie rolls down his window and waves at Saben. Soldier unfastens her seat belt and opens the driver-side door, climbing out into the cold, sunny afternoon, all that clean blue sky overhead like some void her headache is working overtime to fill. She slams her fist down hard against the roof of the Dodge, and Saben sighs and drops the butt of her cigarette to the pavement, grinds it out with the toe of her boot.
"One good reason," Soldier says, "you give me one good goddamn reason why I shouldn't kill you right now and bury the both of you in the same hole."
"I didn't have a choice," Saben says, staring down at the ground instead of looking Soldier in the eye.
"Bull shit," Soldier growls and punches the roof again; this time Odd Willie yells at her to stop, and she yells back at him to get his lazy ass out of the damned car.
"It's fucking complicated," Saben says, glancing right, towards the amusement park's chain-link gates topped with glinting coils of razor wire. "I swear to you both, I did the only thing—"
"You gonna swear it to the Bailiff?" Soldier asks, coming quickly around the front of the Dodge as Odd Willie gets out of the car. "Or maybe you better just swear to the hounds and get it over with? What the hell were you doing in Connecticut in the first damned place?"
"Business," Saben replies. "I had business. I was going to call—"
"Saben, we are supposed to be in fucking Woonsocket right this very fucking second. Right now, all three of us, we're MIA. The Bailiff doesn't know where the hell we are. Those assholes up in Woonsocket don't know where we are."
"Maybe you should let her talk," Odd Willie says, and Soldier glares at him.
"Maybe you should stay the hell out of this," she barks back. "Maybe you should start worrying about your own problems, Mr. Goes-fucking-homicidal-psycho-in-a-goddamned-doughnut-shop-because-an-old-lady-looked-at-you-funny."
"I don't think I need you standing up for me, Willie," Saben says.
"Oh, I think you are gonna find out you're very wrong about that," Willie tells her and laughs. "In fact, I think you have no idea just how—"
"Shut up, Willie," Soldier says, and this time he does. For a moment the only sound is the wind through the high, bare limbs of the trees growing on either side of the Rocky Point gate. Soldier flares her nostrils and takes a deep breath, smelling the nearby bay, pushing back at the headache and the anger crowding her mind, the anger that's going to get her killed someday soon if she doesn't learn to keep it on a shorter leash. She watches Saben and tries to imagine what the Bailiff would say, what he'd tell her to do if he were here. Something simple and obvious, like, You can fret about all the whys and hows and wherefores when you have the time for it, or The son of a bitch is dead, little Soldier. And that grave you've got to dig for him, it don't much care how he got that way. Something like that. Something to get her moving and keep her from kicking Saben's ass until after they've dealt with the mess she's made.
"He's in the trunk?" Soldier asks, pointing at the Impala, and Saben White nods her head.
"Yeah, Soldier, he's in the trunk. I already told Willie he's in the fucking trunk. I can explain this. I really can, if you'll just stop and listen."
Soldier is standing very close to Saben now, close enough to hit her, close enough to knock out a few front teeth or maybe break her nose or both before Odd Willie would even have a chance to ask if that was really such a good idea. Her hands are clenched fists, and Soldier knows how good it would feel, Heaven in the force of the blow, release in the collision of her knuckles with that fragile construction of bone and cartilage and flesh. It would be almost as sweet as whiskey.
"And what good's that going to do us, exactly, me listening to more of your bullshit?" she asks Saben White and then takes a step closer, aware now that she's near enough to make Saben uncomfortable, that she's invaded the invisible, sacred bubble of personal space. To people like her and Saben, people who spend all their short lives waiting for the next confrontation, the next asshole who's come around looking for a piece of you, sometimes that's all it takes, and she silently prays to the nameless gods that Saben will swallow the bait and throw the first punch. "Explaining it to me, is that going to make this guy in your trunk any less dead? Will it get rid of the fucking body for us? Maybe it'll square things with the Bailiff, is that it?"
"Jesus," Odd Willie mutters around his cigarette. "How about let's just dispose of the motherfucker and get it over with. It's freezing out here."
"No, I want her to know how it is," Saben says, risking a glance at Soldier. "I want her to know this time it wasn't my fault."
Soldier spits on the ground at Saben's feet, and a white fleck of saliva spatters the shiny black leather toes of her boots. "Saben, I don't believe you could tell the truth if someone wrote it down for you."
Odd Willie laughs, then shivers and stares up at two big gulls wheeling by overhead. "You guys keep this crap up long enough, we're gonna end up having a chat with the fucking cops, okay?"
"Yeah, so how you want to do this, Willie?" Soldier asks him, not taking her eyes off Saben, thinking she might still get lucky. Saben White's good at not knowing when to shut the hell up, even better at not knowing when she's in over her head.
"Christ. Why are you asking me? I didn't shoot the son of a bitch."
"You feel like digging a hole?" she asks Saben.
"Hell," Willie grumbles. "The ground's probably half fucking frozen," and he stops watching the gulls and kicks at the pavement. "It'd probably take the rest of the day to plant Holy Joe deep enough the dogs wouldn't just come along tonight and dig him up again. Lots of wild dogs out here."
"What about you?" Soldier asks, and steals another inch or two of the space separating her and Saben. "You killed him. Maybe you know what we should do with him."
"Put him in the bay," Saben replies, and Soldier can see how hard she's straining to sound like she has her shit together. "Gut him, fill the bastard with stones, then dump the body off Rocky Point. That's what I was thinking when I drove out here."
"I don't know," Willie says. "I never trust them not to float. Especially priests. Priests and faggots, they'll float on you every goddamn time."
"There've been a lot of fires out here," Saben says, and Odd Willie nods his head, then stares at the glowing tip end of his cigarette.
"She's got that part right," Willie says. "What with all the kids and the bums, it's a wonder there's anything left standing out here. It's a wonder it's not just a heap of charcoal."
"Right, so who's gonna care if there's one more fire?" Saben asks hopefully, looking straight at Soldier now. "We take his teeth and hands—"
"Yeah?" Soldier asks, speaking hardly above a whisper, and she leans close to Saben, sniffing at the air around her. "Is that what we do?" Soldier can smell stale tobacco smoke and expensive leather, cologne and the gun hidden beneath her blazer. The air about Saben stinks like fear and lies and desperation.
"Man, these days burning ain't much better than sinking," Odd Willie says glumly, looking up at the gulls again, "not unless you're dealing with some sort of serious expedient. What with the fucking DNA analysis and pathologists and all that shit, the goddamned cops might as well be magicians. Now, maybe if I had some time to lay my hands on a good accelerant, set up a delay—"
"Could you do it alone?" Soldier asks, and he answers her with a shrug and a scowl.
"Soldier, I heard you had pyromancy," Saben White says, and Soldier moves closer and sniffs at her hair.
"Did you? And where'd you hear a thing like that?"
"After Ipswich—"
"Ipswich was easy," Soldier replies, and now she's so close her lips brush against Saben's right ear. "There weren't any dead Roman fucking Catholic priests to worry about. All I used in Ipswich was a couple of cans of gasoline and a road flare. That's all the pyromancy I know, bitch."
"I'm not scared of you," Saben says, but it's obvious to Soldier that she is, that she's terrified—the way her voice trembles, the sweat standing out on her upper lip, her slightly dilated pupils, the faint electric reek of adrenaline.
"I think I want to see you on your knees," Soldier whispers. She can feel the other woman's body growing tense, fight or flight, resist or submit, and she nips at the top of Saben's right ear, not quite hard enough to draw blood, but hard enough to hurt, catching the soft curve of helix and antihelix between her strong incisors.
"Screw you," Saben says, her voice rattling like empty tin cans strung on baling wire.
Slowly, tentatively, Soldier releases her ear, and when it's free, Saben covers that side of her face with her right hand and turns her head towards Odd Willie.
"Is that what it's going to take?" Soldier asks her. "Do I have to actually fuck you to get your attention, to make you understand who's on top?"
"Willie, call the Bailiff," Saben says. "Tell him I had to do it. Tell him to get her off me."
Willie just shakes his head and flicks the butt of his cigarette at the curb, then takes his silver Zippo and a fresh pack of Winstons out of his coat. He peels away the cellophane wrapper and drops it on the ground. The wind whisks it away at once, and the crumpled plastic snags in a patch of dry brown weeds closer to the gate.
"C'mon, Saben. You know he's not going to do that. You ought to know he can't do that. He might be a lunatic, but he knows the rules. He knows your ass is mine if that's the way I want it. Ain't that right, Odd Willie?"
"Yeah, sure," he replies, and lights another Winston. "I know how it is. But, Soldier, I mean, fuck it, why don't we deal with the good Father Whoever-he-might-be in the trunk there before every cop in this end of Kent County shows up. You can still mess with Saben later, right?"
Soldier ignores him, only half hearing the words anyway, her own heartbeat too loud in her ears now, the taste of Saben's fear too sharp, entirely too immediate.
"Get down on your knees, Saben. Don't you fucking make me tell you again."
"No," Saben says, still watching Odd Willie like maybe there's some hope he's going to change his mind and come to her rescue after all. "I won't do it."
"Fine," Soldier says, "if that's how it's got to be, that's how it's got to be," and she slaps the left side of Saben's face so hard that she staggers and almost falls. Blood spurts from her nose and busted lower lip, falling in vivid crimson droplets across the blacktop. Odd Willie makes a disgusted, impatient sort of noise and retreats a couple of steps towards the Dodge. Saben slumps against her car, stunned, gasping through her bloody lips and nostrils, and Soldier seizes her by the back of her neck and forces her to her knees. She isn't surprised that it requires so little effort. She's learned that one good blow takes the fight out of almost everyone.
"How's that?" Soldier asks, bending over Saben. "Are you comfortable down there?"
"You know, just for the record," Odd Willie says, "it seems to me you got your priorities turned around."
"Duly noted," Soldier replies breathlessly and smiles at Odd Willie. Drawing blood felt even better than she expected it would, and the sight of Saben on her knees, Saben helpless and hurting and humiliated after all the weeks of bullshit, has her a little giddy. Part of her, the sliver of humanity that knows enough to keep its hands clean, reminds Soldier that there are limits to what she can get away with.
Don't worry, she thinks. I'm not gonna kill her. Not today.
"Maybe I should put it in writing," Odd Willie Lothrop says.
"Yeah," Soldier agrees, "maybe you should. There's a pen in the glove compartment. Probably some paper, too."
"But you're not gonna hold this against me?" he asks. "I mean, I know this sort of shit can come back to haunt a guy, taking issues with the powers that be and all."
"Oh, hell no," Soldier says, still smiling, and then she kicks Saben hard in the ribs. There's another gout of blood on the pavement, and Saben White's mouth is opening and closing, opening and closing like a fish caught out of water, fighting to draw breath back into her bruised, deflated chest. "I'm not vindictive," Soldier continues. "You ought to know that by now, Willie. I'd never think of keeping anyone from speaking his mind. I mean, that's the freaking First Amendment. That shit's sacred as George Washington and the goddamn golden rule."
"Never hurts to ask. I figure a guy's gotta watch his own ass," and then he slips quickly back inside the Dodge, shuts his door, and rolls up the window again. Soldier nods once in his direction and turns back to Saben White.
"Still thinking we should get the Bailiff's opinion on this situation?" she asks, taking her cell phone from her jacket and laying it on the blood-spattered ground in front of Saben. "It's easy. All you gotta do is press redial. Of course, I'll break your fucking hand, but nothing's free."
Saben, still gasping, starts to reach for the phone, then pulls her hand back.
"That's a smart girl," Soldier says, squatting down next to Saben White. "Who'd have ever thought it. Now, we're going stop fucking around and come to a mutual understanding, and then we're going to clean up this mess you've made, capisce?"
Saben only nods her head once, then gags and almost vomits. Soldier retrieves her cell phone and returns it to her jacket pocket.
"I..." Saben gasps. "It's...it's not..."
"You need to concentrate on breathing, babe, not talking. We're way past talking now."
Soldier leans forward and closes her jaws firmly around the back of Saben's neck, just below the hairline, just behind the point where her spinal cord enters the foramen magnum. She bites down hard, breaking skin and bruising muscle, sinking her insufficient teeth in as far as they'll go and tasting the warm, coppery sea trapped there inside that body. A low and threatful growl begins deep in Soldier's throat and rises very slowly; Saben White goes rigid, then, gradually, she begins to relax, accepting Soldier's rank, accepting her own position. They both know this drill, have both known it since they were children wrestling with other changelings and the ghoul pups in the tunnels beneath College Hill. It's almost instinct to them both. After only two or three minutes, Soldier releases her, and Saben crawls away, sobbing, still trying to get her breath, trailing blood and spit and snot.
"I'll give you a couple of minutes to pull yourself together," Soldier calls after her. "I wouldn't waste it crying." But the anger is deserting her now, and there's little left in its place but weariness and dread of all the trouble that's yet to come. Her phone call to the Bailiff. Keeping her mouth shut and taking it, whatever it may be, whatever he has to say. Whatever she has coming for letting all this happen on her watch.
She wipes her mouth, and her lips and chin leave a red smudge across the back of her wrist. Her throat's a little sore from the noises it was making only a moment before, the guttural canine sounds that it was never intended for. She thinks of the bottle of whiskey hidden beneath the seat and watches Saben White, who's sitting near the front fender of the Impala now, sobbing, her face hidden in her hands. Her tattoo, the Seal of Solomon, seems very bright beneath the afternoon sun, all those shades of ink shining from her skin like a beacon, like a warning. Soldier imagines cracking open the bottle of Dickel, imagines it filling her mouth, warming her belly, driving back the thought of the Bailiff's voice, and then she even manages to pretend she has that luxury.
The second time that Soldier met the Bailiff, he was wearing the same threadbare seersucker suit as before, the same white tennis shoes, but he offered her sugar cookies and grape soda instead of the foul-tasting, turpentine-scented tea. She'd been sent upstairs again by Madam Terpsichore, and one of the silver-eyed women who kept the house led her through the kitchen and a parlor and down the long hallway to the library where he was waiting at his writing desk. There was an oil lamp with a tall glass chimney sitting near the book he was reading, and it cast the only light in the room besides the waxing three-quarter moon slipping in through the parted draperies. He smiled when he saw her, smiled wide and thanked the silver-eyed woman, whose name was Adelaide.
"You're back so soon?" he asked Soldier, as though the whole thing had been her idea. "Why, it's hardly been a month, has it?"
In fact, it had hardly been a full week, but Soldier figured it would probably be best if she didn't correct him. A plate of cookies and a bottle of soda had been placed on the floor near his chair, and the Bailiff pointed at them and asked if she was hungry.
"Yes sir, I am," she said and sat down near the plate. She was missing her dinner, being sent up into the yellow house to talk with the fat man who claimed that he wasn't the god of men and churches or the Cuckoo, but whom she suspected could be either one and, perhaps, even both. She drank half the grape soda, taking care not to spill any on her blue calico dress, and ate one of the cookies before he said anything else to her.
"Have you been thinking about the things we talked about?" he asked. She told him that she hadn't and finished a second cookie.
"Oh," he said, sounding neither surprised nor angry. "Well, I don't suppose that much matters, does it? I suppose it matters much more what we talk about tonight. Before we begin, do you have any questions, little Soldier?"
And there were things that she wanted to ask him, such as why none of the other children had been sent up to the library, why only her, and whether or not he had a name besides "the Bailiff," which didn't seem like much of a name at all. But before she could think which question to ask him first, which might be most important, in case she got to ask only one, he was already talking again, so she started on her third cookie instead.
"That cup of tea you finished off," he said. "I trust it likely didn't agree with you. It's damned heady stuff, I'll grant you that. Sometimes, it knows the road into a person's dreams. Sometimes, it can leak straight into the soul," and then he tapped himself twice between the eyes with a pudgy index finger. "Unless, of course, you happen to be accustomed to it."
Soldier put down the unfinished cookie and stared up at him; his green eyes seemed even brighter than the last time. She wondered if they were real eyes, or if the Bailiff had somehow lost his own and these were only polished gems.
"I had a dream," she said. "Is that what you're asking, if I had a bad dream after I drank the tea? Were you the one who carried me to bed?"
The Bailiff hesitated a moment, as though confused at having been asked two questions at once. He rubbed at his beard and frowned.
"I had a dream," she said again, recalling the way her head had ached when she woke after her first visit with the Bailiff, waking confused, disoriented, and hurting, and sick to her stomach. She'd been excused from her lessons for the whole day, and Madam Melpomene had even made someone else take her chores. So she'd lain there in her bunk all night long, staring at the underside of the bunk above, afraid to sleep again, thinking about the Bailiff and trying to remember all the missing pieces of the strange, long dream. She knew it had come from the teacup, from whatever she'd swallowed. She didn't need him to tell her that.
"I might have warned you, I suppose," he said, those too-green eyes twinkling in the lamplight.
"Are these real cookies?" she asked him and poked at the one she'd left half-eaten. "Will they give me nightmares, too?"
"Do I look like the sort who would give a little girl tainted sweets?"
"Yes," she replied, and he laughed out loud, his laugh like a thunderclap, and leaned back in his chair. The old wood creaked alarmingly, and she thought for a second that it might burst apart in a hail of varnished splinters.
"The dream, then," he said, when he'd stopped laughing and had wiped his eyes with his handkerchief and put it back into his breast pocket. "Tell me as much of it as you can, little Soldier. Everything you recollect. Take care to omit nothing, no matter how nonsensical or insignificant it might seem to be."
"Nonsensical?" she asked, puzzled by the word because she'd never heard it before.
"Silly," he replied. "Ridiculous. Inane. Absurd."
"I see," she said, though she'd never heard inane before, either.
"Anyway, take care to leave none of it out. The devil is very often in the details, you know."
"Unless he's in this room," she said, deciding against finishing the sugar cookies and grape soda. The Bailiff laughed again, and his chair creaked and popped beneath him.
"Whatever in the hells have you gone and plucked from the wild, wild sea, my dear old Terpsichore," he chuckled, even though Madam Terpsichore wasn't in the room with them. And then to Soldier he said, "Child, you are a rare scrap, indeed. We shall have such delightful conversations, you and I."
"Do you want to hear about my dream or not?" she demanded, and the plate with the remaining cookies squeaked softly against the floor when she pushed it away from her.
"Oh, most assuredly," he said. "More even than I desire my next breath."
I could keep it for myself, she thought. I could make it a secret and keep it forever, and no one would ever know. I'd forget he made me have that dream. I could try—
"Whenever you're ready, dear," he prompted.
"It doesn't make much sense," she warned, but he said that didn't matter. "I don't remember all of it," she added, and he said that didn't matter, either. Of course, he'd told her both these things already, and she could tell he was losing patience. She didn't know what would happen if she made the Bailiff angry, the bald man, the one who might be the god of men and of churches, who might be the Cuckoo, who might only be a demon. She didn't know, and she didn't want to find out, so she took a deep breath, wished that she weren't afraid of drinking the rest of the grape soda, and cleared her throat.
"At first I was in a desert," she said, though that wasn't actually the beginning of it. The true beginning frightened her too much to put into words, so much so that it was better to risk making the bald man angry with her. "I was in a desert, and I wasn't a child. I was grown-up, and there was another woman walking with me. She was also a child of the Cuckoo."
"Did she tell you that?"
"Yes, I think so," Soldier replied. She told the Bailiff that the woman was the most beautiful person that she'd ever seen, dreaming or awake or anywhere in between, Above or Below. She was very, very tall, with long gray-white hair pulled back in dreadlocks to reveal her face and her golden-brown eyes. Her skin had, over the long ages that she'd wandered the sand, been burned black as pitch by the relentless desert sun and might as easily have been ebony or jet as flesh and blood. They'd walked together through the whispering sand, trudging over gigantic dunes that seemed to run on in every direction as far as Soldier could see.
As they walked, the woman told her a story about three witches and a changeling child named Esmeribetheda, an old story that Soldier had known already, though she'd never heard it told quite the way that the black woman told it.
"She was not a traitor," the woman said to Soldier. "That part's a lie. Esmeribetheda had good reasons for the things she did. The djinniyeh in their great domed city had set themselves against the ghul, who were interlopers, who'd come to the deserts from some world far away. Esmeribetheda was to be made a djinniyeh herself, when the ghul were all dead or driven back to that other place."
"How curious," the Bailiff said and lit a briarwood pipe with a kitchen match. He exhaled, and the air in the library smelled of brimstone and cherry tobacco. "And you've told none of this to the hounds?"
"No," Soldier replied, because she hadn't.
"Good. It's best we keep it that way for now. Continue, please."
"Are you sure?" she asked, hoping he'd decided that he'd heard enough and would send her away.
"Of course," he said around the stem of the pipe clutched between his yellow teeth. "You're doing very well. Don't stop now."
So she told him about the ruined walls and empty windows of a city half-buried by the sand. They passed it and then descended a particularly steep dune to stand together on the parched bed of an ancient, vanished lake or sea hidden deep within the endless desert, a vast plain of cracked mudflats and glittering salt and gypsum crystals. All around them lay the crumbling, petrified bones of the monstrous beasts that had lived there long ago, the beasts that had haunted deep places almost from the dawn of time until that last day when the sun had finally consumed the waters, and the things had perished in the mud and blistering heat.
The crooked, scorched trunk of a tree rose from the lake bed. There were symbols Soldier didn't recognize carved into the wood in long vertical columns.
"The ghul murdered her," the woman said, running her fingers across one of the symbols. "That part's true. Her bones turned to dust, and her ghost still murmurs in the night. In time, the djinniyeh became distracted by other things—they're too easily discouraged—and didn't try again. But the hounds moved on, searching for someplace where they'd have no enemies to fear."
And then, Soldier told the Bailiff, the black-skinned woman had gone, and she'd been left alone and standing at the crest of a very high dune, the tallest one yet, the King of Dunes, she thought. She stood there, wrapped in silk and muslin robes that the dark woman had given her, looking down at a place where the desert finally ended, a place where the sea met the land, where the two were forever fighting a war and the sea was forever winning. The yellow-brown-white sand became a green sea capped with silvery waves, and she was starting down the far side of the King of Dunes when she heard something in the sky and looked up.
The Bailiff puffed his pipe and nodded his head.
"I'm not making any of this up," she said.
"I don't think that you are," he replied. "What did you see when you looked at the sky?"
"You don't think that I'm lying to you?"
"Of course not. But tell me now, what did you see?"
What she'd seen were two birds, or flying machines, or dragons, or prehistoric reptiles, or angels—it was difficult to be sure, and perhaps she'd seen all these things. Perhaps, she said, they'd been one thing and then another, changing constantly, and the Bailiff had thought this entirely plausible. The two things, whatever they might have been, were locked in an even fiercer battle than that being waged by the sea upon the sandy shore. One was wreathed all in flame and the other in a whirling veil of shadow. Bits of them were torn away and fell to the ground, sizzling, smoking, melting sand to glass on contact. She was very afraid, seeing them, not knowing what it meant, and Soldier told the Bailiff that she'd tried to reach the shore.
"Why? Did you think you'd be safe there?"
"There was a yellow boat waiting," she told him. "I thought that it might take me to the dark woman again. I thought she might have left it there for me."
"You wanted to be with her?" he asked, and Soldier nodded yes. "Do you wish you were with her now?"
"It was only a dream," Soldier said. "She wasn't a real person. You can't wish to be with a dream." But she knew that was a lie, and she'd missed the dark woman since the moment she'd awakened.
"Are you certain of that, it being only a dream?"
Instead of answering him, Soldier told the Bailiff how she'd lost her footing, had fallen and rolled a long way down the dune, getting sand in her mouth and eyes and nose; when she finally reached the bottom, she was sore and bleeding, dizzy, and her eyes were watering. Overhead, the two things, whatever they might have been, clawed and screamed and tore at each other. She understood that they might do this for all eternity, that they might have been at each other since the world began.
When she finally reached the beach, the boat was gone.
"Had it sailed without you?" the Bailiff asked, relighting his pipe, which had gone out.
"No, it just wasn't there anymore. I'm not sure it was ever really there."
And so, defeated, she'd sat down on the wet sand. There were stranded starfish and oysters and snails and ammonites all around, bits of coral and driftwood. She found a hermit crab huddled inside a snail's shell. The waves cooled her feet, which were bruised and raw from so many days or weeks or months walking in the desert, but the salt water also burned the blisters and the open, weeping sores. She'd become a child again, when she wasn't paying attention, a little girl sitting alone on a faraway beach while titans fought in the blue-white sky overhead. Digging her fingers into the sand, she found a single pearl and hid it in her robes.
"And then I woke up," she said.
"Yes," the Bailiff said. "I see," and then he said nothing else for a while. Soldier sat staring at the floor, at patterns she imagined she could see in the wood's grain, and the Bailiff sat smoking his pipe. She could hear a clock ticking loudly somewhere in the room, though she hadn't noticed a clock before.
Finally, the Bailiff emptied the bowl of his pipe into an ashtray and told her to stand up. "It's getting late," he said. "The sun will be up soon. Off to bed with you, little Soldier."
"Did I tell it right?" she asked.
"If you told the truth, and I think you did. But remember, say nothing of this to the hounds, even if they should ask."
"Lie to them?"
"Just do as I say. I need to think about this. Adelaide will show you back to the basement."
And Soldier almost asked him something else, almost asked what other secrets he was keeping from the ghouls, feeling a sort of thrill that the two of them were sharing a confidence, a conspiracy, that she'd been given instructions to disobey her masters and mistresses.
"Thank you for the cookies," she said, "and for the soda. Next time, orange might be nice."
"Ah, well then, next time, orange soda it will be."
"And arrowroot cookies," she added.
"If you wish," and the Bailiff had already turned away from her. The library door was opening, and Soldier could smell the faint acrid odor that all the silver-eyed people carried with them. Adelaide was coming for her.
"Thank you," Soldier said again, and this time the Bailiff only nodded his bald head and made a harrumphing noise that meant he was busy and she should leave. He was reading a large book lying open on his desk, squinting at the pages by the unsteady light of the oil lantern, peering at the print through the lens of a magnifying glass. There were other books on his desk, and an hourglass, and a bottle of red ink, and a calendar for July 2007, with all the days marked off through the nineteenth. Part of her wanted to ask him if she could stay, if she could sleep upstairs, but then Adelaide was leading her away, back down the long hallway to the parlor, and she was glad that she hadn't asked. He would call her back, if she did as he'd said and kept their secret. He would call her back, and there would be sweets and the smell of his pipe.
"You can find your way from here," Adelaide said.
"Yes," Soldier agreed. "I can. I know my way," and she went back down, through the basement doors, and descended the rickety stairs into darkness.
Soldier took Highway 117 west out of Warwick, shrugging it off like a long, dirty nightmare—the Rolling Stones blaring from the radio, just her and Saben White in the black Dodge sedan, because she left Odd Willie at Rocky Point to get rid of the dead priest and the Impala. She drove on past Apponaug, then took the interstate north, turning back towards Providence. Saben sat up front with her, with her and yet completely apart from her, pressed against the passenger-side door, keeping as much distance between them as possible. And that was fine with Soldier. It meant she'd made her point, so it was just as fine as fine could be, so long as Saben didn't start crying again. It was easily one of the most loathsome sounds that Soldier could imagine, women crying. It never failed to make her angry and always jabbed at the violence that was never very far beneath the uneasy surface of her; sometimes it was almost enough to make her physically ill. She'd once killed a woman for crying, a woman that she might otherwise have let live.
When they're finally clear of Warwick, when Cranston's sprawling bleak and ugly on their left and it won't be much longer before Providence comes in sight, Soldier takes out her cell phone and calls the Bailiff. He doesn't answer, of course. He never answers the phone himself. So first she has to talk to one of his lackeys, some shit-for-brains bootlick calling itself Cacophony who tells her how concerned they've all been, how they've been simply sick to death with worry. Soldier tells the boy to put the Bailiff on the line and then go fuck himself with the sharp end of a broken bottle.
"I was just this very minute talking about you," the Bailiff says. Nothing will ever seem as unnatural to Soldier as the Bailiff's voice coming through a cell phone.
"Guess that's why my ears were burning," Soldier replies and turns down the volume on the radio.
"I got a couple distraught calls from our friends up in Woonsocket," he tells her. "They aren't pleased at having been so neglected this afternoon. I trust it couldn't have been avoided, your dereliction. I trust it was at least a matter of life and limb."
"It was a matter of Saben," Soldier says. "Me and Willie, we were right there, ready to go. Then Saben showed up forty-five minutes late."
"Is that so? Then why am I only finally hearing from you more than two hours late?" he asks. "How, little Soldier, would you care to account for such a marked discrepancy in time?"
"It's a big fucking mess," she says, wishing there were any way at all around the words that have to come next, knowing that there's not, so she gets it over with fast. "She killed a priest somewhere down in Connecticut and brought him back to Warwick in the trunk of her car. We've been trying to deal with it."
"Well, now. That is a mess, isn't it. A priest, you say?" And the Bailiff laughs as though Soldier has just told him that this whole situation was the result of a blown tire or a stripped transmission or Odd Willie running over a fire hydrant. "Well, Miss White's certainly getting a bit more enterprising. Indeed. We'll have to see that she's rewarded for all this ambition."
"I'll reward her right now," Soldier says and glances at Saben. "All you got to do is say the word, and I'll reward her good and proper."
"And what about you?" he asks. "Those two are your responsibility. When they fuck up, you fuck up. I've been quite explicit about that, Soldier. So, then, how shall I reward you?"
"Good damn question," she says before she can think better of it.
"What about the earthly remains?" the Bailiff asks, and it takes her a second to realize he means the priest.
"Odd Willie's having some sort of barbecue. I don't think there'll be anything left to worry about when he's done. Listen, what am I supposed to do now? I can be in Woonsocket in half an hour, tops. Hell, I could be there in twenty-five—"
"No," the Bailiff says, and Soldier hears someone laughing in the background. "I'm afraid that won't be necessary. Our friends don't like to wait around. They're busy men. They've rescheduled for Monday at noon, that is, if you think you can make the time. If not—"
"No, no, Monday at noon," Soldier says, grateful that she'll have some time to recover from Rocky Point and the scene at the Dunkin' Donuts before she has to face Woonsocket. "Monday's totally fucking fine. I'm sorry as hell about this mess, Bailiff. But, please, I want her off my crew, okay? Right now, today."
"I've told you, she's your responsibility," the Bailiff replies. "Should I explain once more precisely what that means?"
Soldier looks over at Saben White again, Saben who's still staring out the window at the other cars or the bare trees along the side of the interstate, Saben with her swollen lip and blackening eye, the dried blood at the nape of her neck. And Soldier wonders if it would really turn out that much worse if she simply took out her gun and blew the bitch's brains out. After all, how much hotter can boiling water get?
How hot is summertime in hell? How hot do you think, little Soldier?
"Yeah, fine. My goddamn responsibility. So, what do you want me to do now?" Soldier asks the cell phone.
Saben White smiles, then, as though she's scored some secret victory. Soldier catches the smile out of the corner of her right eye, and it almost doesn't matter what would happen to her, because whatever it is, she has it coming, and it will surely find her sooner or later, anyway. On the radio, the Rolling Stones give way to Procol Harum, "A Whiter Shade of Pale," and Soldier turns the volume up again.
"I want you home as soon as possible," the Bailiff says. "I want you somewhere safe, somewhere visible, so I don't have to worry about you anymore."
"And what about Saben? What do I do with her?"
"Your responsibility," the Bailiff says for the third time. "Now, stop worrying so much about our wayward, trigger-happy Miss White. Perhaps she had good cause in this transgression. Perhaps there is a greater purpose here that we've yet to comprehend, little Soldier." And again, in the background, that high, girlish laughter.
And, likewise, if behind is in front
Then dirt in truth is clean
Soldier swallows, her mouth so dry there's hardly any spit left at all, and the bottle of whiskey is calling her so loudly now that it's a miracle she can hear either the Bailiff or the radio.
"Willie's a good man, so long as he has the right tools," the Bailiff reminds her. "You did well to leave the good father in his capable hands."
"Do I call you when he's—"
"You just go home," the Bailiff says, that steel tone in his voice that says the matter's settled. "See that Miss White walks the straight and narrow. One priest here or there is hardly a sign of the Apocalypse, but I shouldn't want it to become a habit. Was he Catholic, Greek Orthodox, Russian Orthodox, or perhaps Episcopalian?"
"Catholic, I think." And Saben White nods her head yes. "Yeah, Saben says he was Catholic."
"Well, that makes it all a little easier," the Bailiff says. "Some wheels are much easier to grease than others, if you get my drift."
She doesn't, but isn't about to say so.
"And, Soldier, that bottle you bought this morning," the Bailiff continues, "you best leave that where it is for now. You may give it to me when next we meet. I'll consider it a gift, recompense for this inconvenience." And then there's silence, and she folds the phone shut and tosses it onto the dash. Saben flinches at the noise, and Soldier bites down hard on her lower lip because a little taste of her own blood is better than having to explain to the Bailiff how Saben's door just happened to come flying open, spilling her out onto I-95 at sixty-five miles an hour.
"Don't you worry, sweetie," Soldier says, putting a little more pressure on the accelerator. "It looks like someone's got your ass covered."
And then she notices the clouds building on the horizon, sweeping down towards Providence from the northwest; billowing hills of purple-gray, heavy with snow or freezing rain, and she wonders if Willie's found everything he needs to start his fire. And then the clouds make her think of an ocean, and an immense desert at her back, and monsters warring in the sky.
## THREE
## New York
After the cab ride from the train station, after dinner and half an hour's worth of old Tom and Jerry cartoons on Sadie's little television, Emmie sits on the sofa and watches while her stepmother checks all the locks on the front door of her apartment again.
"Just in case," Sadie would say, or, "It never hurts to be careful," if Emmie were to ask her, but she doesn't ask because she knows better. She knows that Sadie paid extra to have all those locks installed on the door, because Deacon said so. He also told Emmie that it wasn't polite to ask questions about Sadie and her weird thing with the locks, but she'd already suspected as much.
"Something very bad happened," Deacon told her, "a long time ago. That's how her arm got hurt. I've told you that, Emmie. Now she needs to feel safe, that's all."
"Does she have OCD?" Emmie asked him.
"What?" her father asked back, furrowing his brow and scowling at her. "Where the hell did you learn about OCD?"
"From an abnormal-psychology textbook in the library at school. I know about schizophrenia, too, and paranoia—"
"Jesus." Deacon sighed and shook his head. "No, Emmie. She doesn't have OCD. She's just...she's just high-strung, that's all. She needs—"
"To feel safe," Emmie said, even though it annoys him when she finishes his sentences for him. Even when he's drunk and forgets what he was going to say.
"Are we gonna play some Scrabble now?" Emmie asks her stepmother.
"If that's what you want to do, then that's what we're going to do," Sadie replies, apparently satisfied with the assortment of dead bolts and chains on the door, because now she crosses the small living room to check the locks on both the windows again. From the sofa, Emmie can see the rusty iron zigzag of the fire escape, and there's golden light from the street lamps down on St. Mark's.
"I've been reading the dictionary a lot lately," Emmie says, "learning new words."
"New words?" Sadie asks, sounding distracted, staring at something down on the street, her good hand resting on the windowsill. "What kind of new words?"
"Well, Q-words, mostly. Especially Q-words without Us in them," Emmie tells her. She wants to ask Sadie what she sees down on the sidewalk, out in the street, but she doesn't, because Deacon would probably tell her that was rude, as well.
"What's wrong with Q-words with Us in them?" her stepmother asks, not turning away from the window.
"Nothing's wrong with them, except that there aren't enough Us in Scrabble, and I never have one when I have a Q and actually need one."
"Oh," Sadie says.
And then curiosity gets the best of Emmie, which happens a lot more than she'd like to admit. "Sadie, what are you looking at down there?" she asks.
"Nothing," Sadie replies, but she still sounds distracted, like she's thinking about everything in the world but playing Scrabble and what a pain in the ass Q-words can be when you don't have a U. "I was looking at a dog, that's all," she adds. Then she turns her back on the windows and smiles at Emmie Silvey. The smile comes out more than a little forced.
"Q-words like qat," Sadie says.
"Yeah, and qintar and qoph, too."
"Qoph? I don't think I know that one," Sadie admits, then glances over her right shoulder at the windows again.
"It's the nineteenth letter in the Hebrew alphabet," Emmie tells her, and her stepmother nods her head and turns back towards her.
"What about qiviut? Did you learn that one?"
"It's got something to do with a musk ox," Emmie says, and reaches for a magazine lying on the coffee table.
"You'll have to do better than that."
"Look it up." Emmie shrugs, opening the magazine, and maybe she doesn't want to play Scrabble, after all. This whole conversation about Q-words has reminded her how dull it can get sometimes, especially when Sadie's not really thinking about the game and it's too easy to win. "It's got something to do with musk oxes."
"Musk oxen," Sadie corrects her.
"Either way, qiviut has a U, and we were talking about Q-words without Us."
"Yes, you're right," Sadie says, cradling her bad arm, and she nods at the Scrabble box on a bookshelf in the corner. "But if it's Scrabble you want, it's Scrabble you'll get, even though we both know you'll only kick my ass again."
"Maybe not if you'd pay better attention and stop thinking about the locks and about dogs down on the street," Emmie says, immediately wishing that she hadn't, wishing she knew some trick for taking words back. She looks up from the magazine lying open on her lap, and Sadie is staring down at the hardwood floor now, still cradling her crippled left arm.
"I'll spot you," Emmie says, hoping maybe that's the right thing to say now that she's screwed everything up. She tosses the magazine onto the floor and gets up off the sofa, stepping quickly around the corners of the coffee table and a tall stack of books on the floor. "Fifty points. I always spot Deacon fifty points."
"Was that his idea or yours?"
"His," Emmie admits. "He wanted me to spot him a hundred, but I said, 'Uh-uh, no damn way am I spotting you a whole hundred points, Mr. Deacon Silvey.'"
"Good for you." Sadie laughs, and Emmie's relieved. She can tell the laughter is more honest than the forced smile was, that the laugh is real, something that Sadie feels instead of just something she wants Emmie to think she feels. Emmie's at the bookshelf now, standing on tiptoe, fishing the Scrabble box down from the shelf where her stepmother keeps all her tarot cards, copies of the books she's written and a couple more that she hasn't finished yet, a soapstone scarab that she brought back from a trip to Egypt, three blue glass bottles, and her Book of Shadows. Emmie knows about the Book of Shadows—that it's filled with Sadie's spells and rituals—though she's never taken it down, never opened it, has never even touched it. She knows that her stepmother is a witch, has known that for as long as she can remember. Not the sort of witch you see on television shows or in the movies, of course, but the sort that's real, not make-believe. Deacon says it's all a load of horseshit and superstitious nonsense, but Sadie told her that she should wait and decide for herself when she's older. Seeing the Book of Shadows, Emmie's reminded of the strange woman on the train and the Seal of Solomon tattooed on her skin. Perhaps, she thinks, the woman on the train was also a witch.
"Have you ever heard of something called the Seal of Solomon?" Emmie asks, managing to wiggle the Scrabble box out from under several heavy books without knocking anything over.
"Sure," Sadie replies and sits down on the sofa. "What do you want to know?"
"Well, mostly I was wondering if it really has an invisible point on it somewhere?"
Her stepmother glances towards the door, and suddenly she has that anxious expression she gets that Emmie knows means she wants to check all the locks again, that she's afraid she might have missed one, maybe that dead bolt up high that's a bit temperamental sometimes.
"That's what someone told me," Emmie says, "that the Seal of Solomon has seven points, even though you can only see six of them."
"Some things are like that," Sadie tells her, but doesn't look away from the door. "Sometimes you can only see part of a thing, even though it seems like it's all right there in front of you. Sometimes things will seem plain as day, and that's not what they are at all."
"So, it's something magick, like your tarot cards?"
"Everything's magick, Emmie, in one way or another. I've told you that before."
"But you know what I mean. Is it some sort of magickal symbol?" Emmie carries the Scrabble set back over to the coffee table, quickly clearing away all the magazines and papers and books cluttering the table, then sits down on the floor, opens the box, and begins to set up the game.
"Who told you about the Seal of Solomon?" Sadie asks her, finally looking away from the apartment door.
"Someone at school," Emmie says, pausing only a moment before deciding that the lie's probably better than telling Sadie about the woman on the train. "Do you want me to keep score?"
"Sure. You're better at it than me."
"Does it protect people?"
"That depends who and what you believe. Some people think it does."
"The Jews?" Emmie asks, choosing a pencil from the assortment of yellow and black stubs inside the box. She picks the sharpest of the bunch, one that even has a ragged pinkish stump of eraser left on one end.
"Some Jews, yes, and some other people, too. King Solomon is said to have had the star engraved on an iron ring. It gave him the power to command demons."
"I'm not sure I believe in demons," Emmie says and shrugs again. She writes her name at the top of a blank page in the notepad that Sadie keeps inside the Scrabble box with everything else. Then she draws a vertical line, dividing the page neatly in two, and she writes Sadie's name on the other side of the line opposite her own.
"It's just a word," Sadie tells her, and then her stepmother selects a wooden tile from the lid of the box where Emmie has laid them all facedown. "And like all words, it can mean a lot of different things—"
"To different people," Emmie says, and she points at the tile her stepmother has drawn from the box. "What did you get?"
"I got a P. Three points. Let's see you beat that."
Emmie draws a tile, but it's only an E, and she shakes her head and sighs. "Crap. I shouldn't have spotted you anything at all," she says, but she doesn't really care about not getting the first turn. At least Sadie seems to have forgotten about the doors and the locks and the windows for the time being.
"That'll teach you not to underestimate your elders," her stepmother says. Emmie draws six more tiles from the lid of the box and sets them up on the rack so that Sadie can't see what they are. They spell out POCSLEI, and she immediately begins rearranging them in her head, searching for something that will equal a halfway decent score.
"Maybe we should have played chess, instead," she says. "Or Chinese checkers."
"No, no. I'm feeling lucky," Sadie says, but her eyes have drifted back to the apartment door. This time Emmie looks, too, and sees that her stepmother forgot to reset the alarm when they came home from dinner. Emmie was talking, telling Sadie about a new video game. "Tonight I'm gonna take you down," Sadie adds, but the tone in her voice tells Emmie that she's no longer thinking about Scrabble.
"Then you can give me a word. A new word. One that I don't already know. That'll make it fair."
Sadie's pale blue eyes dart back to the game board, then nervously up at Emmie, as though she'd been caught doing something she shouldn't have been doing. "It's already fair, you little brat," she says. "I don't have to give you a word. You're too damned smart as it is."
"And it has to be a magickal word," Emmie adds, then begins rearranging her Scrabble tiles. She doesn't like it when Sadie's eyes look that way, like she's frightened and confused and trying too hard to seem like she's neither one. Now Emmie's tiles spell out IEPOCLS, so she starts over. She glances up at Sadie, and sees that her stepmother's staring at the door again, the red door painted almost the same shade as a ripe pomegranate, festooned with dead bolts and chain locks, the black steel rod she keeps wedged firmly between the door and a thick block of wood nailed directly to the apartment floor. The security pad is mounted just to the right of the door, beneath a framed black-and-white photograph of the three of them taken one summer day at Scarborough Beach, back when Emmie was hardly more than a baby. The alarm's ready light is green, but the armed light is glowing a bright, accusatory red.
"And, also, it has to be a real word," Emmie says, looking away from the door and the neglected alarm, back to her stepmother's face. "No made-up words. It has to be a word I could use to beat you."
"Tetragrammaton," Sadie says. "That's a real word."
"Are you sure? I've never heard it before."
"Well, if you'd heard it, Emmie, then it wouldn't be a new word, now, would it?"
"So, what's it mean?" Emmie asks her, sounding as skeptical as she can manage, hoping to distract her stepmother from the door again.
"It's the Hebrew name for god," Sadie says and chews at her lower lip. "It's supposed to have been part of the Seal of Solomon."
"Well, it has too many letters. You should have given me a shorter word. I'll never have all the letters I need to spell that."
"You didn't say it had to be a short word."
"I should have. I didn't think you'd take advantage of a kid. How do you spell it, anyway?"
And then her stepmother opens her mouth, like she's about to tell Emmie how to spell tetragrammaton, like Emmie's not smart enough to figure it out for herself. But then her eyes move rapidly from the door to the twice-locked windows and back to the keypad beneath the photograph again, and Emmie knows that they've both lost this round. For a second or two, it doesn't matter what Deacon's told her, and she almost asks Sadie why she's so afraid, what it was that ruined her hand and scared her so badly that she's still scared all these years later.
"Hold on," her stepmother says. "I'm sorry. This will only take me a second." And Emmie lets the questions dissolve, unasked, on her tongue while Sadie goes to set the burglar alarm.
"Do you still miss her?" Emmie asked Deacon one day in September, the last day of her vacation before third grade began. They were sitting at the kitchen table, eating the sloppy peanut-butter-and-grape-jelly sandwiches that Deacon had made for lunch, and she'd been thinking about school, because she'd been thinking about hardly anything else for a week, when Deacon had said something about the table being too big for only two people. She always sat at the chair beneath the window, and he always sat directly across from her. When Sadie had lived with them, she'd sat in the chair at Emmie's left. There was a stack of telephone books and old newspapers there now.
Deacon stopped chewing and stared at his plate a moment before answering her. "Sometimes worse than others," he said, and took another bite of his sandwich.
"And this is one of the worse times?" she asked, knowing the answer, and her father nodded his head.
"Don't you mind me," Deacon said, after he'd swallowed and washed down the peanut butter and jelly and white bread with a mouthful of milk. "Now and then I just think too much for my own good, that's all."
"Well, you know, it's not like she was my real mother," Emmie said, though, in fact, she thought it was exactly like that. It seemed like the sort of thing Deacon would want to hear, like something that might make him feel better and get him to thinking about something else.
"She did the best she could," Deacon told her, "under the circumstances. She did a hell of a lot better than—" But then he stopped, took another drink of milk, and wiped his mouth on the back of his hand.
"That's not what I meant," Emmie said, realizing that she'd only managed to make things worse. "I just meant, well, I meant—"
"Finish your lunch," her father said and stood up. "I need to get back to the shop," and then he took his saucer and empty glass over to the sink and rinsed them. Emmie sat looking at the uneaten half of her sandwich, wondering if third grade was when you finally learned when to keep your mouth shut.
Emmie lays her last five tiles down in a vertical line ending at the lower right-hand corner of the Scrabble board. R-E-D-C-A-P, and the P lands on "Triple Word Score." She grins triumphantly at her stepmother and taps her fingers loudly on the edge of the coffee table.
"Emmie, that's two words, not one," Sadie says, licking at her lips and already reaching for the paperback Merriam-Webster's dictionary because they've been playing Scrabble long enough that she knows Emmie won't ever just take her word on these things.
"No, it's not. It's only one word. Go ahead. Look it up. 'Redcap.' It's a sort of mushroom."
"Yeah, and it's also a sort of a fairie," Sadie adds. "I know that, but it's spelled as two words."
"It's also what they used to call Little Red Riding Hood," Emmie says, and Sadie stops turning pages and glances up from the dictionary at Emmie, who's still grinning.
"Go ahead. Look it up. But you're wasting your time. You'll just see that I'm right."
"You don't have to be a snot about it," Sadie tells her.
"Read me what it says. 'A poisonous species of toadstool.' I bet that's exactly what it says."
"Why don't you look it up," Sadie replies and hands her the dictionary. "I'd rather we wasted your time than mine."
"Fine," and Emmie takes the dictionary, which is already turned to the RE's, and she ticks off the words one after another—redbreast, redbrick, redbud, redbug—and there it is, just like she knew it would be, redcap, one word, two syllables. "Redcap," she says. "One word."
Sadie takes the dictionary back and frowns at the page as she reads it. "Okay, but you're wrong about the definition," she tells Emmie, then reads it aloud. "'A baggage porter, as at a railroad station.'"
"I can still use it," Emmie says, adding up her score. "And I'm not wrong, because it is too a kind of mushroom. That dictionary just isn't very good. Maybe you should get a new one, one that knows about mushrooms and fairies."
"Yeah, maybe I should," and Sadie closes the dictionary and sets it aside. "Emmie, why even bother adding the score. We both know you won."
"It's still important. It's still important to know exactly how it turned out."
"Fine, but I have to go to the bathroom right this minute, or I'm gonna pop."
"Yeah, your eyes are starting to turn kind of yellow. Soon they'll be just like mine."
"Your eyes aren't yellow," Sadie says, standing up. "They're amber."
"Wrong," Emmie says. "They're yellow as butter. They're yellow as sunflowers." It's an old argument, one that Emmie's never really understood the reason for. Her eyes are not amber; she's seen amber at the museum, polished chunks of it with ants and ticks and flies trapped inside for millions and millions of years, and that's not the color of her eyes at all. And they aren't golden brown, either, or champagne, or any of the other things that Sadie has tried to call them so that she doesn't have to call them yellow, which is what they actually are. "Yel-low," Emmie says, and then realizes that she's made a mistake in her arithmetic and begins rubbing hard at the notepad with the last stingy scrap of the pencil stub's eraser.
"Whatever. I'm going to pee now," Sadie sighs, sounding a little annoyed, and Emmie watches as she tries to make it past the front door without stopping to check the locks.
And later on, after Emmie has beaten Sadie again and the Scrabble set has finally been put away for the night, after a snack of Double Stuf Oreos and A&W cream soda, after Emmie's brushed her teeth and flossed and washed her hands and face and changed into one of Sadie's T-shirts that she likes to sleep in whenever she visits—after all these things, her stepmother sits on the edge of Emmie's bed in the narrow space between the kitchenette and bathroom. Emmie's bedroom, whenever she comes to Manhattan. It's hardly wider than a large closet, but Emmie likes the closeness. It makes the tiny room feel safe, makes it cozy, and sometimes she thinks she prefers it to her big room upstairs on Angell Street in Providence. A room this small has fewer shadows to wonder about when the lights are out, and there's still a place for her bed and a small chest of drawers, two red plastic milk crates stacked one atop the other to make a combination bookshelf and toy box. There are a couple of posters on the wall, Hello Kitty and a Japanese girl band, and Sadie has painted golden stars on the midnight-blue ceiling, one for each of her stepdaughter's birthdays.
"Do you want a story?" Sadie asks, and Emmie nods yes, because she doesn't want to be sleepy just yet, wants to lie there awake beneath the golden stars listening to Sadie read to her; Sadie reads aloud almost as good as Deacon.
"Read me the end of The Voyage of the 'Dawn Treader,'" Emmie says. "Read me 'The Very End of the World.'"
"Are you sure? I've read you that one at least a dozen times already."
"Not a dozen. Not that many times."
"At least that many times," Sadie says, but she's already taking the paperback copy of The Voyage of the "Dawn Treader" from its place in the topmost milk crate.
"Right near the end," Emmie tells her, "after Reepicheep has gone, when Aslan makes a hole in the sky. I think it's on page two forty-seven."
"You've memorized the page numbers?" Sadie asks, incredulous, and Emmie smiles a hesitant, flustered smile, wondering if maybe she should say it was just a lucky guess.
"Only a few of them," she says instead. "Only for the parts I like the most."
Sadie shakes her head and flips quickly through the dog-eared pages until she comes to page two forty-seven. When her stepmother begins to read, Emmie fixes on the largest of the painted stars, the one that stands for her birthday before last. It has seven points, but none of them are invisible.
"'Oh, Aslan,' said Lucy. 'Will you tell us how to get to your country from our world?'
"'I shall be telling you all the time,' said Aslan. 'But I will not tell you how long or short the way will be; only that it lies across a river. But do not fear that, for I am the great Bridge Builder. And now come; I will open the door in the sky and send you to your own land.'"
"Do you think it was like a black hole?" Emmie asks, but doesn't take her eyes off the star on the ceiling.
"No," Sadie replies. "I think you mean a wormhole. I think maybe Aslan's hole between worlds was more like a wormhole than a black hole. Hardly anything ever escapes a black hole."
"I know that, Sadie," Emmie says, annoyed at being corrected and embarrassed that she didn't get it right in the first place. "I forgot, but you know I know that. I read a book by Stephen Hawking."
"Yeah, I've read that book, too."
"So," Emmie continued, "Aslan made a wormhole for Lucy and Edmund and Eustace."
"I doubt that's what C. S. Lewis meant, but I suppose you could think of it that way."
"Right. A bridge between worlds, a shortcut, and there must have been another one—"
"Do you want me to read any more of this?" Sadie asks, and she glances at the alarm clock on the chest of drawers. "It's getting late. We have a lot to do tomorrow."
"Read just a little more," Emmie says and yawns without covering her mouth.
"Okay, but only a little bit more," Sadie says and turns the page. "'Child,' said Aslan, 'do you really need to know that? Come, I am opening the door in the sky.' Then all in one moment there was a rend—"
"The wormhole," Emmie says, and then she yawns again.
"If that's the way you want to think of it. Yeah, the wormhole."
"They're very unstable things," Emmie says and squints at the birthday star. "That's what Stephen Hawking said. You couldn't just travel through one like that. Even in a spaceship, the gravitational forces would tear you to spaghetti if you tried."
"Emmie, you know this is a fairy tale. Maybe you're getting too old for fairy tales. Maybe science fiction would be better."
"I was just thinking about it, that's all. Go ahead. Read me some more, please."
Sadie blinks and rubs her eyes, and suddenly Emmie misses her father, misses her own room in her own house, and she almost asks Sadie if they can call Deacon, because he's always up late, but Sadie's already started reading the book again.
"Then all in one moment there was a rending of the blue wall (like a curtain being torn) and a terrible white light from beyond the sky, and the feel of Aslan's mane and a Lion's kiss on their foreheads and then—the back bedroom in Aunt Alberta's home in Cambridge."
"That's enough," Emmie says, deciding it's probably best not to call Deacon, even if he is still awake; the wave of homesickness has begun to pass, and she looks at the star again. Sadie closes the book and returns it to its place in the red milk crate.
"Stephen Hawking might be wrong," Emmie says.
"What do you mean?"
"I mean, some scientists think there might be ways to travel through wormholes without being torn apart. I think they might be right."
"Well, then, where do you think the wormholes would lead?" Sadie asks her. "Would they take us to Narnia?"
"Narnia's not a real place," Emmie sighs and rolls over, turning her back on her stepmother and the golden star, turning to face the wall because she knows that if she doesn't fall asleep soon she'll start missing Deacon again. "But they might take us to other places."
"Good places or bad places?"
"That remains to be seen," Emmie says, though she's thinking that her stepmother's old enough to know that places are neither good nor bad. That places are only places, and it's people that make them seem good or bad. "Leave the hall light on, okay?"
"Sure," says Sadie, tucking her in and kissing her on the top of her head. "Now, you'd better get some sleep, pumpkin. We'll go see the dinosaurs tomorrow."
"Night-night," Emmie whispers, and her stepmother switches off the reading lamp and leaves, shutting the bedroom door only halfway. Emmie stares at the wall, imagining a swirling, dinner plate–sized hole in it that might open out into their house on Angell Street. She imagines Deacon peering into the hole to be sure that she's all right and falls asleep to the sound of Sadie's footsteps.
"No," Emmie tells the rat in pirate boots. "I always know when I'm dreaming. It's easy to tell the difference." And when the rat glares skeptically back at her, she explains, "In my dreams, there are no limits."
"No limits to what?" the rat asks her, still glowering, and then it takes a sip of the peach-flavored soda it's been nursing since Emmie sat down at the kitchen table. Deacon's busy somewhere else in the house, and he doesn't know that she's having a conversation about dreams with a rat named Reepicheep. If he did know, he probably wouldn't care, but he might tell her to stop giving away sodas to talking rats, that those things cost money, and money doesn't grow on trees.
"See, that's what I mean," she tells Reepicheep. "Money doesn't grow on trees. So, whenever I see some money growing on a tree, I know that I'm dreaming, because in my dreams there are no limits to what's real and what's not."
The rat shakes its head and scratches at its chin, which is slightly damp and matted with the peach soda that keeps dribbling out of its mouth. "That seems awfully convenient, if you ask me," Reepicheep says.
"I didn't ask you."
"This stuff tastes like cleaning fluid," the rat shudders and stares into the bottle of soda. "Are you trying to poison me or something?"
Emmie's getting bored with the rat, who isn't nearly as interesting to talk to as she thought he might be when she offered him something to drink. She turns and looks out the kitchen window at the house next door and discovers that there's already someone at a window over there watching her. It might be the woman from the train, the woman who got off at the Old Saybrook Station. She waves, but whoever it is, they don't wave back.
"Have you ever been haunted by the ghost of a poisoned rat?" Reepicheep asks her. "Let me just tell you, it's not a pleasant affair."
"Go away now," Emmie says, and the kitchen gradually melts away like snow on a warm winter day, taking the talking rat with it.
"Six of one, half dozen of the other," she says, because it's something that Deacon says whenever he doesn't seem to care one way or another. He says it's something that people from Alabama say all the time. And the graveyard where she's standing now really isn't that much of an improvement over the kitchen and the talking rat and being watched from the house next door, so she says it again—Six of one, half dozen of the other. It's not Swan Point or the Old North Burial Ground or any of the other graveyards where she and Deacon sometimes go for long walks. All the headstones are jagged slabs of charcoal-or rust-colored slate, broken and covered with patches of lichen and moss. Hardly any of the names and dates carved into them are still legible, and that always makes Emmie sad, knowing that someone's dead and buried and so completely forgotten that even their tombstone doesn't know his or her name anymore.
Here's a whole cemetery of forgotten people, she thinks, and sits down in the grass beneath a maple tree, glad that at least it's summer in the dream instead of February.
"I was hoping you'd show up today," her mother says—not Sadie but her real mother, the woman who died the day that she was born—and Emmie looks up to see the tall woman standing over her. There's a rock hammer on her belt, just like the paleontologists Emmie's seen on television. "It's been a while," the woman says.
"I've been busy," Emmie tells her, wishing that the sun weren't so bright right there behind her mother's head, so that she could actually see her face this time. She knows it from photographs, but it's always hidden in the dreams, and sometimes that makes Emmie worry that maybe the woman in the dreams isn't her mother after all.
"You're a busy girl," her mother says.
"I don't believe in ghosts," Emmie says. "I know you're only a dream."
"It doesn't make much difference either way," the woman replies and turns away, the sunlight pouring fiery white about the eclipse of her head and shoulders. "A ghost or a dream, six of one, half dozen of the other."
"It matters," Emmie insists. "It's important to know the difference."
"There's something coming," the woman tells her. "You need to be ready when it gets here. You're a smart girl, aren't you, Emmie? You know the difference between dreams and what's real?"
"Yeah. I was just telling the rat—"
"Then I need you to try just a little bit harder. I need you to see who I am, not who you think I am."
"It's a dream," Emmie protests, and the woman turns back towards her. She doesn't seem so tall now, and the tool belt with the rock hammer's gone. There's a six-pointed star tattooed on the woman's left hand. "You're not my mother," Emmie says, a little disappointed, a little angry even though the dream woman never claimed to be her mother. She squints into the sun, trying harder to get a better look at the woman's face. "You're the woman from the train."
"You're a damned smart girl. But sometimes you talk when you should be listening."
"It's my dream," Emmie says.
"You also need to learn not to make quite so many assumptions."
"What's that supposed to mean?" And Emmie stands up, dusting bits of grass off her jeans, but the woman who's not her mother has already begun to fade away, just like the kitchen and the talking rat. Emmie tries to concentrate and make her stay, but the graveyard is slippery and has already become somewhere else entirely, the sun traded for shadows and mist in the blink of a dreaming eye. An attic, she thinks, because the ceiling's so very low, the vaulted house ribs of huge and rough-hewn crossbeams, cobwebs draped all about like a cartoon haunted house, and the only light is coming from a candlestick that the girl holding her hand is carrying. The girl, who's both older and taller than Emmie, is dragging her through the attic, leading her quickly from one somewhere to somewhere else, and Emmie stops and jerks her hand free.
"Who are you?" she demands. "And where are we going?"
"We can't very well stay here," the older girl says anxiously, impatiently. "They'll hear and come up to check. The clocks are ticking, Emma Jean. They'll find you out."
By the flickering candle, Emmie can see that the girl's skin is brown, skin like almonds or milk stirred into coffee, and her hair's black and cut into a bob. Emmie wonders if the girl's Hispanic or maybe part Narragansett, maybe a little of both. "Who? Who will find me out?" she asks the girl, who's beginning to look more than anxious and impatient—she's beginning to look scared. She tries to take Emmie's hand again, but Emmie snatches it away. "No, answer my question. Who's gonna find out I'm here?"
"It might be your dream, Emma Jean Silvey, but it might be someone else's, too. Did you never stop to consider that?"
"No," Emmie tells her. "That's stupid. This is my dream, and it isn't anybody else's."
The girl looks back the way they've come, and Emmie looks, too, but there's nothing back there except darkness and shadows, spiderwebs and dust. "Who are you so afraid of?" Emmie asks the girl.
"She told you that something's coming, and you need to listen to her. She told you about the horses, too, didn't she? You've already forgotten that, I'll wager?"
"She didn't say anything about any—" Emmie begins, but then she remembers the woman on the train—Saben White, that was her name, at least the name she said was her name—remembers her saying that Emmie should stay away from horses. "I'm in New York City at my stepmom's," Emmie says. "So I really don't think I have to worry too much about horses."
The candlelight glitters in the girl's eyes, eyes that are almost black, eyes that get Emmie to thinking about holes in the sky, holes punched in worlds by lion gods to send children home. The girl looks more frightened than Emmie has ever seen anyone look before.
"You need to calm down," she tells the girl. "You're gonna have a cardiac arrest or a stroke or a panic attack or something if you don't. It's just a dream. I promise. You don't even exist."
"You can hear them, can't you?" the girl asks. "You can hear the clocks ticking?"
"I don't hear anything," Emmie says, but then she stops and listens, and she does hear clocks, dozens of clocks, maybe more, but she can't tell where the sounds are coming from. "Yeah, so what? I can hear clocks ticking."
The girl groans, the same sort of noise that her father makes whenever he's tired of arguing with Emmie about taking a bath or doing her homework or cleaning her room. The end-of-his-rope noise, Sadie calls it. Do it now, or you'll wish you had. That sort of noise.
"You can get lost in dreams," the girl says and motions at the darkness with her candle. "If you're not careful, you can spend forever wandering about in dreams, and it hardly matters if they're yours or someone else's. Maybe this is your dream, Emma Jean, but it's mine, too, and maybe it's the hounds', and maybe it's even old Mother Hydra's, so you'd best hush up a minute and stop thinking things aren't connected just because you can't see the—"
"—shortcuts," Emmie says.
"That's not what I was going to say, but it's near enough. Those clocks shouldn't be ticking. You shouldn't be here. You're getting too far ahead—"
And then the brown-skinned girl's gone, and Emmie's alone in the sun-drenched cemetery again, and there's an empty soda bottle half-buried in the ground in front of every single one of the slate gravestones. There are black ants on the bottles—ants going in, ants coming out. Reepicheep's leather pirate boots are draped across one of the markers, and when she looks again at the lush green grass sprouting from the earth, Emmie can see that the ground has begun to leak something oily and red, not blood, no, but that's the first thing it makes her think of. The air crackles and smells like rain and a dead raccoon she once found out behind their house. There are rough voices drifting across the cemetery, guttural animal sounds that want to be words but haven't quite figured out the trick of it, and she tries not to hear what they're saying.
"She had a plan, your mother did," the leaves of the maple tree rustle in the voice of the girl from the attic. "And now I fear it isn't going so well for her. She made mistakes, and now it isn't going well at all."
"My mother's dead," Emmie tells the girl or the old tree or whomever she's talking to; she's angry now and getting scared, and when she gazes up into the limbs of the old maple, there are eyes there, gazing back down at her, eyes and grinning Cheshire cat jaws with sharp ivory teeth. "My mother's dead," Emmie says. "She's dead and buried, and she isn't planning anything."
"Six of one," the tree whispers. "You said so yourself," and then something slips suddenly down from the branches, slithering snakelike along the trunk of the tree, something almost the same color of red as the stuff oozing from the ground.
"They must have heard the clocks," the girl from the attic says. "I told you they might. They have keen ears, and they're always listening. You know the way back. You'd best start running."
The red thing in the tree laughs and wriggles and vomits a slimy clump of fur and the half-digested bones of a very large rat onto the grass at her feet.
"Now would be better than later," the girl from the attic says.
And there's an ugly tearing sound—a rending, Emmie thinks—as she dreams a wormhole into the fabric of the stinking summer day, a shimmering hole filled with blinding white light. As the red thing from the tree coils about the roots and opens its mouth wide so she can see exactly what it's using for teeth, Emmie tumbles through.
By the time that Emmie's seen all the dinosaurs, saurischian and ornithischian, and all of the fossil mammals, it's almost noon. Sadie buys her a hot dog, a bag of potato chips, and a Snapple from a silver cart parked outside near the bronze statue of Theodore Roosevelt on horseback, flanked by an Indian on one side and a black man on the other. They sit on a granite bench beneath the tall Ionic columns framing the archway leading back into the museum; the sun's warm today, and the wind's not so bad that they can't have lunch outdoors. Emmie tries to remember not to talk while she's chewing, but it's hard, because her head's so full of all the things she's seen in the last few hours.
"Hunter's meeting us in a bit," Sadie says. "You won't mind, will you?"
"No," Emmie manages around a mustardy bite of hog dog. She likes Hunter Fontana, likes her long salt-and-pepper dreadlocks, the spicy-sweet clove smell of the Indonesian cigarettes she sometimes smokes, and the fact that she knows lots of good stories that Emmie's never heard before. Hunter's a witch, too, and a writer, and Deacon says that she's a lesbian. Emmie chews her hot dog, swallows, washes it all down with raspberry-flavored Snapple, then leans back against the bas-relief sculpture of a pair of bighorn sheep.
"I didn't think you would," Sadie says and smiles. She points at what's left of Emmie's hot dog. "Good?" she asks.
"Yep," Emmie says. "I could eat another one."
"We'll see," Sadie tells her and checks her wristwatch.
"I'm sure that I'll still be hungry when I'm finished with this one."
"How about you finish it and find out."
"What do you think I'm doing?" and Emmie takes another big bite. Deacon fixes hot dogs whenever she wants them, because he likes them too, but they're never as good as the ones Sadie buys her outside the Museum of Natural History. Someone at school told her that the Mafia owns all the hot-dog stands in Manhattan, but she doesn't know if she believes that. It seems to her the Mafia would have better things to do than sell hot dogs.
"You can tell Hunter what you were telling me about the prehistoric horses," Sadie says, and Emmie shrugs and swallows.
"I'm not sure she'd be interested."
"Of course she would. You should tell her." Sadie didn't get a hot dog, just a Snapple and a bag of chips. She sips at her bottle of tea and watches the traffic moving up and down Central Park West, or she's watching the brown-and-gray edge of the park itself. Emmie can't be sure which without asking. She likes the park best in summer, when it's green and warm and there's Shakespeare and picnics; in winter, there's something hard and skeletal about it, just like the rest of the city. She finishes her hot dog and decides that she's probably full, after all.
"Tell her about the Protohippius."
"Protohippus," Emmie says, correcting her. "Hippus means 'horse' in Latin. Latin or Greek, I can't remember which."
"Well, so tell her about the Protohippus. Everything that you were telling me."
There's a sudden gust of icy wind then, sweeping down the street to remind them of February, blowing between the high buildings and the park. Emmie turns her face away from it, but it tugs persistently at her clothes and bites the tips of her fingers, snatches away her empty hot-dog wrapper and sends it scuttling like a pale, relish-stained insect across the sidewalk.
"Whoa," Sadie says and laughs. "Maybe we'd better hold on, pumpkin. Another one of those might blow us all the way to New Jersey."
Emmie doesn't laugh, because something about the wind has reminded her of the bad dream—the girl in the attic, the clocks, the old cemetery, and the red thing in the maple tree. She hasn't told her stepmother about the dream, because she'd already forgotten it, mostly, by morning. And besides, Sadie always wants to make more out of dreams than she should. Emmie knows what dreams are, what she believes that they are, and they aren't portents or visions or anything like that. "Brain garbage," Deacon calls them, which seems about right to her.
"I don't want to litter," she says, "even if it's really the wind's fault," and Emmie chases down the hot-dog wrapper before it ends up in the street or across the street in the park. She catches it where it's snagged against the base of a flagpole, then glances back at Sadie, who points at a nearby garbage can.
She told you about the horses, too, didn't she? the dark-skinned girl in the dream asked her. You've already forgotten that, I'll wager? And she had, and she'd forgotten them again when she and Sadie were standing in front of all those fossil horse skeletons—Hyracotherium, Mesohippus, Merychippus, and the perfectly articulated skeleton of a Protohippus, not mounted on welded steel rods like the others but still encased in rock, half-exposed in right profile. Just behind the skeleton's rib cage, underneath its pelvis, were the skull and tiny bones of an unborn foal. Twelve million years ago, the mother horse had died giving birth.
She told you about the horses, too, didn't she?
I know this isn't going to make much sense, said the woman on the train, but you need to stay away from horses.
But Emmie hadn't stayed away from them, had she? And nothing bad had happened. She imagined telling it all to Deacon, imagined him rubbing his stubbly cheeks and nodding his head like it wasn't anything he hadn't heard a thousand times before. "Of course nothing bad happened," he would tell her, "because the woman on the train was crazy—and what have I told you about talking to strangers?—and the dream was just a dream. Brain garbage." And then he'd tap on his forehead with one finger and tell her he was late for work but to try to stop worrying herself about horses and nightmares.
"Hello there," someone says, and when Emmie looks up, Hunter Fontana is standing over her, smiling. Hunter's a lot older than Sadie, and sometimes Emmie pretends that Hunter is her grandmother.
"What's that?" Hunter asks, and she points at the wad of paper in Emmie's hand.
"The stupid wind," Emmie replies. "I'm not a litterbug."
"Where's your stepmom?"
"She's right over there," Emmie tells her and jabs a thumb towards the spot where Sadie's sitting with the bighorn sheep. "We were waiting on you."
"Well, now I'm here. Let's put that in the trash and go see Sadie, shall we?"
Emmie smiles, her uneasiness from the wind already beginning to fade away. "We shall," she says. "She wants me to tell you about the museum, about the fossil horses."
"Eocene, Oligocene, Miocene, or Pliocene?" Hunter asks, following Emmie to the trash. It never surprises Emmie what Hunter seems to know, since she seems to know something about almost everything.
"Miocene," Emmie tells her. "I told her you probably wouldn't want to hear it. I'm sure you know about it already."
"Maybe," Hunter says. "You never can tell." And then Emmie drops the wrapper into the trash can.
"What about the Seal of Solomon?" Emmie asks, the words thought up, strung together, and spilling out of her mouth before she's even sure she wants to start talking about that again. Hunter seems only a little surprised.
"I thought we were talking about fossil horses?"
"Never mind," Emmie says, peering past the rim of the can into the trash, all the bits and pieces that people have thrown away, the bottles and drink cans and paper bags, discarded plastic and glass, aluminum and cardboard. And her hot-dog wrapper, crumpled into a ball and lying on top of everything else. She wrinkles her nose; something in there smells dead, and she wonders if maybe there's a dead pigeon wedged way down at the bottom, or a squirrel, or a rat.
"If you say so," Hunter tells her and takes Emmie's right hand, then leads her back to where Sadie's sitting, finishing her Snapple. "You know she's a weird kid, right?" Hunter asks her stepmother, and Sadie smiles.
"Takes after her dad," she says.
Emmie wishes that they wouldn't talk about her like she wasn't there, and that they wouldn't call her weird, even though she knows it's only a joke. Lesbian witches have no damn business calling anyone else weird, she thinks and turns loose of Hunter's hand.
"Is that a fact?" Hunter asks, sitting down on the bench next to Sadie. "When did Deacon Silvey take an interest in the Kabbalah?"
"Oh, I see. So I guess she's on about the Seal of Solomon again," Sadie says. "I don't know what's up with that. She—"
"You know, I'm standing right here," Emmie says, interrupting her stepmother. "You could just ask me what's up with that."
"Did she tell you about the horses?" Sadie asks Hunter.
"She started to, I think. But then the other thing came up, and she didn't finish."
"You're still doing it," Emmie says, reaching for her backpack lying there on the bench. "It's rude, Sadie. How the heck am I ever going to learn how to act when all the adults I know are rude?"
"How about we go for a walk," Hunter suggests and points at the park. "It's such a nice day, we should take a walk in the park. How's that sound, Emmie?"
"We haven't seen the blue whale yet," Emmie protests, "or the meteorites, either."
"We could go for a walk, then come back to the museum later this afternoon," Sadie says. "Then we can still see the blue whale and the meteorites."
"And the Hall of Vertebrate Origins. And the planetarium," Emmie says.
"Right. And the planetarium."
"Sure," Emmie nods without a drop of enthusiasm. "Whatever."
She's thinking about the dream again, the girl with the candle, the maple tree, and the rat sipping peach soda, and she doesn't want to walk in the park, just wants to go back inside the museum so maybe something interesting will distract her. They haven't even seen the Hall of Minerals or the Hall of Gems. But now Hunter and Sadie are talking about something else, laughing about some secret grown-up thing, and Emmie looks over her shoulder at the busy street, past the people and taxis and a long yellow school bus, to the place where asphalt and concrete give way to a low stone wall that seems to be holding back the grass and trees at the western edge of the park. It's not the same as in the summer, she thinks. There's a whole hungry wilderness in there, hungry and waiting for me. And then she remembers a book she read once about the history of Manhattan, how the island was saltmarshes and forest hundreds of years ago, how the land was drained by Dutch settlers and the Indians driven away so a city could be built here. How Central Park isn't really a wild place at all, but a garden designed by men who wanted to tame nature. Maybe, she thinks, it isn't tame at all. Not really. All those thousands of crooked, bare limbs clawing at the bright blue sky, the poplars and willows, oaks and maples, and suddenly Emmie Silvey is afraid of the park for the first time in her life.
"What about a carriage ride?" Hunter asks, and Emmie glances over at the statue of Roosevelt on his great bronze horse. "I bet that's something you've never done before. How about it, Sadie? My treat."
"What do you think, Emmie?" Sadie asks. "You want to ride in a carriage? Might be fun."
The statue is black and green with verdigris, and Theodore Roosevelt looks more like a Roman god of war riding off to battle than an American president. His head and shoulders are spattered with white smears of bird shit. I know this isn't going to make much sense, but you need to stay away from horses. His bronze mount is an enormous metal beast; no horse was ever that large, not even in prehistoric times. Emmie looks back at her stepmother and Hunter Fontana, trying to think of some way to explain, some way to get them to take her back inside immediately.
"Look, there's one now," Hunter says and points to a hansom cab making its way towards them, drawn by a muscular but weary-looking black mare, her iron hooves clip-clopping loudly on the pavement. The hack is wearing a gray top hat, and there are no riders in the carriage. "Just let me get his attention," Hunter says.
"We haven't seen the blue whale," Emmie says again, because she can think of nothing else to say.
"We will," Sadie assures her. "I promise. But Emmie, this is something you've never done before. You've seen that old whale lots of times."
"Three times," Emmie tells her. "I've only seen it three times." Her mouth has gone very dry, and the cold wind has started blowing again. An empty Coke can rattles past. The carriage has pulled over in front of the museum, parking in front of the statue of Theodore Roosevelt, and the driver pulls back on the mare's leather reins. Wooden wheels lacquered white as snow, white wheels with crimson hubs, and Hunter's already talking with the hack, taking money from her wallet.
"I don't want to go," Emmie whispers. "Please, Sadie, I don't want to do this."
Her stepmother looks confused, her smile fading slowly away to some indistinct concern. "Why not, Emmie?" she asks. "You'll probably love it."
"I can't tell you. I just don't want to do it. Tell Hunter I don't want to do it, please. Make her stop."
"Oh, baby," Sadie says. "No, no, it's okay," and Emmie's crying now, and that's stupid and makes it all even worse. She's too old to be acting this way. She imagines the wind will freeze her tears, and tiny beads of ice will shatter on the sidewalk at her feet, imagines that they will make sounds like wind chimes when they break.
"He can't wait forever," Hunter shouts, and the mare snorts and shakes its head, pulling at the reins. "We'll just make a loop, once around the park, that's all."
"She doesn't want to go," Sadie calls out to Hunter.
"Why not?" Hunter shouts back. "I've already paid the man. Come on, Sadie."
"I can't," Emmie whispers, leaning closer to her stepmother. "I can't do it, Sadie. I can't go. If I do it, something bad's going to happen."
"That's silly," Sadie tells her. "Why would you think—"
"Hurry up!" Hunter yells.
"Just tell him we've changed our minds," Sadie shouts back at her, and she puts her good arm around Emmie and hugs her tightly. "I think she's tired. Thanks, but maybe some other time."
Emmie buries her face in her stepmother's lap, hiding from the wind and the hungry trees and Theodore Roosevelt. "I'm sorry," she sobs. "I'm sorry, Sadie. I'm sorry—"
"Pumpkin, there's nothing for you to be sorry about," Sadie says and begins stroking her hair. "Hunter thought it might be fun, that's all. No one's upset with you."
But then Emmie hears the squeal of tires, rubber burning itself to smoke against the blacktop, and she looks up in time to see the taxi swerve across the yellow dividing line. The mare sees it, too, and tries to bolt, but there's no time, too much weight hitched to her and nowhere to run, and a moment later the horse screams as the car collides with the hansom cab. The taxi's horn, the snap of wood, the wetter snap of splintering bone, the abrupt thud of metal against meat, and the mare goes down in a heap of flesh and tack on the sidewalk, just missing Hunter. Emmie knows from the emptiness in its lolling, dark eyes and the pool of blood spreading thickly across the concrete that it's dead. Sadie screams, her voice not so different from the dead horse's, and Emmie looks away.
Almost forty-five minutes later, and Emmie's sitting on her bed in Sadie's apartment, trying not to hear the things that her stepmother and Hunter Fontana are saying. They went into Sadie's bedroom and shut the door, but Emmie can still hear them. The apartment's too small for privacy.
"—would have been dead."
"Hunter, you don't know that."
"The hell I don't. If the two of you had come when I called—"
—we'd be dead.
They weren't, but the horse was, its ribs caved in and its neck broken, and the man who drove the hansom cab hauled off to the hospital in an ambulance.
"It was a fucking coincidence," Sadie hisses.
"You don't believe that," Hunter says. "You don't believe that for a moment, and we both know it."
"She's just a child."
"I want to go home now," Emmie says, speaking to no one in particular because there's no one there to hear. She's said it several times since they got back to the apartment, and that's why Sadie's packing. "I want to go home now."
"We would be dead," Hunter says again.
But Emmie's pretty sure she's missing the point. If Deacon were here, he'd say she couldn't smell the turds for the sewage or something like that. He always knows what to say when people are missing the point.
She told you that something's coming, and you need to listen to her. She told you about the horses, too....
Just while you're in New York City this time, stay away from horses, okay? I think it's very important.
"She's missing the point," Emmie says, pretending that Sadie's standing there and can hear her. "Hunter's missing the point." Emmie's left shoe has come untied, and she starts to tie it, then stops and just stares at it instead.
"If I'd listened..." Emmie begins, but maybe it's not something she should say aloud, so she finishes the sentence in her head:...the horse would still be alive. If we hadn't gone to the museum, where I knew there would be horses, Hunter never would have made the carriage stop.
"Don't waste your life on regret," the dark-skinned girl from the attic says. Emmie looks up from her untied sneaker, and the girl is standing at the foot of the bed, watching her. "No good ever comes of that."
"If this is only a dream," Emmie tells her, "and I wake up, then the horse will still be alive."
"Six of one," the girl says.
"And I can tell Sadie I don't feel like going to the museum, that I'd rather go to Chinatown instead."
"Half dozen of the other."
"If I'm only dreaming, none of it has to happen."
"Unless it already has," the girl suggests unhelpfully.
"The hell with you," Emmie says, and it feels so good that she says it a second time. "The hell with you."
"You have a very foul mouth on you, Emma Jean Silvey. And, just so you know, you won't ever profit from thinking that second chances are that easy."
"You're not real," Emmie growls at the girl, "so just shut up and leave me alone. And nobody calls me Emma Jean. If you were real, you'd know that."
She can hear Sadie crying now.
"Has anything like this ever happened before?" Hunter asks. "Sadie, the kid saved our goddamn lives today."
"You stop saying that," Sadie snaps back at her. "I don't want to fucking hear it again."
"What the hell are you so afraid of?" Hunter demands.
"You don't fucking know. You don't have any idea what you're talking about."
"If I'm dreaming, then I can change it all," Emmie tells the girl standing at the foot of the bed. The girl's wearing a black dress with a stiff white collar, like something a pilgrim girl might have worn to the first Thanksgiving dinner. She's also wearing black-and-white-striped stockings and old-fashioned black patent leather boots. She frowns and takes a purple Magic Marker from a pocket in her dress and hands it to Emmie.
"Show me," the dark-skinned girl says.
"What do you mean?" Emmie asks, staring at the marker. The label's mostly scraped off and the cap's missing.
"Show me, changeling. You know how it goes. 'I am the great Bridge Builder. And now come; I will open the door in the sky and send you to your own land.'"
"You're a loony bird," Emmie says. She can smell the Magic Marker, and the purple ink smells like African violets and the ocean before a storm, which only goes to prove it's all a dream.
"Show me," the girl says a third time. "Build me a bridge, Emmie. I have long desired one."
And because this is only a dream, and almost anything's better than listening to Sadie and Hunter yelling at each other, yelling the way that Deacon and Sadie did before Sadie left Providence for good, Emmie turns and draws a circle on the wall with the purple marker. The felt tip squeaks loudly against the plaster. When she's done, it's not a bridge or a wormhole or anything else but a big purple circle drawn on Sadie's wall.
"See?" the girl from the attic says. "You're not dreaming," and when Emmie turns to tell her none of this proves a thing one way or the other, she discovers that the girl's gone, and she's alone again.
## FOUR
## Woonsocket
"It wasn't like that," Odd Willie mumbles indignantly, and lights another cigarette off the butt of the one before, crumpling the empty Winston pack and tossing it out the car window. "That sour old cunt, she needed a lesson in the finer fucking points of minding her own business, you know."
"Yeah, well, be that as it may," Soldier says, "don't think you're gonna make a habit out of shit like that." She squints through her sunglasses and the windshield of the Dodge at the cloudy afternoon, the sky still threatening snow. The Bailiff told her to keep driving the old Intrepid for now, even after what Odd Willie pulled at the Dunkin' Donuts and then the scene out at Rocky Point, and that was only one of the dozen or so unwelcome surprises in the last twenty-four hours that have Soldier wondering if maybe she's finally running out of luck, or if maybe the Bailiff's just running out of patience.
"You're wound way too tight, Odd Willie," she says. "You're like the center of a goddamn golf ball."
On the radio, Sam the Sham and the Pharaohs are singing "Lil' Red Riding Hood," and Soldier bumps up the volume a notch. The Dodge rushes past the turnoff for the West Wrentham Road, and the final mile or so of I-22 before Woonsocket stretches out before her, gray and crooked and almost as narrow as her chances of coming out of this thing in one piece. Her eyes ache, and she could have used a few more hours sleep.
"I just can't stand people fucking staring at me," Odd Willie says, and then to Saben White, sitting alone in the backseat, "You make me sound like some kind of goddamn irrational lunatic."
"That's the way Soldier told it to me," Saben replies. "And you're not exactly known for your judicious conduct, Mr. Lothrop." Soldier glares at her in the rearview mirror, but Saben doesn't seem to notice. "You got that nickname somewhere."
"You're one to fucking talk," Odd Willie cackles and shakes his head. "Shit, Saben, I ain't never yet thrown down on a goddamned priest. And, you know, now that I think about it, that's probably why you don't hear people calling me fucking Dead Willie."
"You do what you have to do," Saben says and rubs at the gauze bandage on the back of her neck. "If you knew the whole story—"
"Yeah, well maybe that's the problem," Soldier says. "No one's bothered to let us in on the whole story."
"No one ever tells me jack shit anymore," Odd Willie mutters and sucks sullenly on his cigarette.
Soldier spares another glance at the rearview mirror, at Saben sitting there in the backseat, sitting up very straight and defiant in her neat black clothes, that one fucking outfit probably worth more than Soldier's entire wardrobe. The left side of Saben's face is swollen and splotched various bruised shades of red and purple, her lower lip split badly enough that it took two stitches to close. And the sight of her, all the damage done to that pretty, smug face, makes Soldier feel just a little bit better about almost everything.
"What I want to know," Willie says, "is why these goddamn beaver-beaters up in Woonsocket decided they couldn't wait until Monday to do this thing, and, for that matter, why the fuck the Bailiff let them go and change their minds on him like that?"
"Likely as not, that's none of your goddamn business," Soldier tells him. "Anyway, what difference does it make to you? Did you have a fucking date or something?"
"You can suck my dick, the both of you," Odd Willie snorts and breathes out a cloud of smoke.
"You scared, Odd Willie?" Saben asks, trying to sound amused, but Soldier can hear the deceit.
"Damn straight, and if you aren't too, it's only because what precious little sense you ever had got knocked out of you yesterday. After I got back from Rocky Point last night, I started asking around about Woonsocket."
"What you mean is, you went looking for spook stories," Soldier says, and Odd Willie frowns at her.
"Jesus Christ, you're the one made the place out like it was the goddamned fifth level of Hades or something, so I just asked around, that's all. I talked to Patience Bacon, you know, that little creep used to run with Scarborough Pentecost back before he—"
"I know fucking Patience Bacon."
"Well, yeah, so he says it's never been much of anything, Woonsocket. Leastways, no one's ever given much of a shit what goes on up there. Boston leaves it alone. Providence leaves it alone. Bunch of inbred motherfuckers, he said, half-breeds and quadroons and messed-up shit like that. Total fucking Deliverance territory."
"Yeah, it's a shithole," Soldier says, wishing she'd made Odd Willie drive, because he doesn't talk so damned much when he's driving.
"Patience says it was a big mill town way back when, full of, you know, decent working-class folk," and Willie Lothrop snickers and taps ash onto the floorboard. "Give me your tired, your poor, your huddled motherfuckers yearning to get dicked up the ass by fat-cat mill owners, right?"
"We don't need a history lesson," Saben says. "Some of us know our own backyards."
"Shut the fuck up," Soldier says and smiles at Saben's reflection. "You keep right on talking, Odd Willie. Fancy-pants cunts that ride in the backseat never know half as much as they think they do."
"Hell, Soldier, I don't even remember what I was saying. What's the difference? We do what we're told, like good little henchmen, no matter what's on the other end. Into the Valley of Death rode the six fucking hundred, et cetera, et cetera."
"Whatever you say," Saben mumbles, meaning Soldier or Odd Willie or the both of them at once.
Willie Lothrop rubs at his nose. "Bad fucking mojo up there," he says. "That's what Patience Bacon says. Said that place was rotten to the core before the mills and the beaver-beaters and the fucking hounds showed up, said even the damned Indians knew that place was bad news, but they had the good sense to stay away. You know what Woonsocket means in Algonquin Indian talk? It means the fucking way down to hell; that's what it means."
"No, it doesn't, you moron," Soldier says and laughs. "It means 'place of deep descent,' because of the valley and all the damn waterfalls on the Blackstone River."
Odd Willie glares at her for a second or two and then gives Soldier the finger. "Fine," he says. "I didn't know we had Miss Hollywood fucking Squares in the car with us, Miss I'll-take-goddamn-geography-for-five-hundred-Alex. Jesus fucking crap. You're the one was telling me how messed up the place is."
"I never said it was the gates of hell. It's a rough neighborhood, that's all." Soldier knows better, knows it's more than that, way more than that, but doesn't see the point, at this late date, of telling Odd Willie the truth if he doesn't know it already. He'll see for himself soon enough.
"You want me just to shut the hell up?"
"No," Soldier lies. "I want to hear the rest of it. The history of Woonsocket as told by the illustrious and learned Patience Bacon. By the way, you know he's the same jackass who accidentally locked himself in a mausoleum over at Swan Point, right? He was there three days—"
"You're fucking making fun of me, and I hate that almost as much as crazy old ladies fucking staring at me."
"Oh, come on, Willie. Do it for Saben. She needs something to take her mind off being such a craven pain in my rectum."
Willie sighs and taps more ash onto the floorboard, ignoring the ashtray. "Patience Bacon says it's a bad place, that's all. He says it was always a rough neck of the woods, but after the mills left in the seventies, things kind of went to—"
"Hell?" Saben asks quietly.
"If you gotta put a goddamned name on it, yeah. Things kinda went to hell. It's fucking anarchy up there. Quadroons and octoroons wandering around in broad fucking daylight, buying their crack and crystal and shit right there on the street just like everybody else. The whole town might as well be a goddamn warren—"
"You curse an awful lot, you know that?" Soldier asks him. "You might just be the most foulmouthed bastard I've ever had to work with."
"Fuck you. You know about Ballou?" he asks her.
"Yeah, Willie. I know all about Ballou."
"Crazy son of a whore. Crazy as a shithouse rat. Way I hear it, he's spent the last couple of decades hidden away up here trying to open doors, doors no one's supposed to ever fucking open. No one. Ever. But there he is with his little inbred fuckfest, doing whatever he pleases, and the rest of us be damned. Lots of people go missing in Woonsocket."
"If I lived there, I'd go missing, too," Saben says.
"You might just go missing anyway," Soldier tells her, speaking to the rearview mirror.
"This guy Ballou, he's not even a changeling," Odd Willie says. "He's not a Child of the Cuckoo, and he's not a ghoul, either, just some half-breed mutt, and he's running the whole damn show, right? Descended from some kind of goddamn French Lancelots—"
"Huguenots," Saben White cuts in. "French Huguenots."
Odd Willie looks over his shoulder at her, peering past the rims of his expensive Wayfarers, and she gets the middle finger, too. "Soldier, if you want to pull over, I'd be glad to finish what you started yesterday."
"You'd be glad to fucking try," Saben tells him.
"Shut up, Saben. Interrupt Odd Willie again, and I'm not only going to take him up on his offer, I'm gonna lend a fucking hand."
"Whatever. Fucking Huguenots," Odd Willie says, turning back to face the windshield again. "They started all this textile shit way back in the 1840s, right? And they've been running the place ever since. Fucking half-breed loup-garou sons of bitches. Patience Bacon, he says there's this Precious Blood beaver-beater cemetery with some sort of big marble coliseum, you know, like a ruined Greek temple, and this Ballou fucker and his mongrels have been making human sacrifices there on the Full Worm Moons—"
"—trying to open doors," Saben says. "You're starting to sound like a bad monster movie."
"Oh, that's real fucking funny," Odd Willie sneers back at her.
"Ballou's a thug and a cunt," Soldier says, quoting the Bailiff. "Someone on the outside who's too stupid to know he's never getting in." Never mind how much it annoys Saben, she's tired of listening to Odd Willie. There's enough on her mind already without him making it worse.
"Okay. So, you tell me, why the hell does the Bailiff jump when Ballou says 'boo'?" Odd Willie asks. "Answer me that."
"There's some business we got to settle," Soldier replies. "That's all. So, please, as a personal fucking favor—drop the 'Valley of Death' shtick."
"Yeah, well, what about you, Soldier?" Saben asks. "Are you scared, too? Is he right? Are we really walking into something that bad?"
Soldier shrugs and pushes her sunglasses farther up the bridge of her nose. "Well," she says, "for someone who gets her kicks murdering men of the cloth, I suspect it might seem a little tame."
"I'd tell you," Saben says. "I'd tell you what really happened, if he'd let me."
Soldier glances at the rearview mirror again, and this time Saben's staring back at her. But there's more fear than bravado in her brown eyes, no matter how she's holding herself, no matter how straight her shoulders or high her chin. It's just bluster, and Soldier wonders how Saben White never managed to learn how to back down until after she's been bitten.
"Would you?" Soldier asks her. "Would you really do that? If you could, I mean. If he'd let you."
"I just said that I would."
"I think you're lying. I think it's possible you haven't even told the Bailiff everything. You look like someone with an awfully big secret."
"Did you tell the Bailiff that?" Saben asks, and then she looks down at her hands folded in her lap, the Seal of Solomon tattooed there like something that might actually have the power to protect her from whatever's waiting for them farther down the road. Soldier doesn't answer right away. Instead she listens to Sam the Sham, and the Dodge's tires spinning on the asphalt, the cold breeze rushing into Willie's window, which is still open a crack, and the roar of passing cars.
"I asked you if you told him that," Saben says, and now she's almost whispering. Odd Willie laughs and smokes his Winston.
"No," Soldier tells her. "I didn't say a fucking word." It's not the truth, but Soldier figures even a lie's more than Saben deserves to hear. "Whatever you did, why ever you did it, that's between you and him, because that's the way he wants it."
"Hey, ladies," Odd Willie Lothrop snickers. "How about a fucking joke to ease the tension?" and before Soldier can tell him to shut up, he's already started. "Okay, so, a Catholic boy and a Jewish boy were talking, you see, and the Catholic boy, being an arrogant Irish cocksucker, was like, 'Hey, man, my priest knows more than your rabbi.' And the Jewish boy, being such a wily cocksucker, he was like, 'Yeah? Well, of course he does. You fucking tell him everything.'" Odd Willie giggles and takes another drag off his Winston.
"Shut the hell up," Soldier says.
"Oh, come on. You know that was wicked funny," Odd Willie laughs. "I'm about to piss myself over here. Moreover, if you'll think about it a second, you'll see it was not entirely irrelevant to certain current events," and he turns and winks at Saben.
"Just shut up," Soldier says again, which only makes Willie giggle that much more.
"Thank you," Saben says from the backseat.
Soldier shakes her head and doesn't look at the rearview mirror. "Oh, no. It isn't like that, Saben. It isn't like that at all. Don't you dare start thanking me for anything."
"Yeah," Saben replies, and for a little while, no one says anything else.
"Your responsibility," the Bailiff had told her, no uncertain terms there, so Saturday night, after Rocky Point, Soldier drove Saben White to see the greasy old croaker over on Federal Hill who sews them up and sets broken bones and extracts bullets whenever something goes wrong. The basement where he works smelled like mildew and disinfectant, and Soldier sat in a moldering, duct-taped recliner in one corner while he cleaned up the mess she'd made of Saben's face and gave her an injection of antibiotics for the bite on the back of her neck. She'd heard the doctor had been a field medic in Vietnam, that he'd been sent home on a Section Eight after the My Lai Massacre back in sixty-eight. Now he works for the Bailiff, which means he also works for the ghouls and the Cuckoo, though, if his luck's improved any in the past forty-two years, the croaker knows nothing at all of the powers that held the Bailiff's leash.
"Human teeth are filthy things," the croaker mumbled gravely around his loose dentures while he filled the syringe with amoxicillin. "I'll tell you, I'd rather be bitten by a mangy, three-legged bitch than by a human mouth."
"Maybe it was a dog," Saben said and risked a glance at Soldier. "But I didn't count her legs."
"Just two," Soldier said, and then Saben winced when the old man jabbed her with the needle.
"That bite goes septic, missy, you won't be giving much of a shit if it had six legs and a pogo stick, now, will you?" the croaker asked Saben once he'd emptied the syringe.
Afterwards, Soldier drove across the Seekonk and left Saben on the sidewalk outside a renovated warehouse where she lived in a studio apartment on the third floor. All this time, except for the comments in the doctor's basement, neither of them had talked, because Saben was still too frightened, and there was nothing Soldier wanted to hear and even less she wanted to say. Soldier drove her home, left her on the sidewalk, and that, Soldier thought, was enough babysitting and fucking hand-holding for one night, enough responsibility, and she stopped at Fellini's for a couple of slices of pepperoni pizza on the way home. She drank Coke instead of beer, because a beer or three would only lead to the unopened bottle of whiskey beneath the front seat being opened, and she wasn't such a drunkard that the Bailiff couldn't scare her into staying sober for a few days every now and then. Not yet, anyway.
After the pizza, she went home to her own apartment, a dingy little hole on Lancaster Street. The Bailiff had rented it for her years before, the week she turned sixteen and was allowed to leave the warrens, just someplace to sleep until she could find something better. But Soldier had never really seen the point in looking for another place. It was quiet, and no one there ever fucked with her or got nosy. She had two rooms and a tiny kitchen, a bathroom and a view of the North Burial Ground past the rooftops. Soldier carried the bottle of Dickel in with her and set it on top of the television, then took a hot shower. The water and soap felt good, even if they weren't half clean enough to wash away the anxiety or the wall of angry black thoughts building itself high inside her head, but then, nothing was. Nothing ever was, not even the booze. She lay in bed naked, her wet hair and skin slowly drying in the not-quite-cold, not-quite-warm air of the apartment, and stared at the whiskey bottle on the TV. She imagined it stared back at her, taunting, tempting, reminding her how good it would feel, and all she had to do was crack the seal and take a few swallows. No one would ever know, it promised. Not a single soul, living or dead or anything in between. So then Soldier stared at the low water-stained ceiling instead.
A little while later she dozed, hardly more than half-asleep but dreaming about the night out on the Argilla Road, about Sheldon and the hearse, the cold November rain, and this time it all went more or less the way they'd planned it would, the way the Bailiff had laid it all out for them. At the end, Sheldon helped her burn down Quaker Jameson's roadhouse, and the gaunt and the orchid-skinned demon stood nearby, watching indifferently, and when they were done, the demon taught her a British drinking song that it sang to the tune of "The Star-Spangled Banner"—
The news through Olympus immediately flew;
Where Old Thunder pretended to give himself airs—
If these mortals are suffer'd their scheme to pursue,
There's devil a goddess will stay above stairs.
Hark! Already they cry,
In transports of joy.
A fig for Parnassus! To Rowley's we'll fly;
And there, my good fellows, we'll learn to entwine,
The myrtle of Venus with Bacchus' vine.
He offered to teach her another one, something bawdy the Freemasons sometimes sang to the tune of "God Save the Queen," but she was beginning to see where all this was headed, and she went back upstairs into the yellow house on Benefit Street. One of the silver-eyed women was waiting for her, and Soldier breathed into her palm, checking for the stink of whiskey. And of course she smelled like a fucking distillery, but the silver-eyed woman didn't seem to notice. A clock on the mantel rang the hour, and Soldier wondered if Sheldon had gotten lost in Ipswich again, if maybe he'd spend the whole night wandering about, lost in the rain. The silver-eyed woman, Miss Josephine, who'd lived in the house for more than a hundred years, offered Soldier a glass of brandy, and she said she'd rather have bourbon.
"Such a vulgar spirit," the vampire said disapprovingly, but poured Soldier a tall glass of Wild Turkey and left the bottle on the table so she could help herself if she wanted another. Soldier thanked her, drained the glass, belched, and then sat trying to recall the words to the demon's drinking song.
"I hear it did not go well tonight," the silver-eyed woman said, and when Soldier looked at her again, it was Sheldon Vale sitting on the other side of the wide mahogany dining table. There was a gaping red-black cavity between his eyes, the two overlapping bullet holes she could almost remember putting there. He wiped some of the gore from his face and then stared at the blood and brains and flecks of bone on his hands.
"Shoot first, ask questions later," he said and laughed. It made her think of drowning, that laugh, and she wondered where the silver-eyed woman had gone, and if she'd be coming back. "Of course," Sheldon said, "maybe you think it's better if you skip the questions entirely."
"You'd have done the same damned thing, and you know it," Soldier told him, and the clock on the mantel chimed again, fifteen minutes come and gone like a handful of nothing. The clock was strung together with baling wire and smoke, splintered wood from old tomato crates, three prickly white fish spines where the hands should be. Wheels and spindles carved from human bone, the fat pendulum a dead rat dangling headfirst, its stiff tail and some kite string for the rod. The clock's face had once belonged to a very pretty girl, and Soldier could see that the taxidermist had taken care to give her the finest glass eyes, irises the color of a broken china cup, before he'd tattooed a circle of blue-green Roman numerals onto her skin—XII perfectly centered on her forehead, VI on her chin. Or maybe she'd still been alive when that was done. Maybe they'd only killed her afterwards.
"But you do have questions, don't you?" Sheldon asked. "You've got questions gnawing you apart like maggots. You've got things in you worse than questions. I mean, at least a question is a place to start, right?"
"You're dead and full of shit," she told him and turned away from the clock because she was starting to think that the glass eyes could see her.
"That might be," Sheldon said. "But it doesn't change a word I've said. Do you even remember her?" and he pointed at the clock.
"Do I remember who?"
"The little girl in the attic, Soldier. The Daughter of the Four of Pentacles."
"Maybe I should be the one asking you the fucking questions—"
"So maybe you are. Maybe that's just exactly what you're doing. Do you remember her? The wizard's child?" And he rubbed at his eyes again, then wiped his sticky fingers on the table. The smear looked like chocolate and raspberry tapioca. He took something from the pocket of his jacket, a tarot card, and laid it faceup on the table between them.
"The Four of Pentacles," Sheldon said and tapped the card with an index finger like a magician getting ready for some sleight of hand. Soldier stared at the card a moment, a king seated upon a bench, a pentacle beneath each foot, one balanced atop his crown, and another wrapped up tight in his arms. Behind him there were low, tree-covered hills and the towers and parapets of a medieval city. The man's face was greed, she thought, greed and desperation, and then Sheldon tapped the card again.
"Yeah, so what?" Soldier asked him, losing patience, wondering how he'd made it back from Ipswich so goddamned fast. "It's a tarot card. I fucking hate riddles. You know that. You got something to say, spit it out or go haunt someone else, you backstabbing son of a bitch."
But the dead man only smiled and produced a battered paperback—The Tarot Revealed, by Eden Gray—from the same pocket that had held the card; he read silently from it for a moment, then laid the book on the table beside the card. "Maybe it's time you start asking a few of those questions," he said. "Maybe it's time for you to try and remember."
"Remember what?" she demanded, even though she'd been taught never to raise her voice in the yellow house on Benefit Street. "Just what the hell is it that you want me to try to remember?"
"I've got such a goddamned headache," he said, instead of answering her question. "I should probably take some aspirin and find someplace to lie down."
"They've got these things called graves," Soldier said, glancing from the tarot card back to the clock on the mantel.
"Don't you even want to know why I did it?"
"Not particularly," Soldier told him and poured herself another glass of bourbon. The clock winked at her with one glass eye.
"Well, if that's the way you want it. You'll find out sooner or later, anyway. Right now, though, you'd better wake up," Sheldon said. "Looks like you've got company." And then he was gone, and Soldier was alone in the big dining room. She reached for the glass of whiskey, meaning to toast the clock, but opened her eyes instead, opened her eyes in another place and another time because now she could hear footsteps heavy on the floorboards of her apartment, footsteps and her squeaky front door being eased shut again. Adrenaline like a punch in the heart, brass knuckles to shatter her sternum, and she was awake in an instant, the dream already gone to gray tatters, nothing she'd be able to make sense of later. Soldier slipped her right hand beneath one of the pillows and pulled out the .357 Magnum she kept there, flipped off the safety, and cocked the pistol. The room was dark, even though she'd left the bathroom light burning.
"You would do me a great kindness," the Bailiff said, "if you'd please see fit to point that thing elsewhere."
"Shit," Soldier hissed, easing her finger off the trigger. She could smell her own sweat, the tinny stench of her fear, and she could also smell the Bailiff's cheap cologne like flowers and rubbing alcohol. "You scared the piss out of me, old man," she said, sitting up, returning the gun to its place beneath the pillow.
"I hope you don't mean that literally."
"I did what you said," Soldier told him. "I took her to the doc, and then I took her home."
"Would you like to turn on a light?" the Bailiff asked. "I can't see my hand four inches in front of my face."
Soldier reached for the floor lamp beside the bed, fumbling for the long cord with its frayed gold tassel, and a moment later she was squinting and cursing and shading her eyes against the dim glare of the twenty-five-watt bulb.
"It's right there on the television," she said. "I haven't had a sip. I haven't even opened it. It's all yours. Every goddamn drop, if you still want it."
The Bailiff turned to find the bottle of Dickel where Soldier had left it a few hours earlier. "Really? For me?" he asked, and smiled one of his great carnivorous smiles. "Why, little Soldier girl, how terribly, terribly thoughtful of you. I hope you won't mind if I share this with my boys. They do so appreciate a stiff shot of something now and then."
"I'm sure they do," Soldier yawned and picked up her lighter and a fresh pack of Marlboros from the cluttered nightstand. "Knock yourself out. Pour it down the fucking toilet, for all I care."
"You're naked," the Bailiff said.
"Thank you for noticing."
"Would you like me to give you a moment to dress yourself? The matter that's brought me here, it's nothing so urgent—"
"I'm fine," Soldier said, though the air in the bedroom was much colder than when she'd fallen asleep, and she wished that she'd bothered to bring her ratty green terry-cloth bathrobe to bed with her instead of leaving it hanging on its nail beside the shower stall. "Unless titties offend you, Bailiff."
He sniffed, cocked a bushy eyebrow, and set the bottle of whiskey down again. "It might not be my personal preference, as you should well know, but I have seen more than my share of womanflesh, and I'm not the squeamish sort, besides. So if it suits you—"
"I did what you said," Soldier told him again and lit a cigarette.
"Yes, I know. I just left Miss Saben White not half an hour ago. Somewhat worse for the wear, I might add. At the moment, I'm more concerned with the whereabouts of Mr. William Lothrop and the inconveniently deceased papist."
Soldier took a deep drag off her cigarette, exhaled, and stared through the smoke at the fat man standing near the foot of her bed. Beads of sweat stood out on his bald head, and his long beard was twisted into a short, stiff braid and tied with a rubber band. He was wearing a shiny blue suit of worsted wool.
"I haven't talked to Odd Willie yet," she replied. "He was supposed to call. But you know how he is. He gets caught up in his work. He forgets shit." And Soldier drew little circles in the air around her right ear.
"Yes, he does," the Bailiff said and nodded his head. "But he is a maestro when it comes to incineration, is he not? We can sometimes overlook eccentricities when matters of genius are involved, don't you think?"
"If it suits you," Soldier said, wishing the Bailiff would stop hemming and hawing and get to whatever was on his mind so that maybe she could go back to sleep.
"William is a troubled boy," the Bailiff continued, "but insanity and antisocial behavior have never excluded the likelihood of precocity, little Soldier. You should remember that."
"Fine," she said and took another drag off the Marlboro. "I'll remember that. Have I done something wrong?"
"No, no. Quite the contrary," the Bailiff said. "Why do you ask?" and then he looked about in vain for a place to sit down. Soldier pointed at the foot of the bed, and he eased his bulk onto the mattress. The box springs shrieked like a sockful of stomped mice.
"I asked because I'm not accustomed to waking up with you standing in my goddamn bedroom. That's why I asked."
"Yes, I suppose it was rude of me not to call," he said and tugged at his braided beard. "I'm sorry about that. But I believe, little Soldier, that I had cause not to trust certain information to the telephone."
"And it couldn't have waited until tomorrow?"
"No, as a matter of fact," he said, and then he told her that there had been another call from George Ballou up in Woonsocket, that she'd have to make the trip on Sunday instead of Monday. And then he said other things, and Soldier sat naked on the bed and listened. By the time he was done, the last thing on Soldier's mind was sleep. He took the bottle of Dickel, bade her a good morning and good luck, reminded her that if she fucked it up this time, she'd never have to worry about fucking it up again, kissed her on the left cheek, and then the Bailiff left. Soldier smoked another Marlboro, and when she was done, she picked up her cell and called Odd Willie Lothrop.
In the wan, cloud-filtered light of the winter afternoon, the winding, rugged course of the Blackstone River reminds Soldier of a gigantic coal-colored snake, some vast and immeasurably prehistoric serpent or biblical monster sprawled lazily across the land, and there's the city of Woonsocket held forever tight inside its iridescent coils. The water, which is really only its scales, glints darkly, and, here and there, she can see foam-white scars to mark the various waterfalls and cataracts. Soldier knows there's too much truth in the things Odd Willie said, and maybe when the Algonquin named this place, they meant hell after all. Seeing it now, postcard perfect between the steep, wooded hills and yet surely gone as sick and mean and insane as any town ever will, Soldier could believe just about anything at all about Woonsocket. There have been white men here, of one sort and then another, for almost four hundred years, and they have dug themselves in deep, almost as deep as the hounds and the things that were here before the hounds. The Dodge follows the highway down into the valley, out of the old forest, the trees that she imagines lean in towards the river as if their bare branches might hide it from the terrible gray and purple sky.
Odd Willie giggles anxiously and asks Soldier a question that she only half hears, something about swamp yankees and sundown, but she doesn't answer him, doesn't say a word. She follows the road past shoddy, weatherworn houses and boarded-up discount stores, fast-food joints and fading wooden billboards hawking "chowda" and clam cakes and lobster, out into the crooked streets of the city. The smokestacks and pitched rooftops of abandoned mills, towering church steeples and the brick-and-mortar corpses of long-dead factories rise up before her like some ingenious armor the serpent river has fashioned to keep itself safe, the symbiotic ruins of industry and avarice to guard a sleeping dragon.
"When are you going to get around to telling us what this is all about?" Saben White asks from the backseat.
"It's just a courier job; that's all," Soldier replies and stops at a red light. "As long as no one gets stupid and nothing gets fucked-up, it won't be anything but a simple drop and swap. We give them that leather bag in the trunk, and they give us a package for the Bailiff."
"What the Sam Hill could these yahoos have that the Bailiff wants?" Willie Lothrop asks her and shakes his head. She realizes that he hasn't combed his hair once since they left Providence, hasn't even taken out his pink plastic comb, and that's got to be some sort of record.
"That part's none of our goddamn business. You ought to know that much by now, Willie. That part's between Old Man Ballou and the Bailiff."
"Well, they should have sent two fucking cars," he says.
"Ballou said just one. One car, three occupants," Soldier tells him. "So that's the way we're doing it."
"We've got less than ten minutes," Saben White says, sounding more impatient than anything else, and Soldier nods, but doesn't look at her.
"Believe it or not, I can fucking tell time," Soldier says, then finds a gravelly place at the side of the road and pulls over, the wheels of the Dodge sending up a dense cloud of dust and grit. She lets the motor idle for a moment, as the dust settles again, before she turns the key and switches it off. The car's parked beside an old retaining wall, concrete covered with wild grapevines like a strangling network of dead, dry veins and capillaries.
"I want to talk to Willie for a moment," Soldier says. "Just Willie. You stay where you are, Saben. Unless I call for you, you stay right there where you are."
"Ten minutes," Saben reminds her, and Soldier shows her the middle finger of her right hand.
"Come on, Odd Willie. We need to talk."
"What the hell for?" he asks, reaching for the door handle, reaching so slowly that Soldier gets the impression that maybe he thinks she's going to change her mind and let him stay in the car.
"I need to stretch my legs. I need to clear my head before the drop, that's all."
She opens her door, and the cold air tastes like the highway grime stirred up by the Dodge, like the exhaust from passing cars and trucks, and, beneath that, there's the faintly rotten smell of the Blackstone River. Despite all her years in the tunnels, her schooling by the hounds, Soldier has never really grown accustomed to the smell of rot. Most of the changelings hardly seem to notice, but the smell of rot makes Soldier uneasy, and the smell of wet rot is the worst of them all. Odd Willie follows her along the road's shoulder for fifteen or twenty yards. He lights a cigarette and that helps mask the smell of the river a little, so she lights one of her own, then stops and stares back at the car and Saben sitting in the backseat.
"You think she's getting suspicious?" Odd Willie asks. "You think she knows something's up?"
"Well, you laid it on pretty thick," Soldier says, then looks away from the Dodge, up at the sky; she wonders how the car handles on ice and snow.
"Sometimes it's hard to stop," he says and puffs his cigarette. "Once I get started, it just keeps on coming."
"No, I think she's pissed off, but that's all. She thinks it's just something personal between me and her. And I guess that makes her half right." Soldier flicks ash at the gravel and takes a long drag off her Marlboro. She wants to walk back to the Dodge and put two or three in Saben's skull, but this is the Bailiff's show, and he's the one calling all the shots.
"I'm scared as shit," Odd Willie whispers and giggles softly to himself. "I'm sorry, Soldier, but that's the gods' fucking truth. I never signed on for crap like this."
"You never signed on," she reminds him, as if he might somehow have managed to forget.
"It was just a figure of speech. You know what I fucking meant."
"Sure," Soldier says, looking away from the sky, turning her attention once more to the walls and roofs and streets of Woonsocket laid out below them. "But I've never yet heard of one of us dying of old age. Makes it seem kind of silly to get too worked up over something like this." That's Sheldon talking, she thinks, his words from her mouth, something he said to her once or twice. You ever heard of one of us dying from old age?
"There's always a goddamned first time. I can't see the harm in being ambitious."
"Long as you're there when I need you."
"Shit, Soldier. All I said was I'm scared. I didn't say I was a coward. There's a great fucking difference between the two."
"That's all I wanted to hear," she says, studying the valley—the empty mills that are never really empty, the futility of a dozen ornate church steeples, the roaring white place where the Thundermist Falls were dammed in 1960, five years after a flood had almost destroyed the town. North of the dam, Soldier spots a low tree-shrouded hill rising from the rooftops and squalor, and she points it out to Odd Willie.
"Right there," she says, thinking that in the summer the trees would be lush and green, hiding the cemetery and all its secrets. But in winter, the old trees are little more than the weathered slats of a crooked fence, revealing the patchwork of headstones and less modest monuments to the dead and departed that crown the hill. Markers to signal mortal loss, and the way down to George Ballou.
"Fuck me," Odd Willie says and spits on the road.
"Don't you forget," Soldier tells him, still watching the tall trees surrounding Oak Hill Cemetery. "When the time comes, she's mine. Any of the rest of these assholes, you can take your fucking pick. But she's mine."
"I love it when you talk like Clint Eastwood," he says and snickers, then flicks the rest of his cigarette at a passing car. "It's absolutely fucking beautiful."
"Just don't you get all trigger-happy down there and forget what I'm telling you."
"Well," says Odd Willie, "as much as I'd love to do the honors myself—and I'm not gonna lie to you about that, no, ma'am—I learned a long time ago, you come between some cocksucker and the object of her passionate fucking need for vengeance, and pretty soon the hounds are gonna be shuffling your happy ass off to Mama Hydra—"
"You really believe that stuff?" she asks, interrupting him.
"Like I said last night, it's as good as anything else I've heard."
"Yeah," Soldier says and glances back at the car. Saben hasn't moved. "That other shit we talked about—"
"—stays between me and you," Odd Willie says. "I mean, until such time as they tie me down and some determined asshole comes at me with a pair of wire strippers and an acetylene torch."
"Guess that'll have to do," she says. "Let's get this over with."
Odd Willie Lothrop nods his head, and then he follows Soldier back to the car.
Odd Willie lives three flights up in a rat-infested Federal Hill tenement building that had been new when Calvin Coolidge was president and that should have been torn down long decades ago. It had been condemned, back in the 1990s, but that didn't keep out the squatters or the crackheads or Odd Willie. It didn't keep Soldier out, either. When the Bailiff had gone, she called Willie Lothrop and told him that she was on her way over. He sounded surprised, but not too surprised, and asked Soldier if she wanted to meet somewhere else, a bar or Swan Point or something more pleasant than his place. She told him no, his place was fine, that she just wanted to talk, that it was very important, and he told her to be careful on the stairs. "They're getting worse," he said. "I just about broke my goddamn ankle last week."
"You could move out of that shithole, you know," Soldier said, and she could almost hear him shrug.
"I got roots here," he replied and hung up first.
So, less than half an hour later, she was ducking under the boards nailed across the front doorway of the building, picking her way through the dark lobby cluttered with refuse and filth and broken furniture. She had a flashlight, but didn't turn it on. She could see well enough to find the sagging stairs leading up to Willie's apartment.
"Who the hell are you?" a gruff female voice demanded from the shadows. "You best stay the fuck away from me, bitch, or I'll cut you."
"Will you?" Soldier asked, speaking to the lightless place where the voice had come from, imagining the woman cringing there amid the peeling wallpaper and old cardboard and rat droppings. "Will you really?"
"I ain't messin' around with you, cunt. I'll slice you up so bad ain't no doctor ever been born be able to put you back together again." But the woman sounded more frightened than dangerous. Soldier imagined there were others, less bold, more frightened, cowering in the darkness of the lobby and the hallway leading deep into the ground floor of the building. She crossed the small room, stood at the foot of the stairs and squinted into the narrow stairwell that wound up and up and up, a right-angled whorl like the view from inside some geometrically improbable snail's spiraling shell.
"Don't you waste your time worrying about me," Soldier told the woman. "You got way worse things to worry about than me."
"Don't you tell me what I got to worry over, bitch."
Soldier laid her right hand on the banister and took the first step; she wanted a drink so bad it hurt, so bad she was beginning to feel sick, and she wondered what Odd Willie had upstairs. Odd Willie usually drank cheap tequila and cheaper malt liquor, but right now either one would be fine with her. She took a couple more steps.
"Who you is?" the voice asked. "You ain't no angel. You ain't no servant of the Lord Jesus."
"No," Soldier told the woman, "I'm not that. But you just wait. I expect someone else will be along shortly."
"You some kind of hoodoo," the woman sneered from her hiding place, and Soldier imagined her crossing herself or making some sign to ward off the evil eye. "You some kind of devil."
"Close enough," Soldier replied, and then she left the woman and climbed the stairs to Odd Willie's floor. By the time she reached his front door she was out of breath and thirstier than ever. She had to knock four times before he opened up.
"I was on the can," he explained, zipping his pants, snagging his underwear in the zipper, and having to start over again. Odd Willie was wearing black jeans and a black-and-white Buzzcocks T-shirt. There were several holes in the shirt, and he was barefoot.
"You know there's a fucking crazy lady downstairs?" Soldier asked him, leaning against the doorjamb, trying to catch her breath. "I thought she was gonna try to exorcise me or something."
"Yeah, that's Betty. Don't you worry about her," Odd Willie said and tried his zipper again. "When she's not high, which isn't very often, she's either seeing demons or waiting on the Second Coming or both."
"Charming fucking neighbors you got," Soldier said. "Now, you gonna let me in, or are we going to have to do this right out here in the hallway?"
He apologized and stepped to one side so that Soldier could get past, and then he shut the door and locked it. The apartment smelled like fried food and mildew and stale cigarette smoke, and the only light came from a few flickery fluorescent tubes he'd rigged up overhead.
"Like I said on the phone," Willie started, "it's taken care of. I set the timer for twenty minutes and left him inside the old House of Horrors—"
"That's not why I'm here," Soldier said.
"Oh yeah," Odd Willie said, and giggled, and then he sat down on the mattress lying in the center of the room, the mattress stained and torn and heaped with dirty sheets and a stolen blue U-Haul moving blanket. "You already said that on the phone," he smiled and nodded his head, made a gun with his thumb and forefinger and pressed it to his left temple. "You want something to drink?"
"You fucking know not to ever ask me that question."
"Yeah, but I got a brand-new bottle of Pepe Lopez Gold, hasn't even been opened," Odd Willie said and grinned at her. There was a blotch of something on the front of his shirt that looked like toothpaste, though Soldier was pretty sure Odd Willie Lothrop had never made the acquaintance of a toothbrush.
"Might as well drink paint thinner as that shit," Soldier told him, and glanced about for someplace to sit down. Willie pointed her towards a three-legged chair propped against one wall.
"Hell, it tastes like fucking paint thinner," Odd Willie said. "But it generally does the job."
"I came over because I have to ask you a question," Soldier said, changing the subject because she had enough trouble right now without Odd Willie's cheap-ass tequila. She sat down on the three-legged chair, leaning slightly to one side so it wouldn't tip over.
"You could've just asked when you called."
"It's not that sort of question, Willie."
"Then maybe it's not the sort of question I want to get mixed up with."
"Saben's a spy," Soldier said, and there, it was out, a big enough bomb to shut Odd Willie up for a second or two. "She was screwing Sheldon. The two of them were working for these pricks out in Woonsocket, and they set me up."
"Fuck," Odd Willie Lothrop muttered and ran his fingers through his oily black hair. "Fuck all. Frankly, I never would've thought the cunt had it in her. And, hey, I was asking around about Woonsocket and—"
"The Bailiff said she's been in with that crowd for a long time, almost since the first night the hounds put her on the street," Soldier said and glanced up at the ceiling. In places, the plaster had fallen away, exposing the decaying lath beneath. "There's a kid. Saben got herself knocked up about nine years back. The daddy's part hound, and—get this—he's one of the Woonsocket mongrels."
"Oh, man," Odd Willie groaned and shook his head. "I really don't think I want to hear any more of this shit." He reached for the tequila, which was sitting on the floor near the mattress, cracked the seal and opened the bottle. He tilted it towards Soldier, but she swallowed and shook her head.
"Fuck me," he said again and took a long pull off the pint bottle.
"There's more," Soldier said.
"There always is," Odd Willie told her and wiped his mouth, then screwed the cap back on the bottle of Pepe Lopez and set it down on the floor. "What I want to know is why the hell they didn't keep her out of Woonsocket after that?"
"The Bailiff claims he didn't know."
"And you believe him?"
"I suppose it's possible," Soldier replied.
"Yeah. Right. That bastard knows every time we take a dump, and he didn't know Saben was still getting her kicks in Woonsocket? Did they kill the little squealer?"
"No. They didn't. They gave it to the Cuckoo. That was part of Saben's punishment, and that's where the dead priest comes in. He was a mule."
"Yeah, well, at least I knew that much," Odd Willie said and rubbed at the stubble on his cheeks. "I saw the tattoos before I torched him. He had the wings and the eye, right here," and Odd Willie tapped his chest just below his sternum. "The wings and the eye, all in red."
"Red," Soldier said, wanting to get up and walk out, much too much said already, and she'd hardly even gotten started. She wanted to go back downstairs and maybe have some fun with crazy old Betty the junky, maybe show her a thing or two or three about devils and angels and keeping her goddamn mouth shut. She wanted to be home, or down in the tunnels below Benefit Street. She wanted to be driving, maybe down to Scarborough Beach or Napatree Point, maybe all the way the hell to Stonington. She could sit on the rocks, listen to the surf and watch the cold sunrise. She wanted to be anywhere but Odd Willie's filthy little apartment.
"Yeah," she said, "I guess he would have been red, wouldn't he? The Bailiff said the old bastard had been whoring for the Cuckoo for almost fifty years. That would make him red."
"This is so many flavors of fucked-up," Odd Willie said, and this time when he giggled it made Soldier want to slap him. "So, when'd she start banging Sheldon?"
"Your guess is as good as mine."
There was an alarm clock on the floor, not far from the tequila bottle, and Soldier saw that it was almost four. "You want to get some breakfast?" she asked Odd Willie. "Maybe get some fucking pancakes?"
"Pancakes?" Odd Willie said, like he wasn't so sure what the word meant.
"Never mind," Soldier said.
"So why the hell did she kill the son of a bitch?"
"That was her paycheck for throwing in with Bittern."
"I thought you said she was spying for Woonsocket?"
"It's fucking complicated, Willie," Soldier sighed, and then she rubbed at her eyes, remembering the dream of the yellow house and the clock and Sheldon, wishing she could have gotten more sleep. "But in return for watching Bittern's back, for keeping him posted, she got information. She was trying to find the kid. I'm not sure the Bailiff knows why the hell the priest ended up dead. But if I had to guess, I'd say he's the one who muled out Saben's kid, and she just wanted revenge. Maybe he wouldn't tell her what she wanted to know."
"Okay. So, what's the Bailiff doing about all this?"
"We're going to Woonsocket tomorrow afternoon," Soldier said, and Odd Willie grimaced and then stared at the floor between his bare feet. "When we're done with business, when everything's squared away, then we take care of Saben White."
"Oh, Jesus," he said. "Jesus H. fucking Christ."
"You got a problem with that?" Soldier asked, but Odd Willie just shrugged and laughed again.
"Hey, man, I just do whatever the hell they tell me. They say, 'Yo, Odd Willie! Go fuck yourself,' I start studying up on contortionism. They tell me to blow my own goddamn brains out, and you're gonna find me sucking on the barrel of a gun."
"I'll do her, Willie. I'll pull the trigger. I just want you to know what's going down."
"That's awfully fucking thoughtful of you," he snorted and glanced up at Soldier. "But you said you were here because you wanted to ask me a question, and so far all you've done is delight and beguile me with all this goddamn good news. And, by the by, you do know it's supposed to fucking snow tomorrow, right?"
Soldier leaned forward on the three-legged chair, and the wood creaked and popped. Beneath the unsteady fluorescent lights, Odd Willie's skin made her think of some pallid, waxy cheese. She opened her mouth to ask him for the tequila, all the words she needed lying like ashes on her tongue, her right hand extended, but the Bailiff was there to stop her. The Bailiff standing at the foot of her bed, the Bailiff and his braided beard, and there's a fat, grinning devil that she'd like to introduce to Crazy Betty downstairs, something bad enough to make even the most demented, Jesus-loving crack whores sit up and take notice. Bad enough to keep Soldier's hands off the bottle, the bottle from her lips.
Maybe it's time you start asking a few of those questions, the dream Sheldon had told her, Sheldon Vale with two bullet holes in his head and a single tarot card on Miss Josephine's big table. Maybe it's time for you to try and remember.
"When's the first time you ever saw me, Odd Willie?"
And Willie Lothrop just stared back at her for a moment, and she could tell from the apprehensive shimmer in his eyes and the furrows in his forehead, from the way the corners of his mouth twitched slightly, that he wasn't sure what she was asking him.
"What do you mean?"
"I mean," Soldier said, measuring out her words like some powder or elixir that might heal or might poison, depending on the dose, "when's the very first time that you clearly remember seeing me in the warrens?"
"Fuck all if I know," Willie grunted. "What the hell kind of silly question is that?"
"How old do you think I am?" she asked him.
"None of us knows how old we are," he replied, looking even more confused, and he reached for the tequila again. "Not exactly. None of us knows that, Soldier. The Cuckoo—"
"Willie, I'm asking how old you think I am. Just fucking guess. We were warren mates, weren't we?"
"Sure," Willie replied, replying too quickly as he twisted the plastic cap off the pint of Pepe Lopez. "I mean, yeah, so we must be just about the same age, whatever the—"
"So, what's your earliest memory of me?" she asked again, her mouth as dry as all the deserts that have ever baked beneath the sun. Odd Willie took a drink of the tequila, and she watched his Adam's apple bobbing up and down as he swallowed. "It's a simple fucking question," she said.
Willie burped and set the bottle down without screwing the cap back on. "I don't know," he said. "Hell, Soldier. I'm tired, and I'm drunk, and I don't know what the fuck you're even getting at. We were kids. We fucking grew up together. How am I supposed to remember the first time I saw you?"
"So you remember me as a child?" she asked him, getting in one last question before she lost her nerve.
"To tell you the motherfucking truth," Odd Willie muttered and tapped once at the skin between his shaved eyebrows, "I'm starting to think maybe you exchanged a little more than just pleasant conversation with the freaks downstairs. Maybe Betty gave you a turn at her pipe."
Soldier nodded, then let the chair rock back on all three of its legs, and she took a deep breath and let it out again. She imagined that the last of her resolve went with it, out her nostrils and between her teeth, bleeding away into the ugly too-white light, the last bit of courage she could spare for the night, the night that was almost morning, and she had to hold something back for Woonsocket and George Ballou and Saben White.
"Forget it," she said. "This shit with Saben's got my head all over the place. Do me a favor and forget the whole damn thing, all right?" and Odd Willie shrugged and looked at his dirty feet again.
"Do you pray?" Odd Willie asked.
"You mean Mother Hydra and all that shit?"
"Yeah," Odd Willie said. "That's what I mean."
"No," Soldier told him, because she figured she owed him at least one honest answer. "I don't. What about you, Odd Willie? Are you a true believer? Do you await the cold fucking embrace of the vasty abyss?" and she smiled.
"Don't you fucking laugh at me," he said, almost whispering. He picked up the tequila bottle for the fourth time and began to peel the foil label off the glass.
"I'm not laughing at you. I'm just a faithless old cunt, that's all."
"What the hounds taught us, it makes about as much sense to me as anything else I've heard. And I need something, sometimes. Sometimes I need to at least pretend there's something more."
"Yeah," Soldier said, standing up, watching him peel the bottle and wondering if Odd Willie would still be drunk when they headed for Woonsocket, wondering if it mattered.
"Soldier, I just don't know what the fuck you were asking me," he said and flicked a shred of tequila label at the wall. "I mean, I've known you all my goddamn life."
"I said to forget about it," and he nodded, but she could tell from his expression that the very last thing Odd Willie was going to do was forget about it. "You better get some sleep. It's going to be a long day."
Odd Willie glanced up at her, rubbed his cheeks and smiled one of his crooked, sour smiles. "No shit. Hey, listen. You tell that whore downstairs, you tell her I said she better leave my guests the fuck alone, or she's gonna be looking for another building to occupy. Or worse. You tell her I said that," and Soldier said that she would, and then she showed herself out while Willie sat on the mattress, shaking his head and picking at the shiny label on the half-empty bottle of Pepe Lopez.
The entrance to the cemetery is a narrow paved road flanked on either side by run-down saltbox houses, one of them painted a drab peach and the other the color of a pigeon's egg, and Soldier thinks most people would probably mistake it for a private driveway. Probably no one in Woonsocket, of course, but the people who are only passing through, the outsiders, people who have no business poking about in cemeteries where no one's buried whom they've ever loved or known or to whom they are not even distantly related. There's a sturdy concrete pillar planted on each side of the road, just past the junk-strewn backyards of the houses, and a PRIVATE PROPERTY—NO TRESPASSING sign is hanging from one of them on a loop of wire. It looks to Soldier like someone has been taking shots at the sign with an air rifle, and the metal is crumpled at one corner and streaked with rust. There's no gate, no padlock, just the rusty sign; the warning would either be enough or it wouldn't, but she has a feeling that most people in Woonsocket don't need a warning to steer clear of the place. They'd know better. You could see it in the cast of the sky above the hill, in the way the trees grow a little too wild and a little too close together, and in the way that the patch of land beyond those two pillars seems always shaded by clouds that are nowhere to be found if you actually start looking for them.
"I want to go home," Odd Willie says and lights a cigarette.
"You think that sign's meant for us?" Saben asks. "Maybe we should stop and ask fucking permission." But no one answers her. Soldier cuts the wheel right, turning off George Street onto the road leading into Oak Hill Cemetery, and the Dodge's engine makes a sudden rattling noise deep in its guts.
"Oh, don't you fucking dare," Soldier growls at the car as they pass between the two houses. There's an old yellow sailboat parked behind one of them, behind the pigeon's-egg house, its mast broken and leaning to one side, and its name, the Fly-Away Horse, is painted along the prow in ornate crimson letters that have begun to crack and flake away.
"Patience Bacon said..." Odd Willie begins, but then they're through the pillars, past the pockmarked NO TRESPASSING sign, and his voice trails off as the bare winter limbs flicker like a few tattered frames from an old movie and are replaced by all the lush greens of midsummer. The pale, ice-thin light of the fading day has changed, too, has become the brilliant shafts and bright pools of a June or July afternoon, sunlight spilling through the rustling leaves and falling on the weeds growing along the sides of the road in warm shades of amber and honey. The engine sputters again and dies.
"What the fuck," Odd Willie says, reaching for the Bren Ten 10mm in his shoulder holster.
Soldier curses the Dodge and slaps the steering wheel, turns the key in the ignition switch, but the car doesn't make a sound. The Bailiff promised her that he'd told her everything she needed to know, everything to expect from Ballou and the cemetery, but he hadn't said anything at all about this. She turns the key again, pressing the gas pedal flat against the floor, and again the car remains silent and still.
"Stop freaking out, both of you. It's nothing," Saben says. "It's just a glamour, that's all."
"You can fuck that shit," Odd Willie laughs around the filter of his cigarette, flipping off the safety and cocking the pistol. "Talk to me, Soldier. Tell me what the hell's going on out there. Why didn't you let me in on this?"
"Because I didn't fucking know," Soldier spits back at him. She gives up on the car and turns her head, looking past Saben White to the place where the concrete columns and the rusty sign and the Fly-Away Horse had been a moment or two before. Now there's only the narrow road and the tall green trees bending low over it, and the road seems to go on forever. "Somehow, the Bailiff neglected to mention this part."
"Fuck," Odd Willie says. "Fuck, fuck, fuck."
"It's just a glamour," Saben says again. "We tripped it when we came in, that's all. It can't hurt us."
Odd Willie reaches for his door handle, then pulls his hand back. "Saben, if you don't shut the fuck up—"
"She's right," Soldier says, turning back to the windshield, and she closes her eyes and listens to her heartbeat—too hard, much too fast. She takes a deep breath and tries to remember everything the ghul have ever taught her about glamours and misdirection. But she never cared much for magick and was always a mediocre student, at best. The rituals and incantations too tedious and tricky, success or failure hinging on the most minute turn of a wrist or the slightest inflection. Better to settle things with her fists or a gun or a fucking sharp stick, she told the Bailiff once a long time ago. He frowned and muttered something about never realizing her full potential. Soldier exhales and takes another breath.
"We probably can't dispel this," Saben says, exasperated and impatient. "It's too big. There's too much of it. We shouldn't waste our energy trying."
"A bloody shame to see you squander yourself on account of laziness," the Bailiff said. "Stay ignorant, and someone will almost always benefit from your ignorance."
"You filthy old prick," Soldier whispers, casting the words like a net, her thoughts like impalpable graffiti, so that the Bailiff will have to stumble over them sooner or later. She opens her eyes to see that nothing's changed, nothing at all, the truth of things still hidden somewhere behind the glittering mask that Ballou has fashioned for this hill. But at least her heart's not beating quite so fast, and she's beginning to feel more annoyed than afraid. "Saben, don't you have some grounding with shit like this?" she asks without turning her head or glancing in the rearview mirror.
"No," Saben replies. "I don't handle glamours. I'm fine with simple photomancy, but this is something else. This is something much—"
"I didn't ask you for a lecture."
"He's strong, Soldier," Saben continues, and Odd Willie giggles. "You need to understand that; both of you need to understand that. Don't walk into this thing thinking Ballou's weak just because he's a half-breed."
"Is that your conscience talking?" Soldier asks her. It's getting hot inside the car, summer hot, and Soldier wipes sweat from her forehead and upper lip. Saben doesn't reply, and Soldier leans forward, resting her head against the steering wheel. She's tired of the masquerade, tired of pretending that she doesn't know what Saben's done, sick of waiting for her to die, and Soldier wonders if it would all really turn out that much differently if she just killed Saben right now.
"What next?" Odd Willie asks, and ash from the tip of his cigarette falls into his lap and he brushes it away. "What the fuck do we do now?"
"We get out of the car," Soldier says, because that might be the right answer; it's certainly the only answer she has for Odd Willie or for herself. "We do what we came to do and hope there are no more surprises. We get the bag out of the trunk, and we walk in."
"How about we walk the fuck out?" Odd Willie suggests.
"That might be difficult," Saben says.
"Soldier, will you please tell her to shut up before I shoot her? I'm totally fucking serious."
"We're going to get out of the car," Soldier tells him, and when she checks the rearview mirror again, she sees that Saben's drawn her own gun. "We're going to get out of the car now."
"Fine. Come on, girls," Willie says. "Let's all play follow the fucking leader," and he opens his door and climbs out into the shimmering day. Soldier does the same, and then she tosses Willie the keys across the roof of the car. He misses the catch, drops his cigarette and almost drops the Bren Ten, and has to search about on the ground for a moment to find them again.
"One minute," she says to Odd Willie. "That's all you've got, so stop screwing around," but he tells her to go fuck herself. Soldier takes another deep breath, filling her nostrils with all the scents and tastes of George Ballou's illusion. The summer air smells like dandelions and wild strawberries and is alive with the thrum of cicadas. Soldier looks towards the cemetery proper; the road curves sharply to the right before finishing the short climb to the top of the hill, and there's a low mausoleum not far from the car. Its granite roof is cracked, and the steel door is rusted the color of dried blood.
"Saben, you're going first," Soldier says, and Saben doesn't argue, so maybe she's figured out she's living on borrowed time. "I want you at least ten feet ahead of me and Willie."
"This is fucking insane," Willie grumbles as he tries to open the trunk and drops the keys again. "I'm telling you, Soldier. This is payback for Rocky Point yesterday. And hell, he probably heard about the shit at that Dunkin' Donuts, too. This mess is the Bailiff's way of showing us how fucking pissed—"
"Shut up and open the trunk," Soldier says, watching Saben. "I want the shotguns while you're in there."
"You really think that's such a good idea?" Saben asks. "Ballou might get the wrong impression, you come waltzing into the drop with a couple of scatterguns."
"I don't remember asking you, one way or the other."
Off towards the river, Soldier can hear the hoarse croak of a duck or some other waterbird, and the trees around them are filled with the raucous calls of catbirds and jays.
"You might want to hurry that shit up back there," she tells Odd Willie.
The trunk pops open, and Willie takes out a small black leather valise and drops it on the ground. Then he digs about for a moment, drops a lug wrench next to the valise, and starts cursing.
"What?" Soldier asks.
"There's only one fucking shotgun back here," he replies, even though she knows damn well there were two when they left Providence, the Mossberg twelve-gauge and her Ithaca Mag-10, the Roadblocker she had that night out at Quaker Jameson's.
"That's impossible," she shouts at Odd Willie. "Keep looking."
"I'm telling you, Soldier, it's not fucking back here, and I can keep looking from now till fucking doomsday, and it still won't be back here, so I still won't find it."
"They're watching us," Saben says, glancing up at the limbs of an elm. "They're listening."
"Don't you fucking move, not an inch," Soldier tells her, then walks quickly back to where Odd Willie's standing at the open trunk. In the trees, the birds are getting louder. And it only takes her a couple of seconds to see that he's telling the truth. He's holding the twelve-gauge cradled in the crook of his left arm, and there's nothing else in the trunk but the spare, a box of shells, a roll of duct tape, and the tire jack that goes with the lug wrench.
"It's a fucking setup," Odd Willie says. "Someone fucking set us up, Soldier."
"Don't start that," Soldier tells him, but it's nothing she hasn't already thought of herself. The Bailiff sending her off to Woonsocket with just Odd Willie Lothrop and Saben, not telling her about the glamour. And Saben still so cocky and self-confident when she ought to be broken, Saben smirking half the day like a fucking cat with blood and feathers on its paws. "It doesn't make sense," Soldier says. "It just doesn't make sense."
"Sure it does," Willie says, and then he points the 10mm at Saben. "She's in with Ballou, and she fucking double-crossed us. It all makes perfect sense to me."
"Tell him not to point that gun at me," Saben says, and she's calm, so calm it makes the hairs on the back of Soldier's neck twitch and stand on end.
"Saben, we fucking know," Odd Willie snarls. "The Bailiff told Soldier everything."
"Willie, don't do it," Soldier says, trying hard to think clearly through the shrill scream of the blue jays and catbirds overhead. There are words hidden in there, she thinks, important, powerful words disguised as squawking birds, if she could only tease the lie apart from the truth of things. "It's not that simple. It's just a trick, like the glamour, something else to distract us from what's really going on here."
"Soldier, you told me what she did," Willie says, not lowering his pistol. "You told me—"
"I told you to stop pointing that goddamn gun at Saben."
"You know I'm right. You know she did it."
"At the moment, Willie, I don't know my ass from a goddamn hole in the ground, and I'm not going to tell you again to stop pointing your gun—"
"Then it was the Bailiff," Odd Willie says, and Soldier can see the way his finger's begun to tremble against the trigger. "If it wasn't her, it was the fucking Bailiff did it, because someone fucking set us up."
"Soldier," Saben says, her voice still smooth as ice and milk, and Soldier realizes that the light around her has begun to bend and glimmer. "If he doesn't stop, I'm going to have to make him stop."
"We know about the kid," Odd Willie says, grinning the way he does when he's setting a fire or telling a joke he thinks is funny. "We know about you and Sheldon Vale. We know about the deal you made with these beaver-beater cocksuckers."
Saben's eyelids flutter, and her gun slips from her hand to the ground. "You know what the hounds want you to know," she says. "No more, and no fucking less."
"Did they tell you that you'd get the kid back?" Soldier asks her, trying not to sound scared, and she thinks they might have had a chance, if Willie had left the shotgun lying where she could reach it. If she had the shotgun in her hands right that second, and it was loaded, and there was already a round in the chamber, all that and a lucky shot and they might have a chance of making it back to the entrance of the cemetery. "Is that what they promised you? That you and Sheldon could hide up here in the boonies with your mongrel?"
"Put down the gun, Odd Willie," Saben says, and her eyes roll back to show the whites. "You don't have to die here today." An orb of blue flame has begun to writhe about her left hand, leaking from the Seal of Solomon, licking harmlessly at her skin.
"Fuck you," Odd Willie snarls and pulls the trigger.
The bullet explodes above the Dodge, spraying shrapnel and molten blue droplets. There are specks of blood on Odd Willie's face, and Saben White's wrapped in a writhing sapphire cowl of fire.
All lost, the birds scream from the trees. Poor little soldier girl, all lost now, and she can feel the tiny piece of steel embedded in her throat, a bit of Odd Willie's bullet, can feel the blood pumping from her body and spattering the grass at her feet. All lost, lost, lost now, poor little Soldier girl. Should have stayed where you belonged...
Her knees feel weak, and she reaches for her gun, blinking and squinting because the daylight and the fire from Saben's tattoo are blazing. Her hand closes around the butt of her pistol, but now Saben has turned towards her, and Odd Willie has slumped over dead into the open trunk. The light is blinding her.
Lost, lost, all lost, little girl.
"Let go," Saben says, and the blue flame coils and uncoils like snakes, weaving and unweaving, and...
...All lost, lost, lost now, George Ballou's fairie birds sing, their machete voices carving great slashes in the world. All lost...
...and she sets the glass of Wild Turkey down on the wide mahogany dining table, and looks up at the tall clock on the mantel, the skin of a dead girl's face flayed and tanned and stretched taut, and the fish-spine hands of the clock stop, and then begin to move backwards. She reaches for the whiskey bottle...
...and Soldier opens her eyes, her forehead still leaning on the steering wheel.
"What next?" Odd Willie asks, and ash from the tip of his cigarette falls into his lap, and he brushes it away. "What the fuck do we do now?"
Soldier raises her head, an oily smear of sweat left behind on the steering wheel, and her right hand goes to the spot where the shard of the ruined bullet from Odd Willie's 10mm hasn't yet punched a hole in her jugular. She looks out the windshield, across the hood, and sees the black thing squatting in the road in front of the car, the thing that isn't a man and isn't a hound, either. It grins at her.
"No," Saben White says. "That's an awfully neat trick, Soldier, but it's not gonna save you this time," and the black thing stands up. Odd Willie screams and his gun goes off, blowing out the windshield and deafening Soldier in the instant before the butt of Saben's pistol connects with the base of her skull, and then there's only silence and winter cold and a bottomless, merciful oblivion.
## FIVE
## Angell Street
Emmie is sitting on the floor in her bedroom, her bedroom in Providence, home again after the three-hour train ride from Manhattan. They took the Acela Express, because it was so much faster, but the trip still seemed to take at least twice as long as usual. Sadie hardly spoke the whole way, and Emmie stared out the window at the countryside and the towns and the train stations rushing past. Somewhere in Connecticut, the sky turned stormy and dark; the clouds were like mountains piling up to crush the world. When they finally reached the Providence station, Deacon was there waiting for them. He looked worried and annoyed, but he hugged her tight, kissed her cheek and told her he'd missed her and was glad she was back so soon.
Home again, home again, jiggety-jog.
On Emmie's CD player, Doris Day is singing "Secret Love," and the volume's turned up loud, but she can still hear Sadie and Deacon arguing. They're downstairs in the kitchen, but their voices rise like warm air and get trapped in the heating ducts and travel all the way up to Emmie's room. She's shut the vent in the floor, but the voices are getting in anyway. She's sitting near the window, between her bed and the window, with her back against the wall. I wish they'd stop, she thinks. I wish they'd just shut up and stop and Sadie would go back home, because that's what she's going to do, no matter how much they fight. And then she wishes she were in school, that it wasn't a Sunday afternoon and it wasn't winter break. Then they could fight all they wanted, and she wouldn't have to sit here and listen to it.
"It wouldn't fucking kill you," Deacon says, and then dishes clatter in the sink. "You already have the week off. You could spend it here."
"No," Sadie tells him, the stubborn, angry tone she gets whenever Deacon suggests that she spend a night in the house with them. "You know I won't do that. It's not fair to ask me."
Emmie shuts her eyes and bites at her lower lip. They could fight outside, she thinks. They could walk down the street and fight all day long.
"It's fucking fair when you've already promised to take care of her this week."
"You know I won't stay here. There's no point talking about it, because you know I won't."
Emmie's already done the things she usually does whenever Sadie and Deacon are fighting. She's straightened her room, putting everything in its place because it always seems that creating order ought to help, even though it never has yet. She made sure all her toys and games and clothes were exactly where they should be. She checked to see that all her books were in alphabetical order, and all her CDs, too. She pulled the covers off the bed and then remade it.
"Hell, Deacon, I'll pay for you to hire a sitter."
"I don't want a sitter. You know I don't want strangers in the house. You know how I feel about that."
Emmie opens her eyes again and stares across her room at the closed door to the upstairs hall. If it were a door to somewhere else, anywhere else, she'd open it and step across the threshold and shut it behind her, and then run far enough away that she couldn't possibly hear Sadie and Deacon arguing. Emmie pulls a pillow off the bed—it doesn't matter that it messes up the blankets and sheets, because all the order she made isn't helping—and lays it over the heating vent. But she can still hear them.
"You know how rough things have been at the shop," Deacon says. "And you know damn well I've been counting on having this week to get caught up."
"She wanted to come home. What was I supposed to do? Tell her no?"
"I know it's news to you, but that is a fucking option," Deacon says.
On Emmie's CD player, Doris Day finishes singing "Secret Love" and "Teacher's Pet" begins. Emmie doesn't like that song, and she thinks about getting up and skipping ahead to the next track. She also thinks about making the bed again, just in case, but she stays where she is, listening even though she doesn't want to hear. Her stomach's beginning to hurt, and she has to pee, but their voices would only be louder out in the hall, louder in the bathroom because it's directly above the kitchen.
"It was horrible," Sadie says, bringing up the dead horse again, and Emmie chews her lip a little harder, chewing until she tastes a drop of blood. "That poor horse. Jesus, Deacon, I couldn't make her stay in New York after that. I just couldn't."
"That poor horse," Emmie says, and now she's thinking about the argument that morning between Hunter and Sadie. It was a fucking coincidence, Sadie said, and then there'd been the strange girl who wanted her to draw a bridge. It was getting hard to remember exactly what happened, and Emmie thinks she must have dozed off, listening to Hunter and Sadie, that she was tired and must have fallen asleep. She wishes she could fall asleep now.
"The two of you could rent a car and drive up to Boston or something," Deacon says. "You could spend the week in Boston."
"I can't fucking afford a week in Boston or anywhere else, Deacon, and neither can you. And Emmie doesn't want to be in Boston; she wants to be in her own home where she feels safe. It just about scared the hell out of her. Hunter said—"
"Don't fucking start with fucking Hunter Fontana," Deacon says, and then it sounds like he drops a skillet or a pot or something. "Right now, the last thing I want to have to hear is how that goddamn dyke thinks I should be raising Emmie. And how is it you could afford to pay for a sitter, but not a week in Boston?"
"Jesus, Deacon."
"You know, I've got enough to think about without the sage fucking advice of Hunter Fontana."
"It was a coincidence," Emmie says to no one and shuts her eyes again. "Co-in-ci-dence," she says, taking care to divide and separate each syllable from the other. "It was a coincidence." And even over the noise of her father and stepmother arguing, even over Doris Day, Emmie can hear the terrible, dull shattering sound of the taxi hitting the mare. She remembers the sound perfectly, and she doesn't think there's much difference between hearing it and remembering it. I'm never going to forget that, she thinks. Not ever. She opens her eyes and goes back to staring at the closed bedroom door. The girl from the attic, the brown-skinned girl, is standing near the foot of her bed, but Emmie tries not to notice her. Maybe if she doesn't say anything to the girl, she'll go away.
"That's an awful racket," the girl says. "Do they know you can hear them?"
"I'm not talking to you," Emmie tells her. "You're not real, and I'm not talking to you."
"I like this music," the brown girl says. "I've never heard such music. My father used to bring me music and play it on his Victrola. I liked 'April in Paris,' and 'Mood Indigo,' and—"
"I'm not talking to you," Emmie says again. "Go away."
"—'When the Moon Comes over the Mountain.' I like that one, too, but not as much as I like 'Mood Indigo.'"
Downstairs, Deacon curses and a glass breaks, and Sadie tells him to calm down and stop acting like a damned jackass.
"I could sing it for you," the brown girl offers, "if you'd like me to. I remember all the words."
"I know the words," Emmie says, which is true because she has a CD with Billie Holiday singing "Mood Indigo," one of the few CDs of hers that Deacon likes. "You don't have to sing it to me, because I know it already."
"Besides, it would be very hard to hear me over this awful racket," the girl says and frowns. "Do they know that you can hear them?"
"I don't know," Emmie replies, though she suspects Deacon and Sadie know perfectly well how sound carries in the old house, and they just don't care. "Maybe you should go downstairs and ask them."
"I would," the brown girl says. "But they probably wouldn't be able to see me or hear me. Your father might—"
"I wish you'd shut up and leave me alone."
"—he could do that, a long time ago, hear and see things no one else could see or hear. But he made it all stop. I guess he'd finally seen and heard enough."
Emmie stares at the girl. She seems solid, solid as anything else in the bedroom. She smiles at Emmie and sits down on the bed. The girl's wearing the same black dress with the same stiff white collar that she was wearing the first time Emmie dreamed about her. Maybe that's the only dress she owns, Emmie thinks and then silently scolds herself for thinking of the girl as if she were someone real.
"I didn't say you could do that," Emmie says.
"I was tired of standing," the girl replies. "And it was impolite of you not to offer me a seat."
"Stop it!" Sadie yells at Deacon, and Emmie flinches.
"Does he ever hit her?" the girl with brown skin asks.
Emmie shakes her head, getting angry, and she tries to stop imagining this, wishing now that she could just go back to having to hear the argument downstairs. "No," she says. "Deacon doesn't hit us. He's never hit either one of us. He doesn't do that sort of thing, not even when he's drunk."
"He's drunk a lot," the girl says.
"Yeah, he's drunk a lot. He's an alcoholic, but he doesn't hit us."
"The yelling's just as bad, sometimes," the brown girl says and stares past Emmie at the bedroom window. "You have a very fine house, Miss Emma Jean Silvey. I've never been inside this house before. My father once knew a man who lived across the street from here, a writer—"
"Don't you ever shut up?"
"I spend a lot of time alone," the brown girl says, still staring out the window at the house next door. "Well, that's not precisely true, because mostly it's not precisely time I spend alone. There's only time whenever someone visits me."
"I know about being crazy," Emmie says and glances past the girl at the closed bedroom door. If Sadie opened it, would she see the brown girl in her black dress, or would she see only Emmie sitting on the floor, talking to herself? "I've read books. You're a hallucination. It's probably because of stress, or maybe I have a brain tumor."
"You don't look sick," the girl says. "And I know that you aren't insane."
"How could you possibly know that? You're not even a real person."
Downstairs, the arguing has stopped. Someone slams a door, the front door, and Emmie figures that it's probably Deacon going out for a walk, going out to cool off and maybe have a drink. He'll probably walk for an hour or two, maybe follow Angell west to Hope Street, then turn north and walk all the way up to the Ladd Observatory, because he likes the benches there and being at the top of that hill. He'll stay away from Thayer, because it'll make him think of the shop, and she knows that's one of the things he'll be trying to forget. She's sorry that she couldn't have gone with him, not wanting to be in the old house right now, even with her hallucination for company and Sadie still right downstairs. But it's a relief that they've stopped fighting, relief like the battered silence after a bad storm late at night, and she feels herself start to relax a little.
"Something's coming, Emma Jean," the brown girl says. "Something terrible's coming."
"What are you on about now?"
The girl sighs and looks away from the window, looking down at Emmie. "I don't know everything, and I can't tell you everything that I know."
"Then what good are you?" Emmie asks her and goes back to watching the bedroom door.
"I've found a few flaws, cracks that I think perhaps my father intentionally left for me to find, cracks that the ghouls don't know anything about. I've come here to forewarn you, before it's too late for augury to be any good to you...or to me."
Now Doris Day is singing "Qué Será, Será," and the brown girl looks over her shoulder towards the stereo.
"I read you somewhere," Emmie says. "I made you from different characters in different books, from people in different movies. There's no point trying to convince me otherwise. I'm not a moron, even if I'm going crazy."
"You're not insane," the girl says again. "And you have to trust yourself, because there's going to come a time, very soon, when you can't trust anyone else."
"My father can't afford a shrink," Emmie tells her and kicks at the side of the bed with her sock feet until the girl turns back around again.
"I like that song," the girl says.
"It's an old song. Haven't you ever heard it before?"
"I like it. It's very pretty, but it's not really the truth." And then the brown girl sings along with Doris Day for a couple of lines—Whatever will be, will be. The future's not ours to see—and her voice is high and sweet and reminds Emmie somehow of melting ice.
"What's not true about it?" Emmie asks her, wondering if Sadie will come upstairs, wondering if she'll apologize for the argument and try to explain, hoping that she doesn't.
"I mean that there are some people, and some other things that aren't people, who can see what will be, except there's not only one thing that will be; there's an infinity of things that might be. That's what I mean."
"No one can see the future," Emmie says, finally standing up, and she turns her back on the brown girl, who isn't really there anyway, and stares down at the driveway and the evergreen shrubs and the wooden slat fence dividing their yard from the house next door, the house painted an ugly color like blue Play-Doh. There's a robin down there, though it's too early in the year for robins in Providence, and it hops about a bit before stopping and gazing up at her.
"There's so much that you don't know, Emma Jean Silvey," the brown girl says. "There's so much you're going to have to learn, and you'll have to learn it quickly."
"I know too much already," Emmie says. "I'm just a kid, and I know too much already."
"What's coming, it doesn't care how old you are."
"Nothing's coming, and you're not real. I don't believe in ghosts."
"I'm not a ghost," the brown girl says. "I never claimed that I was a ghost."
"I still don't believe in you," and then the robin flies away, and Emmie looks up at the sky. There are clouds, the sort of clouds that usually mean snow, and she leans forward, flattening her face against the cold glass.
"Later on you'll believe," the girl says, and Emmie sighs because this is getting tiresome. She wonders whether or not she should tell Deacon or Sadie about the girl.
"My father's not a bad person," Emmie says, and her breath makes the window fog.
"No," the brown girl says. "No, he's not. He's only a man who's seen things men weren't meant to see. He's only very tired and very lonely and very worried about you, Emma Jean. He loves Sadie, but he misses his wife very much."
"Shut up," Emmie says, starting to cry even though she doesn't want to, because she knows it isn't going to make her feel any better about the poor dead horse or Sadie or Deacon or anything else. "You don't know about Deacon, and you shouldn't act like you do," she says, turning around to face the brown girl, but she's gone, and now there's no one sitting on the bed. There's a rumpled place in the blankets, a rumpled place where she imagined the brown girl sitting, but that might have happened when Emmie pulled the pillow off the bed.
"Go away," Emmie says, as if the girl were still there to hear her, as if she hadn't gone away already. "Go away, and never come back." And then Emmie lies down on the bed and cries until she finally falls asleep.
And in her dreams, Emmie feels herself rise slowly up from the bed, her soul grown suddenly lighter than air, and she drifts untethered and insubstantial, passing straight through the bedroom ceiling into the shadows of the dusty, disused attic, and then through the pitched roof of the house on Angell Street. She lingers there a moment, a few feet above the chimney, and the sky rushes by overhead like traffic, clouds filled with ice and snow and sudden patches of bright daylight. She isn't afraid of falling, because it's only a dream, though someone at school once told her that if you fall from a high place in a dream and hit bottom before you wake up, you'll die. When she asked Deacon about it, he said it wasn't true, that it was only a silly superstition, so she's not afraid. She looks down and sees that the brown girl is watching her, is standing in the middle of Angell Street gazing up at her.
I'm her balloon, Emmie thinks. I'm her balloon on a string. Though she can't see a string tied to her anywhere. If she lets go, I'll float away forever and be lost.
All lost, lost, lost now... And those were not her thoughts, so Emmie thinks they must be the thoughts of the brown girl, because in dreams she can sometimes hear other people's thoughts.
The girl waves at her, and then a gust of icy wind carries Emmie away, south and east and up and up, over rooftops and treetops. And in only a moment she's floated so high that she can see the Providence and Seekonk rivers going down to all the bays and islands and rocky coves before the ocean. And now she is getting scared, because this is much too high, even for a dream, and if she goes much farther, she might fall into the sea and drown, or end up floating away to some land so distant that she'll never be able to find her way home again. She starts to call out for Deacon, but knows he won't ever hear her, and the air's so cold her words would only freeze and fall to earth like hailstones.
The plink of words on sidewalks and lawns, a sentence heavy enough to shatter a windshield.
"Wake up now," she whispers, and Emmie's beginning to wonder if she'll float all the way out of the atmosphere and die in the near vacuum of space, looking down on the great blue ball of the world, when she realizes that she's standing on a sandy beach near the water, and the brown girl is standing a few feet farther out, the waves lapping about her ankles, soaking her black-and-white-striped stockings. And the air has grown very warm, and a brilliant white sun blazes overhead. It isn't winter here. Maybe, she thinks, it's never been anything but summer here and never will be.
"This is the place where Esmeribetheda saw the warriors in the sky," the brown girl says. "That's one of the parts of the story that the changelings are never taught. They're taught that she was hanged and burned alive because she betrayed the ghul to the djinn, but the ghouls know that's not the truth."
"Where am I?" Emmie asks. "What are you talking about?"
"You're only dreaming," the brown girl says and smiles and kicks at the salty, frothy water. "But you're in the wastes, near the battlegrounds. This place has other names, but I never learned them. Or I've forgotten them. Six of one, half dozen of the other. Either way, you're at the end of the wastes, the wastes at the very end of this world. At least you're not cold anymore."
"No, now I'm hot," Emmie says and sits down on the warm white sand. There's a conch shell nearby, half-buried, and she picks it up and runs her fingers carefully along the sharp spines spaced out along the conch's low spire.
"The Hindu people," the girl says, taking a step towards shore, "they believe that Vishnu, the god of preservation, has a conch shell named Panchajanya, and they believe that Panchajanya represents all the living things that have ever arisen from the life-giving waters of the seas."
"Are you from India?" Emmie asks. "Is that why your skin's so dark?"
The girl doesn't answer the question, but stares up at the cloudless blue-white sky instead. "There's something terrible coming," she says again.
"You told me that already."
"This is one of the places where it began, after the changeling child Esmeribetheda was murdered. After the ghouls fought a great war with the djinn and were driven out of the wastes forever. This is the place where the warriors still battle in the sky to keep balance against the destruction of all creation. This is where the sea meets the sand, where the two are always struggling to expand their earthly empires."
Emmie sets the conch shell back down. "There was a woman on a train," she says. "A woman with the Seal of Solomon tattooed on one of her hands."
The brown girl shakes her head and looks down at her wet feet. "We can't talk about her. Not just yet. Later. Soon."
"There's no point to this, is there? You should just let me wake up."
"I cannot stop you from waking, Emma Jean, but it's not pointless. There are things you need to see."
"Well, I don't need to see sand," Emmie tells her. "I've seen lots and lots of sand before."
"It's not the sand that's important," the brown girl says, and then Emmie's floating again, or maybe this time she's falling. There's no way to be sure exactly which, but now there's a lot more than sand. There are battles worse than anything she's ever imagined, men and horses and bristling hairy creatures that walk like men, blood and fire and the screams of the dying. Swords and spears, arrows and axes, and hungry crows and jackals waiting for their share. There are vast maelstroms in the sky, whirling clouds like hurricanes of flame, and monsters perched at the edges of the maelstroms, monsters tearing one another apart, and Emmie wants to shut her eyes, but it doesn't seem to make much difference. She is seeing these things, because the brown girl is showing them to her. The ruins of a city half-buried in the shifting dunes. The bed of a vanished lake, littered with the skeletons of water monsters. A girl tied to a tree and burned alive. The pages of a storybook, turned before Emmie's eyes, turned behind her eyes, and whenever it gets too scary, she reminds herself that it's only a dream, and she'll probably forget most of it when she wakes up.
"Open your eyes," the brown girl says, so Emmie does, and now she sees they're sitting on the roof of her house on Angell Street, sitting side by side, and there are fat, wet flakes of snow falling from the blue-gray sky.
"That was ridiculous," Emmie says and shivers. "None of that was real. You're not real, and none of those things were real, either."
"A long time ago," the brown girl says, as if Emmie hasn't said a word, "before you were born, Deacon Silvey was fighting a war, too. It wasn't that sort of war, but he was still a soldier, after a fashion, and he fought demons and worse things than demons."
"Bullshit," Emmie says, just like Deacon would say it, and scowls at the brown girl. "Deacon used to work in liquor stores and laundries, and now he owns a used bookstore on Thayer Street. He was never in the army or the navy or anything."
"That doesn't matter. He was still a soldier. He didn't want to be, but he was. He could see things, and so the policemen came to him, and he helped them catch murderers and rapists and thieves."
Emmie stares at the girl, at her dark hair now tangled by the wind and speckled with melting snowflakes. If any of that were true, Deacon would have told her. Deacon's told her lots of stories about before she was born, stories about when he lived in Atlanta and Birmingham, stories about her mother, whose name was Chance, and he's never said anything at all about helping the police.
"You're a liar," she says. "None of that's true."
"Once, he helped them find a woman named Mary English," the brown girl continues, undaunted, "a mule, a madwoman who worked for the Cuckoo, stealing babies to be hidden away in the dark places of the world and raised up by the ghouls. That day, he even saved a girl who should have become a changeling."
"You're full of it, and I'm cold, and I'm going to wake myself up now."
"There's not much time, and there are things you need to understand."
"I need to wake up," Emmie tells her and crosses her arms, trying to stay warm. "I'm gonna freeze to death up here."
The brown girl sticks out her tongue and catches a snowflake. She nods her head and points northeast, to the place where the clouds are coming from. "His daughter will come for you, Emma Jean. She may be dying, and she may not live, but if she does, she'll come for you, because you're a part in this. And there will be others, men who mean you harm, who will kill you if they can, and you have to be ready for them."
"Wake up," Emmie says. She's shut her eyes and is trying to think of nothing but her bed, her room, Doris Day on the CD player, the warmth inside the house. "Wake up, and forget all of this."
"You met your mother on a train," the girl says. "She gave the rest of her life to meet you just that one time. She gave her life to save you."
"Wake up," Emmie says, and she's started crying all over again. The tears are freezing on her cheeks, and they clatter down the steep sides of the roof and break on the driveway. "Wake up now. Wake up now."
"There's a box," the brown girl says. "A cardboard box beneath his bed. When you see that I'm telling you the truth, Emma Jean, you'll know that you can't stay here in this house anymore. Something terrible is coming."
Emmie screams, opens her eyes and swings at the girl, both her hands clenched tightly into fists. The brown girl shrieks and breaks apart into a flock of robins and is swept away by the wind.
"Leave me alone!" Emmie screams after the fleeing birds.
All lost, lost, lost now...
"Just leave me alone," she sobs, all the fight slipping quickly out of her, and the roof opens up like the maw of a hungry creature built of shingles and wood and nails, and it swallows her whole. She slides easily down its black gullet, and when Emmie remembers that the creature was only her house, she opens her eyes. And she's lying in her bed again, and Deacon's holding her. He smells like whiskey and cigarettes, and he's smiling.
"Hey, kiddo," he says. "You were having a bad dream. It's okay. You were just having a bad dream, that's all. How long have you been asleep up here? Do you know it's snowing?"
"Yes," she says, and Emmie starts to tell him about the brown girl and the endless desert beside the sea, the wars and flying high above the city, and then she remembers all the other things that the girl told her, all the crazy things she knows aren't true, but they're still too frightening to think about. Too scary ever to say out loud, and then she's crying again, crying awake and for real this time, and Deacon holds her and whispers soothing things until she stops.
By seven o'clock, there are two inches of snow blanketing Angell Street, and the house is quiet and still, the heavy, e it assumes whenever there's snow. Sadie's gone back to New York; she left while Emmie was asleep, and she told Deacon to tell Emmie that she was sorry she couldn't stick around and say good-bye, but she didn't want to miss her train. She promised to call later. And Deacon's gone to the shop, because Jack's sick or his car wouldn't start or something like that. Emmie told him it was okay, that she would be fine alone, and he said there was a frozen dinner, fried chicken with peas and apple cobbler, and that's what she should have for dinner. "You need something more than just a peanut-butter-and-jelly sandwich," he said, and she promised him she'd eat the frozen dinner, even though she hates fried chicken.
She's downstairs watching the snow fall through the tangerine light of the street lamps, watching as it hides the broad leaves of the rhododendrons growing beside the front porch. The leaves will freeze and begin to curl in on themselves by morning. It's hard to tell where the yard ends and the sidewalk begins, where the sidewalk ends and the road begins. She halfheartedly contemplates getting bundled up and going outside, even though it's after dark. Deacon didn't say that she couldn't, but, then again, she doesn't really feel like playing in the snow.
I could find the place where the sidewalk meets the grass, Emmie thinks. I could find the place where the street begins. And she considers carefully marking them off with sticks or rocks. But the snow would only bury her markers, because the Weather Channel says there'll be at least six more inches before morning. If the world out there is determined to erase all distinctions, she might as well let it. At least there are slushy, dark tire marks on the street, cars going one way or another; later, a truck will be along to salt the road. Emmie glances at the clock above the sofa. It's seven twenty-three, and Deacon won't be closing up the shop until nine, even though he was grumbling about how the snow would probably keep a lot of the customers away. She wanted to say, You never have any customers, anyway, but she didn't. Sometimes someone comes in and buys a book, but mostly people don't come in, and when they do, they usually walk about browsing and reading and leave without buying a thing. "It's not a goddamn library," Deacon says sometimes. "I can't pay the bills with their curiosity."
Emmie leans on the windowsill, resting her chin on her folded hands, wiggling in the chair so the legs squeak against the dingy hardwood floor. It seems pointless, but she wishes Deacon could have stayed home, or that Sadie hadn't needed to rush back to Manhattan. The wind sweeps along Angell Street, making something like a dust devil from the fresh snow; Emmie hears someone coming down the stairs, and she turns to see who it is. But she already knows, because that's what she's really been doing all evening, sitting here waiting for the brown girl to come back.
"I never cared for the snow," the girl says. She's standing near the bottom of the stairs, looking at the television as though she's never seen one before. "It was snowing when my mother died."
"Your mother died, too?" Emmie asks, and of course the brown girl's mother died, too. She had to have, because the brown girl is only some part of Emmie that's decided it wants to be more than a part, that it wants to be a whole person.
"My father tried to save her, but the doctors couldn't help, and he couldn't help, either. It was the influenza, and she developed pneumonia. My mother was a powerful sorceress," the girl says proudly.
"Sure," Emmie says. "Every winter she rode about in the snow offering children hot drinks and Turkish Delight, and that's how she caught her death of cold."
"What's Turkish Delight?" the girl asks.
"Don't be stupid. If I know, then you know, too."
The brown girl stares at the television a moment more, then glances back up the staircase. "You haven't found the box yet, have you?" she asks.
"There isn't any box," Emmie tells her and turns back to the window. "You made that up. I mean, I made that up."
The brown girl sighs and sits down on the stairs.
"I'm going to have to tell Deacon about you," Emmie says. "I'm going to have to tell him, and he's going to make me see a doctor or a psychiatrist or something."
"I wish you'd believe me," the girl says. "There's so little time left."
"I'm sorry that your mother died," Emmie replies, then watches a big yellow SUV moving slowly down Angell Street, its headlights shining through the falling snow.
"My father made a blood sacrifice to Father Kraken and Mother Hydra, and he used his strongest magick and some herbs that had come all the way from Persia and China, but nothing could save her."
"Who are Father Kraken and Mother Hydra supposed to be?" Emmie wonders aloud, trying to remember if she's ever read a book where she saw those names. For a long moment, the girl on the stairs is silent, and when she answers she sounds both amazed and like maybe she feels sorry for Emmie for being such an ignoramus.
"Of all the changelings," the girl says finally, "you may be the only one who would ever have cause to ask such a question. They are the Pillars of the Sea, Emma Jean, the Keepers of the Abyss, Mother and Father, destruction and conception and everything that lies anywhere in between. In the days of void and shadow, before the gods had grown weary of nothingness and pulled the land up from the sea—"
"I get the picture," Emmie says. "God didn't save your mother, but you still believe in him. It doesn't matter to me. I'm an atheist."
"An eight-year-old girl shouldn't be an atheist."
"Well, this one is," Emmie says and looks over her shoulder at the brown girl, who's staring at her between the banister rails. "Anyway, you could at least be consistent."
"What are you talking about?" the brown girl asks and scrunches up her face.
"I mean, kraken is a Norwegian word, which is sometimes used to refer to the giant squid, Architeuthis, or to mythical sea monsters in general. But the Hydra, on the other hand, is a nine-headed monster that Hercules had to kill. Killing the Hydra was the second of the twelve labors of Hercules. So, what I mean is that you're mixing Norse and Greek mythology, and that's inconsistent and stupid. Now, if you'd said Mother Hydra and Father Poseidon, well, that might be a little less silly."
The brown girl watches her for a moment, and Emmie can't read much of anything in her chocolate eyes, if she's insulted or angry or just confused. Then the brown girl shakes her head, the way one of Emmie's teachers might shake his or her head if a student is being particularly dense, and she stands up again.
"The Keepers have many names among the tribes of men," she says. "Names are only tools. Am I supposed to believe that you're a figment of my imagination, because Silvey is derived from a Spanish name, Silva or Silvera, but Emma comes from Old German and Jean is French? Or because Jean is a man's name—"
"That's not the same thing at all," Emmie says, her cheeks flushed, and she stands up so fast that she almost knocks the chair over. "You're not very smart if you don't see the difference."
"But a little while ago you said I am you," the brown girl reminds her, "so how can I be any less bright than you yourself?"
"Shut up," Emmie says, because it's a lot easier than trying to answer the girl's question. As a barb, she adds, "Deacon says that religion is a crutch."
"Why do you call him that?" the girl asks. "Why do you call him Deacon?"
"Because that's his name."
"Why don't you call him Father or Daddy or—"
"Why don't you mind your own business?" Emmie says, and the brown girl shrugs and stands up, straightening her stockings and black dress.
"Maybe it's because you know something, something you don't want to admit."
"That's a lie," Emmie tells her. "I call him Deacon because that's what he wants me to call him, because it's his name."
"We're wasting time we can't afford to squander," the brown girl says. "I was hoping you could find the box on your own, but you haven't even tried."
"I was busy. I was watching the snow."
"I've never much liked the snow," the brown girl says again. "My mother died when it was snowing."
"I might be schizophrenic," Emmie says, half to herself, and thinks about sitting back down in the chair. "If I'm hallucinating and having delusions and all, I might be schizophrenic."
"Or it might be something else entirely. Maybe you're like Deacon once was. Maybe you can see the things that other people can't. You know, it's a sad and unfortunate state of affairs that you have to live in a world where eight-year-olds refuse to believe in anything that they cannot touch or measure, and anyone who happens to see a thing that is invisible to most people is immediately branded a lunatic. Your stepmother still believes in magick."
"Sadie's a writer," Emmie says, repeating something else she's heard Deacon say. "Writers are different. They're supposed to be weird."
The wind is getting stronger, and it buffets the house on Angell Street and moans like a wounded animal as it slides around the corners.
"Come upstairs with me," the brown girl says. "I need to show you something. We don't have much time left."
"What if I want to stay right here and watch the snow?"
"The snow will still be out there when we're done."
Emmie glances about the small living room, surprised at how very dark it's gotten, the only light the shifting glow of the television set and the muted orange of the street lamps coming in through the windows. I should turn on a lamp, she thinks. What if Deacon comes home, and all the lights are out? He might think that something's wrong.
Something is wrong, the brown girl thinks, but Emmie can hear it just as clearly as if she'd said the words aloud. Something's terribly wrong, and we need to hurry.
"That's proof I'm hallucinating you," Emmie says, and then she switches on the old floor lamp near her father's chair, making a small pool of safe cream-colored light, before she follows the girl upstairs.
"Why do you have yellow eyes?" the boy asked, and Emmie told him she didn't know why, that they'd just always been that way, and she didn't have any idea why. She'd been asked the same question lots of times, and sometimes she'd been taunted and called names—yellow eyes, yellow eyes—lots more times than she could remember or cared to ever try. Sometimes, she made up elaborate stories to account for her yellow eyes, or told whoever was asking or bullying her that she had a contagious disease, like hepatitis or the plague, and they really shouldn't come too close. That was usually enough to make them leave her alone. But the day that this boy asked—which was the last day she went to the public elementary school on Camp Street, the last time someone asked before Deacon and Sadie finally moved her to the Wheeler School where hardly anyone ever asks about her eyes—that day, she didn't feel like lying, so she just told him the truth, that she didn't know why her eyes were yellow.
"I used to have an old tomcat with yellow eyes," the boy said. "Maybe you're part cat."
"I'm not part cat," Emmie replied. She was waiting out front, sitting on the stone front steps of the school waiting for Deacon to show up and walk her home. "I have yellow eyes, but I'm not part cat."
"Maybe your mama's a cat," the boy said.
"My mother's dead. She was a paleontologist, not a cat," Emmie told him, wishing that Deacon would hurry up or that the boy would get bored and find someone else to pick on.
"Maybe they just told you she was dead. Maybe they didn't want to tell you she was really a cat. Maybe they're ashamed to have half a cat for a daughter. Do they glow in the dark?"
"Do what glow in the dark?"
"Your eyes, dummy."
"Leave me alone," Emmie said, though she knew that he wasn't going to stop until someone made him stop.
"That would make you a monster," the boy said, "if your mama was a cat. Maybe they don't want you to know you're a monster. Maybe they think if you knew, you'd kill yourself or something."
"Lots of animals have yellow eyes," Emmie said. "Not just cats. Lots of animals have yellow eyes."
The boy smiled and sat down on the step next to her. "Yeah? So maybe she wasn't a cat. Maybe she was one of those other animals," he said. "But you'd still be a monster, because normal people don't have yellow eyes or animals for mothers." The boy was fat, a year older than she was, and he had a short, piggy sort of a pug nose.
"Maybe your mother was a sow," Emmie said, and the boy asked her what the hell a sow was. When Emmie told him it was a mama pig, he called her a freak and a retard and kicked her in the ankle.
By the time one of the teachers was able to pull her off the boy, Emmie had blackened both his eyes, knocked out a front tooth, and bitten him three times on the face. Two of the bites were deep enough that he would have to have stitches, but at least Emmie never had to go to school with him again. When Deacon finally arrived—almost fifteen minutes late because the toilet at the shop had backed up, and he'd had to wait on the plumber—he found her sitting in the grass, surrounded by three teachers, the vice principal, a crossing guard, and several other students. There was crusty, drying blood on her face and hands and the front of her white Curious George T-shirt. Her ankle was starting to swell and turn an angry cloudy color, but she wasn't crying. The boy was still screaming and rolling about on the ground, yelling that Emmie had almost bitten off his goddamn face, and now he was going to die of rabies because she was half cat.
Later, after they'd gone to the emergency room and knew that her ankle wasn't broken, Deacon took her home, and they sat at the kitchen table, eating grape Pop-Tarts and talking.
"He called me a monster," Emmie said. "He said you were ashamed of me because I'm a monster and a freak and a retard and because my mother was a cat."
Deacon chewed his Pop-Tart, washed it down with a mouthful of Coffee Milk, and then he scratched at his head. "And you think that was any way to show him different, acting like a zombie or something, trying to eat off half his face like that? You know you're not a monster. And you know better than to get in fights with assholes that call you names."
"I wouldn't eat his face," Emmie said, picking some of the purple filling out of her Pop-Tart. "You can get worms from eating raw pork."
And then Deacon was laughing so hard that Coffee Milk squirted out both his nostrils, which made Emmie start laughing, and by the time they'd both stopped, Deacon was out of breath and Emmie was beginning to feel a little bit better. He didn't punish her, told her that the swollen ankle was probably punishment enough, but he had to apologize to the boy's parents, and Emmie had to apologize to the boy in the principal's office. There were white bandages on the fat kid's swollen face, and he didn't look at Emmie once the whole time.
Emmie finds the cardboard box beneath Deacon's bed, right where the brown girl said it would be, and she pulls it out and opens it.
"There's very little time," the brown girl says. "You wasted so much of it, piddling about downstairs, staring at the snow."
Emmie nods her head, but she takes her time emptying the box and spreading its contents out on the floor of Deacon's bedroom, the room that used to be his and Sadie's, the room that never was his and her mother's bedroom. There are photographs, newspaper clippings, old letters written by hand, old letters written on a typewriter that punched out holes in all the Es and Os. There's a dried daisy pressed between two sheets of waxed paper. There's a much smaller box filled with ticket stubs from movies, and there's a beaten-up old copy of a Dr. Seuss book, McElligot's Pool. There's an envelope marked WR with a gold ring inside. There's a stack of "Calvin and Hobbes" strips clipped from the funny pages, all of which have something to do with dinosaurs. There's a Bullwinkle key chain with seven keys on it, and, wrapped in tissue paper, there's a reddish brown rock about as big as Emmie's fist with a dime-sized trilobite fossil embedded in it.
These are the pieces of a puzzle, she thinks. Now, if I just knew how to put them all together—
"But you don't," the brown girl says. She's sitting in the window seat, watching Emmie impatiently, "and there isn't time now to figure it out. You should look at the newspaper clippings."
"These are pictures of my mother," Emmie says, setting down the trilobite, returning it to its crumpled tissue, and picking up a stack of Polaroids. She knew her mother's face. Deacon kept a photograph of her on the table beside his bed, and there were others in a scrapbook downstairs. This was the same woman, the same bright green eyes, and in this picture she's smiling and holding a bottle of Coca-Cola. In the next one, her mother isn't smiling. She's sitting on a sofa reading a book, but Emmie can't tell what the book might be. In the third Polaroid, Chance Silvey is standing on a beach, and there's a dead pelican on the sand near her feet.
"You don't grasp how precious time is or how little of it we have left," the girl tells Emmie. "You're wasting it. Look at the clippings."
"But I've never seen these photos," Emmie says, still examining the Polaroid of her mother and the dead pelican. The ocean is flat and silver-blue in the background; the sky's almost the same color.
"That's because he hid all these things from you. Deacon's hidden many things from you."
"That's a damn lie," Emmie snaps at the girl, but she puts down the stack of photos and reaches instead for the nearest bunch of newspaper clippings. They've gone golden brown and brittle around the edges, almost like something that's been burned, but Emmie knows it's just because the paper's old and was so acidic to start with.
"Read the headline," the girl says.
And Emmie does, aloud—"'Atlanta Police, FBI, Used Local Psychic, Source Claims.'" Emmie stops and checks the date—August 23, 1989, twelve years before she was born. Deacon would still have been a young man, and Emmie does the math in her head; in 1989 Deacon would have been only twenty-seven years old. "This was almost twenty-one years ago," Emmie says, and the brown girl frowns.
"That's not the part that's important. Keep reading the article."
And Emmie looks back down at the clipping, but not, she promised herself, because the brown girl said she should. This cardboard box has been filled with secrets her whole life, filled with secrets and hidden all those years right here beneath her father's bed, where she could have found it on her own any day or night, if she'd only bothered to go looking.
"But you didn't find it on your own," the brown girl says haughtily. "If I hadn't come, you probably never would have."
"That's annoying, and I wish you'd cut it out. Stay out of my head. Anyway, they aren't my secrets. They're Deacon's secrets, and this isn't right. I think it's almost like stealing."
"You should keep reading," the brown girl says. "We're quite nearly out of time now. They'll be coming soon, and I need you to understand."
"Then shut up and stop talking to me," Emmie says, and she goes back to reading the article.
"I didn't come here to lie to you. If you'd have believed me, you wouldn't have needed to see the contents of this box."
But Emmie is no longer paying any attention to the girl, too busy with the words printed on that scrap of paper, a source close to the Atlanta PD claiming that the Mary English case had been cracked using a psychic named Deacon Silvey. The woman who was accused of having kidnapped and murdered fifteen children between 1982 and 1989, though no bodies had been found. The newspaper article said that Deacon had helped the police with other cases, and that he'd once been a student at Emory University. It also said that he was there with them when the cops found Mary English in the cellar of an old house in the woods, and that he helped save the life of a four-year-old girl named Jessica Hartwell. When she comes to the last line, Emmie puts the article on the bottom of the stack and begins reading the next clipping—"Inside Douglas County 'House of Horrors.'" There's a photograph of a very old house; the roof's sagging in the middle, and there are vines covering one side of the wide porch.
"Mary English was a woman in the employ of the hounds," the brown girl says. "And the policemen never found the missing children because she didn't kill them. She merely delivered them to the Cuckoo."
"I don't know what the hell you're talking about," Emmie says, staring at a faded photograph of a doorway. An arch has been woven out of old dried vines, maybe some of the vines from the front porch; three human skulls have been nailed above the arch, each one wearing a crown of what appears to be rusted barbed wire, the words LAND OF DREAMS printed in neat black letters on the door underneath.
"Deacon Silvey never should have crossed that threshold," the brown girl says, "for in doing so he interfered in the dealings of the Hounds of Cain. His wife died because he went through that door."
Emmie lays the clippings down, because she doesn't want to read any more, doesn't want to know any more, even though she's still not sure what it all means. "He never told me about any of this stuff," she says quietly, and glances at the stack of Polaroids again.
"He was trying to do something kind, helping the police," the brown girl says. "He was trying to do something good. But he never wanted any part of it."
"But what he did, saving that girl, that got my mother killed?" Emmie asks, and she touches the dead-pelican photo with the tip of her right index finger.
"It's a very complicated story," the brown girl tells her. "There's not time for it now. But that woman is not your mother, Emmie. I know that you've always believed that, that you've been taught that she is, but that doesn't make it true. She's not your mother, and Deacon Silvey isn't your father."
"You're a goddamn liar," Emmie says and pulls her hand back from the Polaroid.
The brown girl stands up and goes to one of the stacks that Emmie has made, the stack of typed letters, and sits down on the floor. She takes an envelope from the stack and hands it to Emmie.
"Your stepmother wrote this to Deacon. You need to read it," she says. "Deacon's wife was pregnant when she died. It's a very long story, Emma Jean, but she died fighting a monster, a much more terrible monster than Mary English had been, a monster that tried to kill Deacon and Sadie and was trying to kill Deacon's wife and her child."
Emmie opens the envelope. The old paper crinkles loudly in the quiet room, all sound stifled and yet made somehow more distinct by the falling snow. There's no date on the letter, but it has Sadie's signature.
"The hounds took Chance Silvey's child for their own. They left you with Deacon, after Chance had died, and he has always believed that you are his daughter. But your mother is a changeling named Saben White."
"Yeah," Emmie says, remembering the woman on the train, the woman with the Seal of Solomon tattooed on her hand, the woman who knew about invisible things that were there even if you couldn't see them. "I met her on the train to New York," she says, and Emmie feels dizzy and sick to her stomach and has to shut her eyes a moment.
"I know it's a lot to have to hear, a lot to take in all at once like this," the brown girl says. "I'd have come to you sooner, had I been capable of doing so."
Emmie opens her eyes again and begins reading the letter. You know that's not Chance's child, her stepmother typed. It isn't over. That isn't Chance's baby, and this all has something to do with Narcissa Snow and whatever she was trying to do. Jesus, I wish you would answer the phone. I wish you would just talk to me. I don't like writing this stuff down in letters and mailing it. I'm always afraid someone else will get the letters.
"This is sometime after she left," Emmie says and chews at her lip.
"Yes," the girl says. "Have you seen enough? Do you believe me now?"
"I don't know what I believe," Emmie replies. "I don't even think I know what's for real anymore."
You can't doubt the blood and DNA tests, Sadie's letter reads. She isn't your daughter, Deke, and she isn't Chance's daughter, either. I talked to the doctor. I know they weren't supposed to talk to me, but they did. Her blood work had them all a little freaked out.
Emmie folds the letter and returns it to the envelope, returns the envelope to the stack on the floor. Then she picks up the trilobite fossil again. The weight of the rock feels good in her hand. There's a tiny handwritten label glued onto the bottom that she didn't notice the first time she picked it up: CRYPTOLITHUS GIGAS, CHICKAMAUGA LMS., RED MTN. CUT, BIRMINGHAM, SUMMER '73.
I won't cry, she thinks. I'll cry later, when she's gone, but I won't cry sitting here in front of her.
"There will be time to think about it later," the brown girl says, standing again. "But now you have to leave this house. The Bailiff is coming, and you must get far away from here before he finds you. I know a place where you'll be safe. You'll need warm clothes—"
"Then who is my father?" Emmie asks, interrupting and ignoring the brown girl. "If it's not Deacon, then who's my father?"
"We can talk about that later, when you're safe. It's nothing you need to know just yet."
"Was he a cat?" Emmie asks, and the brown girl stares at her, then looks over her shoulder at the bedroom window. Outside, the snow seems to be falling even harder than before. "Was he a monster?" Emmie asks.
The brown girl kneels down in front of her, unmindful of the clippings and letters and photographs she's crushing beneath her knees. Emmie wants to slap her, wants to tangle her fingers in the girl's ebony hair, wants to claw her and bite her and kick her until she admits that all of this is a lie. That she wrote the letter, that those aren't even real newspaper articles. Instead, Emmie only bites her lip and sets the trilobite down again.
"Please trust me," the girl says. "If you stay here, in this house, they'll surely find you before the morning, and they will kill you, Emma Jean."
"Who? Who's coming?"
"We need to get you dressed. It's very cold out there. It's still snowing. And it's only going to get colder, I'm afraid, before it gets any warmer."
"I'm going to call Deacon. I'm going to ask him if this is true. He won't lie to me—"
"Listen to me," the girl says, and now she leans close and grasps Emmie's chin in the fingers of her right hand. Suddenly, she looks very old, a shriveled old woman wearing a clever little-girl mask, and her eyes turn as red as ripe cranberries. "If they find you here they will kill you, and if they find Deacon, they'll have to kill him, too. If you leave now, and if they know you've left, they'll have no reason to come here looking for you, and Deacon will be safe. Now, get up and get dressed. We have to leave this house, Emma Jean."
And then another instant passes, and she's only the brown girl again, the girl from the attic, the girl from a dream, and she blinks once at Emmie and lets go of her face. "I'm sorry," she says. "But there's no more time."
"This isn't happening," Emmie says. "I'm asleep, asleep in my bed, and in a moment Deacon will come and wake me up. I'll tell him what I dreamed, and he'll laugh at me. I'll tell him I ran away in the snow with a crazy girl, and we hid so monsters wouldn't kill him, and he'll laugh at me."
"Yes," the brown girl says. "That's exactly what you'll tell him, but first you have to run away. First you have to hide."
"But it's only a dream," Emmie reminds the girl, and then she reaches for the envelope marked WR and takes out the gold wedding ring and slips it on her thumb. But it's too loose and falls off, so she puts it in one of the front pockets of her jeans instead. "I should clean all this up," she says. "Deacon will be mad if he comes home and finds this mess."
"There isn't time. He'll understand."
"But these are his secrets," Emmie says, and she knows that it's not right for them to be spread out all over the bedroom floor for anyone to see.
"Now," the girl says, and Emmie does as she's told, because it's only a dream, after all, and when she's awake again Deacon won't know about the box if she doesn't tell him. The brown girl disappears into the hallway, and Emmie goes to put on warmer clothes and find her coat.
And half an hour later, Emmie is walking east along Waterman Street, trudging forward with the freezing north wind pressing brusquely at her back, driving her, hurrying her along, and the snow swirls about her like fat white fairies. Her breath puffs out in foggy clouds, and the fairies slowly melt whenever they collide with the warm patch of exposed skin around her eyes and the bridge of her nose. This is the way the brown girl said that she had to go, east down Angell and then one street over to Waterman, east on Waterman to Ivy, south on Ivy to East Manning, and then on to Gano Street and the woods at the edge of the river, and there the girl would show her the place where they'd hide until it was safe to come out again. The snow's ankle deep, and she figures the monsters must have sent it to slow her down.
It's smothering everything just to get at me, Emmie thinks and stops walking, looking back to see if the brown girl's still following her. She's been harder to see since they left the house, but Emmie knows she's back there somewhere, because she can still hear her footsteps. From time to time the girl says something, not speaking out loud, just a few words whispered directly into Emmie's head.
Don't stop, Emma Jean. Keep moving.
Or: You can't stop here. It's getting late.
Emmie doesn't know how late it is—eight thirty, almost nine, maybe, and Deacon will be home soon. The streetlights spaced out along Waterman make pools of Creamsicle brilliance in the slippery, uncertain darkness. Overhead the sky is lost in the storm, and the bellies of the low clouds are the same soft orange-white as the light from the street lamps. Emmie walks a few more steps, wipes the melted snow from her eyes, then stops again beneath one of the mercury-vapor lights.
"Where are you?" she shouts back at the spot where she thinks the brown girl might be. "I can't see you."
You don't need to see me. I've told you the way to go.
"I'm freezing my rear end off," Emmie yells back, and she is, even though it's only a dream (she reminds herself), even though she has her good coat—not the pink zebra stripes, but her blue down-filled parka—and her mittens, gloves under the mittens, warm wool socks and her new winter boots, the blue ones from L.L. Bean that Sadie gave her, rubber and nylon and a drawstring to keep the snow from getting inside. She's wearing a sweater and a T-shirt under that, her toboggan cap and muffler, and she knows she ought to be warm. But the cold seems to come right through her clothes, right through her, as though she were no more substantial than the brown girl or the swirling snow.
You can't stop here, the brown girl says urgently. It doesn't matter if you're cold and tired; you can't stop. You can rest when we've reached the tunnel.
"What tunnel?" Emmie asks her. "You didn't say anything about a tunnel." It's getting hard to talk because her teeth have started chattering again and tunnel comes out more like tuh-uh-uh-nel, like a kid Emmie knew in second grade who had to have speech therapy because he stuttered.
You'll see. But you can't stop here.
"I got out of the house. You told me I had to get out so Deacon would be safe, and I'm out, so why do we have to hide?"
Because I also have to keep you safe, Emma Jean, because I also... And then the girl falls silent, and for the first time since they left the house, Emmie feels alone and scared and lost in the disorienting white blur of the storm.
"And because what?" she asks, but the brown girl doesn't answer her. Emmie looks up and down the length of Waterman Street, as far as she can see through the storm, but there's no sign of her anywhere. The houses, the trees, the fire hydrants, the cars parked along the side of the road, everything is rapidly vanishing beneath the snow. Everything is becoming something new, something indistinct and menacing, and Emmie wonders if she could find her way back home or if maybe she really is lost.
It doesn't matter if you get lost in a dream, she reminds herself. You're found again as soon as you wake up.
And then there's a flash of lightning and, only a few seconds later, a rolling, rumbling thunderclap that seems to begin at one end of the street and roll back and forth, back and forth, as if the simple sound of it means to crush everything flat and dead.
"What's happening?" she shouts back at the brown girl, at the murky place where she might have been. "I don't know what I'm supposed to do!" But the thunder steals her frightened voice and grinds it to silent pulp in the snow.
You can't be afraid of them, the brown girl says inside her head. No matter what, you can't be afraid of them. And then she's gone again, and there's only the storm and the snow-covered world and the thunder rolling up and down Waterman Street.
No, Emmie realizes, not thunder. It's not thunder at all. Only something that wants me to think it's thunder, or something that can't help but sound like thunder when it moves. And she remembers a book she has about mastodons and woolly mammoths, towering, shaggy things wandering about in the shadows of ancient glaciers, and Emmie also remembers a scary story that Deacon read her once, a story about the Wendigo, an enormous monster or Indian god that lived on the wind and snatched away people foolish enough to go too deep into the wild places it haunted.
"It's only thunder," she says. "It's only the storm playing tricks with sound, playing tricks on my mind." And then Emmie ducks quickly behind a sugary lump that was a holly bush before the storm began, and she crouches down and waits for the noise to stop. Help me, she thinks, hoping that the brown girl is listening and will hear her thoughts. Help me. Make it stop. Make it stop looking for me.
The brown girl doesn't answer her, and Emmie thinks that maybe it's because she's afraid to, that maybe the thunder monster can hear their thoughts, too, or maybe it's because it's found the girl already and has torn her apart or eaten her or whatever it does. Emmie peers out between the branches of the holly bush, at the abandoned white waste of Waterman Street, and she wonders why there aren't any cars. If there were a car or a truck or just someone walking by on foot, maybe there would be someone who could help her. This time of night there should still be traffic; even with the snow, there should be people coming home from work.
And then she sees it, the stilt-legged, tattletale gray thing that's making the noises she only thought were thunder. It isn't a mammoth or a mastodon—though it has long, shaggy hair—and she's pretty sure it isn't the Wendigo, either. It isn't anything she knows a name for, and maybe it has no name. Maybe you would die, if you even tried to name it. The creature seems to glide effortlessly out of the shadows, drifting along light as a wisp of radiator steam, moving like a whirl of fallen snow stirred up by the wind. But its footsteps are still thunder and lightning, betraying the weight of all the evil trapped deep in its roiling black-hole belly, betraying the gravity of the thing. Its footsteps are the shattering of stone by vast iron hammers, the grinding of pack ice at the bottom of the world, the collision of continental plates, the impossible gait of a thing so heavy that Emmie wouldn't have believed the universe could even contain it, that it doesn't simply tear a hole in space and time and vanish forever.
I'm dreaming, she reminds herself, and when I'm awake, I'll forget it. I'll forget I ever saw it.
It has no eyes that Emmie can see, and when the thing pauses near the holly bush and sniffs at the February night, its breath doesn't steam in the cold air. She doesn't dare move, doesn't breathe, does her best not to even think. Something pink and wet darts from a ragged cleft up high that might be its mouth, and then it bends low and licks at the snow and slush in the middle of the road. It lingers there a moment longer, as if considering whatever it might or might not have just tasted, and then a sudden strong gust of wind seems to catch the creature off guard and blows it away down Waterman Street, back in the direction Emmie's come, tossing it thundering along, weightless as a discarded candy wrapper and heavy as the heart of a collapsing star. Emmie doesn't move, crouching there beneath the holly bush as the snow slowly covers her. The creature's footsteps have become distant, faint fireworks heard from far, far away, and Emmie shuts her eyes tightly and shivers and wishes that this could be the end of the dream.
"It's not so cold anymore," the brown girl says, standing there with Emmie now, and she brushes snow from Emmie's lips and eyelids. "We're lucky," she says. "That was a very wicked place, indeed, but it didn't find us, and it won't come back. They never do."
"A place?" Emmie asks, and then she realizes that the brown girl's right. It is warmer now, only the cold of any snowy winter night, nothing more and nothing less. "What do you mean? I saw it. It wasn't a place. It was a monster."
"There's that silly word again," the brown girl says, and smiles at Emmie. "You use that word entirely too often, Emma Jean Silvey. You're going to have to try harder. Now hurry. We still have a ways to go."
"Will he be safe?" Emmie asks, glancing past the streetlights, peering deep into the night, towards the place where she thinks her house must be. "Will Deacon be safe now that I'm gone?"
"The Bailiff isn't looking for Deacon," the brown girl says. "He's looking for you, dear. Now come on. I don't like the snow. It was snowing when my mother died."
"What about my mother?" Emmie asks her. "Was it snowing when my mother died?"
The brown girl from the attic stares at her a moment, lost in thought or looking at something only she can see, and then she smiles and takes Emmie's hand. "I don't know," she says. "But I'm beginning to think it may turn out that way."
For twenty-eight years, the East Side railroad tunnel has sat half-forgotten beneath the streets of College Hill, a relic from a time when trains and trolleys were still used to carry people through the city. The two teams of workmen, one digging east from Benefit and the other west from Gano, met beneath Cooke Street on April 7, 1908, a full day ahead of schedule. Two hundred thousand cubic yards of hard Paleozoic bedrock were moved in the excavations, and when the workers were finally finished, the tunnel stretched more than five thousand feet from end to end. There were no casualties during the construction, which began in 1906, but there were persistent reports of peculiar noises, especially towards the western end of the dig, and a number of workers complained of foul odors, like rotting meat and ammonia, which seemed to leak in places from the freshly broken rocks. Others said that they often felt that they were being watched, and half a dozen men reported brief but unnerving encounters with doglike "goblins" or "devils" that never left tracks and always managed to lope away into the darkness and vanish before anyone was able to get a good look at them. There was a passing mention of the phantoms in an article published by the Providence Journal, mocking the Portuguese and Italian workers for not having left their superstitions in the old country.
The East Side tunnel was officially opened during a ceremony on November 15, 1908. Dignitaries were in attendance, ribbons were cut, speeches were made, and the stories told by superstitious immigrant workmen were quickly lost from the memory of Providence. For seventy-three years, as the restless world of men changed from one thing to another, passing from gaslights to electric bulbs to the fire of splitting atoms, the tunnel served the purpose for which it had been built. The men and women who passed through it, safe inside their rattling steel carriages, never reported anything strange, and certainly not the watchful eyes of subterranean demons. And then, when the world had at last changed enough that men no longer had any need for the tunnel, it was sealed off at both ends—for the public good—and the earth beneath College Hill became a secret place once again. There were padlocks, and gates were welded shut, but students from RISD and Brown would always break them open again, and parties in the tunnel became a routine occurrence. Sometimes the homeless sought shelter there, but it was wet and inhospitable, and there were usually better places to sleep.
Emmie Silvey crosses Gano Street, aiming for the scraggly patch of woods behind the baseball diamond and the soccer field, but the snow's falling so fast and heavy now that she's having trouble seeing more than a couple of feet in front of her. She follows the brown girl's footprints so that she doesn't end up somewhere she's not supposed to be, only catching a glimpse of the girl now and then, but Emmie can hear her, the girl's voice murmuring in her head, so she knows that she hasn't been left alone in the stormy night.
"It's not much farther," the girl tells her. "We're almost there. I promise."
Emmie's legs are beginning to ache and her lungs hurt, and she almost asks the girl why the woods aren't good enough for hiding. I could sit down, she thinks. I could just walk over to the other side and sit down beneath a tree and wait to wake up.
"It's not fair," Emmie mumbles, tasting the damp wool of her muffler. "Ghosts don't get tired."
I told you that I'm not a ghost, the girl replies. And besides, you don't know that.
"I know Deacon's not going to be happy when he finds out that I've run off in a damn snowstorm. I know that."
He'll understand, the brown girl assures Emmie.
"You think? You don't know Deacon. He's only gonna kill me, if I'm lucky."
It's just a dream, the brown girl reminds her. Surely he won't punish you for what you've done in a dream.
Yeah, Emmie thinks, reaching the far side of Gano Street and pausing beneath the snow-laden limbs of a big maple to catch her breath. It's only a dream, and Deacon is my father, and that woman with the tattoo on the train wasn't my mother. It's only a dream, and there's no box of secrets beneath Deacon's bed.
"I should've stayed in New York," Emmie says. "This is way worse than dead horses."
He would have come for you there, the girl from the attic tells her. The Bailiff goes wherever he needs to go, wherever they need him to be.
"Who's the Bailiff?"
"It's not much farther. We need to keep moving, Emma Jean. The tunnel isn't far." And Emmie realizes that the girl is actually speaking to her now. She looks up, and the brown girl is standing knee-deep in a drift.
"Aren't you even cold?" Emmie asks.
"We have to be going," the girl says. "We have to hurry."
So Emmie keeps quiet and follows her into the dark woods behind the park, too tired to argue anymore, and she's gone this far, so she might as well see what's waiting at the other end of it all. It certainly can't be anything worse than what she saw on Waterman Street. The girl guides her down a trail between the trees and underbrush, a narrow footpath leading them back towards the Seekonk River. The wind rustles through the dry branches, and just as Emmie starts to miss the streetlights, the brown girl begins to sing as she walks. Her voice is making Emmie a little sleepy, but it also helps to take her mind off the cold.
When the mistletoe was green,
Midst the winter's snows,
Sunshine in thy face was seen,
Kissing lips of rose.
Aura Lea, Aura Lea,
Take my golden ring;
Love and light return with thee,
And swallows with the spring.
"What was that?" Emmie asks her, when the girl stops singing. "I've never heard that song before."
"My father taught it to me. It was one of my mother's favorite songs. He sang it almost every single day."
"Is he dead?" Emmie asks. "Is your father dead, too?"
"No," the girl says. "But he's gone away. The hounds sent him away. He's coming back for me soon, though."
They're standing together at what first appears to be a fork in the trail, but then Emmie sees that it's actually a place where the trail ends at an old abandoned railroad track, one side leading off towards the river and the other back west towards Gano Street.
"We go that way," the brown girl says, sounding very certain of herself, and she points left, towards the glow of College Hill.
"But that's where we just were," Emmie protests and kicks at a stump buried in the snow.
"Yes, but nevertheless," the brown girl says, annoyed or insulted that Emmie's questioning her judgement, "that's the way to the tunnel. That way," and she points right, "only leads to the bridge. We don't need the bridge."
"I thought you wanted a bridge."
"Yes, but not that sort of bridge."
"I'm going home," Emmie grumbles.
"No. No, you're not. You know that you're not. Now stop dawdling," and, without another word, the brown girl heads off down the tracks. Emmie waits a moment or two, wondering if the girl bleeds, wondering if she can fight, and then Emmie Silvey takes a deep breath and follows. The brown girl is singing again.
Aura Lea, the bird may flee,
The willow's golden hair
Swing through winter fitfully,
On the stormy air.
Yet if thy blue eyes I see,
Gloom will soon depart;
For to me, sweet Aura Lea
Is sunshine through the heart.
Emmie concentrates on the girl's voice and the sound of her own boots in the snow, the wind and the brittle noise the snowflakes make as they fall. And before long they've come to a place where the tracks end at a rust-red wall of corrugated tin built within a high concrete archway set into the side of the hill.
"Here," the brown girl says. "They tried to seal it shut a long time ago, but nothing ever stays shut. Nothing that should be open."
"The tunnel?" Emmie asks, and she realizes that there's a light here, thin white light tinged faintly blue, light that doesn't cast any shadows that she can see. Emmie can't be sure where it's coming from, but it seems to be shining down from somewhere above their heads, near the top of the concrete arch, but when she looks up, there's only the black and snowy sky, the wall of metal and cement.
"Yes," the brown girl says. "You'll be safe in here till morning. The hounds won't look here. They won't think you'd ever come so near."
In front of the tunnel is a small clearing strewn with junk—the rusted remains of an automobile and a box spring, piles of beer cans, and the rotted carcass of a sofa. The land here is marshy, and there's dark and stagnant water on both sides of the tracks. The water looks almost like ink, ink with little blobs of snow floating about in it, and Emmie thinks of Frodo and Gandalf outside the mines of Moria, the black pool and the guardian of the West Gate.
"You're sure it's safe?" Emmie asks.
"I'm afraid nowhere's safe," the brown girl says. "But they won't look here tonight."
"And tomorrow?"
"We'll deal with that when it comes."
Someone has laid down wooden pallets, end to end, between the steel rails, and they lead to a door set into the wall of rusted tin. The door is standing ajar, and the strange blue-white light doesn't seem to reach inside. There's a red plastic milk crate, half-submerged, floating in the water on the right side of the tracks, and it reminds Emmie of her little room in Sadie's apartment in New York and makes her wish she were there. The brown girl steps gingerly across the pallets. She stops near the floating crate and looks back at Emmie.
"You should hurry," she says. "There are rules."
Emmie nods her head and takes one step forward. The lower part of the tin wall and the concrete of the tunnel are obscured by decades of graffiti, dull metal tagged and spray-painted with a drunken, looping neon tapestry of patterns and symbols and words that mean nothing to her, but it's an unexpected, distracting splash of color among the sleeping brown and gray of the trees and the snow and the stones.
"What kind of rules?" Emmie asks.
"There are always rules, especially in very magickal places," the brown girl tells her. "Don't you read?" And the brown girl takes something from a pocket in her dress, something that glints in the blue light, and Emmie guesses it must be a coin, and the girl from the attic drops it into the milk crate with a faint splash. "Don't worry," she says. "That's enough for both of us."
"Thank you," Emmie says, though she isn't at all sure she should be thankful, and the wooden pallets creak beneath the rubber soles of the boots that Sadie gave her as she walks towards the tunnel. She can see now that there's a black-and-white circle painted on the door, a white circle ringed by interwoven lines of black, and a cracked black heart at the very bottom. Inside, the circle has been painted black, and at its center there's a white horse that's fallen or been knocked down, a horse on its back with its legs in the air. Above the horse, in white lettering, is written, The horse is dead, and below the horse is written, From here we walk.
"What does that mean?" Emmie asks the brown girl, who makes a face like it's the silliest question she's ever heard.
"You were there," she says. "I don't think you need me to explain, Emma Jean." And then she steps through the doorway, and the darkness inside seems to swallow her.
"The horse is dead," Emmie whispers, glancing up at the snowy place where the blue-white light might be coming from, then down at the floating milk crate. "From here we walk." And she thinks once more about turning back, about turning and running all the way home again, before she says Deacon's name aloud, then crosses the threshold into the tunnel.
## II
## Lost Girls
My cocoon tightens, colors tease,
I'm feeling for the air;
A dim capacity for wings
Degrades the dress I wear.
—EMILY DICKINSON
## SIX
## Shadow and Flame
"Fucking dead," Odd Willie moans, somewhere nearby, somewhere in the river-scented darkness and the acrid stench of the mill-befouled Blackstone, and Soldier grits her teeth and shuts her eyes again. No sense staring into that black void until her eyes burn and she begins to imagine taunting, swirling colors that aren't there to see. No light left down here. No light at all. Only pain and the darkness and Odd Willie moaning about being dead. Both of them dangling headdown like fat hogs strung up for the slaughterhouse knife, her wrists tied behind her back, her ankles bound. Soldier's head and spine and shoulders are white fire, but at least she can't feel her hands or feet because she's been hanging here so long, and, besides, the nylon cord and electrical tape have cut off the circulation.
"Fuckers," mumbles Odd Willie.
They may have been hanging here only an hour, but it might have been much longer. She remembers the car and the cemetery, the glamour, the butt of Saben's pistol against her skull, then an indefinite, numb nothingness before being kicked awake and finding herself lying naked on the floor of the ossuary—a floor tiled with polished teeth, walls lined with long bones and skulls, a vaulted ceiling of ribs and vertebrae. And then she and Odd Willie were beaten and tortured and raped by Ballou's goons until the goons got bored and tied them up and left them alone together in the dark.
And then, later, she was dreaming, until Odd Willie's moaning woke her. She was dreaming about the musty library in the yellow house on Benefit Street, and the Bailiff was offering her a candy cane. She took it, translucent sugar the color of amethyst, and it was nothing she'd ever tasted before or since—not quite licorice and not quite peppermint, a bit like eating violets. She sat on a velvet cushion on the floor while he read to her from the big book lying open on his desk.
"She killed us," Odd Willie moans. "We're dead, dead as dog shit, and she fucking killed us."
Soldier opens one eye and tries to guess where his voice is coming from. "We're not dead," she says, though they could be, for all she knows. "Now shut up. My head hurts."
"She killed us both. That crazy, fucking, backstabbing cunt killed us both."
Soldier closes her right eye again, pretty sure Odd Willie must be somewhere to the left of her, anyhow, and tries to recall exactly how the candy tasted.
"Aren't you listening? Aren't you even paying attention?" the Bailiff asks and smiles at her. His green eyes are the kindest eyes that she's ever seen, and Soldier wishes that she could leave the warrens and live up here with him in the house forever. He could be her father, and she could spend every day listening to him read from his books. They could take long walks in the sunlight together, and he could tell her stories whenever she was having trouble getting to sleep.
"My fucking mouth tastes like puke and blood," Odd Willie says and makes a painful spitting sound.
"Someone keeps distracting me," Soldier tells the Bailiff, and he sighs and scratches at his beard.
"Well, you'd best try to ignore her, little soldier girl," he says. "She's trouble good and proper, that one. Lightning in a bottle. I don't mind saying it out loud. They should've slit her throat a long damn time ago, if you were to ask me."
"I didn't ask you anything."
"Too true."
In the darkness, the damp air is cold and smells like mud and mold and earthworms and the poisoned river, and Soldier wishes Odd Willie would shut the hell up. She couldn't hear him in the dream, in the library with the Bailiff, and she couldn't feel all the places where they've hit her or kicked her or sliced her with their knives, their nails and their claws. And then there's a new noise—dry twigs snapping, the living bones of a sparrow cracked open between a cat's teeth, thin ice breaking apart underfoot—and Soldier stares out into the darkness.
"Oh gods," Odd Willie whimpers. "They're coming back. Oh, fuck me, Soldier, fuck me—"
"Shut the fuck up, you fucking pussy," Soldier growls at him, and the effort makes her head pound and her stomach roll.
"Is this shit leading anywhere?" she asks the Bailiff, and he grunts and turns a page.
"Depends what you're asking me," he replies, then begins reading to her again, the poetry of demons transcribed by the hands of dead men, a meandering recollection from the last day of a battle in a city of ghosts that sleeps forever now beneath sand and blistering desert sky.
"What's the point?" she says, interrupting him. "That's what I'm asking you, old man. I'm sick of riddles and dreams and fucking symbolism. If you've got something to say to me, say it or shut up."
The Bailiff snorts and marks the page with a long white feather before closing the book and pushing it away across the wide desk. "The universe," he says, "and all those unfortunate beings locked up within it—cosmic inmates, if you will—have no purpose but the purpose which they imagine for themselves within the spiraling dream of time."
"That's very fucking nice," Soldier says, and she decides that the stick of candy isn't the color of amethyst, after all. "That's almost goddamn poetry. Did you find it scrawled in one of your books? Is it something else you learned from the hounds?"
"It's not a riddle," the child from the attic says, the Daughter of the Four of Pentacles, the girl whose name is never spoken, and now Soldier remembers her standing in front of the Bailiff's desk, running her fingers lovingly back and forth across the worn leather cover of the book. She's smiling, and Soldier thinks that maybe the girl is remembering the time before they locked her away and stopped all the clocks her father built for the hounds.
"You can't be here," Soldier says. The Bailiff doesn't seem to have noticed the girl.
"It wasn't easy," the Daughter of the Four of Pentacles replies, and suddenly she pulls her fingers back from the book, as if it's burned her. "I broke three of his bindings, and now those moments are lost forever. I simply dropped them on the floor and they shattered. It's the most wicked thing I've ever had to do."
"The universe," the Bailiff continues, "and, indeed, all universes, are at war with the consequences of their consciousness. Sometimes they fashion the most ingenious weapons for their regiments. But occasionally the weapons are reluctant to fulfill their purpose. Guns jam. Arrows miss their mark. Shots go wild, as they say."
He doesn't see her, Soldier thinks and opens her eyes again. He doesn't see her, because she wasn't there.
There's light now, warm light from a white candle held tight in the girl's hand, the flame flickering uneasily in the dank air far beneath Woonsocket. "It's simpler for me to move through dreams, usually," she says. "There aren't as many..." And the girl pauses a moment, furrowing her brow as though trying to recall a difficult or unfamiliar word. "...not as many watchers. At least, that's what my father used to call them."
"In the eyes of the hounds," the Bailiff says, "you have become just such a reluctant weapon. And I have failed to set you straight."
"You didn't fail, old man. You never even tried."
"Too true," he says again and smiles. "But that's most likely beside the point."
"Somehow, I doubt the ghul will think so," Soldier tells him, Soldier only five years old and seated on a velvet cushion on the floor, the tall bookshelves rising up all about her, so very tall that the tops are dappled with snow and lost in misty clouds. "You're a traitor," she says and licks the last of the candy off her fingers.
"Now, now," he frowns. "That's an awfully weighty word for such a small girl to be tossing about. Perhaps you should endeavor to avoid it in the future."
"Soldier, that was a long time ago, that day with the Bailiff," the Daughter of the Four of Pentacles says, and wax drips from her candle and puddles on the floor of molars and incisors and canines. "I need you here and now, with me. I need you awake. I have only a few moments left, and I can't come back to you once I've gone."
"She fucking killed us," Odd Willie says so softly that it's almost a whisper, and he laughs and then begins to cough.
"Do you want to die here?" the girl asks.
"No," Soldier says, and the Bailiff glares down at her from his seat at the desk, and she sees that what she mistook for kindness in his green eyes is something else altogether.
"Don't you start believing in fairy stories," he warns her. "It's a little late for that, child. They locked her away in that attic for a reason."
"Ask him why they put you there," the Daughter of the Four of Pentacles says, so much anger and bitterness for the voice of a child, and Soldier realizes that she can't see the girl anymore, only fluttering candlelight and her shadow stretched out across the floor of teeth.
"Where did you go?" Soldier calls out, and her voice echoes dully off the moldering walls of the ossuary.
"I'm here, behind you. Don't make so much noise. These knots are going to take me a moment."
"Don't you dare accept her help, Soldier," the Bailiff commands, "not that one." And he leans forward, towering above Soldier, who's only a child sitting cross-legged on a ratty velvet cushion. Only five and lost in an endless canyon of books and shelves, the taste of something sweet fading on her tongue, and all of heaven become the voice of a madman.
"Did you send me here to die?" Soldier asks him. The Bailiff licks at his lips, and her body sways a little as the wizard's daughter struggles to free her hands. Like the pendulum of a clock, she thinks, like the clock on the mantel in the yellow house on Benefit Street, the clock with the face of a girl.
"Oh, Soldier"—the Bailiff sighs—"don't you know you have meant all the world to me? Should any harm befall you, I could never forgive myself."
"How old am I?" Soldier asks, and the girl stops working at the knots and tells her, No, no, Soldier, it's not time for that question yet. That question will come later. The Bailiff narrows his glittering eyes and leans back in his squeaky chair.
"How old was I when you took me up to the attic?" Soldier asks, feeling sick, and her mind wriggles cold and slippery, a great, squirming fish caught between her fingers. She's blacking out again. "Answer me, you son of a bitch."
"You've had a very trying day," the Bailiff replies, and behind Soldier the library door is opening, the silver-eyed woman come to claim her, to lead her back to the cellar steps and down to the Hounds of Cain. "We'll talk about this another time, when you're not so...excitable."
And then he's gone, and that day is gone, gone and almost forgotten, and Soldier is only hanging in the abscess carved out beneath a hill in Woonsocket, hanging by her ankles while the Daughter of the Four of Pentacles tries to loosen the knots that George Ballou's men have tied.
"Who is she?" Odd Willie mumbles. "Soldier, who the hell is that girl? Where'd she come from?" but Soldier ignores him and shuts her eyes and lets the world fall away.
"I hope you understand that this is an exceedingly special privilege," the Bailiff said as he closed the library doors behind them. "Very few changelings have ever been allowed into the house, and far fewer have seen the second floor, and the number who've visited the attic...well, I can say with confidence that I could count those souls on the fingers of my left foot." She smiled, imagining the mess that the Bailiff's left foot must be; he locked the library with a silver key, then slipped the key into his pants pocket.
"Is it true there's a ghost up there?" she asked.
"A ghost? Well, in a manner of speaking, I suppose. In a manner of speaking, there are many ghosts up there. More than you'd care to know. More than ever you could count."
"I wouldn't care to know any at all," Soldier said, and the Bailiff laughed and led her up a narrow staircase and down a long hallway hung with paintings and peeling Dresden-blue calico wallpaper. Soldier stopped to examine one of the strips of blue wallpaper, which she discovered was damp and sticky with something that smelled like soured milk; the Bailiff scolded her.
"Don't be greedy," he scowled. "And don't take liberties you haven't been offered."
"I just wanted to see," Soldier said, wiping her hand on the front of her dress. "I wanted to know why it was peeling off the wall, that's all."
"But that's a secret, Soldier girl, and this house and all those within its walls will reveal only the secrets they choose to reveal."
"It's only wallpaper. That's a pretty dumb secret."
"That's not for you to say, most especially not when you've already been accorded such grand and all but unprecedented honors." And the Bailiff took her hand firmly in his so that she'd have to walk beside him and couldn't stop to look at any of the portraits or urns or cut-glass doorknobs they'd have to pass to reach the end of the hall. "You wouldn't want to seem ungrateful," the Bailiff said.
"No," she agreed. "I wouldn't," and in only a few more steps they'd reached the far end of the hallway. There was a small wrought-iron table with a green marble top and a peacock green vase of dead roses, and all the wallpaper had been stripped away here to reveal the plaster underneath. The wall was marked with a confusion of red and gold runes that Soldier couldn't read. She was about to ask what they meant, when the Bailiff pointed at the ceiling; she looked up and saw the trapdoor. It had a length of rope for a handle, and there were more runes painted on it.
"I'm scared," she said, and the Bailiff made a concerned sort of face and knelt down beside her.
"I won't lie to you," he said. "There are plenty enough things in the house to be afraid of, and not a few of them are right over our heads. But your coming here today has been arranged. No harm will befall you, child, not so long as you mind your Ps and Qs."
"You'll be with me," she said and held his hand more tightly.
"As long as I can be, but I've told you already, there are places you must go without me. There are places you must see alone."
Soldier stared up at the entrance to the attic, and she wondered if he'd stop her if she tried to run away, if she turned and hurried back past the paintings and down the stairs, to the cellar and the tunnels and the shadows where she belonged.
"What is it, Soldier?" the Bailiff asked. "What is it you're not telling me?"
"The woman in the desert," Soldier replied, "I dreamed of her again yesterday."
"The black woman?" the Bailiff asked, and then he glanced up at the trapdoor.
"Yes, sir. The one who told me about Esmeribetheda and the djinniyeh. We were walking together in the ruined city, and she told me not to come here today."
The Bailiff watched her silently for a moment, then took a deep breath and rubbed at his beard.
"She said that a terrible thing was waiting for me here," Soldier added.
"Did she now?" the Bailiff asked and tugged at his whiskers. "Were those precisely the words she used?"
"Yes sir," Soldier said. "She said the hounds mean to take my life from me."
"But the hounds already have your life, child. You've belonged to them since the Cuckoo brought you from the home of your father and mother."
"Well, I know that," Soldier said, and the Bailiff shook his head. "I told her that."
"And what did she say?"
"That there are lies all about me, that my entire life has been spun from lies, and that the greatest lies, the ones which will take my life, are waiting for me with the ghost in the attic. She said that I can't trust you anymore," and then Soldier looked down, keeping her eyes on the dingy floral pattern of the rug because she didn't want to see his expression.
"And do you believe her?" the Bailiff asked, each word tumbling through the air like iron.
"It was only a dream," Soldier said. "All of it, it was only a dream."
"Are you quite sure?"
"Do you think that matters?" she asked and kicked at the rug with the toe of her shoe.
"And why didn't you tell me this sooner?"
"I wanted to see the attic. I thought if I told you, you might not show me."
"Look at me," he said, and she did. In the dim light of the hallway, his face didn't seem quite so round, and Soldier could see the dark circles beneath his eyes, eyes that had gone the oily color of coal. "We play our assigned parts," he told her, "and that's all we get. My part was to bring you here, and your part is to enter. We all play our parts, little Soldier, nothing more."
"That's exactly what she told me you'd say."
"Is it now? Well, I guess that means she's a right smart nigger lady, doesn't it?"
"Sometimes," Soldier said, "she frightens me."
"I wouldn't worry your pretty head about her any longer," the Bailiff said, standing up, straightening his rumpled seersucker suit, and then he reached for the rope handle on the trapdoor and pulled. The rusted hinges and springs screamed and popped, stiff from long decades of disuse, and Soldier wanted to cover her ears, but she didn't. "Where you're headed, I doubt she'll be able to follow."
Soldier doesn't remember being lowered to the floor, doesn't remember the Daughter of the Four of Pentacles leaving her, doesn't remember the feeling coming back to her hands and feet, and whether or not it was the girl who untied Odd Willie or if maybe she did it herself—but none of that shit matters, none of that shit matters at all.
"I'm so goddamn cold," Odd Willie shivers, and she feels his forehead again. She's pretty sure he has a fever; maybe Ballou's men busted something inside him, and now Odd Willie is bleeding to death. "I'm freezing my fucking dick off."
"I don't know what you expect me to do about it," she says, and Soldier peers into the darkness beyond the muddy alcove where they're waiting, huddled together naked and hurting, waiting to see if anyone's coming back to finish the job, or if they've been left for dead. The alcove is just outside the ossuary. Soldier managed to pry loose the upper half of a femur from the wall, not much against guns and magick, but better than nothing, and she grips it tight in both hands and tries not to feel the cold.
"I'm gonna murder the bitch," Odd Willie says. "I swear, Soldier, if we ever fucking see daylight again, I'm gonna kill her with my goddamn bare hands."
"You're gonna have to get in line behind me," Soldier tells him. "Now, keep your voice down. I'm trying to hear something."
"What? What the hell are you trying to hear?"
"Anything," Soldier replies, "anything but you."
The dark is playing tricks with her eyes, painting sudden movements from the stillness, faking her out, and she ignores it and listens. The darkness is filled with sound—her heartbeat, hers and Willie's breathing, dripping water, a persistent, industrious gnawing that she knows is a rat wearing down its incisors against a piece of bone, and, occasionally, a distant splash. A low murmur that might be voices sifted through the cold and the black, and she leans close to Odd Willie and whispers in his ear.
"I don't think we're under the cemetery anymore," she says. "I think we might be beneath the river."
"Who gives a good goddamn?" Odd Willie says, and then he giggles, and then he moans very loudly.
Soldier lays the femur on the ground at her feet, and then she seizes Odd Willie by the jaw and forces his head around until she's pretty sure he's facing her. He makes a surprised, pissed-off noise, tries to jerk away, and she digs her nails into his face. With her left hand she grabs his testicles.
"What the fuck's your fucking problem?" he mumbles around her fingers.
"Right now, you're my problem, Odd Willie. We're in some pretty serious shit here, but we're not dead, not yet, and I don't intend to end up that way. Do you understand what I'm saying to you?"
"Jesus fucking Christ." Odd Willie grunts, and when he tries to pull free again, she digs her nails deep into the soft, hairy flesh around his scrotum.
"I said, do you understand?"
"Yeah bitch, I fucking understand," he growls back at her, growling like a wounded puppy. "I understand you're as big a psycho as that cunt Saben."
"Be that as it fucking may, Odd Willie, you're gonna keep your mouth shut, and if we're real goddamn lucky, we might still find a way out of this mess."
"Fine. Whatever you say. Now please just let go of my fucking balls, okay?"
"I know you're hurt," she tells him, slowly relaxing her hold on his face and his crotch. "And I know you're scared shitless, because I am, too. But we have to get out of here. We have to find a way out if we want to be sure that Saben gets what she's got coming to her."
"Fine," he says again, and when she lets go, Soldier feels him pull away, pressing himself against the damp stone wall of the tunnel.
"Something's gone wrong, Willie, something the Bailiff didn't see coming, something he couldn't anticipate."
"Fuck that," Odd Willie mutters and giggles very softly. "I think he sent us up here to fucking die. I think this is payback for lots of shit, Soldier. Cocksucker's probably in with Ballou, Saben, the whole stinking lot."
"It doesn't make sense," Soldier says, picking up the femur again. "I've been thinking the same thing, turning it over and over in my head, but it just doesn't add up."
"You fucking grab my balls again," Odd Willie mumbles, "I swear I'm gonna kill you."
"Yeah, whatever," Soldier tells him. "You better start thinking a little less about killing and a little more about keeping your own sorry ass alive and kicking. Remember your place—"
"Fuck that, too. Down here, ain't no places anymore, Soldier. Ain't no rank. No fucking order. Down here—"
"I'm gonna stand up," Soldier says, and she takes a deep breath, even though it hurts like hell. "Can you get up, Odd Willie? Can you walk?"
"I can fucking walk," he replies. "Christ, I'm freezing my ass off."
Soldier nods and listens to the suffocating darkness and the muffled, magnified tunnel noises, trying to fix a direction on what might have been voices, human voices or hound voices or something in between. They seemed to be coming from somewhere far away on her right, so she decides to try to keep to the left for as long as possible. Then Odd Willie vomits again, and she feels it spatter hot across her feet. Soldier waits until he's done, tells him to stay close to the wall, to keep his left hand on the wall and stay right there behind her, and she stands up. Both her knees pop, and she's suddenly so dizzy that she starts to sit back down.
"Just don't you leave me," Odd Willie says. "Whatever happens, don't you dare fucking leave me down here alone."
"I won't. I'm not going to leave you," she promises, though she knows that she will, if that's what it takes to make it out, if Odd Willie can't keep up. Soldier rests her weight against the tunnel wall until the dizziness isn't so bad.
"I'd rather you fucking killed me than leave me down here by myself, Soldier."
"Just stay close. Keep the wall on your left."
Beneath her feet the floor is an obstacle course of uneven, muddy cobblestones and puddles of icy water, and she almost slips twice. When she's gone forty, maybe forty-five yards, Soldier stops and lets Odd Willie catch up to her. She can hear the voices again, louder than before, though still not loud enough to make out whatever it is they're saying. But at least now she's certain that they're somewhere on her right, even if she can't begin to guess how close they might be. She's not even sure how wide or how high this tunnel is; she grips the broken thighbone and takes another step forward.
"You smell that?" Odd Willie asks, and Soldier stops and sniffs at the damp air.
"I smell mud," she says. "Mud and mold, and that's about it. What do you smell, Odd Willie?"
"Smoke. I smell fucking smoke. Woodsmoke. Man, I wish I had a cigarette."
Soldier sniffs at the air again, and this time she catches the faintest whiff of burning, and she also realizes that it's not as stagnant as before. There's a draft here as the air's pulled weakly past them, back towards the ossuary or some side tunnel that they've missed.
"We get out of this shit with our skins, I'll buy you a whole goddamn carton," she tells Odd Willie, and she can hear the clumsy slap of his footsteps, so she knows that he's still following her. "I suppose you were never very good with all the hocus-pocus crap, either," she says.
"I can do a few things," he replies, wheezing a little now, and Soldier wonders if maybe the goons punctured one of his lungs.
"Yeah? Like what, precisely?"
"I'm no damn magician," Odd Willie says, "so don't go getting a hard-on or anything. But I learned a little here and there."
Up ahead there's a sudden loud, clattering commotion, like someone dropping a burlap sack filled with bones, and Soldier stops and lies almost flat on the tunnel floor, instinctively making as small a target of herself as possible; she holds the broken femur under her, gripped close to her chest. She can't hear Odd Willie's big flat feet anymore, so she assumes that he's done the same. There are smaller clatterings, then a whisking sound like a straw broom on wood, and she holds her breath and waits for silence or whatever's coming instead of silence.
Ballou's a maniac, the Bailiff said the night before they left Providence, early Sunday morning, and she was naked then, too. He's a spoiled child fascinated with the smell of his own offal. But he's also a threat, Soldier, a threat that never should have been permitted to become anything more than a nuisance.
You're telling me to kill him? Soldier asked, and the Bailiff smiled and rubbed at his nose.
Nothing too fancy, he said. Try not to make a mess.
Just the two of us? Just me and Odd fucking Willie?
The Bailiff stopped rubbing his nose, which seemed redder than usual, and nodded. We have people up there. There won't be much resistance. We've been softening him from the inside. Trust me, little Soldier. You and young Master Lothrop will be quite sufficient to the task at hand. Nothing too showy, though.
The sweeping sound has stopped, and Soldier exhales slowly, her breath escaping almost silently between her teeth. But she doesn't move, lies dead still on the uneven cobbles and mud and listens to the darkness.
Nothing more than is necessary.
And it's just gonna be me and Odd Willie against the whole damn warren? she asked him again. Odd Willie Lothrop the firebug, the guy with no eyebrows?
The Bailiff sneezed, then took out a white handkerchief and blew his red nose. I shall expect you back by Sunday evening, he said, seven thirty at the latest, and returned the handkerchief to the breast pocket of his jacket. There will be a cleanup crew coming down from Boston sometime around midnight, and I want you out of there long before then.
So, where the hell are they? she thinks, because it's surely long past midnight on Sunday. It might be past midnight on fucking Monday, for all she knows, and Soldier swears on the names of gods she doesn't believe in that she'd trade another half hour as George Ballou's punching bag for just one shot of Dickel. Just one shot of whiskey to make her hands stop shaking.
"It's gone," Odd Willie whispers, and she can hear him getting slowly to his feet. "Whatever it was, it's gone."
Soldier waits another two or three seconds, then rolls over onto her back; a sharp corner of one of the cobblestones digs into a bruised spot beneath her left shoulder blade, bruised muscle and maybe something in there's fractured, as well, and she has to bite the end of her tongue to keep from crying out. The dizziness returns, and she lies still, waiting for it to pass.
"Soldier?" Odd Willie asks, his voice suddenly lost and close to panic. "Can you hear me? Soldier—"
"Yes," she hisses around the pain in her back. "I'm right fucking here. Keep it down. Jesus..."
"I thought you'd left me. I thought maybe you'd gotten away and left me."
"I just fucking said that I wasn't going to leave you," and she shuts her eyes and swallows against the pain and a wave of nausea. She imagines the Bailiff, holed up somewhere with his boys and their drugs, waiting for whatever it is he's started to blow over, the calm after the storm. She pretends he can hear her thoughts, pretends he'd care if he could. I'm not going to die here in the dark and the mud, you bastard. I'm not going to die down here with Odd Willie while that bitch Saben is still alive somewhere.
"I can do this one thing," Odd Willie wheezes. "Magick, I mean. There's this one thing I'm really pretty good with, but it makes me sick as a goddamned poisoned dog. So it's kind of a fucking last resort."
"Willie," Soldier says, "we're naked and beat half to fucking death, lost in the dark, surrounded by people that want us dead, and the only weapon I have is part of a damn rotten bone. Unless you've got a couple of nine-millimeters tucked up your skinny ass, I'm thinking, yeah, maybe this is pretty goddamned last resort."
"Sure," he says, "I know. You think I don't know how fucked we are?" And then he doesn't say anything else, and there's only the sounds of the two of them breathing and the steady plip, plip, plip of water striking water.
"Well, is it a fucking secret?" she asks finally, and she can hear Willie sitting back down again.
"No, it's not a secret," he says. "I just don't like to do it, that's all. So, you know, usually I don't let on."
Soldier opens her eyes, not that it makes any difference, and she sits up, raising herself with her arms and trying to ignore the pain from the bruised shoulder and just about everywhere else. The burning-wood smell's getting stronger, she thinks, either that or she's only noticing it more because now she knows it's there.
"Don't make me ask again," she says and glares at the place she thinks Odd Willie is sitting. "Spit it out."
"There's this...this fucking thing, right, this thing I can call up, summon, conjure, whatever the hell you want to call it. This thing."
"What kind of a thing?"
"Fuck all if I know. Maybe it's a demon, but I kind of doubt it. What kind of demon would take orders from me? Maybe it's a ghost, the ghost of something—"
"What does it do?" Soldier asks him, grasping at straws because they're all she has left to grasp, and Odd Willie giggles.
"Just the usual shit," he wheezes, coughs, then starts again. "Roll over. Fetch. Play fucking dead. Steal some beer. Kill a bunch of motherfuckers, if that's what I tell it to do."
"Fuck." Soldier laughs, wiping some of the mud off her belly and thighs. "Christ, and you were gonna get around to telling me about this little pet of yours when?"
"You don't know how it feels," Odd Willie says. "I've told the Bailiff. It's not a secret."
"Does it feel a whole lot worse than Ballou's dick up your ass? Or a bullet through your fucking skull? Does it feel worse than dying?"
"Yeah," Odd Willie says and coughs again. "Yeah, Soldier, I think maybe it does."
The janitors have been given the strictest instructions in this matter, the Bailiff said. And they're very thorough agents. Nothing alive gets out. Nothing, Soldier. Not even a sewer rat. So, no stragglers, if you catch my drift. Get in, do what I've asked, and then get out. Do not expect help should anything happen to go awry.
The hounds don't even know about this, do they? she asked him, playing a hunch, a queasy feeling in her stomach, and he raised his eyebrows and patted the pocket where he'd put the soiled handkerchief.
You're not a child any longer, he told her. You're quite old enough, I suspect, to know something of the way of things. To be trusted, even, I suppose.
They don't know, she said.
The Bailiff sat up straighter and tugged vigorously at his knotted beard. I manage the affairs of those who manage the affairs of the hounds, he said. And the hounds don't want to know when there are problems. Most of the time it is sufficient that the problems are solved. The hounds don't want to know any more than they have to know. To their way of thinking, they have more pressing matters to attend.
"I asked Madam Terpsichore to tell me what it is," Odd Willie says, "way the hell back when I was just a kid. I asked her twice to tell me. She said I was better off not knowing. She said I'd sleep better that way."
"But it will come?" Soldier asks. "If you call it, it'll come? And you can control it?"
"Yeah. I mean, you know, it always has before. But it's gonna make me sick as shit, Soldier. I might need your help after the—"
"Just fucking do it," she says, daring to raise her voice just enough to get Odd Willie moving. "Say the goddamn magick word or whatever. Do it now, before they come back for us. I already told you I'm not going to leave you down here."
"Yeah, you did," Odd Willie replies, and he sounds tired and ill and frightened. "You told me that. Good thing you're better at killing than you are at lying, Soldier, or the Bailiff would have run out of uses for you a long time ago."
"No shit," she says, slowly getting to her feet, leaning against the tunnel wall because she isn't sure how much longer her legs can support her. And then Odd Willie tries to clear his throat, coughs and suggests that she might want to cover her eyes and turn away until he says otherwise, and Soldier does as she's told.
"You're not a child any longer," the Bailiff said, when the trapdoor had been pulled down to reveal a rickety ladder leading up into the dark attic of the yellow house. The upper end of the ladder was attached somewhere overhead, and it had unfolded as the Bailiff had opened the trapdoor, seeming to Soldier almost like some small bit of magic. "We do not show children our secrets," he said, "as they cannot yet be trusted with the keeping of them."
"You can trust me," Soldier said, and she'd almost forgotten about the dream and the black woman with the long white dreadlocks, the warning that she shouldn't ever enter the attic. "I'm very good at keeping secrets."
"As well I know," the Bailiff said. "Elsewise, you would not be here today. You first, my dear," and he motioned towards the ladder.
"What is it?" she asked him, peering up into the gloom. The warm air drifting down from the attic smelled like cobwebs and dust, neglect and brittle, old paper, all familiar, welcoming smells, smells that Soldier had always known. But there was something else there beneath it all, something spicy and sour that made her wrinkle her nose. "What do they keep hidden up there?"
"You have merely to climb those nine rungs to find out for yourself."
"And you're coming up after me," she said.
"Of course I am," the Bailiff replied, and he smiled reassuringly and gave her an encouraging pat on the back. "As far as I can follow."
Soldier said a short, silent prayer to Mother Hydra, and then she climbed the ladder, though it seemed like many more rungs than nine. She started counting halfway up, and when she'd finally reached the top, she'd counted all the way to twenty-four.
"Go on," the Bailiff shouted up at her. "I can't start up until you're inside. This ladder's seen better days and won't hold us both at once."
So Soldier scrambled over the last rung and stood inside the attic at the edge of the trapdoor, gazing back down at the Bailiff and the floor of the upstairs hallway, a rectangle of light cut into the darkness.
"What do you see up there, Soldier girl?" the Bailiff asked, and she laughed and told him he'd have to come up and find out for himself, just the same as her. The Bailiff started up the ladder, the old wood complaining at the burden of him, and Soldier looked up, squinting into the colorless half-light of the attic. She could make out a few old packing crates and a steamer trunk, stacks of newspapers bundled with twine, and there was the sense that the attic was a very large place, perhaps even somehow larger than the house below, stretching away from her on all sides. But there were no ghosts, at least, not that she could see, and Soldier decided that whatever was in the attic probably wasn't all that interesting, after all. She looked down again, to tell the Bailiff to hurry, and saw that he was no longer coming up the ladder; the Bailiff was raising the trapdoor again, shutting her in. The rusty hinges squealed and screamed, and Soldier opened her mouth, because it seemed that she should scream, too. But there was no scream anywhere inside her, only the fruitless knowledge that there probably should have been, that some other child would have screamed at the thought of being closed up in the attic alone.
"Why?" she asked him, and the Bailiff paused and glanced up at her. He winked once, and, "All is mystery, wonderful mystery," he said, "and life is the revelation." And then he finished closing the trapdoor.
Soldier sat down on the dusty floorboards, waiting for her eyes to adjust so she could walk without tripping over some piece of junk and breaking her neck.
"Then I'm alone," she said, and a small voice—the voice of another girl, but someone who sounded quite a bit older than Soldier—answered her.
"No," the older girl said, "you're not alone, but you should have brought a lantern. They always bring lanterns when they come. It was silly of you to have forgotten."
"I suppose so," Soldier said, and then the brown girl stood up from the milking stool where she'd been sitting, unnoticed, waiting for whoever would come next, her father's gold pocket watch lying open on her lap, the hands frozen at precisely nine twenty-three and thirteen seconds.
"Are you a ghost?" Soldier asked her.
The brown girl, the Daughter of the Four of Pentacles, looked surprised or offended or both at once. "No," she said. "Certainly not. Are you?"
"Not yet," Soldier replied, and the girl took her hand and led her away into the murky depths of the attic.
The crackling, thrumming thing that Odd Willie has called up or down or simply into this world brings light with it, a soft, warm yellow-white light, like morning or butter, that pulses and ripples lazily along the tunnel. Soldier stands with her back to it and stares, transfixed by the miracle of her own shadow, her silhouette an eclipse thrown across rough-hewn rock walls and the paving stones and the sagging timbers barely holding up the low ceiling. She never truly expected to live long enough to see light again, and now she only wants to lie down and let it wash over her bruised, bleeding body, washing away the cold and the pain, the fear and dread knotted deep in her gut. She wants to turn around and let this light stream across her face and fill her eyes, fill her soul, if she has such a thing; she would gladly drown in this light, if it would ever have such a wretched creature as herself.
"Oh," Odd Willie moans. "Oh, Jesus..."
"What next?" she asks, because it's the only thing she can think to say to him, her mind clouded by the light and electric crackle. "What next, Odd Willie?"
"Oh," he says again, and then he begins coughing and the light flickers and dims, and for a moment Soldier's afraid he's going to let it slip away. But the coughing passes, and immediately the light grows brighter.
"Can you talk to it?" she asks him.
"You...you can turn around now if you want to," he tells her and makes an ugly hacking noise.
Do I? Soldier thinks. Do I really want to see it? Do I really want to face all that light?
You've seen a whole lot worse, Soldier girl, and, more than likely, you'll see a lot worse again.
"She won't bite," Odd Willie wheezes, "not unless I ask her to."
Soldier turns her head, and there's Odd Willie on his knees, pale as a corpse and a dark smear of blood and vomit down his chin, his throat, his chest. His eyes are shut, and he looks dead already.
"She doesn't like it here," he says. "That's going to make her hard to hold."
The thing reminds Soldier of a jellyfish, and then it reminds her of something else entirely, and she gives up trying to comprehend it. Her eyes water and burn when she looks directly at it, so she keeps them on Odd Willie instead.
"Tell her to get us the fuck out of here," Soldier says. "The faster she does that, the sooner—"
"She's afraid. Shit, Soldier. I've never seen her afraid before."
"Come on, Willie. What happens next? Show me what this bitch can do for us."
The crackling dynamo hum gets louder and rises an octave or two, and now the heat from the light is becoming uncomfortable. "She's scared," Odd Willie says. "She's asking me to release her."
"No," Soldier yells at him, having to shout to be heard above all the noise the thing is making. "That's not going to fucking happen; you hear me? Not until we're out of here."
"She says it's Monday morning already. Six fifteen on Monday morning. Soldier, we've been down here—"
"It doesn't matter how long we've been down here," Soldier says, and never mind what the Bailiff said about the cleanup crew from Boston, whether they've come and gone, whether they ever even showed up, whether Ballou got them, too. "It only matters that we get out of here now."
"She also says Patience Bacon was right."
"Fuck Patience Bacon. We're not going to die in this stinking sump hole because your astral playmate here has the fucking heebie-jeebies."
Odd Willie takes a deep, hitching breath, and his eyelids flutter. "I'm sorry. Just do it," he tells the thing. "Just do it fast. I won't ever ask you for anything else, not ever again." His eyes roll back to show the whites, and the thing that isn't a jellyfish bobs and sways and begins making a sickly, mewling sort of racket.
There's a trickle of blood from Odd Willie's nostrils, and Soldier turns away again.
"Just kill them," Odd Willie mutters. "Kill them all. Every last goddamn one of them."
And then the thing is gone, a furious stream of fire rolling away down the tunnel, leaving behind steam and a sticky phosphorescent sheen on the floor and walls; at least they won't be in the dark again. The crackling sound is gone, too, but the air smells like ozone and hot metal.
"Can you walk?" Soldier asks Odd Willie, and his eyelids flutter again, and he sways a little, first to one side and then the other. "Hey, come on. Can you fucking hear me? I need to know if you can walk."
Odd Willie opens his eyes and stares up at her, but his expression is blank, distant, almost empty, and Soldier resists the urge to slap him hard. She wants to, but it might break whatever tenuous connection he has with the thing. And the way he looks, it might break him, as well.
"I'm gonna help you up, okay?" she says and stoops down, getting her right arm around him, and Soldier drops the broken femur and lifts Odd Willie Lothrop slowly to his feet. He's heavier than she expected, or she's weaker, and it doesn't help that they're both so slick with sweat and blood and the filth from the ossuary and the tunnel. She's afraid he's going to slide free of her grip. "I wouldn't mind if you helped out a little," she grunts and tries not to lose her balance.
"We should probably follow her," Odd Willie mumbles.
"That's sort of what I had in mind."
"She has a name," he says. "I can't fucking say it, not in words, right, but she has a beautiful name. She was born in the heart of a dying star. She's made of fire, hydrogen, helium, plasma, you know."
"Then George Ballou and his spaniels ought to be a stroll in the fucking park, yeah?"
"She even knows what killed the dinosaurs. I can't tell you, but she fucking knows, I swear."
Soldier shifts her weight, getting a better hold on Odd Willie, and she thinks maybe he's starting to support himself a little. But when she takes a step forward, he stumbles and almost pulls them both down.
"I can't carry you out of here. You're going to have to fucking walk."
"You said you wouldn't leave me," he moans and looks down at his feet. "You promised."
"But I'm a shitty liar, remember? That's what you said. Now, just fucking walk. That's all you got to do." And Odd Willie nods his head, the slow, measured nod of reluctant comprehension, and takes a step on his own, and then another, and another after that. Soldier's still holding him up, but at least she isn't having to try and drag him along an inch at a time.
"Hey, Willie, it ain't no goddamn foot race. Slow the fuck down," she says, and he coughs when he tries to laugh, and more blood leaks from the corners of his mouth.
"No way. You best keep up, bitch," he wheezes.
Good thing you're better at killing than you are at lying, or the Bailiff would have run out of uses for you a long time ago. And Soldier tries not to think about having to leave Odd Willie to die alone in the tunnel. I wouldn't have done that, she thinks. I'm not that big an asshole. I'd take care of him myself before I'd leave him alive down here alone.
"She knows what killed the dinosaurs," he says again. "Hell, she knows what happened to Atlantis. She saw it all. She's seen everything."
"Right now, you just keep moving," Soldier tells him. "You can tell me all about it later, when this shit's over and done with." She glances back at the discarded femur, lying behind them on the cobbles, and wonders if dropping it there was a mistake, if maybe she'll still need it later on. And then Odd Willie's body shudders, and he stops, and his eyelids begin to flutter again.
"Shit on me," he says, and his bladder lets go, spattering the tunnel floor and her feet and legs with hot urine. "She's found them, Soldier. I think she's found them all."
Soldier followed the Daughter of the Four of Pentacles through the attic of the yellow house, beneath great crossbeams carved from pines felled almost two hundred and fifty years before, past sagging shelves crammed with books and manuscripts and scrolls sealed inside baked clay tubes, past broken furniture and dozens more packing crates, an empty iron cage big enough to hold a lion, a bronze bust of Triton balanced on a stone pedestal. Perhaps, she thought, this is the memory of the house. Perhaps this is where it keeps everything that the rest of the house has forgotten or wants to forget. She knew some places were like that. Cemeteries were always like that.
"You knew I was coming?" she asked the girl, who shrugged and paused to blow some of the dust off an elaborate wooden Noah's ark laid out on the floor. The ark was at least ten feet long, and there were hundreds of animals, two by two, carved with an exquisite attention to detail. Soldier stooped for a better look, and the older girl sat down on the floor next to her.
"They told me. They always tell me when someone's coming up. Of course, you're different. Usually they're only bringing me gifts, candy or fruit or a length of red silk ribbon. But you're different."
"This has something to do with my dreams, doesn't it?" Soldier asked her, but the girl didn't reply. She smiled and carefully wiped a spiderweb off the shingled roof of the ark. Soldier sighed and glanced back the way they'd come. She was thirsty and wondered how long they'd been walking, how far they'd gone. This must be the biggest attic in the world, she thought and then tried to remember a word she'd learned from the Bailiff or one of the ghouls (she couldn't recall which), a word for things that are larger on the inside than on the outside.
"This was a gift," the girl said and pointed at the wooden Noah's ark.
"Is it yours? Was it a gift to you?" Soldier asked.
"No, I don't think so. I'm just taking care of it. I take care of almost everything up here."
Soldier leaned closer to the ark, trying to see all the animals more clearly, the brightly painted menagerie filing across the floor and up a long gangplank into the ship. She wished that there were more light. But she could make out two elephants and two giraffes, a pair of hippopotamuses and a pair of moose, ostriches and alligators, horses and a couple of enormous brontosaurs, unicorns and leopards and bison.
"It was made in Italy," the Daughter of the Four of Pentacles said. "There's a signature on the bottom of the ark, and a date—Signior Anastagio Baldassario Moratti, 1888. All of the animals were made in Italy, too."
"They sent me up here just to see a damned toy?" Soldier asked.
"But it's not a toy," the girl said, righting a camel that had fallen over. "It's something else. It's sort of a metaphor, I think."
"What's your name?" Soldier asked the brown-skinned girl. She was growing bored with the Noah's ark and stood up, brushing the dust off her clothes.
"My name is Pearl," the girl said.
"You're the alchemist's daughter, aren't you? The Daughter of the Four of Pentacles?"
"Yes," replied the girl.
"They told me your name was Hester. Everyone down there thinks your name is Hester."
"People often believe silly, mistaken things," the girl said and shook her head. "Did you know that people used to believe that eating tomatoes would kill you, and that the whole Earth was created in only six days?"
"They said that your name was Hester. That's what they all seem to believe, down there."
"We should keep moving," the girl said. "It's never a good idea to tarry here very long. And we've still got a ways to go."
"A ways to go where?"
"Don't be impatient. You'll see," the girl said. "I want it to be a surprise. I never get to surprise anyone anymore. Not since my father went away and all the clocks stopped and—"
"There's no one up here but you?" Soldier asked, not particularly interested in surprises.
"Oh, there are very many people up here," Pearl replied—unless Soldier were to believe everyone downstairs and in the warrens, in which case Hester replied. "I'm not alone. I have company. All the people and things and places caught inside the constructions that my father fashioned. No, I'm not alone, Soldier, not at all. But it's all very complicated, I'm afraid. You're much too young to understand such complex metaphysical—"
"You're not that much older than me," Soldier said, interrupting her.
"I most certainly am," the girl said and then stared at her a moment, taken aback. "Soldier, I'm twelve years old, and you, well, you're still not much more than a baby, are you?"
"If I were only a baby, they never would have sent me up here to see you," Soldier said, angry that the girl would say such a thing, and besides, hadn't the Bailiff just told her that she wasn't a baby anymore, and that's why he could trust her with secrets?
"They've sent me babies before."
"Liar," Soldier said and then she kicked the Noah's ark as hard as she could, and animals fell over and flew through the air and tumbled about this way and that. The great boat swung to starboard and capsized, then rolled away into the shadows, leaving most of the animals behind.
"What did you do that for?" the Daughter of the Four of Pentacles gasped, and when Soldier didn't answer her, the girl just stood staring at her, and there were tears welling up at the corners of her eyes.
"What? Are you going to cry now?" Soldier asked her, and prodded a fallen zebra with the toe of her shoe. She thought about grinding it under her heel until it was only sawdust. "Are you going to cry like a great big twelve-year-old baby?"
"No," the girl replied. "I'm most certainly not going to do anything of the sort," and she bent down and began setting all the animals upright again. "But that was a horrible, horrible thing to do. I've never shown this to anyone before. You were the very first."
Soldier watched while Hester (or Pearl) righted all the animals and then retrieved the ark from where it had rolled. There was a small bashed-in place on the left side near the prow.
"Guess Noah's going to have to fix that before the flood," she said, but the dark-skinned girl ignored her. "It's just an old toy," Soldier told her. "No one cares about all this junk up here. I bet no one even remembers most of it anymore. It's a wonder, really, they haven't forgotten you."
"Follow me," the girl said coldly, when she was finished with the ark, when she'd undone as much of the damage as she could, and she led Soldier deeper and deeper into the attic of the yellow house. After a while they came to a gaping hole in the floor. There was a board laid over it for a bridge, and Pearl crossed it without comment. But Soldier lingered alone at the edge, looking down into the inexplicable cavity. There was nothing in that hole and no bottom to it; as far as she could see, it went on forever.
"Why can't I see anything down there?" she asked, shouting across the gap. "If that's a hole in the ceiling, I should be able to see the upstairs."
"Maybe it's not a hole in the floor," the girl replied. "Maybe it's a hole in something else. Now come on. I'm tired of standing here waiting for you. There are other things I could be doing."
Soldier reached into the pocket of her dress and pulled out one of the wooden animals from the Noah's ark, a wildebeest she'd picked up while the alchemist's daughter was busy with the mess she'd made.
"Hey!" the girl shouted from the far side of the gap. "What are you doing with that? That's not yours—"
"It's not yours, either," Soldier replied. "You told me so yourself."
"No, but it's my responsibility. We have to put it back, right now."
"Oops," Soldier said, letting go of the tempera-brown wildebeest. It fell into the hole and vanished.
"You beastly little brat!" the girl shouted at her, but Soldier wasn't listening; she'd sat down at the edge of the hole and was waiting for the sound the wooden wildebeest would make when it hit bottom. But there was no sound, unless she'd missed it somehow, unless all the noise the alchemist's daughter was making had covered it up. She waited awhile longer, what she figured must have been five minutes, at least, ignoring the brown-skinned girl, who was still shouting at her and stomping back and forth on the other side of the gap.
"I didn't hear it," Soldier said finally, when it had become clear that the hole went down much farther than she'd guessed. "But you were making such a commotion, it's no wonder."
"Cross the bridge, changeling," the girl said, standing there with her hands on her hips, glowering at Soldier from the other side. "Unless, that is, you're afraid that you'll fall. You might, you know. It's not a very wide board, and it isn't very sturdy, and you might end up like that poor animal, falling forever, wondering if you'll ever find the bottom."
"You don't scare me. It's just a hole."
"Yes, that's all it is. A hole. Now come across it."
Soldier set one foot on the board, because that's all it was, a hole, and all holes had bottoms somewhere.
"Don't look down," Pearl said and took a step back from the edge. "That will only make it harder. I'd hate to have to explain this to the Bailiff or Miss Josephine, if you should fall or the board should break and—"
"Stop talking," Soldier said, and she took another step, the board bowing almost imperceptibly beneath her. "How old are you?" she asked the brown-skinned girl again.
"I told you already. I'm twelve. In fact, I'm almost thirteen."
"Everyone down there says you've been up here a very long time. Twelve's not a very long time." And she took another step.
"It's a temporal contrariety," the girl replied. "That's what my father called it. Time's not the same up here. It's a part of his punishment."
Soldier took another step, keeping her eyes straight ahead, keeping her eyes on Pearl; she was almost a third of the way across now. "Because he betrayed the ghouls," she said. "Because he lied to them."
"It's all very complicated."
"Madam Terpsichore said that he was a villain, and that he has been chained in the deepest abyss for his crimes. She said that his daughter is a ghost, and that his wife was killed for his misdeeds."
"My mother died of a fever," Pearl said, and Soldier took another step.
"Did you see her die?"
"I was very young," the Daughter of the Four of Pentacles said, speaking now through clenched teeth. "My mother was a Montauk Indian princess that my father brought to Providence all the way from Long Island. She was a beautiful woman and—"
"Did you see what they did to her?" Soldier asked, and the board popped loudly beneath her, and she almost lost her balance. She glanced down, trying to see the board and her feet, but not the hole. She was halfway across now.
"It was consumption," Pearl said—Pearl, Hester, the brown girl, the alchemist's daughter, the ghostly Daughter of the Four of Pentacles whom Soldier had been taught lived off rats and spiders and devoured anyone who dared wander into her attic.
"Did they wait until she was dead before they ate her?" Soldier asked, and the girl shook her head and stared at the floor. "Did they let you have a taste?"
"You're horrid. You're a monster," she said, and Soldier figured she was probably right, but then maybe the brown girl was only a different kind of monster.
"What year did they take your father away?"
"Why won't you stop? Why won't you leave me alone?"
"Don't you even remember?"
The Daughter of the Four of Pentacles walked back to the edge of the bottomless hole and placed her foot firmly on the end of the board. She leaned forward suddenly, and the board bounced a little, and Soldier had to hold her arms out like a trapeze artist to keep from falling off.
"Don't do that," Soldier yelped. "Are you crazy?"
"August twelfth, 1929," the girl said, and then she made the board bounce again. "They took him away on August twelfth, 1929. It was a Monday night. It was raining."
Soldier pinwheeled her arms once, twice, then stood very still until she was sure she wasn't going to fall. She was good with numbers. Madam Mnemosyne said she was "quite precocious at arithmetic," and it took her only a moment to figure out that the brown girl had been shut away in the attic of the yellow house for more than seventy-eight years.
"You're an old woman," Soldier said. "You might look like you're only twelve, but you're really an old woman."
"I'm not an old woman," Pearl replied. "I'm not an old woman, because there's no time up here anymore, except when the attic door's open. The clocks only tick when the attic door's open. It's a temporal contrariety, and I'm not even thirteen years old yet." And then she bounced the board a third time, and Soldier had to squat down to keep from falling off.
"It would be very easy to make me fall, Hester, if that's really what you want. All you'd have to do is kick that end of the board over the edge."
"That's not why they sent you up here. The Bailiff said that—"
"Then stop jumping on the goddamn board," Soldier shouted as loud as she dared, even the effort of shouting enough to make her wobble and bob, and she knew the wildebeest would be falling forever. She was barely more than halfway across, and if she slipped, she'd still be tumbling after it, head over heels, ass over tits, when the universe had burned itself down to a frozen cinder. Her voice echoed through the ancient timbers, off the distant walls of the cavernous attic, and the Daughter of the Four of Pentacles cursed her, cursed all the changelings and the ghul. And then she turned and ran, vanishing in an instant into the gloom, as though the attic shadows were her true parents, and they would always be there to keep her safe.
In the tunnel beneath Woonsocket, Soldier marks time in footsteps, in Odd Willie's ragged, wheezing breaths, in her own heartbeats. She doesn't know how long they've been following the passageway, not in minutes and seconds, but it's been longer than she expected it would be. Odd Willie is muttering and snickering to himself, and the smell of smoke has grown very strong. It stings her sinuses and makes her eyes water, and it's no longer only the scent of woodsmoke; Soldier knows the smell of roasting human flesh, and the air inside the tunnel reeks of it.
"Yippee damn," Odd Willie croaks. "We're going to a goddamn barbecue."
"How about you shut up and concentrate on walking," Soldier tells him, and just then Odd Willie slips on a bit of the glowing slime and almost falls, almost pulls her down with him. "And watch your fucking step, please."
"Miss Soldier here, she don't like her no barbecue." Odd Willie snickers. "This lady takes it raw, or she takes it not at all."
And Soldier's about to ask him how much farther until they catch up with the familiar, if the thing can tell him that, when the main tunnel makes a sharp turn to the right and the incline grows suddenly much steeper. There are stairs cut deep into the stone here, slick and worn down the middle from centuries of use.
"I think she went thataway," Odd Willie says and points at the stairs, but that much is plain to see from the phosphorescent goo streaking the walls. And then he slides easily from Soldier's grip and sits down hard on the bottommost step, his long legs splayed out in front of him. "I can't climb those things, Soldier. Not for all the pussy in China could I climb those goddamn things."
"Well, I'm not carrying you."
"Then you'll just have to go on without me, Sarge," he says, and Odd Willie Lothrop hacks out a feeble laugh and wipes fresh blood from his lips. "Tell Laura I love her. Give my regards to fucking Broadway. Tell me dear old mum she was the last thought on me mind. But ain't no way I'm climbing those fuckers."
"It's right up there, isn't it? He's right up there, Ballou?"
"And don't you just fucking hate the irony," Odd Willie says, and coughs and wipes his mouth again.
"You should stop talking so much, Odd Willie. You're wasting your strength."
"And just what exactly, my lady sweet, am I supposed to be saving it up for? Are you perhaps hoping for one last go at the ol' arbor vitae here before the bitter end?" and Odd Willie waggles his limp penis at her. "One final ride on the ham bone of love before abandoning poor Willie Lothrop to his cruel, cruel fucking fate?"
"You're a sick fucking fuck," she says and shakes her head, and Soldier wants more than anything to sit down next to him and shut her eyes for maybe five or ten minutes, just long enough to catch her breath. But she's pretty sure if she does that, she'll never get back on her feet and moving again. So, she glances past Odd Willie instead, gazing up the long, uneven stairwell at the place where it ends two or three hundred feet farther on in a bright smudge of restless firelight.
"But we're so goddamn close. It's right up there," Soldier says and realizes that the draft she felt outside the ossuary earlier has changed direction, and now the hungry fire in front of them is drawing the air back towards it. "It's not fucking fair."
"Don't you go getting soft on me. Not this late in the game. I still can't climb those stairs."
"I promised not to leave you," she says, then pauses to swallow, her throat dry and sore and scratchy from the smoke and everything else. "I promised I wouldn't leave you alone down here...not alive."
Odd Willie stops playing with himself, spits, and lies back against the stone steps. "And that was truly very noble of you, my captain. So don't you think I'm not grateful. But I'm afraid you're gonna have to renege on that promise. You kill me, Sparky up there goes bye-bye, and she might not be done with Big Daddy Ballou."
"I'm sorry," Soldier says, ashamed that she feels relief at not having to kill Odd Willie, ashamed and confused. "I'll come back for you if I can. I swear, if there's any way."
"Yeah. And maybe I'll try crawling up those stairs, just as soon as I've found my second wind. Hope springs fucking eternal, right? Now, get the hell out of here, before I change my mind and make you do something unpleasant."
And so she leaves him there and climbs the stairs as quickly as she can. With every few steps, the light at the top of the stairs grows brighter, and it's getting hot, despite the cool, dank air being drawn up from the tunnels below. She doesn't look back, because there isn't any point, no matter how ashamed she might feel. No matter how fucked up things have gotten, and by the time Soldier reaches the brick and mortar landing at the top of the stairs, she has to shield her eyes from the light and can feel the heat of the flames beginning to sear her bare skin. There's a stone archway and then a great chamber spread out past the landing, what she takes to be a natural cavern uncovered during the ghouls' tunneling. There's thick black smoke, oily and choking, and steam, and a blistering curtain of heat that makes it hard to be sure what she's really seeing and what's only a mirage. The crumpled, blazing bodies of several of Ballou's changelings and mongrels lie near the landing. Soldier steps past them into the cavern, which seems to have been long used as a burial chamber, moldering bodies wrapped in winding sheets and skeletons tucked into hundreds of nooks in the walls, and now she's walking on stone that's hot enough to scald her feet if she stands still too long.
In the center of the cavern are the remains of a bonfire, a guttering mound of charred wood heaped seven or eight feet high, and a very tall, broad-shouldered man is standing directly in front of it. Glittering embers swirl around him, filling the air, and he raises his right hand so that Soldier can see what he's holding. It's a stoppered bottle, and even through the smoke and heat haze, she can see that there's something fiery trapped inside. His yellow eyes glimmer triumphantly, and he smiles.
"It was a damn good try," he says, shouting to be heard above the roar of the flames. "I've always said, an honest man gives credit wherever and whenever it's fucking due. Mr. Lothrop is to be commended. I lost some good men before I finally managed to get this beast under control. A goddamned elemental. I ask you, now, who'd have thought such a thing from a twerp like Willie Lothrop?"
Soldier looks around the chamber, and sees that there are others, crouched in the shadows or slinking in through the entrances to other tunnels, emerging through cracks in the walls. Some of them are human, some of them aren't, but most she couldn't say for sure.
"The Bailiff sent me here to kill you," she shouts back at George Ballou.
"Yes, I know. Miss White was very clear on that point. She insisted we should have killed you straight out. Why take chances, she said. Why fuck around? She's usually a cautious woman, but me, I do admit I like a little sport now and again."
"They sent me here to kill you," Soldier says again, and Ballou nods and sets the bottle holding Odd Willie's familiar down on the ground at his feet. The assembling crowd of changelings and half-breeds and ghouls begins to chatter and bark excitedly among themselves.
"Well then," says George Ballou, and he slips a very large hunting knife from a long scabbard on his belt; the blade flashes in the firelight. "Let's start the fucking dance, shall we?"
## SEVEN
## Star
It's a little warmer inside the old tunnel, if only because Emmie's finally out of the storm, out of the wind and snow. But the subterranean air is stale and stinks of neglect and the tiny, long-stemmed mushrooms growing from the old railroad ties. Their caps are neither red nor white, but not exactly pink, either, and Emmie figures she'd probably die if she were to eat one of them. The tunnel is flooded on either side of the tracks, but the brown girl has assured her the water isn't deep. The smooth cement walls rise up around them, converging overhead in a wide arch encrusted with icicles and stained with the soot of forgotten trains. Emmie can see because the brown girl is carrying some sort of fist-sized flashlight or lantern, something she pulled from a pocket of her dress shortly after they entered the darkness. Emmie thinks it looks more like a snow globe with a lightbulb sealed up inside than any sort of flashlight, but when she asked the girl what it was, she wouldn't say, only that it was one of her father's experiments, and she really shouldn't have brought it down from the attic. No one was ever supposed to touch her father's things, she said, not even her. The brown girl holds the light above their heads and walks toe to heel, balanced on one of the steel ties, and the yellow-white light rains down around them. Emmie stays off the rails, because they look slick with moisture and a fuzzy sheen of charcoal-colored mold; the thought of losing her footing and falling into the scummy black water makes her feel queasy. So, instead, she follows the girl by keeping to the half-submerged string of wooden pallets laid down over the railroad ties.
The only sounds are their footsteps and a steady dripping from somewhere nearby, but the tunnel makes the most of it, seizing every footfall and the irregular xylophone beat of water striking water, magnifying, reverberating, and, "I bet this is where the King of Echoes hid from the Queen of Silences," the brown girl says.
Emmie glances back over her left shoulder, but they're far enough inside now that she can no longer see the pale glow of the entrance.
"Where are we going?" she asks the girl. "How much farther is it?"
"Well, truth be told, I've never been down here," the brown girl confesses. "But I can't imagine it could be very much farther. Barnaby said to just keep walking the tracks westward, and we'd come to it, eventually."
"Who's Barnaby? And what are we looking for, anyway?"
"You really do ask an awful lot of questions, Emma Jean Silvey," the brown girl says, and the light in her hand bobs and sways when she almost loses her balance. The shadows on the walls bob and sway, too, and Emmie holds her breath for a moment until she's sure the girl isn't going to fall.
"Be careful! If you drop that thing, if it broke, we'd never find our way out of here in the dark."
"Oh, don't be so melodramatic. All we'd have to do is walk back out the same way we came in. We might misstep and get our feet wet, but I know the way back. So do you, Emma. And besides, if this broke," and she nods at the globe, "we wouldn't have to worry about finding our way back. We wouldn't live that long."
Emmie almost asks her what she meant by that, that they wouldn't live that long, but then decides she doesn't want to know.
"Barnaby's just a ghoul, that's all," the brown girl continues. "They aren't all completely terrible. Some of them are decent enough people. I helped him find something that he'd lost, and, in return, he helped me get out of the house to assist you and Soldier."
"Who's Soldier?" Emmie asks.
"You'll see," the brown girl replies. "We shouldn't get too far ahead of ourselves."
"I'm tired," Emmie says and stops walking. "I'm tired, and I'm sick of riddles, and I'm hungry, too."
"Barnaby said that there'd be a little food. He put it there himself. If the rats haven't found it."
"Deacon's probably looking for me by now," Emmie says, and peers over her shoulder again; there's nothing back there but the dark. "He's probably already called the police."
"You shouldn't worry about the police," the brown girl says. "They won't find us here. They likely couldn't if they tried. But they won't. Try, I mean."
"He'll think I'm lost. He'll think I'm freezing to death out in the snow." And Emmie imagines Deacon searching the big house for her, searching it twice over, top to bottom, then searching the front yard, the backyard, and finally looking up and down the length of Angell Street, banging his fists on doors, talking to people he doesn't know because maybe one of them saw her. She imagines him getting scared, shouting her name over and over until he's hoarse. Then he'd go back home and call the police. He'd call the hospitals. He might call Sadie; he'd think it was her fault.
"There it is," the brown girl says and points at something rising from the gloom and the water at the right side of the tracks.
Emmie stares at it a moment, trying to make sense of the strange, twisting angles and unexpected bulk of the thing. Her first impression is that she must be seeing the rusty red-brown carapace and jointed legs of some gigantic lobster or crab that's dragged itself up from the icy water, which must be much deeper than the brown girl said to ever hide such a thing as that. And now the monster is perched right there at the side of the railroad tracks, waiting to devour them both.
"I told you. I knew that Barnaby wouldn't lie to me," the brown girl says and smiles.
It's only a car, Emmie realizes. Only a shitty old car that someone's dragged in here, her mind slowly making sense of the snarl of rusted metal, slowly recognizing the familiar made unfamiliar by the dim light and years of corrosion. A stripped and burned-out wreck missing its roof, not a giant crustacean after all, not something with pinching claws and blazing eyes set on twitching stalks. Emmie's legs feel weak, and she imagines Deacon trying to tell the police what she looks like, what she might be wearing, Deacon trying to remember what her coat looks like, wondering which coat she's wearing. And maybe the police would ask if he'd been drinking. Maybe they'd figure out he was drunk, or think he was crazy and was only hallucinating that he had a daughter who was lost somewhere in the snow, being chased by monsters and hiding in old railway tunnels.
"I know it doesn't look very inviting," the girl says, "but, sometimes, looks truly can be deceiving, Emma Jean." And then she steps off the rail and over something crushed and folded in upon itself that Emmie thinks might once have been the driver-side door.
"You're gonna cut yourself and get tetanus," Emmie tells her, because that's what Deacon would say—You'll get lockjaw climbing around on something like that—but then the girl's staring back at her from the middle of the wreck, and Emmie reminds herself it's all only a dream. You don't catch diseases in dreams, and even if you do, you don't wake up sick.
"It's got to be here somewhere; I'm sure of it," the brown girl says, holding the ball of light out in front of her as she examines all the crannies and corners and the gaping holes torn in the metal. Emmie isn't exactly sure what the brown girl's looking for, so she waits and watches at the edge of the train track instead of following her inside the wreck.
"What's your name?" she asks.
The brown girl stops looking for whatever it is she's looking for and stares at Emmie a moment without replying.
"Forget it. You don't have to tell me," Emmie says, "not if you don't want to, not if it's a secret or something."
"No, it's not a secret," the girl replies. "It's just not something I have much use for anymore. Sometimes I almost forget about it. Isn't that odd?"
"Well, you know my name," Emmie says and leans closer, and now she can see that someone's gone to the trouble of wrapping what remains of the car's front seat beneath a couple of heavy blue blankets. "I could sit down there," she says, "if you don't mind?"
"Of course," the brown girl says and shakes her head. "Why didn't I think of that?" And it takes Emmie a couple of seconds to realize that of course wasn't an answer to her question. The brown girl gets down on her knees and holds the globe so she can see whatever's under the front seat. When she stands up again, she's grinning and holding a brown paper bag.
"See?" she says. "Barnaby's the most reliable ghoul I ever met." The top of the bag is rolled closed, and the brown girl unrolls it and looks inside. "My name's Pearl," she adds.
"You're not really going to eat something you found in here, are you?" Emmie asks and points again at the seat wrapped in blue wool blankets. "Is it okay if I sit down?"
"If that's what you want to do. I'd wager Barnaby put those blankets there for us, too. He can be very thoughtful."
Emmie steps off the pallet onto the rail, which is at least as slippery as it looked, and then she works her way carefully past the sharp edges of the twisted door frame to stand beside the girl whose name is Pearl.
"There are sandwiches," Pearl says, still staring into the paper bag, "and two pears, and I think that might be...oh...well, ghouls have their own idea of food, you know. But the sandwiches are probably fine, and the pears."
Emmie sits down; the springs beneath the blankets make a soft, crunching sort of sound, and she sinks a few inches into the seat. It feels good, being off her feet after the long walk, better than sitting down has ever felt before, she thinks, and it would be easy to shut her eyes and go right to sleep. She pulls off her mittens and gloves and stuffs them into her coat pocket.
I'm already asleep, she tells herself, so maybe if I shut my eyes, I'd wake up instead.
"I hope you like liverwurst," Pearl says, and Emmie sees that she's taken one of the sandwiches from the bag and is peeling back the waxed paper it's wrapped in. "Because I think that's what we have here. Liverwurst and cheese and horseradish sauce."
"I've never eaten it," Emmie says, leaning back in the seat, "but it sounds disgusting."
"It's just sausage made from pigs' livers, mostly."
"No, thank you," Emmie says and makes a face. "Besides, I don't eat food someone left lying around in a boarded-up train tunnel full of rats and mushrooms and who knows what else."
"Suit yourself," Pearl tells her. "But you said you were hungry," and then she sits down beside Emmie, and the car seat creaks again. The brown girl sets the glowing orb on what's left of the dashboard, puts the bag on the seat between them, and takes a big bite of the sandwich. Emmie ignores her and watches the snow globe thing instead. Up close, it doesn't look like it has a lightbulb inside. It looks like there's a sun trapped within the orb, a star no bigger than a very large jawbreaker. She starts to touch the snow globe, and Pearl mumbles something around a mouthful of liverwurst and bread, and Emmie pulls her hand back and apologizes.
"What's in there?" she asks. Pearl swallows and wipes her mouth. "It almost looks like a star," Emmie says.
"It is a star. That's one of my father's later experiments, and he's extremely proud of it. He was careful to take a star none of the astronomers had ever seen, so no one would miss it. It came from somewhere near the constellation Cassiopeia, I believe."
"That's impossible," Emmie says and squints at the bright thing suspended at the heart of the orb.
"Be that as it may, it's still a star," Pearl tells her and tears away a piece of waxed paper to expose more sandwich.
"Your father put stars inside snow globes?"
"It's not a snow globe," Pearl says, glancing up from her sandwich. "It's all very complicated, but my father moved places and moments. He was studying spatial and temporal translocation for the ghouls," and the way she says translocation makes it sound like the most important word in the world. "But you shouldn't stare at it too long. It could hurt your eyes. You might even go blind."
"Yeah," Emmie says very softly, not quite whispering, "if you say so."
"You might at least have one of the pears," Pearl tells her, looking into the brown bag again. "They're perfectly fine, I promise, and you ought to eat something. You'll need your strength."
"No, Pearl. I need to go home. I need to wake up."
Pearl sighs and takes another bite of the sandwich. Emmie stares at the snow globe again, at the star. "Well, you should at least make up your mind," Pearl says with her mouth full.
"What?"
"Make up your mind. Whether you want to go home or wake up. Whether you're really here in the tunnel with me, or whether you're only having a bad dream."
"This doesn't feel like a dream anymore," Emmie says, and immediately wishes that she hadn't, that she'd kept that thought to herself.
"There's a reason for that, I suspect. But what do I know? I eat garbage and drag frightened little girls away into blizzards when they have perfectly safe, warm houses."
"If I believe you, Pearl, what happens next?" Emmie asks the brown girl, turning away from the snow globe, and her eyes trail yellow-orange afterimages that refuse to go away when she blinks.
"That's not an easy question to answer," Pearl tells her, "not even if you believe, and I don't think that you do, which makes it harder."
"But if I did believe you?"
Pearl drops what's left of her sandwich back into the brown paper bag, then rolls the top closed again. "I hate liverwurst," she says. "My father never makes me eat liverwurst."
"I want to go home," Emmie says. "I want to know if Deacon's safe. I want you to tell me what happens next. I want to know when it'll be safe to go home again."
"You want a terrible lot of things," the brown girl replies. "You may have to settle for less."
"Fine. I want to know what happens next."
There's a loud booming sound then from the darkness waiting at the limits of the snow globe's radiance, and Emmie covers her ears and grits her teeth. She can feel it in her bones, that sound, and it makes her think of the thing she saw on Waterman Street, the thing that wasn't a mammoth or a mastodon or the Wendigo, the thing riding the wind. The booming fades slowly away, but Emmie's ears are still ringing painfully, and something about the sound has left her slightly sick to her stomach.
"Be very, very quiet," Pearl whispers and reaches for the snow globe with the star inside. It seems to glow the smallest bit brighter when she's touching it. "They don't know we're down here, and it can't see or smell through solid rock, and it loathes the tunnel. So, we'll probably be safe if we just stay quiet."
"I want to go home," Emmie says again.
"I know," Pearl tells her and holds the globe up, pushing back the darkness a scant few feet. "I know you do. I want to go home, too, Emma Jean."
"Hold my hand," Emmie says, and the brown girl does, her hand colder than ice, colder than the silty bottom of the deepest sea or the empty heart of a solar system whose star has been snatched away, leaving behind only ice and the endless twinkling night and dead planets frozen straight down to their cores. But Emmie doesn't let go, and they sit together in the wrecked car, hand in hand, shivering and waiting for the booming to come again, waiting for whatever's supposed to happen next...
...and when Emmie opens her eyes again, the cold and damp are gone, and she's staring up into the widest night sky that she's ever seen. There's a cool breeze—cool, but not cold—a wind that smells like cinnamon and jasmine and dust, and when she sits up, she sees that she's lying near the crest of an enormous sand dune. The old railroad tunnel has vanished, the tunnel and the wrecked car and the girl named Pearl. Instead, a vast desert stretches out around her, countless grains of sand to mock the stars overhead, and there's something dark sparkling wetly on the horizon, something she thinks might be water, a river or the sea. The moon is high and white and only a few days from full.
"Damn it," she says, lying back down in the sand, which is still warm from the day before. "I'm not awake. Maybe I'm dead now, and I won't ever wake up again."
"You're awfully young to be so concerned with what is and isn't a dream," someone says, an old woman's voice or only a woman's voice weathered and worn until it seems old. Emmie rolls over onto her left side, and the woman, who doesn't look old at all, is squatting in the sand only a few feet away. Her skin is black—not any shade of brown, but skin as perfectly, truly black as an obsidian arrowhead or a licorice whip—and her amber eyes shimmer dimly. Her eyes remind Emmie of her own, though they're more golden than yellow. The woman is dressed in white muslin, and her hair is white, too, not gray, but white, arranged in long dreadlocks reaching down past her shoulders, framing her high forehead and cheekbones.
"I know the difference," Emmie says, surprised that she's not afraid of the woman, who she's certain wasn't there only a moment before. "I know when I'm dreaming."
The woman smiles and takes a deep breath. "That's a lot to claim," she says. "That's a mighty conceit, child."
"I was here once before, wasn't I?" Emmie asks. "These are the wastes at the end of the world."
"It's true some people have called them that," the woman replies. "And some other people have called them other things. Myself, I've never thought them a waste, and this is hardly the end of the world. The world goes on far beyond this place."
"But I was here before?" Emmie asks her again, growing impatient and noticing that the woman's earrings are the sharp teeth of some tiny animal strung on loops of silver wire. They glint in the bright moonlight.
"Were you?" the black woman asks. "Or was that only a dream? If you know the difference—"
"Yes, it was a dream," Emmie sighs, exasperated and in no mood for games, and she sinks back down onto the warm, welcoming sand. "That's what I meant to say. I was here before in a dream, just this afternoon, when I fell asleep in my bedroom."
"Well, if you say so, since you claim to know the difference."
"Where'd the tunnel go?" Emmie asks. "Where's Pearl?"
"It hasn't gone anywhere. It's right there beneath the hill, where it's always been. And Pearl's where you left her, asleep beside you. Don't worry about Pearl. She's safe, for the time being."
"That's the prettiest sky I've ever seen," Emmie says, though it's more than that, more than pretty, but she can't think of the words that would do it justice. Staring up at all those stars, at the brilliant swath of the Milky Way, makes her a little dizzy, and for a moment Emmie thinks it would be easy to fall, the sand and sky reversed, and all she'd have to do is stop trying not to fall.
"It comforts me," the woman says, glancing up into the night. "When I'm lost, it helps me find my way."
"I'm afraid Pearl's not right," Emmie tells the black-skinned woman. "She's even weirder than me, and that's saying a lot. I think there's something wrong with her."
"She misses her father," the woman suggests.
"I'd be worried about her, if she were real."
Then neither of them says anything for a minute or two, and Emmie lies still, listening to the wind blowing through the dunes. I could lie here forever, she thinks. I could lie here forever, counting stars, and never go back to that smelly tunnel. Never go back to the cold, or my room, or—
"But what about Deacon?" the woman asks. "Wouldn't you miss him?"
"Of course I'd miss him. I'd miss him more than anything. But maybe he'd be better off. Maybe things would be easier for him if I wasn't there, if he didn't have to worry about me all the time."
The woman picks up a handful of sand and then lets it sift out between her fingers; some of it is carried away on the breeze, and some of it falls back onto the dune. "I'm going to ask something of you," she says.
"I don't even know who you are," Emmie replies.
"It's not important who I am. I can't tell you that, not tonight. But I'm still going to ask something of you, something important."
"You mean a favor?"
"You could think of it that way, I suppose."
"Deacon says I shouldn't talk to strangers, so I doubt seriously I'm supposed to go around doing them favors."
"When you first met Pearl," the black woman says, "the day the horse was killed, do you remember what she said to you? She said, 'Build me a bridge, Emmie. I have long desired one.'"
Emmie starts to ask the woman how she knows that, then decides there's no point, that it doesn't matter anyway, and she nods her head instead. "I told her she was a loony bird," Emmie says.
"But she was telling you the truth. You are a bridge builder. Indeed, you are a bridge yourself. You span the distance between humanity and the creatures Pearl calls the Hounds of Cain."
"The ghouls," Emmie mumbles and works the fingers of her left hand into and under the sand.
"Yes," the black-skinned woman says. "And they've been seeking a bridge as well. They have been seeking a bridge desperately for thousands of years. Some among them believe they have found her, but they're mistaken."
"I don't know what you're talking about."
"Child, listen to me," the woman says; her words have become urgent, half whispered, and she's dropped down onto her hands and knees, her lips near Emmie's left ear. For an instant she seems like something more animal than human, some night-colored, reptilian thing that sleeps away the days in caverns beneath the shifting sand and slips out at sunset to stalk the desert. Her breath smells of ashes and sage. "Listen to what I'm saying, child. Your mother is a changeling, and your father was the son of the union of a changeling and a hound. There is a secret locked up in your soul, a terrible secret that would free the ghul, if they could only reach it. The hounds believe the bridge builder is the daughter of Deacon Silvey, but they're mistaken."
"I'm the daughter of Deacon Silvey," Emmie says.
"One among them suspects the truth, I think, and he is the one you must fear above all the rest. He is not a hound, but he keeps their counsel."
"This is a big fat load of baloney," Emmie says. "I'm going to wake up now. And not in that damned tunnel, either. I'm going to wake up in my house, in my bed."
"He'll have you dead, Emmie, if he gets his way."
And Emmie's about to tell the woman to shut the hell up, to stop talking and go away and leave her alone so that she can wake up, when something rises from the sand a few yards from her feet. The keels of its dusky scales scrape roughly across the sand, and its eyes shine bright and red, two holes burned deep into the fabric of the night. Emmie has never imagined a snake even half this big, and she's seen pythons at the zoo, pythons and even an anaconda from Brazil, but this snake makes them look like the worms she finds dying on the sidewalk whenever it rains. The snake, if it is a snake, grins and flicks its long forked tongue, and she can see there are at least a thousand needle teeth set into its jaws.
"Whatever have you found this time, old whore?" the huge snake hisses, dragging more of itself up from the sand, and the sides of its neck open wide like the hood of a cobra. Emmie begins to scramble backwards away from it, but the woman lays a strong hand on her shoulder and tells her not to move.
"You can't ever outrun it, child, though it would be pleased if you were to try. Be still. Be still and wait."
"It's only a dream," Emmie says and shuts her eyes. "It's only a dream, and I'm going to wake up now."
"Share it with me," the snake hisses. "It's a plump little thing. There's plenty enough for us both, and maybe even a few scraps for the jackals and vultures left over."
"Stay away from her," the woman says to the snake. "She's not for eating," and the serpent makes a sound that isn't a laugh, but Emmie knows it was meant to be.
"No?" the snake asks. "That's too, too bad. But tell me, if it isn't for eating, then what's it for? Is it for diversion, perhaps? For play? It's so soft, so pale, so, so young."
"Go, Emmie," the black-skinned woman says. "Go now. This instant. You know the way back across. I don't expect you need me to show you."
And she's surprised to find that she does know the way, sure as she knows all the scars that Deacon says he got the day she was born or the winter-sky color of Sadie's eyes, sure as she knows the way from one side of Angell Street to the other. She could do it blindfolded. She could do it in her sleep.
"Who were you talking to just then?" Pearl asks, and Emmie rubs her eyes and blinks, trying to remember where she is and why she's so cold and hungry. "Where did you go, Emma Jean? What did you see?"
"I didn't go anywhere," she mutters, remembering the railroad tunnel, the long walk through the storm, the desert and the snake and the woman with amber eyes. "I fell asleep, that's all."
"You were talking to someone. I heard you."
"I was dreaming," Emmie says, and then she tells herself that she's still dreaming, that this is only the outer dream wrapped around that darker inner dream. There could be a lot more than two layers, she thinks. It might go on forever, dreams within dreams within dreams, like letting go of the sand and falling up into the sky. "Anyway, it's none of your business what I was dreaming."
Pearl looks annoyed and offended, frowns, then turns away from Emmie and reaches for her father's snow globe, which is still sitting on the dashboard. "I do have my own dreams," she says. "I certainly don't have any need of yours. But we'd better get moving soon, changeling. Something's coming. We shouldn't have slept so long."
"What's coming?" Emmie asks her, sitting up, hugging herself against the dampness and the chill.
"At first I thought it might only be Barnaby," Pearl says, lowering her voice, and then she rubs her bare hands over the surface of the glowing ball, as if to warm them. "But there's more than one of them. The tunnel connects with the warrens, back towards Benefit Street."
"You said we'd be safe—"
"Actually, no. I said that we'd be safe until morning, which it may well be by now, for all I know. By now, Emma, they've probably figured out where we are. If you sit there talking long enough, you're sure to find out."
"Christ, Pearl. Just slow down, okay? I'm not even awake yet. I'm freezing and—"
"Swearing is a sign of a poor upbringing. And, in any case, you probably shouldn't talk so loud. They have very good ears, the hounds. They can hear—"
"My upbringing is something else that's none of your business. My upbringing has been just fine."
"Well, Deacon's a drunkard, and your stepmother's a witch who runs away whenever things get too scary," Pearl says and glances at Emmie.
"And just what the hell was your father, Pearl, putting stars inside that...whatever that is...and leaving you locked up in an attic. People who live in glass houses—"
"Point taken," Pearl says and shrugs. She stares down at the globe, looking deeply into it despite her earlier warnings about damaged eyes and blindness; her face is underlit, washed in the softly pulsating yellow-orange-white light shining from the ball. Emmie squints and rubs her eyes again.
"We're still in the tunnel?" she asks.
"Where else would we be, dear? Can't you hear them?"
Emmie listens, holds her breath and listens for at least a full minute, but she doesn't hear anything at all except the water dripping from the concrete ceiling to the flooded floor of the tunnel. "No," she says. "I can't. Maybe you're wrong." Her hands are cold, and she takes her gloves and mittens from her pocket and puts them on again.
"Maybe your ears are full of wax and fluff," Pearl says and stands up. "There's still a pear left in the bag that Barnaby brought for us, but I'm afraid we don't have time for breakfast, even if you'd deign to eat garbage I found in—"
"What will they do if they catch us?" Emmie asks, standing up and peering into the darkness farther along the railroad tunnel.
"They'll put me back in my attic. They aren't allowed to hurt me. But I can't say precisely what they'll do with you, except that it likely won't be pleasant. It won't be pretty."
"I was dreaming about a woman in a desert," Emmie says, and looks back at the brown girl. "And a giant snake that wanted to eat me. The woman was about to ask me a favor, but the snake came along before she got around to it. Now will you please stop acting like such a bitch?"
Pearl pushes back her bangs, which have fallen across her eyes, and then she glances at Emmie. "A woman with skin as black as a lump of coal?" she asks. "A kind woman with white hair?"
"She didn't seem very kind to me," Emmie says. "But her skin was black and her hair was white. Why? You know who she is?"
"We should go now," Pearl tells her, instead of answering the question. "Maybe if we head back the way we came, they might not try to follow. If the sun's up, they won't follow us outside."
"No. First, you tell me who she is," Emmie says. She still hasn't heard anything but the dripping water and is unconvinced that someone's coming for them. "Tell me what she wants."
"These days, they don't get fresh meat very often," Pearl says, and begins climbing out of the car and back onto the tracks. She moves slowly, the snow globe cradled in the crook of her left arm so both hands are free as she picks her way through the wreckage.
"I could have held that for you," Emmie says and points at the globe.
"No, you couldn't have. There are rules. Hurry along, Emma. They're coming fast now."
"Is she one of them?" Emmie asks, stealing another look into the darkness beyond the car before she follows Pearl. "The black woman in the desert? Is she one of the ghouls?"
"Hardly," Pearl replies. She's standing on the wooden pallets between the rails now, holding the snow globe in her right hand. "Now stop your dillydallying, unless you want to wind up on Miss Josephine's vivisection table."
"Why won't you tell me who she is?"
"There's not time. I can tell you later, unless they catch us, in which case it won't matter."
"I don't know about you, but I'm smart enough to walk and talk at the same time."
"Sticks and stones, dear heart," Pearl says, and then she flips her head to one side and smiles a conceited, insincere smile.
I hate you, too, Emmie thinks, thinking it hard and sharp so maybe Pearl will overhear, wondering if she's ever hated anyone or anything half as much as she's starting to hate this strange girl. Emmie wishes that she'd brought her cell phone, because then she'd know what time it is. If she had her phone, she could call Deacon and tell him where to find her, and he could stop worrying whether she'd been kidnapped or murdered or worse.
"Nothing's coming," she says. "I have pretty good hearing, too. The doctor said so, and if there were anything coming, I'd have heard it by now." But then she does hear something—a wet and breathy sound like panting dogs, the splash of feet moving quickly through the flooded tunnel. Her heart seems to pause between beats, her mouth gone dry and cottony, her left foot dangling halfway between the wrecked car and the slick steel rail.
"You'll have to do better than that," Pearl says and pulls her the rest of the way down to the tracks. Emmie almost falls, her feet slipping, balance lost, and she would have gone down face first in the mud and gravel ballast and splintery, rotting pallets if the brown girl hadn't been there to catch her. "We'll try to go back to the entrance," Pearl says. "Maybe we can get outside and make it across the river. Barnaby said there are places to hide over there."
"Let me go home," Emmie says. Her legs are shaky, and she wants to sit down, wants to cry. "Please, Pearl. Just let me wake up and go home. I can't do this anymore."
"Stop whining and follow me," the brown girl says and begins towing her along the tracks, back the way they entered the railroad tunnel. Emmie stumbles once or twice, the toes of her boots catching on the edges of the pallets, but then she's running on her own. The panting sounds are getting louder, and the splashing feet, too, and Emmie doesn't know why it took her so long to hear them. Maybe, she thinks, I didn't hear them because they weren't there to hear. Maybe it's only Pearl making those noises, making me hear them. But she doesn't stop moving. There's no way to be sure of anything anymore, and she'd rather run from make-believe monsters than find out that she's wrong. The soles of their boots are loud against the boards, her bulky winter boots and the brown girl's old-fashioned lace-up boots.
"We waited too long," Pearl says, out of breath, and she stops and looks hastily back over her shoulder. The noise the ghouls are making is much louder now, and Emmie can hear them calling back and forth to each other in guttural animal voices. "You shouldn't have wasted so much time."
"No, Pearl, we can make it. It's not that far. We should be almost there."
"Then tell me, why can't I see the doorway out? We should be able to see it by now, but it's not there."
"Maybe someone shut it," Emmie says and reaches for Pearl's hand, because they can't stop running now, not with those splashing, yelping things bearing down on them.
"No, they've changed something, Emma Jean. They've changed the tunnel. Maybe there isn't a door out anymore. Maybe the tunnel just loops 'round and 'round. Maybe—"
"That's crazy. They couldn't have changed the whole tunnel. We just haven't gone far enough; that's all."
Pearl sighs, and her breath steams white in the cold air, a sudden rush of mist visible in the light from the snow globe, there and gone in only an instant, and she looks down at the alien sun inside the glass ball.
"They can see it, can't they?" Emmie asks, pointing at the snow globe. "We should hide it, get rid of it or something. If they can see it."
"Of course they can see it. But hiding it wouldn't make any difference. They see better in the dark than in daylight. We're in their element, Emma. And even if they were blind and deaf, they'd still find us by our smell alone."
"Why are we just standing here?" And Emmie takes two or three steps farther along the tracks, but Pearl doesn't move. "Come on," she says. "You don't know the door's shut. You don't know for sure."
"Sometimes," Pearl says, her words wrapped in a veil of white breath, "sometimes, Emma Jean, I think that they've killed him. Sometimes I think my father's never coming back because they've killed him, or they've sent him so far from me he can never find his way home again."
"No place is that far away," Emmie tells her, and one of the pallet boards cracks loudly beneath her feet, and she takes another step. "You can fly all the way around the world in just a couple of days, so nowhere's that far away."
Pearl watches her a moment, her expression enough to say that she's not sure if she knows what Emmie's talking about. "My father never learned to fly," she says. "Once, a demon offered to teach him, but he was never interested."
"Airplanes," Emmie says. "I meant flying in airplanes."
"Oh, aeroplanes. Yes, of course that's what you meant. But there are many, many places where aeroplanes cannot go, Emma Jean Silvey. Most places, in fact."
"Hester? Hester, can you hear me?" a voice calls out from the murk, someone or something half barking, half speaking, and Pearl bites at her lower lip and sighs again.
"That's Barnaby," she says and smiles, but not a relieved or happy smile, a sad, disappointed sort of a smile. "They probably threatened him. He must have lost his nerve. He didn't have that much to start with. Do you know the way back to the desert, back to the black woman?"
"Pearl, that was only a dream."
"You think this is only a dream."
"Hester, you must stop running now. No one wants to see harm come to you, I promise. They'll take you home again; that's all."
"Home again. Back to the attic," Pearl says softly and looks from the glowing ball to Emmie. "If you know how to return to her, perhaps you should go. Sometimes there's no point running away, because you can't run fast enough or far enough to make any difference."
And then Emmie grabs her arm and tries to drag her forward. But Pearl shakes her head and digs her heels in. The ghouls are very close now, close enough that Emmie thinks she can smell them, a stink like spoiled meat and wet dog, like old cheese and boiled cabbage.
"No," Pearl says, wrenching her arm free of Emmie's grip. "I told you, they won't hurt me. They wouldn't dare, and maybe I can stall them here. And if you can find your way back to her...Here. Take this...." And she shoves the snow globe that isn't a snow globe into Emmie's hands. Emmie's so surprised she almost drops it. It's much heavier than she expected and so cold that she can feel it straight through her mittens and gloves, so cold it almost burns her skin.
"But you just said I couldn't touch it," Emmie protests, wanting to give it right back. The cold's bad enough, the cold and the weight, but there's something more coming from the orb, something she could almost mistake for music, the most dreadful music she's ever heard. "You said there were rules."
"For pity's sake, there's not time left to argue," Pearl snaps and pushes her so hard that Emmie almost loses her footing. "Run, you silly little idiot. If you can find the way back, run as fast and far away from this place as you can go."
And then something lopes out of the shadows and into view, something moving on all fours that stops a few feet from Pearl and stands up straight and tall on its long and spindly hind legs. Its matted belly fur is dirty blond in the starlight from the globe, and its thick black lips fold back to reveal teeth the color of antique ivory. Its eyes are the color of fire.
"Where's Barnaby?" Pearl asks the thing, turning away from Emmie. "I heard him. Where is he?"
"Barnaby will be along directly," the thing barks back at her and wrinkles its nose. "Don't you fret about Master Barnaby. But that one there"—and the ghoul points a claw at Emmie—"that one has become a matter of significant concern. Step aside, Hester."
And so Emmie runs, the freezing ball of light clutched tight to her chest, her head quickly filling up with its song. But she's gone only a few feet when one of the pallet boards snaps beneath her, and she pitches forward. Behind her the ghoul has begun to howl, and before her the railroad track has dissolved into a sky full of stars, opening wide to take her back. Emmie shuts her eyes and lets it have her.
"I always knew you'd be back, someday or another," the black-skinned woman says and smiles. "It was really only a matter of when and where."
Emmie's standing with the woman at the very top of an extraordinarily high sand dune, watching as Pearl's snow globe rolls away down the steep slope, throwing up a fine spray of sand as it goes. It slipped from her mittened hands when she opened her eyes and saw the wide blanched sky and the yellow-brown expanse of the desert, the undulating dune fields broken only by a few scattered outcrops of weathered rock. This time the woman is wearing a long-sleeved, ankle-length thobe dyed a very pale blue with tiny red and silver beads embroidered about the neckline. She's barefoot, and there's a short sword or a long dagger tucked into a length of cloth cinched about her waist.
"Crap," Emmie says and begins unwinding the alpaca muffler from around her throat, because the sun here is bright and hot, and she's already sweating underneath her heavy winter clothes. "I wasn't supposed to drop that."
"Don't worry. It won't go very far," the woman says. "No farther than it can roll. We'll fetch it back."
"But if it breaks—"
"It's not going to break. Trust me."
Emmie drops the muffler to the ground at her feet and unzips her coat. "Pearl pushed me," she says, sliding her arms free of the sleeves. "I told her we could make it if she'd run, but she pushed me."
"She must have been very afraid for you," the woman says. "I expect the alchemist's daughter knows well enough what the hounds would have done had they caught you."
Before, in the dark with only the waxing moon for light, Emmie hadn't realized how beautiful the woman was. By daylight, the sun glints off her ebony skin and catches in the dazzling topaz facets of her irises. She might be the most beautiful woman in the world, Emmie thinks and sits down in the sand. She might be an Ethiopian or Egyptian princess disguised as a Bedouin bandit.
"It's a long way down," Emmie says and points at the orb, still rolling towards the bottom of the dune, the place where this dune finally ends and the next begins. "Is that big snake still around here?" she asks and glances up at the woman.
"Oh, don't you bother yourself about him. That was far ago and long away. A lot has changed since then."
Emmie pulls off her gloves, the right and then the left, and she lays them neatly on top of her mittens. "You mean long ago and far away," she says, and then she takes off her sweater, folds it, and puts it on the ground next to her coat and gloves and mittens.
"I think that I know what I mean, child," the woman replies, and squats in the sand next to Emmie. "Anyway, the stone drakes are all but extinct. He was one of the last."
"This is like Narnia, isn't it?" Emmie asks and wipes her sweaty hands on her pants. "Like when Lucy and Susan and Peter and Edmund go back to Cair Paravel, and it's only been a year for them, but in Narnia it's been ages since they left, and everything's changed."
"I certainly wouldn't say that everything's changed," the black-skinned woman tells her. "In fact, it's pretty much the same around here as it has been since the sea dried up and the dunes and salt flats came to take its place. But, yes, the drakes are gone, which is really only a good thing. They were foul, intemperate beasts."
"It wanted to eat me."
"Well, you can't really blame him for that. A creature, even a wicked one, cannot be faulted simply for wanting to eat, and out here, very few creatures can afford to be picky."
Emmie stands and brushes sand and dust off the seat of her pants. She doesn't feel like arguing with the woman about whether or not the snake had a right to try to eat her, and she points at the orb again, which has finally stopped rolling and lies partly buried at the base of the dune. "I should go get it now. I probably shouldn't just leave it lying there like that."
"No," the woman says. "You probably shouldn't."
"You never did tell me what the favor was, what you wanted to ask me to do." And then Emmie starts down the dune before the woman can reply. Each step she takes triggers a small avalanche of sand that spills out before her in gleaming fan-shaped flows. Narrow rivulets break free and meander rapidly towards the bottom, and Emmie thinks it's almost a miracle, all these millions or billions of grains of sand balanced here by the wind, laid one atop the other and nothing to hold them in place but gravity.
"I thought you didn't do favors for strangers," the woman says, starting down the dune after her.
"I've decided Deacon might approve of you," Emmie replies and stops so the black-skinned woman can catch up. "You told the snake it couldn't eat me."
"I see. Then I suppose that was very shrewd of me, wasn't it? Do you still think you're dreaming?"
Emmie stands staring at the streams of sand winding their way downhill, and she notices a small gray lizard that her footsteps have disturbed. It scurries away and burrows into the dune.
"I think maybe it doesn't matter," she says. "If it's all a dream, then I'll wake up sooner or later, unless maybe I'm in a coma or something. And if it's not a dream, I won't wake up. But either way, I think that I should at least try to do the right thing. Just in case."
"Emma Jean, do you believe that you know what the right thing is?"
"I'm only a kid. I know what Deacon and Sadie tell me is right, but sometimes they contradict each other. I know what I've read in books. I know what my teachers say."
"And, more important, you know what you feel in your heart is right," the woman says and continues past her down the dune.
"I'm only a kid," Emmie says again. "So I might be wrong. I might have it all turned around backwards."
"I'm wrong all the goddamned time," the woman laughs and looks over her left shoulder at Emmie. "Don't you start thinking that getting older makes much difference about knowing the right from the wrong."
Emmie watches the woman for a moment, then follows her, and when they reach the foot of the dune, Emmie picks up the orb, but immediately drops it again. Even after baking under the desert sun, the thing's still freezing cold to the touch. "I don't know if I can carry it," she says and blows on her fingers. "I'll get frostbite if I try to hold that thing for very long. I don't know how Pearl was doing it."
"I'm afraid you'll need your gloves after all," the black-skinned woman says, and Emmie looks back up the towering dune to the spot where she left the discarded pile of clothing and groans.
"Crap," she says and sits down beside the orb. There's sweat dripping from her face, speckling the sand, and her T-shirt's already soaked straight through. "I'll have a heart attack."
"I can carry it back to the top for you," the woman says. She kneels beside Emmie and lifts the glowing orb, blows some of the dust and sand away, and then rubs it clean against the front of her blue thobe.
"Doesn't it burn you?" Emmie asks, and the woman shakes her head.
"Getting old might not teach you right from wrong, but it can give you thicker skin."
"You don't look old."
"Which should be a lesson to you, child." Then the woman gazes into the glass ball and licks her lips. "Amazing," she whispers. "Such an amazing and blasphemous thing this is. Perhaps it's best, what the hounds have done with the magician."
"I don't believe in god," Emmie says.
"A thing can be blasphemous," the black-skinned woman tells her, "whether there's a god involved or not. This," she says and taps a fingernail against the orb, "this thing is a perversion, a blasphemy. The mere fact of its existence is a crime against the world. It's wrong, Emmie."
"I thought so, too. But I don't think Pearl agrees."
"Pearl loves her father, and sometimes love can blind us to the truth of things."
Emmie looks back towards the top of the dune again, squinting at the sun reflected off the sand. "I suppose we should get started," she says, "if we have to walk all the way back to the top for my gloves. And if you're still going to ask me for a favor, maybe you should ask me for it now, just in case I do have a heart attack or a heatstroke and die."
The woman nods her head and blinks, looking away from the orb. "There's a changeling woman whom the hounds call Soldier. You have to go to her, Emma Jean Silvey. And you have to carry this thing to her. She's close to death and in great danger, and there's not much time left."
"Pearl said something about her. I asked who Soldier was, but she wouldn't tell me. She said we shouldn't get too far ahead of ourselves."
The woman turns towards the orb again. "Soldier's the true daughter of Deacon Silvey," she says. "She's the one the hounds thought might at last build their bridge away from this world. She's paid an unspeakable price for their beliefs, and your fates, yours and Soldier's, are bound almost inextricably one unto the other."
"Deacon is my father," Emmie says, standing up.
"He loves you," the woman says. "You mean everything to him."
"And you have no right saying he's not my father."
The woman sighs, and for a moment the sun sealed inside the orb dims. "That's the favor I ask of you, child. It's more than anyone should ever ask of another, but these are desperate times. If you do as I've asked, you'll learn the truth of things, and in the end you'll curse me for that knowledge. Would there were any other way."
"She's really dying?" Emmie asks.
"Yes. She's dying and lost in a dark place, and very soon she'll stand before a foe almost as blasphemous as this vile thing," and the woman nods at the glass orb. "I can show you the way down to her, the correct where and when, but it has to be your choice."
"This isn't really like Narnia at all, is it?"
"No," the woman says. "It's not."
"Could you also show me the way home, if I were to say no? Could you show me the way back to Providence?"
"There's no need. You already know the way home."
"Crap," Emmie says again, and then she begins climbing the dune, but the sand shifts and slides out from under the soles of her boots, and three steps only carries her right back to the bottom.
"I know a shortcut," the black-skinned woman says.
"Do you have a name?" Emmie asks her.
"Oh, I've had many names, but I've found it's usually best I keep them to myself," and then the woman takes Emmie's hand in hers, and the desert dissolves, collapsing into stars and empty space the same way that the railroad tunnel dropped away after Pearl pushed her. The scorching desert day is replaced by night, a twinkling indigo sea of constellations, and Emmie thinks about all the questions she wishes she'd asked the black-skinned woman with topaz eyes and waits for the long fall to end.
## EIGHT
## Intersections
And from the starry place, all things are possible, and, perhaps, all things are also probable. Possibility is infinite here, and possibility collides, in spiraling space-time fusillades, with probability at every turn. The unlikely and the never-was become, for fleeting instants, the actual and the inevitable and the black facts of a trillion competing histories, each entirely ignorant of all the others, each confident that it's the only true history. Emmie is wearing her gloves and down coat again, her alpaca muffler and her mittens, and she clutches the glass orb as she slips between everything that was and is and never quite shall be. Sometimes she shuts her eyes, because there are things she cannot comprehend and would rather not see, and sometimes she opens them wide and wishes that she could see more clearly and more fully understand what she's seeing.
The stars, and the almost empty spaces between the stars.
Light and darkness and things that are not exactly either one or the other.
Countless detours on the road the black woman has told her to follow...
~ A girl in an attic, a girl who isn't Pearl, holds a wooden animal out over a wide hole that has no bottom—and she almost drops it in, then decides the wooden animal isn't hers to drop. Instead she sets it down on the dusty floor and steps back from the edge of the abyss.
~ Sadie is reading Emmie The House with a Clock in Its Walls by John Bellairs, but this Sadie has only one hand, and she turns the pages with two shiny steel hooks strapped somehow to the stump of her left wrist. This Sadie has red hair, like her red door, like the red of pomegranates.
~ "What about a carriage ride?" Hunter Fontana asks, and Sadie says no, she's heard terrible things about the way the horses are treated, and the three of them go back inside the museum to see the blue whale instead.
~ A careless elbow, and a paperweight that isn't a paperweight at all rolls off the edge of an alchemist's workbench. There's a breathless moment when the air smells like gardenias and millipedes, and then a blinding flash of light, and the world vanishes, and Emmie is never born and Deacon is never born and the solar system is only an ember.
~ The woman from the train, the woman with the Seal of Solomon tattooed on her hand, presses the barrel of a gun to a priest's head. Her finger tightens on the trigger, and she curses him. The priest is begging for his life, begging her to understand that he had no choice, that he never had a choice. Then he wets himself and sinks to his knees, his tears and urine dripping to the floor of a darkened room. She spits on him, and a few moments later, she places the gun to the soft spot beneath her jaw. The priest screams when she pulls the trigger.
~ Saturday afternoon in the Kingston Station, and Emmie tells Deacon that she's sick to her stomach, that she thinks she's going to vomit, and wants to go home. On the way back to Angell Street, he talks about blackbirds.
~ Another gun, this one in the hands of a tall, pale man in a Dunkin' Donuts, and he shoots an old lady first, then the woman behind the cash register, then...
Concentrate, the woman in the desert tells her. You get lost out there, and you'll never get found again.
But there are so many diverging paths.
There are so many choices.
Emmie tries to look away, fights to keep her eyes on the invisible, intangible current pulling her along towards wherever it is she's meant to be. But faces flicker and moments flash all around her.
~ An albino girl in sunglasses, hitchhiking beneath a scalding Southern sun. At first, Emmie thinks the girl is alone, but then she sees the thing following her. The albino girl thinks it's an angel.
~ Deacon, but he's a young man, and he stands at a rusted iron gate leading into a tunnel, into the limestone heart of a mountain. It's raining, and he lifts a pair of bolt cutters and clamps them shut on the hasp of a lock. The padlock falls away and splashes in the mud at his feet. Someone giggles, and the gate creaks open.
~ A snowy night at the edge of the sea, and a house burns, a very old and haunted house, evil beyond reckoning, and inside a girl named Narcissa Snow dies with her grandfather.
~ On the train to New York, a woman watches Emmie from across the aisle. Emmie wishes that she'd stop, because there's something about the woman that makes her nervous, something that frightens her. The woman smiles at her and starts to stand up, but then someone takes the empty seat next to Emmie, and the woman sits down again. She gets off at Old Saybrook.
~ Deacon's bedroom in their house on Angell Street, and he's taken the box from beneath his bed, his cardboard box of secrets. He carries it downstairs and out the front door to the place where they leave the garbage cans by the curb. Emmie's watching him from a living room window. She's only six years old, and she decides that whatever's in the box, it's probably nothing but old clothes or worn-out shoes or something else she has no interest in. He comes back inside and fixes them hot dogs for dinner, and he tells her a very funny story about how her mother once thought she'd found a dinosaur bone, but it turned out she'd only found part of a fossil tree stump instead.
~ A woman who looks a lot like Chance Silvey is sitting in a hearse at the end of a country road, talking to a man named Sheldon. "Something's wrong," she says and rolls her window down just enough to toss out the butt of a cigarette. "I don't know what, but something's gotten fucked-up." It's raining hard, rain slamming loud against the roof of the hearse, and a flash of lightning reveals the marshes at the edge of the road and a ramshackle building farther out. "Fuck it, Shelly. Let's get the hell out of here," the woman says, and the man named Sheldon shoots her in the face.
~ "She died when you were still just a little baby," Deacon tells Emmie and sips his beer. "Her name was Sadie, and she wanted to write books. I'll be damned if she wouldn't have gotten a kick outta you, kiddo."
Emmie looks back, hoping to see the black-skinned woman and the warm desert sun, hoping it's not too late to turn around and head back that way, hoping she knows how to turn around. But behind her there is only the void, the slipstream of her passage erasing a trillion possible outcomes to a hundred thousand worldlines.
~ Deacon is standing on the front porch of their house talking to two policemen, and he's barefoot despite the snow piled high in the yard and at the edges of the porch. He's been drinking, and he calls one of the policemen a son of a bitch and a cocksucker, and the two officers exchange glances and frown at each other. "We need you to come with us, Mr. Silvey," one of them says. "I'm sorry. I know how hard this must be, but we need you to come down and look at the body, to be sure that it's your daughter."
No, Emmie says and squeezes her eyes shut, her voice lost in the clamor of all these possible existences playing out around her. That's not what happens. That's a lie.
Then you'd better start looking where it is you're putting your feet, the black-skinned woman says, or maybe it's really the girl from the attic who's talking, or her stepmother. There's no way to be sure. But Emmie keeps her eyes closed and drifts in the current, letting it drag her she thinks, and then there's heat and light and the smell of meat cooking reminds her how hungry she is. In an instant, all potential collapses around her into and she opens her eyes in the hollow carved out beneath Woonsocket.
one moment
and
one place,
"I have to admit," George Ballou says, his face a grinning mask of blood and grime and soot, "I was expecting just a wee bit more from ol' Terpsichore's prize bulldog. You've got me kind of fucking disappointed here." He's squatting next to Soldier, who went down fast and hard shortly after Ballou decided that he'd be better off fighting her with incantations than with knives. She's lying naked at his feet, her back and buttocks beginning to blister on the hot stone.
"You're breaking my fucking heart," she mumbles around a busted upper lip, and he slaps her again.
Firelight fills the burial chamber and casts wild, shifting patterns across the high walls as Ballou's mongrels feed a mix of cordwood and corpses to the hungry flames. Some of the men and women and half-breeds have begun a frenzied summoning dance, cavorting around and around and around the fire, shouting and howling, taunting an unseen sky, cackling and recklessly calling out the names of gods that Soldier's been taught never to speak aloud. The very few full-blooded ghouls present stand apart from all the others, lingering in twos and threes near the periphery of the chamber, watching and waiting, patient and curious as all immortal things.
"Now look what you've gone and done," Ballou sighs. "You made me lose my goddamned train of thought. No damn wonder the Bailiff decided he'd be better off without a mouthy little cockshy like you."
"So why don't you kill me, and then maybe I'll shut the fuck up," Soldier says; she keeps her eyes on the dancers and the bonfire because she's already seen more than enough of George Ballou. He's a mountain, a living mountain dressed up in muscle and bone, as if that might be enough to hide the truth, and his eyes blaze bright and vicious as any ghoul's. His long gray hair is pulled back in a ponytail that hangs down past his waist; his huge knuckles are a patchwork of scars and fresh gashes, knuckles like the gnarled roots of an ancient willow tree, if any willow ever bled.
"Down here, Providence, we do things in my time," he replies and leans closer. His breath smells like bad teeth and wintergreen. "Down here, you die when I say you die. And I'm thinking maybe that's gonna be a while longer yet, seeing how you're hardly the hellcat I was led to expect. You might even make a halfway decent meretrice, once the dogs have softened you up a little. We can always use fresh breeding stock down here. And who'll be the wiser, eh?"
"Kill her," one of ghouls barks out above the roar of the fire. "That was the bargain."
George Ballou glares at the ghoul, then glances back down at Soldier. "Is that so?" he asks. "See, I don't remember the exact wording. In fact, I don't seem to remember anything quite that goddamned specific at all."
"Kill her," the ghoul barks again, louder than before.
"You shut the fuck up!" Ballou shouts back. "She'll die. When I'm done with her, she'll die just fine."
"Maybe the leash is a little shorter than you thought," Soldier says, wincing at the heat and all the raw and wounded places where his magick brushed against and through her. He grabs hold of her chin, forcing her head up, forcing her to look into his glittering holocaust eyes.
"Don't you think for a goddamned minute I don't know that game," he snarls, baring his crooked yellow teeth. "That mouth might get you hurt a little more, but it won't get you dead, Providence, not until I'm good and ready for you to be dead."
"Just seems like that collar of yours might be getting tight," she says through clenched teeth, and Ballou slaps her again, this time opening her left cheek with his long nails.
"See what I mean? That hurts, but you're still breathing. Want to try again? Who knows, you might get lucky."
Soldier doesn't answer him. She rolls over on her left side and watches the dancers and the flames and the darting, swooping shadows they're painting on the walls. There are tears streaming from her eyes now, and she begins laughing out loud so she won't start sobbing.
"Yeah, that's what I thought," Ballou snorts. He shakes his head, and beads of sweat and spittle, snot and blood spatter Soldier's naked body. "Damn, but those Benefit Street cunts are gonna wish they'd been choosier about their messiahs. They're gonna wish they'd put their money on a horse that could actually fucking run, aren't they, Providence? Maybe if they'd ever gotten around to teaching you what to do with that special rewind switch you got, if they hadn't been too afraid to try, you might not be in this predicament. Fuck it, you could just send us all back to next fucking Friday and be done with it."
"I don't know what the hell you're talking about," Soldier says and blinks back salt water and pain. "Maybe you've got the wrong mouthy cockshy. Maybe—"
"Oh, no. No, don't you worry your pretty head about that. I've got just exactly who and what I want," he says and gets to his feet. "George Ballou does his homework, yes, ma'am, and your friend Saben, she might be a sloppy killer and a lousy fuck, but she's pretty good at keeping her promises. You're the one, all right, the holy goddamned changeling whore that's supposed to lead them all back to the land of milk and fucking honey. Now, how about you hush up for half a second or so, and I'll see if I can't loosen you up myself." And he unzips his pants.
There's a sudden keening sound from the direction of the bonfire, something loud enough to rise above the din echoing off the walls of the cavern, and a cloud of sparks swirls up from the flames and hangs a few seconds in the smoky air. The dancers cry out triumphantly and begin moving faster, their bare feet slapping ash and paving stones, arms and legs a drunken blur, their matted hair and contorted faces rendering them all but indistinguishable from one another. Soldier doesn't shut her eyes, and she tries not to flinch as a few of the embers settle on her exposed skin.
Ballou leans close again, and his penis slips out of his pants and dangles limply between his legs. It's tattooed to look like the head of a serpent.
"Seems to me like they've done sent you straight on down to hell, Providence," he says, screaming at her to be heard over the keening sound and all the noise the dancers are making. "But maybe you better shut your eyes now. You're a tough piece of snatch, all right, but everybody's got a breaking point, and I'd like you sane and fully fucking cognizant of your situation when I finally do get around to slitting that sweet little throat of yours."
Soldier blinks, and there's the Bailiff watching her from the far side of the bonfire, and it's almost like seeing him from underwater or seeing a ghost of the man, the way the heat makes the light bend and writhe.
Stay ignorant, he says, and someone will almost always benefit from your ignorance, and then she realizes that it's not the Bailiff at all, just one of the Woonsocket ghul, a thin and mangy bitch waiting to see what happens next.
"What do you want from me?" Soldier asks, asking the Bailiff who isn't there, but Ballou thinks she's asking him.
"No more than your masters and mistresses have ever wanted," he says. "Your soul. Your heart, my dear. Every iota of your being. A little sport, while we're at it. Your flesh and bones when there's no more sport to be had."
"I wasn't talking to you." someone will almost always benefit
"Whatever were they thinking, Providence, leaving you all alone, unprepared and unprotected? They've dithered your life away, unable to make a simple decision. It all might have gone another way, but for the turn of a friendly fucking card."
Occasionally, the weapons are reluctant to fulfill their purpose. Guns jam. Arrows miss their mark. Shots go wild....
Ballou touches her, his calloused palm moving slowly across her flat, hard belly, moving down towards the bloodied tangle of her pubic hair, and he nips at her right shoulder with his sharp teeth.
"What do you want, George Ballou?" Soldier asks him. She's noticed that the bonfire is expanding, its circumference steadily increasing like the pupil of some giant's eye dilating as it forces the world into focus—whatever the Woonsocket half-breeds have summoned, waking up, shaking off aeons of sleep. Bright tendrils of fire and glowing ash slither across the floor, licking tentatively at the feet of the dancers. Then Soldier sees that one of the mongrels has the stoppered bottle with Odd Willie's captured elemental inside, a silver-haired woman with a bristling gray mane. She holds the bottle high and casts it into the bonfire.
"Only the respect due to me and mine," Ballou mutters in her ear. "That, and maybe just a little fucking more. Right now, I'm in the mood to watch the whole goddamn world burn, so we'll just have to see how it goes."
Do you want to die here? the Daughter of the Four of Pentacles asks, the words that Soldier dreamed or only imagined, delirious and half-awake, because the girl is locked away in the attic of the yellow house, locked away forever or near enough. But that doesn't change Soldier's answer to the question.
George Ballou bites down on the back of her neck, and his hand slides roughly between her legs.
It's a simple spell, something she doesn't even remember having learned, only a few lines of Tadjik to set an elementary alchemy in motion. She doesn't even have to say the words aloud, just follow through with the pantomime, the silent interplay of her lips and teeth, her palate and her tongue. A simple spell, a very small magick, but there's nothing simple or small about the pain, the searing white ache in her hands as molecular bonds are broken and reforged, and she opens her mouth wide and screams.
The thing the bonfire is becoming screams back at her, but no one else seems to notice.
"Whoa," George Ballou laughs, and then he laps at the damage he's done to her neck. "Slow down, girl. You're getting ahead of me."
When she turns on him, her fingers have become lightning and molten steel, and Soldier screams again and drives both her hands deep into his rib cage until she finds his spine. It snaps like a rotten branch in the hands of a child. For an instant, Ballou only stares at her, surprised and disbelieving, and then his whole body shudders violently and his yellow-orange eyes roll back in his head; Soldier feels his soul slipping free of its tethers, and she lets it go.
Emmie Silvey stands alone in the entrance of the cavern, the stone stairwell at her back, the brown girl's glowing orb cradled in her mittens, and she watches as the naked woman pulls her hands free of the dead man's shattered chest. The woman and the dead man and all the rest of what she sees, the great fire and the monsters skipping and hopping and dancing crazy circles around it, the burning whirlwind rising from the center of the fire towards the high ceiling. And Emmie knows that wherever this is, wherever and whenever she's emerged from the starry place, it's the very worst part of the dream so far. The naked woman glances up at her, looking Emmie directly in the eyes, and for a second Emmie thinks that the woman has knives for fingers.
"Soldier?" Emmie whispers, her voice immediately lost in the pandemonium, swallowed by the mad wails of the dancers and the train-whistle bellow of the fire. The woman looks confused, but she nods her head and then glances down at the dark blood and clots of gore dripping to the floor from the ends of her fingers (which aren't knives after all). Then Emmie sees the two hairy, dog-jawed creatures standing together near the wall, and she knows immediately that they're the same sort of beasts that came for her and Pearl in the old railway tunnel. She takes a step backwards, and the naked woman looks up at her again.
"Help me," Emmie says. "I don't know what to do. I don't understand any of this," and she holds out the glass orb because the black-skinned woman said she was supposed to find someone named Soldier and give it to her.
One of the ghouls shakes its head and makes an ugly mewling sound. It glares at Emmie and bares its long teeth.
"Don't move!" Soldier shouts at her, and then she turns towards the ghouls, and they snarl and whine and huddle against the wall. They're afraid of her, Emmie thinks. They're monsters, but they're still afraid of her.
Soldier has turned to watch the immense and crackling thing dragging itself free of the bonfire, the thing that has begun to coalesce from the fire itself, and she stands silhouetted against the conflagration. Emmie wants to run back down the stairs, vanishing into the darkness behind her. Maybe there would be a place to hide, a cold and lightless place where it won't ever be able to find her. Maybe she could even discover a way out of the dream.
Some of the dancers have started to burn, their thrashing bodies wreathed in tendrils of blue-white flame.
"We have to get out of here!" Emmie screams, and Soldier looks back over her shoulder.
You have to go to her, Emma Jean Silvey. And you have to carry this thing to her.
Don't move, the naked woman says again, but Emmie can't hear the words, can only see the shapes her lips are making. Don't move, but the heat has begun to sting Emmie's eyes and make them water, and she can feel it getting in through all her heavy winter clothes, pushing past her coat and sweater and T-shirt, touching her. She knows that in a minute or only a few more seconds, the fire thing will be done with the dancers, and then it'll eat the woman named Soldier, and the ghouls, and then it will eat her, too.
"We can't stay here!" Emmie screams, squinting against the light and the heat, but Soldier only shakes her head and points at her ears—I can't hear you.
And then the ground begins to lurch and roll, ripples passing quickly through the stone as though it were only mud, the shock waves beginning at the bonfire and racing out towards the walls of the chamber. Emmie almost falls, and the snow globe with a star inside it slips from her clumsy, mittened hands and hits the floor hard enough that a labyrinth of hairline cracks opens up on its glassy surface.
If this broke, Pearl said, we wouldn't have to worry about finding our way back. We wouldn't live that long.
Emmie reaches for the orb, but then the ground begins to shake and roll again, and this time she does fall, goes down hard, and her chin strikes her left knee; she bites her tongue, and her mouth begins to fill with blood. The floor groans and tilts towards the scrabbling fire thing clawing itself free of its bonfire womb as the last of the dancers are incinerated alive, and the orb pulses once, twice, and rolls away towards Soldier.
Emmie begins crawling on her hands and knees, heading for the stairs, and even though there are only a few feet between her and the wide landing, it might as well be a hundred times that far, the way the floor is pitching and rolling about underneath her. All she had to do was give the snow globe to the woman named Soldier, but now she's screwed it up. She's broken the orb, ruined the fragile shell of glass and magick spun by Pearl's father, and soon the star imprisoned inside will escape, and Emmie knows there's no point in running from a star. But she runs anyway, crawls because she can't run; she's been brave too long, and now there's no courage left inside her.
"I want to go home," she says, because maybe that's all it will take, just like Dorothy and Glinda and the ruby slippers. In the desert, hadn't the black-skinned woman told her that she already knew the way, that she didn't need anyone to show it to her? "I want to go home and wake up," she says. "I want to be home with Deacon now. I want to be home."
Another convulsion racks the cavern, and dust and sand and bits of rock and roots pepper the floor all around her. Emmie stops crawling towards the stairs and lies down flat, despite the heat of the stone against her cheek, trying to hold on because she's afraid if she doesn't she's going to start slipping back towards the bonfire and the thing coming out of the bonfire.
"Now," she says. "I want to go home now. I know the way, and I want to go home."
"Kid, I don't know who the hell you are, but I think I know exactly what you mean."
Emmie looks up, and there's a very thin, tall man kneeling next to her. He's naked and filthy, too, just like the woman named Soldier, and he's holding half a long human thighbone clutched tightly in his right hand. He smiles at her, but Emmie doesn't smile back. For all she knows, he's just another crazy person or some other sort of monster. For all she knows, he's someone else who wants to hurt her.
"Stay away from me," she says. "You leave me alone."
"Duck," he says, then puts his free hand on top of Emmie's head and forces her back down onto the hot floor. He swings the thighbone, and there's a dull thud, barely audible above the noise from the fire. Emmie rolls over, and one of the ghouls is lying crumpled at her feet, the left side of its face caved in. There's a rusty carving knife, the wooden handle wrapped in duct tape, lying on the floor beside it.
"Holy mother of crap, I've always wanted to do that," the tall man says and grins. "Just haul off and let one of the ugly motherfuckers have it."
"You saved me. It was going to stab me. It was gonna kill me, but you saved me."
"Don't take it personally," he says. "Now, maybe you can tell me what the hell she thinks she's doing," and he points and Emmie follows his finger, looking past the dead ghoul, and sees that the woman named Soldier is holding the cracked orb and standing much nearer the fire than before. All the dancers are dead now, smoldering, twisted heaps scattered in a charcoal ring, and the whirlwind of sparks and smoke and flame is spinning faster, gathering momentum.
"I don't know," Emmie says, and then the thing that the dancers have summoned turns towards Soldier, and it opens three simmering eyes the color of nothing that Emmie's ever seen, and she turns away.
"Aren't they wonderful?" the girl in the attic of the yellow house on Benefit Street said to Soldier when they'd finally reached the place where the alchemist had worked. They stood in front of the wooden shelves and the tall cases holding hundreds upon hundreds of glass or crystal spheres, most of which seemed to Soldier to be filled with fog or some milky whitish liquid.
"Aren't they simply the most wonderful things you've ever seen?"
"What are they?" Soldier asked, and the Daughter of the Four of Pentacles smiled a secretive smile and shrugged her shoulders.
"I'm not supposed to tell you that. But maybe you could try to guess."
"I don't like guessing games," Soldier said. "If you don't want to tell me, then I don't want to know."
"I'd tell you if I could. They told me not to."
"You could tell me anyway."
"Why? Why should I do that? You've been nothing but dreadful since you climbed up that ladder. And besides, they're not the reason that you're here. I just wanted you to see. My father is so proud of them."
"I think you don't know what they are," Soldier said. "You don't know, and you're lying about being told not to tell because you don't want to look stupid."
"You're impossible," the girl replied. "Of course I know what they are."
"Then prove it."
"I shouldn't even have shown them to you. A brat like you doesn't deserve to see anything this wonderful. Anyway, we should get started soon. There's an awful lot to be done before the Bailiff comes back for you."
"He's never coming back," Soldier sighed and took a step nearer the shelves. One of the spheres had caught her eye. It seemed less foggy than the others, and she thought maybe there was something moving around inside it besides the milky stuff. "He's left me here forever. We have to be sisters now."
"Frankly, I'd rather have a spider for a sister," the alchemist's daughter said and made a face like she'd bitten into something bitter. "No, wait. I like spiders. I'd rather have an old toad for a sister."
"What's in there?" Soldier said, ignoring the insult and peering into the glass ball, at all the specks moving about inside. She thought they must be bugs—ants or fleas, maybe. But then she saw that they weren't bugs at all.
"Hey," she said, "those are people. Those are tiny little people."
"We're wasting time, changeling. I'm a very busy person, you know."
"But that's what they are. They're a bunch of tiny little people in a glass ball."
"Don't you dare touch it," the Daughter of the Four of Pentacles said sharply, because Soldier was reaching for the sphere. "You mustn't ever touch any of them. Not ever, you hear me? It can be very dangerous."
"How'd they get in there?" Soldier asked.
"Never you mind about that. That's not any of your business."
Soldier leaned as close to the sphere as she could without her nose bumping against it, and stared at the scene inside. There were rooms within it, and there were tiny people in most of the rooms—people who seemed no larger than ants or fleas. Some of them appeared to be talking to each other. Some were running about, as if they were being chased by something invisible, something that Soldier couldn't see. Some were sitting alone on miniature chairs or sofas or lying on tiny beds. One was on its knees, praying at an altar festooned with minuscule candles.
"You've seen enough," the Daughter of the Four of Pentacles said and took Soldier by the arm. "You're here for a reason. We need to get started."
"Do they know they're inside a glass ball?" Soldier asked and tried to pull free, but the girl was holding her too tightly.
"Who cares? What difference does it make? Now come on. I have other things to do."
Soldier frowned and glanced at the glass ball one last time, then let the alchemist's daughter lead her away from the shelves and the tall cabinets, towards whatever it was the Bailiff had sent her up into the attic to see or do.
"And what the hell did he put inside you?" Soldier asks, staring down at the orb in her hand. The glass is freezing, so cold that it hurts to hold, but she holds it anyway. A fine mist has begun leaking out through the cracked surface, and the reek of ammonia rises from it to mix with all the hot, burning smells. She can see that there's some sort of fire sealed in behind the glass, a bright mote drifting in a sea of night, and she wonders if it might be something like Odd Willie's elemental. Soldier looks up at whatever it is that Ballou's mongrels and the Woonsocket ghul have midwifed and finds it gazing furiously back down at her. Three gaping holes leading nowhere, holes that must be its eyes, and they open and close, close and open, one after the other in a fierce counterclockwise gyre.
Right now, I'm in the mood to watch the whole goddamn world burn, so we'll just have to see how it goes.
A thin blue wisp flickers from the thing in the bonfire, like the stinging, phosphorescent tentacle of some deep-sea creature, and it briefly grazes the left side of her throat, then disappears again. Soldier winces, but after the beating in the ossuary, after the tunnels and George Ballou, it's only a very small pain.
"I know that you were invited," she says and swallows, her throat so sore and dry it's getting hard to talk. "But I'm afraid we've changed our minds. Here. Take this for your troubles," and Soldier heaves the freezing glass ball at the gyre of blinking eyes. It sinks into the thing, passing straight through flesh that's still mostly flame, and disappears.
"You can go now. We don't want you here."
The three eyes shut in unison and then open again, much wider than before. Chasms of hate, the most perfect expression of hatred that Soldier has ever seen or imagined, malevolence in the absence of any other sentiment that might taint or dilute its purity. Above the thing, the whirlwind breaks apart, scattering cinders across the burial chamber, and she instinctively covers her face with her hands. The floor trembles and tilts again, then seems to drop several inches all at once, and Soldier is hurled to her hands and knees.
This is it, she thinks. This is where I'm going to die, and the thought's not so very terrible, not even so unwelcome, after everything that's happened since she drove through the gates of Oak Hill Cemetery and straight into Ballou's glamour. There is regret, though—that she's not going to live long enough to murder Saben White, that she'll never understand why the Bailiff let her walk into this mess or what the hell George Ballou was up to down here, what he thought he could accomplish by summoning beings he could never hope to control. And other questions, too; the nagging, unanswerable questions she might never have remembered, except for the dream of Sheldon Vale's ghost and a clock with a dead girl's face.
A sound fills the cavern, then, a sound like the world dying and being born again, a sound like tumbling cataracts and falling leaves and the grinding teeth of sleeping gods. Soldier clenches her hands into fists and whispers half-forgotten prayers she's never believed, supplication and benediction, a mumbled litany for Mother Hydra and Father Kraken waiting in their silent, flooded halls at the bottom of the bottom of the sea.
And then the fracturing, rending sound ceases as suddenly as it began, and the thing writhing inside its birth caul of fire howls one last time and breaks apart, dissolving, collapsing into itself, becoming no more than tattered, glimmering shreds of ash and slag. The bonfire is only a bonfire again, and in the instant before Soldier loses consciousness, she can hear Odd Willie calling her name. It sounds like he's at least a thousand miles away.
When it's over, when the brown girl's snow globe has been swallowed by the fire, and the monster has been destroyed or driven away or whatever it is that Soldier has managed to do, Emmie carefully picks her way up the split and buckled sections of flooring to the landing and sits alone at the top of the stairs with her back to the dead bodies and the sputtering remains of the bonfire. She takes off her coat and mittens and gloves, and lets the cool, moist air flowing up the dark stairs wash over her.
The handful of surviving ghouls have fled the chamber through narrow fissures in the walls, and the thin man who told her that his name was Willie is wearing the clothes of the man that Soldier killed, though they're much too big for him and the pants keep sliding down. He's wrapped Soldier in a long linen robe that he stripped from the body of the ghoul he killed with the thighbone, and then it takes him at least half an hour to carry her over the rubble and up to the landing. He lays her next to Emmie and sits down, slicked with sweat, looking sick and gasping for breath.
Emmie brushes hair from Soldier's face. "Is she going to be okay?" she asks him.
"Maybe," Willie Lothrop says and wipes at his sweaty face. "Maybe not. I'm not a doctor, kid. I'm not even sure I'm going to make it."
"My name's Emmie," she says. "The black woman in the desert sent me here."
Odd Willie laughs and wipes at his face again. "Is that a fact? Well, I have no idea what the hell you're talking about, Emmie, but perhaps we'll figure it all out later. Unless we die down here."
"She can't be Deacon's daughter," Emmie says, speaking to herself now instead of the skinny man with no eyebrows. "She's too old. She can't possibly be Deacon's daughter."
"She ain't nobody's daughter, nobody except the hounds. Jesus fucking Christ, did you see that shit?"
"I saw," Emmie says. "I don't know what any of it was, but I saw it."
"Yeah, me either. Me and you both. But it was sure some crazy, fucked-up shit; I know that much."
"She's hurt," Emmie says, then pulls off her right glove and lays her palm against Soldier's forehead. Her skin is pink, like someone with a sunburn, and she has a fever. "I think she needs a doctor."
"Hell, at this point she probably needs a goddamned exorcist. Whatever the shit that thing was, it most certainly wasn't a goddamn hobgoblin. Nobody goes up against something like that and comes out...." And then he stops wiping his face and stares at her a moment or two. "Who the hell are you?"
"I told you. I'm Emmie Silvey. The black woman in the desert sent me to help Soldier."
The man named Willie takes a deep breath and spits. "You ain't one of Ballou's bunch?"
"I've never heard of anyone named Ballou, except for that bear in The Jungle Book."
"Oh, that's fucking cute. That's just precious."
"Well, it's the truth."
"Are you a witch?"
"No, I'm Emmie Silvey. I live in Providence with my father, Deacon. He runs a used bookstore."
Odd Willie laughs again and cracks his knuckles. "Well, Emmie Silvey from Providence. I'm Odd Willie, also from fucking Providence. Now, just let me get my breath, and then we'll see if we can't figure out how to get Soldier out of here. How's that sound to you?"
"Okay," Emmie says, and the skinny man nods and shuts his eyes. She sits with Soldier while he rests, and tries not to think about the monsters or the lost snow globe, Pearl or the woman in the desert, tries not to wonder where she is or how hard it's going to be to get home again. There'll be time for that later. There are strange noises from the darkness beyond the stairs, sounds like animals or people or both, and she tries not to think about those, either.
Emmie opens her eyes, and at first she's unable to remember where she is, and then, remembering, panics that she must have dozed off, and she has no idea how long she might have been asleep. But then she sees that the skinny man is standing over Soldier, who's still unconscious. He's holding a torch, and the landing and archway are bathed in its warm yellow light.
"I fell asleep," she says.
"Yep, you sure as hell did."
Emmie rubs at her eyes, then asks him where he found the torch.
"There were a few pieces of furniture," and he nods towards the devastated chamber. The bonfire has almost burned itself out, and it's mostly dark in there now, only a single shaft of white sunlight getting in from somewhere far overhead. Emmie thinks it might be coming in through a sort of chimney.
"This used to be a table leg," Odd Willie continues. "Toss in a couple of rags and some candle wax, and presto-changeo, abraca-fucking-dabra, voilà, and what's the difference between me and a goddamned Boy Scout, I ask you?"
"Your goddamned dirty mouth," Emmie replies. "I don't think Boy Scouts curse. I think it's against the rules."
"Hey, fuck that," Odd Willie says. "I have it on good authority that Boy Scouts curse like drunken sailors. Hell, there's probably a goddamn merit badge for profanity."
"I'm hungry," she says, wishing she had one of the sandwiches or pears from Pearl's brown bag, and he shrugs.
"Sorry. I can't help you there. First things first. I want you to carry this," and he waves the torch at Emmie; the flames make a loud whooshing noise in the air. "I'll carry Soldier best I can. We need to get out of this place before one of those beaver-beater fucks stops licking its wounds and decides to come back looking for us."
Emmie stands, stretches, puts her coat on again, and then takes the torch from Willie. It's heavier than she expected.
"You gotta be careful with that thing. Hold it out away from you," he says, and she does.
"Can you carry her?" Emmie asks.
"I think so. I'm starting to get some of my strength back. Right now, I only feel like one truck ran over me. She's just real fucking lucky I didn't make her shoot me before she came up here."
"I don't even want to know what you're talking about," Emmie says and holds the torch up as high as she can, trying to get an idea how far down the stone stairs go. She can see that the steps are slick, and there are patches of moss or algae growing on the rock.
"Smart kid," Odd Willie says. "Ignorance is fucking bliss." And when Emmie gets tired of watching the stairs, staring at the murky place at the limit of the torchlight where the stairs blend imperceptibly into the darkness, she turns back to find that he's managed to lift Soldier off the ground and is holding her slung over his left shoulder. Her head's dangling towards the floor, her chin resting against his back and her mouth half-open, and he has both hands clasped together firmly beneath her butt.
"You go first," he says. "And keep that light on the stairs. I gotta be able to see where I'm putting my feet."
"Those steps are slippery," Emmie warns.
"No shit, the steps are slippery. You just make sure I can see where I'm going," and then he curses and shifts Soldier's weight so he has a better grip on her.
So Emmie starts down the stairs, forgetting her mittens and the glove she took off, and Odd Willie follows her. He slips only once, but he doesn't fall. It seems to Emmie that the steps go on almost forever, down and down, curving along the side of the rocky wall. When they finally reach the bottom, Emmie sits on the damp ground, and Willie leans against the side of the tunnel, his eyes shut, gulping air like a goldfish that's jumped out of its bowl.
"How much farther?" Emmie asks.
"How the hell should I know?" he wheezes.
"Left or right?" she asks.
"Do I look like fucking Ranger Rick? Your guess is as good as mine. Flip a coin."
"I don't have a coin," Emmie says and stares at her reflection in a puddle. Her face is streaked with mud and soot, and she's lost her toboggan cap somewhere. Maybe she left it back in the desert; her hair is tangled and sticking out in all directions.
"Well, we can't stop here," Odd Willie says and coughs. "We gotta keep moving, or I'm never going to make it. Left or right, kid. You pick."
Emmie picks left, for no particular reason, and leads Odd Willie Lothrop down the long tunnel, which seems to slope gradually upwards. The air is dank and smells like mud and mold, and there's a chemical odor that burns Emmie's nose and eyes, like the place beneath the sink where Deacon keeps the bottles of Clorox and detergent, Mr. Clean and Drano and Formula 409. When she asks Odd Willie about it, he says it's from the river, that the old textile mills along the Blackstone dump all their toxic sludge right into the river, and that's what she's smelling.
Once, they cross a narrow wooden bridge, pine boards gone slick and punky, and Emmie pauses to look over the edge. But there's only blackness down there, as far as she can see, and the rushing of flowing water far below.
"Shake a leg, kid. Ain't nothing down there you want to see," he says, then grunts as he shifts Soldier from one shoulder to the other. "And, more important, there's nothing down there you want to be seen by."
The tunnel rises, turns left, then right, then left again, and when they come to an archway leading to another chamber or a side tunnel, Willie tells her to keep going straight. There's a skull resting on a small ledge above the arch, and Emmie tells herself it's only the skull of a coyote or a big dog, but she knows better. The next time the tunnel turns right, they're greeted by a gust of cold, fresh air.
"This must be the way," she says. "I think I can even smell the outside. It smells like snow."
"What the hell does snow smell like?" Odd Willie asks, and stops again to get his breath.
"Clean," Emmie tells him. "Snow smells clean."
Odd Willie shuts his eyes and licks at his lips. "Christ, I wish I had a goddamn beer."
"I'd settle for a Cherry Coke," Emmie says, "or some birch beer."
"I thought we were fucking dead back there," and he tightens his hold on Soldier. "If you hadn't come along, you and that goddamn crystal ball of yours—"
"It wasn't mine," Emmie says, interrupting Odd Willie. "It was Pearl's, and it wasn't a crystal ball, either."
"You saved our sorry skins," he says and opens his eyes again. "We owe you one, Emmie Silvey."
"I just want to go home, that's all."
"Then that's where you're gonna go," Odd Willie tells her. "Just as soon as we can get out of this shithole, I'll drive you myself. Hell, I'll even get you that bottle of birch beer. You like Polar?"
"Yeah, Polar's good. Are you ready now?" she asks.
"As I'll ever be, which isn't saying very goddamn much." And she starts walking again, heading in the direction the clean, wintry air is coming from, and he follows.
Odd Willie left Soldier and Emmie in the woods not far from the north side of the cemetery. When she asked where he was going, he told her that he had to steal a car, unless she'd gotten it in her head to walk all the way back to Providence. The day was bright, but it was still snowing—a white sky and whiter ground—and Emmie wondered if it was only the next day, only Monday, or if maybe she'd been gone a long time and it'd been a whole week or more since she left Pearl in the railroad tunnel. Maybe it had been snowing for days. She sat with Soldier, the two of them beneath an old poplar tree that Odd Willie had picked because he said ghouls hated poplar trees. When she asked why, he wouldn't tell her, so she had a feeling he was lying and had only said that so she wouldn't be afraid. Soldier was still unconscious, and her fever was worse. Sweat trickled across her face, and she shivered so hard that her teeth chattered. Emmie wrapped her tighter in the robes that Odd Willie had taken off the ghoul he'd killed, and told her about Aslan defeating the armies of the White Witch. She felt silly doing it, but she was scared, and it was all she could think of to talk about, and it made the time go faster.
And then Willie was back much sooner than she'd expected, back in the cemetery and blowing the horn so she'd know it was only him coming—two short honks, one long. She followed him through the snow as he carried Soldier up the hill again. He laid her in the backseat of the purple Chevrolet Malibu, spread a blanket over her (Emmie didn't ask where the blanket or the car had come from), and then they drove away from Oak Hill and Woonsocket, and no one and nothing tried to stop them.
It's Wednesday, and they've been at the seedy little motel just outside Uxbridge, Massachusetts, for almost two days now. Willie drove north on icy back roads until he found the place, and he told Emmie they'd probably be okay there until Soldier felt better and he could figure out what to do next. The walls are the same shade of green as lime sherbet, and the carpet is orange. Emmie's sitting on one of the queen-sized beds with Soldier, swabbing her bruised and blistered face with a damp washcloth, and Odd Willie's on the other bed, talking on his cell phone, having another argument with the man he calls the Bailiff. The television is on a channel that's been showing nothing but Tom and Jerry cartoons for two hours, but the volume's turned all the way down.
"What I want is some sort of fucking security," he says again. "I've told you that, what, like a hundred goddamn times already? I keep fucking telling you that. I want a fucking guarantee that it's safe to come in. That's what I want."
There's a long pause while Willie chews at a thumbnail and stares at the TV screen. Emmie lays the washcloth back across Soldier's forehead; she's sick of Tom and Jerry and wishes he would change the channel.
"Damn straight, I'm paranoid. Right now, I'm the most paranoid motherfucker in New England. You weren't there, and you have no goddamn idea what we walked into."
Another pause, shorter than the last, and then Odd Willie says he'll call back later and hangs up. "Fucking mouse," he says and points at the television with the antenna of his cell phone. "It's just a game to him."
"He doesn't want to be eaten," Emmie says. "That's all, he just doesn't want to be eaten."
"Then maybe he ought to live somewhere there isn't a damned cat. Lots of people don't have cats. Christ, man, I hate that little brown fucker. He's fucking sadistic; you know that, right? You know what 'sadistic' means? That is a sadistic fucking mouse."
"It's almost suppertime," Emmie says. "You should let me call my father now."
Odd Willie frowns and sits up, swinging his long legs over the edge of the bed. "I told you I can't let you do that. We'll get you home, I swear. There's just some shit has to be cleared up first, so no one else gets hurt."
"Deacon's gonna think I'm dead. He's gonna think someone kidnapped me."
Odd Willie looks at the clock radio on the table between the beds. It's almost seven o'clock. "Well, then, just think how much happier he'll be when you show up safe and sound. Want a hamburger?"
"No. I'm tired of hamburgers," Emmie says and watches Soldier's eyelids flutter. They've been fluttering all day, but she still hasn't opened them.
"Then how about some fucking McNuggets? Those are pretty good."
"They taste like shit," Emmie tells him.
"There's some sort of pizza joint—"
"I don't want pizza. I want to call Deacon. I want you to take me home."
"Kid, you're just gonna have to be cool, all right? When I get this crap straightened out—"
"If you don't get Soldier a doctor soon, she's gonna die," Emmie says. "She's in a coma. Her fever's up to a hundred and two again. The aspirin aren't working."
"I need a fucking cigarette," Willie says and reaches for his shoes, because Emmie won't let him smoke in the room. He got some clothes and shoes that fit from a Salvation Army in Uxbridge; there's also a pair of Levi's, a black sweater, and some scuffed-up old cowboy boots for Soldier, if she ever wakes up. He tosses Emmie the remote and tells her to find something else to watch.
"Hey, you're the one who wouldn't change the channel," she tells him. "I hate Tom and Jerry. I told you I hate Tom and Jerry."
"Well, I hate them worse than you do. You're a kid. I thought kids liked fucking cartoons and shit. Just change the channel. I'll be back in five."
"You know it's almost suppertime," she says again, and Odd Willie rolls his eyes.
"Decide what you want, and I'll go get it. I'll be back in five minutes." And he stands up, puts on the Red Sox jacket that he also found at the Salvation Army store, and leaves Emmie alone in the room with Soldier. Before he pulls the door shut, she sees that it's snowing again. She turns the sound on the television up loud and flips channels until she finds a documentary about the planet Mars, then lies down beside Soldier and stares at the television a few minutes before glancing at the phone sitting useless on the bedside table. Odd Willie disconnected it and hid the cord somewhere. She's tried to find it, whenever he leaves her alone, but she's starting to think he must have hidden it out in the car.
"What's going to happen to me?" she asks Soldier, pretending that Soldier can answer her, pretending that Soldier might know the answer. Soldier's eyelids flutter, her lower lip twitches, and then her face is still again.
"You know I'd help you, if I knew how. Right now, there are lots of things I'd do, if I knew how." Then Emmie stares at her a moment, her face like someone who's only asleep and having a very bad dream.
"You look like my mother," Emmie says. "You look like her a lot." She's been thinking it since she first saw Soldier beneath Woonsocket, but she's been too afraid to actually say it out loud, too afraid of what it may mean to allow herself to even think about it for more than a few seconds at a time. But it's true. Soldier's face isn't exactly her mother's face—the face of the woman in the old photographs that Deacon's shown her and the ones that she found in the box under his bed—but it's close enough. Too close. And Soldier doesn't have yellow eyes. She has green eyes like Deacon. Green eyes like Chance Silvey.
Emmie remembers the gold wedding ring she took from the box, and reaches into her jeans pocket, afraid suddenly that she might have lost it somewhere, in the black woman's desert or one of the tunnels. But the ring's still there, and she takes it out and stares at it for a moment.
On the television, the narrator is talking about a time, billions of years ago, when Mars might have had an ocean.
"Willie's not so bad," she says to Soldier. "He's not nearly as big a creep as he wants everyone to think he is. If he was, he'd have left me down there to die, right? He wouldn't have saved me. He's mostly just scared all the time, and that's why he acts the way he does."
Emmie doesn't have to touch Soldier's skin to feel the heat coming off her; it's almost like lying next to an oven. Emmie thinks about taking her temperature again, slipping the digital thermometer beneath her tongue and watching the numbers go up, waiting to see just how bad it is. Willie bought the thermometer at a CVS the day before, when he bought aspirin and Tylenol, Neosporin and gauze bandages and a tube of something smelly he keeps combing through his hair. She hasn't asked him where he's getting the money. She hasn't really cared.
"You can't die," Emmie whispers. "You can't. If you die—" and then she makes herself stop, because Sadie has told her that words are magick and she should be careful how she uses them. She slips the ring into her pocket again and goes back to watching television. And she holds Soldier's hand, no matter how hot it is, and waits for Willie to come back.
Dreaming, dreaming so long now that Soldier has forgotten that there was ever anything before the dream. There was fire somewhere back towards the beginning, an unimaginable, devouring fire that would have burned away the universe and still been hungry, and now some fraction of the fire is trapped inside her. Perhaps no single fire—Quaker Jameson's roadhouse in flames, the fire Odd Willie set at Rocky Point and then the sizzling being he summoned, George Ballou's bonfire and the beast rising from it, and, finally, a star inside a crystal ball. Fire to destroy and purify, fire to deliver and condemn, and she breathed it all in, and it might never find a way out again.
Dreaming, she has drifted through years and months, days and hours, and sometimes it seems that she only drifts and there's no time at all. She's never entirely alone, because she has the fire that she swallowed for company, and she has the Daughter of the Four of Pentacles, too, who comes and goes, rattling on about things that Soldier only occasionally understands. She has memories and things that can't be memories because they haven't happened yet or will never happen.
"I sent her to you," the alchemist's daughter says, but Soldier isn't at all sure she believes her. "The poor child," the girl continues, "she was lost and alone on the sea in a tiny yellow boat named the Fly-Away Horse, bobbing and lost in a hurricane tempest upon the wild, wild sea. There was a whale that wanted to swallow her alive, and there were monsters, and a phantom mariner who catches mermaids on fishhooks and cooks them in his skillet."
Soldier watches herself sitting on the floor in the Bailiff's study, a child in a blue calico dress eating sugar cookies and drinking grape soda while an old man talks. But here he has silver eyes, like the vampires do, and his words are black and living, and there's a nasty plopping sound as they drip from his lips to lie in a squirming heap on his desk.
She walks the long, narrow path down to the dragon, old Root-nibbler waiting for her on the night of the Full Hunger Moon, the last confirmation before she passes from childhood into the service of the ghouls. At the bottom of the pit, she kneels and thrusts her arms into the twin holes in the earth, one ringed in gold, the other in platinum, and holds her breath, expecting the dragon to take her hands. Expecting to die there with all the ghouls and the other changelings looking on.
"Strictly speaking, this has never happened either," the Daughter of the Four of Pentacles says and shakes her head. "You're the only adult changeling alive who hasn't had to make the passage through the three trials, fire and blades and teeth. The ghul imagine you're much too precious to risk on such dangerous formalities."
"They sent me to Woonsocket to fucking die," she replies, and "No," the girl says. "The Bailiff sent you to Woonsocket. Not the ghouls."
"The Bailiff serves the ghouls. Whatever he does, it's nothing that isn't their will."
"Oh, don't be such a silly sap," the girl says, and then she wanders off again, leaving Soldier alone on the Fly-Away Horse, and the fisherman who only catches mermaids scowls at her and goes back to wrestling with the tiller. The rain's falling so hard now that she can barely see him, even though he's only a few feet away from her, squatting in the stern of the dory. The boat rises and falls on waves so high that their white crests scrape at the low clouds; then it races back down the steep sides of watery canyons, plunging into troughs that carry them almost all the way to the seafloor. The tattered sail, shredded by the wind, flutters uselessly in the gale. She doesn't know the fisherman's name; she isn't even sure that he has one. She paid him a dollar and a handful of dead spiders to get her from one side of Block Island Sound to the other, but now she thinks that they'll both drown before they ever see land again.
"I ought to have known better," the fisherman growls and chews the stem of his soggy corncob pipe. "I ought to have my sorry skull stove in, takin' on the likes of you."
"I was only looking for the way across," she says, surprised that she doesn't have to raise her voice to be heard above the storm.
"Well, I hope you're pleased with yourself," he grunts, but Soldier doesn't know what he means. The voice of the storm is the same as the voice of the fire, the fire from the beginning, and soon it will fall on them like a hammer. The water seethes with the restless coils of great eels and serpents and the eternally searching arms of Mother Hydra and Father Kraken.
"My father caught this boat in 1922," the girl from the attic says, and smiles. "It was one of his earliest experiments with translocation. He was only trying to catch a bit of the sea. He said the boat was pure luck."
Sand blows down from the dunes, which have never been waves, no matter how much they might emulate or envy them. The Fly-Away Horse lies on its starboard side, half-buried in the beach, its mast snapped in two and its rigging and canvas scattered all about, and Soldier is sitting cross-legged in the shade of the boat, hiding from the desert sun. There are dead mackerel and jellyfish and trilobites baking in the noon heat, stranded by the tide. The Daughter of the Four of Pentacles told her to wait here if she wanted to speak with the black woman, the woman who is ancient and unaging and who wanders the desert looking for lost dreamers.
"I never said I was lost," she tells the girl.
"You certainly don't look particularly found," the girl replies and kneels in the sand in front of Soldier. The noonday sun gathers in her hair like honey and washes gently across her almond skin. She doesn't make eye contact with Soldier, pokes at the corpse of a blue-gray starfish instead, flipping it over to expose all the hundreds and hundreds of wriggling tube feet to the sun.
"I don't need your help," Soldier says.
"What I did to you—"
"Is ancient fucking history. Go away. Leave me alone."
"They said they would kill my father if I didn't help them, if I didn't work the contraption. No one else knew how. They said they would send assassins all the way to Weir to find him. They said—"
"Leave me alone," Soldier tells her, and the black-skinned woman kneeling in the sand picks up the dying starfish and sighs.
"It used to make me very sad, finding all the helpless little things that the sea spits out, all the things cast up to die. I used to think, perhaps I could save them all. I'd walk this beach for days and days," and she motions at the shoreline stretching away on either side. "I would find them and carry them back down to the water where they belonged. I would try to give them back to the sea."
"Do you have a point, old woman?" Soldier asks her, and the woman from the desert smiles and shrugs her shoulders. She sets the starfish back down on the sand, then wipes her hand on the hem of her thobe.
"A lot of them died anyway," she says. "That's all I wanted you to know. The world makes orphans of us all, sooner or later. It puts us where and when we don't belong, and even if we manage to find our way home again, we might discover home doesn't want us anymore. That's the truth, and there's not much we can do about it."
Low waves break against the edge of the desert, speaking in a secret language Soldier thinks she might have understood once, a long time ago, before the Bailiff took her up to the attic. A warm wind whispers through the dunes at her back and tugs at the black-skinned woman's pale dreadlocks.
"She was a fine little boat," the woman says and pats the hull of the capsized dory. "A fine little boat, was the Fly-Away Horse. She sailed all the seven seas. She saw typhoons and maelstroms. And she even got you this far."
"And just how the hell far is that?" Soldier asks, and she starts to reach for the starfish, but the sun is much hotter than she expected, and Soldier pulls her hand back into the shade of the wrecked dory.
"Far enough and then some," the woman says, standing up. She shades her amber eyes with her left hand and gazes up at the sky. "Far enough you have to make a choice how this thing's gonna end. That child's mother's coming for her, Soldier, and she means to have the girl."
"What's that to me?"
"Well, now that would seem to be the most important question," the black-skinned woman replies, still watching the wide and simmering sky. "But don't you sit here too long thinking it all over. She's a natural-born sorceress, that one, and she knows well enough how to ride the coattails of a snowstorm. The child is hers, and she believes that you are the last and only thing standing between them. She has a fearful hatred for you."
And then the woman's gone, as are the Fly-Away Horse and the sand and the sea and the blue starfish. The falling snow confuses Soldier for only a moment—a passing dislocation, the half blink of a sleeping eye, a breath—and she's glad for the cold and the winter night after so much sun, lost in fever and grateful for cool air, and the weight of the Winchester shotgun feels good in her hands. She's standing in a parking lot, cars and trucks half-buried in snow, everything veiled in white and countless shades of blue and gray trailing off to black, everything except the soft orange pools from streetlights and the flashing red and green neon of the tall motel sign at the edge of the road. The Daughter of the Four of Pentacles is there, too, walking in circles, catching snowflakes on her tongue. She stops and looks at Soldier.
"We make a fine pair, don't you think?" the girl asks. "Me with too much childhood and you with hardly any at all. That almost makes us sisters, of a sort."
"I'm still dreaming," Soldier says.
"Of course. Otherwise, we wouldn't be having this conversation. They apprehended me in the old railway tunnel. Miss Emma Jean Silvey slipped away, but they caught me—because Barnaby's a coward and a louse—and then they shut me away again. But all that time was wonderful. I aged almost an entire day and night."
Soldier pumps the shotgun, chambering a round, and stares into the swirling, shifting gloom. The storm is filled with shadows and the less distinct shadows of shadows, with almost endless possibility and potential. It is a crucible, like all storms, dreaming or awake.
The Daughter of the Four of Pentacles stops and stares down at the pattern her boots have pressed into the snow and slush—two intersecting triangles to form a six-pointed star. And Soldier realizes that she hasn't been walking in circles at all.
"Saben," she says, and the alchemist's daughter holds one finger up to her lips and frowns.
"Not so loud. She's coming. She's probably already here somewhere," and the girl glances nervously over her shoulder. "There's no need to call her."
"Ballou's dead. Why didn't she just run?"
"She wants her daughter back, and besides, she knows you'd come after her. She knows that if she runs now, she'll never be able to stop running."
At least she got that part right, Soldier thinks and shuts her eyes, or she dreams she shuts her eyes, and she's back in the yellow house on Benefit Street, sitting across the mahogany dining table from the ghost of Sheldon Vale. He lays a tarot card on the table—the Tower—and tells her what it means.
"Saben chose to face me in a dream, didn't she?" Soldier asks him.
"Yeah," he says and taps the card once, twice, three times. "It was her idea, if everything went to shit and she wound up with you on her ass. She understands that you're much weaker here, sleeping. You have to be awake to do that...that thing you do." And he makes a staccato tick-tock-tick-tock noise with his tongue, his index finger up to mimic a clock's second hand moving steadily backwards, and then he winks at her. "No hard feelings, though. We made a fine pair while it lasted, don't you think?"
"Come back," the Daughter of the Four of Pentacles says and shakes Soldier so hard she almost drops the shotgun, almost squeezes the trigger. "You can't keep wandering off like that. She'll kill you if you do. She began the dream, so the labyrinth always works to her advantage."
Two triangles to make a star, you see? Two intersecting dreams, and one angle remains always invisible—the overlap—a shared point in space and time and sentience, the Bailiff said and pulled at his beard.
"You should go now," Soldier tells the girl.
"Are you sure? You may yet have need of me."
"Then I'll find you when I do," Soldier says, and the girl comes apart in a sudden gust of icy wind, becoming briefly something bright and sparkling and even less substantial than the snow.
And Saben White is standing ten or fifteen feet away, on the other side of the star traced in the snow. Her clothes are torn and dirty, and there's dried blood on her face and hands. She isn't alone. One of the Woonsocket mongrels is crouched on either side of her.
"You should've run," Soldier says, and the half-breeds bristle and bare their teeth.
"Haven't you ever loved anything?" Saben asks bitterly. "Haven't you ever once loved something so much that you'd die for it?"
"You can't have the girl, Saben. She belongs to the Cuckoo, just like you do. Just like me. And even if she didn't, after all this bullshit, I still wouldn't let you have her."
"She's my fucking daughter," Saben says, and the mongrel on her left cries out, an ugly, feral shriek that Soldier knows is the nightmare of Saben's loss and denial, and it lunges, bounding through the snow on all fours. Soldier pulls the trigger, and for a heartbeat the roar of the shotgun drowns out every other sound. The half-breed thing goes down in a spray of blood and shredded flesh, becoming only a faceless, broken heap in the snow.
"See what you made me do?" Soldier says, the echo of the Winchester fading away across the shell-shocked winter night.
"After everything they've done to you," Saben says, "you're still willing to fight for them?"
"This isn't about them. You tried to kill me, Saben. Twice now you've tried to kill me. And, unless I'm mistaken, you came here tonight to kill me. That makes three times," and Soldier holds up three fingers.
"I came here for my child."
"Like I said, she's not your child. She belongs to the Cuckoo."
"Soldier, how can you stand there and say that? They've stolen so much from you. You don't even fucking know the things they've taken away from you. Your whole life is lived in a fog they've spun to keep you ignorant. And now the Bailiff—"
"Am I going to have to shoot that one, too?" Soldier asks and points the barrel of the gun at the second Woonsocket mongrel. It snarls and retreats, cowering behind Saben.
"Listen to me, Soldier. You think he didn't know exactly what Ballou was doing? You think he believed you'd ever be coming back to Providence alive?"
"Right now, I don't exactly know. But I figure all that shit's between me and the Bailiff."
"You have to understand," Saben says, "they want to leave the world," and she draws a circle in the air with her left hand as she speaks, a ring of silver fire that hangs suspended above the snow. Soldier takes a step back and pumps the shotgun again.
If you fully comprehend the sum of these angles, and if you can see all the points of convergence simultaneously, then the game may always be turned to your advantage, the Bailiff said, and Soldier stopped nibbling at the cookie and watched what he was doing with his hands. It's a bit like origami, only without the paper. See? A fold here, a bend there—
"Don't make me kill you," Saben White says, and her ring of silver flame grows brighter. She draws another with her right hand.
"Jesus, Saben. You've tried to kill me three goddamn times now, and you want me to think you're feeling merciful?"
—it's simple. Valley fold the left side of the first triangle so the edge falls on the closest crease. Now, Soldier, simply mountain fold the right side of the second intersecting triangle without severing—
"You were in my way before. All you have to do is let me have her. She's my daughter."
There's a tug at the shotgun, the tidal drag of Saben's spell, and Soldier doesn't resist it. A moment later the Winchester slips from her hands and falls to the ground.
—to make a new crease and complete that side of the star. It's up to you where you want the star point to be realized, where you wish it to appear.
"Let me have her, and you won't be in my way anymore."
"Suck my dick," Soldier says and folds the star that the Daughter of the Four of Pentacles trampled into the snow. It's much easier than she remembers—valley fold, mountain fold—and she shuts her eyes again as the geometry of the dream begins to ravel and fray. She feels the sudden eddy when Saben's wheels of silver fire shift into another range of the spectrum. And she hears Saben scream when the fire folds back upon her.
Soldier stands between the desert and the sea, and the black-skinned woman smiles and kisses her softly on the lips. "You see, that wasn't so difficult. That wasn't hard at all."
"But it's not over," Soldier tells her.
"Child, it's never over. You'd better get used to that."
And the Fly-Away Horse moves across a calm green sea, a school of dolphins racing one another at its prow. The old man at the tiller puffs his pipe and tells her about the time he sailed all the way to the Pillars of Hercules.
Deep beneath Providence, on the night of the Full Hunger Moon, Soldier begins the long walk down to the dragon, the walk she never had to make.
These and a thousand other cusps spaced out along a tissue-paper star with seven points, and in the end, she finds her way back past George Ballou's fire in the cavern and the greater fire trapped inside a cracked glass sphere, the twin fires burning at the beginning of the dream, which she sees now are all fires and all dreams. She breathes out an inferno, and in the great emptiness beyond the heat, she can rest and heal and remember herself. And she can hear the girl's voice calling her, the voice of the child who saved her, Saben White's daughter, and she wakes up.
## NINE
## The Bailiff
The clouds have gone, and the sky above the highway is a bright shade of blue, a cold and perfect cloudless blue spread out above the sagging power lines and bare tree branches glistening with ice. The snow is piled high along the sides of the road, and Emmie thinks it'll start melting soon, if it hasn't started already. Odd Willie's driving, and Soldier's riding up front with him, so Emmie has the backseat all to herself. The stolen Chevy sedan glides over brown slush—ice and salt and sand—and the black streaks of asphalt showing through. There's a Beatles song playing on the radio, "Hey Jude," and Odd Willie is humming along to it. Every now and then he smiles at Emmie from the rearview mirror.
"I'm sorry," Soldier says again.
"She was my mother," Emmie replies, but no matter how many times she says the words, it doesn't feel any less unreal. "She was my mother, and she'd come to find me, and you killed her."
Soldier lights another cigarette and rolls her window down an inch or so. She doesn't look much better than she did the night before. Her face seems somehow pale, despite the sunburn that isn't a sunburn, and there are too many bruises and scrapes to even bother counting. The edges of the long gash Ballou made in her left cheek are swollen and scabby. Odd Willie stitched the wound closed with dental floss, but it looks fevery and infected.
"She didn't leave me any fucking choice," Soldier says again, but Emmie's not sure she believes that. Sadie's told her that people always have choices, even when they believe that they don't, that sometimes they just say they don't because it helps them feel better.
"You could have let her come to me like she wanted; you could have let me..." But then Emmie trails off, all these things already said once or twice or three times since they left the motel in Uxbridge, and she knows that repeating them over and over isn't going to do anything to chase away the empty, confused feeling. It isn't going to change what Soldier's done. It isn't going to bring Saben White back from the dead.
"I did what I had to do," Soldier says firmly and exhales. Most of the smoke is sucked out through the open window, but some hangs about her head like a veil.
Odd Willie stops humming and glances up at Emmie again. "You better listen to her, kid," he says. "She knows what she's talking about. Believe me, you're way the hell better off without that bitch."
"She was my mother," Emmie says quietly and shuts her eyes. The bright day is swallowed in darkness, and there are only sounds—the tires against the frozen road, the spray of sand and salt pinging against the wheel wells and the undercarriage of the car, the music from the radio. "Willie, please, just take me home now."
"He can't do that yet," Soldier replies. "We'll do it when we can, but not yet. Maybe after we see the Bailiff, maybe then."
"But none of this has anything to do with me," Emmie tells her, even though she knows that it does, that maybe, somehow, it has an awful lot to do with her. She opens her eyes, and Soldier's watching her across the seat, watching her with Chance Silvey's green eyes.
"Emmie, you have to be patient," Soldier says. "There's too much at stake here. We have to try to do this the right way. I still don't know exactly how you fit into this mess, or where you got hold of that sphere, and I have to find out before you can go home."
"I didn't want to run away," Emmie says and opens her eyes. The day seems even brighter than before. "I didn't want to go into that tunnel. I only did it because Pearl said they'd hurt Deacon if I didn't."
Soldier nods her head. "She was right about that. The way things stand, you're a loose end, and the ghouls don't like loose ends. They'd have come for you and killed anyone else they found with you, because, like I said, they don't like loose ends."
"But now you're taking me to them?"
"I'm gonna find out what the hell's going on, that's all. I'm not going to let anyone hurt you."
And Odd Willie glances at Emmie again from the rearview mirror, a flash of something guarded and uncertain on his face, something she catches despite the sunglasses hiding his eyes. He doesn't believe her, Emmie thinks. He doesn't believe a word she's saying.
"Like I told you," Soldier says, "I think someone tried to kill me and Willie, and until I find out precisely what—"
"I'm not deaf," Emmie snaps at her. "You don't have to keep telling me the same damn lie over and over and over like I'm a retard. I heard you the first time."
"Fine," Soldier says and turns away. She takes another drag off her cigarette and fiddles with the volume knob on the radio. "Hey Jude" ends, and now the dj's talking about a wreck and a traffic jam at the Thurbers Avenue curve.
Last night, Emmie thinks, last night my mother died. Last night Deacon's real daughter killed my real mother.
In the motel room, she waited almost a whole half hour for Willie to come back from having his smoke, and then Soldier started talking, still unconscious but talking in her sleep, muttering about starfish and storms and drowning while her eyelids fluttered and her hands trembled. Emmie thought that maybe she was having a seizure or a stroke or something, and went outside to find Odd Willie. He wasn't standing by the purple Malibu, so she walked from one side of the motel parking lot to the other and back again, calling his name, but he didn't answer, and she couldn't find him anywhere. She gave up and started to go back in the room to see if Soldier was still alive when there was a scream, a woman's scream, and a brilliant flash of blue-white light from the woods directly behind the motel.
Just go back inside, she told herself. Go back inside the room and watch TV and wait for Willie to come back and tell me not to worry. Do that, and everything will be okay.
Instead, she went around to the rear of the motel, pushing her way through the snow and a dense tangle of wild grape-and greenbrier vines, past the sleeping trees. Emmie found Odd Willie sitting alone in a little clearing smoking a cigarette. The night was dark, but the dark has never kept her from seeing, not Emmie Silvey, the girl with yellow eyes, and she clearly saw him sitting there on a rock, and she also saw the thing scattered across the snow, the thing that had once been Saben White. There were tracks everywhere, footprints that Emmie had learned enough to know only looked like the tracks of dogs. Odd Willie sighed and tossed the butt of his cigarette into the woods, then turned to look at her. But she didn't take her eyes off the broken, bloody thing on the snow. It had been folded somehow, white bone and red flesh folded into something like a six-pointed star.
"What...what happened?" Emmie asked him, trying to keep her teeth from chattering.
"Soldier did what Soldier does," Willie replied. "Don't ask me how. I don't know, and I don't want to know how."
Emmie continued to stare at the mess in the snow, and the wind through the trees made her think of whatever it was she'd seen back on Waterman Street. "But Soldier's still in bed," Emmie whispered. "She hasn't left the room."
"You really shouldn't be out here," he said. "You're gonna get sick. You're gonna catch your death of cold. Haven't you ever heard of pneumonia?"
"But she never left the room."
"Yeah, well, she's a talented lady," Odd Willie nodded and stood up, dusting snow off the seat of his pants. "Stop looking at it. That ain't nothing a kid like you ought to see," and he took her hand and led her back towards the motel. When they got to the room, they found Soldier in the bathroom, awake and vomiting into the toilet bowl.
Stop thinking about it, Emmie tells herself. Forget you ever saw it. Forget you ever saw anything at all.
"He's still not fucking answering," Soldier says and throws her cell phone at the dashboard. It beeps loudly, bounces off the vinyl, and lands on the floorboard at her feet.
"It's your decision," Odd Willie tells her.
"Hey, if it were my motherfucking decision, asshole, we'd be all the way in goddamn Mexico by now."
"Then we're going in?"
"What the fuck do you think?"
It's a dream, Emmie tells herself. The worst dream anyone's ever had. And she watches the bright February day, the clean morning sun sparkling off so much ice and snow, the wide blue sky above the rooftops of Providence finally coming into view. She thinks about the black-skinned woman in the desert, and wishes that she knew the way back to her.
They stopped at the Cumberland Farms on Reservoir Avenue because Emmie said that she had to pee, and, she said, if they made her wait any longer, she'd end up wetting both herself and the car seat. "I have a weak bladder," she lied, so Odd Willie cursed and pulled into the parking lot.
"If they've got it, grab me pack of Black Jack gum," Odd Willie says, and Soldier tells him if he wants fucking chewing gum he can damn well get it himself. Then she leads Emmie inside, out of the cold and into the stuffy, crowded warmth of the convenience store. The air stinks of disinfectant and bad coffee, and something about the shadowless white wash of the fluorescent lights hurts Soldier's eyes even more than the sun, even though she's wearing a cheap pair of sunglasses Odd Willie picked up for her in Uxbridge. Her confrontation with Ballou and the fire thing beneath Woonsocket has left her half blind and headachy, and she squints behind the black plastic lenses, squinting through stinging, watery eyes. People at the counter turn to stare—curious, prying, unwelcome eyes for the battered, sunburned woman and the disheveled eight-year-old.
"I'm hungry," Emmie says.
"Yeah, well, you can eat later. We didn't stop so you could get something to eat."
"But as long as we're here—"
Soldier gives her a little push, and the woman at the register frowns and shakes her head.
"In and out," Soldier says, hurrying Emmie down one of the narrow aisles towards the restrooms all the way at the back of the store. "You said you had to piss. Do it, and let's get the hell out of here."
Emmie snatches a bag of ranch-flavored corn chips off one of the racks.
"Put that right back where you got it," Soldier tells her, but Emmie's already opened the bag.
"Is that your plan?" Emmie asks and pops a triangular chip into her mouth. "To starve me to death so you don't have to take me home?"
Soldier looks over her shoulder, and the staring people up front are still watching them. So is a man one aisle over. When she turns back to Emmie, she's gone. The restroom door clicks shut, and Soldier sighs and decides to give her three minutes alone. She turns to face the plate-glass windows and there's Odd Willie, waiting outside in the car, singing along to whatever's playing on the radio and slicking back his hair with his pink comb; he probably couldn't be any less inconspicuous if he set his head on fire.
"What now, Soldier girl?" the Bailiff asks, and Soldier spins around so fast that she almost trips over her own feet. The Bailiff is standing only a few feet away, standing there between her and all the people at the register. He's wearing a gray corduroy suit with a silk vest the color of ripe raspberries, bright yellow galoshes and a wool hunting cap with the left earflap pulled down. He grins and picks at his teeth with a pinkie nail.
"You look ridiculous, old man," Soldier says and takes a step back. "You look like a goddamn circus clown."
"Have you perhaps been anywhere near a mirror lately? From the looks of it, Old Ballou must have put up quite a struggle."
"Yeah, well, he's dead," Soldier says, and the Bailiff flickers and almost fades away entirely. Then, in an instant, he's back again. He stops picking at his teeth and nods his head once or twice.
"And Saben? How's she doing on this fine winter's morn?"
"Why didn't you tell me what we were walking into?" Soldier asks him, and the Bailiff smiles again and shrugs his broad corduroy shoulders.
"Soldier, dear. You know the refrain. Ours is not to question why—"
"Bullshit," Soldier hisses, and the Bailiff holds an index finger up to his lips.
"They can't see me, you know," he says. "Perhaps you should keep your voice down."
Soldier glances around the convenience store, staring back at the half dozen or so people who apparently have nothing better to do than stand around in Cumberland Farms gawking at strangers. The woman behind the register points straight at Soldier, then leans forward and whispers something to the moonfaced man who's just set a box of Slim Jims and a six-pack of bottled springwater on the counter, and he laughs out loud.
"Since when did minding your own damned business cease to be an option?" Soldier shouts at them, and the cashier immediately stops whispering and goes back to pushing keys on the register. The moonfaced man snickers, covers his mouth, and stares down at his shoes.
"You're still a paragon of subtlety," the Bailiff snorts and pulls a white handkerchief from his breast pocket. The edges are embroidered with tiny bluebirds, and he blows his nose loudly.
"Saben's dead. I killed her, too," Soldier tells him. "We've been trying to call you all morning."
"I've been occupied," the Bailiff replies, then stuffs the soiled handkerchief back into his pocket. "As you may know, there's been some trouble with the alchemist's daughter. Do you have Saben's little brat?"
"She's taking a piss."
"How precious. When she's done, bring her to me. You and Master Lothrop, please bring her to me at once."
"You still haven't answered my question."
"No," the Bailiff says. "I suppose I haven't, have I? And perhaps I never shall. We'll just have to see which way the wind blows, as they say."
And then the air before her shimmers and the apparition of the Bailiff dissolves, leaving behind a smell like cinnamon and castor oil.
"Fucking bastard," Soldier whispers, suddenly dizzy and nauseous, a tinfoil aftertaste in the back of her mouth, and she figures that's probably the Bailiff's doing as well. She leans against a sturdy cardboard ziggurat built of red-white-and-blue cartons of Pepsi-Cola, and the things that Saben said to her the night before come rushing back....
They've stolen so much from you. You don't even fucking know the things they've taken away from you. Your whole life is lived in a fog they've spun to keep you ignorant.
Soldier takes a deep breath, swallows, exhales, and then looks at the restroom door. It seems farther away than it did before, and one of the fluorescent bulbs overhead has begun to buzz and flicker.
Listen to me, Soldier. You think he didn't know exactly what Ballou was doing? You think he believed you'd ever be coming back to Providence alive?
"Time's up," Soldier says, and she quickly traces a protective symbol in the air with her left hand before she goes to retrieve Saben White's mongrel daughter.
Emmie's sitting on the closed toilet lid in the Cumberland Farms' women's restroom. She chews the last of the salty corn chips, wishes she had something to drink, and crumples the empty Mylar bag. The woman named Soldier has started knocking at the door, trying to get her to unlock it, trying to draw her out. Emmie drops the crumpled bag to the gray tile floor and then kicks it hard with the toe of her boot; it bounces off the restroom door and rolls under the sink.
"Open the door," Soldier whispers. "We don't have time for this."
Emmie shakes her head no, even though Soldier can't see that she's doing it. She went into the restroom planning to escape through a narrow window, just like she's seen people do in movies. But there aren't any windows in this restroom, narrow or otherwise. There's only the toilet and sink, the mirror above the sink and the gray tile floor. She flushes the toilet again so she can't hear Soldier, and for a few moments there's only the gurgling sound of water racing itself around and around the porcelain bowl.
Six days ago, she sat in Kingston Station with Deacon, waiting for the train that would take her to see Sadie, waiting for New York City and a week without homework, a week of museums and galleries and food that Deacon won't eat and all the other sights and sounds and tastes and smells of the city. Only six short winter days ago, but it seems like it must have been weeks and weeks. And the things that have happened since she boarded the train, since she noticed she was being watched by the woman with the Seal of Solomon tattooed on her hand, have changed everything forever, and nothing is what it was. She's beginning to understand that this is permanent, this shift, that there's no going back, no matter how hard she pretends it's only a dream. Saben White, the girl from the attic, the black-skinned woman in the desert, Soldier and Odd Willie—all of them have torn her loose somehow from everything that was real and true before, and now this is what she has instead. All the hours since Saturday morning and Kingston Station have made her someone and something else, and she's not even sure she knows what she means when she says that she wants to go home.
"Emmie, open this door," Soldier says, louder than before.
"No," Emmie replies flatly. "Go away. Leave me alone."
"You know I can't do that. Now open the damn door."
Or I'll huff and I'll puff and I'll...
"I can find my way home from here," Emmie says. "I can walk or call Deacon to come get me."
"I'm not going to keep asking you...."
Emmie flushes the toilet again, drowning out Soldier's voice. She stares at the silver knob and the restroom door that's the same shade of gray as the floor and tries hard not to think about the bleeding, folded thing lying in the snow behind the motel. Odd Willie took it deeper into the woods before they left and burned it.
"I don't want to be in this story anymore," she says, but then the door opens easy as you please, just like she hadn't bothered to lock it. Soldier steps into the restroom and shuts the door behind her.
"How did you do that?" Emmie asks her, much more angry than surprised. "I locked it. I made sure that I locked it."
"Listen, you smart-mouthed little runt," Soldier says, and, behind her, the door locks itself again. "I don't have time for this crap. I don't know what you know and what you don't know and what you think you know, but, right now, it doesn't fucking matter one way or the other."
"Stop yelling at me," Emmie says very softly, and then she looks down at the gray tiles and pointy toes of Soldier's thrift-store cowboy boots. "You don't have any right to yell at me. I did exactly what she told me to do. I brought you the snow globe thing, because she said you'd die if I didn't, and now I just want to go home."
"Emmie, I can't expect you to comprehend what's happening. Hell, I don't understand half this shit myself. But I haven't got time to hold your hand and coddle you. I need you to do what I tell you to do and stop being such a pain in my ass."
"I know that you and Willie are lying about taking me home. You killed my mother, and now you're going to kill me, too."
Soldier squats down so she's at eye level with Emmie. "Listen," she says. "It's not up to me, whatever happens to you. I'm not going to lie and tell you that I wouldn't kill you if that's what I was supposed to do. But the hounds put you where they did for a reason, and if I hurt you I'd be interfering with their plans, and that, Emmie Silvey, would be the end of me."
"Maybe I understand more than you think I do," Emmie says. "Maybe Pearl and that woman in the desert told me things about you that even you don't know."
"Then maybe you should be so kind as to unburden yourself and fucking enlighten me." And Soldier sounds more than pissed off now. She sounds like Deacon did the night that Sadie left them. She sounds dangerous.
"I'm not getting back into that car with you," Emmie tells her and then looks away again.
"Yes, you will, little girl, even if I have to drag you kicking and screaming all the way."
"But I did what she told me to do. I brought you the star. I brought you the star so you could stop that monster."
"And now you're going to do what I tell you to do."
Emmie squeezes her eyes tightly shut, silently counts to four, then opens them again, something Sadie taught her, a trick to settle her mind whenever she's too scared to think clearly.
"I saw the way all those people out there looked at us," she says, turning to face Soldier again. "If you try to make me go with you, I'll tell them all that you kidnapped me and won't let me go home. I'll tell them you're a murderer."
Soldier shakes her head and smiles, a resigned and weary smile that seems to bleed away some of the fury in her battered face, and for a second Emmie thinks that maybe she's won the game, that maybe Soldier's finally had enough, and now she'll get up and leave Emmie sitting on the toilet. That Soldier and Odd Willie will drive off in their stolen car, and Emmie will never have to see either of them again.
But then Soldier pulls a big black pistol from the waistband of her jeans and holds it only a few inches from Emmie's face.
"You're a smart girl, and you know what this thing is, right? You know what I do with it?"
Emmie nods, her victory dissolving in the dull glint of the restroom light off the barrel of the gun.
"You do that—you start telling people things about me and Odd Willie—"
"And you'll kill me," Emmie says.
"No, I won't kill you. But I will have to kill everyone else in here, every last person who hears what you say. And everyone who sees me killing them, I'll have to kill those people, too. Men, women, children—at this point, I really don't give a rat's ass. And all those deaths, every one of them, will be because you couldn't keep your mouth shut, Emmie. Now, you tell me, do you want that on your conscience? Do you really, truly want to have to think about that fucking day in and day out for the rest of your life?"
"You'd do that?" Emmie asks, though she already knows the answer.
"In a heartbeat. I'll do whatever it takes. I've done worse things."
"Are you proud of that?"
"You should wash your hands," Soldier says instead of answering her question. "You've got that Frito crap all over them. I'll give you a minute to wash up."
"They weren't Fritos," Emmie tells her and looks down at her fingers, dusted with salt and the ranch dressing–flavored powder. "The horse is dead," she says. "From here we walk."
"What the hell's that supposed to mean?" Soldier asks, returning the pistol to its hiding place.
"I thought maybe you might know," Emmie replies, then gets up to wash her hands.
"Why are we going to the museum?" Emmie asks from the backseat of the Malibu. She's sitting directly behind Odd Willie, putting as much space between herself and Soldier as possible. It's the first thing she's said since Soldier led her from the restroom back out to the car. "This isn't even a very good museum. Mostly, they just have a lot of old junk."
"I thought that's why people built museums," Odd Willie snorts. "To have someplace to keep all the old junk."
"You'll see, Emmie," Soldier says and flicks a cigarette butt out the car window as Odd Willie comes to a stop directly in front of the museum building. It seems worn and out of place in the city park, like some peculiar temple to the Victorian scientific enterprise, maybe plucked from the streets of nineteenth-century Paris and meticulously reassembled, stone by stone, in Providence. The steeply pitched roof is covered in black shingles, and the yellow-pink granite facade is decorated with bas-relief Corinthian columns and elaborately carved acroterions. There are stained-glass windows set deeply into the semicircular arch above the entrance, dormers higher up, and a great clock tower front and center.
There are no other cars in the parking lot, and no sign that the museum's even open.
"It's not the museum we've come for," Odd Willie says and kills the engine. "It's where the museum leads."
"The museum leads to the Bailiff?"
"Like I said, you'll see," Soldier tells her.
"I heard your stepmother's some kind of witch," Odd Willie says. "If that's true, she must have told you that things aren't always what they seem. Never judge a book by its fucking cover and all that. So, you might see a run-down, rinky-dink little museum, but me, I see a passageway, a secret avenue—"
"You stay right by me," Soldier cuts in, and she turns around in the seat and glares at Emmie. "None of that shit you pulled back at the gas station, you understand?"
Emmie nods her head but doesn't say anything.
"I'm not fucking kidding with you," Soldier says, and then she turns around again. She pops the clip out of the Smith & Wesson 9mm that Odd Willie picked up in a pawn shop back in Uxbridge and counts the rounds—six, seven, eight—then slides the clip in again. The Model 439 has seen better days, and Soldier wishes she had half the firepower Ballou took off them in Woonsocket. Right now she'd give her eyeteeth for a good shotgun. When she looks up, she sees that Odd Willie's frowning at her.
"What the hell's your problem?" she asks him.
"This is just dumb," he replies, and his eyes drift from Soldier's gun back to the museum building. "I mean, if it was a setup—"
"We don't know that."
"I know we don't know it, but it's what we're both sitting here thinking. Don't you try to lie to me and say that it's not."
"You know I wouldn't dream of lying to you, Odd Willie," Soldier says and sticks the pistol back into the waistband of her jeans.
"Yeah, well, I just don't see what good fucking guns are gonna be. If he's got it in for us, we won't be shooting our way out of there."
"Chance favors the prepared," Soldier says and looks at Emmie again. "Ain't that right, little girl?"
"Don't call me that. You know my name."
"Okay, Emmie. You behave yourself, and that's a deal. Now, listen, when we go through the front doors, there's a staircase on the other side of the lobby—"
"I have been here before," Emmie says and sighs.
"Then you know where the black bear is, right?"
"No. I don't remember a black bear."
"But you just said you'd been here before."
"Yeah, but I still don't remember a bear."
Soldier takes a deep breath and clicks her tongue once against the roof of her mouth.
"I'm sorry. I don't remember a bear," Emmie says again.
"Well, trust me. Halfway up those stairs there's a stuffed black bear, okay? You know, taxidermied."
"Looks like it's got the fucking mange," Odd Willie adds and opens his door, letting in a gust of cold air.
"That's where we're going, Emmie," Soldier continues. "Halfway up those stairs. We're going in the front door and then straight to the bear."
"Why are we going to see a dead bear?" Emmie asks and opens her door, too.
"We're not going to see the goddamned bear. We're going to see the Bailiff and—"
"But you just said—"
"Things aren't always what they seem," Odd Willie tells Emmie again, and he grins at her from the rearview mirror. "Just like the lady said," and he nods at Soldier, "you'll see. It's magick."
"Fine," Emmie replies, "but I don't remember a dead bear," and she starts to get out of the Chevy.
"Hold up," Soldier says. "Give me another second or two." She stares out the smeary windshield at the museum. You're just like everyone else, she tells herself. You're afraid of him. Everyone's afraid of him, everyone except the hounds, and he knows it. He banks on it.
"Jesus, I just want to get this over with," Odd Willie groans and pulls his door shut again.
You're scared half to death of the bastard, Soldier thinks, and he knows it.
"Emmie," she says, keeping her eyes on the hands of the clock mounted high above the museum doors. "You got that glass ball you gave me from a woman in a desert, a black-skinned woman. Not a black woman, but a woman with black skin."
"No, that's not what I said. I got it from Pearl. The woman in the desert told me to bring it to you, but I got it from Pearl. She said it was one of her father's experiments."
Odd Willie giggles to himself and shakes his head.
"Did she tell you her name, the woman in the desert?" Soldier asks.
"She said she had a lot of different names, but she likes to keep them to herself."
Willie Lothrop leans forward, resting his forehead on the steering wheel. "Anytime one of you ladies wants to tell me what the hell you're talking about, I'll be sitting right here."
"And she told you something about me?" Soldier asks Emmie, ignoring Odd Willie. The clock's hour hand is at one, and the minute hand is at four, but the clock's been broken for as long as she can recall.
"Yeah," Emmie says. "She did."
"Well, we'll talk about it later," Soldier tells her. "If that's cool with you. When all this is over, we'll talk about exactly what she told you."
"If I'm not dead yet," Emmie grumbles.
"Just stick close to me." And then Soldier opens her door, and the sky above the old museum in Roger Williams Park seems to shudder and stretch for a moment, like something that's grown too ripe and is almost ready to burst. And then it's only sky again, and she gets out of the car.
"It's a tough row to hoe," the Bailiff said and then leaned back in his squeaky chair. One day, Soldier thought, it's gonna break, that old chair, and he'll end up on the floor. He gazed down at her over the rim of his enormous belly and smiled.
"Being a Child of the Cuckoo?" she asked. "Is that what you mean?"
"I do," he said. "I mean that very thing, little Soldier. The universe is a cruel old cunt, she is. But for the random vagaries of happenstance, you might have been any girl, safe and snug at home with a loving papa and a loving mama to watch over her."
Soldier wondered about it for a moment and chewed thoughtfully at the arrowroot cookie he'd given her after the vampire named Adelaide had led her into the study. "Maybe," she said. "Or I might have ended up like Cinderella, with a wicked stepmother and three wicked stepsisters. Or like Hansel and Gretel. Or the pretty daughter in 'Mother Hulda' who has to spin until her fingers bleed."
"Ah, yes," the Bailiff replied and nodded his head, seeming to consider what Soldier had said. "But all those stories have happy endings."
"That depends who's telling them."
The Bailiff chuckled and tapped the side of his nose. "Too true, too true," he said. "You are wise far beyond your years, child."
"I had another dream last night," Soldier said, because she knew that was probably the reason he'd called for her. He rarely wanted much of anything but to listen to her dreams. "You were in this one."
"Was I now?" he asked and leaned forward again, one bushy eyebrow cocked, and she thought perhaps he looked more worried than curious. "Do tell. Whatever was I doing in one of your dreams?"
"I didn't understand all of it."
"Then let's start with the parts you did understand, and we'll get to the confusing bits later on." He tugged at his beard and licked his thin lips like a hungry dog.
So she told him about a winter day, still many years off, and a lost mongrel girl, and a demon of fire and cinders that she'd slain with nothing more than a crystal ball. There was something about the mongrel's mother, something that had upset her so much she'd forgotten most of it, and, she said, she'd realized, in the dream, that she wasn't a child anymore, that she was a grown woman who could barely even recall ever having been a child.
"But you said that I was a part of this dream," the Bailiff said impatiently. "So far I've not heard one word about me."
"I was coming to that part," she said, annoyed that he was hurrying her, and swallowed another bite of arrowroot cookie. The Bailiff frowned and tugged on his gray beard.
"You wanted to kill me," Soldier told him. "We were standing in a room, and there was incense burning and silk pillows and all these pretty boys who at first I thought were girls. And we weren't here."
"You mean we weren't in this house?"
"No, I mean we weren't here. We were somewhere else. Somewhere you had called me to. Somewhere you went to be away from the hounds."
The Bailiff laughed and took out a handkerchief. He wiped fat droplets of sweat off his face, then laughed again. He'd begun staring at a particularly old book lying open on his desk instead of looking at Soldier.
"Are you feeling well?" Soldier asked, and he wiped his face again.
"Tell me the rest," he said. "Tell me all of it."
But Soldier waited a moment, wishing that she had another cookie or maybe a piece of fudge or something else she'd never tasted.
"Is there a problem?" he asked.
"No," she told him, "not really," because she was pretty sure there was no point, this far along, in asking for fudge. So she told him that in the dream the hounds were leaving, going back to someplace they'd been before they'd found a way into this world. They were all going away forever and leaving him behind, and then he would only be a strange fat man with no work left to do.
"I saw Madam Terpsichore," Soldier said, "who was the last of the ghouls to leave, and she told me to watch after you, that you had become dissipated and decadent and careless, that you'd sunk too deeply into your appetites, forgetting—"
"She said that?" the Bailiff asked, his lower lip trembling, and she saw that his hands were shaking.
"If she hadn't, I wouldn't have told you she had."
"Pray tell, was there anything more?" he asked, laying his palms flat on the desk, on the pages of the old book.
"Yes," she replied, and Soldier watched him and considered whether or not she should say anything more. She'd never seen the Bailiff like this. Even if he wasn't the Cuckoo or the god of men and churches, she'd thought he was something, something powerful enough that he certainly didn't have to be afraid of the dreams of children.
"Well?" he said. "I'm waiting."
Soldier picked a stray speck of cookie off the front of her dress and then continued. "You said to me, 'We have fallen on hard times. Our lords and ladies have all deserted us, and our purpose lies in ruin. A masque which has endured down untold ages is ended here this night, and now we are castaways in our own land.'"
"I said all that?" he asked and wiped his forehead with the damp handkerchief. "Quite the mouthful, don't you think?"
"Yes sir," Soldier told him. "But you said it. Then you cut your throat from ear to ear, and the pretty boys whom I'd thought were girls went down on their hands and knees and lapped your blood from the floor. When they were done, they saved some in a little blue bottle. They locked the bottle in a lead box and dropped it into the sea. When I asked why, one of them told me it was so your soul would go down to the Mother and the Father, and so you'd not come back to haunt them."
"You are an ill wind, indeed," he said, but she wasn't sure what he meant, and since she wasn't finished with the dream, she didn't bother to ask.
"And then," she continued, "I remembered a magic trick I knew, and I..." And she paused, the words eluding her, the language she needed to describe something that seemed all but indescribable.
"Oh, don't stop now," the Bailiff laughed and slammed the big book shut, startling Soldier. "Surely," he said, "there's some snippet of cataclysm yet to be revealed."
He's scared, she thought. He's really scared. And she knew then that the Bailiff was only a man and nothing more.
"I unhappened it," she said, making up a word because she could think of none that fit. "And you were back, and I took the knife away before you could hurt yourself with it. I promised to watch over you, because when I was a little girl you'd been good to me and given me cookies and candy and soda."
The Bailiff dabbed at his sweaty cheeks. "And that's all?" he asked.
"No, but I've mostly forgotten the rest. It wouldn't make any sense if I tried to tell it. Can I have another cookie, please?"
"No," he said, standing up, and the handkerchief slipped from his fingers and fell to the floor at his feet. "You may not have another cookie. We're quite finished for tonight." And then he called out for the silver-eyed woman, who came and led Soldier back down the basement steps to the tunnels.
And the next time the Bailiff called for Soldier, he took her to the long hall on the second floor of the yellow house and opened the foldaway stairs leading up into the attic....
Through the heavy front doors of the museum, into shadows and the musty smell of very old things, past whale bones and mastodon tusks leaning against the foyer walls like the crooked walking canes of leviathans—through the doors and across the lobby, just like Soldier told her, and Emmie goes up the stairs with Odd Willie in front of her and Soldier behind. The girl in the gift shop, just past the entrance, doesn't ask them to pay admission. She doesn't even seem to notice them, and Emmie suspects that's because she can't see or hear them. We're invisible, she thinks, like the seventh point on the Seal of Solomon, but it doesn't seem nearly as remarkable as it ought to. Then she spots the black bear, a neglected, moth-eaten thing rearing up on its hind legs, swiping at the air with its shaggy forepaws, its muzzle frozen permanently in an expression of mock ferocity. The bear is exactly where they said it would be, halfway up the stairs on a wide landing, but she still doesn't remember it ever being there before. There are six stained-glass windows behind it, and the bear is silhouetted against pale kaleidoscope patterns of yellow and red and blue sunlight.
"You know, it's still not too late—" Odd Willie begins, but Soldier interrupts him.
"Not too late for what? You want to go to the hounds, instead? Or maybe you'd rather try to cut and run?"
"Well, to tell you the gods'-honest fucking truth, I was growing kind of fond of that motel room back in Massachusetts."
"We don't know," Soldier says and takes a long, deep breath. "We don't know that he set us up. Until we do, Willie, he's still calling the shots. He's a bastard and a son of a bitch, but he's never betrayed the hounds."
"Not that you know of," Odd Willie sighs. "Maybe those things Saben said last night, maybe she wasn't so far off the mark."
Emmie flinches at the mention of Saben White. She looks at Odd Willie and Soldier and then back to the taxidermied black bear towering over her. There's a Plexiglas barrier surrounding it, so no one can get too close. So no one can pull out its fur or try to snap off a tooth or a claw or something.
"She wanted us dead," Soldier says. "She sold me out to that cocksucker Joey Bittern, and then she sold us both out to Ballou. She'd have done or said anything to get to the kid. I do know that much."
"We could always head for Boston. Ask for sanctuary—"
"Right now, he can hear every single goddamn word we're saying. You think you'd even make it back to the car alive?"
"Killjoy," Odd Willie sighs and shakes his head.
Emmie places her hand flat against the Plexiglas, and it ripples like water. "Is it supposed to do that?" she asks Soldier, who nods her head and places her own hand against the clear plastic, creating a second set of ripples that spreads out and eclipses the ones that Emmie made. Where the ripples cross, there are gentle shimmers of light.
"You stay close to me," Soldier tells Emmie again. "Stay close and don't touch anything."
Soldier takes her hand, and for a second Emmie Silvey pretends she's any child at the museum, and maybe Soldier is only her big sister. Maybe they'll go to Ben & Jerry's after the museum. Maybe they'll go to Johnny Rockets and have chocolate milkshakes and chili-cheese fries. Maybe, she thinks, I have green eyes, just like her. Maybe Deacon's waiting outside to take us both home.
"The horse is dead," she says and squeezes Soldier's hand. "From here we walk."
"Anyone ever told you you're sort of a creepy kid?" Odd Willie asks.
"All the damn time," Emmie tells him. "I don't bother keeping count anymore."
"Just checking." Then Odd Willie brushes the Plexiglas with the fingertips of his left hand, and a third set of ripples spreads rapidly across the barrier. Emmie realizes that she can see through the bear now, through the bear and the stained-glass window behind it. There's some sort of hallway on the other side, past the museum wall, where the park should be, a long hallway dimly lit by bare lightbulbs screwed into sockets strung up along the ceiling.
"That's where the Bailiff is?" Emmie asks, but no one answers her. Soldier takes a step forward, and the rippling, shimmering Plexiglas and the stuffed bear and the wall of the museum parts for her like a theater curtain...
...and before Soldier went up the foldaway stairs, before she climbed into the attic of the yellow house and met the Daughter of the Four of Pentacles, but after the Bailiff lied and promised her that he was coming, too, that he'd be right behind her—in the fleeting moments in between, she almost told him the truth about her dream. She paused at the first step and looked at him and almost said, In my dream, you didn't slit your own throat. I did that.
But there was something slick and sharp and icy in his eyes, something like a snowy January night or the edge of one of Madam Terpsichore's scalpels, and Soldier had decided, even if he might be better off knowing, she'd be better off keeping that part to herself. There are worse things than lies, she thought (and it seemed like a very practical, grown-up thought), and then Soldier said her silent prayer to the Lady of the Abyssal Plains and started up the ladder.
"Oh, but he's in a most peculiar temperament," the rat who isn't Reepicheep, who probably isn't even a real rat at all, says nervously and scratches at a scabby spot behind its right ear.
Emmie looks back the way they've just come, at the bricked-up doorway where there ought to be a museum and a stuffed bear. Coming through, entering the hall, there was only a slight shiver and the faintest passing nausea, and she wonders if the bricks would ripple the way the Plexiglas did were she to walk back and touch them.
"Care to be just a little more specific?" Odd Willie asks the rat, and it frowns and glances towards the shadows at the far end of the hallway.
"Yeah, sure, whatever you say, Mr. Lothrop," the rat sneers. "If you wanted a goddamn turncoat, you should have tried pumping the fucking bear before you came across."
"The bear doesn't talk to changelings," Soldier says. "You know that."
Emmie starts to ask her why the bear won't talk to changelings, and how a stuffed bear can talk to anyone at all, but then she decides both questions are too ridiculous to bother with. She looks past the rat, at the hall, at the faraway place where it ends. The walls are made of doors set one against the other—black doors, crimson doors, weathered gray doors, and doors the color of butterscotch. None of them have knobs or handles, only empty black holes where doorknobs ought to be.
"Anyhow," the rat says and waves a paw in the air dismissively, "you'll see it all for yourselves plenty soon enough now. The Bailiff ain't shy with his little indiscretions, not down here. Follow me, and don't any of you say peep until—"
"We know the drill," Soldier tells the rat. "Let's just get this over with."
The rat shrugs and drops down onto all fours. It sniffs at Emmie's boots and scowls. "Just trying to keep the peace," it snorts. "What precious few crumbs are left of it." And then it turns and walks slowly down the long hall of doors, slipping through the yellow-white pools of light from the bare bulbs overhead and the dark patches dividing them.
"Fucking fleabag twat," Odd Willie mutters and follows the talking rat.
"I know," Emmie says to Soldier. "Stay close to you. Don't touch anything."
"If I hadn't killed her, you'd both be dead by now," Soldier replies, answering a question that Emmie hasn't asked in hours. "No reason you should take my word for it, but it's the truth. I'm not saying that's why I did it. I did it because I didn't have any choice."
"You said that already."
"Well, I thought maybe I should say it again."
"They're getting way ahead of us," Emmie says and points at Odd Willie and the rat. "We're gonna have to hurry to catch up."
"Take my hand," Soldier tells her, and Emmie does, squeezes it tight like she did before they walked through the bear into this place of doors and bare lightbulbs and missing doorknobs, and Soldier leads her down the hallway. Emmie tries to keep her mind off whatever's at the other end by silently counting the doors on her right, and she makes it as far as fifty-eight before they get to the spot where Odd Willie is waiting with the rat. The hallway ends in something that Emmie at first mistakes for a mirror, because she can see herself and Soldier and Odd Willie reflected there in its smooth, glassy surface, but then the rat steps through it and vanishes.
"You guys know what a cliché is?" Emmie asks.
"Hey, I'm not the prick that built this funhouse shithole," Odd Willie says, "so don't go complaining to me."
Emmie stares back at herself a moment, then glances up at Soldier. "We go through a mirror?"
"It's not a mirror," Soldier says. "And I don't have time to explain, so don't even ask."
Odd Willie coughs and smoothes his hair with both his hands. "Stand up straight," he says, and for a second Emmie thinks he's talking to her, so she stands up straight. "Chew with your fucking mouth shut," he continues. "Mind your Ps and Qs, Master Lothrop." And then he steps through the mirror that isn't a mirror and is gone.
"The Bailiff built this place?" Emmie asks.
"No. He's just renting it. A demon built it."
"There's no such thing as demons. I mean, I don't believe in them," Emmie says, even though she's starting to think that maybe she does, that perhaps she believes in lots of things that Deacon's told her are only fairy tales and horseshit. But mostly, she says it just to have something to say, just to buy a few more seconds before Soldier leads her through the looking glass.
"It really doesn't matter what you do and don't believe," Soldier tells her and then chews at her lower lip a moment. "But you'll figure that out, sooner or later."
"If I live that long."
"Yeah, Emmie. That's the way it always works. If you live that long," and then Soldier steps through her own reflection, and there's nothing Emmie can do but follow her.
The passage lasts only a glittering instant, but it fills Soldier's head with the cavernous, shattering, hopeless noise of storms, of waves pounding the shingles of rocky shorelines, of the unanswered prayers of dying men. An instant of the raw chaos and pain woven together centuries ago to fashion this place from the nothingness between worlds, and then she's standing with Odd Willie and Emmie, looking down at the great octagonal chamber, the Bailiff's hiding place, standing on a dais carved from obsidian and petrified cypress logs. It might easily double as the set of some silent-era Rudolph Valentino romance of Arabian horsemen and harem girls, this room, half-lost beneath the clutter of draperies and carpets and silk pillows, Moroccan antiques and Syrian hookahs. The air is warm and smells like blood, like hot wax and frankincense, opium and shisha tobaccos. And there are bodies everywhere. Soldier gags and covers her mouth; astral travel has always made her a little queasy, and the stink of the room isn't helping any.
"Well, well," the Bailiff says, and spreads his arms wide. He's wearing a shabby lime-green bathrobe thrown open to reveal that he's wearing nothing else. There's a kukri in his left hand, its broad blade glinting wetly in the candlelight. "Have a look at what the cat's dragged in." His voice booms, amplified by the countless imperfections and peculiarities of the room's geometry. "I was beginning to think you'd lost your way, Soldier girl. But here you are, safe and sound, almost pretty as a painted picture. Imagine my relief."
"I know you're gonna tell me it's none of my goddamn business," Odd Willie says and points at the corpses of the Bailiff's boy whores scattered about the chamber, and the room turns another trick, making his voice seem small and distant. "But why'd you do this? Why did you kill them?"
"Shall we tell him, do you think?" the Bailiff asks Soldier, and then he laughs and tugs hard at his beard, which is tangled and matted with drying blood and claret wine and slobber. "Do we dare? Shall we be bold and let him in on our little secret?"
"I've done what you asked, old man. I've brought you Saben's daughter," Soldier says, and then she pauses to cough and clear her throat; her mouth tastes like bile and incense. "Was it only so you could murder her, too?"
Emmie tightens her grip on Soldier's right hand, digging her nails into Soldier's palm, and takes a small step backwards.
The Bailiff stops tugging his beard and shakes his head in an unconvincing pantomime of disbelief. "Forgive me, dear, but that seems a most improbable and ironic word to hear from your lips."
"Second thought," Odd Willie says and glances anxiously at Soldier, "maybe you should both keep your secrets. Just pretend like I never asked, and I'll forget I ever saw any of this shit."
"I don't want to be here anymore," Emmie says, and tries to pull Soldier back towards the portal and the hallway of doors, back towards the stuffed black bear and the stolen Chevy waiting for them in the museum's parking lot.
"Be still," Soldier tells her, almost growling the words, not taking her eyes off the Bailiff in his lime-green bathrobe, his fat, hairy belly smeared with the dead boys' blood, the huge knife in his hand.
"He killed them," Emmie says. "Now he's going to kill me, too."
"I'm gonna do it for him," Soldier tells her, "if you don't shut up and stand still."
The Bailiff grins and scratches at his crotch. "It's quite all right, Master Lothrop," he laughs. "Some secrets go sour if you let them sit too long. They go rancid and lose their potency, that dreadful spark that made them secrets in the first place."
"Odd Willie did everything he was told," Soldier says, walking to the edge of the dais. "Whatever quarrel you might have with me and the girl, it doesn't have anything to do with him."
The Bailiff stops scratching at his balls and lets the kukri slip from his fingers and fall to the floor. Then he bows once and claps his pudgy hands together, applause made loud as thunder and sledgehammers on steel rails by the room's deceiving acoustics.
"Brava," he says and bows his bald head again. "You've grown into a fine and gallant woman, Soldier girl, a veritable fucking Hot-spur. But Master Lothrop has his role to play, and we mustn't deprive him of his rightful part in this exquisite comedy of ours. That would be exceedingly unfair. Never rob a man of his place in history."
"He's going to kill us all," Emmie says and tugs on Soldier's arm again.
The Bailiff comes nearer, steps over the body of one of the slaughtered boys, and slips in a pool of blood. For a moment, Soldier thinks he's going to fall, that he's about to land in a heap atop the gore and silk damask and spilled absinthe. But then he finds his balance again and steps gingerly over the remains of a broken sitar and another corpse, neatly decapitated, gutted, its intestines wound tightly about its pale legs.
You're afraid of him, Soldier thinks for the second time that morning, hating herself more than ever, hating her fear and all the years she's allowed her fear of him to steer her life. He knows it. He fucking banks on it. She draws the pawnshop 9mm and aims it at the fat man.
Odd Willie makes a sudden strangled noise, part confusion, part dumbfounded surprise, and turns from the Bailiff to Soldier. "What the hell are you doing?" he asks her. "Come on, now. You know the score. I'm the fucking lunatic with a thinly veiled death wish. You, you're just the sullen drunk with a bad temper."
"Shut up, Odd Willie," Soldier replies. "Take the kid and get the hell out of here. Go to Master Danaüs. Tell him everything."
The Bailiff stops stepping over cushions and dead boys and applauds again.
"Sublime," he says and tugs at his beard. "I did not even dare to hope that you'd ever find the courage to stand up to me, little Soldier girl. What a wondrous, sublime turn of events, here, before the end."
"Soldier," Odd Willie says, and she wants to look him in the eye, wants to make him see that she means what she's said, but she doesn't dare take her eyes off the Bailiff.
"Just fucking do it. Get her the hell out of here."
"You know I can't do that. The Cuckoo—"
"Oh, yes you can," Soldier says and thumbs off the Smith & Wesson's safety. "Yes, you fucking can, Willie, because this doesn't have jack shit to do with the Cuckoo. This is old business between me and the Bailiff, and that's all there is to this."
The Bailiff leans over and lifts a severed head from its place on a large plum-colored pillow, his fingers tangling in ash-blond hair, its blue eyes open and staring forward. He holds it up so the dead boy is facing him. His grin is so broad now that Soldier thinks it's a wonder his skull doesn't split open.
"The question at hand," the Bailiff says, speaking to the severed head, "is whether or not she has also found the courage to pull the trigger. It seems most unlikely, but I can't entirely rule it out, of course, the weight of these particular secrets being what they are."
"Talk to me, old man," Soldier says, tightening her grip on the trigger. "Stop playing games and fucking talk to me."
The Bailiff presses his lips to the left ear of the head and whispers something, then pretends that the dead boy whispers something back to him.
"Only her bless'd girlhood," the Bailiff says and licks blood from the dead boy's chin. "All in all," he says, "it actually wasn't such a very terrible loss. She's overreacting. Some vague hysteria arising from her uterus, most likely. A crying shame she wasn't born with a dick. Then, my lad, perhaps she could have lived up to her name."
"I said talk to me, you fat son of a bitch!" Soldier shouts, and the Bailiff acts startled and drops the head. He puts one hand over his heart, and his mouth is hanging open wide in a perfect, astounded O. The dead boy's blood stains his lips like a smudge of rouge, and Soldier notices that he has an erection.
"She's mad!" the Bailiff wails and rolls his eyes. "Master Lothrop, you are a Child of the Cuckoo, and, as such, you must do your sworn duty."
"Last chance," Soldier says, almost whispering, speaking just loudly enough that she's sure Odd Willie will hear. "Get out of here."
"Now, Master Lothrop!" the Bailiff screams, and the walls shudder and dust sifts down from between the struts of the domed ceiling. "You were delivered unto the Hounds of Cain and have passed through the trials of fire and swords and laid your hands upon the tongue of the dragon. You know your obligation."
Odd Willie draws his own pistol and cocks it, hesitates a moment, and then points it at Soldier's head. "Don't you make me fucking do this," he says. "Not after all that shit in Woonsocket, don't you make me fucking have to pull this trigger."
"He dies, either way," Soldier tells him. "You're not that fast, Odd Willie."
Emmie lets go of Soldier's hand, and the Bailiff takes another step towards the dais. "Don't worry about the girl," he says to Odd Willie. "The girl's a trifling thing. She can be dealt with anytime, at our convenience."
"I don't know what to do," Emmie says, close to tears. "Please, Soldier. Please tell me what I'm supposed to do now."
"Yes," the Bailiff says, and then he sits down on the floor amid the pillows and hookahs and corpses. "Do please tell her. Tell her what you told me the day I took you to meet the alchemist's daughter. Tell her—"
"I'm not afraid of you anymore, old man."
"Don't make me do this," Odd Willie says again. "Just put the fucking gun down, and we'll sort this shit out."
"No," the Bailiff says, speaking to Odd Willie now. "She won't do that. Not our Soldier girl." Then he pauses and surveys the carnage strewn about the octagonal chamber, the naked bodies and broken glass, feathers and torn cloth, the blood splashed across walls and portieres. "My, but I have managed to make quite the mess, haven't I? Have I killed them all?" he asks no one in particular.
"We were both supposed to die in Woonsocket," Soldier says. "Saben made a deal with Ballou, to get her daughter back, but she made a deal with you, too, didn't she?"
The Bailiff shrugs and then lies down beside one of the dead boys. "It hardly matters now, does it? We both know what happens next, dear. We both know..." He trails off and buries his face in the dead boy's long black hair.
Odd Willie presses the barrel of his gun to Soldier's temple. "What do we both know?" he asks her. "What's he talking about?"
"I know a trick," Soldier says. "I was born knowing a trick. When I was a child I had dreams that came true, and I know this fucking trick. And that's what happens next. That's what he thinks happens next."
The Bailiff rolls over on his back again, gazing up at the high ceiling. "She misleads you with eloquent understatement, Master Lothrop. This one here, our pretty little Soldier girl, she knows something more than any mere trick. She knows time, Master Lothrop, and she bends it to her will as easily as you might turn back the hands of a clock. She's poor Professor Einstein's darkest nightmare, a veritable imp of Kronos. Even so, she only half comprehends these things she does—that slick getaway out at Quaker Jameson's, for example."
"You meant for us to die in Woonsocket," Soldier says. "You and Saben White. Just let me hear you say it, old man. Say it so Odd Willie at least knows the truth before he puts a bullet in my brain."
"A man tries as best he can to elude his fate," the Bailiff sighs, all the gleeful fury drained from him now, and he sits up and leans towards them, crouched there on the floor like some obscene Buddha. "Three and half years ago, Soldier, you showed me my fate. You told me about drops of blood sealed in a beautiful china-blue bottle. You told me about a lead box and my path down to the Mother and the Father."
Soldier shakes her head, denial the only thing like sanity she has left, and Odd Willie tells her not to move.
"It was only a dream," she says to the Bailiff. "It was only a dream, and I was only five years old."
"Where does the time go?" the Bailiff laughs and tugs his beard. "See, that's what she's been asking herself, Master Lothrop, even though she was never quite aware of the question."
"I think you've both lost your goddamn marbles," Odd Willie says. "Soldier, put down the motherfucking gun. I've had enough, and I'm not going to tell you again," and he jams the barrel of his pistol against her left cheekbone hard enough to tear the dental-floss stitches and reopen the wound there.
The Bailiff makes a gun with his thumb and index finger and puts it to his own temple. "What was it that I'm supposed to say when the hounds have gone?" he asks Soldier. "Ah, yes. How could I ever forget those words—We have fallen on hard times. Our lords and ladies have all deserted us, and our purpose lies in ruin. A masque which has endured down untold ages is ended here this night, and now we are castaways in our own land. Did I get it right?"
"I don't know," Soldier replies, her voice trembling almost as badly as her hands and the 9mm. "That was a long time ago. I was only a child."
"Oh, but it's such an awful shame to forget such poetry. Even Shakespeare and Blake would have envied such perfect lines as those. Wouldn't you agree, Master Lothrop?"
"I've never cared much for poetry," Odd Willie says.
The Bailiff shrugs, and the green bathrobe slips off his right shoulder. "Soldier, surely you can't blame a man for trying to outwit destiny. Do you honestly believe I could have acted otherwise?"
Soldier slowly lowers her gun, easing back on the trigger, and turns her face away from the bore of Odd Willie's .45. Then she glances down at Emmie Silvey, who's still standing there beside her on the dais. "You asked me," she says to the Bailiff, "and I told you my dream, and then you took me up to the attic."
The Bailiff laughs, his laughter made painfully loud by the demon's architecture. "Finish it, Soldier," he says. "I am a foolish old bastard, and my clumsy schemes and machinations have only made my fate more terrible. Our purpose lies in ruin. The Cuckoo is undone. A masque which has endured down untold ages is ended here. All my world is ended here."
"Take me home," Emmie says, and there are tears streaking her cheeks. "Please, Soldier. Take me home now."
"Close your eyes," Soldier tells Emmie, and she thinks about that stormy night in Ipswich, and she thinks, too, about the cemetery in Woonsocket, the shrill voices of imagined birds singing on a misplaced summer day—All lost, lost, lost now. And, as easy as blinking an eye, Soldier does her trick for Emmie...
...and they're sitting in the car outside the Museum of Natural History in Roger Williams Park, and Soldier asks Emmie if she knows where the stuffed black bear is.
"No. I don't remember a black bear," Emmie tells her.
And then Soldier remembers the Bailiff and his dead boys, the cold muzzle of Odd Willie's gun against her skin, and the words she spoke three and half years ago, when she was only five years old.
"If there's a bear in there somewhere, then I never saw it. I think I wouldn't forget a whole damn bear."
"Never mind," Soldier says, and to Odd Willie, "Something's wrong. We have to go to Benefit Street instead." And he has questions, but only one or two, and it's easy to fake the answers she doesn't know. She tells him what he needs to hear, and then Odd Willie turns the key in the ignition and starts the car again and pulls out of the museum parking lot.
## TEN
## The Yellow House
Soldier can't remember the first time she saw the yellow house on Benefit Street, not the first time that she saw it from the outside. It seems she might have been a grown woman before she ever looked at it the way that other people do, those unsuspecting people of the sunlight who have not been raised in the deep and rotten places of the world, who have never walked the silent halls of the house or climbed the narrow stairs leading into and out of its vast basement. If Soldier had ever paused to consider this juxtaposition—that she knew the terrible heart of the yellow house before she ever glimpsed its concealing face—she might have thought it odd.
There has never been a less haunted house, nor a house more filled with bad memories and restless spirits.
It has a reputation, of course, but then many old houses in this city suffer from unpleasant reputations; too many houses that have stood far too long to escape insanity and murder, suicide and all the less mundane improprieties of men and women. But the "haunted Providence" tours never stop in front of the yellow house, and no medium or investigator of the "paranormal" has ever held a séance in its front parlor or attempted EMF readings in its upstairs bedrooms. There are occasionally sensitive minds who feel a sudden unease whenever they pass by, and some will even cross to the western side of the street to put more distance between themselves and the house. But these people are few, and they rarely spare the house more than a quick, anxious glance.
Mr. H. P. Lovecraft (1890–1937), who made the yellow house the subject of one of his stories, wrote: "Originally a farm or semi-farm building, it followed the average New England colonial lines of the middle eighteenth century—the prosperous peaked-roof sort, with two stories and dormerless attic, and with the Georgian doorway and interior panelling dictated by the progress of taste at that time. It faced south, with one gable end buried to the lower windows in the eastward rising hill, and the other exposed to the foundations toward the street." He also noted, "The general fact is, that the house was never regarded by the solid part of the community as in any real sense 'haunted.' There were no widespread tales of rattling chains, cold currents of air, extinguished lights, or faces at the window. Extremists sometimes said the house was 'unlucky,' but that is as far as even they went." And, fortunately for him and innumerable others, even Lovecraft's excitable and prying imagination never guessed more than a misleading fraction of the truth.
Since its appearance in 1764 (in no conventional sense was it constructed), the great yellow house at 135 Benefit Street has kept its secrets to itself, ever faithful to the iron wills of its architects, just as Soldier and Odd Willie Lothrop and the other changelings have kept their covenants with the Bailiff and the Cuckoo and the Hounds of Cain. The sum of improbable geometries and Cape Ann granite, nails and fallen trees and dim words whispered in forgotten tongues, the house knows its purpose well, as do all those who dwell within its walls and all those who might ever have cause to enter its doors or windows or come slipping up its drainpipes.
"Do you think they're expecting us?" Odd Willie asks, and Soldier shrugs and drops the butt of her cigarette. She grinds it out with the toe of her cowboy boot.
"I'm not sure it much matters," she says. The three of them—Odd Willie, Soldier, and Emmie Silvey—are standing together across the street from the house, standing there on the icy blacktop beside the stolen Chevy Malibu, and Emmie keeps sneezing. Soldier watches the house and knows that it's watching her, in turn, that the silver-eyed ladies and gentlemen inside are waiting for them to cross Benefit and come up the steps to the front door. She knows that somewhere below the yellow house, Madam Terpsichore and Master Danaüs and the other ghul are whispering among themselves, that Madam Mnemosyne sits in her burrow, hunched over her scrying glass, watching the black water for the moments Soldier hasn't yet caught up with. And in the attic, the alchemist's daughter is waiting, too.
"I know this house," Emmie says and sneezes again.
"Gesundheit," Odd Willie mutters and jingles the car keys in one hand.
"I've had dreams about this house," Emmie says and wipes her nose. "I dreamed it isn't really a house at all."
Soldier looks nervously back up Benefit Street. They'd picked up a tail shortly after leaving the museum parking lot, Kennedy and Sea-grave and that dyke bitch Amasa Sprague in the same black hearse that Soldier and Sheldon Vale had used for the long drive up to Ipswich. She knows that Odd Willie saw it, too, but neither of them has said anything about the hearse.
"I used to know the difference between my dreams and being awake," Emmie says. "Now I don't think I ever will again."
"Soldier, we can't just fucking stand here all day," Odd Willie says. "If we're going in, let's please just get it the hell over with."
"Maybe you better hang back," Soldier tells him. "Stay here with the car. I don't think this is about you."
Odd Willie shakes his head and rubs his freezing hands together. "Fuck that. You got me into this shit—you and the Bailiff and Saben and that kid," and he stops talking long enough to point at Emmie. "And now I'm in it, right, and it's too damned late for valiant fucking gestures, and you know that just as well as I do."
"I was just trying to go home," Emmie says. "I'd go home right now, if she'd let me," and she sneezes, then wipes her nose on the sleeve of her coat and glares at Soldier.
"You had a chance to run," Soldier says, "both of you. You could have left me to die back in Woonsocket, or you could have left me in that motel room in Uxbridge."
"Well," Odd Willie sighs, "we can't stand here all goddamn day long waiting for you to take your thumb out of your ass and finally make up your mind. I swear, I think I'm getting fucking hypothermia."
Soldier shuts her eyes, and she's back in Miss Josephine's dining room, sitting across the wide mahogany table from Sheldon's corpse, and he looks sad and smug and angry and very dead, all at the same time.
"You already know what comes next," he says. "All you have to do is cross the street. It's all been arranged. They're waiting for you inside."
And the clock on the mantel, the tall clock with a girl's tattooed face, begins to chime the hour.
Soldier gasps and opens her eyes, and the yellow house is still there, gazing indifferently down at her. An awful, bottomless box of secrets hidden in broad daylight for the past two hundred and forty-six years, and Soldier takes Emmie's hand and crosses the slushy street. A moment later, Odd Willie whispers a well-worn prayer to Mother Hydra and follows her.
Standing in the anteroom of the yellow house, Emmie tries to be polite and not stare at the woman who opened the door, the woman whom Soldier calls Miss Josephine, but she's never known that anyone could be so beautiful and so completely hideous. The woman, who is very tall and dressed as though she's just returned from a funeral in an Edward Gorey book—Miss Underfold in The Other Statue, perhaps—has skin like an antique wax doll. Emmie thinks that if she pressed a finger too roughly against that pale skin, she'd leave an impression behind. But the hardest part about not staring is the woman's shimmering silver eyes, eyes that remind Emmie of an experiment with liquid mercury that she did in science class just before winter break. And mercury is poisonous, she reminds herself and stares at her feet.
"I didn't think anyone would be awake," Soldier says, but Emmie can tell that she's lying. She suspects the silver-eyed woman knows Soldier's lying, too.
"We've all been very concerned," the woman says, and something about her voice makes Emmie flinch. "The household is not quite itself today. There have been the most extraordinary rumors, and sleep has eluded most of us. The Bailiff—"
"Yeah, he said that I should meet him here," Soldier says. "He said that I should go to the attic and wait for him there."
The woman nods her wax-doll head and leans close to Soldier, sniffing the air around her. Then she smiles, and Emmie's sorry that she isn't still staring at her feet and minding her own business, because she knows she won't ever be able to forget that smile, not if she lives to be a hundred and twenty. Like the dead starling, the bird she killed when she was only six, and Deacon said, "Just try not thinking of a white elephant sometime."
"This unfortunate affair," the woman says, her voice all honey and ice and broken bottles, "it is not truly any of my concern. This affair, it's between you and the ones downstairs. We mind our business, and the hounds kindly mind theirs. But the Bailiff, and the attic..." and the woman pauses and sniffs at Soldier again. "You've hurt yourself, my dear," she says and touches one index finger gently to the cut on Soldier's cheek. "You should be more careful."
Odd Willie glances apprehensively at Soldier and shifts from foot to foot. Emmie wonders if he's about to take out his gun and shoot the silver-eyed woman; she wonders if it would make any difference.
"Stop your fidgeting, Master Lothrop," Miss Josephine says and looks directly at him. "If the Bailiff said the two of you should meet him in the attic, then meet him in the attic you shall. As I have said already, this is no proper concern of ours. You and Master Lothrop answer to the Bailiff and to the hounds, not to me nor mine."
"Thank you," Soldier says, and Emmie isn't sure if she sounds relieved or only frightened in another way.
"This one here," Miss Josephine says and turns towards Emmie. "She is also a Child of the Cuckoo, is she not?"
"My name's Emmie Silvey," Emmie says, and the woman nods her head.
"Then I have a message for you."
"She's with us," Soldier says quickly, and the silver-eyed woman flashes her another of those smiles, a smile like a shark only pretending to be a woman, and Soldier takes a step backwards, bumping into Odd Willie.
"Indeed," says Miss Josephine. "Nonetheless, this is an urgent message that I have promised to deliver, and I am ever bound to keep my promises. Unless, dear Soldier, you should object. Unless you believe, possibly, that I am overstepping my rightful boundaries." And Emmie wants to shut her eyes, wants to turn away from that monstrous smile, but she seems to have forgotten how to move.
"Of course not," Soldier says, almost whispering. "I didn't mean to imply that."
"Then be about your business, changeling, and I will be about mine," and she waves her left hand, dismissing Soldier and Odd Willie. Then, before Soldier can protest or Emmie can cry out or Odd Willie can even go back to fidgeting, the smiling, silver-eyed woman places her right thumb firmly against Emmie's forehead. There's sudden cold, an ice age spilling out of the woman and into Emmie, and then a brief electric jolt, and the anteroom dissolves into dusk and the sound of a dry desert wind...
...and the twilight air smells like the dust between grains of sand and, more faintly, the musky sweat of the black-skinned woman.
"I didn't think I was ever going to see you again," Emmie says. "I thought that part of the story was over."
"You ought never second-guess a story," the black-skinned woman says. She's crouched only a few yards down the dune face from Emmie. She's almost completely naked, wearing only a ragged sort of leather breechcloth strung about her waist and a few pieces of jewelry that shine faintly in the glow of the rising moon. Her white dreadlocks are tied back away from her face.
"Not even a story I'm in?"
"Not even then," she replies. "Especially not a story that you're inside."
"It's a very bad place, isn't it?"
"That old house?" the woman asks and begins tracing something in the sand between them. "Dangerous would be a better way of describing it. Think of it as you would think of a nest of hornets. The nest itself is neither good nor evil, not as men reckon such things, but it can hurt you, because of the things that live inside."
"Like that horrible woman."
"Like her. And the others of her kind. And things you haven't seen yet. And things I hope you never will."
The black-skinned woman draws a circle in the sand, then draws a smaller circle inside it, then an even smaller circle at the center of the other two.
"So why am I here again?" Emmie asks, trying to figure out what the image in the sand is meant to be. "She said that she had a message for me, but then she sent me here to you. Why'd she do that?"
"Because she owed me a debt, an ages-old debt that she could never truly repay, and I offered to dismiss it if she would be sure you found your way back here."
Emmie frowns and scoops a small hole in the sand, working the fingers of her right hand in like the spade-shaped snout of some furtive desert creature. Only an inch or two below the surface, the sand is still warm.
"They won't take me home," she says. "I did what you asked, but they won't take me back to Deacon. It's been days and days. He must think I'm dead by now. By now, he must have given up."
The woman stops drawing in the sand and looked at Emmie, shadows hiding her golden eyes. "This path you're on is a roundabout sort of thing, fraught with many twists and turns. But I believe, Emmie, that it will lead you home very soon."
"You believe?" Emmie balks and digs her fingers deeper into the sun-warm sand.
"I'm sorry, but on this path nothing is ever certain. Even when the story seems to be finished, and the book is closed, even then, it may not truly be done."
"We went into the museum, didn't we?" Emmie asks, quick, before she loses her nerve. "We went in, but then she did something so it never happened."
"You're a very observant girl," the black-skinned woman says and goes back to tracing her patterns in the sand. "You're beginning to catch on, which is more than could ever be said for most."
"I thought so," Emmie says, and looks up at the moon and all the stars winking on above the dunes. "It was sort of like having déjà vu, almost. We were about to get out of the car and go into the museum, and then Soldier changed her mind, and, all of a sudden, I had memories of things that hadn't happened."
"It was passed down in her mother's blood, that particular gift, that curse. It's kept her alive many times now, even without Soldier knowing what she was doing, but it could still destroy her...and others."
"It scares me, almost more than anything else," and Emmie gives up trying to find the Big Dipper, because there's something not quite right about these stars, something just unfamiliar enough to be disorienting. "I keep wondering if Soldier's changed things I don't remember. I keep wondering if that first time I saw her, when I gave her Pearl's snow globe, if maybe it wasn't only two days ago, because she keeps changing what happens to make it come out different, and so we keep doing the same things over and over again."
"She's trying to find something, Emmie," the woman says, and draws a crooked triangle to enclose the circles within circles within circles. "Something that was stolen from her. When she finds it, I think the story will end, and you can go home. If she finds it."
"Is that what I'm supposed to do? Help her find what she's looking for?"
"That's a part of it. But there are other things, as well."
"Building bridges," Emmie says very softly. "Building bridges for the ghouls and Pearl."
"You have grievous choices ahead of you, Emmie Silvey, and I do not envy you."
"You know, half the time, I have no idea what you're talking about."
The black-skinned woman laughs softly to herself and stands up, wiping sand from her thighs and legs. "You are a wonder, child. I'd keep you here with me, if I thought the world would ever permit such a thing."
"But I don't want to stay here with you," Emmie tells her, even though part of her does, the part that doesn't really want to know how the story's going to end, the part that's more afraid than homesick. "I want to go home. I want to go home and be with Deacon."
"Then you must watch your steps, and you must make these choices with great care. You must decide if you will help Soldier again, and if you'll build the bridges the hounds desire of you."
"I keep saying that I don't know how to build a damn bridge," Emmie tells her and pulls her hand free of the sand.
The black-skinned woman is silent for a moment. She stands silhouetted against the evening sky, the moon like a bright coin hung above her head. She seems to be admiring whatever it is that she's drawn in the sand, the complex arrangement of lines and curves and curlicues that Emmie can only just make out in the moonlight.
"There was a great war here once," the woman says. "You know that, don't you, Emmie?"
Emmie nods her head. A yellow scorpion as long as her middle finger has crawled out of the sand where her hand was buried only a moment before, and she watches it skitter towards the edge of the shapes drawn on the dune.
"I was dreaming, and Pearl showed me. Or I thought I was dreaming. She said it was a war to drive the ghouls out of the wastelands."
"It was that, and other things. In a sense, it has never actually ended. In a sense, it never will."
The scorpion pauses at the edge of the outermost line the woman has traced in the sand and stirs at the night air with its pincers.
"The warriors still battle in the sky, Emmie. You've seen them, too. Sea and sand, waves washing against the edges of this place, and two demons tearing at one another, two demons perched upon the edges of two maelstroms, one always in flame and the other always in shadow."
"Yeah," Emmie says, still watching the scorpion and starting to feel sleepy. "I saw them. She showed me."
Light has begun to leak from the sand, streaming up from the designs traced into the dune, brilliant, blazing ribbons of crimson and turquoise, amber and tangerine, the deepest, purest blues Emmie's ever seen. The ribbons swirl and dip and swiftly rise, each color racing all the others, twining tightly together, then breaking free again. The scorpion has stopped waving its pincers about, and it vanishes beneath the sand.
"Forgive me," the woman says, stepping into the light. "I have grown too weary of my own secrets. I cannot carry these things forever." And the colors wash over and through her, and Emmie realizes that the ribbons aren't merely light and color, but music, too. Each hue a different voice, and together they form a vast and clamorous symphony, a drowning roar of harmony and discord that rushes forward only to shatter against some shore she can't quite see. The woman holds her arms up, and the ribbons of color and sound wrap themselves about her, playing her ebony flesh like the strings of a thousand guitars and violins and cellos.
"Close your eyes," she says; Emmie can hear her very clearly above the music, can hear her as clearly as if there were no music at all. "Quickly. There's only a little time left."
And Emmie doesn't ask what the woman means about time. She shuts her eyes, but the colors are still there, the colors and the music and the woman standing like a lightning rod planted at the center of it all. The night has disappeared, as have the stars and the moon and the sky, the desert and the dry, dusty smell. If I open my eyes, Emmie thinks, will the world come back, or has she taken it away forever? But then the ribbons of light are moving even faster than before, the music become a perfect cacophony; the black-skinned woman holds out a hand, and Emmie only hesitates a moment or two before she accepts it.
"The hounds tell a story about me," the black-skinned woman says. "They teach it to all the changelings, to keep them close, to keep them slaves, to make them afraid to even wish for what the Cuckoo has taken from them."
And then Emmie is standing somewhere else, somewhere just beyond the high Arabian dunes, the cracked and wind-scoured bed of a lake or inland sea dried up a thousand or a million years ago. There's a dead tree nearby, its charred trunk jutting from the heart of a roaring fire, and the body of a young girl hangs from its lowest limb, a noose pulled tight about her broken neck. Three naked women dance around the fire, and they aren't alone; there are other things dancing with them, feral, loping things with canine faces and bristling fur. The fire licks at the dead girl's feet and legs, and Emmie understands without having to be told—the girl hanging from the tree is the black-skinned woman, and it wasn't the desert sun that burned her dark as pitch. Then the three women become crows and fly away, but the ghouls continue dancing around the burning tree and the body of the girl. Their howls and coarse laughter fill the night.
"They call me a traitor," the woman says from somewhere behind Emmie. "They know it's not that simple, but find half a truth more useful than the whole."
"I'm sorry," Emmie says, not wanting to see any more of it, not wanting to hear the hounds or the wind or the hungry, crackling sound of the flames gnawing the girl's corpse. She would cover her eyes, but she knows this isn't something that she's watching with her eyes and so it wouldn't do any good.
"Look closer, Emmie Silvey, and see through the myth the hounds have fashioned here. Look deep and see all there is to see."
And she's about to say that she has already, and that she's seen more than enough of it, thank you very much, when the night ripples like a stone dropped into still water, breaking some subtle masquerade, an illusion Emmie hadn't even guessed at. There is not one tree. There are dozens of trees studding the floor of the dead sea, each strung with its own body, each rooted in flames. There are hundreds or thousands of dancers, and the sky is filled with crows.
"Oh," Emmie says, a very small sound coughed out like the breath being knocked from her lungs. "Oh."
"I was a traitor," the woman says, "but it was never the hounds that I betrayed."
Overhead, the murder of crows has begun to sing, cawing triumph in their brash crow voices.
"In those days, the ghul were growing weak, their powers diminished by the long war with the djinniyeh, and the Children of the Cuckoo had plotted together and appointed an hour when they would finally rise up, as one, against the hounds."
"But you," Emmie whispers, "you told the ghouls it was going to happen. You told them."
"I loved them," the woman says. "They were all I knew and all that I could comprehend. For all I understood, freedom was nothing more than another wild beast waiting to devour me, no different in my eyes from the jackals and the leopards. So, yes, I went to my mistress, and I warned her."
The air stinks of woodsmoke and searing flesh, and the wings of the crows batter the air until Emmie thinks even the sky will begin to bleed.
"But they killed you, too," Emmie says. "They killed all of you, just to be safe, didn't they? Just to be sure."
"Yes," the black-skinned woman replies, "all of us. Just to be sure. They had to be absolutely certain that they'd found out all the renegades. And then they rewarded my loyalty by naming me the sole conspirator, only me, one foolish girl child who'd defied them because she was selfish and weak and wanted what other children had. They feared what might happen if the truth survived that night, if their slaves knew that once upon a time all the children had turned against the Cuckoo. If changelings knew such a thing were even possible, it might happen again."
Emmie turns away, because there's nothing else she needs to see. "Why are you still here?" she asks.
"They wanted the alchemist to capture this moment, what they'd made of this moment, so that they wouldn't have to rely on mere words and imagination, so that the Children of the Cuckoo could each and every one watch my death again and again and again and all fear the same fate or worse. But time's slippery, Emmie, even for wizards. Even for those who can spin hours and days like spider silk. And I have long guarded this night against them. They will not have it, ever."
"Please send me back now," Emmie whispers, close to tears, and the shade named Esmeribetheda grips her hand and drags her free of the night of fire and crows and dancing ghouls. They stand together at the center of the drawing in the sand, their feet smudging and rearranging the countless grains into something new and necessary. Esmeribetheda holds Emmie close as the ribbons of color and music weave a bright lattice around them.
"What's coming," Esmeribetheda says, "it's a choice that you must make for yourself and for your own reasons and no one else's. But I needed you to see. I needed you to know this."
Emmie, bound in light and trilling crystalline notes, in the horror and sorrow of what she has seen and what the black-skinned woman has told her, feels the cold sinking into her again. The same cold she felt when Miss Josephine touched her forehead, and she knows what's coming and braces herself for the jolt. Esmeribetheda wipes the tears from Emmie's face and then pushes her all the way back to the yellow house on Benefit Street.
Soldier stands on the topmost of the foldaway stairs leading into the attic, and Hester, the brown girl, the Daughter of the Four of Pentacles, is sitting on her milking stool, exactly where Soldier first saw her. And just how long ago was that? Soldier thinks, the last thing she wants to ask herself, but asking it anyway, it and all the questions that come tumbling after it: Was that a long time ago, when I was a child? Was I a child a long time ago? Was it only three years ago, when I was a child?
The girl's holding an antique gold pocket watch, and its ticking seems very, very loud in the stillness of the attic. When the alchemist's daughter sees Soldier, she looks surprised, surprised and perhaps a little disappointed, but not alarmed.
"Oh," she says. "It's only you again. I thought perhaps it was someone else. I thought perhaps you'd died in Woonsocket."
"You were there, weren't you?" Soldier asks her. "You were there, and you helped me. You untied us. How—"
"Did I?" the brown girl smiles. "Well, then, that's my own business, isn't it? I had my reasons. Besides, you haven't even said thank-you."
Odd Willie mumbles something impatient from the stairs behind Soldier, and so she climbs the rest of the way into the attic. He follows, and she sees that he's drawn his gun.
"I really don't think you're going to need that," Soldier tells him and points at the .45.
"Yeah, well. Chance favors the prepared," he replies. "Isn't that what you said back at the museum?"
"Whatever," Soldier says and wonders if there's a way to pull the stairs up after them.
"How did you escape?" the brown girl asks. "I mean, how'd you get out past George Ballou?"
"I killed him. I think maybe I killed them all."
The Daughter of the Four of Pentacles nods her head and asks another question. "Did they catch her?"
"Did they catch who?"
The girl frowns and rolls her eyes, then slips the watch into a pocket of her black dress. "Emma Jean Silvey. Did they catch Emma Jean in the railroad tunnel? She has something that belongs to my father, and I would very much like to get it back."
"The kid's with us," Odd Willie says, staring up at the massive pine beams supporting the roof of the yellow house, half-hidden in the shadows far above their heads. "You know, it's a whole hell of a lot bigger up here than it ought to be."
"Well, I don't see her," the brown girl says.
"Miss Josephine wanted to talk to her," Soldier replies. "You remember me?"
"Of course I remember you. You're Soldier. You're the one the Bailiff brought to see me. The nasty little brat who dropped the wildebeest."
"That happened? That really happened?"
"Unfortunately, yes. That Noah's ark is my responsibility. I'm supposed to keep it safe. It was made in Italy by Signior Anastagio Baldassario Moratti in 1888."
"Is there any way to pull these stairs up from here?" Soldier asks her, and the Daughter of the Four of Pentacles sighs and stands up.
"What possible difference does it make?" she asks, straightening her dress. "Is someone chasing you, too?"
"Jesus goddamn bloody Christ," Odd Willie says and glances back down the stairs. "This kid's even more annoying than the other one."
"Can we shut the trapdoor?" Soldier asks again, starting to think maybe guns aren't such a bad idea after all.
"There's a crank over there, mounted on the wall. But don't turn it too fast or it gets stuck. And don't let go until you feel it catch."
"Thank you very fucking much." Odd Willie snorts, and Soldier squints into the attic gloom until she spots the hand crank.
"If we close it, how's Emmie supposed to find us?" Odd Willie asks her, and Soldier shrugs.
"Maybe it's better if she doesn't. Better for her, anyway."
Odd Willie shakes his head and runs his fingers through his hair, slicking it down flat against his scalp. "I swear to fuck, Soldier, I hope you're planning on telling me what the hell's going on someday real soon. 'Cause I'm getting awfully damned tired of trying to figure it all out for myself."
"Who are you, anyway?" the Daughter of the Four of Pentacles asks Odd Willie, and he snickers nervously and wipes at his nose.
The hand crank squeals and groans and pops, dry-rotted rope and rust and neglect, and Soldier half expects it to come apart at any moment, sending the stairs and the trapdoor crashing to the hallway below. The stairs are still half-extended, and now she's having to strain to turn the crank. "Odd Willie," she says between gritted teeth. "This is Hester—"
"Pearl," the alchemist's daughter corrects her.
"—Hester," Soldier grunts, "this is Odd Willie Lothrop."
"I see," the brown girl says. "I've heard stories about you from Barnaby."
"I never fucked a dead cat," Odd Willie tells her. "If Barnaby said that, he was lying."
"He said it was a live chicken. Either way—"
"—it's a goddamn lie. Soldier, if you shut that all the way, it's going to get awfully damn dark up here."
"You should have brought a lantern," the brown girl says. "Usually they bring lanterns when they come. It was silly of you to have forgotten."
"It was a whole lot sillier, girlie, that we didn't bring a ball gag and a pair of handcuffs," Odd Willie says and goes to help Soldier with the crank.
"Crap. It's locking up on me," she tells him. "The damned thing probably hasn't seen a drop of oil in at least a hundred years."
"Oh, not quite as long as all that," says the brown girl. "My father used it all the time, and I used it, too. He always kept it in good repair."
And Soldier thinks how much easier it would be to give up and let the winch start turning counterclockwise, letting the old rope un-spool, and when the stairs were down again, she could go back to Miss Josephine, could go all the way down to the basement and the tunnels and the ghouls. Then they could figure out what to do with Emmie Silvey, and Soldier would never have to face whatever's waiting for her in the attic. She could find a pint of cheap whiskey and save the truth for people who don't have to go looking for courage and resolve in a goddamn bottle. She could accept her fate and wash her hands of the whole mess, turn Emmie over to Madam Terpsichore and Master Danaüs. She belongs to them, anyway.
Just like me, she thinks. Just like Odd Willie. Then there's a dull crack and a fainter thump from somewhere inside the winch, and it stops turning altogether, leaving the attic stairs suspended halfway between up and down.
"Now you've gone and broken it," the Daughter of the Four of Pentacles says indignantly. "Do you break things everywhere you go, or only when you come up here?"
"Everywhere I go," Soldier replies and takes her hands off the crank handle; her palms are red and tingling. Well, she thinks, I guess that fucking well settles that.
"Piece of shit," Odd Willie says. "It was probably broken to start with."
"It most certainly was not," the brown girl snaps at him. "It worked just fine. It was right as rain, as long as you knew how to handle it."
"Soldier, I'm gonna fucking shoot her if she keeps this up much longer."
Soldier stops staring at her palms and glances at Odd Willie. "You can't. At least not until later on. She's the reason we're stuck up here. She's—"
"—had quite enough of the both of you," the brown girl says. "You're ungrateful and rude, and I don't think you even have permission to be up here. I don't think the hounds even know."
Odd Willie laughs and spits at the foldaway stairs. "Look, kid, I don't care what she says; either shut up or get shot."
The Daughter of the Four of Pentacles glares at him silently for a moment, her mouth hanging half-open in dismay. Then she narrows her eyes and crosses her arms. "You can't shoot me, changeling. No one can hurt me. Ever. There are rules here, Mr. Lothrop."
Odd Willie flips the safety off his gun, then flips it back on again. "Lately," he says, "I've noticed that the rules don't seem to mean a whole hell of a lot."
Soldier looks at the stairs, one last, exasperated glance before she gives up and turns to the alchemist's daughter. "Pearl," she says, "no one's going to shoot you. No one's going to fucking hurt you. But you need to try to be just a little more helpful. The more you cooperate, the sooner you'll be rid of us."
"I don't even know what you want."
"I want you to tell me what happened when I was here before, when the Bailiff sent me up here."
The Daughter of the Four of Pentacles uncrosses her arms and smiles a hard smile that's really more of a smirk. "Maybe I was going to tell you. But now I've changed my mind. Anyway, that's a secret," she says to Soldier. "That's a very important secret that I've promised never to tell anyone—especially not you—because there are rules."
"Then we have a problem," Soldier tells her. "Because me and Mr. Lothrop here are already standing in shit up to our fucking eyeballs, and if you can't—or won't—help me, Pearl, then I might as well let him put a bullet in your face and be done with it."
The brown girl stops smirking and sits back down on her milking stool. "You're even more unpleasant as a grown woman than you were as a child," she says to Soldier.
"Now, I'll buy that," Odd Willie snickers, and then he puts his gun away.
"How old am I?" Soldier asks the girl. "In Woonsocket, you said it wasn't time for that question, that it would come later. And here we are, later. So tell me, how old am I?" But the alchemist's daughter only shrugs and makes a show of twiddling her thumbs.
"That's not a simple question," she says.
"Answer it anyway. We've got the time."
"There's more than one answer."
"But I think you know the one I'm after," Soldier says, gazing into the shadows behind and above the brown girl, the immense attic of the yellow house stretching out before her like a half-remembered nightmare. "The last time I was here, you led me somewhere. I want you to take me there again. On the way, we can talk about what you were doing in Woonsocket."
The brown girl watches her for a moment, then smiles again, and something about that smile makes Soldier want to hit her, something precious and cold and calculated.
"Soldier, haven't you ever learned that you catch more flies with honey than with vinegar?" the girl asks. "Didn't the Bailiff teach you that? You should try using 'please' sometime."
"Fine," Soldier says. "Please, Pearl, will you take me wherever the hell you took me before?"
"Him, too?" And the Daughter of the Four of Pentacles points at Odd Willie.
"Yes, please, him fucking too."
"I don't like guns, Soldier. And I don't like being threatened with them, particularly not by men whose lives I've helped to save. I really don't like being around men like that."
Soldier turns and punches Odd Willie in the stomach as hard as she can. He yelps and doubles over, then sinks to his knees.
"There," she says. "He's very sorry that he pointed a gun at you. He's learned his lesson and promises he'll never do it again."
"You should know, I wasn't trying to help you," the brown girl says. "I was trying to help Emma Jean, and—"
"I don't need an explanation. I just need you to stop fucking around and do what I've asked you to do."
"I thought I should be clear, that's all," and then the alchemist's daughter gets up from her stool again and walks away into the darkness.
"Don't be such a goddamn pussy," Soldier tells Odd Willie, and he gags and calls her a cunt and tells her to fuck off. But she helps him to his feet anyway, and they follow the girl deeper into the attic.
Open your eyes, Esmeribetheda says, and it takes Emmie a moment or two to remember exactly how one does that, opens her eyes, because she's had them closed for so long. Ages, it seems, long ages as she fell through the wrinkled time and space and vacuum cold and starfire that lies between the black-skinned woman's desert and the attic of the yellow house. And it takes her a moment more to realize that Esmeribetheda hasn't traveled with her, that she's come alone into this musty, disorienting place of half-light and shadows. She sits down on the floor, sits down before she falls, her legs weak and trembling, and tries to figure out where she is and what she's supposed to do next.
"Can you still hear me?" she asks Esmeribetheda, but no one answers, only her own echo bouncing back from the darkness. Emmie blinks, realizing that the dim shapes towering around her are pieces of furniture and high rows of shelving. She calls out again, louder than before.
"Can you still hear me?"
This time her echo is more distinct and clearly repeats at least seven times.
"You can't, can you?" Emmie sighs, gazing up at the shelves. There are faintly glinting objects on them that she thinks at first must be fishbowls, row upon row of goldfish bowls of different sizes, until she remembers Pearl's snow globe. A sun inside a crystal sphere, the thing that Soldier used as a weapon to drive away or destroy a demon or maybe something even worse than a demon. And here, here there are a hundred spheres—no, a thousand. At least a thousand.
That's one of my father's later experiments, and he's extremely proud of it. He was careful to take a star none of the astronomers had ever seen....
Staring up at the shelves and all those glinting spheres, Emmie wonders just how many of them contain stars that Pearl's father decided no one would ever miss and what other things he might have trapped.
My father moved places and moments....
There's a dry, fluttering sound then, somewhere overhead to her left, and Emmie squints into the dark, hoping there aren't bats in this place, but fairly certain there are probably lots of them. She gets to her feet again, her legs feeling a little stronger now, and she spots the nub of a candlestick on a nearby table, a long table that's really more like a workbench, crowded with odd mechanical contraptions and glass containers, books and hardened pools of candle wax. She goes to the table and finds a big box of kitchen matches sitting on top of a very ancient-looking book bound in brittle leather; Emmie shakes the box and is relieved to hear wooden matches rattling about inside. The candle stub makes only a small pool of warm yellow light, but it's a welcome thing in this place. She sets the matchbox down a safe distance from the candle's flame, and now she can see the title of the book, Astronomicum Caesarium, stamped into the cover in dingy gold. Emmie begins to open it, disturbing several large silverfish, but then she hears voices and footsteps and looks up to see Pearl and Soldier and Odd Willie appear from behind one of the tall shelves.
Pearl furrows her eyebrows and points an accusing finger at Emmie. "How did you get up here? What are you doing with my father's things?"
"I was just looking at a book," Emmie replies, closing it quickly and stepping quickly away from the table. "That's all. I didn't touch anything."
"Liar. You lit that candle there," Pearl says, "so you touched that, and that means you must have touched the matches, as well."
"Her bark's worse than her bite," Odd Willie says and rubs at his stomach like someone with a bellyache. "Problem is, she never fucking stops barking."
"Shit," Soldier says, stepping past Pearl and walking towards one of the shelves. "This is it. This is the place you brought me before, isn't it? I was here."
"Don't you touch anything," Pearl warns her, instead of answering the question.
Odd Willie plops down on a moldering, threadbare settee, raising a thick cloud of dust, and something inside it cracks loudly.
"What was that?" Pearl says and turns away from Soldier to find Odd Willie coughing and fanning the dust away. "Be careful," she groans. "That belonged to my mother."
"Are we in the attic now?" Emmie asks and watches Soldier examining the crystal spheres on the shelf. "We are, aren't we?"
"We had to take the goddamn stairs," Odd Willie wheezes. "You must have found a shortcut."
"There are no shortcuts," the brown girl says emphatically.
"Oh, hell, there are always shortcuts," Odd Willie croaks and then starts coughing again.
Emmie starts to tell them about Esmeribetheda and the desert, about the burning trees and the crows who were really women (or women who were really crows), but then she decides maybe it's best if she doesn't. Just because she needed to know the secret doesn't mean that everyone else needs to know it, too.
"Miss Josephine showed me the way," she says, and Pearl glares at her suspiciously.
"Where is my father's experiment?" she demands. "You had it when the ghouls came and I lost you in the tunnel. You have to give it back, right this minute."
"She doesn't have it anymore," Soldier says, bending close to a particularly large sphere, one almost as big around as a soccer ball. "She gave it to me. And I can't give it back, because I don't have it anymore, either."
"Why not? Where is it?"
"Hell, maybe. Or Niflheim or Sheol," Soldier tells her. "If you believe in such places." Then she brushes her fingers across the dusty surface of the large sphere, and it begins to glow very softly.
"I told you not to touch anything!" Pearl shouts at her. "The three of you have done enough damage as it is."
"You came looking for me," Emmie says and walks over to stand beside Soldier. "You started this, Pearl. I certainly never wanted to run off in a snowstorm and hide in a tunnel and get kidnapped and chased by monsters."
"There's an entire island in this one," Soldier says.
"It's not an island," Pearl protests. "It's a small continent. My father hardly bothered with islands. That's a place called Lemuria. It's very important, and you're not to touch it again."
"Why did we come here?" Emmie asks Soldier. "Why did you want to come to this attic instead of going to see the Bailiff?"
"Because I think maybe I left something here once," Soldier tells her.
"Do you know what it was, the thing you left?"
"I'm starting to figure it out," Soldier says, and then she turns away from the sphere containing a small, stolen continent, and faces the brown girl.
"Why did you help her?" Soldier asks and nods at Emmie. "And why'd you help me so that I'd wind up helping her, too?"
Pearl frowns and mutters something to herself and picks at a loose thread on her black dress. Maybe she doesn't know, Emmie thinks. Maybe this is just another one of her father's experiments. Maybe we're all stuck inside one of those snow globes. Maybe we're sitting on a shelf somewhere, and her father is watching everything we do.
"You'll have to speak a little louder than that, Hester," Odd Willie says. "Me and Soldier here, we're a little hard of hearing. That's what happens when you play with guns. Next thing you know, you're deaf as a stone."
"I said that I have my reasons."
"The first time I saw you," Emmie says, "you told me you needed me to build a bridge for you. That's why you were helping me, isn't it? Because if they killed me, I couldn't build your bridge."
"You make it sound much more selfish than it is," Pearl tells her, and she picks up something that looks like a clarinet grafted onto a ship's sextant. "I probably would have helped you anyway."
"And what you took from me," Soldier says, "where is it? Where did you put it? Is it here, on one of these shelves?"
Pearl shakes her head, but doesn't say anything.
Emmie feels dizzy and shuts her eyes. This is where it finally ends, Deacon. This is where the story finishes, in the attic of this awful old house.
"You can't do what they thought you could do," Pearl says to Soldier. "I knew it all along, of course, but I had to acquiesce. I had to do whatever the Bailiff and the hounds told me to do. I didn't have a choice. They have my father."
"Soldier's not the bridge builder," Emmie says and opens her eyes, but the dizziness doesn't pass. "I am. I'm the one the hounds need to go back home, but they thought it was Soldier, didn't they?"
"The quadroon daughter of Saben White," Pearl says and smiles and fiddles with a tarnished brass knob on the sextant-clarinet thing. "They made a mistake, a serious miscalculation. If my father had been here, if they hadn't sent him away to Weir and exile, he could have shown them the error of their reckoning. He could have shown them it was you, Emma Jean Silvey, and you'd not have been shuffled off to be raised by that drunkard who is not your father, and none of this would ever have happened."
"Where is it?" Soldier asks her again, and then she takes a step towards the Daughter of the Four of Pentacles.
"Really, what difference does it make?" Pearl asks her. "Surely you don't think you can ever have it back? The Bailiff would never permit me to do any such thing. And the hounds—"
"How about you let me worry about the Bailiff and the hounds. Show me where you've put it, Pearl. Show me now. I'm tired of asking."
Emmie looks back at the crystal spheres lined up neatly on the shelf, the dust and cobwebs, all those imprisoned places and times and lives. And then she looks up at Soldier again, and she knows that the changeling woman isn't bluffing, that she'll kill Pearl, just like she would have killed all those people at the Cumberland Farms, just like she killed Emmie's mother and the creatures below Woonsocket.
"You won't hurt me," Pearl tells Soldier, her dark eyes bright with smug confidence. "You know the rules. You were raised by them, and you won't cross the hounds. You wouldn't ever dare cross the Bailiff."
"I already have," Soldier says very softly, calmly, and draws her pistol from the waistband of her jeans. "So, I hope you appreciate how little I have left to lose."
"Pearl, she's not kidding," Emmie says and steps in between Soldier and the alchemist's daughter. "Just do what she wants, and we'll leave."
"No. It's out of the question. I most certainly will not. My father—"
"Pearl," Emmie snarls, the same way she snarled at the fat kid who said her mother was a cat, snarling like the half-breed sire she's never met. "You want to go to your father. You want to be with him. That's why you saved me, because you believe that I can make the bridge you need to reach him."
Pearl stares back at Emmie and chews her lower lip, her smile faded and her eyes not so bright now. She slowly lays the sextant-clarinet contraption back down on the long table.
"Give her what she wants," Emmie says, "and I'll build a bridge for you so you can be with him again."
"Soldier, I can't let you do this," Odd Willie laughs, a flat laugh that Emmie can tell isn't really a laugh at all. He's still sitting on the old settee, but now his pistol's pointed at Soldier. "Whatever all this shit's about, I'm pretty goddamn sure you're right, and it's not about me, and I have to go on living with the ghouls and the Bailiff when it's over."
"Willie, you don't know—" Soldier begins, but Odd Willie cuts her off.
"Damn straight. And I don't fucking much care, either. I've stood by you this far, because I figured maybe I owed you something after Ballou and Saben and all that crazy shit. You probably saved my life. But, Jesus, this is the wizard's daughter, and I have a pretty goddamn good idea what'll happen to me if I just sit here and let you shoot her."
"You don't know," Soldier says again, not looking at him, her eyes still locked on Pearl, her finger on the trigger. "When I was five, that son of a bitch used to call me up from the tunnels and feed me sweets and ask me about my dreams. And I guess he didn't like what I told him, because one night he sent me up here and this...this self-important little shit, she took my childhood, stole it, just like her father used to steal bits and pieces of the universe and lock them up in all these things," and she motions towards one of the shelves with the hand that isn't holding the gun.
"Well, boo-fucking-hoo." Odd Willie snorts, and then he giggles nervously and wipes his nose again. "Look, I'm sorry, okay? Soldier, I'm sorry as a motherfucker, but I'm still not going to let you kill her. So put the goddamn gun away, 'cause it ain't gonna happen."
"Fuck you," she says, almost whispering now, and shoots Odd Willie in the throat. The attic is filled with thunder that swells and booms beneath the faraway ceiling, and Pearl screams. There's a spurt of blood from the soft spot below his larynx, and he looks surprised as the pistol slips from his hand and clatters to the floor.
"You're not that fast," Soldier says, and Odd Willie Lothrop slumps back against the settee.
Pearl is crouched on the floor, both hands clasped over her ears, and Emmie's head is ringing from the gunshot, the gunshot and the dizziness that's getting worse instead of better. She turns around, and Soldier's pointing the gun at Pearl again.
"If you kill her, you'll never find it," Emmie says, and the ringing's so bad that she can hardly hear herself. "You'll never get it back, not if she's dead."
"Maybe not. But you never know. You never know anything for sure. I might get lucky."
Emmie shakes her head, but the ringing won't go away, and she wonders if it ever will. She glances over her shoulder at Odd Willie sprawled on the settee. "Soldier, he was your friend. You were dying, and he carried you out of that place and took care of you."
"She's a changeling," Pearl sobs from the floor, her hands still covering her ears. "She's a liar and a murderer, a Child of the Cuckoo, and people like her don't have friends. She's a monster, Emma Jean, just like you."
"Pearl," Emmie says, and the room swings and tilts and she has to lean against the table to keep from falling. "Give her what she wants. Don't make her kill you."
Pearl crawls the rest of the way beneath the table, as though it might protect her, leaving little teardrop spatters of salt water and snot on the dusty floorboards.
"I'm only a little girl, Emma Jean. They've had me locked away up here forever and ever, locked up here alone without time or sunlight or anyone to talk to, but I'm still only a child, not an old woman, and I'm afraid of them! I'm afraid of what they'll do to me!"
"You really should listen to her, Hester," Soldier says and squats down so the gun's still aimed at Pearl. "I think maybe she's trying to save your miserable hide."
Pearl's crying so hard that Emmie can't make out whatever she says next, but apparently Soldier can, because she laughs and cocks the pistol again.
Maybe I'm dying, Emmie thinks, wishing that the room would stop rolling drunkenly about like the deck of a boat. Maybe when Esmeribetheda sent me back, she did something wrong. Maybe she messed something up inside my head, and I'm dying. And the thought doesn't frighten her like she's always imagined it should, because maybe when she's dead there will be no one and nothing at all, and she won't even have to bother trying to forget all the horrible, impossible things she's seen in the last six days.
"I'm going to ask you one last time," Soldier says, and that only makes Pearl cry that much louder.
"Stop it," Emmie says, and for a second she wonders if she means the swaying attic or if she means Pearl and Soldier. I'm going to puke, she thinks, but then the nausea passes, and Soldier's looking at her instead of the alchemist's daughter.
"Pearl's right; we're just children," Emmie says, sitting down, holding tightly to one of the table's legs. "Just kids. That's all we are, all three of us. Sure, I'm too weird and too smart and my eyes are yellow, and Soldier's too old, and Pearl's not nearly old enough, but we're all three still just children."
Soldier glances at the gun in her hand, then back at Emmie. "If you have a point—"
"That is my point, Soldier. Can't you see? That is my goddamn point," and she rests her face against the leg of the table. The nausea's back again, worse than before, and she's trying to decide which of the two Soldiers she's seeing is the real one. "Pearl, please, if you still have it, give it back to her. It belongs to her. Give it to her, and I'll build your bridge for you."
"You don't understand," Pearl sobs. "They'll find me anyway. They'll come across the bridge, too, or the ghouls that live in Weir will kill us both, me and my father."
Soldier, both of her, sits down on the floor, halfway between Emmie and Pearl, and looks down at the pistol again.
"Please," Emmie begs the two Pearls huddled under the table. "I'll tear the bridge down once you're across. No one will follow you, I promise. Cross my heart—"
"Hope to die?" Soldier asks, and then she laughs again and rubs at her forehead with the barrel of the gun.
"Yes," Emmie says. "Yes. Cross my heart and hope to die. Give it back to her, and then I'll build your bridge, and then, when you're across, I'll tear it down so no one can come after you or your father."
"They made me do it," Pearl whimpers. "They can make you do things, too, Emma Jean."
"No, they can't. I have a secret place to hide where even they can't follow." Emmie only half understands the things she's saying, but she says them anyway. "I can go there, if I have to. Give it to her, Pearl. Give Soldier what you took from her." And then there are no more words, nothing left to say, and she wants to shut her eyes because she's tired of the tilting and there being two of everything and everyone, but she doesn't because Deacon always said that closing your eyes only makes nausea worse. And he ought to know, she thinks and hangs onto the table leg.
"I'm sorry," Pearl says to Soldier, and she reaches into the pocket of her dress, the same pocket where she keeps the gold watch. She takes out a very small thing, round and smooth and glassy, something the color of buttermilk and no larger than a marble. "I knew how wrong it was, but I was so afraid." And she holds it out to Soldier.
"That's it?" Soldier asks, and Pearl puts the marble into her hand.
"There wasn't much," she says and wipes at her eyes. "You weren't yet even six years old."
Soldier stops rubbing at her forehead with the pistol and rolls the buttermilk marble back and forth in her palm. "What the hell am I supposed to do with it?"
"That's the easy part," Pearl tells her. "Just put it beneath your tongue."
"I think I'm going to pass out," Emmie says, but the words sound so strange, having been said, that she decides she's probably wrong. Anyway, neither Soldier nor Pearl seems to have heard her.
"Put it under my tongue?"
"Yeah," Pearl replies and makes a wet, sniffling noise. "Time will do the rest. My father used to say that time is always trying to repair itself, and that's why it's so hard to change the past."
"That doesn't make sense," Emmie says, trying to sit up straighter. "You just contradicted yourself."
"I'm only telling you what he said. I can't help it if you don't understand." And then Pearl starts weeping again.
"Try it," Emmie tells Soldier. "Put it under your tongue and see what happens."
"What the fuck," Soldier says, and stares at the marble a few seconds more before slipping it into her mouth.
"Emma Jean Silvey, you've promised me." The two Pearls sniffle and blink their four bloodshot eyes at Emmie. "You've promised to build a bridge for me. You've crossed your heart and hoped to die."
"As long as I don't pass out first," Emmie says and lets her head rest against the table leg again.
Soldier tries to say something, but the words come out all wrong because of the marble beneath her tongue.
"Don't swallow it," Pearl tells her. "If you swallow it, it'll only make things worse."
How could things possibly be worse? Emmie thinks, and then Soldier opens her mouth, as if she's about to cry out, and there's a brilliant burst of blue-white electricity, a lightning flash and the throaty rumble of thunder right behind it, real thunder this time, and it's nothing like the rumbling noise the gun made when Soldier shot Odd Willie. Emmie turns her face away, and Pearl screams again, and then Emmie can feel the time flowing around her, flowing from everywhere all at once towards Soldier—the smallest fractions of seconds and countless interminable hours and days that never seem to end.
"You promised me!" Pearl shouts above the din of the storm raging around and inside Soldier, the storm that Soldier has become. I did, Emmie thinks. I did promise, and I have no idea how I'm supposed to build a bridge. But then it comes to her, just the same way she realized that she'd always known how to play a piano, even though no one had ever taught her. The same way she sometimes knows the right answers before people even ask her the questions. It's just something that's there, buried deep inside her and wanting to escape, and Emmie doesn't try to stop it...
...and in his big empty house on Angell Street, the house that Sadie bought for him and Emmie after her first novel sold, Deacon Silvey feels something unpleasant in his sour, alcoholic's belly, something that isn't quite a chill, but isn't anything else he could ever name, either. He sets down the pint bottle of rye whiskey that he's been working on since just after noon and glances at the clock mounted above the kitchen sink. At first, he's not sure which way the black second hand is moving—forwards or backwards—and he tries to recall the last time he got so drunk that he hallucinated.
She's never coming back to me, he thinks. Not after five days. After five days, even the fucking cops know she's never coming back.
The clock ticks forward, and the February wind brushes roughly past the kitchen windows. He wants to cry, but there are no tears left in him. Everything cried out and nothing left inside but whiskey and helpless anger and sorrow and regret.
You're an old man now, Deacon Silvey, the wind mutters. Too old to fight the monsters anymore and too old to find lost children. Too old to do much of anything but get drunk and wish it hadn't all turned out this way.
There's no point arguing with the wind. Only madmen argue with the wind. So he takes the bottle of rye over to the kitchen table and sits down. He thinks about the open cardboard box in the bedroom, the box Emmie must have dragged out from under the bed, all its treasures scattered shamefully across the floor. He wouldn't let Sadie pack those things away again.
I'll call Sadie, he thinks, wishing she were still there with him, there to make him stop drinking, there to keep him from giving up, wishing she hadn't taken the train back to New York the night before. But she'd already given up herself. And she's gone too. And she's not coming back, either.
You're an old man, the wind laughs. You're an old man, and now you're alone. Better start getting used to it.
And Deacon pours himself another glass of whiskey and tries to ignore the cold and vacant feeling coiled in his stomach, the feeling that says the wind is always fucking right, any way you look at it, and he watches the clock eating up the day...
...and Madam Terpsichore pauses on the creaking wooden steps leading up from the great basement of the yellow house on Benefit Street. It's been at least a hundred years since she last climbed those stairs, a long red century since she had any cause to venture from the security of the burrows and tunnels and secret underground meeting places and enter the house itself. She sniffs anxiously at the air, flaring her nostrils, licking her mottled lips, and catches an unfamiliar scent slipped in amongst the comforting cellar smells. Merely a spicy, unexpected hint woven almost imperceptibly between mold and old rot and the pungent white mushrooms that grow in fleshy clumps beneath the stairs.
"Too late," she growls very softly and looks back down at Madam Melpomene, standing at the foot of the cellar stairs. "It's already begun. It's already finished."
Madam Melpomene bares her crooked yellow fangs and scratches furiously at the hard-packed earth with a long thumb claw. "Then we have failed," she says. "We should have killed the wizard's bitch long ago and been done with it. She would have been sweet, at least."
Madam Terpsichore grunts, neither agreeing nor disagreeing, and stares at the basement door, still another dozen steps away. She tries to push back the dim memories of that other world, that place of deep, wooded valleys and steep white mountains rising to scrape the undersides of flickering violet clouds. The towers of slate and granite built a million years before her birth, decaying beneath a dying and indifferent sun. The home her race fled millennia ago, scattering themselves between the stars, coming at last to Weir and Earth and a dozen other alien worlds.
"We've been betrayed," Madam Melpomene says and sighs and then goes back to scratching at the dirt floor.
And Madam Terpsichore, High Dame of the Providence warrens and the last of the ghul alive who even half remembers the passage from that world or the subsequent war with the djinniyeh in this world, slowly turns and begins making her way back down the noisy basement stairs...
...and in her bottom bunk in the warrens below College Hill, the changeling girl named Sparrow Spooner wakes from a dream of the father she'll never have. Mr. H. Elgin Higginson, late of 7 Thomas Street, the witch who lied to her and used her, who never really loved her, who only pretended that he would carry her far away from the Cuckoo and the hounds. The man who cost her two fingers before he was fed alive to Madam Terpsichore's students, before the ghoul snatched his escaping soul and devoured it. Sparrow Spooner lies still in the darkness, surrounded by all the other stolen children and the ghul pups sleeping all around her.
She's trying to find whatever it might have been that woke her, whatever interrupted the familiar, well-worn dream of Mr. Higginson buying her a Dell's lemonade and walking with her under a summer sun while he talked about all the strange and distant places they'd visit together, places like Hollywood and Miami Beach. There was something, a ripple, a wrinkle, the faintest stutter between one moment and the next. Something not quite déjà vu that almost anyone else would have missed, but not Sparrow Spooner, who was first in Discord and Continuity, after all, and she thinks it was not so very different from what she felt that rainy night three months ago when she stood trial and her masters and mistresses spared her life but took her fingers...
...and the Bailiff, who is not the god of men and churches or the Cuckoo or a demon or even much of a magician, who has never been anything more than the hounds needed him to be, sits with the cooling bodies of his murdered harem boys. He finishes wiping blood from the blade of the kukri onto one sleeve of his lime-green bathrobe, and glances up at the dais and the doorway connecting the octagonal chamber with the old museum in Roger Williams Park. It's been almost two hours since he last spoke with Soldier, since he appeared to her in a convenience store and she told him he looked like a circus clown. He was so, so sure that she would come, even after Woonsocket, that Soldier would bring him the child of Saben White, and when the girl was dead and crazy Willie Lothrop had taken care of Soldier, the hounds would never have been the wiser.
"Surely," he says, gazing into the dead blue eyes of one of the boys, "you cannot blame a man, not merely for trying to outwit his destiny."
The boy doesn't reply, though the Bailiff half expects him to.
"Would they spare me?" the Bailiff asks. "If I carved the lot of you up nice and neat as a butcher's window and delivered you ripe and raw, might they at least let me keep my eyes, my tongue, my hands? Might they at least spare my sorry life?"
A sudden breeze stirs the folds of silk and muslin strung from the ceiling, and the Bailiff shivers and pulls his bathrobe closed. He's unaccustomed to breezes in this place, sudden or otherwise. He glances at the dais again, an empty stage readied for players who will never now appear, and then, because time is precious, he begins the task of skinning and dressing the bodies...
...and in her apartment on St. Mark's Place, Sadie Jasper stands inside the sacred circle she's cast upon her bedroom floor. She holds the double-bladed athame in her good hand, her power hand, and strains to see a cleansing blue-white flame dancing about the tip of the dagger. She points the athame down and to the east, then begins walking clockwise, following the inner perimeter of the circle.
"I consecrate this circle to the Dark Mother," she says, her voice trembling, but she's sick of crying, sick of tears and blame and fear, and she takes another step. "To the Dark Mother and the Lord of the Hunt. Here may they manifest...."
There's a noise from the hallway, then, a sound like wind chimes or the sea on a summer's day, and she takes a deep breath and draws a circle in the air with her dagger. "Here may they manifest," she says again, "and bless their child."
She's gone, a hard voice whispers behind her eyes. He asked you to stay and watch her, but you were too afraid, too angry, too weak. You were weak, and you left her there alone, and now she's gone forever.
"This is a time that is not a time," she says, "in a place that is not a place, on a day that is not a day. I stand here at the threshold between the worlds, before the veil...."
From the hall, a sound like distant thunder.
And the voices of lost children.
And the careless, calculated shattering of one moment against the next.
Sadie takes another halting step, but now she's forgotten the protection ritual, the holy words that have never before felt so completely powerless to stand between her and all the real and imagined evils of the universe. Another step and she stumbles and breaks the unfinished circle, as the walls of her bedroom collapse and melt into steam and she finds herself standing on an immense bridge built from fire, a burning, writhing span above some chasm too vast and too deep to even comprehend.
Ahead of her, a young girl with black hair and almond skin is crossing the bridge alone, and she turns and looks back towards Sadie. Pearl, Sadie thinks. Her father calls her Pearl. And the girl's eyes sparkle in the light of the burning bridge. Sadie looks over her right shoulder, then, peering down the bridge into a dark place, a place of shadows and secrets and lies, and there's Emmie, and Sadie can see where the bridge begins.
Then, all of a moment there was a rending of the blue wall (like a curtain being torn) and a terrible white light from beyond the sky, and the feel of Aslan's mane and a Lion's kiss on their foreheads and then...
"They might take us to other places," Emmie said.
"Good places or bad places?" asked Sadie.
Emmie shrugged. "That remains to be seen," she said, as the bridge fades and Sadie finds herself in her bedroom on St. Mark's Place again, standing inside the ruined circle, the athame gripped tightly in her good hand. Out on the street a car horn blares, and Sadie sits down on the floor. When the tears come, she doesn't fight them...
...and when it's over, there's only a very small burn, hardly larger than a penny, perfectly centered between Emmie's yellow eyes, and the dizziness has gone and taken the nausea away with it. Pearl's gone, too, and the air in the attic of the yellow house smells like ozone and blood and dust.
"Is it done?" the little girl named Soldier asks, the child who chose her name from a headstone in Swan Point Cemetery. The gash on her cheek is gone, as are all the other scrapes and bruises, and her skin doesn't look sunburned anymore. Already, her memories of everything that's happened since the Bailiff led her up to the attic have begun to dissolve, coming apart like a cube of sugar in a cup of tea. "Was it that simple?"
Emmie gently touches the burn on her forehead, her mind still filled with the heat of the inferno and the ice of all the not-quite-empty space between worlds, with Pearl's joy and the surprise of seeing Sadie standing there on the bridge, looking back at her.
"Do you think she'll find her father?" Soldier asks. The clothes and boots that Odd Willie brought her from the Salvation Army store in Uxbridge are far too large for a girl who's not quite six years old, and she looks like she's been caught playing dress-up.
"I think she already has," Emmie says and stands up, holding on to the edge of the table, just in case the dizziness decides to come back. "I think he was waiting for her on the other side." She's not entirely sure if that's the truth, but it seems like the right thing to say, here at the end of the story, and even if it's not true, she doesn't think that it's a lie.
"What do we do now?" Soldier asks her and kicks off the ridiculous cowboy boots. "I don't think we should stay here."
"No, we definitely shouldn't stay here," Emmie replies and tries to imagine what comes next. "I have to go home," she says. "I have to go back to Deacon and Sadie."
"And I shall go back down to the hounds. Is that what I'll do, Emmie Silvey?"
"Will they hurt you?"
The child with Soldier's green eyes picks up the pistol lying on the floor, and Emmie immediately bends down and takes it away from her.
"No, I don't think they'll hurt me. I think it will take them a while to understand exactly what's happened, and then, then..." and Soldier trails off and stares at her hands, as if the pistol has left some stain behind that only she can see. "I won't tell them," she says to Emmie. "I won't ever tell them that you're the bridge builder."
"Thank you, Soldier. But do you think they'll let me leave?" Emmie asks her and lays the pistol down on the table beside the clarinet-sextant contraption. "Do you really think they'll let me go home now?"
"I don't know," Soldier says after a moment.
"Then I guess that's what happens next. I guess we find a way out of this place and see if they'll let me go home to Deacon."
"My father," Soldier says and frowns.
"He's my father, too," Emmie tells her and helps Soldier up off the dusty attic floor. When she stands, the too-big jeans slide right back down, and Soldier steps out of them. Fortunately, the sweater reaches all the way to her knees, and Emmie helps her roll up the sleeves so they don't flop down over her hands.
"Maybe I can meet him someday," Soldier says. "Maybe when I'm grown, maybe one day then."
"I think he'd like that," Emmie tells her, and she takes Soldier's left hand, and together they cross from the alchemist's workshop into the greater gloom of the attic, leaving behind the tall shelves of glass spheres filled with stolen places and Odd Willie's corpse and the pawnshop 9mm that killed him. They get lost only once, in a twisting, roundabout maze of dollhouses and dressmaker's dummies, before they reach the milking stool and trapdoor and the foldaway stairs. Miss Josephine and another silver-eyed woman, whose name is Adelaide, are waiting for them there, and at first Emmie's afraid of the pale, waxen women in their high collars and black mourning gowns. But then Miss Josephine smiles and tells her there's a taxicab waiting outside, a driver to take Emmie home to Angell Street, and the four of them go down the stairs, and Adelaide closes the trapdoor.
"I'll try not to forget you," Soldier tells Emmie, though she's already having trouble recalling exactly who the yellow-eyed girl is or when they met.
"A shame about Master Lothrop," Adelaide says. "I expect he'll be missed Below."
"Who?" Soldier asks and blinks up at the vampire.
"Never mind," Emmie says. "It doesn't matter anymore," and then she remembers the gold ring, the ring from the box beneath Deacon's bed, Soldier's mother's wedding ring, and takes it out of her pocket. "I want you to have this," she tells Soldier. "Maybe it'll help you remember me."
"It's pretty," Soldier says, and Emmie lays the ring in her palm.
"What's that?" Adelaide asks, bending close for a better look. "It's rather plain," and then Miss Josephine tells her to be quiet.
"Maybe you'll look at it and dream about me sometime," Emmie says, and then the two women with mercury eyes lead them downstairs.
Except the smaller size, no Lives are round,
These hurry to a sphere, and show, and end.
The larger, slower grow, and later hang—
The Summers of Hesperides are long.
—EMILY DICKINSON
## EPILOGUE
## April
Upstairs in Deacon Silvey's big gray house on Angell Street, Emmie lies in her bed, listening to the rain drumming hard against the roof and the bedroom window. She was dreaming of the desert again, the desert and the black-skinned woman, and then something woke her, probably the rain or a thunderclap. She isn't surprised to find that her father's sitting at the foot of the bed, watching her. He does that a lot these days.
"Were you having a bad dream?" he asks. "You were talking in your sleep."
"I was dreaming," she replies, "but it wasn't a bad dream."
Deacon scratches his chin and nods his head.
"Why aren't you asleep?" she asks him.
"I was, but then the storm woke me, and I thought maybe I should come up and look in on you."
"I'm not afraid of thunderstorms," she says, though he already knows that.
"Yeah, well...maybe I am. Anyway, no harm done."
"No harm done," Emmie says sleepily and rolls over to watch the spring rain streaking the window. The dream is still fresh, and she thinks about taking out the journal that Sadie bought her and writing down all of it she can remember. Sadie says her dreams are magick, the most powerful magick that Sadie's ever seen, and that one day they'll help her to become a very powerful witch, if that's what she wants. Deacon says they're only dreams and to ignore Sadie whenever she starts talking like that, but even he thought the dream journal was a good idea.
"Do you want me to go away?" Deacon asks.
"No," Emmie says. "Sit with me until I fall asleep again," and he says that he will.
I'm much too sleepy to write anything down, Emmie thinks, hoping that she'll still remember the dream in the morning. Climbing up and over the dunes with Esmeribetheda, guarding the memories of the dead, and finally they'd come to a rocky place looking down on the sea.
"They have their suspicions," Esmeribetheda said, and Emmie knew that she meant the ghouls. "They are watching you, always, Emmie. One day, when you're older, they may come for you."
"I won't help them. I already made that decision," Emmie told her and sat down in the shade of a huge red rock. There was a lizard beneath the rock, and it winked at her and then crawled quickly away.
"You may have to make it again."
"That doesn't seem fair," Emmie told her.
"No," the black-skinned woman said. "It doesn't seem fair at all."
And then Emmie woke up to the storm and Deacon at the foot of her bed.
"I should leave," he says. "And you should get back to sleep, kiddo. Tomorrow's a school day."
"It's already tomorrow, Deacon."
"My point exactly," he says. Before he goes, Deacon kisses her on the forehead, presses his rough lips to the small pink scar between her eyes. His breath doesn't smell like beer or whiskey; it hasn't for a long time, now.
"You want me to turn Doris Day back on?"
"No," Emmie says. "That's okay. I think I just want to hear the rain for a while."
"Yeah, the rain's good," Deacon Silvey says, and then he leaves her, and Emmie watches the storm at the window until she finds her way back down to sleep and unfinished dreams.
## About the Author
Caitlín R. Kiernan has written seven novels, including Threshold, Low Red Moon, and Daughter of Hounds, and her short fiction has been collected in four volumes—Tales of Pain and Wonder; From Weird and Distant Shores; To Charles Fort, With Love; and Alabaster. Trained as a vertebrate paleontologist, she lives in Atlanta, Georgia.
Web sites: www.caitlinrkiernan.com
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Els Entorns Personals d'Aprenentatge (PLE, per les seves sigles en anglès Personal Learning Environment) són el conjunt d'eines, aplicacions, serveis web, relacions i/o interaccions que una persona utilitza per a gestionar el seu propi procés d'aprenentatge. Aquests sistemes ajuden als estudiants a prendre el control i gestió del seu propi aprenentatge. Això inclou:
El suport a l'alumnat per a fixar els seus propis objectius d'aprenentatge.
Gestionar el seu propi aprenentatge.
La gestió dels continguts i processos.
Comunicar-se amb altres en el procés d'aprenentatge.
Crear i mantenir una xarxa de contactes amb els quals compartir els aprenentatges. (Xarxa personal d'aprenentatge, PLN).
Context històric
Gràcies al ràpid desenvolupament tecnològic cada dia naixen tecnologies, eines i serveis que obren un món de possibilitats de comunicació i aprenentatge que permet accedir a un coneixement personalitzat.
Paral·lelament a la tecnologia, el desenvolupament de noves teories de l'aprenentatge evolucionen amb el cognitivisme i constructivisme com a base teòrica. D'una banda el cognitivisme i la teoria de l'aprenentatge significatiu el qual assumeix que l'aprenentatge es produeix a través de les connexions entre els individus, els nodes d'informació i les estructures. Aquestes connexions s'estableixen gràcies a les noves tecnologies. I d'altra banda el constructivisme i la teoria de la bastida, o scaffolding en anglès, on els aprenents van progressivament guanyant autonomia d'aprenentatge mitjançant la disminució progressiva de l'actuació del mestre o tutor. Aquests canvis en les formes d'aprenentatge donen com a resultat un canvi en el paper de l'alumne que es converteix en un subjecte actiu i proactiu en el procés d'aprenentatge que autogestiona les seves pròpies eines educatives.
Com a resultat dels esmentats canvis tecnològics, pedagògics i socials l'alumnat nadiu digital dins de la comunitat educativa requereix una actualització del procés d'ensenyament-aprenentatge cap a un entorn més motivador i adaptat al seu nou context educatiu, social i tecnològic el que s'ha traduït en l'aparició dels Entorns Personal d'Aprenentatge.
El 2005, Dan Buckley va definir dues finalitats de l'aprenentatge personalitzat, per una banda "personalització per a l'estudiant on el mestre guía l'aprenentatge i per l'altra "personalització pel estudiant" on l'estudiant desenvolupa habilitats per organitzar el seu propi aprenentatge. La idea principal, al voltant de la qual es desenvolupa el PLE, és el desenvolupament d'un entorn per a la provisió de recursos d'aprenentatge i serveis per a l'estudiant. Amb una filosofia centrada en l'alumne, es diferencia de la primera generació de les tecnologies d'aprenentatge en el fet que aquelles primeres se centraven en la creació d'objectes d'aprenentatge. Aquests objectes d'aprenentatge es van agrupar i distribuir mitjançant Sistema de gestió de l'aprenentatge, anomenats Learning Management Systems (LMS) en anglès. El terme entorn personal d'aprenentatge sorgeix al Regne Unit en relació al moviment de la Web 2.0 i orientat cap al sistema educatiu. Més concretament, el terme va aparèixer en la congrés anual de Joint Information Systems Committee (JISC) el novembre de 2004. Des d'un principi, va néixer com una serie de pràctiques que no són ni homogènies ni compatibles, el propi terme "personal" es converteix en un terme que dona a diverses interpretacions
Components d'un PLE
Un PLE s'organitza al voltant de les eines que permeten l'accés i la relació amb la informació i amb altres persones, més concretament amb les eines "socials" de la Web 2.0 i de les estratègies amb les quals es configura l'ús d'aquestes eines socials Castañeda i Adell (2010). Els PLE permeten integrar tant les tecnologies i les eines com els processos i les pràctiques. Quan una persona aprèn desenvolupa qualitats metacognitives que li permeten fomentar l'enfortiment de les competències digitals.
Els PLE no són homogenis, presenten variacions i la confluència d'aquestes dimensions, que involucren des de l'accés a recursos fins a la presa de decisions, varia al llarg de la vida, donant així al PLE un caràcter dinàmic. No obstant això, Castañeda i Adell, tenint en compte les idees expressades per Attwell el 2007, entenen que un PLE bàsic està format al voltant d'aquelles eines que permeten tres processos cognitius bàsics: llegir, reflexionar i compartir. A continuació, hi ha una descripció detallada de cada apartat i un exemple de les diferents eines, mecanismes o activitats que s'inclouen en cada part.
1. Eines, recursos o fonts d'informació per llegir: fa referència al conjunt d'eines que utilitzem per comunicar-nos amb altres persones. Són els llocs i els mecanismes mitjançant els quals les persones troben informació, ja sigui de manera habitual o excepcional i en diferents formats.
Eines: blogs, wikis, newsletters, canals de vídeo, pàgines web, llista de RSS, etc.
Mecanismes: recerca, curiositat, iniciativa, etc.
Activitats: conferències, lectura, revisió de titulars, visualització d'audiovisuals.
2. Eines, recursos o fonts d'informació per reflexionar: fa referència al conjunt d'eines que permeten transformar la informació. En aquest component del PLE, s'integren les eines i espais en què s'utilitza la informació aconseguida, en les eines utilitzades en el punt 1; són els llocs en els quals es dona sentit i es reconstrueix el coneixement a partir de la reflexió sobre aquesta informació. És en aquests llocs on es reelabora i publica la informació aconseguida.
Eines: blocs, wikis, quaderns de notes, canals de vídeo, presentacions visuals, pàgines web, etc.
Mecanismes: síntesi, reflexió, organització, estructuració, anàlisi, etc.
Activitats: creació d'un diari de treball, fer un mapa conceptual, publicar un vídeo propi, etc.
3. Eines, recursos o fonts d'informació per compartir: fa referència a la PLN (Xarxa Personal d'Aprenentatge) que és el conjunt d'eines, processos mentals i activitats que permeten compartir, reflexionar, discutir i reconstruir coneixements amb altres persones, així com les actituds que propicien i nodreixen aquest intercanvi. Les PLN es poden dividir en tres categories en funció del propòsit. Primer hi ha les eines que permeten aprendre o descobrir sobre una cosa, com ara FLickr, SlideShare o Youtube. Després hi ha les eines en què el propòsit es compartir alguna cosa, com ara Twitter, Diigo entre altres. I finalment, hi ha aquelles eines que serveixen per establir relacions amb altres persones com ara Facebook o LinkedIn.
Eines: eines de programari social, seguiment de l'activitat en xarxa, llocs de xarxes socials.
Mecanismes: assertivitat, capacitat de consens, diàleg, decisió, etc.
Activitats: trobades, reunions, fòrums, discussions, congressos, etc.
Primers passos per crear un PLE
A l'hora de crear un PLE s'haurà de tindre en compte que aquest ha de proporcionar eines i serveis que ens permeten l'accés i la relació amb la informació i amb altres usuaris, en concret les eines socials de la Web 2.0. Per tant, la creació del PLE es farà al voltant de tots aquells recursos que permeten als usuaris informar-se, reflexionar i compartir. D'aquesta manera, s'inclouran 3 tipus d'elements:
Eines i estratègies de lectura que proporcionen una font d'informació. Poden ser mitjans online, biblioteques digitalitzades, blogs, entre d'altres.
Eines i estratègies de reflexió que ajuden a comprendre i interioritzar la informació rebuda, així com a tenir-ne una visió crítica. Com poden ser Blogger, WordPress, Prezi, Powtoon.
Eines i estratègies de relació que permeten compartir la informació obtinguda i reflexionada amb altres usuaris, i de les que s'aprèn. És aquí on es crea la red personal d'aprenentatge (PLN, Personal Learning Network). Podem connectar-nos amb altres usuaris a través de les xarxes socials, Twitter, Facebook o LinkedIn; fòrums; entorns col·laboratius, com els que facilita Google, o altres eines com Skype.
Una proposta concreta de creació de PLE seria aquesta:
Crear un compte Twitter i seguir gent interessant.
Compartir les teves idees amb ells.
Subscriure't via RSS a blogs i/o altres fonts d'informació.
Començar a escriure les teves idees i projectes en el teu propi blog en Blogger.
Etiqueta i comparteix les idees i projectes preferits via serveis de marcadors socials com Memorizame.
Participa en algunes xarxes socials com Facebook o LinkedIn, per tal de compartir mitjançant enllaços socials per configurar la PLN i compartir les experiències.
Gestionar tota la informació mitjançant Symbaloo, Netvibes,...
Alguns exemples de PLE:
Eines 2.0 i 3.0
L'establiment del Web 2.0 va ser un punt d'inflexió en la interacció humana a través d'Internet, el foment de les xarxes socials i el desenvolupament d'entorns més personalitzats per a finalitats educatives. A Internet podem trobar multitud d'eines informàtiques que ens permetran crear i mantenir el nostre entorn personal d'aprenentatge. Es tracta doncs d'una elecció personal en funció de les nostres necessitats i valorant aspectes com la seva utilitat o facilitat d'ús. En el món digital es troben diferents tipus d'eines segons la seva utilitat.
Per l'edició Multimèdia: Totes aquelles eines que serveixen per editar vídeo, foto, àudio, vectorització d'imatges, creació de mapes d'imatge, creació d'animacions Gif o redimensió d'imatges.
Per la creació de Blogs i Llocs web.
Les Xarxes socials; per contactar amb altres persones, serveixen de cercador de continguts d'interès, microblogging, com a gestor de canals RSS, marcador de continguts d'interès.
Per emmagatzematge i transferència d'arxius.
De presentacions.
Enquestes i sondejos.
Publicació digital.
Conversió d'arxius i descàrrega de vídeos.
Eines de col·laboració.
Gràfics i diagrames.
Capturadors de pantalla.
Cercadors.
Marcadors socials: que permeten emmagatzemar, classificar i compartir adreces d'interès amb altres usuaris d'una manera organitzada mitjançant etiquetes.
Així doncs, les eines d'un PLE són tot aquells elements tecnològics que ens permeten buscar, classificar, elaborar i compartir informació i coneixement Aquestes són algunes de les eines més utilitzades per a la gestió d'un PLE:
Blogs/pàgines web: Blogger, Tumblr, Twitter, WordPress
Citacions: EndNote, RefWorks, Zotero
Eines d'emmagatzematge i col·laboració: DropBox, Google Drive, Google Docs, Padlet , SlideShare
Marcadors socials: Diigo, Evernote, Instapaper, Pinboard
Missatgeria instantània: Snapchat, Telegram, WhatsApp
Notícies: RSS, Reddit
Presentacions: Powtoon, Prezi, Screencast-O-Matic, VideoScribe, VoiceThread
Traducció: Google Translate
Vídeo, àudio, imatges: Flickr, Instagram, Pinterest, YouTube, Vimeo
Xarxes socials: Facebook, LinkedIn
Altres: ClickEdu, Viquipèdia
Tot i que els PLE s'han basat inicialment en eines Web 2.0, la integració de les eines Web 3.0 tant en els PLE com les PLN poden ajudar els estudiants a millorar, establir i assolir els seus objectius d'aprenentatge. Les eines Web 3.0 són tecnologies intel·ligents com la intel·ligència artificial, el web semàntic, el Web Geoespaial o el Web 3D. S'ha observat que la personalització del procés d'aprenentatge dels estudiants que aprofiten l'ús de la xarxa semàntica social, utilitzant models que inclouen ontologies, xarxes socials i etiquetatge col·laboratiu els proporciona els recursos i col·laboradors que millor s'adapten a les necessitats individuals d'aprenentatge. També s'han utilitzat les eines web 3.0 en el desenvolupament de PLE orientats a donar suport a la investigació científica.
Fonaments pedagògics
El PLE és un concepte que es fonamenta en idees i principis de les teories de l'aprenentatge connectivista, constructivista i de l'educació oberta i flexible, sobre la manera més adequada en què s'ha d'organitzar l'aprenentatge. Del Connectivisme es rescaten les seves aportacions en relació a les xarxes d'aprenentatge i l'habilitat per establir connexions entre idees i conceptes, la qual cosa constitueix un prerequisit perquè l'estudiant pugui usar, mantenir i desenvolupar les seves xarxes d'aprenentatge i els seus PLE. Del Constructivisme (pedagogia) està present la idea que el coneixement es construeix a partir de les idees pròpies, és a dir, que tota aquella informació que arriba a un individu se suma per construir nous coneixements i qüestionar i millorar els que ja es tenen.
Per altra banda, hi ha molts elements propis de l'Educació oberta i l'educació flexible que queden patents en els PLE. En aquesta línia es poden establir certs paral·lelismes amb les idees de l'educació oberta i la classificació que fa dels determinants administratius i didàctics els quals es podrien fer coincidir en gran manera amb els propis dels PLE.
Accessibilitat i flexibilitat són determinants administratius claus per entendre tan els PLE com l'educació oberta, ja que fins i tot el propi alumne s'administra els temps, horaris, etc., així com la incorporació al procés educatiu que pot gaudir d'una llibertat (això sí, controlada) d'admissió o les facilitats de tindre algun tipus de suport. De la mateixa manera, a nivell didàctic es pot destacar la determinació d'objectius i metes molt especificats en relació al progrés, rendiment i capacitat de cada alumne. Es tracta, per tant d'una filosofia centrada en l'alumnat i en el seu procés d'aprenentatge.
En aquest sentit, cal relacionar estretament aquestes idees amb els fonaments de l'educació flexible. Tot i que autors com Lockwood, marquen diferències entre l'educació oberta i la flexible, en general es té una visió molt aproximada de les dues, ja que una de les principals característiques és la flexibilitat.
Per tant, els entorns d'aprenentatge que neixen amb els PLE han de tenir, de manera inherent, una forta presència de la flexibilitat que permet l'adaptabilitat del discents i del procés educatiu. En aquest sentit, podem trobar diferents tipus de flexibilitat.
Flexibilitat temporal, relativa a l'inici/final del curs, moments d'estudi, d'avaluació o ritmes d'aprenentatge.
Flexibilitat de continguts, relativa als continguts del curs i la seva seqüenciació.
Flexibilitat als requeriments, és a dir, a les condicions per participar-hi.
Flexibilitat relativa a l'enfocament instruccional i als recursos, tenint en compte l'organització social de l'aprenentatge (participants, dimensió dels grups, etc.)
Flexibilitat logística, en relació al temps i al lloc on els suport, recursos, etc. Estan disponibles, com es distribueix la informació.
El perfil d'una persona usuària d'un PLE
Els PLE són útils per als residents digitals o persones que habiten en la xarxa i l'enriqueixen amb les seves col·laboracions, produccions... Aquests primers s'oposen als anomenats visitants digitals que simplement consumeixen el que es produeix en la xarxa, però no hi participen activament.
Residents i visitants digitals
Els «residents digitals» són les persones que viuen en la xarxa i s'hi senten tan còmodes com en les relacions presencials.
Usen les xarxes socials com ara el Twitter o el Facebook per a establir relacions en les seves vides virtuals que no sols es mantenen dins de la xarxa, sinó que també es prolonguen en les seves vides reals.Usen el Blogger per a expressar les seves opinions, per a llegir les opinions de la resta de residents i per a expressar el que hi pensen. Fan videoconferències laborals amb la mateixa desimboltura i confiança amb què conversen cara a cara, parlen per telèfon o es troben en una reunió. No fan diferències entre les conversacions en línia o fora de línia perquè per a ells la xarxa forma part de la realitat i s'hi confon.
Són consumidors proactius o el que s'anomena en anglès prosumer, perquè no sols són consumidors del que es produeix en la xarxa, sinó que també en són productors. Per exemple, tenen perfils plens i actualitzats, comparteixen llistes de música preferida a Spotify, pengen fotos a Flickr, les presentacions Power Points a Slideshare o els escrits a Shribd, piulen les seves impressions sobre un curs de formació i publiquen periòdicament al blog del centre o departament. Se solen associar a la xarxa actual i més social que ha desenvolupat eines més senzilles perquè qualsevol produïsca continguts.
En canvi, els «visitants digitals» són aquelles persones que es connecten de tant en tant a Internet, per revisar el correu, llegir un blog o un web, xatejar amb els coneguts, però continuen preferint la conversa cara a cara, una trucada o les reunions presencials; creuen que els amics de debò són els que es fan petons, que no es pot trobar parella a la xarxa o que les interaccions a través de la xarxa tenen poca credibilitat.
Per a aquestes persones, Internet és una cosa afegida, virtual o diferent de la realitat. Són consumidors i Internet és una cosa afegida, virtual o diferent de la realitat. Són consumidors i s'associen als primers anys de la xarxa, quan produir contingut requeria disposar de programes sofisticats i tenir coneixements tècnics i quan la xarxa era, sobretot, un espai d'institucions i experts que informaven els ciutadans.
Els visitants són consumidors, no són membres de la Xarxa, i aporten poc de valor al fet de pertànyer-hi.
Metàfores de la vida digital
Des que l'any 2001, Mark Prensky difonguera la distinció entre nadius digitals i immigrants digitals i la Teoria dels nadius digitals, molts han estat els autors que n'han parlat i s'han replantejat aquests dos conceptes, fins i tot, el mateix Prensky.
D'acord amb la definició actual que trobem en l'Enciclopèdia Catalana un nadiu digital és un "individu que, havent nascut després del 1990, forma part de la primera generació que ha crescut en un entorn tecnològic i digital normalitzat". Al contrari, un immigrant digital és una "persona nascuda abans de l'existència de la tecnologia digital i que experimenta una certa dificultat per dominar-ne les eines". La situació d'immigrant digital contrasta amb la del nadiu digital que ha nascut en un entorn plenament digitalitzat.
L'any 2009, Mark Prensky afirma que la distinció entre nadius i immigrants digitals, ja no té raó de ser perquè a mesura que ha anat avançant el , tant uns com els altres han crescut en l'era digital. Es planteja que s'ha d'establir una altra manera de fer aquesta distinció entre les persones usuàries d'Internet que prengui com a referent el coneixement digital d'aquestes.
Un any més tard, l'any 2010, durant la intervenció en Re:publica 2010, el psicòleg Peter Kruse ofereix una alternativa suggeridora i raonada a la proposta inicial de Prensky. Parla de residents i visitants, segons el comportament social de les persones. Aquest aspecte ens permet oblidar-nos dels factors de l'edat i l'escolarització que caracteritzava la proposta incial de Prensky. Durant l'any 2011, David S. White i Allison le Cornu publiquen l'article "Visitors and residents. A new typology for online engagement". Després de fer una anàlisi de la proposta de Prensky, els autors proposen dues maneres a través de les quals les persones es relacionen amb la xarxa. Parlen d'un continuum de formes de relacionar-s'hi, no de diferents categories en què es classifiquen les persones.
Des d'aleshores, la metàfora, per tal de considerar quina és la competència digital de cada persona pren com a referent el lloc i l'eina, ja que aquestes representen millor l'ús que es fa de la tecnologia en la societat actual.
Els PLE en les diferents etapes educatives
Existeixen tres estils diferenciats d'aprenentatge que influïxen directament en la manera de crear l'entorn personal d'aprenentatge: How a personal learning environmet can help Primary English teachers to increase their knowledge
Estil d'aprenentatge visual: En l'entorn personal d'aprenentatge d'aquest tipus d'estudiants, és recomanable utilitzar recursos digitals com imatges, videos, documentals i mapes conceptuals.
Estil d'aprenentatge auditiu: Aquests alumnes utilitzen recursos com cançons, gravacions d'audio, programes de ràdio o classes auditives.
Estil d'aprenentatge kinestètic: Aquest tipus d'alumnat necessita jocs de moviment actiu, jocs dinàmics, contacte fisic amb altres persones, simulació i interacció utilitzant xarxes socials.
Un dels principals beneficis que un PLE ofereix és tant la independència de la localització geográfica (García et al.2010) com la de l'etapa de la vida o del nivell d'estudis. Els PLE permeten organitzar la web 2.0 per a desenvolupar experiències i coneixement al llarg de tota la vida, tant professional com empresarial (Attwell & Deitmer, 2012).
PLE en l'educació secundària obligatòria
L'any 2012 es va dur a terme un estudi en un institut de secundària d'Itàlia amb estudiants que afirma que després d'introduir els PLE a classe i de treballar-ho durant un any els alumnes van passar d'un PLE bàsic, format bàsicament per eines o bé recomanades o de moda, centrat en el món de l'oci i orientat a les relacions socials, a un PLE amb molt més potencial educatiu, amb noves eines, amb recomanacions no només d'amistats sinó també de professors i pares. Altres estudis han provat que els PLE en secundària redueixen els problemes de conducta, com que els alumnes prenen consciència del seu aprenentatge i se'n fan responsables, l'ambient de classe millora considerablement.
PLE en l'educació superior
Com en les altres etapes educatives, els PLE que s'utilitzin tant en les etapes de formació universitària com en el món laboral serviràn perquè cadascú pugui establir les seves pròpies prioritats d'aprenentatge, gestionar les seves activitats d'aprenentatge incloent continguts formals i informals, i compartir coneixements amb els companys a través de tota la gamma de recursos socials i de col·laboració. Són diversos els estudis que han estudiat l'impacte d'integrar els PLE a nivell universitari. i també com els PLE poden ajudar a planificar el futur laboral dels estudiants. En tots els casos s'ha arribat a les mateixes conclusions, és a dir, que els PLE ofereixen bones oportunitats de gestió de l'aprenentatge personal i col·laboratiu, però que s'han adaptar totalment a les preferències i necessitats de l'estudiant, i que sovint el desenvolupament i gestió de PLE individualitzats requereix un esforç extra i un mínim de tutorització.
Projectes educatius implementats en Entorns Personals d'Aprenentatge
El projecte Plearn
L'any 2010 el Consell Nacional de Recerca del Canadà es va embarcar en un projecte de PLE anomenat Plearn. El projecte va demostrar amb el seu prototip que un entorn pot ser construït per connectar els serveis tradicionals d'aprenentatge, com ara els arxius en obert i repositoris oberts, i també els programaris i serveis per al disseny i l'explotació de les xarxes socials al si de les organitzacions o l'anomenat social networking. També va demostrar que el PLE és pedagògicament viable, és a dir, que el suport instruccional i social, entesos com a bastides, scaffolding en anglès, els quals són necessaris per a l'aprenentatge, també poden funcionar en un Entorn Personal d'Aprenentatge.
El projecte Plearn mostrà algunes de les noves tecnologies per a la interoperabilitat. Va ser dissenyat al voltant del marc del JavaScript Object Notation (JSON), un sistema de representació que substitueix l'RSS i pot ser utilitzat tant per a la sindicació de contingut com per a l'Aplicació de Programació d'Interfícies (APIs). Això permet als programaris de diferents aplicacions que treballen conjuntament i més de prop. Per finalitzar, permet el reemplaçament de l'SCORM amb una missatgeria més robusta entre els components. El mateix tipus de sistema, anomenat OAuth, el qual permet a una persona utilitzar una eina en línia per als contactes de les xarxes socials com Facebook. Els estudiants poden així combinar les funcions d'un LMS amb les funcions de les xarxes socials.
Sistema de suport a la gestió i a l'aprenentatge o el Learning and Performance Support System (LPSS)
El Sistema de Suport a la Gestió i a l'Aprenentatge, en anglès Learning and Performance Support System (LPSS), com el projecte PLEARN també l'ha impulsat el Consell Nacional de Recerca del Canadà. És un projecte que està pensat per facilitar el desenvolupament i la integració de plataformes personals en línea per controlar la formació, el coneixement i l'aprenentatge, facilitar un aprenentatge de llarga durada, reduir els costos de la formació, disminuir les demandes d'espais físics, permetre un desenvolupament ràpid i en línia, augmentar la implicació dels alumnes, donar suport a l'aprenentatge informal i personal i millorar la productivitat.
El programa aplica aquestes tecnologies a través d'una sèrie de projectes implementats amb socis comercials i tècnics que inclouen altres filials del Govern del Canadà i el NCR. Els programes lliures no sols inclouen el desenvolupament de la tecnologia que pot estar implementada en entorns corporatius, institucionals o governamentals, sinó també en una sèrie de publicacions i llibres blancs que descriuen com funciona, per a què i com s'hi pot connectar a la xarxa d'aprenentatge LPSS. El LPSS està dissenyat i organitzat per mòduls perquè es pugui inserir, per exemple, directament en entorns de treball o contextos corporatius, acoblant-se directament als recursos humans i als requisits de formació. Aquesta interoperabilitat s'aconsegueix mitjançant el personal learning assistant (PLA).
Així com un MOOC es basa en el concepte de sindicació de continguts o RSS per a agrupar recursos des de múltiples proveïdors al voltant d'un sol tema, el LPSS utilitza la mateixa tecnologia anomenada resource repository network (RNN) per permetre a un individu obtenir diferents parts de la seva educació de múltiples proveïdors. En aquest sentit, un LPSS és més semblant a un navegador personal que a un recurs o servei. Com un navegador, cada persona té el seu exemplar de LPSS. Encara que els recursos poden ser compartits, cada usuari del LPSS té la seva pròpia llista d'adreces d'interès o bookmarks i recursos perquè cada LPSS es gestiona individualment.
Referències
Enllaços externs
Personal Learning Environments (PLE) Dan LaSota [Video] [Consultat 29/10/2018]
PLE Entornos Personales de Aprendizaje. Castañeda, Adell. [Video] [Consultat 31/10/2018]
Jordi Adell [Video] [Consultat 11/04/2014]
Entorns personals d'aprenentatge. Humano digital. 5 oct. 2011. [Consultat 31/10/2018]
¿Qué son los entornos personales de aprendizaje? . Aula virtual. Abril de 2012. [Consultat 31/10/2018]
Conecta13. Taller sobre Entornos Personales de Aprendizaje. Aprendizaje social en red. [Consultat 31/10/2018]
Aprenentatge | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 5,321 |
thousands woke up in croatian towns along the border on friday morning... after a night spent sleeping rough because there wasnt enough transport to go around... they're desperate to find a seat on a bus or train...While they slept, croatian authorities closed 7 out of 8 border posts with serbia behind them, and put the army on alert.Croatia had said on wednesday that refugees could pass freely ... but backtracked after chaotic scenes on thursday... as authorities struggled to cope with the sudden influx of some 11,000 people in just 2 days..7. SOUNDBITE (Croatian) Jasmin Budanovski, deputy mayor of Tovarnik:"State institutions have completely failed us - they've deserted us. Look around, only 10 policeman here - to deal with 500 refugees. The only help is coming from 50 people from Ilaca who organised themselves to help these people."Thousands were left to wait for hours in hot weather... this man collapsed and had to be ressucitated...As available transport filled up and frustration mounted amongst the crowds, thousands charged at police lines.. The Croatian Prime Minister said his country didnt want to halt the migrants, but didnt have the resources to PROCESS them.those who finally managed to climb aboard a train to the capital expressed their relief...SOUNDBITE 1 - Mohammad, Syrian migrant « I feel happiness. So I waited for the train 2 days. On the ground, under the sun, without tents, without water, without food… Just on the ground. /DLSSeveral hundred people have made it to reception centres in Zagreb... Others made for the Slovenian border by train, but were stopped in their tracks...Slovenia suspended all rail traffic from Croatia after halting a train carrying migrants at the border... Police forced them to stay on the train, and locked it..UPSOTWater!!The train was finally allowed to pass on friday, and those aboard were taken to a refugee camp in western slovenia...
EN NW PKG UPDATE CROATIA BORDERS CLOSED
Issued on: 18/09/2015 - 10:09 Modified: 18/09/2015 - 17:33
Video by: Catherine CLIFFORD
thousands woke up in croatian towns along the border on friday morning... after a night spent sleeping rough because there wasnt enough transport to go around... they're desperate to find a seat on a bus or train...While they slept, croatian authorit | {
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There has been a growing concern for the improvement of pharmaceutical services provided by healthcare institutions. This concern is also shared by other stakeholders including patients, regulatory organizations, pharmaceutical companies, insurance companies, and research institutions.
Advancing Pharmaceutical Processes and Tools for Improved Health Outcomes presents research-based perspectives on the pharmaceutical industry in today's digitally-fueled world. Focusing on technological innovations for pharmaceutical applications as well as current trends in the industry, this publication is ideally designed for use by pharmacists, medical professionals, administrators in the medical field, health insurance professionals, researchers, and graduate-level students.
This book covers topics like Disease Identification, Information Quality, Open Access, Pharmaceutical Education, Pharmacoinformatics and Policy Planning. | {
"redpajama_set_name": "RedPajamaC4"
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<?php
/**
* LEAGUE MODEL CLASS.
*
* The League Model is the powerhouse of the Fantasy process. It manages many of the
* admin functionality and provides tools and methods to run the league.
* @author Jeff Fox (Github ID: jfox015)
* @version 1.0.4
* @lastModified 04/17/20
*
*/
require_once('./application/libraries/CompiledStats.php');
class league_model extends base_model {
/*--------------------------------
/ VARIABLES
/-------------------------------*/
/**
* SLUG.
* @var $_NAME:String
*/
var $_NAME = 'league_model';
/**
* LEAGUE NAME.
* @var $league_name:String
*/
var $league_name = '';
/**
* LEAGUE DESCRIPTION.
* @var $description:String
*/
var $description = '';
/**
* LEAGUE SCORING TYPE.
* @var $league_type:Int
*/
var $league_type = -1;
/**
* # OF GAMES PLAYED PER TEAM PER WEEK (Head-to-Head Scoring).
* @var $games_per_team:Int
*/
var $games_per_team = 0;
/**
* LEAGUE ACCESS TYPE
* @var $access_type:Int
*/
var $access_type = -1;
/**
* AVATAR.
* @var $avatar:String
*/
var $avatar = '';
/**
* LEAGUE STATUS.
* @var $league_status:Int
*/
var $league_status = 1;
/**
* COMMISSIONER ID.
* @var $commissioner_id:Int
*/
var $commissioner_id = -1;
/**
* # OF REGULAR (NON-PLAYOFF) SCORING PERIODS.
* @var $regular_scoring_periods:Int
*/
var $regular_scoring_periods = 0;
/**
* # OF PLAYOFF ROUNDS.
* @var $playoff_rounds:Int
*/
var $playoff_rounds = 0;
/**
* MAX # OF TEAMS FOR LEAGUE.
* @var $max_teams:Int
*/
var $max_teams = 0;
/**
* ACCEPT TEAM REQUEST.
* @var $accept_requests:Int
*/
var $accept_requests = 0;
/**
* ALLOW PLAYOFF TRANSACTIONS.
* @var $allow_playoff_trans:Int
*/
var $allow_playoff_trans = -1;
/**
* ALLOW PLAYOFF TRADES.
* @var $allow_playoff_trades:Int
*/
var $allow_playoff_trades = -1;
/**
* COMPILED STATS.
* @var $compiledStats:Array
*/
var $compiledStats = array();
/**
* STAT FIELD LIST.
* @var $columnList:Array
*/
var $columnList = array();
/**
* SQL COLUMN NAMES.
* @var $sqlColNames:Array
*/
var $sqlColNames = array();
/*---------------------------------------------
/
/ C'TOR
/ Creates a new instance of league_model
/
/---------------------------------------------*/
function league_model() {
parent::__construct();
$this->tblName = 'fantasy_leagues';
$this->tables['GAMES'] = 'fantasy_leagues_games';
$this->tables['TRANSACTIONS'] = 'fantasy_transactions';
$this->tables['WAIVERS'] = 'fantasy_players_waivers';
$this->tables['WAIVER_CLAIMS'] = 'fantasy_teams_waiver_claims';
$this->tables['TEAMS'] = 'fantasy_teams';
$this->tables['ROSTERS'] = 'fantasy_rosters';
$this->tables['TEAMS_RECORD'] = 'fantasy_teams_record';
$this->tables['TEAMS_SCORING'] = 'fantasy_teams_scoring';
$this->tables['ROSTER_RULES'] = 'fantasy_roster_rules';
$this->tables['SCORING_RULES_BATTING'] = 'fantasy_leagues_scoring_batting';
$this->tables['SCORING_RULES_PITCHING'] = 'fantasy_leagues_scoring_pitching';
$this->tables['TEAM_INVITES'] = 'fantasy_invites';
$this->tables['TEAM_REQUESTS'] = 'fantasy_leagues_requests';
$this->tables['TEAM_TRADES'] = 'fantasy_teams_trades';
$this->tables['TRADES'] = 'fantasy_teams_trades';
$this->tables['TRADES_STATUS'] = 'fantasy_teams_trades_status';
$this->tables['PLAYERS'] = 'fantasy_players';
$this->fieldList = array('league_name','description','league_type','games_per_team','access_type','league_status','regular_scoring_periods','max_teams','playoff_rounds','accept_requests','allow_playoff_trans','allow_playoff_trades');
$this->conditionList = array('avatarFile','new_commisioner');
$this->readOnlyList = array('avatar','commissioner_id');
$this->textList = array('description');
$this->columns_select = array('id','league_type','description','league_name','max_teams','access_type','avatar','commissioner_id','league_status','regular_scoring_periods','allow_playoff_trans','allow_playoff_trades');
$this->addSearchFilter('league_type','Scoring Type','leagueType','leagueType');
$this->addSearchFilter('access_type','Public/Private','accessType','accessType');
$this->addSearchFilter('league_status','Status','leagueStatus','leagueStatus');
parent::_init();
}
/*--------------------------------------------------
/
/ PUBLIC FUNCTIONS
/
/-------------------------------------------------*/
/**
* APPLY DATA.
*
* Applies custom data values to the object.
*
* @return TRUE on success, FALSE on failure
*
*/
public function applyData($input,$userId = -1) {
$success = parent::applyData($input,$userId);
if ($success) {
if ($input->post('new_commisioner')) {
if ($this->ownerCanBeCommish($input->post('new_commisioner'))) {
$this->commissioner_id = $input->post('new_commisioner');
}
}
if (isset($_FILES['avatarFile']['name']) && !empty($_FILES['avatarFile']['name'])) {
$success = $this->uploadFile('avatar',PATH_LEAGUES_AVATAR_WRITE,$input,'avatar',$this->league_name);
}
}
return $success;
}
/**
* DELETE ROSTERS.
* <p>
* Deletes all rosters for the specified league_id. If no id is passed, the current league id of the loaded bbject is used.
* </p>
* <p><b>NOTE:</b> To delete rosters for a given team, use the team_model->deleteRosters function instead.
* </p>
* @param $league_id {int} The League Id
* @return {Boolean} TRUE on success
*
* @since 1.0.6
* @access public
* @see application -> models -> team_model -> deleteRosters
*/
public function deleteRosters($league_id = false, $scoring_period_id = false) {
if ($league_id === false) { $league_id = $this->id; }
$this->db->where('league_id',$league_id);
if ($scoring_period_id !== false) {
$this->db->where('scoring_period_id',$scoring_period_id);
}
$this->db->delete($this->tables['ROSTERS']);
return true;
}
/**
* DELETE TEAM RECORDS.
* <p>
* Deletes all records for the specified league_id. If no id is passed, the current league id of the loaded bbject is used.
* </p>
* <p><b>NOTE:</b> To records trades for a given team, use the team_model->deleteRecords function instead.
* </p>
* @param $league_id {int} The League Id
* @return {Boolean} TRUE on success
*
* @since 1.0.6
* @access public
* @see application -> models -> team_model -> deleteRecords
*/
public function deleteRecords($league_id = false) {
return $this->deleteLeagueData($this->tables['TEAMS_RECORD'],$league_id);
}
/**
* DELETE SCHEDULE.
* <p>
* Deletes all schedules for the specified league_id. If no id is passed, the current league id of the loaded object is used.
* </p>
* <p><b>NOTE:</b> Because scheduling involves more than one team per game, there is no equivilent function for removing scheduled
* games for individual teams in the team_model.
* </p>
* @param $league_id {int} The League Id
* @return {Boolean} TRUE on success
*
* @since 1.0.6
* @access public
*/
public function deleteSchedule($league_id = false) {
return $this->deleteLeagueData($this->tables['GAMES'],$league_id);
}
/**
* DELETE TEAM SCORING.
* <p>
* Deletes all scoring for the specified league_id. If no id is passed, the current league id of the loaded bbject is used.
* </p>
* <p><b>NOTE:</b> To delete scoring for a given team, use the team_model->deleteScoring function instead.
* </p>
* @param $league_id {int} The League Id
* @return {Boolean} TRUE on success
*
* @since 1.0.6
* @access public
* @see application -> models -> team_model -> deleteRecords
*/
public function deleteScoring($league_id = false) {
return $this->deleteLeagueData($this->tables['TEAMS_SCORING'],$league_id);
}
/**
* DELETE TRADES.
* <p>
* Deletes all trades for the specified league_id. If no id is passed, the current league id of the loaded bbject is used.
* </p>
* <p><b>NOTE:</b> To delete trades for a given team, use the team_model->deleteTrades function instead.
* </p>
* @param $league_id {int} The League Id
* @return {Boolean} TRUE on success
*
* @since 1.0.6
* @access public
* @see application -> models -> team_model -> deleteTrades
*/
public function deleteTrades($league_id = false) {
return $this->deleteLeagueData($this->tables['TEAM_TRADES'],$league_id);
}
/**
* DELETE TRANSACTIONS.
* <p>
* Deletes all transactions for the specified league_id. If no id is passed, the current league id of the loaded bbject is used.
* </p>
* <p><b>NOTE:</b> To delete transactions for a given team, use the team_model->deleteTransactions function instead.
* </p>
* @param $league_id {int} The League Id
* @return {Boolean} TRUE on success
*
* @since 1.0.6
* @access public
* @see application -> models -> team_model -> deleteTransactions
*/
public function deleteTransactions($league_id = false) {
return $this->deleteLeagueData($this->tables['TRANSACTIONS'],$league_id);
}
/**
* DELETE WAIVER CLAIMS.
* <p>
* Deletes all waiver claims for the specified league_id. If no id is passed, the current league id of the loaded bbject is used.
* </p>
* <p><b>NOTE:</b> To delete waiver claims for a given team, use the team_model->deleteWaiverClaims function instead.
* </p>
* @param $league_id {int} The League Id
* @return {Boolean} TRUE on success
*
* @since 1.0.6
* @access public
* @see application -> models -> team_model -> deleteWaiverClaims
*/
public function deleteWaiverClaims($league_id = false) {
return $this->deleteLeagueData($this->tables['WAIVER_CLAIMS'],$league_id);
}
// SPECIAL QUERIES
/**
* GET LEAGUE NAME.
* <p>
* Return the name of the league specified (or leaded).</p>
* @param $league_id {int} The League Id
* @return {String} League Name
*
* @since 1.0
* @access public
*/
public function getLeagueName($league_id = false) {
if ($league_id === false) { $league_id = $this->id; }
$this->db->select('league_name');
$this->db->from($this->tblName);
$this->db->where("id",$league_id);
$league_name = '';
$query = $this->db->get();
if ($query->num_rows() > 0) {
$row = $query->row();
$league_name = $row->league_name;
}
$query->free_result();
return $league_name;
}
/**
* HAS TEAMS.
* <p>Return true if the league contains teams and FALSE if not.</p>
* @param $league_id {int} The League Id
* @return {Boolean} TRUE if teams exist, FALSE if no
*
* @since 1.0
* @access public
*/
public function hasTeams($league_id = false) {
if ($league_id === false) { $league_id = $this->id; }
$this->db->select('id');
$this->db->from($this->tables['TEAMS']);
$this->db->where("league_id",$league_id);
$count = $this->db->count_all_results();
//echo("League ".$league_id." team count = ".$count."<br />");
if ($count != 0) {
return true;
} else {
return false;
}
}
/**
* HAS VALID ROSTERS.
* <p>Simple test if teams in a legaue have players on rosters. it does not, however,
* test if they are valid against a league roster rules.</p>
* @param $league_id {int} The League Id
* @return {Boolean} TRUE if teams have rosters, FALSE if no
*
* @since 1.0
* @access public
*/
public function hasValidRosters($league_id = false) {
// VALIDATE ROSTER COUNTS
if ($league_id === false) { $league_id = $this->id; }
$this->db->select('fantasy_rosters.player_id');
$this->db->from($this->tables['TEAMS']);
$this->db->join('fantasy_rosters','fantasy_rosters.team_id = fantasy_teams.id','left');
$this->db->where("fantasy_teams.league_id",$league_id);
$count = $this->db->count_all_results();
if ($count != 0) {
return true;
} else {
return false;
}
}
/**
* GET SCORING TYPE.
* <p>Returns the leagues scoring type.</p>
* @param $league_id {int} The League Id
* @return {int} Scoring type ID
*
* @since 1.0.5
* @access public
*/
public function getScoringType($league_id = false) {
if ($league_id === false && $this->id != -1) {
return $this->league_type;
} else {
$type = -1;
$this->db->select('league_type');
$this->db->from($this->tblName);
$this->db->where("id",$league_id);
$query = $this->db->get();
if ($query->num_rows() > 0) {
$row = $query->row();
$type = $row->league_type;
}
$query->free_result();
return $type;
}
}
/**
* OWNER CAN BE COMMISSIONER.
* <p>Tests if the passed user ID can be commissioner of the specified (or leaded) league (I.E. if they are already
* commissioner or not).</p>
* @param $userId {int} The user Id
* @param $league_id {int} OPTIONAL - The League Id
* @return {Boolean} TRUE if user can be commissioner, FALSE if not
*
* @since 1.0
* @access public
*/
public function ownerCanBeCommish($userId = false, $league_id = false) {
if ($userId === false || $userId == -1) { return false; }
if ($league_id === false) { $league_id = $this->id; }
if ($userId != $this->commissioner_id && !$this->userIsCommish($userId)) {
return true;
} else {
return false;
}
}
/**
* USER IS COMMISSIONER.
* <p>Tests f the passed user ID is the leagues commissioner.</p>
* @param $userId {int} The user Id
* @param $league_id {int} OPTIONAL - The League Id
* @return {Boolean} TRUE if user is commissioner, FALSE if not
*
* @since 1.0
* @access public
*/
public function userIsCommish($userId = false, $league_id = false) {
if ($userId === false || $userId == -1) { return false; }
$this->db->select('id');
$this->db->from($this->tblName);
$this->db->where("commissioner_id",$userId);
if ($league_id !== false) {
$this->db->where("id",$league_id);
}
$count = $this->db->count_all_results();
if ($count != 0) {
return true;
} else {
return false;
}
}
/**
* USER LEAGUE COUNT.
* <p>Tests f the passed user ID is the leagues commissioner.</p>
* @param $userId {int} The user Id
* @param $league_id {int} OPTIONAL - The League Id
* @return {Boolean} TRUE if user is commissioner, FALSE if not
*
* @since 1.0
* @access public
*/
public function userLeagueCount($userId = false) {
if ($userId === false || $userId == -1) { return false; }
$this->db->select('id');
$this->db->from($this->tblName);
$this->db->where("commissioner_id",$userId);
return $this->db->count_all_results();
}
/**
* GET COMMISSIONER ID.
* <p>Returns a leagues commissioenr ID.</p>
* @param $league_id {int} The League Id
* @return {int} Commissioenr ID, -1 if not found
*
* @since 1.0.6
* @access public
*/
public function getCommissionerId($userId = false, $league_id = false) {
if ($league_id === false) { $league_id = $this->id; }
$commishId = -1;
$this->db->select('commissioner_id');
$this->db->where("id",$league_id);
$query = $this->db->get($this->tblName);
if ($query->num_rows() > 0) {
$row = $query->row();
$commishId = $row->commissioner_id;
}
$query->free_result();
return $commishId;
}
/**
* GET OWNER IDS.
* <p>Returns a list of user ids who own teams in the passed (or loaded) league.</p>
* @param $league_id {int} OPTIONAL - The League Id
* @return {Boolean} TRUE if user is commissioner, FALSE if not
*
* @since 1.0
* @access public
*/
public function getOwnerIds($league_id = false) {
$owners = array();
if ($league_id === false) { $league_id = $this->id; }
$this->db->select('owner_id');
$this->db->from($this->tables['TEAMS']);
$this->db->where("league_id",$league_id);
$query = $this->db->get();
if ($query->num_rows() > 0) {
foreach($query->result() as $row) {
array_push($owners,$row->owner_id);
}
}
$query->free_result();
return $owners;
}
/**
* GET OWNER INFO.
* <p>Returns an array of owner details including the owners id, first and last name.</p>
* <p><b>NOTE:</b> The <code>$showTeam</code> arg can be used to return the team name instead of owner name. The owner ID is
* returned regardless.</p>
* @param $league_id {int} OPTIONAL - The League Id
* @param $showTeam {Boolean} TRUE ot show the team name, FALSE to show owner name
* @return {Boolean} TRUE if user is commissioner, FALSE if not
*
* @since 1.0
* @access public
*/
public function getOwnerInfo($league_id = false, $showTeam = false) {
$owners = array();
if ($league_id === false) { $league_id = $this->id; }
$this->db->select('fantasy_teams.id, teamname, teamnick, firstName, lastName');
$this->db->from($this->tables['TEAMS']);
$this->db->join('users_meta','users_meta.userId = fantasy_teams.owner_id');
$this->db->where("league_id",$league_id);
$query = $this->db->get();
if ($query->num_rows() > 0) {
foreach($query->result() as $row) {
$ownerName = $row->firstName." ".$row->lastName;
if ($showTeam) {
$ownerName .= " - ".$row->teamname." ".$row->teamnick;
}
$owners = $owners + array($row->id=>$ownerName);
}
}
$query->free_result();
return $owners;
}
public function getDetailedOwnerInfo($league_id = false, $showTeam = false) {
$owners = array();
if ($league_id === false) { $league_id = $this->id; }
$this->db->select('fantasy_teams.id as team_id, teamname, teamnick, users_core.id as owner_id, email, username');
$this->db->from($this->tables['TEAMS']);
$this->db->join('users_core','users_core.id = fantasy_teams.owner_id');
$this->db->where("league_id",$league_id);
$query = $this->db->get();
if ($query->num_rows() > 0) {
foreach($query->result() as $row) {
$owners = $owners + array($row->team_id=>array('owner_id'=>$row->owner_id,'email'=>$row->email, 'username'=>$row->username,
'teamname'=>$row->teamname,'teamnick'=>$row->teamnick));
}
}
$query->free_result();
return $owners;
}
/**
* USER HAS ACCESS
* Returns TRUE Or FALSE if the passed user has access to the LEague (I.E. is a team owner).
* Modified version of isLeagueMember to add access restriction checking
* @param $user_id {int} The USER ID to check for ownership of a team
* @param $league_id {int} League ID var, if FALSE, defaults to models ID
* @return {int} Count value, 0 if no teams open
*
* @since 1.0.3 PROD
*
*/
public function userHasAccess($user_id = false, $league_id = false) {
$access = false;
if ($user_id === false || $user_id == -1) { return false; }
if ($league_id === false) { $league_id = $this->id; }
if ($league_id === false || $league_id == -1) { return false; }
//print "user id = ".$user_id."<ber />";
$ownerIds = $this->getOwnerIds($league_id);
//print "has access? = ".(in_array($user_id,$ownerIds) ? "yes":"no")."<ber />";
$access = ($this->access_type != -1 || ($this->access_type == -1 && (sizeof($ownerIds) > 0 && in_array($user_id,$ownerIds))));
return $access;
}
/**
* IS LEAGUE MEMBER
* Returns TRUE Or FALSE if the passed user has access to the LEague (I.E. is a team owner)
* @param $user_id {int} The USER ID to check for ownership of a team
* @param $league_id {int} League ID var, if FALSE, defaults to models ID
* @return {int} Count value, 0 if no teams open
*
* @since 1.0.6 Beta
* @changelog Changed from userHasAccess to isLeagueMmember
*
*/
public function isLeagueMember($user_id = false, $league_id = false) {
$access = false;
if ($user_id === false || $user_id == -1) { return false; }
if ($league_id === false) { $league_id = $this->id; }
//print "user id = ".$user_id."<ber />";
$ownerIds = $this->getOwnerIds($league_id);
//print "has access? = ".(in_array($user_id,$ownerIds) ? "yes":"no")."<ber />";
$access = (sizeof($ownerIds) > 0 && in_array($user_id,$ownerIds));
return $access;
}
/**
* GET OPEN TEAM COUNT
* Function that gets the number of unowned teams for the given league ID
* @param $league_id {int} League ID var, if FALSE, defaults to models ID
* @return {int} Count value, 0 if no teams open
*
* @since 1.0.3 PROD
* @see $this->getLeagueList()
*/
public function getOpenTeamCount($league_id = false) {
$count = 0;
if ($league_id === false) { $league_id = $this->id; }
$this->db->select('COUNT(id) as teamCount');
$this->db->from($this->tables['TEAMS']);
$this->db->where("league_id",$league_id);
$this->db->where("owner_id", -1);
$query = $this->db->get();
$row = $query->row();
$count = $row->teamCount;
$query->free_result();
return $count;
}
/**
* GET PLAYOFF SETTINGS
* Function that gets Playoff settings for the passed league
* @param $league_id {int} League ID var, if FALSE, defaults to models ID
* @return {Array} Array of settings value
*
* @since 1.0.3 PROD
*/
public function getPlayoffSettings($league_id = false) {
$length = array();
if ($league_id === false) { $league_id = $this->id; }
$this->db->select('regular_scoring_periods, playoff_rounds, allow_playoff_trans,allow_playoff_trades');
$this->db->from($this->tblName);
$this->db->where("id",$league_id);
$query = $this->db->get();
$row = $query->row();
$length = array('regular_scoring_periods'=>$row->regular_scoring_periods, 'playoff_rounds'=>$row->playoff_rounds,
'allow_playoff_trans'=>$row->allow_playoff_trans, 'allow_playoff_trades'=>$row->allow_playoff_trades,
'total_periods'=>intval($row->regular_scoring_periods) + intval($row->playoff_rounds));
$query->free_result();
return $length;
}
/*----------------------------------------------------------------------
/
/ INVITES AND REQUESTS
/
/----------------------------------------------------------------------*/
public function getLeagueInvite($invite_id = false, $league_id = false, $to_email = false, $status_id = false) {
$inviteObj = (object) array('id' => -1);
if ($invite_id === false && ($league_id === false && $to_email === false)) {
$this->errorCode = 1;
$this->statusMess = "No required ID was provided.";
return false;
}
$this->db->select('*');
if ($invite_id !== false) {
$this->db->where('id',$invite_id);
}
if ($league_id !== false) {
$this->db->where('league_id',$league_id);
}
if ($to_email !== false) {
$this->db->where('to_email',$to_email);
}
if ($status_id !== false) {
$this->db->where('status_id',$status_id);
}
$query = $this->db->get('fantasy_invites');
//echo($this->db->last_query()."<br />");
if ($query->num_rows() > 0) {
$inviteObj = $query->row();
}
$query->free_result();
return $inviteObj;
}
/**
* GET LEAGUE INVITES
* Function that returns of list of league team invites. Can be all invites only those
* still in PENDING status.
* @param $onlyPending {int} Only return pending and not completed invites
* @param $league_id {int} League ID var, if FALSE, defaults to models ID
* @return {int} Count value, 0 if no teams open
* @since 1.0.6 Beta
*
*/
public function getLeagueInvites($onlyPending = false, $league_id = false, $email = false) {
$invites = array();
if ($league_id === false) { $league_id = $this->id; }
$this->db->select($this->tables['TEAM_INVITES'].'.id, to_email, send_date, team_id, teamname, teamnick, inviteStatus, status_id, '.$this->tables['TEAM_INVITES'].'.league_id');
$this->db->from($this->tables['TEAM_INVITES']);
$this->db->join('fantasy_teams','fantasy_teams.id = fantasy_invites.team_id','left');
$this->db->join('fantasy_invites_status','fantasy_invites_status.id = fantasy_invites.status_id','left');
$this->db->where("fantasy_invites.league_id",$league_id);
if ($email !== false) {
$this->db->where($this->tables['TEAM_INVITES'].".to_email",$email);
}
if ($onlyPending !== false) {
$this->db->where('status_id', INVITE_STATUS_PENDING);
}
$this->db->where($this->tables['TEAM_INVITES'].".league_id",$league_id);
$this->db->order_by('status_id, team_id','asc');
$query = $this->db->get();
//echo($this->db->last_query()."<br />");
if ($query->num_rows() > 0) {
foreach($query->result() as $row) {
array_push($invites,array('to_email'=>$row->to_email, 'send_date'=>$row->send_date,
'team_id'=>$row->team_id,'team'=>$row->teamname." ".$row->teamnick,
'inviteStatus'=>$row->inviteStatus, 'status_id'=>$row->status_id,
'id'=>$row->id, 'league_id'=>$row->league_id,
));
}
}
$query->free_result();
return $invites;
}
/**
* UPDATE INVITE
* Changes the status and details of an open team invitation.
* @param $invite_id {int} The REQUEST ID
* @param $response {int} The new REQUEST STATUS TYPE
* @return {Int} Count of rows changed
*
*/
public function updateInvitation($invite_id = false, $response = false) {
if ($invite_id === false || $response === false) {
return false;
}
$this->db->select('*');
$this->db->where('id',$invite_id);
$query = $this->db->get($this->tables['TEAM_INVITES']);
if ($query->num_rows() == 0) {
$this->errorCode = 1;
$this->statusMess = 'No invite matching the passed ID was found in the system.';
return false;
} else {
$row = $query->row();
if ($response == INVITE_STATUS_ACCEPTED) {
$this->load->model('user_auth_model');
$data = array('owner_id'=>$this->user_auth_model->getUserIdByEmail($row->to_email));
$this->db->where('id',$row->team_id);
$this->db->update($this->tables['TEAMS'],$data);
if ($this->db->affected_rows() == 0) {
$this->errorCode = 2;
$this->statusMess = 'The team owner update could not be saved to the database.';
return false;
}
}
$this->db->flush_cache();
$this->db->where('id',$invite_id);
$this->db->update($this->tables['TEAM_INVITES'],array('status_id'=>intval($response)));
//echo($this->db->last_query()."<br />");
$rows_changed = $this->db->affected_rows();
if ($rows_changed == 0) {
$this->errorCode = 3;
$this->statusMess = 'The update could not be saved at this time.';
return false;
} // END if
} // END if
return $rows_changed;
}
/**
* GET LEAGUE REQUESTS
* Function that returns the number of requests for teams for a League. Can be filtered to only show
* PENDING requesta, requests by a specific user ID or a single request passing the REQUEST ID.
* @param $onlyPending {int} (OPTIONAL) Only return pending and not completed invites
* @param $league_id {int} League ID var, if FALSE, defaults to models ID
* @param $request_id {int} (OPTIONAL) REQUEST ID if passed
* @param $user_id {int} (OPTIONAL) The Requesting USER ID
* @return {int} Count value, 0 if no teams open
* @since 1.0.6 Beta
* @changelog 1.0.3 PROD - Updated to support passing a USER ID value.
*
*/
public function getLeagueRequests($onlyPending = false, $league_id = false, $request_id = false, $user_id = false) {
$requests = array();
if ($league_id === false) { $league_id = $this->id; }
$this->db->select($this->tables['TEAM_REQUESTS'].'.id, user_id, username, date_requested, team_id, teamname, teamnick, requestStatus, status_id');
$this->db->from($this->tables['TEAM_REQUESTS']);
$this->db->join('users_core','users_core.id = '.$this->tables['TEAM_REQUESTS'].'.user_id','left');
$this->db->join('fantasy_teams','fantasy_teams.id = '.$this->tables['TEAM_REQUESTS'].'.team_id','left');
$this->db->join('fantasy_leagues_requests_status','fantasy_leagues_requests_status.id = '.$this->tables['TEAM_REQUESTS'].'.status_id','left');
$this->db->where($this->tables['TEAM_REQUESTS'].'.league_id',$league_id);
if ($request_id !== false) {
$this->db->where($this->tables['TEAM_REQUESTS'].'.id',$request_id);
}
if ($onlyPending !== false) {
$this->db->where('status_id', REQUEST_STATUS_PENDING);
}
if ($user_id !== false) {
$this->db->where('user_id', $user_id);
}
$this->db->order_by('status_id, team_id','asc');
$query = $this->db->get();
//echo($this->db->last_query()."<br />");
if ($query->num_rows() > 0) {
foreach($query->result() as $row) {
array_push($requests,array('id'=>$row->id,'user_id'=>$row->user_id, 'username'=>$row->username,'date_requested'=>date_format(date_create($row->date_requested),"m/d/Y"),
'requestStatus'=>$row->requestStatus,'status_id'=>$row->status_id,'team_id'=>$row->team_id,'team'=>$row->teamname." ".$row->teamnick));
}
}
$query->free_result();
return $requests;
}
/**
* GET TEAM REQUEST
* Looks up and returns a single request details
* @param $request_id {int} The request ID
* @return {Obj} Request Object
* @since 1.1 PROD
*
*/
public function getLeagueRequest($request_id = false) {
$requestObj = false;
if ($request_id === false) { return false; $statusMess = "No Request ID provided."; }
$this->db->select('*');
$this->db->where('id',$request_id);
$query = $this->db->get($this->tables['TEAM_REQUESTS']);
if ($query->num_rows() > 0) {
$requestObj = $query->row();
}
$query->free_result();
return $requestObj;
}
/**
* TEAM REQUEST
* Checks if the passed user has a team request already PENDING and if so, rejects the request,
* If no other requests are open the the given League, the request is logged.
* @param $team_id {int} The TEAM ID for the request
* @param $user_id {int} The Requesters USER ID
* @param $league_id {int} League ID var, if FALSE, defaults to models ID
* @return {int} Count value, 0 if no teams open
* @since 0.6 Beta
* @changelog 1.1 PROD Removed error for Denied Requests.
*
*/
public function teamRequest($team_id = false, $user_id = false, $league_id = false) {
if ($league_id === false) {
$league_id = $this->id;
}
if ($team_id === false || $user_id === false) {
$this->errorCode = 3;
$this->statusMess = 'Required team and/or user ID parameters not recieved.';
return false;
}
$this->lang->load('league');
$this->db->select('id, status_id');
//$this->db->where('team_id',$team_id);
$this->db->where('user_id',$user_id);
$this->db->where('league_id',$league_id);
$this->db->where('(status_id = '.REQUEST_STATUS_PENDING.' OR status_id = '.REQUEST_STATUS_ACCEPTED.')');
$query = $this->db->get($this->tables['TEAM_REQUESTS']);
//echo($this->db->last_query()."<br />");
if ($query->num_rows() > 0) {
$row = $query->row();
$this->errorCode = 2;
switch($row->status_id) {
case REQUEST_STATUS_PENDING:
$mess = $this->lang->line('league_request_status_pending');
break;
case REQUEST_STATUS_ACCEPTED:
$mess = $this->lang->line('league_request_status_accepted');
break;
//case REQUEST_STATUS_DENIED:
// $mess = $this->lang->line('league_request_status_denied');
// break;
}
$this->statusMess = $mess;
return false;
}
$requestData = array('team_id'=>$team_id,'user_id'=>$user_id,'league_id'=>$league_id);
$this->db->insert($this->tables['TEAM_REQUESTS'],$requestData);
if ($this->db->affected_rows() == 0) {
$this->errorCode = 1;
$this->statusMess = 'The request data was not saved to the database.';
return false;
}
return true;
}
/**
* UPDATE REQUEST
* Changes the status and details of an open request.
* @param $request_id {int} The REQUEST ID
* @param $response {int} The new REQUEST STATUS TYPE
* @param $league_id {int} League ID var, if FALSE, defaults to models ID
* @return {Boolean} TRUE on success, FALSE on failure
*
*/
public function updateRequest($request_id = false, $response = false, $league_id = false) {
if ($league_id === false) {
$league_id = $this->id;
}
if ($request_id === false || $response === false) {
return false;
}
$this->db->select('*');
$this->db->where('id',$request_id);
$query = $this->db->get($this->tables['TEAM_REQUESTS']);
if ($query->num_rows() == 0) {
$this->errorCode = 1;
$this->statusMess = 'No request matching the passed ID was found in the system.';
return false;
} else {
$row = $query->row();
$cleanDb = true;
$newStatus = 0;;
switch($response) {
case REQUEST_STATUS_ACCEPTED:
$data = array('owner_id'=>$row->user_id);
$this->db->where('id',$row->team_id);
$this->db->update($this->tables['TEAMS'],$data);
if ($this->db->affected_rows() == 0) {
$this->errorCode = 2;
$this->statusMess = 'The team owner update could not be saved to the database.';
return false;
}
$newStatus = REQUEST_STATUS_ACCEPTED;
break;
case REQUEST_STATUS_WITHDRAWN:
$newStatus = REQUEST_STATUS_WITHDRAWN;
break;
case REQUEST_STATUS_DENIED:
$newStatus = REQUEST_STATUS_DENIED;
break;
case REQUEST_STATUS_REMOVED:
$newStatus = REQUEST_STATUS_REMOVED;
break;
default:
$newStatus = REQUEST_STATUS_UNKNOWN;
break;
} // END switch
$this->db->flush_cache();
$this->db->where('id',$request_id);
$this->db->update($this->tables['TEAM_REQUESTS'],array('status_id'=>$newStatus));
if ($this->db->affected_rows() == 0) {
$this->errorCode = 3;
$this->statusMess = 'The update could not be saved at this time.';
return false;
} // END if
} // END if
return true;
}
/**
* DELETE TEAM REQUESTS.
* This function clear the team request queue for a given league. It can be filtered down to an individual team or
* user as well.
*
* @param $league_id (int) The league identifier
* @param $user_id (int) OPTIONAL user identifier
* @param $team_id (int) OPTIONAL Team identifier
* @return (int) Affected Row count
*
* @since 1.0.6
* @see controllers->league->clearRequestQueue()
*/
public function deleteTeamRequests($league_id = false, $user_id = false, $team_id = false) {
if ($league_id === false) { $league_id = $this->id; }
$this->db->where("league_id",$league_id);
if ($user_id !== false) {
$this->db->where("user_id",$user_id);
}
if ($team_id !== false) {
$this->db->where("team_id",$team_id);
}
$this->db->delete($this->tables['TEAM_REQUESTS']);
return $this->db->affected_rows();
}
/**
* REMOVE TEAM REQUESTS.
* This function changes Team Request to REMOVED to take them out of an active state but DOES NOT
* delete them.
*
* @param $league_id (int) The league identifier
* @param $user_id (int) OPTIONAL user identifier
* @param $team_id (int) OPTIONAL Team identifier
* @return (int) Affected Row count
*
* @since 1.1 PROD
*
*/
public function removeTeamRequests($league_id = false, $user_id = false, $team_id = false) {
if ($league_id === false) { $league_id = $this->id; }
$this->db->where("league_id",$league_id);
if ($user_id !== false) {
$this->db->where("user_id",$user_id);
}
if ($team_id !== false) {
$this->db->where("team_id",$team_id);
}
$this->db->where("status_id",REQUEST_STATUS_PENDING);
$data = array("status_id" => REQUEST_STATUS_REMOVED);
$this->db->update($this->tables['TEAM_REQUESTS'], $data);
return $this->db->affected_rows();
}
/**
* DELETE TEAM INVITES.
* This function clear the team invitiations for a given league. It can be filtered down to an individual team or
* user as well.
*
* @param $league_id (int) The league identifier
* @param $user_id (int) OPTIONAL user identifier
* @param $team_id (int) OPTIONAL Team identifier
* @return (int) Affected Row count
*
* @since 0.6 Beta
* @see controllers->league->clearRequestQueue()
*/
public function deleteTeamInvites($league_id = false, $user_id = false, $team_id = false) {
if ($league_id === false) { $league_id = $this->id; }
$userMail = '';
if ($user_id !== false) {
$userMail = getEmail($user_id);
}
$this->db->flush_cache();
$this->db->where("league_id",$league_id);
if (!empty($userMail)) {
$this->db->where("to_email",$userMail);
}
if ($team_id !== false) {
$this->db->where("team_id",$team_id);
}
$this->db->delete($this->tables['TEAM_INVITES']);
return true;
}
/**
* REMOVE TEAM INVITES.
* This function changes the status of an invite to REMOVED. It DOES NOT delete the invite.
*
* @param $league_id (int) The league identifier
* @param $user_id (int) OPTIONAL user identifier
* @param $team_id (int) OPTIONAL Team identifier
* @return (int) Affected Row count
*
* @since 1.1 PROD
*/
public function removeTeamInvites($league_id = false, $user_id = false, $team_id = false) {
if ($league_id === false) { $league_id = $this->id; }
$userMail = '';
if ($user_id !== false) {
$userMail = getEmail($user_id);
}
$this->db->flush_cache();
$this->db->where("league_id",$league_id);
if (!empty($userMail)) {
$this->db->where("to_email",$userMail);
}
if ($team_id !== false) {
$this->db->where("team_id",$team_id);
}
$this->db->where("status_id",INVITE_STATUS_PENDING);
$data = array("status_id" => INVITE_STATUS_REMOVED);
$this->db->update($this->tables['TEAM_INVITES'], $data);
return true;
}
/*------------------------------------------------------------------
/
/ LEAGUE SEARCH AND INFORMATION
/
/-----------------------------------------------------------------*/
/**
* GET LEAGUES.
* Returns a list of public leagues.
* @param $type - 1 = Public, -1 = all (admin only)
* @param $status - 1 = Public, -1 = all (admin only)
* @return array of league information
*/
public function getLeagues($type=1, $status = false) {
$leagues = array();
$this->db->select($this->tblName.'.id, league_name, description, avatar, shortDesc, commissioner_id, league_status, access_type, league_type, leagueType, max_teams, regular_scoring_periods, games_per_team, playoff_rounds');
$this->db->join('fantasy_leagues_types','fantasy_leagues_types.id = '.$this->tblName.'.league_type','left');
if ($type != -1) $this->db->where('access_type',1);
if ($status !== false) $this->db->where('league_status', $status);
$query = $this->db->get($this->tblName);
//echo("getLeagues, query->num_rows = ".$query->num_rows()."<br />");
if ($query->num_rows() > 0) {
foreach ($query->result() as $row) {
$commish = resolveUsername($row->commissioner_id);
$leagues = $leagues + array($row->id=>array('league_name'=>$row->league_name,'avatar'=>$row->avatar,'max_teams'=>$row->max_teams,
'league_status'=>$row->league_status,'commissioner_id'=>$row->commissioner_id,'commissioner'=>$commish,
'league_type_desc'=>$row->shortDesc,'league_type_lbl'=>$row->leagueType,'league_type'=>$row->league_type,
'description'=>$row->description,'access_type'=>$row->access_type,
'regular_scoring_periods'=>$row->regular_scoring_periods,'games_per_team'=>$row->games_per_team,
'playoff_rounds'=>$row->playoff_rounds));
}
}
$query->free_result();
//echo($this->db->last_query()."<br />");
return $leagues;
}
/**
* GET LEAGUE LISTING.
* Returns a list of all leagues on the site. The array ciontains extended information such as if it is accepting owners,
* if so, if it has openings as well.
*
* @param $user_id - The user ID to check against
* @return leagues array, empty if not league found on failure
* @since 1.0.3 PROD
*
*/
public function getLeagueList($user_id = false, $active = false, $public = false) {
$leagues = array();
$openCount = 0;
$pendingRequests = array();
$select = $this->tblName.'.id, league_name, description, avatar, league_status, leagueStatus, max_teams, accept_requests, shortDesc, commissioner_id, username, access_type, league_type, leagueType';
if ($user_id !== false) {
$select .= ', (SELECT COUNT(id) FROM fantasy_teams WHERE league_id = '.$this->tblName.'.id AND owner_id = '.$user_id.') as teamsOwned';
}
$this->db->select($select);
$this->db->join('fantasy_leagues_types','fantasy_leagues_types.id = '.$this->tblName.'.league_type','left');
$this->db->join('fantasy_leagues_status','fantasy_leagues_status.id = '.$this->tblName.'.league_status','left');
$this->db->join("users_core","users_core.id = ".$this->tblName.".commissioner_id", "left");
if ($active !== false) {
$this->db->where("league_status", 1);
}
if ($public !== false) {
$this->db->where("access_type", 1);
}
$this->db->order_by("league_status", "asc");
$query = $this->db->get($this->tblName);
if ($query->num_rows() > 0) {
foreach ($query->result() as $row) {
$openCount = $this->getOpenTeamCount($row->id);
if ($user_id !== false) {
$pendingRequests = $this->getLeagueRequests(true, $row->id, false, $user_id);
}
array_push($leagues,array('league_id'=>$row->id,'league_name'=>$row->league_name,'description'=>$row->description,'avatar'=>$row->avatar,'access_type'=>$row->access_type,
'league_status'=>$row->league_status,'accept_requests'=>$row->accept_requests,'max_teams'=>$row->max_teams,
'shortDesc'=>$row->shortDesc,'commissioner'=>$row->username, 'commissioner_id'=>$row->commissioner_id,
'league_type_desc'=>$row->shortDesc,'league_type_lbl'=>$row->leagueType,'league_status_lbl'=>$row->leagueStatus,
'league_type'=>$row->league_type,'openCount'=>$openCount,'teamsOwned'=>$row->teamsOwned, 'pendingRequests'=>$pendingRequests));
}
}
//echo($this->db->last_query()."<br />");
$query->free_result();
return $leagues;
}
public function loadLeagueTeams($league_id = false) {
if ($league_id === false) { $league_id = $this->id; }
$teamNames = array();
$this->db->select("id, teamname, teamnick");
$this->db->where("league_id",$league_id);
$query = $this->db->get($this->tables['TEAMS']);
if ($query->num_rows() > 0) {
foreach($query->result() as $row) {
$teamNames[$row->id] = $row->teamname." ".$row->teamnick;
}
}
$query->free_result();
return $teamNames;
}
public function loadGameData($game_id = false, $team_model, $excludeList = array(), $league_id = false, $debug = false) {
if ($league_id === false) { $league_id = $this->id; }
if ($game_id === false) return false;
// FIRST GET THE TEAMS INVOLVED
$teams = array();
$scoring_period = 0;
$this->db->select('home_team_id, away_team_id, scoring_period_id');
$this->db->where('id',$game_id);
$query = $this->db->get('fantasy_leagues_games');
if ($query->num_rows() > 0) {
$row = $query->row();
$teams = array('home'=>$row->home_team_id, 'away'=>$row->away_team_id);
$scoring_period = $row->scoring_period_id;
}
$query->free_result();
// LOAD RELEVANT SCORING CATEGORIES
$scoring_rules = $this->getScoringRules($this->id);
// NOW GET EACH TEAMS ROSTERS
$rosters = array('home'=>array(),'away'=>array());
foreach ($teams as $key => $team_id) {
$statsCompiled = array(1=>array(), -1=>array());
// GET ACTIVE BATTERS
if ($team_model->load($team_id)) {
$team_data = array('id'=>$team_id,'team_name'=>$team_model->teamname." ".$team_model->teamnick,
'players_active'=>array(),'players_reserve'=>array(),'stats_active'=>'','stats_reserve'=>'');
$statuses = array(1, -1,2);
foreach ($statuses as $status) {
$player_list = array();
$players = $team_model->getBatters($scoring_period, false,$status) + $team_model->getPitchers($scoring_period, false,$status);
foreach ($players as $player_id => $player_data) {
//echo("PLayer position = ".$player_data['player_position']."<br />");
if ($player_data['player_position'] != 1) {
$type = "batting";
$pos = $player_data['player_position'] ;
} else {
$type = "pitching";
$pos = $player_data['player_role'];
}
// GET PLAYER DATA
$select = "";
foreach($scoring_rules[$type] as $cat => $val) {
if ($select != '') { $select.=","; } // END if
$select .= strtolower(get_ll_cat($cat, true));
}
// SUBQUERY FOR FANTASY TOTALS
$select .= ",(SELECT total FROM fantasy_players_scoring WHERE player_id = ".intval($player_data['id'])." AND
league_id = ".intval($league_id)." AND scoring_period_id = ".intval($scoring_period)." AND
scoring_type = ".$scoring_rules['scoring_type'].") AS total ";
// GET ALL PLAYERS SCORING FOR TEAMS ROSTER
$player_stats = array();
$this->db->flush_cache();
$this->db->select($select);
$this->db->where("player_id",intval($player_data['id']));
$this->db->where("scoring_period_id",intval($scoring_period));
$query = $this->db->get("fantasy_players_compiled_".$type);
//echo($this->db->last_query()."<br />");
if ($query->num_rows() > 0) {
$player_stats = $query->row();
} // END if
$query->free_result();
$pRow = false;
$stats = "";
$total = 0;
if (sizeof($excludeList) == 0 || (sizeof($excludeList) > 0 && !in_array($team_id,$excludeList))) {
if (sizeof($player_stats) > 0) {
//$pRow = $player_stats->row();
$colCount = 0;
foreach($scoring_rules[$type] as $cat => $val) {
$colName = strtolower(get_ll_cat($cat, true));
if ($player_stats->$colName != 0) {
if (!empty($stats)) $stats .= ", ";
$stats .= $player_stats->$colName." ".strtoupper(get_ll_cat($cat));
//if (($colCount+1) != sizeof($scoring_rules[$type])) { $stats.=", "; }
$useStatus = $status;
if ($status == 2) {
$useStatus = -1;
}
if (isset($statsCompiled[$useStatus][$cat])) {
$statsCompiled[$useStatus][$cat] += $player_stats->$colName;
} else {
$statsCompiled[$useStatus][$cat] = $player_stats->$colName;
}
}
$colCount++;
}
$total = $player_stats->total;
}
//$pQuery->free_result();
}
$player_list = $player_list + array($player_data['id']=>array('name'=>$player_data['first_name']." ".$player_data['last_name'],
'stats'=>$stats,'total'=>$total,'position'=>$pos,
'injury_is_injured'=>$player_data['injury_is_injured'],
'injury_dl_left'=>$player_data['injury_dl_left'], 'injury_left'=>$player_data['injury_left'], 'injury_dtd_injury'=>$player_data['injury_dtd_injury'],
'injury_id'=>$player_data['injury_id'],'injury_career_ending'=>$player_data['injury_career_ending']));
}
if ($status == 1) $team_data['players_active'] = $player_list;
else $team_data['players_reserve'] = $team_data['players_reserve'] + $player_list;
}
foreach ($statsCompiled as $statusType => $compiled_stats) {
$statsStr = "";
foreach ($compiled_stats as $cat => $value) {
if (!empty($statsStr)) $statsStr .= ", ";
$statsStr .= $value." ".strtoupper(get_ll_cat($cat));
}
if ($statusType == 1) { $typeStr = 'stats_active'; } else { $typeStr = 'stats_reserve'; }
$team_data[$typeStr] = $statsStr;
}
$rosters[$key] = $team_data;
}
}
return $rosters;
}
public function getMemberCount($league_id = false) {
if ($league_id === false) { $league_id = $this->id; }
$count = 0;
$this->db->where('league_id',$league_id);
$this->db->from($this->tables['TEAMS']);
$count = $this->db->count_all_results();
return $count;
}
public function getDivisionNames($league_id = false) {
if ($league_id === false) { $league_id = $this->id; }
$divisions = array();
$this->db->select('id, division_name');
$this->db->where('league_id',$league_id);
$query = $this->db->get('fantasy_divisions');
if ($query->num_rows() > 0) {
foreach ($query->result() as $row) {
$divisions = $divisions + array($row->id=>$row->division_name);
}
}
$query->free_result();
return $divisions;
}
/**
*
* GET TEAM DETAILS
*
* @param $league_id {int} League ID
* @param $selectBox {Boolean} Return data as select box data or full array
* @param $noOwner {Boolean} Return only teams without owners
* @return {Array} Array of team arrays objects
*
* @since 0.1 Beta
* @changelog 1.1 PROD Added team draft settings to full return list
*
*/
public function getTeamDetails($league_id = false, $selectBox = false, $noOwner = false) {
if ($league_id === false) { $league_id = $this->id; }
$this->db->select($this->tables['TEAMS'].'.id, teamname, teamnick, '.$this->tables['TEAMS'].'.avatar, owner_id, username, firstName, lastName, email, auto_draft, auto_list, auto_round_x');
$this->db->join('users_core','users_core.id = '.$this->tables['TEAMS'].'.owner_id','left');
$this->db->join('users_meta','users_meta.userId = '.$this->tables['TEAMS'].'.owner_id','left');
$this->db->where('league_id',$league_id);
if ($noOwner !== false) {
$this->db->where('(owner_id = 0 OR owner_id = -1)');
}
$this->db->order_by('id','asc');
$query2 = $this->db->get($this->tables['TEAMS']);
$teams = array();
if ($selectBox != false) { $teams = array('-1'=>""); }
if ($query2->num_rows() > 0) {
//echo("Teams for league".$league_id." = <br/>");
foreach ($query2->result() as $trow) {
if ($selectBox != false) {
$teams = $teams + array($trow->id=>$trow->teamname." ".$trow->teamnick);
} else {
$ownerName = "No owner";
if ($trow->owner_id != -1) {
$ownerName = (!empty($trow->firstName) && !empty($trow->lastName)) ? $trow->firstName." ".$trow->lastName : $trow->username;
if ($this->commissioner_id != -1 && $trow->owner_id == $this->commissioner_id) {
$ownerName .= " (Commisioner)";
}
}
$teams = $teams + array($trow->id=>array('teamname'=>$trow->teamname,'teamnick'=>$trow->teamnick,'avatar'=>$trow->avatar,
'owner_id'=>$trow->owner_id,'owner_name'=>$ownerName,'owner_email'=>$trow->email,
'auto_draft'=>$trow->auto_draft,'auto_list'=>$trow->auto_list,'auto_round_x'=>$trow->auto_round_x
));
}
}
}
$query2->free_result();
return $teams;
}
public function getTeamIdList($league_id = false) {
if ($league_id === false) { $league_id = $this->id; }
$this->db->select('id');
$this->db->where('league_id',$league_id);
$this->db->order_by('id','asc');
$query2 = $this->db->get($this->tables['TEAMS']);
$teams = array();
if ($query2->num_rows() > 0) {
foreach ($query2->result() as $trow) {
array_push($teams, $trow->id);
}
}
$query2->free_result();
return $teams;
}
public function getLeagueDetails($league_id = false) {
if ($league_id === false) { $league_id = $this->id; }
$league = array();
$this->db->select($this->tblName.'.id,league_name,accessType,description,'.$this->tblName.'.avatar,leagueType,max_teams,league_status,leagueStatus,commissioner_id,games_per_team,regular_scoring_periods,playoff_rounds,accept_requests,firstName, lastName, username');
$this->db->join('fantasy_leagues_status','fantasy_leagues_status.id = '.$this->tblName.'.league_status','left');
$this->db->join('fantasy_leagues_types','fantasy_leagues_types .id = '.$this->tblName.'.league_type','left');
$this->db->join('fantasy_leagues_access','fantasy_leagues_access.id = '.$this->tblName.'.access_type','left');
$this->db->join('users_core','users_core.id = '.$this->tblName.'.commissioner_id','left');
$this->db->join('users_meta','users_meta.userId = '.$this->tblName.'.commissioner_id','left');
$this->db->where($this->tblName.'.id',$league_id);
$query = $this->db->get($this->tblName);
$teams = array();
if ($query->num_rows() > 0) {
$league = $query->row_array();
$commish = (!empty($league['firstName']) && !empty($league['lastName'])) ? $league['firstName']." ".$league['lastName'] : $league['username'];
$league['commissioner'] = $commish;
}
$query->free_result();
return $league;
}
public function getFullLeageDetails($league_id = false, $noOwner = false) {
if ($league_id === false) { $league_id = $this->id; }
$divisions = array();
$this->db->select('id, division_name');
$this->db->where('league_id',$league_id);
$this->db->order_by('division_name','asc');
$query = $this->db->get('fantasy_divisions');
if ($query->num_rows() > 0) {
foreach ($query->result() as $row) {
$this->db->flush_cache();
$this->db->select('fantasy_teams.id, teamname,teamnick,fantasy_teams.owner_id,firstName, lastName, username, email,fantasy_teams.avatar,auto_draft,auto_list,auto_round_x');
$this->db->join('users_core','users_core.id = fantasy_teams.owner_id','left');
$this->db->join('users_meta','users_meta.userId = fantasy_teams.owner_id','left');
$this->db->where('league_id',$league_id);
$this->db->where('division_id',$row->id);
if ($noOwner !== false) {
$this->db->where('(owner_id = 0 OR owner_id = -1)');
}
$this->db->order_by('teamname, teamnick','asc');
$query2 = $this->db->get($this->tables['TEAMS']);
$teams = array();
if ($query2->num_rows() > 0) {
foreach ($query2->result() as $trow) {
$ownerName = "No owner";
if ($trow->owner_id != -1) {
$ownerName = (!empty($trow->firstName) && !empty($trow->lastName)) ? $trow->firstName." ".$trow->lastName : $trow->username;
if ($this->commissioner_id != -1 && $trow->owner_id == $this->commissioner_id) {
$ownerName .= " (Commisioner)";
}
}
$teams = $teams + array($trow->id=>array('teamname'=>$trow->teamname,'teamnick'=>$trow->teamnick,
'owner_id'=>$trow->owner_id,'owner_name'=>$ownerName ,
'owner_aim'=>'','owner_email'=>$trow->email,
'avatar'=>$trow->avatar,'auto_draft'=>$trow->auto_draft,
'auto_list'=>$trow->auto_list,'auto_round_x'=>$trow->auto_round_x));
}
}
$query2->free_result();
$divisions = $divisions + array($row->id=>array('division_name'=>$row->division_name,'teams'=>$teams));
}
}
$query->free_result();
return $divisions;
}
public function getLeagueStandings($curr_period_id = false,$league_id = false) {
if ($league_id === false) { $league_id = $this->id; }
$scoring_type = $this->getScoringType();
switch ($scoring_type) {
case LEAGUE_SCORING_ROTO:
case LEAGUE_SCORING_ROTO_5X5:
case LEAGUE_SCORING_ROTO_PLUS:
$rules = $this->getScoringRules();
$teams = array();
$this->db->flush_cache();
$this->db->select($this->tables['TEAMS'].'.id, teamname, teamnick, avatar, value_0 as val_0, value_1 as val_1, value_2 as val_2, value_3 as val_3, value_4 as val_4,
value_5 as val_5, value_6 as val_6, value_7 as val_7, value_8 as val_8, value_9 as val_9, value_10 as val_10, value_11 as val_11, total as total');
$this->db->join($this->tables['TEAMS'],'fantasy_teams_scoring.team_id = '.$this->tables['TEAMS'].'.id','right outer');
$this->db->where('fantasy_teams.league_id',$league_id);
if ($curr_period_id !== false) {
$this->db->where('scoring_period_id',$curr_period_id);
} // END if
$this->db->order_by('total','desc');
$query = $this->db->get('fantasy_teams_scoring');
//print($this->db->last_query()."<br />");
$teams = array();
if ($query->num_rows() > 0) {
foreach ($query->result() as $row) {
$catCount = 0;
$types = array('batting','pitching');
foreach($types as $type) {
foreach($rules[$type] as $cat => $val) {
$colName = 'val_'.$catCount;
if (!isset($row->$colName)) {
}
}
}
$teams = $teams + array($row->id=>array('teamname'=>$row->teamname,'teamnick'=>$row->teamnick,'avatar'=>$row->avatar,
'value_0' => $row->val_0, 'value_1' => $row->val_1, 'value_2' => $row->val_2, 'value_3' => $row->val_3, 'value_4' => $row->val_4,
'value_5' => $row->val_5, 'value_6' => $row->val_6, 'value_7' => $row->val_7, 'value_8' => $row->val_8, 'value_9' => $row->val_9,
'value_10' => $row->val_10, 'value_11' => $row->val_11, 'total' => $row->total));
}
}
$query->free_result();
return $teams;
break;
case LEAGUE_SCORING_HEADTOHEAD:
default:
$divisions = array();
$this->db->select('id, division_name');
$this->db->where('league_id',$league_id);
$query = $this->db->get('fantasy_divisions');
if ($query->num_rows() > 0) {
foreach ($query->result() as $row) {
$this->db->flush_cache();
$this->db->select('fantasy_teams.id, teamname, teamnick, avatar, g, w, l, pct');
$this->db->join('fantasy_teams_record','fantasy_teams_record.team_id = fantasy_teams.id','left');
$this->db->where('fantasy_teams.league_id',$league_id);
$this->db->where('fantasy_teams.division_id',$row->id);
if ($curr_period_id !== false) {
$this->db->where('scoring_period_id',$curr_period_id);
} // END if
$this->db->order_by('pct','desc');
$query2 = $this->db->get($this->tables['TEAMS']);
$teams = array();
if ($query2->num_rows() > 0) {
foreach ($query2->result() as $trow) {
$teams = $teams + array($trow->id=>array('teamname'=>$trow->teamname,'teamnick'=>$trow->teamnick,'avatar'=>$trow->avatar,
'g'=>$trow->g,'w'=>$trow->w,'l'=>$trow->l,'pct'=>$trow->pct));
}
}
$query2->free_result();
$divisions = $divisions + array($row->id=>array('division_name'=>$row->division_name,'teams'=>$teams));
}
}
$query->free_result();
return $divisions;
break;
}
}
/**
* CREATE LEAGUE SCHEDULE.
* Builds a scheudle for all teams based on the number of teams, number of scoring periods
* and the number of games per team.
* @param $league_id - If not specified, no league filter is applied.
* @return [JSON] "OK" on success, error message on failure
*
* @since 1.0
* @version 2.0
*/
public function createLeagueSchedule($league_id = false, $debug = false) {
if ($league_id === false) { $league_id = $this->id; }
/*-------------------------------------------------------
/ 1.0 GET ALL TEAMS
/------------------------------------------------------*/
$teams = array();
$this->db->select("id");
$this->db->where("league_id",$league_id);
$query = $this->db->get($this->tables['TEAMS']);
if ($query->num_rows() > 0) {
foreach($query->result() as $row) {
array_push($teams,$row->id);
}
}
$query->free_result();
/*-------------------------------------------------------
/ 1.1 DELETE ALL GAMES FOR THIS LEAGUE (IF THEY EXIST)
/------------------------------------------------------*/
$this->db->flush_cache();
$this->db->where('league_id',$league_id);
$this->db->delete('fantasy_leagues_games');
/*-------------------------------------------------------
/ 1.2 SET BASIC DATA VARS
/------------------------------------------------------*/
$matchups = (sizeof($teams)/2) * $this->games_per_team;
$gamesNeeded = $matchups*$this->regular_scoring_periods;
$success = false;
$outMess = "";
$data = array();
/*-------------------------------------------------------
/ 1.2.1 IF DEBUGGING OUTPUT THE BASICS
/------------------------------------------------------*/
//$this->debug =false;
if ($this->debug) {
echo("Team count = ".sizeof($teams)."<br />");
echo("Games per team = ".$this->games_per_team."<br />");
echo("matchups per period = ".$matchups."<br />");
echo("this->regular_scoring_periods = ".$this->regular_scoring_periods."<br />");
echo("Total games needed = ".$gamesNeeded."<br />");
}
/*-------------------------------------------------------
/
/ EDIT 1.0.3 PROD
/ Switch from unreliable automatic generation of games
/ to using a pre-built schedule XML file. The file
/ format is based off the LSDL format used by OOTP.
/
/-----------------------------------------------------*/
/*-------------------------------------------------------
/ 2.0 IF LOAD THE SCHEDULE XML FILE
/------------------------------------------------------*/
$schedFile = DIR_WRITE_PATH.PATH_MEDIA_WRITE."schedules".URL_PATH_SEPERATOR."ootpfls_t_".sizeof($teams).".lsdl";
if ($this->debug) { echo "XML file path = ".$schedFile."<br />\n"; }
if (!file_exists($schedFile)){
$outMess = "The Schedule file could not be found in path:".$schedFile."<br />\n";
} else {
$fr = fopen($schedFile,"r") or $outMess = "Failed to load the file ".$schedFile;
$xmlIn = fread($fr,filesize($schedFile));
fclose($fr);
if ($this->debug) {
echo "XML data loaded!<br />\n";
//echo "XML data = ".$xmlIn."<br />\n";
}
/*--------------------------------------------
/ 2.1 CREATE A NEW SIMPLE XML ELEMENT
/-------------------------------------------*/
$xml = new SimpleXMLElement($xmlIn);
if ($xml === false) {
$outMess = "Failed loading XML: ";
foreach(libxml_get_errors() as $error) { $outMess .= "<br>". $error->message; }
if ($this->debug) { echo($outMess."<br />\n"); }
} else {
/*--------------------------------------------
/ 2.2 DEFINE CHILD NODES AND START INDEX
/-------------------------------------------*/
$games = $xml->children();
$gameList = $games->children();
// DEFINE A STARTING INDEX FOR PULLING THE REQUIRED AMOUNT OF GAMES
$loopCount = 0;
$startIndex = 0;
$endIndex = (count($gameList) - $gamesNeeded)-1;
while (true) {
$startIndex = random_int(0, $endIndex);
// ASSERT START INDEX IS DIVISIBLE BY THE NUMBER OF GAMES PER GAME GROUP (1/2 the teams in league)
if (($startIndex % (sizeof($teams)/2)) == 0 || $loopCount > 1000) {
break;
}
$loopCount++;
}
if ($this->debug) {
echo "XML parsed!<br />\n";
echo("Root node name = ".$xml->getName()."<br />\n");
echo("number of top level nodes = ".count($xml->children())."<br />\n");
echo("Root games node name = ".$games->getName()."<br />\n");
echo("Number of Games = ".count($gameList)."<br />\n");
echo("Game sets available = ".(count($gameList)/$matchups)."<br />\n");
echo("Start Index for games block = ".$startIndex."<br />\n");
echo("Highest piossible new start Index = ".$endIndex."<br />\n");
}
/*-------------------------------------------------
/ 2.3 LOOP THROUGH GAME SETS AND MAKE MATCHUPS
/------------------------------------------------*/
$s = 1;
$totalGames = 0;
do {
if ($this->debug) {
echo("------------------------------------------------");
echo("Creating games for scoring period ".$s."<br />\n");
echo("------------------------------------------------");
}
$gamesCreated = 0;
while($gamesCreated < $matchups) {
// START EXTRACTING THE GAME BLOCk
$game = $gameList[$startIndex];
// GET GAME ATTRIBUTES
$attrs = $game->attributes();
$home_team_id = $teams[$attrs['home']-1];
$away_team_id = $teams[$attrs['away']-1];
if ($this->debug) {
echo("Start Index = ".$startIndex."<br />\n");
echo "Current set = ".$attrs['day']."<br />\n";
echo "Home team = ".$attrs['home']."<br />\n";
echo "Away team = ".$attrs['away']."<br />\n";
echo "Home team ID = ".$home_team_id."<br />\n";
echo "Away team ID = ".$away_team_id."<br />\n";
}
// PUSH THE GAME INTO THE GAME LIST
array_push($data,array('league_id'=>$league_id,'home_team_id'=>$home_team_id,
'away_team_id'=>$away_team_id,'scoring_period_id'=>$s));
$gamesCreated++;
$startIndex++;
$totalGames++;
if ($this->debug) {
echo("------------------------------------------------<br />");
echo("Games created for scoring period ".$s." = ".$gamesCreated."<br />\n");
echo("------------------------------------------------<br />");
}
} // END while($gamesCreated < $matchups)
$s++;
} while ($s < ($this->regular_scoring_periods + 1)); // END do...while
if ($this->debug) {
echo "Games needed for league league ".$league_id." is ".$gamesNeeded."<br />\n";
echo "Total games created for league ".$league_id." is ".$totalGames."<br />\n";
echo "Total game records in data array ".count($data)."<br />\n";
foreach($data as $query) {
foreach($query as $key => $val) {
echo($key." = ".$val."<br />");
}
}
} // END if
if (count($data) == $gamesNeeded) {
$this->db->flush_cache();
//$this->db->insert_batch($this->tables['GAMES'], $data);
foreach($data as $query) {
$this->db->flush_cache();
$this->db->insert($this->tables['GAMES'],$query);
} // END foreach
$rows = $this->db->affected_rows();
if ($this->debug) {
echo "Rows written = ".$rows."<br />\n";
}
$success = true;
$outMess = "OK";
} else {
$success = false;
$outMess = "Not enough games created vs games needed. Check the integrity of the schedule XML file in <code>"".DIR_WRITE_PATH.PATH_MEDIA_WRITE."schedules"</code> and try again";
} // END if
} // END if
} // END if
return $outMess;
} // END function
/**
* GET LEAGUE SCHEDULE.
* Returns either the entire schdule for the specified league OR only games for a
* specific team (if specified).
* @param $team_id - If not specified, the schedule for the entire league is returned.
* @param $excludeScores - Set to TRUE to not return score information
* @param $league_id - If not specified, no league filter is applied.
* @return schedule array, false on failure
*/
public function getLeagueSchedule($team_id = false, $excludeScores = false, $league_id = false) {
if ($league_id === false) { $league_id = $this->id; }
$teamNames = $this->loadLeagueTeams($league_id);
$schedule = array();
$score_period_id = 0;
$this->db->flush_cache();
$this->db->select('away_team_id, away_team_score, home_team_score, home_team_id, scoring_period_id');
$this->db->where('league_id',$league_id);
if ($team_id != false) {
$this->db->where('(home_team_id = '.$team_id.' OR away_team_id = '.$team_id.')');
}
$this->db->order_by('scoring_period_id','asc');
$query = $this->db->get($this->tables['GAMES']);
if ($query->num_rows() > 0) {
foreach($query->result() as $row) {
if ($score_period_id != $row->scoring_period_id) {
$score_period_id = $row->scoring_period_id;
$schedule[$score_period_id] = array();
}
$homeTeam = $teamNames[$row->home_team_id];
$awayTeam = $teamNames[$row->away_team_id];
$stats = array('home_team'=>$homeTeam, 'away_team'=>$awayTeam);
if (!$excludeScores && (isset($row->home_team_score) && isset($row->away_team_score))) {
$stats = $stats + array('home_team_score'=>$row->home_team_score,'away_team_score'=>$row->away_team_score);
}
array_push($schedule[$score_period_id],$stats);
}
}
$query->free_result();
return $schedule;
}
/**
* GET LEAGUE GAME COUNT.
* Returns a count of the number of games scheduled for the passed league
* @param $team_id - If not specified, the schedule for the entire league is returned.
* @param $excludeScores - Set to TRUE to not return score information
* @param $league_id - If not specified, no league filter is applied.
* @return schedule array, false on failure
*/
public function getLeagueGameCount($league_id = false) {
if ($league_id === false) { $league_id = $this->id; }
$count = 0;
$this->db->select('COUNT(id) as gameCount');
$this->db->where('league_id',$league_id);
$query = $this->db->get($this->tables['GAMES']);
$row = $query->row();
$count = $row->gameCount;
$query->free_result();
return $count;
}
/**
* GET LEAGUE TRANSACTIONS.
* Returns either the entire schdule for the specified league OR only games for a
* specific team (if specified).
* @param $team_id - If not specified, the schedule for the entire league is returned.
* @param $excludeScores - Set to TRUE to not return score information
* @param $league_id - If not specified, no league filter is applied.
* @return schedule array, false on failure
*/
public function getLeagueTransactions($limit = -1, $startIndex = 0, $team_id = false, $league_id = false) {
if ($league_id === false) { $league_id = $this->id; }
$transactions = array();
$teamNames = $this->loadLeagueTeams($league_id);
$this->db->select("trans_date, team_id, added, dropped, claimed, tradedTo, tradedFrom, trade_team_id, trans_owner, effective");
$this->db->where("league_id",$league_id);
if ($team_id !== false) {
$this->db->where('team_id',$team_id);
}
if ($limit != -1 && $startIndex == 0) {
$this->db->limit($limit);
} else if ($limit != -1 && $startIndex > 0) {
$this->db->limit($startIndex,$limit);
}
$this->db->order_by('trans_date','desc');
$query = $this->db->get($this->tables['TRANSACTIONS']);
if ($query->num_rows() > 0) {
$transTypes = array('added','dropped','claimed','tradedTo','tradedFrom');
if (!function_exists('getFantasyPlayersDetails')) {
$this->load->helper('roster');
}
foreach($query->result() as $row) {
$transArrays = array();
foreach ($transTypes as $field) {
//echo($field."<br />");
$transArrays[$field] = array();
if (isset($row->$field) && !empty($row->$field) && strpos($row->$field,":")) {
$fieldData = unserialize($row->$field);
if (is_array($fieldData) && sizeof($fieldData) > 0) {
//echo("size of ".$field." data = ".sizeof($fieldData)."<br />");
$playerDetails = getFantasyPlayersDetails($fieldData);
foreach ($fieldData as $playerId) {
//echo($field." player id = ".$playerId."<br />");
$transStr = '';
if (isset($playerDetails[$playerId])) {
$pos = $playerDetails[$playerId]['position'];
if ($pos == 1) { $pos = $playerDetails[$playerId]['role']; }
$transStr .= get_pos($pos);
$transStr .= " ".anchor('/players/info/league_id/'.$league_id.'/player_id/'.$playerId,$playerDetails[$playerId]['first_name']." ".$playerDetails[$playerId]['last_name']);
} // END if
//echo($transStr."<br />");
if (!empty($transStr)) { array_push($transArrays[$field], $transStr); }
} // END foreach
} // END if
} // END if
} // END foreach
if (!function_exists('getScoringPeriodByDate')) {
$this->load->helper('admin');
}
$trade_team_name = "";
if ($row->trade_team_id > 0) {
$trade_team_name = $teamNames[$row->trade_team_id];
}
array_push($transactions,array('trans_date'=>$row->trans_date, 'team_id'=>$row->team_id,
'added'=>$transArrays['added'], 'dropped'=>$transArrays['dropped'],
'claimed'=>$transArrays['claimed'], 'tradedTo'=>$transArrays['tradedTo'], 'tradedFrom'=>$transArrays['tradedFrom'],
'trans_owner'=>$row->trans_owner, 'effective'=>$row->effective,
'trade_team_id'=>$row->trade_team_id,'trade_team_name'=>$trade_team_name));
}
}
$query->free_result();
unset($query);
return $transactions;
}
/**
* GET WAIVER CLAIMS.
* Returns pending waiver claims for the specified league.
* @param $team_id - If not specified, the schedule for the entire league is returned.
* @param $league_id - If not specified, no league filter is applied.
* @return schedule array, false on failure
*/
public function getWaiverClaims($limit = -1, $startIndex = 0, $team_id = false, $league_id = false) {
if ($league_id === false) { $league_id = $this->id; }
$claims = array();
$this->db->select($this->tables['WAIVER_CLAIMS'].".id, ".$this->tables['WAIVER_CLAIMS'].".team_id, teamname, teamnick, ".$this->tables['WAIVER_CLAIMS'].".player_id, first_name, last_name");
$this->db->join("fantasy_teams","fantasy_teams.id = ".$this->tables['WAIVER_CLAIMS'].".team_id", "left");
$this->db->join("fantasy_players","fantasy_players.id = ".$this->tables['WAIVER_CLAIMS'].".player_id", "left");
//$this->db->join("fantasy_players_waivers","fantasy_players_waivers.player_id = fantasy_players.id", "left");
$this->db->join("players","fantasy_players.player_id = players.player_id", "left");
$this->db->where($this->tables['WAIVER_CLAIMS'].".league_id",$league_id);
if ($team_id !== false) {
$this->db->where($this->tables['WAIVER_CLAIMS'].'.team_id',$team_id);
}
if ($limit != -1 && $startIndex == 0) {
$this->db->limit($limit);
} else if ($limit != -1 && $startIndex > 0) {
$this->db->limit($startIndex,$limit);
}
$this->db->order_by('teamname, last_name','asc');
$query = $this->db->get($this->tables['WAIVER_CLAIMS']);
//echo($this->db->last_query()."<br />");
if ($query->num_rows() > 0) {
foreach($query->result() as $row) {
array_push($claims,array('id'=>$row->id,'team_id'=>$row->team_id, 'teamname'=>$row->teamname." ".$row->teamnick,
'player_id'=>$row->player_id, 'player_name'=>$row->first_name." ".$row->last_name));
}
}
$query->free_result();
unset($query);
return $claims;
}
/**
* GET SCORING RULES.
* Returns the scoring rules for a specific league (if they exist) or the
* global scoring rules in they don't
* @param $league_id - If not specified, no league filter is applied.
* @return Rules array on success, false on failure
*/
public function getScoringRules($league_id = false, $scoring_type = false, $returnDefault = true) {
if ($league_id === false) { $league_id = $this->id; } // END if
if ($scoring_type === false) { $scoring_type = $this->league_type; } // END if
$rules = array('batting'=>array(),'pitching'=>array());
$default = false;
foreach($rules as $key => $data) {
// TEST for custom scoring rules
// if not present, use the default rules for the league (league_id 0)
$this->db->where('league_id',$league_id);
$this->db->where('scoring_type',$scoring_type);
$this->db->from('fantasy_leagues_scoring_'.$key);
$count = $this->db->count_all_results();
if ($count == 0 && $returnDefault === true) {
$league_id = 0;
} // END if
$this->db->select('*');
$this->db->where('league_id',$league_id);
$this->db->where('scoring_type',$scoring_type);
$query = $this->db->get('fantasy_leagues_scoring_'.$key);
if ($query->num_rows() > 0) {
$score_type =
$cats = array();
foreach ($query->result() as $row) {
for ($i = 0; $i < 12; $i++) {
$columnT = "type_".$i;
$columnV = "value_".$i;
if ($row->$columnT != -1) {
$cats = $cats + array($row->$columnT=>$row->$columnV);
} // END if
} // END for
} // END foreach
$rules[$key] = $cats;
} // END if
$query->free_result();
unset($query);
} // END foreach
$rules['league_id'] = $league_id;
$rules['scoring_type'] = $scoring_type;
return $rules;
}
/**
* SET SCORING RULES
* This function accepts a form input object and applies the passed values to
* the scoring rules tables.
* @param $input CodeIgniter form input object
* @param $league_id Optional league ID. Defaults to "0" if no id is passed.
*/
public function setScoringRules($input, $league_id = false) {
if ($league_id === false) { $league_id = 0; }
$this->db->where('league_id', $league_id);
$this->db->where('scoring_type', $input->post('scoring_type'));
$this->db->delete($this->tables['SCORING_RULES_BATTING']);
$this->db->where('league_id', $league_id);
$this->db->where('scoring_type', $input->post('scoring_type'));
$this->db->delete($this->tables['SCORING_RULES_PITCHING']);
$types = array('batting','pitching');
foreach($types as $type) {
$lineCount = 0;
$data = array('league_id'=>$league_id,'scoring_type'=>$input->post('scoring_type'));
while ($lineCount < 11) {
if ($input->post($type.'_type_'.$lineCount) && $input->post($type.'_type_'.$lineCount) != -1) {
$data = $data + array('type_'.$lineCount=>$input->post($type.'_type_'.$lineCount),
'value_'.$lineCount=>$input->post($type.'_value_'.$lineCount));
}
$lineCount++;
}
$this->db->insert($this->tables['SCORING_RULES_'.strtoupper($type)],$data);
//echo($this->db->last_query()."<br />");
}
return true;
}
public function getAllScoringRulesforSim($league_id = false) {
$scoring_rules = array();
$scoring_types = loadSimpleDataList('leagueType');
// ASSEMBLE UNIQUE SCORING RULES FOR LEAGUES
if ($league_id !== false) {
$scoring_list = array();
foreach($scoring_types as $typeId => $typeName) {
$scoring_list[$typeId] = $this->getScoringRules($league_id,$typeId);
}
$scoring_rules[$league_id] = $scoring_list;
} else {
$leagues = $this->getLeagues();
//echo($this->_NAME." = ".$this->db->last_query()."<br />");
$returnDefault = true;
foreach ($leagues as $id => $data) {
$scoring_list = array();
foreach($scoring_types as $typeId => $typeName) {
if (isset($scoring_rules[0]) && sizeof($scoring_rules[0]) > 0) {
$returnDefault = false;
}
$scoring_list[$typeId] = $this->getScoringRules($id,$typeId,$returnDefault);
}
if ($scoring_list !== false) {
$scoring_rules = $scoring_rules + array($id=>$scoring_list);
}
}
}
return $scoring_rules;
}
/**
* GET Roster RULES.
* Returns the roster rules for a specific league (if they exist) or the
* global roster rules in they don't
* @param $league_id - If not specified, no league filter is applied.
* @return Rules array on success, false on failure
*/
public function getRosterRules($league_id = false) {
if ($league_id === false) { $league_id = $this->id; }
$rules = array();
$this->db->select();
$this->db->from('fantasy_roster_rules');
$this->db->where('league_id',$league_id);
if ($this->db->count_all_results() == 0) {
$league_id = 0;
$this->db->where('league_id',$league_id);
}
$this->db->select();
$this->db->order_by('position', 'asc');
$query = $this->db->get('fantasy_roster_rules');
//echo($this->db->last_query()."<br />");
if ($query->num_rows() > 0) {
foreach($query->result() as $row) {
//echo($row->position."<br />");
$rules = $rules + array($row->position=>array('position'=>$row->position,
'active_min'=>$row->active_min,'active_max'=>$row->active_max));
}
}
return $rules;
}
/**
* SET ROSTER RULES
* This function accepts a form input object and applies the passed values to
* the roster rules table.
* @param $input CodeIgniter form input object
* @param $league_id Optional league ID. Defaults to "0" if no id is passed.
*/
public function setRosterRules($input, $league_id = false) {
if ($league_id === false) { $league_id = 0; }
$this->db->where('league_id', $league_id);
$this->db->delete($this->tables['ROSTER_RULES']);
$lineCount = 0;
while ($lineCount < 10) {
$data = array();
if ($input->post('pos'.$lineCount) && $input->post('pos'.$lineCount) != -1) {
$data = array('league_id'=>$league_id,
'position'=>$input->post('pos'.$lineCount),
'active_min'=>$input->post('min'.$lineCount),
'active_max'=>$input->post('max'.$lineCount));
$this->db->insert($this->tables['ROSTER_RULES'],$data);
}
$lineCount++;
}
$types = array(100=>'active', 101=>'reserve',102=>'injured');
foreach($types as $code => $label) {
$this->db->insert($this->tables['ROSTER_RULES'],array('league_id'=>$league_id,
'position'=>$code,
'active_min'=>$input->post('total_'.$label.'_min'),
'active_max'=>$input->post('total_'.$label.'_max')));
}
return true;
}
/**
* VALIDATE ROSTER
*
* This function tests the provided roster against the leagues roster rules.
* @param $roster Team Roster Array
* @param $league_id Optional league ID. Defaults to model value if no id is passed.
* @return {Boolean} TRUE if valid, FALSE if not
* @since 1.0
*
*/
public function validateRoster($roster,$league_id = false) {
//echo("Validate Roser<br />");
$valid = true;
if ($league_id === false) { $league_id = $this->id; }
$errors = "";
$rules = $this->getRosterRules($league_id);
$errCount = 0;
$activePos = array();
$activeCount = 0;
$reserveCount = 0;
$injuredCount = 0;
//echo("Roster size = ".sizeof($roster)."<br />");
foreach($roster as $player_info) {
//echo("Player ".$player_info['player_name']."<br />");
if ($player_info['player_status'] == 1) {
if ($player_info['player_position'] != 1) {
$pos = $player_info['player_position'];
} else {
$pos = $player_info['player_role'];
}
//echo("Player pos = ".get_pos($pos)."<br />");
if (!isset($activePos[$pos])) {
$activePos[$pos] = 1;
} else {
$activePos[$pos] += 1;
}
$activeCount++;
} else if ($player_info['player_status'] == -1) {
$reserveCount++;
} else if ($player_info['player_status'] == 2) {
$injuredCount++;
}
}
foreach($rules as $position => $ruleData) {
if ($position < 100) {
if (isset($activePos[$position])) {
if ($activePos[$position] < $ruleData['active_min']) {
//echo("Count for ".strtoupper(get_pos($position))." is below the minimum of ".$ruleData['active_min']."<br />");
$valid = false;
if ($errors != '') { $errors .= "<br />"; }
$errors .= "The position ".strtoupper(get_pos($position))." has ".$activePos[$position]." active players. At least ".$ruleData['active_min']." ".($ruleData['active_min']>1 ? "are" : "is")." required.";
$errCount++;
} else if ($activePos[$position] > $ruleData['active_max']) {
$valid = false;
if ($errors != '') { $errors .= "<br />"; }
$errors .= "The position ".strtoupper(get_pos($position))." has ".$activePos[$position]." active players. At most, ".$ruleData['active_max']." ".($ruleData['active_min']>1 ? "are" : "is")." allowed.";
$errCount++;
}
} else {
if ($ruleData['active_min'] > 0) {
$valid = false;
if ($errors != '') { $errors .= "<br />"; }
$errors .= "The position ".strtoupper(get_pos($position))." has 0 active players. At least ".$ruleData['active_min']." ".($ruleData['active_min']>1 ? "are" : "is")." required.";
$errCount++;
}
}
} else {
if ($position == 100) {
//$active = sizeof($activeCount);
if ($activeCount < $ruleData['active_min']) {
$valid = false;
if ($errors != '') { $errors .= "<br />"; }
$errors .= "There are only ".$activeCount." active players. At least ".$ruleData['active_min']." ".($ruleData['active_min']>1 ? "are" : "is")." required.";
$errCount++;
} else if ($activeCount > $ruleData['active_max']) {
$valid = false;
if ($errors != '') { $errors .= "<br />"; }
$errors .= "There are ".$activeCount." active players. At most, ".$ruleData['active_max']." ".($ruleData['active_max']>1 ? "are" : "is")." allowed.";
$errCount++;
}
}
if ($position == 101) {;
if ($reserveCount < $ruleData['active_min']) {
$valid = false;
if ($errors != '') { $errors .= "<br />"; }
$errors .= "There are only ".$reserveCount." reserve players. At least ".$ruleData['active_min']." ".($ruleData['active_min']>1 ? "are" : "is")." required.";
$errCount++;
} else if ($reserveCount > $ruleData['active_max']) {
$valid = false;
if ($errors != '') { $errors .= "<br />"; }
$errors .= "There are ".$reserveCount." reserve players. At most, ".$ruleData['active_max']." ".($ruleData['active_max']>1 ? "are" : "is")." allowed.";
$errCount++;
}
}
if ($position == 102) {;
if ($injuredCount < $ruleData['active_min']) {
$valid = false;
if ($errors != '') { $errors .= "<br />"; }
$errors .= "There are only ".$injuredCount." inured reserrve players. At least ".$ruleData['active_min']." ".($ruleData['active_min']>1 ? "are" : "is")." required.";
$errCount++;
} else if ($injuredCount > $ruleData['active_max']) {
$valid = false;
if ($errors != '') { $errors .= "<br />"; }
$errors .= "There are ".$injuredCount." injured reserve players. At most, ".$ruleData['active_max']." ".($ruleData['active_max']>1 ? "are" : "is")." allowed.";
$errCount++;
}
}
}
}
if (!$valid) {
$this->errorCode = 1;
}
$this->errorCount = $errCount;
$this->statusMess = $errors;
return $valid;
}
/**
* VALIDATE LEAGUE ROSTERS
*
* Validate the rosters of all teams in a league.
* Mainly a simple helper that calls validateRoster() for the passed teams.
*
* @param $scoring_period {Array}
* @param $league_id {Int}
* @param $excludeList {Array} Array of treams to exclude from validation
* @return {Array} Array of validation reports for each team
* @since 1.0.3 PROD
*/
public function validateRosters($scoring_period = false, $league_id = false, $excludeList = array()) {
if ($league_id === false) { $league_id = $this->id; }
if ($excludeList === false) { $excludeList = array(); }
if ($scoring_period === false) { array('id'=>1, 'date_start'=>EMPTY_DATE_STR, 'date_end'=>EMPTY_DATE_STR,'manual_waivers'=>-1); }
$validation = array();
if ($league_id === false) { $league_id = $this->id; }
$message = "";
$allValid = true;
if (!function_exists('getBasicRoster')) {
$this->load->helper('roster');
}
$teams = $this->getTeamDetails($league_id);
if (sizeof($teams) > 0) {
foreach($teams as $team_id => $details) {
if (!in_array($team_id, $excludeList, false)) {
$playoffSettings = $this->getPlayoffSettings($league_id);
$total_periods = intval($playoffSettings['regular_scoring_periods']) + intval($playoffSettings['playoff_rounds']);
if ($scoring_period['id'] <= $total_periods) {
$roster = getBasicRoster($team_id, $scoring_period);
$valid = $this->validateRoster($roster, $league_id);
} else {
$valid = true;
}
array_push($validation, array('team_id'=>$team_id, 'details'=>$details, 'rosterValid'=>(($valid)?1:-1), 'issueCount'=>$this->errorCount, 'validationDetails'=>$this->statusMess));
if (!$valid) $allValid = false;
} else {
array_push($validation, array('team_id'=>$team_id, 'details'=>$details, 'rosterValid'=>100, 'issueCount'=>0, 'validationDetails'=>'Validation Skipped'));
}
}
} else {
array_push($validation, array('team_id'=>'-1', NULL, -1, 1, "The League has no teams."));
}
if (!$allValid) {
$this->errorCode = 1;
$this->$errorCount = $errorCount;
$this->statusMess = "error|One or more rosters are currently invalid!";
} else {
$this->statusMess = "success|All rosters are currently valid!";
}
return $validation;
}
/**
* GET GAMES FOR PERIOD
*
* Returns the games for a league for the passed scoring period
*
* @param $period_id {Int} The scoring period ID
* @param $excludeList {Array} Sets game count to 0 for any passed IDs
* @param $league_id {Int} Optional league ID. Defaults to model value if no id is passed.
* @return {Array} An array of games
* @since 1.0
*
*
*/
public function getGamesForPeriod($period_id = false, $excludeList = array(), $league_id = false) {
if ($period_id === false) { $period_id = 1; }
if ($league_id === false) { $league_id = $this->id; }
$games = array();
$this->db->select('id, home_team_id, home_team_score, away_team_id, away_team_score');
$this->db->where('league_id',intval($league_id));
$this->db->where('scoring_period_id',intval($period_id));
$query = $this->db->get('fantasy_leagues_games');
//print($this->db->last_query()."<br />");
if ($query->num_rows() > 0) {
foreach ($query->result() as $row) {
// RESOLVE TEAM NAMES
$homeTeamName = resolveTeamName($row->home_team_id);
$awayTeamName = resolveTeamName($row->away_team_id);
$home_score = $row->home_team_score;
$away_score = $row->away_team_score;
if (sizeof($excludeList) > 0) {
if (in_array($row->home_team_id, $excludeList)) {
$home_score = 0;
}
if (in_array($row->away_team_id, $excludeList )) {
$away_score = 0;
}
}
$games = $games + array($row->id=>array('home_team_id'=>$row->home_team_id, 'home_team_name'=>$homeTeamName, 'home_team_score'=>$home_score,
'away_team_id'=>$row->away_team_id, 'away_team_name'=>$awayTeamName, 'away_team_score'=>$away_score));
}
}
$query->free_result();
return $games;
}
public function getAvailableScoringPeriods($league_id = false) {
if ($league_id === false) { $league_id = $this->id; }
$periods = array();
$this->db->flush_cache();
$this->db->select('scoring_period_id');
$this->db->where('league_id',$league_id);
$this->db->group_by('scoring_period_id');
$query = $this->db->get('fantasy_players_scoring');
if ($query->num_rows() == 0) {
$league_id = 0;
}
$query->free_result();
$this->db->flush_cache();
$this->db->select('scoring_period_id');
$this->db->where('league_id',$league_id);
$this->db->group_by('scoring_period_id');
$query = $this->db->get('fantasy_players_scoring');
if ($query->num_rows() > 0) {
foreach ($query->result() as $row) {
array_push($periods,$row->scoring_period_id);
}
}
$query->free_result();
asort($periods);
return $periods;
}
public function getAvailableRosterPeriods($league_id = false, $total_periods = false) {
if ($league_id === false) { $league_id = $this->id; }
$periods = array();
$this->db->flush_cache();
$this->db->select('scoring_period_id');
$this->db->where('league_id',$league_id);
$this->db->group_by('scoring_period_id');
$query = $this->db->get('fantasy_rosters');
if ($query->num_rows() == 0) {
$league_id = 0;
}
$query->free_result();
$this->db->flush_cache();
$this->db->select('scoring_period_id');
$this->db->where('league_id',$league_id);
$this->db->group_by('scoring_period_id');
$query = $this->db->get('fantasy_rosters');
$count = 0;
if ($query->num_rows() > 0) {
if ($total_periods === false) $maxcount = $query->num_rows(); else $maxcount = $total_periods;
foreach ($query->result() as $row) {
if ($count < $maxcount) {
array_push($periods,$row->scoring_period_id);
$count++;
}
}
}
$query->free_result();
asort($periods);
return $periods;
}
public function getAvailableStandingsPeriods($league_id = false) {
if ($league_id === false) { $league_id = $this->id; }
$periods = array();
$this->db->flush_cache();
$this->db->select('scoring_period_id');
$this->db->where('league_id',$league_id);
$this->db->group_by('scoring_period_id');
$query = $this->db->get('fantasy_teams_record');
if ($query->num_rows() > 0) {
foreach ($query->result() as $row) {
array_push($periods,$row->scoring_period_id);
}
}
$query->free_result();
asort($periods);
return $periods;
}
/*--------------------------------------------------------------------------------------------
/
/
/ SIM PROCESSING SUPPORT
/
/
/--------------------------------------------------------------------------------------------*/
/**
* COPY ROSTERS.
* Copies all rosters for a league from the current scoring period to a new one.
*
* @param $old_scoring_period {int} Old Scoring period ID
* @param $new_scoring_period {int} New Scoring period ID
* @param $league_id {int} League ID
* @return {Boolean} TRUE on success, FALSE on error
*
* @since 1.0
* @access public
*
*/
public function copyRosters($old_scoring_period, $new_scoring_period, $league_id = false) {
if ($league_id === false) { $league_id = $this->id; } // END if
$this->db->select('*');
$this->db->from('fantasy_rosters');
$this->db->where('league_id',$league_id);
$this->db->where('scoring_period_id',$old_scoring_period);
$pQuery = $this->db->get();
if ($pQuery->num_rows() > 0) {
foreach($pQuery->result() as $row) {
$data = array('league_id'=>$league_id, 'team_id'=>$row->team_id, 'player_id'=>$row->player_id,
'player_position'=>$row->player_position, 'scoring_period_id'=>$new_scoring_period, 'player_role'=>$row->player_role, 'player_status'=>$row->player_status);
$this->db->insert('fantasy_rosters',$data);
} // END foreach
} // END if
$pQuery->free_result();
return true;
}
/**
* UPDATE LEAGUE SCORING
*
* @param $scoring_period The scoring period to update against
* @param $league_id The league to process, defaults to $this->id if no value passed
* @param $ootp_league_id OOTP League ID value, defaults to 100 if no value passed
* @return Summary String
* @since 1.0
* @version 1.1 (Revised OOTPFL 1.0.4)
*
*/
public function updateLeagueScoring($scoring_period, $league_id = false, $ootp_league_id = 100) {
$error = false;
if ($league_id === false) { $id = $this->id; } // END if
/*--------------------------------------
/ 1.0 SET BASIC VARS
/-------------------------------------*/
$league_name = $this->getLeagueName($league_id);
$scoring_type = $this->getScoringType($league_id);
/*--------------------------------------
/ 1.1 LOAD LANGUAGE IF NOT ALREADY EXISTS
/-------------------------------------*/
$noteam = $this->lang->line('sim_no_teams');
if (empty($noteam)) {
$this->lang->load('admin');
} // END if
unset($noteam);
$summary = str_replace('[LEAGUE_NAME]',$league_name,$this->lang->line('sim_league_processing'));
/*------------------------------------------------
/ 2.0 PROCEED IF THE LEAGUE IS POPULATED
/-----------------------------------------------*/
if ($this->hasTeams($league_id)) {
/*----------------------------------------------------
/ 2.1 GET TEAM LIST & DETAILS
/---------------------------------------------------*/
$teams = $this->getTeamDetails($league_id);
$summary .= str_replace('[TEAM_COUNT]',sizeof($teams),$this->lang->line('sim_team_count'));
/*----------------------------------------------------
/ 2.2 VALIDATE ROSTERS AND GENERATE EXCLUSION LISTS
/---------------------------------------------------*/
if (!function_exists('getBasicRoster')) {
$this->load->helper('roster');
} // END if
$excludeList = array();
$valSum = "";
foreach($teams as $team_id => $teamData) {
if (!$this->validateRoster(getBasicRoster($team_id, $scoring_period), $league_id )) {
array_push($excludeList,$team_id);
$valSum .= str_replace('[LEAGUE_NAME]',$league_name,$this->lang->line('sim_roster_validation_error'));
$valSum = str_replace('[TEAM_NAME]',$teamData['teamname']." ".$teamData['teamnick'],$valSum);
} // END if
} // END foreach
if (!empty($valSum)) {
$summary .= $this->lang->line('sim_roster_validation_title').$valSum.$this->lang->line('sim_roster_validation_postfix');
} // END if
/*--------------------------------------
/ 2.3 SCORING AND RECORDS
/-------------------------------------*/
$scoring_type = $this->getScoringType($league_id);
$scoring_rules = $this->getScoringRules($league_id,$scoring_type);
$this->loadPlayerScoring($scoring_period,$league_id, $scoring_rules, $scoring_type);
$summary .= $this->lang->line('sim_process_h2h');
// IF RUNNING ON THE FINAL DAY OF THE SIM
$summary .= $this->updateTeamScoring($scoring_period, $league_id, $excludeList, $scoring_rules, $scoring_type);
$summary .= $this->updateTeamRecords($scoring_period, $league_id, $excludeList, $scoring_rules, $scoring_type);
/*--------------------------------------
/ 2.4 ROSTERS
/-------------------------------------*/
// COPY CURRENT ROSTERS TO NEXT SCORING PERIOD
$summary .= $this->lang->line('sim_process_copy_rosters');
$this->copyRosters($scoring_period['id'], ($scoring_period['id'] + 1), $league_id);
/*--------------------------------------
/ 2.5 TRADES
/-------------------------------------*/
// IF TRADE EXPIRATIONS ARE ENABLED, PROCESS EXPIRING TRADES
if ((isset($this->params['config']['useTrades']) && $this->params['config']['useTrades'] == 1 && $this->params['config']['tradesExpire'] == 1)) {
$summary .= $this->lang->line('sim_process_trades');
$summary .= $this->expireOldTrades($league_id, true, $this->debug);
} // END if
// INCREMENT REMAINING OFFERED TRADES FOR THE SCORING PERIOD TO THE NEXT ONE
$summary .= $this->lang->line('sim_increment_trades');
$summary .= $this->incrementTrades($scoring_period['id'], $league_id, 'same', $this->debug);
/*--------------------------------------
/ 2.6 WAIVERS
/-------------------------------------*/
// IF ENABLED, PROCESS WAIVERS
if ((isset($this->params['config']['useWaivers']) && $this->params['config']['useWaivers'] == 1)) {
$summary .= $this->lang->line('sim_process_waivers');
$summary .= $this->processWaivers(($scoring_period['id'] + 1), $league_id, 'same', $this->debug);
} // END if
} else {
$this->errorCode = 1;
$summary .= $this->lang->line('sim_no_teams');
} // END if
/*--------------------------------------
/ 3.0 APPLY SUMMARY AND RETURN
/-------------------------------------*/
// APPLY SUMMARY TO RETURN ADMIN CONTROLLER
if ($this->errorCode == -1) {
$summary = $this->lang->line('success').$summary;
} else {
$summary = $this->lang->line('error').$summary;
} // END if
return $summary;
}
/**
* LOAD PLAYER SCORING.
* This function loads the compiled scoring results from the players copiled stats tables and stores them to local member
* variables for use in later processing. This uses memory, but saves calling the stats from the DB for each subsequent
* league being processed.
*
* This function will also compile head to head points for all active players across all head-to head leagues.
*
* @param $scoring_period The scoring period to update against
* @param $league_id The league to process, defaults to $this->id if no value passed
* @param $excludeList A list of team IDs that are excempt from scoring updates due to an illegal roster
* @param $scoring_rules Array of scoring categories and values
* @param $scoring_type The leagues scoring type (rotisserie or head to head)
* @param $debug Trace debugging flag
* @return Summary String
* @since 1.0.6
* @version 1.0
*
*/
protected function loadPlayerScoring($scoring_period = false, $league_id = false, $scoring_rules = false, $scoring_type = false, $debug = false) {
if ($league_id === false) { $league_id = $this->id; }
if ($scoring_period === false) return false;
//------------------------------------
// 2.2 LOAD COMPILED SCORING STATS
// EXACT STAT CATEGORIES TO LOAD IS
// BASED ON LEAGUE SCORING RULES
//-----------------------------------
if (sizeof($this->compiledStats) == 0) {
$this->compiledStats = array('batting'=>array(),'pitching'=>array());
}
if (sizeof($this->columnList) == 0) {
$this->columnList = array('batting'=>array(),'pitching'=>array());
}
if (sizeof($this->sqlColNames) == 0) {
$this->sqlColNames = array('batting'=>array(),'pitching'=>array());
}
$playerTypes = array('batting','pitching');
foreach($playerTypes as $playerType) {
if (sizeof($this->compiledStats[$playerType]) == 0) {
$select = "*";
// GET ALL PLAYERS SCORING, LIMITED TO JUST ROSTER PLAYERS FOR ROTISSERIE
$player_stats = array();
$this->db->flush_cache();
$this->db->select($select);
$this->db->where("scoring_period_id",intval($scoring_period['id']));
$query = $this->db->get("fantasy_players_compiled_".$playerType);
if ($debug) {
echo($this->db->last_query()."<br />");
}
if ($query->num_rows() > 0) {
$this->columnList[$playerType] = $query->list_fields();
foreach($query->result() as $row) {
$this->compiledStats[$playerType][$row->player_id] = $row;
}
} // END if
$query->free_result();
}
if ($scoring_type == LEAGUE_SCORING_HEADTOHEAD) {
foreach($this->compiledStats[$playerType] as $player_id => $statsRow) {
$totalVal = 0;
foreach($scoring_rules[$playerType] as $cat => $val) {
$colName = strtolower(get_ll_cat($cat, true));
if (isset($statsRow->$colName)) {
if (isset($score_vals[$cat])) {
$score_vals[$cat] += $statsRow->$colName;
} else {
$score_vals[$cat] = $statsRow->$colName;
} // END if
$totalVal += ($statsRow->$colName * $val);
} // END if
} // END foreach
// APPLY VALUES TO THE STATS AND SAVE THEM TO THE PLAYERS SCORING TABLE
$this->db->flush_cache();
$this->db->select('id');
$this->db->where('player_id',$player_id);
$this->db->where('scoring_period_id',$scoring_period['id']);
$this->db->where('league_id',$league_id);
$this->db->where('scoring_type',$scoring_type);
$tQuery = $this->db->get('fantasy_players_scoring');
$data = array();
$data['total'] = $totalVal;
if ($tQuery->num_rows() == 0) {
$this->db->flush_cache();
$data['player_id'] = $player_id;
$data['scoring_period_id'] = $scoring_period['id'];
$data['scoring_type'] = $scoring_type;
$data['league_id'] = $league_id;
$this->db->insert('fantasy_players_scoring',$data);
} else {
$this->db->flush_cache();
$this->db->where('player_id',$player_id);
$this->db->where('scoring_period_id',$scoring_period['id']);
$this->db->where('league_id',$league_id);
$this->db->where('scoring_type',$scoring_type);
$this->db->update('fantasy_players_scoring',$data);
} // END if
unset($data);
$tQuery->free_result();
}
}
}
return true;
}
/**
* UPDATE TEAM SCORING
*
* @param $scoring_period The scoring period to update against
* @param $league_id The league to process, defaults to $this->id if no value passed
* @param $excludeList A list of team IDs that are excempt from scoring updates due to an illegal roster
* @param $scoring_rules Array of scoring categories and values
* @param $scoring_type The leagues scoring type (rotisserie or head to head)
* @param $debug Trace debugging flag
* @return Summary String
* @since 1.0.4
* @version 1.0
*
*/
public function updateTeamScoring($scoring_period, $league_id = false, $excludeList = array(), $scoring_rules = false, $scoring_type = false, $debug = false) {
if ($league_id === false) { $league_id = $this->id; }
$summary = $this->lang->line('sim_process_scoring');
$summary .= "League Scoring Type = ".$scoring_type."<br />";
/*---------------------------------------
/
/ 1.0 LOAD TEAMS
/
/--------------------------------------*/
$teams = $this->getTeamIdList($league_id);
if (sizeof($teams) > 0) {
$summary .= str_replace('[TEAM_COUNT]',sizeof($teams),$this->lang->line('sim_process_scoring_teams'));
$summary = str_replace('[LEAGUE_ID]',$league_id,$summary);
/*---------------------------------------
/
/ 1.1 TEAM LOOP
/
/--------------------------------------*/
foreach($teams as $team_id) {
$teamRoster = array();
$team_score = 0;
$score_vals = array();
//echo("Team Id = ".$team_id."<br />");
$team_stats = array('batting'=>array(),'pitching'=>array());
//------------------------------------
// 1.1.1 GET PLAYERS FOR TEAM
//-----------------------------------
// DIVIDE ROSTER ARRAY INTO BATTERS AND PITCHERS
// REQUIRED FOR CONVERTING SCORING RULES TO SCORING RESULTS
// ONLE GET ROSTERS AND CORES IF THIS TEAM IF IT HAS A VALID ROSTER
if (sizeof($excludeList) == 0 || (sizeof($excludeList) > 0 && !in_array($team_id, $excludeList))) {
$this->db->select("player_id,player_position");
$this->db->where("team_id",intval($team_id));
$this->db->where("player_status",1);
$this->db->where("scoring_period_id",intval($scoring_period['id']));
$query = $this->db->get("fantasy_rosters");
if ($query->num_rows() > 0) {
foreach($query->result() as $row) {
$type = "batting";
if ($row->player_position == 1) {
$type = "pitching";
}
if (isset($this->compiledStats[$type][$row->player_id])) {
array_push($team_stats[$type],$this->compiledStats[$type][$row->player_id]);
}
} // END foreach
} // END if
$query->free_result();
} // END if sizeof($excludeList)
$playerTypes = array('batting','pitching');
foreach($playerTypes as $playerType) {
//------------------------------------
// 2.3 PROCESS STATS VS SCORING RULES
//-----------------------------------
foreach($team_stats[$playerType] as $row) {
//$totalVal = 0;
switch ($scoring_type) {
case LEAGUE_SCORING_ROTO:
case LEAGUE_SCORING_ROTO_5X5:
case LEAGUE_SCORING_ROTO_PLUS:
foreach($this->columnList[$playerType] as $field) {
if (isset($score_vals[$field])) {
$score_vals[$field] += $row->$field;
} else {
$score_vals[$field] = $row->$field;
} // END if
} // END foreach
break;
case LEAGUE_SCORING_HEADTOHEAD:
default:
foreach($scoring_rules[$playerType] as $cat => $val) {
$colName = strtolower(get_ll_cat($cat, true));
if (isset($row->$colName)) {
if (isset($score_vals[$cat])) {
$score_vals[$cat] += $row->$colName;
} else {
$score_vals[$cat] = $row->$colName;
} // END if
// UPDATE THE PLAYERS SCORING TOTAL
$team_score += $row->$colName * $val;
} // END if
} // END foreach
break;
} // END switch
} // END foreach $player_stats
} // END foreach $playerTypes
//------------------------------------
// 2.4 UPDATE TEAM SCORING RESULTS
//-----------------------------------
switch ($scoring_type) {
case LEAGUE_SCORING_ROTO:
case LEAGUE_SCORING_ROTO_5X5:
case LEAGUE_SCORING_ROTO_PLUS:
// EDIT - 1.0.6
$teamStatObj = new CompiledStats();
// FIRST LOAD EXISTING STATS, IF THEY EXIST
$this->db->flush_cache();
$this->db->select("stats_compiled");
$this->db->where('team_id',$team_id);
$use_period = $scoring_period['id'];
if ($scoring_period['id'] > 1) {
$use_period = $scoring_period['id'] - 1;
}
$this->db->where('scoring_period_id',$use_period);
$this->db->where('league_id',$league_id);
$tQuery = $this->db->get('fantasy_teams_scoring');
$teamStats = NULL;
if ($tQuery->num_rows() > 0) {
$row = $tQuery->row();
$teamStatObj->unserialize($row->stats_compiled);
}
$tQuery->free_result();
// SAVE COMPILED TEAM STATS TO A COMPILED STAT OBJECT FOR LATER USE
$teamStatObj->updateStats($score_vals);
// APPLY VALUES TO THE STATS AND SAVE THEM TO THE TEAM SCORING TABLE
$colCount = 0;
$team_vals = array();
if (sizeof($score_vals) > 0) {
$player_types = array('batting','pitching');
foreach ($player_types as $type) {
foreach($scoring_rules[$type] as $cat => $val) {
$team_vals['value_'.$colCount] = $teamStatObj->getCompiledStats($cat);
$colCount++;
} // END foreach
if ($colCount > 0 && $colCount < 6) { $colCount = 6; }
} // END foreach
// SAVE VALUES TO THE DB FOR THIS TEAM
} else {
for ($z = 0; $z < 12; $z++) {
$team_vals['value_'.$z] = 0;
}
}
$team_vals['stats_compiled'] = $teamStatObj->serialize();
if (sizeof($team_vals) > 0) {
$this->db->flush_cache();
$this->db->select('id');
$this->db->where('team_id',$team_id);
$this->db->where('scoring_period_id',$scoring_period['id']);
$this->db->where('league_id',$league_id);
$tQuery = $this->db->get('fantasy_teams_scoring');
if ($tQuery->num_rows() == 0) {
$this->db->flush_cache();
$team_vals['team_id'] = $team_id;
$team_vals['scoring_period_id'] = $scoring_period['id'];
$team_vals['league_id'] = $league_id;
$this->db->insert('fantasy_teams_scoring',$team_vals);
} else {
$this->db->flush_cache();
$this->db->where('team_id',$team_id);
$this->db->where('scoring_period_id',$scoring_period['id']);
$this->db->where('league_id',$league_id);
$this->db->update('fantasy_teams_scoring',$team_vals);
} // END if
$tQuery->free_result();
} // END if
break;
case LEAGUE_SCORING_HEADTOHEAD:
default:
// LOOK UP AND UPDATE THE SCORES OF ANY GAMES THIS TEAM IS PLAYING IN
if ($team_score != 0) {
$this->db->flush_cache();
$this->db->select('id, away_team_id, home_team_id');
$this->db->where('(away_team_id = '.$team_id.' OR home_team_id = '.$team_id.')');
$this->db->where('scoring_period_id',$scoring_period['id']);
$this->db->where('league_id',$league_id );
$query = $this->db->get('fantasy_leagues_games');
if ($query->num_rows() > 0) {
foreach($query->result() as $row) {
$score = array();
if ($row->away_team_id == $team_id) {
$col = 'away_team';
} else {
$col = 'home_team';
} // END if
//echo("update col = ".$col."<br />");
$score[$col.'_score'] = $team_score;
$this->db->flush_cache();
$this->db->where('id',$row->id);
$this->db->update('fantasy_leagues_games',$score);
//echo($this->db->last_query()."<br />");
} // END foreach
} // END if
$query->free_result();
} // END if
break;
} // END switch
} // END foreach teams
} // END if sizeof($teams) > 0
return $summary;
}
/**
* UPDATE TEAM RECORDS
*
* @param $scoring_period The scoring period to update against
* @param $league_id The league to process, defaults to $this->id if no value passed
* @param $excludeList A list of team IDs that are excempt from scoring updates due to an illegal roster
* @param $scoring_rules Array of scoring categories and values
* @param $scoring_type The leagues scoring type (rotisserie or head to head)
* @param $debug Trace debugging flag
* @return Summary String
* @since 1.0.4
* @version 1.0
*
*/
public function updateTeamRecords($scoring_period, $league_id = false, $excludeList = array(), $scoring_rules = false, $scoring_type = false, $debug = false) {
if ($league_id === false) { $league_id = $this->id; }
$summary = $this->lang->line('sim_process_records');
// GET ALL TEAMS
$teams = array();
$point_max = 0;
switch ($scoring_type) {
case LEAGUE_SCORING_ROTO:
case LEAGUE_SCORING_ROTO_5X5:
case LEAGUE_SCORING_ROTO_PLUS:
$fields = array();
$this->db->select();
$this->db->where("league_id",$league_id);
$this->db->where("scoring_period_id",$scoring_period['id']);
$query = $this->db->get($this->tables['TEAMS_SCORING']);
if ($query->num_rows() > 0) {
$fields = $query->list_fields();
} // END if
$point_max = $query->num_rows() - sizeof($excludeList);
$query->free_result();
//compile batting stats
//print("Compiling team standings for league ".$league_id."<br />");
//print("point_max = ".$point_max."<br />");
$i = 0;
$types = array('batting','pitching');
foreach($types as $type) {
foreach ($scoring_rules[$type] as $cat => $val) {
$order = 'desc';
switch ($cat) {
case 4:
case 30:
case 36:
case 37:
case 39:
case 40:
case 41:
case 42:
case 50:
case 59:
case 60:
case 61:
$order = 'asc';
break;
default:
break;
}
$point_count = $point_max;
$this->db->select('team_id');
$this->db->where("value_".$i." <> -1");
$this->db->where("league_id",$league_id);
$this->db->where("scoring_period_id",$scoring_period['id']);
$this->db->order_by("value_".$i, $order);
$query = $this->db->get($this->tables['TEAMS_SCORING']);
//print($this->db->last_query()."<br />");
//print("num results = ".$query->num_rows()."<br />");
if ($query->num_rows() > 0) {
foreach ($query->result() as $row) {
if (isset($teams[$row->team_id])) {
$teams[$row->team_id] += $point_count;
} else {
$teams[$row->team_id] = $point_count;
} // END if
// No negative numbers
if ($point_count > 0)
$point_count--;
} // END foreach
} // END if
$query->free_result();
$i++;
}
if ($i < 6) { $i = 6; }
}
//foreach ($fields as $field) {
//if (strpos($field,'value_') !== false) {
// BUILD QUERY TO RANK TEAM IDS BY THIS FIELD
//} // END if
//} // END switch
foreach ($teams as $id => $total) {
$this->db->flush_cache();
$this->db->set('total',$total);
$this->db->where('team_id',$id);
$this->db->where("league_id",$league_id);
$this->db->where("scoring_period_id",$scoring_period['id']);
$this->db->update($this->tables['TEAMS_SCORING']);
//print($this->db->last_query()."<br />");
} // END foreach
break;
case LEAGUE_SCORING_HEADTOHEAD:
default:
$this->db->select("fantasy_teams.id, g, w, l");
$this->db->join("fantasy_teams_record","fantasy_teams_record.team_id = fantasy_teams.id","left");
$this->db->where("fantasy_teams.league_id",$league_id);
$query = $this->db->get($this->tables['TEAMS']);
if ($query->num_rows() > 0) {
foreach($query->result() as $row) {
$games = 0;
$wins = 0;
$losses = 0;
$this->db->flush_cache();
$this->db->select('id, away_team_id, away_team_score, home_team_id, home_team_score');
$this->db->where('(away_team_id = '.$row->id.' OR home_team_id = '.$row->id.')');
$this->db->where('scoring_period_id',$scoring_period['id']);
$this->db->where('league_id',$league_id );
$gQuery = $this->db->get('fantasy_leagues_games');
if ($gQuery->num_rows() > 0) {
foreach($gQuery->result() as $gRow) {
if ($gRow->away_team_id == $row->id) {
$teamScore = $gRow->away_team_score;
$oppScore = $gRow->home_team_score;
} else {
$teamScore = $gRow->home_team_score;
$oppScore = $gRow->away_team_score;
}
if ($teamScore > $oppScore) {
$wins++;
} else {
$losses++;
}
$games++;
}
}
$gQuery->free_result();
$games += $row->g;
$wins += $row->w;
$losses += $row->l;
$perc = 0;
if ($games > 0) {
$perc = ($wins/$games);
}
$data = array("w"=>$wins,"l"=>$losses,"g"=>$games,'pct'=>$perc);
$this->db->flush_cache();
$this->db->select('id');
$this->db->where('team_id',$row->id);
$this->db->where("league_id",$league_id);
$this->db->where("scoring_period_id",$scoring_period['id']);
$tQuery = $this->db->get('fantasy_teams_record');
if ($tQuery->num_rows() == 0) {
$this->db->flush_cache();
$data['team_id'] = $row->id;
$data['league_id'] = $league_id;
$data['scoring_period_id'] = $scoring_period['id'];
$this->db->insert('fantasy_teams_record',$data);
} else {
$this->db->flush_cache();
$this->db->where('team_id',$row->id);
$this->db->where("league_id",$league_id);
$this->db->where("scoring_period_id",$scoring_period['id']);
$this->db->update('fantasy_teams_record',$data);
}
}
}
$query->free_result();
break;
}
return $summary;
}
/**
* GET TRADES IN LEAGUE REVIEW.
* Retrieves a list of all trades under league review It retreives trades within their active review period
* and approves all trade that have passed their review period deadline.
* @param $period_id {int} The scoring period to process waivers for.
* @param $league_id {int} If not specified, no league filter is applied.
* @param $expireDays {int} Number of days in the league review period, tested vs the trade response date
* @param $rosterPeriod {int} Whether to apply this function to the period specified in period_id or the next (period_id + 1)
* @param $debug {Boolean} TRUE to enabled tracing, FALSE to disable
* @return {Boolean} TRUE on success, FALSE on error
*
* @since 1.0.7
* @access public
*
*/
public function getTradesInLeagueReview($period_id = false, $league_id = false, $expireDays = false, $rosterPeriod = 'same', $debug = false) {
if ($period_id === false || $expireDays === false) {
$this->errorCode = 1;
$this->statusMess = "Period ID or protest expiration days parameters were not received.";
return false;
} // END if
if ($league_id === false) { $league_id = $this->id; } // END if
$tradeList = array();
$this->db->select('id, send_players, receive_players, team_1_id, team_2_id, response_date');
$this->db->where('league_id',$league_id);
$this->db->where('status',TRADE_PENDING_LEAGUE_APPROVAL);
$query = $this->db->get($this->tables['TRADES']);
if ($query->num_rows() > 0) {
foreach ($query->result() as $row) {
$expireDate = (strtotime($row->response_date) + ((60*60*24) * $expireDays));
if (time() < $expireDate) {
array_push($tradeList,$row);
} else {
if (!function_exists('processTrade')) { $this->load->helper('roster'); }
$processResponse = processTrade($row->id, TRADE_APPROVED, $this->lang->line('league_protest_expired_trade_approved'),$league_id);
if ($processResponse) {
$offeringTeamOwner = getTeamOwnerId($row->team_1_id);
$receivingTeamOwner = getTeamOwnerId($row->team_2_id);
$commishId = $this->getCommissionerId($league_id);
logTransaction(NULL, NULL, NULL, $row->send_players, $row->receive_players,
$commishId, $offeringTeamOwner,false, $period_id,
$league_id, $row->team_1_id, $offeringTeamOwner, $row->team_2_id);
logTransaction(NULL, NULL, NULL, $row->receive_players, $row->send_players,
$commishId, $receivingTeamOwner, false, $period_id,
$league_id, $row->team_2_id, $offeringTeamOwner, $row->team_1_id);
$types = array('offering','receiving');
$tradeTypes = loadSimpleDataList('tradeStatus');
foreach($types as $type) {
if ($type == 'offering') {
$msg = $this->lang->line('league_trade_approved_league_offering_team');
$msg = str_replace('[ACCEPTING_TEAM_NAME]', getTeamName($row->team_2_id), $msg);
$msg = str_replace('[USERNAME]', getUsername($offeringTeamOwner), $msg);
$msg = str_replace('[URL_LINEUP]', anchor('/team/info/'.$row->team_1_id,'adjust your lineup'),$msg);
$email = getEmail($offeringTeamOwner);
} else {
$msg = $this->lang->line('league_trade_approved_league_accepting_team');
$msg = str_replace('[OFFERING_TEAM_NAME]', getTeamName($row->team_1_id), $msg);
$msg = str_replace('[USERNAME]', getUsername($receivingTeamOwner), $msg);
$msg = str_replace('[URL_LINEUP]', anchor('/team/info/'.$row->team_2_id,'adjust your lineup'),$msg);
$email = getEmail($receivingTeamOwner);
} // END if
$msg .= $this->lang->line('email_footer');
$msg = str_replace('[COMMENTS]', $this->lang->line('league_protest_expired_trade_approved'),$msg);
$msg = str_replace('[LEAGUE_NAME]', $this->league_model->league_name,$msg);
$data['messageBody']= $msg;
//print("email template path = ".$this->config->item('email_templates')."<br />");
$data['leagueName'] = $this->getLeagueName($league_id);
$data['title'] = $this->lang->line('league_email_title_trade_response');
$message = $this->load->view($this->config->item('email_templates').'general_template', $data, true);
// SEND MESSAGES
// SEND TO TEAM ONE
$error = !sendEmail($email,getEmail($this->params['config']['primary_contact']),
$this->params['config']['site_name']." Administrator",$this->getLeagueName($league_id).' Fantasy League - Trade Update - Offer '.$tradeTypes[TRADE_COMPLETED],
$message,'','email_trd_');
} // END foreach
} // END if
} // END if
} // END foreach
} else {
$this->errorCode = 2;
$this->statusMess = "No trades under league review were found.";
} // END if
$query->free_result();
return $tradeList;
}
/**
* INCREMENT TRADES.
* Sets all offered trades scoring period to +1.
* @param $period_id {int} The scoring period to process waivers for.
* @param $league_id {int} If not specified, no league filter is applied.
* @param $rosterPeriod {int} Whether to apply this function to the period specified in period_id or the next (period_id + 1)
* @param $debug {Boolean} TRUE to enabled tracing, FALSE to disable
* @return {Boolean} TRUE on success, FALSE on error
*
* @since 1.0.6
* @access public
*
*/
public function incrementTrades($period_id = false, $league_id = false, $rosterPeriod = 'same', $debug = false) {
if ($period_id === false) { return false; }
if ($league_id === false) { $league_id = $this->id; }
$error = false;
$summary = '';
// COLLECT TRADES THAT ARE IN OFFERED STATE FOR THIS SCORING PERIOD
$this->db->select($this->tables['TRADES'].".id");
$this->db->where($this->tables['TRADES'].".league_id",$league_id);
$this->db->where("(".$this->tables['TRADES'].".status = ".TRADE_OFFERED." OR ".$this->tables['TRADES'].".status =".TRADE_PENDING_COMMISH_APPROVAL
." OR ".$this->tables['TRADES'].".status =".TRADE_PENDING_LEAGUE_APPROVAL.")");
$this->db->where($this->tables['TRADES'].".in_period",$period_id);
$query = $this->db->get($this->tables['TRADES']);
if ($debug) {
print($this->db->last_query()."<br />");
}
$summary .= str_replace('[COUNT]',$query->num_rows(),$this->lang->line('sim_process_trades_to_increment_count'));
if ($query->num_rows() > 0) {
$trade_id_str = "(";
foreach ($query->result() as $row) {
if ($trade_id_str != "(") { $trade_id_str .= ","; }
$trade_id_str .= $row->id;
}
$rowCount = 0;
$trade_id_str .= ")";
if ($trade_id_str != "()") {
// UPDATE THE TRADE
$this->db->where('id IN '.$trade_id_str);
$this->db->set('in_period',($period_id+1));
$this->db->update($this->tables['TRADES']);
if ($debug) {
print($this->db->last_query()."<br />");
}
$rowCount = $this->db->affected_rows();
}
$dbSummary = str_replace('[COUNT]',$rowCount,$this->lang->line('sim_increment_trades_count'));
$summary .= str_replace('[PERIOD_ID]',($period_id+1),$dbSummary);
}
$query->free_result();
if ($error) {
$summary .= $this->lang->line('sim_increment_trades_error');
}
return $summary;
}
/**
* EXPIRE OLD TRADES.
* Sets all trades that were effective in the specified period_id to <code>EXPIRED</code> status.
* @param $period_id {int} The scoring period to process waivers for.
* @param $league_id {int} If not specified, no league filter is applied.
* @param $rosterPeriod {int} Whether to apply this function to the period specified in period_id or the next (period_id + 1)
* @param $debug {Boolean} TRUE to enabled tracing, FALSE to disable
* @return {Boolean} TRUE on success, FALSE on error
*
* @since 1.0.6
* @access public
*
*/
public function expireOldTrades($league_id = false, $processSimExpirations = true, $debug = false) {
if ($league_id === false) { $league_id = $this->id; } // END if
$error = false;
$summary = '';
// COLLECT TRADES THAT ARE IN OFFERED STATE FOR THIS SCORING PERIOD
$this->db->select($this->tables['TRADES'].".id, offer_date, status, team_1_id, team_2_id, tradeStatus, in_period, expiration_days");
$this->db->join($this->tables['TRADES_STATUS'],$this->tables['TRADES_STATUS'].".id = ".$this->tables['TRADES'].".status", "right outer");
$this->db->where($this->tables['TRADES'].".league_id",$league_id);
$this->db->where($this->tables['TRADES'].".status",TRADE_OFFERED);
$this->db->where($this->tables['TRADES'].".expiration_days > -1"); // IGNORE TRADES MARKED AS NON-EXPIRING
if ($processSimExpirations === false) {
$this->db->where($this->tables['TRADES'].".expiration_days < 500"); // IGNORE NEXT SIM TRADES FOR CRON-ESQUE CHECKS
}
$query = $this->db->get($this->tables['TRADES']);
if ($debug) {
print($this->db->last_query()."<br />");
} // END if
$summary .= str_replace('[COUNT]',$query->num_rows(),$this->lang->line('sim_process_trades_to_expire_count'));
if ($query->num_rows() > 0) {
$ownersByTeam = $this->getDetailedOwnerInfo($league_id);
$this->lang->load('team');
$trade_id_str = "(";
$owner_info = array('receiving'=>array(),'offering' => array());
foreach ($query->result() as $row) {
$expireDate = strtotime(EMPTY_DATE_TIME_STR);
if ($row->expiration_days < 500) {
$expireDate = (strtotime($row->offer_date) + ((60*60*24) * $row->expiration_days));
}
if ($row->expiration_days == 500 || $expireDate > time()) {
if ($trade_id_str != "(") { $trade_id_str .= ","; } // END if
$trade_id_str .= $row->id;
array_push($owner_info['offering'],array("email"=>$ownersByTeam[$row->team_1_id]['email'],'username'=>$ownersByTeam[$row->team_1_id]['username'],
"receiving_team"=>$ownersByTeam[$row->team_2_id]['teamname']. " ".$ownersByTeam[$row->team_2_id]['teamnick']));
array_push($owner_info['receiving'],array("email"=>$ownersByTeam[$row->team_2_id]['email'],'username'=>$ownersByTeam[$row->team_2_id]['username'],
"offering_team"=>$ownersByTeam[$row->team_1_id]['teamname']. " ".$ownersByTeam[$row->team_1_id]['teamnick']));
}
} // END foreach
$trade_id_str .= ")";
if ($trade_id_str != "()") {
// UPDATE THE TRADE
$this->db->where('id IN '.$trade_id_str);
$this->db->set('status',TRADE_EXPIRED);
$this->db->set('response_date',date('Y-m-d h:m:s',time()));
if (!empty($comments)) {
$this->db->set('response',$this->lang->line('team_trade_auto_expired'));
} // END if
$this->db->update($this->tables['TRADES']);
$summary .= str_replace('[COUNT]',$this->db->affected_rows(),$this->lang->line('sim_process_trades_count'));
if ($debug) {
print($this->db->last_query()."<br />");
} // END if
$tradeTypes = loadSimpleDataList('tradeStatus');
$emailCount = 0;
foreach ($owner_info as $infoType => $infoData) {
foreach ($infoData as $ownerData) {
$msg = '';
switch ($infoType) {
case 'offering':
$msg = $this->lang->line('team_trade_expired_offering_team').$this->lang->line('email_footer');
$msg = str_replace('[ACCEPTING_TEAM_NAME]', $ownerData['receiving_team'], $msg);
break;
case 'receiving':
$msg = $this->lang->line('team_trade_expired').$this->lang->line('email_footer');
$msg = str_replace('[OFFERING_TEAM_NAME]', $ownerData['offering_team'], $msg);
break;
default:
$msg = $this->lang->line('team_trade_expired').$this->lang->line('email_footer');
$msg = str_replace('[ACCEPTING_TEAM_NAME]', 'Unknown Team', $msg);
$msg = str_replace('[OFFERING_TEAM_NAME]', 'Unknown Team', $msg);
break;
} // END switch
$msg = str_replace('[USERNAME]', $ownerData['username'], $msg);
$data['messageBody']= $msg;
$data['leagueName'] = $this->league_name;
$data['title'] = $this->lang->line('team_email_title_trade_response');
$message = $this->load->view($this->config->item('email_templates').'general_template', $data, true);
// SEND MESSAGES
$error = !sendEmail($ownerData['email'],getEmail($this->params['config']['primary_contact']),
$this->params['config']['site_name']." Administrator",$this->league_name.' Fantasy League - Trade Update - Offer '.$tradeTypes[TRADE_EXPIRED],
$message,'','email_trd_msg_');
$emailCount++;
} // END foreach
} // END foreach
$summary .= str_replace('[COUNT]',$emailCount,$this->lang->line('sim_process_trades_emails'));
}
} // END if
$query->free_result();
if ($error) {
$summary .= $this->lang->line('sim_process_trades_error');
} // END if
return $summary;
}
/**
* PROCESS WAIVERS.
* Processes pending waiver claims for the specified league and clears waivers for the selected scoring period.
* @param $period_id {int} The scoring period to process waivers for.
* @param $league_id {int} If not specified, no league filter is applied.
* @param $rosterPeriod {int} Whether to apply this function to the period specified in period_id or the next (period_id + 1)
* @param $debug {Boolean} TRUE tO enabled tracing, FALSE to disable
* @return {Boolean} TRUE on success, FALSE on error
*
* @since 1.0
* @access public
*/
public function processWaivers($period_id = false, $league_id = false, $rosterPeriod = 'same', $debug = false) {
if ($period_id === false) { return; }
if ($league_id === false) { $league_id = $this->id; }
$summary = '';
// GET LEAGUE TEAM ID LIST
if (!function_exists('getPlayersOnWaivers')) {
$this->load->helper('roster');
}
$playersOnWaivers = getPlayersOnWaivers($period_id, $league_id);
$claims = $this->getWaiverClaims(-1,0,false,$league_id);
$waiverOrder = getWaiverOrder($league_id, true);
//if ($debug) {
$summary .= "# of players on waivers = ".sizeof($playersOnWaivers)."<br />";
$summary .= "# of claims by teams = ".sizeof($claims)."<br />";
$summary .= "waiver order = ".sizeof($waiverOrder)."<br />";
//}
foreach($playersOnWaivers as $player) {
// SEE IF THERE IS A WAIVER CLAIM FOR THIS PLAYER
$numClaims = 0;
$claimList = array();
$claimCount = 1;
foreach($claims as $claim) {
$summary .= "claim ".$claimCount." team = ".$claim['team_id'].", player = ".$claim['player_id']."<br />";
if ($claim['player_id'] == $player['player_id']) {
// CLAIMS FOUND
$numClaims++;
array_push($claimList, $claim['team_id']);
}
} // END foreach
$summary .= "current player = ".$player['player_id']."<br />";
$summary .= "# of claims for player ".$player['player_id']." = ".$numClaims."<br />";
if ($numClaims > 0) {
$index = 0;
foreach($waiverOrder as $team_id) {
if (in_array($team_id, $claimList)) {
//if ($debug) {
$summary .= "claim found for player ".$player['player_id']." by team = ".$team_id."<br />";
//}
// CLAIM THIS PLAYER FOR TEAM
$this->db->set('team_id',$team_id);
$this->db->set('player_id',$player['player_id']);
$this->db->set('league_id',$league_id);
if ($player['position'] == 7 || $player['position'] == 8 || $player['position'] == 9) { $player['position'] = 20; }
if ($player['role'] == 13) { $player['role'] = 12; }
$this->db->set('player_position',$player['position']);
$this->db->set('player_role',$player['role']);
if ($rosterPeriod != 'same') {
$period_id--;
} // END if
$this->db->set('scoring_period_id',$period_id);
$this->db->insert('fantasy_rosters');
if (!function_exists('updateOwnership')) {
$this->load->helper('roster');
} // END if
$ownership = updateOwnership($player['player_id']);
$pData = array('own'=>$ownership[0],'start'=>$ownership[1]);
$this->db->flush_cache();
$this->db->where('id',$claim['player_id']);
$this->db->update('fantasy_players',$pData);
// LOG THE TRANSACTION
logTransaction(NULL, NULL, array($player['player_id']),NULL, NULL, -1,1, false,$period_id,$league_id,$team_id,-1);
// REMOVE TEAM FROM WAIVER ORDER ARRAY AND PUT IT AT THE END
$waiveTeam = array_splice($waiverOrder,$index, 1);
array_push($waiverOrder,$waiveTeam[0]);
break;
} // END if
$index++;
} // END foreach
// REMOVE ALL WAIVER CLAIMS FOR THIS PLAYER
if (!$debug) {
$this->db->where('player_id',$player['player_id']);
$this->db->where('league_id',$league_id);
$this->db->delete($this->tables['WAIVER_CLAIMS']);
} // END if
$summary .= "cleared = ".$this->db->affected_rows()." of ".$numClaims." waiver claims for this player<br />";
} // END if
// REMOVE PLAYER FROM WAIVERS
if (!$debug) {
$this->db->where('player_id',$player['player_id']);
$this->db->where('league_id',$league_id);
$this->db->where('waiver_period',$period_id);
$this->db->delete($this->tables['WAIVERS']);
} // END if
$summary .= "cleared = ".$this->db->affected_rows()." of ".$numClaims." waiver records for player ".$player['player_id']."<br />";
}
// UPDATE THE WAIVER ORDER OF THE TEAMS IN THE LEAGUE
$waiverList = array();
$rank = 1;
$summary .= "New waiver order:<br />";
foreach($waiverOrder as $waiveTeam) {
$this->db->set('waiver_rank',$rank);
$this->db->where('id',$waiveTeam);
$this->db->update($this->tables['TEAMS']);
//if ($debug) {
$summary .= $rank." = ".$waiveTeam.'<br />';
//}
$rank++;
}
return $summary;
}
/**
* DENY WAIVER CLAIM.
* Called when a league commissioner denies a wa9iver claim.
*
* @param $claim_id (Integer) The waiver claim ID. Function returns false if not passed.
* @return {Array} Claim object, false on failure
*
* @since 1.0.5
* @access public
*/
public function denyWaiverClaim($claim_id = false) {
if ($claim_id === false) { return false; }
$claim = false;
$this->db->select($this->tables['WAIVER_CLAIMS'].".id, ".$this->tables['WAIVER_CLAIMS'].".team_id, teamname, teamnick, ".$this->tables['WAIVER_CLAIMS'].".player_id, first_name, last_name, waiver_period");
$this->db->join("fantasy_teams","fantasy_teams.id = ".$this->tables['WAIVER_CLAIMS'].".team_id", "left");
$this->db->join("fantasy_players","fantasy_players.id = ".$this->tables['WAIVER_CLAIMS'].".player_id", "left");
$this->db->join("fantasy_players_waivers","fantasy_players_waivers.player_id = fantasy_players.id", "right outer");
$this->db->join("players","fantasy_players.player_id = players.player_id", "right outer");
$this->db->where($this->tables['WAIVER_CLAIMS'].".id",$claim_id);
$query = $this->db->get($this->tables['WAIVER_CLAIMS']);
if ($query->num_rows() > 0) {
$row = $query->row();
$claim = array('id'=>$row->id,'team_id'=>$row->team_id, 'teamname'=>$row->teamname." ".$row->teamnick,
'player_id'=>$row->player_id, 'player_name'=>$row->first_name." ".$row->last_name,
'waiver_period'=>$row->waiver_period);
$query->free_result();
$this->db->where("id",$claim_id);
$this->db->delete($this->tables['WAIVER_CLAIMS']);
}
$query->free_result();
unset($query);
return $claim;
}/**
* REMOVE FROM WAIVERS.
* Called when a league commissioner removes a waiver claim entirerly,
* incluing removing the player fromt he Waiver Wire.
*
* @param $player_id (Integer) The PLayer ID (REQUIRED)
* @param $league_Id (Integer) The League ID (OPTIONAL)
* @return {Boolean} TRUE on success, FALSE on failure
*
* @since 1.0.5
* @access public
*/
public function removeFromWaivers($player_id = false, $league_Id = false) {
if ($player_id === false) { return false; }
if ($league_Id === false) { $league_Id = $this->id; }
$complete = false;
$this->db->flush_cache();
$this->db->where("league_Id",$league_Id);
$this->db->where("player_id",$player_id);
$this->db->delete($this->tables['WAIVER_CLAIMS']);
$this->db->flush_cache();
$this->db->where("league_Id",$league_Id);
$this->db->where("player_id",$player_id);
$this->db->delete($this->tables['WAIVERS']);
return $complete;
}
/*------------------------------------------------------------------
/
/ PRIVATE/PROTECTED FUNCTIONS
/
/-----------------------------------------------------------------*/
protected function deleteLeagueData($table = false, $league_id = false) {
if ($table === false) { return false; }
if ($league_id === false) { $league_id = $this->id; }
$this->db->where('league_id',$league_id);
$this->db->delete($table);
return true;
}
/*---------------------------------------
/ DEPRECATED FUNCTIONS
/--------------------------------------*/
/**
* AUTO DRAFT.
* Runs automatic roster drafts for a given league.
*
* @param $max_rounds Maximum number of rounds to run
* @param $curr_year The current league year
* @param $league_id The fatntasy league ID, defaults to $id property if nothing is passed
* @return Error string or "OK" on success
* @deprecated
*/
/*public function _auto_draft($max_rounds,$curr_year, $league_id = false) {
$errors = "";
if ($league_id === false) { $league_id = $this->id; }
// GET ALL TEAMS
$teams = array();
$this->db->select("id");
$this->db->where("league_id",$league_id);
$query = $this->db->get($this->tables['TEAMS']);
if ($query->num_rows() > 0) {
foreach($query->result() as $row) {
array_push($teams,$row->id);
$this->db->where("team_id",$row->id);
$this->db->delete("fantasy_rosters");
//echo("Roster rows deleted = ".$this->db->affected_rows()."<br />");
}
}
$last_year = date('Y',strtotime($curr_year)-(60*60*24*365));
$query->free_result();
$pos_batters = array(2,3,4,5,6,7,8,9,25);
for ($i = 0; $i < sizeof($pos_batters); $i++) {
$batters = array();
if ($pos_batters[$i] == 25) {
$sqlPos = $pos_batters[rand(0,(sizeof($pos_batters)-2))];
$pickedBatters = array();
$this->db->select("player_id");
$this->db->where('league_id',$league_id);
$query = $this->db->get("fantasy_rosters");
if ($query->num_rows() > 0) {
foreach($query->result() as $row) {
array_push($pickedBatters,$row->player_id);
}
}
$query->free_result();
} else {
$sqlPos = $pos_batters[$i];
}
$this->db->select("fantasy_players.id");
$this->db->join("players_career_batting_stats",'players_career_batting_stats.player_id = fantasy_players.player_id','left');
$this->db->join("players",'players.player_id = fantasy_players.player_id','left');
$this->db->where("(fantasy_players.player_status = 1 OR fantasy_players.player_status = 3)");
$this->db->where('players.position',$sqlPos);
$this->db->where('players.retired',0);
$this->db->where('players_career_batting_stats.year',$last_year);
if(isset($pickedBatters) && !empty($pickedBatters)) $this->db->where_not_in('fantasy_players.id',$pickedBatters);
$this->db->order_by("players_career_batting_stats.war",'desc');
$query = $this->db->get("fantasy_players");
//echo("sql = ".$this->db->last_query()."<br />");
if ($query->num_rows() > 0) {
$count = 0;
foreach($query->result() as $row) {
array_push($batters,$row->id);
//echo("batter id = ".$row->id."<br />");
$count++;
if ($count >= sizeof($teams)) break;
}
shuffle($batters);
foreach($teams as $team_id) {
if ($pos_batters[$i] == 7 || $pos_batters[$i] == 8 || $pos_batters[$i] == 9) {
$pos = 20;
} else {
$pos = $pos_batters[$i];
}
$data = array('player_id'=>$batters[0],'league_id'=>$league_id,'team_id'=>$team_id,'scoring_period_id'=>1,'player_position'=>$pos,
'player_role'=>-1,'player_status'=>1);
$this->db->insert('fantasy_rosters',$data);
array_shift($batters);
}
}
}
$pos_pitchers = array(11=>5,12=>2);
foreach ($pos_pitchers as $pos => $draftCount) {
$pitchers = array();
$this->db->select("fantasy_players.id");
$this->db->join("players_career_pitching_stats",'players_career_pitching_stats.player_id = fantasy_players.player_id','left');
$this->db->join("players",'players.player_id = fantasy_players.player_id','left');
$this->db->where("fantasy_players.player_status",1);
$this->db->where('players.position',1);
if ($pos == 12) {
$this->db->where('players.role',13);
} else {
$this->db->where('players.role',$pos);
}
$this->db->where('players.retired',0);
$this->db->where('players_career_pitching_stats.year',$last_year);
$this->db->order_by("players_career_pitching_stats.war",'desc');
$query = $this->db->get("fantasy_players");
//echo("last query = ".$this->db->last_query()."<br />");
if ($query->num_rows() > 0) {
$count = 0;
foreach($query->result() as $row) {
array_push($pitchers,$row->id);
$count++;
if ($count >= (sizeof($teams)*$draftCount)) break;
}
shuffle($pitchers);
for ($i = 0; $i < $draftCount; $i++) {
foreach($teams as $team_id) {
$data = array('player_id'=>$pitchers[0],'league_id'=>$league_id,'team_id'=>$team_id,'scoring_period_id'=>1,'player_position'=>1,
'player_role'=>$pos,'player_status'=>1);
$this->db->insert('fantasy_rosters',$data);
array_shift($pitchers);
}
}
}
}
if (empty($errors)) $errors = "OK"; else $errors = $errors;
return $errors;
}*/
/**
* UPDATE LEAGUE SCORING
* Runs scoring against each leagues scoring rules for all players.
*
* @param $scoring_period The scoring period to compile
* @param $league_id The fatntasy league ID, defaults to $id property if nothing is passed
* @param $ootp_league_id The OOTP League ID to run stats from
* @return TRUE on success, FALSE on ERROR
* @deprecated Use $this->updateLeagueScoring() instead
*/
/*private function _updateLeagueScoring($scoring_period, $excludeList = array(), $league_id = false) {
if ($league_id === false) { $league_id = $this->id; }
// LOAD RELEVANT SCORING CATEGORIES
$rules = $this->getScoringRules($league_id);
if (isset($rules) && sizeof($rules) > 0) {
// UPDATE SCORING FOR ALL PLAYERS FOR THIS PERIOD
$player_list = array();
$this->db->flush_cache();
$this->db->select("fantasy_players.id, fantasy_players.player_id, position, role, player_status");
$this->db->join("players","players.player_id = fantasy_players.player_id","left");
$this->db->where("player_status",1);
$query = $this->db->get("fantasy_players");
//echo($this->db->last_query()."<br />");
if ($query->num_rows() > 0) {
//echo("Number of players found = ".$query->num_rows()."<br />");
foreach($query->result() as $row) {
// BUILD QUERY TO PULL CURRENT GAME DATA FOR THIS PLAYER
if ($row->position != 1) {
$type = "batting";
$table = "players_game_batting";
} else {
$type = "pitching";
$table = "players_game_pitching_stats";
}
$select = "";
foreach($rules[$type] as $cat => $val) {
if ($select != '') { $select.=","; }
$select .= strtolower(get_ll_cat($cat, true));
}
$this->db->flush_cache();
$this->db->select($select);
$this->db->join($table,'games.game_id = '.$table.'.game_id','left');
$this->db->where($table.'.player_id',$row->player_id);
$this->db->where("DATEDIFF('".$scoring_period['date_start']."',games.date)<=",0);
$this->db->where("DATEDIFF('".$scoring_period['date_end']."',games.date)>=",0);
$gQuery = $this->db->get('games');
//echo("Num of games found for player ".$row->player_id." = ".$gQuery->num_rows() .", status = ".$row->player_status." <br/>");
//echo($this->db->last_query()."<br />");
if ($gQuery->num_rows() > 0) {
$score_vals = array();
$totalVal = 0;
foreach ($gQuery->result() as $sRow) {
$colCount = 0;
// APPLY VALUES TO THE STATS AND SAVE THEM TO THE SCORING TABLE
foreach($rules[$type] as $cat => $val) {
$fVal = 0;
$colName = strtolower(get_ll_cat($cat, true));
if (isset($score_vals['value_'.$colCount])) {
$score_vals['value_'.$colCount] += $sRow->$colName;
} else {
$score_vals['value_'.$colCount] = $sRow->$colName;
}
if ($sRow->$colName != 0) {
$totalVal += $sRow->$colName * $val;
}
$colCount++;
}
}
$score_vals['total'] = $totalVal;
//echo("Player ".$row->player_id." total = ".$totalVal.", status = ".$row->player_status." <br/>");
//if ($row->player_status == 1) { $team_score += $totalVal; }
//echo("Team ".$team_id." total = ".$team_score."<br/>");
if (sizeof($score_vals) > 0) {
$this->db->flush_cache();
$this->db->select('id');
$this->db->where('player_id',$row->id);
$this->db->where('scoring_period_id',$scoring_period['id']);
$this->db->where('league_id',$rules['league_id']);
$tQuery = $this->db->get('fantasy_players_scoring');
if ($tQuery->num_rows() == 0) {
$this->db->flush_cache();
$score_vals['player_id'] = $row->id;
$score_vals['scoring_period_id'] = $scoring_period['id'];
$score_vals['league_id'] = $rules['league_id'];
$this->db->insert('fantasy_players_scoring',$score_vals);
} else {
$this->db->flush_cache();
$this->db->where('player_id',$row->id);
$this->db->where('scoring_period_id',$scoring_period['id']);
$this->db->where('league_id',$rules['league_id']);
$this->db->update('fantasy_players_scoring',$score_vals);
}
$tQuery->free_result();
}
}
$gQuery->free_result();
}
}
}
// GET ALL TEAMS
$teams = array();
$this->db->select("id");
$this->db->where("league_id",$league_id);
$query = $this->db->get($this->tables['TEAMS']);
if ($query->num_rows() > 0) {
foreach($query->result() as $row) {
array_push($teams,$row->id);
}
}
$query->free_result();
foreach($teams as $team_id) {
//echo("Team Id = ".$team_id."<br />");
// GET PLAYERS FOR TEAM
$teamRoster = array();
$team_score = 0;
// ONLY GET ROSTERS AND CORES IF THIS TEAM IF IT HAS VALID ROSTERS
if (sizeof($excludeList) == 0 || (sizeof($excludeList) > 0 && !in_array($team_id, $excludeList))) {
$this->db->select("player_id");
$this->db->where("team_id",intval($team_id));
$this->db->where("player_status",1);
$this->db->where("scoring_period_id",intval($scoring_period['id']));
$query = $this->db->get("fantasy_rosters");
if ($query->num_rows() > 0) {
foreach($query->result() as $row) {
array_push($teamRoster,$row->player_id);
}
}
$query->free_result();
$this->db->flush_cache();
$this->db->distinct();
$this->db->where_in("player_id",$teamRoster);
$this->db->where("fantasy_players_scoring.scoring_period_id",intval($scoring_period['id']));
$query = $this->db->get("fantasy_players_scoring");
//echo($this->db->last_query()."<br />");
if ($query->num_rows() > 0) {
foreach($query->result() as $row) {
$team_score += $row->total;
}
}
$query->free_result();
}
// LOOK UP AND UPDATE THE SCORES OF ANY GAMES THIS TEAM IS PLAYING IN
$this->db->flush_cache();
$this->db->select('id, away_team_id, home_team_id');
$this->db->where('(away_team_id = '.$team_id.' OR home_team_id = '.$team_id.')');
$this->db->where('scoring_period_id',$scoring_period['id']);
$this->db->where('league_id',$league_id );
$query = $this->db->get('fantasy_leagues_games');
//echo($this->db->last_query()."<br />");
//echo("scoring period id = ".$scoring_period['id']."<br />");
//echo("Number of games found for team ".$team_id." = ".$query->num_rows()."<br />");
if ($query->num_rows() > 0) {
foreach($query->result() as $row) {
$score = array();
if ($row->away_team_id == $team_id) {
$col = 'away_team';
} else {
$col = 'home_team';
}
//echo("update col = ".$col."<br />");
$score[$col.'_score'] = $team_score;
$this->db->flush_cache();
$this->db->where('id',$row->id);
$this->db->update('fantasy_leagues_games',$score);
//echo($this->db->last_query()."<br />");
}
}
$query->free_result();
}
return false;
}*/
/**
* GET OPEN LEAGUES.
* Returns a list of league available to the current player.
* @param $user_id - The user ID to check against
* @return leagues array, empty if not league found on failure
*
* @deprecated 1.0.3 PROD onward, use $this->getLeagueList() instead
*/
/*public function getOpenLeagues($user_id = false) {
$leagues = array();
$select = $this->tblName.'.id,league_name,commissioner_id,username,leagueType, max_teams, (SELECT COUNT(id) FROM fantasy_teams WHERE league_id = '.$this->tblName.'.id AND (owner_id = 0 OR owner_id = -1)) as openCount';
if ($user_id !== false) {
$select .= ', (SELECT COUNT(id) FROM fantasy_teams WHERE league_id = '.$this->tblName.'.id AND owner_id = '.$user_id.') as teamsOwned';
}
$this->db->select($select);
$this->db->join("fantasy_leagues_types","fantasy_leagues_types.id = ".$this->tblName.".league_type", "left");
$this->db->join("users_core","users_core.id = ".$this->tblName.".commissioner_id", "left");
$this->db->where('accept_requests',1);
$this->db->where('league_status',1);
$query = $this->db->get($this->tblName);
if ($query->num_rows() > 0) {
foreach($query->result() as $row) {
if ($row->openCount > 0 && ($user_id === false || ($user_id !== false && isset($row->teamsOwned) && $row->teamsOwned == 0))) {
array_push($leagues,array('id'=>$row->id,'league_name'=>$row->league_name, 'max_teams'=>$row->max_teams,
'leagueType'=>$row->leagueType, 'openings'=>$row->openCount,
'commissioner_id'=>$row->commissioner_id,'commissioner_name'=>$row->username));
} // END if
} // END foreach
} // END if
$query->free_result();
unset($query);
return $leagues;
}*/
} | {
"redpajama_set_name": "RedPajamaGithub"
} | 5,640 |
Total Leak Concepts
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Currently offering state distribution rights for Canada and all states except California, Florida, Louisiana, Eastern Texas, Georgia, Alabama, Massachusetts, Connecticut, Rhode Island and New Hampshire.
Home Based Franchise Opportunities
Below is a quick comparison of Total Leak Concepts against other Home Based franchises and business opportunities. For an in depth comparison of Total Leak Concepts with another opportunity including start-up costs and fees, business experience requirements, training & support and financing options click on the name of the company to view the detailed comparison.
Advertising Fee
Development Costs
A Painting Fiesta $25,000 $80,000-$110,000
AirMD $45,000 to 4% $250/mo. $53,750-$75,750
Ambit Energy N/A $429
Apex Fun Run $49,500 8% $80,500-$113,200
AR WORKSHOP $20,000-$30,000 6% - 7% $71,449-$125,666
Aussie Pet Mobile $60,000 8-4% 4-2% $153,250-$161,225
Bar-B-Clean $30,000 8% $36,950-$50,070
BBQ Cleaner, The N/A $20,000
Bloomin' Blinds $40,000 3%-6% 2% $62,570-$137,425
Body and Brain $10,000 $19,680-$36,300
Bottle & Bottega N/A $94,300-$146,100
Boulder Designs $20,000-$35,000 $195-400/mo. $54,730-$119,530
Bubble Bus, The N/A $87,000-$108,000
Busy Bees Pottery & Arts Studio $19,995 5% 1%Nat'l+2% Local $65,295-$128,895
Card My Yard $4,000 $4,725-$13,800
Cascade Cycleboats N/A $110,000
Cell Boys, The $7,500 5% 1% $20,700-And Up
Charter Financial N/A $195
Ciggies Distributing N/A 0 0 $9,995
Clothes Bin $40,000 $156,145-$199,965
Concrete Technology Inc. N/A 0 0 $5,000-$40,000
Conklin Roofing N/A $799-$2,950
Conserva Irrigation $49,500 5%-8% 1.5% $81,800-$102,250
Crock A Doodle $20,000 0 2% of Gross Sales $75,000-$120,000
CURB APPEAL USA N/A $8,888-$28,888
Driver Please N/A $5,000
Driverseat $13,000 $389/mo. $37,400-$142,650
Driveway Company, The $54,900 $83,465-$156,555
Dry Boys, The $30,000 7% 2% $74,000-$116,000
EasyClean N/A 0 0 $10,000-And Up
Executive Image Building Services N/A $7,500-And Up
FamilyIQ $179-$1,495 0 0 $179-$1,495
Fence Dynamics $30,000 6% 2%Nat'l+2% Local $110,900-$154,100
Flying Locksmiths, The $75,000-$300,000 8% $182,596-$428,396
Fun News $1,995 0 0 $1,995
Germinator Mobile Sanitizing $20,000 $28,405-$51,500
Get Moving Supplies N/A $80/mo. $99
Got Produce? $45,000 $410,600-$3,046,200
Handpiece Express N/A $35,000
HempE Distribution N/A $148
HMI Concrete raising N/A $50,000
Hobby Quest, The N/A $6,000
Hotels Etc. N/A 0 0 $495-$50,000
Hunting Lease Network $15,000 5% $23,500-$39,000
Improve Success N/A $2,400-$24,000
Intrigue Media N/A Percentage of sales $45,000-And Up
IslandTime Treasures $7,500 10% 2% $11,000-$26,350
iTRIPVACATIONS $30,000 5% $107,070-$145,875
JunkAway $30,000 6% 4% $76,000-$163,100
Just Let Me Do It N/A $64,000-$180,000
Local Door Coupons $24,900 9% 2% $26,100-$41,500
MAGIS Fundraising $25,000 $25,000-And Up
Midtown Chimney Sweeps $20,900 7% 3% $38,688-$105,210
Monster Tree Service $49,500 6.5% down to 5% after $1M in sales $397,435-$550,226
Mosquito Police $20,000 10% $49,600-$95,900
Neurosculpting Institute, The $7,500 $10,100-$130,650
Next Day Access $48,000 $103,025-$135,200
NextGen Great Sealcoating $15,000 7.5% 1% $72,850-$179,100
Ovenu $17,900 $295/ month $33,500-$43,400
Painting with a Twist $25,000 6% 2% $120,500-$260,000
Park Select $40,000 10% - 6% 1% $55,000-And Up
Party For a Living $24,900 $37,000-$51,600
Patch Boys, The $25,000 7% 2% $48,900-$75,900
Pinch A Penny $50,000 6% 4% $289,375-$411,200
Pinot's Palette $22,500-$27,500 6% 2% $96,800-$246,700
PlumbingPro $147,500 $327,700-$341,000
Pool Scouts $20,000 8% 2% $80,400-$153,300
Poolwerx $35,000 7% 3% $93,000-$309,500
Precision Concrete Cutting $63,000 8% $3500/year $135,000-And Up
Preferred HealthStaff $35,000 0 0 $62,120-$94,661
Premier Pools & Spas $30,000 2.25% up to 1% $38,000-$97,500
PureNRGFx N/A $99-$1,500
Rapid Drug Test N/A $19,500-$25,000
RD Fresh N/A $5,000-And Up
Reliable Appliance & Dryer Vent Service N/A $500-$1,000
Royal Restrooms $20,000-$30,000 6% $107,900-$200,200
RVPlusYou LLC N/A $10,000
Seal King $20,000 7% $58,050-$107,900
Sears Tile and Grout Cleaning N/A $11,025-$191,550
Sign Gypsies $3,500 $3,550-$9,300
SirVent Chimney and Venting Services $30,000 3% to 8% 3% Nat'l + 8-12% local $97,882-$180,482
Spectrum Towels & Scrubs N/A 0 0 $125-$4,450
SportsXpress $15,000 10% which includes a 2% towards training fund $15,000-And Up
STEARCLEAR $37,500-$93,750 20% $53,400-$143,050
Storkdelivery.com N/A $999
Superior Mosquito Defense $5,000 5% 1% $17,400-$46,200
Systems Paving $14,800-$17,800 6% $40,000-$100,000
Teleteria N/A $6,000
Trupanion N/A $375-$1,500
U Got Stink? $2,600-$15,625 $18,400-$83,515
USA Mobile Drug Testing N/A $73,500-$99,400
velofix Group of Companies $25,000 8% 2% $169,200-$202,700
Waterway $30,000 0 0 $100,000-And Up
WT Cafe $25,000-$40,000 6% $72,000-$125,000
To see more fees, make screen wider / rotate
Franchise Fee Royalty Advertising Fee Development Costs
A Painting Fiesta
$25,000 $80,000-$110,000
AirMD
$45,000 to 4% $250/mo. $53,750-$75,750
N/A $429
Apex Fun Run
$49,500 8% $80,500-$113,200
AR WORKSHOP
$20,000-$30,000 6% - 7% $71,449-$125,666
Aussie Pet Mobile
$60,000 8-4% 4-2% $153,250-$161,225
Bar-B-Clean
$30,000 8% $36,950-$50,070
BBQ Cleaner, The
N/A $20,000
$40,000 3%-6% 2% $62,570-$137,425
$10,000 $19,680-$36,300
Bottle & Bottega
N/A $94,300-$146,100
$20,000-$35,000 $195-400/mo. $54,730-$119,530
Bubble Bus, The
Busy Bees Pottery & Arts Studio
$19,995 5% 1%Nat'l+2% Local $65,295-$128,895
Card My Yard
$4,000 $4,725-$13,800
N/A $110,000
Cell Boys, The
$7,500 5% 1% $20,700-And Up
Charter Financial
Ciggies Distributing
N/A 0 0 $9,995
$40,000 $156,145-$199,965
Concrete Technology Inc.
N/A 0 0 $5,000-$40,000
Conklin Roofing
N/A $799-$2,950
Conserva Irrigation
$49,500 5%-8% 1.5% $81,800-$102,250
Crock A Doodle
$20,000 0 2% of Gross Sales $75,000-$120,000
CURB APPEAL USA
N/A $8,888-$28,888
Driver Please
N/A $5,000
Driverseat
$13,000 $389/mo. $37,400-$142,650
Driveway Company, The
Dry Boys, The
$30,000 7% 2% $74,000-$116,000
EasyClean
N/A 0 0 $10,000-And Up
Executive Image Building Services
N/A $7,500-And Up
FamilyIQ
$179-$1,495 0 0 $179-$1,495
Fence Dynamics
$30,000 6% 2%Nat'l+2% Local $110,900-$154,100
Flying Locksmiths, The
$75,000-$300,000 8% $182,596-$428,396
Fun News
$1,995 0 0 $1,995
Germinator Mobile Sanitizing
Get Moving Supplies
N/A $80/mo. $99
Got Produce?
$45,000 $410,600-$3,046,200
Handpiece Express
HempE Distribution
HMI Concrete raising
Hobby Quest, The
Hotels Etc.
N/A 0 0 $495-$50,000
Hunting Lease Network
Improve Success
Intrigue Media
N/A Percentage of sales $45,000-And Up
IslandTime Treasures
$7,500 10% 2% $11,000-$26,350
iTRIPVACATIONS
$30,000 5% $107,070-$145,875
JunkAway
Just Let Me Do It
Local Door Coupons
$24,900 9% 2% $26,100-$41,500
MAGIS Fundraising
$25,000 $25,000-And Up
Midtown Chimney Sweeps
Monster Tree Service
$49,500 6.5% down to 5% after $1M in sales $397,435-$550,226
Mosquito Police
$20,000 10% $49,600-$95,900
Neurosculpting Institute, The
$7,500 $10,100-$130,650
Next Day Access
NextGen Great Sealcoating
$15,000 7.5% 1% $72,850-$179,100
Ovenu
$17,900 $295/ month $33,500-$43,400
$25,000 6% 2% $120,500-$260,000
Park Select
$40,000 10% - 6% 1% $55,000-And Up
Party For a Living
Patch Boys, The
Pinch A Penny
Pinot's Palette
$22,500-$27,500 6% 2% $96,800-$246,700
PlumbingPro
$147,500 $327,700-$341,000
Precision Concrete Cutting
$63,000 8% $3500/year $135,000-And Up
Preferred HealthStaff
$35,000 0 0 $62,120-$94,661
Premier Pools & Spas
$30,000 2.25% up to 1% $38,000-$97,500
PureNRGFx
N/A $99-$1,500
Rapid Drug Test
N/A $19,500-$25,000
RD Fresh
Reliable Appliance & Dryer Vent Service
Royal Restrooms
$20,000-$30,000 6% $107,900-$200,200
RVPlusYou LLC
Seal King
Sears Tile and Grout Cleaning
Sign Gypsies
$3,500 $3,550-$9,300
SirVent Chimney and Venting Services
$30,000 3% to 8% 3% Nat'l + 8-12% local $97,882-$180,482
Spectrum Towels & Scrubs
N/A 0 0 $125-$4,450
SportsXpress
$15,000 10% which includes a 2% towards training fund $15,000-And Up
STEARCLEAR
$37,500-$93,750 20% $53,400-$143,050
Storkdelivery.com
Superior Mosquito Defense
$5,000 5% 1% $17,400-$46,200
Systems Paving
$14,800-$17,800 6% $40,000-$100,000
Teleteria
U Got Stink?
$2,600-$15,625 $18,400-$83,515
velofix Group of Companies
$30,000 0 0 $100,000-And Up
WT Cafe | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 9,021 |
In Primer 10 we opened a new window using the "window.open" command. That window was then filled with another HTML document we named in the instance.
Here, we're going to create a new window function where the new window and all of its contents will be carried along in the same HTML document. It is literally the equivalent of two pages in one.
The effect is exactly the same as in Primer 10: The same sized window opened and it contained the same two links. The difference is that it was all done with one page. To see the script on this page in action, click here.
The main script, the part that contains the function, is placed in between the <HEAD> and </HEAD> tags, as are most functions.
The function is named "openindex()" in the normal fashion. Then the fancy parentheses go in to surround the following commands.
The format is familiar. The only real difference is that there is no URL denoted. See the empty double quotes? That tells the browser to look to the script to find the new window information. It's very similar to not placing a URL in the command that closes the window. It wouldn't close if it had something to load. Same here. It wouldn't look to the script if it had something to load.
This format should also look somewhat familiar. The command is saying that on the variable "OpenWindow" (the new window) this link of text should be written to the document.
Look back up at the full script. That format is followed again and again and again writing lines of text. There's no reason why there cannot be hundreds of lines of text creating a fully functioning HTML document. Mine is small because it's a primer example.
Remember: When you are writing HTML inside a "document.write" command, you cannot use double quotes to surround subcommands. Use single. If you don't, error.
Finally, the function is called for in the BODY command through an "onLoad" Event Handler.
For today's assignment, you'll create a window that opens using a function. Please give the document that appears in the window a green background. In addition, make the TITLE command read "Hello user name - Here is your window! You can gather the user's name through a prompt. Of course, make a link that closes the window. | {
"redpajama_set_name": "RedPajamaC4"
} | 3,281 |
December 31, 2013 — A study published in the Journal of the American Heart Association revealed findings that may impact diagnostic strategies and clinical decision-making for patients with suspected coronary artery disease (CAD).
The study, "Clinical Implications of Referral Bias in the Diagnostic Performance of Exercise Testing for Coronary Artery Disease," examined the issue of referral bias in assessments of diagnostic performance for exercise testing with echocardiography (echo) or myocardial perfusion imaging (MPI).
Referral bias occurs when patients with an abnormal stress test result are referred to cardiac catheterization at a higher rate than patients with normal stress test results. While clinically appropriate, failing to adjust for this difference in referral rates when measuring test performance can significantly distort the observed diagnostic characteristics of exercise testing.
The study authors conducted a meta-analysis of 21 studies examining CAD referral patterns in 49,006 patients. After systematically reviewing cardiac catheterization referral rates and aggregating them to adjust pooled estimates of exercise test performance, the authors found that adjusting for referral bias significantly reduced test sensitivity and increased test specificity.
"We found that the sensitivity of exercise testing is much lower than previously reported. Wider recognition of this among clinicians — particularly primary care physicians and hospitalists — may influence how healthcare professionals use exercise echo and MPI to rule-in versus rule-out disease," said Joseph Ladapo, M.D., Ph.D., assistant professor, departments of population health and medicine, NYU Langone Medical Center, and lead author.
The study found exercise echocardiography and myocardial perfusion imaging are considerably less sensitive and more specific for CAD after adjustment for referral. Given these findings, researchers said future work should assess the comparative ability of these and other tests to rule?in versus rule?out coronary artery disease.
A K23 Career Development Award from the National Heart, Lung, and Blood Institute (NHLBI) supported Ladapo's work. CardioDx also helped fund the study. | {
"redpajama_set_name": "RedPajamaC4"
} | 5,960 |
Contents
FOREWORD
THE PHILOSOPHICAL SAILOR
ACKNOWLEDGMENTS
PART 1 PASSING THROUGH PAIN AND FEAR IN THE PLACE OF PERPETUAL UNDULATION
CHAPTER 1 SHIPS OF WOOD AND MEN OF IRON
CHAPTER 2 WINNING PHILOSOPHY
CHAPTER 3 "HARD A' LEE"
Preparing the Boat to Sail
Casting Off
Some Existentialist Reflections
Cruising
The Social Dimensions of Sailing
"Hard a' Lee" or Coming About
Sailing Close-Hauled
Noticing the "Presence of the Absence" (Heavy Sailing Ahead)
The Broad Reach
Practical Wisdom
Capsizing
Human Experience
Returning to the Pier
Pleasure, Elegance, and Truth
Final Tasks
CHAPTER 4 SOLO SAILING AS SPIRITUAL PRACTICE
A Phenomenology of Moral Presence at Sea
Tragic Comedy (or Comic Tragedy): The Paradox of Sailing
PART 2 THE MEANING OF THE BOAT: THREE SCHOOLS OF THOUGHT
CHAPTER 5 BUDDHA'S BOAT
CHAPTER 6 FREEDOM OF THE SEAS
Cheerful Resignation
Self-Sufficiency
Murphy was an Optimist: Negative Visualization
Agency and Control
Fate, Freedom, and Sailing
CHAPTER 7 SAILORS OF THE THIRD KIND
PART 3 BEAUTY AND OTHER AESTHETIC ASPECTS OF THE SAILING EXPERIENCE
CHAPTER 8 WHAT THE RACE TO MACKINAC MEANS
CHAPTER 9 SAILING, FLOW, AND FULFILLMENT
The Key: Losing Oneself
Windsurfing
Performance, Psychology, and Embedded Cognition
Windsurfing and Flow
CHAPTER 10 ON THE CREST OF THE WAVE
Ahoy!
The Sublime Poetry of Sail and Wind
Poseidon's Wrath
She Moves
One is Free... on a Boat?
CHAPTER 11 NAVIGATING WHAT IS VALUABLE AND STEERING A COURSE IN PURSUIT OF HAPPINESS
What's So Great About Sailing?
Aristotle, Virtues, and Flourishing
Is Sailing Virtuous?
Is Sailing More Virtuous Than Other Pursuits?
Conclusion
PART 4 PHYSICS AND METAPHYSICS FOR THE PHILOSOPHICAL SAILOR
CHAPTER 12 DO YOU HAVE TO BE (AN) EINSTEIN TO UNDERSTAND SAILING?
Introduction
Don't Laugh at "Slow" Sailing: Average Versus Instantaneous Motion
Motion Relative to What? – Galilean Relativity
But There are No Fixed Reference Frames – Special Relativity
General Relativity – Can it Really Matter?
CHAPTER 13 PARADOXES OF SAILING
Appendix: Analysis of the Wind-Powered Boat
CHAPTER 14 THE NECESSITY OF SAILING
Of Greek Gods, the Judaeo-Christian God, and the Sea
A Ship Bound for India
Beyond the Pillars of Hercules
CHAPTER 15 THE CHANNEL
NOTES ON CONTRIBUTORS
**VOLUME EDITOR**
PATRICK GOOLD is Associate Professor of Philosophy at Virginia
Wesleyan College. His current research focuses on defining
rationality. He is co-editor with Steven Emmanuel of the Blackwell
anthology _Modern Philosophy from Descartes to Nietzsche_. Patrick is
passionate about sailing, and, in addition to maintaining a small
daysailer and a cruising boat of his own, frequently crews on the boats
of others. The bays and sounds of Virginia and North Carolina are his
home waters but he has sailed the length of the East Coast of the
United States from Hilton Head to Long Island Sound, made a
Bermuda crossing, done club racing in Brittany, and cruised in the
Lesser Antilles.
**SERIES EDITOR**
FRITZ ALLHOFF is an Associate Professor in the Philosophy
department at Western Michigan University, as well as a senior research
fellow at the Australian National University's Centre for Applied
Philosophy and Public Ethics. In addition to editing the _Philosophy for
Everyone_ series, he is also the volume editor or co-editor for several
titles, including _Wine and Philosophy_ (Wiley-Blackwell, 2007), _Whiskey_
_and Philosophy_ (with Marcus P. Adams, Wiley-Blackwell, 2009), and
_Food and Philosophy_ (with Dave Monroe, Wiley-Blackwell, 2007). His
academic research interests engage various facets of applied ethics,
ethical theory, and the history and philosophy of science.
### **PHILOSOPHY FOR EVERYONE**
Series editor: Fritz Allhoff
Not so much a subject matter, philosophy is a way of thinking. Thinking not just about the Big Questions, but about little ones too. This series invites everyone to ponder things they care about, big or small, significant, serious... or just curious.
** _Running & Philosophy: A Marathon for the Mind_**
Edited by Michael W. Austin
** _Wine & Philosophy: A Symposium on Thinking and Drinking_** ****
Edited by Fritz Allhoff
** _Food & Philosophy: Eat, Think and Be Merry_**
Edited by Fritz Allhoff and Dave Monroe
** _Beer & Philosophy: The Unexamined Beer Isn't Worth Drinking_**
Edited by Steven D. Hales
** _Whiskey & Philosophy: A Small Batch of Spirited Ideas_**
Edited by Fritz Allhoff and Marcus P. Adams
** _College Sex – Philosophy for Everyone: Philosophers With Benefits_**
Edited by Michael Bruce and Robert M. Stewart
** _Cycling – Philosophy for Everyone: A Philosophical Tour de Force_** ****
Edited by Jesús Ilundáin-Agurruza and Michael W. Austin
** _Climbing – Philosophy for Everyone: Because It's There_**
Edited by Stephen E. Schmid
** _Hunting – Philosophy for Everyone: In Search of the Wild Life_**
Edited by Nathan Kowalsky
** _Christmas – Philosophy for Everyone: Better Than a Lump of Coal_**
Edited by Scott C. Lowe
** _Cannabis – Philosophy for Everyone: What Were We Just Talking About?_**
Edited by Dale Jacquette
** _Porn – Philosophy for Everyone: How to Think With Kink_** ****
Edited by Dave Monroe
** _Serial Killers – Philosophy for Everyone : Being and Killing_**
Edited by S. Waller
** _Dating – Philosophy for Everyone: Flirting With Big Ideas_**
Edited by Kristie Miller and Marlene Clark
** _Gardening – Philosophy for Everyone: Cultivating Wisdom_**
Edited by Dan O'Brien
** _Motherhood – Philosophy for Everyone: The Birth of Wisdom_**
Edited by Sheila Lintott
** _Fatherhood – Philosophy for Everyone: The Dao of Daddy_** ****
Edited by Lon S. Nease and Michael W. Austin
** _Coffee – Philosophy for Everyone: Grounds for Debate_**
Edited by Scott F. Parker and Michael W. Austin
** _Fashion – Philosophy for Everyone: Thinking with Style_**
Edited by Jessica Wolfendale and Jeanette Kennett
** _Yoga – Philosophy for Everyone: Bending Mind and Body_**
Edited by Liz Stillwaggon Swan
** _Blues – Philosophy for Everyone: Thinking Deep About Feeling Low_**
Edited by Abrol Fairweather and Jesse Steinberg
** _Tattoos – Philosophy for Everyone: I Ink, Therefore I Am_**
Edited by Robert Arp
** _Sailing – Philosophy for Everyone: Catching the Drift of Why We Sail_**
Edited by Patrick Goold
This edition first published 2012
© 2012 John Wiley & Sons, Inc
Wiley-Blackwell is an imprint of John Wiley & Sons, formed by the merger of Wiley's global Scientific, Technical and Medical business with Blackwell Publishing.
_Registered Office_
John Wiley & Sons, Ltd., The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, UK
_Editorial Offices_
350 Main Street, Malden, MA 02148-5020, USA
9600 Garsington Road, Oxford, OX4 2DQ, UK
The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, UK
For details of our global editorial offices, for customer services, and for information about how to apply for permission to reuse the copyright material in this book please see our website at www.wiley.com/wiley-blackwell.
The right of Patrick Goold to be identified as the author of the editorial material in this work has been asserted in accordance with the UK Copyright, Designs and Patents Act 1988.
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by the UK Copyright, Designs and Patents Act 1988, without the prior permission of the publisher.
Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books.
Designations used by companies to distinguish their products are often claimed as trademarks. All brand names and product names used in this book are trade names, service marks, trademarks or registered trademarks of their respective owners. The publisher is not associated with any product or vendor mentioned in this book. This publication is designed to provide accurate and authoritative information in regard to the subject matter covered. It is sold on the understanding that the publisher is not engaged in rendering professional services. If professional advice or other expert assistance is required, the services of a competent professional should be sought.
_Library of Congress Cataloging-in-Publication Data_
Sailing : philosophy for everyone : catching the drift of why we sail / edited by
Patrick Goold; Foreword by John Rousmaniere.
p. cm.
Includes bibliographical references.
ISBN 978-0-470-67185-6 (pbk.)
1. Sailing–Philosophy. I. Goold, Patrick Allen.
GV811.S25515 2012
797′.124–dc23
2012009766
A catalogue record for this book is available from the British Library.
**JOHN ROUSMANIERE**
# FOREWORD
# The Craft and the Mystery
This welcome collection of essays about the examined life under sail touches many of my nerve endings. A topic I call "the meaning of the boat" has been high on my agenda for many years, and I have long been fascinated by the passionate connection that so many of us feel with boats and the sea. Who reading this does not agree or at least sympathize with E. B. White's declaration, "With me, I cannot not sail"?1 For most sailors, this intense engagement is much more than a mere intellectual decision. It is a transforming connection between mind and heart, thought and belief, boat and sea.
This is what Joseph Conrad referred to when he wrote of "our fellowship in the craft and the mystery of the sea" in his seafaring memoir, _The Mirror of the Sea_ , one of the crucial texts of meaning-of-the-boat studies.2 Here is a fine match: _craft_ (what might be called "the physics of sailing") and _mystery_ ("the metaphysics of sailing"). Craft we all know – or at least we know we _should_ know. It is the skills and equipment needed to get a boat from one place to another. Mystery, however, is a little more complex.
Recently, during a panel discussion of an upcoming race across the Atlantic, the moderator, Gary Jobson, asked me to describe my most vivid memory of transatlantic sailing. I could have mentioned the thrill of starting a race to Spain, or exuberant days of fast running before westerly gales in seas seemingly as high as the boat was long. Stretching the subject a little, I might have said something about a brutal beat out to Fastnet Rock in a force-ten storm, or carefully skirting Bermuda's reef after four days at sea, sailing the boat like a dinghy as we fought to win a Newport Bermuda Race. I might well have recalled many of those memories of great excitement, but, somewhat to my surprise, my mind went immediately to an altogether different moment.
Deep into a moonless night during my first long Atlantic voyage, a perfect calm enveloped the big ketch. The skipper came on deck, took a look around, and cut the engine. He didn't have to explain why; we understood. The boat carried her way for a few minutes as the bow wave trickled into silence, and our little world was inhabited by stillness. The only sound was the occasional flutter of empty sails or confused birds. The single sign of reckoning time was the slow march of constellations across the great dome of darkness overhead. We could have been anywhere, at any moment.
After a while – I can't say how long because minutes and hours were abstractions – someone switched on the spreader lights, and we tiptoed to the rail and peered down many fathoms into the clear, magical sea. Suspended between those two worlds in that moment, decades ago, I felt more connected to the eternal mysteries than any prayer or song or poem has ever allowed.
I am reminded of this magical moment by a photograph on my study wall. An anonymous sailor, his back to us, stands on the deck of a sailboat becalmed on a still dawn, peering ahead at the rising sun. Is he searching for land? For wind? Or for himself?
FIGURE F.1 Photo used by permission of Mystic Seaport.
Many sailors of all levels of ability have told me that they have had similar moments afloat, when time stood still and they discovered another world. The mystery of the sea is shared by all sailors, even (perhaps especially) the most technically gifted masters of the craft. The man who took this photograph was one of the most successful ocean racing sailors who ever lived, Carleton Mitchell. The high naval official Samuel Pepys was taking a row on the Mediterranean in 1683 when he was overwhelmed by one of these moments. He later wrote in a journal, "I know nothing that can give a better notion of infinity and eternity than the being upon the sea in a little vessel without anything in sight but yourself within the whole hemisphere."3 Pepys was no flake but a tough-minded inspector of warships whose outbursts (as anyone who has read his diary knows) tended to be sexual, not spiritual; yet on this day the sea took on a whole new meaning for him.
More than two hundred years later, a self-promoting New York City magazine editor and ocean sailor named Thomas Fleming Day explained why he founded a race to Bermuda in this way: "Sailors wanted to get a smell of the sea and forget for the time being that there is such a thing as God's green earth in the universe."4 In short, they were seeking another world. So was an exceptionally experienced English writer-sailor, Maurice Griffiths, who laid out his feelings upon heading out in a small cruising boat in these words:
I found my pulse beating with suppressed excitement as I threw the mooring buoy overboard. It seemed as if that simple action had severed my connection with the life on shore; that I had thereby cut adrift the ties of convention, the unrealities and illusions of cities and crowds; that I was free now, free to go where I chose, to do and to live and to conquer as I liked, to play the game wherein a man's qualities count for more than his appearance.5
A few years ago the champion long-distance racer Ellen MacArthur wrote in her log as she neared the finish of a solo transatlantic race:
I now feel so wonderfully in tune with the boat and the sea that I know I shall really miss this once the race is over. At night I watch the sun go down and in the morning the sky is there above me, a wonderful feeling of space and timelessness.6
And a pioneer British ocean racer of the 1920s, George Martin, noted that there are times when, "except for the knowledge of contact with the deck, one seemed to have passed right out of the world."7
One point to make about these visions is that nobody should ever feel embarrassed to have them. Not only are they common – sometimes so much so as to be commonplace – but they are paths to valuable truths. In fact, they carry considerable philosophical weight. The adjective often applied to them is "numinous," a term originated by Rudolph Otto (1869–1937), a German theologian and authority in the field of comparative religions. In his influential book _The Idea of the Holy_ , Otto defined a numinous event as a non-rational, non-sensory experience or feeling whose primary and immediate object is outside the self.8 That object might be called God or "the wholly other." The experience of confrontation, the " _mysterium tremendum_ ," inspires great awe, distance, humility, and even fear.
Two decades before Otto, the American philosopher and psychologist William James (1842–1910) described this engagement with the sacred slightly differently. Collecting reports of mystical states, he noted patterns. Some included a sense of déjà vu. In others, everything seemed to carry special meaning. A common theme was oneness with the absolute, which sometimes had a maritime setting; one of James' sources described a mystical state in which he was "immersed in the infinite ocean of God." James emphasized that, while these states may appear non-rational, they can influence and even team up with our thinking. Their existence, James wrote, "absolutely overthrows the pretension of non-mystical states to be the sole and ultimate dictators of what we may believe."9
A sailor living an examined life could do a lot worse than read up on William James. He is mentioned briefly in this volume in Chapter 10, but more really should be said about this great thinker and man who so well appreciated the call of the sea. James did a little sailing as a young man in Newport, and evidently learned something because he employed a clever nautical metaphor to make a point about one of his favorite causes, free will. How can human beings be automatons, he asked, if, when they are passengers in a sailboat, they freely volunteer to take the helm or help with the reefing when the wind comes up?10 James' own sport of choice was mountain hiking. That and his own psychology and intellectual work gave him an intimate understanding of the needs and life of the thoughtful adventurer. He urged opening ourselves up to making choices, pushing aside dogmatic determinism, and trusting that feeling can be a reliable adjunct to thinking (all good options for sailors, I would say). Finding meaning and mental health in vigorous adventures, he propounded a notion familiar to any sailor who has ever been at one and the same time wet, worn out, and exhilarated (and what sailor hasn't?): "It is indeed a remarkable fact that sufferings and hardships do not, as a rule, abate the love of life; they seem, on the contrary, usually to give it a keener zest."11
The philosopher Charles Taylor has described James as "our great philosopher of the cusp," explaining, "He tells us more than anyone else about what it's like to stand in that open space and feel the winds pulling you now here, now there."12 The analogy is apt; few people face more decisions than the typical sailor on an average day. Should we reef now? Tack now? Anchor here? Heave to? Do nothing? There is no room for indecisiveness or complacency either in William James or in the best-sailed boats.
I have long wondered whether James knew about the extraordinary Richard T. McMullen, a pioneer cruising sailor who was never happier than when things were toughest. He first headed out in 1850 and was still at it forty-two years later, when he died at the helm while singlehanding. In his collection of sea stories, _Down Channel_ , first published in 1869, he laid down a personal philosophy of vigorous yacht cruising that he summarized in the phrase, "my hard sailing habits." He loved a hard beat to windward in a strong breeze, an activity that he called "terrible but very grand." McMullen memorably said of good sailing that it "is not unlike the pleasure human nature has invariably found in successfully gathering roses off thorns." There may be the occasional bloody finger, but there are great rewards.13
There is much in these pages to prove that the joy in nautical rose-picking and the old sense of mystery both remain alive among sailors. Nicholas Hayes, in Chapter 8, for instance, describes the second day in a typical Chicago to Mackinac race as "a day of transcendence and transformation. Sailors will tell you that every person who starts this race will finish as someone new." Yet mystery and romance cannot suffice on their own. It has been wisely said that, in a boat at sea, "piety is no substitute for seamanship."14 Romantics who have fallen in love with "the dream" head out with too little knowledge, and they and their rescuers suffer for it. The most famous example of inappropriate romance is the poet Percy Bysshe Shelley. So incompetent a seaman that he bragged about it, he built himself a boat that was "fast as a witch" but utterly unseaworthy, and he drowned in it. We certainly don't want to encourage more Shelleys by suggesting that all it takes to sail around the world (much less around the harbor) is one numinous experience.15
Mystery and craft, feelings and intellect – they function at their best when they team up to work together. As Joseph Conrad, himself a consummate mariner, knew very well, a numinous experience is not at all incompatible with competence and technology. In fact, they often nurture each other, making sailing even more of an obsession than it already is.
**NOTES**
1 E. B. White, "The sea and the wind that blows," in Peter Neill (Ed.), _American Sea Writing: A Literary Anthology_ (New York: Library of America, 2000), p. 612.
2 Joseph Conrad, _Mirror of the Sea_ (Doylestown, PA: Wildside Press, 2003), p. 177.
3 Claire Tomalin, _Samuel Pepys, The Unquali_ _fied Self_ (New York: Knopf, 2003), p. 330.
4 John Rousmaniere, _Berth to Bermuda: 100 Years of the World's Classic Ocean Race_ (Mystic, CT: Mystic Seaport and Cruising Club of America, 2006), p. 17.
5 Maurice Griffiths, _Magic of the Swatchways_ (Dobbs Ferry, NY: Sheridan, 1997 [1932]), p. 50.
6 Ellen MacArthur, _Taking on the World_ (New York: McGraw-Hill/International Marine, 2003), pp. 137–138.
7 E. G. Martin, _Deep Water Cruising_ (New York: Yachting, 1928), p. 99.
8 Rudolf Otto, _The Idea of the Holy_ (New York: Oxford University Press, 1958), pp. 10–11.
9 William James, _Varieties of Religious Experience_ (New York: Viking Penguin, 1982 [1902]), p. 427.
10 Robert D. Richardson, _William James: In the Maelstrom of American Modernism, a Biography_ (Boston, MA: Houghton Mifflin Harcourt, 2007), p. 199.
11 William James, "Is life worth living?" In _Will to Believe and Other Essays in Popular Philosophy_ (New York: Dover, 1959), p. 47.
12 Charles Taylor, "Risking belief," _Commonweal_ , March 8 2002, pp. 14–17.
13 Richard T. McMullen, _Down Channel_ (London: Rupert Hart-Davis, 1949 [1869]), pp. 147, 272.
14 Donald Wharton, "Biblical archetypes." In John Rousmaniere (Ed.), _Oxford Encyclopedia of Maritime History_ , vol. 1 (New York: Oxford University Press, 2007), p. 285.
15 John Rousmaniere, _After the Storm: True Stories of Disaster and Recovery at Sea_ (New York: McGraw-Hill/International Marine, 2002), p. 12.
**PATRICK GOOLD**
# THE PHILOSOPHICAL SAILOR
# An Introduction to _Sailing – Philosophy for Everyone_
Why philosophize about sailing? Because people sail! Serious people – that is, people serious about life – pour their time and treasure into sailing. They drag their families into it, or abandon friends and family to do it. They even risk their lives. Why do they do this? Can there be any sense to it? Every activity is a possible subject for philosophical reflection, the more so the more deliberately and passionately the activity is pursued and the more demanding it is. Committed action pre-supposes goals, values, and meanings. These give it its structure. Philosophical reflection wants to explicate these goals, in order to comprehend them, to see how they hang together with one another and with the larger set of commitments that the actor shares with others, and, finally, to interpret them as a signpost pointing us toward wisdom. Philosophical reflection focuses attention on what matters in the activity and on the fit and finish of its implicit ideals. Connections with the larger human drama are made. Lines of improvement in the activity might be suggested. Meaning and value grow. The activity becomes a practice, with a code of conduct (perhaps unspoken), an unfolding tradition with recognizable heroes and reformers, a sense of camaraderie with others similarly engaged, and more or less formal modes of organization to protect it. And so it could be with sailing. Philosophical sailors have more fun sailing. Sailing becomes bigger for them and more real.
It might be, too, that sailing could add life to one's philosophy. Do philosophers who sail find themselves liberated as philosophers? Do traditional categories take on a different appearance when viewed from "a place of perpetual undulation"?1 Could seamanship under sail, a species of practical wisdom with ancient roots, serve as a model for wisdom in general? Philosophy can help us grasp the meanings, aims, and satisfactions of sailing. Perhaps reflecting on sailing can also help us grasp the meanings, aims, and satisfactions of philosophy itself. Philosophers might find a path to more _philos_ in their philosophy. Philosophy might become something bigger for them and more real. The reader will find suggestions to this effect in the essays that follow.
What is sailing? Traditionally, Socrates set the form of the fundamental philosophical question: _Ti esti_? – What is it? What is courage? What is piety? What is knowledge? What is love? For Socrates, the path to wisdom leads through questions of this form. He rejects mere examples as an answer. He wants a statement of the essence of justice. His way of constructing such a statement set the pattern for philosophical reflection ever since: one listens to how those around one use the term, maps what inferences are sanctioned by this use, and then tests the set for consistency. Theoretically, this process continues until no further inconsistencies appear. Breathe easily! None of the essays below takes this approach. It is worthwhile, however, to take a moment to consider the _ti esti_ question with regard to our subject.
What is sailing _really_? Is there a timeless essence that can be put into words? I doubt it. Try something like "moving a boat through water by means of the wind." Counterexamples will not be long in coming: ice-boating, windsurfing, sailing stones in Death Valley, an imagined spaceship harnessing solar winds, and so on. More importantly, even if we set aside these possible counterexamples, the definition will not be able to capture what sailing means to us now. The function of sailing has changed. Sailing vessels were once movers of goods and passengers, tools for the exploitation of fisheries, the only way to cross seas, and powerful weapons of war. Other machines have taken over these functions. Something other than the necessities of commerce and war prompt contemporary men and women to sail. Ignoring this different function would lump together practices that ought to remain distinct. It would put us at a level of abstraction at which something essential is lost.
What then has sailing become? The genus "hobby" cannot contain it without remainder. A hobby is an activity done regularly in one's leisure time for pleasure. While some sailors may be hobbyists, many are not. Sailing has its professionals as well as those strange birds the circumnavigators and long-term cruisers – people for whom sailing is not a leisure-time activity but the focus of their lives. The physical effort required in these more strenuous forms of sailing, as well as the risks involved, sit uneasily with the ideas of pleasure and of toys or pursuits lacking seriousness, part of the connotation of "hobby."
Sport, which the _American Heritage Dictionary_ defines as "an activity involving physical exertion and skill in which an individual or team competes against another or others for entertainment," does better perhaps. But, of course, sailing need not involve competition. This clause might be a mistake in the definition of sport since hunting and mountain climbing are universally considered to be sports but seldom involve competition. We might try looking inward for the competition in these cases, to some sort of self-overcoming, but this would be an unusual use of the word. The awful word "entertainment" also connotes an ultimate lack of seriousness that is inadequate to the circumnavigator or long-term cruiser as well as the hunter and the mountaineer. If these activities entertain, once again it must be in some very special sense of the word.
Many do call sailing a sport. The National Sailing Hall of Fame circulates a brochure that describes what the Hall is and what it does.2 On the second page of that document a statement by Walter Cronkite contains these sentences: "Baseball, basketball, football, tennis, soccer, golf and lacrosse all have halls of fame. I believe the sport of sailing should have a similar facility that will focus attention on our sport and recognize its heroes as role models for our youth." On the whole, however, the National Sailing Hall of Fame, in this publication and on its website, uses the word "sport" very sparingly. In fact, the word "sport" occurs only two more times in the brochure. More often "the art of sailing" or "the science of sailing" are referred to. There is a gallery in the Hall of Fame dedicated to "The Spectacle of Sailing." Well, is it also a sport?
This simple question takes us back full circle: What is sailing? And what is a sport? The first, we have seen, is not as straightforward to answer as one might hope. Even if we stipulate away the funky counterexamples and stick to boats moved by the wind, the second question cannot be answered unequivocally. In this sense, if you are racing sailboats you are sailing, and if you are cruising sailboats you are sailing. James Cook was sailing when he explored the South Pacific in _Endeavour_ , and so was the previous master of that ship when he took it from Grimsby to Hull, delivering coal. If sailing is anything done to move a boat using the wind, then clearly at least some sailing is not a sport. Getting clear about what sailing means to us today may cut away Cook and the collier. It leaves other unclear cases like that of the long-term cruising couple. They are clearly sailing; are they engaged in a sport?
This little exercise is salutary. Concepts that we might otherwise deploy in a thoughtless and reified way are thematized – set in motion, so to speak – and, instead of bricks out of which to build a final answer, they become hypotheses that we must test continually against our own intellectual experience. We are freer now to philosophize, philosophy being, as Adorno has said, "thought in a perpetual state of motion." To put it another way, philosophy is the attempt to think in a disciplined way about something when we don't know exactly what it is we are thinking about. We cannot begin until we have realized that we don't know. To ordinary consciousness this loss of certainty is a step backward. To the philosopher, and to the reflective sailor, it is progress; now one may begin.
There is another traditional philosophical question: What is the ultimate purpose of the thing in question? Aristotelians use the terms _telos_ , or final cause, to name this motive. What is it that draws the soul to sailing? The four essays in Part I of this volume reflect on the fact that effort, uncertainty, and sometimes pain and danger are inescapable parts of sailing. If we are thinking in the direction of pleasure or entertainment, toward which the categories of hobby and sport would point us, then this is a particularly salient question for sailing: Where is the fun of it? If pleasure or entertainment is not the final goal, what is? How does one explain a pursuit that commits one to so much discomfort and even danger?
A common thread runs through the answers the first four essays give to these questions. Facing the uncertainty and the technical challenge of sailing can lead to self-knowledge and to increased virtue. One finds out what one is made of and, perhaps, takes heart, becoming more confident, more open to experience and more self-directed.
Jack Stillwaggon, in "Ships of Wood and Men of Iron," recounts a voyage he took as a young man on a replica of Captain Bligh's _Bounty_. Jack encountered considerable danger and discomfort and yet, despite the passage of years, this voyage has remained a high point in his life. Reflecting on this, Jack asks about the point of making such journeys: "Do we strive to experience the past as amateur historians or are we atoning for our frivolous modern lifestyle? Are we like the actor who takes up a political cause to prove he can do more than just play make-believe? Does the fact that we endured a risky trial and survived add importance to an otherwise silly existence?" His answer is that we face the perils of the sea in order to prove something to ourselves about our abilities and our personal worth. To face such a challenge successfully is life-changing. "Like the rock climber who maintains an inner self-confidence because of having scaled challenging heights, or the marathon runner who completes a race, I was more at peace with myself for reaching new horizons and I expected more from myself from that point onward."
In "Winning Philosophy," Gary Jobson takes up the themes of courage and self-confidence from the perspective of a long and distinguished career racing sailboats. He describes the exhilaration and the anxiety of competition and offers tips for overcoming the latter. "Winning takes inner strength and even courage. The big question for every competitive sailor is how to generate the will to overcome fear and to win." While Gary speaks directly from his long experience, using no philosophical vocabulary, his essay answers the question: What must I do to acquire the virtues I need to succeed at sailing? Strength of character and the power of self-transformation have been the aim of philosophical exercises since the beginning of philosophy. Gary's essay links up in especially illuminating ways with the essay of Greg and Tod Bassham on stoicism and sailing and with the Aristotelian approach that Jesse Steinberg and Michael Stuckart take in their essay.
Crista Lebens, in "'Hard a' Lee,'" agrees with Jack and Gary that the opportunity to face down danger and to win the respect of others is part of sailing's allure. She sees this, however, as one element of a larger pursuit, the pursuit of _eudaimonia_. This ancient Greek word she translates as "human flourishing." Perhaps it is Crista's different sailing experience that gives her this perspective: "I have sailed almost exclusively in small craft on inland lakes with family and friends. I have raced in informal afternoon regattas that are about as competitive as a game of touch football in the park. The life lessons I have learned from sailing are drawn from my experiences of these enjoyable afternoons." Themes of sociability, beauty, and enjoyment emerge here. They will be developed at length in Part III.
In "Solo Sailing as Spiritual Practice," Richard Hutch rounds out this opening section on the motives and rewards of sailing with an analysis based on his own experience as an open-ocean sailor and on his study of the writings of a number of single-handed ocean sailors. Richard introduces the notion of "moral presence" and argues that this is what these sailors seek. They sometimes describe their goal in terms of personal quest (as we see in Jack's essay), sometimes in terms of a cosmic quest, and sometimes as the pursuit of technical finesse. Richard shows how these varying self-descriptions describe different pathways to moral presence.
The confluence of the views in these four essays is striking. It is even more striking when one reflects on the very different experiences of sailing that ground them: a crewman on a square-rigger, high-level racing of dinghies and large-keel boats, clubbing on a C-Scow in Lake Michigan, and single-handed ocean sailing.
The essays of Part II develop this fundamental idea that sailing can be a pathway to moral fulfillment by looking at it through the lens of particular historical schools of philosophy.
Sailing his catboat is for James Whitehill a Zen practice. In "Buddha's Boat," he says that "Buddhist tradition holds the view that there are countless ways to the difficult goal of an awakened, flourishing self. Sailing, I believe, can be one of those ways or practices." James describes how archery and meditation have been taught by traditional Zen practitioners as a means to enlightenment and then develops, out of his own carefully observed experience of sailing catboats around Cape Cod, an account of how he approaches sailing in a similar spirit.
In "Freedom of the Seas," Greg and Tod Bassham find in the teachings of Cicero, Seneca, Epictetus, Marcus Aurelius, and the Stoic school a guide to the most successful sailing. "For sailors," they write, "Stoicism has much to offer as a means to create and sustain the sense of joy we seek in sailing." They explain the Stoic metaphysics of fate and agency, the complementary virtues of cheerful resignation and self-sufficiency, and the all-important technique of negative visualization, and demonstrate their value for the sailor.
Steven Horrobin, in "Sailors of the Third Kind," takes his inspiration from Spinoza, leading the reader on a metaphysical journey the goal of which is to break the hold on him or her of a substantialist view of things. A sailor of the third kind is one who recognizes that everything is process: not just the weather and the waves but persons, boats, everything. Citing Spinoza, Steven describes this person in process of becoming as "in essence a move of the universal super-process toward its own self-realization, with persons achieving greater or lesser moves toward representing, in themselves and their own being, an approximation of the whole." For Spinoza, "the good was manifest in the direction of movement, within a person, toward the accurate reflection of the whole of nature, in microcosm." Heavy stuff! But Steven connects it to his experience of a certain sort of sailing and of sailors he calls sailors "of the third kind." Those who have perused the literature of ocean sailing, or who have much acquaintance with those who do such sailing, will recognize this sort of sailor at once. And, anyone who spends time maintaining a boat will be moved by Steven's eloquent depiction of the boat as a process unfolding in interaction with other surrounding processes.
The essays in Part III explore in detail various aspects of the "flourishing" or "fulfillment" to be found in sailing. In "What the Race to Mackinac Means," Nicholas Hayes gives a compelling account of competing in the Chicago to Mackinac Island Race. His racers encounter natural beauty, camaraderie, "lessons about work, reward, luck, and injustice," and a new sense of "our relationship with time." "Everything is now," says Nick near the end of his essay.
If it is not the same sort of consciousness, Nick's "nowness" is closely related to the one that Steve Matthews unpacks in his essay on the phenomenon he calls "flow." In "Sailing, Flow, and Fulfillment," Steve draws on both Daoism and the work of contemporary psychologists to explain the special value he finds in sailing, and especially in the sort of sailing of particular interest to him, windsurfing. Here, I think, specific features of the "boat" are important. Steve emphasizes the tight connection between the windsurfer and his board, a feeling of oneness with the kit.
Though I feel something similar in sailing my small skiff in a fresh breeze, it is very different sailing my five-ton yawl. There the feeling is more one of sovereignty. Set the course, trim the sails, and, once the boat is comfortable, set the tiller pilot and step back. The joy in this sort of sailing arises from the felt difference between the boat and me. The boat is an orderly world and I am its contemplating spirit. This contemplation does have an "everything is now" aspect to it, but for me at least it also has a time-soaked quality. When I sail, I feel the continuity with a long tradition. These differences in the experiences of various sorts of sailing are worth noting. We must not pull our conceptual nets too taut, lest we arrest "the motion of thought/and its restless iteration."3
Jesús Ilundáin-Agurruza, Luísa Gagliardini Graça, and José Jáuregui-Olaiz, in "On the Crest of the Wave," rhapsodize on many of the themes touched on in this book. They make playful use of common expressions with roots in the sailing world in order to evoke, as well as describe, the special aesthetic aspects of sailing. Slocum, Gautier, MacArthur, and other heroic sailors are brought in to testify to the natural beauty encountered in sailing, awe in the face of the sublime, the cultivated beauty of boats and of the precise and pertinent motions of the practiced sailor, and the joy of the constrained freedom in seamanship.
Jesse Steinberg and Michael Stuckart, in "Navigating What is Valuable and Steering a Course in Pursuit of Happiness," the final essay in this section, consider practical, moral, and aesthetic values as they relate to sailing. They provide a brief taxonomy of what is valuable and then take up Aristotle's notion of "human flourishing" again. They argue that sailing has all the necessary elements for achieving a happy and full life. Indeed, they conclude, "Although it is certainly not the only activity that cultivates virtues, we think that sailing is an especially rich activity in this regard. That is, sailing is remarkably replete with opportunities to be virtuous. Sailing enables and even prompts one to be courageous, careful, curious, knowledgeable, decisive, and clever, and to have a whole host of other virtues."
Part IV shifts from the realms of psychology and values to physics and metaphysics. Sebastian Kuhn's delightful essay, "Do You Have to be (an) Einstein to Understand Sailing," unpacks the notion of relativity and its connection to sailing. "You will find," he writes, "maybe without realizing it, you are making use of 'relativity' in one way or another every time you weigh anchor. Even the General Theory of Relativity, long considered the most arcane of Einstein's ideas, has a direct impact on something as mundane as navigation." Sebastian's conceptual change-ups push one's thought out of the conventional ruts and invite one to consider a bigger picture. For example, the hoary comparison of the slowness of sailing with the speed of power-boating is reformulated by replacing instantaneous speed with average velocity (the total distance between the start and finish position divided by the elapsed time). "By this reckoning," Sebastian writes, "all weekend cruisers, whether laid-back sailors or high-speed motorboaters, have the same average velocity – namely zero! This is because they tend to end up right where they started." Later in the essay, playing with the notion of the reference frame, Sebastian invites us to look at the two sorts of cruiser using the fixed star as the frame. "Compared to that frame, both the most languid sailboat and the most souped-up powerboat move with velocities that are quite impressive – and practically indistinguishable (thirty kilometers per second, or 58,000 knots if you count the sun as the origin of this frame)." Sebastian's reminder that we don't just calculate but that we calculate within a particular frame of reference helps to keep our thinking in perpetual motion. Are there different frames available? What if we used one of those?
In "Paradoxes of Sailing," John D. Norton considers three puzzles about sailing. How is it possible to sail into the wind, and faster than the wind? These are two. The other draws on a common conceit of sailors that a sailboat "makes its own wind." The puzzle then is that, if this is so, why does it need any other wind? Considering these puzzles, John makes some interesting connections with two issues in recent philosophy, one concerning the metaphysics of causation and the other the notion of the thought experiment and the value of thought experiments as a mode of investigating nature. The issues are large and John's conclusions are substantive. His position on thought experiments is particularly interesting. He describes the Platonist view that grasping a thought experiment is like grasping the ideal triangle that is the meaning behind the crude drawing on the chalkboard. We grasp something beyond what is presented to our senses. John, however, sees the thought experiment as no more than a picturesque way of drawing out the implication of assumptions behind its constructions. What assumptions we want to agree to will ultimately be a matter of empirical investigation. Sailing provided the original thought experiment.
In the penultimate essay, "The Necessity of Sailing," Tamar and Nathan Rudavsky look at some of the ways in which sailing has provided philosophers with analogies to explain their views of freedom, fate, and determinism. They move from Aristotle and Maimonides all the way to Nietzsche. It is fascinating to see how philosophy is saturated with sailing metaphors and sailing images, just as ordinary speech is seen to be in Chapter 10. As in earlier essays, themes of freedom, fate, and determinism loom large.
The final essay, "The Channel," by Hilaire Belloc, serves as an epilogue to the volume. It appeared in 1913 as a chapter in the curious collection of travel writings entitled _Hills and Sea_. Belloc's writings about sailing are out of print and somewhat hard to find; they deserve to be better known. Belloc had a lust for life that drove him often to the sea. His views on sailing were "conservative," as were most of his views. He disdained the modern and comfortable and often chose to do things the hard way because only that way allowed him to suck the marrow out of the experience. His nostalgia for a past where life was bigger and more vibrant has something Greek about it. Sailing is a surrogate for him for life in the Golden Age. Raising as he does the suspicion that modern life distracts us with shadows, he is a true philosopher. In this nostalgia and in this choice of surrogate he speaks, I think, to the condition of many of us who sail.
**NOTES**
1 Wallace Stevens, "The Place of the Solitaires." In _Harmonium_ (New York: Knopf, 1991), p. 60.
2 National Sailing Hall of Fame, _The National Sailing Hall of Fame and Sailing Center_ (n.d., <http://www.nshof.org/images/booklet15.pdf>).
3 Stevens, "The Place of the Solitaires."
# ACKNOWLEDGMENTS
Many people have had a hand in helping me to become a sailor, from the college classmate who first taught me to sail to current friends who share their time and knowledge with me. With their help I have found much pleasure and an added dimension to life. I hope this book, in a small way, pays forward their generosity to me.
Many thanks to Fritz Allhoff, the editor of the _Philosophy for Everyone_ series, for his invaluable assistance in putting this volume into proper shape. He has been a most patient and helpful guide.
Thanks are due to my colleague in the Philosophy Department here at Virginia Wesleyan College, Cathal Woods, who made some excellent suggestions for improving the introduction, and to my daughter, Emma Goold, who helped to prepare the manuscript.
Much of my work on the volume was done during a sabbatical from teaching granted me by Virginia Wesleyan College.
# PART 1
# PASSING THROUGH PAIN AND FEAR IN THE PLACE OF PERPETUAL UNDULATION
**JACK STILLWAGGON**
# CHAPTER 1
# SHIPS OF WOOD AND MEN OF IRON
# Voyaging the Old-Fashioned Way and Seeking Meaning in Adversity
The ship slid down the face of the wave, listing to port at the trough, shuddering as if she would break up, then, lurching upright, she lifted once again to the crest of the next mountain of seawater. Rain and wind pelted us as we tried to hold a steady course, but the rain was insignificant in comparison to the spray caused by the bow smashing into the sea. Globs of water the size of a man's fist came speeding toward us, followed by smaller droplets in a heavy spray. Waves thirty-five feet high lifted behind and rolled toward our stern without relief. We pitched, we rolled, we scurried to our tasks with "one hand for the sailor and one for the ship" to do our duty and not lose our lives.
No, not a scene from centuries past on a whaler or on a clipper rounding the Horn; it was a "pleasure" trip on the replica of _HMS Bounty_ that brought me on this perilous journey. I was seventeen years old and the youngest member of the crew. How does someone that age get to crew on such a ship? My father piloted the _Bounty_ into New York Harbor after her global voyage to promote the 1962 film _Mutiny on the Bounty_ , starring Marlon Brando and Trevor Howard. The _Bounty_ was still owned by MGM Studios and was to spend two years berthed at the Flushing Bay Marina in sight of Shea Stadium just outside the grounds of the 1964–1965 World's Fair in Flushing Meadow. My father and two generations before him were tugboat men in New York Harbor. He advanced to piloting in the northeastern United States and around the world. His accomplishments included being chosen to pilot Queen Elizabeth's royal yacht while she visited America, so he was the first choice for MGM for this prestigious assignment. His influence got me the job.
On arrival in Flushing Bay, the MGM public relations man aboard was disappointed at the lack of wind and the _Bounty_ 's listless sails as she approached the dock under the power of her two huge diesel engines. When my father understood MGM's concern, he solved the problem by shifting the _Bounty_ into reverse. Stunning photographs were taken of the majestic square rigger on a bright sunny day with all sails set and filled with a breeze. Of course, sailors questioned why there was a wake in front of the ship in the photograph and not behind, but the landlubbers never knew. Ah, the magic of Hollywood!
I had worked on the _Bounty_ at the World's Fair when I was sixteen years old. Few visitors came because we were located outside the park grounds and only the truly determined found their way to us. We were supposed to be tour guides but were transformed into maintenance men to repair her decks and spars, going aloft with a brush tied to the wrist of one hand and carrying a bucket of log oil in the other. I was young and fearless and climbed the ratlines without concern. MGM decided not to remain for the second summer of the Fair but to put in at Jacobson's Shipyard in Oyster Bay, Long Island for a winter re-fit. I sailed the _Bounty_ there and Hugh Boyd, the ship's boatswain, asked me and another _Bounty_ sailor to work weekends to get her into shape.
What a surprise when Boatswain Boyd called me to invite me to sail her to Florida in June of 1965! I could make the journey from New York to Port Everglades, Florida but not the last leg around the Tortugas and up the Gulf coast to St. Petersburg, where a new park was to be built to showcase the ship. I had to return to take my final exams of senior year in high school. As the youngest on board, I received the dubious honor of being designated "the Captain of the Head." That meant that, in addition to my duties aloft, on watch, and at the helm, I was to clean the toilet used by twenty-one crewmen.
Remember that I was not "shanghaied" or pressed into service by a foreign power. I volunteered and was thrilled to accompany the crew, which included some VIPs, such as my watch officer, Julian Roosevelt, a successful businessman and Olympic sailor. What causes modern men and women purposely to leave comfort and security behind to venture out into danger and discomfort? Even a family camping trip can involve inclement weather, insects, faulty equipment, and possibly a close encounter with ferocious animals. Why not stay home and enjoy a DVD of the great outdoors? Some campers mitigate the inconveniences by using air-conditioned motor homes, televisions, and bug zappers. Still, why not just stay home? There is some motivation in us to push the envelope and to experience greater challenges. Most of the time, we seem to seek the greatest comfort level possible and to look for cures for any ailment. Yet, we also eschew modern tools to prove something to ourselves or to others. _Certo ergo sum_? (I struggle, therefore I am?)
I was aware of the struggle and danger when I eagerly accepted the offer to ship out with the _Bounty_ crew two hundred miles off the East Coast from New York down to southern Florida. My great-grandfather died in the frigid winter water of Newtown Creek by the East River in New York City. He slipped on ice and fell into the water from his tugboat at night. Some years later, his son, my grandfather, had to abandon his tugboat as the load suddenly shifted on the barge he was towing on a short hawser in the East River, not far from where his father had died. My grandfather, Captain Walter Stillwaggon, yelled to his sleeping deckhand to get off the boat. Without hesitation or the need for greater clarification, the deckhand leaped overboard and was saved. In almost the same spot as his father's and grandfather's maritime accidents, my father, Captain Jim Stillwaggon, then an eighteen-year-old deckhand, had his foot crushed between a tug and barge. He had a successful career in spite of that terrible injury, using an artificial limb, playing a very good game of golf, and even climbing Jacob's Ladders well into his seventies in the Caribbean, Wales, South America, and the Middle East. That provided us children with an important lesson in overcoming adversity and remaining positive.
So, in 1965, I knew the dangers, but they excited rather than discouraged me. The risks presented themselves soon after we left the dock. There was a small ceremony at our departure with the families of some of the sailors on hand to wish us well. Local news reporters filmed the event. The crew could not stop to enjoy the festivities as we took orders from the mates and the boatswain to load up supplies and stow them below. We motored from Oyster Bay on Long Island under the bridges that connect Queens with the Bronx, through the infamous Hell Gate and down the East River to the harbor. As we left Sandy Hook behind, my father completed his piloting of the _Bounty_ and boarded a small vessel to return to the safety of terra firma. Almost as if on cue, the swells started to lift and roll our 180-foot-long, 412-gross-ton ship. What power the sea has that even large ships are not immune to its strength and capricious nature!
A heavy wood and metal block was swinging and smashing against the main mast crosstree and shrouds. It had to be belayed so the boatswain looked around for a junior crewman who could spring into action and lash down the heavy block. I was recruited for the task. With a leather string in my teeth, I climbed aloft. Although I was on my guard, the block hit my head with a glancing blow. I was dazed but held on until I felt well enough to grab the pendulous block and lash it to the shroud.
As we lost sight of land, the separation from the security of home and hearth was jolting. A squall had moved in and an ominous look came over all we could see. The June sky looked like winter and the waves built to incredible heights. The wind whistled through the miles of rigging on board our ship. Soon we all regretted the sumptuous breakfast provided by our Greek cook – wet scrambled eggs, spicy sausages, pastries... all you could eat. Between the constant rolling and pitching as we motored southward and the morning feast, seasickness was rampant. I had never before, nor have I ever since, suffered from _mal de mer_. One of the _Mayflower_ crew who was on board with us jokingly pointed out that I should not fear dying from seasickness as I was "too sick to die." I drew little comfort from his attempts to cheer me and ate no food for thirty-six hours. Just seeing Boatswain Boyd at dawn eating a green apple as he danced on the deck playing his concertina made me ill again.
I had the four-to-eight watch so I saw dawn and dusk each day. Bow watch was a horror as the bow dug deep into each trough with green water running down the decks. We needed to be aware of the flotsam and jetsam that can stave in a wooden hull like ours. Telephone poles or even a semi-submerged container can put an end to a ship and her sailors in a moment. Steering was even more of a challenge as the _Bounty_ slid down thirty-five-foot waves like a surfboard. She was being pushed, not making headway under sail, so her bow pointed ten or twenty degrees port or starboard off course. Mr. Roosevelt, first mate and my watch officer, sternly urged me to "hold the course, helmsman, hold the course steady." I steered this way and that, hoping to average our 175-degree bearing, but it was beyond me. The compass rose was spinning in the binnacle and I was still getting sick.
On the first full day at sea I awoke to the sound of an alarm. It was the claxon blast you hear in movies when an attack is expected. Ours was a fire alarm. That is a sound that can strike fear into the most intrepid heart. Fire at sea! There is no escape route. It is an odd feeling to look all around you and see nothing but a horizon. Instead of providing a sense of vastness, it gave me the opposite impression. It was as if our existence had been reduced to a small disk, like a toy boat left by a child on a round table top. I went to a fire station but learned later that it was not the one assigned to me. That error might have saved my life. My fire station partner, a crewman from the _Mayflower_ , passed out as we extinguished the fire. He remained in his cot for five days without moving. We learned later that he had inhaled sulfur fumes from the caulking used to seal the teak deck.
We had three fires in the first three days. Two were from dry-rotted wood that was next to the engine exhaust conduits. The third was electrical, from faulty wiring. We worried each time as sparks landed on wood or as flaming ashes rose up through our hemp lines and canvas sails.
I wasn't privy to the discussions our captain and officers must have had about our misfortunes. Did they consider turning back? How would the owners and organizers react? Would they suspect that our officers had given up prematurely? Did our officers take further risks to prove their courage? I did overhear one such conversation between the first and second mates. It was about the _Mayflower_ crewman. They wondered whether he should be left to die on board or be transferred by helicopter to die on shore. His mates from the Mayflower believed he had a sister in Georgia and surely that would be a better place to die with family nearby. That was a dose of grim reality for a seventeen-year-old to hear. Days later, when the man emerged on deck looking well, it was as if Lazarus had returned from the dead.
Imagine the sailors of olden days who had no radio, no inflatable rafts, and no Coast Guard. Turning around would not have been acceptable. Whaling ships out of Bedford and Nantucket left for four-year voyages. There was no opting out for them. When you made your mark on the crew list before shipping out, you were on board come what may. They had little choice and lived in harder times.
Ishmael in _Moby Dick_ seemed to voyage for adventure more than fortune or need. For most, though, fear of poverty, the law, or oppression drove young men to the sea. What a difference from us who leave our modern conveniences behind and long to experience what life was like when deprivation was the rule and people were expected to toil endlessly. We leave our norm of non-stop entertainment and self-satisfying behavior to take on the role of Richard Henry Dana in his experience "before the mast." Do we strive to experience the past as amateur historians or are we atoning for our frivolous modern lifestyle? Are we like the actor who takes up a political cause to prove he can do more than just play make-believe? Does the fact that we endured a risky trial and survived add importance to an otherwise silly existence?
Most people do not climb Mt. Everest, kayak through Oceania, or sail offshore on a replica of an eighteenth-century sailing ship, but many attempt to step out of their secure environments to expand their horizons and test their boundaries. Do we fully understand our motivations in doing such things? Think of the soldier we praise for his or her patriotism. Is there another motivation, though, in addition to love of one's country? Is that person also trying to test his mettle? The soldier wants to know whether he has what it takes to earn his red badge of courage. So many combat veterans say that, when the first bullet whizzes past your face close enough to feel the heat, your mind orders your body to flee the danger. The soldier stays, though, partially through training and discipline but also to prove to himself and to his comrades that he has fortitude.
Most of the crew on the _Bounty_ had met only shortly before the departure. Even so, we quickly came together as a team and worked well together. Our officers and boatswain get some of the credit. But there must have been another reason. Perhaps we wanted to prove we were worthy of each other's trust. One example of bravery stunned us. We had finally emerged from a couple of days and nights of stormy weather and had set the fore and aft sails. The engine finally was shut down and the rolling stopped as we heeled over into a comfortable pitch and the seawater turned from green to blue as we entered the Gulf Stream. "All hands on deck" was called out as we tacked from port to starboard and back again hours later. Our peaceful resting in the sunshine was abruptly shattered as a poorly secured sheet slipped from its belaying pin and the main topmast staysail luffed wildly. The boatswain sent us aloft but no one could figure out how to gain control of this large canvas sail. Suddenly, a young sailor leaped from the shrouds, aiming for the flapping staysail. His weight collapsed the sail and he pinned the bulk of it to the main mast, holding on with all his might. Although shocked for a moment, we responded to the boatswain's charge to "help that man!" We left the relative safety of the shrouds for the flimsy-looking monkey ropes that spanned the gap from the shrouds to the mast and, like clumsy trapeze artists, tried to add our weight to his and to secure the sail until the sheet could be tamed once again. When the job was done, it was pats on his back and cheers all around. There was no boasting on his part and the brave deed was not mentioned again. His act was so selfless that it stands out as unique. What if a blast of wind had lifted the sail just as he jumped? We would have been asking ourselves again whether a sailor had family ashore to whom to deliver the body.
We went back to work knowing that something special had been shared. Were we feeling more like the "men of iron" who came this way before us? There is a sense of achievement that comes from taking risks and persevering. We feel reassured to be accepted by others we admire and respect. General Eisenhower went to the staging area where the first paratroops to land in Normandy were about to embark on their aircraft. He wondered what he could possibly talk about to these men who were headed into grave danger. He asked one of the men whether he liked jumping out of planes. The paratrooper replied, "No sir, I don't. But I like to hang out with fellows who do." This spirit must lie in all of us to one degree or another.
It has been said that "the mass of men lead lives of quiet desperation," yet we strive to break out of the mold that has been made for us and to do more or to be different. We might not like jumping out of planes or climbing a hundred feet above a rolling deck, but may feel that we must do these things to feel truly alive. Now, four days out, we were experiencing the best of the old sailing days. A whale was spotted nearby. Flying fish hopped on board – surprised at their dry landing, I'm sure. We tossed them back and noted that we couldn't be far from the Bahamas. The sun felt warm and dried us out after the first few days of thorough soaking. The food was good and tasty after days of thinking I would never be able to eat again. Life seemed fuller now that we had gone to the edge and made it back. There was time in between watches and meals to just experience life and be glad for it.
Do we value life more when it seems like we might have lost it? For those who have tasted adventure, it is particularly irksome to hear the teenager who has everything say, "I'm bored." Do they need a sea voyage on an ancient vessel to cure their boredom? Whence this modern curse of ennui? Can you imagine the sailor of old who felt fortunate to receive some discarded sail cloth to fashion a pair of trousers, or maybe even mittens to prevent frostbite as he went aloft while rounding Cape Horn? Was he bored? Ha! I don't think those sailors even knew the word. Work consumed much of their waking hours and rum helped them sleep. Their lives might have been miserable for much of the time but the high points probably seemed higher by comparison. Think of the original crew of _HMS Bounty_ leaving foggy Britain behind, enduring short rations and flogging underway, but then arriving in paradise. Perfect weather, willing women, and endless feasting! To what extent did that reward counterbalance all the dreariness and suffering of the rest of their lives?
Is that why those seven days on the _Bounty_ come back to my memory so clearly? In spite of the danger and discomfort, it was a high point. It stands out and is cherished. You have seen the veterans on parade with their versions of their old uniforms. Often they wear hats that show what unit they belonged to or the location where they served. They have such pride now but, when they were on active duty, most of them could not wait to get out of the service. They counted the days with "twenty-one and a wake up" or kidded a pal who had more time to serve with sayings such as "I'm so short that down looks like up to me." So, why do they shift gears later and remember their service so fondly? They recognize that, in spite of the danger and lack of luxury, those years were a high point. They were involved in something important. They were needed by their teammates and maybe were even recognized for an accomplishment. The negatives are forgotten as the lens of time focuses their memories more favorably and they say, "Those were the days." Do we all selectively reshape our experiences that way? Are the insults and slights allowed to fade and the low points relegated to a dustbin, while the high points are refined so that our victories become more glorious over time? I suppose it is harmless. It seems less a deceit than an artifice to help us to cope.
Being too young to drink legally meant that I stood watch the night we docked in Port Everglades, Florida. The others went into town and celebrated. It was fun to see them return. One group piled out of a taxi, drunk as skunks. I could not figure out why the cab remained where it had stopped until I realized that there was no cabbie. I guess my shipmates had borrowed the cab. Even the boatswain arrived after the 0100h curfew he had set. He apologized for making me overstay my watch. Can you imagine this accomplished sailor who had sailed the seven seas saying sorry to me, the youngest man on board? There were no apologies necessary from the boatswain. I was just seventeen years old so his apology made me feel like one of the men. Maybe I was becoming one of the "men of iron." "Ships of wood and men of iron," as they used to say.
The next day, I packed my duffel bag and stepped ashore. The solid ground felt funny and seemed to roll like a deck at sea. As we assembled on deck to say our goodbyes, the captain presented me with a framed certificate praising me for my able-bodied sailing and sober habits. I was given a one-dollar bill for volunteering, which is Canadian practice and was done because the _Bounty_ was built in Nova Scotia.
I was glad to be heading home yet very sad to see the _Bounty_ sail away. I watched until the main royal disappeared over the horizon. To end the crucible and part with my shipmates after all we had been through together was a wrenching change.
Still, the journey lives on as part of who I am and what I believe I can do. I had left the safety of home and hearth and endured a trial. I had survived and earned respect from men whom I admired. I had struggled in the face of adversity and valued life a little more than before. Like the rock climber who maintains an inner self-confidence because of having scaled challenging heights, or the marathon runner who completes a race, I was more at peace with myself for reaching new horizons and I expected more from myself from that point onward.
**GARY JOBSON**
# CHAPTER 2
# WINNING PHILOSOPHY
# Developing Patience, Inner Strength, and an Eye for the Good Lanes
In sports the final result is never a sure thing. For competitors and spectators this uncertainty makes every event exciting. This is why we tend to cheer for the underdog. We prefer a close contest with both teams having a chance to win. Being part of a race or game that comes right down to the wire is a thrill. For competitors the uncertainty can be frightening as well as exhilarating. Winning takes inner strength and even courage. The big question for every competitive sailor is how to generate the will to overcome fear and to win.
I have found many effective techniques in my fifty-plus years of racing at every level of sailing. The first step is to avoid worrying about the final outcome, and instead think about the task at hand. I work on one thing at a time and measure my progress along the way. A well-organized practice session gives competitors confidence later during the heat of battle. Before important events I spend time visualizing the race, my boat, my competitors, the surroundings, and the water. This makes me relaxed. If I sense nervousness coming on I go back to my normal pre-race routine. And, if it all goes bad, I have learned to take a deep breath and quietly say to myself, "Just for fun let's see how we can turn this around." Unexpected things happen on the race course. Both good and bad experiences are valuable tools in helping to overcome adversity. By thinking hard and being creative, success is always a possibility. In the end, every victory is well earned. All athletes know there are many defeats backing up a championship.
As mentioned, everyone loves cheering for the underdog. Respect is earned when the situation looks helpless but the underdog prevails. Three special events come to mind from over the years as examples of amazing feats in sailing.
Imagine being expected to win the Olympic trials. The qualifying series is seven races with one throw-out against the cream of America's sailors. Race one ends with a disappointing fifth. In race two, the mast goes over the side. Now, five races are left and the odds of winning are looking slim. For many sailors the end would come soon. But not for Buddy Melges, Bill Bentsen, and Bill Allen. With a second string mast and a lot of grit, the trio bounced back by taking five straight firsts. They went on to the Olympics in 1972 but the story did not end there. The trials were sailed on windy San Francisco Bay with the idea of emulating the breezy conditions expected in Kiel, Germany. But there was a surprise. The winds of Kiel went light. How many times have you been at a regatta and heard the words, "The winds aren't normally like this here." Melges and his crew had trained in heavy winds. But somehow the Americans found one zephyr after another to defeat four-time Olympic gold medalist Paul Elvstrom. Melges and crew took home the gold medal. America has never produced a better sailor. Buddy's ability to adapt to unexpected changes sets him apart from all other sailors.
By the age of forty, Ted Turner had won every major ocean race, the America's Cup, and dozens of small boat titles. Turner understood comebacks, having suffered through a miserable summer on the super-slow _Mariner_ in 1974 and returned to defend the America's Cup in 1977. But that victory wasn't complete since one race still stood out in his mind – the Fastnet. Aboard his Sparkman & Stephans 61-foot _Tenacious_ , Turner assembled a crack crew of seasoned ocean racers. The preceding Cowes Week was windy and _Tenacious_ won many of the daily prizes. But the magnitude of the 1979 Fastnet race was not anticipated.
The first three hundred miles to Fastnet Rock, eight miles off the coast of Ireland, were easy. But soon after rounding the rock the wind built to forty, then to fifty, and finally to over a frightening sixty knots. The _Tenacious_ felt the fury of the storm. Turner kept the crew focused and calm. It was as if Turner actually thrived on the massive waves and powerful winds. With no visibility, feeling the waves and wind proved to be the solution. Turner mastered the storm, emerging well into the lead to take the first overall prize out of 303 boats. To this day, Turner will tell you it was his most satisfying victory. For me it is a real thrill to have shared both Ted's America's Cup and Fastnet victories.
Being the first American skipper to lose the America's cup brought a lot of criticism Dennis Conner's way, but actually he sailed brilliantly in 1983. The score could easily have been 4–0, advantage _Australia II_. But Conner outfoxed the Aussies time after time. In one race he actually crossed the Australians on port tack. This is something that should never happen in a match race. It is the equivalent of fumbling a football on the opening kickoff and the other team scoring a touchdown.
In the end, Conner failed to cover on the final run of the last race. The error would have ended many careers, but Conner knew he had the ability to make a comeback. He and his syndicate head, Malin Burnham, pioneered corporate sponsorship in sailing. Conner had been beaten by _Australia II_ with an innovative wing keel, so he launched a huge research project to develop a fast boat. And they won the next Cup! Overnight Conner became a national hero. Adding to the pressure on Conner was live, worldwide television coverage with onboard cameras. Conner's strongest assets in that campaign were his personal resolve to erase the loss of 1983 and his ability to play the wind shifts. Throughout the 1987 series, he cleverly chose between covering and playing the wind. It was his finest hour.
In each of these three cases, American sailing champions knew they had a chance to overcome adversity. It was not a question of luck, it was a question of resolve. I, too, have learned to come back after sailing disappointing races.
A particular collegiate regatta was going to be intense for me. Only two boats from the Middle Atlantic District would qualify for the Intercollegiate Singlehanded Championship. The elimination series was held on my home waters off the SUNY Maritime College in a new boat, the Laser. This nifty dinghy was new to all of us. I had only sailed the Laser once. At six-foot-two and 180 pounds I was the right size for the boat. But after two races I was in twelfth place out of sixteen boats. Not good! Back at the dock my coach, Graham Hall, said in a very calm way, "you look like you are trying too hard and thinking about qualifying." Hall continued, "you're out of it right now so just go back out there and try to make one improvement at a time." "But what about boat speed?" I asked. "Oh, don't worry about that, just go have fun!" "Fun?" I thought. Somehow Hall loosened me up. It was at this precise moment that I regained my confidence and my sailing career changed dramatically. I went out and had fun, by winning the next race. In fact, I won the next ten races in a row.
Suddenly, sixteen years of sailing clicked. Looking back through my college logbooks I recorded two thousand races in four years. There were many fundamentals that I refined through competitive drills and lessons learned from mistakes. Every single day I spent time working on boat handling by sailing continuous figure eights, practicing timed runs, and tacking and jibing endlessly. Even to this day I find crisp roll tacks one of the highlights of life. Making figure eights by rounding a set pair of buoys gives you practice jibing, tacking, rounding windward, rounding leeward, and accelerating. Precise maneuvering is an essential asset when you are around other boats. These drills gave me an edge and to this day I savor every roll tack.
At the Intercollegiate National Championship two months later, I again had trouble in the first two races. Now we were racing Lasers on Mission Bay, near San Diego. Observing my problem, Graham Hall suggested just one tactical adjustment: "don't be the first to tack after every start. Wait until you have a clear lane. Use a boat to leeward and ahead to push away traffic." He continued, "think of a running back using a blocker to gain more yardage."
The philosophical shift for me was to be more patient and look for good lanes. For the first time I understood how to use competitors as "blockers." I went on to win ten of the next fourteen races to take the championship.
The most important lesson Graham Hall taught me was to go out on the racecourse and have fun. Another was the importance of good coaching and helping others become successful sailors. Improving is easier if someone helps you. The combination of coaching, practice, and a positive attitude worked well for me.
Taking command is a big responsibility and often lonely. Many people would rather not be in charge. But, for others, leadership is the essence of life. And for many sailors the most rewarding leadership opportunity is aboard a boat. After the fact, a leader can be a hero or a goat. Unfortunately, there is rarely anything in between. Thanks to the experience of racing with many champion skippers over the years and also having had the command myself from time to time, I've learned many valuable lessons in the art of leadership.
To be a good leader, once in charge, be in charge. Trust your instincts! Second-guessing a decision can cause a breakdown in the command structure. The fewer words spoken, the better. Consistency and confidence are essential. Work to avoid arrogance. Plan in advance. Once a plan is decided, execute it with precision. Let everyone on board know your plan. Mistakes are avoided with early preparation. Hold a crew meeting after leaving the dock. Gather everyone on deck so they can easily see and hear the leader. Prompt attendance is mandatory. Talk about safety routine, job assignments, a stated purpose for the day, the anticipated weather, the course to be sailed, and any potential surprises that may occur. Forecasting situations in advance allows the crew to be mentally prepared. By being an oracle of the future, the leader builds confidence that he or she is on the correct course and knows how to handle a troublesome situation. During bad times always remain calm. If the leader is in control, the rest of the crew will follow his or her example.
Communicate clearly! During the crew meeting publicly state every crewmember's job during "all hands." Take the guesswork out of the equation. As I've mentioned earlier, if events become tense, go back to your normal routine. The crew will feel more comfortable.
Deploy your crew carefully! Be sure the right people are resting at the optimum times. Keep a steady pace going and avoid burnout. You cannot have an "all hands" situation every single hour for two days running, for example. It simply will not work. Set up a buddy system on board. In this way two people can share any single task. A common goal brings a crew together. If there is a problem, a second person provides a backup or can help with a solution. Remember the old saying, "many hands make light work."
Keep no secrets! Be open with your crew. It helps to say nice things about the people you sail with. Word will spread. Everyone responds positively to a job well done and the proper acknowledgment. Good leadership is hard. It is learned over time. The best leaders understand all the functions on their boat and the capability of their crews. By setting a common goal and organizing the team by giving specific assignments, the crew will naturally want to work together.
Strike a balance between selling your ideas and being a dictator! Although there is strength in collective wisdom, ruling by committee rarely works. Yacht races and naval battles are won by leaders, not committees. I believe in collective action. Asking for advice ahead of time helps people feel they are part of the decision-making process. But during the heat of battle there is no time to consult. Take charge! Be authoritative and clear! Hesitation creates uncertainty and undermines confidence. America's Cup champion Dennis Conner often polls his crew before the start of a race about any ideas they might have on the best strategy. But Conner will point out that, once the race starts or after an event has occurred, he doesn't want to hear "we should have." The time to speak up is prior to the event, not after the fact. There is no room for Monday-morning quarterbacking on board a boat.
Striking the right balance will require some creativity at times. I recall a regatta in Maine in which one crewmember, sitting on the windward rail, constantly questioned the decisions of our afterguard. It was a tough series. The currents in Maine are complex and the wind was shifting dramatically. At times we looked great, at other times we did not. The key to doing well is to believe in your convictions and to stick with your game plan. Crisscrossing the racecourse simply puts you further behind. But, after every tack, the crewmember on the windward rail sarcastically questioned every call. "Why did we do that?" or "Well that was a big mistake." Our afterguard elected to ignore the patter. As it turned out, we had a good race, finishing second.
Overnight I thought about the young man on the rail and what to do about him. The next race the wind filled in well before the start. It was going to be a good day. After our crew meeting, but before the preparatory signal, I invited the young rail rider aft and said, "We have three options. We can sail to the right side of the course, up the middle, or on the left side of the course. Since you seemed to have all the answers yesterday, which side should we sail on? And remember – we're in the running for first place for the week, so your decision could make the difference." The young man stuttered a little bit and sweat started to form on his brow, but I just waited. Finally, after two minutes he admitted, "I'm not really qualified to make this call," at which point I said, "Well, when we do make decisions, it would be helpful if you were supportive as opposed to second-guessing." We did not hear a negative word out of him for the remainder of the race.
At the risk of sounding boastful, we won a big Etchells regatta with 1–1–1–2–5–1–4–1 finishes. The next day one of my friends called to offer his Bravo Zulu: "Nice going except for the fifth," he said. It made me chuckle because that was our best, and most important, race. Sometimes when you pop into the lead after a good start or fortunate wind shift winning is easy. Racing when you are well back in the pack is the real challenge and often where regattas are won.
At the start of race five (on the second day), we got off to an adequate start and rounded the windward mark in second. But the wind shifted, and we were late to maneuver into the center of the course. Every boat behind us did take advantage of the wind shift and suddenly we were last. Ouch! To make matters worse, our rival, who was in second in the regatta at that point, was winning the race. My crew, Jud Smith, was brilliant as we approached the leeward turning mark; he declared, "Here's what we are going to do. We are going to round last, and then start the race over." His words were delivered with humor and helped me recover from blowing a good position. By the time we made the turn for the next beat to windward on the five-leg course, our Etchells fleet had caught up to a fleet of thirty-eight J22s sailing ahead of us. J22s are smaller and slower than Etchells. It is hard to sail through a fleet that is slower because they disturb the wind and chop up the waves. And, adding to the confusion, the wind strength was dropping and it was getting very shifty. In other words, there was lots of opportunity to pass some boats.
There is always a tough choice between sailing in disturbed wind and staying on a big lift, or tacking away to sail in clear wind but toward the wrong side of the course. We decided to work the lifts and ignore the boats ahead. It was a good strategy. Within a few minutes we had passed five of the twelve boats in our Etchells fleet. At the next windward mark we jibed immediately and headed downwind. On the run we were able to pass another boat. Meanwhile, our rival got tangled up with the J22 fleet and missed a few wind shifts. One more leg to windward and we moved into fifth while our rival ended up in seventh place. A positive attitude, confusion from another fleet, and weird wind shifts saved the race.
College sailors learn that maintaining a good, consistent average is the key to success. In 2009 and 2010 the winning team at the College Nationals Championship averaged fifth place out of eighteen teams. For me, the real challenge is being able to clear my head when adversity strikes. Imagine being in the top two places in five consecutive races and suddenly being in last. It is easy to be upset and start complaining. That is why Jud's words when we lost the lead were both soothing and helpful. After recovering mentally, the next step is to make a plan.
The best time to plan your strategy during a race is on the prior leg. Think through, in advance, which side of the course has more wind. If you see a boat make a substantial gain, the pattern often repeats itself. Once, in a cat boat regatta on Barnegat Bay in New Jersey, on the last leg to windward, I noticed one boat sail from last into fourth. No one else seemed to see this big gain. About fifteen minutes later a second race was started and I sailed off in the direction where the big gain was made. Happily, the wind was stronger and there was some favorable current over in this area. We led at the first mark and held on to win. Staying alert paid off for us. More often than not, however, sailing on the closest course to the next mark is the best strategy. In long-distance races, using just one strategy of simply staying on the rhumb line gives good odds of winning.
There is the age-old question of when to split with the fleet and take a flyer. In my experience flyers rarely pay. Ask yourself this question: Should you split with the fleet to make a big advance and risk losing many boats, or chip away at the fleet by trying to pass one boat a time? It's no accident that America's two Olympic medalists in 2008, Anna Tunnicliffe and Zach Railey, won their medals without winning a single race. The lesson learned is to avoid taking big chances.
If you adhere to the philosophy of going for a good, consistent average then I suggest staying with the fleet. Before the Olympic Games, Zach Railey told me his goal was to simply try to pass one boat at a time without taking big risks. The cat boat race was an exception to this rule. I was confident that one side of the course was clearly favored. I had seen it during the previous race. This is when you should be courageous and take a chance with a flyer. Watch for new opportunities and take advantage of them! Anna Tunnicliffe made a flyer pay in the last race of the 2008 Olympics. She worked the right side of the course on the first leg, and was back in eighth out of ten boats. On the next leg she headed left. Over her shoulder she noticed a strong puff well to windward and beyond the lay line. She went with her gut instinct, tacked for the wind, got the puff, and surged into third place at the next mark. The move earned Anna a gold medal.
Maintaining a good attitude during a race is essential. In that Etchells regatta we had two moments that could have turned bad, but a quick acknowledgment of our mistakes calmed our crew down. The first case was jumping the gun at the start. Of course, we did not think we were over, but whoever does? The first step is to get bad thoughts out of your head so you can make a quick recovery. After restarting we noticed that we were on a big port lift, while the rest of the fleet was still sailing on starboard tack. A few minutes later the wind headed and we tacked. Instantly we were back among the leaders. Had we spent time and energy complaining we would never have noticed the wind pattern and would not have made the comeback. A few races later we made the mistake of splitting with our two closest rivals on points. Sure enough we paid dearly as the rivals found better wind and took the lead. Once again Jud simply said, "My mistake." That was it and we went to work to make up ground.
Try to work on one thing at a time. Compare your boat's speed with the other boats around you. Make one adjustment at a time to see what works. One person on the crew should watch what the competition is doing. Make mental notes on who is gaining. When the weather gets sloppy take the attitude that there is opportunity to pass. A good attitude during times of adversity will help you earn that good, consistent average. And sometimes the worst finish in a regatta might just be your best.
At a collegiate National Sailing Championship I noticed how smoothly the teams maneuvered their dinghies. Every dock landing was perfect, roll tacks were crisp, boat speed was at peak performance, and, best of all, attitudes were focused. Is talent like this natural or does it come from within? While almost every college sailor has the fundamentals mastered, I wondered what it takes to achieve that extra margin of excellence that produces champions. A mentor can begin the process.
Inspiration by mentors starts by setting a high example. We need to encourage current sailing stars to reach out and help others. This is the responsibility of success. Give a protégé a reason to work hard by getting them to get goals. Often a brief talk followed by an encouraging letter initiates the process. At this stage, mention that you will be following the aspiring sailor's progress. Occasional checkups will reinforce the message. When the young sailor starts asking questions, you will know that forward progress has begun. The next step is to keep records so lessons learned will be remembered.
Once the first seed is sown, the next step comes from within. A young sailor must start with the desire and then put in a worthy effort. There is no substitute for scheduling and participating in disciplined practice sessions.
Encourage your protégé to ask for help, read about techniques, and experiment on the water. When setting goals, make the priority learning over winning. Use high but attainable short-term goals to measure progress (I define short-term as under two months). Offer case studies as examples of how successful sailors have achieved long-term goals.
Many young sailors give up during times of adversity. They take errors hard. You can help overcome these problems by pointing out specific problems. During a regatta, ask one simple question at a time to allow a sailor to answer for himself what could be done better on the racecourse. Too much input creates confusion. As a mentor, always be calm, as if you expect improvement. This philosophy builds confidence. It's okay to be a cheerleader, but help to flatten out the inevitable rollercoaster ride by not getting too excited when winning or depressed during defeat. Remember that winning is the elimination of errors. Teach sailors to thrive on working out of bad positions after a slow start, making a penalty turn, or being on the wrong side of a wind shift. It takes work to keep one's attitude focused. During the heat of battle, a sailor must be calm and never sweat the small stuff when things go wrong.
There is a fine but dangerous line between helpful coaching and pushing too hard. This problem is frequently found in the Optimist dinghy class. If expectations are too high for early ages, a young sailor may reject sailing. Always balance sailing with other activities. Good grades are the key to opening doors. Make studies a priority. Sailing should never be a young person's only sport. The attributes and lessons learned in other sports are often analogous to sailing. The discipline instilled in a team sport is particularly helpful in sailing with a crew.
Once the seed of desire starts growing, organize a purposeful regatta schedule. Sail both larger and smaller boats to broaden experience. Singlehanded sailing always sharpens your skills. To prepare for maxi racing, for example, I sail Lasers to get me in tune with the wind and water. Sailing on different waters also builds experience. Sailing today has become too specialized. In the long term, variety keeps interest up and builds skills. One type of sailing supports another. Rising sailors exhibit many common attributes. They physically match up well with their boat. During a race, the body is fluid with a boat's motion. Actions are deliberate. Being in good physical condition enhances performance.
Many sailors try to over-sail a race and make mistakes such as tacking too frequently. Teach your sailor to develop a game plan and stick to it. Use common sense. Avoid locking into a match race battle while you are fleet racing. I like to refer to other boats by sail numbers to keep emotions out. It always pays to avoid protests even if it means occasionally giving a little extra room.
Progress may be slow and sailors might even regress sometimes, but keep plugging. As a mentor, if you detect burnout or rejection, it is okay to take a break from sailing, or change boats, crews, or venues. Young sailors should learn to work with the media. Few sailors are good communicators. Learning to be enthusiastic, helpful, and comfortable with the media at an early age will inspire more people to take up sailing.
Help young sailors by opening doors and providing thoughtful encouragement. There are no limits to the skill level sailors can achieve if they have the desire and put in a balanced effort. Look for a young sailor with potential, then help to build the desire and both of you will end up winning.
Winning in sailing takes inner strength. Years of experience help to build confidence. A shortcut to learning is to ask good questions and take notes. If you follow the stories and ideas outlined above, your time on the water will be more successful.
**CRISTA LEBENS**
# CHAPTER 3
# "HARD A' LEE"
# Why the Work of Sailing Can Be Great Fun
People choose to sail for many reasons, and certainly the physical and intellectual challenge is one of them. One could interpret the pleasure a sailor finds in such a challenge as a throwback to some evolutionary process that favored quick responses to threats or risks. Or the sailor could be read as a real-life existentialist facing the void. Both interpretations add depth to the understanding of sailing as a human endeavor, but both are a bit reductionist and miss the pleasurable dimension of undertaking an engaging physical activity on a beautiful summer day. One need not have one's survival threatened to experience the joy of sailing, and the knowledge one gains from experiences such as sailing exceeds the explanation focused on natural selection. Both ideas can be incorporated into an understanding of sailing as a human endeavor that enriches our inner life beyond simply a sharpening of skills and leadership abilities. The nature of this pursuit lies in the distinction between active engagement with reality and passive experience.
With a nod to Aristotle, I will take the reader on a journey to interpret sailing as the pursuit of "eudaimonia": human flourishing. We prepare the boat for the sail; face the risks of casting off and tacking, the challenge of sailing close-hauled, and the pleasure of cruising on a broad reach; and finally return to the pier without dashing the boat on the merciless shoreline. Throughout, the sailor gains a kind of self-knowledge about her skill and abilities as well as an appreciation of the relationship between the true and the beautiful as found in the elegance of a well-skippered boat. While the experience of the sublime may have a biological basis, I wish to explore sailing as an activity that leads us to human flourishing.
I have sailed almost exclusively in small craft on inland lakes with family and friends. I have raced in informal afternoon regattas that are about as competitive as a game of touch football in the park. The life lessons I have learned from sailing are drawn from my experiences of these enjoyable afternoons. For those interested in the details, I will be describing my experiences sailing a C-Scow boat, originally designed by Johnson Boat Works of White Bear Lake, Minnesota. The C-Scow is a flat-bottomed boat, about twenty feet long, with a single sail and two bilgeboards. I have also sailed on X-class, E-class, and J-class boats by the same manufacturer; on keelboats; and on the single-handed "Tech" designed by MIT. I prefer the smaller boats to keel boats because they are what I am most accustomed to, because they are more flexible and accessible (I can be in the water within an hour of deciding to sail), and because they test my sailing abilities at the limit to which I aspire. All of these characteristics make the smaller boat more appealing to me, despite the lure of the keel boat, where, as one experienced sailor put it, you won't spill your Mai Tai.
# Preparing the Boat to Sail
You and your mates must judge the wind to be strong enough for a good sail without being unsafe. The ideal wind is steady rather than gusty. If the gusts are strong, you will need to recruit a few more people to provide enough counterbalance for the boat. Recruiting may involve convincing people who have not sailed that they can do it and that _it will be fun!_
Next you should get "provisions," which can include the beverage of one's choice and some snacks. Then complete other preparations – including gathering personal gear, lifejackets, and the sail – and get down the bank to the boat lift. Transfer and stow all the gear, attach the sail to the mast and boom, recruit any last-minute crew, and paddle the boat out to the buoy, the point from which you will cast off.
As these preparations are being made, you must also coach any new sailors on points of safety, which mostly involve wearing one's lifejacket and watching out for the boom, especially before casting off and after mooring at the buoy at the end of the cruise. Finally you are ready to go: raise the sail, bring the boat around to head cross wind, and prepare to cast off.
# Casting Off
This procedure, along with catching the buoy at the end of the cruise, is one of the two tricky times during the sail when experience is especially important. A lot is happening at once: the sail is flapping in the wind, which can make it hard to hear other crew members and is rather alarming to new sailors; the boom can whack an unsuspecting person in the head with little warning; and each member of the crew needs to either know his task or stay out of the way. Once the skipper has brought the boat around to the correct angle to the wind, which takes a bit of effort, the sail will fill with wind and the painter (the line that ties the boat to the buoy) is cast off, allowing the boat to take off. This process can be a bit awkward because the boat is not easily maneuvered with the sail up and still moored to the buoy. If one casts off too early, the boat will drift downwind, possibly all the way onto the rocks. If one casts off too late, the boat may no longer be at the proper angle to the wind and the process must begin again. The crewmember at the painter should give the skipper enough slack to be able to direct the boat properly before letting go. Once the sails fill with wind, the boat should start to move forward. The sailing cruise begins.
# Some Existentialist Reflections
In Lee Werth's existentialist analysis of solo ocean-crossing sailors, he uses the example of Robin Graham, who at age sixteen set out to circumnavigate the globe and recounted his journey in the book _Dove_ , named after his boat. Werth draws on Kierkegaard's concept of being "in untruth" as characteristic of "the crowd," those who live their lives as followers. As Werth describes it, "Robin had great difficulty tolerating those who were 'in untruth.' He risked his life too often and was himself too much 'in truth' to easily endure pretence."1
I have not experienced "blue water sailing" on the open ocean. I have not taken a solo journey on land or sea that would compare to the journeys Werth considers. But I think that sailing nearly any type of boat lends itself to the possibility of living "in truth." Sailing requires a level of skill that cannot be faked. If one takes a boat out on a sail and is not competent to do so, soon that fact will become apparent to all, including to one's self, even though some may try to blame it on faulty equipment. Many sports demand a fair degree of skill to be enjoyable, but sailing without skill is not only unpleasant but can be dangerous. When the wind and the water have forced the realization that one is out of one's league, the best to hope for is to get back safely and without damaging the boat. Sometimes competence is less about skill than good judgment. Having the ability to choose wisely whether and when to sail, with whom to sail, and under what conditions (environmental and physical) is as important as having the know-how to execute key moves such as coming about (changing the direction of the boat) and landing the boat.
The risk may be much less when going for an afternoon sail on an inland lake than when attempting a solo crossing of an ocean, but it is still there. And that risk provides a moment of truth about one's abilities that one must confront. This is part of what makes sailing a meaningful activity.
# Cruising
Now the fun part begins! After casting off, during which sometimes directions must be uttered directly and tersely, it helps to remind new sailors that we have now reached the fun part. The crew arranges themselves on either side of the boat to distribute the weight evenly. With three or four sailors, everyone can sit on the "upwind" side – the side of the boat that the wind comes across. If numbers are larger, some crew may sit on the "downwind" side, which is less desirable because you must be ready to duck under the sail if the skipper needs to release it suddenly in the event of a large gust. Also, you end up leaning forward into the middle of the boat as it heels over rather than back out over the water, as one does on the upwind side. The crew usually keeps their spot for the length of the tack; that is, until the skipper changes tack (the angle of the boat in relation to the wind, whether upwind or downwind) by either "coming about" or "jibing." But that comes later. Now is the time to crack open a tasty beverage or grab a snack.
If the wind is fairly light, the boat will pick up speed and glide smoothly across the lake. If the wind is strong and gusty, the crew may be in for a wild ride! The boat will heel over and the upwind crewmembers must lean out over the water to counterbalance the weight of the mast and sail against the wind. No Mai Tai under these conditions! Some adventurous types may attempt to stand on the bilgeboard if the wind and the skipper are strong enough to keep the boat up on edge. It's important to remember that this is fun! As the downwind crew leans in under the boom, they will remember why they were advised to wear bathing suits. "C-Scow sailors have soggy buns!" as the bumper sticker says. They'll get their time up top soon enough.
More experienced members of the crew may work lines for bilgeboards, trim the sail, and make other adjustments, and also assist the skipper in looking out for other boats and the occasional swimmer. At this point, in my family, we may decide whether we want to try sailing into the bay of the lake – tricky because the wind shifts and often dies – or whether we want to crisscross the main lake for the length of the sail. Then conversation shifts to the boat, strategies for speeding it up, dreams of a family member buying another boat so we can race, and pondering the style of boat we should get them to buy. Occasionally more serious topics arise, such as "what do you teach in your philosophy classes?" But mostly the conversation is light and enjoyable, renewing family bonds and building friendships.
# The Social Dimensions of Sailing
Another element of risk that ought to be considered in contemporary life is the way in which perception of risk is gendered. Werth, in writing about solo sailors, notes that "A waiting 'cheerleader' is not a bad incentive. Too often, solitary sailing leads to solitary living; few wives can bear to 'only stand and wait.' Even fewer genuinely wish to make ocean passages with their mates."2 His statement underscores the point that sailing has been, and at the professional level still is, a (white, upper-class) male-dominated sport. It is worth noting that originally sailing was no sport at all, but for centuries a dangerous occupation undertaken not for enjoyment but for trade or military purposes. Sailing as a sport was the realm of the upper classes until the mid-twentieth century. I cannot speak for ocean sailing but, as far as inland lakes go, the availability of affordable boats for day sailing and community clubs that do outreach to school-aged kids make it a more democratic sport.
Recently (in the last summer or so) I asked my sister-in-law if she felt like she was taking a "risk" going sailing with my brother and me. She said she did, and so did I – not a great risk, but enough to make it a bit more exciting than, say, an afternoon bike ride. I found it interesting that my brother did not share that same sense of risk. I'm sure the sail is enjoyable and exciting to him too, but perhaps more as an exercise in skill and less as an experience where something dangerous could happen and someone could get hurt. (I want to emphasize that we are generally careful sailors, especially as we are all now middle-aged.) My point is that the element of risk can be, in part, a subjective thing. The perception of the risk one takes when going out on the water is definitely not the same for everyone. We who sail across the lake on a warm summer afternoon do not confront the void, the very real possibility of one's own death, as does the solo ocean-going sailor. But we face some degree of risk and the moment of truth about one's abilities.
# "Hard a' Lee" or Coming About
Eventually one reaches the opposite shoreline of the lake and it is time to turn the boat around. This is most often done by coming about. To warn the crew, the skipper calls out "ready about!" and the crew prepares to do their respective tasks – adjusting a line, moving, or simply ducking. Then, when everyone is ready, the skipper pushes the tiller (the long handle attached to the rudder that steers the boat) firmly and quickly to the downwind (lee) side of the boat, calling out "hard a' lee" (or "hard to the lee"). This movement turns the boat directly into the wind for a moment and the sail loses power, which slows the boat down a bit, but the pushing motion of the tiller moves the boat past that point to catch the wind on the other side of the sail. The change in direction may be moderate or it may be a switch from sailing mostly into the wind to sailing downwind. If the skipper doesn't push the tiller with enough speed and firmness, the boat will end up "in irons" and begin to drift backward. If the boat swings around too far, a strong wind may catch the sail and capsize the boat without warning. As the ancient Greeks advised, the middle path is the one to follow, and it takes experience to find it.
# Sailing Close-Hauled
The puzzle that non-sailors have about sailing is that it seems to be a lot of work, and often newcomers are told to do things in a rather brusque manner, and to do them correctly or someone might get hurt. Aside from casting off, landing, and coming about, these experiences are most likely to happen when sailing close-hauled, which means that the boat is heading as closely into the wind as possible without losing all power. This tack is not particularly speedy and not particularly enjoyable, but may be necessary to round a mark when racing or simply to return home. It is a good test of one's ability to trim the sail and make other adjustments to maximize the ability to "head up."
If one sails too closely into the wind, one ends up "in irons" with a flapping sail and no forward motion. The pressure of the waves starts to push the boat backward, which can be a disconcerting experience, especially if the shoreline is nearby. The way to get out of this is to "back sail," which means the skipper pushes the tiller downwind, steering backward. This will bring the boat broadside to the wind, and the sail should power up and the boat begin to move forward. It takes confidence to execute a move that seems to be the opposite of what one would do instinctively, and some experience to know when the boat is at the correct angle to the wind. Adjust too soon and you'll end up in irons again, this time possibly closer to shore. It takes experience, trust in one's abilities, and steady nerves. This is _not_ the fun part.
# Noticing the "Presence of the Absence" (Heavy Sailing Ahead)
The Existentialists sometimes ask what, or who, is not present? I have, for several years, been a member of Hoofers Sailing Club, affiliated with the University of Wisconsin-Madison. Community members are welcome to join. The club makes special efforts to welcome anyone interested in sailing, regardless of experience or ability. To that end the club offers programs to make sailing accessible to low-income kids and persons with disabilities. While I have noticed members who are people of color around the pier, few of those people are African American. This is an absence I want to note. For people to feel comfortable sailing, they must feel comfortable around the water. Typically that means they must be able to swim. I cannot imagine the terror that one would feel standing on a pier without some confidence in one's ability to swim to shore should one accidently fall in the water. And the ability to swim is a classed and raced ability – that is, historically, people of color, especially African Americans, have been banned from many public swimming pools, and not just in the South. Parents often teach their kids the basics of swimming. To take lessons one must put a value on the skill and have access to a pool or a public beach, and one must pay for swimming lessons. The absence of the ability to swim is also passed down from generation to generation. The economics of access to public beaches or pools mean that some people are excluded from a sport such as sailing before they even reach the pier. I say this because I notice the absence of my friends, who have experienced these conditions, and I miss their presence when I go sailing. Sailing, then, is one dimension of the human experience that is not open to them. This bit of analysis is kind of like sailing close-hauled. It is not the most enjoyable part of the trip, but it is necessary if one is to consider sailing as an activity that makes those of us who have access to this experience, in some sense, more human.
# The Broad Reach
No, this is not tough women stretching or anything similar that you may be thinking of. "Broad reach" is the sailing term for a tack where the wind is slightly behind the boat but the boat is not heading completely downwind. This is one of the tacks that makes the boat go the fastest. If speed is what you are going for, this is a fun tack! Depending upon the strength of the wind, the waves may crash over the bow, the boat may heel over enough to allow standing on the upper bilgeboard, and the crew on the downwind side may be anxious to move up. But mostly this is an enjoyable tack where one experiences the pleasure of working with wind and wave to move the boat across a sparkling lake on a pleasant afternoon with one's companions.
# Practical Wisdom
Aristotle introduced the concept of _phronesis_ : practical wisdom, or wisdom that guides action. John (Michael) Atherton explores the role of _phronesis_ in what he calls "outdoor kinetic experiences" (OKEs) and likens it to a kind of body-knowledge. Atherton draws on Peter Todd's concept of ecological rationality to explain knowledge-gathering and decision-making in risky situations: "Ecological rationality involves sampling the environment and making educated guesses under conditions where time is limited and dangers are real... Move fast or die."3 Such a skill has a clear evolutionary advantage. Atherton draws out the philosophical implications of such a skill:
OKEs expand our repertoire of knowledge-gathering skills that we may have previously ignored because in familiar places we may have had time to conduct full inquiries and make robust decisions based on the best information. If our cognitive system is, as Gigerenzer claims, one designed to find multiple ways to know, where we must decide and act quickly and without all relevant information, then OKEs connect us with a neglected aspect of our thinking, our knowledge-gathering abilities, and finally our epistemology.4
Epistemology is the study of knowledge – how we acquire it, how we justify it, and so on. Our understanding of knowledge is expanded by the inclusion of this dimension of our knowledge-gathering abilities; that is, the ability to make good decisions under pressure. This is a skill that requires both understanding and practice. In sailing, one can read about good techniques, but one must put that knowledge into practice until it becomes a kind of body-wisdom before one is said to be skilled. Or, as Yogi Berra put it, "In theory there is no difference between theory and practice. In practice there is." The experience of sailing will quickly teach one the difference between theory and practice as well as the need for both.
# Capsizing
On a reach, especially if one is directly perpendicular to the wind, a sudden gust could cause the boat to capsize. This does not happen often and is usually avoidable; however, in sailing a scow, one should be prepared for the possibility – that is, one should be ready to wind up "in the drink." In my experience, capsizing happens in one of two ways. The first is when, while moving slowly, a sudden gust comes up and possibly a line jams or some other malfunction prevents the skipper from responding to avoid the boat capsizing. The second circumstance in which capsizing is likely is when the skipper sails in a high wind, making the boat heel over to the maximum degree. This provides the chance for the crew to ride the bilgeboard, but sometimes the boat heels over just a little too far for the skipper to bring it back. If one wants to maximize speed, this is not good racing form because the boat actually loses momentum, but it is fun. Things happen fast up until the moment you know the boat is going over, and then everything slows down. The wind is no longer driving the boat fast, the sail catches the surface of the water, and eventually the boat comes to a stop with the sail in the water. Regardless of the circumstances that lead to this situation, the first priority is to keep the crew safe from falling or being hit (everyone should have put their lifejackets on before casting off). The second priority is to keep the boat from "turtling," which is where the sail sinks below the surface of the water and the boat ends up completely upside down with the mast pointing into the lake. This makes it more difficult to right the boat and could break the mast.
To right the boat, at least one person must climb on the upper bilgeboard (now perpendicular to the water) and lean out using her weight to counterbalance the weight of the mast. It helps if another person, ideally in another boat, can position himself at the end of the mast, lift it out of the water, and give a push to right the boat. Once the boat is upright, the remaining crew can climb back in and reposition themselves and the provisions and begin to discuss "what happened."
Capsizing is not fun. Though it is a rare occurrence, it is not unheard of. Mostly the crew can take measures to be safe and avoid injury, but when something goes wrong someone may get hurt. While capsizing is not fun, pushing the boat to the edge without capsizing can be a great deal of fun.
# Human Experience
Sailing provides opportunities for learning via engagement in outdoor kinetic experiences, but it also offers more than that. I have had some experience, though not extensive, with sports such as kayaking and cycling. While the latter two sports do require a certain degree of skill and technical know-how, sailing presents a much greater demand on one's abilities, both physical and intellectual. And, I maintain, there is pleasure to be found in that demand. Albert Borgmann, philosopher of technology, examines the meaning of everyday objects in modern life. Drawing on Csikszenmihalyi and Rochberg-Halton's study of the significance of material possessions in contemporary (1970s American) urban life, Borgmann focuses on these authors' analysis of music – specifically, the difference between a stereo and a musical instrument in terms of the meaning and significance people attribute to them as possessions. Borgmann draws a distinction between what he calls a "device" and a "thing":
The stereo as a device contrasts with the instrument as a thing. A thing, in the sense in which I want to use the term, has an intelligible and accessible character and calls forth skilled and active human engagement. A thing requires practice while a device invites consumption.5
One must interact thoughtfully and intentionally with a thing and develop skill in this interaction. Sailing a boat requires such an interaction; it calls forth skilled and active human engagement. At the other end of the spectrum, a powerboat is the epitome, in Borgmann's scheme, of a device calling forth consumption. Paddling a kayak, riding a bike, and hiking fall somewhere in between, but these activities do not call forth the degree of engagement that skippering or even crewing on a sailboat require.
# Returning to the Pier
By this time the crew may be getting a bit tired, or may need reprovisioning. In any case, eventually the cruise must come to an end. As the boat approaches the buoy, the skipper and the crew must assign duties. One crew member must prepare to catch the buoy as the boat moves toward it; another must prepare to lower the sail as soon as the buoy is caught. Other experienced crew may stand ready to raise the bilgeboards, thus reducing the power of the wind to push the boat forward. First-time sailors will be warned to keep low and once again watch out for the boom. The goal is to stop forward movement as close to the buoy as possible to ease the task of catching the buoy while maintaining enough momentum to reach the buoy rather than drifting backward toward the rocky shoreline.
The approach is made from downwind so that the boat can be turned directly into the wind within about ten feet or so of the buoy. This move deliberately puts the boat "in irons," which will slow it down but (one hopes) not so much as to lose momentum entirely. On a light-wind day this can be a relaxed set of maneuvers, but a high-wind day calls for experience and clarity regarding each task. The boat may be moving at a relatively high speed, necessary to keep from drifting downwind, and the skipper must bring the boat about at the right angle so as to be aiming directly into the wind and toward the buoy – not the pier and not the shoreline. On a nice, sunny day, non-sailing family members, friends, and neighbors are likely to be casually observing this sequence of events as they sip cool beverages. The crew member catching the buoy must be able to keep track of it as the boat comes about, and be ready to grab it at the right time, get the painter around it fast, hang on until the boat stops, and then tie a good knot that will hold the boat in place. The crew member lowering the sail must wait until the painter is definitely around the buoy and held securely, then drop the sail as fast as possible to ease up on the pull of the painter held by the other crewmember. Once both these tasks are accomplished, the crew can take a breath and relax for a moment. Friends, family, and neighbors may cheer, depending upon the quality of entertainment provided (which is preferable to their running to the rescue). The sail can be secured or removed, extraneous items can be gathered, and the boat can be left on the buoy for another sail that day or moved to the lift if that is the last sail of the day.
# Pleasure, Elegance, and Truth
Sailing can evoke the connection between truth and beauty, where beauty is a kind of elegance. The well-trimmed sail is not only aesthetically more pleasing than a sail flapping about but results in greater speed. The correct way to trim a sail is also more pleasing to the senses. The smoothly executed come-about is safer and more enjoyable to experience and to observe. The truth of one's skills is demonstrated in the aesthetic qualities of the sailing experience. If one seeks to develop this form of excellence, one will continue to practice this skill. Sailing, then, offers a chance to cultivate habits that lead to eudaimonia, or human flourishing. But, if one wants to be passively entertained, one will quickly turn to a form of activity that "calls forth consumption" rather than engagement. This is actually not a judgment on those who opt for a less demanding form of entertainment. Perhaps it is a question we may ask regarding the quality of our lives. What kind of life permits us to fully engage our human abilities and test our skills in the way that sailing (or practicing a musical instrument) provides? What kind of life leaves us with no more energy than what it takes to be passively entertained? And what kind of life is accessible to whom?
# Final Tasks
Once the boat has been put away on the rack for the night, the sail should be folded up and stored indoors. It should be folded like an accordion, following previous fold lines. This maintains the stiffness of the sail over time. Once the sail and other gear are stored, the cruise is over, but the storytelling has only just begun. Now for the Mai Tai!
**NOTES**
1 Lee F. Werth, "The paradox of single-handed sailing (case studies in Existentialism)," _The Journal of American Culture_ 10:1 (1992), p. 70.
2 Ibid., p. 69.
3 John Michael Atherton, "Philosophy outdoors: First person physical." In Mike McNamee (Ed.), _Philosophy, Risk and Adventure Sports_ (New York: Routledge, 2007), p. 49.
4 Ibid.
5 Albert Borgmann, " The moral significance of the material culture," _Inquiry_ 35:3/4 (1992), p. 296.
**RICHARD HUTCH**
# CHAPTER 4
# SOLO SAILING AS SPIRITUAL PRACTICE
# A Phenomenology of Mastery and Failure at Sea
Wisdom is facing "life itself in the face of death itself."1 Being in peril at sea can be deadly, but it can also be redemptive, a personal process of getting life into proper perspective. The sea is a microcosm of nature's indifference, impartiality, and caprice in regard to human will. How might we come to understand ourselves amid a universe of indifference, impartiality, and caprice? From the autobiographies of sailors who have gone on long offshore voyages by themselves, along with my own personal experiences of sailing, I will try to draw out the essence of the lived experience of facing the sea alone.
# A Phenomenology of Moral Presence at Sea
I would like to suggest that what I call "moral presence" serves as the general form of the lived experience of sailing for most sporting sailors. What is this? Moral presence is an existential counterpoint to the breakdown of technical self-reliance at sea, or failures of sporting performance. Moral presence is a personal stance in which a willingness to fail when nothing otherwise can be done is implicit in a person's sense of the moment. Making this implicit sense explicit is a manifestation of moral presence, or an assertion of human integrity in the face of despair. Moral presence appears in sailors' lives in three characteristic ways, and it becomes evident in their autobiographical reflections on risk-management and self-worth when they are engaged in their sport. The three ways are:
* Technical finesse
* Cosmic quest
* Personal test
Each will be elaborated by way of illustrations from a number of sailors' lives at sea and their thoughts about their nautical experiences.
First, moral presence may appear as the _technical finesse_ of Robin Knox-Johnston, who skillfully avoided possible excesses of technical hubris and won the Golden Globe Race of 1968–1969, the first "solo" and unassisted sailing race around the world. For him, the kind of spirituality that he found at sea always gave rise to pragmatic outcomes, or a "God helps those who help themselves" point of view. As he put it after rounding Cape Horn and taking stock of the readiness of his yacht, _Suhaili_ , to make the final run up the Atlantic to the finish line in England,
The sea and ships are great levellers. There is certainly no room on a small boat for a person who is incompetent or won't pull his own weight... All share the same risks in a storm, and no earthly influence will select you above the rest to be saved if the ship founders... Their whole existence depended on their ability to come to terms with the wind and sea, and to use these forces to drive their ship... It is not surprising that most [seamen] thought more of their counterparts ashore about the cause of these forces, and not in the least surprising to me that so many were superstitious or developed unshakeable religious beliefs, and sometimes both. I have found myself thinking deeply on the matter when out in rough weather on a small boat. It is all very well for someone sitting in an office to explain logically how the waves can build up before the wind, for we have discovered the natural laws that control this, but to a seaman, the explanation of these laws does not always seem sufficient... the rules are there, the physical laws that we have slowly learned. If we obey them we have a chance of survival.2
There is "action through non-action" here, the Taoist concept of _wu-wei_. Moral presence is a similar inner or "spiritual" posture assumed by sailors as they learn to get used to the sea. Sailors may engender a capacity to affirm personal integrity as a hedge against incipient despair in perilous circumstances at sea. If they are lucky, then sailors come to learn that there is much that can be accomplished by taking action balanced by "non-action" after all. Knox-Johnston writes,
It is no use knowing that your boat is heading towards the eye of a storm and praying to God to see you through it safely. That's not his job. It's your task to steer the boat away from the eye, and you are asking too much if you expect the boat to survive when you deliberately ignore the rules. My own philosophy is developed about the phrase, "The Lord helps those who help themselves." It is no good lying in your bunk, listening to the rising wind and feeling the boat beginning to strain and praying for God to take in reef. No one but a fool would expect anything to happen. One has to get up and reef the sails oneself before the boat's movement will ease... When everything has been done that you know you can do, you put your trust in your Superior Being, and just hope that what you have done is right... Because of this belief, throughout the voyage I never really felt I was completely alone, and I think a man would have to be inhumanly confident and self-reliant if he were to make this sort of voyage without a faith in God.3
A statement like this one testifies to Knox-Johnston's "detached concern for life itself in the face of death itself," albeit with occasional recourse to religious thoughts and behaviors.4 The achievement of such wisdom is a spiritual endeavor.
Second, moral presence may invoke a large and purposeful _cosmic quest_ , as it did for Bernard Moitessier, who felt driven to circumnavigate the globe as long as his food, water, and stamina allowed, opting out of the Golden Globe of 1968–1969 entirely. If it can be said that Knox-Johnston carried forward a mainly competitive style of sailing as racing, then Moitessier represents a more leisurely style of sailing as cruising. For Moitessier, spirituality meant drawing close to the elemental sea and believing it to be infused with a life-giving mystical force, a power that could be made one's own. For example, there is no better illustration of his resistance to keeping and using high technology on board his yacht, _Joshua_ , than his means of communicating his progress during the race.
The staff of London's _Sunday Times_ were eager to get as much publicity for their newspaper as possible by featuring frequent stories about Moitessier's voyage. They offered the Frenchman equipment that he had never before owned. The skipper of _Joshua_ came to resent such offers, but mellowed toward key staff once they took into account his views about just how much equipment was needed. Moitessier wrote,
I stopped resenting the staff of the _Sunday Times_... Robert, the head of the team, would have liked me to ship a big transmitter with batteries and generator. They offered it gratis... so [I] could send them two weekly messages. The big cumbersome contraptions were not welcome. [My] peace of mind, and thereby [my] safety was more important, so [I] preferred not to accept them... Steve,... from the Press Service, loaded [me] with film, as well as watertight Nikonos cameras....5
While a transmitter and batteries were one thing, a fancy camera was a manageable concession. The problem arose with timing: how would pictures be able to be sent to the newspaper in lieu of radio broadcasts in order for the Press Service to report on the race day by day, week by week? Moitessier reached back into the arsenal of his youth in French colonial Hanoi for a solution; namely, his tried, true, and trusty slingshot, along with a packet of fresh and properly sized wide rubber bands:
I preferred my old, quiet friend the slingshot to two or three hundred pounds of noisy radio equipment, but [Steve] could feel the "how" and "why" and helped me to find good rubber bands, supplying me with aluminium cans to contain messages I would shoot onto passing ships. A good slingshot is worth all the transmitters in the world! And it is so much better to shift for yourself, with the two hands God gave you and a pair of elastic bands. I will try to send them messages and film for their rag. It would make them so happy... and me too.6
One reads with delight how perplexed and amazed sailors on passing ships were to see Moitessier on his yacht shooting his message- and film-filled cans into the air in the direction of their ships, which then accurately fell on board in one shot and clinked across the deck to be retrieved by crew. News of _Joshua_ would then be wired back to London by the captains, and the photographs would be passed on to the nearest British Consul and posted to the _Sunday Times_ by diplomatic courier. Moitessier's slingshot was technically efficient but primitive. Moderating any urge to get carried away by modern marine technology was the priority of sustaining a mystical sense of purpose and relationship with the natural elements.
The matter of a camera and film took Moitessier's memory back several years when he sailed from Tahiti to Alicante, Spain with his wife, Françoise: "we never dared take pictures of the sea before the Horn, and least of all our big gale in the Pacific. Not because of danger or fatigue, but because we felt, in a confused sort of way, that it would have been a kind of desecration." 7 He was convinced that destiny controls the moral nature of men and women as does the stars of the horoscope, but also that destiny allows a person a range of options with which to play out one's moral nature in history, from event to event: "Destiny deals the cards, but we play them." 8 He often waxed lyrical about his solitude and likened himself to a seagull:
I felt such a need to rediscover the wind and the high sea, nothing else counted at that moment... All _Joshua_ and I wanted was to be left alone with ourselves... You do not ask a tame seagull why it needs to disappear from time to time toward the open ocean. It goes, that's all, and it is as simple as a ray of sunshine, as normal as the blue of the sky. 9
Moitessier saw himself as a sea mystic in tune with the elemental forces that bathed him in natural wonder.
Third, moral presence may also come in the form of a _personal test_ for sailors who are realistically unsure of their skills, but who also are bold enough to embark on an offshore voyage alone in order to do their best in the face of such a challenge. The Australian Kay Cottee lacked self-confidence at the outset, but recouped heaps of it during her circumnavigation of 1988. (This established her as the first woman to complete a single-handed non-stop circumnavigation and the first woman to circumnavigate non-stop west to east, south of the five southernmost capes. She set records for the fastest time for a solo circumnavigation by a woman, the fastest speed for a solo circumnavigation by a woman, the longest period alone at sea by a woman, and the greatest non-stop distance covered by a solo woman.) Nearing Cape Horn is a major emotional, technical, and symbolic event in the life of any solo sailor, more so than approaching and rounding any of the other four capes along the way. Cottee's autobiography comes to a pinnacle of feeling that is mostly sublime in an aesthetical sense, but also physically demanding. Her description of the beauty of the heaving Southern Ocean is perhaps one of the most captivating portrayals of the sea at its elemental best and worst, and a powerful personal epiphany or primordial experience of (in Rudolf Otto's famous phrase) _mysterium tremendum et fascinans_.10 The _mysterium tremendum et fascinans_ is not an ordinary lived experience but an extraordinary event, one that does not immediately make sense as it is beyond rationality and too powerful to contain. _Mysterium_ means "mystery" in the sense of the unknown and unknowable; it is the "sacred." _Tremendum_ is that aspect of the sacred that induces awe and fear. _Fascinans_ is the sacred as evoking fascination, allure, and a sense of the sublime. Like a moth before a flame, a person who undergoes an experience of _mysterium tremendum et fascinans_ must reckon with a coincidence of opposites, a paradoxical event that could well create, destroy, and re-make the self all at the same time. Just 585 miles from Cape Horn and in winds of fifty-five knots, Cottee had a personal epiphany:
It wasn't easy trying to slow down the boat. The further south we went the higher were the seas, as in the south latitudes of the Southern Ocean there is no land mass to break the speed and size of the waves as they hurtle round the globe. By 1600 hours we were under bare poles, towing the sea anchor, still doing a steady 7-plus knots and surfing up to 12 knots on the breakers. I tried setting the storm jib again and backing it with the helm down to put the boat into irons. After all my efforts I remember standing below, looking out of the clear companionway slide and watching the sea anchor, towed behind on the end of 10 metres of chain plus 70 metres of line, bouncing down the face of the next wave after the one we were riding. I estimated the waves to be approximately 20 metres high and breaking with a nice curl. When we were in the troughs I looked up, and despite my fears of being pitchpoled I was captured by the beauty of the aquamarine colours of the sun shining through the peak of the next approaching wave.11
Here the _mysterium_ presents its more destructive _tremendum_ face. Cottee's physical tension at the time was unshakable. With the Horn approaching, she wrote,
The tension was really getting to me. My shoulders felt stiff and my neck hurt badly since I had put it out a few days ago winching the storm jib up. There were tingles down my spine and my hands continually went numb. I was increasingly worried that I couldn't relax enough to get my neck to click back into place. If I settled into that position, permanent damage could be done. But no show of relaxing tonight, with land only a few miles away.12
Cottee's protracted personal epiphany progressed from a sense of destructive power that could not be controlled, even when it became embodied in the contorted vertebrae of her neck, to a fresh sense of excitement, wonder, and personal accomplishment:
After all the stories I had read about this ocean graveyard, here we were, only a few miles from it. I had thought it would give me a spooky feeling, considering the number of sailors who had been dashed to death on the treacherous black cliffs, or drowned in the mountainous seas. But my prime emotion was excitement and I had a great sense of accomplishment that I had reached what I then perceived to be the major obstacle in the voyage.13
Hiding deep within herself was a fear of rounding Cape Horn, but she now was doing it in the light of day and fully conscious of how far beyond that inner fear she was then growing. As a fitting ritual gesture to commemorate the powerful event that had come good, Cottee opened the Cape Horn present that had been given to her by her mother, a bottle of her favorite "Joy" perfume. "I sprayed myself with the lovely scent, then put on some lipstick, before sitting down to a delicious belated lunch of fresh bread with crab and mayonnaise and the remainder of the bottle of Grange."14
Cottee used the sea and a fine balance between action and non-action to test herself not only as a self-reliant technical sailor but also as a moral being in formation. Terrifying experience was transformed into fascination. This is a spiritual event, or a means to ultimate personal transformation. She wrote:
I was very lucky on this particular day because the sun was shining, and as the next huge wave rolled up behind the boat blocking out the sun, the sun shone through the top of it and the colours of the sunlight refracting through the water were just magnificent. I immediately thought that not many people would get to see a sight like it from that angle and how beautiful the waves were. So, after that, the bigger the waves got the more beautiful they became, and that's when I realised you definitely **can** change your thoughts if you put enough practice and conviction into it.15
At the end of the day, Cottee's lesson is one of inner growth, or an inner reframing of self-worth as a moral being above all else. Lesson learned, Kay Cottee was on her way home a changed woman, perhaps with a maturity that would allow her to go home but not as she had imagined she would, only to fall again into dependency on others for her self-esteem. She became free from all that.
# Tragic Comedy (or Comic Tragedy): The Paradox of Sailing
The human face of the sport is what makes sailing so compelling. In the end, it could be said that the sport of sailing is drama writ large, a combination of comedy and tragedy all in one. Only around the edges of the drama is the technology of yachts and their equipment a topic worthy of consideration. The comic aspect of the sport would certainly include the often obscenely exorbitant financial costs of state-of-the-art boats, equipment, and, in highly competitive racing, professional crews. For the average weekend sailor, the costs of marina berths, annual haul-outs, regular anti-fouling of the boat's hull, and the not infrequent need for repairs are laughable to outside observers. Some standard jokes in this regard go like this: "Yachting is like standing in a cold shower and tearing up $100 bills," "Yachting is like throwing money into a great hole in the water," and "There are two wonderful days in yachting – the one when you buy the boat and then the day when the boat is finally sold." Indeed, even committed sailors often wonder why in the world they ever got into sailing in the first place and what keeps them "hanging in there." The comic irony of yachting escapes no one, and it always elicits a chuckle.
The tragic dimension of the human drama of sailing takes in all of the existential peril that is faced both by competitive-racing and leisurely cruising sailors. Peril is taken on board and reckoned with miles offshore over the course of time on the high seas, far from effective immediate assistance. Weekend sailing is hardly high drama, though it can become so. For offshore sailors, tragedies can be woven out of the jagged separations of leaving family and friends behind, psychological oscillations between profound solitude and unsettling hallucinations inhabited by companions of varying sorts, real anxieties about the boat striking whales or shipping containers that have fallen off freighters and float just below the surface of the ocean, being unable to control the yacht in wind and waves, and worry about one's mental health and not knowing whether a voyage will succeed or end in tragedy. Whether the human drama of going to the sea in a yacht is more or less comic or tragic must be discerned case by case from reports of sailors themselves. Most single-handed circumnavigators of the globe attest to having been profoundly changed by what they did and all that happened at sea. If sailing is drama writ large, combining comedy and tragedy all at once, then a naïve or short-sighted reliance on the technology of sailing and on an over-reaching will, or hubris, can make a voyage into a fool's paradise.
Perhaps the drama of sailing arises from the worldview of the sailor, a worldview that is more like the pre-modern earth-centered vision of Ptolemy than the modern cosmic outlook of Copernicus and moon-based astronauts. The pre-modern worldview always implied that when human beings found themselves in difficulties appeals could be put to higher powers or God for divine help. In pre-Christian ancient Greek drama, such appeals were represented on stage by a simple technical device called a _deus ex machina_ , or "god from a machine." Attached to the set on which dramatic performances would be staged was a small crane. The crane was used to lower props as needed into scenes of the play being acted out below. Included among various props were "gods." They represented the chief divine and semi-divine players of Greek mythology. Representations of Zeus, Prometheus, Demeter, and so on could be attached to the hoist and lowered into dramatic performances, and this was done usually at moments of heightened comic or tragic feeling in the audience. The gods could always be relied on by mere mortals to save the day or condemn them to tragic fates. The stage-bound mortals always happened to find themselves in difficulties that evoked crying and tears of sorrow as well as peals of laughter and joy from onlookers. The result was a catharsis for the audience in the amphitheatre – an emotional release and insight into the drama of being human amid forces beyond one's control.
Do single-handed circumnavigating sailors (and others) persist in hoping that something like a _deus ex machina_ will help them to sort out their perilous human dramas at sea? In other words, how might sailors with an otherwise "blind faith" in technology and personal hubris learn to abandon hope that such modern gods will appear and be effective when they think they are most needed? And, so, sailors play their parts on the dramatic stage of the sea. Some soar to heights of technical skill and over-extended hubris, thinking nothing of the difficulties and the dangers, until things begin falling apart and nothing goes exactly according to plan, and when it would be comical if that was all there was to it. Do the "wheels fall off" the _deus ex machina_ of technical self-reliance and hubris? Other sea-bound actors venture bravely into the whirlpool of such comedy, where they enter into a maelstrom of desperation that leads to tragedy, perhaps even to death. Such was the tragic fate of Donald Crowhurst, who also set out in the Golden Globe Race of 1968–1969 but who broke down mentally and eventually disappeared. If, somehow, the _deus ex machina_ of technology and personal hubris can be restored, if the "wheels" can be put back on again, then sailors can continue to believe that survival may be possible. However, a general sense of futility persists in all of it. Such is the existential nature of the human drama that grips most single-handed sailors, and perhaps grips other sporting people too.
It is not the technological device of the "god machine" itself that is so important, and not the often-vain reach of hubris. The most important thing is the human end served by the _deus ex machina_ , whatever form that technological device may take in critical moments at sea. (Does the radio suddenly work again? Does rain replenish empty water tanks? Do flooded bilges pump dry? Does the chopper locate the damaged yacht? Is a long-missing bottle of brandy finally found?) Between tragedy and comedy, the most important thing that is served by technology at sea is the human drama played out to a final catharsis by the sailor. Such a catharsis is a spiritual operation that transforms the seemingly mirror-like waters of the sea into an impregnable "other" in the modern period, neither friend nor foe by dint of its indifference in a person's life. As single-handed sailor and author Jonathan Raban puts it,
In a secular world, it is this sacral quality of the sea that survives most vividly in poetry of our own time. The sea lies on the far margin of society, and it is – as nothing else is – serious and deep. The last line of Derek Walcott's epic narrative poem, Omeros, has Achilles (a West Indian fisherman who, in Alcott's poem as in Homer's _Iliad_ , is the prototype of a busy, mortal man) walking away from the end of the story: When he left the beach the sea was still going on.16
So, too, lives pass on the backdrop of seemingly timeless nature.
The sea is made to reflect not only its impartial and perilous nature but also all that the sailors have witnessed of themselves out there unaided and mirrored by the water, either liking or worrying about what they observe themselves to be, especially when peril is imminent and the "chips are down." What values arise? Does one develop a capacity to affirm integrity over the temptation to surrender to despair? Bearing such witness to oneself pressed into personal peril in sport is a stepping stone to spiritual insight, or to _wisdom_. Such insight leads to a life in which doing nothing may often be the most effective action to take, action that represents a zenith of human value. This is especially so when the ever-veiled cosmic stakes that underscore human existence are taken into account, terms for living that embrace failure and defeat and, nonetheless, make for exhilaration and a revitalization of life come what may.
**NOTES**
1 Erik Erikson, _Insight and Responsibility: Lectures on the Ethical Implications of Psychoanalytic Insight_ (New York: W. W. Norton, 1964), p. 133.
2 Robin Knox-Johnston, _A World of My Own: The Single-Handed, Non-Stop Circumnavigation of the World in Suhaili_ (New York: Morrow, 1969–1970), pp. 172–173.
3 Ibid.
4 Erikson, _Insight and Responsibility_ , p. 133.
5 Bernard Moitessier, _The Long Way_ , trans. William Rodamur (London: Adlard Coles, Ltd., 1973), p. 5.
6 Ibid., p. 6.
7 Ibid., pp. 26–27.
8 Ibid., p. 33.
9 Ibid., p. 3.
10 Rudolf Otto, _The Idea of the Holy_ , trans. John W. Harvey (New York: Oxford University Press, 1958).
11 Kay Cottee, _Kay Cottee,_ _First Lady: A History-Making Voyage Around the World_ (South Melbourne, VIC: Macmillan, 1989), p. 192.
12 Ibid., p. 74.
13 Ibid., p. 77.
14 Ibid.
15 Ibid.
16 Jonathan Raban (Ed.), _The Oxford Book of the Sea_ (Oxford and New York: Oxford University Press, 1992), p. 32.
# PART 2
# THE MEANING OF THE BOAT
## THREE SCHOOLS OF THOUGHT
**JAMES WHITEHILL**
# CHAPTER 5
# BUDDHA'S BOAT
# The Practice of Zen in Sailing
We sailors can know magical hours. Interludes of peace, freedom, and uncommon joy just seem to happen now and then around sailboats, blessing both experienced and novice sailors. Perhaps you are at the helm of a well-tuned blue-water boat, running with a fresh breeze under a big sky toward a far horizon of the imagination. Or the magic unfolds as you sit in a dinghy on a quiet mooring in a hidden cove, simply waiting for the wind to rise or the tide to change.
For some sailors, however, it is not enough for the magic to "just happen." For them, it is not enough to sail around a pond or across a great ocean for the sheer fun of it, or to race around the buoys for the fellowship and intensity of competition, or to steal a few hours to escape in a sailing dinghy from the dulling routines and distractions of life on land. No, for these few mariners, to sail is to self-consciously seek the mystery dimension of their being, "to rediscover the Time of the Very Beginnings, where each thing is simple."1 So wrote Bernard Moitessier, the legendary French blue-water sailor-writer, in _The Long Way_ , his account of sailing in 1968 on _Joshua_ in the first solo, non-stop, round-the-world race. Moitessier, perhaps more vividly than most, viewed going forth under sail as a meditation, a yoga, a spiritual path liberating him from the "dragon" of civilization. For the Moitessiers among us, sailing may hold the promise of a mystic journey to inner peace in communion with what he called the "singing" of the sea.
Once upon a younger time, I felt a yearning for the mystic, deep sailing magic seemingly made possible only during a long, solitary ocean voyage. Until a few years ago, I imagined myself making a blue-water crossing of the Pacific or, more modestly, signing up for an offshore, week-long cruise with a sailing school, in order to see what happened to me far from land, a thousand miles out, at night, silent and alone in the cockpit at the wheel of a seaworthy sailboat. Would I hear the singing of the sea, as did Moitessier... and Slocum, and Melville, and Odysseus? Could I become one with the sea's songs, in tune with sublime rhythms rising and falling in a salty symbiosis of myself and the sea?
But these deep-water fantasies are gone from me nowadays. I no longer have the time, will, or resources to captain or crew on a sea-going boat. Even more responsible for my letting go of Odyssean dreams are decades of experience as a practitioner and teacher of Zen Buddhism. Experience and studies in Zen combine to remind me that there are many paths to the boundless, nameless origin of one's self, to awakening in the "Time of the Very Beginnings." Hours upon hours of Zen practice in meditation halls, as well as in a number of the Zen arts such as tea ceremony, assure me that I can taste peace, freedom, and communion in a teacup of a boat, playing with waves on a small patch of shallow water, during an afternoon's brief sail under a haiku sky.
Still, I recognize inklings of my aging self in the words of E. B. White, who wrote in his seventies: "I know well enough that I have lost touch with the wind and, in fact, do not like the wind any more. It jiggles me up, the wind does, and what I really love are windless days, when all is peace."2 I, too, think of limiting my physical risks, but not to White's point where I fear the moment when the wind presses on my sail. I still welcome what springs up in me with the wind's first wisp of breath on my skin: a cellular gladness, and then gratitude that I am being awakened and moved by something generous and invisible. "Buddha's breath," a friend calls it.
I retired from college teaching about six years ago and moved to Cape Cod with my wife. We looked forward to good sailing on the Cape. Because docking slips and moorings on the Cape are either too expensive or not available for a sea-going boat, we thought that a small boat would precisely fit our situation. We could trailer the boat and launch from the many town landings on the Cape, and go daysailing in protected waters.
I like small boats. My first boat was a Sunfish. But I was getting too old to tolerate a capsizing boat and less inclined to enjoy speed or racing. The motivation to race, to focus my attention and will outward in a competition with others, had eased up and turned inward, almost without me noticing. I needed a boat in which I could hope to finish well in a new inner contest with myself.
So, in this mood I began my retirement on Cape Cod. I felt a nagging urge to approach sailing in a different, more intimate and perhaps more ultimate way, in keeping with this stage of my life. Recalling the ancient Buddhist imagery for the journey to Nirvana, of a raft crossing the currents of ordinary, unsatisfactory life to the distant shore of a fully awakened life, I came about into a new self-narrative of sailing: I needed a Buddha's boat, if by sailing I meant to aim for, and hopefully reach closer to, that far, secret shore in myself.
The search for a Buddha's boat began and ended quickly. I bought a Cape Cod catboat. It suited my obvious need for a stable, small sailboat that would fit on a trailer, not capsize easily, and be appropriately designed for the shallow, protected waters around Cape Cod. I also liked the idea of having only one sail to the boat, so I could sail more simply and alone. For these good reasons, I bought a fourteen-foot gaff-rigged Compass Classic Cat, built at what is now the Pleasant Bay Boat and Spar Company in Orleans, Massachusetts. It has two hundred pounds of ballast in the skeg keel, a seven-foot beam for additional stability, and only ten inches of draft with the centerboard up.
I named it _Garuda_. According to William Snaith,
Each man uses his boat in his own way to fill certain wants. There are as many roads to Nirvana as boats and men. Nowhere does this show up as precisely as in choosing a name for the darling of his heart... The choice of a boat's name is the semantic key to his dream.3
In Hindu-Buddhist mythology, Garuda is a bird–man hybrid who flies the great god Vishnu back and forth to Heaven, protects the many Buddhas through the eons, and subdues sea dragons by eating them. Luckily, _Garuda_ was a happy choice for me as a boat and it has become a very good teacher of a different way of sailing.
I decided to organize my learning to sail solo on _Garuda_ along the lines and according to stages I had encountered in learning and teaching other Zen practices, especially sitting meditation and Zen archery. While the Cape Cod catboat design began over one hundred years ago as a shallow draft workboat mainly for near-shore fishing in protected waters and has evolved into a pleasure craft for a dedicated group of owners and builders,4 for me it would become a Buddha's boat.
Buddhist tradition holds the view that there are countless ways to the difficult goal of an awakened, flourishing self. Sailing, I believe, can be one of those ways or practices. As such it is best approached as a practice of both _outer work_ and _inner work_ , a useful division of focus seen in Zen arts such as tea ceremony, swordsmanship, calligraphy, flower arrangement, and meditation itself.
Within both outer and inner aspects of Zen-oriented practices, at every stage of learning and unlearning, we see methods that coordinate two mutually supportive overall aims: _concentrating attention_ and _purifying character_ , which is the cultivation of specific virtues across an arc of what I call "self-forgetting," the penultimate goal of Zen. Both aims are realized by training body and mind together. Could sailing become a Zen practice deepening and opening my body–mind awareness, while also nurturing virtues such as patience, courage, equanimity, humility, gratitude, reverence, and selflessness?
_Outer work_ in a Zen practice of sailing focuses on sailing _technique_ and _performance_. Technique includes learning how to sail, the language and terms of sailing, and hours of practice leading to a more intuitive and body-centered habit of maintaining attention to particular tasks, as well as a broader, peripheral awareness of the environment in which sailing occurs. Working on technique ranges from attending to micro-adjustments in body–boat balance and sail shape to scanning other boats, water, wind, and sky to improve one's powers of anticipation in a flexible, open awareness.
With practice regarding how to get things done and how the boat works, the Zen sailor, as would any sailor, will increase her powers of observation and learn to sustain her attention in ways that improve not only her sailing but also her mental, physical, and emotional capacity. She will learn to concentrate on the job at hand in the moment. This often deepens into an absorption in the task that is also a type of _self-forgetting_ , developing as her concentration shifts from a stressed problem-solving body–mind orientation to a more relaxed, intuitive one.
Learning to sail is a form of physical learning. Certain senses, without one's noticing at first, become better-tuned to boat response and speed. Which senses are these? One is hearing, because the boat's hull sounds different as it slows or speeds up. Another is touch, because the apparent wind and its direction register on the face, neck, and tiller arm as a pressure that correlates with changes in speed. Finally there is the oft-ignored but vital sense called proprioception, because our inner sensations of bodily position and movement, especially of balance, become sensitive to the boat's tenor – the continuous, often subtle movement and tension of the boat, mainly felt through "the seat of the pants." While tempted by inclination to take a calculating, left-brain approach to learning, sailing in a Zen mood taught me to trust my intuition enough to "just sail" with and in a more open, subtle mind and body. Sailing solo accelerated my learning to sail _Garuda_ by simplifying the environment and intensifying self-observation.
Performance, the second level of outer work in a Zen practice, goes beyond a collection of techniques, such as learning how to trim the sails or work the tiller. Performance refers to "putting it all together" into a flowing form, to fashioning a ritual or symphony of sailing practices. Performance includes sequential routines and sub-routines of technique, but also body balancing and cognitive-emotional tuning. The analogy for me is with Zen archery, where formal, ritualized shooting is taught. The rituals of _kyudo_ (the Way of the Bow) are known as _kata_ , precise body, bow, and arrow routines made up of eight discrete movements or steps in a single, flowing act of establishing body posture, mental concentration, and the bow and arrow for the explosive, sudden of act of shooting. Each of the eight steps includes many technical details and nuances of posture, hand and eye position, and breathing rhythms that are practiced until they become consistent, intuitive, or "natural."
As a ritual, a Zen performance implies a transformation of the person into an actor, one who submits to and acts "without thinking" according to a well-practiced form, who moves "per the form" or "performance" of his art, be it archery, making and drinking tea, or sailing.
A sailing teacher, and the sailor himself, sees what others see outwardly in a sailing performance, but usually with more precision. She looks at technical mastery and fluid performance in a sailor as signs of talent, dedication, patience, and right effort. But sailing becomes Zen sailing for the teacher and the student when they believe that sailing is more than the skillful performance of guiding a wind-driven boat efficiently and safely through water. Zen sailing is grounded in a relation of the sailor with herself, and so the work must now move inward.
The _inner work_ of Zen sailing builds upon the outer achievement of acquiring a useful level of technical skills and coordinated performance. The goal now becomes widely inclusive and deep within: to better bind together the boat, the sailor's self, and the natural world in which they move and have their being. With growing competence in tasks such as hauling up and setting a sail to the wind, there will come a more relaxed attitude, relaxed shoulders and hands, a slower breath, and subconscious motor learning. These are signs of meditative sailing.
Even before stepping into the boat and setting sail, however, a sailor can make himself ready for sailing in a Zen way. Sailing includes preparations for sailing, in which opportunities abound for developing Zen awareness and what we might call the sailor's trance.
In Zen calligraphy, for example, the expert calligrapher may take up to an hour making his black ink by rubbing a solid ink stick round and round on a two-level stone with water in the lower level. What is happening is the preparation of the calligrapher for the action of dipping a bamboo brush in the ink and, in some Zen painting, tracing out shapes of Chinese characters or everyday objects on absorbent rice paper in a lightning flash of intense, flowing, intuitive brush strokes. No touchups are allowed. It is a matter of "one brush stroke, one life." But behind what appears to be a spontaneous act of free expression may lie hundreds and hundreds of practice sessions of the same painting subject.
Similar preparations, by means of simple, repetitive, physical acts that focus the mind's attitude and exercise the body's supple readiness to act expressively, can be found across the spectrum of Zen arts. The archetypal model of this preparation is the attentive counting of breaths in the Zen meditation hall. My own experience is that it takes me about ten minutes of counting breaths before I can achieve a useful state of relaxed alertness and physical loosening of tensions. This seemingly simple skill and capacity came to fruition for me only after years and years of Zen practice. However, by what means I know not, it comes to some in a few minutes of practice.
In sailing _Garuda_ , I usually do attentive breathing, slowly lengthening and deepening my exhales, during the ten minutes it takes me to fire up my little Honda outboard engine and motor slowly from my mooring in Arey's Pond to the mouth of the Namequoit River, where I set sail for Pleasant Bay. Sometimes I prepare for sailing just by sitting in the boat at the mooring and doing sitting meditation for ten minutes, or a bit more if need be.
Preparatory rituals before sailing in a Zen mood can involve practical exercises such as cleaning and polishing the boat for ten minutes or so, even if the boat was already clean and Bristol enough. Tony Davis, owner of Arey's Pond Boatyard, uses a set routine before taking new catboat owners or students out for a lesson. His ritual is to go through a preparatory checking of the weather, wind, tide, boat equipment, and sail reefing decisions in a mostly set order. This well-worn routine, he has found, brings him to the relaxed and alert state he needs to teach with presence of mind and to model a correct attitude in sailing.
Right performance or ritual efficacy can be an end in itself for some sailors. But, in sailing as an _inner work_ of relating oneself to oneself as one seeks, however indirectly, the plenitude of one's being, sailing rituals become a transitional or liminal zone where we shed our social roles and alter our internal self-narrative. Ritual, somewhat paradoxically if we think of it as limiting and restricting, can help a sailor make the shift toward a freer, more expansive, and more visceral way of being in the world. The demands of sailing well, especially in small boats, entail setting aside the habitual body and mind of the businessman or high school student or professor of philosophy. Sailing well frees the sailor from the constraints of his normal role(s) _by means_ of the sailor's role, as he submits to the limits and pressures of his boat and the surprises of wind and water, while suspended in the moment between the earth below and the great sky above.
The Zen sailor comes to the belief that the inner work only now making its appearance in his sailing practice is a "profound and far-reaching contest with himself."5 Now embedded in what he has learned, yet also going beyond technique and performance, he "will see with other eyes and measure with other measures."6
The _first stage of inner work_ focuses on becoming aware of, nurturing, and directing an experiential energy or power, called _ki_ in Japanese, _ch'i_ in Chinese, and by other names elsewhere. Some describe _ki_ holistically as an energy matrix or field running through all things and connecting them somehow. But I mean by _ki_ something much less metaphysical. By _ki_ I know merely what I have experienced myself: a visceral or "nervous" activation accompanying certain breathing exercises inextricably combined with attention to movements in posture. I observe the energy's warmth, strength, or something like magnetism in and around my body. Specifically, my more ordinary experiences of _ki_ consist of warmth in my hands, an instant easing like an opening or hollowing in my arms, shoulders, or legs, and a feeling of localized power not dependent on stressing muscles. Put another way, _ki_ tends to appear when I am relaxed mentally and physically, focused in a special way on a body part or a stage of breathing in and out, and _expecting ki to appear_.
_Ki_ can enter Zen sailing in at least two ways. The most productive way probably comes after a sailor has become comfortable with sailing technique and performance. If you are at the level where both have melted into a flowing, flexible, adaptive ritual of mindful and enjoyable sailing, you have learned a way of staying "in the groove." You experience "flow" in your sailing that is "on a different, more intense, yet easeful level." At that level the sailor does not have to think much about fundamentals and has developed preparatory or performing rituals that do not require constant monitoring. She has reached the stage where she almost automatically moves the tiller to leeward ("into the sail" is my mnemonic) when a jibe threatens, and she tells her crew to move or moves her own body to balance the boat _before_ even thinking about it. In this phase of sailing, cultivating _ki_ will almost certainly empower the experienced sailor's abilities, increasing endurance, deepening relaxation, sharpening the senses, and helping her to respond precisely to circumstances without reflection or hesitation.
The other, somewhat different, occasion for activating _ki_ is in the beginning of learning the outer work of sailing; that is, while learning techniques and the forms of sailing practices. Simple breathing or moving exercises to awaken _ki_ can be useful here in relaxing and strengthening a novice sailor who struggles with physical fatigue, mental wandering, nervousness, or frustration. In the case of a sailor facing these hindrances, it is usually best to stop sailing and take ten minutes for breathing and stretching exercises that awaken _ki_ ; or, even better is to do basic _ki_ exercises as preparation before thinking about or getting oneself into the boat.
In teaching Zen meditation to students, I used a basic warm-up set of _ki_ movements in a simple standing posture before static sitting called _zazen_ (literally, "sitting meditation"). For eight to ten minutes, the students could tune themselves to the right balance of relaxation and tension by concentrating on slowing and deepening their breathing and by easing away from their worries. That was for many the key to sitting effectively on their meditation cushions for thirty-minute sessions of _zazen_.
A brief sketch of a breathing exercise and posture adjustments may be useful at this point. The simplest, most effective breathing exercise is based on a long sigh. While sitting erect or standing comfortably, take in a breath normally, through the nose, but expand your ribs a little more than usual, pause about one second, then slowly exhale through your mouth, letting your shoulders relax and your facial muscles loosen (as in a slight grin). Imagine that your exhaling breath is flying or sailing to the end of the cosmos, if that helps to lengthen it. Repeat this mindful rhythm ten to twenty times, closing your eyes for the first five to ten exhales. Over the next ten or so breaths, gradually open your eyes and take in the visual environment without fixing your gaze too much on one thing.
This exercise usually produces a noticeable, often pleasant relaxation in body and mind. The next step is to keep up the slow, sighing exhale while focusing on one visual object or one sound, such as the sound of your exhale. If your experience is, for example, "looking at the coil of line" or "listening to the _hooooo hum_ of my exhale," things are going well.
Now the critical shift: activating awareness of _ki_. On a long exhale, if you are focusing on a visual object, open your eyes a little bit more as if to take in slightly more of the scene, then right away push your inner feeling of breathing pressure lower into your belly until you feel that you are seeing with your whole torso, not just your eyes in your head. That's it! If you are focusing your "self" on a sound or "just listening," let your torso or even your belly hear the sound.
This belly awareness or perception from the belly is one very effective key to activating _ki_ energy. Belly awareness brings about a wider zone of sensory perceptions, a panoramic mental awareness, and a lower center of effort in your actions. It calms the nerves. I trust that you will find this new way of awakened acting to be more balanced, intuitive, rooted, and powerful than others, and much less ego-centered. When combined with deeply embedded skill and experience in sailing, drinking tea, or even folding diapers, this new way opens up a new orientation of the self, even if sustained for less than a minute. The sailor, now sailing with his belly, not his head, sails without sailing.
When practice makes this awakening power increasingly available to the sailor, such that it flows softly into the technique and sailing performance of a sailor more and more on self-command, he may begin to glimpse the second, most inward work of the life of Zen sailing: _self-forgetting_. Self-forgetting is the term I prefer to terms such as "egolessness" or "selflessness." Self-forgetting refers to what most often is a process, with nudges, little jumps, and, once in awhile, awesome leaps. In self-forgetting interludes we shed the usual attitude or sense that it is "I" who is sailing this boat. I and the boat are unified, without the habitual sharp physical, emotional, or cognitive separations and distinctions of "normal" experience.
In my view, sometimes too much is said and written of the process leading to and experienced in self-forgetting. Following the tradition in many writings about the Zen arts, a sailor might speak of "awakening to her true Self," "realizing the Buddha-mind," or "achieving _satori_ ," but I prefer an ascetic, modest silence. Describing the experience of self-forgetting in grand terms that point to or even promise a peak experience is potentially misleading, since a special experience is neither the conclusion nor the goal of a Zen practice.
The silence I keep to is due to the "mystic," ineffable, tacit quality of the experience of self-forgetting. Silence is a way to avoid giving hope that there is any direct, easy path to self-forgetting. Self-forgetting is approached indirectly, not by choosing to be humble and selfless but by anchoring oneself again and again to the disciplines and constraints of technique, performance standards, and energy work on one's body, breath, and _ki_. Paradoxically, the state of self-forgetting, which is characterized by creative improvisation, liberation from ego-driven desires and delusions, and a wide, inclusive awareness, arises most commonly within the mundane, repetitive regularity of practicing mindfully the formalities of sailing and everyday life.
To students of Zen tradition it should come as no surprise that liberation from the ego's control of our conventional self-narrative can come from entering Zen-like training in self-forgetting within and through a formal practice of sailing. Dogen Kigen, the thirteenth-century Japanese Zen master and philosopher, famously wrote that studying Zen is "studying yourself; studying yourself is forgetting yourself; and forgetting yourself is being authenticated by all things."7 The even more ancient _Dhammapada_ applauds this introspection and inner effort at self-mastery in clear terms: "Greater in battle than the man who would conquer a thousand-thousand men, is he who would conquer just one –himself."8
The second of Dogen's stages, where "studying yourself is forgetting yourself,"9 lies at the heart of Zen practice. While "studying yourself" means attending to and noting your reactions to all outer work in sailing, as well as the more inward meditative work of developing _ki_ energy, in Zen even this intentional mindfulness eventually is allowed or encouraged to "drop away" in what I am calling self-forgetting.
This "dropping away" of self-scrutiny and interior monologue comes about by means of exhausting even the introspective mood, through long periods of meditation or by hard, sustained physical and mental labor (which can happen in sailing even in a small boat on quiet waters). It can happen in an afternoon suddenly and dramatically, but also in decades of subtle change in dedicated, patient practice. It can be stimulated by a sailing instructor's word that turns your awareness in a new direction or by a puff of wind on your ear. It can be shallow or deep, momentary or lasting for hours. It can make you laugh or weep. You are never the same again, yet you are more your self than before. You have uncovered what Dogen and the earliest Zen tradition call your "original face."
The last stage of Dogen's rubric on Zen training, where "forgetting yourself is being confirmed by all things," is profound and yet simple. "Forgetting yourself" or dropping off narrow ego-oriented attitudes and actions is a kind of emptying of the self; its desires, fears, and delusions; its thirst for control; its hope for the weather to change and the wind to spring up or die back; and its worry about boatyard bills, crew competence, and a short sailing season. Awa Kenzo, the archery teacher in Eugen Herrigel's _Zen in the Art of Archery_ , tells Herrigel that at the point of maximum tension, when the bow is fully stretched before loosing the shot, when everything is at stake and the archer waits in a suspension of all striving, he most learn "to wait properly... by letting go of yourself, by leaving yourself and all things yours behind you so decisively that nothing more is left of you but a purposeless tension."10
The "presence" of mind and body in Zen sailing, one's liberation from the ego's grasping for control of experience, and that "purposeless tension" allow boat, water, wave, cloud, and sky to advance and give shape to the sailor's awareness with an unimpeded richness and beauty. All things breathe "yes!" They confirm a Zen-trained sailor in her being.
By not reaching for or clinging to particular expectations of reality while sailing, the sailor is free, with no mind hindrances, and so she acts with creative improvisation in response to what is dynamically real in the moment. Subtly yet firmly, she relies on the skills, habits, routines, and energy developed in hours of mindful sailing practice. Listening to the whispers of his boat and the "shhh" of wind on her original face, she sails in nirvana.
**NOTES**
1 Bernard Moitessier, _The Long Way_ , trans. William Rodamur (Dobbs Ferry, NY: Sheridan House, 1995), p. 105.
2 E. B. White, "The sea and the wind that blows." In _Essays of E. B. White_ (New York: Harper & Row, 1977), p. 207.
3 William Snaith, "On the wind's way." In Christopher Caswell (Ed.), _The Greatest Sailing Stories Ever Told_ (Guilford, CT: The Lyons Press, 2004), p. 113.
4 Stan Grayson, _Cape Cod Catboats_ (Marblehead, MA: Devereux Books, 2002) is the best book source on the history and rebirth of the Cape Cod catboat.
5 Eugen Herrigel, _Zen in the Art of Archery_ (New York: Vintage Books, 1989), pp. 65–66.
6 Ibid.
7 Cited in David Loy, _The Great Awakening: A Buddhist Social Theory_ (Boston, MA: Wisdom Publications, 2003), p. 118.
8 Thanissaro Bhikku, _Dhammapada: A Translation_ (Barre, MA: Dhamma Dana Publications, 1998), p. 31.
9 Cited in Loy, _The Great Awakening_ , p. 118.
10 Herrigel, _Zen in the Art of Archery_ , pp. 31–32.
**GREGORY BASSHAM AND TOD BASSHAM**
# CHAPTER 6
# FREEDOM OF THE SEAS
# The Stoic Sailor
Sailing is, for many of us, about freedom. Casting off the lines to shore is a figurative and sometimes literal release from the oppressor's wrong, the proud man's contumely, the insolence of office, and a thousand other insults to our sense of autonomy. On a passage, with the wind drawing in the sails and the wake lengthening behind us, we feel as much freedom, perhaps, as the human condition allows. The paradox, however, is that sailing necessarily requires subjecting our perceived autonomy to elemental forces far beyond our control, among them winds, weather, tides, and storm.
To a significant extent, the art of seamanship resides in a clear-eyed grasp of our _agency_ : understanding the fine lines between what we can control, what we can influence but not control, and the vast world that is beyond our control. The annals of exploration and modern recreational sailing are replete with tales of sailors who came to grief from overestimating their agency. Yet, as many tales can be told of those who lingered on the shore, or puttered safely about the bay, or unnecessarily called the Coast Guard for rescue from failure to appreciate the true scope of their agency.
Many philosophical perspectives speak to agency, but one of the more ancient, and pertinent to the joys and sorrows of seamanship, is the classical philosophy of Stoicism. Stoicism has a reputation as a cheerless, pessimistic philosophy useful only to those who must endure pain, torture, or the other slings and arrows of outrageous fortune. It has little superficial appeal to those of us living in industrial nations with social safety nets, good health and nutrition, material prosperity, democratic institutions, and the rule of law. But, as William B. Irvine showed in his recent book, _A Guide to the Good Life: The Ancient Art of Stoic Joy_ , Stoicism is ultimately concerned with the joy of living.1 For sailors, Stoicism has much to offer as a means to create and sustain the sense of joy we seek in sailing.
Stoicism was founded by Zeno of Citium in ancient Greece around 300 BCE, but it had its greatest impact several centuries later during the Roman period, when it powerfully shaped the lives and characters of leading Roman philosophers and statesmen such as Cicero (106–43 BCE), Seneca ( _circa_ 4 BCE–65 CE), Epictetus ( _circa_ 55–135 CE), and Marcus Aurelius (121–180 CE). Philosophy, for the ancient Stoics, was not a theoretical discipline or body of speculative doctrines. It was a way of life, a practical guide to the art of living. Like Socrates, the Stoics believed that "care of the soul" – virtue, excellence of character – should be our primary concern. In fact, the Stoics believed that virtue is the only thing that is strictly "good," all other so-called goods (pleasure, wealth, reputation, health, relationships, even life itself) being at best "preferred." The ancient Stoics believed that the universe is pervaded and wisely governed by God (the divine Logos); that human souls are "sparks" or "fragments" of God; that all events are inexorably fated to occur as they do; that virtue is the sole human good; that virtue consists in "living according to nature"; and that living according to nature – for rational beings such as ourselves – consists in willing the universal good, having the right motives and intentions, standing strong and unbowed in the face of adversity, and accepting with equanimity and even thankfulness whatever misfortunes life throws in one's path. The Stoics developed a number of practical techniques to achieve virtue and enduring inner peace. Let's look at three of these techniques – cheerful resignation, self-sufficiency, and negative visualization – and their application to sailing and seamanship.
# Cheerful Resignation
As noted above, the ancient Stoics believed that whatever happens is fated to happen by God, the all-wise and benevolent Logos. The Stoics deduced from this that everything happens for the best and that all "discord" is but "harmony not understood." This doesn't mean that nothing bad ever happens to individuals. God's primary concern is with the welfare of the universe as a whole, and what's good for the whole is not necessarily what's good for each of its parts. Can individuals, then, justly complain when they suffer personal tragedy or misfortune? Not at all, for reason – that holy spark of divinity that lies within each of us – requires that we abandon self-centered desires and egocentric perspectives and look at things from the point of view of the universe. From that cosmic, God's-eye perspective, there can be no grounds for murmurings or complaint. Whatever happens to us in this life, good or bad, we must accept cheerfully as part of God's wise and beneficent plan.
In explaining this Stoic attitude of cheerful resignation, Epictetus famously compares life to a play:
Remember that you are an actor in a play the character of which is determined by the author – if short, then in a short one; if long, then in a long one. If it should be his pleasure that you should enact a poor man, see that you act it well; or a cripple, or a ruler, or a private citizen. For this is your business, to act well the given part; but to choose it belongs to God.2
Elsewhere, Epictetus writes:
True instruction is this – learning to desire that things should happen as they do... I must die; must I die groaning too? I must be exiled; does anyone keep me from going smiling, and cheerful, and serene?... "Then I will fetter you." What do you say, man? Fetter me? You will fetter my leg, but not even Zeus himself can get the better of my free will.3
The freedom of the seas entails the possibility, even the probability over a long career in sailing, of experiencing storms or other conditions that reduce our scope of action to its lowest point. The boat is hove to, the tiller lashed, all secured above and below, and there seems nothing more one can do but resign oneself to fate. That sense of resignation from the need for physical action can be a tremendous psychological relief. But there is always one thing more we can do, and should: maintain a cheerful composure.
Minnesota schoolteacher Gerry Spiess certainly needed – and exhibited – the Stoic virtues during his epic 1979 crossing of the North Atlantic in his home-made plywood ten-foot sailboat, _Yankee Girl_. Tossed for days in a howling storm, Spiess' tiny boat was repeatedly flooded and was eventually capsized by a rogue wave. Miraculously, Spiess managed to right _Yankee Girl_ before the next wave rolled the boat all the way over. As the sun set, an exhausted Spiess slipped back into the hatch and gripped the rails, preparing to face another night of the North Atlantic's fury. "At that moment," he wrote,
I wanted more than anything to give up – to get away from the agony, to escape the fear. I wanted to close my eyes and open them again and be somewhere else – back home in Minnesota, in safety and security, with my family and friends.
I bowed my head. I would cry out to God for his help.
But then, surprisingly, I found myself hesitating...
God had given me all of the resources I needed to survive this storm and any other that came along. It was up to me to use them – and not to ask for more.
Instead of pleading for help, I said a prayer of thanks. I was alive, and my boat was whole. That was enough.4
One of the most difficult passages on record was that of the _James Caird_ , a twenty-two-foot converted lifeboat sailed by Sir Ernest Shackleton, Captain Frank Worsley, and a four-man crew on an eight-hundred-mile voyage through the winter storms of the Southern Ocean, from Elephant Island to South Georgia Island. Under the most trying conditions imaginable, the crew of the _James Caird_ struggled against nearly hopeless odds to find and safely land on South Georgia, driven by the need to seek rescue for their twenty-two fellow sailors stranded on Elephant Island. Throughout the ordeal, the crew maintained a disciplined composure and even the ability to laugh, as Worsley recounted in his attempts to fix their broken cooker: "My subsequent antics with the crumpled-up thing that now bears a faint resemblance to a lady's hat that I am endeavoring to trim, sends everyone into yells of laughter, in which, after a while, I cannot help joining too."5 During the worst moments, attempting to claw off the bleak western cliffs of South Georgia Island against hurricane-force winds, the crew did not give in to despair. Worsley related:
As we looked at that hellish rock-bound coast, with its roaring breakers, we wondered, impersonally, at which spot our end was to come. The thoughts of the others I did not know – mine were regret for having brought my diary and annoyance that no one would ever know we had got so far. At intervals we lied [to each other], saying: "I think she'll clear it."6
Sterling seamanship and a providential wind shift allowed the _Caird_ to clear the rocks, and after sixteen days at sea the crew landed safely in a remote inlet. Following an equally epic overland journey to reach a manned whaling station, all of the expedition's sailors were later rescued, in no small part due to the Stoic virtues employed by Shackleton, Worsley, and others, including the virtue of remaining calm and even cheerful in the worst of circumstances.
# Self-Sufficiency
Like Socrates, the Stoics believed that "no evil can happen to a good man." Why? Because only what harms the soul is evil, and nothing can harm the soul except vice, which no good man commits. A good man, therefore, is self-sufficient. His happiness and well-being lie entirely within his own control. All that is good – an upright heart and an invincible will – is within his power. All that is evil – anything that corrupts the soul – can be avoided through an act of will. Thus, "indifferents" such as death, pain, poverty, and loss have no terrors for him. Like the English poet William Earnest Henley, he can proclaim:
It matters not how strait the gate,
How charged with punishments the scroll,
I am the master of my fate:
I am the captain of my soul.7
Even with modern technology and conveniences – EPIRBs, satellite phones, and a full-service boatyard in every port – the ethos of sailing has demanded and will always demand the maximum practical self-sufficiency. A landsman can travel many thousands of miles by automobile without having the slightest idea how to change the oil or a tire, but no sailor on even the shortest passage should be ignorant of how to splice a line, bleed air out of a fuel line, or unclog the head. Stoicism also stresses the virtue of self-sufficiency, but with an emphasis on the psychological: simply, our essential happiness and tranquility should depend as little as possible on the approval or actions of others, and still less on the functioning of plumbing facilities on board.
One of the many reasons sailors go to sea is to enjoy companionship with like-minded souls, and to spend as little time as possible among the harpies of the shore. Yet, the social life on board a small boat can be fraught with tension, misunderstandings, and festering annoyances. Extended cruises can be particularly hard on marriages and family relations, as attested by the number of boats for sale in tropical ports at divorce-rate prices. As Irvine notes, in difficult social situations the Stoic philosophers recommend first reflecting on our own shortcomings and doing what we can to eliminate our own annoying behavior. The Stoics note that becoming annoyed or angry at another's behavior will almost always be more detrimental to us than the behavior itself, and simply make the situation worse. If all else fails, we can always reflect on the brevity of life and the triviality of human differences _sub specie aeternitatis_ ("from the standpoint of eternity") and do what we can to maintain the attitude of cheerful tolerance and emotional self-sufficiency so necessary for life on board a small vessel.
# Murphy was an Optimist: Negative Visualization
The third, and perhaps most important, Stoic technique is what William Irvine calls "negative visualization," which is essentially the practice of periodically and systematically imagining what could go wrong in life and the loss of something dear to you (possessions, status, loved ones, life itself). The point of such an exercise is not to develop a morose sense of fatalism but quite the opposite: to free ourselves as much as possible of the fear of loss, and hence to increase the capacity for experiencing joy.
Negative visualization has immediate practical implications for sailors. For example, by imagining a scenario in which we experience sudden loss of auxiliary engine power when docking in a crowded marina, we are prompted to prepare ourselves for that eventuality by having the sails ready to set, the anchor ready to deploy, and a provisional plan for using either to avoid collision or grounding. By imagining the loss of our rudder at sea, we are prompted to prepare ourselves, both to prevent that loss to the extent we can (by proper maintenance and periodic inspections) and also by learning how to jury-rig a replacement if the rudder is someday lost despite our precautions. As sailors, we can take negative visualization one step further and actually _practice_ deprivation, for example by shutting off the engine and practicing how to dock under sail, or by learning to navigate by sextant as a backup to the modern electronic navigation instruments at our disposal. As Seneca remarks,
Everyone approaches with more courage a hazard to which he has long squared himself, and resists harsh circumstances by contemplating them in advance. But the man without preparation panics at even the lightest troubles. We must see to it that nothing comes to us unexpectedly, and since novelty makes all things more burdensome, constant meditation will guarantee that you are not a raw recruit for any misfortune.8
Perhaps even more important than the physical preparations prompted by negative visualization is the effect on our emotional and mental states. By engaging in systematic contemplation of what could go wrong while sailing, we are far more likely to react calmly, quickly, and therefore effectively to emergencies and difficult situations, even those that we have not specifically contemplated or prepared against. As Seneca writes,
Unexpected disasters weigh more heavily; novelty adds weight to calamities, and there is no mortal man who has not felt more grief at something that left him in amazement. So we should make sure nothing is unforeseen; we must send our mind ahead to face everything and think not of whatever usually happens but whatever can happen.9
At a less practical but no less important level, negative visualization allows us to better appreciate what we do possess and all that has gone right in life's voyage. It is a common human failing to take much for granted (good health, a sound boat, pleasant weather, and so on). It is somewhat paradoxical, but, if we imagine the loss of those things we tend to take for granted, we can truly appreciate them, perhaps for the first time, and take real joy in their presence. Stated differently, by undergoing the difficult emotional work of visualizing the loss of something dear to us or that we take for granted, we can diminish our fear of loss and the crippling effects of fear.
As Marcus Aurelius noted, fear is often rooted in false values and lack of perspective. We fear death, but death is part of nature, causes us no harm (since, as he supposes, there is either a happy afterlife or we don't exist to be harmed), and is necessary for the universal good. We fear loss of creature comforts, power, reputation, pleasure, and youthful good looks, but what are these from the perspective of eternity? Nothing but smoke and bling. All that endures is goodness and truth, for then we participate in something eternal and god-like.
Aside from injury or loss of life, what sailors dread most is to lose their sailboat to storm or reef. The legendary circumnavigator Bernard Moitessier built and lost three boats in his lifetime, including _Joshua_ , the steel ketch that he helmed in the first round-the-world, non-stop, single-handed race, in 1968. After seven months of solo racing, Moitessier famously turned back from the finish line in Plymouth, England, and sailed non-stop half again around the world to Tahiti, for a total of thirty-five thousand miles alone under sail. After a decade of activism and further adventures in the South Pacific, Moitessier and _Joshua_ were anchored off Cabo San Lucas when an unexpected storm drove a number of boats onto the beach, including _Joshua_. With no funds to salvage his beloved ketch, Moitessier gave the hulk to a friend and for the third time in his life moved on from shipwreck. His life in close communion with the seas had given him the perspective necessary to absorb such losses with something approaching tranquility.
# Agency and Control
The Stoic sailor is distinguished, above all, by an accurate understanding of what seamanship can and cannot accomplish. In 2008, veteran offshore sailor Skip Allen was returning solo from Hawaii on board _Wildflower_ , the Wylie 27 he built in 1975, after winning the single-handed TransPac race. For three days he rode out a gale off the coast of California, running before the wind and twenty-five-to-thirty-five-foot waves under stormsail and autopilot, trailing a drogue. The autopilot continued to work flawlessly, but if it failed a broach was inevitable before Allen could regain control of the boat. Exhausted, and with the gale forecasted to strengthen and extend for at least three more days, Allen had a choice: stay with his still-seaworthy boat or radio for rescue from a freighter in the vicinity. He was extremely loathe to abandon his companion of thirty-four years. Still, the sixty-year-old reminded himself that he was responsible not only for his own life but also for taking care of an elderly family member. As Allen described it in an online forum, "I cried, pounded my fist, looked out through the hatch numerous times at the wave mountains, remembered all the good times I had shared with _Wildflower_ , and came to a decision."10 He radioed for rescue. In a final act of seamanship, Allen opened a seawater intake on his beloved boat before leaping onto the freighter's ladder, so that _Wildflower_ would sink beneath the waves rather than drift as a menace to navigation.
Allen's story illustrates a key distinction that Stoics make between what is within our control (limited mostly to our goals, reactions, desires, and other internal states) and what is not within our control (pretty much everything else). The latter category can be further divided into things that we can influence but not completely control (for example, how well our crew performs during a sailing race) and things we have little or no influence over (the tide, wind direction, how other racers perform, and so on), resulting in what Irvine calls the "trichotomy of control." Stoics devote most of their mental energy and discipline to the first category (control of our goals, reactions, desires, and internal states), knowing that success in that regard will increase the amount of influence we can bring to bear on external events not within our control. Stoics strive to remain emotionally indifferent to the universe of events completely beyond our control.
As an experienced offshore sailor, Allen was as prepared as possible for the gale, but events beyond his control forced him to make a difficult emotional decision. He focused on what he could control (his goals and desires) and made what was almost certainly the correct judgment in the circumstances: to abandon his vessel. Importantly, though, he continued to influence events to the maximum extent, and even in the act of abandoning _Wildflower_ exercised his agency and seamanship to ensure his beloved boat would endanger no others.
Sometimes there is nothing to be done but endure, and it is then that Stoicism can help most. In the 1996 Vendée Globe single-handed race around the world, Raphael Dinelli's boat was capsized and dismasted in hurricane-force winds in the cold Southern Ocean, over five hundred miles from the nearest land. With the boat awash and slowly sinking, liferaft torn away, he stood in the cockpit for twenty-four hours, dancing like a madman to stave off hypothermia. "I wasn't afraid," he remembered. He was angry that death was coming, but knew he "had to keep fighting mentally, because if you don't fight ceaselessly, you're finished."11
The next day an Australian long-range search-and-rescue plane dropped him a liferaft, and he clambered it into minutes before his boat finally succumbed to the waves. Inside the raft was a message that Pete Goss, a fellow Vendée Globe racer, was ten hours away, beating upwind in gale-force winds to reach him. Goss was Dinelli's only hope of rescue. A bottle of champagne bobbed to the surface from the wreck, and Dinelli grabbed it with numb fingers. Exposure was taking its toll. He spent the night in the raft, frozen and paralyzed, only the hope of seeing Goss in the morning keeping him alive.
The next morning the plane reappeared and flashed its lights. Dinelli thought, "That's it, Pete Goss must have broken his mast or something, he's not coming, I'll never last another day."12 But the plane was guiding Goss to the bobbing raft, almost invisible in the immense waves, and ten minutes later Goss' sailboat appeared. Dinelli carefully handed up the bottle of champagne, and Goss heaved the hypothermic sailor aboard. Against all odds, Dinelli had endured.
# Fate, Freedom, and Sailing
Ironically, it is because so much of our lives is "fated" and beyond our control that we have the potential to experience true freedom – a freedom of the mind that, once gained, no man or externality can ever take away from us. Imagine a sailboat coasting along in light winds; the tide turns and the speed over ground slows to zero. Many sailors would curse, fire up the iron genny, and bash against the tidal current, even if in no particular hurry. A Stoic sailor might smile, toss out the anchor, sit in the cockpit reading his Epictetus, and cheerfully wait for the tide to turn again. As Epictetus notes, accepting what is necessary with inner calm is the true secret to freedom and contentment in this storm-tossed world:
Remember... that if you attribute freedom to things by nature dependent, and take what belongs to others for your own, you will be hindered, you will lament, you will be disturbed, you will find fault both with gods and men. But if you take for your own only that which is your own... then no one will ever compel you, no one will restrict you, you will find fault with no one, and you will accuse no one, you will do nothing against your will.13
Sailing has an almost unique capacity to teach us patience, and to reward us for practicing the Stoic virtues. We cannot control the winds or tide, sometimes reefs appear off our bow, and always our brief voyage on this planet is over too soon. But, with the help of the Stoic virtues, our seamanship, and a keen understanding of our agency, we can shape a course free of fear and full of joy.
**NOTES**
1 William B. Irvine, _A Guide to the Good Life: The Ancient Art of Stoic Joy_ (New York: Oxford University Press, 2008).
2 Epictetus, "The Enchiridion." In _Epictetus: The Discourses and Enchiridion_ , trans. Thomas Wentworth Higginson (New York: Walter J. Black, 1944), p. 337.
3 Ibid., p. 376.
4 Gerry Spiess, " North Atlantic storm." In Michael Bartlett and Joanne A. Fishman (Eds.), _The Sailing Book_ (Westminster, MD: Arbor House, 1982), p. 118.
5 Frank A. Worsley, _Shackleton's Boat Journey_ (New York: W. W. Norton & Company, 1977), p. 87.
6 Ibid., p. 144.
7 William Earnest Henley, "Invictus." In Roy Cook (Ed.), _101 Famous Poems_ (New York: McGraw-Hill, 2003), p. 95.
8 Seneca, _Selected Letters_ , trans. Elaine Fantham (New York: Oxford University Press, 2010), p. 228.
9 Ibid., pp. 188–189.
10 Skip Allen, " _Wild_ _flower_ ship log," _Sailing Anarchy Forums_ (2003, <http://forums.sailinganarchy.com/index.php?showtopic=78146>).
11 Raphael Dinelli, "How we met: Pete Goss and Raphael Dinelli: Interviews with Philip Sweeney," _The Independent_ (May 18, 1997, http://www.independent.co.uk/arts-entertainment/how-we-met-pete-goss-and raphaeldinelli-1262298.html).
12 Ibid.
13 Epictetus, "The Enchiridion," p. 331.
**STEVEN HORROBIN**
# CHAPTER 7
# SAILORS OF THE THIRD KIND
# Sailing and Self-Becoming in the Shadow of Heraclitus
_It was so old a ship – who knows, who knows?_
_And yet so beautiful, I watched in vain_
_To see the mast burst open with a rose,_
_And the whole deck put on its leaves again._
J. E. Flecker, "The Old Ships"
_Full fathom five thy father lies;_
_Of his bones are coral made;_
_Those are pearls that were his eyes;_
_Nothing of him that does fade,_
_But doth suffer a sea-change_
_Into something rich and strange._
Shakespeare, Ariel's Song, _The Tempest_
Several years ago, when I was leaving port on a long voyage of uncertain destination, which has yet to reach its end, an old sailor and mariner, self-confessed one-time smuggler, and engineering mystic growled to me through his bushy horseshoe moustache: "If you ever get to that horizon, give it a good kicking, from me!" I laughed and took a photo of him, standing on the ramshackle Scottish dockside, heavy arms folded, leaning against a large fuel drum between an equally ancient wooden fishing boat, in fading yellow, blue, and red, and a modern hull of brushed aluminum. He didn't smile as he watched me slip away, but the moment, I felt, was one of mutual recognition and understanding. A good line; but what could it have meant? I think he knew I understood it, which is why he gave it to me. Where was I going? Toward the horizon, certainly. And from that horizon I would raise shorelines, mountains, lights, cities, bays, relationships, skills, struggles, happiness, and near despair. Below the horizon my bow would dip, and my anchor dive, many times. But each destination I raise, every one he raised in his long life at sea, each brief bond of the anchor chain is never the end of the journey. It is never the "home port" at which my cutter sloop and I will remain, and end. For a mariner, for a sailor of the third kind, the horizon is a visible avatar of the source of the process of being itself. To reach it would be a kind of death. For a sailor of the third kind, a day out racing round the cans is all very well, and a trip across the bay with the promise of a pint or good company happy enough. But the bow swung round and pointed to an open horizon on the first hours of a voyage whose destination is more an arbitrary placeholder, a cleat or transient binding point for one end of the first theoretical line of a great circle route: that is a moment of pure joy.
People covet boats, and sail, for many reasons. In this essay I will attempt to sort these sailors into three broad classes. The categories described are generalized and to a certain extent must be caricatures of the real situation. I do not consider that each and every sailor fits neatly into one or other category to the exclusion of the rest, nor do I consider that the characteristics so described are the only ones that may exist. However, the conceit contains, I feel, an approximation of the truth, and is a tool for gaining some understanding of these sailors, and in particular the sailors who most approximate the third. This third will also include those who simply aspire but never have the opportunity to set out toward that unreachable horizon.
First, there are those who become involved with sailing because of its social iconography, a glamorous thread in Western culture and, most especially relevantly for this category, in Western marketing culture. For at least some of these, the mere fact of vague association or involvement is sufficient, sublimated into sailing clothes and shoes, with civilized hands, or a boat owned and polished and prized but rarely used. The endless use of the symbolism of sailing by high-end clothes manufacturers, luxury watchmakers, jewelers, car brands, and other similar products not remotely connected to the practical realities of sailing, has this category of sailor, and aspirant, as its object. The exclusive yacht club is the temple and ultimate showplace for the cultural aspirants of this category, a place wherein the sailors of the second kind might find a welcome, if not a home, and those of the third likely no welcome at all, in the unlikely case that one were to be desired. The hardships and sacrifices of the sea are unlikely to be relished by members of this category, and, while an afternoon's champagne sail along some warm shoreline and a bob at anchor in an idyllic tropical bay may be on the menu, hard-won lonely anchorages alongside rough, uninhabited places are unlikely ever to be experienced or, perhaps, desired. Scars of use on the hands and mind of the sailor, and on the fabric of the boat, are undesirable signs of an experience that is alien to the mandarin cleanness of this world, wherein the realities and passion of sailing are tamed, lacquered over, and reduced to mere ciphers for a culture that is as far from them as are the symbols of heraldry from the events that first gave them meaning. The boat owned by one of this first kind of sailor is likely to be trussed and tethered like a beautiful slave to one or another marina pontoon, with only brief afternoons with impressed guests or direct trips to the next well-chosen showplace. For some of these, the more prestigious the boat – the bigger and more expensive – the better: the boats themselves and the trappings of them are exalted beyond their obvious function to being trophies and symbols of power, reaching their ultimate expression in the mega-yachts, floating pyramids of would-be latter-day pharaohs.
Then there are the sailors of the second kind: those who sail to race. These sailors are no strangers to hardship, and maximum use of the vessel in its most practical form is the name of their game. For the ocean sailors among them, the fastest, shortest possible route, regardless of intensity of prevailing conditions or difficulty of sea-states, often sailing _toward_ heavy weather for wind advantage rather than prudently away from it, as might a sailor of the third kind, is the order of the day. Sailing style often subjugates prudence, and what sailors of the third kind would regard as "good seamanship" loses out to the exigencies of speed. Single-handed racers, perforce, often barrel down rough wavescapes at maximum speed asleep with no watch save radar, which is all but ineffective in such conditions, as the radar reflections off the waves, or sea clutter, hide almost everything significant. The use of automatic identification systems, which transmit a boat's position, course, and speed to others in the area, with computerized alarms, has substantially reduced this risk, but it still exists, as many ships and smaller vessels still lack such systems. Tolerances of materials and hull construction are finely calculated compromises whose object is speed and performance over durability and strength in the face of the unexpected, which calculations, as the likes of Pete Goss and Tony Bullimore have found, all too often err on the side of performance. Sail changes on the larger fully manual rigs are often accomplished under severe duress for both crew and boat, perhaps changing a number two to a number three yankee headsail in a rising gale in cold high-latitude waters in the dark, with no bearing away or course deviation to reduce risk or wear and stress on flesh and equipment. There is huge skill and heroic effort and courage involved in these practices, and the money and competition involved in this area of sailing are responsible for driving innovation in many areas, though not always resulting in the most voyage-worthy crafts. Sailors of the second and third kinds share the overriding imperative of practicality, but for those of the second kind this practicality has a single absolute focus: speed, and this speed is not the sole end in itself, but rather is the means to a further end as well. These sailors might take delight in the technical mastery of the latest rig or in the feeling of speed, but ultimately they are in it to compete with others for a definite goal; a social, human prize: prestige. The soul of this goal is land-based, I, as a sailor of the third kind, would argue. At its core is the demonstration, to others, of how short a time the boat can be pushed to remain at sea. The boat becomes a tool for the goal of producing a thrill and exhibiting a skill. The shore-based prize-giving and league tables are the true manifestation of this kind of relationship. Its history is rooted in the land-market demands of commerce. The great tea-clippers are the forerunners of this style of sail.
I once had the experience of sailing with an amateur crew on an amateur one-design round-the-world race. This particular part of it was from Vancouver to Panama City. Somewhere south of the Sea of Cortez, we were nearly becalmed, ghosting along under the lightweight spinnaker at perhaps three or four knots, and uncatchably behind the leader on that leg. Suddenly, fine on the port bow, we spotted massive whaleblows, the size and shape of tall birch trees, around a mile away. Having seen many whales of different species and their blows, it seemed to me the most likely source for these prodigious forms was the blue whale: very rare to see, and the largest species, land or sea, ever to have lived on Earth. I was bouncing with joy and excitement, not least because the day was so calm; we should have been able to see the five or so individuals very clearly in the water. I expected an immediate change of course, to feather by them, at least, if not a diversion to take a more lingering look to honor one of life's great opportunities. I was sorely disappointed. The skipper was not for altering course, not even by the handful of degrees it would have required; and, to my astonishment, of the thirteen remaining crew, not one besides myself wished to pause or alter course. "We are racing," was the universal answer. And, despite my protestations, that was the end of the matter. We did, in the end, ghost past them – perhaps two hundred meters away – on a four-thousand-mile passage, which for those non-sailors among you likely amounted to no difference at all. It is unlikely that such an opportunity will arise again for any one of us then aboard.
At that moment I realized I would never participate in an ocean race again. There have been many other such incidents over the course of my sailing life, but this one stands out as the starkest example of the difference between racers and sailors of the third kind. These two different kinds of sailors, while not always mutually exclusive in a character, are nonetheless quite distinct. I do not disdain these sailors of the second kind. They often accomplish very arduous and remarkable things. I wish them well in their endeavors, but I simply am not, and cannot be, one of them.
So what, then, of the last category? One may borrow, perhaps, a little of the form of words of Immanuel Kant in understanding the difference here. For the sailors of the first and second kind, the boat and the act of sailing are only ever means to some further end: an end that is self-contained and definite, but of a closed, complete, and largely land-based kind – say a prize, or a place on a record, or a favorable impression made. For the sailor of the third kind, the boat itself and its function, the life with it and within it, its relationship to the wind, the land, the ocean and its living beings, the sun, moon, stars, and horizon – that process, focused through the sailor's self, is an end _in itself_. And what is to be found in that end? For these sailors, the act of sailing, especially long-distance living-cruising, represents a kind of singularity in which the beingness of human existence and the being-in of universal existence are as close to unified as it may be possible to be. This will no doubt seem a rather opaque and perhaps suspiciously airy assertion, so I will attempt to clarify.
The branch of philosophy that focuses on the questions "What sorts of things are there? What is real?" is called ontology. It is built on the Greek word _ontos_ , which means "being." Ontologists take one of two fundamentally distinct approaches in answering these questions. These approaches may be described as the substantialist and the process views of reality. For the substantialists, who walk in the shadow of the ancient Greek philosopher Parmenides, the world is made up of substantial things that persist and perdure through time, that have events happen to them, and that cause events, but that are themselves discrete, unitary, unique, and in some fundamental sense unchanging. The ancient philosopher Democritus, the founder of the concept of atoms, defended just such a position. For the processists, followers in the tradition of the Greek philosopher Heraclitus, in a deep sense there are no "things" at all. Every "thing" is simply a passing momentary fluctuation of the universal substance, which is for these philosophers the _only_ substance. Every piece of the world, every "thing" that appears stable and solid and unitary and real in itself, is no more real and unitary and stable and solid than the whitecaps that wink above the crests of waves. It is simply a matter of time, and perspective. Consider showing a photograph of whitecaps to one who has never seen waves – they may appear to be solid "things" in the manner of objects. The same may be said if one could only see them in "slow motion" as it were, so their rise and collapse might take days or weeks, years or millennia. It is for this reason that we can name cosmic events such as the Horsehead or Crab Nebulas and speak of them as though they were "things" and not wave-like processes, which is what they are. For the substantialists, the person is, in some fundamental way, a whole and unitary being, with at least one utterly changeless core element that is reidentifiable as such at any moment of its personal history and that does not, in itself, change. For these philosophers, persons are, in some sense, complete beings who interact with the other substances and processes of the world but who ultimately remain above or outside them. The traditional concept of an immortal soul is one way to begin to understand this substantialist view. Sailors of the first and second kinds, so focused upon precise and definite, self-contained and discrete objects and goals are, I feel, more likely to subscribe to a substantialist ontology, whether they know it or not. Wholly open-ended voyaging, with the journey itself as the end, is most likely unappealing to minds committed to this view of the world.
For the processists, the whole of Nature is one vast super-process, and out of this, like everything else, persons arise, as sub-processes of the whole, distinct from it in the way that individual waves are distinct from each other and the ocean's surface but ultimately changeful and restless in their very nature, ultimately and indivisibly part of the whole. For the adherents to this view, the person is never present, whole, and complete in any one moment, and no person is unchanging. To be unchanging is not to be, just as to be a wave without change and movement is meaningless. To be a person is to be a process, and to be a process is to be ever in a state of becoming. For the great seventeenth-century philosopher Spinoza, this process of becoming was in essence a move of the universal super-process toward its own self-realization, with persons achieving greater or lesser moves toward representing, in themselves and their own being, an approximation of the whole. For him, the good was manifest in the direction of movement, within a person, toward the accurate reflection of the whole of nature, in microcosm. In this reflection, in this personal realization, the universe comes to reflect and understand, to re-present its own self. This line of thinking was further clarified in the works of the twentieth-century philosopher Hans Jonas. I would argue that the sailors of the third kind are most likely to be those who, knowingly nor not, live and self-become in the shadow of Heraclitus.
The sea is a rough and violent place, and a boat, properly used in long-distance travel, is a rough and violent being. It is hard on the body and mind and can break people and their relationships. It strips away trivialities and it bonds and exposes the practical, intellectual, and spiritual elements of the self to the open systems of the earth and the wider universe through the tight discipline of the boat's processes. Even tied up to a dock, a sea boat is in a state analogous to a constant saltwater earthquake. Quite apart from the brute mechanical violence of the boat's existence at the juncture of wind, wave, and land – jarring, loosening, and weakening – the salt fingers of the sea extend through atomization into the air and penetrate the hull into every aspect of the boat's systems, requiring constant vigilance, and husbandry from its occupant. Metals laid against each other, by design or accident (a stray screw in the bilge, perhaps), if not equivalent on the galvanic table, form electric currents in the mode of batteries and result in galvanic corrosion, turning the hardest metallic structures to dust. Even stainless steel is not immune to this process. Sacrificial anodes placed at the ends of these sequences to protect them must be constantly maintained, and the circuits that lead to them perpetually checked. The electrical systems of the boat, fuelled by its power systems and batteries, add vigor to this destruction and may, by stray current facilitated by pervasive salt, destroy key parts of the boat's integrity in weeks or days. On deck, fair weather is no refuge as the wind, water, and salt mollify and abrade, and the sun irradiates the sails, plastics, covers, and lines, weakening them constantly and insidiously even through cloud. A long-distance mariner is forced thereby into a constant dance of maintenance, repair, and replacement. Nothing escapes or is immune, and a lack of complete vigilance and constant labor threatens the integrity of the whole. A single failed component, however small, as every experienced mariner knows, may rapidly compound in non-linear ways and lead to sudden disaster.
There is an old philosophical chestnut known as the Ship of Theseus paradox. Essentially, the conundrum is as follows. Say one has a ship – the ship of Theseus, perhaps – wooden for simplicity, wherein parts wear or rot out and must be replaced piecemeal. At every stage of this process, one is tempted to continue calling this ship by the same name, but gradually it is all changed, until no part of the original remains. Is it still the same ship? A modern wrinkle on this problem of identity, popularized by the philosopher Derek Parfit, is to imagine building a second ship in parallel, alongside the first, from its discarded parts. In the end, which is the "true" ship? Now, a mariner may well be justified in noting to Professor Parfit that the ship built of worn-out and discarded parts will be useless and unable to function as a ship, so perhaps the practical goes some way to resolving this philosophical thought experiment. But of course the question is serious, and interesting. However, it is only really a puzzle to those who accept a substantialist view of reality. The puzzle depends upon the idea that identity consists of a particular set of definite, hard, reidentifiable parts, which constitute the whole, discrete, singular being. By contrast, there is to the process philosopher far less of a conundrum. For, identity to a processist consists only in the particular nature or signature of a process. For a ship to be a particular ship, it needs only to hold the continuous original function and general form of that ship. That it may be true that another ship may be constructed of perhaps not-quite worn-out parts of the original process is no real problem, since a process view of identity does not require that processes do not branch. Indeed, it requires that they do. A whole series of ships constructed in this way would in an important sense be, indeed, the same ship, in very much the way that different owners of different yachts in the same class of yacht – say, a copy of Moitessier's famous _Joshua_ – may instantly recognize something of their own vessel in the other, and something of themselves in the other owner. There is a commonality, as they are aspects of a shared local process. This recognition points toward the underlying super-process, from which all processes, or modes of being (as Spinoza would have said), extend, and to which they are inextricably connected. In this way, to a processist, identity is never complete in the way that a substantialist would wish (i.e., unchanging and discrete), and the problem really diminishes, or disappears. The final detail of particularity really comes more from time and place than anything more fundamental: my _Joshua_ , at this or that time, was or is here, with me, and not there, where that other _Joshua_ was or is, with you. Inevitably these share some of their identity, and more so than they would, say, with a Beneteau Oceanis, and perhaps its owner.
It is no coincidence that the classic form of this conundrum is presented by reference to a ship. There is something about living on the land that gives advantage to the mode of thinking of Parmenides and the substantialists. Simply, it is that the land appears stable and fixed, replete with stable, fixed, unitary, and discrete objects: in a word, substantial. Equally, the opposite is true. The violence of the sea, its accelerated processes, and very fluidity impel the mind toward a more process-oriented view of reality. To a processist the former is, of course, an illusion, supplied by the relative perspective of duration. The processes of the land-based life are simply more stable, less violent relative to the lifespan and daily activities of a person. To see this, one merely needs to add time. The whole continent of Africa, if viewed in a long-term perspective of hundreds, thousands, and millions of years, may easily be seen to be far more fluid, to be in fact not substantial in the traditional mode of solidity and changelessness but rather a boiling mass of process and change, wherein everyday objects, buildings, and even the continent itself melt and flow away as fluidly as do waves on the ocean's surface. In this way a life on the ocean supplies not a complete categorical change but rather a change of perspective, and one arguably toward the underlying reality of things. In respect of the short life of the sailor, the violence of the oceanic life and the threatened, frail processes of his or her boat reveal more closely the underlying reality of the nature of Nature, the total super-process. And the life lived as a long-distance mariner is one in which the being of that mariner blends, perforce, into the being of the violent processes of his or her craft: the days absorbed in the management of passages, and of the management of the fragile confluence that is the boat's own process on the ocean wave.
A salient feature of the voyaging life is its time-consuming nature. A transoceanic passage, which for a passenger on a jetliner may take a matter of hours with little effort at all, will cost a sailing mariner weeks of concentrated effort. I would wager that sailors of the third kind are not likely to be easily bored. For these persons, the time-dilation of the act of sailing has rather the opposite effect that one might suppose: far from making long periods drag, it quickens the perception of the violent, passionate flux of the world. In this way, in all its restless quietnesses, brief intensities, and seeming drudgery, there is something akin to the monastic. But it is not a regulated, calm withdrawal from the world. Rather, it is a move outward, toward the world in all its wonderful and sometimes dangerous caprice: more anchorite than monk.
Ocean voyaging generates an openness to change: change of weather, change of plans, sudden whales, and changes of life, relationship, status, and fortune – the unexpected that arises swiftly from the blank apparent sameness of the ocean. It mandates sensitivity to subtleties that would otherwise be missed. A wavescape, a cloudscape, some new feature on the water is never merely that but is always to be watched and understood. Each new landfall rises like the lid of a chest holding treasure and threat in unequal and uncertain measure. This watchful unexpectedness alters the mariner's perspective of the world. The relative time of the processes themselves appears foreshortened. All is fluid, and any event, any identity, appears as a fragile nexus, a brief disequilibrium against the constant flow of universal stuff toward equilibrium. This may seem counterintuitive, of course, since the common understanding of equilibrium is one of stability. But the stability of equilibrium, in physics, is absolutely inimical to the requirements of salient, complex processes, such as those that define any living being. All living beings are projections _away_ from equilibrium, are highly precarious towers of disequilibrium, held together in a brief relationship by effort and work afforded by free energy from the sun. The boat is a being just like the living body. It is held in high disequilibrium, through constant effort and energy, above the state toward which it would rapidly fall if so allowed. On the ocean, equilibrium is represented all around by that very blank, apparent sameness – a windy surface thinly hiding the sunless, windless abyss, the ever-threatened fall toward darkness, silence, stasis, and death. Above this rides, by the free energy of the wind, a tiny and wildly improbably being, inhabited by, and an extension of, a wildly improbable being whose very improbability is what gives it life and meaning. All the phases of matter are present: a fragment of solid land in the hand, atmospheric gas in the sail, and the keel in a liquid ocean. The sky, the sea, and the land, tides, pressures, and precipitation, the stars, sun, and moon, the universe beyond, the whole history of human endeavor, culture, and technical achievement manifest in the boat's construction. These are fully connected through the act of navigation, and the intimate acquaintance and husbandry of the sailor. They are captured by the hull, keel, and sail, and focused through the mariner's hand at the helm.
The mariner, whose hands bleed daily in working to maintain the nexus of disequilibrium, the complex beauty of the relationship of the boat, mind, and wider universe, is drawn into an intense relationship with these processes. In so doing, such mariners place themselves in a situation closer to, and more open to, an understanding of this naked super-process than most in the human sphere. As Spinoza might have it, they are arguably closer to a kind of universal self-representation. And for that they sail toward the horizon they will never reach.
Or perhaps they are just curmudgeonly misfits who are grumpy about not getting to see whales, or whatever else, as and when they wished!
# PART 3
# BEAUTY AND OTHER AESTHETIC ASPECTS OF THE SAILING EXPERIENCE
**NICHOLAS HAYES**
# CHAPTER 8
# WHAT THE RACE TO MACKINAC MEANS
Seen from the International Space Station through the night sky, the most conspicuous patch of ink-black darkness in the lower United States is the 22,400 square surface miles of Lake Michigan. Lake Superior is larger, but its edges are less defined by electric light, because fewer people live around it. Lake Michigan appears to hang from Canada like a smooth dollop of cool, dark honey from the edge of a jar. If we use this contrast to consider the human condition, we quickly conclude that we hug the shores and seem never to breach them. But, during a few nights every July, the night population on Lake Michigan surges from none to about one person for every seven square miles, in an apparent rush to be off-grid. It is not enough to pierce the sky, but the surface is specked with red, green, and white navigation lights that can only be seen by sailors.
The occasion is a race from Chicago to Mackinac Island, in which about three thousand sailors on about three hundred sailboats cross a starting line near Chicago and use wind to travel three hundred miles north-by-northeast.
To a casual observer watching the sailors prepare their boats, the harbor might seem like a circus grounds; on one end, shimmering stars and starlets in perfect painted luxury porta-homes served by bright-white uniformed entourages; on the other, the hardy, hardened, pockmarked tractor-trailer drivers doing double duty as midway amusement attendees. In the middle, the observer finds an American microcosm: moms and grandmothers, teachers and taxi drivers, frat-boys and fullbacks, cat-lovers and Cubs-fans, sojourners and sportsmen, eggheads and environmentalists, rookies and retirees. One might conclude that sailing can appeal to anyone but that it self-selects: the boats appear to reflect the people and the people seem to reflect the boats.
There are boats that seem like "passenger trains," designed for long work. Engines (sails) will overcome friction and weight to move the train (a boat) down the track (the course) at a relatively predictable speed. Passenger trains can be designed for different duties: some like city work, some like the long haul, some favor volume over comfort, and some favor comfort over speed. The locomotive can be made more efficient or designed to pull more weight, and the cars can be made sleeker or more posh. There are old trains and new ones. Some are brightly painted, others utilitarian. But, architecturally speaking, passenger trains don't differ from each other significantly in form or function. Timing and execution are what separate the good from the bad. Running a passenger train well requires that it is well-known by its engineer, well-serviced by its mechanic, and well-timed by its conductor, and that it keeps moving all along its route. Then, it might reach its destination faster than another passenger train.
There are the "Model Ts." Not Model T, as in old and square, but Model T as in all the same, or one design – Henry Ford's central idea. Sailors seeking competitive purity and parity often rally others with a common interest and select boats that are identical – perfect matches – to race on a level time playing field. Many races, this one included, start and score these fleets without handicap (time allowance for differences) except in the overall fleet. A student studying group think might suggest a thesis to explain the Model T team behavior, where apparently same-age, same-income, same-education, same-sex teams clad in same-colored T-shirts, start at precisely the same moment and race against more same-age, same-income, same-education, same-sex teams on the same boats. It's not so simple; only one Model T finishes first in each Model T fleet. So someone did something very different.
There are the "booster rockets." These are the boats designed to break sailing barriers when the conditions allow; they are super-charged and made practically weightless with exotic composites and vast sail engines. Sailing one means combining the stamina of a young Ethiopian marathoner, the brazenness of a test pilot, and the sleep deprivation of a medical resident. Booster rockets can't take extra crew because they can't afford the weight, and there isn't any place to put them, so there is never a break from intense sailing. Booster rockets are driven to peak performance – to the bleeding edge of safe speeds – almost all the time, but have a narrower window of usefulness and safety margin than a passenger train or a Model T. In extreme conditions or in the hands of the wrong sailors, a booster rocket is a large liability – too powerful and angry to tame and too tiny to protect its occupants. Booster rocket sailors gamble that they'll reach the finish just in time to eject.
There are the "collegiates." Not collegiates as in college sailors, but collegiates as in boats and teams explained with the college athletics metaphor. These are the sailors who make years-long commitments, joining the same teams at the same stadium (sailboat) every season, and who are well-rehearsed and collectively set upon winning the annual tournament (this race). There are small school and large school and Big Ten, Big East, and Ivy League collegiates, usually seen in the size and expense of the program. Collegiate boats are branded and built to attract fan power, and they do. Shore-side, family, friends, and retired teammates track their progress like fans. Collegiate teammates forever identify with the team mascot and the teams take on personas as enduring as the Hoosiers, Aggies, or Badgers.
There are the "retreads." These appear to be tired old boats, scuffed and chalky, skippered by mostly scuffed and chalky sailors. These are the boats that return year after year even though their dated designs and older sails make for long, often slow and labored racing. They might finish well in the competition with the perfect weather conditions, but climate statistics and patterns almost never play in their favor. Retread sailors must be in it for the time together, since the time will inevitably be long. They pack ample food. They bring extra clothes. They seem to be doing the same thing each year, only comparatively slower with time. However, one finds a kind of churning usefulness in this group, new tread each year, so to speak. Notably, aboard the retreads are many if not most of the newly invited: friends, neighbors, nieces, and grandkids whose lives are about to change forever.
This is not a circus, and these are not circus players. A deeper understanding of Mac race sailors requires a closer look at the experience that they share.
The start resembles Chicago Loop traffic, with dense congestion, jostling, horns, shouts, and lane-changes. Each sailor is alert to the positions of jockeying competitors fighting for an express lane. But, as the city skyline falls away, the fleet scatters and stretches out, and the shadows lengthen with the setting sun, the sailor's attention narrows. Awareness shifts from the noise of daily living and the energy of the armada to the mates who will share close quarters and a journey through the longest, darkest darkness in the United States.
This shift of focus toward the center comes clear, often, at the first dinner, regardless of conditions. The day has been spent in trains and cabs, on docks and decks, stringing lines and hauling bags of sails and boxes of food, planning assignments and practicing positions, balancing intense physical outbursts with breaks to breathe and to relish Chicago's extraordinary man-made and natural vistas in every direction. The warm bean casserole hits the gut and concentrates the sailor's attention. It stuns. From a warm belly, one feels a need to balance comfort and dampness, anticipation and fear, nourishment and a tinge of nausea, cold and hot, light and dark, and noise and silence. Flavors pop: a _haute cuisine_ of grocery store green beans and dried boxed noodles. Any degree over room temperature is ideal. Everyone wants seconds. The burliest ingest thirds and fourths.
When the bowls are empty and the sun gone, watches begin. Familiar jobs are handed to others, sometimes with relief or even glee, sometimes with trepidation. If you are on the first watch, you signal trust to your crewmates to keep all safe. Invariably you find it impossible to decompress in your berth on the first night. You hear every sound, feel every bounce, and wait for your chance to sail again. If you stay on deck, you learn that the Lake Michigan night is nothing like you expected.
Old-timers will tell you that every night on this lake in this race is different from all others. I can only describe the ones I've seen; I can't guess at the ones that I haven't.
There is the "darkest night," when clouds and fog cover stars and the moon never shows, and when the wind can't coax a drop of luminescence from the water. There is no horizon. There are no neighbors, or, if there are, you can't see them. There are no sounds outside of the whistling in the rig and sails, the water gurgling as it jumps out of hull's way, and the watch crew murmuring. On these nights, the dim glow of instruments, candles, and compasses creates a sphere of visibility that collapses around the crew space as the night matures. Once every few minutes a sailor is compelled to shoot a flashlight toward the masthead just to break the bubble of light and make sure the sails are still above. All sense of direction is lost, but the compass stays true so everyone strains to watch it. On the darkest nights the crew huddles and talks just enough to keep the boat at speed and on course. Long stretches of silence seem as important to moving forward as is the wind.
There is the "stormy night," when the noise grows until it deafens and the motion worsens until it punishes. The off-watch tends to split up, out of necessity. Since the cabin rocks and rolls and stinks of wet sails and urine, only a few can take it down below. The rest trade their chance at warm and dry to ride a bucking rail, facing cold continuous spray in between dousings from the breakers. The driver and trimmers swerve to avoid the biggest waves, protecting themselves and the boat from the force. The boat strains and pounds and complains loudly. Thoughts go to the design and construction of the keel-bolts and the mast-step and the other places that are bending and twisting. Trust the boat. Trust the people. Trust.
There is the "breathless night," when high pressure somewhere north of Chicago spreads lifelessness into the atmosphere over the lake. Sails go limp and boats slow to a stop and bob, as if the champion racehorse has left the gate at full gallop and come up lame in the first turn. Nobody is more disappointed than the jockey, but his job is now damage control. The difference, of course, is that, while the horse may be hobbled and out of the horse race, the wind might return anytime and the sailing crew is still in the sailboat race. These are the longest nights of all – maddening, worrying, and out of direct control. The team tames a persistent collective frustration, trying to remain focused as if the race might return to full health at any moment. Inevitably it does and the calmest and most patient and balanced crew will be in the best place when the night is over.
There is the "velvet night," the one where the wind is high and firm and the water is smooth and slippery. It takes rare weather to make velvet: just enough nearby isobars to force the wind from one side to the other side of the lake but not enough to force the surface of the lake into protest. On this night every change is perceptible. A slightly stronger puff, an inch of jib-trim, an ounce of pressure on the tiller, the new center of balance when someone off watch turns in a bunk below. Unlike the breathless night, when it's hard to feel anything and everyone whispers, the velvet night fine-tunes everyone's senses. Thus, the crew tends to chat non-stop about the boat, the competition, and the conditions. On a velvet night, the crew goes into tactical and technical high gear, trying new setups, trimmings, or balances to coax the most speed from breeze and boat. Rushes of feedback produce rushes of trial and error to achieve what some call "grooving," that precious time when everyone and everything are in balance.
There is the "brightest night," when the clouds move off with the rain and leave a massive silver moon, the Milky Way, and, for the luckiest, the aurora borealis, to light the way. On this night, every ripple on the surface of the water calls for one second of attention. The sounds of wind and water are crisp and clear, and the sailboat wants all the sail it can handle as it jets down a sparkling runway of moonlight. It wants to take off, and sometimes seems to do it, a few waves at a time. On nights like these, sailors tend to unzip their jackets, to open necks and chests to the crispness, no matter the cold. Small teams execute tacks and jibes with precision and vigor, as if fully and professionally crewed. On-watches often don't want to go off-watch. The arrival of morning goes unnoticed, and the sailors forget to celebrate the new day. They just keep sailing.
Morning brings light and with it finer focus to these people – this space, place, and time. This will be the day that starts and ends sailing; nothing else: no trains, cabs, skylines, or docks. It is the ultimate sailor's day; a rarity understood by everyone aboard as so special that they talk about how many days like it they've enjoyed and how few others will get to experience even one.
I've heard some call this day the "endless one." Depending on your route, it can start and end without a shoreline. Water slips by. Time both speeds up and slows down. As the clock turns, some nap, some eat, some chat, watches go on and off, but the sailing never stops.
Others call it "getting into the rhythm." Sails go up and down with the wind. Trimmers trim and drivers drive to the sensations of speed and power. Teams consider new information about the race and the boat and talk about new ideas and possibilities to make them better. Change is constant but subtle. Everyone is aware of it but nobody acts too quickly. Most eyes look forward, peering into the future, trying to make sense of what might happen.
It is as if everyone on board is in the throes of study at the same time and at the same speed and level of intensity. The core lessons are highly personal, even secret, flavored by nuanced individual perspective, held in the mouth, swirled by the tongue, and swallowed slowly, to ensure that they can be recalled later as something truly grand. Some learn about teamwork over time, a long-winded social experiment, so to speak. Some focus on the competition, trying to decipher what everyone on the boats nearby is thinking and planning. Others are absorbed in nature. A few see God.
For me, this is the "day of hope." Not hope for the race or the boat. Hope for everything. It is the day when I am reminded that I am not in control of anything except myself and that I don't own anything except my time.
It is a day of transcendence and transformation. Sailors will tell you that every person who starts this race will finish as someone new. Perhaps it might be more true to say that a real person appears for a few days. As with boats and races, so it is with sailors; there are recognizable types.
There is the "tweener," the offshore rookie with the smallest sailing vocabulary and the most naivety, but who finds his or her own way to fit in and add value as the race evolves. Tweeners might start slowly but surprise everyone with their contributions, like the eleven-year-old who offers an insightful adult comment at a serious dinner-table discussion. Tweeners are tweeners just once or twice. They provide the rest of the team with a sense of purpose, connection, and pride.
There is the "superhero," mild-mannered and subdued onshore but emerging as a massive force of positive physical and emotional energy on the boat. He exudes strength and optimism, handling sails and winching loads beyond what seems possible. You'll hear this one cheering for the next puff or wave or see him pumping and flexing at the sky. The superhero's suit doesn't come off until the race is over, and then he returns to drab. Superheroes remind the rest of the team that this is something far out of the ordinary.
There is the "teammate," who stays put, working one job non-stop without complaint. But it's more than just consistency, continuity, and dependability. The teammate knows what everyone else is doing, and she knows how her work helps or hinders the others. The teammate is keyed to the dance of the boat and its people and to the adjustments necessary to make it and them more graceful or more fluid or powerful. She selectively shares new ideas and supports them with a strong case, usually with perfect timing. The teammate makes everyone else successful.
There is the "yogi," who strips to shorts, spreads his chest, lengthens his neck and back, and inhales deeply on every puff, grabbing some of the excess energy spilling from the rig. Yogis don't just push and pull lines or sails. The Yogi feels the line and the sail mingling its forces into his own. Yogis can be among the most valuable crewmates, as they often notice subtle wind shifts or hidden favorable currents, long before anyone else. Yogis have the rest of the crew trying to figure out how they stay warm while nearly naked and what they see in the dark or hear in the silence. Perhaps mysticism matters.
There is the "minister," who sits furthest forward on the rail on the stormy night, taking waves for off-watch crewmates, or who clips-in to stand over a steamy stove boiling water for hot cocoa. The minister is calm and reflective while noting the glory in the wind and water even when the wind and water are punishing. The minister tends to all and points out the sublime, the scarcity of velvet, the end of the breathless night, or the rhythm of the endless day. The minister links the sailor to nature.
There is the "salt," whose expression is unchanged no matter the conditions or the situation. The salt has sailed enough to know every boat he is on. When something doesn't feel right, the salt scans the symptoms and knows just the right prescription. When something breaks, the salt knows exactly how to fix it. He has tricks nobody else knows and tools nobody else owns, but a way of speaking plainly, clearly, and slowly so that one day they might.
I have never been on a boat in this race with a "skipper." I'm sure there are some, but I've been lucky to avoid them. My sense is that modern leaders have come to know that leadership is a group concern so they stopped pretending to lead long ago and started actually doing it. It takes a serious group thing like this to challenge us with a workable moral imperative and find real collaboration. It is a matter of assembling skills and perspectives that can work in concert, even as the venue and the program change unpredictably. So, "skippering" in the traditional sense – that is, a director calling shots and the rest doing as directed – is a model that is becoming obsolete on this race. In its place is a flexible group that finds its own organic response and engages both with itself and within the place and time to match whatever nature throws.
Finishing fast in this sailing race means that leaders and teams and nature have come together, regardless of the sailboat that they share, for a couple of very intense days, constructing an experience that deserves its own name too.
There is the "four-in-one." This is the race that stops and starts, figuratively and literally. The first leg is a long open-water sprint to somewhere on the Michigan shoreline, where the weather changes and the second leg begins; the fleet starts a day-long battle to find pockets of shore breeze. A boat can easily win the first leg and lose the second. The third leg begins when the boats make a right-hand turn at a mid-lake light at Gray's Reef to sprint again down a narrowing channel that ends at the Mackinac Bridge. Either shore can be treacherous. A boat can win the first and the second legs and easily lose the third. The fourth leg seems like nothing, but it can be everything. It's the final handful of miles from the bridge to the finish line off the south shore of Mackinac Island. For whatever reason – geography, location, or time of day – this stretch often has its own weather. A boat can easily win the first, second, and third legs and lose the last. In this race, passenger trains can pass booster rockets and retreads can pass passenger trains in the last mile. Winners of a four-in-one often split on whether it was their doing or their destiny. Losers learn large, humbling lessons about work, reward, luck, and injustice.
There is the "crawl," the race that starts and ends slowly and seems to take a lifetime. Stubborn high pressure hangs over the length of the course, leaving boats with little more than a betting game. Will zephyrs of sea or shore breeze favor one place or another? Both advantage and equilibrium are made null by where one happens to be when a puff touches the water. Will nighttime be better near or away from shore? What will Sunday or Monday or Tuesday bring and where shall we try to be when they come? A Model T in a rut is not the same as a Model T on a paved road. This is the race that has crews talking in whispers, trimmers trimming in millimeters, and drivers driving to touch, for hours and hours and hours. Winners sigh, relieved, and welcome everyone else to the finish with a "job well done." Losers that finish win too.
There is the "breakup." This is the race that scares some away. It can start or finish like any other, but somewhere along the way the gods unleash a fury and sailors, at least for a short while, wonder what in hell made them do this. The breakup is so named because when it happens it batters and bruises, and everything and everyone seem scattered about and wide-eyed in its wake. It usually comes with warning, such as lightning and dark clouds on the horizon, but when it hits all the preplanning and organization are both out the window and vital at the same time. Squalls change everything about this experience. Booster rockets become coffins and passenger trains become refuges. Ministers, tweeners, superheroes, yogis, teammates, and even salts are forced to confront their own emotions, the emotions of the others on the boat, and, if they believe in God, His or Hers too. The end is usually understated and contemplative, no matter who wins.
There is the "surf." This is the race that everyone wants, and nobody wants to end. The south-by-southwest breeze is on and strong from the beginning and it never quits. The starting line is less about who gets there first and more about who gets there with their spinnaker drawing and the boat at full speed. After that, all horizons are a mass of color, sailors scanning and coveting the largest, most powerful sails. A stern wave sits just behind every boat, nudging northbound. This is the race where every boat (even the retread) is a thoroughbred and every sailor a jockey. Crews coax the most from every stride. You'll hear them cheering her like a friend asking for a bit more effort. You'll see them petting and slapping her topsides to make sure she's good for more. And then, when the finish line comes and goes and the sails come down, they thank her in unison with a hardy cheer for the ride of a lifetime, surf bums forever.
However the Race to Mackinac unfolds, it unfolds as it wants to and keeps the plot to itself until the very end. But the people are wholly intentional. Newcomers seem to know that, even if they have only one chance at this adventure, it will cement as a vivid and indelible memory. For others, it is a lifelong addiction. There are more than three hundred "old goats," veterans of at least twenty-five races each. A single race can shape an individual's identity. A lifetime of these events can define the identity of a family or many families.
The question is why? Why do these people do it and why does it matter? A classical philosopher might say that we are witnessing a modern hybrid combining a number of philosophic theories. My amateurish interpretation finds overtones of skepticism in the highly charged moments of the gale-force breakup, or in losing a four-in-one after winning the first three legs, or in the endless waiting of the breathless night; and it is hard to miss the themes of enlightenment in the yogi trimming, in the tweener speaking truth, in the design of the booster rocket, or in every glorious minute of the surf.
The deeper answer, I think, lies in the awareness shift itself, which is both obvious and subtle at the same time. Clearly, when one boards a small sailing boat with a few friends and family and heads onto open water, there will be times of deeper contemplation, stressful worry, wild exuberance, and intense dismay. Every sailor knows that the self is both put at ease and put at risk at the same time. In these times, we are becoming aware of something on the edges of our being that we can't see or feel on shore or in the glow of incandescent lights. We have slipped into a place just beyond our physical selves but not too far away; a place where our emotional selves meet the most real and primitive world.
We're pressed into intimate emotional and physical contact with strange and wonderful forces, both massive and minuscule, where we meet, at least for a few days, our relationship with time.
We might become aware of it gradually, first as the fleet spreads apart and then when we taste the first mouthful of bean salad. We might slam into it headfirst, when the night is darkest or stormy; or our ministers, our super-heroes, our yogis, or any other of our teammates might help us into it. It might take a while to get going on a breathless night. It might only take minutes on a velvet night. Almost without exception, it takes complete hold by the day of hope. By then, we have forgotten that we are not onshore. We don't care whose boat or what kind of boat it is, if we are on or off watch, or what we packed. Normal daily happenings don't happen and daily objects lose their purpose.
There is no waiting, because what will happen happens. There is no reminiscing, because what happened happened. There is only one thing.
Everything is now.
The Race to Mackinac usually starts on a Saturday and ends on a Tuesday. Some faster boats finish earlier. If it's a crawl, it might take longer; if it's a surf, shorter. Regardless of on which day the race ends, when it ends, there is wistfulness, a palatable sense of regret and finality that consumes the fleet and the island to which it is moored.
Awareness, perhaps drawn out by the ubiquitous electric lamp, is broadening and diluting again, distracted outward and to all things; the days ahead, the trip home or to the next place, the news of the world. Gear is loaded onto carts and into taxis and eventually into buses and planes. Boats are sent home with skeletal delivery crews. We pay for parking and fall in line with traffic.
Within a day, not more and usually less, everything is back to normal, but normal isn't normal anymore.
Where there is light, everything is tomorrow and yesterday. Having come from the darkest darkness, we long for now again.
**STEVE MATTHEWS**
# CHAPTER 9
# SAILING, FLOW, AND FULFILLMENT
_We have all experienced times when, instead of being buffeted by anonymous forces, we do feel in control of our actions, masters of our own fate. On the rare occasions that it happens, we feel a sense of exhilaration, a deep sense of enjoyment that is long cherished and that becomes a landmark in memory for what life should be like. This is what we mean by_ optimal experience _. It is what the sailor holding a tight course feels when the wind whips through her hair, when the boat lunges through the waves like a colt – sails, hull, wind, and sea humming a harmony that vibrates in the sailor's veins._
Mihály Csíkszentmihályi1
Csíkszentmihályi's words are inspiring, and they remind us that the experience of sailing has many great qualities. In this essay I want to focus on one such quality inherent in that range of feelings we associate with an experience described as "flow." Csíkszentmihályi describes flow as a state that arises in people involved in some skilled activity who become fully immersed in it; they reach a state of "intrinsic motivation" and loss of self-awareness; their actions seem to occur spontaneously so that they seem to become simultaneously a passive witness to their own highly skilled agency. There are skilled movements and maneuvers in sailing in which the equipment becomes, as we say, "an extension of oneself." Under these conditions the sailor has usually reached such a level of proficiency that the state of flow just described may obtain. Moments of flow are relatively rare, and are highly prized by those who know what to look for. Losing oneself in the activity in this way is one of its high points, a point that makes it thereby significant and meaningful. Excellence in sailing confers a kind of fulfillment we rarely attain. It is, for this reason, an ideal worth striving for.
I am going to unpack these concepts – sailing, flow, and fulfillment – but with a twist, for I am going to use as my central case windsurfing. How does it fit with sailing, then? Well, windsurfers refer to what they do as sailing, and to themselves as sailors. Windsurfing shares with yachting features such as the derivation of energy from the relative movement of wind and water, a common vocabulary of sailing terms, some common equipment, and a healthy respect for the elements. Techniques such as the extreme hiking in some catamaran and dinghy sailing classes are reminiscent of what takes place when a windsurfer planes. For my purposes the differences between the windsurfer and the yachtsperson sailing a large sloop do not matter. For, the kind of philosophical exploration I have in mind here of the experience of windsurfing will shed light on an important value of _sailing_ insofar as this activity requires skill and concentration and bodily movement that can lead to a state in which one loses oneself in its doing.
# The Key: Losing Oneself
I am standing atop Point Danger, a famous windsurfing site in Victoria, Australia. It consists of a narrow jut of land and reef extending through the breakers. If nature could have designed a windsurfer's playground, this place would be it, with flat water, waves, a bay on one side, and big ocean swells on the other. Its best asset is that on the leeward side of the reef smooth flat conditions prevail even in the biggest swells and the strongest winds, making for ease of entry through the breakers. The wind is cross-shore, twenty-five knots. I rig a 5.0 – in windsurfing this refers to a sail size equivalent to five square meters – and today I am using a 75 liter sailboard, relatively small, but the surf is running at about five feet, so small and light is best. Soon I am down to the water's edge, my heart rate is up, and the adrenalin primes me for what's to come. As I walk the board and rig through knee-deep water, the wind swiftly fills the sail and I quickly ease off my back hand to release the pressure; then I slowly bring it back in to ensure the center of effort of the sail is balanced against the tension in my arms. All is ready. I bring the sail in closer, place my back foot on the board, push off, and I am away. Within seconds the board and rig are at full speed. I hook my harness to the rig and instantly everything is effortless. I am leaning back, the board is planing at twenty knots, and I take aim at the first ramp – a small shoaling wave. Just before takeoff I bear away to ensure maximum lift and thrust, and abruptly I am airborne. The jump is long and high, and I land cleanly and smoothly before the next wave is upon me. I am perfectly powered and perfectly in control.
I am a windsurfer; windsurfing is part of my identity. Its value is, ironically, derived from the fact that, when I am immersed in the activity of sailing itself, I forget who I am, perhaps even that I am a windsurfer. For, especially at those moments of extreme concentration or (physical) effort, nothing but the experience itself is present to my mind. It is as if the memory cords linking me-now, carving across the face of a wave, to all else in the past, have been severed. My future self too seems cut off from the current experience – no room for thoughts about tomorrow's work day. And the focus of my current self on the complexities of the action at hand is total: nothing is left over for thoughts that go beyond it. Action and awareness have become unified, and during that period I forget who I am. For that period at least, I am thereby released from the stresses that come from the intrusion of self-conscious thoughts arising from the tedium of life or its worries.
The description above of sailing at Point Danger is not one of the phenomenology as it occurred then but rather of something salient now from memories of the event. My stream of consciousness at the time contained fewer words; it involved a loss of a sense of self as marked off against the action taking place in the scene described. From beach start to jump, and beyond, my conscious resources were pointed wholly toward the activity. As Csíkszentmihályi has put it,2 in flow I am in a zone bounded by two alternative possible states: boredom and anxiety. When an activity has ceased to challenge me, I soon lose interest in it, but, when an activity pushes me beyond the limits of my capability, my apprehension disrupts my performance. When bored, my self becomes visible – what am I to do with my self? When anxious, my self, again, becomes visible – how am I to overcome failure? In the state between these points in the cases of interest I become _in_ visible.
# Windsurfing
The experience of flow is itself possible because of years of training. That training is aimed at achieving skilled action in which all the movements and adjustments made while sailing occur spontaneously, non-self-consciously, and without (mental) effort. In cases where the action involves a relation to a piece of substantial equipment, a special condition must obtain: the skilled action requires the equipment and the human person to act in unison. The skis become part of the skier, the racing car and the driver are fused, the violinist and her instrument play as one, and so on. When all is in flow, the windsurfer too does not mark points of distinction between sailor and gear. I am convinced that this point is vital to understanding flow in the case of windsurfing, or any sailing activity in which hiking requires straps, or trapeze, for leverage points to effect a better center of gravity. The more points on the sailor's body there are connected to the equipment, the more he becomes a part of the structure itself. This seems to enhance the sense of the merging of action and awareness.
In order to see how this works, it might be useful at this point to describe the windsurfer's equipment, or "kit," and provide a little more general background that relates it to sailing. Windsurfing is the modification of two kinds of watercraft: the rig of a dinghy and the board of wave surfing. The rig, when set up, comprises a monofilm and dacron-trimmed sail that is "downhauled" to produce a curved aerodynamic shape. When the sail fills, it creates an airfoil for extra power, speed, and stability. The sail is outhauled at the clew, which attaches to the end of a wishbone boom. The rig attaches to the sailboard on a universal joint, so that it then becomes a handheld single piece that moves in any direction. Boards vary in length, width, and volume as appropriate to the style of windsurfing and the conditions. There are now about seven recognized styles, but the fundamental distinction is between short boards and long boards. The former are typically used in wave or freestyle environments; the latter in flat water, for high-speed or slalom sailing.
Learning to windsurf takes skill, persistence, and a strong will to push through the disappointments. That is because it takes near circus-like skills to execute even some of the basic moves well. _The_ basic move is something called the carve jibe: a downwind turn through 180 degrees that, when correctly performed in flat water, is a tack involving no loss of momentum or speed. Most windsurfers cannot perform this basic maneuver fully correctly. When it is performed correctly it is pure joy. I have completed thousands of jibes and the sensation that accompanies a (relatively rare!) perfectly executed one performed in high winds is as rewarding now as it ever was.
In the late 1970s, windsurfers began to move from sailing in predominantly flat water and lighter winds to waves and stronger conditions, and this presented a problem: how to stay connected to the board and rig without getting hurt or too tired. The solution: foot straps that unite the body with the board, and a harness that unites the body to the rig (via harness lines attached to the boom). These additions formed the sailor and kit into a more integrated unit. The kit became an extension of the sailor. When learning, one has the sense of riding on the board and controlling it; in straps and harness there is more a sense of simply riding on the water. Let's call this idea "integration." Integration implies three things in relation to performance, psychology, and a philosophical point about what has come to be called "embedded cognition."
# Performance, Psychology, and Embedded Cognition
First, with respect to performance, the straps and harness enable more control and stability, and more speed with control. The sailor is locked into, and so part of, the kit itself. Greater control is available because fewer mental resources are being devoted to staying attached to the kit. In addition, there is the mundane fact that sailing without a harness in strong winds is hard work and the pain of hanging on is a distraction. Csíkszentmihályi lists the steps essential to transform mere physical action from a mechanical process to one in which flow may obtain. Among these he cites the need to be able to concentrate and to make finer and finer distinctions with respect to the perfection of a skill, and that such skill development must be flexible enough to fit differing external conditions. In the present case, one needs equipment and techniques that eliminate "noise" – that is, those things irrelevant to achieving a performance ideal. Integration helps to eliminate noise, and with it in place one is much better located to achieve the final step Csíkszentmihályi nominates: "to keep raising the stakes if the activity becomes boring"; or, in other words, to keep exploring new challenges as mastery of old ones is gained.3
Second, psychologically, the main effects of more closely merging the human body with the sailing gear are greater commitment and confidence. Commitment in this context is both metaphorical and mental. Although a windsurfer can exit the footstraps, or unhook from the harness quite readily, there is no guarantee of escape in all situations. Once hooked in, the windsurfer is then physically bound by those circumstances ("committed") to respond as one with the kit of which they are a part. Learning to sail in straps and harness is, needless to say, a testing and potentially dangerous time, and inevitably all novices at this point experience the dreaded catapult. This occurs when the off-balance sailor loses control of the powered-up sail _to which he or she is attached_ , resulting in an often violent centrifugal action. Hence, there is a need for _mental_ commitment. Once through these learning stages, however, when body and kit form a single unit, the result is less strain and pain and so more confidence to move up through skill levels.
The third point to follow from integration is one about cognition extending out beyond the mind traditionally conceived. The French philosopher René Descartes was a kind of early cognitive scientist interested in understanding the nature of the mind and its activities. Descartes famously concluded that minds were _essentially_ different from the things in the physical world, but it did worry him that the distinction between mind and body was not straightforward. He wrote:
Nature likewise teaches me by these sensations of pain, hunger, thirst, etc., that I am not only lodged in my body as a pilot in a vessel, but that I am besides so intimately conjoined, and as it were intermixed with it, that my mind and body compose a certain unity.4
There is sometimes an emphasis on keeping mind and body well distinct, but in the context of action this is unjustified even in the light of Descartes' own writings. Recent work in the philosophy of mind construes some mental aspects as neither purely in the head nor purely constituted by observable physical movements. Rather, the thesis of embodied cognition is that some mental processes are realized quite literally by involving parts of the body that extend beyond the brain and nervous system. An action, according to this view, is a cognitive act and the body is an essential element of that act.
There is, even more interestingly, a thesis in the philosophy of mind that takes this point further to regard elements of one's environment – such as tools, prostheses, or use-objects generally – as constituting an ineliminable aspect of one's cognitive activities. One version of this has come to be known as the "extended mind" thesis. In this view, a cognitive activity has to be thought of as embedded within a part of the environment that facilitates that activity. The idea, according to Clark and Chalmers, is this:
The human organism is linked with an external entity in a two-way interaction, creating a _coupled system_ that can be seen as a cognitive system in its own right. All the components in the system play an active causal role, and they jointly govern behavior in the same sort of way that cognition usually does. If we remove the external component the system's behavioral competence will drop, just as it would if we removed part of its brain. Our thesis is that this sort of coupled process counts equally well as a cognitive process, whether or not it is wholly in the head.5
And so we might import these last considerations into our current question: a windsurfer's kit and a human sailor form a coupled system. Let's just call it a windsurfer. When windsurfing, the first person pronoun "I" extends its reference to include this coupled system. This is part of a normal pattern of language use anyway. For example, after a collision while driving my car I might say, "I hit something on the road today." Similarly, after a day sailing, I might say, "I pulled off a perfect jibe today." The referent of "I" now has windsurfer parts, and that is how I think about it, that is how it feels, and that, with respect to technique, is the best way to conceive of performing all of the windsurfing maneuvers.
Dant and Wheaton have discussed what looks to be a form of the embedded thesis. It is worth comparing their account with what Clark and Chalmers say.
Unlike, say, formula one racing where there are mechanical controls, the control of the sailboard is wholly achieved by fine-tuning the orientation of the body to the object. At speed, this ability to achieve control must become as if it was intuitive; it must happen without conscious thought so that the equipment becomes like a prosthetic extension of the sailor's body. The sailor must perceive and respond to the environment of wind and water through the equipment of board and sail. Touch and proprioception must work through the various bits of equipment rather than on them.6
# Windsurfing and Flow
I arrive at the site full of anticipation. A sea breeze is building, and already – it's still only midday – there are small white flecks standing out against an azure sea. At the peak of summer there are only ever a handful of days like this. The wind is going to be strong and smooth. I rig a 5.4, knowing full well it will need to be changed down; by four o'clock I'll be in 4.2 territory since by then the wind will be blowing at over twenty-five knots. I'm rigged up and at the water's edge. I don't hesitate. I ease the board into the water and jump on. It's a perfect beach start, and within seconds I'm hooked in, feet in the straps, and I'm planing. It doesn't take long before the troubles at breakfast and the traffic snarl encountered on the way to the site are forgotten; it's as if a lid has snapped shut on the past. Everything moves into the present. I'm surprised by the strength of the wind and the acceleration reminds me of a fairground ride. I can afford a brief smile, but now it's time to concentrate. As I emerge out of the chop, the first swells rise up ahead of me and I have to focus for the first big jump of the day. The start of the day's sailing is all-important to gauge the temperament of the elements. I take off, and it's a big jump. Automatically I tuck my back foot up under my torso and draw the sail down toward the board so that I am now configured into the shape of an arrow. It's an exhilarating jump and landing, perfectly executed. By now I am completely consumed by the need to concentrate. As I continue through the activity, my sense of what is taking place oscillates between feeling like an active participant and feeling like a witness to an action taking place under my care.
Csíkszentmihályi has described the components of the phenomenology of enjoyment gleaned from studies involving "thousands of individuals from many different walks of life."7 In simplified form, he says that optimal experience is possible, conditional on there being a challenging and skilful task viewed as achievable through an act of control and concentration with clear goals and feedback. The involvement in the activity has a time-altering quality, and leaves no room for worries about everyday life so that concern for the self disappears, yet, he says "paradoxically the sense of self emerges stronger after the flow experience is over."8 This last point is a crucial focus for this essay, and I shall address it in the final section. I won't consider all of Csíkszentmihályi's components, but it will be worth considering skill, concentration, goals, and feedback.
The intensity of a flow experience is partly a function of the skill level required, built from years of effort, together with the merging of action and awareness. It is a common observation for those in the midst of a highly skilled performance that as soon as one pays attention either to the _collection_ of the technical elements of the action or, worse yet, to oneself performing those elements, failure is not far away. This is one reason why some musicians, for example, take beta blocker drugs to screen out the noise of self-consciousness. Flow requires two features that appear in tension: great skills about which one must avoid thinking in a direct way. The hint of paradox is dispelled once we recognize that the skills themselves are "built into" the agent who performs them, much as the complex behavior of a sophisticated machine, such as a jet airliner, is built into it during the design phase. Sometimes this is called muscle memory, but much more is at stake than that suggests. (After all, sleepwalking can occur because of muscle memory.) In flow, automatic action is accompanied by monitoring of the movements as they occur. The monitoring self during the period of flow is "checking" to make sure the action lives up to an ideal it recognizes. So long as the action being performed maintains the ideal – or something that reaches a threshold for it – the self in flow remains volitionally inert, but always ready to step in and make corrections if needed.
I am construing flow as applied to sailing as something essentially phenomenological, and this raises a question about what counts as a single flow-infused experience. One way into the question is to consider the role that one's goals play in deriving the goods that come from the flow experience. Csíkszentmihályi says that "the reason it is possible to achieve such complete involvement in a flow experience is that goals are usually clear, and feedback immediate."9 He goes on to cite the examples of tennis and chess. Competition games such as these are heavily rule-governed and so goals are very clear. In non-competition windsurfing, and sailing generally, the lack of such rules means that the goals of the activity are subject to the creativity of the participant. Nevertheless, such goals are indeed present. It is useful in the non-competition cases to distinguish between synchronic and diachronic goals. Synchronic (or at-a-time) goals are those for which the end of the activity is perceived within the space of working memory, a period of time no longer than about thirty seconds. Thus, hurtling toward the shoreline, some fifty yards away, I form the intention to jibe at its edge. This now becomes the focus of all attention, and the jibe is duly executed. Diachronic goals (those involving an extended period of time) cannot, in all their specific detail, be fitted into working memory. Thus, my plan to windsurf in and around this particular site for the next few hours is nebulous. There can't really be any particular piece of feedback from executing the plan that the goal has been reached. So, flow is not a quality that attaches to the diachronic case.
However, it is arguable that flow does not appear to obtain in the case I have described because a vague or nebulous plan can't thereby generate the feedback that would signify the reaching of the goal. However, Csíkszentmihályi discusses cases from solo ocean cruising in which a plan might extend for weeks. He writes:
Jim Macbeth, who did a study of flow in ocean cruising, comments on the excitement a sailor feels when, after days of anxiously scanning the empty reaches of water, he discerns the outline of the island he had been aiming for... [one legendary cruiser said,] "I... experienced a sense of satisfaction coupled with some astonishment that my observations of the very distant sun from an unsteady platform and the use of some simple tables [had] enabled a small island to be found with certainty after an ocean crossing."10
What should we say here? Is this just a case of someone deriving a lot of satisfaction from a hard-won goal? Or is there real flow involved here? I think we should be reluctant to claim these as cases of flow. First of all, true flow experiences are relatively rare, but deriving satisfaction from a goal completed is relatively common. Second, the feedback one receives in flow must be directed back into the action with which one is involved, and that feature is absent here. Third, flow experiences involve skilled actions, but in these cases the agent is experiencing the end point of a set of skilled actions. Finally, flow involves the loss of self, yet in the case we just saw the self is present to the self: the ocean cruiser is engaged in some well-earned self-celebration. Now, this is not to underplay the value of self-recognition within the circumstances of a great accomplishment. But it is to underscore a distinction between the components of enjoyment on the one hand and the components of flow on the other.
## **Sailing, Flow, and Fulfillment**
In this final section I present some brief reflections on the connection between flow and its value. The claim is of course not that sailing is peculiarly enabling of flow. Any activity in which flow may obtain is potentially valuable. And the aim here is not to identify a set of causal factors linking flow with the values it brings. That is a job for psychologists. The task is to identify the conceptual links between flow as described in the context of sailing and some features of human happiness. To do this we need to return to the experience itself. What is it about the structure of agency within flow that makes it a valuable experience?
Csíkszentmihályi says:
A typical day is full of anxiety and boredom. Flow experiences provide the flashes of intense living against this dull background... [A] person in flow is completely focused. There is no space in consciousness for distracting thoughts, irrelevant feelings... When a person's entire being is stretched in the full functioning of body and mind, whatever one does becomes worth doing for its own sake; living becomes its own justification.11
Though I would want to qualify this in many ways, it does convey an important insight about human fulfillment and action _as its own end_. Those who become habituated to passive activities – such as watching television, or more generally being a lifeless consumer of leisure – typically do so aiming at hedonic pleasures causally downstream from the activity. If pleasure is the ultimate aim, paradoxically it never really comes. If the mediating source is something as banal as watching television, it too is unlikely to deliver the goal.
Thus, the claim has to be that we derive value from an activity when the activity itself is its own end. Flow activities of course have this quality, and, though speculative, I would want to claim that the level of value at stake varies commensurate with the level of challenge and skill involved in the activity. The main point is that flow activities generate such value because they focus the agent away from herself and "into" the action itself (an embedded cognition), and they can do this with great intensity. Another way to describe the way our selves can become immersed in the action is in terms of _intrinsic motivation_. What does this mean? It needs to be understood in connection with the notion with which we began this essay: losing oneself in the activity.
Csíkszentmihályi quotes the words of a famous long-distance ocean sailor: "So one forgets oneself, one forgets everything, seeing only the play of the boat with the sea, the play of the sea around the boat, leaving aside everything not essential to that game."12 In moments of flow like these, the motivational structure of the agent might be thought mysterious. Is the concept of an action performed by an agent who forgets herself even intelligible? I think it is. We need a conception of motivated action that is neither that which an animal performs (such as a stalking cat) nor that performed ordinarily by a person focused on what he himself is doing (e.g., shaking a person's hand while being introduced), but somehow combines elements of both. David Velleman gives a characterization of the difficulty of what we are seeking to describe. He says:
The more conscious we become of a motive, the more it becomes the object of our thought; and the more it becomes the object of our thought, the less we think from the perspective of its subject; and the less we think from the perspective of the motive's subject, the less engrossed we are in the activities that it motivates.13
In flow we are in a state of effortless action while simultaneously remaining completely engrossed in what we are doing. Because we are engrossed, we are not thinking _about_ what we are doing, we are just doing it; we are, as they say, "in the zone," a bit like the stalking cat. Yet we seem simultaneously to be _watching_ the action, and in flow this is different to becoming conscious either of ourselves or of our motives. Rather, we partially dissociate into a pleasant state of watchful readiness in which the disposition to correct ourselves isn't triggered. Flow states are pleasant partly just because we do not need to correct ourselves. The best sailing days involve great, often highly technical, accomplishments that are effortless.
Velleman also warns of the need to avoid becoming conscious of our motives. And this is a critical part of flow. To put it another way, the trick is to avoid becoming aware of what we are trying to do while we are doing it, since that is an extraneous thought that interferes with the doing. The right motivational structure, then, is a balancing act between the doing itself and an awareness of the doing that omits the reason for it.
In this essay I have applied the concept of flow to windsurfing (sailing). My motive has been to understand how it applies, and the sense in which flow-in-sailing may constitute something of intrinsic worth in a meaningful life. Because sailing at this micro level involves the close connection between sailor and gear, we see that flow is the result of both working as one. This integration idea is essential to the success of flow, because in correctly performed windsurfing maneuvers a single action occurs and is carried out in one seamless "arc," not in two separated movements. I suspect also that integration partly explains the loss of self-awareness that accompanies flow and is integral to what is valuable about that state. In this connection it is worth recalling the concept of embedded cognition – in "prosthetically" enhanced skilled action we lose ourselves into the world, to speak loosely. This has a beautiful, uncanny feeling.
I have characterized the value of flow-in-sailing in terms of the experience itself, as an activity involving intrinsic motivation. The key to this idea is that the activity constitutes its own end, and so the value is always within reach, unlike the many meaningless actions we must perform as mere stepping stones to something else (filling in some tedious form, for example). In flow we are in that valuable state of realizing an ideal we have strived for, while at the same time acting out movements that constitute that very ideal. When contemplating the things that make for a fulfilling life, a sailor may count states of flow among those that significantly contribute to it.
**NOTES**
1 Mihály Csíkszentmihályi, _Flow: The Psychology of Optimal Experience_ (New York: Harper Collins, 1990), p. 3.
2 Csíkszentmihályi, _Flow_ , p. 74.
3 Ibid., p. 97.
4 René Descartes, "Meditation VI." In John Cottingham (Ed. and trans.), _Meditations on_ _First Philosophy: With Selections from the Objections and Replies_ (Cambridge: Cambridge University Press, 1996), p. 13.
5 Andy Clark and David Chalmers, "The extended mind," _Analysis_ 58:1 (1998), p. 13.
6 Tim Dant and Belinda Wheaton, "Sailing a board: An extreme form of material and embodied interaction?" _Anthropology Today_ 23:6 (2007), p. 10.
7 Csíkszentmihályi, _Flow_ , p. 4.
8 Ibid., p. 49.
9 Ibid., p. 54.
10 Ibid., p. 55.
11 Mihály Csíkszentmihályi, _Finding_ _Flow: The Psychology of Engagement with Everyday Life_ (New York: Basic Books, 1997), pp. 30–32.
12 Csíkszentmihályi, _Flow_ , p. 63.
13 David Velleman, "The way of the wanton." In Kim Atkins and Catriona Mackenzie (Eds.), _Practical Identity and Narrative Agency_ (New York: Routledge, 2008), p. 180.
**JESÚS ILUNDÁIN-AGURRUZA, LUÍSA GAGLIARDINI GRAÇA, AND JOSÉ ÁNGEL JÁUREGUI-OLAIZ**
# CHAPTER 10
# ON THE CREST OF THE WAVE
# The Sublime, Tempestuous, Graceful, and Existential Facets of Sailing
# Ahoy!
There is an old French proverb, "he who would go to sea for pleasure would go to hell for a pastime." Proverbs let on more than the popular "wisdom" with which they are often credited. They serve as heuristic shortcuts to more or less successful common-sense solutions. They also reveal the communal psyche of a people and their culture. In Japan, where the _katana_ was the samurai's soul, the sword terminology and maxims dominate Japanese expressions. In seafaring nations such as England, Portugal, or Spain, nautical idioms command the helm. So, while the Japanese speak of the final and decisive stage of an endeavor as _tsuba-zeriai_ (from a phrase meaning a dogged fight to the end where sword guards – _tsubas_ – hook), we say that things come to the bitter end. We needn't get overly dramatic. We're still moored!
So, let the French saying unfurl for a bit, and allow its implications to luff in your mind. What does it say about us sailing-obsessed people? It intimates a certain lunacy, and, indeed, heading out to sea does require an inclination for the intrepid. It also speaks of women and men of a particular mettle: persons who enjoy themselves when others are taken aback. Additionally, it suggests that whoever coined the saying forswore sailing due to a nasty bout of seasickness (many of us sailors have paid culinary tribute to Neptune at one wave or another, but by and large this is forgotten with the promise of another thrilling close-hauled run).
What's in the offing? Well, we've charted a four-leg course for our essay that explores the rich aesthetic waters of a life on sails, specifically its sublime, tempestuous, graceful, and existential facets. All come aboard!
# The Sublime Poetry of Sail and Wind
Some describe sailing as endless tedium punctuated by moments of sheer panic. Anyone who's sailed enough has lived through both. The middle ground offers a better understanding of the experience, as usually happens in philosophy. Let's look for the center of effort on our conceptual canvas, then.
The natural settings where we may sail and the very emotions these arouse make the notion of "beauty" as inadequate as describing a majestic great white shark as a big sardine. The Greek mercenaries' reaction in _Anabasis_ , as their leader Xenophon narrated it, gives a better idea. After fighting their way back home across the Persian desert, they broke into ecstatic cries of "The sea! The sea!" when beholding the azure Aegean Sea. Indeed, that first glimpse of the water after an arduous tribulation, whether through enemy-infested lands or a long week at the office, is a magical moment that takes the wind out of our lungs to put it behind the sails we long for. Once we find ourselves hands on tiller or wheel, we begin to weave the poetry in motion of sailing: a harmonious synergy of water, wind, sail, and hull that transports us to a realm where, as our mate Gagliardini puts it, "we encounter peace, silence, even the infinite." Of course, sailing one's rig in places like the Balearic Islands, the Aegean, or the Algarve Coast, as she has, goes a long way toward finding this kind of transcendence. In everyday life this is as likely as fair days in Cape Horn, but even the most modest of sailboats and the tiniest bay or lake can lead to memorable experiences. The exhilaration sailing can bring has diaphanous mystical overtones. As Ellen MacArthur wrote in her Vendeé Globe log,
This morning as the sun rose the sky went pink in the west, and... the striking orange glow looked like it hid a million treasures as it peeked out from behind the scattered clouds... suspended almost as if pre-arranged in the sky... There were layers and layers of them... It really is so beautiful... as I stood in the cockpit I watched _in wonder_... My eyes began to fill with tears as I marveled at this intense beauty.1
Philosophy begins in wonder. And, it's no wonder that poets have lauded the sea's beauty. But, as Joshua Slocum, the first solo circumnavigator (who cried under similar circumstances) makes clear, "Poets have sung of beacon-light and pharos, but did ever poet behold a great light flash up before his path on a dark night in the midst of a choral sea? If so he knew the meaning of his song."2 We should write of what we know, and sailing enables us to experience some of nature's most amazing facets.
To increase the intensity, we need only point our prow toward the sublime. "It's just awesome!" we gasp – words futilely try to express sailing's wondrous ineffability. The shortest tack is to describe this as _awe_ , an emotion that mixes admiration, respect, and intimidation. Even Slocum, a prosaically witty writer, cannot help but wax poetic: "During these days a feeling of awe crept over me... The ominous, the insignificant, the great, the small, the wonderful, the commonplace – all appeared before my mental vision in magical succession."3 Of course, sailing can be quite trying: sweltering heat, ice-numbing water, or the bone-jarring battering by waves; however, this is _precisely_ what – after the fact – enhances the sublimity of the experience.
Watch for flying jibs and anticipate stinging spray as we stay the course into the sublime! We simplify to a two-reefed Marconi, the illuminating, if complex, rigging that a famous landlubber, Immanuel Kant– overly cautious as ever – deployed to analyze the sublime. These reefs fasten the ideas of transcending boundaries and courting risks. First, there are unassailable restrictions: hulls have maximum attainable speeds; our capacity to imagine is limited. We're awash with the feeling that we can't fathom the sheer size of something like the ocean's immensity. Magellan, his second-in-command Elcano, and their mates surely felt this on encountering the vastness of the misleadingly named Pacific Ocean. Ellen MacArthur expresses the mood well: "all movement, sounds, and thoughts seemed very small against the magnitude of the ocean."4 Second, the pursuit of danger is another aspect of and way to elicit the sublime – but, unlike a penguin's casual stroll amid hungry polar bears, it should not be foolhardy. After all, reckless sailors are sunken sailors, eventually. The sublime plays with danger amid natural phenomena that have an absolute power over us. Here's our motto for an intellectually badass tattoo: "sublimity, the ultimate playground!"
Structurally, the sublime isn't unlike the two opposing forces of wind on the sail and water on the keel, which push the boat forward as we seek the point of balance. In the case of the sublime, there is a disturbing feeling, elicited either by the realization of the limits of our ability to imagine or sense the greatness of forces that overpower us, or the fear of these forces actually acting upon us. But this is matched by a pleasant emotion: the delight we experience upon realizing that we are equipped to handle these feelings through our skills and reason. One of us, Ilundáin, remembers the roaring excitement of sailing in twenty knots of wind with gusts to thirty, water flirting with the cockpit, and having to stand on the hull's port topside to avoid capsizing. The sublime rides on the point of balance between wave and hull, as emotions drift between the fear that risk arouses and the joy in our skill in meeting the challenge. It's a dynamic, paradoxical process in which harmony results from the power struggle between (our) intelligently harnessed skills and (natural) might – our insignificant "bit o' wood" frolicking in the enormous sea. And the result is exhilarating. Good ol' Slocum again recounts: "These rolling waves thrilled me as they tossed my ship, passing quickly under her keel. This was grand sailing."5 But, when the forces break the point of balance and go to one extreme or the other, the result is either boredom, that accursed lack of wind, or sheer panic. Out of control and at the mercy of fate, one's skills are irrelevant to cope with either scenario.
Sailing can be beautiful. Better yet, it can be awe-full. Awe is an emotion best reserved for moments of worship. Our sailboat is our altar and the ocean our church. On a boat, we worship best by cultivating our skills and being humbly aware of our limits and the forces we face, else we may have to pray to the sea gods. But we're fast gliding down to our next leg. Brace yourselves. We're about to face...
# Poseidon's Wrath
Amid a nasty storm, Pompeo Magno harangued his sailors: _Navigare necesse est, vivere non est necesse_ ("To navigate is necessary, to live is not necessary"). Bringing supplies from Africa to their beloved Rome was essential, even at the risk of their lives. But, to understand those sailors who undertake similar or worse risks not for Rome, glory, riches, or the promise of a warm embrace but the fun of it, we had better interpret the above as "a life without sailing is not (much of) a life." Explaining this "cares to the wind" attitude means cruising far offshore. With gale winds forecast, we recommend life vests for the next few paragraphs. Batten down the hatches!
Luckily our skipper, John Dewey, is an accomplished seaman and philosopher. His notion of an experience anchors sailing's revealing and transformative potential to imbue meaningful, life-enriching insights not only when enjoying the harmony of beauty or sublimity but when the lifeline stanchions live up to their name and sailing becomes utterly terrifying.
Amid the innumerable waves – moments – in our lives, some stand out like mountainous rogue waves. Dewey explains: "Experience in this vital sense is defined by those situations and episodes that we spontaneously refer to as being 'real experiences'; those things of which we say in recalling them, 'that _was_ an experience.' "6 This could be as dramatic as a blown gasket that almost sinks you as the hull fills with water, as José Luis Ugarte recounts about his experiences sailing in the Antarctic,7 or something banal yet epiphany-worthy such as _that_ first Optimist sail that inoculated us with the incurable sailing virus.
Any sailor worth his sextant knows that idyllic waters can turn into a nightmare at Poseidon's whim. As the lips of the waves kiss the gunwale, tongue and all, it doesn't take much to turn this sensual awesomeness into well-justified fear. As Dewey says,
Then, there is that storm one went through in crossing the Atlantic – a storm that seemed in its fury, as it was experienced, to sum up in itself all that a storm can be, complete in itself, standing out because marked out from what went before and what came after.8
This could well have been a description of the tempest Ilundáin's maternal grandmother faced as a little girl ship-bound for Cuba. Even the captain was on his knees praying, as it was pointless to attempt to command the helm. And, being aware that between you and the sea there is but a thin bit of hull is small comfort. Insightfully, skipper Dewey argues that this type of complete and mature experience also has "an element of suffering, in its large sense" because "otherwise there would be no taking in of what preceded."9 MacArthur evokes this when she describes the sickening feeling of barely missing an iceberg, compounded by the terrifying idea of hitting a boat-sinking ice chunk while surfing down waves at over twenty knots in the southern seas.10 But, as with the sublime, this enriches both the experience and our lives: we can feel _both_ terror and exhilaration.
**FIGURE 10.1** The _Snowgoose_ in happier times, before its stormy voyage. Author's copyright.
Aboard his sloop _Snowgoose_ (see Figure 10.1), another of us, Jáuregui, found out that the might of a pissed-off sea is nothing to trifle with. The plan was to sail from south of Barcelona to Alicante. In marked disagreement with the forecast, a gale sprung up at night. Murphy was certainly a stowaway: the electronic systems, lights, automatic pilot, _everything_ failed (it never rains but it pours). Jáuregui's ill-chosen crewmate was below decks in a drunken stupor (besides the importance of redundancy, Jáuregui learned to better vet enrollees). So, Jáuregui spent the night shining a flashlight on the sails to alert passing merchants of their presence (they were in one of the busiest Mediterranean shipping corridors). Chilled to the bone, desperately trying to control rudder and storm jib, just ahead he saw... well, you'll have to wait to find out (we'll return to this adventure in a bit). But let's say right here that they made it (else there'd be no tale to tell). After reaching port in a sorry shape and kissing the ground à-la-Pope, the welcome was less than comforting: Jáuregui's wife, who didn't yet know of their troubles, snapped, "You guys stink!"
What's the payout of being between the devil and the deep blue sea, other than the stink of fear? For Dewey, an aesthetic experience is _enriching_ and not a mere happening. It marks a before and an after. The windfall for us results from how we reflect on our (mis)adventures. An experience is like a Socratic spyglass that gives us a privileged insight into ourselves. Revealingly, the first question that Vendeé Globe veteran and team member of MacArthur's _Kingfisher_ project Alain Gautier asked her after she finished second was what she had learned about herself and how she had changed during the regatta.11 Her discerning reflections are worth many years of armchair pondering. Our tribulations, which test our mettle, can be deep _and_ wide sources of self-knowledge and invaluable opportunities to cultivate our character, once we have identified and learned to handle our talents and limitations. And this goes as much post-traverse as before leaving the dock: when Slocum set out he "felt that there could be no turning back, and that [he was] engaging in an adventure the meaning of which [he] thoroughly understood."12 We need to know _why_.
Being an outdoors activity, hazardous surprises are unavoidable in sailing. Sailors' lives are enhanced precisely because they are willing to face risks in their quest to refine skills that expand the richness of their experiences, be they fun, beautiful, sublime, or scary. These challenges are an intrinsic part of sailing. How they are met reveals much of who we are. "Where, after all, would be the poetry of the sea were there no wild waves?"13
# She Moves
Ilundáin's professional deformation as an aesthetician means that he spends countless hours looking at (and lusting after) boats (or anything that floats, actually), whether on docks (the object of many a pilgrimage) or boat magazines (more titillating than any racy centerfold). Although he can justify this as part of his job, he prefers to rationalize it to his wife thus: "If I'm going to lust... much better if she is a sailboat!"
The union of form and function in a sailboat, which combines to be more than the sum of its parts, is unrivaled among human creations. The contours, perfectly streamlined; the materials, strong and light; the design, simplicity itself, nothing superfluous; and the striking elegance of the boat in motion. Alexander Rost makes the case for this superlative blend:
The necessity for beauty, born from purpose, did not come to the sea by accident. Nature itself had proclaimed it. Wind and waves, and friction and lateral resistance, to mention the technical aspects, were the inescapable factors... the old philosopher's maxim that nature does not progress in and thus does not tolerate deviations from her logic, was and still is the most important law in the building of yachts.14
The feather is the counterpart in the animal kingdom: both are perfectly suited to their task, can offer the visual virtuosity of artworks, and, unsurprisingly, also take advantage of the wind. Literary great Joseph Conrad corroborates this regarding sailboats: "They are the birds of the sea, whose swimming is like flying, and resemble more a natural function than the handling of man-invented appliances."15 Moreover, in these days of environmental urgency, the very premise of sailboats – motion by means of wind – is more pertinent, even indispensable, than ever. Small surprise, Slocum describes the _Aquidneck_ , a boat he once owned, as "a little bark which of all man's handiwork seemed to me nearest to perfection of beauty."16
For all its visual appeal, where the sailboat truly stands out is in action. Then _she moves_! And, she enables us to _live_ truly unique kinesthetic and existential moments. Here we take aesthetics beyond the wharf of convention.
Consider what sailing feels like: the wind on your face, your tiller or wheel transmitting the pressure of the wind from the sail, the rudder pushing back, the water's lapping noise, your moist (or soaked!) skin, the sea's fragrance. But this physical aspect of sailing belies the fact that sailing experiences, in all their variety, rival religious ones (to echo William James' abundant work on the latter), having as they do just as much awe and a lot less animosity. As Dave Franzell writes, "each type of boat creates its own sailing experience."17 Indeed, the material conditions – the size, shape, actual materials used – of the boat make possible the "what-it-is-like-to-sail" _that_ kind of boat. Cutters, yawls, sloops, ketches, schooners; one mast, two, or more; centerboard or keel; clinker or carvel construction; wood, aluminum, or composite; single-handed or crewed; designed for long offshore hauls, speed, regattas, or casual and coastal cruising – each of these and combinations thereof act like distinct deck light-prisms that kaleidoscopically particularize our experiences.
We could say that a sailor is to his or her sailboat as a snail is to its shell, but the image is hardly inspiring. Let us instead fly a spinnaker and catch more inspiration and depth, and say that there is a symbiosis between body and boat: "The sailor becomes one with his vessel, just as the rider becomes one with his horse."18 Reading Chichester, Tabarly, Knox-Johnston, Slocum, or MacArthur makes this patent, but Joseph Conrad articulates it best:
The genuine masters of their craft – I say this confidently from my experience of ships – have thought of nothing but of doing their very best by the vessel under their charge. To forget one's self, to surrender all personal feeling in the service of that fine art, is the only way for a seaman to the faithful discharge of his trust.19
This Zen-like focus and dedication to one's charge is no mere attachment: it is held tightly by the capstan that cultivates our skills.
Sailing can be intimidating to newcomers, but for most getting started is easy. "Let's be honest, shall we?" writes Don Casey in the foreword to _Things I Wish I'd Known Before I Started Sailing_ ,
Basic sailing requires about as much skill as, say, riding a bicycle. Borrow or rent a small boat and in a hilarious half hour of pulling strings and pushing on that stick thing, you will become skillful enough to be able to sail back to the dock with almost no risk of injury or serious property damage.20
However, for some it is not as easy to master "the stick thing" as it is to ride a bike. Just ask Ilundáin's college sailing partner, his brother Jaime. His impressive capsizing record had the head instructor appoint him to the permanent role of ballast.
The challenge lies in getting better at it. There is an ethos that seeks to cultivate excellence and combines sailing's aesthetics and ethics into a _performance_ : it's about being the best sailor we can be. MacArthur exhibits the requisite attitude: "I am very much in love with the sport, and I desperately want to be better at it."21 Improving requires making of it a passion in the full sense: no effort spared, the goal is the continued refinement of abilities while lucidly aware that port is never reached. A most arcane skill is "reading" the invisible wind. _The Princess Bride's_ Inigo Montoya asks, when noticing they're quickly being overtaken, "I wonder if he's using the same wind we are using." The wind-games played in regattas are nothing short of wizardry.
Besides boat-handling abilities, navigational skills are crucial once we venture beyond spitting distance from shore. The potential for things to go wrong, as we saw with Jáuregui, is great. Back to his plight! Begging for safe harbor, he saw... the saving red light of what _had_ to be, by his calculations, Vinaroz Harbor. Repeatedly radioing the harbormaster – "Vinaroz Harbor, this is _Snowgoose_ , do you copy?" – he set a course toward the light. Other than the red beacon light, he could see about as much as a deaf bat. Aware of the rocky bottom, he sounded for depth – twenty feet, fifteen, twelve... way too shallow for comfort. Suddenly, a lucky break in the clouds showed him a cliff towering above him. He realized, frantically pulling all to port, that it was no lighthouse after all... it was the red light of a bordello!
Way back in 1898, Slocum belittled our reliance on technology: "In our newfangled notions of navigation it is supposed that a mariner cannot find his way without [a sea chronometer]."22 Those were other times. In today's GPS world, knowing how to work the parallel rules and compass rose is one of those underappreciated skills. Until we need them, of course. After all, in life and on water, points of reference and signs are most ambiguous.
# One is Free... on a Boat?
Ready about? Hard a' lee! As we head back to port, we run a leg in which the currents of aesthetics and the existential fuse.
"For the first time in my life I felt totally free."23 Ellen MacArthur reflects a view endorsed by many: sailing is a vehicle to freedom. But our aspirations of liberty seem to be firmly tied to the cleat of circumstance (and without much pomp). Let's board the play _Rosencrantz and Guildenstern are Dead_ , where the pirate flag of fate flies. Sidekick Rosencrantz remarks of the bilge into which he plummets, "Nice bit of planking, that." Chum Guildenstern adds,
Yes, I am very fond of boats myself. I like the way they're – contained. You don't have to worry about which way you go, or whether you go at all – the question doesn't arise, because you are on a _boat_ , aren't you?... I think I'll spend most of my life on boats.
Upon further consideration, he clarifies: "One is free on a boat. For a time. Relatively." Rosencrantz asks, "What's it like?" and Guildenstern replies tersely, "Rough."24
Ballast heavy words! Indeed, life is like a boat. One may be relatively free. As Guildenstern realizes, we are "free to move, speak, extemporize, and yet. We have not been cut loose. Our truancy is defined by one fixed star, and our drift represents merely a slight change of angle to it."25 Can we direct that drift? If you can tie a bowline knot, chances are you've sailed long enough to realize that onshore and offshore we face impersonal forces that don't give a bosun's whistle what our plans are. Our idea of free will is fouled – tangled – at both ends of the rope. Our physiology and psychology entangle the "inside" subjective end with neurological processes that are tied to objective causal chains and the below-decks – that is, unconscious – origin of many thought processes and seemingly spontaneous actions. There isn't room to freely swing a cat here. Circumstances (social, political, environmental) knot the "outside" end. Their dynamics, from when and where we are born to complex societal and natural elements, escape our control in much the same way that we cannot command winds or currents. We're running with three sheets to the wind – out of control. This threatens to capsize meaning.
Instead of a mutiny of denial, we propose as a jury-rig to – paradoxically yet sensibly – work with(in) these restrictions. Partially, the bind results from rigging this as a matter of mutually exclusive extremes: either free will or determinism, the will is either the lazy guy on this tack or a working sheet carrying the whole load. Oftentimes we are at the mercy of the elements: our life's circumstances, inclinations, and talents. But, sometimes, we can learn how to gain control. This begins by accepting the said limitations and situation and the direction of the wind, then learning to adjust and work with them. Actually, we need _some_ resistance, for a complete absence of restrictions – pure chance – also throws us overboard into determinism's unforgiving waters. Thus, while sailing directly upwind is impossible, we can beat to _any_ point should we be skilled enough. We're not unfettered, but becoming apprised to this gives us the opportunity to gain a measure of freedom. For a time. Relatively. As the old saying has it: "You cannot change the direction of the wind, but you can adjust your sails."
But there's a devil to pay even for this relative freedom. Our ability to choose is but a capacity to earn and learn, not an inalienable right to a wide berth of choices. We become free to the extent that we cultivate our talents. Accordingly, there are better sailors and freer people (than others). The weatherly (nice) side to this lies in the creative freedom of beautiful performances, which sailing strikingly exemplifies. Coming full circle to run downwind, the aforementioned sailing skills that enable us to experience beauty, the sublime, and even the terrifying become relevant again. The discipline and rigor needed to develop our abilities allow us to sail in more challenging conditions _and_ to perform beautifully. The economy of movement and the elegance of accomplished sailors' movements are honed by the constraints of training. Here the ethos of discipline and aesthetics blend into a liberating performance in the wake of which we divine the traces of restrained autonomy. Freedom by way of constraint, if you will. By cultivating ways to actually _control_ our impulses, movements, and thoughts, we gain a measure to beautifully navigate the "choice" of, and how to get to, our destination.
To dock now, a few words to the wind and the wise. Sailing's crow's nest affords a distinctive, advantageous philosophic perspective on life. Perched on it, we discern and make use of sailing's transformative potential while pointing out the flotsam and jetsam of the mundane. We can spot what's important for a joyful life should our temperament lean toward the naval _and_ avoid the gunk threatening to mar our hull. Compared to a safe, lackluster _dry_ life, for many sailing is much more than a sport, vehicle, or métier: it is one of the most beautiful, daring, jubilant ways to literally and philosophically sail our existence's familiar _and_ uncharted possibilities. Maybe _navigare necesse est_ after all.
Godspeed!
**NOTES**
1 Ellen MacArthur, _Taking on the World: A Sailor's Extraordinary Solo Race Around the Globe_ (New York: McGraw-Hill, 2005), p. 296; emphasis added.
2 Joshua Slocum, _Sailing Alone Around the World_ (New York: Dover Publications, 1956), p. 192.
3 Ibid., p. 26.
4 MacArthur, _Taking on the World_ , p. 131.
5 Slocum, _Sailing Alone Around the World_ , p. 55.
6 John Dewey, _Art as Experience_ (New York: Perigee Books, 1980), p. 36.
7 José Luis Ugarte, _El último desa_ _fio: La más dura regata de altura, narrada por el único participante español_ (Barcelona: Editorial Juventud, 1997), pp. 67–76.
8 Dewey, _Art as Experience_ , p. 36.
9 Ibid., p. 41.
10 MacArthur, _Taking on the World: A Sailor's Extraordinary Solo Race Around the Globe_ , p. 299.
11 bid., p. 20.
12 Slocum, _Sailing Alone Around the World_ , p. 11.
13 Ibid., p. 192.
14 Frank Grube and Gerhard Richter (Eds.), _The Big Book of Sailing: The Sailors, the Ships and the Sea_ (Hauppauge, NY: Barron's Educational Series, 1978), p. 41.
15 Ibid., p. 10.
16 Slocum, _Sailing Alone Around the World_ , p. 3.
17 David Franzel, _Sailing: The Basics_ (Guilford, CT: The Lyons Press, 2003), p. 1.
18 Grube and Richter, _The Big Book of Sailing_ , p. 6.
19 Ibid., 10.
20 John Vigor, _Things I Wish I'd Known Before I Started Sailing_ (Dobbs Ferry, NY: Sheridan House, 2005), p. xiii.
21 MacArthur, _Taking on the World_ , p. 319.
22 Slocum, _Sailing Alone Around the World_ , p. 15.
23 MacArthur, _Taking on the World_ , p. 14.
24 Tom Stoppard, _Rosencrantz and Guildenstern Are Dead_ (New York: Grove Press, 1967), pp. 100–101.
25 Ibid., p. 101.
**JESSE STEINBERG AND MICHAEL STUCKART**
# CHAPTER 11
# NAVIGATING WHAT IS VALUABLE AND STEERING A COURSE IN PURSUIT OF HAPPINESS
There are a number of different senses of "value," or of what is worthwhile. Some things are valuable in a practical sense. Given a desire not to run aground, one should steer clear of that shoal. Some things are valuable in a moral sense. One has a moral obligation not to pollute the environment. Some things are more aesthetically valuable than others. A ketch is more attractive than a tug boat. In this essay, we explore the nature of value and how one ought to live one's life through the lens of sailing. After discussing the concept of value and various things that are valuable about sailing and sailboats, we consider what Aristotle had to say about the pursuit of happiness and what it is to flourish. We argue that sailing has all the necessary elements for achieving a fully happy life.
The concept of value is hard to pin down. It's not easy to say exactly what it is to be valuable or to be good. If precise definitions are hard to come by, perhaps we should proceed by thinking about clear examples of valuable things. Just as one can know a boat when one sees one despite being unable to provide a satisfactory definition of "boat," one can tell what sorts of things are valuable or ought to be pursued even if one can't quite say what "value" is exactly.
We began with such a list. Some things ought to be pursued because they are of practical value. Other things are good because they are beautiful or have aesthetic value. Some things are valuable largely in virtue of the worth that an individual or group puts on them. For example, a coin is just a disc of metal, but it is _worth_ something – more than just the same amount of metal shaped into a sphere. And we value certain traits in people such as honesty, patience, kindness, generosity, humor, and intelligence.
Of course, we could go on for days listing things that are valuable. In thinking about the myriad valuable things, a distinction presents itself. Philosophers use the phrases "instrumentally valuable" and "intrinsically valuable." Instrumentally valuable things are valuable because of other values that they promote. Money is an instrument used to buy goods and services, so it has instrumental value in the sense that it is valuable in virtue of the other things that it enables us to obtain. Repainting a sailboat might be wise given the benefits of preserving the boat and enhancing its beauty. So repainting one's boat has instrumental value.
Teak used as a boat-building material is another example of something with instrumental value. It has instrumental value as a wood that resists the deleterious effects of weather and moisture. Teak decks are durable, non-slip surfaces that allow us to move more safely about a boat under adverse conditions, and teak handrails provide safety when we move about. But teak is also a beautiful wood when finished. Most sailors probably give some thought from time to time about whether or not to spend the time on their boats to create the often-admired brightwork that characterizes some sailboats. Of course, many sailors eschew teak handrails and trim for stainless because they see the latter as having more instrumental value. Stainless serves the same purposes, but doesn't require the work to maintain.
Lots of equipment on sailboats has instrumental value. And it's interesting to note that what is seen as having the most instrumental value has changed over time. Not long ago, paper charts, the compass, and perhaps the sextant were essential pieces of valuable equipment for navigating. These were things that were viewed as being quite useful tools. But that's not the case for many sailors today, despite frequent warnings about the consequences of going without them. These objects have been replaced by GPS. And how about autohelm? Clearly the wheel or tiller has instrumental value around the dock. So does a mooring ball. But an autohelm enables the single-handed sailor to work away from the helm or to hold a course when there are other priorities to attend to.
Most sailors probably place greater value on those pieces of equipment that relate more directly to the sailing performance of their boats. This includes everything that works on a boat, from the shape and size of its keel to its prop (is it folding?), its winches (how big are they and are they self-tailing, multi-speed winches?), and so on. It also includes the materials out of which the equipment is made. What fabrics are used in the sails? What is the fiber composition of the halyards and sheets? Of what are the spars made? Are the boat's materials older and heavier or newer and lighter?
While a sailboat and its hardware obviously have instrumental value, it is sailing itself that ultimately attracts people to the sport. With some reason, many sailors disdain power-boaters because of what they don't know. In their minds, after all, power-boaters only need to know how to operate a gearshift, throttle, and steering wheel. Sailors, on the other hand, have to compute wind and water conditions, the direction in which they want to go, the best possible combinations of sail and trim, the sailing characteristics of their particular boat, and myriad other factors to produce the best possible effect in terms of speed, safety, and comfort. Perhaps more than anything else, it is the knowledge required to produce the ideal sailing experience that has such instrumental value to the sailor. It is the harnessing of nature and the maximal use of available technology that is so valued. And, when one's boat enters that groove of which sailors speak, we know that we are close to that beautiful perfection – that marvelous _je ne sais quoi_ – that is at the heart of sailing a boat. And this knowledge and these skills provide additional examples of things that are instrumentally valuable.
Perhaps paradoxically, the pleasure and value derived by many sailors actually increase under adverse conditions. Confronting challenging elements increases the need for good decisions. When made correctly, the instrumental value of the act increases. Mastering malfunctioning technology also increases the perceived rewards of sailing. Being self-sufficient, being able to jury-rig a piece of faulty equipment, being able to solve an operational problem all are satisfying to the sailor, are the stuff of sailing lore. They are an important part of sailing culture and are yet further examples of traits that are quite valuable.
So much for instrumental values; what about intrinsic values? These are said to be valuable "in and of themselves." This means that a thing that is intrinsically valuable is good in virtue of it being the very thing that it is and not simply because of the other valuable things it brings. In other words, it is valuable for its own sake. Some philosophers have argued that _pleasure_ is an intrinsic value. When we ask why repainting a boat is good or why making sails out of a certain material is good, we appeal to the other valuable things that these bring. But, when we ask why pleasure is good, we don't appeal to other valuable things. Rather, pleasure is good for its own sake. It's good because of the very thing that it is! A philosopher named William Frankena (1908–1994) gave a rather extensive list of things that are intrinsically valuable:
Life, consciousness, and activity; health and strength; pleasures and satisfactions of all or certain kinds; happiness, beatitude, contentment, etc.; truth ; knowledge and true opinion of various kinds, understanding, wisdom; beauty, harmony, proportion in objects contemplated; aesthetic experience; morally good dispositions or virtues; mutual affection, love, friendship, cooperation; just distribution of goods and evils; harmony and proportion in one's own life; power and experiences of achievement; self-expression; freedom; peace, security; adventure and novelty; good reputation, honor, esteem, etc.1
These things appear to be valuable in their own right. They aren't valuable simply because of other valuable things that they bring, but rather are valuable in and of themselves. Interestingly enough, when discussing examples of instrumental values above, some of these intrinsic values came up. The pleasure we get from seeing the teak of our boats shine in the sun, having the knowledge to jury-rig a piece of faulty equipment, and being able to successfully navigate through a tough spot are each examples of things that are valuable in and of themselves. They are good not just for other things but for their own sake too. There are thus plenty of examples of things that are _both_ instrumentally and intrinsically valuable. And, as you can probably tell, we think there are an extraordinary number of valuable things having to do with sailing and sailboats.
# What's So Great About Sailing?
With the distinction between instrumental and intrinsic value in hand, let's look more closely at the sorts of things involving sailing and sailboats that are valuable. Of course, many sailors enjoy sailing for its aesthetic value. First, there is the water. We have all seen incredible photographs of the water in various places around the globe, and the magnificent colors it can be. Many of us have seen those colors first-hand. Their beauty at first light, mid-day, and sunset provide sailors with a limitless palette of colors and beauty. Many of us especially enjoy the way that the waves sparkle in the morning or evening, as they reflect the rays of a rising or setting sun. Add the colors of the sun and the terrain, and the aesthetic possibilities are virtually limitless. Clouds, too, are an endless source of aesthetic enjoyment. They come in all sorts of shapes, sizes, and hues, and they convey a multitude of subtle and powerful impressions and moods.
Whether beauty (aesthetic value) is objective or relative is an issue we'll here avoid. Although it's a fascinating question whether beauty is just "in the eye of the beholder," we can't hope to settle this issue in this chapter. It's clear that there is a great deal of variation in the design of sailboats and in the opinions that people have about these assorted boats. As we have said, sailboats are a multifarious lot – they come in various sizes and shapes, they are made of different materials, and so on. In addition, some people find a classic Hinckley or an Island Packet, for example, quite beautiful, while others may opt for the newer designs of Jeanneau or Hanse. One might wonder what constitutes the most aesthetic line of a bow, the shear, or a transom? What makes for a beautiful trunk? The cultural and personal differences in aesthetic judgments are clear, but one thing is certain: there's much to find beautiful in sailboats.
Most sailors, we think, probably attach significant aesthetic value to a sailboat based upon the materials out of which it is constructed. We peruse the wooden boat calendars nostalgically when we take a break from holiday shopping. When was the last time we looked through a calendar dedicated to fiberglass resting at anchor or underway? Of course, one popular sailing magazine does feature "classic plastic" sailboats. But we don't really glamorize boats that aren't wood. No one would seriously compare or prefer gleaming stainless steel to perfectly varnished brightwork.
The sails of a boat obviously have instrumental value (they help to keep the boat moving), but they also have aesthetic value in many cases. This can be true in the case of mainsails and jibs, where perhaps the aesthetic value of a sail is tied to its instrumentality in terms of sail fabrics ranging from the blinding white of new Dacron sails against a blue sky to more exotic fibers whose aesthetic value is tied to their potent instrumentality.
The above thoughts on boat materials raise questions about what might be called "practical value." While many sailors would agree that wooden boats are beautiful boats, they wouldn't want to own one because of the work associated with those boats. One might think that it's better to look at someone else's wooden boat than to own one. Many other sailors opt for the wooden boat; in this case, the practical value of fiberglass is disdained for the aesthetic value of wood. These sailors often spend much of the sailing season with their boats on dry land, working to make their boats more aesthetically pleasing. What they value is owning a beautiful boat, and they derive a great deal of pleasure and satisfaction from the labor expended to keep it that way. Other sailors prefer to be on a lower-maintenance, fiberglass boat. For them, it's more practical. They want to be on the water, sailing. The sailing is more valuable to them. They may still want a full-fledged aesthetic experience, but they want it – and find it – on the water. And this leads us to yet another example of something valuable related to sailing. It's not just the boats that are attractive but also the environment in which people sail. The upshot of all this is that, regardless of what one values, one can derive a great deal of satisfaction and find a tremendous amount of beauty, pleasure, and excitement in sailing. Sailing can be exhilarating and, at other times, relaxing. It can be intellectually or cognitively stimulating and it can be an activity via which to "zone out" and let the cares of life on land wash away. Sailing is indeed a pursuit chock-full of instrumental and intrinsic value.
# Aristotle, Virtues, and Flourishing
In a book called _Nicomachean Ethics_ , Aristotle focused on the concept of eudaimonia. This is Greek for "happiness" or, perhaps more accurately, "flourishing." Aristotle argued that, just as caring for an apple tree will help it grow, blossom, and bear fruit, certain ways of living one's life promote flourishing, and thus lead one to live a full and happy life. He claimed that one must _cultivate virtues_ in order to flourish in this way. But, you might be wondering, what exactly does he mean by "virtue"? Aristotle thought of virtues as patterns of behavior sort of like character traits. A virtue on this understanding is a tendency to act, desire, and feel in certain ways in certain situations. It is important to emphasize that Aristotle did not think that a virtue is an unthinking habit or tendency that one has. Rather, he thought that being virtuous involves some careful thought and results from a rational cultivation of habits that reflect appropriate response to whatever situation is at hand.2
Some examples will help to shed light on his theory. The list of virtues is quite large (and somewhat controversial). These seem like virtues: being clever, courageous, generous, hard-working, honest, kind, sympathetic, and wise. We might add things like candor, commitment, confidence, determination, enthusiasm, humility, integrity, tact, vigilance, and so on. Aristotle thought that each of the virtues are "golden means" between two vicious (as in "vices") extremes. For example, being generous is to give what is appropriate and not to be overly generous (such that one ends up in the poor house) and not overly miserly. So, each virtue involves responding appropriately to the relevant situation. Of course, whether or not we should characterize someone as kind or honest involves evaluating the person not just on the basis of one action but rather on the basis of how the person behaves over an extended period of time. In order for a person to be viewed as possessing the virtue of honesty, he or she needs to be honest on numerous occasions or, more generally, to have an honest character.
The upshot is this: the trick for leading a full and happy life is to live a virtuous life. The greater the proportion of virtues one exemplifies, the happier one's life will be. Indeed, Aristotle did not think that one just had to be, say, courageous and then one would be happy. Instead, one must cultivate and exemplify as many of the different virtues as one can. The virtuous person, then, is someone who has harmonized each of the virtues. That is, the virtues must be woven into the very fabric of a person's life.
# Is Sailing Virtuous?
The obvious question, then, is whether sailing is virtuous in Aristotelian terms. We think it is on many grounds, but will discuss only a few. Certainly, Aristotle's notion of the virtue of courage applies.
Courage is the mean between being foolhardy and being cowardly. A courageous person thinks, acts, and feels in the "right way" in response to danger. That is, the courageous person is not cowardly or overly concerned with his or her wellbeing. This is one vicious extreme regarding one's response to danger and so we rightly think that being cowardly is a vice. But the courageous person is also not overly unconcerned for his or her wellbeing. Being foolhardy or rash (the other vicious extreme on this scale) is also a vice.
So, the virtuous sailor is neither one who is overly afraid (cowardly) to go out in certain adverse conditions, such as a day when the wind is up a bit more than usual, nor one who is too eager to go sailing when a courageous sailor might wisely opt to stay ashore. This really applies if one is the skipper of a boat, and responsible for its crew and passengers and the boat itself. We all check weather maps before we go out. We check our equipment to make sure that we are properly prepared. We think about the sailing experience and the physical welfare of our guests. These are all factors that can ultimately place us in the position of the golden mean on the continuum between being afraid and being foolhardy. Interestingly, on the same day, it might be foolhardy to go out with one crew and unnecessarily fearful not to go out with another. This last point demonstrates the context sensitivity of what counts as displaying a virtue such as courage.
This all applies more obviously to blue-water sailing, where longer periods of time on the water are involved, where assistance might be less at hand, and where more trying circumstances might be encountered. And, sadly, the threats to sailors' safety are not just natural. There are pirates. So, is it foolhardy to go blue-water cruising or racing? We think of sailors in the Volvo Ocean Race or the Vendée Globe as courageous. Didn't Kipling title one of his books _Captains Courageous_?3 There are certainly many couples who cruise the world. But would it be fair to label others "afraid" for not wanting to go out in twenty-knot winds if they thought that it was beyond their abilities? Would they be guilty of being cowardly if they didn't go out? Would they be foolhardy if they did?
This essential question becomes more heated when we consider the case of young teens who want to circumnavigate the world solo. They're clearly not cowards, but are they foolhardy or courageous? We give acclaim to the young people who do it but castigate the parents of some youngsters who want to do it. When is young too young? How many people's scales of fear–courage–foolhardiness apply differently to young sailors according to gender? The point is that what is virtuous or vicious depends upon the context one finds oneself in and the sort of person one is – including things like what abilities one has.
Consider another virtue: being hard-working. Weber discussed this at length when he linked the rise of capitalism to the Protestant ethic.4 If the virtue of work lies between two extremes and one of them is laziness, we assume that the other is something akin to being a workaholic. Is the work of sailing virtuous? Do sail trimming and all of the other forms of work that go into making a sailboat move more efficiently or comfortably constitute an Aristotelian virtue? Haven't many of us heard sailors denigrate power-boating by saying that there is nothing to do except drive? Aren't we saying that we value the work of sailing, and that it makes our sport better? At the same time, we are also aware that we can regulate our workload. On a leisurely sail, we can "set it and forget it" – trim the sails, set the autohelm, and enjoy. If we're racing, that all changes dramatically. Here the workloads can be overwhelming to some. But, as sailors, we have the ability to choose the level of work that we want to perform. The whole point here is that sailing allows one to exemplify various virtues, including being hard-working. And, just like many things in life, one has to work at being a virtuous sailor. Most choose a level of work between being lazy and being a workaholic. The sailor who does it right reaps the rewards of being virtuous.
Aristotle also talks about the virtue of being clever. How many great sailing stories focus upon that virtue – that ability to solve problems with limited resources? As Captain Ron so prophetically said, "If it's gonna happen, it'll happen out there."5 It seems that power-boaters are more vulnerable to problems when they arise. Sailors seem to revel in the virtue of being able to adapt equipment to solve mechanical misfortune. All sailors are familiar with the idea of using a spinnaker pole as a mast or boom. One of us has had to transform an unused port frame into an alternator bracket. Is it not virtuous to be able to do this? We think it is.
More routinely, isn't navigation largely about being clever? How do we get from point A to point B in the fastest way possible? What is the weather forecast? What tides and currents are there? Where are they? In what direction is the wind blowing and how strong? What sails (and sizes) should be used ideally? Should they be shortened? What are the most efficient points of sail of our particular boat? The best answers combine together to produce the best possible sail. Depending on one's definition of "clever," we think that sailing requires people to be clever and to enjoy being that way.
# Is Sailing More Virtuous Than Other Pursuits?
One might contend that similar points can be made about other pursuits, such as power-boating, playing golf, or bowling. That is, one might think that these other activities similarly require one to cultivate, and allow one to exemplify, virtues – and so it's not clear that sailing is any better than these other pursuits in terms of contributing to a happy life. In response to this, we would urge that sailing requires a much broader range of virtues (if one is going to sail well) compared with these other types of pursuits. For example, one need not be courageous in order to bowl and golfers don't need to be able to cleverly jury-rig their clubs on the links.
In addition, these other pursuits typically don't require the exercise of as great a depth of skill or virtue as is required of sailors. Although being a good bowler requires some hard work and commitment, being a competent sailor requires a great deal more. Golfers might derive satisfaction from seeing beautiful greens, rolling fairways, and clouds on the horizon, but sailing has all of these aesthetic features and more. These differences are undoubtedly attributable to the fact that sailing is so much more complex and rich than these other sorts of activities. And this can be spelled out in terms of the multifaceted nature of the virtues (including cognitive abilities, decision-making, improvisational talents, emotional control, and so on) that need to be developed and executed by the proficient sailor. Lastly, in keeping with the points made above, one might consider the full range of activities in which one might engage, from those most benign to those considered "extreme" by being associated with the thrill of exposing oneself to significant risk. We would suggest that sailing fits nicely into this spectrum as a "golden mean" between these two extremes and so represents an ideal pursuit for cultivating virtues.
# Conclusion
In the preceding sections, we have discussed some of the virtues of sailing. There are clearly numerous others, and we'll leave it to the reader's imagination to think of all the ways in which that sailing can be a virtuous pursuit. Although it is certainly not the only activity that cultivates virtues, we think that sailing is an especially rich activity in this regard. That is, sailing is remarkably replete with opportunities to be virtuous. Sailing enables and even prompts one to be courageous, careful, curious, knowledgeable, decisive, and clever, and to have a whole host of other virtues. Assuming that Aristotle was right that one must be virtuous to be happy, it follows that sailing is a wonderful route to happiness. It's certainly not the only course, but we find sailing to be one of the best for those of us who are fortunate enough to have access to bodies of water suitable for sailing.
**NOTES**
1 William Frankena, _Ethics_ , second edition (Englewood Cliffs: Prentice Hall, 1973), pp. 87–88.
2 Aristotle, _The Nicomachean Ethics_ , trans. David Ross (Oxford: Oxford University Press, 1998).
3 Rudyard Kipling, _Captains Courageous_ (Garden City, NY: Doubleday, 1897).
4 Max Weber, _The Protestant Ethic and the Spirit of Capitalism_ , trans. Talcott Parsons (New York: Scribner, 1958).
5 John Dwyer, ___Captain Ron_ _, dir. Thom Eberhardt_ (Burbank, CA: Touchstone Pictures, 1992).
# PART 4
# PHYSICS AND METAPHYSICS FOR THE PHILOSOPHICAL SAILOR
**SEBASTIAN KUHN**
# CHAPTER 12
# DO YOU HAVE TO BE (AN) EINSTEIN TO UNDERSTAND SAILING?
# Introduction
Albert Einstein (see Figure 12.1) is widely admired as the smartest – and the "coolest" – scientist ever. After all, _Time_ magazine elected him the "person of the century" for the 1900s, and practically everyone knows his name (and associates it with "smart"). Among his many endearing traits, his love of sailing ranks right up there as a measure of his greatness. His groundbreaking discoveries, the theories of special and general relativity, revolutionized our understanding of space and time and yielded the universally recognized equation _E_ = _mc_ 2. But do _you_ have to be an Einstein to get to a full and deep understanding of sailing? I mean this not in the usual sense ("being smart") but quite literally: do the laws of special and general relativity have any bearing on the ancient art (and modern practice) of sailing?
If you restrict yourself to practical matters (how to get from A to B in the fastest and safest way by use of a sailing vessel, or how to win the weeknight race around the cans), Einstein's theories of relativity have little immediate value. Ostensibly, this is so because they mostly concern themselves with extreme situations – speeds close to the speed of light (not exactly what comes to mind when describing the average cruise) and bizarre aspects of the universe such as black holes and the Big Bang. A closer look, however, reveals that the philosophical underpinnings of Einstein's insights have a lot to do with the basics of sailing. Pondering these underpinnings may not necessarily make you a better sailor but hopefully will provide something worth mulling over while ghosting along in a zephyr (or while eagerly awaiting the return of fair sailing weather in the middle of winter). You will find that, maybe without realizing it, you are making use of "relativity" in one way or another every time you weigh anchor. Even the general theory of relativity, long considered the most arcane of Einstein's ideas, has a direct impact on something as mundane as navigation. In this article, I hope to provide you with some guideposts along the path from everyday concepts familiar to most sailors all the way to Einstein's take on issues such as time, space, and motion.
**FIGURE 12.1** Einstein in his sloop _Tümmler_ , a gift from well-to-do friends that was custom-built for him. Photo used by permission of _Die Yacht_.
# Don't Laugh at "Slow" Sailing: Average Versus Instantaneous Motion
A well-known jibe defines sailing as "the art of going slowly nowhere, at great expense and personal discomfort." And, sure enough, if you discount tricked-out Open 70s, America's Cup trimarans, and novelties such as the flying moth, most leisure sailors with limited pocketbooks rarely crack the ten-knot barrier. Conversely, our power-boating friends easily reach twenty, thirty, or even forty knots on the water. No wonder they like to poke fun at the comparatively slow progress of a typical sailing cruiser. However, it all depends on your definition of velocity – if you aren't talking about instantaneous speed, but average velocity, we sailors have no reason to feel inferior.
By definition, average velocity takes the total displacement during a given time interval and divides it by the amount of time elapsed. ("Displacement" is a physics term here, not a nautical one. It means the distance between the start and finish position.) By this reckoning, all weekend cruisers, whether laid-back sailors or high-speed motorboaters, have the same average velocity – namely zero! This is because they tend to end up right where they started – in their marina slip or on the boat trailer from which their vessel was launched. In other words, the total distance between start and finish is zero. And zero distance divided by _any_ number of elapsed hours yields zero velocity.
If we average over longer and longer time spans, sailors do comparatively better and better with respect to average velocity – few power-boaters venture on long blue-water cruises to distant locales, while many a sailor has traveled "over the horizon" to fulfill a life-long dream. This distinction is not just an accident – the energy source for the propulsion of a sailboat is the ubiquitous wind, which makes possible unlimited travel for years, while the fuel contained in the tank of an ordinary powerboat usually can't get it across a major ocean. (Full-displacement boats such as trawlers come closest in long-distance cruising capability, albeit at the price of nearly "sailing-like" speed.)
Still, whether averaged over long time periods or measured instantaneously, sailboat velocities are hardly in the range that physicists call "relativistic." The most counterintuitive effects of special relativity (time dilation, length contraction, and so on; see below) only become obvious when speeds approach the universal limit of 300,000 km/s (583 million knots) – the speed of light. The speed of even the fastest powerboat pales in comparison. But read on! The connection is real, even if it is more subtle.
# Motion Relative to What? – Galilean Relativity
So far, I have been rather casual in my use of terms such as "distance" and "displacement." As a physicist, I need to define my terms more precisely. Without that, it is not even clear what we mean by our position at a given time, let alone the distance between two positions. (For now, I will pretend that the other variable entering into velocity, elapsed time, is clearly defined – after all, we have high-precision clocks and watches to measure it. But, as we will see later, this is not quite unproblematic either!)
For physicists, the fundamental concept needed here is that of a "reference frame." We can visualize this as an infinite frame, like the frame of a house, made of wooden beams, all rigidly attached at ninety-degree angles to each other. A more abstract concept would simply assume a set of points in space, all with fixed and immutable distances from each other. One of these points is given a special importance – we call it the "origin" of the reference frame, and designate it with "0" distance. Any position in space can then be uniquely designated by giving both its distance from that origin and the direction (relative to the "beams") in which you would have to move to get there.
In one dimension, the nautical equivalent would be a set of mile markers; for example, the ones along the Intracoastal Waterway along the US East Coast, with the origin designated by "mile marker 0" in Portsmouth, Virginia. Snowbirds who want to know where they are during their annual journey south on the Intracoastal Waterway can simply look at the nearest mile marker. In three dimensions, however, things are a little more complicated. For sailors, two dimensions usually suffice, and the system of latitudes and longitudes serves the same purpose as the mile markers in the one-dimensional example (with the origin on the intersection of the Greenwich Meridian and the equator, in the Atlantic south of Ghana). Note that this latter coordinate system is a bit tricky – for instance, you cannot easily calculate the distance between two points simply by knowing their longitudes and latitudes, because you need to take into account the curvature of the surface of our planet. Similarly, it is not obvious what path to follow for the shortest possible distance between two points. If you start from the East Coast of the United States and want to reach a point exactly due east on, say, the European continent, it turns out that following a route due east at constant latitude (the rhumb line) will _not_ give you the shortest distance – that's why all transatlantic flights tend to cross over rather northerly places such as Greenland. This complication of ordinary Euclidean geometry by Earth's curved surface has a direct analog in Einstein's general theory of relativity, in which the whole of four-dimensional space-time turns out to be curved (see below).
It is important to realize that any "fixed" reference frame is only fixed _relative to something else_ , and in fact the choice of reference frame is quite arbitrary. For instance, a reference frame "fixed" to the surface of the patch of water on which you happen to sail is obviously of significant importance – after all, without any propulsion by wind or iron genny you will be at rest relative to that reference frame. The faster you want to go relative to the surrounding water, the more force needs to be brought to bear on your vessel. And, as sailors know, it is the motion of the surrounding air relative to _this_ reference frame that determines this force. If you want to move in the direction the wind blows you, your speed will depend directly on the wind velocity relative to the water. On any point of sail other than straight downwind, you need the balance of two forces to give you net propulsion in the forward direction (see also Chapter 13). One force is generated by the motion of air relative to your above-water foils (the sails) and the other by the motion of water relative to your underwater foils (keel or centerboard and, to a lesser extent, rudder). Successful propulsion requires that these two relative motions are not the same. If they are, all you get is drag, which will bring the boat to rest relative to both air and sea in short order.
The water surface is just one reference frame to keep in mind while sailing. If you plan to arrive at a certain destination (within some given time frame), you are obliged to pay close attention to the other reference frame we already encountered – the one fixed to the solid surface of Earth. Even if you really don't care where wind and waves carry you, you still need to keep track of where you are relative to this reference frame – if only to avoid shoals or other obstacles to safe navigation. And the number of possible (and relevant) reference systems doesn't stop there – your own boat is an obvious third example, which you automatically invoke when talking about the direction of another vessel ("abeam," "abaft," or "ahead"), or when you tell your crew to "go forward to raise the jib." Finally, if you are a blue-water cruiser from the old school (or just like to keep your options open), you will even have learned to consider the reference frame of the stars, which can help you to navigate on the open ocean. And, compared to _that_ frame, both the most languid sailboat and the most souped-up powerboat move with velocities that are quite impressive – and practically indistinguishable (thirty kilometers per second, or 58,000 knots if you count the sun as the origin of this frame). Keep in mind that none of these various reference frames is intrinsically any more correct or fundamental than any other one; they are all equally valid, if not necessarily equally practical, for a given purpose.
Awareness of these reference frames and their relative motion helps to explain many common situations encountered while sailing. Examples abound: While the wind may truly come out of the west, if you are moving due north, it will _appear_ to come somewhere from a northwesterly direction relative to your boat, and this relative (apparent) wind direction determines the shape and position of your sails. Vice versa, if you and your boat are floating down a river, with a breeze of exactly the same velocity as that of the water, you will feel completely becalmed – the _relative_ wind velocity is zero. So, your velocity relative to the water would be zero, as well, but you would be moving at a good clip over ground. Inversely, if the wind velocity _relative to ground_ were zero, you would notice a breeze blowing up the river, and if you wanted to speed up your trip you could even use this breeze to tack back and forth, making way downstream relative to the flow of the river (and therefore be even faster than the river). Conversely, if you have both current and wind against you, even as an excellent sailor you may not be able to make any headway above ground!
In the end, one has to keep at least three or four of these different reference frames in mind to fully understand the magic of sailing. And one should be able to easily convert positions and velocities measured in one system to any of the others. In principle, this problem was solved a long time ago, by Galileo Galilei and other early giants of physics. Not only were they able to derive the laws describing the transformation of positions and velocities from one frame to another (e.g., velocities must be added as vectors) but they also realized an enormously important principle (nowadays called "Galilean relativity"): The laws of nature (for instance, Newton's famous three force laws) do not change no matter which (inertial1) reference frame one chooses to describe one's observations. While the velocity measured relative to one frame may turn out to be quite different from that relative to another one, its rate of change (the acceleration) is the same in both of them, and directly proportional to the net force acting. This principle has allowed us to make sense of the world around us, to send space probes to the planets, and to devise many of the wonders of technology. The principle is perfectly adequate to understand the basic laws of sailing (even how to design a faster racing yacht). It has just one small flaw: Unbeknownst (and unknowable) to Galileo and his contemporaries, it is woefully incomplete, and the rules for transforming velocities from one system to another are simply (if subtly) wrong!
# But There are No Fixed Reference Frames – Special Relativity
The trouble with Galilean relativity is not that it is inherently wrong – quite the opposite: It is not encompassing enough. What Galileo couldn't have known is that the laws governing electric and magnetic interactions (i.e., electromagnetism, which hadn't been discovered yet) must be the same in all reference frames – but they directly contradict the simple rule for the addition of velocities. Let me explain.
Electromagnetic phenomena are of course a constant companion of any mariner, sailing or otherwise. Magnetic compasses have been crucial navigation tools for centuries, and the power of electric currents in the form of lightning strikes has inspired awe and fear in seafarers from the ancient past to this day. Modern sailboats typically make much more comprehensive use of electromagnetic processes (depending on how much money the owner likes to spend on electronics). From the electrical system powering navigation and cabin lights (and windlasses, microwaves, air conditioners, and other "necessities" of cruising life) to the multiple receivers and emitters of radio waves (VHF, radar, GPS, and cell and satellite phones), electromagnetism seems indispensable for most sailors. However, only in the nineteenth century was a complete and unified understanding of all the different manifestations of electricity and magnetism accomplished, with the crowning achievement of the four differential equations now named after Maxwell. Among the astonishing consequences of these equations, scientists were able to _predict_ that varying electric and magnetic fields should be able to propagate together through vast expanses of space, traveling in the form of electromagnetic waves (just like water waves travel on the open sea). Even more astonishingly, the speed of propagation for these waves turned out to be a number already fairly well known – 300,000 km/s, the speed of light. This insight not only explained light as just one more electromagnetic phenomenon but also predicted the existence of, and subsequently allowed the generation of, a vast array of other electromagnetic waves – radio, radar, infrared, and ultraviolet up to X-rays and gamma rays. All of these waves were predicted to propagate with the same speed. Subsequent experiments confirmed these ideas with increasing precision, and many ingenious devices (such as the ones mentioned above) are based on them.
There is only one problem. As we've just learned, velocities must always be measured relative to a reference frame, and the result should change if we change reference frames. Surely, if light travels at 300,000 km/s relative to the frame of a fixed star, an observer on Earth (using a ground-based frame) should measure either 300,030 km/s while Earth is moving toward that star (in spring, say) or 299,970 km/s in the opposite case (in fall). This is totally equivalent to the head wind one would experience in our earlier example of a sailboat floating on a fast river during a calm day. But the laws of electromagnetism don't allow such a change of wave velocity with the reference frame – and all experiments devised to find it have failed. No matter where the light comes from, and no matter your own motion, the velocity measured always turns out to be exactly the same: 300,000 km/s. If you don't find that weird, you haven't thought this through. The radio wave carrying a Mayday call arrives at your boat with exactly the same (relative) velocity, whether you are moving at a good clip or are at anchor. The same is true with the light from navigation aids, the radar beam that gets bounced back to you from an obstacle, and any other electromagnetic wave. Even the signals from the GPS satellites that help you to determine your position arrive with that same unchanging velocity, no matter how fast those satellites are moving relative to the ground (in fact, several kilometers per second). This is the same as if a fifteen-knot breeze from the north were to always feel exactly the same to a sailor, whether she were sailing close-hauled, at a beam reach, or downwind! If the laws of physics (including electromagnetism) really are the same in all reference frames, something else has to give. That something turns out to be our very notion of time and space. To understand why, we have to go back to our definition of a reference frame.
Remember the blithe assumption that one can measure positions and distances relative to a reference frame by using "rigid wooden beams" or "fixed markers"? In fact, the first idea is utterly impractical for obvious reasons, while the second has more subtle flaws. Sailors understand more readily than landlubbers the ultimate impermanence of things that seem to be quite solid: Channel markers can be moved or succumb to waves and weather, floating markers can move, sand and clay bottoms can shift, and so on. On a long enough time scale, the inexorable drift of the continental plates means that literally nothing stays in exactly the same place (even if measured from one "fixed" point on Earth to the next). And Earth is the most permanent frame there is – how about the reference frame of the water itself? Currents not only change with time and tides but also vary from one location to the next. What is fixed here relative to what? Does it even make sense to talk about _the_ reference frame of the water surface? Ultimately, which method of measuring positions (and times!) can be considered truly reliable?
When Einstein pondered this conundrum, he realized that the apparent _problem_ in the laws of electromagnetism actually yielded the (only) solution: Once we _know_ that the speed of light (or other electromagnetic waves) is the same under all circumstances, we can use this fact to measure the distance between any two points simply by measuring the time light takes to travel from one to the other. To determine a three-dimensional position in space, in general one needs to measure the distance from (at least) three points with known coordinates. If each of these three points were to send out a short light pulse simultaneously, we would only have to measure the arrival times of these three pulses to fix our position. There is one remaining problem, though: How can we be sure that our own (on-board) clock is exactly synchronized with the clocks that determine the start time of the three light pulses? Otherwise, we couldn't reliably measure the travel time. Well, it turns out you _can't_ be sure unless you include a _fourth_ sender at yet another position. Admittedly, this is getting a bit complicated, but I hope it makes sense: To determine the four coordinates of a position in space and point in time simultaneously, you need four pieces of independent information – which the arrival time of the four light pulses can provide.
In the early 1900s, when Einstein was developing these theories (while working at a Swiss patent office), this whole arrangement of light senders and receivers would have appeared extremely contrived and utterly impractical – for one, to get good accuracy of position you would have had to measure the elapsed times to a precision of the order of one billionth of a second (one nanosecond)! This is because light travels so insanely fast – by a whole foot in that one nanosecond. Such precision was completely out of reach at the beginning of the twentieth century. Einstein therefore considered this method of measuring position and time as a purely mental exercise – a _Gedankenexperiment_. Amazingly, a hundred years later his method is exactly how most of us measure where we are during our sailing trips (and even on the car ride to the marina)! And all of the complicated calculations necessary can be done by a little electronic box no larger than a transistor radio – the GPS receiver, which requires only a modicum of operator training (as compared to celestial navigation). Of course, the details are quite a bit more complicated than I could possibly explain here, but the basic idea is sound: By measuring the time it takes for electromagnetic waves (radio, in this case) to reach the GPS receiver from any of a number of satellites that are constantly sending them out, we can not only calculate our position (and plot it on an electronic chart) but also calculate the actual time of day (your GPS knows it a lot better than your beautiful brass clock down below!).
But Einstein's idea goes way beyond a convenient new navigation method – he realized that this method (and assuming a fixed velocity of light) are the _only_ reliable way to synchronize clocks and measure distances. This has profound consequences that lead to all the confusing paradoxes that come with his special theory of relativity: First, the length of an object contracts if measured from a reference frame in which it moves. For example, your boat, as measured from shore, is shorter than its nominal overall length, if only by one trillionth of one-thirty-second of an inch (assuming a seventy-seven-foot yacht moving at eight knots). Second, time appears to pass more slowly for a clock that is moving relative to the observer (time dilation). After a twenty-four-hour day of sailing at those same eight knots, even the most perfect onboard clock will show eight trillionths of a second less elapsed time than an onshore one. Third, the notion of two things happening simultaneously no longer can be assumed to be universal. Ultimately, this last consequence is the biggest jolt to our understanding. It is also the reason why the other two only appear to be paradoxical.
To understand this, imagine yourself exactly at the middle point between two GPS satellites (let's pretend for now that they are at rest relative to Earth, which is of course far from true). Both of them send out a radio pulse at exactly the same time. If you are at anchor, you will receive those two waves at exactly the same (later time), and, since you (or, rather, your GPS receiver) know about the relative position of your boat and those satellites and the constant speed of radio waves, you can indeed conclude that they must have been emitted simultaneously. However, what happens if you are making eight knots toward one satellite and away from the other? Since it takes a little while (a fraction of a second) for the radio waves to reach your position, you will have moved away from the point exactly in the middle of the two satellites and will have moved closer to one than the other by the time you receive the pulses. However, since you must assume that the speed of radio waves is still 300,000 km/s in either direction, you will have to conclude that the satellite ahead of you emitted its signal a smidgen earlier than the one behind you. And you are not wrong with that conclusion! The whole notion of simultaneity depends on your reference frame, and a moving boat has a different one from a boat at anchor.
Of course, I hasten to add that all of these seemingly paradoxical consequences are miniscule in magnitude unless your boat is going close to the speed of light, so you can safely ignore them while cruising. However, without careful consideration of special relativity, your GPS system wouldn't work. And there are other consequences of Einstein's discovery: The mass ( _m_ ) of an object is not a constant, but depends on its state of motion and, more generally, its energy ( _E_ ): _m_ = _E_ / _c_ 2. (If you remember a little algebra, you will quickly realize that this equation has the exact same meaning as the famous one quoted in the introduction.) Again, the increase resulting from this relationship to the mass of a five-ton yacht underway is so miniscule to be entirely negligible. However, only recently a group of physicists made the stunning discovery that, quite apart from nuclear power and other more well-known applications of this famous equation, even such mundane devices as lead batteries (ubiquitous in today's yachts) would not work, or at least not as well without these relativistic effects.2
# General Relativity – Can it Really Matter?
It took Einstein only a few years to develop his special theory of relativity. However, his greatest achievement (and arguably the greatest achievement of any single scientist, ever), the general theory, occupied him for a decade. To this day, many people (though fewer and fewer physicists) consider this ultimate theory of gravity to be beautiful but thoroughly mysterious and utterly irrelevant for daily life (including sailing). But, once again, it turns out that, while its consequences are miniscule, they are not exactly null, and they are becoming more and more measurable. As philosopher-sailors, we shouldn't be content ignoring something just because it is incredibly small – maybe we should be more like obsessive racers, who will fret over the tiniest barnacle adhering to the otherwise slick underbody of their sailing machines. The plain truth is that even the special theory of relativity has shortcomings.
I alluded to one problem already earlier: the ambiguous nature of notions such as distance and direction if your reference frame is not strictly rectilinear and orthogonal ("flat" in geometry parlance). As sailors, we are familiar with one such coordinate system, that of longitudes and latitudes on the surface of the (emphatically not flat) Earth. We have already discussed some of the non-intuitive consequences of measuring positions and directions using these coordinates. We could of course avoid these problems by using an ordinary, three-dimensional (rectilinear or Cartesian) coordinate system instead – breaking down any position into distances and directions measured relative to the center of the Earth. Such a system would regain the ordinary definitions of direction and distance for the price of awkwardness – you could not even tell immediately whether any position given within this system was on the surface of planet Earth or hundreds of meters in the air (or below the water).
What the general theory of relativity tells us, though, is that the very space itself that we take for granted (and time with it) cannot be described in terms of a purely flat geometry, even if we try. Space-time is intrinsically warped or curved by the presence of mass and energy. It is this warping that leads to all the well-known effects of gravity (e.g., falling and buoyancy) as well as the more exotic consequences of Einstein's theory – black holes and Big Bang cosmology. And it has consequences for our very notion of distance and direction that cannot be cured by changing to a different coordinate system.
As an example, consider two ships starting out simultaneously (with the same speed) from the equator, separated by sixty nautical miles and moving exactly parallel toward the north (along fixed lines of longitude – say 0 and 1 degrees). Our everyday notion of parallel lines would imply that these two vessels will never be in danger of collision – but, unfortunately, on a curved space (Earth's surface) this isn't true: both will eventually reach the North Pole (ignoring intervening land masses) and therefore arrive at the same point at the same time – the definition of a collision. Similarly, light rays from one and the same distant star can bend left and right around a massive object (say, a galaxy cluster) that they have to pass on their way to an observer on Earth. This can lead to that same star appearing at two different points in the sky at once, an effect called "gravitational lensing." (For an illustration, think of two ships starting in different directions from the South Pole and reuniting at the North Pole.) Yet both light rays (and both ships) simply follow the shortest possible path in a curved geometry – what we would normally call a straight line.
And it is not only space that is affected by this "irreducible" curvature; time also becomes warped in a sense. For instance, it turns out that time elapses (slightly) more slowly close to a massive object than further away from it. Again, the effect is small, but with an ultraprecise pair of clocks it has recently been demonstrated for a change of only one foot in height above Earth (the massive object in this case). As a consequence, I can rightfully claim that you will age ever so slightly more quickly while aloft on your mast than down in the cockpit – although only by one trillionth of a second for each quarter hour. And, once again, it turns out that the effect is big enough to be corrected for when using GPS satellites – thousands of miles above our heads – to fix our position on Earth, where time is elapsing just a tad more slowly. So, in the end, it turns out that Einstein does have something to tell us about our favorite pastime – maybe one of the reasons he enjoyed sailing.
**NOTES**
1 Technically, inertial reference frames are those special frames where an object at rest will remain at rest as long as no force acts on it. Most (or all) of the frames we have considered so far are not exactly inertial ones – which explains the sloshing back and forth of water in the bilge or of the tides on the ocean. However, this distinction is of minor practical importance for our purposes, and turns out to be illusory anyway in light of the general theory of relativity.
2 If you want to know the details: The increase in mass of the electrons in lead atoms due to their very high motional energy as they swirl around the nucleus leads to stronger binding, which in turn increases significantly the energy released by moving these electrons from cathode to anode – in other words, the battery voltage.
**JOHN D. NORTON**
# CHAPTER 13
# PARADOXES OF SAILING
# The Physics of Sailing and the Import of Thought Experiments
Paradoxes have long been a driving force in philosophy. They compel us to think more clearly about what we otherwise take for granted. In antiquity, Zeno insisted that a runner could never complete the course because he'd first need to go half way, and then half way again, and so on indefinitely. Zeno also argued that matter could not be infinitely divisible, else it would be made of parts of no size at all. Even infinitely many nothings combined still measure nothing. These simple thoughts have forced us to develop ever more careful and sophisticated accounts of space, time, motion, continuity, and measure. And modern versions of these paradoxes continue to vex us.
This engine of paradox has continued to power us to this day. Relatively recently, Einstein fretted over a puzzle. How was it possible that all inertially moving observers would find the same speed for light? Surely if one of them was chasing rapidly after the light that observer would find the light slowed. But Einstein's investigations into electricity and magnetism assured him that the light would not slow. He resolved this paradox with one of the most influential conceptual analyses of the twentieth century. He imagined clocks, synchronized by light signals, and concluded that whether two events are judged simultaneous will depend upon the motion of the observer judging.
What is distinctive about these philosophical paradoxes is that they are not mere expressions of practical limitations. The difficulties they expose lie within the very ideas themselves. Zeno's worry was not that a real runner might fail to complete a long race because of tiredness. His concern related to the very idea of any runner, no matter how accomplished or idealized, completing any race, no matter how short. The difficulty lies in the ideas of space, time, and motion. In his paradox of measure, Zeno was not concerned that we might never find a real knife capable of slicing matter indefinitely finely. His concern was that matter must be such that infinite division lies beyond even the sharpest knife, whose edge has been honed to the perfection of an ideal mathematical point.
Now let us consider sailing. There are many difficult technical problems associated with sailing. If a sailboat is to be even minimally serviceable, its design must conform to an engineering lore that has grown through the centuries. In general, the problems this tradition solves do not rise to the level of paradox. However, there are some puzzles attached to sailing that are more fundamental than a particular engineering challenge. In this essay I will consider three. They do not have the importance of the paradoxes of Zeno and Einstein. Indeed, as I shall try to show, their diagnosis and resolution is a short and, I hope, entertaining diversion. However, they are foundational paradoxes, for they challenge no particular sailboat but the very idea of sailboats powered by the wind. They are:
* If a sailboat is powered by the wind, how can it sail into the wind?
* If a sailboat is powered by the wind, how can it sail faster than the wind?
* If a sailboat "makes its own wind" when it moves, why does it need any other wind?
The first two will be familiar to sailors, most of whom will have made their peace with them. The third is less straightforward. That sailboats "make their own wind" is commonly said by sailors, but few explicitly pursue the thought to its paradoxical end. We shall do so now.
While profound philosophical morals will not be found in these paradoxes, I will suggest that they connect nicely to two issues in recent philosophy. The first two paradoxes will lead us directly into a conundrum concerning causal metaphysics. The third will lead us to ponder an intriguing mode of investigation of nature, the thought experiment.
Here are the first two paradoxes again, spelled out in greater detail. They are treated together since they involve essentially the same issues.
1. The cause of a sailboat's motion is the motion of the wind.
2. The effect of a cause cannot be greater than or contrary to the cause.
3. Therefore, a wind-powered sailboat cannot sail faster than the wind or into the wind.
But this conclusion is contradicted by the reality:
4. Real sailboats routinely sail into the wind, and sailboats designed for speed can sail faster than the wind.
Those who are not sailors will likely find the argumentation leading to the conclusion (point three) convincing, at least initially. Its plausibility depends upon a limited experience of what the wind can do. It calls to mind dry leaves blown about by wind. The wind may lift them, but it will not move them faster than or contrary to its own motion. The conclusion is also correct for some cases of sailing, such as old-fashioned square-rigged sailing ships running before the wind. Then their sails function like big bags catching the wind. The boat will be blown in the direction of the wind, near enough, and, as long as it sails in that direction, the ship will never move faster than the wind.
However, the conclusion is incorrect for almost any sailboat that can align its sails in a fore–aft direction. This is especially so for the most common type of small sailboat now used recreationally. A Bermuda-rigged sailboat has a single mast with two triangular sails, a jib and a mainsail, oriented in the fore–aft direction. Such sailboats routinely sail into the wind, and, if designed for speed, easily sail faster than the wind when sailing across the wind.
Where the above analysis (points one to three) fails for such boats is that it mischaracterizes the causal processes. The motion of the wind is not the immediate cause of the motion of the boat. A more immediate cause is the force with which the wind presses on the sails. For even light winds, this force can be considerable. In what are called "moderate breezes" on the Beaufort scale of wind (thirteen to seventeen miles per hour), the wind generates pressures of around one pound per square foot on the sails. Small modern sailboats, under twenty feet in length, can carry two hundred square feet of sail, and older designs often carried significantly more.1 So, the wind exerts a considerable force of many hundreds of pounds on the sails. This force now acts independently of the motion of the wind that produced it. A few hundred pounds of force pressing on the sails will lead the sailboat to heel over, just as if someone were to attach a rope to the center of effort of the sail and pull it.
Some of this force can be directed toward the bow of the boat and drives it over the water. How much boat speed results from a given force depends almost entirely on the design of the boat's hull and, as a result, the resistance the water provides to its motion. Once that forward-directed force is fixed, so is the motion of the hull. It makes no difference if the force comes from the press of the wind, oars, and paddles or a motor-powered propeller. The force contains no coded record of the speed of the wind that produced it for the sailboat to read covertly and respect!
A small boat of the familiar monohull design can easily be driven up to a maximum speed that cannot be passed by greater forces generated by sails. For small boats this maximum speed is commonly less than the speed of the wind. But that is purely an accident of the hull design. If the hull is designed for speed, nothing prevents the boat achieving speeds greater than the wind. Two-hulled catamarans present considerably less resistance that monohull boats. If sailing across the wind, they do not lose the press of the wind when they move fast. Then, well-designed catamarans are easily able to sail faster than the wind. The wind can provide considerable force; their hulls provide little resistance, so off they go!
To see how a sailboat can gain against the wind, we need to consider the different "points of sail" of a sailboat. These are the different ways in which a sailboat can proceed in relation to the wind. They are shown in Figure 13.1.
When a sailboat is on a run, the wind blows directly from its stern. Then the sails function like bags just catching the wind. On this point of sail the fastest the boat can move is the speed of the wind. As the boat approaches the speed of the wind, the boat's motion cancels out the speed of the wind, so that the wind felt on the boat by the sails diminishes. When the boat is close to the speed of the wind, the air on deck becomes calm. The experience is not unlike being carried by the wind in a balloon. One's speed over the ground may be quite high, but in the balloon's basket the air will be still.
All this changes when the boat sails across the wind on a beam reach. On this point of sail, the sails are let out so that they deflect the wind toward the rear of the boat. The resulting pressure on the sails yields a force, "Fwind," pointed diagonally forward, as shown in the first diagram of Figure 13.2.
**FIGURE 13.1** Points of sail. Author's copyright.
**FIGURE 13.2** Resolution of forces on the sail. Author's copyright.
If the boat's hull was simply a tub, then this force would move the boat in that diagonal direction. However, an essential part of hull design is to make it as resistant as possible to sideways motion. This is usually effected with a centerboard in small boats and a broad, flat keel in bigger boats. The force on the sails, Fwind, can be divided into two components, as shown in Figure 13.2. One, "Fdrive," is parallel to the boat's motion, and the other, "Fbeam," is perpendicular to it. The high resistance to sideways motion means that the sideways force, Fbeam, produces little or no motion, whereas the low resistance to forward motion means the forward force, Fdrive, produces motion forward. Hence the boat is driven across the wind.
Only a small modification to the above analysis shows how sailboats can sail into the wind. When a sailboat is close-hauled, as shown in the second diagram of Figure 13.2, the wind still produces a force on the sails. That force, Fwind, can once again be decomposed into two parts, Fdrive and Fbeam. Since the sails are now pulled in closer to the centerline of the boat, the component Fdrive is smaller in relation to Fbeam. However, the hull will still prevent Fbeam producing sideways motion, so that Fdrive will drive the boat forward.
This forward motion will now gain against the wind. It is common for Bermuda-rigged sailboats to be able to sail at 45 degrees to the wind. As a result, if a close-hauled sailboat tacks repeatedly – that is, zig-zags across the wind – it can follow a track whose average course points directly into the wind.
In sum, the first two paradoxes are resolved by denying the second premise, that the motion of the wind, as a cause of the motion of the boat, cannot have an effect greater than or contrary to itself. When powering a sailboat, the motion of the wind can produce faster motions in the sailboat and motions directed against the wind.
To a philosopher, what is important in this last analysis is the centrality of causal notions. In the abstract, it seemed entirely unremarkable to expect that the effect of a cause cannot be greater than or contrary to the cause. Yet this simple causal truism was wrong and generated the first two paradoxes.
We see here in miniature one of the dominant and, in my view, most important facts about our investigations into causation. At any moment in history, we have held to a repertoire of facts about causation that we believed to be necessities. They are assertions that, shielded from deeper reflection and a broader exposure to experience, seem unassailable. However, when we think more and learn more about the world, we find we must abandon them.
Until the seventeenth century, it was widely accepted on Aristotle's authority that a final cause, the goal toward which a process moved, was as important as the efficient cause, that which initiated the process. In that century, the advent of mechanical philosophy was premised on the denunciation of final causes. However, we had by no means then "got it right." The century's hero, Isaac Newton, felt he had such an unassailable grasp on causation that he could, in 1692, denounce causal action at a distance as "so great an absurdity, that I believe no man, who has in philosophical matters a competent faculty of thinking, can ever fall into it."2 Yet, by the nineteenth century, Newton's gravitation was widely accepted to be precisely this, unmediated action at a distance. In that century, the notion of causation was stripped down to its barest essentials. It came to be equated with determinism, the simple fact that the present state fixes the future. This pure and apparently secure notion of causation fell. It was overturned with the advent of modern quantum theory in the 1920s. According to that theory, the present cannot fix the future. The best we can have are probabilities for a range of different futures.
These are just a few episodes in the history of our failure to grasp what causality demands. It is important that we see just what this failure has been. It has not been our failure to discern what has always concerned causal thinking: how it is that things in the world are connected. The little history just unfolded is a story of our coming to understand better and better how things are connected. The failures of the story were our efforts to discern ahead of science what sorts of connections new science must reveal.
How are we to interpret this long history of failure? There are causality optimists who think that the best response to failure is to try and try again. Eventually, they hope, we will hit upon the true causal principles that govern the world and all possible sciences. My own view is rather different and represents a minority view among theorists of causation.3 It is that we need to learn that efforts to legislate causal principles ahead of experience are doomed to failure at the hands of new investigations.
As a result, I believe that the familiar causal talk is very different from what it seems. One could be forgiven for imagining that science is exploring a realm governed by some general law of causality that rules from the metaphysical heights above all sciences and to which all sciences must defer. In my view, something like the reverse is correct. Science is revealing to us deeper truths about the interconnectedness of things in the world than we could have ever imagined. In order to facilitate our understanding of it, we graft causal talk onto those discoveries. The repeated cycle of the failure and revival of causal talk is really a history of the elasticity of causal terms and our eagerness to apply them to whatever science may deliver. We do not have and will never have a factual principle of causality to which all sciences, known and as yet unknown, must conform.
Sailors commonly remark that sailboats create their own wind. The effect is a familiar one. If you pedal a bicycle at 10 mph on a calm day, you will find yourself pedaling into a 10 mph headwind created by your motion. Exactly the same thing happens with a sailboat. A sailboat travelling at 10 mph is sailing into a 10 mph wind it has created. Of course, sailors never see this headwind in isolation. The wind they see, the apparent wind, is always the vector sum of the created wind and the true wind. So, if the sailboat is on a beam reach in 10 mph winds, the two winds combine to yield a 14 mph wind coming at an angle of 45 degrees to the bow. The two velocities are at right angles and so must be summed by Pythagoras' theorem:
**FIGURE 13.3** True and apparent wind for a close-hauled sailboat. Author's copyright.
Figure 13.3 shows how the true wind and apparent wind are combined when a sailboat is close-hauled. It will be important for later discussion to note that the effect of the created wind is to move the direction of the apparent wind closer to the bow.
Thus far we have no paradox. By their motions, sailboats create wind. Our quest is for paradoxes and there does seem to be an intriguing paradox lurking in shadows. It arises from the essential difference between the cases of a bicycle and a sailboat. A bicycle is powered by your muscles; a sailboat is powered by the very thing created, the wind. Here is the paradox:
5. A moving sailboat creates its own wind.
6. A moving sailboat is powered by the wind.
7. Therefore, a moving sailboat is in part self-powered and is thus, in part, a perpetual-motion machine.
But this conclusion is contradicted by the following:
8. Perpetual-motion machines (self-powered devices) are impossible.
The concern is that a sailboat is, in part, realizing a device whose impossibility underlies one of the most important laws of physics, the conservation of energy. For it appears to be achieving just what perpetual-motion-machine makers have long sought. Their goal is a device that derives the power to run from its own internal operations. They have tried many designs. For example, they have equipped an electric car with a generator so that as the car moves the generator is turned. The generator produces electricity that, supposedly, now fully powers the car's electric motor. This simple design and its thousand and one variants have all failed. Is a sailboat the thousand and second variant that has finally succeeded?
It is not too hard to see that the traditional design of a sailboat acquires no added motive force from the created wind. Qualitatively, the result comes from combining effects that work in opposing directions. The force that drives the sailboat comes from the speed of the wind over the sails. So, an increase in the speed of the wind over the sails will increase the force on the sails. It doesn't matter whether the wind is the true wind or the apparent wind. Sails cannot distinguish the two. The force on the sails is determined by the speed of the wind at the sails, however it arises.
If that were the only effect, then we would be well on our way to realizing the paradox just sketched. However, there is a counteracting effect. As Figure 13.3 shows, the effect of adding the created wind to the true wind is to move the direction of the wind closer to the bow. As a result, the angle between the wind direction and the sails decreases; the wind now comes closer to blowing parallel to the sails' surfaces. This diminished angle reduces the wind-generated force on the sails in two ways. First, the volume of air scooped up by the sails diminishes since the profile of the sails facing the wind is smaller. Second, the force-generating deflection of the wind is now passes through a smaller angle. Figuratively, the wind strikes a more glancing blow onto the sails and thus exerts a weaker force on them.
These effects have been described only qualitatively. However, when they are combined, the effects that diminish the force overwhelm the one that increases it, so that the net effect is a loss of motive force. To see quantitatively that this is so, one needs to construct a careful mathematical model of the interaction of the wind with the sails. The result follows after some elaborate juggling of trigonometric functions. I will not reproduce them here, since the details of the calculations are tedious and not any more illuminating than the reciting of the qualitative effects above.
One can, however, get a sense that the apparent wind cannot drive a sailboat merely by recalling an experience familiar to every sailor. Imagine the sailboat sitting becalmed in completely dead air. If the boat is given a small push, perhaps from a paddle or a hand on a dock, it will move forward. That motion will create wind. However, the wind will blow straight down the centerline of the boat and, therefore, will be unusable by the sails as a way of generating any forward-directed motive force. The boat will gently slow to a halt, just as the generator–dynamo self-powered car cannot sustain an initial push.
What this last analysis shows is that a particular design of sailboat, the common Bermuda rig, is unable to realize the perpetual-motion machine of the paradox. Does that settle the matter? Might another design fare better? Might an improved design of sailboat be able to extract energy from the created wind and thus realize a perpetual-motion machine? Here the decision is not so straightforward. The normal response to a proposal for a perpetual-motion machine is that it is impossible because it would violate the law of conservation of energy. However, in addition it is customary to complete the refutation by pinpointing where the design fails. The generator–electric motor car, for example, fails because the slightest loss of energy due to friction means that the generator cannot supply as much energy as the electric motor demands.
What complicates the question is that a sailboat has an external source of energy, the kinetic energy of the true wind, as well as the possibility of the internally created energy of the created wind. Any analysis must disentangle the two. If a sailboat generates more energy when it is moving faster and thus experiences a greater apparent wind, which is the source of the extra energy? Is it merely more energy harvested licitly from the kinetic energy of the true wind? Or are we generating more energy from the created wind in violation of the law of the conservation of energy?
What we should like to develop is a general sense that the created-wind perpetual-motion machine will always be defeated by internally counteracting effects. The greater apparent wind will deliver greater energy, but all gains will be lost by some other effect that essentially arises in connection with the created wind. To see that things will always work out this way is hard if we examine the functioning of any real sailboat or even any real wind-powered device. For all such devices are beset by many inefficiencies, such as frictional energy losses or incomplete extractions of wind energy. If a boat functions better when sailing into the wind, is it truly because of some sort of perpetual-motion effect, or is it simply the result of the reduction of inefficiencies?
**FIGURE 13.4** Wind-turbine-powered boat. Velocities with respect to water. Author's copyright.
The way to escape this problem is to consider an imaginary, wind-powered boat in which all the inefficiencies are idealized away. In this thought experiment, we consider a device that is perfectly efficient in extracting energy from the wind and is beset by no dissipative processes. For concreteness, we will imagine that our boat extracts energy from the wind with a large system of wind turbines and that this energy then powers its propeller. Any idealized system capable of extracting all the energy from the wind could be used; the turbine system is used simply because it is easy to visualize and compute. Its operation is shown in Figure 13.4.
The boat sails at vector velocity **v** into a true wind with vector velocity **w**. The wind turbines are perfectly efficient, so that the wind turbine extracts all the kinetic energy of the wind that enters its throat. This means that the wind enters the turbine throat at velocity **w** and, as the boat moves off, it discharges a wake of entirely quiescent air; that is, air with zero velocity.
**FIGURE 13.5** Wind-turbine-powered boat. Velocities with respect to boat. Author's copyright.
How does this moving boat appear to a sailor on its deck? We merely add a velocity **v** to each of the velocities, with the result shown in Figure 13.5. The boat is now at rest and the water beneath the boat moves at velocity **v** toward the stern. The air discharged by the wind turbine is at rest with respect to the water, so it also moves at **v** toward the stern. Finally, the air entering the turbine moves at an apparent velocity of **w + v**. This added velocity **v** is the wind created by the boat's motion.
We use these velocities to compute the energy the turbine extracts from the wind, for the turbine has no way of distinguishing true from apparent wind. All it knows is that it scoops up air at velocity **w + v** and discharges it at **v**. It turns out that the resulting energy extraction is greater than the kinetic energy of the true wind passing through the turbine (see the Appendix for the calculations). We interpret the extra energy as supplied by the created wind and write:
(1)
If this were the entirety of the analysis, then we would have achieved a device that generates energy from nothing. However, it is not. The total energy of (1) is not available to power the boat. There is a consumption of energy that arises inescapably as part of the operation of the wind turbine. In order to extract energy from the wind, the turbine must take rapidly moving air and slow it down. That means that the turbine must apply a force to the wind. This is an ineliminable resistance force against which the boat must work. Moving against this force consumes energy. It turns out that this energy consumption matches exactly the extra, created energy:
(2)
Combining (1) and (2), we recover
(3)
Hence, the extra energy we thought we had gained from the created wind is exactly consumed as the energy needed move the boat against the wind. That is, the net energy extracted is just the kinetic energy extracted from the true wind. The boat is not a perpetual-motion machine that is powered even in part by its own self-created energy.
In sum, we learn for the highly idealized wind-powered boat of the thought experiment that it can extract energy from the wind created by its own motion. However, exactly that extra energy is consumed by an inescapable counteracting effect. The result seems quite general. There is nothing in the thought experiment that specifically requires a wind turbine to extract the energy. Any device will be subject to essentially the same analysis. Making the boat more realistic by removing the idealization of perfect efficiency and no dissipative frictional effects will not help. It will carry us further from the possibility of a perpetual-motion machine. We now develop the sense that extracting net energy from the created wind is an appealing but impossible illusion.
For a philosopher interested in epistemology – the study of how we get to know things – this last conclusion is fascinating. The thought experiment has taught us something important about the operation of wind-powered vehicles like sailboats that is much harder to recover from experiment. We could have conducted a series of tests on a variety of sailboats to see whether we could gain net energy from the created wind. Presumably each test would have told us that we could not, in that case. However we would always have been left wondering whether our failure to extract net energy from the created wind merely resulted from our lack of ingenuity in finding the clever design of boat that could do it. The thought experiment, however, indicates that our failure is a matter of principle. The quest for a better design can end.
Merely thinking about examples so idealized as to be unrealizable gives us a more secure and more general understanding of physical possibility than real experiments. How is that possible? This is the central problem of the epistemology of thought experiments. This problem has attracted a flourishing philosophical literature. I'll mention two extreme views in this literature. One is defended by my colleague Jim Brown of the University of Toronto4 and the other by me.
Brown is a Platonist and he urges that something in the right sort of thought experiment enables us to tap into a Platonic realm in which the laws of nature reside. The thought experiment lets us "see" the laws in a way that mere material experiments cannot. If this seems far-fetched, it might be helpful to recall the case that is the model for Platonic thought, mathematics. Draw an equilateral triangle – one with three equal sides – on a piece of paper and measure its angles. To within the accuracy of measurement, you will find that the angles are the same. Repeat the exercise for several more triangles. The results will be the same. That is no surprise. You fully expected it to be so, to the extent that any slight differences in your measurements would be dismissed as errors. But how did you get this knowledge that trumps actual experience? It is because thought affords you a deeper understanding of triangles than mere measurement can bring. Your mind can grasp the ideal triangles of the Platonic realm of which the triangles you drew are but poor imitations.
My view is the opposite of Brown's. It is deflationary and finds nothing epistemically remarkable in thought experiments. While they certainly have great rhetorical powers, epistemically they can do nothing more than ordinary argumentation. They are, I maintain, merely picturesque argumentation. As a result, you get nothing more out of a thought experiment than what you put into it as assumptions and what can be wrestled from those assumptions by deductive or inductive argumentation. In the thought experiment concerning wind-powered boats, what was assumed was the Newtonian mechanics of frictionless fluids. That theory conforms to the conservation of energy. As a result, it was a foregone conclusion that it would not allow the creation of energy from nothing. The only novelty was to see precisely how the theory blocked its creation. We did not learn anything that transcended the assumptions made. Had we made different assumptions, such as some concocted mechanics that did not respect energy conservation, we could have arrived at a thought experiment that vindicates the free creation of energy.
To see how Brown and I have sought to settle our differences and for an entry into the literature on thought experiments, see James Brown, "Why thought experiments transcend empiricism" and James Norton, "Why thought experiments do not transcend empiricism."5
# Appendix: Analysis of the Wind-Powered Boat
Air will enter the inlet of the turbine with cross-sectional area6 **A** in at density rin and velocity **w + v**. It is discharged at the outlet with cross-sectional area **A** out at density rout and velocity **v**. Conservation of mass requires
(A0)
Considering velocities in the boat frame of reference, the turbine scoops up air with energy density (1/2) rin | **w + v** |2 at a volumetric rate **A** in · ( **w + v** ), and discharges air with energy density (1/2) rout | **v** |2 at a volumetric rate **A** out · **v**. Hence, the total power – that is, the total rate at which energy is delivered by the turbine – is
Applying equation (A0), this becomes7
(A1)
This equation corresponds to equation (1) of the main text. The first term represents the rate of delivery of kinetic energy by the true wind. The true wind, moving at speed | **w** |, has kinetic energy density at the inlet of (1/2) rin | **w** |2 and arrives at a volumetric rate ( **A** in · ( **w + v** )). The second term is the energy delivered by the created wind, which has an apparent energy density at the inlet of rin ( **w** · **v** ).
To operate, the turbine scoops up air with a momentum density rin ( **w + v** ) and discharges air with the reduced momentum density rout **v**. To slow the air, the turbine must apply a force to the air equal to the rate of change of momentum:
Applying (A0), this expression reduces to
Since the boat moves at velocity **v** , energy is consumed in working against this force at the rate
(A2)
This corresponds to equation (2) of the main text. The net power available is just the difference
(A3)
This equation corresponds to equation (3) of the main text.
**NOTES**
1 Norman L. Skene, _Elements of Yacht Design_ (Dobbs Ferry, NY: Sheridan House, 2001), p. 92.
2 Isaac Newton, "Four letters to Richard Bentley." In Milton K. Munitz (Ed.), _Theories of the Universe_ (New York: Free Press, 1957), pp. 211–219.
3 See John D. Norton, "Causation as folk science," _Philosophers' Imprint_ 3:4 (2003, <http://quod.lib.umich.edu/p/phimp/3521354.0003.004>). Reprinted in Huw Price and Richard Corry (Eds.), _Causation and the Constitution of Reality: Russell's Republic Revisited_ (Oxford: Oxford University Press, 2007), pp. 11–44. See also John D. Norton, "Do the causal principles of modern physics contradict causal anti-fundamentalism?" In Peter K. Machamer and Gereon Wolters (Eds.), _Thinking about Causes: From Greek Philosophy to Modern Physics_ (Pittsburgh, PA: University of Pittsburgh Press, 2007).
4 James R. Brown, "Why thought experiments transcend empiricism." In Christopher Hitchcock (Ed.), _Contemporary Debates in the Philosophy of Science_ (Malden, MA: Blackwell, 2004), pp. 23–43.
5 Brown, "Why thought experiments transcend empiricism," pp. 23–43 and John D. Norton, "Why thought experiments do not transcend empiricism." In Hitchcock (Ed.), _Contemporary Debates in the Philosophy of Science_ , pp. 44–66.
6 The vector **A** in has a magnitude equal to the cross-sectional area of the inlet and a direction normal to the cross-section. The same is true for **A** out.
7 Using (1/2) [| **w + v** |2 − | **v** |2] = (1/2) [| **w** |2 \+ 2 **w** · **v** \+ | **v** |2 − | **v** |2] = (1/2) | **w** |2 \+ **w** · **v**.
**TAMAR M. RUDAVSKY AND NATHANIEL RUDAVSKY-BRODY**
# CHAPTER 14
# THE NECESSITY OF SAILING
# Of Gods, Fate, and the Sea
The shallow rocks and unpredictable winds of the Aegean make for some of the world's most dangerous sailing. The sea routes connecting Greek cities with their colonies, from the shores of the Black Sea to Sicily and further to the remote edges of the Mediterranean, were long and perilous. Before the Greeks learned to philosophize they learned to fear the sea. They knew the uncertainty of the winds and of the gods, neither of which were to be trusted.
From this uncertainty were born the twins philosophy and theology. Theology is the study of the gods, and if gods controlled the sea perhaps there were ways to appease them. Perhaps there were ways in which the uncertain forces of the world could be controlled, or at least encouraged. Philosophy, the study of the self, postulates that another way to master external forces is to master oneself, one's mind. If the highest good was a state of mind, then the dangers of the external world counted for very little. The question kept coming back to individual freedom. If the gods willed a storm at sea, did that mean they willed the traveler's destruction, and that death was inevitable? If a fortune teller foresaw a bad end at sea, could the individual choose to go another way? Could the inner freedom of the philosopher render one impervious to the sea?
The distinction between an inner philosophical freedom and an outer freedom in the world would last as long as the ancient world. Only with the victory of a single God – and the monotheistic philosophers recognized the same God, even if their Jewish, Christian, and Muslim co-religionists did not – would inside and outside meld the two freedoms into one. So the questions were rearranged. What was the difference between postulating a god who knew all shipwrecks in advance (like the ancient fortune teller) and recognizing one who caused shipwrecks as part of his natural ordering of the world? How much freedom, how much room to maneuver do individuals have in shaping their destiny in a world ordered by such an all-knowing or all-powerful God? If choosing to live a just life was the only personal freedom, what were the risks for a just man setting sail for a distant port?
With the regularization of travel, stricter safety standards, and modern shipbuilding in the eighteenth and nineteenth centuries, sea travel lost much of its menacing power to drive humans to God or to philosophy. But the sea took on further meaning, signifying the world of explorers pushing forward to new horizons. Nietzsche was one of the last of the great sea philosophers. He spent most of his life on the coast, fascinated by sailing. For Nietzsche, the sea provided a palpable image for the inner struggles and inner freedoms of the "new individual." In this brief essay, we explore three moments in the history of philosophy that exemplify human struggles with sailing. Our story begins with the ancient obsession with foretelling sea journeys and then turns to medieval grappling with God's foreknowledge. We end with Nietzsche's enthralled use of sailing toward open horizons as an image for human freedom.
# Of Greek Gods, the Judaeo-Christian God, and the Sea
The Greek gods that we know were given their form by Hesiod and Homer. Hesiod, in the rich farmlands of Boetia of the eighth century, took a dim view of sailing. "If the desire for stormy seagoing seizes upon you," he sings to his brother, then at least sail during
the timely season for men to voyage.
You will not
break up your ship, nor will the sea drown
its people, unless
Poseidon, the shaker of earth,
of his own volition,
or Zeus, the king of the immortals wishes
to destroy it,
for with these rests authority for all outcomes,
good or evil.
At that time the breezes can be judges
and the sea is not trouble1
Sailing, for this singer of harvests and sowings, is the lot of traders who take to the waves to escape debt or earn riches. Hesiod's gods, whose intricate family relations he describes elsewhere, have little personal to offer. Zeus sends a hard-blowing south wind and heavy rain in autumn and good weather in early summer. He and Poseidon destroy a ship at will. Their temperaments are natural forces, responsible for all things and accountable to none, against which Hesiod can only counsel wariness and good timing. How far from the gods of Homer, who pick and choose their mortal loves. Of course, Hesiod is singing advice for a farmer whose greatest act of heroism was to bribe local judges into giving him more than his share of his father's land. In the end he can only council caution. "You will find it hard to escape coming to grief," he sings, "and even so, men in their short-sightedness do undertake it; for acquisition means life to miserable mortals; but it is an awful thing to die among the waves."2
While Hesiod's gods restricted themselves to autumn gales and nourishing spring rains, they lived intensely human private lives among themselves. The first Greeks to reject such a cosmos we call the pre-Socratic philosophers. Among them, Pherecydes of Syros discussed the creation of the world and its first principles, the transmigration of souls, and other topics we can only guess at, since his great book, the _Heptamychos_ , comes down to us in fragments. He is also said to have constructed a sundial and predicted eclipses. Pythagoras of Samos taught about the mystical properties of numbers and founded a religious sect that made important discoveries in mathematics. Both were also said to have predicted shipwrecks as well as earthquakes, storms, and the outcomes of battles. And, since Pythagoras' students were sworn to keep his teachings secret, the first one to spill the beans perished in a shipwreck. Or so the story goes. Don't dismiss this as mere legend. The pre-Socratics were intoxicated with their new ideas and the force of the world they were trying to explain. It is hardly surprising that some should have perished by those forces and others have co-opted them, seduced by their powers.
One of the best challenges to the old gods is said to have occurred at Samothrace, the island where the famous statue of Winged Victory (now at the Louvre) once reigned over the harbor. Under her wings, travelers arrived to visit the sanctuary of the great gods. These were deep gods – deep in time, since they were worshiped before the Greeks settled their lands, and also deep in the earth, since the Greeks assimilated them to chthonic deities: gods of the soil and underworld. They were known as protectors of sailors, and their cult flourished for centuries before and after the flourishing of the Greek world.
This story comes down to us in the writings of Cicero, a later Roman philosopher:
While Diagoras, who is called the atheist, being at Samothrace, one of his friends showed him several pictures of people who had endured very dangerous storms; "See," says he, "you who deny a providence, how many have been saved by their prayers to the Gods." "Aye," says Diagoras, "I see those who were saved, but where are those painted who were shipwrecked?"3
Diagoras was a sophist, a contemporary of Socrates. He had a reputation for getting into trouble, both for denying the gods and for more mundane political reasons.
Also with Diagoras we see the end of the mystical. Another time when his ship was caught in a storm the sailors wanted to throw him overboard, believing his impiety had called misfortune down on them. "What about the other ships caught in the same storm?" the thinker asked them "if they believed Diagoras was also aboard those ships?"4
We turn finally to "the philosopher," Aristotle, the most influential philosopher in the ancient world. In his logical work, _On Interpretation_ , Aristotle uses the example of a sea battle to show that, while logical propositions about the present must necessarily be either true or false, propositions about future events are logically open, since the world is not determined. Though he seems to be talking only about logic, he is in fact criticizing fortune telling. His true intentions become clear when he says, "events will not take place or fail to take place because it was stated that they would or would not take place, nor is this any more the case if the prediction dates back 10,000 years."5 In this important passage (one that would be dissected by generations of medieval philosophers), Aristotle tries to salvage human freedom by arguing that the future is truly indeterminate – although it is necessarily the case that a sea-fight battle will either occur or not, neither possibility is necessary until the future actually unfolds.
Consider further Aristotle's famous conclusion about man's happiness. A generation earlier, the Athenian statesman Solon had warned the king Croesus, "Count no man happy until he is dead." Aristotle transformed that into "he who is happy... is active in accordance with complete virtue and is sufficiently equipped with external goods, not for some chance period but throughout a complete life."6 Philosophy may provide protection against the uncertainties of life, but only for the mind. Accidents still determine the span of life and the ultimate trajectory of happiness. A few centuries later, the Roman Cicero would put the two (fortune telling and the sea) together in his treatise _On Fate_ , in which he considers propositions such as "If anyone is born at the rising of the dog-star, he will not die at sea."7
But soon, a far greater death was announced on a calm sea. The sailor Thamus heard a voice call across the water as he passed the island of Paxi off the west coast of Greece: "When you reach Palodes, take care to proclaim that the Great God Pan is dead."8 The occasion for such a lament, according to Christian commentators, was the coming of Jesus of Nazareth: a new god had come to take the place of all the old ones.
The Acts of the Apostles tells how Paul was arrested and sent to Rome as a prisoner for the emperor, guarded by the centurion Julius. Midway across the Mediterranean, rough seas forced them to take shelter in a harbor on the southern coast of Crete, but the sailors were impatient to sail. Paul warned them to wait for spring. Nevertheless, the crew pushed off, and a storm tossed them about fiercely for fourteen days. In the end it was Paul who rallied spirits with comforting words and sound advice. He encouraged them to eat bread because they would need strength to attempt a beaching on unknown shores. At last, too, a prophecy:
Paul stood up among them and said, "You should have taken my advice, gentlemen, not to sail from Crete; then you would have avoided this damage and loss. But now I urge you not to lose heart; not a single life will be lost, only the ship. For last night there stood by me an angel of the God whose I am and whom I worship. 'Do not be afraid, Paul,' he said; 'it is ordained that you shall appear before the emperor; and be assured, God has granted you the lives of all who are sailing with you.' So keep up your courage."9
God had his sights on the long view – bringing Paul to Rome. These sailors were saved because Paul was on board. Perhaps this was the first message of Christianity, a god who saved all who came in contact with his grace. Still, the sailors could have spared themselves some trouble had they listened to Paul's first warning. With the advent of Christianity, the gods of the Greek world were dying. Of course, Christianity took much from the Jews as well, and with the spread of Islam the three religions developed together in a newly monotheistic world. From now on the gods were one God, accidents were God's will, fate was God's knowledge. But the sea was the same sea.
# A Ship Bound for India
David ben Maimon found himself in the Sudanese port of 'Aydhāb with nothing to buy. In spring of the year 1169 he had sailed up the Nile from Fustat ("old Cairo") to Qūs, on the edge of the desert. He and his travel companion had been separated from their caravan and had crossed the desert alone, a foolhardy risk; they reached 'Aydhāb only after "disastrous" hardships. "God had willed that we be saved." Yet in the city there was nothing to buy except indigo. "So I thought about what I had endured in the desert; then it appeared to me an easy thing to embark on a sea voyage. I took Mansur as my travel companion."10 David decided to trust his fate to sea, to the monsoon winds, the India passage.
David was writing to his brother, and the letter was fortuitously preserved in the _Cairo Genizeh_ , where Cairo's Jewish community discarded their used documents. When this treasure trove was rediscovered in 1896, the lives of Cairo's medieval Jews were opened up for us in amazing detail. Many were traders, sailing to every shore of the Mediterranean. They also joined in the India trade, navigating the seasonal monsoon winds from the Red Sea, the Persian Gulf, and the southern coast of Arabia to India's Malabar coast and Sri Lanka. The stakes were luxury goods, precious stones, spices, gold, ivory, metalwork, and fine fabrics; raw materials – iron and copper and tin, and hardwoods; and foodstuffs – wheat, rice, and cheese. Many were learned, and a few were scholars.
But David's brother was more than another learned merchant. He was Moses ben Maimon, known as Maimonides, the most important Jewish philosopher of the Middle Ages. In 1169 he was still young, but soon he would be recognized as leader of Egypt's Jews. Later he would be appointed court physician to Saladin's son and successor, al-Afdad. Ever since the family had arrived in the East as refugees from war-torn Andalusia, David had supported his brother's studies.
Now poised between the desert and the sea, David reassured his brother that God would look out for the family's finances. He asked him to comfort "the little one," his wife: his ship was to embark around Ramadan. He did not return. Eight years later the philosopher Maimonides recalled his grief in a letter:
The most terrible blow which befell me... was the death of the most perfect and righteous man, who was drowned while traveling in the Indian Ocean. For nearly a year after I received the sad news, I lay ill on my bed struggling with fever and despair. Eight years have since passed, and I still mourn, for there is no consolation. What can console me?... My one joy was to see him. Now my joy has been changed into darkness; he has gone to his eternal home, and has left me prostrated in a strange land.11
Left to support both families, Maimonides began practicing medicine professionally, an activity that would leave him exhausted and short of time all his life. Still, his reputation as a religious thinker grew as he took on the philosophical dilemmas of his Islamic masters, Averroes, Avicenna, al-Farabi, and ibn-Bajja. These thinkers were steeped in Aristotle and had worked hard to make Greek philosophy compatible with their monotheistic religion. Maimonides was to devote much of his life to the same dilemma.
In the third book of his monumental _Guide to the Perplexed_ , he asks a very direct question: when a ship founders, why does a just man drown? First he summarizes and rejects five theories held by his predecessors. Epicurus and his followers denied that God had any role in the world. Aristotle believed that divine providence extends only to the realm of the celestial spheres, not to the terrestrial world. The Islamic Ash'arites claimed that every event in the world is predestined by God, denying any contingency. The Islamic Mu'tazilites believed in limited free will for humans. Finally, traditional Jewish law accorded humans perfect freedom, while at the same time insisting that everything that happens them is the result of God's justice. In this last view, God is absolutely just, and all calamities are deserved. Does it suggest that Maimonides' brother deserved to die at sea?
The philosopher rejects these five opinions, and describes his own: "Divine providence watches only over the individuals belonging to the human species... but regarding all the other animals, and all the more, the plants and other things, my opinion is that of Aristotle."12 Although providence in the terrestrial world extends to all and only human beings, it does not necessarily reflect divine retribution or reward. For Aristotle, a storm was part of the natural order, but the deaths of those on a ship that foundered were caused by to chance and undeserved. Both the Mu'tazilites and Ash'arites would have agreed that the deaths were undeserved, but the result of divine will. According to Jewish law, the drownings were deserved, but we cannot understand the ways of God. Maimonides agrees with Aristotle that the foundering of a ship is "due to pure chance." In his view, divine providence is consequent upon the perfection of human intellect, and intervenes only in the decision to board the ship:
The fact that the people in the ship went on board and that the people in the house were sitting in it is, according to our opinion, not due to chance but to divine will in accordance with the deserts of those people as determined by his judgments, the rule of which cannot be determined by our intellects.13
Maimonides' successors contested his position. Yet the summation of his argument, which some philosophers have found problematic or even naive, turns out to be deeply moving. David, when he arrived in 'Aydhāb, chose to go on to India. The traditional question for the untutored reader is how God could let a just man like David die undeservedly. David's choice to embark upon a sailing ship represents not only the perils of sailing and the chance of shipwreck but also a struggle with the very fatalistic powers of the sea. That very choice, not just the drowning itself, must be accounted for in a theory of divine providence. Even if it implicates God, the fateful choice must be given meaning, so that David's death and then his life have meaning in a world of merchant scholars where religious study and contemplation were the highest goods – all the while accepting from the philosophers that we live in a chaotic natural world.
# Beyond the Pillars of Hercules
So far we have been unkind to the sea. We have dragged up shipwrecks and skirted storms, and reported deaths both foretold and unexpected. Sailing itself connotes death, destruction, and the power of the gods and of God. But the sea has other moods: calm as a breeze fluttering the leaves of a plane tree on a summer day, an image familiar to Greeks; whipped up by the terse gusts of a sunny morning; gray and expansive. For every traveler who felt the weight of uncertainty as he boarded his ship, another was filled with the promise of fresh mornings under sail. Such a one was Ulysses, the conniving hero who ended up in the eighth circle of Dante's hell for crimes of fraud. At Dante's wish, the hero recounts his own death.14 After returning to Ithaka from his wanderings, not even his love for Penelope could calm Ulysses' lust for adventure: "So I set forth on the open deep with but a single ship and that handful of shipmates who had not deserted me."15 Ulysses and his companions sailed west, through the straits of Gibraltar, the very pillars Hercules had placed as a western barrier to human travel. None who passed these pillars had ever returned. There, to raise the spirits of his rowers, he gave a famous speech:
Oh brothers, I said, who in the course
of a hundred thousand perils, at last
have reached the west, to such brief wakefulness
of our senses as remains to us
do not deny yourselves the chance to know –
following the sun – the world where no one lives.
Consider how your souls were sown:
you were not made to live like brutes or beasts
but to pursue virtue and knowledge.16
Reinvigorated, they sailed southwest across the Atlantic for five months until a mountain loomed before them: Purgatory. But a whirlwind rose up, capsizing their ship, and "the sea closed over us."17 Dante seems to have invented this story himself, maybe inspired by the Visconti brothers, who passed Gibraltar in 1291 in search of India, and disappeared. It makes a nice counterpoint to the travelers we have seen terrified of a bad end at sea. Ulysses, challenging heaven and earth, faces the sea head on. Adventure for him arises out of the passion for knowledge and discovery.
It took Columbus to discover land across the western ocean and return. For Friedrich Nietzsche, writing in the nineteenth century, the sea carried a double message that reflected the theme of discovery. To question the Christian metaphysics and morality that weighed on Europe was to set off on a voyage to the unknown. Columbus' voyages to the New World became a symbol for this daring philosopher. "Finally the horizon seems clear again," Nietzsche announced in _The Gay Science_ , a work written to prepare his readers for his destruction of Western morality. "Finally our ships may set out again, set out to face any danger; every daring of the lover of knowledge is allowed again; the sea, _our_ sea, lies open again; maybe there has never been such an 'open sea.' "18
Nietzsche's works are replete with references to the sea, to sailing to new horizons, to destroying the old gods in order to usher in a new moral order. "We have left the land behind and boarded the ship! We have burned our bridges – more than that, we have demolished the land behind us," he tells us in _The Gay Science_.19 In this work, a "madman," presumably a spokesperson for Nietzsche himself, comes running to the marketplace one morning decrying the death of God. But the crowds do not want to hear him, do not have the patience to listen to his words, and they mock him. Finally the madman grows silent and announces, "I come too early... it is not the right time for me, yet," reiterating Nietzsche's own view that most people are simply not ready to hear and accept his message.20 Nietzsche realizes that such a journey is not for everyone. It is the journey of a lifetime, not of impetuous youth. As a young man, he had already noted that
to dare to launch out on the sea of doubt without compass or guide is death and destruction for undeveloped heads; most are struck down by storms, few discover new lands. From the midst of this immeasurable ocean of ideas one will often long to be back on firm land.21
Live dangerously, Nietzsche exhorts us: "Build your cities by Vesuvius! Send your ships into unexplored seas!"22 In other words, be prepared to take risks.
In _Thus Spake Zarathustra_ , written toward the end of his life, Nietzsche returns to the motif of sailing into uncharted seas as an exhortation to follow the words of Zarathustra the prophet. "Ah, this sombre, sad sea, below me! Ah, this sombre nocturnal vexation! Ah, fate and sea! To you must I now GO DOWN!" says Zarathustra as he begins his journey to share his wisdom with those on Earth.23 His message is addressed "To you, the daring venturers and adventurers, and whoever hath embarked with cunning sails upon frightful seas, – To you the enigma-intoxicated, the twilight-enjoyers, whose souls are allured by flutes to every treacherous gulf."24 In other words, to those who are willing to take risks, to embark upon rough waters without fear, Zarathustra offers the promise of a new code of values, a trans-valuation of morality, a replacement of the "old" tablets of the law for "new" tablets. This trans-valuation, or replacement, is itself described in sea-worthy language:
O my brethren, when I enjoined you to break up the good, and the tables of the good, then only did I embark man on his high seas.
And now only cometh unto him the great terror, the great outlook, the great sickness, the great nausea, the great sea-sickness.
False shores and false securities did the good teach you; in the lies of the good were ye born and bred. Everything hath been radically contorted and distorted by the good.
But he who discovered the country of "man," discovered also the country of "man's future." Now shall ye be sailors for me, brave, patient!
Keep yourselves up betimes, my brethren, learn to keep yourselves up! The sea stormeth: many seek to raise themselves again by you.25
With these powerful words, Nietzsche connects the raw energy of the sea and the danger of sailing to distant shores with the denial of fate, and the rejection of conventional morality. Only the brave sea-worthy individual can discover new lands and, in so doing, create a new covenant.
With the words of Zarathustra, we are not so far from Aristotle, who also tried to mitigate the hazards of life by denying their force in the face of a philosophical life, nor from Maimonides, who made contemplation the only way to include oneself under divine providence. But, when Nietzsche snatches fate from the force of the sea and the will of the gods and places it in the hands of the individual, he makes contemplation and philosophy the way to freedom, not because they protect one from the shocks of adversity but because they can be used to forge a path through it. The gods have died, humans push forward in their small boats on a new quest, but the sea is still the same sea.
**NOTES**
1 Hesiod, _The Works and Days; Theogony The Shield of Herakles_ , trans. Richmond Lattimore (Ann Arbor, MI: Ann Arbor Paperbacks, 1991), ll. 665 ff.
2 Ibid., ll. 683 ff.
3 Cicero, _Cicero's Treatise on the Nature of the Gods_ , trans. Charles D. Yonge (London: G. Bell, 1878), pp. 137–138.
4 Ibid., p. 138.
5 Aristotle, _Complete Works of Aristotle: The Revised Oxford Translation_ , vol. 1, ed. and trans. Jonathan Barnes (Oxford: Oxford University Press, 1974), 18b38.
6 Aristotle, _Complete Works of Aristotle_ , 1101a10.
7 Cicero, _Cicero's Treatise on the Nature of the Gods_ , p. 138.
8 Plutarch, "The Obsolescence of Oracles," _Moralia_ , book 5:17.
9 Acts 27:21–25.
10 Solomon D. Goitein (Ed. and trans.), _Letters of Medieval Jewish Traders_ (Princeton, NJ: Princeton University Press, 1973).
11 Moses Maimonides, _A Maimonides Reader_ , ed. Isadore Twersky (New York: Behrman House, 1972), pp. 4–5.
12 Moses Maimonides, _The Guide of the Perplexed_ , ed. and trans. Shlomo Pines (Chicago, IL: Chicago University Press, 1963), p. 471.
13 Ibid., p. 472.
14 Dante Aligheiri, _Inferno_ , trans. Robert Hollander and Jean Hollander (New York: Doubleday, 2000), Canto 26, p. 483.
15 Ibid.
16 Ibid.
17 Ibid.
18 Friedrich Nietzsche, _The Gay Science_ , ed. Bernard Williams, trans. Josephine Nauckhoff (Cambridge, UK: Cambridge University Press, 2001), p. 199.
19 Ibid., p. 124.
20 Ibid.
21 Julian Young, _Friedrich Nietzsche: A Philosophical Biography_ (Cambridge, UK: Cambridge University Press, 2010), p. 35.
22 Friedrich Nietzsche, _The Gay Science_ , p. 283.
23 Friedrich Nietzsche, _Thus Spake Zarathustra_ , trans. Thomas Common (New York: Modern Library, 1921), p. 45.
24 Ibid., p. 46.
25 Ibid., p. 56.
**HILAIRE BELLOC**
# CHAPTER 15
# THE CHANNEL
# An Old Drama by which the Soul of a Healthy Man is Kept Alive1
Friends of mind, friends all, and you also, publishers, colonials and critics, do you know that particular experience for which I am trying to find words? Do you know that glamour in the mind which arises and transforms our thought when we see the things that the men who made us saw – the things of a long time ago, the origins? I think everybody knows that glamour, but very few people know where to find it.
Every man knows that he has in him the power for such revelations and every man wonders in what strange place he may come upon them. There are men also (very rich) who have considered all the world and wandered over it, seeking those first experiences and trying to feel as felt the earlier men in a happier time – yet these few rich men have not so felt and have not so found the things which they desire. I have known men who have thought to find them in the mountains, but would not climb them simply enough and refused to leave their luxuries behind, and so lost everything, and might as well have been walking in a dirty town at home for all the little good that the mountains did to them. And I know men who have thought to find this memory and desire in foreign countries, in Africa, hunting great beasts such as our fathers hunted; yet even these have not relit those old embers, which if they lie dead and dark in a man make his whole soul dusty and useless, but which if they be once rekindled can make him part of all the centuries.
Yet there is a simple and an easy way to find what the men who made us found, and to see the world as they saw it, and to take a bath, as it were, in the freshness of beginnings; and that is to go to work as cheaply and as hardly as you can, and only as much away from men as they were away from men, and not to read or to write or to think, but to eat and drink and use the body in many immediate ways, which are at the feet of every man. Every man who will walk for some days carelessly, sleeping rough when he must, or in poor inns, and making for some one place direct because he desires to see it, will know the thing I mean. And there is a better way still of which I shall now speak: I mean, to try the seas in a little boat not more than twenty-five feet long, preferably decked, of shallow draught, such as can enter into all creeks and havens, and so simply rigged that by oneself, or with a friend at most, one can wander all over the world.
Certainly every man that goes to sea in a little boat of this kind learns terror and salvation, happy living, air, danger, exultation, glory, and repose at the end; and they are not words to him, but, on the contrary, realities which will afterwards throughout his life give the mere words a full meaning. And for this experiment there lies at our feet, I say, the Channel.
It is the most marvelous sea in the world – the most suited for these little adventures; it is crammed with strange towns, differing one from the other; it has two opposite people upon either side, and hills and varying climates, and the hundred shapes and colors of the earth, here rocks, there sand, here cliffs, and there marshy shores. It is a little world. And what is more, it is a kind of inland sea.
People will not understand how narrow it is, crossing it hurriedly in great steamships; nor will they make it a home for pleasure unless they are rich and can have great boats; yet they should, for on its water lies the best stage for playing out the old drama by which the soul of a healthy man is kept alive. For instance, listen to this story:
The sea being calm, and the wind hot, uncertain, and light from the east, leaving oily gaps on the water, and continually dying down, I drifted one morning in the strong ebb to the South Goodwin Lightship, wondering what to do. There was a haze over the land and over the sea, and through the haze great ships a long way off showed, one or two of them, like oblong targets which one fires at with guns. They hardly moved in spite of all their canvas set, there was so little breeze. So I drifted in the slow ebb past the South Goodwin, and I thought: "What is all this drifting and doing nothing? Let us play the fool, and see if there are no adventures left."
So I put my little boat about until the wind took her from forward, such as it was, and she crawled out to sea.
It was a dull, uneasy morning, hot and silent, and the wind, I say, was hardly a wind, and most of the time the sails flapped uselessly.
But after eleven o'clock the wind first rose, and then shifted a little, and then blew light but steady; and then at last she heeled and the water spoke under her bows, and still she heeled and ran, until in the haze I could see no more land; but even so far out there were no seas, for the light full breeze was with the tide, the tide ebbing out as strong and silent as a man in anger, down the hidden parallel valleys of the narrow sea. And I held this little wind till about two o'clock, when I drank wine and ate bread and meat at the tiller, for I had them by me, and just afterwards, still through a thick haze of heat, I saw Grisnez, a huge ghost, right up against and above me; and I wondered, for I had crossed the Channel, now for the first time, and knew now what if felt like to see new land.
Though I knew nothing of the place, I had this much sense, that I said to myself, "The tide is right down Channel, racing through the hidden valleys under the narrow sea, so it will all go down together and all come up together, and the flood will come on this foreign side much at the same hour that it does on the home side." My boat lay to the east and the ebb tide held her down, and I lit a pipe and looked at the French hills and thought about them and the people in them, and England which I had left behind, and I was delighted with the loneliness of the sea; and still I waited for the flood.
But in a little while the chain made a rattling noise, and she lay quite slack and swung oddly; and then there were little boiling and eddying places in the water, and the water seemed to come up from underneath sometimes, and altogether it behaved very strangely, and this was the turn of the tide. Then the wind dropped also, and for a moment she lollopped about, till at last, after I had gone below and straightened things, I came on deck to see that she had turned completely round, and that the tide at last was making up my way, towards Calais, and her chain was taut and her nose pointed down Channel, and a little westerly breeze, a little draught of air, came up cool along the tide.
When this came I was very glad, for I saw that I could end my adventure before night. So I pulled up the anchor and fished it, and then turned with the tide under me, and the slight half-felt breeze just barely filling the mainsail (the sheet was slack, so powerless was the wind), and I ran up along that high coast, watching eagerly every new thing; but I kept some way out for fear of shoals, till after three good hours under the reclining sun of afternoon, which glorified the mist, I saw, far off, the roofs and spires of a town, and a low pier running well out to sea, and I knew that it must be Calais. And I ran for these piers, careless of how I went, for it was already half of the spring flood tide, and everything was surely well covered for so small a boat, and I ran up the fairway in between the piers, and saw Frenchmen walking about, and a great gun peeping up over its earthwork, and plenty of clean new masonry. And a man came along and showed me where I could lie; but I was so strange to the place that I would not take a berth, but lay that night moored to an English ship.
And when I had eaten and drunk and everything was stowed away and darkness had fallen, I went on deck, and for a long time sat silent, smoking a pipe and watching the enormous lighthouse of Calais, which is built right in the town, and which turns round and round above one all night long.
And I thought: "Here is a wonderful thing! I have crossed the Channel in this little boat, and I know now what the sea means that separates France from England. I have strained my eyes for shore through a haze. I have seen new lands, and I feel as men do who have dreamt dreams."
But in reality I had had very great luck indeed, and had had no right to cross, for my coming back was to be far more difficult and dreadful, and I was to suffer many things before again I could see tall England, close by me, out of the sea.
But how I came back, and of the storm, and of its majesty, and of how the boat and I survived, I will tell you another time, only imploring you to do the same; not to tell of it, I mean, but to sail it in a little boat.
**NOTE**
1 "The Channel" from _Hills and Sea_ by Hilaire Belloc. Reprinted by permission of Peters Fraser & Dunlop on behalf of the estate of Hilaire Belloc.
# NOTES ON CONTRIBUTORS
**GREGORY BASSHAM** is Professor and Chair of the Philosophy Department at King's College (Pennsylvania), where he specializes in philosophy of law and critical thinking. He co-edited _The Lord of the Rings and Philosophy_ (2003) and _The Ultimate Harry Potter and Philosophy_ (2010), and is co-author of _Critical Thinking: A Student's Introduction_. A veteran armchair sailor, Gregory has rounded the three Capes on many voyages of imagination.
**TOD BASSHAM** is a lawyer, writer, and sailor in Portland, Oregon. He races and cruises obsessively on the Columbia River, the Salish Sea, and wherever else he can cadge a berth. He is blessed with a tolerant wife and two forgiving children, and still laughs with childish delight when the wind fills the sails.
**HILAIRE BELLOC,** who lived from 1870 to 1953, was a prolific writer of poetry, essays, novels, histories, and political and social commentary. He was also an ardent yachtsman and his experiences at sea were a frequent subject of his essays.
**LUÍSA GAGLIARDINI GRAÇA** is an avid Portuguese sailor who likes to keep her country's great tradition of navigation alive. She holds a masters in sport sciences with emphasis on sport aesthetics from the Faculty of Sport, University of Porto (Portugal), where she is currently working on her PhD as she focuses on the intersection between aesthetics and ethics in sport. She has presented her work numerous times at conferences such as the International Association for the Philosophy of Sport and the European Association for the Philosophy of Sport. She is a licensed captain, and has sailed extensively in the Mediterranean, the Atlantic, and the North Sea, whether as crew in a tall ship race or a more manageable sloop or catamaran – so long as it floats, she'll sail it. Her next sailing adventure will be in the Aegean Sea. Ideally, she'd like to explore the whole world by sea... and she is working on that, only that it will have to wait until the winds or the whims of the sea deities are more favorable.
**PATRICK GOOLD** is Associate Professor of Philosophy at Virginia Wesleyan College. His current research focuses on defining rationality. He is co-editor with Steven Emmanuel of the Blackwell anthology _Modern Philosophy from Descartes to Nietzsche_. Patrick is passionate about sailing, and, in addition to maintaining a small daysailer and a cruising boat of his own, frequently crews on the boats of others. The bays and sounds of Virginia and North Carolina are his home waters but he has sailed the length of the East Coast of the United States from Hilton Head to Long Island Sound, made a Bermuda crossing, done club racing in Brittany, and cruised in the Lesser Antilles.
**NICHOLAS HAYES** is the author of _Saving Sailing, The Story of Choices, Families, Time Commitments, and How We Can Create a Better Future_ (Crickhollow Books, 2009). _Saving Sailing_ won an Independent Publisher 2010 Living Now Silver Medal, was a ForeWord 2009 Book of the Year Award finalist. Nick is a veteran of fifteen Mackinac Races and of uncounted other offshore sailing races in three decades of racing on Lake Michigan. He lives in Shorewood, Wisconsin, with his wife Angela and their daughters Kate and Elizabeth, all decorated sailors. The Hayes family campaign their B-32 sailboat _Syrena_ out of the South Shore Yacht Club and sail together for fun as often as they can.
**STEVEN HORROBIN** has sailed oceangoing yachts since the age of fifteen, completing his first long sea voyages in the later 1980s. Prior to returning to the study of philosophy, he was a playwright, actor, theater manager, and designer. His PhD, at the University of Edinburgh, was on the subject of the nature of the value of life to persons conceived as processes, and he has published, edited, tutored, and lectured in the subjects of general philosophy, bioethics, meta-ethics, and ontology in books, academic journals, and universities internationally for much of the past decade. He is a commercially certified Yachtmaster and Cruising Instructor and a passionate diver and freediver.
**RICHARD HUTCH** completed his MA and PhD at the University of Chicago, his BD at Yale University, and his BA at Gettysburg College in the United States. Since 1978, he has been Reader in Religion and Psychological Studies in the School of History, Philosophy, Religion, and Classics at the University of Queensland in Australia. For the past fifteen years he has skippered his twenty-seven-foot yacht, _Impulse_ , off the coast of Brisbane, not only trying his hand at racing on Moreton Bay but also cruising over short runs up and down the coasts of Queensland and New South Wales. He has written five academic books including one based on his sailing experiences, _Lone Sailors and Spiritual Insights: Cases of Sport and Peril at Sea_ (2005). His love of sailing was triggered during his boyhood years in the United States, where he sailed small boats on Barnegat Bay in New Jersey, where he grew up.
**JESÚS ILUNDÁIN-AGURRUZA** is an Associate Professor of Philosophy and Allen and Pat Kelley Faculty Scholar at Linfield College in Oregon. He hails from landlocked Pamplona, Spain, where he learned to yearn for the sea. His specialties are the philosophy of sport, aesthetics, and metaphysics. The horizons of his teaching and research span the philosophy of literature, Asian philosophy, ethics, and the philosophy of mind. He has published in journals such as _Sports, Ethics, and Philosophy_ and _Proteus_ ; he co-edited Wiley-Blackwell's _Cycling – Philosophy for Everyone_ ; and has chapters in anthologies on sports and risk, soccer, childhood and sports, the Olympics, hunting, and others (some in Spanish). Of lesser sailing experience than his writing crewmates, he matches their love for sailboats and the azure.
**JOSÉ ÁNGEL JÁUREGUI-OLAIZ** , from Pamplona (Spain), did his best to follow in the wake of Juan Sebastián Elcano, who was among the first crew to circumnavigate the Earth and was born about forty nautical miles northwest of Pamplona as the seagull flies. He is a former Second-Class Pilot in the Spanish Merchant Navy who has sailed and cruised the Atlantic Ocean, the Mediterranean, the Cantabric Sea, and North Sea in all kinds of weather both for pleasure and professionally. Aboard his now-gone sloop _Snowgoose_ he spent a literally tempestuous yet happy honeymoon. Nowadays retired as a seaman, he takes care of the family business and voraciously reads sailing books (enjoying narratives of solo exploits the most). Any chance he gets finds him on the waves. His current dream is to save enough money for another sailboat where he and his crew – wife Chus, daughter Marina, and son Lorenzo – may humor him pretending to follow his orders.
**GARY JOBSON** is the president of US Sailing, the national governing body of sailing in the United States. He has authored seventeen sailing books and is Editor at Large of _Sailing World_ and _Cruising World_ magazines. His newest book is _Gary Jobson: An American Sailing Story_. He has been ESPN's sailing commentator since 1985. Gary has won many championships in one-design classes, the America's Cup with Ted Turner, the infamous Fastnet Race, and many of the world's ocean races. He currently races a Swan 42, _Mustang_ , and an Etchells, _Whirlwind_. Also an active cruising sailor, Gary has led ambitious expeditions to the Arctic, Antarctica, and Cape Horn. In 1999 he was honored with the Nathanael G. Herreshoff Trophy, US Sailing's most prestigious award. This trophy is awarded annually to an individual who has made an outstanding contribution to the sport of sailing in the United States. A fuller account of Gary's many awards and accomplishments can be found on his website: www.jobsonsailing.com.
**SEBASTIAN KUHN** teaches physics at all levels (from large undergraduate survey courses to highly specialized graduate classes) at Old Dominion University (Norfolk, Virginia). He also probes the innermost life of nuclei, smashing atoms at the Thomas Jefferson National Accelerator Facility (Newport News, Virginia). He obtained his doctorate in physics in 1986 (University of Bonn, Germany) and had stints in Berkeley and Stanford before moving to the East Coast. In his spare time, he loves to cruise on the Chesapeake Bay with his wife Kathrin (and occasionally his daughter and son) on an aging Vega 27″ sloop. He'd be happy to discuss physics and sailing with anyone – contact information can be found at <http://odu.edu/~skuhn>.
**CRISTA LEBENS** is an Associate Professor of Philosophy teaching a wide range of courses, including feminist philosophy and philosophy of race, to students at the University of Wisconsin-Whitewater. Her area of specialty is social ontology. Crista is active in the Midwest Society for Women in Philosophy, and attends the California Roundtable on Philosophy and Race as often as possible. She began sailing on an old wooden X-Scow (the _Firefly_ ) at age eight, and was told to stay down in the cockpit and watch out for the boom. She liked watching the water creep up the deck as the boat "high-sided" and soon learned to sail from her father, a business owner and former Marine pilot who, after retiring from the reserves, decided that sailing was an enjoyable alternative to flying. Crista finds herself out on the water less than she'd like these days, but still notes whether or not the wind is good for sailing. She continues to be fascinated by the notion that the sail on a boat works on the same principle as the wing of an airplane. Crista says, "This one's for you, Dad."
**STEVE MATTHEWS** is Senior Research Fellow at the Plunkett Centre (a center for medical ethics), at the Australian Catholic University. His research areas include personal identity, agency and autonomy, philosophy of psychiatry, and applied ethics. His work has appeared in a range of journals including _Philosophical Studies_ , _The Monist_ , and _American Philosophical Quarterly_. He has been sailing since 1999.
**JOHN D. NORTON** is Director of the Center for Philosophy of Science and past Chair of the Department of History and Philosophy of Science, University of Pittsburgh. His research is in the history and philosophy of physics, with special attention to Einstein's work in special and general relativity; and he also works in general topics in the philosophy of science, including inductive inference, causation, and thought experiments. He thinks philosophy while sailing the rivers around Pittsburgh's point and wonders why there are no others sailing there regularly.
**JOHN ROUSMANIERE** (pronounced "Room-an-ear," with a silent "s") was educated at Columbia University (bachelor's and master's degrees in history) and Union Theological Seminary (M Div). He taught history and writing, then edited _Yachting_ magazine for several years, eventually becoming a freelance writer. He is the author of more than two dozen books on sailing, yachting history, and New York history. His sailing books include _The Annapolis Book of Seamanship_ , _A Berth to Bermuda: One Hundred Years of the World's Classic Ocean Race_ , _Fastnet Force 10: The Deadliest Storm in the History of Modern Sailing_ , _The Golden Pastime: A New History of Yachting_ , _In a Class by Herself: The Yawl Bolero_ , and _Sailing at Fishers: A History of the Fishers Island Yacht Club_. John has sailed on most of the world's oceans, logging some 35,000 miles on the water.
**TAMAR M. RUDAVSKY** is Professor of Philosophy at The Ohio State University. She specializes in medieval Jewish philosophy and has edited or co-edited three volumes: _Divine Omniscience and Omnipotence in Medieval Philosophy: Islamic, Jewish, and Christian Perspectives_ (1984), _Gender and Judaism: The Transformation of Tradition_ (1995), and, along with Steven Nadler, the _Cambridge History of Jewish Philosophy: From Antiquity through the Seventeenth Century_ (2009). Her volume _Time Matters: Time, Creation and Cosmology in Medieval Jewish Philosophy_ appeared in 2000. Her most recent book (2010) on Maimonides appeared in Wiley-Blackwell's "Great Minds" series. She is also the author of numerous articles and encyclopedia entries, and her major research continues to focus on issues connected to philosophical cosmology in medieval Jewish and scholastic thought. She learned to sail in the Charles River in Boston, and has continued sailing in Ohio.
**NATHANIEL RUDAVSKY-BRODY** graduated from Northwestern University with a degree in mathematics. He continued his studies in mathematics and medieval literature. He currently works as a translator in Brussels. He learned to sail on Lake Michigan, where even in good weather he preferred staying close to shore.
**JESSE STEINBERG** teaches philosophy at the University of Pittsburgh at Bradford. He has published articles on a variety of topics including ethics, metaphysics, and philosophy of mind. Jesse grew up in California and thoroughly enjoyed the Pacific Ocean. Now that he lives in Pennsylvania, time on the water is sadly reduced, though he has enjoyed sailing with co-author Michael Stuckart on Lake Ontario immensely.
**JACK STILLWAGGON** comes from a seafaring family. Three generations before him were tugboat men in New York harbor. His grandfather captained troopships during World War II in both the Pacific and European theaters. His great-grandfather was captain of a schooner that ran out of Liverpool for exotic ports of call in South America. Jack chose a business career instead of a life on the water but did serve as a "soldier of the sea" as an officer in the Marines. Since that time, he has owned both sailboats and powerboats and tries very hard to never lose sight of land. Jack has spent over thirty years in business in general management. He currently advises private equity firms, facilitating smooth transitions of the CEOs of acquired companies into new working environments for greater growth and strong partnerships.
**MICHAEL STUCKART** teaches anthropology at the University of Pittsburgh at Bradford. As an anthropologist, he specializes in traditional arts, tourist arts, religion and culture, and Latin America. He grew up in Huntington, Long Island and has sailed for over fifty years. His major blue-water experience involved crossing the Gulf of Mexico, and he enjoys single-handed sailing on the Great Lakes.
**JAMES WHITEHILL** is Professor Emeritus of Religion and Philosophy at Stephens College, Columbia, Missouri, where he taught Asian religions, Zen Buddhism, and philosophical ethics for thirty-six years. He directed the College's Japan Studies Program for thirty years and was awarded a Fulbright senior lecturer fellowship to Japan in 1991–1992. He co-founded and directed the Columbia Zen Center, a _zazenkai_ , and practiced in Soto and Rinzai Zen monasteries. His publications include a book on meditation practice, _Enter the Quiet_ , and several articles and chapters on Buddhist and Zen ethics. He sailed since his high-school days, and has owned small boats over the decades, auspiciously named _Miko, Hotei, Blue Lotus_ , and, since his retirement to Cape Cod, _Garuda_.
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from django.db import models
from django.db.models import Q
from datetime import timedelta
import operator
from msgvis.apps.base.models import MappedValuesQuerySet
from msgvis.apps.corpus import models as corpus_models
from msgvis.apps.groups import models as groups_models
from msgvis.apps.dimensions import registry
from msgvis.apps.corpus import utils
import re
from django.db import connection
MAX_CATEGORICAL_LEVELS = 10
def find_messages(queryset):
"""If the given queryset is actually a :class:`.Dataset` model, get its messages queryset."""
if isinstance(queryset, corpus_models.Dataset):
queryset = queryset.message_set.all()
return queryset
def get_field_name(text):
pattern = re.compile('(?<=__)\w+')
results = pattern.search(text)
if results:
return results.group()
return None
def fetchall(sql):
sql = utils.convert_boolean(sql)
cursor = connection.cursor()
cursor.execute(sql)
desc = cursor.description
return [
row[0]
for row in cursor.fetchall()
]
def fetchall_table(sql):
sql = utils.convert_boolean(sql)
cursor = connection.cursor()
cursor.execute(sql)
desc = cursor.description
return [
dict(zip([col[0] for col in desc], row))
for row in cursor.fetchall()
]
def group_messages_by_dimension_with_raw_query(query, dimension, callback):
queryset = corpus_models.Message.objects.raw(query)
message_id = corpus_models.Message._meta.model_name + "_id" #message_id
fieldname = get_field_name(dimension.field_name)
key = dimension.key
related_mgr = getattr(corpus_models.Message, dimension.key)
if hasattr(related_mgr, "RelatedObjectDoesNotExist"):
related_table = related_mgr.field.rel.to._meta.db_table
related_id = related_mgr.field.rel.to._meta.model._meta.model_name + "_id"
if related_id == "person_id":
related_id = "sender_id"
elif related_id == "messagetype_id":
related_id = "type_id"
final_query = "SELECT B.%s AS `%s`, count(*) AS `value` FROM (%s) AS A, `%s` AS B WHERE A.%s=B.id GROUP BY B.%s ORDER BY `value` DESC" %(fieldname, key, query, related_table, related_id, fieldname)
else:
if hasattr(related_mgr, "field"):
through_table = related_mgr.through._meta.db_table # e.g., corpus_message_hashtags
related_table = related_mgr.field.rel.to._meta.db_table # e.g., corpus_hashtag
related_id = related_mgr.field.rel.to._meta.model._meta.model_name + "_id" # e.g., hashtag_id
elif hasattr(related_mgr, "related"):
through_table = related_mgr.related.field.rel.through._meta.db_table # e.g., enhance_messageword
related_table = related_mgr.related.model._meta.db_table # e.g., enhance_word
related_id = related_mgr.related.model._meta.model_name + "_id" # e.g., word_id
final_query = "SELECT B.%s AS `%s`, count(*) AS `value` FROM (%s) AS A, `%s` AS B, `%s` AS C WHERE A.id=C.%s AND B.id=C.%s GROUP BY B.%s ORDER BY `value` DESC" %(fieldname, key, query, related_table, through_table, message_id, related_id, fieldname)
return callback(final_query)
def group_messages_by_words_with_raw_query(query, callback):
pattern = re.compile(r'T\d+.`text`')
results = pattern.search(query)
if results:
table = results.group()
query = query.replace("`corpus_message`.`id`, `corpus_message`.`dataset_id`, `corpus_message`.`original_id`, `corpus_message`.`type_id`, `corpus_message`.`sender_id`, `corpus_message`.`time`, `corpus_message`.`language_id`, `corpus_message`.`sentiment`, `corpus_message`.`timezone_id`, `corpus_message`.`replied_to_count`, `corpus_message`.`shared_count`, `corpus_message`.`contains_hashtag`, `corpus_message`.`contains_url`, `corpus_message`.`contains_media`, `corpus_message`.`contains_mention`, `corpus_message`.`text`",
"%s AS words, count(*) AS value" %(table))
query += "GROUP BY `words` ORDER BY `value` DESC"
return callback(query)
class DataTable(object):
"""
This class knows how to calculate appropriate visualization data
for a given pair of dimensions.
"""
def __init__(self, primary_dimension, secondary_dimension=None):
"""
Construct a DataTable for one or two dimensions.
Dimensions may be string dimension keys or
:class:`msgvis.apps.dimensions.models.CategoricalDimension` objects.
:type primary_dimension: registry.models.CategoricalDimension
:type secondary_dimension: registry.models.CategoricalDimension
:return:
"""
# Look up the dimensions if needed
if isinstance(primary_dimension, basestring):
primary_dimension = registry.get_dimension(primary_dimension)
if secondary_dimension is not None and isinstance(secondary_dimension, basestring):
secondary_dimension = registry.get_dimension(secondary_dimension)
# a dirty way
if secondary_dimension is not None and hasattr(secondary_dimension, 'key') and secondary_dimension.key == "groups":
secondary_dimension = None
self.primary_dimension = primary_dimension
self.secondary_dimension = secondary_dimension
self.mode = "default"
def set_mode(self, mode):
self.mode = mode
def render(self, queryset, desired_primary_bins=None, desired_secondary_bins=None):
"""
Given a set of messages (already filtered as necessary),
calculate the data table.
Optionally, a number of primary and secondary bins may be given.
The result is a list of dictionaries. Each
dictionary contains a key for each dimension
and a value key for the count.
"""
if not self.secondary_dimension:
# If there is only one dimension, we should be able to fall back
# on that dimension's group_by() implementation.
queryset = self.primary_dimension.group_by(queryset,
grouping_key=self.primary_dimension.key,
bins=desired_primary_bins)
return queryset.annotate(value=models.Count('id'))
else:
# Now it gets nasty...
primary_group = self.primary_dimension.get_grouping_expression(queryset,
bins=desired_primary_bins)
secondary_group = self.secondary_dimension.get_grouping_expression(queryset,
bins=desired_secondary_bins)
if primary_group is None or secondary_group is None:
# There is no data to group
return queryset.values()
queryset, internal_primary_key = self.primary_dimension.select_grouping_expression(
queryset,
primary_group)
queryset, internal_secondary_key = self.secondary_dimension.select_grouping_expression(
queryset,
secondary_group)
# Group the data
queryset = queryset.values(internal_primary_key,
internal_secondary_key)
# Count the messages
queryset = queryset.annotate(value=models.Count('id'))
# We may need to remap some fields
mapping = {}
if internal_primary_key != self.primary_dimension.key:
mapping[internal_primary_key] = self.primary_dimension.key
if internal_secondary_key != self.secondary_dimension.key:
mapping[internal_secondary_key] = self.secondary_dimension.key
if len(mapping) > 0:
return MappedValuesQuerySet.create_from(queryset, mapping)
else:
return queryset
def render_others(self, queryset, domains, primary_flag, secondary_flag, desired_primary_bins=None, desired_secondary_bins=None):
"""
Given a set of messages (already filtered as necessary),
calculate the data table.
Optionally, a number of primary and secondary bins may be given.
The result is a list of dictionaries. Each
dictionary contains a key for each dimension
and a value key for the count.
"""
# check if any of the dimensions is categorical
if not primary_flag and not secondary_flag:
return None
if not self.secondary_dimension and self.primary_dimension.is_categorical() and primary_flag:
# If there is only one dimension, we should be able to fall back
# on that dimension's group_by() implementation.
queryset = queryset.exclude(utils.levels_or(self.primary_dimension.field_name, domains[self.primary_dimension.key]))
domains[self.primary_dimension.key].append(u'Other ' + self.primary_dimension.name)
return [{self.primary_dimension.key: u'Other ' + self.primary_dimension.name, 'value': queryset.count()}]
elif self.secondary_dimension:
# both dimensions are categorical
if self.primary_dimension.is_categorical() and self.secondary_dimension.is_categorical():
original_queryset = queryset
others_results = []
if primary_flag:
domains[self.primary_dimension.key].append(u'Other ' + self.primary_dimension.name)
if secondary_flag:
domains[self.secondary_dimension.key].append(u'Other ' + self.secondary_dimension.name)
# primary others x secondary others
if primary_flag and secondary_flag:
queryset = queryset.exclude(utils.levels_or(self.primary_dimension.field_name, domains[self.primary_dimension.key]))
queryset = queryset.exclude(utils.levels_or(self.secondary_dimension.field_name, domains[self.secondary_dimension.key]))
others_results.append({self.primary_dimension.key: u'Other ' + self.primary_dimension.name,
self.secondary_dimension.key: u'Other ' + self.secondary_dimension.name,
'value': queryset.count()})
# primary top ones x secondary others
if secondary_flag:
queryset = original_queryset
queryset = queryset.filter(utils.levels_or(self.primary_dimension.field_name, domains[self.primary_dimension.key]))
queryset = queryset.exclude(utils.levels_or(self.secondary_dimension.field_name, domains[self.secondary_dimension.key]))
queryset = self.primary_dimension.group_by(queryset,
grouping_key=self.primary_dimension.key)
queryset = queryset.annotate(value=models.Count('id'))
results = list(queryset)
for r in results:
r[self.secondary_dimension.key] = u'Other ' + self.secondary_dimension.name
others_results.extend(results)
# primary others x secondary top ones
if primary_flag:
queryset = original_queryset
queryset = queryset.exclude(utils.levels_or(self.primary_dimension.field_name, domains[self.primary_dimension.key]))
queryset = queryset.filter(utils.levels_or(self.secondary_dimension.field_name, domains[self.secondary_dimension.key]))
queryset = self.secondary_dimension.group_by(queryset,
grouping_key=self.secondary_dimension.key)
queryset = queryset.annotate(value=models.Count('id'))
results = list(queryset)
for r in results:
r[self.primary_dimension.key] = u'Other ' + self.primary_dimension.name
others_results.extend(results)
return others_results
# primary categorical and secondary quantitative
elif self.primary_dimension.is_categorical() and primary_flag and not self.secondary_dimension.is_categorical():
queryset = queryset.exclude(utils.levels_or(self.primary_dimension.field_name, domains[self.primary_dimension.key]))
domains[self.primary_dimension.key].append(u'Other ' + self.primary_dimension.name)
queryset = self.secondary_dimension.group_by(queryset,
grouping_key=self.secondary_dimension.key,
bins=desired_secondary_bins)
queryset = queryset.annotate(value=models.Count('id'))
results = list(queryset)
for r in results:
r[self.primary_dimension.key] = u'Other ' + self.primary_dimension.name
return results
# primary quantitative and secondary categorical
elif not self.primary_dimension.is_categorical() and self.secondary_dimension.is_categorical() and secondary_flag:
queryset = queryset.exclude(utils.levels_or(self.secondary_dimension.field_name, domains[self.secondary_dimension.key]))
domains[self.secondary_dimension.key].append(u'Other ' + self.secondary_dimension.name)
queryset = self.primary_dimension.group_by(queryset,
grouping_key=self.primary_dimension.key,
bins=desired_primary_bins)
queryset = queryset.annotate(value=models.Count('id'))
results = list(queryset)
for r in results:
r[self.secondary_dimension.key] = u'Other ' + self.secondary_dimension.name
return results
def domain(self, dimension, queryset, filter=None, exclude=None, desired_bins=None):
"""Return the sorted levels in this dimension"""
if filter is not None:
queryset = dimension.filter(queryset, **filter)
if exclude is not None:
queryset = dimension.exclude(queryset, **exclude)
domain = dimension.get_domain(queryset, bins=desired_bins)
labels = dimension.get_domain_labels(domain)
return domain, labels
def groups_domain(self, dimension, queryset_all, group_querysets, desired_bins=None):
"""Return the sorted levels in the union of groups in this dimension"""
if dimension.is_related_categorical():
query = ""
for idx, queryset in enumerate(group_querysets):
if idx > 0:
query += " UNION "
query += "(%s)" %(utils.quote(str(queryset.query)))
domain = group_messages_by_dimension_with_raw_query(query, dimension, fetchall)
else:
queryset = queryset_all
domain = dimension.get_domain(queryset, bins=desired_bins)
labels = dimension.get_domain_labels(domain)
return domain, labels
def filter_search_key(self, domain, labels, search_key):
match_domain = []
match_labels = []
for i in range(len(domain)):
level = domain[i]
if level is not None and level.lower().find(search_key.lower()) != -1 :
match_domain.append(level)
if labels is not None:
match_labels.append(labels[i])
return match_domain, match_labels
def generate(self, dataset, filters=None, exclude=None, page_size=100, page=None, search_key=None, groups=None):
"""
Generate a complete data group table response.
This includes 'table', which provides the non-zero
message frequency for each combination of primary and secondary dimension values,
respecting the filters.
It also includes 'domains', which provides, for both
primary and secondary dimensions, the levels of the
dimension irrespective of filters (except on those actual dimensions).
"""
if (groups is None):
queryset = dataset.message_set.all()
# Filter out null time
queryset = queryset.exclude(time__isnull=True)
if dataset.start_time and dataset.end_time:
range = dataset.end_time - dataset.start_time
buffer = timedelta(seconds=range.total_seconds() * 0.1)
queryset = queryset.filter(time__gte=dataset.start_time - buffer,
time__lte=dataset.end_time + buffer)
unfiltered_queryset = queryset
# Filter the data (look for filters on the primary/secondary dimensions at the same time
primary_filter = None
secondary_filter = None
if filters is not None:
for filter in filters:
dimension = filter['dimension']
queryset = dimension.filter(queryset, **filter)
if dimension == self.primary_dimension:
primary_filter = filter
if dimension == self.secondary_dimension:
secondary_filter = filter
primary_exclude = None
secondary_exclude = None
if exclude is not None:
for exclude_filter in exclude:
dimension = exclude_filter['dimension']
queryset = dimension.exclude(queryset, **exclude_filter)
if dimension == self.primary_dimension:
primary_exclude = exclude_filter
if dimension == self.secondary_dimension:
secondary_exclude = exclude_filter
domains = {}
domain_labels = {}
max_page = None
queryset_for_others = None
# flag is true if the dimension is categorical and has more than MAX_CATEGORICAL_LEVELS levels
primary_flag = False
secondary_flag = False
# Include the domains for primary and (secondary) dimensions
domain, labels = self.domain(self.primary_dimension,
unfiltered_queryset,
primary_filter, primary_exclude)
# paging the first dimension, this is for the filter distribution
if primary_filter is None and self.secondary_dimension is None and page is not None:
if search_key is not None:
domain, labels = self.filter_search_key(domain, labels, search_key)
start = (page - 1) * page_size
end = min(start + page_size, len(domain))
max_page = (len(domain) / page_size) + 1
# no level left
if len(domain) == 0 or start > len(domain):
return None
domain = domain[start:end]
if labels is not None:
labels = labels[start:end]
queryset = queryset.filter(utils.levels_or(self.primary_dimension.field_name, domain))
else:
if (self.mode == 'enable_others' or self.mode == 'omit_others') and \
self.primary_dimension.is_categorical() and len(domain) > MAX_CATEGORICAL_LEVELS:
primary_flag = True
domain = domain[:MAX_CATEGORICAL_LEVELS]
queryset_for_others = queryset
queryset = queryset.filter(utils.levels_or(self.primary_dimension.field_name, domain))
if labels is not None:
labels = labels[:MAX_CATEGORICAL_LEVELS]
domains[self.primary_dimension.key] = domain
if labels is not None:
domain_labels[self.primary_dimension.key] = labels
if self.secondary_dimension:
domain, labels = self.domain(self.secondary_dimension,
unfiltered_queryset,
secondary_filter, secondary_exclude)
if (self.mode == 'enable_others' or self.mode == 'omit_others') and \
self.secondary_dimension.is_categorical() and \
len(domain) > MAX_CATEGORICAL_LEVELS:
secondary_flag = True
domain = domain[:MAX_CATEGORICAL_LEVELS]
if queryset_for_others is None:
queryset_for_others = queryset
queryset = queryset.filter(utils.levels_or(self.secondary_dimension.field_name, domain))
if labels is not None:
labels = labels[:MAX_CATEGORICAL_LEVELS]
domains[self.secondary_dimension.key] = domain
if labels is not None:
domain_labels[self.secondary_dimension.key] = labels
# Render a table
table = self.render(queryset)
if self.mode == "enable_others" and queryset_for_others is not None:
# adding others to the results
table_for_others = self.render_others(queryset_for_others, domains, primary_flag, secondary_flag)
table = list(table)
table.extend(table_for_others)
results = {
'table': table,
'domains': domains,
'domain_labels': domain_labels
}
if max_page is not None:
results['max_page'] = max_page
else:
domains = {}
domain_labels = {}
max_page = None
queryset_for_others = None
# flag is true if the dimension is categorical and has more than MAX_CATEGORICAL_LEVELS levels
primary_flag = False
secondary_flag = False
primary_filter = None
secondary_filter = None
primary_exclude = None
secondary_exclude = None
queryset = dataset.message_set.all()
queryset = queryset.exclude(time__isnull=True)
if dataset.start_time and dataset.end_time:
range = dataset.end_time - dataset.start_time
buffer = timedelta(seconds=range.total_seconds() * 0.1)
queryset = queryset.filter(time__gte=dataset.start_time - buffer,
time__lte=dataset.end_time + buffer)
if filters is not None:
for filter in filters:
dimension = filter['dimension']
queryset = dimension.filter(queryset, **filter)
if dimension == self.primary_dimension:
primary_filter = filter
if dimension == self.secondary_dimension:
secondary_filter = filter
if exclude is not None:
for exclude_filter in exclude:
dimension = exclude_filter['dimension']
queryset = dimension.exclude(queryset, **exclude_filter)
if dimension == self.primary_dimension:
primary_exclude = exclude_filter
if dimension == self.secondary_dimension:
secondary_exclude = exclude_filter
queryset_all = queryset
#queryset = corpus_models.Message.objects.none()
group_querysets = []
group_labels = []
#message_list = set()
for group in groups:
group_obj = groups_models.Group.objects.get(id=group)
if group_obj.order > 0:
group_labels.append("#%d %s"%(group_obj.order, group_obj.name))
else:
group_labels.append("%s"%(group_obj.name))
queryset = group_obj.messages
# Filter out null time
queryset = queryset.exclude(time__isnull=True)
if dataset.start_time and dataset.end_time:
range = dataset.end_time - dataset.start_time
buffer = timedelta(seconds=range.total_seconds() * 0.1)
queryset = queryset.filter(time__gte=dataset.start_time - buffer,
time__lte=dataset.end_time + buffer)
unfiltered_queryset = queryset
# Filter the data (look for filters on the primary/secondary dimensions at the same time
if filters is not None:
for filter in filters:
dimension = filter['dimension']
queryset = dimension.filter(queryset, **filter)
if exclude is not None:
for exclude_filter in exclude:
dimension = exclude_filter['dimension']
queryset = dimension.exclude(queryset, **exclude_filter)
group_querysets.append(queryset)
#########################################################################################################################
# deal with union distribution
# This is due to union of queries in django does not work...
# super ugly. Refactoring is required.
# Include the domains for primary and (secondary) dimensions
domain, labels = self.groups_domain(self.primary_dimension,
queryset_all, group_querysets)
# paging the first dimension, this is for the filter distribution
if primary_filter is None and self.secondary_dimension is None and page is not None:
if search_key is not None:
domain, labels = self.filter_search_key(domain, labels, search_key)
start = (page - 1) * page_size
end = min(start + page_size, len(domain))
max_page = (len(domain) / page_size) + 1
# no level left
if len(domain) == 0 or start > len(domain):
return None
domain = domain[start:end]
if labels is not None:
labels = labels[start:end]
else:
if (self.mode == 'enable_others' or self.mode == 'omit_others') and \
self.primary_dimension.is_categorical() and len(domain) > MAX_CATEGORICAL_LEVELS:
primary_flag = True
domain = domain[:MAX_CATEGORICAL_LEVELS]
if labels is not None:
labels = labels[:MAX_CATEGORICAL_LEVELS]
domains[self.primary_dimension.key] = domain
if labels is not None:
domain_labels[self.primary_dimension.key] = labels
if self.secondary_dimension:
domain, labels = self.groups_domain(self.secondary_dimension,
queryset_all, group_querysets)
if (self.mode == 'enable_others' or self.mode == 'omit_others') and \
self.secondary_dimension.is_categorical() and \
len(domain) > MAX_CATEGORICAL_LEVELS:
secondary_flag = True
domain = domain[:MAX_CATEGORICAL_LEVELS]
if labels is not None:
labels = labels[:MAX_CATEGORICAL_LEVELS]
domains[self.secondary_dimension.key] = domain
if labels is not None:
domain_labels[self.secondary_dimension.key] = labels
#########################################################################################################################
group_tables = []
for queryset in group_querysets:
queryset_for_others = queryset
if (self.mode == 'enable_others' or self.mode == 'omit_others') and \
self.primary_dimension.is_categorical():
queryset = queryset.filter(utils.levels_or(self.primary_dimension.field_name, domains[self.primary_dimension.key]))
if self.secondary_dimension:
if (self.mode == 'enable_others' or self.mode == 'omit_others') and \
self.secondary_dimension.is_categorical():
if queryset_for_others is None:
queryset_for_others = queryset
queryset = queryset.filter(utils.levels_or(self.secondary_dimension.field_name, domains[self.secondary_dimension.key]))
# Render a table
if self.primary_dimension.key == "words":
table = group_messages_by_words_with_raw_query(utils.quote(str(queryset.query)), fetchall_table)
else:
table = self.render(queryset)
if self.mode == "enable_others" and queryset_for_others is not None:
# adding others to the results
table_for_others = self.render_others(queryset_for_others, domains, primary_flag, secondary_flag)
table = list(table)
table.extend(table_for_others)
group_tables.append(table)
if self.secondary_dimension is None:
final_table = []
for idx, group_table in enumerate(group_tables):
for item in group_table:
item['groups'] = groups[idx]
final_table.extend(group_table)
domains['groups'] = groups
domain_labels['groups'] = group_labels
results = {
'table': final_table,
'domains': domains,
'domain_labels': domain_labels
}
else:
tables = []
for idx, group_table in enumerate(group_tables):
tables.append({
'group_id': groups[idx],
'group_name': group_labels[idx],
'table': group_table
})
results = {
'tables': tables,
'domains': domains,
'domain_labels': domain_labels
}
if max_page is not None:
results['max_page'] = max_page
return results
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,115 |
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The magazine Medical Science
Published: 2013-10-14 00:00 | Updated: 2014-10-29 10:28
ADHD drug effective for people with dependency
People with ADHD and substance dependence rarely respond as they should to ADHD medication. A randomised study from Karolinska Institutet now shows that it is possible to obtain the desired efficacy by administering the drug in higher doses. The results of the study are published in the scientific journal Addiction.
ADHD is much more common in people who use drugs than in the population at large. ADHD can be treated with methylphenidate, a CNS stimulant used for both children and adults. However, no previous studies have been able to show that methylphenidate is effective against ADHD in people with dependence. One possible reason for this lack of potency is that the doses tested were too low to have an effect on people with ADHD and long-standing substance dependence, since such individuals have often developed a tolerance to CNS stimulants.
In this latest study, researchers examined the effect of the medication on prison inmates with ADHD and amphetamine dependenc, using doses up to double those administered in previous studies. What they found was that the experimental group had fewer relapses into drug use, displayed fewer ADHD symptoms and adhered to their treatment regimens for longer than the placebo group.
"We've shown for the first time that ADHD in these patients is treatable," says lead-author Dr Maija Konstenius at the Department of Clinical Neuroscience. "Moreover, the treatment led to fewer relapses to drug use, which is a very significant finding since a return to crime is often linked to drug abuse in this group."
The present study was a randomised placebo-controlled clinical trial involving 54 incarcerated men diagnosed with ADHD and with amphetamine dependence. The participants were randomly assigned to either of two groups, one that received methylphenidate (Concerta) and one that received an identical placebo (inactive pill) for a total of 24 weeks. The participants began their treatment two weeks before their release from prison and then continued it during non-custodial care. They also received psychological therapy to help them handle the risk of relapsing into drug use.
The study was conducted in partnership with the Stockholm County Council and the Swedish Prison and Probation Service and was financed by grants from the Swedish Research Council and other bodies. Study leader was Professor Johan Franck, Karolinska Institutet.
Methylphenidate for attention deficit hyperactivity disorder and drug relapse in criminal offenders with substance dependence: a 24-week randomized placebo-controlled trial.
Konstenius M, Jayaram-Lindström N, Guterstam J, Beck O, Philips B, Franck J
Addiction 2014 Mar;109(3):440-9
Webb Admin 2014-10-29
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\section{Introduction}
The study of information inequalities is a subfield of information theory that
describes linear constraints on the entropies of
finite collections of jointly distributed discrete random variables.
Historically, the known information inequalities were orignally all special
cases of Shannon's conditional mutual information inequality $I(X;Y|Z)\ge 0$,
but later were generalized to other types of inequalities, called non-Shannon inequalities.
Information inequalities have been shown to be useful for computing upper bounds
on the network coding capacities of certain networks.
Analagously,
the study of linear rank inequalities is a topic of linear algebra,
which describes linear constraints on the dimensions of collections of subspaces of finite dimensional vector spaces.
In fact, the set of all information inequalities can be viewed as subclass of
the set of all linear rank inequalities.
Information inequalities hold over all collections of a certain number of random variables.
In constrast, linear rank inequalities may hold over only certain vector spaces,
such as those whose scalars have particular field characteristics.
In this paper,
we present two new linear rank inequalities over finite fields,
which are not information inequalities,
and with the peculiar property
that they only hold for certain fields,
depending on the associated vector space.
The first inequality is shown to hold over all vector spaces when the field characteristic is anything but three
(Theorem~\ref{thm:T8}),
but does not always hold when the field characteristic is three
(Theorem~\ref{thm:T8-char3}).
In contrast,
the second inequality is shown to hold over all vector spaces when the field characteristic is three
(Theorem~\ref{thm:nonT8}),
but does not always hold when the field characteristic is not three
(Theorem~\ref{thm:non-T8-non-char3}).
We also show how these inequalities can be used to obtain bounds on the capacities of
certain networks
(Corollaries~\ref{cor:T8-capacity} and \ref{cor:nonT8-capacity}).
It will be assumed that the reader has familiarity with linear algebra,
finite fields, information theory, and network coding.
Nevertheless, we will give some brief tutorial descriptions of these topics for completeness.
\subsection{Background}
In 2000, Ahlswede, Cai, Li, and Yeung introduced the field of Network Coding~\cite{networkcoding1}
and showed that coding can outperform routing in
directed acyclic networks
\footnote{In what follows, by ``network'' we shall always mean a directed acyclic network.}
There are presently no known
algorithms to determine the capacity or the linear capacity of a given network.
In fact, it is not even known if such algorithms exist.
Information inequalities are linear inequalities that hold for all jointly
distributed random variables,
and Shannon inequalities are information
inequalities of a certain form~\cite{shannon}.
Both are defined in Section~\ref{section:LRI}.
It is known~\cite{Yeung-book} that all information
inequalities containing three or fewer variables are Shannon inequalities.
The first ``non-Shannon'' information inequality was of four variables and was
published in 1998 by Zhang and Yeung~\cite{Zhang-Yeung}.
Since 1998, various other non-Shannon inequalities have been found,
for example, by
Ln\v{e}ni\v{c}ka~\cite{Lnenicka},
Makarychev, Makarychev, Romashchenko, and Vereshchagin~\cite{MMRV},
Zhang~\cite{Zhang13},
Zhang and Yeung~\cite{Zhang-Yeung14},
Dougherty, Freiling, and Zeger~\cite{DFZ2}, and
Mat\'{u}\v{s}~\cite{Matus}.
Additionally, in 2007, Mat\'{u}\v{s} demonstrated an infinite collection of independent
non-Shannon information inequalities~\cite{Matus}
and there were necessarily an infinite number of such inequalities.
In 2008, Xu, Wang, and Sun~\cite{XuWangSun} also gave an infinite list of inequalities
but did not establish their necessity.
There is a close connection between information inequalities and network coding~\cite{ChanGrant}.
Capacities of some networks have been computed by finding
matching lower and upper bounds~\cite{nonshannon}.
Lower bounds have been found by deriving coding solutions.
Upper bounds have been found by using
information inequalities and treating the sources as independent random variables
that are uniformly distributed over the alphabet.
One ``holy grail'' problem of network coding is to develop an algorithm to compute the coding capacity of an arbitrary network.
If such an algorithm exists,
information inequalities may potentially play a role in the solution.
It has been shown that linear codes are insufficient for network coding in
general~\cite{insufficient}.
However, linear codes may be desirable to use in practice due to ease of analysis
and implementation.
It has been shown that the coding capacity
is independent of the alphabet size~\cite{Cannons-Dougherty-Freiling-Zeger05}.
However, the linear coding capacity is dependent on alphabet size,
or more specifically the field characteristic.
In other words, one can potentially achieve a higher rate of linear
communication by choosing one characteristic over another.
To provide
upper bounds for the linear coding capacity for a particular field one can look
at linear rank inequalities~\cite{rateregions-journal}.
Linear rank inequalities are
linear inequalities that are always satisfied by rank
\footnote{Throughout this paper,
we will use the terminology ``rank'' of a subspace to mean the dimension of the subspace
(i.e. the rank of a matrix whose columns are a basis for the subspace),
in order to parallel the terminology of matroid theory.
}
of subspaces of a vector space.
All information inequalities are linear rank inequalities but not all
linear rank inequalities are information inequalities.
The first example of a
linear rank inequality that is not an information inequality was found by
Ingleton~\cite{ingleton}.
Information inequalities can provide an upper bound
for the capacity of a network, but this upper bound would hold for all alphabets.
Therefore, to determine the linear coding capacity over a certain
characteristic one would have to consider linear rank inequalities.
All linear rank inequalities up to and including five variables are known
and none of these depend on the vector spaces' field characteristics~\cite{5var}.
The set of all linear rank inequalities for six variables has not yet been determined.
Characteristic-dependent linear rank inequalities
are given, for example, in
~\cite{Blasiak-Kleinberg-Lubetzky}
and ~\cite{rateregions-journal}.
An inequality is given in~\cite{rateregions-journal} which is valid for characteristic two and another inequality is given which is valid
for every characteristic except for two.
These inequalities are then used to
provide upper bounds for the linear coding capacity of two networks.
In the present paper, we give two characteristic-dependent linear rank inequalities on eight variables.
One is valid for characteristic three and the other is valid for
every characteristic except for three.
These inequalities are then used to
provide upper bounds for the linear coding capacity of two networks.
It is our intention that the techniques presented here may prove useful or
otherwise motivate further progress in determining network capacities.
\subsection{Matroids}
In this section a very brief review of matroids is given which
will enable discussion in subsequent sections of a matroid-based
method for constructing a particular network that helps in the
derivation of the linear rank inequalities presented in this paper.
A matroid is an abstract structure that captures a notion of ``independence" that is found in
finite dimensional vector spaces, graphs, and various other mathematical topics.
We will follow the notation and results of~\cite{matroidBook}.
\begin{defn}
A \textit{matroid}, $M$, is a pair $(E,I)$, where $E$ is a
finite set and $I$ is a set of subsets of $E$ that satisfies the
following properties:
\begin{enumerate}
\item[(I1)] $\emptyset \in I$.
\item[(I2)] $\forall A,B\subset E$, if $A \subseteq B \in I$, then $A \in I$.
\item[(I3)] $\forall A,B\subset E$,
if $A,B \in I$ and $|A| > |B|$,
then $\exists u \in A\setminus B$ such that $B \cup \{u\} \in I$.
\end{enumerate}
\end{defn}
The sets in $I$ are called \textit{independent sets}.
If a subset of $E$ is not in $I$, then it is called \textit{dependent}.
An example of a matroid is obtained from linear algebra.
Let $F$ be a finite field and let $V(m,F)$ be the vector space of all $m$-dimensional vectors whose
components are elements of $F$.
Suppose $A$ is an $m\times n$ matrix over $F$.
Let $E = \{1,\ldots,n\}$ and $I$ be the set of all $X \subseteq E$ such
that the multiset of columns of $A$ indexed by the elements of $X$ is linearly
independent in the vector space $V(m,F)$.
Then $M = (E,I)$ is a matroid called the \textit{vector matroid} of $A$.
A matroid is said to be \textit{representable} over the field $F$
if it is isomorphic to some vector matroid over $V(m,F)$.
For example, if $F$ is the binary field and
$$A = \bordermatrix{& a & b & c & d & e\cr
& 1 & 0 & 0 & 1 & 1\cr
& 0 & 1 & 0 & 0 & 1}$$
where $a,b,c,d,e$ denote the columns of $A$ from left to right,
then $M = (E,I)$ is a vector matroid of $A$,
where $E = \{a,b,c,d,e\}$ and
\begin{align*}
I &= \{\emptyset, \{a\}, \{b\}, \{d\}, \{e\}, \{a,b\}, \{a,e\}, \{b,d\}, \{b,e\}, \{d,e\}\}.
\end{align*}
A \textit{base} is a maximal independent set.
Let $B(M)$ denote the set of all bases of a matroid $M$.
In our example,
$$B(M) = \{\{a,b\}, \{a,e\}, \{b,d\}, \{b,e\}, \{d,e\}\}.$$
It is well known that all the bases of a matroid are of the same cardinality.
If we let $X \subseteq E$ and $I|X = \{i \subseteq X: i \in I\}$,
then it is easy to see that $(X,I|X)$ is a matroid.
The \textit{rank} of $X$, denoted by $r(X)$, is defined to be the cardinality of a base in $M|X$.
In our example, $r(M) = 2$.
A \textit{circuit} is a minimal dependent set.
The circuits in our example are $\{\{c\}, \{a,d\}, \{a,b,e\}, \{b,d,e\}\}$.
\subsection{Information Theory and Linear rank Inequalities}\label{section:LRI}
In this section we will use the information theoretic concepts of entropy and mutual information
to define and use the linear algebraic concept of linear rank inequalities.
Connections between information inequalities and linear rank inequalities is also discussed.
Let $A,B,C$ be collections of discrete random variables over a finite alphabet
$\mathcal{X}$, and let $p$ be the probability mass function of $A$.
The
\textit{entropy} of $A$ is defined by
\begin{align*}
H(A) &= -\sum_u p(u)\log_{|\mathcal{X}|}p(u).
\end{align*}
The \textit{conditional entropy} of $A$ given $B$ is
\begin{align}
H(A|B) &= H(A,B) - H(B)\label{h1},
\end{align}
the \textit{mutual information} between $A$ and $B$ is
\begin{align}
I(A;B) &= H(A) - H(A|B) = H(A) + H(B) - H(A,B),\label{h2}
\end{align}
and the \textit{conditional mutual information} between $A$ and $B$ given $C$ is
\begin{align}
I(A;B|C) &= H(A|C) - H(A|B,C) = H(A,C) + H(B,C) - H(C) - H(A,B,C)\label{h3}.
\end{align}
We will make use of the following basic information-theoretic facts \cite{Yeung-book}:
\begin{align}
0 &= H(\emptyset) \label{h4}\\
0 &\le H(A) = H(A|\emptyset)\label{h5}\\
0 &\le H(A|B)\label{h6}\\
0 &\le I(A;B)\label{h7}\\
H(A,B|C) &\le H(A|C) + H(B|C)\label{h8}\\
H(A|B,C) &\le H(A|B) \leq H(A,C|B)\label{h9}\\
I(A;B,C) &= I(A;B|C) + I(A;C). \label{h12}
\end{align}
The equations (\ref{h5})-(\ref{h9}) were originally given by Shannon in 1948
\cite{shannon}, and can all be obtained from the single inequality $I(A;B|C) \geq 0$.
\begin{defn}
Let $q$ be a positive integer, and let $S_1,\ldots,S_k$ be subsets of
$\{1,\ldots,q\}$.
Let $\alpha_i \in \mathbb{R}$ for $1\leq i \leq k$.
A linear
inequality of the form
\begin{align}
\alpha_1 H(\{A_i:i\in S_1\}) + \dots + \alpha_k H(\{A_i:i\in S_k\}) &\ge 0
\label{inequality-format}
\end{align}
is called an \textit{information inequality} if it holds for all jointly distributed random variables $A_1,\ldots,A_q$.
\end{defn}
As an example, taking $q = 2$, $S_1 = \{1\}$, $S_2 = \{2\}$, $S_3 = \emptyset$,
$S_4 = \{1,2\}$, $\alpha_1 = \alpha_2 = 1$, $\alpha_4 = -1$, and using
(\ref{h8}) shows that $H(A_1) + H(A_2) - H(A_1,A_2) \ge 0$ is an information
inequality.
\bigskip
A \textit{Shannon information inequality} is any information inequality that can be expressed as a finite sum of the form
\begin{align*}
& \sum_i \alpha_i I(A_i;B_i|C_i) \ge 0
\end{align*}
where each $\alpha_i$ is a nonnegative real number.
Any information inequality that cannot
be expressed in the form above will be called a \textit{non-Shannon information inequality}.
Linear rank inequalities are closely
related to information inequalities.
In fact, in order to describe linear rank inequalities
we will borrow notation from information theory to use in the context of
linear algebra in the following manner.
Suppose $A$ and $B$ are subspaces of a given vector space $V$,
and let $\langle A, B\rangle$ denote the span of $A\cup B$.
We will let $H(A)$ denote the rank of $A$,
and let $H(A,B)$ denote the rank of $\langle A, B\rangle$.
The meanings of some other information theoretic notation in the context of linear algebra
then follows from~\eqref{h1}-\eqref{h3}.
Specifically, note that
the conditional entropy notation $H(A|B)$
denotes the excess rank of subspace $A$ over that of subspace $A\cap B$,
or equivalently, the codimension of $A\cap B$ in $A$;
and the mutual information notation $I(A;B)$ denotes the rank of $A\cap B$.
A \textit{linear rank inequality}
over a vector space $V$
is a linear inequality
of the form in \eqref{inequality-format},
that is satisfied
by every assignment of subspaces of $V$ to the variables $A_1, \dots, A_q$.
All information inequalities are
linear rank inequalities over all finite vector spaces,
but not all linear rank inequalities are information inequalities.
For background material on these concepts, the reader is referred to
Hammer, Romashchenko, Shen, and Vereshchagin \cite{rank}.
The first known example of a linear rank inequality
over all finite vector spaces
that is not an information inequality is the \textit{Ingleton inequality}~\cite{ingleton}:
\begin{align*}
I(A;B) &\le I(A;B|C) + I(A;B|D) + I(C;D).
\end{align*}
To see that the Ingleton inequality is not an information inequality,
let $A,B,C,D$ be binary random variables, and let $X = (A,B,C,D)$ with probabilities:
\begin{align*}
P(X=0000) &= 1/4\\
P(X=1111) &= 1/4\\
P(X=0101) &= 1/4\\
P(X=0110) &= 1/4.
\end{align*}
Then the Ingleton inequality fails since:
\begin{align*}
\underbrace{I(A;B)}_{(5-\log_2 27)/2}
&- \underbrace{I(A;B|C)}_{0} - \underbrace{I(A;B|D)}_{0} - \underbrace{I(C;D)}_{0}
> 0.
\end{align*}
\subsection{Network Coding}
In this section,
we will briefly
review some concepts of network coding.
This will enable the discussion later in this paper of our construction of linear rank inequalities
using networks constructed from two particular matroids (T8 and non-T8).
For more details on network coding, see \cite{networkcodingbook}.
A \textit{network} is a finite, directed, acyclic multigraph with messages and demands.
Network \textit{messages} are arbitrary vectors of $k$ symbols over a finite alphabet $\mathcal{A}$.
Each network edge carries a vector of $n$ symbols from $\mathcal{A}$.
Each message originates at
a particular node called the \textit{source node} for that message and is
required by one or more \textit{demand nodes}.
When we draw a network,
a message variable appearing above a node indicates the message is generated by such nod
\footnote{
We note that in Figures~\ref{fig:T8} and \ref{fig:nonT8},
for convenience,
we label source messages above nodes lying in both the top and bottom layers in each diagram.
This is meant to indicate that there is, in fact, a separate (but hidden) distinct node for each such
source message,
whose out-edges go directly to the nodes labeled by the source message in the top and bottem layers.
},
and
a message variable appearing below a node indicates the message is demanded by such node,
For a given network, the values of $k$ and $n$ can be chosen in order to implement certain
codes and to obtain certain throughput $k/n$.
The inputs to a network node are the vectors carried on its in-edges as well as
the messages, if any, generated at the node.
The outputs of a network node are the
packets carried on its out-edges as well as any demanded messages at the node.
Each output of a node must be a function only of its inputs.
A \textit{coding solution} for the
network is an assignment of such functions to the network edges.
When the values of $k$ and $n$ need
to be emphasized, the coding solution will be called a $(k,n)$-coding solution.
The \textit{capacity} of a network is defined as:
\begin{align*}
\mathcal{C} &= \sup\{ k/n : \exists\mbox{ a $(k,n)$-coding solution}\}.
\end{align*}
A solution is called a \textit{linear solution},
if the alphabet $\mathcal{A}$ is a finite field
and the edge functions are linear
(i.e. linear combinations of their input vectors where the coefficients are matrices over the field).
The \textit{linear capacity} is defined the same
as the capacity but restricting solutions to be linear.
It is also easily verified that if $x$ is a message,
then $H(x) = k$,
and if $x$ is a vector carried by an edge, then $H(x) \leq n$.
\begin{figure}
\begin{center}
\psfrag{n1}{\LARGE $n_{1}$}
\psfrag{n2}{\LARGE $n_{2}$}
\psfrag{n3}{\LARGE $n_{3}$}
\psfrag{n4}{\LARGE $n_{4}$}
\psfrag{n5}{\LARGE $n_{5}$}
\psfrag{n6}{\LARGE $n_{6}$}
\includegraphics[scale=.75]{butterfly.eps}
\end{center}
\caption{\label{fig:butterfly} The Butterfly network with source messages $x$ and $y$,
generated by source nodes $n_1$ and $n_2$, respectively.
Demand nodes $n_5$ and $n_6$ demand messages $y$ and $x$, respectively.
}
\end{figure}
Let us illustrate a method for finding capacity bounds by examining the well-known Butterfly network,
depicted in Figure~\ref{fig:butterfly}.
We assume the network messages $x$ and $y$
are independent, $k$-dimensional, random vectors with uniformly distributed components.
Then in any solution it must be the case that
\begin{align}
H(y|x,z) &= 0 \label{butterfly:1}
\end{align}
since $y$ is a function of $x$ and $z$,
and also that
\begin{align}
2k &= H(x) + H(y)\\
&= H(x,y) & \text{[from indep. of x and y]}\nonumber\\
&\leq H(x,y,z) & \text{ [from \eqref{h9}]}\nonumber\\
&= H(x,z) + H(y|x,z) & \text{[from \eqref{h1}]}\nonumber\\
&= H(x,z) & \text{[from (\ref{butterfly:1})]}\nonumber\\
&\le H(x) + H(z) & \text{[from \eqref{h8}]}\nonumber\\
&\le k + n. \label{rv-inequality}
\end{align}
This implies $2k \le k + n$,
or equivalently $k/n \le 1$.
Since this bound holds for all choices of $k$ and $n$,
the coding capacity must be at most $1$.
On the other hand,
a solution with $k = n = 1$ is obtained by taking
$z = x+y$ over any finite field alphabet,
so the coding capacity is at least $1$.
Thus the coding capacity for the Butterfly network is the same as the linear coding
capacity which is exactly equal to $1$.
The inequalities in \eqref{rv-inequality} were based on random variables $x,y,z$.
Later, in the proofs of Corollaries~\ref{cor:T8-capacity} and~\ref{cor:nonT8-capacity},
we will obtain bounds on the capacities of networks by using linear rank inequalities,
instead of information inequalities.
In those cases, certain vector subspaces will be used instead of random variables,
but the procedure will appear similar.
\newpage
\section{Preliminaries}
In this section, we given some technical lemmas which will be useful for proving the main results of the paper.
If $A$ is a subspace of vector space $V$,
and $\overline{A}$ is a subspace of $A$,
then we will use the notation
$\Codim_A(\overline{A}) = \Dim(A) - \Dim(\overline{A})$
to represent the codimension
of $\overline{A}$ in $A$.
We will omit the subscript when
it is obvious from the context which space the codimension is with respect to.
\begin{lem}\label{lemma1}
\cite{rateregions-journal} Let $V$ be a finite dimensional vector space with subspaces
$A$ and $B$.
Then the subspace $A \cap B$ has codimension at most
$\Codim(A) + \Codim(B)$ in $V$.
\end{lem}
\begin{proof}
We know $H(A) + H(B) - I(A; B) = H( A,B) \le H(V)$.
Then adding $H(V)$ to both
sides of the inequality gives $H(V) - I(A; B) \le H(V)-H(A) + H(V) - H(B)$.
Thus, $\Codim(A\cap B) \le \Codim(A) + \Codim(B)$.
\end{proof}
\begin{lem}\label{lemma2}
\cite{rateregions-journal} Let $A$ and $B$ be vector spaces over the same finite scalar field and with subspaces
$\overline{A}$ and $\overline{B}$, respectively.
Let $f:A \rightarrow B$ be a
linear function such that $f(A\backslash \overline{A}) \subseteq B\backslash \overline{B}$.
Then the codimension of $\overline{A}$ in $A$ is at most the
codimension of $\overline{B}$ in $B$.
\end{lem}
\begin{proof}
Suppose
a base for $A$ consists of a base for $\overline{A}$
together with the vectors $a_1, \ldots, a_n$.
Let $\gamma_1,\ldots,\gamma_n$ be field elements which are not all zero.
Then $\gamma_1 a_1+ \cdots + \gamma_n a_n\in A\setminus\overline{A}$,
so
$\gamma_1f(a_1)+ \cdots + \gamma_nf(a_n) = f(\gamma_1 a_1+ \cdots + \gamma_n a_n) \in B\setminus\overline{B}$.
Thus, the vectors
$f(a_1),\ldots,f(a_n)$ are linearly independent over the subspace $\overline{B}$,
and therefore
$\Codim_A(\overline{A})
= n \le
\Codim_B(\overline{B})$.
\end{proof}
\begin{lem}\label{lemma3}
\cite{rateregions-journal}
Let $A$ and $B$ be vector spaces over the same finite scalar field,
let $\overline{B}$ be a subspace of $B$,
and let $f:A \rightarrow B$ be a linear function.
Then $f(t) \in \overline{B}$ on a subspace of $A$ of codimension at most the codimension of
$\overline{B}$.
\end{lem}
\begin{proof}
Let $\overline{A} = \{t \in A: f(t) \in \overline{B}\}$.
Then $f(A\backslash \overline{A}) \subseteq B\backslash \overline{B}$
and the result follows from Lemma~\ref{lemma2}.
\end{proof}
\begin{lem}\label{lemma4}
\cite{rateregions-journal} Let $V$ be a finite dimensional vector space and let
$A_1,\ldots,A_k,B$ be subspaces of $V$.
Then for $i = 1,\dots,k$,
there exist linear functions $f_i:B \rightarrow A_i$ such that $f_1 + \cdots + f_k = I$ on
a subspace of $B$ of codimension $H(B|A_1,\ldots,A_k)$.
\end{lem}
\begin{proof}
Let $W$ be a subspace of $B$ defined by $W = \langle A_1, \ldots, A_k\rangle \cap B$.
The subspace on which this lemma holds is $W$.
If $H(W) = 0$ , then
the lemma would be trivially true.
So, assume that $H(W) > 0$, and let
$\{w_1,\ldots,w_n\}$ be a basis for $W$.
For each $j = 1,\ldots, n$, choose
$x_{i,j} \in A_i$ for $i = 1,\ldots, k$ such that $w_j = x_{1,j} + \dots + x_{k,j}$.
For each $i = 1,\ldots,k$, define a linear mapping $g_i : W \rightarrow A_i$ so that $g_i(w_j) = x_{i,j}$ for all $i$ and $j$.
Then extend
$g_i$ arbitrarily to $f_i:B\rightarrow A_i$.
Now we have linear functions
$f_1,\ldots,f_k$ such that $f_1 + \cdots + f_k = I$ on $W$.
The dimension of
$W$ is $H(W) = I(A_1,\ldots,A_k;B)$, so the codimension of $W$ is $H(B) - I(A_1,\ldots,A_k;B) = H(B|A_1,\ldots,A_k)$.
\end{proof}
\begin{lem}\label{lemma5}
\cite{rateregions-journal} Let $V$ be a finite-dimensional vector space and let $A,B$,
and $C$ be subspaces of $V$.
Let $f:A\rightarrow B$ and $g:A \rightarrow C$ be
linear functions such that $f+g = 0$ on $A$.
Then $f = g = 0$ on a subspace of
$A$ of codimension at most $I(B;C)$.
\begin{proof}
Let $K$ be the kernel of $f$.
Clearly, $f$ maps $A$ into $B\cap C$ and since $f$ is linear
the rank of its domain is at most the sum of the ranks of its kernel and range, so
\begin{align*}
\Codim(K) &=H(A) - H(K) \le I(B;C).
\end{align*}
\end{proof}
\end{lem}
\begin{lem}\label{lemma6}
\cite{rateregions-journal} Let $V$ be a finite dimensional vector space and let
$A,B_1,\dots,B_k$ be subspaces of $V$.
For each $i = 1,\ldots,k$ let $f_i:A \rightarrow B_i$ be a linear function such that $f_1 + \cdots + f_k = 0$
on $A$.
Then $f_1 = \cdots = f_k = 0$ on a subspace of $A$ of
codimension at most $H(B_1) + \cdots + H(B_k) - H(B_1,\ldots,B_k)$.
\end{lem}
\begin{proof}
First we apply Lemma~\ref{lemma5} to $f_1$ and $(f_2 + \dots + f_k)$ to get
$f_1 = (f_2 + \dots + f_k) = 0$ on a subspace $A_1$ of $A$ of codimension at
most $I(B_1; B_2,\ldots,B_k) = H(B_1) + H(B_2,\ldots,B_k) - H(B_1,B_2,\ldots,B_k)$.
Then apply Lemma~\ref{lemma5} to $f_2$ and $(f_3 + \dots + f_k)$ to get $f_2 = (f_3 + \dots + f_k) = 0$ on a subspace $A_2$ of
$A_1$ of codimension at most $I(B_2; B_3,\ldots,B_k) = H(B_2) + H(B_3,\ldots,B_k) - H(B_2,B_3,\ldots,B_k)$.
Continue on until we apply Lemma
\ref{lemma5} to $f_{k-1}$ and $f_k$ to get $f_{k-1} = f_k = 0$ on a subspace
$A_{k-1}$ of $A_{k-2}$ of codimension at most $I(B_{k-1}; B_k) = H(B_{k-1}) + H(B_k) - H(B_{k-1},B_k)$.
Now $A_{k-1}$ is a subspace of $A$ of codimension at
most $H(B_1) + \dots + H(B_k) - H(B_1,\ldots,B_k)$, on which $f_1 = f_2 = \dots
= f_k = 0$.
\end{proof}
\begin{lem}\label{lem Inj}
Let $A,B,C,D,E$ be subspaces of a vector space $V$ and let $f_R,f_L,g_R$,
and $g_L$ be functions such that $f_R:A \rightarrow C, f_L:C \rightarrow A, g_R:B \rightarrow D,$ and $g_L:D \rightarrow E$.
If $f_Lf_R = I$ on $A$ and
$g_Lg_R$ is injective on $B$, then $g_Lf_R$ is injective on $f_L(f_RA \cap g_RB)$.
\end{lem}
\begin{proof}
Let $x,y \in f_L(f_RA \cap g_RB)$.
We know $f_R f_L=I$ on $f_RA$ because $f_R f_L(f_R(w)) = f_R(f_L f_R(w)) = f_R(w)$
for all $w\in A$.
Since $x\in f_L(f_RA \cap g_RB)$, we know
$f_R(x) \in f_Rf_L(f_RA \cap g_RB) = f_RA \cap g_RB$,
which implies $f_R(x) = g_R(b_x)$ for some $b_x\in B$.
Similarly, we know $f_R(y) = g_R(b_y)$ for some
$b_y \in B$.
So, we have $g_Lg_R(b_x) = g_Lf_R(x)$ and $g_Lg_R(b_y) = g_Lf_R(y)$.
If we assume $g_Lf_R(x) = g_Lf_R(y)$, then we have $g_Lg_R(b_x) = g_Lg_R(b_y)$.
Since $g_Lg_R$ is injective on $B$, we know $b_x = b_y$.
Thus
$f_R(x) = g_R(b_x) = g_R(b_y) = f_R(y)$, which implies $f_Lf_R(x) = f_Lf_R(y)$.
Since $f_Lf_R = I$ on $A$, we know $x = y$.
Thus $g_Lf_R$ is
injective on $f_L(f_RA \cap g_RB)$.
\end{proof}
\newpage
\section{A Linear Rank Inequality for fields of characteristic other than 3}
In this section, we use the known T8 matroid to construct a ``T8 network'',
and then in turn we use the T8 network to guide a construction of a
``T8 linear rank inequality'' that is shown to hold for all vector spaces
having finite scalar fields of characteristic not equal to $3$.
Then we show that the T8 inequality does not necessarily hold when such scalar
fields have characteristic $3$.
Finally, we determine the exact coding capacity of the
T8 network and its linear coding capacity over finite field alphabets
of characteristic $3$,
as well as a linear capacity upper bound for finite field alphabets
whose characteristic is not $3$.
The T8 matroid \cite{matroidBook} is a vector matroid which
is represented by the following matrix,
where column dependencies are over characteristic 3:
$$\bordermatrix{& A & B & C & D & W & X & Y & Z \cr
& 1 & 0 & 0 & 0 & 0 & 1 & 1 & 1\cr
& 0 & 1 & 0 & 0 & 1 & 0 & 1 & 1\cr
& 0 & 0 & 1 & 0 & 1 & 1 & 0 & 1\cr
& 0 & 0 & 0 & 1 & 1 & 1 & 1 & 0 }. $$
The T8 matroid is representable over a field if and only if the field is
of characteristic 3.
Figure~\ref{fig:T8} is a network whose dependencies and
independencies are consistent with the T8 matroid.
It was designed by the
construction process described in \cite{nonshannon}, and we will refer to it as
the T8 network.
Theorem~\ref{thm:T8} uses the T8 network as a guide to
derive a linear rank inequality valid for every characteristic except for 3.
We refer to the inequality in the following theorem as the \textit{T8 linear rank inequality}.
\begin{figure}
\begin{center}
\psfrag{n1}{\LARGE $n_{1}$}
\psfrag{n2}{\LARGE $n_{2}$}
\psfrag{n3}{\LARGE $n_{3}$}
\psfrag{n4}{\LARGE $n_{4}$}
\psfrag{n5}{\LARGE $n_{5}$}
\psfrag{n6}{\LARGE $n_{6}$}
\psfrag{n7}{\LARGE $n_{7}$}
\psfrag{n8}{\LARGE $n_{8}$}
\psfrag{n9}{\LARGE $n_{9}$}
\psfrag{n10}{\LARGE $n_{10}$}
\psfrag{n11}{\LARGE $n_{11}$}
\psfrag{n12}{\LARGE $n_{12}$}
\psfrag{n13}{\LARGE $n_{13}$}
\psfrag{n14}{\LARGE $n_{14}$}
\psfrag{n15}{\LARGE $n_{15}$}
\includegraphics[width=13.5cm]{T8-network.eps}
\end{center}
\caption{ The T8 network has source messages $A,B,C,$ and $D$ generated at hidden source nodes with
certain hidden out-edges pointing to corresponding displayed nodes
$n_1$, $n_3$, $n_5$, and $n_9$--$n_{15}$
(which are labeled by incoming messages above such nodes).
The nodes $n_9$--$n_{15}$ each demand one message, as labeled below such nodes.
}
\label{fig:T8}
\end{figure}
\begin{thm}\label{thm:T8}
Let $A,B,C,D,W,X,Y$, and $Z$ be subspaces of a vector space $V$ whose scalar field is finite
and of characteristic other than $3$.
Then the following is a linear rank inequality over $V$:
\begin{align*}
H(A) &\le 8H(Z) + 29H(Y) + 3H(X) + 8H(W) -6H(D) -17H(C) - 8H(B) - 17H(A)\\
&\ \ \ + 55H(Z|A,B,C) + 35H(Y|W,X,Z) + 50H(X|A,C,D) + 49H(W|B,C,D)\\
&\ \ \ + 18H(A|B,D,Y) + 7H(B|D,X,Z) + H(B|A,W,X) + 7H(C|D,Y,Z) \\
&\ \ \ + 7H(C|B,X,Y) + 3H(C|A,W,Y) + 6H(D|A,W,Z)\\
&\ \ \ + 49(H(A) + H(B) + H(C) + H(D) - H(A,B,C,D)).
\end{align*}
\end{thm}
\begin{proof}
By Lemma~\ref{lemma4} we get linear functions:
\begin{center}
\begin{tabular}{ccc}
$f_1:Z \rightarrow A$, & $f_2:Z \rightarrow B$, & $f_3:Z \rightarrow C$,\\
$f_4:W \rightarrow B$, & $f_5:W \rightarrow C$, & $f_6:W \rightarrow D$,\\
$f_7:X \rightarrow A$, & $f_8:X \rightarrow C$, & $f_9:X \rightarrow D$,\\
$f_{10}:Y \rightarrow Z$, & $f_{11}:Y \rightarrow W$, & $f_{12}:Y \rightarrow X$,\\
$f_{13}:A \rightarrow B$, & $f_{14}:A \rightarrow D$, & $f_{15}:A \rightarrow Y$,\\
$f_{16}:D \rightarrow Z$, & $f_{17}:D \rightarrow W$, & $f_{18}:D \rightarrow A$,\\
$f_{19}:C \rightarrow Z$, & $f_{20}:C \rightarrow Y$, & $f_{21}:C \rightarrow D$,\\
$f_{22}:B \rightarrow Z$, & $f_{23}:B \rightarrow X$, & $f_{24}:B \rightarrow D$,\\
$f_{25}:C \rightarrow Y$, & $f_{26}:C \rightarrow X$, & $f_{27}:C \rightarrow B$,\\
$f_{28}:C \rightarrow Y$, & $f_{29}:C \rightarrow W$, & $f_{30}:C \rightarrow A$,\\
$f_{31}:B \rightarrow W$, & $f_{32}:B \rightarrow X$, & $f_{33}:B \rightarrow A$
\end{tabular}
\end{center}
such that
\begin{align}
f_1 + f_2 + f_3 &= I \mbox{ on a subspace of $Z$ of codimension $H(Z|A,B,C)$}\label{eq:1}\\
f_4 + f_5 + f_6 &= I \mbox{ on a subspace of $W$ of codimension $H(W|B,C,D)$}\label{eq:2}\\
f_7 + f_8 + f_9 &= I \mbox{ on a subspace of $X$ of codimension $H(X|A,C,D)$}\label{eq:3}\\
f_{10} + f_{11} + f_{12} &= I \mbox{ on a subspace of $Y$ of codimension $H(Y|W,X,Z)$}\label{eq:4}\\
f_{13} + f_{14} + f_{15} &= I \mbox{ on a subspace of $A$ of codimension $H(A|B,D,Y)$}\label{eq:5}\\
f_{16} + f_{17} + f_{18} &= I \mbox{ on a subspace of $D$ of codimension $H(D|A,W,Z)$}\label{eq:6}\\
f_{19} + f_{20} + f_{21} &= I \mbox{ on a subspace of $C$ of codimension $H(C|D,Y,Z)$}\label{eq:7}\\
f_{22} + f_{23} + f_{24} &= I \mbox{ on a subspace of $B$ of codimension $H(B|D,X,Z)$}\label{eq:8}\\
f_{25} + f_{26} + f_{27} &= I \mbox{ on a subspace of $C$ of codimension $H(C|B,X,Y)$}\label{eq:9}\\
f_{28} + f_{29} + f_{30} &= I \mbox{ on a subspace of $C$ of codimension $H(C|A,W,Y)$}\label{eq:10}\\
f_{31} + f_{32} + f_{33} &= I \mbox{ on a subspace of $B$ of codimension $H(B|A,W,X)$}.\label{eq:11}
\end{align}
Now let
\begin{align*}
f_A &\triangleq f_7f_{12} + f_1f_{10}\\
f_B &\triangleq f_4f_{11} + f_2f_{10}\\
f_C &\triangleq f_8f_{12} + f_5f_{11} + f_3f_{10}\\
f_D &\triangleq f_9f_{12} + f_6f_{11}.
\end{align*}
Combining the functions we obtained from Lemma~\ref{lemma4} gives new functions:
\begin{align*}
f_A\Compose f_{15} &: A \rightarrow A \\
f_B\Compose f_{15} + f_{13} &: A \rightarrow B\\
f_C\Compose f_{15} &: A \rightarrow C\\
f_D\Compose f_{15} + f_{14} &: A \rightarrow D.
\end{align*}
Using \eqref{eq:1} - \eqref{eq:5}, Lemma~\ref{lemma1}, and Lemma~\ref{lemma3}
we know the sum of these functions is equal to $I$ on a subspace of $A$ of
codimension at most $H(Z|A,B,C) + H(W|B,C,D) + H(X|A,C,D) + H(Y|W,X,Z) + H(A|B,D,Y)$.
Applying Lemma~\ref{lemma6} and Lemma~\ref{lemma1} to
$f_A\Compose f_{15} - I$, $f_B\Compose f_{15} + f_{13}$, $f_C\Compose f_{15}$, and $f_D\Compose f_{15} + f_{14}$
we get a subspace $\overline{A}$ of $A$ of codimension at most
\begin{align*}
\Delta_{\overline{A}} &= H(Z|A,B,C) + H(W|B,C,D) + H(X|A,C,D) + H(Y|W,X,Z) + H(A|B,D,Y)\\
& + H(A) + H(B) + H(C) + H(D) - H(A,B,C,D)
\end{align*}
on which
\begin{align}
f_A\Compose f_{15} &= I \label{eq:Abar1}\\
f_B\Compose f_{15} + f_{13} &= 0 \label{eq:Abar2}\\
f_C\Compose f_{15} &= 0 \label{eq:Abar3}\\
f_D\Compose f_{15} + f_{14} &= 0. \label{eq:Abar4}
\end{align}
To see how the T8 network is used as a guide,
consider receiver node $n_9$, which demands $A$.
Let
$M_1$,
$M_7$,
$M_{10}$,
$M_{12}$,
$M_{15}$
be matrices corresponding to the transformations along the edges
$(A,Z)$,
$(A,X)$,
$(Z,Y)$,
$(X,Y)$,
$(Y,A)$,
respectively.
Using algebra to solve for $A$ one deduces that
\begin{align*}
M_{15}M_{10}M_{1} + M_{15}M_{12}M_{7} &= I.
\end{align*}
Equation \eqref{eq:Abar1} was designed to model this property.
Similarly, we get a subspace $\overline{B}$ of $B$ of codimension at most
\begin{align*}
\Delta_{\overline{B}} &= H(Z|A,B,C) + H(X|A,C,D) + H(B|D,X,Z)\\
&\ \ \ + H(A) + H(B) + H(C) + H(D) - H(A,B,C,D)
\end{align*}
on which
\begin{align}
f_7\Compose f_{23} + f_1\Compose f_{22} &= 0 \label{eq:Bbar1}\\
f_2\Compose f_{22} &= I \label{eq:Bbar2}\\
f_8\Compose f_{23} + f_3\Compose f_{22} &= 0 \label{eq:Bbar3}\\
f_{24} + f_9\Compose f_{23} &= 0. \label{eq:Bbar4}
\end{align}
We get a subspace $\widehat{B}$ of $B$ of codimension at most
\begin{align*}
\Delta_{\widehat{B}} &= H(W|B,C,D) + H(X|A,C,D) + H(B|A,W,X)\\
&\ \ \ + H(A) + H(B) + H(C) + H(D) - H(A,B,C,D)
\end{align*}
on which
\begin{align}
f_{33} + f_{7}\Compose f_{32} &= 0 \label{eq:Bhat1}\\
f_4\Compose f_{31} &= I \label{eq:Bhat2}\\
f_8\Compose f_{32} + f_5\Compose f_{31} &= 0 \label{eq:Bhat3}\\
f_9\Compose f_{32} + f_6\Compose f_{31} &= 0. \label{eq:Bhat4}
\end{align}
We get a subspace $\overline{C}$ of $C$ of codimension at most
\begin{align*}
\Delta_{\overline{C}} &= 2H(Z|A,B,C) + H(W|B,C,D) + H(X|A,C,D) + H(Y|W,X,Z) + H(C|D,Y,Z)\\
&\ \ \ + H(A) + H(B) + H(C) + H(D) - H(A,B,C,D)
\end{align*}
on which
\begin{align}
f_A\Compose f_{20} + f_1\Compose f_{19} &= 0 \label{eq:Cbar1}\\
f_B\Compose f_{20} + f_2\Compose f_{19} &= 0 \label{eq:Cbar2}\\
f_C\Compose f_{20} + f_3\Compose f_{19} &= I \label{eq:Cbar3}\\
f_D\Compose f_{20} + f_{21} &= 0. \label{eq:Cbar4}
\end{align}
We get a subspace $\widehat{C}$ of $C$ of codimension at most
\begin{align*}
\Delta_{\widehat{C}} &= H(Z|A,B,C) + H(W|B,C,D) + 2H(X|A,C,D) + H(Y|W,X,Z) + H(C|B,X,Y)\\
&\ \ \ + H(A) + H(B) + H(C) + H(D) - H(A,B,C,D)
\end{align*}
on which
\begin{align}
f_A\Compose f_{25} + f_7\Compose f_{26} &= 0 \label{eq:Chat1}\\
f_B\Compose f_{25} + f_{27} &= 0 \label{eq:Chat2}\\
f_C\Compose f_{25} + f_8\Compose f_{26} &= I \label{eq:Chat3}\\
f_D\Compose f_{25} + f_9\Compose f_{26} &= 0. \label{eq:Chat4}
\end{align}
We get a subspace $\widetilde{C}$ of $C$ of codimension at most
\begin{align*}
\Delta_{\widetilde{C}} &= H(Z|A,B,C) + 2H(W|B,C,D) + H(X|A,C,D) + H(Y|W,X,Z) + H(C|A,W,Y)\\
&\ \ \ + H(A) + H(B) + H(C) + H(D) - H(A,B,C,D)
\end{align*}
on which
\begin{align}
f_A\Compose f_{28} + f_{30} &= 0 \label{eq:Ctilde1}\\
f_B\Compose f_{28} + f_4\Compose f_{29} &= 0 \label{eq:Ctilde2}\\
f_C\Compose f_{28} + f_5\Compose f_{29} &= I \label{eq:Ctilde3}\\
f_D\Compose f_{28} + f_6\Compose f_{29} &= 0. \label{eq:Ctilde4}
\end{align}
We get a subspace $\overline{D}$ of $D$ of codimension at most
\begin{align*}
\Delta_{\overline{D}} &= H(Z|A,B,C) + H(W|B,C,D) + H(D|A,W,Z)\\
&\ \ \ + H(A) + H(B) + H(C) + H(D) - H(A,B,C,D)
\end{align*}
on which
\begin{align}
f_{18} + f_{1}\Compose f_{16} &= 0 \label{eq:Dbar1}\\
f_4\Compose f_{17} + f_2\Compose f_{16} &= 0 \label{eq:Dbar2}\\
f_5\Compose f_{17} + f_3\Compose f_{16} &= 0 \label{eq:Dbar3}\\
f_6\Compose f_{17} &= I. \label{eq:Dbar4}
\end{align}
First notice that \eqref{eq:Abar1} implies
\begin{align}
&f_{15} \mbox{ is injective on }\overline{A}. \label{p2:f15}
\end{align}
We need to define a subspace of $\overline{A}$ on which $f_{13}$ and $f_{14}$
are injective.
The justifications can be found on \eqref{p2:f13} and \eqref{p2:f14}.
Let
\begin{align*}
\overline{C}^\ast &\triangleq f_3(f_{19}(\overline{C} \cap f_{20}^{-1}f_{15}\overline{A}) \cap f_{22}\overline{B}) \subseteq \overline{C}\\
\widetilde{C}^\ast &\triangleq f_5(f_{29}(\widetilde{C} \cap f_{28}^{-1}f_{15}\overline{A}) \cap f_{17}\overline{D}) \subseteq \widetilde{C}\\
\overline{A}^\ast &\triangleq f_A(f_{15}\overline{A} \cap f_{20}\overline{C}^\ast \cap f_{28}\widetilde{C}^\ast) \subseteq \overline{A}.
\end{align*}
To justify why $\overline{C}^\ast \subseteq \overline{C}$, by \eqref{eq:Abar3}
we know $f_Cf_{15} = 0$ on $\overline{A}$
and by \eqref{eq:Cbar3} we know $f_Cf_{20} + f_3f_{19} = I$.
Thus for each $\overline{c} \in \overline{C} \cap f_{20}^{-1}f_{15}\overline{A}$,
we have $f_Cf_{20} = 0$ on $\overline{C}$
which gives
\begin{align}
f_3f_{19} &= I \ \mbox{ on\ $\overline{C} \cap f_{20}^{-1}f_{15}\overline{A}$}. \label{p2:f3f19}
\end{align}
Using (\ref{eq:Abar3}) and (\ref{eq:Ctilde3}) we have
\begin{align}
f_5f_{29} &= I \ \mbox{ on\ $\widetilde{C} \cap f_{28}^{-1}f_{15}\overline{A}$}. \label{p2:f5f29}
\end{align}
Using (\ref{eq:Abar3}) and (\ref{eq:Chat3}) we have
\begin{align}
f_8f_{26} &= I \ \mbox{ on\ $\widehat{C} \cap f_{25}^{-1}f_{15}\overline{A}$}. \label{p2:f8f26}
\end{align}
We are now going to show $f_{13}$ is injective on $\overline{A}^\ast$.
First we
need to apply Lemma~\ref{lem Inj} to show $f_2f_{19}$ is injective on
$\overline{C}^\ast$ and then again to show $f_Bf_{15}$ is injective on
$\overline{A}^\ast$.
By (\ref{eq:Bbar2}) and (\ref{p2:f3f19}), we know
$f_2f_{22}$ is injective on $\overline{B}$ and $f_3f_{19} = I$ on $\overline{C} \cap f_{20}^{-1}f_{15}\overline{A}$.
So, we can apply Lemma~\ref{lem Inj} by
letting $g_L = f_2$, $g_R = f_{22}$, $f_L = f_3$, and $f_R = f_{19}$ to get
that $f_2f_{19}$ is injective on $\overline{C}^\ast$.
Then using
(\ref{eq:Cbar2}), we know $f_Bf_{20}$ is injective on $\overline{C}^\ast$.
Now
we can apply Lemma~\ref{lem Inj} again by using the fact that $f_Af_{15} = I$
on $\overline{A}$ and by letting $g_L = f_B$, $g_R = f_{20}$, $f_L = f_A$, and
$f_R = f_{15}$ to get $f_Bf_{15}$ is injective on $\overline{A}^\ast$.
Thus by
(\ref{eq:Abar2}),
\begin{align}
& f_{13} \mbox{ is injective on $\overline{A}^\ast$.}\label{p2:f13}
\end{align}
Similarly, we are going to show $f_{14}$ is injective on
$\overline{A}^\ast$.
We will first apply Lemma~\ref{lem Inj} to show
$f_6f_{29}$ is injective on $\widetilde{C}^\ast$ and then again to show
$f_Df_{15}$ is injective on $\overline{A}^\ast$.
By (\ref{eq:Dbar4}) and
(\ref{p2:f5f29}), we know $f_6f_{17}$ is injective on $\overline{D}$ and
$f_5f_{29} = I$ on $\widetilde{C} \cap f_{28}^{-1}f_{15}\overline{A}$.
So, we
can apply Lemma~\ref{lem Inj} by letting $g_L = f_6$, $g_R = f_{17}$,
$f_L = f_5$, and $f_R = f_{29}$ to get that $f_6f_{29}$ is injective on
$\widetilde{C}^\ast$.
Then using (\ref{eq:Ctilde4}), we know $f_Df_{28}$ is
injective on $\widetilde{C}^\ast$.
Now we can apply Lemma~\ref{lem Inj} again
by using the fact that $f_Af_{15} = I$ on $\overline{A}$ and by letting
$g_L = f_D$, $g_R = f_{28}$, $f_L = f_A$, and $f_R = f_{15}$ to get $f_Df_{15}$ is
injective on $\overline{A}^\ast$.
Thus by (\ref{eq:Abar4}),
\begin{align}
& f_{14}\mbox{ is injective on $\overline{A}^\ast$.}\label{p2:f14}
\end{align}
Now we are going to find an upper bound for $\Codim_A(\overline{A}^\ast)$.
First
we need to find upper bounds for $\Codim_C(\overline{C}^\ast)$ and
$\Codim_C(\widetilde{C}^\ast)$.
Using (\ref{p2:f15}) to show
$\Dim(f_{15}\overline{A}) = \Dim(\overline{A})$, and again using Lemma
\ref{lemma1} and Lemma~\ref{lemma3}, we have
\begin{align}
\Codim_C(\overline{C}^\ast) &= H(C) - \Dim(\overline{C}^\ast)\nonumber\\
&= H(C) - \Dim(f_3(f_{19}(\overline{C} \cap f_{20}^{-1}f_{15}\overline{A}) \cap f_{22}\overline{B})) \nonumber\\
&= H(C) - \Dim(f_{19}(\overline{C} \cap f_{20}^{-1}f_{15}\overline{A}) \cap f_{22}\overline{B})\nonumber\\
&= H(C) - H(Z) + \Codim_Z(f_{19}(\overline{C} \cap f_{20}^{-1}f_{15}\overline{A}) \cap f_{22}\overline{B})\nonumber\\
&\le H(C) - H(Z) + \Codim_Z(f_{19}(\overline{C} \cap f_{20}^{-1}f_{15}\overline{A})) + \Codim_Z(f_{22}\overline{B})\nonumber\\
&= H(C) - H(Z) + H(Z) - \Dim(f_{19}(\overline{C} \cap f_{20}^{-1}f_{15}\overline{A})) + H(Z) - \Dim(f_{22}\overline{B})\nonumber\\
&= H(C) + H(Z) - \Dim(\overline{C} \cap f_{20}^{-1}f_{15}\overline{A}) - \Dim(\overline{B})\nonumber\\
&= H(C) + H(Z) - H(C) + \Codim_C(\overline{C} \cap f_{20}^{-1}f_{15}\overline{A}) - H(B) + \Codim_B(\overline{B})\nonumber\\
&= H(Z) - H(B) + \Codim_C(\overline{C} \cap f_{20}^{-1}f_{15}\overline{A}) + \Codim_B(\overline{B})\nonumber\\
&\le H(Z) - H(B) + \Delta_{\overline{C}} + \Codim_C( f_{20}^{-1}f_{15}\overline{A}) + \Delta_{\overline{B}}\nonumber\\
&\le H(Z) - H(B) + \Delta_{\overline{C}} + \Codim_Y( f_{15}\overline{A}) + \Delta_{\overline{B}}\nonumber\\
&\le H(Z) - H(B) + \Delta_{\overline{C}} + H(Y) - \Dim( f_{15}\overline{A}) + \Delta_{\overline{B}}\nonumber\\
&= H(Z) - H(B) + \Delta_{\overline{C}} + H(Y) - \Dim(\overline{A}) + \Delta_{\overline{B}}\nonumber\\
&= H(Z) - H(B) + \Delta_{\overline{C}} + H(Y) - H(A) + \Codim_A(\overline{A}) + \Delta_{\overline{B}}\nonumber\\
&\le H(Z) - H(B) + H(Y) - H(A) +\Delta_{\overline{C}} + \Delta_{\overline{A}} + \Delta_{\overline{B}}
\end{align}
\begin{align}
\Codim_C(\widetilde{C}^\ast) &= H(C) - \Dim(\widetilde{C}^\ast)\nonumber\\
&= H(C) - \Dim(f_5(f_{29}(\widetilde{C} \cap f_{28}^{-1}f_{15}\overline{A}) \cap f_{17}\overline{D}))\nonumber\\
&= H(C) - \Dim(f_{29}(\widetilde{C} \cap f_{28}^{-1}f_{15}\overline{A}) \cap f_{17}\overline{D})\nonumber\\
&= H(C) - H(W) + \Codim_W(f_{29}(\widetilde{C} \cap f_{28}^{-1}f_{15}\overline{A}) \cap f_{17}\overline{D})\nonumber\\
&\le H(C) - H(W) + \Codim_W(f_{29}(\widetilde{C} \cap f_{28}^{-1}f_{15}\overline{A})) + \Codim_W(f_{17}\overline{D})\nonumber\\
&= H(C) - H(W) + H(W) - \Dim(f_{29}(\widetilde{C} \cap f_{28}^{-1}f_{15}\overline{A})) + H(W) - \Dim(f_{17}\overline{D})\nonumber\\
&= H(C) + H(W) - \Dim(\widetilde{C} \cap f_{28}^{-1}f_{15}\overline{A}) - \Dim(\overline{D})\nonumber\\
&= H(C) + H(W) - H(C) + \Codim_C(\widetilde{C} \cap f_{28}^{-1}f_{15}\overline{A}) - H(D) + \Codim_D(\overline{D})\nonumber\\
&= H(W) - H(D) + \Codim_C(\widetilde{C} \cap f_{28}^{-1}f_{15}\overline{A}) + \Codim_D(\overline{D})\nonumber\\
&\le H(W) - H(D) + \Delta_{\widetilde{C}} + \Codim_C( f_{28}^{-1}f_{15}\overline{A}) + \Delta_{\overline{D}}\nonumber\\
&\le H(W) - H(D) + \Delta_{\widetilde{C}} + \Codim_Y( f_{15}\overline{A}) + \Delta_{\overline{D}}\nonumber\\
&= H(W) - H(D) + \Delta_{\widetilde{C}} + H(Y) - \Dim( f_{15}\overline{A}) + \Delta_{\overline{D}}\nonumber\\
&= H(W) - H(D) + \Delta_{\widetilde{C}} + H(Y) - \Dim( \overline{A}) + \Delta_{\overline{D}}\nonumber\\
&= H(W) - H(D) + \Delta_{\widetilde{C}} + H(Y) - H(A) + \Codim_A(\overline{A}) + \Delta_{\overline{D}}\nonumber\\
&\le H(W) - H(D) + H(Y) - H(A) + \Delta_{\widetilde{C}} + \Delta_{\overline{A}} + \Delta_{\overline{D}}.
\end{align}
In the justification for (\ref{p2:f13}), we concluded that $f_Bf_{20}$ is
injective on $\overline{C}^\ast$, which implies $f_{20}$ is injective on
$\overline{C}^\ast$.
In the justification for (\ref{p2:f14}), we concluded that
$f_Df_{28}$ is injective on $\widetilde{C}^\ast$, which implies $f_{28}$ is
injective on $\widetilde{C}^\ast$.
These facts combined with (\ref{p2:f15})
will be used to arrive on line (\ref{app:inj1}).
\begin{align}
\Codim_A(\overline{A}^\ast) &= H(A) - \Dim(f_A(f_{15}\overline{A} \cap f_{20}\overline{C}^\ast \cap f_{28}\widetilde{C}^\ast))\nonumber\\
&= H(A) - \Dim(f_{15}\overline{A} \cap f_{20}\overline{C}^\ast \cap f_{28}\widetilde{C}^\ast)\nonumber\\
&= H(A) - H(Y) + \Codim_Y(f_{15}\overline{A} \cap f_{20}\overline{C}^\ast \cap f_{28}\widetilde{C}^\ast)\nonumber\\
&\le H(A) - H(Y) + \Codim_Y(f_{15}\overline{A}) + \Codim_Y(f_{20}\overline{C}^\ast) + \Codim_Y(f_{28}\widetilde{C}^\ast)\nonumber\\
&= H(A) - H(Y) + H(Y) - \Dim(f_{15}\overline{A}) + H(Y) - \Dim(f_{20}\overline{C}^\ast) \nonumber\\
&\ \ \ + H(Y) - \Dim(f_{28}\widetilde{C}^\ast)\nonumber\\
&= H(A) + 2H(Y) - \Dim(\overline{A}) - \Dim(\overline{C}^\ast) - \Dim(\widetilde{C}^\ast)\label{app:inj1}\\
&= H(A) + 2H(Y) - H(A) + \Codim_A(\overline{A}) - H(C) + \Codim_C(\overline{C}^\ast) \nonumber\\
&\ \ \ - H(C) + \Codim_C(\widetilde{C}^\ast)\nonumber\\
&= 2H(Y) - 2H(C) + \Codim_A(\overline{A}) + \Codim_C(\overline{C}^\ast) + \Codim_C(\widetilde{C}^\ast)\nonumber\\
&\le 2H(Y) - 2H(C) + \Delta_{\overline{A}} \nonumber\\
&\ \ \ + H(Z) - H(B) + H(Y) - H(A) +\Delta_{\overline{C}} + \Delta_{\overline{A}} + \Delta_{\overline{B}}\nonumber\\
&\ \ \ + H(W) - H(D) + H(Y) - H(A) + \Delta_{\widetilde{C}} + \Delta_{\overline{A}} + \Delta_{\overline{D}}\nonumber\\
&= H(W) + 4H(Y) + H(Z) - 2H(A) - H(B) -2H(C) - H(D) \nonumber\\
&\ \ \ + 3\Delta_{\overline{A}} + \Delta_{\overline{B}} + \Delta_{\overline{C}} + \Delta_{\widetilde{C}}+ \Delta_{\overline{D}}\nonumber\\
&\triangleq \Delta_{\overline{A}^\ast}. \label{app:Abarstar}
\end{align}
Let $t \in A$.
We will next make a collection of assumptions on $t$ in (\ref{p2:ass1})--(\ref{p2:ass6}).
Each such assumption gives rise to an upper bound on the codimension of a particular subspace of $A$.
The justification of these upper bounds will be given in what follows.
Ultimately, we will show that these assumptions imply that $3t=0$ and
thus for field characteristics other than $3$,
no nonzero $t$ can satisfy this condition.
This in turn implies that the codimension of the intersection of the subspaces of $A$ in the upper bounds
of (\ref{p2:ass1})--(\ref{p2:ass6})
must be at least as big as the dimension of $A$,
which then yields the desired inequality.
\begin{align}
&\mbox{We will assume } t \in \overline{A}^\ast.
\mbox{ This is true on a subspace of $A$ of codimension at most } \Delta_{\overline{A}^\ast}. \label{p2:ass1}\\
&\mbox{We will assume } f_{10}f_{15}t \in f_{19}(\overline{C} \cap f_{20}^{-1}f_{15}\overline{A}^\ast).
\mbox{ This is true on a subspace of $A$ of } \nonumber\\
&\qquad\qquad \mbox{ codimension at most } H(Z) - H(C) + H(Y) - H(A) + \Delta_{\overline{C}} + \Delta_{\overline{A}^\ast}.\label{p2:ass2}\\
&\mbox{We will assume } f_{11}f_{15}t \in f_{29}(\widetilde{C} \cap f_{28}^{-1}f_{15}\overline{A}^\ast).
\mbox{ This is true on a subspace of $A$ of } \nonumber\\
&\qquad\qquad \mbox{ codimension at most } H(W) -H(C) + H(Y) - H(A) + \Delta_{\widetilde{C}} + \Delta_{\overline{A}^\ast}.\label{p2:ass3}\\
&\mbox{We will assume } f_{12}f_{15}t \in f_{26}(\widehat{C} \cap f_{25}^{-1}f_{15}\overline{A}^\ast).
\mbox{ This is true on a subspace of $A$ of } \nonumber\\
&\qquad\qquad \mbox{ codimension at most } H(X)- H(C) + H(Y) - H(A) + \Delta_{\widehat{C}} + \Delta_{\overline{A}^\ast}.\label{p2:ass4}\\
&\mbox{We will assume } f_{10}f_{15}t \in f_{22}(\overline{B} \cap f_{23}^{-1}f_{26}(\widehat{C} \cap f_{25}^{-1}f_{15}\overline{A}^\ast)).
\mbox{ This is true on a subspace } \nonumber\\
&\qquad\qquad \mbox{ of $A$ of codimension at most }\nonumber\\
&\qquad\qquad H(Z) - H(B) + H(X) - H(C) + H(Y) - H(A) + \Delta_{\overline{A}^\ast} + \Delta_{\overline{B}} + \Delta_{\widehat{C}}.\label{p2:ass5}\\
&\mbox{We will assume } f_{11}f_{15}t \in f_{31}(\widehat{B} \cap f_{32}^{-1}f_{26}(\widehat{C} \cap f_{25}^{-1}f_{15}\overline{A}^\ast)).
\mbox{ This is true on a subspace } \nonumber\\
&\qquad\qquad\mbox{ of $A$ of codimension at most }\nonumber\\
& \qquad \qquad H(W) - H(B) + H(X) - H(C) + H(Y) - H(A) + \Delta_{\overline{A}^\ast} + \Delta_{\widehat{B}} + \Delta_{\widehat{C}}. \label{p2:ass6}
\end{align}
To justify (\ref{p2:ass2}), first we know $f_{19}$ is injective on
$\overline{C} \cap f_{20}^{-1}f_{15}\overline{A}^\ast$ by
(\ref{p2:f3f19}).
Then by Lemma~\ref{lemma3}, we know
$f_{10}f_{15}t \in f_{19}(\overline{C} \cap f_{20}^{-1}f_{15}\overline{A}^\ast)$
on a subspace of $A$ of codimension at most
$H(Z) - H(C) + \Codim_C(\overline{C} \cap f_{20}^{-1}f_{15}\overline{A}^\ast)$.
By Lemma~\ref{lemma1}, we know
\begin{align*}
\Codim_C(\overline{C} \cap f_{20}^{-1}f_{15}\overline{A}^\ast) &\le \Delta_{\overline{C}} + \Codim_C(f_{20}^{-1}f_{15}\overline{A}^\ast).
\end{align*}
Then using Lemma~\ref{lemma3} and (\ref{p2:f15}), we know
\begin{align}
\Codim_C(\overline{C} \cap f_{20}^{-1}f_{15}\overline{A}^\ast) &\le \Delta_{\overline{C}} + \Codim_Y(f_{15}\overline{A}^\ast) \nonumber\\
&= \Delta_{\overline{C}} + H(Y) - \Dim(f_{15}\overline{A}^\ast) \nonumber\\
&= \Delta_{\overline{C}} + H(Y) - \Dim(\overline{A}^\ast) \nonumber\\
&\le \Delta_{\overline{C}} + H(Y) - H(A) + \Delta_{\overline{A}^\ast}.
\end{align}
So, we have $f_{10}f_{15}t \in f_{19}(\overline{C} \cap f_{20}^{-1}f_{15}\overline{A}^\ast)$ on a subspace of $A$ of codimension at most
$H(Z) - H(C) + H(Y) - H(A) + \Delta_{\overline{C}} + \Delta_{\overline{A}^\ast}$.
To justify (\ref{p2:ass3}), first we know $f_{29}$ is injective on
$\widetilde{C} \cap f_{28}^{-1}f_{15}\overline{A}^\ast$ by
(\ref{p2:f5f29}).
Then by Lemma~\ref{lemma3},
we know $f_{11}f_{15}t \in f_{29}(\widetilde{C} \cap f_{28}^{-1}f_{15}\overline{A}^\ast)$ on a subspace of
$A$ of codimension at most $H(Z) - H(C) + \Codim_C(\widetilde{C} \cap f_{28}^{-1}f_{15}\overline{A}^\ast)$.
By Lemma~\ref{lemma1}, we know
\begin{align*}
\Codim_C(\widetilde{C} \cap f_{28}^{-1}f_{15}\overline{A}^\ast) &\le \Delta_{\widetilde{C}} + \Codim_C(f_{28}^{-1}f_{15}\overline{A}^\ast).
\end{align*}
Then using Lemma~\ref{lemma3} and (\ref{p2:f15}), we know
\begin{align}
\Codim_C(\widetilde{C} \cap f_{28}^{-1}f_{15}\overline{A}^\ast) &\le \Delta_{\widetilde{C}} + \Codim_Y(f_{15}\overline{A}^\ast) \nonumber\\
&= \Delta_{\widetilde{C}} + H(Y) - \Dim(f_{15}\overline{A}^\ast) \nonumber\\
&= \Delta_{\widetilde{C}} + H(Y) - \Dim(\overline{A}^\ast) \nonumber\\
&\le \Delta_{\widetilde{C}} + H(Y) - H(A) + \Delta_{\overline{A}^\ast}.
\end{align}
So, we have $f_{11}f_{15}t \in f_{29}(\widetilde{C} \cap f_{28}^{-1}f_{15}\overline{A}^\ast)$ on a subspace of $A$ of codimension at
most $H(Z) - H(C) + H(Y) - H(A) + \Delta_{\widetilde{C}} + \Delta_{\overline{A}^\ast}$.
To justify (\ref{p2:ass4}), first we know $f_{26}$ is injective on
$\widehat{C} \cap f_{25}^{-1}f_{15}\overline{A}^\ast$ by (\ref{p2:f8f26}).
Then by Lemma
\ref{lemma3}, we know $f_{12}f_{15}t \in f_{26}(\widehat{C} \cap f_{25}^{-1}f_{15}\overline{A}^\ast)$
on a subspace of $A$ of codimension at
most $H(Z) - H(C) + \Codim_C(\widehat{C} \cap f_{25}^{-1}f_{15}\overline{A}^\ast)$.
By Lemma~\ref{lemma1}, we know
\begin{align*}
\Codim_C(\widehat{C} \cap f_{25}^{-1}f_{15}\overline{A}^\ast) &\le \Delta_{\widehat{C}} + \Codim_C(f_{25}^{-1}f_{15}\overline{A}^\ast)
\end{align*}
Then using Lemma~\ref{lemma3} and (\ref{p2:f15}), we know
\begin{align}
\Codim_C(\widehat{C} \cap f_{25}^{-1}f_{15}\overline{A}^\ast) &\le \Delta_{\widehat{C}} + \Codim_Y(f_{15}\overline{A}^\ast) \nonumber\\
&= \Delta_{\widehat{C}} + H(Y) - \Dim(f_{15}\overline{A}^\ast) \nonumber\\
&= \Delta_{\widehat{C}} + H(Y) - \Dim(\overline{A}^\ast) \nonumber\\
&\le \Delta_{\widehat{C}} + H(Y) - H(A) + \Delta_{\overline{A}^\ast}.
\end{align}
So, we have $f_{12}f_{15}t \in f_{26}(\widehat{C} \cap f_{25}^{-1}f_{15}\overline{A}^\ast)$ on a subspace of $A$ of codimension at
most $H(Z) - H(C) + H(Y) - H(A) + \Delta_{\widehat{C}} + \Delta_{\overline{A}^\ast}$.
To justify (\ref{p2:ass5}), we first know $f_{22}$ is injective on
$\overline{B} \cap f_{23}^{-1}f_{26}(\widehat{C} \cap f_{25}^{-1}f_{15}\overline{A}^\ast)$ by (\ref{eq:Bbar2}).
Then by Lemma
\ref{lemma3}, we know
$f_{10}f_{15}t \in f_{22}(\overline{B} \cap f_{23}^{-1}f_{26}(\widehat{C} \cap f_{25}^{-1}f_{15}\overline{A}^\ast))$
on a subspace of $A$ of codimension at most
$H(Z) - H(B) + \Codim_B(\overline{B} \cap f_{23}^{-1}f_{26}(\widehat{C} \cap f_{25}^{-1}f_{15}\overline{A}^\ast))$.
Now
again we are going to use Lemma~\ref{lemma1}, Lemma~\ref{lemma3}, and
(\ref{p2:f15}).
Also on line (\ref{p2:condf26}) we will use the fact that
$f_{26}$ is injective on $\widehat{C} \cap f_{25}^{-1}f_{15}\overline{A}^\ast$
from (\ref{p2:f8f26}).
\begin{align}
\Codim_B(\overline{B} &\cap f_{23}^{-1}f_{26}(\widehat{C} \cap f_{25}^{-1}f_{15}\overline{A}^\ast)) \le \Delta_{\overline{B}} + \Codim_B(f_{23}^{-1}f_{26}(\widehat{C} \cap f_{25}^{-1}f_{15}\overline{A}^\ast))\nonumber\\
&\le \Delta_{\overline{B}} + \Codim_X(f_{26}(\widehat{C} \cap f_{25}^{-1}f_{15}\overline{A}^\ast))\nonumber\\
&= \Delta_{\overline{B}} + H(X) - \Dim(f_{26}(\widehat{C} \cap f_{25}^{-1}f_{15}\overline{A}^\ast))\nonumber\\
&= \Delta_{\overline{B}} + H(X) - \Dim(\widehat{C} \cap f_{25}^{-1}f_{15}\overline{A}^\ast)\label{p2:condf26}\\
&\le \Delta_{\overline{B}} + H(X) - H(C) + \Codim_C(\widehat{C}) + \Codim_C(f_{25}^{-1}f_{15}\overline{A}^\ast)\nonumber\\
&\le \Delta_{\overline{B}} + H(X) - H(C) + \Delta_{\widehat{C}} + \Codim_Y(f_{15}\overline{A}^\ast)\nonumber\\
&= \Delta_{\overline{B}} + H(X) - H(C) + \Delta_{\widehat{C}} + H(Y) - \Dim(f_{15}\overline{A}^\ast)\nonumber\\
&= \Delta_{\overline{B}} + H(X) - H(C) + H(Y) + \Delta_{\widehat{C}} - \Dim(\overline{A}^\ast)\nonumber\\
&= \Delta_{\overline{B}} + H(X) - H(C) + H(Y) + \Delta_{\widehat{C}} - H(A) + \Codim_A(\overline{A}^\ast)\nonumber\\
&\le \Delta_{\overline{B}} + H(X) - H(C) + H(Y) - H(A) + \Delta_{\widehat{C}} + \Delta_{\overline{A}^\ast}.\label{p2:condf262}
\end{align}
So, we have $f_{10}f_{15}t \in f_{22}(\overline{B} \cap f_{23}^{-1}f_{26}(\widehat{C} \cap f_{25}^{-1}f_{15}\overline{A}^\ast))$ on a
subspace of $A$ of codimension at most
$H(Z) - H(B) + H(X) - H(C) + H(Y) - H(A) + \Delta_{\overline{A}^\ast} + \Delta_{\overline{B}} + \Delta_{\widehat{C}}$.
To justify (\ref{p2:ass6}), we first know $f_{31}$ is injective on
$\widehat{B} \cap f_{32}^{-1}f_{26}(\widehat{C} \cap f_{25}^{-1}f_{15}\overline{A}^\ast)$
by (\ref{eq:Bhat2}).
Then by Lemma~\ref{lemma3}, we know
$f_{11}f_{15}t \in f_{31}(\widehat{B} \cap f_{32}^{-1}f_{26}(\widehat{C} \cap f_{25}^{-1}f_{15}\overline{A}^\ast))$
on a subspace of $A$ of codimension at
most $H(W) - H(B) + \Codim_B(\widehat{B} \cap f_{32}^{-1}f_{26}(\widehat{C} \cap f_{25}^{-1}f_{15}\overline{A}^\ast))$.
Now again we are going to use Lemma
\ref{lemma1} and Lemma~\ref{lemma3},
\begin{align}
\Codim_B(\widehat{B} \cap f_{32}^{-1}f_{26}(\widehat{C} \cap f_{25}^{-1}f_{15}\overline{A}^\ast)) &\le \Delta_{\widehat{B}} + \Codim_B(f_{32}^{-1}f_{26}(\widehat{C} \cap f_{25}^{-1}f_{15}\overline{A}^\ast))\nonumber\\
&\le \Delta_{\widehat{B}} + \Codim_X(f_{26}(\widehat{C} \cap f_{25}^{-1}f_{15}\overline{A}^\ast))\nonumber\\
&\le \Delta_{\widehat{B}} + H(X) - H(C) + H(Y) - H(A) + \Delta_{\widehat{C}} + \Delta_{\overline{A}^\ast}. \nonumber
\end{align}
The last line was derived by copying the argument from (\ref{p2:condf262}).
So, we have
$f_{11}f_{15}t \in f_{31}(\widehat{B} \cap f_{32}^{-1}f_{26}(\widehat{C} \cap f_{25}^{-1}f_{15}\overline{A}^\ast))$
on a subspace of $A$ of codimension at most
$H(W) - H(B) + H(X) - H(C) + H(Y) - H(A) + \Delta_{\overline{A}^\ast} + \Delta_{\widehat{B}} + \Delta_{\widehat{C}}$.
From (\ref{p2:ass2}) and (\ref{p2:ass5}) we know
$\exists \overline{c} \in \overline{C}, \overline{b} \in \overline{B}$ such that
\begin{align}
& f_{10}f_{15}t = f_{19}\overline{c} = f_{22}\overline{b} \mbox{ where } f_{20}\overline{c} \in f_{15}\overline{A}^\ast \mbox{ and } f_{23}\overline{b} \in f_{26}(\widehat{C} \cap f_{25}^{-1}f_{15}\overline{A}^\ast). \label{p2:f10}
\end{align}
From (\ref{p2:ass3}) and (\ref{p2:ass6}) we know
$\exists \widetilde{c} \in \widetilde{C}, \widehat{b} \in \widehat{B}$ such that
\begin{align}
& f_{11}f_{15}t = f_{29}\widetilde{c} = f_{31}\widehat{b} \mbox{ where } f_{28}\widetilde{c} \in f_{15}\overline{A}^\ast \mbox{ and } f_{32}\widehat{b} \in f_{26}(\widehat{C} \cap f_{25}^{-1}f_{15}\overline{A}^\ast). \label{p2:f11}
\end{align}
From (\ref{p2:ass4}) we know
$\exists \widehat{c} \in \widehat{C}$ such that
\begin{align}
& f_{12}f_{15}t = f_{26}\widehat{c} \mbox{ where } f_{25}\widehat{c} \in f_{15}\overline{A}^\ast. \label{p2:f12}
\end{align}
From (\ref{eq:Abar1}) and (\ref{eq:Abar2}), we know
\begin{align}
f_Bf_{15} &= -f_{13} \mbox{ on $\overline{A}$} \nonumber\\
f_B &= -f_{13}f_A \mbox{ on $f_{15}\overline{A}$}. \label{p2:fB}
\end{align}
From (\ref{eq:Abar1}) and (\ref{eq:Abar4}), we know
\begin{align}
f_Df_{15} &= -f_{14} \mbox{ on $\overline{A}$} \nonumber\\
f_D &= -f_{14}f_A \mbox{ on $f_{15}\overline{A}$}. \label{p2:fD}
\end{align}
From (\ref{eq:Abar1}) we have
\begin{align*}
f_7f_{12}f_{15}t + f_1f_{10}f_{15}t &= t.
\end{align*}
Then (\ref{p2:f12}), (\ref{p2:f10}), (\ref{eq:Chat1}), and (\ref{eq:Cbar1}) give
\begin{align}
f_7f_{12}f_{15}t + f_1f_{10}f_{15}t &= t \nonumber\\
f_7f_{26}\widehat{c} + f_1f_{19}\overline{c} &= t\nonumber\\
-f_Af_{25}\widehat{c} - f_Af_{20}\overline{c} &= t \nonumber\\
f_Af_{25}\widehat{c} + f_Af_{20}\overline{c} &= -t. \label{p2:I1}
\end{align}
From (\ref{eq:Abar2}) we have
\begin{align*}
f_4f_{11}f_{15}t + f_2f_{10}f_{15}t &= -f_{13}t.
\end{align*}
Then (\ref{p2:f11}), (\ref{p2:f10}), (\ref{eq:Ctilde2}), and (\ref{eq:Cbar2}) give
\begin{align}
f_4f_{11}f_{15}t + f_2f_{10}f_{15}t &= -f_{13}t \nonumber\\
f_4f_{29}\widetilde{c} + f_2f_{19}\overline{c} &= -f_{13}t\nonumber\\
-f_Bf_{28}\widetilde{c} -f_Bf_{20}\overline{c} &= -f_{13}t. \nonumber
\end{align}
By (\ref{p2:f11}) and (\ref{p2:f10}), we know
$f_{28}\widetilde{c} \in f_{15}\overline{A}^\ast$ and $f_{20}\overline{c} \in f_{15}\overline{A}^\ast$.
Now by (\ref{p2:fB}), we have
\begin{align}
-f_Bf_{28}\widetilde{c} -f_Bf_{20}\overline{c} &= -f_{13}t \nonumber\\
f_{13}f_Af_{28}\widetilde{c} + f_{13}f_Af_{20}\overline{c} &= -f_{13}t. \nonumber
\end{align}
Then using (\ref{eq:Abar1}), we know
$f_Af_{28}\widetilde{c} \in \overline{A}^\ast$ and $f_Af_{20}\overline{c} \in \overline{A}^\ast$.
By
(\ref{p2:f13}), we have
\begin{align}
f_{13}f_Af_{28}\widetilde{c} + f_{13}f_Af_{20}\overline{c} &= -f_{13}t \nonumber\\
f_Af_{28}\widetilde{c} + f_Af_{20}\overline{c} &= -t. \label{p2:I2}
\end{align}
From (\ref{eq:Abar4}) we have
\begin{align*}
f_9f_{12}f_{15}t + f_6f_{11}f_{15}t &= -f_{14}t.
\end{align*}
Then (\ref{p2:f12}), (\ref{p2:f11}), (\ref{eq:Ctilde4}), and (\ref{eq:Chat4}) give
\begin{align}
f_9f_{12}f_{15}t + f_6f_{11}f_{15}t &= -f_{14}t \nonumber\\
f_9f_{26}\widehat{c} + f_6f_{29}\widetilde{c} &= -f_{14}t\nonumber\\
-f_Df_{25}\widehat{c} + -f_Df_{28}\widetilde{c} &= -f_{14}t. \nonumber
\end{align}
By (\ref{p2:f12}) and (\ref{p2:f11}), we know
$f_{25}\widehat{c} \in f_{15}\overline{A}^\ast$ and $f_{28}\widetilde{c} \in f_{15}\overline{A}^\ast$.
Now by (\ref{p2:fD}), we have
\begin{align}
-f_Df_{25}\widehat{c} + -f_Df_{28}\widetilde{c} &= -f_{14}t \nonumber\\
f_{14}f_Af_{25}\widehat{c} + f_{14}f_Af_{28}\widetilde{c} &= -f_{14}t. \nonumber
\end{align}
Then using (\ref{eq:Abar1}), we know
$f_Af_{25}\widehat{c} \in \overline{A}^\ast$ and $f_Af_{28}\widetilde{c} \in \overline{A}^\ast$.
By (\ref{p2:f14}), we have
\begin{align}
f_{14}f_Af_{25}\widehat{c} + f_{14}f_Af_{28}\widetilde{c} &= -f_{14}t \nonumber\\
f_Af_{25}\widehat{c} + f_Af_{28}\widetilde{c} &= -t. \label{p2:I3}
\end{align}
From (\ref{eq:Cbar1}) and (\ref{p2:f3f19}), we know
\begin{align}
f_1f_{19} &= -f_Af_{20} \mbox{ on $\overline{C}$} \nonumber\\
f_1 &= -f_Af_{20}f_3\mbox{ on $f_{19}(\overline{C} \cap f_{20}^{-1}f_{15}\overline{A}^\ast)$}. \label{p2:f1}
\end{align}
From (\ref{eq:Chat1}) and (\ref{p2:f8f26}), we know
\begin{align}
f_7f_{26} &= -f_Af_{25} \mbox{ on $\widehat{C}$} \nonumber\\
f_7 &= -f_Af_{25}f_8\mbox{ on $f_{26}(\widehat{C} \cap f_{25}^{-1}f_{15}\overline{A}^\ast)$}. \label{p2:f7}
\end{align}
From (\ref{eq:Bbar1}), we have
\begin{align}
f_7f_{23}\overline{b} + f_1f_{22}\overline{b} &= 0. \nonumber
\end{align}
By (\ref{p2:f10}), we know $f_{23}\overline{b} \in f_{26}(\widehat{C} \cap f_{25}^{-1}f_{15}\overline{A}^\ast)$.
By (\ref{p2:f10}), we also know
$f_{22}\overline{b} = f_{19}\overline{c}$, which implies
$f_{22}\overline{b} \in f_{19}(\overline{C} \cap f_{20}^{-1}f_{15}\overline{A}^\ast)$.
Now we can
apply (\ref{p2:f1}) and (\ref{p2:f7}) to give us
\begin{align}
f_7f_{23}\overline{b} + f_1f_{22}\overline{b} &= 0 \nonumber\\
-f_Af_{25}f_8f_{23}\overline{b} - f_Af_{20}f_3f_{22}\overline{b} &= 0. \nonumber
\end{align}
Now using (\ref{eq:Bbar3}), (\ref{p2:f10}), and (\ref{p2:f3f19}), we have
\begin{align}
-f_Af_{25}f_8f_{23}\overline{b} - f_Af_{20}f_3f_{22}\overline{b} &= 0 \nonumber\\
f_Af_{25}f_3f_{22}\overline{b} - f_Af_{20}f_3f_{22}\overline{b} &= 0 \nonumber\\
f_Af_{25}f_3f_{22}\overline{b} &= f_Af_{20}f_3f_{22}\overline{b} \nonumber\\
f_Af_{25}f_{3}f_{19}\overline{c} &= f_Af_{20}f_3f_{19}\overline{c}\nonumber\\
f_Af_{25}\overline{c} &= f_Af_{20}\overline{c}.\label{p2:f20f25}
\end{align}
From (\ref{eq:Chat4}) and (\ref{p2:f8f26}), we know
\begin{align}
f_9f_{26} &= -f_Df_{25} \mbox{ on $\widehat{C}$} \nonumber\\
f_9 &= -f_Df_{25}f_{8} \mbox{ on $f_{26}(\widehat{C} \cap f_{25}^{-1}f_{15}\overline{A}^\ast)$}.\label{p2:f9}
\end{align}
From (\ref{eq:Ctilde4}) and (\ref{p2:f5f29}), we know
\begin{align}
f_6f_{29} &= -f_Df_{28} \mbox{ on $\widetilde{C}$} \nonumber\\
f_6 &= -f_Df_{28}f_{5} \mbox{ on $f_{29}(\widetilde{C} \cap f_{28}^{-1}f_{15}\overline{A}^\ast)$}.\label{p2:f6}
\end{align}
From (\ref{eq:Bhat4}), we have
\begin{align}
f_9f_{32}\widehat{b} + f_6f_{31}\widehat{b} &= 0. \nonumber
\end{align}
From (\ref{p2:f11}) we know $f_{31}\widehat{b} = f_{29}\widetilde{c}$ so
$f_{31}\widehat{b} \in f_{29}(\widetilde{C} \cap f_{28}^{-1}f_{15}\overline{A}^\ast)$.
From (\ref{p2:f11}) we also know that
$f_{32}\widehat{b} \in f_{26}(\widehat{C} \cap f_{25}^{-1}f_{15}\overline{A}^\ast)$,
so (\ref{p2:f9}) and (\ref{p2:f6}) give us
\begin{align}
f_9f_{32}\widehat{b} + f_6f_{31}\widehat{b} &= 0 \nonumber\\
-f_Df_{25}f_8f_{32}\widehat{b} - f_Df_{28}f_5f_{31}\widehat{b} &= 0. \nonumber
\end{align}
From (\ref{p2:f11}), we know $f_{32}\widehat{b} \in f_{26}(\widehat{C} \cap f_{25}^{-1}f_{15}\overline{A}^\ast)$.
From (\ref{p2:f8f26}), we know $f_8f_{26} = I$ on $\widehat{C} \cap f_{25}^{-1}f_{15}\overline{A}^\ast$.
So
$f_8f_{32}\widehat{b} \in f_{25}^{-1}f_{15}\overline{A}^\ast$, which implies
$f_{25}f_8f_{32}\widehat{b} \in f_{15}\overline{A}^\ast$.
By (\ref{p2:f11}) and
(\ref{p2:f5f29}), we know
$f_{28}f_5f_{31}\widehat{b} = f_{28}f_5f_{29}\widetilde{c} = f_{28}\widetilde{c} \in f_{15}\overline{A}^\ast$.
Now we can apply (\ref{p2:fD}) to give us
\begin{align}
-f_Df_{25}f_8f_{32}\widehat{b} - f_Df_{28}f_5f_{31}\widehat{b} &= 0 \nonumber\\
f_{14}f_Af_{25}f_8f_{32}\widehat{b} + f_{14}f_Af_{28}f_5f_{31}\widehat{b} &= 0. \nonumber
\end{align}
Since we already established that
$f_{25}f_8f_{32}\widehat{b} \in f_{15}\overline{A}^\ast$ and $f_{28}f_5f_{31}\widehat{b} \in f_{15}\overline{A}^\ast$,
by (\ref{eq:Abar1}) and (\ref{p2:f14}) we know
\begin{align}
f_{14}f_Af_{25}f_8f_{32}\widehat{b} + f_{14}f_Af_{28}f_5f_{31}\widehat{b} &= 0 \nonumber\\
f_Af_{25}f_8f_{32}\widehat{b} + f_Af_{28}f_5f_{31}\widehat{b} &= 0.\nonumber
\end{align}
Now by (\ref{eq:Bhat3})
\begin{align}
f_Af_{25}f_8f_{32}\widehat{b} + f_Af_{28}f_5f_{31}\widehat{b} &= 0\nonumber\\
-f_Af_{25}f_5f_{31}\widehat{b} + f_Af_{28}f_5f_{31}\widehat{b} &= 0\nonumber\\
f_Af_{25}f_5f_{31}\widehat{b} &= f_Af_{28}f_5f_{31}\widehat{b}. \nonumber
\end{align}
By (\ref{p2:f11}) and (\ref{p2:f5f29}), we have
\begin{align}
f_Af_{25}f_5f_{31}\widehat{b} &= f_Af_{28}f_5f_{31}\widehat{b} \nonumber\\
f_Af_{25}f_5f_{29}\widetilde{c} &= f_Af_{28}f_5f_{29}\widetilde{c} \nonumber\\
f_Af_{25}\widetilde{c} &= f_Af_{28}\widetilde{c}. \label{p2:f28f25}
\end{align}
Now adding (\ref{p2:I1}), (\ref{p2:I2}), and (\ref{p2:I3}), we have
\begin{align*}
-3t &= 2(f_Af_{20}\overline{c} + f_Af_{25}\widehat{c} + f_Af_{28}\widetilde{c}).
\end{align*}
Now using (\ref{p2:f20f25}) and (\ref{p2:f28f25}) we have
\begin{align}
-3t &= 2(f_Af_{25}\overline{c} + f_Af_{25}\widehat{c} + f_Af_{25}\widetilde{c})\nonumber\\
-3t &= 2f_Af_{25}(\overline{c} + \widehat{c} + \widetilde{c}).\nonumber
\end{align}
By (\ref{p2:f3f19}), (\ref{p2:f5f29}), and (\ref{p2:f8f26}) we know
\begin{align}
-3t &= 2f_Af_{25}(f_3f_{19}\overline{c} + f_8f_{26}\widehat{c} + f_5f_{29}\widetilde{c}).\nonumber
\end{align}
By (\ref{p2:f10}), (\ref{p2:f11}), (\ref{p2:f12}), and (\ref{eq:Abar3}), we have
\begin{align}
-3t &= 2f_Af_{25}(f_3f_{10}f_{15}t + f_8f_{12}f_{15}t + f_5f_{11}f_{15}t)\nonumber\\
-3t &= 2f_Af_{25}(0)\nonumber\\
3t &= 0.
\end{align}
Thus if the field is of characteristic other than 3, then no nonzero $t$ can
satisfy conditions (\ref{p2:ass1})--(\ref{p2:ass6}).
Therefore the sum of the
codimensions given in the assumptions must be at least the dimension of $A$.
So
we have a linear rank inequality for fields of characteristic other than 3:
\begin{align*}
H(A) &\le \Delta_{\overline{A}^\ast} + H(Z) - H(C) + H(Y) - H(A) + \Delta_{\overline{C}} + \Delta_{\overline{A}^\ast} \\
&\ \ \ + H(W) - H(C) + H(Y) - H(A) + \Delta_{\widetilde{C}}+ \Delta_{\overline{A}^\ast} \\
&\ \ \ + H(X) - H(C) + H(Y) - H(A) + \Delta_{\widehat{C}} + \Delta_{\overline{A}^\ast} \\
&\ \ \ + H(Z) - H(B) + H(X) - H(C) + H(Y) - H(A) + \Delta_{\overline{A}^\ast} + \Delta_{\overline{B}} + \Delta_{\widehat{C}}\\
&\ \ \ + H(W) - H(B) + H(X) - H(C) + H(Y) - H(A) + \Delta_{\overline{A}^\ast} + \Delta_{\widehat{B}} + \Delta_{\widehat{C}}\\
&= 2H(Z) + 5H(Y) + 3H(X) + 2H(W) - 5H(A) -2H(B) - 5H(C)\\
&\ \ \ + 6\Delta_{\overline{A}^\ast} + \Delta_{\overline{B}} + \Delta_{\widehat{B}} + \Delta_{\overline{C}} + \Delta_{\widetilde{C}} + 3\Delta_{\widehat{C}}\\
&= 2H(Z) + 5H(Y) + 3H(X) + 2H(W) - 5H(A) -2H(B) - 5H(C)\\
&\ \ \ + 6( H(W) + 4H(Y) + H(Z) - 2H(A) - H(B) -2H(C) - H(D) )\\
&\ \ \ + 6( 3\Delta_{\overline{A}} + \Delta_{\overline{B}} + \Delta_{\overline{C}} + \Delta_{\widetilde{C}}+ \Delta_{\overline{D}} )+ \Delta_{\overline{B}} + \Delta_{\widehat{B}} + \Delta_{\overline{C}} + \Delta_{\widetilde{C}} + 3\Delta_{\widehat{C}}\\
&= 8H(Z) + 29H(Y) + 3H(X) + 8H(W) -6H(D) -17H(C) - 8H(B) - 17H(A)\\
&\ \ \ + 18\Delta_{\overline{A}} + 7\Delta_{\overline{B}} + \Delta_{\widehat{B}} + 7\Delta_{\overline{C}} + 7\Delta_{\widetilde{C}} + 3\Delta_{\widehat{C}} + 6\Delta_{\overline{D}}\\
&= 8H(Z) + 29H(Y) + 3H(X) + 8H(W) -6H(D) -17H(C) - 8H(B) - 17H(A)\\
&\ \ \ + 55H(Z|A,B,C) + 35H(Y|W,X,Z) + 50H(X|A,C,D) + 49H(W|B,C,D)\\
&\ \ \ + 18H(A|B,D,Y) + 7H(B|D,X,Z) + H(B|A,W,X) + 7H(C|D,Y,Z) \\
&\ \ \ + 7H(C|B,X,Y) + 3H(C|A,W,Y) + 6H(D|A,W,Z)\\
&\ \ \ + 49(H(A) + H(B) + H(C) + H(D) - H(A,B,C,D)).
\end{align*}
\end{proof}
The next theorem demonstates that the inequality in Theorem~\ref{thm:T8}
does not in general hold for vector spaces with finite fields of characteristic $3$.
\begin{thm}\label{thm:T8-char3}
There exists a vector space $V$ with a finite scalar field of characteristic $3$
such that the T8 inequality in Theorem~\ref{thm:T8}
is not a linear rank inequality over $V$.
\end{thm}
\begin{proof}
Let $V$ be the vector space of $4$-dimensional vectors whose components are from the field $GF(3)$,
and define the following subspaces of $V$:
\begin{align*}
A = \langle (1,0,0,0)\rangle &\ \ \ B = \langle (0,1,0,0)\rangle\\
C = \langle (0,0,1,0)\rangle &\ \ \ D = \langle (0,0,0,1)\rangle\\
W = \langle (0,1,1,1)\rangle &\ \ \ X = \langle (1,0,1,1)\rangle\\
Y = \langle (1,1,0,1)\rangle &\ \ \ Z = \langle (1,1,1,0)\rangle.
\end{align*}
We have:
\begin{align}
0 &= H(Z|A,B,C)\Comment{$(1,1,1,0)=(1,0,0,0)+(0,1,0,0)+(0,0,1,0)$}\nonumber\\
&= H(W|B,C,D)\Comment{$(0,1,1,1)=(0,1,0,0)+(0,0,1,0)+(0,0,0,1)$}\nonumber\\
&= H(X|A,C,D)\Comment{$(1,0,1,1)=(1,0,0,0)+(0,0,1,0)+(0,0,0,1)$}\nonumber\\
&= H(Y|W,X,Z)\Comment{$(1,1,0,1)=2^{-1}\cdot((0,1,1,1)+(1,0,1,1)+(1,1,1,0))$}\nonumber\\
&= H(A|B,D,Y)\Comment{$(1,0,0,0)=(1,1,0,1)-(0,1,0,0)-(0,0,0,1)$}\nonumber\\
&= H(D|A,W,Z)\Comment{$(0,0,0,1)=(0,1,1,1)+(1,0,0,0)-(1,1,1,0)$}\nonumber\\
&= H(C|D,Y,Z)\Comment{$(0,0,1,0)=(1,1,1,0)+(0,0,0,1)-(1,1,0,1)$}\nonumber\\
&= H(B|D,X,Z)\Comment{$(0,1,0,0)=(1,1,1,0)+(0,0,0,1)-(1,0,1,1)$}\nonumber\\
&= H(C|B,X,Y)\Comment{$(0,0,1,0)=(1,0,1,1)+(0,1,0,0)-(1,1,0,1)$}\nonumber\\
&= H(C|A,W,Y)\Comment{$(0,0,1,0)=(0,1,1,1)+(1,0,0,0)-(1,1,0,1)$}\nonumber\\
&= H(B|A,W,X)\Comment{$(0,1,0,0)=(0,1,1,1)+(1,0,0,0)-(1,0,1,1)$}.\nonumber\\
\label{eq:zero-entropies-T8}
\end{align}
Note that the characteristic $3$ assumption is used above
in showing $H(Y|W,X,Z)=0$,
by using the fact that the ranks of $Y$ and $Y\cap \langle W, X, Z\rangle$ are both $1$,
since
$(1,1,0,1) = 2^{-1}\cdot ( (0,1,1,1) + (1,0,1,1) + (1,1,1,0))$,
which holds for scalar fields of characteristic $3$
(in fact, for all characteristics except $2$).
We know $H(A)=H(B)=H(C)=H(D)=H(W)=H(X)=H(Y)=H(Z)=1$.
Also, we have
$$H(A) + H(B) + H(C) + H(D) = H(A,B,C,D).$$
So, if the inequality in
Theorem~\ref{thm:T8} were to hold over $V$,
then we would have
\begin{align*}
1
&= H(A)\\
&\le 8H(Z) + 29H(Y) + 3H(X) + 8H(W) -6H(D) -17H(C) - 8H(B) - 17H(A)\\
&= 8 + 29 + 3 + 8 -6 -17 - 8 - 17\\
&= 0
\end{align*}
which is impossible.
\end{proof}
Consider a network over finite field $F$ with a $(k,n)$ linear code.
The \textit{vector space associated with any message} is defined to be $F^k$.
The \textit{vector space associated with any edge} is defined to be the set of all
possible vectors from $F^n$ that can be carried on that edge
(i.e. taking into account the linear code).
Since each output of a network node is a function of the node's inputs,
the conditional entropy of the vector carried by a node's out-edge,
given the entropies of the vectors carried by the node's in-edges, is zero,
assuming the network messages are uniform random vectors.
The following lemma extends this idea from random variables to vector spaces
and will be useful for the proof of Corollary~\ref{cor:T8-capacity}.
\begin{lem}
Suppose a network has a node with an out-edge (or demand) $x$
and in-edges and messages (in some order) $y_1, \dots, y_m$.
Suppose the network has a finite field alphabet and a linear code.
Let us view $X,Y_1, \dots, Y_m$ as the vector spaces associated with
$x,y_1, \dots, y_m$, respectively.
Then we have $H(X|Y_1, \dots, Y_m)=0$.
\label{lem:conditional-dimensions}
\end{lem}
\begin{proof}
The vector carried on the node's out-edge (or demand) $x$ is a linear combination
of the vectors carried on the node's in-edges and the node's messages
$y_1, \dots, y_m$.
Thus, every vector appearing on the node's out-edge (or demand) lies in the span
of the subspaces $Y_1, \dots, Y_m$.
This implies
$\Dim (X) = \Dim (X \cap \langle Y_1, \dots, Y_m\rangle)$,
or equivalently,
$H(X|Y_1, \dots, Y_m)=0$.
\end{proof}
The following corollary uses the T8 linear rank inequality to derive
capacities and a capacity bound on the T8 network.
Note that although the T8 network itself was used as a guide in obtaining the
T8 linear rank inequality,
subsequently using the inequality to bound the network capacity is not circular reasoning.
The proof of Corollary~\ref{cor:T8-capacity} below makes use of the T8 linear rank inequality,
and resembles the example shown earlier in \eqref{rv-inequality} for computing the capacity of the Butterfly network
using information inequalities and random variables.
\begin{cor}
For the T8 network,
the linear coding capacity is at most $48/49$
over any finite field alphabet of characteristic not equal to $3$.
The linear coding capacity over finite field alphabets of characteristic $3$
and the coding capacity are both equal to $1$.
\label{cor:T8-capacity}
\end{cor}
\begin{proof}
Let $F$ be a finite field alphabet.
Consider a $(k,n)$ linear solution of the T8 network over $F$,
such that the characteristic of $F$ is not $3$.
Let $A$, $B$, $C$, $D$ be message random variables in the T8 network,
that are uniformly distributed over vectors in $F^k$.
Let $W$, $X$, $Y$, $Z$ be the resulting random variables
associated with the corresponding labeled edges of T8 in Figure~\ref{fig:T8}.
Equations \eqref{eq:zero-entropies-T8} now hold
with random variables $A,B,C,D,W,X,Y,Z$
(i.e. not as subspaces as in Theorem~\ref{thm:T8-char3})
by Lemma~\ref{lem:conditional-dimensions}:
\begin{align*}
0 &= H(Z|A,B,C)\Comment{$(n_1,n_2)$}\nonumber\\
&= H(W|B,C,D)\Comment{$(n_3,n_4)$}\nonumber\\
&= H(X|A,C,D)\Comment{$(n_5,n_6)$}\nonumber\\
&= H(Y|W,X,Z)\Comment{$(n_4,n_7)$}\nonumber\\
&= H(A|B,D,Y)\Comment{$n_{ 9}$}\nonumber\\
&= H(D|A,W,Z)\Comment{$n_{10}$}\nonumber\\
&= H(C|D,Y,Z)\Comment{$n_{11}$}\nonumber\\
&= H(B|D,X,Z)\Comment{$n_{12}$}\nonumber\\
&= H(C|B,X,Y)\Comment{$n_{13}$}\nonumber\\
&= H(C|A,W,Y)\Comment{$n_{14}$}\nonumber\\
&= H(B|A,W,X)\Comment{$n_{15}$}
\end{align*}
and since the vector spaces $A,B,C,D$ are associated with independent random variables,
we have
$$H(A) + H(B) + H(C) + H(D) = H(A,B,C,D)$$
so the T8 inequality in Theorem~\ref{thm:T8} reduces to
\begin{align*}
H(A) &\le 8H(Z) + 29H(Y) + 3H(X) + 8H(W) -6H(D) -17H(C) - 8H(B) - 17H(A).
\end{align*}
Now since
$H(A) {=} H(B) {=} H(C) {=} H(D) {=} k$ and
$H(W) {=} H(X) {=} H(Y) {=} H(Z) \le n$,
we have
\begin{align*}
k &\le 8n + 29n + 3n + 8n - 6k -17k - 8k - 17k\\
k/n &\le 48/49.
\end{align*}
So, the linear coding capacity over every characteristic except for 3 is at most $48/49 < 1.$
The T8 network has a scalar linear solution over characteristic 3 by using the following
edge functions
(here we are using the notations $A,B,C,D,W,X,Y,Z$ to denote edge variables rather than vector spaces):
\begin{align*}
Z &= A + B + C\\
W &= B + C + D\\
X &= A + C + D\\
Y &= W + X + Z.
\end{align*}
and decoding functions:
\begin{align*}
n_{ 9}: A &= (2^{-1}\cdot Y) - B - D\\
n_{10}: D &= W - Z + A\\
n_{11}: C &= Z - (2^{-1}\cdot Y) + D\\
n_{12}: B &= Z - X + D\\
n_{13}: C &= X - (2^{-1}\cdot Y) + B\\
n_{14}: C &= W - (2^{-1}\cdot Y) + A\\
n_{15}: B &= W - X + A
\end{align*}
Thus the linear coding capacity for characteristic 3 is at least 1.
We know the coding capacity is at most 1 because every path from
source $A$ to node $n_9$ passes through the single edge $(n_7,n_8)$.
Since the coding capacity is at least as large as the linear coding
capacity for characteristic 3,
we conclude that the coding capacity is exactly equal to 1.
\end{proof}
\newpage
\section{A Linear Rank Inequality for Fields of Characteristic 3}
In the T8 matroid, $W+X+Y+Z=(3,3,3,3)$, which equals $(0,0,0,0)$
in characteristic 3.
We define the \textit{non-T8 matroid} to be the T8 matroid except that
we force the T8's characteristic 3 circuit $\{W,X,Y,Z\}$ to be a base in the non-T8 matroid.
Figure~\ref{fig:nonT8}
is a network
that we call the \textit{non-T8 network},
whose dependencies and independencies are consistent with the non-T8 matroid.
The non-T8 network was designed by the construction process described in \cite{nonshannon}.
Theorem~\ref{thm:nonT8} uses the non-T8 network as a guide to derive a linear rank
inequality valid for characteristic 3.
The new linear rank inequality can then be used to prove the non-T8 network
has linear capacity less than 1 if the field characteristic is 3.
\begin{figure}
\begin{center}
\psfrag{n1}{\Large $n_{1}$}
\psfrag{n2}{\Large $n_{2}$}
\psfrag{n3}{\Large $n_{3}$}
\psfrag{n4}{\Large $n_{4}$}
\psfrag{n5}{\Large $n_{5}$}
\psfrag{n6}{\Large $n_{6}$}
\psfrag{n7}{\Large $n_{7}$}
\psfrag{n8}{\Large $n_{8}$}
\psfrag{n9}{\Large $n_{9}$}
\psfrag{n10}{\Large $n_{10}$}
\psfrag{n11}{\Large $n_{11}$}
\psfrag{n12}{\Large $n_{12}$}
\psfrag{n13}{\Large $n_{13}$}
\psfrag{n14}{\Large $n_{14}$}
\psfrag{n15}{\Large $n_{15}$}
\includegraphics[width=13.5cm]{non-T8-network.eps}
\end{center}
\caption{ The Non-T8 Network
has source messages $A,B,C,$ and $D$ generated at hidden source nodes with
certain hidden out-edges pointing to corresponding displayed nodes
$n_1$, $n_3$, $n_5$, $n_7$, and $n_9$--$n_{14}$
(which are labeled by incoming messages above such nodes).
The nodes $n_9$--$n_{15}$ each demand one message, as labeled below such nodes.
}
\label{fig:nonT8}
\end{figure}
\begin{thm}\label{thm:nonT8}
Let $A,B,C,D,W,X,Y$, and $Z$ be subspaces of a vector space $V$ whose scalar field is finite
and of characteristic 3.
Then the following is a linear rank inequality over $V$:
\begin{align*}
H(A) &\le 9H(Z) + 8H(Y) + 5H(X) + 6H(W) - 4H(D) - 12H(C) - 11H(B) -H(A)\\
&\ \ \ +19H(Z|A,B,C) + 17H(Y|A,B,D) + 13H(X|A,C,D) + 11H(W|B,C,D)\\
&\ \ \ +H(A|W,X,Y,Z) + H(A|B,W,X) + 7H(B|D,X,Z) + 4H(B|C,X,Y)\\
&\ \ \ +7H(C|D,Y,Z) + 5H(C|A,W,Y) + 4H(D|A,W,Z)\\
&\ \ \ + 29( H(A) + H(B) + H(C) + H(D) - H(A,B,C,D) ).
\end{align*}
\end{thm}
\begin{proof}
By Lemma~\ref{lemma4} we get linear functions:
\begin{center}
\begin{tabular}{ccc}
$f_1:W \rightarrow B$, & $f_2:W \rightarrow C$, & $f_3:W \rightarrow D$, \\
$f_4:X \rightarrow A$, & $f_5:X \rightarrow C$, & $f_6:X \rightarrow D$, \\
$f_7:Y \rightarrow A$, & $f_8:Y \rightarrow B$, & $f_9:Y \rightarrow D$, \\
$f_{10}:Z \rightarrow A$, & $f_{11}:Z \rightarrow B$, & $f_{12}:Z \rightarrow C$, \\
$f_{13}:A \rightarrow B$, & $f_{14}:A \rightarrow W$, & $f_{15}:A \rightarrow X$, \\
$f_{16}:C \rightarrow A$, & $f_{17}:C \rightarrow W$, & $f_{18}:C \rightarrow Y$, \\
$f_{19}:B \rightarrow C$, & $f_{20}:B \rightarrow X$, & $f_{21}:B \rightarrow Y$, \\
$f_{22}:D \rightarrow W$, & $f_{23}:D \rightarrow A$, & $f_{24}:D \rightarrow Z$, \\
$f_{25}:B \rightarrow X$, & $f_{26}:B \rightarrow D$, & $f_{27}:B \rightarrow Z$, \\
$f_{28}:C \rightarrow Y$, & $f_{29}:C \rightarrow Z$, & $f_{30}:C \rightarrow D$,
\end{tabular}
\begin{tabular}{cccc}
$f_{31}:A \rightarrow W$, & $f_{32}:A \rightarrow X$, & $f_{33}:A \rightarrow Y$, & $f_{34}:A \rightarrow Z$
\end{tabular}
\end{center}
such that
\begin{align}
f_1 + f_2 + f_3 &= I \mbox{ on a subspace of $W$ of codimension $H(W|B,C,D)$}\label{p3:eq:1}\\
f_4 + f_5 + f_6 &= I \mbox{ on a subspace of $X$ of codimension $H(X|A,C,D)$}\label{p3:eq:2}\\
f_7 + f_8 + f_9 &= I \mbox{ on a subspace of $Y$ of codimension $H(Y|A,B,D)$}\label{p3:eq:3}\\
f_{10} + f_{11} + f_{12} &= I \mbox{ on a subspace of $Z$ of codimension $H(Z|A,B,C)$}\label{p3:eq:4}\\
f_{13} + f_{14} + f_{15} &= I \mbox{ on a subspace of $A$ of codimension $H(A|B,W,X)$}\label{p3:eq:5}\\
f_{16} + f_{17} + f_{18} &= I \mbox{ on a subspace of $C$ of codimension $H(C|A,W,Y)$}\label{p3:eq:6}\\
f_{19} + f_{20} + f_{21} &= I \mbox{ on a subspace of $B$ of codimension $H(B|C,X,Y)$}\label{p3:eq:7}\\
f_{22} + f_{23} + f_{24} &= I \mbox{ on a subspace of $D$ of codimension $H(D|A,W,Z)$}\label{p3:eq:8}\\
f_{25} + f_{26} + f_{27} &= I \mbox{ on a subspace of $B$ of codimension $H(B|D,X,Z)$}\label{p3:eq:9}\\
f_{28} + f_{29} + f_{30} &= I \mbox{ on a subspace of $C$ of codimension $H(C|D,Y,Z)$}\label{p3:eq:10}\\
f_{31} + f_{32} + f_{33} + f_{34} &= I \mbox{ on a subspace of $A$ of codimension $H(A|W,X,Y,Z)$}.\label{p3:eq:11}
\end{align}
Now combining some functions we obtained from Lemma \ref{lemma4} gives four new functions:
\begin{align*}
f_{4} f_{32} + f_{7} f_{33} + f_{10} f_{34} &: A
\rightarrow A \\
f_{1} f_{31} + f_{8} f_{33} + f_{11} f_{34} &: A
\rightarrow B \\
f_{2} f_{31} + f_{5} f_{32} + f_{12} f_{34} &: A
\rightarrow C \\
f_{3} f_{31} + f_{6} f_{32} + f_{9} f_{33} &: A
\rightarrow D.
\end{align*}
Using (\ref{p3:eq:1})--(\ref{p3:eq:4}), (\ref{p3:eq:11}), Lemma~\ref{lemma1},
and Lemma~\ref{lemma3} we know the sum of these four functions is equal to $I$ on a
subspace of $A$ of codimension at most
$H(W|B,C,D) + H(X|A,C,D) + H(Y|A,B,D) + H(Z|A,B,C) + H(A|W,X,Y,Z)$.
Now applying Lemma~\ref{lemma6} and Lemma~\ref{lemma1} to the functions
$f_{4}\Compose f_{32} + f_{7}\Compose f_{33} + f_{10}\Compose f_{34} - I$,
$f_{1}\Compose f_{31} + f_{8}\Compose f_{33} + f_{11}\Compose f_{34}$,
$f_{2}\Compose f_{31} + f_{5}\Compose f_{32} + f_{12}\Compose f_{34}$,
and
$f_{3}\Compose f_{31} + f_{6}\Compose f_{32} + f_{9}\Compose f_{33}$,
we get a subspace $\widehat{A}$ of $A$ of codimension at most
\begin{align*}
\Delta_{\widehat{A}} &= H(W|B,C,D) + H(X|A,C,D) + H(Y|A,B,D) + H(Z|A,B,C) + H(A|W,X,Y,Z)\\
& + H(A) + H(B) + H(C) + H(D) - H(A,B,C,D)
\end{align*}
on which
\begin{align}
f_{4}\Compose f_{32} + f_{7}\Compose f_{33} + f_{10}\Compose f_{34} &= I \label{p3:eq:Ahat1}\\
f_{1}\Compose f_{31} + f_{8}\Compose f_{33} + f_{11}\Compose f_{34} &= 0 \label{p3:eq:Ahat2}\\
f_{2}\Compose f_{31} + f_{5}\Compose f_{32} + f_{12}\Compose f_{34} &= 0 \label{p3:eq:Ahat3}\\
f_{3}\Compose f_{31} + f_{6}\Compose f_{32} + f_{9}\Compose f_{33} &= 0. \label{p3:eq:Ahat4}
\end{align}
Similarly, we get a subspace $\overline{A}$ of $A$ of codimension at most
\begin{align*}
\Delta_{\overline{A}} &= H(W|B,C,D) + H(X|A,C,D) + H(A|B,W,X)\\
& + H(A) + H(B) + H(C) + H(D) - H(A,B,C,D)
\end{align*}
on which
\begin{align}
f_4\Compose f_{15} &= I \label{p3:eq:Abar1}\\
f_{13} + f_1\Compose f_{14} &= 0 \label{p3:eq:Abar2}\\
f_2\Compose f_{14} + f_5\Compose f_{15} &= 0 \label{p3:eq:Abar3}\\
f_3\Compose f_{14} + f_6\Compose f_{15} &= 0. \label{p3:eq:Abar4}
\end{align}
We get a subspace $\overline{B}$ of $B$ of codimension at most
\begin{align*}
\Delta_{\overline{B}} &= H(X|A,C,D) + H(Y|A,B,D)+ H(B|C,X,Y)\\
& + H(A) + H(B) + H(C) + H(D) - H(A,B,C,D)
\end{align*}
on which
\begin{align}
f_4\Compose f_{20} + f_7\Compose f_{21} &= 0 \label{p3:eq:Bbar1}\\
f_8\Compose f_{21} &= I \label{p3:eq:Bbar2}\\
f_{19} + f_5\Compose f_{20} &= 0 \label{p3:eq:Bbar3}\\
f_6\Compose f_{20} + f_9\Compose f_{21} &= 0. \label{p3:eq:Bbar4}
\end{align}
We get a subspace $\widehat{B}$ of $B$ of codimension at most
\begin{align*}
\Delta_{\widehat{B}} &= H(X|A,C,D) + H(Z|A,B,C) + H(B|D,X,Z)\\
& + H(A) + H(B) + H(C) + H(D) - H(A,B,C,D)
\end{align*}
on which
\begin{align}
f_4\Compose f_{25} + f_{10}\Compose f_{27} &= 0 \label{p3:eq:Bhat1}\\
f_{11}\Compose f_{27} &= I \label{p3:eq:Bhat2}\\
f_{5}\Compose f_{25} + f_{12}\Compose f_{27} &= 0 \label{p3:eq:Bhat3}\\
f_{6}\Compose f_{25} + f_{26} &= 0. \label{p3:eq:Bhat4}
\end{align}
We get a subspace $\overline{C}$ of $C$ of codimension at most
\begin{align*}
\Delta_{\overline{C}} &= H(W|B,C,D) + H(Y|A,B,D) + H(C|A,W,Y)\\
& + H(A) + H(B) + H(C) + H(D) - H(A,B,C,D)
\end{align*}
on which
\begin{align}
f_{16} + f_7\Compose f_{18} &= 0 \label{p3:eq:Cbar1}\\
f_{1}\Compose f_{17} + f_{8}\Compose f_{18} &= 0 \label{p3:eq:Cbar2}\\
f_2\Compose f_{17} &= I \label{p3:eq:Cbar3}\\
f_3\Compose f_{17} + f_9\Compose f_{18} &= 0. \label{p3:eq:Cbar4}
\end{align}
We get a subspace $\widehat{C}$ of $C$ of codimension at most
\begin{align*}
\Delta_{\widehat{C}} &= H(Y|A,B,D) + H(Z|A,B,C) + H(C|D,Y,Z)\\
& + H(A) + H(B) + H(C) + H(D) - H(A,B,C,D)
\end{align*}
on which
\begin{align}
f_7\Compose f_{28} + f_{10}\Compose f_{29} &= 0 \label{p3:eq:Chat1}\\
f_{8}\Compose f_{28} + f_{11}\Compose f_{29} &= 0 \label{p3:eq:Chat2}\\
f_{12}\Compose f_{29} &= I \label{p3:eq:Chat3}\\
f_9\Compose f_{28} + f_{30} &= 0. \label{p3:eq:Chat4}
\end{align}
We get a subspace $\overline{D}$ of $D$ of codimension at most
\begin{align*}
\Delta_{\overline{D}} &= H(W|B,C,D) + H(Z|A,B,C) + H(D|A,W,Z)\\
& + H(A) + H(B) + H(C) + H(D) - H(A,B,C,D)
\end{align*}
on which
\begin{align}
f_{23} + f_{10}\Compose f_{24} &= 0 \label{p3:eq:Dbar1}\\
f_{1}\Compose f_{22} + f_{11}\Compose f_{24} &= 0 \label{p3:eq:Dbar2}\\
f_2\Compose f_{22} + f_{12}\Compose f_{24} &= 0 \label{p3:eq:Dbar3}\\
f_3\Compose f_{22} &= I. \label{p3:eq:Dbar4}
\end{align}
Let $\widehat{B}^\ast = f_{11}(f_{27}\widehat{B} \cap f_{29}\widehat{C}) \subseteq \widehat{B}$.
Considering (\ref{p3:eq:Bhat2}) and
(\ref{p3:eq:Chat3}), we can apply Lemma~\ref{lem Inj} to show that
$f_{12}f_{27}$ is injective on $\widehat{B}^\ast$.
By (\ref{p3:eq:Bhat3}), we
know
\begin{align}
& f_{5}f_{25}\mbox{ is injective on $\widehat{B}^\ast$.}\label{p3:f5f25}
\end{align}
Let $\widehat{C}^\ast = f_{12}( f_{29}\widehat{C} \cap f_{27}\widehat{B}) \subseteq \widehat{C}$.
Considering again (\ref{p3:eq:Bhat2}) and
(\ref{p3:eq:Chat3}), we can apply Lemma~\ref{lem Inj} to show that
$f_{11}f_{29}$ is injective on $\widehat{C}^\ast$.
By (\ref{p3:eq:Chat2}), we know
\begin{align}
& f_{8}f_{28}\mbox{ is injective on $\widehat{C}^\ast$.}\label{p3:f8f28}
\end{align}
Let $\overline{A}^\ast = f_4(f_{15}\overline{A} \cap f_{25}\widehat{B}^\ast) \subseteq \overline{A}$.
Considering (\ref{p3:eq:Abar1}) and (\ref{p3:f5f25}),
we can apply Lemma~\ref{lem Inj} to show that $f_5f_{15}$ is injective on
$\overline{A}^\ast$.
By (\ref{p3:eq:Abar3}), we know $f_2f_{14}$ is injective
on $\overline{A}^\ast$ which implies
\begin{align}
&\mbox{$f_{14}$ is injective on $\overline{A}^\ast$.}\label{p3:f14}
\end{align}
Let $\overline{C}^\ast = f_2(f_{17}\overline{C} \cap f_{22}\overline{D}) \subseteq \overline{C}$.
Considering (\ref{p3:eq:Cbar3}) and
(\ref{p3:eq:Dbar4}),we can apply Lemma~\ref{lem Inj} to show that
$f_3f_{17}$ is injective on $\overline{C}^\ast$.
Then by (\ref{p3:eq:Cbar4}),
we know
\begin{align}
& \mbox{ $f_9f_{18}$ is injective on $\overline{C}^\ast$.}\label{p3:f9f18}
\end{align}
Let $\overline{B}^\ast = f_8(f_{21}\overline{B} \cap f_{18}\overline{C}^\ast) \subseteq \overline{B}$.
Considering (\ref{p3:eq:Bbar2}) and \eqref{p3:f9f18},
we can apply Lemma~\ref{lem Inj} to show that
\begin{align}
& f_9f_{21}\mbox{ is injective on $\overline{B}^\ast$.}\label{p3:f9f21}
\end{align}
By (\ref{p3:eq:Bbar4}), we know
\begin{align}
&\mbox{$f_6f_{20}$ is injective on $\overline{B}^\ast$}\label{p3:f6f20}
\end{align}
which implies
\begin{align}
&\mbox{$f_{20}$ is injective on $\overline{B}^\ast$.}\label{p3:f20}
\end{align}
Let us define the functions
\begin{align*}
g_{14} &= (f_{14} | \overline{A}^\ast)^{-1}\\
g_{20} &= (f_{20} | \overline{B}^\ast)^{-1}
\end{align*}
where
$f_{14} | \overline{A}^\ast$
and
$f_{20} | \overline{B}^\ast$
are the restrictions of the functions $f_{14}$ and $f_{20}$
to the sets
$\overline{A}^\ast$
and
$\overline{B}^\ast$,
respectively.
Now, considering
(\ref{p3:eq:Bbar2}), (\ref{p3:eq:Bhat2}), (\ref{p3:eq:Cbar3}), and (\ref{p3:eq:Chat3}) we have
\begin{align}
f_1 &= -f_8f_{18}f_2 \mbox{ on $f_{17}\overline{C}$} & \Comment{(\ref{p3:eq:Cbar2})} \label{p3:f1}\\
f_2 &= -f_5f_{15}g_{14} \mbox{ on $f_{14}\overline{A}^\ast$} & \Comment{(\ref{p3:eq:Abar3})} \label{p3:f2}\\
f_3 &= -f_6f_{15}g_{14} \mbox{ on $f_{14}\overline{A}^\ast$ and } f_3 = -f_9f_{18}f_2 \mbox{ on $f_{17}\overline{C}$ }
& \Comment{(\ref{p3:eq:Abar4}), (\ref{p3:eq:Cbar4})} \label{p3:f3}\\
f_4 &= -f_7f_{21}g_{20} \mbox{ on $f_{20}\overline{B}^\ast$} & \Comment{(\ref{p3:eq:Bbar1})} \label{p3:f4}\\
f_6 &= -f_9f_{21}g_{20} \mbox{ on $f_{20}\overline{B}^\ast$} & \Comment{(\ref{p3:eq:Bbar4})} \label{p3:f6}\\
f_7 &= -f_4f_{20}f_8 \mbox{ on $f_{21}\overline{B}$} & \Comment{(\ref{p3:eq:Bbar1})} \label{p3:f7}\\
f_9 &= -f_6f_{20}f_8 \mbox{ on $f_{21}\overline{B}$} & \Comment{(\ref{p3:eq:Bbar4})} \label{p3:f9}\\
f_{10} &= -f_4f_{25}f_{11} \mbox{ on $f_{27}\widehat{B}$ and }f_{10} = -f_7f_{28}f_{12} \mbox{ on $f_{29}\widehat{C}$}
& \Comment{(\ref{p3:eq:Bhat1}), (\ref{p3:eq:Chat1})} \label{p3:f10}\\
f_{11} &= -f_8f_{28}f_{12} \mbox{ on $f_{29}\widehat{C}$} & \Comment{(\ref{p3:eq:Chat2})} \label{p3:f11}\\
f_{12} &= -f_5f_{25}f_{11} \mbox{ on $f_{27}\widehat{B}$}. & \Comment{(\ref{p3:eq:Bhat3})} \label{p3:f12}
\end{align}
Next, we provide upper bounds for the codimensions of
$\overline{A}^\ast$,
$\widehat{B}^\ast$,
$\overline{B}^\ast$,
$\widehat{C}^\ast$,
and
$\overline{C}^\ast$.
From
(\ref{p3:eq:Bhat2}), we know $f_{11}$ is injective on $f_{27}\widehat{B}$ and
$f_{27}$ is injective on $\widehat{B}$.
These facts will be used to arrive on
lines (\ref{p3:Bhatstar:1}) and (\ref{p3:Bhatstar:2}).
From
(\ref{p3:eq:Chat3}), we know $f_{29}$ is injective on $\widehat{C}$, which will
also be used to arrive on line (\ref{p3:Bhatstar:2}).
Lemma~\ref{lemma1}
will be used to arrive on (\ref{p3:Bhatstar:3}).
\begin{align}
\Codim_B\widehat{B}^\ast &= H(B) - \Dim(\widehat{B}^\ast)\nonumber\\
&= H(B) - \Dim(f_{11}(f_{27}\widehat{B} \cap f_{29}\widehat{C}))\nonumber\\
&= H(B) - \Dim(f_{27}\widehat{B} \cap f_{29}\widehat{C})\label{p3:Bhatstar:1}\\
&= H(B) - H(Z) + \Codim_Z(f_{27}\widehat{B} \cap f_{29}\widehat{C})\nonumber\\
&\le H(B) - H(Z) + \Codim_Z(f_{27}\widehat{B}) + \Codim_Z( f_{29}\widehat{C})\label{p3:Bhatstar:3}\\
&= H(B) - H(Z) + H(Z) - \Dim(f_{27}\widehat{B}) + H(Z) - \Dim( f_{29}\widehat{C})\nonumber\\
&= H(B) + H(Z) - \Dim(\widehat{B}) - \Dim(\widehat{C})\label{p3:Bhatstar:2}\\
&\le H(B) + H(Z) - H(B) + \Delta_{\widehat{B}} - H(C) + \Delta_{\widehat{C}}\\
&\le H(Z) - H(C) + \Delta_{\widehat{B}} + \Delta_{\widehat{C}}\\
&\triangleq \Delta_{\widehat{B}^\ast}. \nonumber
\end{align}
From (\ref{p3:eq:Abar1}), we know $f_4$ is injective on $f_{15}\overline{A}$
and $f_{15}$ is injective on $\overline{A}$.
These facts will be used on lines
(\ref{p3:Abarstar:1}) and (\ref{p3:Abarstar:2}).
From (\ref{p3:f5f25}), we know
$f_{25}$ is injective on $\widehat{B}^\ast$, which will also be used to arrive
on line (\ref{p3:Abarstar:2}).
Lemma~\ref{lemma1} will be used to arrive on
(\ref{p3:Abarstar:3}).
\begin{align}
\Codim_A\overline{A}^\ast &= H(A) - \Dim(\overline{A}^\ast)\nonumber\\
&= H(A) - \Dim( f_4(f_{25}\widehat{B}^\ast \cap f_{15}\overline{A}) )\nonumber\\
&= H(A) - \Dim( f_{25}\widehat{B}^\ast \cap f_{15}\overline{A} )\label{p3:Abarstar:1}\\
&= H(A) - H(X) + \Codim_X( f_{25}\widehat{B}^\ast \cap f_{15}\overline{A} )\nonumber\\
&\le H(A) - H(X) + \Codim_X( f_{25}\widehat{B}^\ast) + \Codim_X( f_{15}\overline{A} )\label{p3:Abarstar:3}\\
&= H(A) + H(X) - \Dim( f_{25}\widehat{B}^\ast) - \Dim( f_{15}\overline{A} )\nonumber\\
&= H(A) + H(X) - \Dim( \widehat{B}^\ast) - \Dim( \overline{A} )\label{p3:Abarstar:2}\\
&\le H(A) + H(X) - H(B) + \Delta_{\widehat{B}^\ast} - H(A) + \Delta_{\overline{A}} \nonumber\\
&= H(X) - H(B) + H(Z) - H(C) + \Delta_{\widehat{B}} + \Delta_{\widehat{C}} + \Delta_{\overline{A}} \nonumber\\
&\triangleq \Delta_{\overline{A}^\ast}. \nonumber
\end{align}
From (\ref{p3:eq:Cbar3}), we know $f_2$ is injective on $f_{17}\overline{C}$
and $f_{17}$ is injective on $\overline{C}$.
These facts will be used to arrive
on lines (\ref{p3:Cbarstar:1}) and (\ref{p3:Cbarstar:2}).
From
(\ref{p3:eq:Dbar4}), we know $f_{22}$ is injective on $\overline{D}$, which
will also be used on line (\ref{p3:Cbarstar:2}).
Lemma~\ref{lemma1} will be
used to arrive on (\ref{p3:Cbarstar:3}).
\begin{align}
\Codim_C\overline{C}^\ast &= H(C) - \Dim(\overline{C}^\ast)\nonumber\\
&= H(C) - \Dim( f_2(f_{17}\overline{C} \cap f_{22}\overline{D}) )\nonumber\\
&= H(C) - \Dim( f_{17}\overline{C} \cap f_{22}\overline{D} )\label{p3:Cbarstar:1}\\
&= H(C) - H(W) + \Codim_W( f_{17}\overline{C} \cap f_{22}\overline{D} )\nonumber\\
&\le H(C) - H(W) + \Codim_W( f_{17}\overline{C}) + \Codim_W( f_{22}\overline{D} )\label{p3:Cbarstar:3}\\
&= H(C) - H(W) + H(W) - \Dim( f_{17}\overline{C}) + H(W) - \Dim( f_{22}\overline{D} )\nonumber\\
&= H(C) + H(W) - \Dim( \overline{C}) - \Dim( \overline{D} )\label{p3:Cbarstar:2}\\
&\le H(C) + H(W) - H(C) + \Delta_{\overline{C}} - H(D) + \Delta_{\overline{D}} \nonumber\\
&= H(W) - H(D) + \Delta_{\overline{C}} + \Delta_{\overline{D}} \nonumber\\
&\triangleq \Delta_{\overline{C}^\ast}. \nonumber
\end{align}
From (\ref{p3:eq:Bbar2}), we know $f_8$ is injective on $f_{21}\overline{B}$
and $f_{21}$ is injective on $\overline{B}$.
These facts will be used to arrive
on lines (\ref{p3:Bbarstar:1}) and (\ref{p3:Bbarstar:2}).
From
(\ref{p3:f9f18}), we know $f_{18}$ is injective on $\overline{C}^\ast$, which
will also be used on line (\ref{p3:Bbarstar:2}).
Lemma~\ref{lemma1} will be
used to arrive on (\ref{p3:Bbarstar:3}).
\begin{align}
\Codim_B\overline{B}^\ast &= H(B) - \Dim(\overline{B}^\ast)\nonumber\\
&= H(B) - \Dim( f_8(f_{21}\overline{B} \cap f_{18}\overline{C}^\ast) )\nonumber\\
&= H(B) - \Dim( f_{21}\overline{B} \cap f_{18}\overline{C}^\ast )\label{p3:Bbarstar:1}\\
&= H(B) - H(Y) + \Codim_Y( f_{21}\overline{B} \cap f_{18}\overline{C}^\ast )\nonumber\\
&\le H(B) - H(Y) + \Codim_Y( f_{21}\overline{B}) + \Codim_Y( f_{18}\overline{C}^\ast )\label{p3:Bbarstar:3}\\
&= H(B) - H(Y) + H(Y) - \Dim( f_{21}\overline{B}) + H(Y) - \Dim( f_{18}\overline{C}^\ast )\nonumber\\
&= H(B) + H(Y) - \Dim( \overline{B}) - \Dim( \overline{C}^\ast )\label{p3:Bbarstar:2}\\
&\le H(B) + H(Y) - H(B) + \Delta_{\overline{B}} - H(C) + \Delta_{\overline{C}^\ast} \nonumber\\
&= H(Y) - H(C) + \Delta_{\overline{B}} + \Delta_{\overline{C}^\ast} \nonumber\\
&= H(Y) - H(C) + H(W) - H(D) + \Delta_{\overline{C}} + \Delta_{\overline{D}} + \Delta_{\overline{B}} \nonumber\\
&\triangleq \Delta_{\overline{B}^\ast}. \nonumber
\end{align}
From (\ref{p3:eq:Chat3}), we know $f_{12}$ is injective on $f_{29}\widehat{C}$
and $f_{29}$ is injective on $\widehat{C}$.
These facts will be used to arrive
on lines (\ref{p3:Chatstar:1}) and (\ref{p3:Chatstar:2}).
From
(\ref{p3:eq:Bhat2}), we know $f_{27}$ is injective on $\widehat{B}$, which will
also be used on line (\ref{p3:Chatstar:2}).
Lemma~\ref{lemma1} will be used
to arrive on (\ref{p3:Chatstar:3}).
\begin{align}
\Codim_C\widehat{C}^\ast &= H(C) - \Dim(\widehat{C}^\ast)\nonumber\\
&= H(C) - \Dim( f_{12}(f_{27}\widehat{B} \cap f_{29}\widehat{C}) )\nonumber\\
&= H(C) - \Dim( f_{27}\widehat{B} \cap f_{29}\widehat{C} )\label{p3:Chatstar:1}\\
&= H(C) - H(Z) + \Codim_Z( f_{27}\widehat{B} \cap f_{29}\widehat{C} )\nonumber\\
&\le H(C) - H(Z) + \Codim_Z( f_{27}\widehat{B}) + \Codim_Z( f_{29}\widehat{C} )\label{p3:Chatstar:3}\\
&= H(C) - H(Z) + H(Z) - \Dim( f_{27}\widehat{B}) + H(Z) - \Dim( f_{29}\widehat{C} )\nonumber\\
&= H(C) + H(Z) - \Dim( \widehat{B}) - \Dim( \widehat{C} )\label{p3:Chatstar:2}\\
&\le H(C) + H(Z) - H(B) + \Delta_{\widehat{B}} - H(C) + \Delta_{\widehat{C}}\nonumber\\
&= H(Z) - H(B) + \Delta_{\widehat{B}} + \Delta_{\widehat{C}}\nonumber\\
&\triangleq \Delta_{\widehat{C}^\ast}. \nonumber
\end{align}
Let $t \in A$.
Now, we will assume $t$ satisfies conditions (\ref{p3:ass1})--(\ref{p3:ass7}).
The justifications can be found below.
\begin{align}
& t \in \widehat{A}\mbox{ ; this is true on a subspace of $A$ of codimension at most } \Delta_{\widehat{A}}\label{p3:ass1}\\
& f_{32}t \in f_{20}\overline{B}^\ast \cap f_{25}\widehat{B}^\ast \mbox{ ; this is true on a subspace of $A$ of codimension at most }\nonumber\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad 2H(X) - 2H(B) + \Delta_{\overline{B}^\ast} + \Delta_{\widehat{B}^\ast}\label{p3:ass3}\\
& f_{33}t \in f_{28}\widehat{C}^\ast \cap f_{21}\overline{B}^\ast \mbox{ ; this is true on a subspace of $A$ of codimension at most }\nonumber\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad 2H(Y) - H(B) - H(C) + \Delta_{\overline{B}^\ast} + \Delta_{\widehat{C}^\ast}\label{p3:ass4}\\
& f_{34}t \in f_{29}\widehat{C}^\ast \cap f_{27}\widehat{B}^\ast \mbox{ ; this is true on a subspace of $A$ of codimension at most } \nonumber\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad 2H(Z) - H(C) - H(B) + \Delta_{\widehat{C}^\ast} + \Delta_{\widehat{B}^\ast}\label{p3:ass5}\\
& f_{18}f_2f_{31}t \in f_{21}\overline{B}^\ast \cap f_{28}\widehat{C}^\ast \mbox{ ; this is true on a subspace of $A$ of codimension at most } \nonumber\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad 2H(Y) - H(B) - H(C) + \Delta_{\overline{B}^\ast} + \Delta_{\widehat{C}^\ast}\label{p3:ass6}
\end{align}
Now, we need to make two assumptions on $t$ simultaneously.
\begin{align}
& f_{31}t \in f_{17}\overline{C} \cap f_{14}\overline{A}^\ast
\mbox{ and }f_{15}g_{14}f_{31}t \in f_{20}\overline{B}^\ast \cap f_{25}\widehat{B}^\ast ; \nonumber\\
& \mbox{this is true on a subspace of $A$ of codimension at most}\nonumber\\
&\qquad\qquad 2H(X) - 2H(B) + 2H(W) - H(C) - H(A) + \Delta_{\overline{C}} + \Delta_{\overline{A}^\ast} + \Delta_{\overline{B}^\ast} + \Delta_{\widehat{B}^\ast}\label{p3:ass7}
\end{align}
To justify (\ref{p3:ass3}), first we know $f_{20}$ is injective on
$\overline{B}^\ast$ by (\ref{p3:f6f20}).
Then by Lemma~\ref{lemma3}, we
know $f_{32}t \in f_{20}\overline{B}^\ast$ on a subspace of $A$ of codimension
at most $H(X) - H(B) + \Codim_B(\overline{B}^\ast) \le H(X) - H(B) + \Delta_{\overline{B}^\ast}$.
By (\ref{p3:f5f25}), we also know $f_{25}$ is
injective on $\widehat{B}^\ast$.
Then by Lemma~\ref{lemma3}, we know
$f_{32}t \in f_{25}\widehat{B}^\ast$ on a subspace of $A$ of codimension at
most $H(X) - H(B) + \Codim_B(\widehat{B}^\ast) \le H(X) - H(B) + \Delta_{\widehat{B}^\ast}$.
Then using Lemma~\ref{lemma1}, we have $f_{32}t \in f_{20}\overline{B}^\ast \cap f_{25}\widehat{B}^\ast$ on a subspace of $A$
of codimension at most $2H(X) - 2H(B) \Delta_{\overline{B}^\ast} + \Delta_{\widehat{B}^\ast}$.
Conditions (\ref{p3:ass4})--(\ref{p3:ass6}) can be justified similarly.
To justify (\ref{p3:ass7}), first we know $f_{17}$ is injective on
$\overline{C}$ by (\ref{p3:eq:Cbar3}).
Then by Lemma~\ref{lemma3}, we know
$f_{31}t \in f_{17}\overline{C}$ on a subspace of $A$ of codimension at most
$H(W) - H(C) + \Codim_C(\overline{C}) \le H(W) - H(C) + \Delta_{\overline{C}}$.
By (\ref{p3:f14}), we also know $f_{14}$ is injective
on $\overline{A}^\ast$.
Then by Lemma~\ref{lemma3}, we know $f_{31}t \in f_{14}\overline{A}^\ast$ on a subspace of $A$ of codimension at most $H(W) -
H(A) + \Codim_A(\overline{A}^\ast) \le H(W) - H(A) + \Delta_{\overline{A}^\ast}$.
Then using Lemma~\ref{lemma1}, we have
\begin{align*}
&\mbox{$f_{31}t \in f_{17}\overline{C} \cap f_{14}\overline{A}^\ast$}
\end{align*}
on a subspace, $S$, of $A$ of codimension at most $2H(W) - H(C) - H(A) + \Delta_{\overline{C}} + \Delta_{\overline{A}^\ast}$.
Since $f_{14}$ is
injective on $\overline{A}^\ast$, the function $f_{15}g_{14}f_{31}$ is
defined on $S$.
Using the same technique as before we can show that
\begin{align*}
f_{15}g_{14}f_{31}t \in f_{20}\overline{B}^\ast \cap f_{25}\widehat{B}^\ast
\end{align*}
on a subspace, $\overline{S}$, of codimension with respect to $S$ at most
$2H(X) - 2H(B) + \Delta_{\overline{B}^\ast} + \Delta_{\widehat{B}^\ast}$.
Thus
both conditions are true on $\overline{S}$, which has codimension with respect
to $A$ at most
$\Codim_{S}\overline{S} + \Codim_A{S} \le 2H(X) - 2H(B) + 2H(W) - H(C) - H(A)
+ \Delta_{\overline{C}} + \Delta_{\overline{A}^\ast} + \Delta_{\overline{B}^\ast} + \Delta_{\widehat{B}^\ast}$.
Our final goal is to show that $t = 3x$ for some $x$ so that we may conclude
that $t = 0$ if the characteristic is 3.
We will accomplish this by using
(\ref{p3:eq:Ahat1}) and by proving that $f_4f_{32}t = f_7f_{33}t =
f_{10}f_{34}t$.
\begin{claim}\label{p3:claim1}
$f_4f_{32}t = f_{10}f_{34}t$
\end{claim}
\begin{proof}
First we must show that $f_{28}f_{12}f_{34}t = f_{21}g_{20}f_{32}t$.
By (\ref{p3:eq:Ahat2}), we know
\begin{align*}
f_8f_{33}t &= -f_{11}f_{34}t - f_1f_{31}t.
\end{align*}
Then by using (\ref{p3:f11}) and condition (\ref{p3:ass5}), we have
\begin{align*}
f_8f_{33}t &= f_8f_{28}f_{12}f_{34}t - f_1f_{31}t.
\end{align*}
Now, by using (\ref{p3:f1}) and condition (\ref{p3:ass7}), we have
\begin{align*}
f_8f_{33}t &= f_8f_{28}f_{12}f_{34}t + f_8f_{18}f_2f_{31}t.
\end{align*}
By (\ref{p3:f8f28}), we know $f_8$ is injective on $f_{28}\widehat{C}^\ast$.
By condition (\ref{p3:ass4}), we know $f_{33}t \in f_{28}\widehat{C}^\ast$.
By condition (\ref{p3:ass6}), we know $f_{18}f_2f_{31}t \in f_{28}\widehat{C}^\ast$.
By condition (\ref{p3:ass5}), we know $f_{34}t \in f_{29}\widehat{C}^\ast$.
Using (\ref{p3:eq:Chat3}), we know $f_{12}f_{34}t \in \widehat{C}^\ast$.
Thus, we have
\begin{align}
f_{33}t &= f_{28}f_{12}f_{34}t + f_{18}f_2f_{31}t. \label{p3:claim1:1}
\end{align}
By (\ref{p3:eq:Ahat4}), we have
\begin{align*}
f_9f_{33}t &= -f_6f_{32}t - f_3f_{31}t.
\end{align*}
Then by using (\ref{p3:f6}) and condition (\ref{p3:ass3}), we have
\begin{align*}
f_9f_{33}t &= f_9f_{21}g_{20}f_{32}t - f_3f_{31}t.
\end{align*}
Now, by using (\ref{p3:f3}) and condition (\ref{p3:ass7}), we have
\begin{align*}
f_9f_{33}t &= f_9f_{21}g_{20}f_{32}t + f_9f_{18}f_2f_{31}t.
\end{align*}
By (\ref{p3:f9f21}), we know $f_9$ is injective on $f_{21}\overline{B}^\ast$.
By condition (\ref{p3:ass4}), we know $f_{33}t \in f_{21}\overline{B}^\ast$.
By condition (\ref{p3:ass3}), we know $f_{32}t \in f_{20}\overline{B}^\ast$ so $f_{21}g_{20}f_{32}t \in f_{21}\overline{B}^\ast$.
By condition (\ref{p3:ass6}), we know $f_{18}f_2f_{31}t \in f_{21}\overline{B}^\ast$.
Thus, we have
\begin{align}
f_{33}t &= f_{21}g_{20}f_{32}t + f_{18}f_2f_{31}t. \label{p3:claim1:2}
\end{align}
Now, setting (\ref{p3:claim1:1}) and (\ref{p3:claim1:2}) equal to each other, we have
\begin{align}
f_{21}g_{20}f_{32}t &= f_{28}f_{12}f_{34}t. \label{p3:claim1:3}
\end{align}
By (\ref{p3:f4}) and condition (\ref{p3:ass3}), we know
\begin{align*}
f_4f_{32}t &= -f_7f_{21}g_{20}f_{32}t.
\end{align*}
Using (\ref{p3:claim1:3}), we have
\begin{align*}
f_4f_{32}t &= -f_7f_{28}f_{12}f_{34}t.
\end{align*}
Then using (\ref{p3:f10}) and condition (\ref{p3:ass5}), we know
\begin{align*}
f_4f_{32}t &= f_{10}f_{34}t.
\end{align*}
\end{proof}
\begin{claim}\label{p3:claim2}
$f_7f_{33}t = f_{10}f_{34}t$.
\end{claim}
\begin{proof}
First we must show that $f_{25}f_{11}f_{34}t = f_{20}f_8f_{33}t$.
By (\ref{p3:eq:Ahat3}), we know
\begin{align*}
f_5f_{32}t &= -f_{12}f_{34}t - f_2f_{31}t.
\end{align*}
Then by using (\ref{p3:f12}) and condition (\ref{p3:ass5}), we have
\begin{align*}
f_5f_{32}t &= f_5f_{25}f_{11}f_{34}t - f_2f_{31}t.
\end{align*}
Now, by using (\ref{p3:f2}) and condition (\ref{p3:ass7}), we have
\begin{align*}
f_5f_{32}t &= f_5f_{25}f_{11}f_{34}t + f_5f_{15}g_{14}f_{31}t.
\end{align*}
By (\ref{p3:f5f25}), we know $f_5$ is injective on $f_{25}\widehat{B}^\ast$.
By condition (\ref{p3:ass3}), we know $f_{32}t \in f_{25}\widehat{B}^\ast$.
By condition (\ref{p3:ass5}), we know $f_{34}t \in f_{27}\widehat{B}^\ast$.
Now, using (\ref{p3:eq:Bhat2}), we know $f_{11}f_{34}t \in \widehat{B}^\ast$.
By condition (\ref{p3:ass7}), we know $f_{15}g_{14}f_{31}t \in f_{25}\widehat{B}^\ast$.
Thus, we have
\begin{align}
f_{32}t &= f_{25}f_{11}f_{34}t + f_{15}g_{14}f_{31}t. \label{p3:claim2:1}
\end{align}
By (\ref{p3:eq:Ahat4}), we have
\begin{align*}
f_6f_{32}t &= -f_9f_{33}t - f_3f_{31}t.
\end{align*}
Then using (\ref{p3:f9}) and condition (\ref{p3:ass4}), we have
\begin{align*}
f_6f_{32}t &= f_6f_{20}f_{8}f_{33}t - f_3f_{31}t.
\end{align*}
Now, by using (\ref{p3:f3}) and condition (\ref{p3:ass7}), we have
\begin{align*}
f_6f_{32}t &= f_6f_{20}f_{8}f_{33}t + f_6f_{15}g_{14}f_{31}t.
\end{align*}
By (\ref{p3:f6f20}), we know that $f_{6}$ is injective on $f_{20}\overline{B}^\ast$.
By condition (\ref{p3:ass3}), we know $f_{32}t \in f_{20}\overline{B}^\ast$.
By condition (\ref{p3:ass4}), we know $f_{33}t \in f_{21}\overline{B}^\ast$.
Now, using (\ref{p3:eq:Bbar2}), we know $f_8f_{33}t \in \overline{B}^\ast$.
By condition (\ref{p3:ass7}), we know $f_{15}g_{14}f_{31}t \in f_{20}\overline{B}^\ast$.
Thus, we have
\begin{align}
f_{32}t &= f_{20}f_{8}f_{33}t + f_{15}g_{14}f_{31}t. \label{p3:claim2:2}
\end{align}
Now, setting (\ref{p3:claim2:1}) and (\ref{p3:claim2:2}) equal to each other, we have
\begin{align}
f_{25}f_{11}f_{34}t &= f_{20}f_{8}f_{33}t. \label{p3:claim2:3}
\end{align}
By (\ref{p3:f7}) and condition (\ref{p3:ass4}), we know
\begin{align*}
f_7f_{33}t &= -f_4f_{20}f_{8}f_{33}t.
\end{align*}
Using (\ref{p3:claim2:3}), we have
\begin{align*}
f_7f_{33}t &= -f_4f_{25}f_{11}f_{34}t.
\end{align*}
Then using (\ref{p3:f10}) and condition (\ref{p3:ass5}), we know
\begin{align*}
f_7f_{33}t &= f_{10}f_{34}t.
\end{align*}
\end{proof}
Now, by \eqref{p3:eq:Ahat1} and the two claims,
we have
\begin{align*}
t &= f_4f_{32}t + f_7f_{33}t + f_{10}f_{34}t\\
&= f_{10}f_{34}t + f_{10}f_{34}t + f_{10}f_{34}t\\
&= 3f_{10}f_{34}t.
\end{align*}
Thus if the field has characteristic 3, then
\begin{align}
t &= 0.
\end{align}
No nonzero $t$ can satisfy all of the conditions (\ref{p3:ass1})--(\ref{p3:ass7}),
so we must have
\begin{align*}
H(A) &\le \Delta_{\widehat{A}} + 2H(W) - H(C) - H(A) + \Delta_{\overline{C}} + \Delta_{\overline{A}^\ast} \\
&\ \ \ + 2H(X) - 2H(B) + \Delta_{\overline{B}^\ast} + \Delta_{\widehat{B}^\ast}\\
&\ \ \ + 2H(Y) - H(B) - H(C) + \Delta_{\overline{B}^\ast} + \Delta_{\widehat{C}^\ast}\\
&\ \ \ + 2H(Z) - H(C) - H(B) + \Delta_{\widehat{C}^\ast} + \Delta_{\widehat{B}^\ast}\\
&\ \ \ + 2H(Y) - H(B) - H(C) + \Delta_{\overline{B}^\ast} + \Delta_{\widehat{C}^\ast}\\
&\ \ \ + 2H(X) - 2H(B) + \Delta_{\overline{B}^\ast} + \Delta_{\widehat{B}^\ast}\\
&= 2H(Z) + 4H(Y) + 4H(X) + 2H(W) - 4H(C) - 7H(B) - H(A) \\
&\ \ \ + \Delta_{\overline{A}^\ast} + 4\Delta_{\overline{B}^\ast} + 3\Delta_{\widehat{B}^\ast} + 3\Delta_{\widehat{C}^\ast} + \Delta_{\widehat{A}}+ \Delta_{\overline{C}}\\
&= 2H(Z) + 4H(Y) + 4H(X) + 2H(W) - 4H(C) - 7H(B) - H(A) \\
&\ \ \ + H(X) - H(B) + H(Z) - H(C) + \Delta_{\widehat{B}} + \Delta_{\widehat{C}} + \Delta_{\overline{A}}\\
&\ \ \ + 4( H(Y) - H(C) + H(W) - H(D) + \Delta_{\overline{C}} + \Delta_{\overline{D}} + \Delta_{\overline{B}} )\\
&\ \ \ + 3( H(Z) - H(C) + \Delta_{\widehat{B}} + \Delta_{\widehat{C}} )\\
&\ \ \ + 3( H(Z) - H(B) + \Delta_{\widehat{B}} + \Delta_{\widehat{C}} )\\
&\ \ \ + \Delta_{\widehat{A}}+ \Delta_{\overline{C}}\\
&= 9H(Z) + 8H(Y) + 5H(X) + 6H(W) - 4H(D) - 12H(C) - 11H(B) -H(A)\\
&\ \ \ + \Delta_{\widehat{A}} + \Delta_{\overline{A}} + 7\Delta_{\widehat{B}} + 4\Delta_{\overline{B}} + 7\Delta_{\widehat{C}} + 5\Delta_{\overline{C}} + 4\Delta_{\overline{D}} \\
&= 9H(Z) + 8H(Y) + 5H(X) + 6H(W) - 4H(D) - 12H(C) - 11H(B) -H(A)\\
&\ \ \ + H(W|B,C,D) + H(X|A,C,D) + H(Y|A,B,D) + H(Z|A,B,C) \\
&\ \ \ + H(A|W,X,Y,Z)+ H(W|B,C,D) + H(X|A,C,D) + H(A|B,W,X) \\
&\ \ \ + 7( H(X|A,C,D) + H(Z|A,B,C) + H(B|D,X,Z) )\\
&\ \ \ + 4( H(X|A,C,D) + H(Y|A,B,D)+ H(B|C,X,Y) )\\
&\ \ \ + 7( H(Y|A,B,D) + H(Z|A,B,C) + H(C|D,Y,Z) )\\
&\ \ \ + 5( H(W|B,C,D) + H(Y|A,B,D) + H(C|A,W,Y) )\\
&\ \ \ + 4( H(W|B,C,D) + H(Z|A,B,C) + H(D|A,W,Z) )\\
&\ \ \ + 29( H(A) + H(B) + H(C) + H(D) - H(A,B,C,D) )\\
&= 9H(Z) + 8H(Y) + 5H(X) + 6H(W) - 4H(D) - 12H(C) - 11H(B) -H(A)\\
&\ \ \ +19H(Z|A,B,C) + 17H(Y|A,B,D) + 13H(X|A,C,D) + 11H(W|B,C,D)\\
&\ \ \ +H(A|W,X,Y,Z) + H(A|B,W,X) + 7H(B|D,X,Z) + 4H(B|C,X,Y)\\
&\ \ \ +7H(C|D,Y,Z) + 5H(C|A,W,Y) + 4H(D|A,W,Z)\\
&\ \ \ + 29( H(A) + H(B) + H(C) + H(D) - H(A,B,C,D) ).
\end{align*}
\end{proof}
The next theorem demonstates that the inequality in Theorem~\ref{thm:nonT8}
does not in general hold for vector spaces with finite fields of characteristic other than $3$.
\begin{thm}\label{thm:non-T8-non-char3}
For each prime number $p\ne 3$
there exists a vector space $V$ with a finite scalar field of characteristic $p$
such that the non-T8 inequality in Theorem~\ref{thm:nonT8}
is not a linear rank inequality over $V$.
\end{thm}
\begin{proof}
Let $V$ be the vector space of $4$-dimensional vectors whose components are from $GF(p)$,
and define the following subspaces of $V$:
\begin{align*}
A = \langle (1,0,0,0)\rangle & B = \langle (0,1,0,0)\rangle\\
C = \langle (0,0,1,0)\rangle & D = \langle (0,0,0,1)\rangle\\
W = \langle (0,1,1,1)\rangle & X = \langle (1,0,1,1)\rangle\\
Y = \langle (1,1,0,1)\rangle & Z = \langle (1,1,1,0)\rangle.
\end{align*}
We have:
\begin{align}
0
&= H(W|B,C,D) & \Comment{$(0,1,1,1)=(0,1,0,0)+(0,0,1,0)+(0,0,0,1)$}\nonumber\\
&= H(X|A,C,D) & \Comment{$(1,0,1,1)=(1,0,0,0)+(0,0,1,0)+(0,0,0,1)$}\nonumber\\
&= H(Y|A,B,D) & \Comment{$(1,1,0,1)=(1,0,0,0)+(0,1,0,0)+(0,0,0,1)$}\nonumber\\
&= H(Z|A,B,C) & \Comment{$(1,1,1,0)=(1,0,0,0)+(0,1,0,0)+(0,0,1,0)$}\nonumber\\
&= H(A|B,W,X) & \Comment{$(1,0,0,0)=(1,0,1,1)+(0,1,0,0)-(0,1,1,1)$}\nonumber\\
&= H(C|A,W,Y) & \Comment{$(0,0,1,0)=(0,1,1,1)+(1,0,0,0)-(1,1,0,1)$}\nonumber\\
&= H(B|C,X,Y) & \Comment{$(0,1,0,0)=(1,1,0,1)+(0,0,1,0)-(1,0,1,1)$}\nonumber\\
&= H(D|A,W,Z) & \Comment{$(0,0,0,1)=(0,1,1,1)+(1,0,0,0)-(1,1,1,0)$}\nonumber\\
&= H(B|D,X,Z) & \Comment{$(0,1,0,0)=(1,1,1,0)+(0,0,0,1)-(1,0,1,1)$}\nonumber\\
&= H(C|D,Y,Z) & \Comment{$(0,0,1,0)=(1,1,1,0)+(0,0,0,1)-(1,1,0,1)$}\nonumber\\
&= H(A|W,X,Y,Z) & \Comment{$(1,0,0,0)=3^{-1}( (1,0,1,1){+}(1,1,0,1){+}(1,1,1,0) {-} 2(0,1,1,1))$}.
\label{eq:zero-entropies-nonT8}
\end{align}
We know $H(A)=H(B)=H(C)=H(D)=H(W)=H(X)=H(Y)=H(Z)=1$,
Also, we have
$$H(A) + H(B) + H(C) + H(D) = H(A,B,C,D).$$
So, if the inequality in
Theorem~\ref{thm:nonT8} were to hold over $V$,
then we would have
\begin{align*}
1
&= H(A) \\
&\le 9H(Z) + 8H(Y) + 5H(X) + 6H(W) - 4H(D) - 12H(C) - 11H(B) -H(A)\\
&= 9 + 8 + 5 + 6 -4 -12 - 11 - 1\\
&= 0
\end{align*}
which is impossible.
\end{proof}
\begin{cor}
For the non-T8 network,
the linear coding capacity is at most $28/29$
over any finite field alphabet of characteristic equal to $3$.
The linear coding capacity over finite field alphabets of characteristic not $3$
and the coding capacity are all equal to $1$.
\label{cor:nonT8-capacity}
\end{cor}
\begin{proof}
Let $F$ be a finite field alphabet.
Consider a $(k,n)$ linear solution of the non-T8 network over $F$,
such that the characteristic of $F$ is $3$.
Let $A$, $B$, $C$, $D$ be message random variables in the T8 network,
that are uniformly distributed over vectors in $F^k$.
Let $W$, $X$, $Y$, $Z$ be the resulting random variables
associated with the corresponding labeled edges of T8 in Figure~\ref{fig:nonT8}.
Equations \eqref{eq:zero-entropies-nonT8} now hold
with random variables $A,B,C,D,W,X,Y,Z$ are taken as random variables
(i.e. not as subspaces as in Theorem~\ref{thm:non-T8-non-char3})
by Lemma~\ref{lem:conditional-dimensions}:
\begin{align*}
0
&= H(W|B,C,D) & \Comment{$(n_1,n_2)$}\nonumber\\
&= H(X|A,C,D) & \Comment{$(n_3,n_4)$}\nonumber\\
&= H(Y|A,B,D) & \Comment{$(n_5,n_6)$}\nonumber\\
&= H(Z|A,B,C) & \Comment{$(n_7,n_8)$}\nonumber\\
&= H(A|B,W,X) & \Comment{$n_{ 9}$}\nonumber\\
&= H(C|A,W,Y) & \Comment{$n_{10}$}\nonumber\\
&= H(B|C,X,Y) & \Comment{$n_{11}$}\nonumber\\
&= H(D|A,W,Z) & \Comment{$n_{12}$}\nonumber\\
&= H(B|D,X,Z) & \Comment{$n_{13}$}\nonumber\\
&= H(C|D,Y,Z) & \Comment{$n_{14}$}\nonumber\\
&= H(A|W,X,Y,Z) & \Comment{$n_{15}$}
\end{align*}
and since the source message $A,B,C,D$ are independent random variables,
we have
$$H(A) + H(B) + H(C) + H(D) = H(A,B,C,D)$$
so the non-T8 inequality in Theorem~\ref{thm:nonT8} reduces to
\begin{align*}
H(A) &\le 9H(Z) + 8H(Y) + 5H(X) + 6H(W) - 4H(D) - 12H(C) - 11H(B) -H(A).
\end{align*}
Now, since
$H(A) = H(B) = H(C) = H(D) = k$ and
$H(W) = H(X) = H(Y) = H(Z) \le n$,
we have
\begin{align*}
k &\le 9n + 8n + 5n + 6n - 4k -12k - 11k - k\\
k/n &\le 28/29.
\end{align*}
So, the linear coding capacity over characteristic 3 is at most $28/29 < 1.$
The non-T8 network
has a scalar linear solution over every characteristic except for 3 by using the
following edge functions
(here we are using the notations $A,B,C,D,W,X,Y,Z$ to denote edge variables rather than vector spaces):
\begin{align*}
W &= B + C + D\\
X &= A + C + D\\
Y &= A + B + D\\
Z &= A + B + C
\end{align*}
and decoding functions:
\begin{align*}
n_{ 9}: A &= X - W + B\\
n_{10}: C &= W - Y + A\\
n_{11}: B &= Y - X + C\\
n_{12}: D &= W - Z + A\\
n_{13}: B &= Z - X + D\\
n_{14}: C &= Z - Y + D\\
n_{15}: A &= 3^{-1} \cdot ( X + Y + Z - 2W ).
\end{align*}
We know the coding capacity is at most 1 because there is a unique path from
source $A$ to node $n_9$ (through node $n_4$).
Since the coding capacity is at least as large as the linear coding
capacity for characteristics other than 3,
we conclude that the coding capacity is exactly equal to 1.
\end{proof}
\newpage
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 35 |
est la formule brute de plusieurs isomères.
Acide gras
acide dihomo-gamma-linolénique ou acide cis-icosa-8,11,14-triénoïque, numéro CAS
acide de Mead ou acide cis-5,8,11-eicosatriénoïque, numéro CAS | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 6,187 |
package atlas
import (
"fmt"
"strings"
)
// ParseSlug parses a slug of the format (x/y) into the x and y components. It
// accepts a string of the format "x/y" ("user/name" for example). If an empty
// string is given, an error is returned. If the given string is not a valid
// slug format, an error is returned.
func ParseSlug(slug string) (string, string, error) {
if slug == "" {
return "", "", fmt.Errorf("missing slug")
}
parts := strings.Split(slug, "/")
if len(parts) != 2 {
return "", "", fmt.Errorf("malformed slug %q", slug)
}
return parts[0], parts[1], nil
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 6,258 |
Q: Using ArcGIS python window to change definition query on layer? I post a very simple script to the python window to change the definition query on a layer. I save the map and load a new template then try to run the query again and nothing happens. I have recently upgraded to 10.4 and when running 10.3 I never had this kind of issue. I have tried deleting all variables at the end of the script. Nothing works. It will only run one time and I have to close out the ArcMap session before it will work again.
A: I ran in to the same issue with my script and all I had to do was add:
arcpy.env.overwriteOutput = True
and I haven't had to exit/re-enter since.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,143 |
{"url":"https:\/\/en.wikipedia.org\/wiki\/Cache_algorithms","text":"# Cache replacement policies\n\n(Redirected from Cache algorithms)\n\nIn computing, cache algorithms (also frequently called cache replacement algorithms or cache replacement policies) are optimizing instructions\u2014\u200b\u200bor algorithms\u200b\u200bthat a computer program or a hardware-maintained structure can follow in order to manage a cache of information stored on the computer. When the cache is full, the algorithm must choose which items to discard to make room for the new ones.\n\n## Overview\n\nThe average memory reference time is[1]\n\n${\\displaystyle T=m\\times T_{m}+T_{h}+E}$\n\nwhere\n\n${\\displaystyle T}$ = average memory reference time\n${\\displaystyle m}$ = miss ratio = 1 - (hit ratio)\n${\\displaystyle T_{m}}$ = time to make a main memory access when there is a miss (or, with multi-level cache, average memory reference time for the next-lower cache)\n${\\displaystyle T_{h}}$= the latency: the time to reference the cache when there is a hit\n${\\displaystyle E}$ = various secondary effects, such as queuing effects in multiprocessor systems\n\nThere are two primary figures of merit of a cache: The latency, and the hit rate. There are also a number of secondary factors affecting cache performance.[1]\n\nThe \"hit ratio\" of a cache describes how often a searched-for item is actually found in the cache. More efficient replacement policies keep track of more usage information in order to improve the hit rate (for a given cache size).\n\nThe \"latency\" of a cache describes how long after requesting a desired item the cache can return that item (when there is a hit). Faster replacement strategies typically keep track of less usage information\u2014or, in the case of direct-mapped cache, no information\u2014to reduce the amount of time required to update that information.\n\nEach replacement strategy is a compromise between hit rate and latency.\n\nHit rate measurements are typically performed on benchmark applications. The actual hit ratio varies widely from one application to another. In particular, video and audio streaming applications often have a hit ratio close to zero, because each bit of data in the stream is read once for the first time (a compulsory miss), used, and then never read or written again. Even worse, many cache algorithms (in particular, LRU) allow this streaming data to fill the cache, pushing out of the cache information that will be used again soon (cache pollution).[2]\n\nOther things to consider:\n\n\u2022 Items with different cost: keep items that are expensive to obtain, e.g. those that take a long time to get.\n\u2022 Items taking up more cache: If items have different sizes, the cache may want to discard a large item to store several smaller ones.\n\u2022 Items that expire with time: Some caches keep information that expires (e.g. a news cache, a DNS cache, or a web browser cache). The computer may discard items because they are expired. Depending on the size of the cache no further caching algorithm to discard items may be necessary.\n\nVarious algorithms also exist to maintain cache coherency. This applies only to situation where multiple independent caches are used for the same data (for example multiple database servers updating the single shared data file).\n\n## Policies\n\n### B\u00e9l\u00e1dy's Algorithm\n\nThe most efficient caching algorithm would be to always discard the information that will not be needed for the longest time in the future. This optimal result is referred to as B\u00e9l\u00e1dy's optimal algorithm\/simply optimal replacement policy or the clairvoyant algorithm. Since it is generally impossible to predict how far in the future information will be needed, this is generally not implementable in practice. The practical minimum can be calculated only after experimentation, and one can compare the effectiveness of the actually chosen cache algorithm.\n\nAt the moment when a page fault occurs, some set of pages is in memory. In the example once the sequence of '5' '0' '1' is accessed by Frame 1, Frame 2, Frame 3 respectively and when 2 is accessed, it will replace value '5' which is in frame 1 since it predicts that value '5' is not going to be accessed in the near future. So this is the reason B\u00e9l\u00e1dy's Algorithm cannot be implemented in general purpose operating system\n\n### First In First Out (FIFO)\n\nUsing this algorithm the cache behaves in the same way as a FIFO queue. The cache evicts the first block accessed first without any regard to how often or how many times it was accessed before.\n\n### Last In First Out (LIFO)\n\nUsing this algorithm the cache behaves in the exact opposite way as a FIFO queue. The cache evicts the block accessed most recently first without any regard to how often or how many times it was accessed before.\n\n### Least Recently Used (LRU)\n\nDiscards the least recently used items first. This algorithm requires keeping track of what was used when, which is expensive if one wants to make sure the algorithm always discards the least recently used item. General implementations of this technique require keeping \"age bits\" for cache-lines and track the \"Least Recently Used\" cache-line based on age-bits. In such an implementation, every time a cache-line is used, the age of all other cache-lines changes. LRU is actually a family of caching algorithms with members including 2Q by Theodore Johnson and Dennis Shasha,[3] and LRU\/K by Pat O'Neil, Betty O'Neil and Gerhard Weikum.[4]\n\nThe access sequence for the below example is A B C D E D F.\n\nIn the above example once A B C D gets installed in the blocks with sequence numbers (Increment 1 for each new Access) and when E is accessed, it is a miss and it needs to be installed in one of the blocks. According LRU Algorithm, since A has the lowest Rank(A(0)), E will replace A.\n\n### Most Recently Used (MRU)\n\nDiscards, in contrast to LRU, the most recently used items first. In findings presented at the 11th VLDB conference, Chou and DeWitt noted that \"When a file is being repeatedly scanned in a [Looping Sequential] reference pattern, MRU is the best replacement algorithm.\"[5] Subsequently other researchers presenting at the 22nd VLDB conference noted that for random access patterns and repeated scans over large datasets (sometimes known as cyclic access patterns) MRU cache algorithms have more hits than LRU due to their tendency to retain older data.[6] MRU algorithms are most useful in situations where the older an item is, the more likely it is to be accessed.\n\nThe access sequence for the below example is A B C D E C D B.\n\nHere, A B C D are placed in the cache as there is still space available. At the 5th access E, we see that the block which held D is now replaced with E as this block was used most recently. Another access to C and at the next access to D, C is replaced as it was the block accessed just before D and so on.\n\n### Pseudo-LRU (PLRU)\n\nFor CPU caches with large associativity (generally >4 ways), the implementation cost of LRU becomes prohibitive. In many CPU caches, a scheme that almost always discards one of the least recently used items is sufficient. So many CPU designers choose a PLRU algorithm which only needs one bit per cache item to work. PLRU typically has a slightly worse miss ratio, has a slightly better latency, uses slightly less power than LRU and lower overheads compared to LRU.\n\nIn the following example, it is clearly shown how Bits work as a binary tree of 1-bit pointers that point to the less recently used subtree. Following the pointer chain to the leaf node identifies the replacement candidate. Upon an access all pointers in the chain from the accessed way's leaf node to the root node are set to point to subtree that does not contain the accessed way.\n\nThe access sequence is A B C D E.\n\nThe principle here is simple to understand if we only look at the arrow pointers. When there is an access to a value say 'A' and the we cannot find it in the cache then load it from memory and place it at the block where the arrows are pointing go from top to bottom and when you place that block make the arrows point away from that block go from bottom to top. In the above example we see how 'A' was placed followed by 'B', 'C and 'D'. Then as the cache became full 'E' replaced 'A' as that was where the arrows were pointing at that time. On the next access block where 'C' is being held will be replaced.\n\n### Random Replacement (RR)\n\nRandomly selects a candidate item and discards it to make space when necessary. This algorithm does not require keeping any information about the access history. For its simplicity, it has been used in ARM processors.[7] It admits efficient stochastic simulation.[8]\n\nThe access sequence for the below example is A B C D E B D F\n\n### Segmented LRU (SLRU)\n\nSLRU cache is divided into two segments, a probationary segment and a protected segment. Lines in each segment are ordered from the most to the least recently accessed. Data from misses is added to the cache at the most recently accessed end of the probationary segment. Hits are removed from wherever they currently reside and added to the most recently accessed end of the protected segment. Lines in the protected segment have thus been accessed at least twice. The protected segment is finite, so migration of a line from the probationary segment to the protected segment may force the migration of the LRU line in the protected segment to the most recently used (MRU) end of the probationary segment, giving this line another chance to be accessed before being replaced. The size limit on the protected segment is an SLRU parameter that varies according to the I\/O workload patterns. Whenever data must be discarded from the cache, lines are obtained from the LRU end of the probationary segment.[9]\n\n### Least-Frequently Used (LFU)\n\nCounts how often an item is needed. Those that are used least often are discarded first. This works very similar to LRU except that instead of storing the value of how recently a block was accessed, we store the value of how many times it was accessed. So of course while running an access sequence we will replace a block which was used least number of times from our cache. E.g., if A was used (accessed) 5 times and B was used 3 times and others C and D were used 10 times each, we will replace B.\n\n### LFU with Dynamic Aging (LFUDA)\n\nA variant called LFU with Dynamic Aging (LFUDA) that uses dynamic aging to accommodate shifts in the set of popular objects. It adds a cache age factor to the reference count when a new object is added to the cache or when an existing object is re-referenced. LFUDA increments the cache ages when evicting blocks by setting it to the evicted object\u2019s key value. Thus, the cache age is always less than or equal to the minimum key value in the cache.[10] Suppose when an object was frequently accessed in the past and now it becomes unpopular, it will remain in the cache for a long time thereby preventing the newly or less popular objects from replacing it. So this Dynamic aging is introduced to bring down the count of such objects thereby making them eligible for replacement. The advantage of LFUDA is it reduces the cache pollution caused by LFU when cache sizes are very small (256MB). When Cache sizes are large (more than 250GB) few replacement decisions are sufficient and cache pollution will not be a problem.\n\n### Low Inter-reference Recency Set (LIRS)\n\nA page replacement algorithm with an improved performance over LRU and many other newer replacement algorithms. This is achieved by using reuse distance as a metric for dynamically ranking accessed pages to make a replacement decision. LIRS effectively address the limits of LRU by using recency to evaluate Inter-Reference Recency (IRR) for making a replacement decision. The algorithm was developed by Song Jiang and Xiaodong Zhang.\n\nIn the above figure, \"x\" represents that a block is accessed at time t. Suppose if block A1 is accessed at time 1 then Recency will become 0 since this is the first accessed block and IRR will be 1 since it predicts that A1 will be accessed again in time 3. In the time 2 since A4 is accessed, the recency will become 0 for A4 and 1 for A1 because A4 is the most recently accessed Object and IRR will become 4 and it will go on. At time 10, the LIRS algorithm will have two sets LIR set = {A1, A2} and HIR set = {A3, A4, A5}. Now at time 10 if there is access to A4, miss occurs. LIRS algorithm will now evict A5 instead of A2 because of its largest recency.\n\nConstantly balances between LRU and LFU, to improve the combined result.[11] ARC improves on SLRU by using information about recently-evicted cache items to dynamically adjust the size of the protected segment and the probationary segment to make the best use of the available cache space. Adaptive replacement algorithm is explained with the example.[12]\n\n### Clock with Adaptive Replacement (CAR)\n\nCombines the advantages of Adaptive Replacement Cache (ARC) and CLOCK. CAR has performance comparable to ARC, and substantially outperforms both LRU and CLOCK. Like ARC, CAR is self-tuning and requires no user-specified magic parameters. It uses 4 doubly linked lists: two clocks T1 and T2 and two simple LRU lists B1 and B2. T1 clock stores pages based on \"recency\" or \"short term utility\" whereas T2 stores pages with \"frequency\" or \"long term utility\". T1 and T2 contain those pages that are in the cache, while B1 and B2 contain pages that have recently been evicted from T1 and T2 respectively. The algorithm tries to maintain the size of these lists B1\u2248T2 and B2\u2248T1. New pages are inserted in T1 or T2. If there is a hit in B1 size of T1 is increased and similarly if there is a hit in B2 size of T1 is decreased. The adaptation rule used has the same principle as that in ARC, invest more in lists that will give more hits when more pages are added to it.\n\n### Multi Queue (MQ) caching algorithm|Multi Queue (MQ)\n\nThe Multi Queue Algorithm or MQ was developed to improve the performance of second level buffer cache for e.g. a server buffer cache. It is introduced in a paper by Zhou, Philbin, and Li.[13]The MQ cache contains an m number of LRU queues: Q0, Q1, ..., Qm-1. Here, the value of m represents a hierarchy based on the lifetime of all blocks in that particular queue. For example, if j>i, blocks in Qj will have a longer lifetime than those in Qi. In addition to these there is another history buffer Qout, a queue which maintains a list of all the Block Identifiers along with their access frequencies. When Qout is full the oldest identifier is evicted. Blocks stay in the LRU queues for a given lifetime, which is defined dynamically by the MQ algorithm to be the maximum temporal distance between two accesses to the same file or the number of cache blocks, whichever is larger. If a block has not been referenced within its lifetime, it is demoted from Qi to Qi\u22121 or evicted from the cache if it is in Q0. Each queue also has a maximum access count; if a block in queue Qi is accessed more than 2i times, this block is promoted to Qi+1 until it is accessed more than 2i+1 times or its lifetime expires. Within a given queue, blocks are ranked by the recency of access, according to LRU.[14]\n\nWe can see from Fig. how the m LRU queues are placed in the cache. Also see from Fig. how the Qout stores the block identifiers and their corresponding access frequencies. a was placed in Q0 as it was accessed only once recently and we can check in Qout how b and c were placed in Q1 and Q2 respectively as their access frequencies are 2 and 4. The queue in which a block is placed is dependent on access frequency(f) as log2(f). When the cache is full, the first block to be evicted will be the head of Q0 in this case a. If a is accessed one more time it will move to Q1 below b.\n\n### Pannier: Container-based caching algorithm for compound objects\n\nPannier [15] is a container-based flash caching mechanism that identifies divergent (heterogeneous) containers where blocks held therein have highly varying access patterns. Pannier uses a priority-queue based survival queue structure to rank the containers based on their survival time, which is proportional to the live data in the container. Pannier is built based on Segmented LRU (S2LRU), which segregates hot and cold data. Pannier also uses a multi-step feedback controller to throttle flash writes to ensure flash lifespan.\n\n## References\n\n1. ^ a b Alan Jay Smith. \"Design of CPU Cache Memories\". Proc. IEEE TENCON, 1987. [1]\n2. ^ Paul V. Bolotoff. \"Functional Principles of Cache Memory\". 2007.\n3. ^ http:\/\/www.vldb.org\/conf\/1994\/P439.PDF\n4. ^ O'Neil, Elizabeth J.; O'Neil, Patrick E.; Weikum, Gerhard (1993). \"The LRU-K Page Replacement Algorithm for Database Disk Buffering\". Proceedings of the 1993 ACM SIGMOD International Conference on Management of Data. SIGMOD '93. New York, NY, USA: ACM: 297\u2013306. doi:10.1145\/170035.170081. ISBN\u00a00-89791-592-5.\n5. ^ Hong-Tai Chou and David J. DeWitt. An Evaluation of Buffer Management Strategies for Relational Database Systems. VLDB, 1985.\n6. ^ Shaul Dar, Michael J. Franklin, Bj\u00f6rn \u00de\u00f3r J\u00f3nsson, Divesh Srivastava, and Michael Tan. Semantic Data Caching and Replacement. VLDB, 1996.\n7. ^ ARM Cortex-R series processors manual\n8. ^ An Efficient Simulation Algorithm for Cache of Random Replacement Policy [2]\n9. ^ Ramakrishna Karedla, J. Spencer Love, and Bradley G. Wherry. Caching Strategies to Improve Disk System Performance. In Computer, 1994.\n10. ^ https:\/\/arxiv.org\/pdf\/1001.4135.pdf\n11. ^ Nimrod Megiddo and Dharmendra S. Modha. ARC: A Self-Tuning, Low Overhead Replacement Cache. FAST, 2003.","date":"2017-01-24 07:11:00","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 6, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.4459952116012573, \"perplexity\": 2125.7475805249524}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-04\/segments\/1484560284352.26\/warc\/CC-MAIN-20170116095124-00559-ip-10-171-10-70.ec2.internal.warc.gz\"}"} | null | null |
{"url":"https:\/\/proofwiki.org\/wiki\/Mathematician:Bryant_Tuckerman","text":"# Mathematician:Bryant Tuckerman\n\n## Mathematician\n\nAmerican mathematician who discovered, on March $4$th, $1971$, the $24$th Mersenne prime: $2^{19 \\, 937} - 1$.\n\nAmerican\n\n## History\n\n\u2022 Born: November 28, 1915\n\u2022 Died: May 19, 2002\n\n## Theorems and Inventions\n\n\u2022 Published a proof in $1968$ that an odd perfect number has to be greater than $10^{36}$.\n\n## Publications\n\n\u2022 1968:\u00a0Odd Perfect Numbers: A Search Procedure, and a New Lower Bound of $10^{36}$ (Not. Amer. Math. Soc.\u00a0Vol. 15:\u00a0p. 226)","date":"2022-01-27 15:12:31","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.4152308702468872, \"perplexity\": 7133.960454347911}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-05\/segments\/1642320305266.34\/warc\/CC-MAIN-20220127133107-20220127163107-00174.warc.gz\"}"} | null | null |
**The Essential Turing**
Alan M. Turing
# **The Essential Turing**
**Seminal Writings in Computing, Logic, Philosophy,
Artificial Intelligence, and Artificial Life
_plus_ The Secrets of Enigma**
_Edited by_ **B. Jack Copeland**
Great Clarendon Street, Oxford OX2 6DP
Oxford University Press is a department of the University of Oxford.
It furthers the University's objective of excellence in research, scholarship,
and education by publishing worldwide in
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With offices in
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Published in the United States
by Oxford University Press Inc., New York
© In this volume the Estate of Alan Turing 2004
Supplementary Material © the several contributors 2004
The moral rights of the author have been asserted
Database right Oxford University Press (maker)
First published 2004
All rights reserved. No part of this publication may be reproduced,
stored in a retrieval system, or transmitted, in any form or by any means,
without the prior permission in writing of Oxford University Press,
or as expressly permitted by law, or under terms agreed with the appropriate
reprographics rights organization. Enquiries concerning reproduction
outside the scope of the above should be sent to the Rights Department,
Oxford University Press, at the address above.
You must not circulate this book in any other binding or cover
and you must impose this same condition on any acquirer.
British Library Cataloguing in Publication Data
Data available
Library of Congress Cataloging in Publication Data
Data available
ISBN 0–19–825079–7
ISBN 0–19–825080–0 (pbk.)
10 9 8 7 6 5 4 3
Typeset by Kolam Information Services Pvt. Ltd, Pondicherry, India
Printed in Great Britain
on acid-free paper by Biddles Ltd., King's Lynn, Norfolk
## Acknowledgements
Work on this book began in 2000 at the Dibner Institute for the History of Science and Technology, Massachusetts Institute of Technology, and was completed at the University of Canterbury, New Zealand. I am grateful to both these institutions for aid, and to the following for scholarly assistance: John Andreae, Friedrich Bauer, Frank Carter, Alonzo Church Jnr, David Clayden, Bob Doran, Ralph Erskine, Harry Fensom, Jack Good, John Harper, Geoff Hayes, Peter Hilton, Harry Huskey, Eric Jacobson, Elizabeth Mahon, Philip Marks, Elisabeth Norcliffe, Rolf Noskwith, Gualtiero Piccinini, Andrés Sicard, Wilfried Sieg, Frode Weierud, Maurice Wilkes, Mike Woodger, and especially Diane Proudfoot. This book would not have existed without the support of Turing's literary executor, P. N. Furbank, and that of Peter Momtchiloff at Oxford University Press.
B.J.C.
## **Contents**
**Alan Turing 1912–1954**
_Jack Copeland_
**Computable Numbers: A Guide**
_Jack Copeland_
1. On Computable Numbers, with an Application to the Entscheidungsproblem ( _1936_ )
2. On Computable Numbers: Corrections and Critiques
_Alan Turing, Emil Post, and Donald W. Davies_
3. Systems of Logic Based on Ordinals ( _1938_ ), including excerpts from Turing's correspondence, 1936–1938
4. Letters on Logic to Max Newman ( _c_.1940)
**Enigma**
_Jack Copeland_
5. History of Hut 8 to December 1941 ( _1945_ ), featuring an excerpt from Turing's 'Treatise on the Enigma'
_Patrick Mahon_
6. Bombe and Spider ( _1940_ )
7. Letter to Winston Churchill ( _1941_ )
8. Memorandum to OP-20-G on Naval Enigma ( _c.1941_ )
**Artificial Intelligence**
_Jack Copeland_
9. Lecture on the Automatic Computing Engine ( _1947_ )
10. Intelligent Machinery ( _1948_ )
11. Computing Machinery and Intelligence ( _1950_ )
12. Intelligent Machinery, A Heretical Theory ( _c.1951_ )
13. Can Digital Computers Think? ( _1951_ )
14. Can Automatic Calculating Machines Be Said to Think? ( _1952_ )
_Alan Turing, Richard Braithwaite, Geoffrey Jefferson, and Max Newman_
**Artificial Life**
_Jack Copeland_
15. The Chemical Basis of Morphogenesis ( _1952_ )
16. Chess ( _1953_ )
17. Solvable and Unsolvable Problems ( _1954_ )
Index
## **Alan Turing 1912–1954**
_Jack Copeland_
Alan Mathison Turing was born on 23 June 1912 in London;he died on 7 June 1954 at his home in Wilmslow, Cheshire. Turing contributed to logic, mathematics, biology, philosophy, cryptanalysis, and formatively to the areas later known as computer science, cognitive science, Artificial Intelligence, and Artificial Life.
Educated at Sherborne School in Dorset, Turing went up to King's College, Cambridge, in October 1931 to read Mathematics. He graduated in 1934, and in March 1935 was elected a Fellow of King's, at the age of only 22. In 1936 he published his most important theoretical work, 'On Computable Numbers, with an Application to the Entscheidungsproblem Decision Problem]' ([Chapter 1, with corrections in Chapter 2). This article described the abstract digital computing machine—now referred to simply as the universal Turing machine—on which the modern computer is based. Turing's fundamental idea of a universal stored-programme computing machine was promoted in the United States by John von Neumann and in England by Max Newman. By the end of 1945 several groups, including Turing's own in London, were devising plans for an electronic stored-programme universal digital computer—a Turing machine in hardware.
In 1936 Turing left Cambridge for the United States in order to continue his research at Princeton University. There in 1938 he completed a Ph.D. entitled 'Systems of Logic Based on Ordinals', subsequently published under the same title (Chapter 3, with further exposition in Chapter 4). Now a classic, this work addresses the implications of Gödel's famous incompleteness result. Turing gave a new analysis of mathematical reasoning, and continued the study, begun in 'On Computable Numbers', of uncomputable problems—problems that are 'too hard' to be solved by a computing machine (even one with unlimited time and memory).
Turing returned to his Fellowship at King's in the summer of 1938. At the outbreak of war with Germany in September 1939 he moved to Bletchley Park, the wartime headquarters of the Government Code and Cypher School (GC & CS). Turing's brilliant work at Bletchley Park had far-reaching consequences. 'I won't say that what Turing did made us win the war, but I daresay we might have lost it without him', said another leading Bletchley cryptanalyst. Turing broke Naval Enigma—a decisive factor in the Battle of the Atlantic—and was the principal designer of the 'bombe', a high-speed codebreaking machine. The ingenious bombes produced a flood of high-grade intelligence from Enigma. It is estimated that the work done by Turing and his colleagues at GC & CS shortened the war in Europe by at least two years. Turing's contribution to the Allied victory was a state secret and the only official recognition he received, the Order of the British Empire, was in the circumstances derisory. The full story of Turing's involvement with Enigma is told for the first time in this volume, the material that forms Chapters 5, , and having been classified until recently.
In 1945, the war over, Turing was recruited to the National Physical Laboratory (NPL) in London, his brief to design and develop an electronic digital computer—a concrete form of the universal Turing machine. His design (for the Automatic Computing Engine or ACE) was more advanced than anything else then under consideration on either side of the Atlantic. While waiting for the engineers to build the ACE, Turing and his group pioneered the science of computer programming, writing a library of sophisticated mathematical programmes for the planned machine.
Turing founded the field now called 'Artificial Intelligence' (AI) and was a leading early exponent of the theory that the human brain is in effect a digital computer. In February 1947 he delivered the earliest known public lecture to mention computer intelligence ('Lecture on the Automatic Computing Engine' (Chapter 9)). His technical report 'Intelligent Machinery' (Chapter 10), written for the NPL in 1948, was effectively the first manifesto of AI. Two years later, in his now famous article 'Computing Machinery and Intelligence' (Chapter 11), Turing proposed (what subsequently came to be called) the Turing test as a criterion for whether machines can think. _The Essential Turing_ collects together for the first time the series of five papers that Turing devoted exclusively to Artificial Intelligence (Chapters 10, , , , ). Also included is a discussion of AI by Turing, Newman, and others (Chapter 14).
In the end, the NPL's engineers lost the race to build the world's first working electronic stored-programme digital computer—an honour that went to the Computing Machine Laboratory at the University of Manchester in June 1948. The concept of the universal Turing machine was a fundamental influence on the Manchester computer project, via Newman, the project's instigator. Later in 1948, at Newman's invitation, Turing took up the deputy directorship of the Computing Machine Laboratory (there was no Director). Turing spent the rest of his short career at Manchester University. He was elected a Fellow of the Royal Society of London in March 1951 (a high honour) and in May 1953 was appointed to a specially created Readership in the Theory of Computing at Manchester.
It was at Manchester, in March 1952, that he was prosecuted for homosexual activity, then a crime in Britain, and sentenced to a period of twelve months' hormone 'therapy'—the shabbiest of treatment from the country he had helped save, but which he seems to have borne with amused fortitude.
Towards the end of his life Turing pioneered the area now known as Artificial Life. His 1952 article 'The Chemical Basis of Morphogenesis' (Chapter 15) describes some of his research on the development of pattern and form in living organisms. This research dominated his final years, but he nevertheless found time to publish in 1953 his classic article on computer chess (Chapter 16) and in 1954 'Solvable and Unsolvable Problems' (Chapter 17), which harks back to 'On Computable Numbers'. From 1951 he used the Computing Machine Laboratory's Ferranti Mark I (the first commercially produced electronic stored-programme computer) to model aspects of biological growth, and in the midst of this groundbreaking work he died.
Turing's was a far-sighted genius and much of the material in this book is of even greater relevance today than in his lifetime. His research had remarkable breadth and the chapters range over a diverse collection of topics—mathematical logic and the foundations of mathematics, computer design, mechanical methods in mathematics, cryptanalysis and chess, the nature of intelligence and mind, and the mechanisms of biological growth. The chapters are united by the overarching theme of Turing's work, his enquiry into (as Newman put it) 'the extent and the limitations of mechanistic explanations'.
### _Biographies of Turing_
Gottfried, T., _Alan Turing: The Architect of the Computer Age_ (Danbury, Conn.: Franklin Watts, 1996).
Hodges, A., _Alan Turing: The Enigma_ (London: Burnett, 1983).
Newman, M. H. A., 'Alan Mathison Turing, 1912–1954', _Biographical Memoirs of Fellows of the Royal Society_ , 1 (1955), 253–63.
Turing, S., _Alan M. Turing_ (Cambridge: W. Heffer, 1959).
## Computable Numbers: A Guide
_Jack Copeland_
**Part I The Computer**
1. Turing Machines _6_
2. Standard Descriptions and Description Numbers _10_
3. Subroutines _12_
4. The Universal Computing Machine _15_
5. Turing, von Neumann, and the Computer _21_
6. Turing and Babbage _27_
7. Origins of the Term 'Computer Programme' _30_
**Part II Computability and Uncomputability**
8. Circular and Circle-Free Machines _32_
9. Computable and Uncomputable Sequences _33_
10. Computable and Uncomputable Numbers _36_
11. The Satisfactoriness Problem _36_
12. The Printing and Halting Problems _39_
13. The Church-Turing Thesis _40_
14. The _Entscheidungsproblem_ _45_
'On Computable Numbers, with an Application to the Entscheidungsproblem' appeared in the _Proceedings of the London Mathematical Society_ in 1936. This, Turing's second publication, contains his most significant work. Here he pioneered the theory of computation, introducing the famous abstract computing machines soon dubbed 'Turing machines' by the American logician Alonzo Church. 'On Computable Numbers' is regarded as the founding publication of the modern science of computing. It contributed vital ideas to the development, in the 1940s, of the electronic stored-programme digital computer. 'On Computable Numbers' is the birthplace of the fundamental principle of the modern computer, the idea of controlling the machine's operations by means of a programme of coded instructions stored in the computer's memory.
In addition Turing charted areas of mathematics lying beyond the scope of the Turing machine. He proved that not all precisely stated mathematical problems can be solved by computing machines. One such is the _Entscheidungsproblem_ or 'decision problem'. This work—together with contemporaneous work by Church —initiated the important branch of mathematical logic that investigates and codifies problems 'too hard' to be solvable by Turing machine.
In this one article, Turing ushered in both the modern computer and the mathematical study of the uncomputable.
### Part I The Computer
#### 1. Turing Machines
A Turing machine consists of a scanner and a limitless memory-tape that moves back and forth past the scanner. The tape is divided into squares. Each square may be blank or may bear a single symbol—'0' or '1', for example, or some other symbol taken from a finite alphabet. The scanner is able to examine only one square of tape at a time (the 'scanned square').
The scanner contains mechanisms that enable it to _erase_ the symbol on the scanned square, to _print_ a symbol on the scanned square, and to _move_ the tape to the left or right, one square at a time.
In addition to the operations just mentioned, the scanner is able to alter what Turing calls its ' _m_ -configuration'. In modern Turing-machine jargon it is usual to use the term 'state' in place of ' _m_ -configuration'. A device within the scanner is capable of adopting a number of different states ( _m_ -configurations), and the scanner is able to alter the state of this device whenever necessary. The device may be conceptualized as consisting of a dial with a (finite) number of positions, labelled 'a', 'b', 'c', etc. Each of these positions counts as an _m_ -configuration or state, and changing the _m_ -configuration or state amounts to shifting the dial's pointer from one labelled position to another. This device functions as a simple memory. As Turing says, 'by altering its _m_ -configuration the machine can effectively remember some of the symbols which it has "seen" (scanned) previously' (p. 59). For example, a dial with two positions can be used to keep a record of which binary digit, 0 or 1, is present on the square that the scanner has just vacated. (If a square might also be blank, then a dial with three positions is required.)
The operations just described—erase, print, move, and change state—are the _basic_ (or _atomic_ ) operations of the Turing machine. Complexity of operation is achieved by chaining together large numbers of these simple basic actions. Commercially available computers are hard-wired to perform basic operations considerably more sophisticated than those of a Turing machine—add, multiply, decrement, store-at-address, branch, and so forth. The precise list of basic operations varies from manufacturer to manufacturer. It is a remarkable fact, however, that despite the austere simplicity of Turing's machines, they are capable of computing anything that any computer on the market can compute. Indeed, because they are abstract machines, with unlimited memory, they are capable of computations that no actual computer could perform in practice.
##### _Example of a Turing machine_
The following simple example is from Section 3 of 'On Computable Numbers' (p. 61). The once-fashionable Gothic symbols that Turing used in setting out the example—and also elsewhere in 'On Computable Numbers'—are not employed in this guide. I also avoid typographical conventions used by Turing that seem likely to hinder understanding (for example, his special symbol 'ə', which he used to mark the beginning of the tape, is here replaced by '!').
The machine in Turing's example—call it M—starts work with a blank tape. The tape is endless. The problem is to set up the machine so that if the scanner is positioned over any square of the tape and the machine set in motion, the scanner will print alternating binary digits on the tape, 0 1 0 1 0 1..., working to the right from its starting place, and leaving a blank square in between each digit:
In order to do its work, **M** makes use of four states or _m_ -configurations. These are labelled ' **a** ', ' **b** ', ' **c** ', and ' **d** '. (Turing employed less familiar characters.) **M** is in state **a** when it starts work.
The operations that **M** is to perform can be set out by means of a table with four columns (Table 1). 'R' abbreviates the instruction 'reposition the scanner one square to the right'. This is achieved by moving the tape one square to the left. 'L' abbreviates 'reposition the scanner one square to the left', 'P[0]' abbreviates 'print 0 on the scanned square', and likewise 'P[1]'. Thus the top line of Table 1 reads: if you are in state **a** and the square you are scanning is blank, then print 0 on the scanned square, move the scanner one square to the right, and go into state **b**.
A machine acting in accordance with this table of instructions—or _programme_ —toils endlessly on, printing the desired sequence of digits while leaving alternate squares blank.
Turing does not explain how it is to be brought about that the machine acts in accordance with the instructions. There is no need. Turing's machines are abstractions and it is not necessary to propose any specific mechanism for causing the machine to act in accordance with the instructions. However, for purposes of visualization, one might imagine the scanner to be accompanied by a bank of switches and plugs resembling an old-fashioned telephone switchboard. Arranging the plugs and setting the switches in a certain way causes the machine to act in accordance with the instructions in Table 1. Other ways of setting up the 'switchboard' cause the machine to act in accordance with other tables of instructions. In fact, the earliest electronic digital computers, the British Colossus (1943) and the American ENIAC (1945), were programmed in very much this way. Such machines are described as 'programme-controlled', in order to distinguish them from the modern 'stored-programme' computer.
**Table 1**
As everyone who can operate a personal computer knows, the way to set up a stored-programme machine to perform some desired task is to open the appropriate programme of instructions stored in the computer's memory. The stored-programme concept originates with Turing's _universal_ computing machine, described in detail in Section 4 of this guide. By inserting different programmes into the memory of the universal machine, the machine is made to carry out different computations. Turing's 1945 technical report 'Proposed Electronic Calculator' was the first relatively complete specification of an electronic stored-programme digital computer (see Chapter 9).
##### _E-squares and F-squares_
After describing **M** and a second example of a computing machine, involving the start-of-tape marker '!' (p. 62), Turing introduces a convention which he makes use of later in the article (p. 63). Since the tape is the machine's general-purpose storage medium—serving not only as the vehicle for data storage, input, and output, but also as 'scratchpad' for use during the computation—it is useful to divide up the tape in some way, so that the squares used as scratchpad are distinguished from those used for the various other functions just mentioned.
Turing's convention is that every alternate square of the tape serves as scratchpad. These he calls the 'E-squares', saying that the 'symbols on E-squares will be liable to erasure' (p. 63). The remaining squares he calls 'F-squares'. ('E' and 'F' perhaps stand for 'erasable' and 'fixed'.)
In the example just given, the 'F-squares' of **M** 's tape are the squares bearing the desired sequence of binary digits, 0 1 0 1 0 1... In between each pair of adjacent F-squares lies a blank E-square. The computation in this example is so simple that the E-squares are never used. More complex computations make much use of E-squares.
Turing mentions one important use of E-squares at this point (p. 63): any F-square can be 'marked' by writing some special symbol, e.g. '*', on the E-square immediately to its right. By this means, the scanner is able to find its way back to a particular string of binary digits—a particular item of data, say. The scanner locates the first digit of the string by finding the marker '*'.
##### _Adjacent blank squares_
Another useful convention, also introduced on p. 63, is to the effect that the tape must never contain a run of non-blank squares followed by two or more adjacent blank squares that are themselves followed by one or more non-blank squares. The value of this convention is that it gives the machine an easy way of finding the last non-blank square. As soon as the machine finds two adjacent blank squares, it knows that it has passed beyond the region of tape that has been written on and has entered the region of blank squares stretching away endlessly.
##### _The start-of-tape marker_
Turing usually considers tapes that are endless in one direction only. For purposes of visualization, these tapes may all be thought of as being endless to the right. By convention, each of the first two squares of the tape bears the symbol ' **!** ', mentioned previously. These 'signposts' are never erased. The scanner searches for the signposts when required to find the beginning of the tape.
#### 2. Standard Descriptions and Description Numbers
In the final analysis, a computer programme is simply a (long) stream, or row, of characters. Combinations of characters encode the instructions. In Section 5 of 'On Computable Numbers' Turing explains how an instruction table is to be converted into a row of letters, which he calls a 'standard description'. He then explains how a standard description can be converted into a single number. He calls these 'description numbers'.
Each line of an instruction table can be re-expressed as a single 'word' of the form **q** _i_ S _j_ S _k_ M **q** _l_. **q** _i_ is the state shown in the left-hand column of the table. S _j_ is the symbol on the scanned square (a blank is counted as a type of symbol). S _k_ is the symbol that is to be printed on the scanned square. M is the direction of movement (if any) of the scanner, left or right. **q** _l_ is the next state. For example, the first line of Table 1 can be written: **a** -0R **b** (using '-' to represent a blank). The third line is: **c** -1 R **d**.
The second line of the table, which does not require the contents of the scanned square (a blank) to be changed, is written: **b** \--R **c**. That is to say we imagine, for the purposes of this new notation, that the operations column of the instruction table contains the redundant instruction P[-]. This device is employed whenever an instruction calls for no change to the contents of the scanned square, as in the following example:
It is imagined that the operations column contains the redundant instruction P[x], enabling the line to be expressed: **d** xxL **c**.
Sometimes a line may contain no instruction to move. For example:
The absence of a move is indicated by including 'N' in the instruction-word: **d** *1 N **c**.
Sometimes a line may contain an instruction to erase the symbol on the scanned square. This is denoted by the presence of 'E' in the 'operations' column:
Turing notes that E is equivalent to P[-]. The corresponding instruction-word is therefore **m** *-R **n**.
Any table of instructions can be rewritten in the form of a stream of instruction-words separated by semicolons. Corresponding to Table 1 is the stream:
**a** -0R **b** ; **b** \--R **c** ; **c** -1R **d** ; **d** \--R **a** ;
This stream can be converted into a stream consisting uniformly of the letters A, C, D, L, R, and N (and the semicolon). Turing calls this a _standard description_ of the machine in question. The process of conversion is done in such a way that the individual instructions can be retrieved from the standard description.
The standard description is obtained as follows. First, '-' is replaced by 'D', '0' by 'DC', and '1' by 'DCC'. (In general, if we envisage an ordering of all the printable symbols, the _n_ th symbol in the ordering is replaced by a 'D' followed by _n_ repetitions of 'C'.) This produces:
**a** DDCR **b** ; **b** DDR **c** ; **c** DDCCR **d** ; **d** DDR **a** ;
Next, the lower case state-symbols are replaced by letters. ' **a** ' is replaced by 'DA', ' **b** ' by 'DAA', ' **c** ' by 'DAAA', and so on. An obvious advantage of the new notation is that there is no limit to the number of states that can be named in this way.
The standard description corresponding to Table 1 is:
DADDCRDAA; DAADDRDAAA; DAAADDCCRDAAAA; DAAAADDRDA;
Notice that occurrences of 'D' serve to mark out the different segments or regions of each instruction-word. For example, to determine which symbol an instruction-word says to print, find the third 'D' to the right from the beginning of the word, and count the number of occurrences of 'C' between it and the next D to the right.
The standard description can be converted into a number, called a _description number_. Again, the process of conversion is carried out in such a way that the individual instructions can be retrieved from the description number. A standard description is converted into a description number by means of replacing each 'A' by '1', 'C' by '2', 'D' by '3', 'L' by '4', 'R' by '5', 'N' by '6', and ';' by 7. In the case of the above example this produces:
31332531173113353111731113322531111731111335317.
Occurrences of '7' mark out the individual instruction-words, and occurrences of '3' mark out the different regions of the instruction-words. For example: to find out which symbol the third instruction-word says to print, find the second '7' (starting from the left), then the third '3' to the right of that '7', and count the number of occurrences of '2' between that '3' and the next '3' to the right. To find out the exit state specified by the third instruction-word, find the last '3' in that word and count the number of occurrences of '1' between it and the next '7' to the right.
Notice that _different_ standard descriptions can describe the behaviour of one and the same machine. For example, interchanging the first and second lines of Table 1 does not in any way affect the behaviour of the machine operating in accordance with the table, but a different standard description—and therefore a different description number—will ensue if the table is modified in this way.
This process of converting a table of instructions into a standard description or a description number is analogous to the process of _compiling_ a computer programme into 'machine code'. Programmers generally prefer to work in so-called high-level languages, such as Pascal, Prolog, and C. Programmes written in a high-level language are, like Table 1, reasonably easy for a trained human being to follow. Before a programme can be executed, the instructions must be translated, or compiled, into the form required by the computer (machine code).
The importance of standard descriptions and description numbers is explained in what follows.
#### 3. Subroutines
Subroutines are programmes that are used as components of other programmes. A subroutine may itself have subroutines as components. Programmers usually have access to a 'library' of commonly used subroutines—the programmer takes ready-made subroutines 'off the shelf' whenever necessary.
Turing's term for a subroutine was 'subsidiary table'. He emphasized the importance of subroutines in a lecture given in 1947 concerning the Automatic Computing Engine or ACE, the electronic stored-programme computer that he began designing in 1945 (see Chapter 9 and the introduction to Chapter 10):
Probably the most important idea involved in instruction tables is that of standard _subsidiary tables_. Certain processes are used repeatedly in all sorts of different connections, and we wish to use the same instructions... every time... We have only to think out how [a process] is to be done once, and forget then how it is done.
In 'On Computable Numbers'—effectively the first programming manual of the computer age—Turing introduced a library of subroutines for Turing machines (in Sections 4 and 7), saying (p. 63):
There are certain types of process used by nearly all machines, and these, in some machines, are used in many connections. These processes include copying down sequences of symbols, comparing sequences, erasing all symbols of a given form, etc.
Some examples of subroutines are:
**cpe** ( _A_ , _B_ , _x_ , _y_ ) (p. 66):
' **cpe** ' may be read 'compare for equality'. This subroutine compares the string of symbols marked with an _x_ to the string of symbols marked with a _y_. The subroutine places the machine in state _B_ if the two strings are the same, and in state _A_ if they are different. Note: throughout these examples, ' _A_ ' and ' _B_ ' are variables representing any states; ' _x_ ' and ' _y_ ' are variables representing any symbols.
**f** ( _A_ , _B_ , _x_ ) (p. 63):
' **f** ' stands for 'find'. This subroutine finds the leftmost occurrence of _x_. **f** ( _A_ , _B_ , _x_ ) moves the scanner left until the start of the tape is encountered. Then the scanner is moved to the right, looking for the first _x_. As soon as an _x_ is found, the subroutine places the machine in state _A_ , leaving the scanner resting on the _x_. If no _x_ is found anywhere on the portion of tape that has so far been written on, the subroutine places the machine in state _B_ , leaving the scanner resting on a blank square to the right of the used portion of the tape.
**e** ( _A_ , _B_ , _x_ ) (p. 64):
' **e** ' stands for 'erase'. The subroutine **e** ( _A_ , _B_ , _x_ ) contains the subroutine **f** ( _A_ , _B_ , _x_ ). **e** ( _A_ , _B_ , _x_ ) finds the leftmost occurrence of symbol _x_ and erases it, placing the machine in state _A_ and leaving the scanner resting on the square that has just been erased. If no _x_ is found the subroutine places the machine in state _B_ , leaving the scanner resting on a blank square to the right of the used portion of the tape.
##### _The subroutine_ f( _A, B, x_ )
It is a useful exercise to construct **f** ( _A_ , _B_ , _x_ ) explicitly, i.e. in the form of a table of instructions. Suppose we wish the machine to enter the subroutine **f** ( _A_ , _B_ , _x_ ) when placed in state **n** , say. Then the table of instructions is as shown in Table 2. (Remember that by the convention mentioned earlier, if ever the scanner encounters two adjacent blank squares, it has passed beyond the region of tape that has been written on and has entered the region of blank squares stretching away to the right.)
As Turing explains, **f** ( _A_ , _B_ , _x_ ) is in effect built out of two further subroutines, which he writes **f** 1( _A_ , _B_ , _x_ ) and **f** 2( _A_ , _B_ , _x_ ). The three rows of Table 2 with an ' **m** ' in the first column form the subroutine **f** 1( _A_ , _B_ , _x_ ), and the three rows with ' **o** ' in the first column form **f** 2( _A_ , _B_ , _x_ ).
##### _Skeleton tables_
For ease of defining subroutines Turing introduces an abbreviated form of instruction table, in which one is allowed to write expressions referring to subroutines in the first and fourth columns (the state columns). Turing calls these abbreviated tables 'skeleton tables' (p. 63). For example, the skeleton table corresponding to Table 2 is as in Table 3.
**Table 2**
**Table 3**
Turing's notation for subroutines is explained further in the appendix to this guide ('Subroutines and _m_ -functions').
#### 4. The Universal Computing Machine
In Section 7 of 'On Computable Numbers' Turing introduces his 'universal computing machine', now known simply as the universal Turing machine. The universal Turing machine is the stored-programme digital computer in abstract conceptual form.
The universal computing machine has a single, fixed table of instructions (which we may imagine to have been set into the machine, once and for all, by way of the switchboard-like arrangement mentioned earlier). Operating in accordance with this table of instructions, the universal machine is able to carry out _any_ task for which an instruction table can be written. The trick is to put an instruction table—programme—for carrying out the desired task onto the tape of the universal machine.
The instructions are placed on the tape in the form of a standard description—i.e. in the form of a string of letters that encodes the instruction table. The universal machine reads the instructions and carries them out on its tape.
##### _The universal Turing machine and the modern computer_
Turing's greatest contributions to the development of the modern computer were:
• The idea of controlling the function of a computing machine by storing a programme of symbolically encoded instructions in the machine's memory.
• His demonstration (in Section 7 of 'On Computable Numbers') that, by this means, a _single_ machine of _fixed structure_ is able to carry out every computation that can be carried out by any Turing machine whatsoever, i.e. is universal.
Turing's teacher and friend Max Newman has testified that Turing's interest in building a stored-programme computing machine dates from the time of 'On Computable Numbers'. In a tape-recorded interview Newman stated, 'Turing himself, right from the start, said it would be interesting to try and make such a machine'. (It was Newman who, in a lecture on the foundations of mathematics and logic given in Cambridge in 1935, launched Turing on the research that led to the universal Turing machine; see the introduction to Chapter 4.) In his obituary of Turing, Newman wrote:
The description that [Turing] gave of a 'universal' computing machine was entirely theoretical in purpose, but Turing's strong interest in all kinds of practical experiment made him even then interested in the possibility of actually constructing a machine on these lines.
Turing later described the connection between the universal computing machine and the stored-programme digital computer in the following way (Chapter 9, pp. 378 and 383):
Some years ago I was researching on what might now be described as an investigation of the theoretical possibilities and limitations of digital computing machines. I considered a type of machine which had a central mechanism, and an infinite memory which was contained on an infinite tape...It can be shown that a single special machine of that type can be made to do the work of all... The special machine may be called the universal machine; it works in the following quite simple manner. When we have decided what machine we wish to imitate we punch a description of it on the tape of the universal machine. This description explains what the machine would do in every configuration in which it might find itself. The universal machine has only to keep looking at this description in order to find out what it should do at each stage. Thus the complexity of the machine to be imitated is concentrated in the tape and does not appear in the universal machine proper in any way... [D]igital computing machines such as the ACE... are in fact practical versions of the universal machine. There is a certain central pool of electronic equipment, and a large memory. When any particular problem has to be handled the appropriate instructions for the computing process involved are stored in the memory of the ACE and it is then 'set up' for carrying out that process.
Turing's idea of a universal stored-programme computing machine was promulgated in the USA by von Neumann and in the UK by Newman, the two mathematicians who, along with Turing himself, were by and large responsible for placing Turing's abstract universal machine into the hands of electronic engineers.
By 1946 several groups in both countries had embarked on creating a universal Turing machine in hardware. The race to get the first electronic stored-programme computer up and running was won by Manchester University where, in Newman's Computing Machine Laboratory, the 'Manchester Baby' ran its first programme on 21 June 1948. Soon after, Turing designed the input/output facilities and the programming system of an expanded machine known as the Manchester Mark I. (There is more information about the Manchester computer in the introductions to Chapters 4, , and , and in 'Artificial Life'.) A small pilot version of Turing's Automatic Computing Engine first ran in 1950, at the National Physical Laboratory in London (see the introductions to Chapters 9 and ).
By 1951 electronic stored-programme computers had begun to arrive in the market place. The first model to go on sale was the Ferranti Mark I, the production version of the Manchester Mark I (built by the Manchester firm Ferranti Ltd.). Nine of the Ferranti machines were sold, in Britain, Canada, the Netherlands, and Italy, the first being installed at Manchester University in February 1951. In the United States the first UNIVAC (built by the Eckert-Mauchly Computer Corporation) was installed later the same year. The LEO computer also made its debut in 1951. LEO was a commercial version of the prototype EDSAC machine, which at Cambridge University in 1949 had become the second stored-programme electronic computer to function. 1953 saw the IBM 701, the company's first mass-produced stored-programme electronic computer. A new era had begun.
##### _How the universal machine works_
The details of Turing's universal machine, given on pp. 69–72, are moderately complicated. However, the basic principles of the universal machine are, as Turing says, simple.
Let us consider the Turing machine **M** whose instructions are set out in Table 1. (Recall that **M** 's scanner is positioned initially over any square of **M** 's endless tape, the tape being completely blank.) If a standard description of **M** is placed on the universal machine's tape, the universal machine will _simulate_ or _mimic_ the actions of **M** , and will produce, on specially marked squares of its tape, the output sequence that **M** produces, namely:
0 1 0 1 0 1 0 1 0 1...
The universal machine does this by reading the instructions that the standard description contains and carrying them out on its own tape.
In order to start work, the universal machine requires on its tape not only the standard description but also a record of **M** 's intial state ( **a** ) and the symbol that **M** is initially scanning (a blank). The universal machine's own tape is initially blank except for this record and **M** 's standard description (and some ancillary punctuation symbols mentioned below). As the simulation of **M** progresses, the universal machine prints a record on its tape of:
• the symbols that **M** prints
• the position of **M** 's scanner at each step of the computation
• the symbol 'in' the scanner
• **M** 's state at each step of the computation.
When the universal machine is started up, it reads from its tape **M** 's initial state and initial symbol, and then searches through **M** 's standard description for the instruction beginning: 'when in state **a** and scanning a blank...' The relevant instruction from Table 1 is:
The universal machine accordingly prints '0'. It then creates a record on its tape of **M** 's new state, **b** , and the new position of **M** 's scanner (i.e. immediately to the right of the '0' that has just been printed on **M** 's otherwise blank tape). Next, the universal machine searches through the standard description for the instruction beginning 'when in state **b** and scanning a blank...'. And so on.
How does the universal machine do its record-keeping? After **M** executes its first instruction, the relevant portion of **M** 's tape would look like this—using ' **b** ' both to record **M** 's state and to indicate the position of the scanner. All the other squares of **M** 's tape to the left and right are blank.
The universal machine keeps a record of this state of affairs by employing three squares of tape (pp. 62, 68):
The symbol ' **b** ' has the double function of recording **M** 's state and indicating the position of **M** 's scanner. The square immediately to the right of the state-symbol displays the symbol 'in' **M** 's scanner (a blank).
What does the universal machine's tape look like before the computation starts? The standard description corresponding to Table 1 is:
DADDCRDAA; DAADDRDAAA; DAAADDCCRDAAAA; DAAAADDRDA;
The operator places this programme on the universal machine's tape, writing only on F-squares and beginning on the second F-square of the tape. The first F-square and the first E-square are marked with the start-of-tape symbol ' **!** '. The E-squares (shaded in the diagram) remain blank (except for the first).
On the F-square following the final semicolon of the programme, the operator writes the end-of-programme symbol ' **::** '. On the next F-square to the right of this symbol, the operator places a record of **M** 's initial state, **a** , and leaves the following F-square blank in order to indicate that **M** is initially scanning a blank. The next F-square to the right is then marked with the punctuation symbol ' **:** '. This completes the setting-up of the tape:
What does the universal machine's tape look like as the computation progresses? In response to the first instruction in the standard description, the universal machine creates the record ' **0b-:** ' (in describing the tape, '-' will be used to represent a blank) on the next four F-squares to the right of the first ' **:** '. Depicting only the portion of tape to the right of the end-of-programme marker ' **::** ' (and ignoring any symbols which the universal machine may have written on the E-squares in the course of dealing with the first instruction), the tape now looks like this:
Next the universal machine searches for the instruction beginning 'when in state **b** and scanning a blank...'. The relevant instruction from Table 1 is
This instruction would put **M** into the condition:
So the universal machine creates the record ' **0-c-:** ' on its tape:
Each pair of punctuation marks frames a representation (on the F-squares) of **M** 's tape extending from the square that was in the scanner at start-up to the furthest square to the right to have been scanned at that stage of the computation.
The next instruction is:
This causes the universal machine to create the record ' **0-1d-:** '. (The diagram represents a single continuous strip of tape.)
And so on. Record by record, the outputs produced by the instructions in Table 1 appear on the universal machine's tape.
Turing also introduces a variation on this method of record-keeping, whereby the universal machine additionally prints on the tape a second record of the binary digits printed by **M**. The universal machine does this by printing _in front of_ each record shown in the above diagram a record of any digit newly printed by **M** (plus an extra colon):
These single digits bookended by colons form a representation of what has been printed by **M** on the F-squares of its tape.
Notice that the record-keeping scheme employed so far requires the universal machine to be able to print each type of symbol that the machine being simulated is able to print. In the case of **M** this requirement is modest, since **M** prints only '0', '1', and the blank. However, if the universal machine is to be able to simulate each of the infinitely many Turing machines, then this record-keeping scheme requires that the universal machine have the capacity to print an endless variety of types of discrete symbol. This can be avoided by allowing the universal machine to keep its record of **M** 's tape in the same notation that is used in forming standard descriptions, namely with 'D' replacing the blank, 'DC' replacing '0', 'DCC' replacing '1', 'DA' replacing ' **a** ', 'DAA' replacing ' **b** ', and so on.
The universal machine's tape then looks like this (to the right of the end-of-programme symbol ' **::** ' and not including the second record of digits printed by **M** ):
In this elegant notation of Turing's, 'D' serves to indicate the start of each new term on the universal machine's tape. The letters 'A' and 'C' serve to distinguish terms representing **M** 's states from terms representing symbols on **M** 's tape.
##### _The E-squares and the instruction table_
The universal machine uses the E-squares of its tape to mark up each instruction in the standard description. This facilitates the copying that the universal machine must do in order to produce its records of **M** 's activity. For example, the machine temporarily marks the portion of the current instruction specifying **M** 's next state with ' _y_ ' and subsequently the material marked ' _y_ ' is copied to the appropriate place in the record that is being created. The universal machine's records of **M** 's tape are also temporarily marked in various ways.
In Section 7 Turing introduces various subroutines for placing and erasing markers on the E-squares. He sets out the table of instructions for the universal machine in terms of these subroutines. The table contains the detailed instructions for carrying out the record-keeping described above.
In Section 2.4 of Chapter 2 Turing's sometime colleague Donald Davies gives an introduction to these subroutines and to Turing's detailed table of instructions for the universal machine (and additionally corrects some errors in Turing's own formulation).
#### 5. Turing, von Neumann, and the Computer
In the years immediately following the Second World War, the Hungarian-American logician and mathematician John von Neumann—one of the most important and influential figures of twentieth-century mathematics—made the concept of the stored-programme digital computer widely known, through his writings and his charismatic public addresses. In the secondary literature, von Neumann is often said to have himself invented the stored-programme computer. This is an unfortunate myth.
From 1933 von Neumann was on the faculty of the prestigious Institute for Advanced Study at Princeton University. He and Turing became well acquainted while Turing was studying at Princeton from 1936 to 1938 (see the introduction to Chapter 3). In 1938 von Neumann offered Turing a position as his assistant, which Turing declined. (Turing wrote to his mother on 17 May 1938: 'I had an offer of a job here as von Neumann's assistant at $1500 a year but decided not to take it.' His father had advised him to find a job in America, but on 12 April of the same year Turing had written: 'I have just been to see the Dean [Luther Eisenhart] and ask him about possible jobs over here; mostly for Daddy's information, as I think it unlikely I shall take one unless you are actually at war before July. He didn't know of one at present, but said he would bear it all in mind.')
It was during Turing's time at Princeton that von Neumann became familiar with the ideas in 'On Computable Numbers'. He was to become intrigued with Turing's concept of a universal computing machine. It is clear that von Neumann held Turing's work in the highest regard. One measure of his esteem is that the only names to receive mention in his pioneering volume _The Computer and the Brain_ are those of Turing and the renowned originator of information theory, Claude Shannon.
The Los Alamos physicist Stanley Frankel—responsible with von Neumann and others for mechanizing the large-scale calculations involved in the design of the atomic and hydrogen bombs—has recorded von Neumann's view of the importance of 'On Computable Numbers':
I know that in or about 1943 or'44 von Neumann was well aware of the fundamental importance of Turing's paper of 1936 'On computable numbers...', which describes in principle the 'Universal Computer' of which every modern computer (perhaps not ENIAC as first completed but certainly all later ones) is a realization. Von Neumann introduced me to that paper and at his urging I studied it with care. Many people have acclaimed von Neumann as the 'father of the computer' (in a modern sense of the term) but I am sure that he would never have made that mistake himself. He might well be called the midwife, perhaps, but he firmly emphasized to me, and to others I am sure, that the fundamental conception is owing to Turing—insofar as not anticipated by Babbage, Lovelace, and others. In my view von Neumann's essential role was in making the world aware of these fundamental concepts introduced by Turing and of the development work carried out in the Moore school and elsewhere.
In 1944 von Neumann joined the ENIAC group, led by Presper Eckert and John Mauchly at the Moore School of Electrical Engineering (part of the University of Pennsylvania). At this time von Neumann was involved in the Manhattan Project at Los Alamos, where roomfuls of clerks armed with desk calculating machines were struggling to carry out the massive calculations required by the physicists. Hearing about the Moore School's planned computer during a chance encounter on a railway station (with Herman Goldstine), von Neumann immediately saw to it that he was appointed as consultant to the project. ENIAC—under construction since 1943—was, as previously mentioned, a programme-controlled (i.e. not stored-programme) computer: programming consisted of rerouting cables and setting switches. Moreover, the ENIAC was designed with only one very specific type of task in mind, the calculation of trajectories of artillery shells. Von Neumann brought his knowledge of 'On Computable Numbers' to the practical arena of the Moore School. Thanks to Turing's abstract logical work, von Neumann knew that by making use of coded instructions stored in memory, a single machine of fixed structure could in principle carry out _any_ task for which an instruction table can be written.
Von Neumann gave his engineers 'On Computable Numbers' to read when, in 1946, he established his own project to build a stored-programme computer at the Institute for Advanced Study. Julian Bigelow, von Neumann's chief engineer, recollected:
The person who really... pushed the whole field ahead was von Neumann, because he understood logically what [the stored-programme concept] meant in a deeper way than anybody else... The reason he understood it is because, among other things, he understood a good deal of the mathematical logic which was implied by the idea, due to the work of A. M. Turing... in 1936–1937.... Turing's [universal] machine does not sound much like a modern computer today, but nevertheless it was. It was the germinal idea... So... [von Neumann] saw... that [ENIAC] was just the first step, and that great improvement would come.
Von Neumann repeatedly emphasized the fundamental importance of 'On Computable Numbers' in lectures and in correspondence. In 1946 von Neumann wrote to the mathematician Norbert Wiener of 'the great positive contribution of Turing', Turing's mathematical demonstration that 'one, definite mechanism can be "universal"'. In 1948, in a lecture entitled 'The General and Logical Theory of Automata', von Neumann said:
The English logician, Turing, about twelve years ago attacked the following problem. He wanted to give a general definition of what is meant by a computing automaton... Turing carried out a careful analysis of what mathematical processes can be effected by automata of this type... He... also introduce[d] and analyse[d] the concept of a 'universal automaton'... An automaton is 'universal' if any sequence that can be produced by any automaton at all can also be solved by this particular automaton. It will, of course, require in general a different instruction for this purpose. _The Main Result of the Turing Theory_. We might expect a priori that this is impossible. How can there be an automaton which is at least as effective as any conceivable automaton, including, for example, one of twice its size and complexity? Turing, nevertheless, proved that this is possible.
The following year, in a lecture delivered at the University of Illinois entitled 'Rigorous Theories of Control and Information', von Neumann said:
The importance of Turing's research is just this: that if you construct an automaton right, then any additional requirements about the automaton can be handled by sufficiently elaborate instructions. This is only true if [the automaton] is sufficiently complicated, if it has reached a certain minimal level of complexity. In other words... there is a very definite finite point where an automaton of this complexity can, when given suitable instructions, do anything that can be done by automata at all.
Von Neumann placed Turing's abstract 'universal automaton' into the hands of American engineers. Yet many books on the history of computing in the United States make no mention of Turing. No doubt this is in part explained by the absence of any explicit reference to Turing's work in the series of technical reports in which von Neumann, with various co-authors, set out a logical design for an electronic stored-programme digital computer. Nevertheless there is evidence in these documents of von Neumann's knowledge of 'On Computable Numbers'. For example, in the report entitled 'Preliminary Discussion of the Logical Design of an Electronic Computing Instrument' (1946), von Neumann and his co-authors, Burks and Goldstine—both former members of the ENIAC group, who had joined von Neumann at the Institute for Advanced Study—wrote the following:
3.0. _First Remarks on the Control and Code_ : It is easy to see by formal-logical methods, that there exist codes that are in abstracto adequate to control and cause the execution of any sequence of operations which are individually available in the machine and which are, in their entirety, conceivable by the problem planner. The really decisive considerations from the present point of view, in selecting a code, are more of a practical nature: Simplicity of the equipment demanded by the code, and the clarity of its application to the actually important problems together with the speed of its handling of those problems.
Burks has confirmed that the first sentence of this passage is a reference to Turing's universal computing machine.
##### _The situation in 1945–1946_
The passage just quoted is an excellent summary of the situation at that time. In 'On Computable Numbers' Turing had shown _in abstracto_ that, by means of instructions expressed in the programming code of standard descriptions, a single machine of fixed structure is able to carry out any task that a 'problem planner' is able to analyse into effective steps. By 1945, considerations _in abstracto_ had given way to the practical problem of devising an equivalent programming code that could be implemented efficiently by means of thermionic valves (vacuum tubes).
A machine-level programming code in effect specifies the basic operations that are available in the machine. In the case of Turing's universal machine these are move left one square, scan one symbol, write one symbol, and so on. These operations are altogether too laborious to form the basis of efficient electronic computation. A _practical_ programming code should not only be universal, in the sense of being adequate in principle for the programming of any task that can be carried out by a Turing machine, but must in addition:
• employ basic operations that can be realized simply, reliably, and efficiently by electronic means;
• enable the 'actually important problems' to be solved on the machine as rapidly as the electronic hardware permits;
• be as easy as possible for the human 'problem planner' to work with.
The challenge of designing a practical code, and the underlying mechanism required for its implementation, was tackled in different ways by Turing and the several American groups.
##### _Events at the Moore School_
The 'Preliminary Discussion of the Logical Design of an Electronic Computing Instrument' was not intended for formal publication and no attempt was made to indicate those places where reference was being made to the work of others. (Von Neumann's biographer Norman Macrae remarked: 'Johnny borrowed (we must not say plagiarized) anything from anybody.' The situation was the same in the case of von Neumann's 1945 paper 'First Draft of a Report on the EDVAC'. This described the Moore School group's proposed stored-programme computer, the EDVAC. The 'First Draft' was distributed (by Goldstine and a Moore School administrator) before references had been added—and indeed without consideration of whether the names of Eckert and Mauchly should appear alongside von Neumann's as co-authors. Eckert and Mauchly were outraged, knowing that von Neumann would be given credit for everything in the report—their ideas as well as his own. There was a storm of controversy and von Neumann left the Moore School group to establish his own computer project at Princeton. Harry Huskey, a member of the Moore School group from the spring of 1944, emphasizes that the 'First Draft' should have contained acknowledgement of the considerable extent to which the design of the proposed EDVAC was the work of other members of the group, especially Eckert.
In 1944, before von Neumann came to the Moore School, Eckert and Mauchly had rediscovered the idea of using a single memory for data and programme. (They were far, however, from rediscovering Turing's concept of a universal machine.) Even before the ENIAC was completed, Eckert and Mauchly were thinking about a successor machine, the EDVAC, in which the ENIAC's most glaring deficiencies would be remedied. Paramount among these, of course, was the crude wire'n'plugs method of setting up the machine for each new task. Yet if pluggable connections were not to be used, how was the machine to be controlled without a sacrifice in speed? If the computation were controlled by means of existing, relatively slow, technology—e.g. an electro-mechanical punched-card reader feeding instructions to the machine—then the high-speed electronic hardware would spend much of its time idle, awaiting the next instruction. Eckert explained to Huskey his idea of using a mercury 'delay line':
Eckert described a mercury delay line to me, a five foot pipe filled with mercury which could be used to store a train of acoustic pulses... [O]ne recirculating mercury line would store more than 30 [32 bit binary] numbers... My first question to Eckert: thinking about the pluggable connections to control the ENIAC, 'How do you control the operations?' 'Instructions are stored in the mercury lines just like numbers,' he said. Of course! Once he said it, it was so obvious, and the _only_ way that instructions could come available at rates comparable to the data rates. That was the _stored program computer_.
Following his first visit to the ENIAC in 1944, von Neumann went regularly to the Moore School for meetings with Eckert, Mauchly, Burks, Goldstine, and others. Goldstine reports that 'these meetings were scenes of greatest intellectual activity' and that 'Eckert was delighted that von Neumann was so keenly interested' in the idea of the high-speed delay line memory. It was, says Goldstine, 'fortunate that just as this idea emerged von Neumann should have appeared on the scene'.
Eckert had produced the means to make the abstract universal computing machine of 'On Computable Numbers' concrete! Von Neumann threw himself at the key problem of devising a practical code. In 1945, Eckert and Mauchly reported that von Neumann 'has contributed to many discussions on the logical controls of the EDVAC, has prepared certain instruction codes, and has tested these proposed systems by writing out the coded instructions for specific problems'. Burks summarized matters:
Pres [Eckert] and John [Mauchly] invented the circulating mercury delay line store, with enough capacity to store program information as well as data. Von Neumann created the first modern order code and worked out the logical design of an electronic computer to execute it.
Von Neumann's embryonic programming code appeared in May 1945 in the 'First Draft of a Report on the EDVAC'.
So it was that von Neumann became the first to outline a 'practical version of the universal machine' (the quoted phrase is Turing's; see p. 16). The 'First Draft' contained little engineering detail, however, in particular concerning electronics. Turing's own practical version of the universal machine followed later the same year. His 'Proposed Electronic Calculator' set out a detailed programming code—very different from von Neumann's—together with a detailed design for the underlying hardware of the machine (see Chapter 9).
#### 6. Turing and Babbage
Charles Babbage, Lucasian Professor of Mathematics at the University of Cambridge from 1828 to 1839, was one of the first to appreciate the enormous potential of computing machinery. In about 1820, Babbage proposed an 'Engine' for the automatic production of mathematical tables (such as logarithm tables, tide tables, and astronomical tables). He called it the 'Difference Engine'. This was the age of the steam engine, and Babbage's Engine was to consist of more accurately machined forms of components found in railway locomotives and the like—brass gear wheels, rods, ratchets, pinions, and so forth.
Decimal numbers were represented by the positions of ten-toothed metal wheels mounted in columns. Babbage exhibited a small working model of the Engine in 1822. He never built the full-scale machine that he had designed, but did complete several parts of it. The largest of these—roughly 10 per cent of the planned machine—is on display in the London Science Museum. Babbage used it to calculate various mathematical tables. In 1990 his 'Difference Engine No. 2' was finally built from the original design and this is also on display at the London Science Museum—a glorious machine of gleaming brass.
In 1843 the Swedes Georg and Edvard Scheutz (father and son) built a simplified version of the Difference Engine. After making a prototype they built two commercial models. One was sold to an observatory in Albany, New York, and the other to the Registrar-General's office in London, where it calculated and printed actuarial tables.
Babbage also proposed the 'Analytical Engine', considerably more ambitious than the Difference Engine. Had it been completed, the Analytical Engine would have been an all-purpose mechanical digital computer. A large model of the Analytical Engine was under construction at the time of Babbage's death in 1871, but a full-scale version was never built.
The Analytical Engine was to have a memory, or 'store' as Babbage called it, and a central processing unit, or 'mill'. The behaviour of the Analytical Engine would have been controlled by a programme of instructions contained on punched cards, connected together by ribbons (an idea Babbage adopted from the Jacquard weaving loom). The Analytical Engine would have been able to select from alternative actions on the basis of outcomes of previous actions—a facility now called 'conditional branching'.
Babbage's long-time collaborator was Ada, Countess of Lovelace (daughter of the poet Byron), after whom the modern programming language ADA is named. Her vision of the potential of computing machines was in some respects perhaps more far-reaching even than Babbage's own. Lovelace envisaged computing that went beyond pure number-crunching, suggesting that the Analytical Engine might compose elaborate pieces of music.
Babbage's idea of a general-purpose calculating engine was well known to some of the modern pioneers of automatic calculation. In 1936 Vannevar Bush, inventor of the Differential Analyser (an analogue computer), spoke in a lecture of the possibility of machinery that 'would be a close approach to Babbage's large conception'. The following year Howard Aiken, who was soon to build the digital—but not stored-programme and not electronic—Harvard Automatic Sequence Controlled Calculator, wrote:
Hollerith... returned to the punched card first employed in calculating machinery by Babbage and with it laid the groundwork for the development of... machines as manufactured by the International Business Machines Company, until today many of the things Babbage wished to accomplish are being done daily in the accounting offices of industrial enterprises all over the world.
Babbage's ideas were remembered in Britain also, and his proposed computing machinery was on occasion a topic of lively mealtime discussion at Bletchley Park, the wartime headquarters of the Government Code and Cypher School and birthplace of the electronic digital computer (see 'Enigma' and the introductions to Chapters 4 and ).
It is not known when Turing first learned of Babbage's ideas. There is certainly no trace of Babbage's influence to be found in 'On Computable Numbers'. Much later, Turing generously wrote (Chapter 11, p. 446):
The idea of a digital computer is an old one. Charles Babbage... planned such a machine, called the Analytical Engine, but it was never completed. Although Babbage had all the essential ideas, his machine was not at that time such a very attractive prospect.
Babbage had emphasized the generality of the Analytical Engine, claiming that 'the conditions which enable a finite machine to make calculations of unlimited extent are fulfilled in the Analytical Engine'. Turing states (Chapter 11, p. 455) that the Analytical Engine was _universal_ —a judgement possible only from the vantage point of 'On Computable Numbers'. The Analytical Engine was not, however, a stored-programme computer. The programme resided externally on punched cards, and as each card entered the Engine, the instruction marked on that card would be obeyed.
Someone might wonder what difference there is between the Analytical Engine and the universal Turing machine in that respect. After all, Babbage's cards strung together with ribbon would in effect form a tape upon which the programme is marked. The difference is that in the universal Turing machine, but not the Analytical Engine, there is no fundamental distinction between programme and data. It is the absence of such a distinction that marks off a stored-programme computer from a programme-controlled computer. As Gandy put the point, Turing's 'universal machine is a stored-program machine [in that], unlike Babbage's all-purpose machine, the mechanisms used in reading a program are _of the same kind_ as those used in executing it'.
#### 7. Origins of the Term 'Computer Programme'
As previously mentioned, Turing's tables of instructions for Turing machines are examples of what are now called computer programmes. When he turned to designing an electronic computer in 1945 (the ACE), Turing continued to use his term 'instruction table' where a modern writer would use 'programme' or 'program'. Later material finds Turing referring to the actual process of writing instruction tables for the electronic computer as 'programming' but still using 'instruction table' to refer to the programme itself (see Chapter 9, pp. 388, 390–91).
In an essay published in 1950 Turing explained the emerging terminology to the layman (Chapter 11, p. 445): 'Constructing instruction tables is usually described as "programming". To "programme a machine to carry out the operation A" means to put the appropriate instruction table into the machine so that it will do A.'
Turing seems to have inherited the term 'programming' from the milieu of punched-card plug-board calculators. (These calculators were electromechanical, not electronic. Electro-mechanical equipment was based on the _relay_ —a small electrically driven mechanical switch. Relays operated much more slowly than the thermionic valves (vacuum tubes) on which the first electronic computers were based; valves owe their speed to the fact that they have no moving parts save a beam of electrons—hence the term 'electronic'.) Plug-board calculators were set up to perform a desired sequence of arithmetical operations by means of plugging wires into appropriate sockets in a board resembling a telephone switchboard. Data was fed into the calculator from punched cards, and a card-punching device or printer recorded the results of the calculation. An early example of a punched-card machine was constructed in the USA by Herman Hollerith for use in processing statistical data gathered in the 1890 census. By the mid-twentieth century most of the world's computing was being done by punched-card calculators. Gradually the technology was displaced by the electronic computer.
When Turing joined the National Physical Laboratory in 1945 there was a large room filled with punched-card calulating equipment. David Clayden, one of the engineers who built the ACE, describes the punched-card equipment and the terminology in use at that time:
When I started at NPL in 1947 there was a well established punched card department, mainly Hollerith. The workhorse of punched card equipment is the 'Reproducer', which has a broadside card reader and a broadside card punch. By taking a stack of cards from the punch and putting them into the reader, it is possible to do iterative calculations. All functions are controlled by a plugboard on which there are two sets of 12 × 80 sockets, one for the reader and one for the punch. In addition there is a relay store [i.e. memory]. The plugboard can be connected in many ways (using short plugleads) in order to perform many functions, including addition, subtraction, and multiplication. The plug-boards are removable. NPL had a stack of them and called them 'programme' boards.
Turing's own preference for 'instruction table' over the noun 'programme' was not shared by all his colleagues at the NPL. Mike Woodger, Turing's assistant from 1946, says: '"Programme" of course was an ordinary English word meaning a planned sequence of events. We adopted it naturally for any instruction table that would give rise to a desired sequence of events.' The noun 'programme' was in use in its modern sense from the earliest days of the ACE project. A report (probably written by Turing's immediate superior, Womersley) describing work done by Turing and his assistants during 1946 stated: 'It is intended to prepare the instructions to the machine [the ACE] on Hollerith cards, and it is proposed to maintain a library of these cards with programmes for standard operations.' By the early 1950s specially printed ruled sheets used at the NPL for writing out programmes bore the printed heading 'ACE Pilot Model Programme'.
A document written by Woodger in 1947 used the single 'm' spelling: 'A Program for Version H'. Woodger recalls: 'We used both spellings carelessly for some years until Goodwin (Superintendent of Mathematics Division from 1951) laid down the rule that the "American" spelling should be used.' It is possible that the single 'm' spelling first came to the NPL via the American engineer Huskey, who spent 1947 with the ACE group. Huskey was responsible for 'Version H', a scaled-down form of Turing's design for the ACE (see Chapter 10).
Like Turing, Eckert and Mauchly, the chief architects of ENIAC, probably inherited the terms 'programming' and 'program' from the plug-board calculator. In 1942, while setting out the idea of a high-speed electronic calculator, Mauchly used the term 'programming device' (which he sometimes shortened to 'program device') to refer to a mechanism whose function was to determine how and when the various component units of a calculator shall perform. In the summer of 1946 the Moore School organized a series of influential lectures entitled 'Theory and Techniques for Design of Electronic Digital Computers'. In the course of these, Eckert used the term 'programming' in a similar sense when describing the new idea of storing instructions in high-speed memory: 'We... feed those pieces of information which relate to programming from the memory.' Also in 1946, Burks, Goldstine, and von Neumann (all ex-members of the Moore School group) were using the verb-form 'program the machine', and were speaking of 'program orders' being stored in memory. The modern nominalized form appears not to have been adopted in the USA until a little later. Huskey says, 'I am pretty certain that no one had written a "program" by the time I left Philadelphia in June 1946.'
### Part II Computability and Uncomputability
#### 8. Circular and Circle-Free Machines
Turing calls the binary digits '0' and '1' symbols 'of the first kind'. Any symbols that a computing machine is able to print apart from the binary digits—such as '2', '*', ' _x_ ', and blank—Turing calls 'symbols of the second kind' (p. 60). He also uses the term 'figures' for symbols of the first kind.
A computing machine is said by Turing to be _circular_ if it never prints more than a finite number of symbols of the first kind. A computing machine that will print an infinite number of symbols of the first kind is said to be _circle-free_ (p. 60). For example, a machine operating in accordance with Table 1 is circle-free. (The terms 'circular' and 'circle-free' were perhaps poor choices in this connection, and the terminology has not been followed by others.)
A simple example of a circular machine is one set up to perform a single calculation whose result is an integer. Once the machine has printed the result (in binary notation), it prints nothing more.
A circular machine's scanner need not come to a halt. The scanner may continue moving along the tape, printing nothing further. Or, after printing a finite number of binary digits, a circular machine may work on forever, printing only symbols of the second kind.
Many real-life computing systems are circle-free, for example automated teller machine networks, air traffic control systems, and nuclear reactor control systems. Such systems never terminate by design and, barring hardware failures, power outages, and the like, would continue producing binary digits forever.
In Section 8 of 'On Computable Numbers' Turing makes use of the circular/ circle-free distinction in order to formulate a mathematical problem that cannot be solved by computing machines.
#### 9. Computable and Uncomputable Sequences
The sequence of binary digits printed by a given computing machine on the F-squares of its tape, starting with a blank tape, is called the _sequence computed by the machine_. Where the given machine is circular, the sequence computed by the machine is finite. The sequence computed by a circle-free machine is infinite.
A sequence of binary digits is said to be a _computable_ sequence if it is the sequence computed by some circle-free computing machine. For example, the infinite sequence 010101... is a computable sequence.
Notice that although the finite sequence 010, for example, is the sequence computed by some machine, this sequence is _not_ a _computable_ sequence, according to Turing's definition. This is because, being finite, 010 is not the sequence computed by any circle-free machine. According to Turing's definition, no finite sequence is a computable sequence. Modern writers usually define 'computable' in such a way that every finite sequence is a computable sequence, since each of them can be computed (e.g. by means of an instruction table that simply prints the desired sequence). Turing, however, was not much interested in finite sequences.
The focus of Turing's discussion is his discovery that not every infinite sequence of binary digits is a computable sequence. That this is so is shown by what mathematicians call a _diagonal_ argument.
##### _The diagonal argument_
Imagine that all the computable sequences are listed one under another. (The order in which they are listed does not matter.) The list stretches away to infinity both downwards and to the right. The top left-hand corner might look like this:
Let's say that this list was drawn up in the following way (by an infinite deity, perhaps). The first sequence on the list is the sequence computed by the machine with a description number that is smaller than any description number of any other circle-free machine. The second sequence on the list is the one computed by the circle-free machine with the _next smallest_ description number, and so on. Every computable sequence appears somewhere on this list. (Some will in fact be listed twice, since sometimes different description numbers correspond to the same sequence.)
To prove that not all infinite binary sequences are computable, it is enough to describe one that does not appear on this list. To this end, consider the infinite binary sequence formed by moving diagonally down and across the list, starting at the top left:
The twist is to transform this sequence into a different one by switching each '0' lying on the diagonal to '1' and each '1' to '0'. So the first digit of this new sequence is formed by switching the first digit of the first sequence on the list (producing 1); the second digit of the sequence is formed by switching the second digit of the second sequence on the list (producing 0); the third digit is formed by switching the third digit of the third sequence on the list (producing 1); and so on. Turing calls this sequence ' **β** ' (p. 72).
A moment's reflection shows that **β** cannot itself be one of the listed sequences, since it has been constructed in such a way that it differs from each of these. It differs from the first sequence on the list at the first digit. It differs from the second sequence on the list at the second digit. And so on. Therefore, since every computable sequence appears somewhere on this list, **β** is not among the computable sequences.
##### _Why the computable sequences are listable_
A sceptic might challenge this reasoning, saying: 'Perhaps the computable sequences _cannot_ be listed. In assuming that the computable sequences can be listed, one, two, three, and so on, you are assuming in effect that each computable sequence can be paired off with an integer (no two sequences being paired with the same integer). But what if the computable sequences cannot be paired off like this with the integers? Suppose that there are just _too many_ computable sequences for this to be possible.' If this challenge were successful, it would pull the rug out from under the diagonal argument.
The response to the challenge is this. Each circle-free Turing machine produces just one computable sequence. So there cannot be more computable sequences than there are circle-free Turing machines. But there certainly cannot be more circle-free Turing machines than there are integers. This is because every Turing machine has a description number, which _is_ an integer, and this number is not shared by any other Turing machine.
This reasoning shows that each computable sequence can be paired off with an integer, one sequence per integer. As Turing puts this, the computable sequences are 'enumerable' (p. 68).
The totality of infinite binary sequences, however, is _non-enumerable_. Not all the sequences can be paired off with integers in such a way that no integer is allocated more than one sequence. This is because, once _every_ integer has had an infinite binary sequence allocated to it, one can 'diagonalize' in the above way and produce an extra sequence.
##### _Starting with a blank tape_
Incidentally, notice the significance, in Turing's definition of _sequence computed by the machine_ , of the qualification 'starting with a blank tape'. If the computing machine were allowed to make use of a tape that had already had an infinite sequence of figures printed on it by some means, then the concept of a computable sequence would be trivialized. Every infinite binary sequence would become computable, simply because any sequence of digits whatever— e.g. β—could be present on the tape before the computing machine starts printing.
The following trivial programme causes a machine to run along the tape printing the figures that are already there!
(The third line is required to deal with blank E-squares, if any.)
#### 10. Computable and Uncomputable Numbers
Prefacing a binary sequence by '0' produces a real number expressed in the form of a binary decimal. For example, prefacing the binary sequence 010101... produces 0.010101... (the binary form of the ordinary decimal 0.363636...). If _B_ is the sequence of binary digits printed by a given computing machine, then 0. _B_ is called the _number computed by the machine_.
Where the given machine is circular, the number computed by the machine is always a rational number. A circle-free machine may compute an irrational number (π, for example).
A number computed by a circle-free machine is said to be a _computable number_. Turing also allows that any number 'that differs by an integer' from a number computed by a circle-free machine is a computable number (p. 61). So if _B_ is the infinite sequence of binary digits printed by some circle-free machine, then the number computed by the machine, 0. _B_ , is a computable number, as are all the numbers that differ from 0. _B_ by an integer: 1. _B_ , 10. _B_ , etc.
In Section 10 of 'On Computable Numbers', Turing gives examples of large classes of numbers that are computable. In particular, he proves that the important numbers π and _e_ are computable.
Not all real numbers are computable, however. This follows immediately from the above proof that not all infinite binary sequences are computable. If _S_ is an infinite binary sequence that is uncomputable, then 0. _S_ is an uncomputable number.
#### 11. The Satisfactoriness Problem
In Section 8 of 'On Computable Numbers' Turing describes two mathematical problems that cannot be solved by computing machines. The first will be referred to as the _satisfactoriness problem_.
##### _Satisfactory descriptions and numbers_
A standard description is said to be _satisfactory_ if the machine it describes is circle-free. (Turing's choice of terminology might be considered awkward, since there need be nothing at all unsatisfactory, in the usual sense of the word, about a circular machine.)
A number is satisfactory if it is a description number of a circle-free machine. A number is unsatisfactory if either it is a description number of a circular machine, or it is not a description number at all.
The satisfactoriness problem is this: decide, of any arbitrarily selected standard description—or, equivalently, any arbitrarily selected description number— whether or not it is satisfactory. The decision must be arrived at in a finite number of steps.
##### _The diagonal argument revisited_
Turing approaches the satisfactoriness problem by reconsidering the above proof that not every binary sequence is computable.
Imagine someone objecting to the diagonal argument: 'Look, there must be something wrong with your argument, because β evidently _is_ computable. In the course of the argument, you have in effect given instructions for computing each digit of β, in terms of counting out digits and switching the relevant ones. Let me try to describe how a Turing machine could compute β. I'll call this Turing machine BETA. BETA is similar to the universal machine in that it is able to simulate the activity of any Turing machine that one wishes. First, BETA simulates the circle-free machine with the smallest description number. BETA keeps up the simulation just as far as is necessary in order to discover the first digit of the sequence computed by this machine. BETA then switches this digit, producing the first digit of β. Next, BETA simulates the circle-free machine with the next smallest description number, keeping up the simulation until it finds the second digit of the sequence computed by this machine. And so on.'
The objector continues: 'I can make my description of BETA specific. BETA uses only the E-squares of its tape to do its simulations, erasing its rough work each time it begins a new simulation. It prints out the digits of β on successive F-squares. I need to take account of the restriction that, in order for it to be said that β is the sequence computed by BETA, BETA must produce the digits of β _starting from a blank tape_. What BETA will do first of all, starting from a blank tape, is find the smallest description number that corresponds to a circle-free machine. It does this by checking through the integers, one by one, starting at 1. As BETA generates the integers one by one, it checks each to see whether it is a description number. If the integer is not a description number, then BETA moves on to the next. If the integer is a description number, then BETA checks whether the number is satisfactory. Once BETA finds the first integer to describe a circle-free machine, it uses the instructions contained in the description number in order to simulate the machine. This is how BETA finds the first digit of β. Then BETA continues its search through the integers, until it finds the next smallest description number that is satisfactory. This enables BETA to calculate the second digit of β. And so on.'
Turing tackles this objection head on, proving that no computing machine can possibly do what BETA is supposed to do. It suffices for this proof to consider a slightly simplified version of BETA, which Turing calls . is just like BETA except that does not switch the digits of the list's 'diagonal' sequence. is supposed to write out (on the F-squares) the successive digits not of β but of the 'diagonal' sequence itself: the sequence whose first digit is the first digit of the first sequence on the list, whose second digit is the second digit of the second sequence on the list, and so on. Turing calls this sequence β′. If no computing machine can compute β′, then there is no such computing machine as BETA— because if there were, a machine that computes β′ could be obtained from it, simply by deleting the instructions to switch each of the digits of the diagonal sequence.
##### _What happens when meets itself?_
Turing asks: _what happens when_ , _as_ _searches through the integers one by one, it encounters a number describing_ _itself?_ Call this description number K.
must first check whether K is a description number. Having ascertained that it is, must test whether K is satisfactory. Since is supposed to be computing the endless binary sequence β′, itself must be circle-free. So must pronounce K to be satisfactory.
In order to find the next digit of β′, must now simulate the behaviour of the machine described by K. Since is that machine, must simulate its _own_ behaviour, starting with its very first action. There is nothing wrong with the idea of a machine starting to simulate its own previous behaviour (just as a person might act out some episode from their own past). first simulates (on its E-squares) the series of actions that it performed up to and including writing down the first digit of β′, then its actions up to and including writing down the second digit of β′, and so on.
Eventually, however, 's simulation of its own past reaches the point where began to simulate the behaviour of the machine described by K. What must do now? must simulate the series of actions that it performed when simulating the series of actions that culminated in its writing down the first digit of β′, and then simulate the series of actions that it performed when simulating the series of actions that culminated in its writing down the second digit of β′, and so on! is doomed to relive its past forever.
From the point when it began simulating itself, writes only on the E-squares of its tape and never adds another digit to the sequence on its F-squares. Therefore, does _not_ compute β′. computes some finite number of digits of β′ and then sticks.
The problem lies with the glib assumption that and BETA are able to determine whether each description number is satisfactory.
##### _No computing machine can solve the satisfactoriness problem_
Since, as has just been shown, no computing machine can possibly do what was introduced to do, one of the various tasks that is supposed to carry out must be impossible for a computing machine. But all the things that is supposed to do _apart_ from checking for satisfactoriness—decide whether a number is a description number, extract instructions from a description number, simulate a machine that follows those instructions, and so on—are demonstrably things that can be done by the universal machine.
By a process of elimination, then, the task that it is impossible for a computing machine to carry out must be that of determining whether each description number is satisfactory or not.
#### 12. The Printing and Halting Problems
##### _The printing problem_
Some Turing-machine programmes print '0' at some stage in their computation; all the remaining programmes never print '0'. Consider the problem of deciding, given any arbitrarily selected programme, into which of these two categories it falls. This is an example of the printing problem.
The printing problem (p. 73) is the problem of determining whether or not the machine described by any arbitrarily selected standard description (or, equivalently, any arbitrarily selected description number) ever prints a certain symbol ('0', for example). Turing proves that if the printing problem were solvable by some computing machine, then the satisfactoriness problem would be too. Therefore neither is.
##### _The halting problem_
Another example of a problem that cannot be solved by computing machines, and a close relative of the printing problem, is the _halting problem_. This is the problem of determining whether or not the machine described by any arbitrarily selected standard description eventually halts—i.e. ceases moving altogether— when started on a given tape (e.g. a blank tape).
The machine shown in Table 1 is rather obviously one of those that never halt—but in other cases it is not at all obvious from a machine's table whether or not it halts. Simply watching the machine run (or a simulation of the machine) is of little help, for what can be concluded if after a week or a year the machine has not halted? If the machine does eventually halt, a watching human—or Turing machine—will sooner or later find this out; but in the case of a machine that has not yet halted, there is no systematic method for deciding whether or not it is going to halt.
The halting problem was so named (and, it appears, first stated) by Martin Davis. The proposition that the halting problem cannot be solved by computing machine is known as the 'halting theorem'. (It is often said that Turing stated and proved the halting theorem in 'On Computable Numbers', but strictly this is not true.)
#### 13. The Church–Turing Thesis
##### _Human computers_
When Turing wrote 'On Computable Numbers', a computer was not a machine at all, but a human being. A computer—sometimes also spelt 'computor'—was a mathematical assistant who calculated by rote, in accordance with a systematic method. The method was supplied by an overseer prior to the calculation. Many thousands of human computers were employed in business, government, and research establishments, doing some of the sorts of calculating work that nowadays is performed by electronic computers. Like filing clerks, computers might have little detailed knowledge of the end to which their work was directed.
The term 'computing machine' was used to refer to small calculating machines that mechanized elements of the human computer's work. These were somewhat like today's non-programmable hand-calculators: they were not automatic, and each step—each addition, division, and so on—was initiated manually by the human operator. A computing machine was in effect a homunculus, calculating more quickly than an unassisted human computer, but doing nothing that could not in principle be done by a human clerk working by rote. For a complex calculation, several dozen human computers might be required, each equipped with a desk-top computing machine.
In the late 1940s and early 1950s, with the advent of electronic computing machines, the phrase 'computing machine' gave way gradually to 'computer'. During the brief period in which the old and new meanings of 'computer' coexisted, the prefix 'electronic' or 'digital' would usually be used in order to distinguish machine from human. As Turing stated, the new electronic machines were 'intended to carry out any definite rule of thumb process which could have been done by a human operator working in a disciplined but unintelligent manner'. Main-frames, laptops, pocket calculators, palm-pilots—all carry out work that a human rote-worker could do, if he or she worked long enough, and had a plentiful enough supply of paper and pencils.
It must be borne in mind when reading 'On Computable Numbers' that Turing there used the word 'computer' in this now archaic sense. Thus he says, for example, 'Computing is normally done by writing certain symbols on paper' (p. 75) and 'The behaviour of the computer at any moment is determined by the symbols which he is observing, and his "state of mind"' (p. 75).
The Turing machine is an idealization of the human computer (p. 59): 'We may compare a man in the process of computing a real number to a machine which is only capable of a finite number of conditions... called " _m_ -configurations". The machine is supplied with a "tape"...' Wittgenstein put the point in a striking way: 'Turing's "Machines". These machines are _humans_ who calculate.'
In the primary sense, a computable number is a real number that can be calculated by a human computer—or in other words, a real number that a human being can calculate by means of a systematic method. When Turing asserts that 'the "computable" numbers include all numbers which would naturally be regarded as computable' (p. 74), he means that each number that is computable in this primary sense is also computable in the technical sense defined in Section 2 of 'On Computable Numbers' (see Section 10 of this introduction).
##### _The thesis_
_Turing's thesis_ , that
the 'computable' numbers include all numbers which would naturally be regarded as computable,
is now known as the _Church–Turing thesis_.
Some other ways of expressing the thesis are:
1. The universal Turing machine can perform any calculation that any human computer can carry out.
2. Any systematic method can be carried out by the universal Turing machine.
The Church–Turing thesis is sometimes heard in the strengthened form:
Anything that can be made completely precise can be programmed for a universal digital computer.
However, this strengthened form of the thesis is false. The printing, halting, and satisfactoriness problems are completely precise, but of course cannot be programmed for a universal computing machine.
##### _Systematic methods_
A systematic method—sometimes also called an _effective_ method and a _mechanical_ method—is any mathematical method of which all the following are true:
• the method can, in practice or in principle, be carried out by a human computer working with paper and pencil;
• the method can be given to the human computer in the form of a _finite_ number of instructions;
• the method demands neither insight nor ingenuity on the part of the human being carrying it out;
• the method will definitely work if carried out without error;
• the method produces the desired result in a finite number of steps; or, if the desired result is some _infinite_ sequence of symbols (e.g. the decimal expansion of π), then the method produces each individual symbol in the sequence in some finite number of steps.
The term 'systematic' and its synonyms 'effective' and 'mechanical' are terms of art in mathematics and logic. They do not carry their everyday meanings. For example: if some type of machine were able to solve the satisfactoriness problem, the method it used would not be systematic or mechanical in _this_ sense. (Turing is sometimes said to have proved that _no machine_ can solve the satisfactoriness problem. This is not so. He demonstrates only that his idealized human computers—Turing machines—cannot solve the satisfactoriness problem. This does not in itself rule out the possibility that some other type of machine might be able to solve the problem.)
Turing sometimes used the expression _rule of thumb_ in place of 'systematic'. If this expression is employed, the Church–Turing thesis becomes (Chapter 10, p. 414):
LCMs can do anything that could be described as 'rule of thumb' or 'purely mechanical'.
'LCM' stands for 'logical computing machine', a term that Turing seems to have preferred to the (then current) 'Turing machine'.
Section 9 of 'On Computable Numbers' contains a bouquet of arguments for Turing's thesis. The arguments are persuasive, but do not offer the certainty of mathematical proof. As Turing says wryly of a related thesis in Chapter 17 (p. 588): 'The statement is... one which one does not attempt to prove. Propaganda is more appropriate to it than proof.'
Additional arguments and other forms of evidence for the thesis amassed. These, too, left matters short of absolute certainty. Nevertheless, before long it was, as Turing put it, 'agreed amongst logicians' that his proposal gives the 'correct accurate rendering' of talk about systematic methods (Chapter 10, p. 414). There have, however, been occasional dissenting voices over the years (for example, Kalmár and Péter).
##### _The converse of the thesis_
The _converse_ of the Church–Turing thesis is:
Any number, or binary sequence, that can be computed by the universal Turing machine can be calculated by means of a systematic method.
This is self-evidently true—the instruction table on the universal machine's tape is itself a specification of a systematic method for calculating the number or sequence in question. In principle, a human being equipped with paper and pencil could work through the instructions in the table and write out the digits of the number, or sequence, without at any time exercising ingenuity or insight ('in principle' because we have to assume that the human does not throw in the towel from boredom, die of old age, or use up every sheet of paper in the universe).
##### _Application of the thesis_
The concept of a systematic method is an informal one. Attempts—such as the above—to explain what counts as a systematic method are not rigorous, since the requirement that the method demand neither insight nor ingenuity is left unexplicated.
One of the most significant achievements of 'On Computable Numbers'—and this was a large step in the development of the mathematical theory of computation—was to propose a rigorously defined expression with which the informal expression 'by means of a systematic method' might be replaced. The rigorously defined expression is, of course, 'by means of a Turing machine'.
The importance of Turing's proposal is this. If the proposal is correct—i.e. if the Church–Turing thesis is true—then talk about the existence or non-existence of systematic methods can be replaced throughout mathematics and logic by talk about the existence or non-existence of Turing-machine programmes. For instance, one can establish that there is no systematic method at all for doing such-and-such a thing by proving that no Turing machine can do the thing in question. This is precisely Turing's strategy with the _Entscheidungsproblem_ , as explained in the next section.
##### _Church's contribution_
In 1935, on the other side of the Atlantic, Church had independently proposed a different way of replacing talk about systematic methods with formally precise language (in a lecture given in April of that year and published in 1936). Turing learned of Church's work in the spring of 1936, just as 'On Computable Numbers' was nearing completion (see the introduction to Chapter 4).
Where Turing spoke of numbers and sequences, Church spoke of mathematical _functions_. ( _x_ 2 and _x_ \+ _y_ are examples of mathematical functions. 4 is said to be the _value of the function x_ 2 for _x_ = 2.) Corresponding to each computable sequence _S_ is a computable function _fx_ (and vice versa). The value of _fx_ for _x_ = 1 is the first digit of _S_ , for _x_ = 2, the second digit of _S_ , and so on. In 'On Computable Numbers' Turing said (p. 58): 'Although the subject of this paper is ostensibly the computable _numbers_ , it is almost equally easy to define and investigate computable functions... I have chosen the computable numbers for explicit treatment as involving the least cumbrous technique.'
Church's analysis was in terms of his and Stephen Kleene's concept of a _lambda-definable_ function. A function of positive integers is said to be lambda-definable if the values of the function can be calculated by a process of repeated substitution.
Thus we have alongside Turing's thesis
_Church's thesis_ : every function of positive integers whose values can be calculated by a systematic method is lambda-definable.
Although Turing's and Church's approaches are different, they are nevertheless equivalent, in the sense that every lambda-definable function is computable by the universal machine and every function (or sequence) computable by the universal machine is lambda-definable. Turing proved this in the Appendix to 'On Computable Numbers' (added in August 1936).
The name 'Church–Turing thesis', now standard, seems to have been introduced by Kleene, with a flourish of bias in favour of his mentor Church: 'So Turing's and Church's theses are equivalent. We shall usually refer to them both as _Church's thesis_ , or in connection with that one of its... versions which deals with "Turing machines" as _the Church-Turing thesis_.'
Although Turing's and Church's theses are equivalent in the logical sense, there is nevertheless good reason to prefer Turing's formulation. As Turing wrote in 1937: 'The identification of "effectively calculable" functions with computable functions is possibly more convincing than an identification with the λ-definable [lambda-definable] or general recursive functions.' Church acknowledged the point:
As a matter of fact, there is... equivalence of three different notions: computability by a Turing machine, general recursiveness in the sense of Herbrand–Gödel–Kleene, and λ-definability in the sense of Kleene and [myself]. Of these, the first has the advantage of making the identification with effectiveness in the ordinary (not explicitly defined) sense evident immediately... The second and third have the advantage of suitability for embodiment in a system of symbolic logic.
The great Kurt Gödel, it seems, was unpersuaded by Church's thesis until he saw Turing's formulation. Kleene wrote:
According to a November 29, 1935, letter from Church to me, Gödel 'regarded as thoroughly unsatisfactory' Church's proposal to use λ-definability as a definition of effective calculability... It seems that only after Turing's formulation appeared did Gödel accept Church's thesis.
Hao Wang reports Gödel as saying: 'We had not perceived the sharp concept of mechanical procedures sharply before Turing, who brought us to the right perspective.'
Gödel described Turing's analysis of computability as 'most satisfactory' and 'correct... beyond any doubt'. He also said: 'the great importance of... Turing's computability... seems to me... largely due to the fact that with this concept one has for the first time succeeded in giving an absolute definition of an interesting epistemological notion.'
#### 14. The _Entscheidungsproblem_
In Section 11 of 'On Computable Numbers', Turing turns to the _Entscheidungsproblem_ , or _decision problem_. Church gave the following definition of the _Entscheidungsproblem_ :
By the Entscheidungsproblem of a system of symbolic logic is here understood the problem to find an effective method by which, given any expression Q in the notation of the system, it can be determined whether or not Q is provable in the system.
The decision problem was brought to the fore of mathematics by the German mathematician David Hilbert (who in a lecture given in Paris in 1900 set the agenda for much of twentieth-century mathematics). In 1928 Hilbert described the decision problem as 'the main problem of mathematical logic', saying that 'the discovery of a general decision procedure is a very difficult problem which is as yet unsolved', and that the 'solution of the decision problem is of fundamental importance'.
##### _The Hilbert programme_
Hilbert and his followers held that mathematicians should seek to express mathematics in the form of a complete, consistent, decidable formal system—a system expressing 'the whole thought content of mathematics in a uniform way'. Hilbert drew an analogy between such a system and 'a court of arbitration, a supreme tribunal to decide fundamental questions—on a concrete basis on which everyone can agree and where every statement can be controlled'. Such a system would banish ignorance from mathematics: given any mathematical statement, one would be able to tell whether the statement is true or false by determining whether or not it is provable in the system. As Hilbert famously declared in his Paris lecture: 'in mathematics there is no _ignorabimus_ ' (there is no _we shall not know_ ).
It is important that the system expressing the 'whole thought content of mathematics' be _consistent_. An inconsistent system—a system containing contradictions—is worthless, since _any_ statement whatsoever, true or false, can be derived from a contradiction by simple logical steps. So in an inconsistent system, absurdities such as 0 = 1 and 6 ≠ 6 are provable. An inconsistent system would indeed contain all true mathematical statements—would be _complete_ , in other words—but would in addition also contain all false mathematical statements!
Hilbert's requirement that the system expressing the whole content of mathematics be _decidable_ amounts to this: there must be a systematic method for telling, of each mathematical statement, whether or not the statement is provable in the system. If the system is to banish ignorance totally from mathematics then it must be decidable. Only then could we be confident of always being able to tell whether or not any given statement is provable. An undecidable system might sometimes leave us in ignorance.
The project of expressing mathematics in the form of a complete, consistent, decidable formal system became known as 'proof theory' and as the 'Hilbert programme'. In 1928, in a lecture delivered in the Italian city of Bologna, Hilbert said:
In a series of presentations in the course of the last years I have... embarked upon a new way of dealing with fundamental questions. With this new foundation of mathematics, which one can conveniently call proof theory, I believe the fundamental questions in mathematics are finally eliminated, by making every mathematical statement a concretely demonstrable and strictly derivable formula...
[I]n mathematics there is no _ignorabimus_ , rather we are always able to answer meaningful questions; and it is established, as Aristotle perhaps anticipated, that our reason involves no mysterious arts of any kind: rather it proceeds according to formulable rules that are completely definite—and are as well the guarantee of the absolute objectivity of its judgement.
Unfortunately for the Hilbert programme, however, it was soon to become clear that most interesting mathematical systems are, if consistent, _incomplete_ and _undecidable_.
In 1931, Gödel showed that Hilbert's ideal is impossible to satisfy, even in the case of simple arithmetic. He proved that the formal system of arithmetic set out by Whitehead and Russell in their seminal _Principia Mathematica_ is, if consistent, incomplete. That is to say: if the system is consistent, there are true statements of arithmetic that are not provable in the system—the formal system fails to capture the 'whole thought content' of arithmetic. This is known as Gödel's _first incompleteness theorem_.
Gödel later generalized this result, pointing out that 'due to A. M. Turing's work, a precise and unquestionably adequate definition of the general concept of formal system can now be given', with the consequence that incompleteness can 'be proved rigorously for _every_ consistent formal system containing a certain amount of finitary number theory'. The definition made possible by Turing's work is this (in Gödel's words): 'A formal system can simply be defined to be any mechanical procedure for producing formulas, called provable formulas.'
In his incompleteness theorem, Gödel had shown that no matter how hard mathematicians might try to construct the all-encompassing formal system envisaged by Hilbert, the product of their labours would, if consistent, inevitably be incomplete. As Hermann Weyl—one of Hilbert's greatest pupils— observed, this was nothing less than 'a catastrophe' for the Hilbert programme.
##### _Decidability_
Gödel's theorem left the question of decidability open. As Newman summarized matters:
The Hilbert decision-programme of the 1920's and 30's had for its objective the discovery of a general process... for deciding... truth or falsehood... A first blow was dealt at the prospects of finding this new philosopher's stone by Gödel's incompleteness theorem (1931), which made it clear that truth or falsehood of _A_ could not be equated to provability of _A_ or not- _A_ in any finitely based logic, chosen once for all; but there still remained in principle the possibility of finding a mechanical process for deciding whether _A_ , or not- _A_ , or neither, was formally provable in a given system.
The question of decidability was tackled head on by Turing and, independently, by Church.
On p. 84 of 'On Computable Numbers' Turing pointed out—by way of a preliminary—a fact that Hilbertians appear to have overlooked: if a system is complete then it follows that it is also decidable. Bernays, Hilbert's close collaborator, had said: 'One observes that [the] requirement of deductive completeness does not go as far as the requirement of decidability.' Turing's simple argument on p. 84 shows that there is no conceptual room for the distinction that Bernays is claiming.
Nevertheless, the crucial question was still open: given that in fact simple arithmetic is (if consistent) _in_ complete, is it or is it not decidable? Turing and Church both showed that no consistent formal system of arithmetic is decidable. They showed this by proving that not even the _functional calculus_ —the weaker, purely logical system presupposed by any formal system of arithmetic—is decidable. The Hilbertian dream of a completely mechanized mathematics now lay in total ruin.
##### _A tutorial on first-order predicate calculus_
What Turing called the functional calculus (and Church, following Hilbert, the _engere Funktionenkalkül_ ) is today known as _first-order predicate calculus_ (FOPC). FOPC is a formalization of deductive logical reasoning.
There are various different but equivalent ways of formulating FOPC. One formulation presents FOPC as consisting of about a dozen formal rules of inference. (This formulation, which is more accessible than the Hilbert–Ackermann formulation mentioned by Turing on p. 84, is due to Gerhard Gentzen.)
The following are examples of formal rules of inference. The symbol ' ' indicates that the statement following it can be concluded from the statements (or statement) displayed to its left, the premisses.
(i) _X_ , if _X_ then _Y_ _Y_
(ii) _X_ and _Y_ _X_
(iii) _X_ , _Y_ _X_ and _Y_
So if, for example, ' _X_ ' represents 'It is sunny' and ' _Y_ ' represents 'We will go for a picnic', (i) says:
'We will go for a picnic' can be concluded from the premisses 'It is sunny' and 'If it is sunny then we will go for a picnic'.
(ii) says:
'It is sunny' can be concluded from the conjunctive premiss 'It is sunny and we will go for a picnic'.
Turing uses the symbol '→' to abbreviate 'if then' and the symbol '&' to abbreviate 'and'. Using this notation, (i)–(iii) are written:
(i) _X_ , _X_ → _Y_ _Y_
(ii) _X_ & _Y_ _X_
(iii) _X_ , _Y_ _X_ & _Y_
Some more rules of the formal calculus are as follows. _a_ represents any object, _F_ represents any property:
(iv) _a_ has property _F_ there is an object that has property _F_
(v) each object has property _F_ a has property _F_
In Turing's notation, in which ' _a_ has property _F_ ' is abbreviated ' _F_ ( _a_ )', these are written:
(iv) _F_ ( _a_ ) (∃ _x_ ) _F_ ( _x_ )
(v) ( _x_ ) _F_ ( _x_ ) _F_ ( _a_ )
'(∃ _x_ )' is read: 'there is an object (call it _x_ ) which...'. So '(∃ _x_ ) _F_ ( _x_ )' says 'there is an object, call it _x_ , which has property _F_ ′. '( _x_ )' is read: 'each object, _x_ , is such that...'. So '( _x_ ) _F_ ( _x_ )' says 'each object, _x_ , is such that _x_ has property _F_ '.
Set out in full, FOPC contains not only rules like (i)–(v) but also several rules leading from statements containing ' ' to other statements containing ' '. One such rule is the so-called 'cut rule', used in moving from lines (2) and (3) to (4) in the proof below.
Turing calls '(∃ _x_ )' and '( _x_ )' _quantors_ ; the modern term is _quantifiers_. A symbol, such as ' _F_ ', that denotes a property is called a _predicate_. Symbols denoting relationships, for example '<' (less than) and '=' (identity), are also classed as predicates. The symbol ' _x_ ' is called a _variable_.
(FOPC is _first-order_ in the sense that the quantifiers of the calculus always involve variables that refer to individual objects. In _second-order_ predicate calculus, on the other hand, the quantifiers can contain predicates, as in '(∃ _F_ )'. The following are examples of second-order quantification: 'Jules and Jim have some properties in common,' 'Each relationship that holds between _a_ and _b_ also holds between _c_ and _d_.')
Using the dozen or so basic rules of FOPC, more complicated rules of inference can be proved as _theorems_ ('provable formulas') of FOPC. For example:
_Theorem_ ( _x_ )( _G_ ( _x_ ) → _H_ ( _x_ )), _G_ ( _a_ ) (∃ _x_ ) _H_ ( _x_ )
This theorem says: 'There is an object that has property _H_ ' can be concluded from the premisses 'Each object that has property _G_ also has property _H_ ' and ' _a_ has property G'.
The proof of the theorem is as follows:
(1) ( _x_ )( _G_ ( _x_ ) → _H_ ( _x_ )) _G_ ( _a_ ) → _H_ ( _a_ ) | (rule (v))
---|---
(2) _G_ ( _a_ ), ( _G_ ( _a_ ) → _H_ ( _a_ )) _H_ ( _a_ ) | (rule (i))
(3) _H_ ( _a_ ) (∃ _x_ ) _H_ ( _x_ ) | (rule (iv))
(4) _G_ ( _a_ ), ( _G_ ( _a_ ) → _H_ ( _a_ )) (∃ _x_ ) _H_ ( _x_ ) | (from (2) and (3) by the cut rule)
(5) ( _x_ )( _G_ ( _x_ ) → _H_ ( _x_ )), _G_ ( _a_ ) (∃ _x_ ) _H_ ( _x_ ) | (from (1) and (4) by the cut rule)
The cut rule (or rule of transitivity) says in effect that whatever can be concluded from a statement _Y_ (possibly in conjunction with additional premisses _P_ ) can be concluded from any premiss(es) from which _Y_ can be concluded (together with the additional premisses _P_ , if any). For example, if _Y_ _Z_ and _X_ _Y_ , then _X_ _Z_. In the transition from (1) and (4) to (5), the additional premiss _G_ ( _a_ ) in (4) is gathered up and placed among the premisses of (5).
So far we have seen how to prove further inference rules in FOPC. Often logicians are interested in proving not inference rules but single statements unbroken by commas and ' '. An example is the complex statement
_not_ ( _F_ ( _a_ ) & _not_ (∃ _x_ ) _F_ ( _x_ )),
which says 'It is not the case that _both F_ ( _a_ ) and the denial of (∃ _x_ ) _F_ ( _x_ ) are true'; or in other words, you are not going to find _F_ ( _a_ ) true without finding (∃ _x_ ) _F_ ( _x_ ) true.
To say that a single statement, as opposed to an inference rule, is provable in FOPC is simply to say that the result of _prefixing_ that statement by ' ' can be derived by using the rules of the calculus. Think of a ' ' with no statements on its left as indicating that the statement on its right is to be concluded as a matter of 'pure logic'—no premisses are required.
For example, the theorem
_not (F(a) & not (∃x) F(x))_
can be derived using rule (iv) and the following new rule.
This rule is read:
If _Y_ can be concluded from _X_ , then it can be concluded that not _both X_ and the denial of _Y_ are true.
Much of mathematics and science can be formulated within the framework of FOPC. For example, a formal system of arithmetic can be constructed by adding a number of arithmetical axioms to FOPC. The axioms consist of very basic arithmetical statements, such as:
( _x_ )( _x_ \+ 0 = _x_ )
and
( _x_ )( _y_ )(S _x_ = S _y_ → _x_ = _y_ ),
where 'S' means 'the successor of'—the successor of 1 is 2, and so on. (In these axioms the range of the variables ' _x_ ' and ' _y_ ' is restricted to numbers.) Other arithmetical statements can be derived from these axioms by means of the rules of FOPC. For example, rule (v) tells us that the statement
1 + 0 = 1
can be concluded from the first of the above axioms.
If FOPC is undecidable then it follows that arithmetic is undecidable. Indeed, if FOPC is undecidable, then so are very many important mathematical systems. To find decidable logics one must search among systems that are in a certain sense _weaker_ than FOPC. One example of a decidable logic is the system that results if all the quantifier rules—rules such as (iv) and (v)—are elided from FOPC. This system is known as the _propositional calculus_.
##### _The proof of the undecidability of FOPC_
Turing and Church showed that there is no systematic method by which, given any formula Q in the notation of FOPC, it can be determined whether or not Q is provable in the system (i.e. whether or not Q). To put this another way, Church and Turing showed that the _Entscheidungsproblem_ is _unsolvable_ in the case of FOPC.
Both published this result in 1936. Church's demonstration of undecidability proceeded via his lambda calculus and his thesis that to each effective method there corresponds a lambda-definable function. There is general agreement that Turing was correct in his view, mentioned above (p. 45), that his own way of showing undecidability is 'more convincing' than Church's.
Turing's method makes use of his proof that no computing machine can solve the printing problem. He showed that if a Turing machine could tell, of any given statement, whether or not the statement is provable in FOPC, then a Turing machine could tell, of any given Turing machine, whether or not it ever prints '0'. Since, as he had already established, no Turing machine can do the latter, it follows that no Turing machine can do the former. The final step of the argument is to apply Turing's thesis: if no Turing machine can perform the task in question, then there is no systematic method for performing it.
In detail, Turing's demonstration contains the following steps.
1. Turing shows how to construct, for any computing machine **m** , a complicated statement of FOPC that says 'at some point, machine **m** prints 0'. He calls this formula 'Un( **m** )'. (The letters 'Un' probably come from 'undecidable' or the German equivalent 'unentscheidbare'.)
2. Turing proves the following:
( _a_ ) If Un( **m** ) is provable in FOPC, then at some point **m** prints 0.
( _b_ ) If at some point **m** prints 0, then Un( **m** ) is provable in FOPC.
3. Imagine a computing machine which, when given any statement Q in the notation of FOPC, is able to determine (in some finite number of steps) whether or not Q is provable in FOPC. Let's call this machine HILBERT'S DREAM. 2( _a_ ) and 2( _b_ ) tell us that HILBERT'S DREAM would solve the printing problem. Because if the machine were to indicate that Un( **m** ) is provable then, in view of 2( _a_ ), it would in effect be indicating that **m** does print 0; and if the machine were to indicate that the statement Un( **m** ) is not provable then, in view of 2( _b_ ), it would in effect be indicating that **m** does not print 0. Since no computing machine can solve the printing problem, it follows that HILBERT'S DREAM is a figment. _No computing machine is able to determine in some finite number of steps, of each statement Q, whether or not Q is provable in FOPC_.
4. If there were a systematic method by which, given any statement Q, it can be determined whether or not Q is provable in FOPC, then it would follow, by Turing's thesis, that there is such a computing machine as HILBERT'S DREAM. Therefore there is no such systematic method.
##### _The significance of undecidability_
Poor news though the unsolvability of the _Entscheidungsproblem_ was for the Hilbert school, it was very welcome news in other quarters, for a reason that Hilbert's illustrious pupil von Neumann had given in 1927:
If undecidability were to fail then mathematics, in today's sense, would cease to exist; its place would be taken by a completely mechanical rule, with the aid of which any man would be able to decide, of any given statement, whether the statement can be proven or not.
As the Cambridge mathematician G. H. Hardy said in a lecture in 1928: 'if there were... a mechanical set of rules for the solution of all mathematical problems... our activities as mathematicians would come to an end.'
#### _Further reading_
Barwise, J., and Etchemendy, J., _Turing's World: An Introduction to Computability Theory_ (Stanford, Calif.: CSLI, 1993). (Includes software for building and displaying Turing machines.)
Boolos, G. S., and Jeffrey, R. C., _Computability and Logic_ (Cambridge: Cambridge University Press, 2nd edn. 1980).
Copeland, B. J., 'Colossus and the Dawning of the Computer Age', in R. Erskine and M. Smith (eds.), _Action This Day_ (London: Bantam, 2001).
Epstein, R. L., and Carnielli, W. A., _Computability: Computable Functions, Logic, and the Foundations of Mathematics_ (Belmont, Calif.: Wadsworth, 2nd edn. 2000).
Hopcroft, J. E., and Ullman, J. D., _Introduction to Automata Theory, Languages, and Computation_ (Reading, Mass.: Addison-Wesley, 1979).
Minsky, M. L., _Computation: Finite and Infinite Machines_ (Englewood Cliffs, NJ: Prentice-Hall, 1967).
Sieg, W., 'Hilbert's Programs: 1917–1922', _Bulletin of Symbolic Logic_ , 5 (1999), 1–44.
Sipser, M., _Introduction to the Theory of Computation_ (Boston: PWS, 1997).
### Appendix
#### _Subroutines and M-Functions_
Section 3 of this guide gave a brief introduction to the concept of a _skeleton table_ , where names of subroutines are employed in place of letters referring to states of the machine. This appendix explains the associated idea of an _m-function_ , introduced by Turing on p. 63. _m_ -functions are subroutines with _parameters_ —values that are plugged into the subroutine before it is used.
The example of the 'find' subroutine **f** makes this idea clear. The subroutine **f** ( _A, B, x_ ) is defined in Section 3 (Tables 2 and 3). Recall that f( _A, B, x_ ) finds the leftmost _x_ on the tape and places the machine in _A_ , leaving the scanner resting on the _x_ ; or if no _x_ is found, places the machine in _B_ and leaves the scanner resting on a blank square to the right of the used portion of the tape. ' _A_ ', ' _B_ ', and ' _x_ ' are the parameters of the subroutine. Parameter ' _x_ ' may be replaced by any symbol (of the Turing machine in question). Parameters ' _A_ ' and ' _B_ ' may be replaced by names of states of the machine. Alternatively, Turing permits ' _A_ ' and ' _B_ ' (one or both) to be replaced by a name of a subroutine. For example, replacing ' _A_ ' by the subroutine name ' **e** 1( _C_ )' produces:
**f** ( **e** 1( _C_ ), _B_ , _x_ )
This says: find the leftmost _x_ , let the scanner rest on it, and go into subroutine **e** 1( _C_ ); or, if there is no _x_ , go into _B_ (leaving the scanner resting on a blank square to the right of the used portion of the tape).
The subroutine **e** 1( _C_ ) simply erases the scanned square and places the machine in _C_ , leaving the scanner resting on the square that has just been erased. (' _C_ ' is another parameter of the same type as ' _A_ ' and ' _B_ '.) Thus the subroutine **f** ( **e** 1( _C_ ), _B_ , _x_ ) finds the leftmost occurrence of the symbol _x_ and erases it, placing the machine in _C_ and leaving the scanner resting on the square that has just been erased (or if no _x_ is found, leaves the scanner resting on a blank square to the right of the used portion of the tape and places the machine in _B_ ). Since in this case nothing turns on the choice of letter, the name of the subroutine may also be written ' **f** ( **e** 1( _A_ ), _B_ , _x_ )'.
The subroutine **f** ( **e** 1( _A_ ), _B_ , _x_ ) is one and the same as the subroutine **e** ( _A_ , _B_ , _x_ ) (Section 3). The new notation exhibits the structure of the subroutine.
More examples of _m_ -functions are given below. While the use of _m_ -functions is not strictly necessary for the description of any Turing machine, _m_ -functions are very useful in describing large or complex Turing machines. This is because of the possibilities they offer for generalization, reusability, simplification, and modularization. Generalization is achieved because tasks of a similar nature can be done by a single _m_ -function, and modularization because a complex task can be divided into several simpler _m_ -functions. Simplification is obtained because the language of _m_ -functions submerges some of the detail of the language of instruction-words—i.e. words of the form qiSjSkMq1—so producing transparent descriptions of Turing machines. Reusability arises simply because we can employ the same _m_ -function in different Turing machines.
Although it is difficult (if not impossible) to indicate the exact role that Turing's concept of an _m_ -function played in the development of today's programming languages, it is worth emphasizing that some characteristics of _m_ -functions are present in the subroutines of almost all modern languages. Full use was made of the idea of parametrized subroutines by Turing and his group at the National Physical Laboratory as they pioneered the science of computer programming during 1946. A contemporary report (by Huskey) outlining Turing's approach to programming said the following:
The fact that repetition of subroutines require[s] large numbers of orders has led to the abbreviated code methods whereby not only standard orders are used but special words containing parameters are converted into orders by an interpretation table. The general idea is that these describe the entries to subroutines, the values of certain parameters in the subroutine, how many times the subroutine is to be used, and where to go after the subroutine is finished.
Rather than give a formal definition of an _m_ -function we present a series of illustrative examples.
First, some preliminaries. An _alphabet_ A is some set of symbols, for example {-, 0, 1, 2}, and a _word_ of alphabet **A** is a finite sequence of non-blank symbols of **A**. The blank symbol, represented '-', is used to separate different words on the tape and is part of the alphabet, but never occurs within words. The following examples all assume that, at the start of operation, there is a single word _w_ of the alphabet on an otherwise blank tape, with the scanner positioned over any symbol of _w_. The symbols of _w_ are written on adjacent squares, using both E-squares and F-squares, and _w_ is surrounded by blanks (some of the examples require there to be at least one blank in front of _w_ and at least three following _w_ ).
Let **M** be a Turing machine with alphabet **A** = {-, 0, 1, 2}. The following instructions result in **M** printing the symbol '1' at the end of _w_ , replacing the first blank to the right of _w_ :
**q** 100R **q** 1, **q** 111R **q** 1, **q** 122R **q** 1, **q** 1-1N **q** 2
The first three instructions move the scanner past the symbols '0', '1', and '2', and once the scanner arrives at the first blank square to the right of _w_ , the fourth instruction prints '1' (leaving **M** in state **q** 2).
If the symbols '3', '4',..., '9' are added to the alphabet, so A = {-, 0, 1, :::,9}, then the necessary instructions for printing '1' at the end of _w_ are lengthier:
**q** 100R **q** 1, **q** 111R **q** 1,..., **q** 199R **q** 1, **q** 1-1N **q** 2
The _m_ -function **add(S** , α) defined by Table 4 carries out the task of printing one symbol 'α' at the end of any word _w_ of _any_ alphabet (assuming as before that the machine starts operating with the scanner positioned over one or another symbol of _w_ and that _w_ is surrounded by blanks).
Table 4 is the skeleton table for the _m_ -function **add(S** , α). (Skeleton tables are like tables of instructions but with some parameters to be replaced by concrete values.) Table 4 has two parameters, 'α' and ' **S** '. The second parameter ' **S** ' is to be replaced by the state or _m_ -function into which the machine is to go once **add** ( **S** , α) completes its operation, and the first parameter 'α' is to be replaced by whatever symbol it is that we wish to be printed at the end of the word.
Both sets of instruction-words shown above can now be replaced by a simple call to the _m_ -function **add** ( **S** , α), where **S** = **q** 2 and α = 1.
If instead of adding '1' at the end of a word from alphabet **A** = {-, 0, 1,..., 9}, we wanted to add a pair of symbols '5' and '4', then the instruction-words would be:
**q** 100R **q** 1, **q** 111R **q** 1,..., **q** 199R **q** 1, **q** 1-5R **q** 2, **q** 2-4N **q** 3
These instruction-words can be replaced by the _m_ -function **add** ( **add** ( **q** 3, 4), 5). This _m_ -function finds the end of the word and writes '5', going into _m_ -function **add** ( **q** 3, 4), which writes '4' and ends in state **q** 3.
Another example: suppose that '5' and '4' are to be printed as just described, and then each occurrence of the symbol '3' is to be replaced by '4'. The _m_ -function **add** ( **add** ( **change** ( **q** **n** , 3, 4), 4), 5) carries out the required task, where the _m_ -function **change** ( **S** , α, β) is defined by Table 5. The _m_ -function **change** 1( **S** , α, β) is a subroutine inside the _m_ -function **change** ( **S** , α, β).
_m_ -functions can employ _internal variables_. Although internal variables are not strictly necessary, they simplify an _m_ -function's description. Internal variables are not parameters of the _m_ -function—we do not need to replace them with concrete values before the _m_ -function is used. In the following example, the internal variable 'δ' refers to whatever symbol is present on the scanned square when the machine enters the _m_ -function **repeat** 1( **S** ).
Suppose we wish to print a repetition of the first symbol of _w_ at the end of _w_. This can be achieved by the _m_ -function **repeat** ( **S** ) defined by Table 5. (The _m_ -function **add** ( **S** , δ) is as given by Table 4.)
Every _m_ -function has the form: **name** ( **S** 1, **S** 2,..., **S** **n** , α1, α2,..., α _m_ ), where **S** 1, **S** 2,..., **S** **n** refer either to states or to _m_ -functions, and _α_ 1, _α_ 2,..., _α_ _m_ denote symbols. Each _m_ -function is a Turing machine with parameters. To convert an _m_ -function's skeleton table to a Turing-machine instruction table, where each row is an instruction-word of the form **q** iSjSkM **q** 1, it is necessary to know the context in which the _m_ -function is to be used, namely, the underlying Turing machine's alphabet and states. It is necessary to know the alphabet because of the use in skeleton tables of expressions such as 'does not contain !', 'not α', 'neither α nor -', 'any'. Knowledge of the underlying machine's states is necessary to ensure that the _m_ -function begins and ends in the correct state.
**Table 4**
**Table 5**
**Table 6**
The economy effected by _m_ -functions is illustrated by the fact that if the _m_ -functions are eliminated from Turing's description of his universal machine, nearly 4,000 instruction-words are required in their place.
### **CHAPTER 1
On Computable Numbers, with an Application to the Entscheidungsproblem ( _1936_ )**
_Alan Turing_
The "computable" numbers may be described briefly as the real numbers whose expressions as a decimal are calculable by finite means. Although the subject of this paper is ostensibly the computable _numbers_ , it is almost equally easy to define and investigate computable functions of an integral variable or a real or computable variable, computable predicates, and so forth. The fundamental problems involved are, however, the same in each case, and I have chosen the computable numbers for explicit treatment as involving the least cumbrous technique. I hope shortly to give an account of the relations of the computable numbers, functions, and so forth to one another. This will include a development of the theory of functions of a real variable expressed in terms of computable numbers. According to my definition, a number is computable if its decimal can be written down by a machine.
In 9, 10 I give some arguments with the intention of showing that the computable numbers include all numbers which could naturally be regarded as computable. In particular, I show that certain large classes of numbers are computable. They include, for instance, the real parts of all algebraic numbers, the real parts of the zeros of the Bessel functions, the numbers π, _e_ , etc. The computable numbers do not, however, include all definable numbers, and an example is given of a definable number which is not computable.
Although the class of computable numbers is so great, and in many ways similar to the class of real numbers, it is nevertheless enumerable. In 8 I examine certain arguments which would seem to prove the contrary. By the correct application of one of these arguments, conclusions are reached which are superficially similar to those of Gödel. These results have valuable applications. In particular, it is shown ( 11) that the Hilbertian Entscheidungsproblem can have no solution.
In a recent paper Alonzo Church has introduced an idea of "effective calculability", which is equivalent to my "computability", but is very differently defined. Church also reaches similar conclusions about the Entscheidungsproblem. The proof of equivalence between "computability" and "effective calculability" is outlined in an appendix to the present paper.
#### 1. Computing machines
We have said that the computable numbers are those whose decimals are calculable by finite means. This requires rather more explicit definition. No real attempt will be made to justify the definitions given until we reach 9. For the present I shall only say that the justification lies in the fact that the human memory is necessarily limited.
We may compare a man in the process of computing a real number to a machine which is only capable of a finite number of conditions _q_ 1, _q_ 2,..., _q R_ which will be called " _m_ -configurations". The machine is supplied with a "tape" (the analogue of paper) running through it, and divided into sections (called "squares") each capable of bearing a "symbol". At any moment there is just one square, say the _r_ -th, bearing the symbol ( _r_ ) which is "in the machine". We may call this square the "scanned square". The symbol on the scanned square may be called the "scanned symbol". The "scanned symbol" is the only one of which the machine is, so to speak, "directly aware". However, by altering its _m_ -configuration the machine can effectively remember some of the symbols which it has "seen" (scanned) previously. The possible behaviour of the machine at any moment is determined by the _m_ -configuration _q n_ and the scanned symbol ( _r_ ). This pair _q n_, ( _r_ ) will be called the "configuration": thus the configuration determines the possible behaviour of the machine. In some of the configurations in which the scanned square is blank ( _i.e_. bears no symbol) the machine writes down a new symbol on the scanned square: in other configurations it erases the scanned symbol. The machine may also change the square which is being scanned, but only by shifting it one place to right or left. In addition to any of these operations the _m_ -configuration may be changed. Some of the symbols written down will form the sequence of figures which is the decimal of the real number which is being computed. The others are just rough notes to "assist the memory". It will only be these rough notes which will be liable to erasure.
It is my contention that these operations include all those which are used in the computation of a number. The defence of this contention will be easier when the theory of the machines is familiar to the reader. In the next section I therefore proceed with the development of the theory and assume that it is understood what is meant by "machine", "tape", "scanned", etc.
#### 2. Definitions
##### _Automatic machines_
If at each stage the motion of a machine (in the sense of 1) is _completely_ determined by the configuration, we shall call the machine an "automatic machine" (or _a_ -machine).
For some purposes we might use machines (choice machines or _c_ -machines) whose motion is only partially determined by the configuration (hence the use of the word "possible" in 1). When such a machine reaches one of these ambiguous configurations, it cannot go on until some arbitrary choice has been made by an external operator. This would be the case if we were using machines to deal with axiomatic systems. In this paper I deal only with automatic machines, and will therefore often omit the prefix _a_ -.
##### _Computing machines_
If an _a_ -machine prints two kinds of symbols, of which the first kind (called figures) consists entirely of 0 and 1 (the others being called symbols of the second kind), then the machine will be called a computing machine. If the machine is supplied with a blank tape and set in motion, starting from the correct initial _m_ -configuration, the subsequence of the symbols printed by it which are of the first kind will be called the _sequence computed by the machine_. The real number whose expression as a binary decimal is obtained by prefacing this sequence by a decimal point is called the _number computed by the machine_.
At any stage of the motion of the machine, the number of the scanned square, the complete sequence of all symbols on the tape, and the _m_ -configuration will be said to describe the _complete configuration_ at that stage. The changes of the machine and tape between successive complete configurations will be called the _moves_ of the machine.
##### _Circular and circle-free machines_
If a computing machine never writes down more than a finite number of symbols of the first kind, it will be called _circular_. Otherwise it is said to be _circle-free_.
A machine will be circular if it reaches a configuration from which there is no possible move, or if it goes on moving, and possibly printing symbols of the second kind, but cannot print any more symbols of the first kind. The significance of the term "circular" will be explained in 8.
##### _Computable sequences and numbers_
A sequence is said to be computable if it can be computed by a circle-free machine. A number is computable if it differs by an integer from the number computed by a circle-free machine.
We shall avoid confusion by speaking more often of computable sequences than of computable numbers.
#### 3. Examples of computing machines
I. A machine can be constructed to compute the sequence 010101.... The machine is to have the four _m_ -configurations and is capable of printing "0" and "1". The behaviour of the machine is described in the following table in which " _R_ " means "the machine moves so that it scans the square immediately on the right of the one it was scanning previously". Similarly for " _L_ ". " _E_ " means "the scanned symbol is erased" and " _P_ " stands for "prints". This table (and all succeeding tables of the same kind) is to be understood to mean that for a configuration described in the first two columns the operations in the third column are carried out successively, and the machine then goes over into the _m_ -configuration described in the last column. When the second column is left blank, it is understood that the behaviour of the third and fourth columns applies for any symbol and for no symbol. The machine starts in the _m_ -configuration with a blank tape.
If (contrary to the description in 1) we allow the letters _L_ , _R_ to appear more than once in the operations column we can simplify the table considerably.
II. As a slightly more difficult example we can construct a machine to compute the sequence 001011011101111011111.... The machine is to be capable of five _m_ -configurations, viz. and of printing The first three symbols on the tape will be "әә0"; the other figures follow on alternate squares. On the intermediate squares we never print anything but " _x_ ". These letters serve to "keep the place" for us and are erased when we have finished with them. We also arrange that in the sequence of figures on alternate squares there shall be no blanks.
To illustrate the working of this machine a table is given below of the first few complete configurations. These complete configurations are described by writing down the sequence of symbols which are on the tape, with the _m_ -configuration written below the scanned symbol. The successive complete configurations are separated by colons.
This table could also be written in the form
in which a space has been made on the left of the scanned symbol and the _m_ -configuration written in this space. This form is less easy to follow, but we shall make use of it later for theoretical purposes.
The convention of writing the figures only on alternate squares is very useful: I shall always make use of it. I shall call the one sequence of alternate squares _F_ -squares and the other sequence _E_ -squares. The symbols on _E_ -squares will be liable to erasure. The symbols on _F_ -squares form a continuous sequence. There are no blanks until the end is reached. There is no need to have more than one _E_ -square between each pair of _F_ -squares: an apparent need of more _E_ -squares can be satisfied by having a sufficiently rich variety of symbols capable of being printed on _E_ -squares. If a symbol **β** is on an _F_ -square _S_ and a symbol **α** is on the _E_ -square next on the right of _S_ , then _S_ and **β** will be said to be _marked_ with **α**. The process of printing this **α** will be called marking **β** (or _S_ ) with **α**.
#### 4. Abbreviated tables
There are certain types of process used by nearly all machines, and these, in some machines, are used in many connections. These processes include copying down sequences of symbols, comparing sequences, erasing all symbols of a given form, etc. Where such processes are concerned we can abbreviate the tables for the _m_ -configurations considerably by the use of "skeleton tables". In skeleton tables there appear capital German letters and small Greek letters. These are of the nature of "variables". By replacing each capital German letter throughout by an _m_ -configuration and each small Greek letter by a symbol, we obtain the table for an _m_ -configuration.
The skeleton tables are to be regarded as nothing but abbreviations: they are not essential. So long as the reader understands how to obtain the complete tables from the skeleton tables, there is no need to give any exact definitions in this connection.
Let us consider an example:
If we were to replace throughout by (say), by _r_ , and **α** by _x_ , we should have a complete table for the _m_ -configuration . is called an " _m_ -configuration function" or " _m_ -function".
The only expressions which are admissible for substitution in an _m_ -function are the _m_ -configurations and symbols of the machine. These have to be enumerated more or less explicitly: they may include expressions such as ( _e_ , _x_ ); indeed they must if there are any _m_ -functions used at all. If we did not insist on this explicit enumeration, but simply stated that the machine had certain _m_ -configurations (enumerated) and all _m_ -configurations obtainable by substitution of _m_ -configurations in certain _m_ -functions, we should usually get an infinity of _m_ -configurations; _e.g_., we might say that the machine was to have the _m_ -configuration and all _m_ -configurations obtainable by substituting an _m_ -configuration for in ( ). Then it would have ... as _m_ -configurations.
Our interpretation rule then is this. We are given the names of the _m_ -configurations of the machine, mostly expressed in terms of _m_ -functions. We are also given skeleton tables. All we want is the complete table for the _m_ -configurations of the machine. This is obtained by repeated substitution in the skeleton tables.
##### _Further examples_
(In the explanations the symbol "→" is used to signify "the machine goes into the _m_ -configuration....")
The last example seems somewhat more difficult to interpret than most. Let us suppose that in the list of _m_ -configurations of some machine there appears _e_ , ( _x_ ) (= , say). The table is
Or, in greater detail:
In this we could replace _e_ 1( , , _x_ ) by and then give the table for (with the right substitutions) and eventually reach a table in which no _m_ -functions appeared.
The last line stands for the totality of lines obtainable from it by replacing **β** by any symbol which may occur on the tape of the machine concerned.
The first symbol marked **α** and the first marked **β** are compared. If there is neither **α** nor **β** , → . If there are both and the symbols are alike, → . Otherwise →
( , , , **α** , **β** ) differs from ( , , , **α** , **β** ) in that in the case when there is similarity the first **α** and **β** are erased.
( , , **α** , **β** ). The sequence of symbols marked **α** is compared with the sequence marked **β**. → if they are similar. Otherwise → . Some of the symbols **α** and **β** are erased.
#### 5. Enumeration of computable sequences
A computable sequence **γ** is determined by a description of a machine which computes **γ**. Thus the sequence 001011011101111... is determined by the table on p. [62], and, in fact, any computable sequence is capable of being described in terms of such a table.
It will be useful to put these tables into a kind of standard form. In the first place let us suppose that the table is given in the same form as the first table, for example, I on p. [61]. That is to say, that the entry in the operations column is always of one of the forms _E_ : _E_ , _R_ : _E_ , _L_ : _P_ **α** : _P_ **α** , _R_ : _P_ **α** , _L_ : _R_ : _L_ : or no entry at all. The table can always be put into this form by introducing more _m_ -configurations. Now let us give numbers to the _m_ -configurations, calling them _q_ 1,..., _q R_, as in 1. The initial _m_ -configuration is always to be called _q_ 1. We also give numbers to the symbols _S_ 1,..., _S m_ and, in particular, blank = _S_ 0, 0 = _S_ 1, 1 = _S_ 2. The lines of the table are now of form
Lines such as
are to be written as
and lines such as
to be written as
In this way we reduce each line of the table to a line of one of the forms ( _N_ 1), ( _N_ 2), ( _N_ 3).
From each line of form ( _N_ 1) let us form an expression _q iSjSkLqm_; from each line of form ( _N_ 2) we form an expression _q iSjSkRqm_; and from each line of form ( _N_ 3) we form an expression _q iSjSkNqm_.
Let us write down all expressions so formed from the table for the machine and separate them by semi-colons. In this way we obtain a complete description of the machine. In this description we shall replace _q i_ by the letter " _D_ " followed by the letter " _A_ " repeated _i_ times, and _S j_ by " _D_ " followed by " _C_ " repeated _j_ times. This new description of the machine may be called the _standard description_ (S.D). It is made up entirely from the letters " _A_ ", " _C_ ", " _D_ ", " _L_ ", " _R_ ", " _N_ ", and from ";".
If finally we replace " _A_ " by "1", " _C_ " by "2", " _D_ " by "3", " _L_ " by "4", " _R_ " by "5", " _N_ " by "6", and ";" by "7" we shall have a description of the machine in the form of an arabic numeral. The integer represented by this numeral may be called a _description number_ (D.N) of the machine. The D.N determine the S.D and the structure of the machine uniquely. The machine whose D.N is _n_ may be described as ( _n_ ).
To each computable sequence there corresponds at least one description number, while to no description number does there correspond more than one computable sequence. The computable sequences and numbers are therefore enumerable.
Let us find a description number for the machine I of 3. When we rename the _m_ -configurations its table becomes:
Other tables could be obtained by adding irrelevant lines such as
Our first standard form would be
The standard description is
_DADDCRDAA_ ; _DAADDRDAAA_ ; _DAAADDCCRDAAAA_ ; _DAAAADDRDA_ ;
A description number is
31332531173113353111731113322531111731111335317
and so is
3133253117311335311173111332253111173111133531731323253117
A number which is a description number of a circle-free machine will be called a _satisfactory_ number. In 8 it is shown that there can be no general process for determining whether a given number is satisfactory or not.
#### 6. The universal computing machine
It is possible to invent a single machine which can be used to compute any computable sequence. If this machine is supplied with a tape on the beginning of which is written the S.D of some computing machine , then will compute the same sequence as . In this section I explain in outline the behaviour of the machine. The next section is devoted to giving the complete table for .
Let us first suppose that we have a machine which will write down on the _F_ -squares the successive complete configurations of . These might be expressed in the same form as on p. [62], using the second description, (C), with all symbols on one line. Or, better, we could transform this description (as in 5) by replacing each _m_ -configuration by " _D_ " followed by " _A_ " repeated the appropriate number of times, and by replacing each symbol by " _D_ " followed by " _C_ " repeated the appropriate number of times. The numbers of letters " _A_ " and " _C_ " are to agree with the numbers chosen in 5, so that, in particular, "0" is replaced by " _DC_ ", "1" by " _DCC_ ", and the blanks by " _D_ ". These substitutions are to be made after the complete configurations have been put together, as in (C). Difficulties arise if we do the substitution first. In each complete configuration the blanks would all have to be replaced by " _D_ ", so that the complete configuration would not be expressed as a finite sequence of symbols.
If in the description of the machine II of 3 we replace by " _DAA_ ", "ә" by " _DCCC_ ", " " by " _DAAA_ ", then the sequence (C) becomes:
(This is the sequence of symbols on _F_ -squares.)
It is not difficult to see that if can be constructed, then so can . The manner of operation of could be made to depend on having the rules of operation ( _i.e_. the S.D) of written somewhere within itself ( _i.e_. within ); each step could be carried out by referring to these rules. We have only to regard the rules as being capable of being taken out and exchanged for others and we have something very akin to the universal machine.
One thing is lacking: at present the machine prints no figures. We may correct this by printing between each successive pair of complete configurations the figures which appear in the new configuration but not in the old. Then (C1) becomes
It is not altogether obvious that the _E_ -squares leave enough room for the necessary "rough work", but this is, in fact, the case.
The sequences of letters between the colons in expressions such as (C1) may be used as standard descriptions of the complete configurations. When the letters are replaced by figures, as in 5, we shall have a numerical description of the complete configuration, which may be called its description number.
#### 7. Detailed description of the universal machine
A table is given below of the behaviour of this universal machine. The _m_ -configurations of which the machine is capable are all those occurring in the first and last columns of the table, together with all those which occur when we write out the unabbreviated tables of those which appear in the table in the form of _m_ -functions. _E.g_., appears in the table and is an _m_ -function. Its unabbreviated table is (see p. [66])
Consequently 1 is an _m_ -configuration of .
When is ready to start work the tape running through it bears on it the symbol ә on an _F_ -square and again ә on the next _E_ -square; after this, on _F_ -squares only, comes the S.D of the machine followed by a double colon "::" (a single symbol, on an _F_ -square). The S.D consists of a number of instructions, separated by semi-colons.
Each instruction consists of five consecutive parts
(i) " _D_ " followed by a sequence of letters " _A_ ". This describes the relevant _m_ -configuration.
(ii) " _D_ " followed by a sequence of letters " _C_ ". This describes the scanned symbol.
(iii) " _D_ " followed by another sequence of letters " _C_ ". This describes the symbol into which the scanned symbol is to be changed.
(iv) " _L_ ", " _R_ ", or " _N_ ", describing whether the machine is to move to left, right, or not at all.
(v) " _D_ " followed by a sequence of letters " _A_ ". This describes the final _m_ -configuration.
The machine is to be capable of printing " _A_ ", " _C_ ", " _D_ ", "0", "1", " _u_ ", " _v_ ", " _w_ ", " _x_ ", " _y_ ", " _z_ ". The S.D is formed from ";", " _A_ ", " _C_ ", " _D_ ", " _L_ ", " _R_ ", " _N_ ".
##### _Subsidiary skeleton table_
##### _The table for_
. Taking the long view, the last instruction relevant to the last configuration is found. It can be recognised afterwards as the instruction following the last semi-colon marked _z_. → .
#### 8. Application of the diagonal process
It may be thought that arguments which prove that the real numbers are not enumerable would also prove that the computable numbers and sequences cannot be enumerable. It might, for instance, be thought that the limit of a sequence of computable numbers must be computable. This is clearly only true if the sequence of computable numbers is defined by some rule.
Or we might apply the diagonal process. "If the computable sequences are enumerable, let **α** _n_ be the _n_ -th computable sequence, and let _ϕ_ _n_ ( _m_ ) be the _m_ -th figure in **α** _n_. Let **β** be the sequence with 1 − _ϕ n_( _n_ ) as its _n_ -th figure. Since **β** is computable, there exists a number _K_ such that 1 – _ϕ n_( _n_ ) = _ϕ K_ ( _n_ ) all _n_. Putting _n_ = _K_ , we have 1 = 2 _ϕ K_ ( _K_ ), _i.e_. 1 is even. This is impossible. The computable sequences are therefore not enumerable."
The fallacy in this argument lies in the assumption that **β** is computable. It would be true if we could enumerate the computable sequences by finite means, but the problem of enumerating computable sequences is equivalent to the problem of finding out whether a given number is the D.N of a circle-free machine, and we have no general process for doing this in a finite number of steps. In fact, by applying the diagonal process argument correctly, we can show that there cannot be any such general process.
The simplest and most direct proof of this is by showing that, if this general process exists, then there is a machine which computes **β**. This proof, although perfectly sound, has the disadvantage that it may leave the reader with a feeling that "there must be something wrong". The proof which I shall give has not this disadvantage, and gives a certain insight into the significance of the idea "circle-free". It depends not on constructing **β** , but on constructing **β** ′, whose _n_ -th figure is _ϕ n_( _n_ ).
Let us suppose that there is such a process; that is to say, that we can invent a machine which, when supplied with the S.D of any computing machine will test this S.D and if is circular will mark the S.D with the symbol " _u_ " and if it is circle-free will mark it with " _s_ ". By combining the machines , and we could construct a machine to compute the sequence **β** ′. The machine may require a tape. We may suppose that it uses the _E_ -squares beyond all symbols on _F_ -squares, and that when it has reached its verdict all the rough work done by is erased.
The machine has its motion divided into sections. In the first _N_ − 1 sections, among other things, the integers 1, 2,..., _N_ − 1 have been written down and tested by the machine . A certain number, say _R_ ( _N_ − 1), of them have been found to be the D.N's of circle-free machines. In the _N_ -th section the machine tests the number _N_. If _N_ is satisfactory, _i.e_., if it is the D.N of a circle-free machine, then _R_ ( _N_ ) = 1 + _R_ ( _N_ − 1) and the first _R_ ( _N_ ) figures of the sequence of which a D.N is _N_ are calculated. The _R_ ( _N_ )-th figure of this sequence is written down as one of the figures of the sequence **β** ′ computed by . If _N_ is not satisfactory, then _R_ ( _N_ ) = _R_ ( _N_ − 1) and the machine goes on to the ( _N_ \+ 1)-th section of its motion.
From the construction of we can see that is circle-free. Each section of the motion of comes to an end after a finite number of steps. For, by our assumption about , the decision as to whether _N_ is satisfactory is reached in a finite number of steps. If _N_ is not satisfactory, then the _N_ -th section is finished. If _N_ is satisfactory, this means that the machine ( _N_ ) whose D.N is _N_ is circle-free, and therefore its _R_ ( _N_ )-th figure can be calculated in a finite number of steps. When this figure has been calculated and written down as the _R_ ( _N_ )-th figure of **β** ′, the _N_ -th section is finished. Hence is circle-free.
Now let _K_ be the D.N of . What does do in the _K_ -th section of its motion? It must test whether _K_ is satisfactory, giving a verdict " _s_ " or " _u_ ". Since _K_ is the D.N of and since is circle-free, the verdict cannot be " _u_ ". On the other hand the verdict cannot be " _s_ ". For if it were, then in the _K_ -th section of its motion would be bound to compute the first _R_ ( _K_ − 1) + 1 = _R_ ( _K_ ) figures of the sequence computed by the machine with _K_ as its D.N and to write down the _R_ ( _K_ )-th as a figure of the sequence computed by . The computation of the first _R_ ( _K_ ) − 1 figures would be carried out all right, but the instructions for calculating the _R_ ( _K_ )-th would amount to "calculate the first _R_ ( _K_ ) figures computed by and write down the _R_ ( _K_ )-th". This _R_ ( _K_ )-th figure would never be found. _I.e_., is circular, contrary both to what we have found in the last paragraph and to the verdict " _s_ ". Thus both verdicts are impossible and we conclude that there can be no machine .
We can show further that _there can be no machine_ _which, when supplied with the S.D of an arbitrary machine_ , _will determine whether_ _ever prints a given symbol_ (0 _say_ ).
We will first show that, if there is a machine , then there is a general process for determining whether a given machine prints 0 infinitely often. Let 1 be a machine which prints the same sequence as , except that in the position where the first 0 printed by stands, 1 prints . 2 is to have the first two symbols 0 replaced by , and so on. Thus, if were to print
_A B A_ 0 1 _A A B_ 0 0 1 0 _A B_...,
then 1 would print
_A B A_ 1 _A A B_ 0 0 1 0 _A B_...
and 2 would print
_A B A_ 1 _A A B_ 0 1 0 _A B_....
Now let , be a machine which, when supplied with the S.D of , will write down successively the S.D of , of 1, of 2,... (there is such a machine). We combine with and obtain a new machine, . In the motion of first is used to write down the S.D of , and then tests it,: 0: is written if it is found that never prints 0; then writes the S.D of 1, and this is tested,: 0: being printed if and only if 1 never prints 0, and so on. Now let us test with . If it is found that never prints 0, then prints 0 infinitely often; if prints 0 sometimes, then does not print 0 infinitely often.
Similarly there is a general process for determining whether prints 1 infinitely often. By a combination of these processes we have a process for determining whether prints an infinity of figures, _i.e_. we have a process for determining whether is circle-free. There can therefore be no machine .
The expression "there is a general process for determining..." has been used throughout this section as equivalent to "there is a machine which will determine...". This usage can be justified if and only if we can justify our definition of "computable". For each of these "general process" problems can be expressed as a problem concerning a general process for determining whether a given integer _n_ has a property _G_ ( _n_ )[ _e.g. G_ ( _n_ ) might mean " _n_ is satisfactory" or " _n_ is the Gödel representation of a provable formula"], and this is equivalent to computing a number whose _n_ -th figure is 1 if _G_ ( _n_ ) is true and 0 if it is false.
#### 9. The extent of the computable numbers
No attempt has yet been made to show that the "computable" numbers include all numbers which would naturally be regarded as computable. All arguments which can be given are bound to be, fundamentally, appeals to intuition, and for this reason rather unsatisfactory mathematically. The real question at issue is "What are the possible processes which can be carried out in computing a number?"
The arguments which I shall use are of three kinds.
( _a_ ) A direct appeal to intuition.
( _b_ ) A proof of the equivalence of two definitions (in case the new definition has a greater intuitive appeal).
( _c_ ) Giving examples of large classes of numbers which are computable.
Once it is granted that computable numbers are all "computable", several other propositions of the same character follow. In particular, it follows that, if there is a general process for determining whether a formula of the Hilbert function calculus is provable, then the determination can be carried out by a machine.
I. [Type ( _a_ )]. This argument is only an elaboration of the ideas of 1. Computing is normally done by writing certain symbols on paper. We may suppose this paper is divided into squares like a child's arithmetic book. In elementary arithmetic the two-dimensional character of the paper is sometimes used. But such a use is always avoidable, and I think that it will be agreed that the two-dimensional character of paper is no essential of computation. I assume then that the computation is carried out on one-dimensional paper, _i.e_. on a tape divided into squares. I shall also suppose that the number of symbols which may be printed is finite. If we were to allow an infinity of symbols, then there would be symbols differing to an arbitrarily small extent. The effect of this restriction of the number of symbols is not very serious. It is always possible to use sequences of symbols in the place of single symbols. Thus an Arabic numeral such as 17 or 999999999999999 is normally treated as a single symbol. Similarly in any European language words are treated as single symbols (Chinese, however, attempts to have an enumerable infinity of symbols). The differences from our point of view between the single and compound symbols is that the compound symbols, if they are too lengthy, cannot be observed at one glance. This is in accordance with experience. We cannot tell at a glance whether 9999999999999999 and 999999999999999 are the same.
The behaviour of the computer at any moment is determined by the symbols which he is observing, and his "state of mind" at that moment. We may suppose that there is a bound _B_ to the number of symbols or squares which the computer can observe at one moment. If he wishes to observe more, he must use successive observations. We will also suppose that the number of states of mind which need be taken into account is finite. The reasons for this are of the same character as those which restrict the number of symbols. If we admitted an infinity of states of mind, some of them will be "arbitrarily close" and will be confused. Again, the restriction is not one which seriously affects computation, since the use of more complicated states of mind can be avoided by writing more symbols on the tape.
Let us imagine the operations performed by the computer to be split up into "simple operations" which are so elementary that it is not easy to imagine them further divided. Every such operation consists of some change of the physical system consisting of the computer and his tape. We know the state of the system if we know the sequence of symbols on the tape, which of these are observed by the computer (possibly with a special order), and the state of mind of the computer. We may suppose that in a simple operation not more than one symbol is altered. Any other changes can be split up into simple changes of this kind. The situation in regard to the squares whose symbols may be altered in this way is the same as in regard to the observed squares. We may, therefore, without loss of generality, assume that the squares whose symbols are changed are always "observed" squares.
Besides these changes of symbols, the simple operations must include changes of distribution of observed squares. The new observed squares must be immediately recognisable by the computer. I think it is reasonable to suppose that they can only be squares whose distance from the closest of the immediately previously observed squares does not exceed a certain fixed amount. Let us say that each of the new observed squares is within _L_ squares of an immediately previously observed square.
In connection with "immediate recognisability", it may be thought that there are other kinds of square which are immediately recognisable. In particular, squares marked by special symbols might be taken as immediately recognisable. Now if these squares are marked only by single symbols there can be only a finite number of them, and we should not upset our theory by adjoining these marked squares to the observed squares. If, on the other hand, they are marked by a sequence of symbols, we cannot regard the process of recognition as a simple process. This is a fundamental point and should be illustrated. In most mathematical papers the equations and theorems are numbered. Normally the numbers do not go beyond (say) 1000. It is, therefore, possible to recognise a theorem at a glance by its number. But if the paper was very long, we might reach Theorem 157767733443477; then, further on in the paper, we might find "... hence (applying Theorem 157767733443477) we have...". In order to make sure which was the relevant theorem we should have to compare the two numbers figure by figure, possibly ticking the figures off in pencil to make sure of their not being counted twice. If in spite of this it is still thought that there are other "immediately recognisable" squares, it does not upset my contention so long as these squares can be found by some process of which my type of machine is capable. This idea is developed in III below.
The simple operations must therefore include:
( _a_ ) Changes of the symbol on one of the observed squares.
( _b_ ) Changes of one of the squares observed to another square within _L_ squares of one of the previously observed squares.
It may be that some of these changes necessarily involve a change of state of mind. The most general single operation must therefore be taken to be one of the following:
( _A_ ) A possible change ( _a_ ) of symbol together with a possible change of state of mind.
( _B_ ) A possible change ( _b_ ) of observed squares, together with a possible change of state of mind.
The operation actually performed is determined, as has been suggested on p. [75], by the state of mind of the computer and the observed symbols. In particular, they determine the state of mind of the computer after the operation is carried out.
We may now construct a machine to do the work of this computer. To each state of mind of the computer corresponds an " _m_ -configuration" of the machine. The machine scans _B_ squares corresponding to the _B_ squares observed by the computer. In any move the machine can change a symbol on a scanned square or can change any one of the scanned squares to another square distant not more than _L_ squares from one of the other scanned squares. The move which is done, and the succeeding configuration, are determined by the scanned symbol and the _m_ -configuration. The machines just described do not differ very essentially from computing machines as defined in 2, and corresponding to any machine of this type a computing machine can be constructed to compute the same sequence, that is to say the sequence computed by the computer.
II. [Type ( _b_ )].
If the notation of the Hilbert functional calculus is modified so as to be systematic, and so as to involve only a finite number of symbols, it becomes possible to construct an automatic machine , which will find all the provable formulae of the calculus.
Now let **α** be a sequence, and let us denote by _G_ **α** ( _x_ ) the proposition "The _x_ -th figure of **α** is 1", so that − _G_ **α** ( _x_ ) means "The _x_ -th figure of **α** is 0". Suppose further that we can find a set of properties which define the sequence a and which can be expressed in terms of _G_ α( _x_ ) and of the propositional functions _N_ ( _x_ ) meaning " _x_ is a non-negative integer" and _F_ ( _x, y_ ) meaning " _y_ = _x_ \+ 1". When we join all these formulae together conjunctively, we shall have a formula, say, which defines **α**. The terms of must include the necessary parts of the Peano axioms, viz.,
which we will abbreviate to _P_.
When we say " defines **α** ", we mean that − is not a provable formula, and also that, for each _n_ , one of the following formulae (A _n_ ) or (B _n_ ) is provable.
where _F_ ( _n_ ) stands for _F_ ( _u_ , _u_ ′) & _F_ ( _u_ ′, _u_ ′′) &... _F_ ( _u_ ( _n_ − 1), _u_ ( _n_ )).
I say that **α** is then a computable sequence: a machine α to compute **α** can be obtained by a fairly simple modification of .
We divide the motion of **α** into sections. The _n_ -th section is devoted to finding the _n_ -th figure of **α**. After the ( _n_ − 1)-th section is finished a double colon:: is printed after all the symbols, and the succeeding work is done wholly on the squares to the right of this double colon. The first step is to write the letter " _A_ " followed by the formula (A _n_ ) and then " _B_ " followed by (B _n_ ). The machine **α** then starts to do the work of , but whenever a provable formula is found, this formula is compared with (A _n_ ) and with (B _n_ ). If it is the same formula as (A _n_ ), then the figure "1" is printed, and the _n_ -th section is finished. If it is (B _n_ ), then "0" is printed and the section is finished. If it is different from both, then the work of is continued from the point at which it had been abandoned. Sooner or later one of the formulae (A _n_ ) or (B _n_ ) is reached; this follows from our hypotheses about a and , and the known nature of . Hence the _n_ -th section will eventually be finished. **α** is circle-free; **α** is computable.
It can also be shown that the numbers **α** definable in this way by the use of axioms include all the computable numbers. This is done by describing computing machines in terms of the function calculus.
It must be remembered that we have attached rather a special meaning to the phrase " defines **α** ". The computable numbers do not include all (in the ordinary sense) definable numbers. Let δ be a sequence whose _n_ -th figure is 1 or 0 according as _n_ is or is not satisfactory. It is an immediate consequence of the theorem of 8 that δ is not computable. It is (so far as we know at present) possible that any assigned number of figures of δ can be calculated, but not by a uniform process. When sufficiently many figures of δ have been calculated, an essentially new method is necessary in order to obtain more figures.
III. This may be regarded as a modification of I or as a corollary of II.
We suppose, as in I, that the computation is carried out on a tape; but we avoid introducing the "state of mind" by considering a more physical and definite counterpart of it. It is always possible for the computer to break off from his work, to go away and forget all about it, and later to come back and go on with it. If he does this he must leave a note of instructions (written in some standard form) explaining how the work is to be continued. This note is the counterpart of the "state of mind". We will suppose that the computer works in such a desultory manner that he never does more than one step at a sitting. The note of instructions must enable him to carry out one step and write the next note. Thus the state of progress of the computation at any stage is completely determined by the note of instructions and the symbols on the tape. That is, the state of the system may be described by a single expression (sequence of symbols), consisting of the symbols on the tape followed by (which we suppose not to appear elsewhere) and then by the note of instructions. This expression may be called the "state formula". We know that the state formula at any given stage is determined by the state formula before the last step was made, and we assume that the relation of these two formulae is expressible in the functional calculus. In other words, we assume that there is an axiom which expresses the rules governing the behaviour of the computer, in terms of the relation of the state formula at any stage to the state formula at the preceding stage. If this is so, we can construct a machine to write down the successive state formulae, and hence to compute the required number.
#### 10. Examples of large classes of numbers which are computable
It will be useful to begin with definitions of a computable function of an integral variable and of a computable variable, etc. There are many equivalent ways of defining a computable function of an integral variable. The simplest is, possibly, as follows. If **γ** is a computable sequence in which 0 appears infinitely often, and _n_ is an integer, then let us define ξ( **γ** , _n_ ) to be the number of figures 1 between the _n_ -th and the ( _n_ \+ 1)-th figure 0 in **γ**. Then _ϕ_ ( _n_ ) is computable if, for all _n_ and some **γ** , _ϕ_ ( _n_ ) = ξ( **γ** , _n_ ). An equivalent definition is this. Let _H_ ( _x_ , _y_ ) mean _ϕ_ ( _x_ ) = _y_. Then, if we can find a contradiction-free axiom _ϕ_ , such that _ϕ_ → _P_ , and if for each integer _n_ there exists an integer _N_ , such that
and such that, if _m_ _ϕ_ ( _n_ ), then, for some _N_ ′,
then _ϕ_ may be said to be a computable function.
We cannot define general computable functions of a real variable, since there is no general method of describing a real number, but we can define a computable function of a computable variable. If _n_ is satisfactory, let **γ** _n_ be the number computed by ( _n_ ), and let
unless **γ** _n_ = 0 or **γ** _n_ = 1, in either of which cases **α** _n_ = 0. Then, as _n_ runs through the satisfactory numbers, **α** _n_ runs through the computable numbers. Now let _ϕ_ ( _n_ ) be a computable function which can be shown to be such that for any satisfactory argument its value is satisfactory. Then the function _f_ , defined by _f_ ( **α** _n_ ) = **α** _ϕ(n)_ , is a computable function and all computable functions of a computable variable are expressible in this form.
Similar definitions may be given of computable functions of several variables, computable-valued functions of an integral variable, etc.
I shall enunciate a number of theorems about computability, but I shall prove only (ii) and a theorem similar to (iii).
(i) A computable function of a computable function of an integral or computable variable is computable.
(ii) Any function of an integral variable defined recursively in terms of computable functions is computable. _I.e_. if _ϕ_ ( _m_ , _n_ ) is computable, and _r_ is some integer, then _η_ ( _n_ ) is computable, where
(iii) If _ϕ_ ( _m_ , _n_ ) is a computable function of two integral variables, then _ϕ_ ( _n_ , _n_ ) is a computable function of _n_.
(iv) If _ϕ_ ( _n_ ) is a computable function whose value is always 0 or 1, then the sequence whose _n_ -th figure is _ϕ_ ( _n_ ) is computable.
Dedekind's theorem does not hold in the ordinary form if we replace "real" throughout by "computable". But it holds in the following form:
(v) If _G_ ( **α** ) is a propositional function of the computable numbers and
and there is a general process for determining the truth value of _G_ (α), then there is a computable number such that
In other words, the theorem holds for any section of the computables such that there is a general process for determining to which class a given number belongs.
Owing to this restriction of Dedekind's theorem, we cannot say that a computable bounded increasing sequence of computable numbers has a computable limit. This may possibly be understood by considering a sequence such as
On the other hand, (v) enables us to prove
(vi) If **α** and **β** are computable and **α** < **β** and _ϕ_ ( **α** ) < 0 < _ϕ_ ( **β** ), where _ϕ_ ( **α** ) is a computable increasing continuous function, then there is a unique computable number **γ** , satisfying **α** < **γ** < **β** and _ϕ_ ( **γ** ) = 0.
##### _Computable convergence_
We shall say that a sequence **β** _n_ of computable numbers _converges computably_ if there is a computable integral valued function _N_ ( _ε_ ) of the computable variable _ε_ , such that we can show that, if _ε_ > 0 and _n_ > _N ε_ ) and _m_ > _N ε_ ), then .
We can then show that
(vii) A power series whose coefficients form a computable sequence of computable numbers is computably convergent at all computable points in the interior of its interval of convergence.
(viii) The limit of a computably convergent sequence is computable.
And with the obvious definition of "uniformly computably convergent":
(ix) The limit of a uniformly computably convergent computable sequence of computable functions is a computable function. Hence
(x) The sum of a power series whose coefficients form a computable sequence is a computable function in the interior of its interval of convergence.
From (viii) and we deduce that p is computable.
From we deduce that **π** is computable.
From (vi) we deduce that all real algebraic numbers are computable.
From (vi) and (x) we deduce that the real zeros of the Bessel functions are computable.
##### _Proof of (ii)_
Let _H_ ( _x_ , _y_ ) mean " _η_ ( _x_ ) = _y_ ", and let _K_ ( _x_ , _y_ , _z_ ) mean " _ϕ_ ( _x_ , _y_ ) = _z_ ". _ϕ_ is the axiom for _ϕ_ ( _x_ , _y_ ). We take _η_ to be
I shall not give the proof of consistency of _η_. Such a proof may be constructed by the methods used in Hilbert and Bernays, _Grundlagen der Mathematik_ (Berlin, 1934), p. 209 _et seq_. The consistency is also clear from the meaning.
Suppose that, for some _n, N_ , we have shown
then, for some M,
and
Hence for each _n_ some formula of the form
is provable. Also, if _M_ ′ ≥ _M_ and _M_ ′ ≥ _m_ and _m_ _η_ ( _u_ ), then
and
The conditions of our second definition of a computable function are therefore satisfied. Consequently _η_ is a computable function.
##### _Proof of a modified form of (iii)_
Suppose that we are given a machine , which, starting with a tape bearing on it **ә ә** followed by a sequence of any number of letters " _F_ " on _F_ -squares and in the _m_ -configuration _b_ , will compute a sequence **γ** _n_ depending on the number _n_ of letters " _F_ ". If _ϕ n_( _m_ ) is the _m_ -th figure of **γ** _n_ , then the sequence **β** whose _n_ -th figure is _ϕ n_( _n_ ) is computable.
We suppose that the table for has been written out in such a way that in each line only one operation appears in the operations column. We also suppose that **Ξ** , **Θ** , , and do not occur in the table, and we replace **ә** throughout by **Θ** , 0 by , and 1 by . Further substitutions are then made. Any line of form
we replace by
and any line of the form
and we add to the table the following lines:
and similar lines with for and 1 for 0 together with the following line
We then have the table for the machine ′ which computes **β**. The initial _m_ -configuration is _c_ , and the initial scanned symbol is the second **ә**.
#### 11. Application to the Entscheidungsproblem
The results of 8 have some important applications. In particular, they can be used to show that the Hilbert Entscheidungsproblem can have no solution. For the present I shall confine myself to proving this particular theorem. For the formulation of this problem I must refer the reader to Hilbert and Ackermann's _Grundzüge der Theoretischen Logik_ (Berlin, 1931), chapter 3.
I propose, therefore, to show that there can be no general process for determining whether a given formula of the functional calculus **K** is provable, _i.e_. that there can be no machine which, supplied with any one of these formulae, will eventually say whether is provable.
It should perhaps be remarked that what I shall prove is quite different from the well-known results of Gödel. Gödel has shown that (in the formalism of Principia Mathematica) there are propositions such that neither nor − is provable. As a consequence of this, it is shown that no proof of consistency of Principia Mathematica (or of **K** ) can be given within that formalism. On the other hand, I shall show that there is no general method which tells whether a given formula is provable in **K** , or, what comes to the same, whether the system consisting of **K** with − adjoined as an extra axiom is consistent.
If the negation of what Gödel has shown had been proved, _i.e_. if, for each , either or − is provable, then we should have an immediate solution of the Entscheidungsproblem. For we can invent a machine which will prove consecutively all provable formulae. Sooner or later will reach either or − . If it reaches , then we know that is provable. If it reaches − , then, since **K** is consistent (Hilbert and Ackermann, p. 65), we know that is not provable.
Owing to the absence of integers in **K** the proofs appear somewhat lengthy. The underlying ideas are quite straightforward.
Corresponding to each computing machine we construct a formula Un ( ) and we show that, if there is a general method for determining whether Un ( ) is provable, then there is a general method for determining whether ever prints 0.
The interpretations of the propositional functions involved are as follows:
_R Sl_ ( _x_ , _y_ ) is to be interpreted as "in the complete configuration _x_ (of ) the symbol on the square _y_ is _S_ ".
_I_ ( _x_ , _y_ ) is to be interpreted as "in the complete configuration _x_ the square _y_ is scanned".
_K qm_( _x_ ) is to be interpreted as "in the complete configuration _x_ the _m_ -configuration is _q m_.
_F_ ( _x_ , _y_ ) is to be interpreted as " _y_ is the immediate successor of _x_ ".
Inst { _q iSjSkLql_} is to be an abbreviation for
Inst { _q iSjSkRql_} and Inst { _q iSjSkNql_}
are to be abbreviations for other similarly constructed expressions.
Let us put the description of into the first standard form of 6. This description consists of a number of expressions such as " _q iSjSkLql_" (or with _R_ or _N_ substituted for _L_ ). Let us form all the corresponding expressions such as Inst { _q iSjSkLql_} and take their logical sum. This we call Des ( ).
The formula Un ( ) is to be
[ _N_ ( _u_ )&... & Des ( )] may be abbreviated to _A_ ( ).
When we substitute the meanings suggested on [pp. 84–85] we find that Un ( ) has the interpretation "in some complete configuration of , _S_ 1 ( _i.e_. 0) appears on the tape". Corresponding to this I prove that
( _a_ ) If _S_ 1 appears on the tape in some complete configuration of , then Un ( ) is provable.
( _b_ ) If Un ( ) is provable, then _S_ 1 appears on the tape in some complete configuration of .
When this has been done, the remainder of the theorem is trivial.
LEMMA 1. _If S_ 1 _appears on the tape in some complete configuration of_ , _then_ Un ( ) _is provable_.
We have to show how to prove Un ( ). Let us suppose that in the _n_ -th complete configuration the sequence of symbols on the tape is _S r (n, 0)_, _S r(n, 1)_,..., _S r(n, n)_, followed by nothing but blanks, and that the scanned symbol is the _i_ ( _n_ )-th, and that the _m_ -configuration is _q k(n)_. Then we may form the proposition
which we may abbreviate to _CC n_.
As before, _F_ ( _u_ , _u_ ′) & _F_ ( _u_ ′, _u_ ′′) &... & _F_ ( _u_ ( _r_ −1), _u_ ( _r_ )) is abbreviated to _F_ ( _r_ ).
I shall show that all formulae of the form _A_ ( )& _F_ ( _n_ ) → _CC n_ (abbreviated to _CF n_) are provable. The meaning of _CF n_ is "The _n_ -th complete configuration of is so and so", where "so and so" stands for the actual _n_ -th complete configuration of . That _CF n_ should be provable is therefore to be expected.
_CF_ 0 is certainly provable, for in the complete configuration the symbols are all blanks, the _m_ -configuration is _q_ 1, and the scanned square is _u, i.e_. _CC_ 0 is
( _y_ ) _R S0_( _u_ , _y_ ) & _I_ ( _u_ , _u_ ) & _K q1_( _u_ ).
_A_ ( ) → _CC_ 0 is then trivial.
We next show that _CF n_ → _CF n+1_ is provable for each _n_. There are three cases to consider, according as in the move from the _n_ -th to the ( _n_ \+ 1)-th configuration the machine moves to left or to right or remains stationary. We suppose that the first case applies, _i.e_. the machine moves to the left. A similar argument applies in the other cases. If _r(n_ , _i_ ( _n_ )) = _a_ , _r(n_ \+ 1, _i_ ( _n_ \+ 1)) = _c_ , _k(i_ ( _n_ )) = _b_ , and _k(i_ ( _n_ \+ 1)) = _d_ , then Des ( ) must include Inst { _q aSbSdLqc_} as one of its terms, _i.e_.
Des ( ) → Inst { _q aSbSdLqc_}.
Hence
_A_ ( ) & _F_ ( _n_ +1) → Inst { _q aSbSdLqc_} & _F_ ( _n_ +1).
But
Inst { _q aSbSdLqc_} & _F_ ( _n_ +1) → ( _CC n_ → _CC n+1_)
is provable, and so therefore is
_A_ ( )& _F_ ( _n_ +1) → ( _CC n_ → _CC n+1_)
and
( _A_ ( ) & _F_ ( _n_ ) → _CC n_) → ( _A_ ( ) & _F_ ( _n_ +1) → _CC n+1_),
_i.e_.
_CF_ n → _CF_ n+1.
_CF n_ is provable for each _n_. Now it is the assumption of this lemma that _S_ 1 appears somewhere, in some complete configuration, in the sequence of symbols printed by ; that is, for some integers _N_ , _K_ , _CC N_ has _R S1_( _u_ ( _N_ ), _u_ ( _K_ )) as one of its terms, and therefore _CC N_ → _R S1_ ( _u_ ( _N_ ), _u_ ( _K_ )) is provable. We have then
and
We also have
where _N_ ′ = max ( _N_ , _K_ ). And so
_i.e_. Un ( ) is provable.
This completes the proof of Lemma 1.
LEMMA 2. _If_ Un ( ) _is provable, then S_ 1 _appears on the tape in some complete configuration of_ .
If we substitute any propositional functions for function variables in a provable formula, we obtain a true proposition. In particular, if we substitute the meanings tabulated on pp. [84–85] in Un ( ), we obtain a true proposition with the meaning " _S_ 1 appears somewhere on the tape in some complete configuration of ".
We are now in a position to show that the Entscheidungsproblem cannot be solved. Let us suppose the contrary. Then there is a general (mechanical) process for determining whether Un ( ) is provable. By Lemmas 1 and 2, this implies that there is a process for determining whether ever prints 0, and this is impossible, by 8. Hence the Entscheidungsproblem cannot be solved.
In view of the large number of particular cases of solutions of the Entscheidungs problem for formulae with restricted systems of quantors, it is interesting to express Un ( ) in a form in which all quantors are at the beginning. Un ( ) is, in fact, expressible in the form
where contains no quantors, and _n_ = 6. By unimportant modifications we can obtain a formula, with all essential properties of Un ( ), which is of form (I) with _n_ = 5.
_Added_ 28 _August_ , 1936.
##### Appendix
_Computability and effective calculability_
The theorem that all effectively calculable ( **λ** -definable) sequences are computable and its converse are proved below in outline. It is assumed that the terms "well-formed formula" (W.F.F.) and "conversion" as used by Church and Kleene are understood. In the second of these proofs the existence of several formulae is assumed without proof; these formulae may be constructed straightforwardly with the help of, _e.g_., the results of Kleene in "A theory of positive integers in formal logic", _American Journal of Math_ , 57 (1935), 153–173, 219–244.
The W.F.F. representing an integer _n_ will be denoted by _N n_. We shall say that a sequence **γ** whose _n_ -th figure is _ϕ_ γ( _n_ ) is **λ** -definable or effectively calculable if 1 + _ϕ_ γ( _u_ ) is a **λ** -definable function of _n_ , _i.e_. if there is a W.F.F. _M_ **γ** such that, for all integers _n_ ,
_i.e_. { _M_ γ ( _N _n__ ) is convertible into **λ** _xy_. _x_ ( _x_ ( _y_ )) or into **λ** _xy_. _x_ ( _y_ ) according as the _n_ -th figure of **λ** is 1 or 0.
To show that every **λ** -definable sequence **γ** is computable, we have to show how to construct a machine to compute **γ**. For use with machines it is convenient to make a trivial modification in the calculus of conversion. This alteration consists in using _x_ , _x_ ′, _x_ ′′,... as variables instead of _a, b, c_ ,.... We now construct a machine which, when supplied with the formula _M_ **γ** , writes down the sequence **γ**. The construction of is somewhat similar to that of the machine which proves all provable formulae of the functional calculus. We first construct a choice machine 1, which, if supplied with a W.F.F., _M_ say, and suitably manipulated, obtains any formula into which _M_ is convertible. 1 can then be modified so as to yield an automatic machine 2 which obtains successively all the formulae into which _M_ is convertible (cf. foot-note p. [77]). The machine includes 2 as a part. The motion of the machine when supplied with the formula _M_ **γ** is divided into sections of which the _n_ -th is devoted to finding the _n_ -th figure of **γ**. The first stage in this _n_ -th section is the formation of { _M_ **γ** }( _N n_). This formula is then supplied to the machine 2, which converts it successively into various other formulae. Each formula into which it is convertible eventually appears, and each, as it is found, is compared with
and with
If it is identical with the first of these, then the machine prints the figure 1 and the _n_ -th section is finished. If it is identical with the second, then 0 is printed and the section is finished. If it is different from both, then the work of 2 is resumed. By hypothesis, { _M_ **γ** }( _N n_) is convertible into one of the formulae _N_ 2 or _N_ 1; consequently the _n_ -th section will eventually be finished, _i.e_. the _n_ -th figure of **γ** will eventually be written down.
To prove that every computable sequence **γ** is **λ** -definable, we must show how to find a formula _M_ **γ** such that, for all integers _n_ ,
{ _M_ **γ** }( _N n_) conv _N_ 1+ _ϕ_ **γ** (n).
Let be a machine which computes **γ** and let us take some description of the complete configurations of by means of numbers, _e.g_. we may take the D.N of the complete configuration as described in 6. Let ( _n_ ) be the D.N of the _n_ -th complete configuration of . The table for the machine gives us a relation between ( _n_ \+ 1) and ( _n_ ) of the form
( _n_ \+ 1) = _ρ_ **γ** ( ( _n_ )),
where _ρ_ **γ** is a function of very restricted, although not usually very simple, form: it is determined by the table for . _ρ_ **γ** is **λ** -definable (I omit the proof of this), _i.e_. there is a W.F.F. _A_ **γ** such that, for all integers _n_ ,
{ _A_ **γ** }( _N_ (n)) conv _N_ (n+1).
Let _U_ stand for
**λ** _u_ [{{ _u_ }( _A_ **γ** )}( _N r_)],
where _r_ = (0); then, for all integers _n_ ,
{ _U_ **γ** }( _N n_) conv _N_ (n).
It may be proved that there is a formula _V_ such that
Let _W_ **γ** stand for
**λ** _u_ [{{ _V_ }({ _A_ **γ** }({ _U_ **γ** }( _u_ )))} ({ _U_ **γ** }( _u_ ))],
so that, for each integer _n_ ,
{{ _V_ }( _N_ (n+1))} ( _N_ (n)) conv { _W_ **γ** }( _N n_),
and let _Q_ be a formula such that
{{ _Q_ }( _W_ **γ** )} ( _N s_) conv _N r(z)_,
where _r_ ( _s_ ) is the _s_ -th integer _q_ for which { _W_ **γ** }( _N q_) is convertible into either _N_ 1 or _N_ 2. Then, if _M_ **γ** stands for
**λ** _w_ [{ _W_ **γ** }({{ _Q_ }( _W_ γ)}( _w_ ))],
it will have the required property.
### **CHAPTER 2
On Computable Numbers: Corrections and Critiques**
_Alan Turing, Emil Post, and Donald W. Davies_
#### Introduction
##### _Jack Copeland_
This chapter contains four items:
2.1 On Computable Numbers, with an Application to the Entscheidungs-problem. A Correction. _Alan Turing_
2.2 On Computable Numbers, with an Application to the Entscheidungs-problem. A Critique. _Emil Post_
2.3 Draft of a Letter from Turing to Alonzo Church Concerning the Post Critique
2.4 Corrections to Turing's Universal Computing Machine _Donald W. Davies_
As is not uncommon in work of such complexity, there are a number of mistakes in 'On Computable Numbers' (Chapter 1). Turing corrected some of these in his short note 2.1, published in the _Proceedings of the London Mathematical Society_ a few months after the original paper had appeared.
The mathematician Emil L. Post's critique of 'On Computable Numbers' was published in 1947 and formed part of Post's paper 'Recursive Unsolvability of a Problem of Thue'. Post is one of the major figures in the development of mathematical logic in the twentieth century, although his work did not gain wide recognition until after his death. (Born in 1897, Post died in the same year as Turing.)
By 1936 Post had arrived independently at an analysis of computability substantially similar to Turing's. Post's 'problem solver' operated in a 'symbol space' consisting of 'a two way infinite sequence of spaces or boxes'. A box admitted 'of but two possible conditions, i.e., being empty or unmarked, and having a single mark in it, say a vertical stroke'. The problem solver worked in accordance with 'a fixed unalterable set of directions' and could perform the following 'primitive acts': determine whether the box at present occupied is marked or not; erase any mark in the box that is at present occupied; mark the box that is at present occupied if it is unmarked; move to the box to the right of the present position; move to the box to the left of the present position.
Later, Post considerably extended certain of the ideas in Turing's 'Systems of Logic Based on Ordinals' (Chapter 3), developing the important field now called _degree theory_.
In his draft letter to Church, Turing responded to Post's remarks concerning 'Turing convention-machines'. It is doubtful whether Turing ever sent the letter. The approximate time of writing can be inferred from Turing's opening remarks: Kleene's review appeared in the issue of the _Journal of Symbolic Logic_ dated September 1947 (12: 90–1) and Turing's 'Practical Forms of Type Theory' appeared in the same journal in June 1948.
In his final year at university Donald Davies (1924–2000) heard about Turing's proposed Automatic Computing Engine and the plans to build it at the National Physical Laboratory in London (see Chapter 9). Davies immediately applied to join the National Physical Laboratory and in September 1947 became a member of the small team surrounding Turing. Davies played a leading role in the development and construction of the pilot model of the Automatic Computing Engine, which ran its first programme in May 1950. From 1966 he was head of the computer science division at the National Physical Laboratory. He originated the important concept of 'packet switching' used in the ARPANET, forerunner of the Internet. From 1979 Davies worked on data security and public key cryptosystems.
'On Computable Numbers' contained a number of what would nowadays be called programming errors. Davies described Turing's reaction when he drew Turing's attention to some of these:
I was working more or less under [Turing's] supervision... I had been reading his famous work on computable numbers... and I began to question some of the details of his paper. In fact I... found a number of quite bad programming errors, in effect, in the specification of the machine that he had written down, and I had worked out how to overcome these. I went along to tell him and I was rather cock-a-hoop... I thought he would say 'Oh fine, I'll send along an addendum' [to the London Mathematical Society]. But in fact he was very annoyed, and pointed out furiously that really it didn't matter, the thing was right in principle, and altogether I found him extremely touchy on this subject.
In Section 4 of his 'Corrections to Turing's Universal Computing Machine' Davies mends the errors that he discovered in 1947. He emphasizes that—as Turing said—these programming errors are of no significance for the central arguments of 'On Computable Numbers'.
Davies's lucid commentary forms an excellent introduction to 'On Computable Numbers'.
#### 2.1
On Computable Numbers, with an Application to the Entscheidungsproblem. A Correction. (1937)
##### _Alan Turing_
In a paper entitled "On computable numbers, with an application to the Entscheidungsproblem" [Chapter 1] the author gave a proof of the insolubility of the Entscheidungsproblem of the "engere Funktionenkalkül". This proof contained some formal errors which will be corrected here: there are also some other statements in the same paper which should be modified, although they are not actually false as they stand.
The expression for Inst { _q iSjSkLql_} on p. [85] of the paper quoted should read
( _x_ , _y_ , _x_ ′, _y_ ′) {( _R_ _S_ _j_ ( _x_ , _y_ ) & _I_ ( _x_ , _y_ ) & _K_ _qi_ ( _x_ ) & _F_ ( _x_ , _x_ ′) & _F_ ( _y_ ′, _y_ )) → ( _I_ ( _x_ ′, _y_ ′) & _R_ _Sk_ ( _x_ ′, _y_ ) & _K_ _ql_ ( _x_ ′) & _F_ ( _y_ ′, _z_ ) v [( _R_ _S_ 0( _x_ , _z_ ) → _R_ _S_ 0( _x_ ′, _z_ )) & _R_ _S_ 1( _x_ , _z_ ) → _R_ _S_ 1( _x_ ′, _z_ )) &... & _R_ _S_ _M_ ( _x_ , _z_ ) → _R_ _S_ _M_ ( _x_ ′, _z_ ))])},
_S_ 0, _S_ 1,..., _S_ _M_ being the symbols which can print. The statement on p. [86], [lines 24–25], viz.
"Inst { _q aSbSdLqc_} & _F_ ( _n_ +1) → ( _CC_ _n_ → _CC_ _n_ +1)
is provable" is false (even with the new expression for Inst { _q aSbSdLqc_}): we are unable for example to deduce _F_ ( _n_ \+ 1) → (– _F_ ( _u_ , _u_ ″)) and therefore can never use the term
_F_ ( _y_ ′, _z_ ) v [( _R_ _S_ 0 ( _x_ , _z_ ) → _R_ _S_ 0 ( _x_ ′, _z_ )) &... & ( _R_ _S_ _M_ ( _x_ , _z_ ) → _R_ _S_ _M_ ( _x_ ′, _z_ ))]
in Inst { _q aSbSdLqc_}. To correct this we introduce a new functional variable _G_ [ _G_ ( _x_ , _y_ ) to have the interpretation " _x_ precedes _y_ "]. Then, if _Q_ is an abbreviation for
( _x_ )(∃ _w_ )( _y_ , _z_ ) { _F_ ( _x_ , _w_ ) & ( _F_ ( _x_ , _y_ ) → _G_ ( _x_ , _y_ )) & ( _F_ ( _x_ , _z_ ) & _G_ ( _z_ , _y_ ) → _G_ ( _x_ , _y_ )) & [ _G_ ( _z_ , _x_ ) v ( _G_ ( _x_ , _y_ ) & _F_ ( _y_ , _z_ )) v ( _F_ ( _x_ , _y_ ) & _F_ ( _z_ , _y_ )) → (– _F_ ( _x_ , _z_ ))]}
the corrected formula Un ( ) is to be
(∃ _u_ ) _A_ ( ) → (∃ _s_ )(∃ _t_ ) _R_ _S_ 1( _s_ , _t_ ),
where _A_ ( ) is an abbreviation for
_Q_ &( _y_ ) _R_ _S_ 0( _u_ , _y_ )& _I_ ( _u_ , _u_ )& _K_ _q_ 1( _u_ ) & Des ( ).
The statement on page [86] (line [24]) must then read
Inst { _q_ _a_ _S_ _b_ _S_ _d_ _Lq_ _c_ } & _Q_ & _F_ ( _n_ \+ 1) → _CC_ _n_ → _CC_ _n_ +1),
and [lines 19–20] should read
For the words "logical sum" on p. [85], line [13], read "conjunction". With these modifications the proof is correct. Un ( ) may be put in the form (I) (p. [87]) with _n_ = 4.
Some difficulty arises from the particular manner in which "computable number" was defined (p. [61]). If the computable numbers are to satisfy intuitive requirements we should have:
_If we can give a rule which associates with each positive integer n two rationals a_ _n_ , _b_ _n_ satisfying _a_ _n_ ≤ _a_ _n_ \+ 1 < _b_ _n_ \+ 1 ≤ _b_ _n_ , _b_ _n_ _a_ _n_ < 2– _n_ , _then there is a computable number α for which a_ _n_ ≤ _α_ ≤ _b_ _n_ _each n_. (A)
A proof of this may be given, valid by ordinary mathematical standards, but involving an application of the principle of excluded middle. On the other hand the following is false:
_There is a rule whereby, given the rule of formation of the sequences a n_, _b n in (A) we can obtain a_ D.N. _for a machine to compute_ α. (B)
That (B) is false, at least if we adopt the convention that the decimals of numbers of the form _m_ /2 _n_ shall always terminate with zeros, can be seen in this way. Let be some machine, and define _c_ _n_ as follows: if has not printed a figure 0 by the time the _n_ -th complete configuration is reached 3 if 0 had first been printed at the _m_ -th complete configuration ( _m_ ≤ _n_ ). Put _a_ _n_ = _c_ _n_ –2– _n_ –2, _b_ _n_ = _c_ _n_ \+ 2– _n_ – 2. Then the inequalities of (A) are satisfied, and the first figure of α is 0 if ever prints 0 and is 1 otherwise. If (B) were true we should have a means of finding the first figure of α given the D.N. of : _i.e_. we should be able to determine whether ever prints 0, contrary to the results of §8 of the paper quoted. Thus although (A) shows that there must be machines which compute the Euler constant (for example) we cannot at present describe any such machine, for we do not yet know whether the Euler constant is of the form _m_ /2 _n_.
This disagreeable situation can be avoided by modifying the manner in which computable numbers are associated with computable sequences, the totality of computable numbers being left unaltered. It may be done in many ways of which this is an example. Suppose that the first figure of a computable sequence γ is _i_ and that this is followed by 1 repeated _n_ times, then by 0 and finally by the sequence whose _r_ -th figure is _c_ _r_ ; then the sequence γ is to correspond to the real number
If the machine which computes γ is regarded as computing also this real number then (B) holds. The uniqueness of representation of real numbers by sequences of figures is now lost, but this is of little theoretical importance, since the D.N.'s are not unique in any case.
The Graduate College, Princeton, N.J., U.S.A.
#### 2.2 On Computable Numbers, with an Application to the Entscheidungsproblem. A Critique. (1947)
##### _Emil Post_
The following critique of Turing's "computability" paper [Chapter 1] concerns only pp. [58–74] thereof. We have checked the work through the construction of the "universal computing machine" in detail, but the proofs of the two theorems in the section following are there given in outline only, and we have not supplied the formal details. _We have therefore also left in intuitive form the proofs of the statements on recursiveness, and alternative procedures, we make below_.
One major correction is needed. To the instructions for p. [70], add the line: None _PD, R_ , _P_ α, _R_ , _R_ , _R_ . This is needed to introduce the representation _D_ of the blank scanned square when, as at the beginning of the action of the machine, or due to motion right beyond the rightmost previous point, the complete configuration ends with a , and thus make the of p. [71] correct. We may also note the following minor slips and misprints in pp. [58–74]. Page [63], to the instructions for add the line: None _L_ ; p. [67] and p. [68], the S.D should begin, but not end, with a semicolon; p. [69], omit the first _D_ in (C2); p. [70], last paragraph [above skeleton table], add ":" to the first list of symbols; pp. [71–72], replace by ; p. [71], in the instruction for , should be 1; p. [71], in the second instruction for , replace the first _R_ by _L_ ; p. [71], in the first instruction for , replace by . A reader of the paper will be helped by keeping in mind that the "examples" of pages [63–66] are really parts of the table for the universal computing machine, and accomplish what they are said to accomplish not for all possible printings on the tape, but for certain ones that include printings arising from the action of the universal computing machine. In particular, the tape has ə printed on its first two squares, the occurrence of two consecutive blank squares insures all squares to the right thereof being blank, and, usually, symbols referred to are on " _F_ -squares", and obey the convention of p. [63].
Turing's definition of an arbitrary machine is not completely given in his paper, and, at a number of points, has to be inferred from his development. In the first instance his machine is a "computing machine" for obtaining the successive digits of a real number in dyadic notation, and, in that case, starts operating on a blank tape. Where explicitly stated, however, the machine may start operating on a tape previously marked. From Turing's frequent references to the beginning of the tape, and the way his universal computing machine treats motion left, we gather that, unlike our tape, this tape is a one-way infinite affair going right from an initial square.
Primarily as a matter of practice, Turing makes his machines satisfy the following convention. Starting with the first square, alternate squares are called _F_ -squares, the rest, _E_ -squares. In its action the machine then never directs motion left when it is scanning the initial square, never orders the erasure, or change, of a symbol on an _F_ -square, never orders the printing of a symbol on a blank _F_ -square if the previous _F_ -square is blank, and, in the case of a computing machine, never orders the printing of 0 or 1 on an _E_ -square. This convention is very useful in practice. However the actual performance, described below, of the universal computing machine, coupled with Turing's proof of the second of the two theorems referred to above, strongly suggests that Turing makes this convention part of the definition of an arbitrary machine. We shall distinguish between a Turing machine and a Turing convention-machine.
By a uniform method of representation, Turing represents the set of instructions, corresponding to our quadruplets, which determine the behavior of a machine by a single string on seven letters called the standard description (S.D) of the machine. With the letters replaced by numerals, the S.D of a machine is considered the arabic representation of a positive integer called the description number (D.N) of the machine. If our critique is correct, a machine is said to be circle-free if it is a Turing computing convention-machine which prints an infinite number of 0's and 1's. And the two theorems of Turing's in question are really the following. There is no Turing convention-machine which, when supplied with an arbitrary positive integer _n_ , will determine whether _n_ is the D.N of a Turing computing convention-machine that is circle-free. There is no Turing convention-machine which, when supplied with an arbitrary positive integer _n_ , will determine whether _n_ is the D.N of a Turing computing convention-machine that ever prints a given symbol (0 say).
In view of [Turing (1937)], these "no machine" results are no doubt equivalent to the recursive unsolvability of the corresponding problems. But both of these problems are infected by the spurious Turing convention. Actually, the set of _n_ 's which are D.N's of Turing computing machines as such is recursive, and hence the condition that _n_ be a D.N offers no difficulty. But, while the set of _n_ 's which are not D.N's of convention-machines is recursively enumerable, the complement of that set, that is, the set of _n_ 's which are D.N's of convention-machines, is not recursively enumerable. As a result, in both of the above problems, neither the set of _n_ 's for which the question posed has the answer yes, nor the set for which the answer is no, is recursively enumerable.
This would remain true for the first problem even apart from the convention condition. But the second would then become that simplest type of unsolvable problem, the decision problem of a non-recursive recursively enumerable set of positive integers [(Post 1944)]. For the set of _n_ 's that are D.N's of unrestricted Turing computing machines printing 0, say, is recursively enumerable, though its complement is not. The Turing convention therefore prevents the early appearance of this simplest type of unsolvable problem.
It likewise prevents the use of Turing's second theorem in the... unsolvability proof of the problem of Thue. For in attempting to reduce the problem of Turing's second theorem to the problem of Thue, when an _n_ leads to a Thue question for which the answer is yes, we would still have to determine whether _n_ is the D.N of a Turing convention-machine before the answer to the question posed by _n_ can be given, and that determination cannot be made recursively for arbitrary _n_. If, however, we could replace the Turing convention by a convention that is recursive, the application to the problem of Thue could be made. An analysis of what Turing's universal computing machine accomplishes when applied to an arbitrary machine reveals that this can be done.
The universal computing machine was designed so that when applied to the S.D of an arbitrary computing machine it would yield the same sequence of 0's and 1's as the computing machine as well as, and through the intervention of, the successive "complete configurations"—representations of the successive states of tape versus machine—yielded by the computing machine. This it does for a Turing convention-machine. For an arbitrary machine, we have to interpret a direction of motion left at a time when the initial square of the tape is scanned as meaning no motion. The universal computing machine will then yield again the correct complete configurations generated by the given machine. But _the space sequence of_ 0' _s and_ 1' _s printed by the universal computing machine will now be identical with the time sequence of those printings of_ 0' _s and_ 1' _s by the given machine that are made in empty squares_. If, now, instead of Turing's convention we introduce the convention that the instructions defining the machine never order the printing of a 0 or 1 except when the scanned square is empty, or 0, 1 respectively, and never order the erasure of a 0 or 1, Turing's arguments again can be carried through. And this "(0, 1) convention", being recursive, allows the application to the problem of Thue to be made. Note that if a machine is in fact a Turing convention-machine, we could strike out any direction thereof which contradicts the (0, 1) convention without altering the behavior of the machine, and thus obtain a (0, 1) convention-machine. But a (0, 1) convention-machine need not satisfy the Turing convention. However, by replacing each internal-configuration _q i_ of a machine by a pair _q i_, _q i ′_ to correspond to the scanned square being an _F_ -or an _E_ -square respectively, and modifying printing on an _F_ -square to include testing the preceding _F_ -square for being blank, we can obtain a "( _q_ , _q ′_ ) convention" which is again recursive, and usable both for Turing's arguments and the problem of Thue, and has the property of, in a sense, being equivalent to the Turing convention. That is, every ( _q_ , _q ′_ ) convention-machine is a Turing convention-machine, while the directions of every Turing convention-machine can be recursively modified to yield a ( _q_ , _q ′_ ) convention-machine whose operation yields the same time sequence and spatial arrangement of printings and erasures as does the given machine, except for reprintings of the same symbol in a given square.
These changes in the Turing convention, while preserving the general outline of Turing's development and at the same [time] admitting of the application to the problem of Thue, would at least require a complete redoing of the formal work of the proof of the second Turing theorem. On the other hand, very little added formal work would be required if the following changes are made in the Turing argument itself, though there would still remain the need of extending the equivalence proof of [Turing (1937)] to the concept of unsolvability. By using the above result on the performance of the universal computing machine when applied to the S.D of an arbitrary machine, we see that Turing's proof of his first theorem, whatever the formal counterpart thereof is, yields the following theorem. There is no Turing convention-machine which, when supplied with an arbitrary positive integer _n_ , will determine whether _n_ is the D.N of an arbitrary Turing machine that prints 0's and 1's in empty squares infinitely often. Now given an arbitrary positive integer _n_ , if that _n_ is the D.N of a Turing machine , apply the universal computing machine to the S.D of to obtain a machine *. Since * satisfies the Turing convention, whatever Turing's formal proof of his second theorem is, it will be usable intact in the present proof, and, via the new form of his first theorem, will yield the following usable result. There is no machine which, when supplied with an arbitrary positive integer _n_ , will determine whether _n_ is the D.N of an arbitrary Turing machine that ever prints a given symbol (0 say).
These alternative procedures assume that Turing's universal computing machine is retained. However, in view of the above discussion, it seems to the writer that Turing's preoccupation with computable numbers has marred his entire development of the Turing machine. We therefore suggest a redevelopment of the Turing machine based on the formulation given in ['Recursive Unsolvability of a Problem of Thue']. This could easily include computable numbers by defining a computable sequence of 0's and 1's as the _time sequence_ of printings of 0's and 1's by an arbitrary Turing machine, provided there are an infinite number of such printings. By adding to Turing's complete configuration a representation of the act last performed, a few changes in Turing's method would yield a universal computing machine which would transform such a time sequence into a space sequence. Turing's convention would be followed as a matter of useful practice in setting up this, and other, particular machines. But it would not infect the theory of arbitrary Turing machines.
##### _References_
Church, A. 1936. 'An Unsolvable Problem of Elementary Number Theory', _American Journal of Mathematics_ , 58, 345–363.
Post, E. L. 1943. 'Formal Reductions of the General Combinatorial Decision Problem', _American Journal of Mathematics_ , 65, 197–215.
Post, E. L. 1944. 'Recursively Enumerable Sets of Positive Integers and their Decision Problems', _Bulletin of the American Mathematical Society_ , 50, 284–316.
Turing, A. M. 1937. _Computability and λ-definability, Journal of Symbolic Logic_ , 2, 153–163.
#### 2.3 Draft of a Letter from Turing to Alonzo Church Concerning the Post Critique
Dear Professor Church,
I enclose corrected proof of my paper 'Practical forms of type theory' and order for reprints.
Seeing Kleene's review of Post's paper (on problem of Thue) has reminded me that I feel I ought to say a few words somewhere to clear up the points which Post has raised about 'Turing machines' and 'Turing convention machines' [see 2.2]. Post observes that my initial description of a machine differs from the machines which I describe later in that the latter are subjected to a number of conventions (e.g. the use of E and F squares). These conventions are nowhere very clearly enumerated in my paper and cast a fog over the whole concept of a 'Turing machine'. Post has enumerated the conventions and embodied them in a definition of a 'Turing convention machine'.
My intentions in this connection were clear in my mind at the time the paper was written; they were not expressed explicitly in the paper, but I think it is now necessary to do so. It was intended that the 'Turing machine' should always be the machine without attached conventions, and that all general theorems about machines should apply to this definition. To the best of my belief this was adhered to. On the other hand when it was a question of describing particular machines a host of conventions became desirable. Clearly it was best to choose conventions which did not restrict the essential generality of the machine, but one was not called upon to establish any results to this effect. If one could find machines obeying the conventions and able to carry out the desired operations, that was enough. It was also undesirable to keep any fixed list of conventions. At any moment one might wish to introduce a new one.
#### 2.4 Corrections to Turing's Universal Computing Machine
##### _Donald W. Davies_
#### 1. Introduction
In 1947 I was working in a small team at the National Physical Laboratory in London, helping to build one of the first programmed computers. This had been designed by Turing. (See Chapter 9.)
When I first studied Turing's 'On Computable Numbers, with an Application to the Entscheidungsproblem', it soon became evident to me that there were a number of trivial errors, amounting to little more than typographic errors, in the design of his universal computing machine U. A closer look revealed a—nowadays typical—programming error in which a loop led back to the wrong place. Then I became aware of a more fundamental fault relating to the way U describes the blank tape of the machine it is emulating. Perhaps it is ironic, as well as understandable, that the first emulation program for a computer should have been wrong. I realized that, even though the feasibility of the universal computing machine was not in doubt, the mistakes in Turing's exposition could puzzle future readers and plague anyone who tried to verify Turing's design by implementing his universal machine in practice.
When I told Turing about this he became impatient and made it clear that I was wasting my time and his by my worthless endeavours. Yet I kept in mind the possibility of testing a corrected form of U in the future. It was to be nearly fifty years before I finally did this.
I could not implement exactly Turing's design because this generates a profusion of states when the 'skeleton tables' are substituted by their explicit form (to a depth of 9). Also the way in which U searches for the next relevant instruction involves running from end to end of the tape too many times. Features of Turing's scheme which greatly simplify the description also cause the explicit machine to have many symbols, a considerable number of states and instructions, and to be extremely slow. Turing would have said that this inefficiency was irrelevant to his purpose, which is true, but it does present a practical problem if one is interested in verifying an actual machine. Some fairly simple changes to the design reduce this problem. The final part of this paper outlines a redesign of the universal machine which was tested by simulation and shown to work. There can be reasonable confidence that there are no further significant errors in Turing's design, but a simulation starting directly from Turing's 'skeleton tables' would clinch the matter.
By and large I use Turing's notation and terminology in what follows. Where my notation differs from Turing's the aim has been to make matters clearer. In particular, Turing's Gothic letters are replaced by roman letters. I sometimes introduce words from modern computer technology where this makes things clearer. (There is no special significance to the use of boldface type—this is used simply for increased clarity.)
##### 2. The Turing Machine T
Turing required a memory of unlimited extent and a means of access to that memory. Access by an address would not provide unlimited memory. In this respect the Turing machine goes beyond any existing real machine.
His method, of course, was to store data in the form of symbols written on a tape of unlimited length. Specifically, the tape had a beginning, regarded as its left-hand end, marked with a pair of special symbols 'e e' that can easily be found. To the right of these symbols there are an unlimited number of symbol-positions or 'squares' which can be reached by right and left movements of the machine, shown as 'R' and 'L' respectively in the machine's instructions. By repeated R and L movements any square can be accessed.
Let us consider how the machine's instructions are composed. We are not concerned yet with U, the universal machine, but with a specific Turing machine T—the 'target machine'—which will later be emulated by U.
An instruction for T consists of five parts. The purpose of the first two parts is to address the instruction. These give the state of the machine (I shall call this M) and the symbol S that the machine is reading in the scanned square. This state-symbol pair M-S determines the next operation of the machine. Turing called M-S a 'configuration'. For each such pair that can occur (finitely many, since there is a finite number of states and of symbols), the next operation of the machine must be specified in the instructions, by notations in the remaining three parts. The first of these is the new symbol to be written in place of the one that has been read, and I call this S′. Then there is an action A, which takes place after the writing of the new symbol, and this can be a right shift R, a left shift L, or N, meaning no movement. Finally the resultant state M′ is given in the last of the five parts of the instruction.
To summarize: an instruction is of the form MSS′ AM′. The current machine configuration is the pair M-S, which selects an appropriate instruction. The instruction then specifies S **′** , A, and M **′** , meaning that the symbol S **′** is written in place of S. The machine then moves according to action A (R, L, or N) and finally enters a new state M **′**. A table of these instructions, each of five parts, specifies the entire behaviour of the target machine T. The table should have instructions for all the M-S pairs that can arise during the operation of the machine.
There are some interesting special cases. One of the options is that S **′** = S, meaning that the symbol that was in the scanned square remains in place. In effect, no writing has occurred. Another is that S **′** = blank, meaning that the symbol S which was read has been erased.
As a practical matter, note that the part of the tape which has been used is always finite and should have well-defined ends, so that the machine will not run away down the tape. As already mentioned, Turing specifies that at the left-hand end the pair of special symbols 'e e' is printed. These are never removed or altered. On the extreme right of the used part of the tape there must be a sequence of blank squares. Turing arranges that there will never be two adjacent blanks anywhere in the used part of the tape, so that the right-hand end of the used portion can always be found. This convention is necessary to make U work properly, as is explained in detail later. However, Turing does not necessarily follow this convention in specifying target machines T. This can result in misbehaviour.
The purpose of T is to perform a calculation, therefore T must generate numbers. For this reason the symbols it can print include 0 and 1, which are sufficient to specify a binary result. By convention these two symbols are never erased but remain on the tape as a record of the result of the computation. It happens that they are treated in a special way in the emulation by U, in order to make the result of the calculation more obvious, as I shall explain in due course.
##### 3. The Basic Plan of U
The universal machine U must be provided with the table of instructions of the Turing machine T that it is to emulate. The instructions are given at the start of U's tape, separated by semicolons ';'. At the end of these instructions is the symbol '::' (which is a single symbol). Later I shall settle the question of whether a semicolon should be placed _before_ the first instruction or _after_ the last one.
Following '::' is the workspace, in which U must place a complete description, in U's own symbols, of machine T. This description consists of all the symbols on T's tape, the position of the machine on that tape, and the state of the machine. I call such a description an _image_ or _snapshot_ of machine T. As T goes through its computational motions, more and more snapshots are written in U's workspace, so that an entire history of T gradually appears.
In order to make this evolving representation of T's behaviour possible, U must not change any of the images on its tape. U simply adds each new image to the end of the tape as it computes it.
The first action of U is to construct the initial image of T, in the space immediately following the '::' symbol which terminates the set of instructions. Subsequently, U writes new images of T, separated by colons ':', each image representing a successive step in the evolution of T's computation.
Whenever T is asked to print a 0 or 1, the image in which this happens is followed by the symbols '0 :' or '1 :' respectively. This serves to emphasize the results of the computation. For example, U's tape might look like this, with the instruction set followed by successive images of T. The 'output' character 0 printed in image 3 is highlighted by printing it again after the image, and likewise in the case of the 'output' character 1 that is printed in image 5.
e e ; inst 1 ; inst 2 ;... inst n :: image 1 : image 2 : image 3 : 0 : image 4 : image 5 : 1 : etc.
Note that each instruction _begins_ with a semicolon. This differs from Turing, who has the semicolon _after_ each instruction, so that the instructions end with the pair of characters '; ::'. I found this departure from Turing's presentation necessary, as I shall explain later.
The symbols on the tape up to and including the '::'—i.e. T's instructions—are given to U before it starts operating. When U commences its operations, it writes out the first image and then computes successive images from this, using the instructions, and intersperses the images with 'outputs' as required.
To complete this description of the basic plan of U I must specify how the symbols are spaced out in order to allow for _marking_ them with other symbols. Looking in more detail at the start of the tape we would find squares associated in pairs. The left square of each pair contains a symbol from the set:
A C D L R N 0 1 ; :: :
These symbols are never erased by U; they form a permanent record of the instructions and the images of T. Thus the first few squares of a tape might contain these symbols representing an instruction, where '–' refers to the blank symbol, meaning an empty square:
The blank squares leave room for symbols from the set _u v w x y z_ , which are reserved for use as markers and serve to mark the symbol to their immediate left. For example, at one stage of the operation of U the parts of this instruction are marked out as follows:
**e** D − A − D − D _u_ C _u_ R _u_ D _y_ A _y_ A _y_ ; –...
Here the ' _u_ ' and ' _y_ ' mark D C R and D A A respectively.
Unlike the symbols A C D L R N 0 1 ; :: : which are never erased, the letters _u v w x y z_ are always temporary markings and are erased when they have done their job.
The positions of the A C D L R N 0 1 ; :: : symbols will be called 'non-erasing positions'. Note that the left hand 'e' occupies such a position also.
##### 4. Notation for States, Symbols, and Actions in U's Instructions and in Images of T
The states, symbols, and actions which U represents on its tape are those of T, which U is emulating. We do not know how many states and symbols have to be emulated, yet the set of symbols of U is limited by its design. For economy in U's symbols, the _n_ th state of T is represented on U's tape by DAA... A with _n_ occurrences of letter 'A'. The blank spaces between these letters are not shown here but are important for the operation of U and will always be assumed. The _n_ th symbol of T is shown on U's tape by the string DCC... C with _n_ occurrences of letter 'C'. It appears from an example in Turing's text that D by itself (with a marking square to its right, as always) can be one of these 'symbol images'. In fact I shall choose to make this the 'blank' symbol.
Read in accordance with these conventions, the symbols in example (1), above, form the instruction: 'when in state 1—reading a blank—write symbol 0—move right—change to state 2'.
With the semicolons as spacers we now have a complete notation for _instructions_ on U's tape. Next we need a notation for an _image_ of machine T, which is also made up from these state and symbol images, DAA... and DCC... respectively.
The keys to the next action of T are its present state and the symbol it is reading, which have been stored in the instruction table as the pair M-S. So this same combination is used in the image, by listing in correct sequence all the symbols on T's tape and inserting the state image immediately in front of the symbol which machine T's emulation is currently reading. The placing of the state image indicates which square is currently scanned. So a tape image might look like this:
symbol 1, symbol 2, symbol 3, state, symbol 4, symbol 5 :
The current state is given at the position indicated. This emulated tape of T records that T is scanning symbol 4. The combination 'state, symbol 4' which appears in this string is the M-S pair that U must look up, by searching for it in the instruction table. To get the image of the next state of T, state and symbol 4 may have to be changed and the state image, also changed, may have to be moved to the right, or to the left, or not moved. These are the processes which occur in one step of evolution of the machine T. It will be done by building a completely new image to the right of the last ':'. (Consequently the tape space is rapidly used up.)
##### 5. Notation for Machine U
In principle, machine U should be specified in the same formal notation as machine T, but this would be bulky and tedious to read and understand, so Turing used a much more flexible notation. A compiler could be built to take the 'higher-level' notation of machine U's specification as given by Turing and generate the complete set of instructions for U.
The statements which make up Turing's specification of U are similar to procedures with parameters (more correctly they are like macros) and they have four parts. As with the instructions of T, a state and symbol select which statement is to apply, except that the symbol can now be a logical expression such as 'not C', which in the explicit specification would require as many instructions as there are symbols other than 'C'. There are also statements that copy symbols from one part of the (real) tape to another, and this requires one instruction for each allowed symbol variety. The actions, which in the explicit notation can only write a symbol and optionally move one place left or right, are expanded here to allow multiple operations such as 'L, P _u_ , R, R, R', specifying that symbol ' _u_ 'is being printed to the left of the starting point and the machine ends up two places to the right of that point. Turing requires that these specifications be 'compiled' into the standard five-part instructions. I will call these procedures 'routines'. Their parameters are of two kinds, the states that the operations lead to, which are shown as capital letters, and symbol values, which are shown in lower case.
The specification of machine U consists of a collection of 'subroutines' which are then used in a 'program' of nine routines that together perform the evolution of T. We shall first describe the subroutines, then the main program.
##### 6. Subroutines
In principle, the action of a routine depends on the position of the machine when the routine is invoked, i.e. the square on the tape which is being scanned. Also, the position of the scanned square at the end of a routine's operation could be significant for the next operation to come. But Turing's design avoids too much interaction of this kind. It can be assumed that the positions are not significant for the use of the subroutines in U unless the significance is described here. Certain subroutines which are designed to find a particular symbol on the tape (such as **f** ( _A_ , _B_ , _a_ ) and **q** ( _A_ , _a_ ) and others) leave the machine in a significant position. **Con** ( _A_ , _a_ ) has significant starting and finishing positions.
Where a routine uses other routines, these are listed (for convenience in tracing side-effects). Also listed are those routines that use the routine in question. It would be possible to make several of the routines much more efficient, greatly reducing the amount of machine movement, but I have not made such changes here. (For efficiency, routines could be tailored for each of their uses and states and symbols could easily be coded in binary. But this would be a redesign.)
**f** ( _A, B, a_ )
The machine moves left until it finds the start of the tape at an 'e' symbol. Then it moves right, looking for a symbol ' _a_ '. If one is found it rests on that symbol and changes state to _A_. If there is no symbol ' _a_ ' on the whole tape it stops on the first blank non-erasing square to the right of the used portion of tape, going into state _B_. In general terms, this routine is looking for the leftmost occurrence of the symbol ' _a_ '. The special case f( _A_ , _B_ , e) will find the leftmost occurrence of 'e', which is in a non-erasing position. Uses no routines. Used by **b** , **e** ( _A_ , _B_ , _a_ ), **f** ′( _A_ , _B_ , _a_ ), **cp** ( _A_ , _B_ , _E_ , _a_ , _b_ ), **sh** , and **pe** ( _A_ , _b_ ).
**f** ′( _A, B, a_ )
As for f( _A_ , _B_ , _a_ ) except that if a symbol ' _a_ ' is found, the machine stops one square to the left, over the square which is marked by the ' _a_ '. Uses f( _A_ , _B_ , _a_ ) and 1( _A_ ). Used by cp( _A_ , _B_ , _E_ , _a_ , _b_ ), sim, and c( _A_ , _b_ , _a_ ).
1( _A_ )
Simply shifts one square to the left. Uses none. Used by f **′** ( _A_ , _B_ , _a_ ), **mk** , and **inst**.
**e** ( _A_ )
The marks are erased from all marked symbols, leaving the machine in state _A_. Uses none. Used by **ov**.
**e** ( _A, B, a_ )
The machine finds the leftmost occurrence of symbol ' _a_ ', using routine f( _A_ , _B_ , _a_ ), then erases it, resting on the blank symbol and changing to state _A_. If there is no such symbol to erase, it stops on the non-erasing square to the right of the used portion of tape in state _B_ , as for **f** ( _A_ , _B_ , _a_ ). Uses **f** ( _A_ , _B_ , _a_ ). Used by **kmp** , **cpe** ( _A_ , _B_ , _E_ , _a_ , _b_ ), and **e** ( _B_ , _a_ ).
**e** ( _B, a_ )
Erases all occurrences of the symbol ' _a_ ' on the tape, leaving the machine in state _B_. Uses e( _A_ , _B_ , _a_ ). Used by **sim**.
**pe** ( _A, b_ )
Prints the symbol ' _b_ ' in the first blank non-erasing position at the end of the sequence of symbols. Uses **f** ( _A_ , _B_ , _a_ ). Used by **pe** 2( _A_ , _a_ , _b_ ) and c( _A_ , _B_ , _a_ ).
**pe** 2( _A, a, b_ )
Prints the symbol ' _a_ ' and then ' _b_ ' in the first blank non-erasing positions. Uses **pe** ( _A_ , _b_ ). Used by **sh**.
**q** ( _A_ )
Moves to the next non-erasing position after the used portion of tape and goes to state _A_. Uses none. Used by **q** ( _A_ , _a_ ).
**q** ( _A, a_ )
Finds the last occurrence of symbol ' _a_ ' and stops there in state _A_. Uses **q** ( _A_ ). Used by **anf** , **mk** , and **inst**. If the symbol does not exist, the machine will run off the tape to the left. However, in its use in **anf** , **mk** , and **inst** , it will find a colon or ' _u_ ' on which to stop.
**c** ( _A, B, a_ )
Finds the leftmost symbol marked with ' _a_ ' and copies it at the end of the tape in the first non-erasing square available, then goes to state _A_. If no symbol ' _a_ ' is found goes to state _B_. The symbols which this routine (and those that use it) will be required to copy are 'D', 'C', and 'A'. This means that each invocation needs three different states. Uses **f** ′( _A_ , _B_ , _a_ ) and **pe** ( _A_ , _b_ ). Used by **ce** ( _A_ , _B_ , _a_ ).
**ce** ( _A, B, a_ )
Copies at the end of the tape in the first non-erasing square the leftmost symbol marked by ' _a_ ', then goes to state _A_ with the single (leftmost) marking ' _a_ ' erased. If there is no symbol ' _a_ ' on the tape, goes to state _B_. Uses **c** ( _A_ , _B_ , _a_ ) and **e** ( _A_ , _B_ , _a_ ). Used by **ce** ( _B_ , _a_ ).
**ce** ( _B, a_ )
Copies in the correct sequence, at the end of the tape in non-erasing squares, all the symbols on the tape that are marked with ' _a_ ', at the same time erasing each ' _a_ ', and then goes to state _B_. If there are no symbols marked with ' _a_ ' it goes straight to _B_. Uses **ce** ( _A_ , _B_ , _a_ ). Used by **ce** 2( _B_ , _a_ , _b_ ), **ce** 3( _B_ , _a_ , _b_ , _c_ ), **ce** 4( _B_ , _a_ , _b_ , _c_ , _d_ ), and **ce** 5( _B_ , _a_ , _b_ , _c_ , _d_ , _e_ ).
**ce** 5( _B, a, b, c, d, e_ )
Copies in the sequence given, at the end of the tape in non-erasing squares, all the symbols marked with ' _a_ ', then those marked with ' _b_ ', then ' _c_ ', ' _d_ ', and ' _e_ ' in turn, ending in state _B_. Uses **ce** 4( _B_ , _a_ , _b_ , _c_ , _d_ ) and **ce** ( _B_ , _a_ ) and **ce** 4 uses **ce** 3 which uses **ce** 2 and all of these use **ce** ( _B_ , _a_ ). Only **ce** 5( _B_ , _a_ , _b_ , _c_ , _d_ , _e_ ) is used elsewhere, by **inst**.
**cp** ( _A, B, E, a, b_ )
Compares the leftmost symbols marked by ' _a_ ' and ' _b_ '. First it finds the symbol marked ' _a_ ', by using the routine **f** ′( _A_ , _B_ , _a_ ). It enters a different state **cp** 2( _A_ , _B_ , _x_ ) according to the symbol ' _x_ ' it finds there. There are three possible symbols: 'D', 'C', and 'A'. Then it finds the symbol marked ' _b_ ' in the same way. If they are the same, the resultant state is _A_ , if not the state becomes _B_. If one of these marked symbols is found, but not the other, the outcome is state _B_. If neither is found, the outcome is _E_. Uses **f** ( _A_ , _B_ , _a_ ) and **f** ′( _A_ , _B_ , _a_ ). Used by **cpe** ( _A_ , _B_ , _E_ , _a_ , _b_ ). (I have changed Turing's state-symbol ' _U_ ' to ' _B_ ' to avoid confusion.)
**cpe** ( _A, B, E, a, b_ )
Action as for **cp** ( _A_ , _B_ , _E_ , _a_ , _b_ ) followed by, if the marked symbols are the same, the erasure of both the markings ' _a_ ' and ' _b_ '. Uses **cp** ( _A_ , _B_ , _E_ , _a_ , _b_ ) and **e** ( _A_ , _B_ , _a_ ). Used by **cpe** ( _B_ , _E_ , _a_ , _b_ ).
**cpe** ( _B, E, a, b_ )
Comparison of two marked sequences. First it compares the leftmost symbols marked with ' _a_ ' and ' _b_ '. If they differ the state _B_ is reached and the process stops. If they are both absent, state _E_. Otherwise both marked symbols are erased and the process is repeated with the next leftmost marked symbols. So, if the whole sequence of symbols marked with ' _a_ ' equals the sequence marked with ' _b_ ' (or both are absent) the result is state _E_ and all markings ' _a_ ' and ' _b_ ' have been removed. If the sequences differ, state _B_ is reached and some of the markings have been removed. Uses **cpe** ( _A_ , _B_ , _E_ , _a_ , _b_ ). Used by **kmp**.
**con** ( _A, a_ )
This routine's action depends on where it starts on the tape. It leaves the machine in a significant position after its action is completed. The purpose is to mark with symbol ' _a_ ' the M-S pair next on the right of the start position. The routine must start on a non-erasing square. It seeks a pattern such as:
D − A − A −... A − D − C − C... C −
and will replace all the blanks (or any other symbols in these places) by the symbol ' _a_ '. There may be as few as one 'A' after the first 'D', representing the state, and optionally no 'C' following the second 'D', representing the symbol. The starting point can be on the first 'D' of this pattern, or earlier if no other 'A' symbols intervene. For example, in an image there is only one state, so the first 'A' symbol can be sought from anywhere to its left in the image string. The final state is _A_ and the position is two non-erasable squares to the right of the last marked symbol. (Turing refers to this as 'the last square of C' but his own example shows a symbol image with no Cs. Turing's comment 'C is left unmarked' does not seem to make sense.) Uses none. Used by **anf** , **kom** , **sim** , and **mk**. The use of con( _A_ , _a_ ) in **sim** and **mk** employs the final position it reaches.
#### 7. Operation of the Universal Machine U
The routines which comprise the operation of U are entered in succession, except for the process of searching for the relevant instruction, which has its own loop. The initial state of the machine is b, remembering that the instruction table for the Turing machine which U is emulating must already be on the tape. The starting position is immaterial.
**b**
The beginning of U's operation. It writes the symbols ': D A' in the non-erasing squares after the symbol :: that signifies the end of the instructions which are already on the tape. This is the image of the initial state of the emulated machine T and consists of the coding for 'state 1' with no following symbol images, meaning that the initial tape of **T** is blank.
This operation uses **f** ( _A_ , _B_ , _a_ ).
**anf**
Presumably this is from the German _Anfang_ , or beginning. It is the start of the process of generating the next image of **T**. The process will return to **anf** when one new image has been appended on **U** 's tape, so that building of successive images continues. Its action is **q** ( **anf** 1, :) which finds the last ':' and then anf1 uses **con** ( **kom** , _y_ ) to mark the last M-S pair on the tape with ' _y_ ' and go to state **kom**. Thus the machine state and scanned symbol of the last image on the tape have been marked. Initially this results in marking just the 'D' 'A', but there is an error in this design because absence of a symbol image means that con will fail, since it looks only for symbols 'A' or 'D' when in its internal state con1. This is easily corrected: see Section 8.
This operation uses **con** ( _A_ , _a_ ) and **q** ( _A_ , _a_ ).
**kom**
From its position in the image region, the machine moves left, looking for either ';' or ' _z_ '. It will find ';' at the start of the last instruction, provided that the termination of the instruction area is shown as just '::' and not '; ::' (the latter is implied by Turing). Also, if all instructions are to be available, the first instruction must begin with ';'. The correct designation of a single instruction should be ';' followed by the five parts, and not as shown on p. 68 of Turing's treatment. With these changes, **kom** will ignore any ';' which is marked with ' _z_ '. ' _z_ ' signifies that the instruction which follows has already been tried. The rightmost unmarked ';' having been found, this symbol is marked with ' _z_ ' and the routine **con** ( **kmp** , _x_ ) is used to mark the following M-S pair with ' _x_ '. Each time that **kom** is used, the next instruction to the left will be processed.
If no instruction matches the current state and symbol of **T** , meaning that the machine is badly defined, the search for a colon will run off the left end of the tape. This bug is fixed in Section 9.
This operation uses **con** ( _A_ , _a_ ).
**kmp**
The action of **kmp** is shown by Turing as **cpe** ( **e** ( **kom** , _x, y_ ), **sim** , _x, y_ ). This will compare the sequences marked by ' _x_ ' and ' _y_ ' to discover if the marked instruction actually applies to the M-S pair shown in the current image of **T**. If it does, the state becomes **sim** , which is the start of building the next image. If not, there is a problem with partial erasure of the markings, so these are erased by the **e** ( **kom** , _x, y_ ) operation and we try again, this time trying the next instruction to the left of those tried so far, which have the ' _z_ ' marking. However, there is a bug, explained and corrected in Section 8.
This operation uses **cpe** ( _B_ , _E_ , _a_ , _b_ ) and e( _A_ , _B_ , _a_ ).
**sim**
This routine marks the parts of the leftmost instruction marked with ' _z_ ', which now applies to the next of **T** 's operations to be performed. The leftmost marked colon is located by **f** ′( **sim** 1, **sim** 1, _z_ ) and then **sim** 1 is **con** ( **sim** 2,), which marks the M-S pair with blanks, since this is not required again in this round. The routine con leaves the machine scanning the next non-erasing square to the right of the 'D' in the coding for the new symbol to be written by **T**. The 'D' is marked with ' _u_ ' as are any Cs which follow, so that the new symbol image is marked. The marking with ' _u_ ' continues, marking to the left of each square being examined until an 'A' is found, when the marking is changed to ' _y_ '. Consequently, both the new symbol image and the action (L, R, or N) have been marked with ' _u_ ' and the new state image has been marked with ' _y_ ', which continues until the ';' or '::' terminating the instruction has been reached. Then the ' _z_ ' markings are all removed, since the relevant parts of the instruction for the next stage of T's operation have been marked. The line on p. 71 for state **sim** 2 and symbol 'not A' has an error; see Section 10.
This operation uses **e** ( _B_ , _a_ ), con( _A_ , _a_ ), and **f** ′( _A_ , _B_ , _a_ ).
**mk**
The last ':' is found by means of an operation which should read **q** ( **mk** 1, :) (see Section 10). Moving right from there, the first 'A' is found, which is at the start of the state image, then two non-erasable squares to the left of this point is the end of the preceding symbol image. The start of the whole image might have been found (':'), if this was the initial state, which finishes this part of the marking. If a 'C' is found, this is marked with ' _x_ ' and so are previous Cs moving backwards until a 'D' is found and marked. From this point backward, symbols in non-erasable squares are marked with ' _v_ ' until the ':' that begins the image is found. In this way the symbol image preceding the one currently scanned (if any) has been marked with ' _x_ ' and all the earlier symbols in the image (if any) are marked with ' _v_ '.
In the second stage of marking, the **con** ( _A_ , _a_ ) routine is used to find the M-S pair, marking it with a blank and ending two non-erasable squares beyond the last marked 'D' or 'C' of the symbol image. Two left shifts leave the machine at the start of the symbol image which immediately follows the scanned symbol. From there, all symbols are marked with ' _w_ ' until the end is reached and finally a ':' is placed at the end of the old image, ready for copying out the new image later.
It may be useful to summarize the marking which is now ready for copying out the new image. The instruction has been marked with ' _u_ ' for both the new symbol and the action (R, L, or N) then with ' _y_ ' for the new state. The image has been marked in three places. Starting from the left, its symbol images are marked with ' _v_ ' until the one before the scanned symbol, which is marked with ' _x_ '. Then the state and symbol scanned, the current M-S pair, are unmarked, because they will be superseded. To the right of the scanned symbol, all symbol images are marked with ' _w_ '.
This operation uses **con** ( _A_ , _a_ ), 1( _A_ ), and q( _A_ ,).
**sh**
Before the copying of these marked regions to make a new image, the output of machine **T** is made visible on the tape. The part marked with ' _u_ ' begins with the new symbol (S') to be written on **T** 's tape, which may be 0, coded as D C, or 1, coded as D C C. It is immediately preceded in the instruction by the old symbol that is being overwritten (S). The 0 or 1 is considered as being written if it replaces a space, which is coded as D, so **sh** looks for D–D _u_ C _u_ or D–D _u_ C _u_ C _u_ , using the ' _u_ ' marking to locate this combination. According to which it finds, it writes either ': 0' or ': 1' at the end of the tape.
The convention that 0 or 1 is _output_ if it replaces a space implies that the target machine **T** conforms to the same convention as **U** , which is not to overwrite most symbols, only treating special markers in this way. This is in fact an unnecessary restriction on the programming of **T**. The restriction means that **U** is not really universal. On the other hand, the use of 0 and 1 as special symbols indicating output goes beyond the fundamental definition of a Turing machine. We must assume that the **sh** operation is designed for use in the special case of a target Turing machine with output, the output action consisting of writing 0 or 1 in a square which previously held a space.
This operation uses **f** ( _A_ , _B_ , _a_ ) and **pe** 2( _A_ , _a_ , _b_ ).
**inst**
This routine creates the new image of **T** 's machine and tape and writes it at the end of **U** 's tape, after the final colon. The operation **q** (1( **inst** 1), _u_ ) finds the last ' _u_ ' on the tape and the square that it marks, which contains **L** , **R** , or N. This single ' _u_ ' is deleted because the action symbol L, R, or N must not be copied into the new image. According to the action symbol found, the parts are assembled in one of these sequences, where the machine state ' _y_ ' is shown in bold.
In each case the symbols _preceding_ the active part are marked ' _v_ ' and are copied first (they may not exist). The symbols _following_ the previously scanned symbol are marked ' _w_ ' and are copied last. Without a movement of machine **T** , the new state, marked ' _y_ ', is followed by the new symbol, marked ' _u_ '; and the previous symbol, marked ' _x_ ', stays in its place. With left movement, the new state is placed before the previous symbol. With right movement, the new state is placed after the new symbol, marked ' _u_ '.
The tape image should always end with a 'blank' symbol, which is simply D. Any rewritten symbols within the used portion of **T** 's tape which are deleted will have been overwritten with 'D', but at the end, if the section marked with _w_ s was empty, the action R may leave the state image at the end of the tape. This will cause a matching failure during the next cycle of the emulation if comparison occurs with the M-S pair of an instruction that has a symbol value of 'blank', represented by D. Repairs are made in Section 8.
This operation uses **1** ( _A_ ), **q** ( _A_ , _a_ ), **f** ( _A_ , _B_ , _a_ ), and **ce** 5( _B_ , _a_ , _b_ , _c_ , _d_ , _e_ ).
**ov**
This final operation **e** ( **anf** ) clears all markings and returns to **anf** to begin once again the process of generating a new tape image. Since the ' _z_ ' markings were cleared by **sim** and the **ce** 5 operation clears all its markings, there seems no need for **ov** , but it does no harm.
This operation uses **e** ( _A_ ).
#### 8. The Interesting Errors
The first phase of an evolution of **T** is to find the relevant instruction. This is done by marking the current state-symbol pair of **T** with ' _y_ ' and the state-symbol pair of an instruction with ' _x_ ', then using the **cpe** operation to compare the marked strings. The process of **cpe** deletes some of the ' _x_ ' and ' _y_ ' markings. When comparison fails on one instruction the machine moves on to the next. This comparing process is shown in the table for U on p. 71 of Turing's paper as
**kmp** **cpe** ( **e** ( **kom** , _x_ , _y_ ), **sim** , _x_ , _y_ )
The **e** operation is intended to delete all the remaining ' _x_ ' and ' _y_ ' markings. In fact this is not quite how erasure works as defined on p. 64 and the correct form would be **e** ( **e** ( **kom** , _x_ ), _y_ ). But there is a more serious error in returning to kom, since the essential ' _y_ ' marking will not be restored. Returning to anf will repair this error. The correct definition of **kmp** should be
**kmp** **cpe** ( **e** ( **e** ( **anf** , _x_ ), _y_ ), **sim** , _x_ , _y_ )
To introduce the second of these interesting errors, it is instructive to look at the penultimate step in the copying out of the evolved new snapshot of **T**. This has been reduced, by Turing's clever scheme of skeleton tables, to a choice of one of three copy instructions on p. 72, such as, for example
**inst** 1( **R** ) **ce** 5( **ov** , _v_ , _x_ , _u_ , _y_ , _w_ )
This copies five marked areas from the current instruction and the last image of **T** in the sequence to create the new image, for the case where the machine moves right. The part marked ' _y_ ' is the new machine state, ' _v_ ' and ' _x_ ' form the string of **T** -symbols to the left, ' _u_ ' is the newly printed symbol, and ' _w_ ' the string of **T** -symbols to the right. Because ' _u_ ' replaces an existing symbol, the number of symbols (including blanks) on T's conceptual tape has not changed→ The same is true for left and null movement. There must be something wrong in an emulation in which the emulated machine can never change the number of symbols on its tape.
The image shows just the occupied part of **T** 's tape, and this is conceptually followed by an unlimited set of blank symbols, which are the tape as yet unused. The number of symbols in the image of **T** will increase by moving right from the last occupied square and writing on the blank square. After a move right onto blank tape, there will be no string marked ' _w_ ', so machine state ' _y_ ' will be the last thing in the image.
This will lead to a failure of the emulation at the next evolution because the state-symbol pair of the image, due to be marked with ' _y_ ' during the search for the relevant instruction, is incomplete.
The remedy is to print a new blank symbol for **T** at the end of the image, when the move has been to the right and there is no **T** -symbol there. The necessary corrections, on p. 72, to the table for **U** are:
In the case of a move right, after copying the parts of the previous image, the operation **q** finds the last 'A' on the tape, which is the end of the state-symbol copied from markings _y_. If there is a **T** -symbol to its right, there is no problem. After two right moves of **U** , if a 'D' is found there is a **T** -symbol but, if not, by printing 'D', a new blank tape square is added to the image of **T**. In this way T's conceptual tape is extended and the state-symbol pair is made complete.
The error perhaps arose because the endless string of blank symbols on **U** 's tape was taken as sufficient for the purpose of **T**. But for the emulation a blank square is shown as D. Machines **U** and **T** represent a blank tape differently.
There is a corresponding error in the way the initial state of **T** is placed on **U** 's tape. It should contain the **U** -symbols: DAD with suitable spaces between them, representing **T** 's initial state DA followed by the scanned symbol D, a blank. The correction on p. 70 is:
**b** 1 R, R, P:, R, R, PD, R, R, PA, R, R, PD **anf**
#### 9. Diagnostics
With experience of writing programs it is second nature to build in diagnostics. Whether they are needed in **U** is arguable. Since **U** is a conceptual tool, its requirements are determined by its use in the argument of Turing's paper. For testing the design of **U** , diagnostics are certainly needed.
There may be a need for two kinds of failure indication in the program of **U**. Suppose that **T** has a deficient set of instructions, meaning that its latest image has a state-symbol pair which does not appear among the instructions. I believe that Turing would class this as a _circular_ machine. The effect on the operation of **U** is that the search for the relevant instruction fails with **U** moving left beyond the left-hand end of its tape, and continuing to move left indefinitely. Perhaps this is acceptable for the purpose for which **U** was intended, but it seems anomalous that a deficiency in **T** should cause **U** to misbehave. It can be avoided by adding a line to the definition of **kom** on p. 71:
**kom** e fail1 (deficient T instructions)
then changing the next line to respond to symbols _not z nor_ ; _nor_ **e**.
If **T** moves right without limit, this will be emulated correctly, but moving left beyond the limits of its tape is a problem. The way **U** works will cause the next image of T to appear as if no shift had occurred. There is no way to represent **T** as scanning a square to the left of its starting position. This means that the subsequent behaviour of T will differ from what its instructions imply. I think this might affect the use that Turing made of the machine in the main part of the paper.
The changes to deal with this problem are:
#### 10. Trivial Errors and Corrections
1. There is potential confusion in the use of the symbol **q** for different states in two places, and it is also confused with state **g**. The best resolution is as follows.
We can treat the use of **q** in the example on p. 62 as casual, without permanent significance. The same might be said of its use on p. 64, which is unrelated. But from there onwards the examples will form part of the definition of **U** , so the symbols have global significance.
On p. 66, the states **q** and **q** 1 appear but, in their subsequent uses in **U** , they have been replaced by **g** , for example in the definitions of **anf** , **m** k, and **inst**. I have retained the notation **q** , while remembering that previous uses of this symbol are unrelated.
2. The skeleton tables for **re** and **cr** on p. 65, which comprise five different states, are redundant, serving no illustrative purpose and not being used again.
3. On p. 68, the format of the instruction table of **T** , as written on the tape of **U** , is described. Instructions are separated by semicolons. An example DADDCRDDA;DAA... DDRDA; is given. As already explained, this is misleading, because each instruction should be preceded by a semicolon. The example should begin with a semicolon, not end with one.
4. In the explanation of the skeleton table for **con** (p. 70), 'C' is one of the symbols being read and marked. But the words refer to 'the sequence C of symbols describing a configuration'. The final remarks '... to the right of the last square of C. C is left unmarked.' use 'C' in the second sense. It would otherwise seem as if the final symbol 'C' was left unmarked, but this is not so. To clarify, replace by 'the sequence S' and '... last square of S. Configuration S is left unmarked.'
5. On p. 71, a line for **sim** 2 should read:
6. On p. 71, the line for **mk** should read:
7. On p. 72, the line for **inst** 1(N) should read:
#### 11. A Redesign of the Universal Machine
To verify, as far as this is possible, that there are no remaining errors in the amended version of Turing's program for **U** , it would be best to generate the explicit machine instruction table by substitutions and repetitions, then run this machine with one or more examples of a machine **T** and find if the emulations behaved as they should. But the complexity and slowness of the explicit form of Turing's **U** makes this difficult.
Therefore I made some changes to the design of **U** before constructing a simulation of a Turing machine, loading the instructions for **U** , producing a tape image for a machine **T** and running the program. After some corrections to my version of **U** , the simulation behaved correctly. In this section the main features of the redesign are described.
The new version of **U** follows Turing's methods quite closely. The substitution process introduced with the skeleton tables had been nested to a depth of 9, causing a proliferation of states and instructions in the explicit machine. To avoid this, no skeleton tables were used in the new version and this allowed the procedures to be optimized for each application. The downside is that the 'low-level' description of **U** which results takes up more space than the original and is harder to understand and check for accuracy. There are 147 states and 295 instructions in the new version, an enormous reduction.
The representation of **T** 's states and symbols in a monadic notation such as DAAAA was replaced by a binary notation. This was an easy change that reduced the length of the workspace used. Because nearly all the time is used moving from end to end of the workspace, this is worthwhile. The small cost of the change is that there are four **U** -symbols to represent states and symbols instead of three.
The classic Turing machine can move right or left or stay put in each operation. To simplify **U** a little, the third option was removed, so that a left or right movement became mandatory. For consistency, **U** was also run on a machine **T** with this characteristic. In the whole of **U** 's program, a compensating movement became necessary only a few times, so it is not a significant restriction.
Turing's skeleton tables show, for the scanned symbol, such words as _any_ , or _not A_. When translated into a list of discrete symbols for the explicit machine these generate many instructions. By introducing a 'wild card' notation and searching instructions in a definite sequence, this proliferation can be avoided. A form of instruction was added which, in its written form, had an asterisk for both the scanned symbol and the written symbol. This acted on any scanned symbol and did not overwrite it. The way that **U** worked would have made it possible to read a wild card (i.e. any) scanned symbol and write over it or to read a specific symbol and leave it unchanged, but these were never needed in practice. The wild card scanned symbol should only be actioned after all other possibilities (for this particular state) have been tested. Therefore instructions now have a defined sequence and must be tested accordingly. **U** always did test instructions in sequence but never made use of that fact.
Testing all instructions in sequence to find a match is very time-consuming because it requires marking, then comparing square by square, running from instruction space to work space. It was largely avoided by writing in **U** 's instruction table an _offset_ which indicated where the next instruction could be found. This indication led to a section of instructions dealing with a given state; after this, sequential testing took place. This was a shortcut to speed up **U** and was not envisaged as a feature of all Turing machines, since it would greatly complicate **U**. Technically it was a little more complex than I have described, but it has no effect on the design of **U** , being merely a chore for the programmer and a detail of the computer program which interprets those instructions.
**U** spends some of its time searching for a region on the tape where it will begin work. To make this easier, additional markers were introduced, for the action symbol (L or R) and for the start of the current snapshot. Also, the end of the workspace was marked, and this marking was placed in one of the squares normally reserved for permanent symbols. Since it had to be overwritten when the workspace extended, this broke one of Turing's conventions.
Finally the two failure-indications described earlier were incorporated, one for a deficient **T** -instruction set and the other for **T** running its machine left, beyond the usable tape.
##### _11.1 Testing the redesigned machine_
A computer program, which I shall call **T** *, was written which would simulate the underlying Turing machine, using a set of instructions in its own special code, which had one byte per symbol or state. This code was chosen for convenience of writing **U** 's instructions. It incorporated the wild card feature and the offset associated with each instruction, but the offset did not alter the way it responded to its instructions, only making it faster. When the design of **U** is complete and its instructions have been loaded, **T** * will behave as the universal machine **U**.
A simple editor was written to help the user write and amend the instruction tables for **T** * and prepare a starting tape for **T** * which holds the coded instructions for the emulated Turing machine **T**.
For **T** , the example given by Turing on p. 62 was used. It prints a sequence of increasing strings of ones, such as 001011011101111011111... This program in its explicit form would have 23 instructions and 18 states. To make it simpler, it was rewritten without the 'alternate squares' principle and it then had 12 instructions and 6 states. It may be interesting to see how the wild card feature operates by studying this example, shown below.
As a first step, the example was loaded into the program space of **T** * and run, thus testing the mechanism of **T** * as well as the example in the table below.
Then the example was coded for the initial part of the tape of **T** *, so that it would cause **U** to emulate it as the target **T**. The program of **U** was loaded in many stages, debugging each by testing its part in the whole operation of **U**. Two serious program errors were found. One was in the operation sh which prints the output of **T** between the snapshots of **T** 's evolution. The other was in the correction to Turing's scheme which wrote a blank symbol (D) at the end of the tape. It had been inserted at the wrong place. With these and several minor errors corrected the redesigned **U** performed as expected and the evolution of **T** agreed with expectation and with its earlier running, directly on **T** *. Only this one example of **T** was tried, but it probably does test the universal machine fully. The full results are given below.
Because of the differences between the version of **U** that was tested and Turing's design with my corrections, the testing must be regarded as incomplete. A compiler could be written to take the design in the form of skeleton tables and generate the explicit machine, which could then be run to emulate examples of the target machine. This would be extraordinarily slow.
##### 12. The Program for T
The instructions for **T** are given in the standard five-part form: state, scanned symbol, written symbol, movement, and resultant state. The images are shown for the first eleven moves, in the standard form with the state-symbol (a to e, printed bold) preceding the scanned symbol.
The blank space symbol is a hyphen and the other symbols are 0, 1, _x_ , and _y_.
The program writes a block of _x_ s followed by a _y_ , then converts the _x_ s successively to 1s and the _y_ to a 0, while writing the next block of _x_ s and a _y_ , increasing the number of _x_ s by one.
##### 13. Results of the Test
Here is a copy of the symbols on the tape of **T** * after 22 evolutions of **U**. The part up to the symbol % represents the 12 instructions for **T**. Then follow the 23 images, separated by colons. Whenever **U** prints a 0 or 1, this is also an output of **T**. To make this explicit (following Turing's practice) the strings '1 :' or '0 :' are inserted into the tape (bold in our table). So the whole set of evolutions shown has printed '0010'. The tape shown is printed on alternate spaces, except for the initial 'ee'. The final F is a device of my own to make it easy to find the end of the written area of tape.
As an aid to understanding this tape, here are the symbols and states of **T** in **U** 's notation:
The first few snapshots therefore read:
s – :0:0 a :0 _y_ b :0 c _y x_ : d 0 _y x_ :0 e _y x_ :0: 00 a _x_ :00 a _x_ : 00 _x y_ b :00 _x_ c _y x_ :00 d _x y x_ :1:001 b _y x_ :001 _y_ b _x_ :001 _y x_ b : 001 _y_ c _x x_ :001 c _y x x_ :
The final configuration of the above tape is 0010 _x x_ a :
##### 14. The Corrected Tables for U: Summary
The table for f( _A_ , _B_ , _a_ ) is unchanged on p. 63.
On p. 64, **e** ( _A_ , _B_ , _a_ ) and **e** ( _B_ , _a_ ) are unchanged, but note that the state **q** used in the explanation of **e** ( _B_ , _a_ ) is a local notation, unrelated to the states of that name on p. 66.
On p. 65, **pe** ( _A_ , _b_ ), **l** ( _A_ ), **f ′** ( _A_ , _B_ , _a_ ), and **c** ( _A_ , _B_ , _a_ ) are unchanged, but **r** ( _A_ ) and **f** ″( _A_ , _B_ , _a_ ), defined on that page, are not used again.
On pp. 65–66, **ce** ( _A_ , _B_ , _a_ ), **ce** ( _B_ , _a_ ), **cp** ( _A_ , _B_ , _C_ , _a_ , _b_ ), **cpe** ( _A_ , _B_ , _C_ , _a_ , _b_ ), and **cpe** ( _A_ , _B_ , _a_ , _b_ ) are unchanged but **re** ( _A_ , _B_ , _a_ , _b_ ), **re** ( _B_ , _a_ , _b_ ), **cr** ( _A_ , _B_ , _a_ ), and **cr** ( _B_ , _a_ ) are not used again.
On p. 66, **q** ( _A_ ), **pe** 2( _A_ , _a_ , _b_ ), and **e** ( _A_ ) are unchanged. Also, **ce** 2( _B_ , _a_ , _b_ ) and **ce** 3( _B_ , _a_ , _b_ , _c_ ) are defined, but it is **ce** 5( _B_ , _a_ , _b_ , _c_ , _d_ , _e_ ), derived in an analogous way, which is actually used, in the inst function.
On p. 70, **con** ( _A_ , _a_ ) is unchanged, but the remark that 'C is left unmarked' is confusing and is best ignored.
In the table for **U** , which begins on p. 70, the state **b** 1 should have the following action: R, R, P :, R, R, PD, R, R, PA, R, R, PD, in order to print ': D A D' on the F squares, so that a blank symbol D is available for matching with an instruction.
On p. 70 the table for **anf** should lead to **q** ( **anf** 1, :).
If the set of instructions for the target machine **T** is deficient, so that a state-symbol pair is created which has no matching instruction, machine **U** will attempt to search beyond the left-hand end of its tape. What happens then is undefined. To make it definite, **kom** (p. 71) can be augmented by the line:
which indicates the failure, and the last line will be:
The table for kmp (p. 71) should read:
since e( _A_ , _B_ , _a_ ) should return to **anf** , to restore the markings deleted by **cpe**.
On p. 71, **sim** 2 with scanned symbol 'not A' should have the action L, P _u_ , R, R, R.
The first line of **mk** (p. 71) should lead to **q** ( **mk** 1, :). On this same page, **sh** is unchanged.
On p. 72, **inst** should lead to **q** ( **l** ( **inst** 1), _u_ ) and the line for **inst** 1(N) should read
The instruction for **inst** 1(L) (p. 72) could try to move the target machine left beyond its end of tape, but there is no way for **U** to represent this condition, so **T** will seem not to move. To make this kind of error explicit, these changes can be made:
To correct the fundamental flaw that a right movement inst1(R) (p. 72) could move the state-symbol to the right of all other symbols, making a future match with an instruction impossible, the following change is needed:
Finally, **ov** (p. 72) is unchanged.
### **CHAPTER 3
Systems of Logic Based on Ordinals ( _1938_ )**
_Alan Turing_
#### Introduction
##### _Jack Copeland_
#### The Princeton Years, 1936–38
On 23 September 1936 Turing left England on a vessel bound for New York. His destination was Princeton University, where the Mathematics Department and the Institute for Advanced Study combined to make Princeton a leading centre for mathematics. Turing had applied unsuccessfully for a Visiting Fellowship to Princeton in the spring of 1935. When a year later he learned of Church's work at Princeton on the _Entscheidungsproblem_ , which paralleled his own (see 'Computable Numbers: A Guide'), Turing 'decided quite definitely' to go there. He planned to stay for a year.
In mid-1937 the offer of a Visiting Fellowship for the next academic year persuaded him to prolong his visit, and he embarked on a Ph.D. thesis. Already advanced in his academic career, Turing was an unusual graduate student (in the autumn of 1937, he himself was appointed by Cambridge University to examine a Ph.D. thesis). By October 1937 Turing was looking forward to his thesis being 'done by about Christmas'. It took just a little longer: 'Systems of Logic Based on Ordinals' was accepted on 7 May 1938 and the degree was awarded a few weeks later. The following year the thesis was published in the _Proceedings of the London Mathematical Society_.
'Systems of Logic Based on Ordinals' was written under Church's supervision. His relationship to Turing—whose formalization of the concept of an effective procedure and work on the _Entscheidungsproblem_ was 'possibly more convincing' than Church's own—was hardly the usual one of doctoral supervisor to graduate student. In an interview given in 1984, Church remarked that Turing 'had the reputation of being a loner' and said: 'I forgot about him when I was speaking about my own graduate students—truth is, he was not really mine.' Nevertheless Turing and Church had 'a lot of contact' and Church 'discussed his dissertation with him rather carefully'. Church's influence was not all for the good, however. In May 1938 Turing wrote:
My Ph.D. thesis has been delayed a good deal more than I had expected. Church made a number of suggestions which resulted in the thesis being expanded to an appalling length. I hope the length of it won't make it difficult to get it published.
Moreover, Turing elected to couch 'Systems of Logic Based on Ordinals' in the notation of Church's lambda calculus, so making his work much less accessible than it might otherwise have been. (By that time even Church's student Kleene, who had contributed importantly to the development of the lambda calculus, had turned away from it. Kleene said: 'I myself, perhaps unduly influenced by rather chilly receptions from audiences around 1933–35 to disquisitions on Λ-definability, chose, after general recursiveness had appeared, to put my work in that format.') In a letter written not long after Turing's death, Turing's friend Robin Gandy said: 'Alan considered that his paper on ordinal logics had never received the attention it deserved (he wouldn't admit that it was a stinker to read).'
Notwithstanding its notational obscurity, 'Systems of Logic Based on Ordinals' is a profound work of first rank importance. Among its achievements are the exploration of a means of circumventing Gödel's incompleteness theorems; the introduction of the concept of an 'oracle machine', thereby opening the field of relative computability; and, in the wake of the demolition of the Hilbert programme (by Gödel, Turing, and Church), an analysis of the place of intuition in mathematics and logic.
Turing's two years at Princeton are the best documented of his life, thanks to a series of letters that he wrote to Sara Turing. (Of the fifty-five letters that he sent her from 1932 until his death, twenty-seven are from the Princeton period.) The following excerpts give a glimpse of his time there. All were written from the Graduate College, Princeton University.
#### Excerpts from Turing's Letters Home
##### _6 October 1936_
I reached here late last Tuesday evening. We were practically in New York at 11:00 a.m. on Tuesday but what with going through quarantine and passing the immigration officers we were not off the boat until 5:30 p.m. Passing the immigration officers involved waiting in a queue for over two hours with screaming children round me. Then, after getting through the customs I had to go through the ceremony of initiation to the U.S.A., consisting of being swindled by a taxi driver. I considered his charge perfectly preposterous, but as I had already been charged more than double English prices for sending my luggage, I thought it was possibly right. However, more knowing people say it was too much....
The mathematics department here comes fully up to expectations. There is a great number of the most distinguished mathematicians here. J. v. Neumann, Weyl, Courant, Hardy, Einstein, Lefschetz, as well as hosts of smaller fry. Unfortunately there are not nearly so many logic people here as last year. Church is here of course, but Gödel, Kleene, Rosser and Bernays who were here last year have left. I don't think I mind very much missing any of these except Gödel. Kleene and Rosser are, I imagine, just disciples of Church and have not much to offer that I could not get from Church. Bernays [I] think is getting rather 'vieux jeu': that is the impression I get from his writing, but if I were to meet him I might get a different impression.
The graduate students include a very large number who are working in mathematics, and none of them mind talking shop. It is very different from Cambridge in that way.
I have seen Church two or three times and I get on with him very well. He seems quite pleased with my paper and thinks it will help him to carry out a programme of work he has in mind. I don't know how much I shall have to do with this programme of his, as I am now developping [ _sic_ ] the thing in a slightly different direction, and shall probably start writing a paper on it in a month or two. After that I may write a book.
The proofs have been sent direct to me here. They arrived last Saturday, and I have just finished them and sent them off. It should not be long now before the paper comes out. I have arranged for the reprints to be sent to you, and will get you, if you would not mind, to send out the ones that are to go to people in Europe, and to send some of the remainder on to me....
These Americans have various peculiarities in conversation which catch the ear somehow. Whenever you thank them for any thing they say 'You're welcome'. I rather liked it at first, thinking that I was welcome, but now I find it comes back like a ball thrown against a wall, and become positively apprehensive. Another habit they have is to make the sound described by authors as 'Aha'. They use it when they have no suitable reply to a remark, but think that silence would be rude.
Maurice Pryce has just got a Fellowship at Trinity.
##### _14 October 1936_
I have just discovered a possible application of the kind of thing I am working on at present. It answers the question 'What is the most general kind of code or cipher possible', and at the same time (rather naturally) enables one to construct a lot of particular and interesting codes. One of them is pretty well impossible to decode without the key and very quick to encode. I expect I could sell them to H. M. Government for quite a substantial sum, but am rather doubtful about the morality of such things....
Church had me out to dinner the other night. Considering that the guests were all university people I found the conversation rather disappointing. They seem, from what I can remember of it, to have discussed nothing but the different States that they came from. Description of travel and places bores me intensely.
I had a nasty shock when I got into Church's house. I think I had told you that Church was half blind in one eye. Well I saw his father in the house and he was quite blind (and incidentally very deaf). I should have thought very little of it had it not been for Church being rather blind himself. Any hereditary defects of that kind give me the shudders.
Hardy is here for this term. At first he was very standoffish or possibly shy. I met him in Maurice Pryce's rooms the day I arrived, and he didn't say a word to me. But he is getting much more friendly now.
##### _3 November 1936_
Church has just suggested to me that I should give a lecture to the Mathematical Club here on my Computable Numbers. I hope I shall be able to get an opportunity to do this, as it will bring the thing to people's attention a bit....
I have got one or two things on hand at present not connected with my work in logic, but in theory of groups. One of them is something I did about a year ago and left in cold storage, and which Baer thinks is quite useful; but of course am not taking these things so seriously as the logic.
Tonight is the evening of election day and all results are coming out over the wireless ('radio' they say in the native language). My method of getting the results is to go to bed and read them in the paper next morning.
##### _11 November 1936_
One of the Commonwealth Fellows, Francis Price (not to be confused with Maurice Pryce or Bobby Price) arranged a hockey match the other day between the Graduate College and Vassar, a women's college (amer.)/university (engl.) some 130 miles away. He got up a team of which only half had ever played before. We had a couple of practice games and went to Vassar in cars on Sunday. It was raining slightly when we arrived, and what was our horror when we were told the ground was not fit for play. However we persuaded them to let us play a pseudo-hockey game in their gymn. at wh. we defeated them 11–3. Francis is trying to arrange a return match, which will certainly take place on a field.
##### _22 November 1936_
I am sending you some cuttings about Mrs Simpson as representative sample of what we get over here on this subject. I don't suppose you have even heard of her, but some days it has been 'front page stuff' here.
The hockey here has become a regular fixture three days a week. It's great fun.
##### _1 December 1936_
I spent a good deal of my time in New York pottering about Manhattan getting used to their traffic and subways (underground). I went to the Planetarium....
I am giving my lecture to the Maths Club tomorrow.
##### _3 December 1936_
I am horrified at the way people are trying to interfere with the King's marriage. It may be that the King should not marry Mrs Simpson, but it is his private concern. I should tolerate no interference by bishops myself and I don't see that the King need either.
##### _11 December 1936_
I suppose this business of the King's abdication has come as rather a shock to you. I gather practically nothing was known of Mrs Simpson in England till about ten days ago. I am rather divided on my opinion of the whole matter. At first I was wholly in favour of the King retaining the throne and marrying Mrs Simpson, and if this were the only issue it would still be my opinion. However I have heard talk recently which seems to alter it rather. It appears that the King was extremely lax about state documents leaving them about and letting Mrs Simpson and friends see them. There had been distressing leakages. Also one or two other things of same character, but this is the one I mind about most.
##### _December 1936 (no day)_
... Talking of Christmas reminds me that as a small child I was quite unable to predict when it would fall, I didn't even realise that it came at regular intervals.
##### _1 January 1937_
I have been away with Maurice skiing in New Hampshire....
I am sorry that Edward VIII has been bounced into abdicating. I believe the Government wanted to get rid of him and found Mrs Simpson a good opportunity. Whether they were wise to try to get rid of him is another matter. I respect Edward for his courage. As for the Archbishop of Canterbury I consider his behaviour disgraceful. He waited until Edward was safely out of the way and then unloaded a whole lot of quite uncalled-for abuse. He didn't dare do it whilst Edward was King. Further he had no objections to the King having Mrs Simpson as a mistress, but marry her, that wouldn't do at all. I don't see how you can say that Edward was guilty of wasting his ministers time and wits at a critical moment. It was Baldwin who opened the subject.
There was rather bad attendance at the Maths Club for my lecture on Dec 2. One should have a reputation if one hopes to be listened to. The week following my lecture G. D. Birkhoff came down. He has a very good reputation and the room was packed. But his lecture wasn't up to standard at all. In fact everyone was just laughing about it afterwards.
##### _27 January 1937_
I have just finished a paper in group theory; not a very exciting one this time. I shall send it off in a day or two to the L. M. S or possibly to the 'Annals of Mathematics', which is the Princeton mathematical journal.
There was a problem in the 'Caliban' volume of the N S & N a few weeks ago set by Eddington. It was phrased in Alice through the Looking Glass language and called 'Looking Glass Zoo'. The solution picked out for publication was also in looking glass language and sent by 'Champ', i.e. Champernowne. It started off 'There couldn't have been more than three girls' reflected Humpty Dumpty 'because a girl is always the square root of minus one, and there are only 12 of those, they taught us that at school'....
Maurice and Francis Price arranged a party with a Treasure Hunt last Sunday. There were 13 clues of various kinds, cryptograms, anagrams and others completely obscure to me. It was all very ingenious, but I am not much use at them.
##### _11 February 1937_
The printers for the L. M. S have been rather inefficient, sending the reprints straight on to me instead of looking at the address I had filled up on their form. Unfortunately I had not kept a second copy of all the addresses I gave you, so as they are rather tiresome to find I am sending some of the reprints back to you to deal with if you can find time before you go.... I have dealt with
All King's addresses
Littlewood
Wittgenstein
Newman
Atkins
Eperson
I am told that Bertrand Russell is inclined to be ashamed of his peerage, so the situation calls for tact. I suggest that the correct address for an earl be used on envelope, but that you mark the reprint itself 'Bertrand Russell' on the top right hand corner of the cover.
##### _22 February 1937_
I went to the Eisenhart's regular Sunday tea yesterday, and there they took me in relays to try and persuade me to stay another year. Mrs Eisenhart mostly put forward social or semi-moral semi sociological reasons why it would be a good thing to have a second year. The Dean weighed in with hints that the Procter Fellowship was mine for the asking (this is worth $2,000 p.a.). I said I thought King's would probably prefer that I return, but gave some vague promise that I would sound them on the matter. Whether I want to stay is another matter. The people I know here will all be leaving, and I don't much care about the idea of spending a long summer in this country.... I think it is most likely I shall come back to England.
I have had two letters asking for reprints, one from Braithwaite at King's and one from a proffessor [ _sic_ ] in Germany... They seemed very much interested in the paper. I think possibly it is making a certain amount of impression. I was disappointed by its reception here. I expected Weyl who had done some work connected quite closely with it some years ago at least to have made a few remarks about it.
##### _15 March 1937_
I only wrote to the Provost last week so don't expect to hear from him just yet. I was rather diffident and apologetic and told him most probably I should be coming back.
##### _29 March 1937_
I have been sent a notice of lecturers in mathematics to be appointed next term, by Philip Hall. Maurice and I are both putting in for it, though I don't suppose either of us will get it: I think it is a good thing to start putting in for these things early, so as to get one's existence recognised. It's a thing I am rather liable to neglect. Maurice is much more conscious of what are the right things to do to help his career. He makes great social efforts with the mathematical bigwigs....
I am now working out some new ideas in logic. Not so good as the computable numbers, but quite hopeful.
##### _18 April 1937_
The temperature here is going right up already. It's almost like June now. Tennis has started. They play on courts of dry clay. Easier on the feet (and probably on the pocket) than our hard courts, but not very quick at recovering from showers.
There was a return hockey match on Sunday against Vassar, who came over to us this time. We defeated them quite easily, but I think only because we could run faster.
I shall certainly be coming back in July.
##### _19 May 1937_
I have just made up my mind to spend another year here, but I shall be going back to England for most of summer in accordance with previous programme. Thank you very much for your offer of help with this: I shall not need it, for if I have this Procter as the Dean suggests I shall be a rich man, and otherwise I shall go back to Cambridge. Another year here on the same terms would be rather an extravagance. I don't think there can be any reasonable doubt I shall have the Procter: the Dean would hardly have made any remarks about it unless they meant something....
My boat sails June 23. I might possibly do a little travelling here before the boat goes, as there will be very little doing here during the next month and it's not a fearfully good time of year for work. More likely I shall not as I don't usually travel for the sake of travelling.
##### _15 June 1937_
Have just been back from Cousin Jack's a couple of days. I went up north with Maurice in his car and Maurice stayed a night with Cousin Jack, and made a good impression there. I enjoyed the time I spent at Cousin Jack's. He is an energetic old bird. He has a little observatory with a telescope that he made for himself. He told me all about the grinding of mirrors.... I think he comes into competition with Aunt Sybil for the Relations Merit Diploma. Cousin Mary is a little bit of a thing you could pick up and put in your pocket. She is very hospitable and rather timid: she worships Cousin Jack. Cousin Mary's sister Annie also lives in the house. I forgot her surname very soon after I was told it, which put me at rather a disadvantage.
I am just starting in on packing etc.
##### _4 October 1937_
Journey completed without any mishap more serious than loss of my fountain pen a few hours after getting on board....
A vast parcel of manuscript arrived for me from the L. M. S secretary the other day. It was a paper for me to referee; 135 pages. Also have just heard from Bernays.
##### _19 October 1937_
The refereeing business rather petered out. The author's mathematical technique was hopelessly faulty, and his work after about p. 30 was based on so many erroneous notions as to be quite hopeless. So I had to send it back and say so. Rather distressing as the man has apparently been working on it for 18 months or so.
I am working on my Ph.D thesis now. Should have it done by about Christmas.
Scholz of Münster sent me a photolithoprinted reprint the other day, containing the gist of my paper in the L. M. S, apparently as 'vorgetragt' in Münster. It was most delightfully done, with most excellent translations into the German of the expressions I had used.
##### _2 November 1937_
Have just been playing in a hockey match, the first we have had this year. The team is not so good as last year's, our two brilliant players from New College no longer being in Princeton. I have found I get involved with making a good deal of arrangements for these games, but it has not yet got to the point of being really tiresome....
I am getting rather more competent with the car...
##### _23 November 1937_
I had a letter the other day from the Secretary of the Faculty Board of Mathematics at Cambridge asking if I would be a Ph.D examiner: the candidate is the same man whose paper I refereed for L. M. S. After some hesitation I decided to take this thing on. I thought it might be rather unsuitable for me to be connected with it twice, but I talked to Newman (who is here for a term) about it, and he thought such scruples were rather foolish....
There is a mysterious woman in Virginia who has invited me to stay for Christmas. She gets the names of Englishmen living in the Graduate College from Mrs Eisenhart.
##### _c. New Year_
Did I tell you that a very nice man called Martin (the i is mute in this country) asked me to go and stay with him in South Carolina before Christmas. We drove down from here in two days and then I stayed there for two or three days before I came back to Virginia to stay with Mrs Welbourne. It was quite as far south as I had ever been—about 34o. The people seem to be all very poor down there still, even though it is so long since the civil war.
Mrs Welbourne and her family were all very agreeable, though I didn't make much conversational progress with any of them.
Two short papers of mine have just come out in the Journal of Symbolic Logic.
##### _7 March 1938_
I went to 'Murder in the Cathedral' last Saturday with Will James. Was very much impressed with it. It was very much easier to understand when acted than when read. Most particularly this was so with the choruses....
I can't say for certain yet when I will be back. I haven't yet booked a passage. Most probably it will be about the same time and I shall be going up for the Long again.
##### _12 April 1938_
Have found out now about my Fellowship: it has been renewed. When Daddy wrote me about getting a job here I thought it was time to get King's to say something definite, so I sent them a cable. I can't think why they didn't let me know before. They are usually rather strong on formal notifications: it all seems rather out of character.
I have just been to see the Dean and ask him about possible jobs over here; mostly for Daddy's information, as I think it unlikely I shall take one unless you are actually at war before July. He didn't know of one at present, but said he would bear it all in mind.
Have just been down to Washington and Annapolis with Will James. Will went to visit some people who are running St John's College Annapolis, and we both went to lunch there. They have a scheme in operation for teaching people by making them read a vast syllabus of 100% concentrated classics. Kant's 'Critique of Pure Reason' is a fairly typical example. The trouble about it is that they are so deep that any one of them really needs several years study to be understood. Presumably their undergraduates will only get something very superficial out of them.
We also went and listened to the Senate for a time. They seemed very informal. There were only six or eight of them present and few of them seemed to be attending.
##### _7 May 1938_
There was quite a good performance of 'H. M. S. Pinafore' and 'Trial by Jury' here last week-end. The 'Pinafore' didn't seem to be so good as when we saw it in Hertford (and picked up measles). 'Trial by Jury' was very good: I think I like it better than any other Gilbert and Sullivan.
My Ph.D. thesis has been delayed a good deal more than I had expected. Church made a number of suggestions which resulted in the thesis being expanded to an appalling length. I hope the length of it won't make it difficult to get it published. I lost some time too when getting it typed by a professional typist here. I took it to a firm which was very well spoken of, but they put a very incompetent girl onto it. She would copy things down wrong on every page from the original, which was almost entirely in type. I made long lists of corrections to be done and even then it would not be right....
The thesis has just been accepted to-day.
I expect to leave here at the beginning of July. Shall probably go direct to Cambridge.
I had an offer of a job here as von Neumann's assistant at $1,500 a year but decided not to take it.
##### The Purpose of Ordinal Logics
Turing explained the purpose of his 'ordinal logics' in Section 11 of 'Systems of Logic Based on Ordinals'. He first distinguished between what he called 'intuition' and 'ingenuity' (a distinction that is discussed again in Chapter 4 in his letters to Newman):
Mathematical reasoning may be regarded rather schematically as the exercise of a combination of two faculties, which we may call _intuition_ and _ingenuity_. The activity of the intuition consists in making spontaneous judgments which are not the result of conscious trains of reasoning. These judgments are often but by no means invariably correct (leaving aside the question what is meant by 'correct'). Often it is possible to find some other way of verifying the correctness of an intuitive judgment. We may, for instance, judge that all positive integers are uniquely factorizable into primes; a detailed mathematical argument leads to the same result. This argument will also involve intuitive judgments, but they will be less open to criticism than the original judgment about factorization. I shall not attempt to explain this idea of 'intuition' any more explicitly.
The exercise of ingenuity in mathematics consists in aiding the intuition through suitable arrangements of propositions, and perhaps geometrical figures or drawings. It is intended that when these are really well arranged the validity of the intuitive steps which are required cannot seriously be doubted.
The parts played by these two faculties differ of course from occasion to occasion, and from mathematician to mathematician. This arbitrariness can be removed by the introduction of a formal logic. The necessity for using the intuition is then greatly reduced by setting down formal rules for carrying out inferences which are always intuitively valid. When working with a formal logic, the idea of ingenuity takes a more definite shape. In general a formal logic will be framed so as to admit a considerable variety of possible steps in any stage in a proof. Ingenuity will then determine which steps are the more profitable for the purpose of proving a particular proposition. (p. 192)
The intuition/ingenuity distinction is illustrated by the Gentzen-style formal logic described in 'Computable Numbers: A Guide' (see the subsection 'A tutorial on first-order predicate calculus'). Most people see intuitively that, for example, the rule
is valid. In order to grasp that the rule is valid, it is necessary only to reflect on the rule's meaning, namely:
If _Y_ can be concluded from _X_ , then it can be concluded that not _both X_ and the denial of _Y_ are true.
With only a dozen or so basic rules, this formal logic places very little demand on intuition. Once one has accepted these few rules as valid, proofs can be constructed in the system without the need for any further exercise of intuition. Nevertheless, constructing proofs can place considerable strain on one's ingenuity. To form a proof one must—playing always by the rules—devise a chain of propositions culminating in the proposition that is to be proved. As Turing remarked, at each point in the chain there are always various possibilities for the next move. Typically, most of these possible moves are of no help at all, and it may require significant ingenuity to find a sequence of moves that leads to the desired conclusion.
Important though ingenuity is in practice, it is in principle unnecessary so long as unlimited time and paper are available. This perhaps surprising fact is clear from 'On Computable Numbers'. Once intuition has supplied the materials from which proofs are to be constructed—the basic inference rules, in the case of the logical system under discussion—then a suitably programmed Turing machine is able to grind out all the valid proofs of the system one by one. No ingenuity is required to apply the rules of the system blindly, making legal move after legal move. If a proposition is provable in the system then a machine operating in this 'blind' fashion will sooner or later prove it (so long as the machine is programmed in such a way that no legal moves are missed):
We are always able to obtain from the rules of a formal logic a method of enumerating the propositions proved by its means. We then imagine that all proofs take the form of a search through this enumeration for the theorem for which a proof is desired. In this way ingenuity is replaced by patience. (p. 193)
Intuition, on the other hand, cannot be replaced by patience. This is a lesson of Gödel's incompleteness results.
In pre-Gödel times it was thought by some that it would probably be possible to carry this programme [the setting down of formal rules] to such a point that all the intuitive judgments of mathematics could be replaced by a finite number of these rules. The necessity for intuition would then be entirely eliminated. (pp. 192–3)
Turing is here referring to the Hilbert programme. The Hilbert programme aimed to bring mathematics to order by setting down a finite system of formal rules (a 'concrete basis on which everyone can agree') by means of which all the infinitely many intuitively true mathematical statements could be proved _without_ further appeals to intuition—without 'mysterious arts', as Hilbert put it. (See further 'Computable Numbers: A Guide'.) Following Gödel, it was clear that this cannot be done. No matter which rules are selected, there will always be statements that a mathematician can see intuitively are true but which cannot be proved using the rules.
While it had been shown that intuition cannot be replaced by such a system of rules, there remained the question whether it might nevertheless be possible to _circumscribe_ the use made of intuition, so that the mathematician is only required to use intuition in judging the truth of (an unlimited number of) propositions of a very specific form. This would not achieve the elimination of intuition desired by Hilbertians, but would achieve something in that direction. The truths of mathematics could be derived, not by using a set of formal rules alone, but by using a set of formal rules together with intuitive judgements of that very specific form. To those wary of intuition, this is certainly preferable to the uncontrolled use of 'mysterious arts'. Although not eliminated, the use of intuition would at least be brought under strict control.
This successor to the defeated Hilbert programme is the subject of investigation of 'Systems of Logic Based on Ordinals':
In our discussions...we have gone to the opposite extreme [to Hilbertians] and eliminated not intuition but ingenuity, and this in spite of the fact that our aim has been in much the same direction. We have been trying to see how far it is possible to eliminate intuition, and leave only ingenuity. (p. 193)
In consequence of the impossibility of finding a formal logic which wholly eliminates the necessity of using intuition, we naturally turn to 'non-constructive' systems of logic with which not all the steps in a proof are mechanical, some being intuitive.... What properties do we desire a non-constructive logic to have if we are to make use of it for the expression of mathematical proofs? We want it to show quite clearly when a step makes use of intuition, and when it is purely formal. The strain put on the intuition should be a minimum. (ibid.)
The following extension of the Gentzen-style formal logic just discussed is a simple example of a logical system that incorporates, as well as formal rules, intuitive judgements of a very specific kind. In this extended system, each step in a proof is either a derivation from previous statements in the proof, using a formal rule (as before), or else—and this is the new part—is the assertion of a (true) proposition of form 'Un(m)'. 'Un(m)' is a construction saying 'At some point, Turing machine **m** prints 0' (see Section 11 of 'On Computable Numbers', and the subsection 'The proof of the undecidability of FOPC' of 'Computable Numbers: A Guide'). Steps of this second sort are non-mechanical, in the sense that no Turing machine—no effective procedure—is able to determine, for all propositions of the form 'Un(m)', which are true and which false. Nevertheless, when one is proving theorems in this system, one is allowed to include, at any point in a proof where it is helpful to do so, an intuitively true proposition of the form 'Un(m)'. No other type of intuitive step is permitted in the system—the strain put on the intuition is 'a minimum'. The result is a system whose theorems go far beyond what is provable in the first-order predicate calculus alone.
The question of _how much_ mathematics can be captured by a system in which the use of intuition is strictly controlled in this manner is the crucial one for the success or otherwise of this post-Hilbertian programme.
#### Ordinal Logics and Gödel's Incompleteness Theorem
To say that a system is complete with respect to some specified set of formulae _S_ , e.g. the set of arithmetical truths, is to say that every formula in _S_ is provable in the system. Gödel showed that not all arithmetical truths can be proved in the formal system of arithmetic set out by Whitehead and Russell in _Principia Mathematica_. Turing's work enabled this result to be extended to _any mechanical procedure_ for producing truths of arithmetic (see Section 14 of 'Computable Numbers: A Guide').
Gödel established his incompleteness result by showing how to construct an arithmetical formula—call it _G_ —that is not provable in the system and yet is true. In order to show that _G_ is true, Gödel appealed to the way he constructed _G_. _G_ is of such a nature that _G_ in effect says that it itself is not provable in the system—and so, since _G_ is _not_ provable, what _G_ says is true.
Can the incomplete formal system of arithmetic be made complete by adding _G_ to it as a new axiom (thereby making it the case—trivially—that _G_ is provable in the system)? No. This is because, once _G_ is added, producing a new system, the Gödel construction can be applied once again to produce a true formula _G_ 1 that is unprovable in the new system. And when _G_ 1 is itself added as a new axiom, producing a further system, there is a true but unprovable _G_ 2, and so on _ad infinitum_.
Following Turing's notation (p. 146), let the system of arithmetic that forms the starting point of this infinite progression be called _L_. The result of adding _G_ to _L_ is called _L_ 1; the result of adding _G_ 1 to _L_ 1 is _L_ 2, and so on. Taken together, the systems in the infinite progression _L_ , _L_ 1, _L_ 2, _L_ 3,... form a non-constructive logic of the sort described by Turing in the above quotation. New axioms are seen to be true by intuition, but otherwise only ingenuity (or patience) is required in proving theorems in any of the systems.
There are _a lot_ of systems in the progression _L_ , _L_ 1, _L_ 2, _L_ 3,... Saying merely that there are infinitely many oversimplifies matters. Not only is there a system for each one of the infinitely many finite ordinal numbers 1, 2, 3,... There is a system that contains the theorems of _every one_ of the systems _L i_, where _i_ is a finite ordinal. This system is called _L_ **ω** ( **ω** being the first 'transfinite' ordinal number). The system _L_ **ω** is 'bigger' than any one of the systems _L i_ in the sense that, no matter which _L i_ is considered, _L_ **ω** includes all the theorems of _L i_, but not vice versa. If _P_ 1 is the set of provable formulae of _L_ 1, _P_ 2 of _L_ 2, and so on, then _P_ **ω** is the union of all the sets _P_ 1, _P_ 2, _P_ 3,... But even _L_ **ω** has a true but unprovable _G_ **ω**. Adding _G_ **ω** to _L_ **ω** produces _L_ **ω** \+ 1, adding _G_ **ω** \+ 1 to _L_ **ω** \+ 1 produces _L_ **ω** \+ 2, and so on and so on. The progression of systems _L_ , _L_ 1, _L_ 2, _L_ 3,..., _L_ **ω** , _L_ **ω** \+ 1, _L_ **ω** \+ 2,... is an example of an ordinal logic.
As Turing noted in the first paragraph of 'Systems of Logic Based on Ordinals', each _L i_ is 'more complete' than its predecessor: some of the true formulae unprovable in _L_ are provable in the less incomplete _L_ 1, and so on. This raises the possibility of our being able to construct a progression of systems—an ordinal logic—that is complete, the systems in the progression proving between them all the truths of arithmetic. If so, then not every systematic formulation of arithmetic falls prey to Gödel's theorem. Such an ordinal logic would 'avoid as far as possible the effects of Gödel's theorem', Turing said (p. 178):
Gödel's theorem shows that such a system cannot be wholly mechanical; but with a complete ordinal logic we should be able to confine the non-mechanical steps... (p. 180)
In his investigations Turing considered sequences of systems in which the non-mechanical steps consist, not of recognizing that Gödel-formulae (the _G_ s) are true, but of recognizing that certain formulae are what he called _ordinal_ formulae. The concept of an ordinal formula is defined in terms of operations of the lambda calculus (p. 162). Roughly, a formula of the lambda calculus is an ordinal formula if it represents a (constructive) ordinal number. The important point is that there is no effective procedure for determining, of any given formula of the calculus, whether or not it is an ordinal formula; Turing proved this on p. 170.
Turing gave examples of ordinal logics of three different types, the logic **Λ** _P_ (p. 177), the logic **Λ** _H_ (p. 178), and his 'Gentzen type' ordinal logics (Section 12). (The ' _P_ ' in ' **Λ** _P_ ' refers to Gödel's 1931 system _P_ , equivalent to the system of arithmetic given by Whitehead and Russell in _Principia Mathematica_. Seemingly ' _H_ ' was for Hilbert.)
#### Ordinal Logics and Proof-Finding Machines
In one of his letters to Newman (Chapter 4), Turing outlined the relationship between an ordinal logic and a hierarchy of theorem-proving Turing machines (p. 215):
One imagines different machines allowing different sets of proofs, and by choosing a suitable machine one can approximate 'truth' by 'provability' better than with a less suitable machine, and can in a sense approximate it as well as you please.
If one wants a particular true statement to be proved by a Turing machine (in the sense described earlier, where 'ingenuity is replaced by patience'), then, since no single Turing machine can prove every true arithmetical statement, one must pick a suitable machine, a machine that actually is able to prove the statement in question. The selection of a suitable machine typically 'involves intuition' (p. 215).
The intuition involved in choosing a suitable proof-finding machine is, Turing went on to say in the letter, 'interchangeable' with the intuition required for selecting a system, from among a progression of systems, in which the statement is provable. Furthermore, if rather than following the rules of a particular logical system, one were to prove the statement free-style, then this too would require intuition, and the necessary intuition would be interchangeable with that required for choosing a suitable proof-finding machine or for choosing a suitable logical system. (See further the introduction to Chapter 12.)
#### Completeness of Ordinal Logics
Let _l_ 1, _l_ 2, _l_ 3,... be any progression of logical systems indexed by (expressions for) ordinals. To say that _l_ 1, _l_ 2, _l_ 3,... is complete with respect to some set of formulae _S_ is to say that for each formula _x_ in _S_ , there is _some_ ordinal α such that _x_ is provable in _l_ α.
Turing proved the following 'completeness theorem': his ordinal logic Λ _P_ is complete with respect to the set of all true formulae of the form 'for every integer _x_ , _f_ ( _x_ ) = 0', where _f_ is a primitive recursive function (pp. 187–190). Many mathematically interesting theorems are of this form. In modern terminology, formulae of this form are called ' formulae'. Turing referred to them as being of the form ' _f_ ( _x_ ) vanishes identically' (by ' _f_ ( _x_ ) vanishes' is meant ' _f_ ( _x_ ) = 0').
This completeness theorem shows that L _P_ circumvents Gödel's incompleteness result in the way discussed above. Solomon Feferman, who in the 1960s continued Turing's work on ordinal logics, commented on the theorem: '[This] partial completeness result... could have been regarded as meeting the... aim of "overcoming" the incompleteness phenomena discovered by Gödel, since these only concerned true but unprovable statements.'
In his letter to Newman from the Crown (Chapter 4), Turing pointed out that this 'completeness theorem... is of course completely useless for the purpose of actually producing proofs' (p. 213). Why this is so is explained by means of an example in 'Systems of Logic Based on Ordinals' (p. 191): Although [the completeness theorem] shows, for instance, that it is possible to prove Fermat's last theorem with **Λ** _P_ (if it is true) yet the truth of the theorem would really be assumed by taking a certain formula as an ordinal formula.
Nevertheless, as Turing went on to say in the letter, the completeness theorem does succeed perfectly well in its purpose of providing 'an insurance against certain sorts of "Gödel incompleteness theorems" being proved about the ordinal logic'.
Not all true arithmetical statements are of form (this is why the completeness theorem is only a partial result). Turing was especially concerned with formulae of the form ( _x_ )(∃ _y_ ) _f_ ( _x_ , _y_ ) = 0 (where _f_ is a primitive recursive function). In modern terminology, formulae of this form are called formulae'. Turing called (true) formulae of this form 'number-theoretic theorems' (p. 152). (The choice of this term is curious; he defended it in a footnote to p. 152.) In Section 5, Turing explained why he regarded number-theoretic theorems as having 'an importance which makes it worth while to give them special consideration' (mentioning also, on p. 155, that a number of unsolved mathematical problems are number-theoretic). Turing conjectured that **Λ** _P_ is complete with respect to the set of all true formulae, but said 'I cannot at present give a proof of this' (p. 187).
Unfortunately Turing's conjecture that **Λ** _P_ is complete with respect to true formulae was proved incorrect by Feferman, in work published in 1962. Is there nevertheless _some_ ordinal logic that is complete with respect to this wider class of truths? A negative answer would represent a spectacular incompleteness result. Commenting on his refutation of Turing's conjecture, Feferman said:
A general incompleteness theorem for recursive progressions... would have been dramatic proof of the far-reaching extent of incompleteness phenomena. However, the situation has not turned out in this way.... [A]ll true sentences of elementary number theory are provable in the recursive progression based on [a principle studied by Shoenfield].
At the end of Section 11, Turing gave some reasons for being dissatisfied with the logics **Λ** _H_ and **Λ** _P_ , and moved on to the Gentzen-type ordinal logics of Section 12. He said of the last of his three examples of Gentzen-type logics, **Λ** _G_ 3, that it 'appears to be adequate for most purposes', adding 'How far this is the case can, of course, only be determined by experiment' (p. 202).
#### Oracle Machines
In Section 4, 'A type of problem which is not number-theoretic', Turing introduced the concept of an _o-machine_. An _o_ -machine is like a Turing machine except that the machine is endowed with an additional basic operation of a type that no Turing machine can simulate. For example, the new operation may be that of displaying the answer to any question of the form 'Is Turing machine m circle-free?' (A circle-free machine is one that prints an infinite number of binary digits; see Sections 8 and 11 of 'Computable Numbers: A Guide'.) Turing called the new operation the 'oracle'. He did not go into the nature of the oracle: it works by 'some unspecified means' (p. 156).
The question 'Is Turing machine m circle-free?' may be presented to the oracle simply by writing out, on successive F-squares of the _o_ -machine's tape, the description number of the Turing machine in question (marking the E-squares at the start and finish of the description number with some special symbol, e.g. '@'). As in the case of an ordinary Turing machine, the behaviour of an _o_ -machine is governed by a table of instructions. Among the states of the _o_ -machine is a state that is used to call in the oracle. When an instruction in the table places the machine in that state, the marked description is 'referred to the oracle' (p. 156). The oracle determines by unspecified means whether or not the Turing machine so numbered is circle-free, and delivers its pronouncement by shifting the machine into one or other of two states, one indicating the affirmative answer and the other the negative.
Turing's aim in Section 4, paralleling his aim in 'On Computable Numbers', was to prove the existence of mathematical problems that cannot be solved by _o_ -machine. Just as no Turing machine can decide, of arbitrarily selected Turing-machine description numbers, which are numbers of circle-free machines, no _o_ -machine can decide, of arbitrarily selected _o_ -machine description numbers, which are numbers of circle-free _o_ -machines. Turing showed this by reworking the argument that he gave in 'On Computable Numbers' (p. 72ff of 'On Computable Numbers' and p. 157 of 'Systems of Logic Based on Ordinals').
The connection with number-theoretic problems is via an equivalence pointed out by Turing in Section 3: 'every number-theoretic theorem is equivalent to the statement that a corresponding [Turing] machine is circle free' (p. 154). In the light of this equivalence, an oracle for deciding whether or not Turing machines are circle-free is in effect an oracle for deciding whether or not statements are number-theoretic theorems. (Indeed, Turing introduced _o_ -machines in terms of an oracle for 'solving number-theoretic problems' (p. 156).)
Given that an _o_ -machine is able to solve all number-theoretic problems, the _o_ -machine satisfactoriness problem—the problem of deciding whether arbitrarily selected _o_ -machine description numbers are numbers of circle-free machines—is an example of a type of problem that is not number-theoretic.
Turing has shown, then, that there are types of mathematical truth that cannot be proved by means of an effective method augmented by pronouncements of the oracle. If the project is to formalize mathematics by means of Hilbertian inference rules augmented by a strictly circumscribed use of intuition, then the intuitive steps cannot be limited to true propositions of the form 'Turing machine m is circle-free'. Equivalently, the intuitive steps cannot be limited to number-theoretic theorems.
In Section 9, Turing generalized his oracle-machine argument, considering ordinal logics that 'prove more general theorems than number-theoretic ones' (p. 179). An ordinal logic itself provides the materials for the formulation, via the oracle-machine construction, of a problem that it cannot solve. Consider an oracle for a class of problems more extensive than the number-theoretic problems, and let the notion of an _o_ -machine be broadened to include machines availing themselves of this oracle. It can be shown, by the same form of argument as before, that there are problems 'which cannot be solved by a uniform process even with the help of this oracle' (p. 180). Thus there are true mathematical statements not provable in an ordinal logic in which all the intuitive steps correspond to pronouncements of the oracle.
This negative result of Turing's exposes the limitations of the post-Hilbertian programme that is his focus, just as his attack in 'On Computable Numbers' exposed the limitations of Hilbert's own programme. Turing tempered this negative result with the pragmatic view that an appropriate ordinal logic (perhaps **Λ** _G_ 3) would be 'adequate for most purposes' (p. 202).
#### Generalized Recursion Theory and Degrees of Unsolvability
The oracle-machine concept has in fact had a considerably greater impact on mathematical logic than the parent concept of an ordinal logic. Turing's introduction of the _o_ -machine led to the development of the rich and important field of mathematical logic known as generalized recursion theory. Generalized recursion theory extends the concept of an algorithm. In his textbook _Degrees of Unsolvability_ Joseph Shoenfield introduces the field in this way:
We extend the notion of an algorithm to allow the use of oracles. The extension consists of allowing a new instruction to appear in the algorithm. This new instruction tells us that the next computation step is to be the value given by the oracle for the argument obtained at the last computation step.
Much of the credit for laying the foundations of generalized recursion theory belongs to Emil Post, who in the 1940s further developed what Turing had begun in 'Systems of Logic Based on Ordinals'. In 1944 Post introduced the term 'Turing-reducible' into the literature, saying that the decision problem for a (recursively enumerable) set of positive integers _A_ —i.e. the problem of correctly answering the question 'Is _x_ in _A_?' for each positive integer _x_ —is Turing-reducible to the decision problem for a (recursively enumerable) set of positive integers _B_ when the first problem is solvable by an _o_ -machine with an oracle for solving the second. If the decision problem for _A_ is Turing-reducible to the decision problem for _B_ , _A_ is said to be computable _relative_ to _B_.
In the same article Post introduced the concept of _degrees of unsolvability_. It is natural to think of the problems that are solvable by an _o_ -machine as being _harder_ than those solvable by a Turing machine unaided by an oracle. Problems solvable by an unaided Turing machine are said to be of the lowest degree. Problems of equal hardness are said to be of the same degree.
_o_ -machines whose oracle answers questions of the form 'Is Turing machine m circle-free?' are sometimes called _first-order_ oracle machines. Problems that are solvable by an _o_ -machine whose oracle answers questions of the form 'Is first-order _o_ -machine **m** circle-free?'—a second-order _o_ -machine—are of a higher degree than (harder than) problems solvable by first-order _o_ -machines. Problems that are solvable by a third-order _o_ -machine are of a higher degree again than problems solvable by a second-order _o_ -machine, and so on.
Post wrote:
For unsolvable problems the concept of reducibility leads to the concept of _degree of unsolvability_ , two unsolvable problems being of the same degree of unsolvability if each is reducible to the other, one of lower degree of unsolvability than another if it is reducible to the other, but that other is not reducible to it, of incomparable degrees of unsolvability if neither is reducible to the other.
Post called problems other than those of the lowest degree 'unsolvable' (Turing comments on this terminology in Chapter 17). He pointed out that, in Section 4 of 'Systems of Logic Based on Ordinals', Turing proved 'a result which immediately generalizes to the result that for any "recursively given" unsolvable problem there is another of higher degree of unsolvability'.
In the modern literature the degrees of unsolvability generated by Turing-reducibility are known simply as _Turing degrees_.
#### _Further reading_
Barendregt, H. P., _The Lambda-Calculus, its Syntax and Semantics_ (Amsterdam: North-Holland, 1984).
Davis, M., _Computability and Unsolvability_ (New York: McGraw-Hill, 1958).
Feferman, S., 'Turing in the Land of O(z)', in R. Herken (ed.), _The Universal Turing Machine: A Half-Century Survey_ (Oxford: Oxford University Press, 1988).
Kleene, S. C., _Mathematical Logic_ (New York: Wiley, 1967).
Rogers, H., _Theory of Recursive Functions and Effective Computability_ (New York: McGraw-Hill, 1967).
Shoenfield, J. R., _Degrees of Unsolvability_ (Amsterdam: North-Holland, 1971).
Simpson, S. G., 'Degrees of Unsolvability: A Survey of Results', in J. Barwise (ed.), _Handbook of Mathematical Logic_ (Amsterdam: North-Holland, 1977).
Soare, R. I., _Recursively Enumerable Sets and Degrees_ (Berlin: Springer-Verlag, 1987).
## **Systems of Logic Based on Ordinals**
Introduction _146_
1. The conversion calculus. Gödel representations _147_
2. Effective calculability. Abbreviation of treatment _150_
3. Number-theoretic theorems _152_
4. A type of problem which is not number-theoretic _156_
5. Syntactical theorems as number-theoretic theorems _157_
6. Logic formulae _158_
7. Ordinals _161_
8. Ordinal logics _170_
9. Completeness questions _178_
10. The continuum hypothesis. A digression _191_
11. The purpose of ordinal logics _192_
12. Gentzen type ordinal logics _194_
Index of definitions _202_
Bibliography _203_
The well-known theorem of Gödel (Gödel [1], [2]) shows that every system of logic is in a certain sense incomplete, but at the same time it indicates means whereby from a system _L_ of logic a more complete system _L ′_ may be obtained. By repeating the process we get a sequence _L_ , _L_ 1 = _L_ ′, _L_ 2 = _L_ 1′,... each more complete than the preceding. A logic _L_ **ω** may then be constructed in which the provable theorems are the totality of theorems provable with the help of the logics _L_ , _L_ 1, _L_ 2,... We may then form _L_ 2 **ω** related to _L_ **ω** in the same way as _L_ **ω** was related to _L_. Proceeding in this way we can associate a system of logic with any constructive ordinal. It may be asked whether a sequence of logics of this kind is complete in the sense that to any problem _A_ there corresponds an ordinal α such that _A_ is solvable by means of the logic _L_ α. I propose to investigate this question in a rather more general case, and to give some other examples of ways in which systems of logic may be associated with constructive ordinals.
#### 1. The calculus of conversion. Gödel representations
It will be convenient to be able to use the "conversion calculus" of Church for the description of functions and for some other purposes. This will make greater clarity and simplicity of expression possible. I give a short account of this calculus. For detailed descriptions see Church [3], [2], Kleene [1], Church and Rosser [1].
The formulae of the calculus are formed from the symbols {, }, (,), [, ], **_λ_** , δ, and an infinite list of others called variables; we shall take for our infinite list _a_ , _b_ ,..., _z_ , _x_ ′, _x_ ″,... Certain finite sequences of such symbols are called _well-formed formulae_ (abbreviated to W.F.F.); we define this class inductively, and define simultaneously the free and the bound variables of a W.F.F. Any variable is a W.F.F.; it is its only free variable, and it has no bound variables. δ is a W.F.F. and has no free or bound variables. If **M** and **N** are W.F.F. then { **M** }( **N** ) is a W.F.F., whose free variables are the free variables of **M** together with the free variables of **N** , and whose bound variables are the bound variables of **M** together with those of **N**. If **M** is a W.F.F. and **V** is one of its free variables, then **_λ_ V**[ **M** ] is a W.F.F. whose free variables are those of **M** with the exception of **V** , and whose bound variables are those of **M** together with **V**. No sequence of symbols is a W.F.F. except in consequence of these three statements.
In metamathematical statements we use heavy type letters to stand for variable or undetermined formulae, as was done in the last paragraph, and in future such letters will stand for well-formed formulae unless otherwise stated. Small letters in heavy type will stand for formulae representing undetermined positive integers (see below).
A W.F.F. is said to be in normal form if it has no parts of the form { ** _λ_ V**[ **M** ]}( **N** ) and none of the form {{δ}( **M** )}( **N** ), where **M** and **N** have no free variables.
We say that one W.F.F. is _immediately convertible_ into another if it is obtained from it either by:
(i) Replacing one occurrence of a well-formed part **_λ_ V**[ **M** ] by **_λ_ U**[ **N** ], where the variable **U** does not occur in **M** , and **N** is obtained from M by replacing the variable **V** by **U** throughout.
(ii) Replacing a well-formed part { ** _λ_ V**[ **M** ]}( **N** ) by the formula which is obtained from **M** by replacing **V** by **N** throughout, provided that the bound variables of **M** are distinct both from **V** and from the free variables of **N**.
(iii) The process inverse to (ii).
(iv) Replacing a well-formed part {{ **δ** }( **M** )}( **M** ) by
**_λ_** _f_ [ ** _λ_** _x_ [{ _f_ }({ _f_ }( _x_ ))]]
if **M** is in normal form and has no free variables.
(v) Replacing a well-formed part {{δ}( **M** )}( **N** ) by
**_λ_** _f_ [ ** _λ_** _x_ [{ _f_ }( _x_ ))]]
if **M** and **N** are in normal form, are not transformable into one another by repeated application of (i), and have no free variables.
(vi) The process inverse to (iv).
(vii) The process inverse to (v).
These rules could have been expressed in such a way that in no case could there be any doubt about the admissibility or the result of the transformation [in particular this can be done in the case of process (v)].
A formula **A** is said to be _convertible_ into another **B** (abbreviated to " **A** conv **B** ") if there is a finite chain of immediate conversions leading from one formula to the other. It is easily seen that the relation of convertibility is an equivalence relation, _i.e_. it is symmetric, transitive, and reflexive.
Since the formulae are liable to be very lengthy, we need means for abbreviating them. If we wish to introduce a particular letter as an abbreviation for a particular lengthy formula we write the letter followed by "→" and then by the formula, thus
_I_ → **_λ_** _x_ [ _x_ ]
indicates that _I_ is an abbreviation for **_λ_** _x_ [ _x_ ]. We also use the arrow in less sharply defined senses, but never so as to cause any real confusion. In these cases the meaning of the arrow may be rendered by the words "stands for".
If a formula **F** is, or is represented by, a single symbol we abbreviate { **F** }( **X** ) to **F** ( **X** ). A formula {{ **F** }( **X** )}( **Y** ) may be abbreviated to
{ **F** }( **X** , **Y** ),
or to **F** ( **X** , **Y** ) if **F** is, or is represented by, a single symbol. Similarly for {{{ **F** }( **X** )}( **Y** )}( **Z** ), etc. A formula **_λ_ V**1[ ** _λ_ V**2... [ ** _λ_ V** _r_ [ **M** ]]...] may be abbreviated to **_λ_ V**1 **V** 2... **V** _r_. **M**.
We have not as yet assigned any meanings to our formulae, and we do not intend to do so in general. An exception may be made for the case of the positive integers, which are very conveniently represented by the formulae **_λ_** _fx_. _f_ ( _x_ ), **_λ_** _fx_. _f_ ( _f_ ( _x_ )),... In fact we introduce the abbreviations
and we also say, for example, that **_λ_** _fx_. _f_ ( _f_ ( _x_ )), or in full
**_λ_** _f_ [ ** _λ_** _x_ [{ _f_ }({ _f_ }( _x_ ))]],
represents the positive integer 2. Later we shall allow certain formulae to represent ordinals, but otherwise we leave them without explicit meaning; an implicit meaning may be suggested by the abbreviations used. In any case where any meaning is assigned to formulae it is desirable that the meaning should be invariant under conversion. Our definitions of the positive integers do not violate this requirement, since it may be proved that no two formulae representing different positive integers are convertible the one into the other.
In connection with the positive integers we introduce the abbreviation
_S_ → **_λ_** _ufx_. _f_ ( _u_ ( _f_ , _x_ )).
This formula has the property that, if n represents a positive integer, _S_ ( **n** ) is convertible to a formula representing its successor.
Formulae representing undetermined positive integers will be represented by small letters in heavy type, and we adopt once for all the convention that, if a small letter, _n_ say, stands for a positive integer, then the same letter in heavy type, **n** , stands for the formula representing the positive integer. When no confusion arises from so doing, we shall not trouble to distinguish between an integer and the formula which represents it.
Suppose that _f_ ( _n_ ) is a function of positive integers taking positive integers as values, and that there is a W.F.F. F not containing δ such that, for each positive integer _n_ , **F** ( **n** ) is convertible to the formula representing _f_ ( _n_ ). We shall then say that _f_ ( _n_ ) is **_λ_** - _definable_ or _formally definable_ , and that **F** _formally defines f_ ( _n_ ). Similar conventions are used for functions of more than one variable. The sum function is, for instance, formally defined by **_λ_** _abfx_. _af_ , _b_ ( _f_ , _x_ )); in fact, for any positive integers _m_ , _n_ , _p_ for which _m_ \+ _n_ = _p_ , we have
{ ** _λ_** _abfx_. _a_ ( _f_ , _b_ ( _f_ , _x_ ))} ( **m** , **n** ) conv **p**.
In order to emphasize this relation we introduce the abbreviation
**X** \+ **Y** → { ** _λ_** _abfx_. _a_ ( _f_ , _b_ ( _f_ , _x_ ))} ( **X** , **Y** )
and we shall use similar notations for sums of three or more terms, products, etc.
For any W.F.F. **G** we shall say that **G** _enumerates_ the sequence **G** (1), **G** (2),... and any other sequence whose terms are convertible to those of this sequence.
When a formula is convertible to another which is in normal form, the second is described as a _normal form_ of the first, which is then said to _have a normal form_. I quote here some of the more important theorems concerning normal forms.
(A) _If a formula has two normal forms they are convertible into one another by the use of_ (i) _alone_. (Church and Rosser [1], 479, 481.)
(B) _If a formula has a normal form then every well-formed part of it has a normal form_. (Church and Rosser [1], 480–481.)
(C) _There is_ ( _demonstrably_ ) _no process whereby it can be said of a formula whether it has a normal form_. (Church [3], 360, Theorem XVIII.)
We often need to be able to describe formulae by means of positive integers. The method used here is due to Gödel (Gödel [1]). To each single symbol _s_ of the calculus we assign an integer _r_ [ _s_ ] as in the table below.
If _s_ 1, _s_ 2,..., _s k_ is a sequence of symbols, then (where _p_ _k_ is the _k_ -th prime number) is called the _Gödel representation_ (G.R.) of that sequence of symbols. No two W.F.F. have the same G.R.
Two theorems on G.R. of W.F.F. are quoted here.
(D) _There is a_ W.F.F. "form" _such that if a is the_ G.R. _of a_ W.F.F. A _without free variables, then_ form ( _a_ ) conv A. (This follows from a similar theorem to be found in Church [3], 53 66. Metads are used there in place of G.R.)
(E) _There is a_ W.F.F. Gr _such that, if_ A _is a_ W.F.F. _with a normal form without free variables, then_ Gr(A) conv a, _where a is the_ G.R. _of a normal form of_ A. [Church [3], 53, 66, as (D).]
#### 2. Effective calculability. Abbreviation of treatment
A function is said to be "effectively calculable" if its values can be found by some purely mechanical process. Although it is fairly easy to get an intuitive grasp of this idea, it is nevertheless desirable to have some more definite, mathematically expressible definition. Such a definition was first given by Gödel at Princeton in 1934 (Gödel [2], 26), following in part an unpublished suggestion of Herbrand, and has since been developed by Kleene [2]). These functions were described as "general recursive" by Gödel. We shall not be much concerned here with this particular definition. Another definition of effective calculability has been given by Church (Church [3], 356–358), who identifies it with l-definability. The author has recently suggested a definition corresponding more closely to the intuitive idea (Turing [1], see also Post [1]). It was stated above that "a function is effectively calculable if its values can be found by some purely mechanical process". We may take this statement literally, understanding by a purely mechanical process one which could be carried out by a machine. It is possible to give a mathematical description, in a certain normal form, of the structures of these machines. The development of these ideas leads to the author's definition of a computable function, and to an identification of computability with effective calculability. It is not difficult, though somewhat laborious, to prove that these three definitions are equivalent (Kleene [3], Turing [2]).
In the present paper we shall make considerable use of Church's identification of effective calculability with **_λ_** -definability, or, what comes to the same thing, of the identification with computability and one of the equivalence theorems. In most cases where we have to deal with an effectively calculable function, we shall introduce the corresponding W.F.F. with some such phrase as "the function _f_ is effectively calculable, let _F_ be a formula **_λ_** defining it", or "let _F_ be a formula such that _F_ (n) is convertible to... whenever n represents a positive integer". In such cases there is no difficulty in seeing how a machine could in principle be designed to calculate the values of the function concerned; and, assuming this done, the equivalence theorem can be applied. A statement of what the formula _F_ actually is may be omitted. We may immediately introduce on this basis a W.F.F. ϖ with the property that
ϖ( **m** , **n** ) conv **r** ,
if _r_ is the greatest positive integer, if any, for which _m_ **r** divides _n_ and _r_ is 1 if there is none. We also introduce Dt with the properties
Dt( **n** , **n** ) conv 3,
Dt( **n** \+ **m** , **n** ) conv 2,
Dt( **n** , **n** \+ **m** ) conv 1.
There is another point to be made clear in connection with the point of view that we are adopting. It is intended that all proofs that are given should be regarded no more critically than proofs in classical analysis. The subject matter, roughly speaking, is constructive systems of logic, but since the purpose is directed towards choosing a particular constructive system of logic for practical use, an attempt at this stage to put our theorems into constructive form would be putting the cart before the horse.
Those computable functions which take only the values 0 and 1 are of particular importance, since they determine and are determined by computable properties, as may be seen by replacing "0" and "1" by "true" and "false". But, besides this type of property, we may have to consider a different type, which is, roughly speaking, less constructive than the computable properties, but more so than the general predicates of classical mathematics. Suppose that we have a computable function of the natural numbers taking natural numbers as values, then corresponding to this function there is the property of being a value of the function. Such a property we shall describe as "axiomatic"; the reason for using this term is that it is possible to define such a property by giving a set of axioms, the property to hold for a given argument if and only if it is possible to deduce that it holds from the axioms.
Axiomatic properties may also be characterized in this way. A property ψ of positive integers is axiomatic if and only if there is a computable property φ of two positive integers, such that ψ( _x_ ) is true if and only if there is a positive integer _y_ such that φ( _x_ , _y_ ) is true. Or again ψ is axiomatic if and only if there is a W.F.F. F such that ψ( _n_ ) is true if and only if **F** ( **n** ) conv 2.
#### 3. Number-theoretic theorems
By a _number-theoretic theorem_ we shall mean a theorem of the form "θ( _x_ ) vanishes for infinitely many natural numbers _x_ ", where θ( _x_ ) is a primitive recursive function.
We shall say that a problem is number-theoretic if it has been shown that any solution of the problem may be put in the form of a proof of one or more number-theoretic theorems. More accurately we may say that a class of problems is number-theoretic if the solution of any one of them can be transformed (by a uniform process) into the form of proofs of number-theoretic theorems.
I shall now draw a few consequences from the definition of "number theoretic theorems", and in section 5 I shall try to justify confining our consideration to this type of problem.
An alternative form for number-theoretic theorems is "for each natural number _x_ there exists a natural number _y_ such that φ( _x_ , _y_ ) vanishes", where φ( _x_ , _y_ ) is primitive recursive. In other words, there is a rule whereby, given the function θ( _x_ ), we can find a function φ( _x_ , _y_ ), or given φ( _x_ , _y_ ), we can find a function θ( _x_ ), such that "θ( _x_ ) vanishes infinitely often" is a necessary and sufficient condition for "for each _x_ there is a _y_ such that φ( _x_ , _y_ ) = 0". In fact, given θ( _x_ ), we define
φ( _x_ , _y_ ) = θ( _x_ ) + α( _x_ , _y_ ),
where α( _x_ , _y_ ) is the (primitive recursive) function with the properties
If on the other hand we are given φ( _x_ , _y_ ) we define θ( _x_ ) by the equations
θ1(0) = 3,
where ϖ _r_ ( _x_ ) is defined so as to mean "the largest _s_ for which _r_ _s_ divides _x_ ". The function σ( _x_ ) is defined by the equations σ(0) = 0, σ( _x_ \+ 1) = 1. It is easily verified that the functions so defined have the desired properties.
We shall now show that questions about the truth of the statements of the form "does _f_ ( _x_ ) vanish identically", where _f_ ( _x_ ) is a computable function, can be reduced to questions about the truth of number-theoretic theorems. It is understood that in each case the rule for the calculation of _f_ ( _x_ ) is given and that we are satisfied that this rule is valid, _i.e_. that the machine which should calculate _f_ ( _x_ ) is circle free ([p. 60]). The function _f_ ( _x_ ), being computable, is general recursive in the Herbrand–Gödel sense, and therefore, by a general theorem due to Kleene, is expressible in the form
where means "the least _y_ for which is true" and ψ( _y_ ) and φ( _x_ , _y_ ) are primitive recursive functions. Without loss of generality, we may suppose that the functions _ϕ_ , _ψ_ take only the values 0, 1. Then, if we define ρ( _x_ ) by the equations 3.1 and
it will be seen that _f_ ( _x_ ) vanishes identically if and only if ρ( _x_ ) vanishes for infinitely many values of _x_.
The converse of this result is not quite true. We cannot say that the question about the truth of any number-theoretic theorem is reducible to a question about whether a corresponding computable function vanishes identically; we should have rather to say that it is reducible to the problem of whether a certain machine is circle free and calculates an identically vanishing function. But more is true: every number-theoretic theorem is equivalent to the statement that a corresponding machine is circle free. The behaviour of the machine may be described roughly as follows: the machine is one for the calculation of the primitive recursive function θ( _x_ ) of the number-theoretic problem, except that the results of the calculation are first arranged in a form in which the figures 0 and 1 do not occur, and the machine is then modified so that, whenever it has been found that the function vanishes for some value of the argument, then 0 is printed. The machine is circle free if and only if an infinity of these figures are printed, _i.e_. if and only if θ( _x_ ) vanishes for infinitely many values of the argument. That, on the other hand, questions of circle freedom may be reduced to questions of the truth of number-theoretic theorems follows from the fact that θ( _x_ ) is primitive recursive when it is defined to have the value 0 if a certain machine prints 0 or 1 in its ( _x_ \+ 1)-th complete configuration, and to have the value 1 otherwise.
The conversion calculus provides another normal form for the number-theoretic theorems, and the one which we shall find the most convenient to use. Every number-theoretic theorem is equivalent to a statement of the form "A(n) is convertible to 2 for every W.F.F. **n** representing a positive integer", A being a W.F.F. determined by the theorem; the property of **A** here asserted will be described briefly as " **A** is dual". Conversely such statements are reducible to number-theoretic theorems. The first half of this assertion follows from our results for computable functions, or directly in this way. Since θ( _x_ – 1) + 2 is primitive recursive, it is formally definable, say, by means of a formula G. Now there is (Kleene [ **1** ], 232) a W.F.F. with the property that, if **T** ( **r** ) is convertible to a formula representing a positive integer for each positive integer _r_ , then ( **T** , **n** ) is convertible to _s_ , where _s_ is the _n_ -th positive integer _t_ (if there is one) for which T(t) conv 2; if **T** ( **t** ) conv 2 for less than _n_ values of _t_ then ( **T** , **n** ) has no normal form. The formula **G** ( **G** , **n** )) is therefore convertible to 2 if and only if θ( _x_ ) vanishes for at least _n_ values of _x_ , and is convertible to 2 for every positive integer _x_ if and only if θ( _x_ ) vanishes infinitely often. To prove the second half of the assertion, we take Gödel representations for the formulae of the conversion calculus. Let _c_ ( _x_ ) be 0 if _x_ is the G.R. of 2 ( _i.e_. if _x_ is 23 . 310 . 5 . 73 . 1128 . 13 . 17 . 1910 . 232 . 29 . 31 . 3710 . 412 . 43 . 4728 . 532 . 592 . 612 . 672) and let _c_ ( _x_ ) be 1 otherwise. Take an enumeration of the G.R. of the formulae into which **A** ( **m** ) is convertible: let _a_ ( _m, n_ ) be the _n_ -th number in the enumeration. We can arrange the enumeration so that _a_ ( _m, n_ ) is primitive recursive. Now the statement that **A** ( **m** ) is convertible to 2 for every positive integer _m_ is equivalent to the statement that, corresponding to each positive integer _m_ , there is a positive integer _n_ such that _c_ ( _a_ ( _m_ , _n_ )) = 0; and this is number-theoretic.
It is easy to show that a number of unsolved problems, such as the problem of the truth of Fermat's last theorem, are number-theoretic. There are, however, also problems of analysis which are number-theoretic. The Riemann hypothesis gives us an example of this. We denote by ξ( _s_ ) the function defined for by the series and over the rest of the complex plane with the exception of the point _s_ = 1 by analytic continuation. The Riemann hypothesis asserts that this function does not vanish in the domain It is easily shown that this is equivalent to saying that it does not vanish for that it does not vanish inside any rectangle , where _T_ is an integer greater than 2. Now the function satisfies the inequalities
and we can define a primitive recursive function ξ( _l_ , _l_ ′, _m_ , _m_ ′, _N_ , _M_ ) such that
and therefore, if we put
ξ( _l_ , _M_ , _m_ , _M_ , _M_ 2 \+ 2, _M_ ) = _X_ ( _l_ , _m_ , _M_ ),
we have
provided that
If we define _B_ ( _M, T_ ) to be the smallest value of _X_ ( _l, m, M_ ) for which
then the Riemann hypothesis is true if for each _T_ there is an _M_ satisfying
_B_ ( _M_ , _T_ ) > 122 _T_.
If on the other hand there is a _T_ such that, for all _M_ , _B_ ( _M_ , _T_ ) ≤ 122 _T_ , the Riemann hypothesis is false; for let _l M_, _m M_ be such that
_X_ ( _l_ _M_ , _m_ _M_ , _M_ ) ≤ 122 _T_ ,
then
Now if _a_ is a condensation point of the sequence ( _l_ _M_ \+ _im_ _M_ )/ _M_ then since ξ( _s_ ) is continuous except at _s_ = 1 we must have ξ( _a_ ) = 0 implying the falsity of the Riemann hypothesis. Thus we have reduced the problem to the question whether for each _T_ there is an _M_ for which
_B_ ( _M_ , _T_ ) > 122 _T_.
_B_ ( _M_ , _T_ ) is primitive recursive, and the problem is therefore number-theoretic.
#### 4. A type of problem which is not number-theoretic
Let us suppose that we are supplied with some unspecified means of solving number-theoretic problems; a kind of oracle as it were. We shall not go any further into the nature of this oracle apart from saying that it cannot be a machine. With the help of the oracle we could form a new kind of machine (call them _o_ -machines), having as one of its fundamental processes that of solving a given number-theoretic problem. More definitely these machines are to behave in this way. The moves of the machine are determined as usual by a table except in the case of moves from a certain internal configuration . If the machine is in the internal configuration and if the sequence of symbols marked with _l_ is then the well-formed formula **A** , then the machine goes into the internal configuration or according as it is or is not true that **A** is dual. The decision as to which is the case is referred to the oracle.
These machines may be described by tables of the same kind as those used for the description of _a_ -machines, there being no entries, however, for the internal configuration . We obtain description numbers from these tables in the same way as before. If we make the convention that, in assigning numbers to internal configurations, are always to be _q_ 2, _q_ 3, _q_ 4, then the description numbers determine the behaviour of the machines uniquely.
Given any one of these machines we may ask ourselves the question whether or not it prints an infinity of figures 0 or 1; I assert that this class of problem is not number-theoretic. In view of the definition of "number-theoretic problem" this means that it is not possible to construct an _o_ -machine which, when supplied with the description of any other _o_ -machine, will determine whether that machine is _o_ -circle free. The argument may be taken over directly from Turing [1], § 8. We say that a number is _o_ -satisfactory if it is the description number of an _o_ -circle free machine. Then, if there is an _o_ -machine which will determine of any integer whether it is _o_ -satisfactory, there is also an _o_ -machine to calculate the values of the function 1 – _φ_ _n_ ( _n_ ). Let _r_ ( _n_ ) be the _n_ -th _o_ -satisfactory number and let _φ_ _n_ ( _m_ ) be the _m_ -th figure printed by the _o_ -machine whose description number is _r_ ( _n_ ). This _o_ -machine is circle free and there is therefore an _o_ -satisfactory number _K_ such that _φ_ _K_ ( _n_ ) = 1 – _φ_ _n_ ( _n_ ) for all _n_. Putting _n_ = _K_ yields a contradiction. This completes the proof that problems of circle freedom of _o_ -machines are not number-theoretic.
Propositions of the form that an _o_ -machine is _o_ -circle free can always be put in the form of propositions obtained from formulae of the functional calculus of the first order by replacing _some_ of the functional variables by primitive recursive relations. Compare foot-note [5] on page [152].
#### 5. Syntactical theorems as number-theoretic theorems
I now mention a property of number-theoretic theorems which suggests that there is reason for regarding them as of particular importance.
Suppose that we have some axiomatic system of a purely formal nature. We do not concern ourselves at all in interpretations for the formulae of this system; they are to be regarded as of interest for themselves. An example of what is in mind is afforded by the conversion calculus (§1). Every sequence of symbols " **A** conv **B** ", where **A** and **B** are well formed formulae, is a formula of the axiomatic system and is provable if the W.F.F. **A** is convertible to **B**. The rules of conversion give us the rules of procedure in this axiomatic system.
Now consider a new rule of procedure which is reputed to yield only formulae provable in the original sense. We may ask ourselves whether such a rule is valid. The statement that such a rule is valid would be number-theoretic. To prove this, let us take Gödel representations for the formulae, and an enumeration of the provable formulae; let _φ_ ( _r_ ) be the G.R. of the _r_ -th formula in the enumeration. We may suppose _φ_ ( _r_ ) to be primitive recursive if we are prepared to allow repetitions in the enumeration. Let ψ( _r_ ) be the G.R. of the _r_ -th formula obtained by the new rule, then the statement that this new rule is valid is equivalent to the assertion of
( _r_ )(∃ _s_ )[ψ( _r_ ) = φ( _s_ )]
(the domain of individuals being the natural numbers). It has been shown in §3 that such statements are number-theoretic.
It might plausibly be argued that all those theorems of mathematics which have any significance when taken alone are in effect syntactical theorems of this kind, stating the validity of certain "derived rules" of procedure. Without going so far as this, I should assert that theorems of this kind have an importance which makes it worth while to give them special consideration.
#### 6. Logic formulae
We shall call a formula **L** a _logic formula_ (or, if it is clear that we are speaking of a W.F.F., simply a _logic_ ) if it has the property that, if **A** is a formula such that **L** ( **A** ) conv 2, then **A** is dual.
A logic formula gives us a means of satisfying ourselves of the truth of number-theoretic theorems. For to each number-theoretic proposition there corresponds a W.F.F. **A** which is dual if and only if the proposition is true. Now, if **L** is a logic and **L** ( **A** ) conv 2, then **A** is dual and we know that the corresponding number-theoretic proposition is true. It does not follow that, if **L** is a logic, we can use **L** to satisfy ourselves of the truth of _any_ number-theoretic theorem.
If **L** is a logic, the set of formulae **A** for which **L** ( **A** ) conv 2 will be called the _extent_ of **L**.
It may be proved by the use of (D), (E), p. [150], that there is a formula _X_ such that, if **M** has a normal form, has no free variables and is not convertible to 2, then _X_ ( **M** ) conv 1, but, if M conv 2, then _X_ ( **M** ) conv 2. If **L** is a logic, then **_λ_** _x_. _X_ **L** ( _x_ ) is also a logic whose extent is the same as that of **L** , and which has the property that, if **A** has no free variables, then
{ ** _λ_** _x_. _X_ ( **L** ( _x_ ))}( **A** )
either is always convertible to 1 or to 2 or else has no normal form. A logic with this property will be said to be _standardized_.
We shall say that a logic **L** ′ is _at least as complete as_ a logic L if the extent of L is a subset of the extent of **L** ′. The logic **L** ′ is _more complete than_ **L** if the extent of **L** is a proper subset of the extent of **L** ′.
Suppose that we have an effective set of rules by which we can prove formulae to be dual; _i.e_. we have a system of symbolic logic in which the propositions proved are of the form that certain formulae are dual. Then we can find a logic formula whose extent consists of just those formulae which can be proved to be dual by the rules; that is to say, there is a rule for obtaining the logic formula from the system of symbolic logic. In fact the system of symbolic logic enables us to obtain a computable function of positive integers whose values run through the Gödel representations of the formulae provable by means of the given rules. By the theorem of equivalence of computable and **_**_λ_**_** -definable functions, there is a formula **J** such that **J** (1), **J** (2),... are the G.R. of these formulae. Now let
Then I assert that _W_ ( **J** ) is a logic with the required properties. The properties of imply that ( **C** , 1) is convertible to the least positive integer **n** for which **C** ( **n** ) conv 2, and has no normal form if there is no such integer. Consequently ( **C** , 1, _I_ , 2) is convertible to 2 if **C** ( **n** ) conv 2 for some positive integer _n_ , and it has no normal form otherwise. That is to say that _W_ ( **J** , **A** ) conv 2 if and only if **δ** ( **J** ( **n** ), **A** ) conv 2, some _n, i.e_. if **J** ( **n** ) conv **A** some _n_.
There is conversely a formula _W_ ′ such that, if **L** is a logic, then _W_ ′( **L** ) enumerates the extent of **L**. For there is a formula _Q_ such that _Q_ ( **L** , **A** , **n** ) conv 2 if and only if **L** ( **A** ) is convertible to 2 in less than _n_ steps. We then put
Of course, _W_ ′( _W_ ( **J** )) normally entirely different from **J** and _W_ ( _W_ ′( **L** )) from **L**.
In the case where we have a symbolic logic whose propositions can be interpreted as number-theoretic theorems, but are not expressed in the form of the duality of formulae, we shall again have a corresponding logic formula, but its relation to the symbolic logic is not so simple. As an example let us take the case where the symbolic logic proves that certain primitive recursive functions vanish infinitely often. As was shown in §3, we can associate with each such proposition a W.F.F. which is dual if and only if the proposition is true. When we replace the propositions of the symbolic logic by theorems on the duality of formulae in this way, our previous argument applies and we obtain a certain logic formula **L**. However, **L** does not determine uniquely which are the propositions provable in the symbolic logic; for it is possible that "θ1( _x_ ) vanishes infinitely often" and "θ2( _x_ ) vanishes infinitely often" are both associated with "A is dual", and that the first of these propositions is provable in the system, but the second not. However, if we suppose that the system of symbolic logic is sufficiently powerful to be able to carry out the argument on pp. [154–5] then this difficulty cannot arise. There is also the possibility that there may be formulae in the extent of L with no propositions of the form "θ( _x_ ) vanishes infinitely often" corresponding to them. But to each such formula we can assign (by a different argument) a proposition _p_ of the symbolic logic which is a necessary and sufficient condition for A to be dual. With _p_ is associated (in the first way) a formula **A** ′. Now **L** can always be modified so that its extent contains **A** ′ whenever it contains **A**.
We shall be interested principally in questions of completeness. Let us suppose that we have a class of systems of symbolic logic, the propositions of these systems being expressed in a uniform notation and interpretable as number-theoretic theorems; suppose also that there is a rule by which we can assign to each proposition _p_ of the notation a W.F.F. **A** _p_ which is dual if and only if _p_ is true, and that to each W.F.F. **A** we can assign a proposition _p_ A which is a necessary and sufficient condition for **A** to be dual. _p_ A _p_ is to be expected to differ from _p_. To each symbolic logic _C_ we can assign two logic formulae **L** _C_ and **L** _C_ ′. A formula **A** belongs to the extent of **L** _C_ if _p_ A is provable in _C_ , while the extent of **L** _C_ ′ consists of all **A** _p_ , where _p_ is provable in _C_. Let us say that the class of symbolic logics is complete if each true proposition is provable in one of them: let us also say that a class of logic formulae is complete if the set-theoretic sum of the extents of these logics includes all dual formulae. I assert that a necessary condition for a class of symbolic logics _C_ to be complete is that the class of logics **L** _C_ is complete, while a sufficient condition is that the class of logics **L** _C_ ′ is complete. Let us suppose that the class of symbolic logics is complete; consider _p_ A, where **A** is arbitrary but dual. It must be provable in one of the systems, _C_ say. A therefore belongs to the extent of **L** _C_ , _i.e_. the class of logics **L** _C_ is complete. Now suppose the class of logics **L** _C_ ′ to be complete. Let _p_ be an arbitrary true proposition of the notation; **A** _p_ must belong to the extent of some **L** _C_ ′, and this means that _p_ is provable in _C_.
We shall say that a single logic formula L is complete if its extent includes all dual formulae; that is to say, it is complete if it enables us to prove every true number-theoretic theorem. It is a consequence of the theorem of Gödel (if suitably extended) that no logic formula is complete, and this also follows from ( _C_ ), p. [150], or from the results of Turing [ **1** ], §8, when taken in conjunction with §3 of the present paper. The idea of completeness of a logic formula is not therefore very important, although it is useful to have a term for it.
Suppose **Y** to be a W.F.F. such that **Y** ( **n** ) is a logic for each positive integer _n_. The formulae of the extent of **Y** ( **n** ) are enumerated by _W_ ( **Y** ( **n** )), and the combined extents of these logics by
If we put
then **Γ** ( **Y** ) is a logic whose extent is the combined extent of
**Y** (1), **Y** (2), **Y** (3),...
To each W.F.F. **L** we can assign a W.F.F. _V_ ( **L** ) such that a necessary and sufficient condition for **L** to be a logic formula is that _V_ ( **L** ) is dual. Let Nm be a W.F.F. which enumerates all formulae with normal forms and no free variables. Then the condition for **L** to be a logic is that **L** Nm( **r** ), **s** ) conv 2 for all positive integers _r, s, i.e_. that
is dual. We may therefore put
#### 7. Ordinals
We begin our treatment of ordinals with some brief definitions from the Cantor theory of ordinals, but for the understanding of some of the proofs a greater amount of the Cantor theory is necessary than is set out here.
Suppose that we have a class determined by the propositional function _D_ ( _x_ ) and a relation _G_ ( _x, y_ ) ordering its members, _i.e_. satisfying
The class defined by _D_ ( _x_ ) is then called a _series_ with the ordering relation _G_ ( _x, y_ ). The series is said to be _well ordered_ and the ordering relation is called an _ordinal_ if every sub-series which is not void has a first term, _i.e_. if
The condition 7.2 is equivalent to another, more suitable for our purposes, namely the condition that every descending subsequence must terminate; formally
The ordering relation _G_ ( _x, y_ ) is said to be similar to _G_ **′** ( _x_ , _y_ ) if there is a one–one correspondence between the series transforming the one relation into the other. This is best expressed formally, thus
Ordering relations are regarded as belonging to the same ordinal if and only if they are similar.
We wish to give names to all the ordinals, but this will not be possible until they have been restricted in some way; the class of ordinals, as at present defined, is more than enumerable. The restrictions that we actually impose are these: _D_ ( _x_ ) is to imply that _x_ is a positive integer; _D_ ( _x_ ) and _G_ ( _x, y_ ) are to be computable properties. Both of the propositional functions _D_ ( _x_ ), _G_ ( _x, y_ ) can then be described by means of a single W.F.F. **Ω** with the properties:
**Ω** ( **m** , **n** ) conv 4 unless both _D_ ( _m_ ) and _D_ ( _n_ ) are true,
**Ω** ( **m** , **m** ) conv 3 if _D_ ( _m_ ) is true,
**Ω** ( **m** , **n** ) conv 2 if _D_ ( _m_ ), _D_ ( _n_ ), _G_ ( _m_ , _n_ ), ∼ ( _m_ = _n_ ) are true,
**Ω** ( **m** , **n** ) conv 1 if _D_ ( _m_ ), _D_ ( _n_ ), ∼ _G_ ( _m_ , _n_ ), ∼ ( _m_ = _n_ ) are true.
In consequence of the conditions to which _D_ ( _x_ ), _G_ ( _x_ , _y_ ) are subjected, **Ω** must further satisfy:
( _a_ ) if **Ω** ( **m** , **n** ) is convertible to 1 or 2, then **Ω** ( **m** , **m** ) and **Ω** ( **n** , **n** ) are convertible to 3,
( _b_ ) if **Ω** ( **m** , **m** ) and **Ω** ( **n** , **n** ) are convertible to 3, then **Ω** ( **m** , **n** ) is convertible to 1, 2, or 3,
( _c_ ) if **Ω** ( **m** , **n** ) is convertible to 1, then **Ω** ( **n** , **m** ) is convertible to 2 and conversely,
( _d_ ) if **Ω** ( **m** , **n** ) and **Ω** ( **n** , **p** ) are convertible to 1, then **Ω** ( **m** , **p** ) is also,
( _e_ ) there is no sequence _m_ 1, _m_ 2,... such that **Ω** ( **m** _i_ +1, **m** _i_ ) conv 2 for each positive integer _i_ ,
( _f_ ) **Ω** ( **m** , **n** ) is always convertible to 1, 2, 3, or 4.
If a formula **Ω** satisfies these conditions then there are corresponding propositional functions _D_ ( _x_ ), _G_ ( _x, y_ ). We shall therefore say that **Ω** is an _ordinal formula_ if it satisfies the conditions ( _a_ )–( _f_ ). It will be seen that a consequence of this definition is that Dt is an ordinal formula; it represents the ordinal v. The definition that we have given does not pretend to have virtues such as elegance or convenience. It has been introduced rather to fix our ideas and to show how it is possible in principle to describe ordinals by means of well-formed formulae. The definitions could be modified in a number of ways. Some such modifications are quite trivial; they are typified by modifications such as changing the numbers 1, 2, 3, 4, used in the definition, to others. Two such definitions will be said to be equivalent; in general, we shall say that two definitions are equivalent if there are W.F.F. **T** , **T** ′ such that, if **A** is an ordinal formula under one definition and represents the ordinal a, then **T** ′( **A** ) is an ordinal formula under the second definition and represents the same ordinal; and, conversely, if **A** ′ is an ordinal formula under the second definition representing α, then **T** ( **A** ′) represents a under the first definition. Besides definitions equivalent in this sense to our original definition, there are a number of other possibilities open. Suppose for instance that we do not require _D_ ( _x_ ) and _G_ ( _x, y_ ) to be computable, but that we require only that _D_ ( _x_ ) and _G_ ( _x_ , _y_ ) & _x_ < _y_ are axiomatic. This leads to a definition of an ordinal formula which is (presumably) not equivalent to the definition that we are using. There are numerous possibilities, and little to guide us in choosing one definition rather than another. No one of them could well be described as "wrong"; some of them may be found more valuable in applications than others, and the particular choice that we have made has been determined partly by the applications that we have in view. In the case of theorems of a negative character, one would wish to prove them for each one of the possible definitions of "ordinal formula". This programme could, I think, be carried through for the negative results of §9, 10.
Before leaving the subject of possible ways of defining ordinal formulae, I must mention another definition due to Church and Kleene (Church and Kleene [1]). We can make use of this definition in constructing ordinal logics, but it is more convenient to use a slightly different definition which is equivalent (in the sense just described) to the Church–Kleene definition as modified in Church [ **4** ].
Introduce the abbreviations
We define first a partial ordering relation "<" which holds between certain pairs of W.F.F. [conditions (1)–(5)].
(1) If **A** conv **B** , then **A** < **C** implies **B** < **C** and **C** < **A** implies **C** < **B**.
(2) **A** < Suc ( **A** ).
(3) For any positive integers _m_ and _n_ , **_λ_** _ufx_. **R** ( **n** ) < **_λ_** _ufx_. **R** ( **m** ) implies **_λ_** _ufx_. **R** ( **n** ) < **_λ_** _ufx_. _u_ ( **R** ).
(4) If **A** < **B** and **B** < **C** , then **A** < **C**. (1)–(4) are required for any W.F.F. **A** , **B** , **C** , **_λ_** _ufx_. **R**.
(5) The relation **A** < **B** holds only when compelled to do so by (1)–(4).
We define C-K ordinal formulae by the conditions (6)–(10).
(6) If **A** conv **B** and **A** is a C-K ordinal formula, then B is a C-K ordinal formula.
(7) _U_ is a C-K ordinal formula.
(8) If **A** is a C-K ordinal formula, then Suc ( **A** ) is a C-K ordinal formula.
(9) If **_λ_** _ufx_. **R** ( **n** ) is a C-K ordinal formula and
**_λ_** _ufx_. **R** ( **n** ) < **_λ_** _ufx_. **R** _S_ ( **n** ))
for each positive integer _n_ , then **_λ_** _ufx_. _u_ ( **R** ) is a C- **K** ordinal formula.
(10) A formula is a C-K ordinal formula only if compelled to be so by (6)–(9).
The representation of ordinals by formulae is described by (11)–(15).
(11) If **A** conv **B** and **A** represents α, then **B** represents α.
(12) _U_ represents 1.
(13) If **A** represents α, then Suc ( **A** ) represents α + 1.
(14) If **_λ_** _ufx_. **R** ( **n** ) represents α _n_ for each positive integer _n_ , then **_λ_** _ufx_. _u_ ( **R** ) represents the upper bound of the sequence α1, α2, α3,...
(15) A formula represents an ordinal only when compelled to do so by (11)–(14).
We denote any ordinal represented by **A** by ΞA without prejudice to the possibility that more than one ordinal may be represented by **A**. We shall write **A** ≤ **B** to mean **A** < **B** or **A** conv **B**.
In proving properties of C-K ordinal formulae we shall often use a kind of analogue of the principle of transfinite induction. If φ is some property and we have:
then φ( **A** ) for each C-K ordinal formula **A**. To prove the validity of this principle we have only to observe that the class of formulae **A** satisfying φ( **A** ) is one of those of which the class of C-K ordinal formulae was defined to be the smallest. We can use this principle to help us to prove:
(i) Every C-K ordinal formula is convertible to the form **_λ_** _ufx_. **B** , where **B** is in normal form.
(ii) There is a method by which for any C-K ordinal formula, we can determine into which of the forms _U_ , Suc ( ** _λ_** _ufx_. **B** ), **_λ_** _ufx_. _u_ ( **R** ) (where _u_ is free in **R** ) it is convertible, and by which we can determine **B** , **R**. In each case **B** , **R** are unique apart from conversions.
(iii) If **A** represents any ordinal, ΞA is unique. If ΞA, ΞB exist and **A** < **B** , then Ξ **A** < Ξ **B**.
(iv) If **A** , **B** , **C** are C-K ordinal formulae and **B** < **A** , **C** < **A** , then either **B** < **C** , **C** < **B** , or **B** conv **C**.
(v) A formula A is a C-K ordinal formula if:
(A) _U_ ≤ **A** ,
(B) If **_λ_** _ufx_. _u_ ( **R** ) ≤ **A** and _n_ is a positive integer, then
**_λ_** _ufx_. **R** ( **n** ) < **_λ_** _ufx_ : **R** (S( _n_ )),
(C) For any two W.F.F. **B** , **C** with **B** < **A** , **C** < **A** we have **B** < **C** , **C** < **B** , or **B** conv **C** , but never **B** < **B** ,
(D) There is no infinite sequence **B** 1, **B** 2,... for which
**B** **r** < **B** **r** – 1 < **A**
for each _r_.
(vi) There is a formula _H_ such that, if **A** is a C-K ordinal formula, then _H_ ( **A** ) is an ordinal formula representing the same ordinal. _H_ ( **A** ) is not an ordinal formula unless **A** is a C-K ordinal formula.
_Proof of_ (i). Take φ( **A** ) to be " **A** is convertible to the form **_λ_** _ufx_. **B** , where **B** is in normal form". The conditions ( _a_ ) and ( _b_ ) are trivial. For ( _c_ ), suppose that **A** conv **_λ_** _ufx_. **B** , where **B** is in normal form; then
Suc ( **A** ) conv **_λ_** _ufx_. _f_ ( **B** )
and _f_ ( **B** ) is in normal form. For ( _d_ ) we have only to show that _u_ ( **R** ) has a normal form, _i.e_. that **R** has a normal form; and this is true since **R** (1) has a normal form.
_Proof of_ (ii). Since, by hypothesis, the formula is a C-K ordinal formula we have only to perform conversions on it until it is in one of the forms described. It is not possible to convert it into two of these three forms. For suppose that **_λ_** _ufx_. _f_ ( **A** ( _u_ , _f_ , _x_ )) conv **_λ_** _ufx_. _u_ ( **R** ) and is a C-K ordinal formula; it is then convertible to the form **_λ_** _ufx_. **B** , where **B** is in normal form. But the normal form of **_λ_** _ufx_. _u_ ( **R** ) can be obtained by conversions on R, and that of **_λ_** _ufx_. _f_ ( **A** ( _u_ , _f_ , _x_ )) by conversions on **A** ( _u_ , _f_ , _x_ ) (as follows from Church and Rosser [ **1** ], Theorem 2); this, however, would imply that the formula in question had two normal forms, one of form **_λ_** _ufx_. _u_ ( **S** ) and one of form **_λ_** _ufx_. _f_ ( **C** ), which is impossible. Or let _U_ conv **_λ_** _ufx_. _u_ ( **R** ), where **R** is a well formed formula with _u_ as a free variable. We may suppose **R** to be in normal form. Now _U_ is **_λ_** _ufx_. _u_ **_λ_** _y_. _f_ ( _y_ ( _I_ , _x_ ))). By (A), p. [149], **R** is identical with **_λ_** _y_. _fy_ ( _I_ , _x_ ) ð þ, which does not have _u_ as a free variable. It now remains to show only that if
Suc ( ** _λ_** _ufx_. **B** ) conv Suc ( ** _λ_** _ufx_. **B** ′) and **_λ_** _ufx_. _u_ ( **R** ) conv **_λ_** _ufx_. _u_ ( **R** ′), then **B** conv **B** ′ and **R** conv **R** ′.
but both of these formulae can be brought to normal form by conversions on **B** , **B** ′ and therefore **B** conv **B** ′. The same argument applies in the case in which **_λ_** _ufx_. _u_ ( **R** ) conv **_λ_** _ufx_. _u_ ( **R** ′).
_Proof of_ (iii). To prove the first half, take φ( **A** ) to be "ΞA is unique". Then 7.5 ( _a_ ) is trivial, and ( _b_ ) follows from the fact that _U_ is not convertible either to the form Suc ( **A** ) or to **_λ_** _ufx_. _u_ ( **R** ), where **R** has _u_ as a free variable. For ( _c_ ): Suc ( **A** ) is not convertible to the form **_λ_** _ufx_. _u_ ( **R** ); the possibility that Suc ( **A** ) represents an ordinal on account of (12) or (14) is therefore eliminated. By (13), Suc ( **A** ) represents α′ + 1 if **A** ′ represents α′ and Suc ( **A** ) conv Suc ( **A** ′). If we suppose that **A** represents a, then **A** , **A** ′, being C-K ordinal formulae, are convertible to the forms **_λ_** _ufx_. **B** , **_λ_** _ufx_. **B** ′; but then, by (ii), **B** conv **B** ′, _i.e_. **A** conv **A** ′, and therefore α = α′ by the hypothesis φ( **A** ). Then ΞSuc( **A** ) = α′ + 1 is unique. For ( _d_ ): **_λ_** _ufx_. _u_ ( **R** ) is not convertible to the form Suc ( **A** ) or to _U_ if **R** has _u_ as a free variable. If **_λ_** _ufx_. _u_ ( **R** ) represents an ordinal, it is so therefore in virtue of (14), possibly together with (11). Now, if **_λ_** _ufx_. _u_ ( **R** ) conv **_λ_** _ufx_. _u_ ( **R** ′), then **R** conv **R** ′, so that the sequence **_λ_** _ufx_. **R** (1), **_λ_** _ufx_. **R** (2),... in (14) is unique apart from conversions. Then, by the induction hypothesis, the sequence α1, α2, α3,... is unique. The only ordinal that is represented by **_λ_** _ufx_. _u_ ( **R** ) is the upper bound of this sequence; and this is unique.
For the second half we use a type of argument rather different from our transfinite induction principle. The formulae **B** for which **A** < **B** form the smallest class for which:
It will be sufficient to prove that the class of formulae **B** for which either Ξ **B** does not exist or Ξ **A** < Ξ **B** satisfies the conditions 7.6. Now
If Ξ ** _λ_** _ufx_. **R** ( **n** ) does not exist, then Ξ ** _λ_** _ufx_. _u_ (R) does not exist, and therefore **_λ_** _ufx_. _u_ ( **R** ) is in the class. If Ξ ** _λ_** _ufx_. R(n) exists and is greater than ΞA, and **_λ_** _ufx_. **R** ( **n** ) < **_λ_** _ufx_. **R** ( **m** ), then
Ξ ** _λ_** _ufx_. _u_ (R) ≥ Ξ ** _λ_** _ufx_. **R** ( **n** ) > Ξ **A** ,
so that **_λ_** _ufx_. _u_ ( **R** ) belongs to the class.
_Proof of_ (iv). We prove this by induction with respect to **A**. Take φ( **A** ) to be "whenever **B** < **A** and **C** < **A** then **B** < **C** or **C** < **B** or **B** conv **C** ′". φ( _U_ ) follows from the fact that we never have **B** < _U_. If we have φ( **A** ) and **B** < Suc ( **A** ), then either **B** < **A** or **B** conv **A** ; for we can find D such that **B** ≤ **D** , and then **D** < Suc ( **A** ) can be proved without appealing either to (1) or (5); (4) does not apply, so we must have **D** conv **A**. Then, if **B** < Suc ( **A** ) and **C** < Suc ( **A** ), we have four possibilities,
In the first case **B** conv **C** , in the second **C** < **B** , in the third **B** < **C** , and in the fourth the induction hypothesis applies.
Now suppose that **_λ_** _ufx_. **R** ( **n** ) is a C-K ordinal formula, that
**_λ_** _ufx_. **R** ( **n** ) < **_λ_** _ufx_. **R** ( _S_ (n)) and φ( **R** ( **n** )),
for each positive integer _n_ , and that **A** conv **_λ_** _ufx_. _u_ ( **R** ). Then, if **B** < **A** , this means that **B** < **_λ_** _ufx_. **R** ( **n** ) for some _n_ ; if we have also **C** < **A** , then **B** < **_λ_** _ufx_. **R** ( **q** ), **C** < **_λ_** _ufx_. **R** ( **q** ) for some _q_. Thus, for these **B** and **C** , the required result follows from φ( ** _λ_** _ufx_. **R** ( **q** )).
_Proof of_ (v). The conditions ( _C_ ), ( _D_ ) imply that the classes of interconvertible formulae **B** , **B** < **A** are well-ordered by the relation "<". We prove (v) by (ordinary) transfinite induction with respect to the order type a of the series formed by these classes; (α is, in fact, the solution of the equation 1 + α = Ξ **A** , but we do not need this). We suppose then that (v) is true for all order types less than α. If **E** < **A** , then **E** satisfies the conditions of (v) and the corresponding order type is smaller: **E** is therefore a C-K ordinal formula. This expresses all consequences of the induction hypothesis that we need. There are three cases to consider:
In case ( _x_ ) we must have **A** conv _U_ on account of ( **A** ). In case ( _y_ ) there is a formula **D** such that **D** < **A** , and **B** ≤ **D** whenever **B** < **A**. The relation **D** < **A** must hold in virtue of either (1), (2), (3), or (4). It cannot be in virtue of (4); for then there would be **B** , **B** < **A** , **D** < **B** contrary to (C), taken in conjunction with the definition of D. If it is in virtue of (3), then a is the upper bound of a sequence α1, α2,... of ordinals, which are increasing by reason of (iii) and the conditions **_λ_** _ufx_. **R** ( **n** ) < **_λ_** _ufx_. **R** _S_ ( **n** ) in (B). This is inconsistent with α = β + 1. This means that (2) applies [after we have eliminated (1) by suitable conversions on **A** , **D** ] and we see that **A** conv Suc ( **D** ); but, since **D** < **A** , **D** is a C-K ordinal formula, and **A** must therefore be a C-K ordinal formula by (8). Now take case ( _z_ ). It is impossible for **A** to be of the form Suc ( **D** ), for then we should have **B** < **D** whenever **B** < **A** , and this would mean that we had case ( _y_ ). Since _U_ < **A** , there must be an **F** such that **F** < **A** is demonstrable either by (2) or by (3) (after a possible conversion on **A** ); it must of course be demonstrable by (3). Then **A** is of the form **_λ_** _ufx_. _u_ ( **R** ). By (3), (B) we see that **_λ_** _ufx_. **R** ( **n** ) < **A** for each positive integer _n_ ; each **_λ_** _ufx_. **R** ( **n** ) is therefore a C-K ordinal formula. Applying (9), (B) we see that **A** is a C-K ordinal formula.
_Proof of_ (vi). To prove the first half, it is sufficient to find a method whereby from a C-K ordinal formula **A** we can find the corresponding ordinal formula **Ω**. For then there is a formula _H_ 1 such that _H_ 1 (a) conv **p** if _a_ is the G.R. of **A** and _p_ is that of **Ω**. _H_ is then to be defined by
_H_ → **_λ_** _a_. form ( _H_ 1(Gr( _a_ ))).
The method of finding **Ω** may be replaced by a method of finding **Ω** ( **m** , **n** ), given **A** and any two positive integers _m_ , _n_. We shall arrange the method so that, whenever **A** is not an ordinal formula, either the calculation of the values does not terminate or else the values are not consistent with **Ω** being an ordinal formula. In this way we can prove the second half of (vi).
Let Ls be a formula such that Ls( **A** ) enumerates the classes of formulae **B** , **B** < **A** [ _i.e_. if **B** < **A** there is one and only one positive integer _n_ for which Ls( **A, n** ) conv **B** ]. Then the rule for finding the value of **Ω** ( **m** , **n** ) is as follows:
First determine whether _U_ ≤ **A** and whether **A** is convertible to the form **r** (Suc, _U_ ). This terminates if **A** is a C-K ordinal formula.
If **A** conv **r** (Suc, _U_ ) and either _m_ > _r_ \+ 1 or _n_ > _r_ \+ 1, then the value is 4. If _m_ < _n_ ≤ _r_ \+ 1, the value is 2. If _n_ < _m_ ≤ _r_ \+ 1, the value is 1. If _m_ = _n_ ≤ _r_ \+ 1, the value is 3.
If **A** is not convertible to this form, we determine whether either **A** or Ls( **A, m** ) is convertible to the form **_λ_** _ufx_. _u_ ( **R** ); and if either of them is, we verify that **_λ_** _ufx_. **R** ( **n** ) < **_λ_** _ufx_. **R** ( _S_ (n). We shall eventually come to an affirmative answer if **A** is a C-K ordinal formula.
Having checked this, we determine concerning _m_ and _n_ whether Ls( **A, m** ) < Ls( **A, n** ), Ls( **A, n** ) < Ls( **A, m** ), or _m_ = _n_ , and the value is to be accordingly 1, 2, or 3.
If **A** is a C-K ordinal formula, this process certainly terminates. To see that the values so calculated correspond to an ordinal formula, and one representing ΞA, first observe that this is so when ΞA is finite. In the other case (iii) and (iv) show that ΞB determines a one–one correspondence between the ordinals β,1 ≤ β ≤ ΞA, and the classes of interconvertible formulae **B** , **B** < **A**. If we take _G_ ( _m_ , _n_ ) to be Ls( **A, m** ) < Ls( **A, n** ), we see that _G_ ( _m_ , _n_ ) is the ordering relation of a series of order type ΞA and on the other hand that the values of **Ω** ( **m** , **n** ) are related to _G_ ( _m_ , _n_ ) as on p. [162].
To prove the second half suppose that **A** is not a C-K ordinal formula. Then one of the conditions (A)–(D) in (v) must not be satisfied. If ( **A** ) is not satisfied we shall not obtain a result even in the calculation of **Ω** (1, 1). If (B) is not satisfied, we shall have for some positive integers _p_ and _q_ ,
Ls( **A, p** ) conv **_λ_** _ufx_. _u_ ( **R** )
but not **_λ_** _ufx_. **R** ( **q** ) < **_λ_** _ufx_. **R** ( _S_ ( **q** )). Then the process of calculating **Ω** ( **p** , **q** ) will not terminate. In case of failure of (C) or (D) the values of **Ω** ( **m** , **n** ) may all be calculable, but if so conditions ( _a_ )–( _f_ ), p. [162], will be violated. Thus, if **A** is not a C-K ordinal formula, then _H_ ( **A** ) is not an ordinal formula.
I propose now to define three formulae Sum, Lim, Inf of importance in connection with ordinal formulae. Since they are comparatively simple, they will for once be given almost in full. The formula Ug is one with the property that Ug( **m** ) is convertible to the formula representing the largest odd integer dividing _m_ : it is not given in full. _P_ is the predecessor function; _PS_ ( **m** ) conv **m** , _P_ (1) conv 1.
The essential properties of these formulae are described by:
it is then convertible to x.
If **Ω** , **Ω ′** are ordinal formulae representing α, β respectively, then Sum( **Ω** , **Ω ′** ) is an ordinal formula representing α + β. If **Z** is a W.F.F. enumerating a sequence of ordinal formulae representing α1, α2,..., then Lim(Z) is an ordinal formula representing the infinite sum α1 \+ α2 \+ α3... If **Ω** is an ordinal formula representing a, then Inf( **Ω** ) enumerates a sequence of ordinal formulae representing all the ordinals less than α without repetitions other than repetitions of the ordinal 0.
To prove that there is no general method for determining about a formula whether it is an ordinal formula, we use an argument akin to that leading to the Burali-Forti paradox; but the emphasis and the conclusion are different. Let us suppose that such an algorithm is available. This enables us to obtain a recursive enumeration **Ω** 1, **Ω** 2,... of the ordinal formulae in normal form. There is a formula **Z** such that **Z** ( **n** ) conv **Ω** **n**. Now Lim ( **Z** ) represents an ordinal greater than any represented by an **Ω** **n** , and it has therefore been omitted from the enumeration.
This argument proves more than was originally asserted. In fact, it proves that, if we take any class _E_ of ordinal formulae in normal form, such that, if **A** is any ordinal formula, then there is a formula in _E_ representing the same ordinal as **A** , then there is no method whereby one can determine whether a W.F.F. in normal form belongs to _E_.
#### 8. Ordinal logics
An ordinal logic is a W.F.F. **Λ** such that **Λ** ( **Ω** ) is a logic formula whenever **Ω** is an ordinal formula.
This definition is intended to bring under one heading a number of ways of constructing logics which have recently been proposed or which are suggested by recent advances. In this section I propose to show how to obtain some of these ordinal logics.
Suppose that we have a class _W_ of logical systems. The symbols used in each of these systems are the same, and a class of sequences of symbols called "formulae" is defined, independently of the particular system in _W_. The rules of procedure of a system _C_ define an axiomatic subset of the formulae, which are to be described as the "provable formulae of _C_ ". Suppose further that we have a method whereby, from any system _C_ of _W_ , we can obtain a new system _C_ ′, also in _W_ , and such that the set of provable formulae of _C_ ′ includes the provable formulae of _C_ (we shall be most interested in the case in which they are included as a proper subset). It is to be understood that this "method" is an effective procedure for obtaining the rules of procedure of _C_ ′ from those of _C_.
Suppose that to certain of the formulae of _W_ we make number-theoretic theorems correspond: by modifying the definition of formula, we may suppose that this is done for all formulae. We shall say that one of the systems _C_ is _valid_ if the provability of a formula in _C_ implies the truth of the corresponding number-theoretic theorem. Now let the relation of _C_ ′ to _C_ be such that the validity of _C_ implies the validity of _C_ ′, and let there be a valid system _C_ 0 in _W_. Finally, suppose that, given any computable sequence _C_ 1, _C_ 2,... of systems in _W_ , the "limit system", in which a formula is provable if and only if it is provable in one of the systems _C j_, also belongs to _W_. These limit systems are to be regarded, not as functions of the sequence given in extension, but as functions of the rules of formation of their terms. A sequence given in extension may be described by various rules of formation, and there will be several corresponding limit systems. Each of these may be described as _a_ limit system of the sequence.
In these circumstances we may construct an ordinal logic. Let us associate positive integers with the systems in such a way that to each _C_ there corresponds a positive integer _m C_, and that _m C_ completely describes the rules of procedure of _C_. Then there is a W.F.F. **K** , such that
**K** ( **m** _C_ ) conv **m** _C_
for each _C_ in _W_ , and there is a W.F.F. Θ such that, if **D** ( **r** ) conv **m** _Cr_ for each positive integer _r_ , then Θ( **D** ) conv **m** _C_ , where _C_ is a limit system of _C_ 1, _C_ 2,... With each system _C_ of _W_ it is possible to associate a logic formula **L** _C_ : the relation between them is that, if _G_ is a formula of _W_ and the number-theoretic theorem corresponding to _G_ (assumed expressed in the conversion calculus form) asserts that **B** is dual, then **L** _C_ ( **B** ) conv 2 if and only if _G_ is provable in _C_. There is a W.F.F. **G** such that
**G** ( **m** _C_ ) conv **L** _C_
for each _C_ of _W_. Put
**N** → **_**Λ**_** _a_. **G** ( _a_ ( **Θ** , **K** , **m** _C_ 0)).
I assert that **N** ( **A** ) is a logic formula for each C-K ordinal formula **A** , and that, if **A** < **B** , then **N** ( **B** ) is more complete than **N** ( **A** ), provided that there are formulae provable in _C_ ′ but not in _C_ for each valid _C_ of _W_.
To prove this we shall show that to each C-K ordinal formula **A** there corresponds a unique system _C_ [ **A** ] such that:
(i) **A** (Θ, **K** , **m** _C_ 0) conv **m** _C_ [ **A** ],
and that it further satisfies:
(ii) _C_ [ _U_ ] is a limit system of _C_ 0′, _C_ 0′,...,
(iii) _C_ [Suc ( **A** )] is ( _C_ [ **A** ])′,
(iv) _C_ [ ** _ **Λ**_** _ufx_. _u_ ( **R** )] is a limit system of _C_ [ ** _ **Λ**_** _ufx_. **R** (1)], _C_ [ ** _ **Λ**_** _ufx_. **R** (2)],...,
**A** and **_**Λ**_** _ufx_. _u_ ( **R** ) being assumed to be C-K ordinal formulae. The uniqueness of the system follows from the fact that _m C_ determines _C_ completely. Let us try to prove the existence of _C_ [ **A** ] for each C-K ordinal formula **A**. As we have seen (p. 164) it is sufficient to prove
( _a_ ) _C_ [ _U_ ] exists,
( _b_ ) if _C_ [ **A** ] exists, then _C_ [Suc ( **A** )] exists,
( _c_ ) if _C_ [ ** _ **Λ**_** _ufx_. **R** (1)], _C_ [ ** _ **Λ**_** _ufx_. **R** (2)],... exist, then _C_ [ ** _ **Λ**_** _ufs_. _u_ ( **R** )] exists.
_Proof of_ ( _a_ ).
{ ** _ **Λ**_** _y_. **K** ( _y_ ( _I_ , **m** _C_ 0))} (n) conv **K** ( **m** _C_ 0) conv **m** _C_ 0′
for all positive integers _n_ , and therefore, by the definition of **Θ** , there is a system, which we call _C_ [ _U_ ] and which is a limit system of _C_ 0′, _C_ 0′,..., satisfying
**Θ** ( ** _ **Λ**_** _y_. **K** ( _y_ ( _I_ , **m** _C_ 0))) conv **m** _C_ [ _U_ ].
But, on the other hand,
_U_ ( **Θ** , **K** , **m** _C_ 0) + conv **Θ** ( ** _ **Λ**_** _y_. **K** ( _y_ ( _I_ , **m** _C_ 0))).
This proves ( _a_ ) and incidentally (ii).
_Proof of_ ( _b_ ).
Hence _C_ [Suc ( **A** )] exists and is given by (iii).
_Proof of_ ( _c_ ).
by hypothesis. Consequently, by the definition of Θ, there exists a _C_ which is a limit system of
and satisfies
Θ({ ** _ **Λ**_** _ufx_. _u_ ( **R** )}(Θ, K, m _C_ 0)) conv **m** _C_.
We define _C_ [ ** _ **Λ**_** _ufx_. _u_ ( **R** )] to be this _C_. We then have (iv) and
This completes the proof of the properties (i)–(iv). From (ii), (iii), (iv), the fact that _C_ 0 is valid, and that _C_ ′ is valid when _C_ is valid, we infer that _C_ [ **A** ] is valid for each C-K ordinal formula **A** : also that there are more formulae provable in _C_ [ **B** ] than in _C_ [ **A** ] when **A** < **B**. The truth of our assertions regarding **N** now follows in view of (i) and the definitions of **N** and **G**.
We cannot conclude that **N** is an ordinal logic, since the formulae **A** are C-K ordinal formulae; but the formula _H_ enables us to obtain an ordinal logic from **N**. By the use of the formula Gr we obtain a formula Tn such that, if **A** has a normal form, then Tn( **A** ) enumerates the G.R.'s of the formulae into which **A** is convertible. Also there is a formula Ck such that, if _h_ is the G.R. of a formula _H_ ( **B** ), then Ck( **h** ) conv **B** , but otherwise Ck( **h** ) conv _U_. Since _H_ ( **B** ) is an ordinal formula only if **B** is a C-K ordinal formula, Ck (Tn( **Ω** , **n** )) is a C-K ordinal formula for each ordinal formula **Ω** and each integer _n_. For many ordinal formulae it will be convertible to _U_ , but, for suitable **Ω** , it will be convertible to any given C-K ordinal formula. If we put
**Λ** → **_**Λ**_** _wa_. **Γ** ( ** _ **Λ**_** _n_. **N** (Ck(Tn( _w_ , _n_ ))), _a_ ,)
Λ is the required ordinal logic. In fact, on account of the properties of **Γ** , **Λ** ( **Ω** , **A** ) will be convertible to 2 if and only if there is a positive integer _n_ such that
**N** (Ck(Tn( **Ω** , **n** )), **A** ) conv 2.
If **Ω** conv _H_ ( **B** ), there will be an integer _n_ such that Ck(Tn( **Ω** , **n** )) conv **B** , and then
**N** (Ck(Tn( **Ω** , **n** )), **A** ) conv **N** ( **B** , **A** ).
For any _n_ , Ck(Tn( **Ω** , **n** )) is convertible to _U_ or to some **B** , where **Ω** conv _H_ ( **B** ). Thus **Λ** ( **Ω** , **A** ) conv 2 if **Ω** conv _H_ ( **B** ) and **N** ( **B** , **A** ) conv 2 or if **N** ( _U_ , **A** ) conv 2, but not in any other case.
We may now specialize and consider particular classes _W_ of systems. First let us try to construct the ordinal logic described roughly in the introduction. For _W_ we take the class of systems arising from the system of _Principia Mathematica_ by adjoining to it axiomatic (in the sense described on p. [151]) sets of axioms. Gödel has shown that primitive recursive relations can be expressed by means of formulae in _P_. In fact, there is a rule whereby, given the recursion equations defining a primitive recursive relation, we can find a formula such that
is provable in _P_ if _F_ ( _m_ 1,..., _m_ _r_ ) is true, and its negation is provable otherwise. Further, there is a method by which we can determine about a formula whether it arises from a primitive recursive relation in this way, and by which we can find the equations which defined the relation. Formulae of this kind will be called _recursion formulae_. We shall make use of a property that they possess, which we cannot prove formally here without giving their definition in full, but which is essentially trivial. Db[ _x_ 0, _y_ 0] is to stand for a certain recursion formula such that Db[ _f_ ( _m_ )0, _f_ ( _n_ )0] is provable in _P_ if _m_ = 2 _n_ and its negation is provable otherwise. Suppose that , are two recursion formulae. Then the theorem which I am assuming is that there is a recursion relation such that we can prove
The significant formulae in any of our extensions of _P_ are those of the form
where is a recursion formula, arising from the relation _R_ ( _m_ , _n_ ) let us say. The corresponding number-theoretic theorem states that for each natural number _m_ there is a natural number _n_ such that _R_ ( _m_ , _n_ ) is true.
The systems in _W_ which are not valid are those in which a formula of the form 8.2 is provable, but at the same time there is a natural number, _m_ say, such that, for each natural number _n_ , _R_ ( _m_ , _n_ ) is false. This means to say that is provable for each natural number _n_. Since 8.2 is provable, is provable, so that
are all provable in the system. We may simplify 8.3. For a given _m_ we may prove a formula of the form in _P_ , where is a recursion
formula. Thus we find that a necessary and sufficient condition for a system of _W_ to be valid is that for no recursion formula are all of the formulae
provable. An important consequence of this is that, if
are recursion formulae, if
is provable in _C_ , and _C_ is valid, then we can prove in _C_ for some natural numbers _r_ , _a_ , where 1 ≤ _r_ ≤ _n_. Let us define to be the formula
and then define recursively by the condition that is and be Now I say that
is provable for 1 ≤ _r_ ≤ _n_. It is clearly provable for _r_ = 1: suppose it to be provable for a given _r_. We can prove
( _y_ 0)(∃ _x_ 0) **Db** [ _x_ 0, _y_ 0]
and
( _y_ 0)(∃ _x_ 0) **Db** [ _f x_ 0, _f y_ 0]
from which we obtain
and
These together with 8.1 yield
which is sufficient to prove 8.6 for _r_ \+ 1. Now, since 8.5 is provable in _C_ , must also be provable, and, since _C_ is valid, this means that must be provable for some natural number _m_. From 8.1 and the definition of we see that this implies that is provable for some natural numbers _a_ and _r_ , 1 ≤ _r_ ≤ _n_.
To any system _C_ of _W_ we can assign a primitive recursive relation _P_ _C_ ( _m_ , _n_ ) with the intuitive meaning " _m_ is the G.R. of a proof of the formula whose G.R. is _n_ ". We call the corresponding recursion formula Proof _C_ _x_ 0, _y_ 00, _f_ ( _n_ )0] is provable when _P_ _C_ ( _m_ , _n_ ) is true, and its negation is provable otherwise). We can now explain what is the relation of a system _C_ ′ to its predecessor _C_. The set of axioms which we adjoin to _P_ to obtain _C_ ′ consists of those adjoined in obtaining _C_ , together with all formulae of the form
where m is the G.R. of .
We want to show that a contradiction can be obtained by assuming _C_ ′ to be invalid but _C_ to be valid. Let us suppose that a set of formulae of the form 8.4 is provable in _C_ ′. Let be those axioms of _C_ ′ of the form 8.7 which are used in the proof of . We may suppose that none of them is provable in _C_. Then by the deduction theorem we see that
is provable in _C_. Let be (∃ _x_ 0) Proof Then from 8.8 we find that
is provable in _C_. It follows from a result which we have just proved that either is provable for some natural number _c_ , or else Proof _C_ [ _f_ ( _n_ )0, _f_ ( _ml_ )0] is provable in _C_ for some natural number _u_ and some _l_ , 1 ≤ _l_ ≤ _k_ : but this would mean that is provable in _C_ (this is one of the points where we assume the validity of _C_ ) and therefore also in _C_ ′, contrary to hypothesis. Thus must be provable in _C_ ′; but we are also assuming to be provable in _C_ ′. There is therefore a contradiction in _C_ ′. Let us suppose that the axioms of the form 8.7, when adjoined to _C_ are sufficient to obtain the contradiction and that none of these axioms is that provable in _C_. Then
is provable in _C_ , and if is (∃ _x_ 0) Proof then
is provable in _C_. But, by repetition of a previous argument, this means that is provable for some _l_ , 1 ≤ _l_ ≤ _k_ ′, contrary to hypothesis. This is the required contradiction.
We may now construct an ordinal logic in the manner described on pp. [171–3]. We shall, however, carry out the construction in rather more detail, and with some modifications appropriate to the particular case. Each system _C_ of our set _W_ may be described by means of a W.F.F. _M C_ which enumerates the G.R.'s of the axioms of _C_. There is a W.F.F. _E_ such that, if _a_ is the G.R. of some proposition , then _E_ ( _M C_, **a** ) is convertible to the G.R. of
If **a** is not the G.R. of any proposition in _P_ , then _E_ ( _M C_, **a** ) is to be convertible to the G.R. of 0 = 0. From _E_ we obtain a W.F.F. _K_ such that _K_ ( _M_ _C_ ,2 **n** \+ 1) conv _M_ _C_ ( **n** ), _K_ ( _M_ _C_ , 2 **n** ) conv _E_ ( _M_ _C_ , **n** ). The successor system _C_ ′ is defined by _K_ ( _M_ _C_ ) conv _M_ _C_ ′. Let us choose a formula _G_ such that _G_ ( _M_ _C_ , **A** ) conv 2 if and only if the number-theoretic theorem equivalent to " **A** is dual" is provable in _C_. Then we define **Λ** _P_ by
This is an ordinal logic provided that _P_ is valid.
Another ordinal logic of this type has in effect been introduced by Church. Superficially this ordinal logic seems to have no more in common with **Λ** _P_ than that they both arise by the method which we have described, which uses C-K ordinal formulae. The initial systems are entirely different. However, in the relation between _C_ and _C_ ′ there is an interesting analogy. In Church's method the step from _C_ to _C_ ′ is performed by means of subsidiary axioms of which the most important (Church [2], p. 88, **l** _m_ ) is almost a direct translation into his symbolism of the rule that we may take any formula of the form 8.4 as an axiom. There are other extra axioms, however, in Church's system, and it is therefore not unlikely that it is in some respects more complete than **Λ** _P_.
There are other types of ordinal logic, apparently quite unrelated to the type that we have so far considered. I have in mind two types of ordinal logic, both of which can be best described directly in terms of ordinal formulae without any reference to C-K ordinal formulae. I shall describe here a specimen **Λ** _H_ of one of these types of ordinal logic. Ordinal logics of this kind were first considered by Hilbert (Hilbert [1], 183 ff.), and have also been used by Tarski (Tarski [1], 395 ff.); see also Gödel [1], foot-note 48 _a_.
Suppose that we have selected a particular ordinal formula **Ω**. We shall construct a modification _P_ **Ω** of the system _P_ of Gödel (see foot-note [16] on p. [173]. We shall say that a natural number _n_ is a _type_ if it is either even or 2 _p_ 1, where **Ω** ( **p** , **p** ) conv 3. The definition of a variable in _P_ is to be modified by the condition that the only admissible subscripts are to be the types in our sense. Elementary expressions are then defined as in _P_ : in particular the definition of an elementary expression of type 0 is unchanged. An elementary formula is defined to be a sequence of symbols of the form , where are elementary expressions of types _m_ , _n_ satisfying one of the conditions ( _a_ ), ( _b_ ), ( _c_ ).
( _a_ ) _m_ and _n_ are both even and _m_ exceeds _n_ ,
( _b_ ) _m_ is odd and _n_ is even,
( _c_ ) _m_ = 2 _p_ – 1, _n_ = 2 _q_ – 1, and **Ω** ( **p, q** ) conv 2.
With these modifications the formal development of _P_ **Ω** is the same as that of _P_. We want, however, to have a method of associating number-theoretic theorems with certain of the formulae of _P_ **Ω**. We cannot take over directly the association which we used in _P_. Suppose that _G_ is a formula in _P_ interpretable as a number-theoretic theorem in the way described in the course of constructing **Λ** _P_ (p. [174]). Then, if every type suffix in _G_ is doubled, we shall obtain a formula in _P_ **Ω** which is to be interpreted as the same number-theoretic theorem. By the method of §6 we can now obtain from _P_ **Ω** a formula _L_ **Ω** which is a logic formula if _P_ **Ω** is valid; in fact, given **Ω** there is a method of obtaining _L_ **Ω** , so that there is a formula **Λ** _H_ such that **Λ** _H_ ( **Ω** ) conv _L_ **Ω** for each ordinal formula **Ω**.
Having now familiarized ourselves with ordinal logics by means of these examples we may begin to consider general questions concerning them.
#### 9. Completeness questions
The purpose of introducing ordinal logics was to avoid as far as possible the effects of Gödel's theorem. It is a consequence of this theorem, suitably modified, that it is impossible to obtain a complete logic formula, or (roughly speaking now) a complete system of logic. We were able, however, from a given system to obtain a more complete one by the adjunction as axioms of formulae, seen intuitively to be correct, but which the Gödel theorem shows are unprovable in the original system; from this we obtained a yet more complete system by a repetition of the process, and so on. We found that the repetition of the process gave us a new system for each C-K ordinal formula. We should like to know whether this process suffices, or whether the system should be extended in other ways as well. If it were possible to determine about a W.F.F. in normal form whether it was an ordinal formula, we should know for certain that it was necessary to make extensions in other ways. In fact for any ordinal formula **Λ** it would then be possible to find a single logic formula **L** such that, if **Λ** ( **Ω** , **A** ) conv 2 for some ordinal formula **Ω** , then **L** ( **A** ) conv 2. Since L must be incomplete, there must be formulae **A** for which L( **Ω** , **A** ) is not convertible to 2 for any ordinal formula **Ω**. However, in view of the fact, proved in §7, that there is no method of determining about a formula in normal form whether it is an ordinal formula, the case does not arise, and there is still a possibility that some ordinal logics may be complete in some sense. There is a quite natural way of defining completeness.
_Definition of completeness of an ordinal logic_. We say that an ordinal logic **Λ** is complete if corresponding to each dual formula **A** there is an ordinal formula **Ω** **A** such that **Λ** ( **Ω** **A** , **A** ) conv 2.
As has been explained in §2, the reference in the definition to the existence of **Ω** **A** for each **A** is to be understood in the same naïve way as any reference to existence in mathematics.
There is room for modification in this definition: we might require that there is a formula **X** such that **X** ( **A** ) conv **Ω** **A** , **X** ( **A** ) being an ordinal formula whenever **A** is dual. There is no need, however, to discuss the relative merits of these two definitions, because in all cases in which we prove an ordinal logic to be complete we shall prove it to be complete even in the modified sense; but in cases in which we prove an ordinal logic to be incomplete, we use the definition as it stands.
In the terminology of §6, **Λ** is complete if the class of logics **Λ** ( **Ω** ) is complete when **Ω** runs through all ordinal formulae.
There is another completeness property which is related to this one. Let us for the moment describe an ordinal logic **Λ** as _all inclusive_ if to each logic formula **Λ** there corresponds an ordinal formula **Ω** ( **L** ) such that **Λ** ( **Ω** ( **L** )) is as complete as **L.** Clearly every all inclusive ordinal logic is complete; for, if **A** is dual, then **δ** ( **A** ) is a logic with **A** in its extent. But, if **Λ** is complete and
Ai → **_**_**Λ**_**_** _kw_. **Γ** ( _ ** _ **Λ**_** ra_. **δ** (4, **δ** (2, _k_ ( _w_ , _V_ (Nm(r)))) + **δ** (2,Nm( _r_ , _a_ )))),
then Ai( **Λ** ) is an all inclusive ordinal logic. For, if **A** is in the extent of **Λ** ( **Ω** **A** ) for each **A** , and we put **Ω** ( **L** ) → **Ω** _V_ ( **L** ), then I say that, if **B** is in the extent of **L** , it must be in the extent of Ai( **Λ** , **Ω** ( **L** )). In fact, we see that Ai( **Λ** , **Ω** _V_ ( **L** ), **B** ) is convertible to
**Γ** ( **Λ** _ra_. **δ** (4, **δ** (2, **Λ** ( **Ω** _V_ ( **L** ), _V_ (Nm( _r_ )))) + **δ** (2,Nm( _r_ , _a_ ))), **B** ).
For suitable _n_ , Nm( **n** ) conv **L** and then
**Λ** ( **Ω** _V_ ( **L** ), _V_ (Nm( **n** ))) conv 2,
Nm( **n** , **B** ) conv 2,
and therefore, by the properties of **Γ** and **δ**
Ai( **Λ** , **Ω** _V_ ( **L** ), **B** ) conv 2.
Conversely Ai( **Λ** , **Ω** _V_ ( **L** ), **B** ) can be convertible to 2 only if both Nm( **n** , **B** ) and **Λ** ( **Ω** _V_ ( **L** ), _V_ (Nm( **n** ))) are convertible to 2 for some positive integer _n_ ; but, if **Λ** ( **Ω** _V_ ( **L** ), _V_ (Nm( **n** ))) conv 2, then Nm( **n** ) must be a logic, and, since Nm( **n** , **B** ) conv 2, **B** must be dual.
It should be noticed that our definitions of completeness refer only to number-theoretic theorems. Although it would be possible to introduce formulae analogous to ordinal logics which would prove more general theorems than number-theoretic ones, and have a corresponding definition of completeness, yet, if our theorems are too general, we shall find that our (modified) ordinal logics are never complete. This follows from the argument of §4. If our "oracle" tells us, not whether any given number-theoretic statement is true, but whether a given formula is an ordinal formula, the argument still applies, and we find that there are classes of problem which cannot be solved by a uniform process even with the help of this oracle. This is equivalent to saying that there is no ordinal logic of the proposed modified type which is complete with respect to these problems. This situation becomes more definite if we take formulae satisfying conditions ( _a_ )–( _e_ ), ( _f_ ′) (as described at the end of §12) instead of ordinal formulae; it is then not possible for the ordinal logic to be complete with respect to any class of problems more extensive than the number-theoretic problems.
We might hope to obtain some intellectually satisfying system of logical inference (for the proof of number-theoretic theorems) with some ordinal logic. Gödel's theorem shows that such a system cannot be wholly mechanical; but with a complete ordinal logic we should be able to confine the non-mechanical steps entirely to verifications that particular formulae are ordinal formulae.
We might also expect to obtain an interesting classification of number-theoretic theorems according to "depth". **A** theorem which required an ordinal **α** to prove it would be deeper than one which could be proved by the use of an ordinal **β** less than **α**. However, this presupposes more than is justified. We now define
_Invariance of ordinal logics_. An ordinal logic **Ω** is said to be _invariant up to_ an ordinal **α** if, whenever **Ω** , **Ω** ′ are ordinal formulae representing the same ordinal less than **α** , the extent of **Λ** ( **Ω** ) is identical with the extent of **Λ** ( **Ω** ′). An ordinal logic is _invariant_ if it is invariant up to each ordinal represented by an ordinal formula.
Clearly the classification into depths presupposes that the ordinal logic used is invariant.
Among the questions that we should now like to ask are
( _a_ ) Are there any complete ordinal logics?
( _b_ ) Are there any complete invariant ordinal logics?
To these we might have added "are all ordinal logics complete?"; but this is trivial; in fact, there are ordinal logics which do not suffice to prove any number-theoretic theorems whatever.
We shall now show that ( _a_ ) must be answered affirmatively. In fact, we can write down a complete ordinal logic at once. Put
and
Comp → **_**_**Λ**_**_** _wa_. **δ** ( _w_ , Od( _a_ )).
I shall show that Comp is a complete ordinal logic.
For if, Comp( **Ω** , **A** ) conv 2, then
**Ω** ( **m** , **n** ) has a normal form if **Ω** is an ordinal formula, so that then
has a normal form; this means that r( _I_ , **A** ( **m** )) conv 2 some _r, i.e_. **A** ( **m** ) conv 2. Thus, if Comp( **Ω** , **A** ) conv 2 and **Ω** is an ordinal formula, then **A** is dual. Comp is therefore an ordinal logic. Now suppose conversely that **A** is dual. I shall show that Od( **A** ) is an ordinal formula representing the ordinal **ω**. For
_i.e_. Od( **A** ) is an ordinal formula representing the same ordinal as Dt. But
Comp(Od( **A** ), **A** ) conv **δ** (Od( **A** ), Od( **A** )) conv 2.
This proves the completeness of Comp.
Of course Comp is not the kind of complete ordinal logic that we should really wish to use. The use of Comp does not make it any easier to see that **A** is dual. In fact, if we really want to use an ordinal logic, a proof of completeness for that particular ordinal logic will be of little value; the ordinals given by the completeness proof will not be ones which can easily be seen intuitively to be ordinals. The only value in a completeness proof of this kind would be to show that, if any objection is to be raised against an ordinal logic, it must be on account of something more subtle than incompleteness.
The theorem of completeness is also unexpected in that the ordinal formulae used are all formulae representing **ω**. This is contrary to our intentions in constructing **Λ** _P_ for instance; implicitly we had in mind large ordinals expressed in a simple manner. Here we have small ordinals expressed in a very complex and artificial way.
Before trying to solve the problem ( _b_ ), let us see how far **Λ** _P_ and **Λ** _H_ are invariant. We should certainly not expect **Λ** _P_ to be invariant, since the extent of **Λ** _P_ ( **Ω** ) will depend on whether **Ω** is convertible to a formula of the form _H_ ( **A** ): but suppose that we call an ordinal logic **Λ** "C-K invariant up to α" if the extent of **Λ** ( _H_ ( **A** )) is the same as the extent of **Λ** ( _H_ ( **B** )) whenever **A** and **B** are C-K ordinal formulae representing the same ordinal less than α. How far is **Λ** _P_ C-K invariant? It is not difficult to see that it is C-K invariant up to any finite ordinal, that is to say up to **ω**. It is also C-K invariant up to **ω** \+ 1, as follows from the fact that the extent of
is the set-theoretic sum of the extents of
**λ** _P_ ( _H_ ( **λ** _ufx_. **R** (1))), **λ** _P_ ( _H_ ( **λ** _ufx_. **R** (2))),...
However, there is no obvious reason for believing that it is C-K invariant up to **ω** \+ 2, and in fact it is demonstrable that this is not the case (see the end of this section). Let us find out what happens if we try to prove that the extent of
**Λ** _P_ ( _H_ (Suc ( **λ** _ufx_. _u_ ( **R** 1))))
is the same as the extent of
**Λ** _P_ ( _H_ (Suc ( **λ** _ufx_. _u_ ( **R** 2)))),
where **λ** _ufx_. _u_ ( **R** 1) and **λ** _ufx_. _u_ ( **R** 2) are two C-K ordinal formulae representing **ω**. We should have to prove that a formula interpretable as a number-theoretic theorem is provable in _C_ [Suc ( **λ** _ufx_. _u_ ( **R** 1))] if, and only if, it is provable in _C_ [Suc ( **λ** _ufx_. _u_ ( **R** 2))]. Now _C_ [Suc ( **λ** _ufx_. _u_ ( **R** 1))] is obtained from _C_ [ **λ** _ufx_. _u_ ( **R** 1)] by adjoining all axioms of the form
where _m_ is the G.R. of , and _C_ [Suc ( **λ** _ufx_. _u_ ( **R** 2))] is obtained from _C_ [ **λ** _ufx_. _u_ ( **R** 2)] by adjoining all axioms of the form
The axioms which must be adjoined to _P_ to obtain _C_ [ **λ** _ufx_. _u_ ( **R** 1)] are essentially the same as those which must be adjoined to obtain the system _C_ [ **λ** _ufx_. _u_ ( **R** 2)]: however the _rules of procedure which have to be applied before these axioms can be written down are in general quite different in the two cases_. Consequently 9.1 and 9.2 are quite different axioms, and there is no reason to expect their consequences to be the same. **A** proper understanding of this will make our treatment of question ( _b_ ) much more intelligible. See also footnote [17] on page [173].
Now let us turn to **_Λ_** _H_. This ordinal logic is invariant. Suppose that **Ω** , **Ω** ′ represent the same ordinal, and suppose that we have a proof of a number-theoretic theorem _G_ in _P_ **Ω**. The formula expressing the number-theoretic theorem does not involve any odd types. Now there is a one–one correspondence between the odd types such that if 2 _m_ − 1 corresponds to 2 _m_ ′ − 1 and 2 _n_ − 1 to 2 _n_ ′ − 1 then **Ω** ( **m** , **n** ) conv 2 implies **Ω** ′(m′, n′) conv 2. Let us modify the odd type-subscripts occurring in the proof of _G_ , replacing each by its mate in the one–one correspondence. There results a proof in _P_ **Ω** ′ with the same end formula _G_. That is to say that if _G_ is provable in _P_ **Ω** it is provable in _P_ **Ω** ′. **Λ** _H_ is invariant.
The question ( _b_ ) must be answered negatively. Much more can be proved, but we shall first prove an even weaker result which can be established very quickly, in order to illustrate the method.
I shall prove that an ordinal logic **Λ** cannot be invariant and have the property that the extent of **Λ** ( **Ω** ) is a strictly increasing function of the ordinal represented by **Ω**. Suppose that **Λ** has these properties; then we shall obtain a contradiction. Let **A** be a W.F.F. in normal form and without free variables, and consider the process of carrying out conversions on **A** (1) until we have shown it convertible to 2, then converting **A** (2) to 2, then **A** (3) and so on: suppose that after _r_ steps we are still performing the conversion on **A** ( **m** _r_ ). There is a formula Jh such that Jh( **A** , **r** ) conv **m** _r_ for each positive integer _r_. Now let **Z** be a formula such that, for each positive integer _n_ , **Z** ( **n** ) is an ordinal formula representing **ω** n, and suppose **B** to be a member of the extent of **Λ** (Suc (Lim( **Z** ))) but not of the extent of **Λ** (Lim( **Z** )). Put
**K** * → **λ** _a_. **Λ** (Suc (Lim ( **λ** _r_. **Z** (Jh( _a_ , _r_ )))), **B** ;
then **K** * is a complete logic. For, if **A** is dual, then
Suc(Lim( **λ** _r_. **Z** (Jh( **A** , _r_ ))))
represents the ordinal **ω ω** \+ 1, and therefore **K** * ( **A** ) conv 2; but, if **A** ( **c** ) is not convertible to 2, then
Suc(Lim( **λ** _r_. **Z** (Jh( **A** , _r_ )))
represents an ordinal not exceeding **Ω** c \+ 1, and **K** *( **A** ) is therefore not convertible to 2. Since there are no complete logic formulae, this proves our assertion.
We may now prove more powerful results.
_Incompleteness theorems_. (A) If an ordinal logic **Λ** is invariant up to an ordinal α, then for any ordinal formula **Ω** representing an ordinal **β** , **β** < **α** , the extent of **Λ** ( **Ω** ) is contained in the (set-theoretic) sum of the extents of the logics **Λ** ( **P** ), where **P** is finite.
(B) If an ordinal logic **Λ** is C-K invariant up to an ordinal α, then for any C-K ordinal formula **A** representing an ordinal **β** , **β** < **α** , the extent of **Λ** ( _H_ ( **A** )) is contained in the (set-theoretic) sum of the extents of the logics **Λ** ( _H_ ( **F** )), where **F** is a C-K ordinal formula representing an ordinal less than **ω** 2.
_Proof of_ ( **A** ). It is sufficient to prove that, if **Ω** represents an ordinal γ, ω ≤ γ < α, then the extent of **Λ** ( **Ω** ) is contained in the set-theoretic sum of the extents of the logics **Λ** ( **Ω** ′), where **Ω** ′ represents an ordinal less than γ. The ordinal γ must be of the form γ0 \+ _ρ_ , where _ρ_ is finite and represented by **P** say, and γ0 is not the successor of any ordinal and is not less than **ω**. There are two cases to consider; γ0 = ω and γ0 ≥ 2 **ω**. In each of them we shall obtain a contradiction from the assumption that there is a W.F.F. **B** such that **Λ** ( **Ω** , **B** ) conv 2 whenever **Ω** represents γ, but is not convertible to 2 if **Ω** represents a smaller ordinal. Let us take first the case γ0 ≥ 2 **ω**. Suppose that γ0 = **ω** \+ γ1, and that **Ω** 1 is an ordinal formula representing γ1. Let **A** be any W.F.F. with a normal form and no free variables, and let _Z_ be the class of those positive integers which are exceeded by all integers _n_ for which **A** ( **n** ) is not convertible to 2. Let _E_ be the class of integers 2 _p_ such that **Ω** ( **p** , **n** ) conv 2 for some _n_ belonging to _Z_. The class _E_ , together with the class _Q_ of all odd integers, is constructively enumerable. It is evident that the class can be enumerated with repetitions, and since it is infinite the required enumeration can be obtained by striking out the repetitions. There is, therefore, a formula En such that En( **Ω** , **A** , **r** ) runs through the formulae of the class _E_ \+ _Q_ without repetitions as _r_ runs through the positive integers. We define
Rt → **λ** _wamn_. Sum(Dt, _w_ , En( _w_ , _a_ , _m_ ), En( _w_ , _a_ , _n_ )).
Then Rt( **Ω** 1, **A** ) is an ordinal formula which represents γ0 if **A** is dual, but a smaller ordinal otherwise. In fact
Rt( **Ω** 1, **A** , **m** , **n** ) conv {Sum(Dt, **Ω** 1)}(En( **Ω** 1, **A** , **m** ), En( **Ω** 1, **A** , **n** )).
Now, if **A** is dual, _E_ \+ _Q_ includes all integers _m_ for which
{Sum(Dt, **Ω** 1)}( **m** , **m** ) conv 3.
(This depends on the particular form that we have chosen for the formula Sum.) Putting "En( **Ω** 1, **A** , **p** ) conv **q** " for _M_ ( _p, q_ ), we see that condition 7.4 is satisfied, so that Rt( **Ω** 1, **A** ) is an ordinal formula representing γ0. But, if **A** is not dual, the set _E_ \+ _Q_ consists of all integers _m_ for which
{Sum(Dt, **Ω** 1)}( **m, r** ) conv 2,
where _r_ depends only on **A**. In this case Rt( **Ω** 1, **A** ) is an ordinal formula representing the same ordinal as Inf(Sum(Dt, **Ω** 1), **r** ), and this is smaller than γ0. Now consider **K** :
**K** → **λ** _a_. **Λ** (Sum(Rt( **Ω** 1, **A** ), **P** ), **B** ).
If **A** is dual, **K** ( **A** ) is convertible to 2 since Sum(Rt( **Ω** 1, **A** ), **P** ) represents **γ**. But, if **A** is not dual, it is not convertible to 2, since Sum(Rt( **Ω** 1, **A** ), **P** ) then represents an ordinal smaller than **γ**. In **K** we therefore have a complete logic formula, which is impossible.
Now we take the case **γ** 0 = **ω**. We introduce a W.F.F. Mg such that if _n_ is the D.N. of a computing machine , and if by the _m_ -th complete configuration of the figure 0 has been printed, then Mg( **n** , **m** ) is convertible to **λ** _pq_. Al(4( _P_ , 2 _p_ \+ 2 _q_ ), 3, 4) (which is an ordinal formula representing the ordinal 1), but if 0 has not been printed it is convertible to **λ** _pq_. _p_ ( _q_ , _I_ , 4) (which represents 0). Now consider
**M** → **λ** _n_. **Λ** (Sum(Lim(Mg( _n_ )), **P** ), **B** ).
If the machine never prints 0, then Lim( **λ** _r_. Mg( **n** , _r_ )) represents **Ω** and Sum(Lim(Mg( **n** )), **P** ) represents **γ**. This means that **M** ( **n** ) is convertible to 2. If, however, never prints 0, Sum(Lim(Mg( **n** )), **P** ) represents a finite ordinal and **M** ( **n** ) is not convertible to 2. In **M** we therefore have means of determining about a machine whether it ever prints 0, which is impossible (Turing [ **1** ], 8). This completes the proof of ( **A** ).
_Proof of_ (B). It is sufficient to prove that, if **C** represents an ordinal **γ** , **ω** 2 ≤ **γ** < **α** , then the extent of **Λ** ( _H_ ( **C** )) is included in the set-theoretic sum of the extents of **Λ** ( _H_ (G)), where **G** represents an ordinal less than **γ**. We obtain a contradiction from the assumption that there is a formula **B** which is in the extent of **Λ** ( _H_ ( **G** )) if **G** represents **γ** , but not if it represents any smaller ordinal. The ordinal **γ** is of the form **δ** \+ **ω** 2 \+ , where < **ω** 2. Let **D** be a C-K ordinal formula representing **δ** and **λ** _ufx_. **Q** ( _u_ , _f_ , **A** ( _u_ , _f_ , _x_ )) one representing **α** \+ whenever **A** represents **α**.
We now define a formula Hg. Suppose that **A** is a W.F.F. in normal form and without free variables; consider the process of carrying out conversions on **A** (1) until it is brought into the form 2, then converting **A** (2) to 2, then **A** (3), and so on. Suppose that at the _r_ -th step of this process we are doing the _n r_-th step in the conversion of **A** ( **m** _r_ ). Thus, for instance, if **A** is not convertible to 2, _m r_ can never exceed 3. Then Hg( **A** , **r** ) is to be convertible to **λ** _f_. _f_ ( **m** _r_ , **n** _r_ ) for each positive integer _r_. Put
Sq → **λ** _dmn_. _n_ (Suc, _m_ ( **λ** _aufx_. _u_ ( **λ** _y_. _y_ (Suc, _a_ ( _u_ , _f_ , _x_ ))), _d_ ( _u_ , _f_ , _x_ ))),
**M** → **λ** _aufx_. Q( _u_ , _f_ , _u_ ( **λ** _y_. Hg( _a_ , _y_ , Sq( **D** )))),
**K** 1 → **λ** _a_. **Λ** ( **M** ( _a_ ), **B** ),
then I say that **K** 1 is a complete logic formula. Sq( **D** , **m** , **n** ) is a C-K ordinal formula representing **δ** \+ **_m_ ω** \+ **_n_** , and therefore Hg( **A** , **r** , Sq( **D** )) represents an ordinal _ζ r_ which increases steadily with increasing _r_ , and tends to the limit **δ** \+ **ω 2** if **A** is dual. Further
Hg( **A** , **r** , Sq( **D** )) < Hg( **A** , _S_ ( **r** ), Sq( **D** )
for each positive integer _r_. Therefore **λ** _ufx_. _u_ ( **λ** _y_. Hg( **A** , _y_ , Sq( **D** ))) is a C-K ordinal formula and represents the limit of the sequence _ζ_ 1, _ζ_ 2, _ζ_ 3,... This is **δ** \+ **ω** 2 if **A** is dual, but a smaller ordinal otherwise. Likewise **M** ( **A** ) represents γ if **A** is dual, but is a smaller ordinal otherwise. The formula **B** therefore belongs to the extent of **Λ** ( _H_ ( **M** ( **A** ))) if and only if **A** is dual, and this implies that **K** 1 is a complete logic formula, as was asserted. But this is impossible and we have the required contradiction.
As a corollary to (A) we see that Λ _H_ is incomplete and in fact that the extent of Λ _H_ (Dt) contains the extent of **Λ** _H_ ( **Ω** ) for any ordinal formula **Ω**. This result, suggested to me first by the solution of question ( _b_ ), may also be obtained more directly. In fact, if a number-theoretic theorem can be proved in any particular _P_ **Ω** , it can also be proved in _P_ λ _mn_. _m_ ( _n_ , _I_ , 4). The formulae describing number-theoretic theorems in _P_ do not involve more than a finite number of types, type 3 being the highest necessary. The formulae describing the number-theoretic theorems in any _P_ **Ω** will be obtained by doubling the type subscripts. Now suppose that we have a proof of a number-theoretic theorem _G_ in _P_ **Ω** and that the types occurring in the proof are among 0, 2, 4, 6, _t_ 1, _t_ 2, _t_ 3,... We may suppose that they have been arranged with all the even types preceding all the odd types, the even types in order of magnitude, and the type 2 _m_ − 1 preceding 2 _n_ − 1 if **Ω** ( **m** , **n** ) conv 2. Now let each _t_ _r_ be replaced by 10 + 2 _r_ throughout the proof of _G_. We thus obtain a proof of _G_ in _P_ λ _mn_. ( _n_ , _I_ , 4).
As with problem ( _a_ ), the solution of problem ( _b_ ) does not require the use of high ordinals [ _e.g_. if we make the assumption that the extent of **Λ** ( **Ω** ) is a steadily increasing function of the ordinal represented by **Ω** we do not have to consider ordinals higher than **ω** \+ 2]. However, if we restrict what we are to call ordinal formulae in some way, we shall have corresponding modified problems ( _a_ ) and ( _b_ ); the solutions will presumably be essentially the same, but will involve higher ordinals. Suppose, for example, that Prod is a W.F.F. with the property that Prod( **Ω** 1, **Ω** 2) is an ordinal formula representing α1α2 when **Ω** 1, **Ω** 2 are ordinal formulae representing α1, α2 respectively, and suppose that we call a W.F.F. a 1-ordinal formula when it is convertible to the form Sum(Prod( **Ω** , Dt), **P** ), where **Ω, P** are ordinal formulae of which **P** represents a finite ordinal. We may define 1-ordinal logics, 1-completeness and 1-invariance in an obvious way, and obtain a solution of problem ( _b_ ) which differs from the solution in the ordinary case in that the ordinals less than **ω** 2 take the place of the finite ordinals. More generally the cases that I have in mind are covered by the following theorem.
Suppose that we have a class V of formulae representing ordinals in some manner which we do not propose to specify definitely, and a subset _U_ of the class _V_ such that:
(i) There is a formula **Φ** such that if **T** enumerates a sequence of members of _U_ representing an increasing sequence of ordinals, then **Φ** ( **T** ) is a member of _U_ representing the limit of the sequence.
(ii) There is a formula **E** such that **E** ( **m** , **n** ) is a member of _U_ for each pair of positive integers _m_ , _n_ and, if it represents _m_ , _n_ , then _m_ , _n_ < _m_ ′, _n_ ′ if either _m_ < _m_ ′ or _m_ = _m_ ′, _n_ < _n_ ′.
(iii) There is a formula **G** such that, if **A** is a member of _U_ , then **G** ( **A** ) is a member of _U_ representing a larger ordinal than does **A** , and such that **G** ( **E** ( **m** , **n** )) always represents an ordinal not larger than _m_ , _n_ +1.
We define a _V_ -ordinal logic to be a W.F.F. **Λ** such that **Λ** ( **A** ) is a logic whenever **A** belongs to _V_. **Λ** is _V_ -invariant if the extent of **Λ** ( **A** ) depends only on the ordinal represented by **A**. Then it is not possible for a _V_ -ordinal logic **Λ** to be _V_ -invariant and have the property that, if **C** 1 represents a greater ordinal than **C** 2( **C** 1 and **C** 2 both being members of _U_ ), then the extent of **Λ** ( **C** 1) is greater than the extent of **Λ** ( **C** 2).
We suppose the contrary. Let **B** be a formula belonging to the extent of **Λ** (( **Φ** ( **λ** _r_. **E** ( _r_ , 1)))) but not to the extent of **Λ** ( **Φ** ( **λ** _r_. **E** ( _r_ , 1))),
and let
**K** ′ → **λ** _a_. **Λ** ( **G** ( **Φ** ( **λ** _r_. Hg( _a_ , _r_ , **E** ), **B** ).
Then **K** ′ is a complete logic. For
Hg( **A, r, E** ) conv **E** ( **m** _r_ , **n** _r_ ).
**E** ( **m** _r_ , **n** _r_ ) is a sequence of _V_ -ordinal formulae representing an increasing sequence of ordinals. Their limit is represented by **Φ** ( **λ** _r_. Hg( **A** , _r_ , **E** )); let us see what this limit is. First suppose that **A** is dual: then _m r_ tends to infinity as _r_ tends to infinity, and **Φ** ( **λ** _r_. Hg( **A** , _r_ , **E** )) therefore represents the same ordinal as **Φ** ( **λ** _r_. **E** ( _r_ , 1)). In this case we must have
**K** ′( **A** ) conv 2.
Now suppose that **A** is not dual: _m r_ is eventually equal to some constant number, _a_ say, and **Φ** ( **λ** _r_. Hg( **A** , _r_ , **E** )) represents the same ordinal as **Φ** ( **λ** _r_. **E** ( **a** , _r_ )), which is smaller than that represented by **Φ** ( **λ** _r_. **E** ( _r_ , 1)). **B** cannot therefore belong to the extent of **Λ** ( **G** ( **Φ** ( **λ** _r_. Hg( **A** , _r_ , **E** )))), and **K** ′( **A** ) is not convertible to 2. We have proved that **K** ′ is a complete logic, which is impossible.
This theorem can no doubt be improved in many ways. However, it is sufficiently general to show that, with almost any reasonable notation for ordinals, completeness is incompatible with invariance.
We can still give a certain meaning to the classification into depths with highly restricted kinds of ordinals. Suppose that we take a particular ordinal logic L and a particular ordinal formula **Ψ** representing the ordinal a say (preferably a large one), and that we restrict ourselves to ordinal formulae of the form Inf( **Ψ** , a). We then have a classification into depths, but the extents of all the logics which we so obtain are contained in the extent of a single logic.
We now attempt a problem of a rather different character, that of the completeness of **Λ** _P_. It is to be expected that this ordinal logic is complete. I cannot at present give a proof of this, but I can give a proof that it is complete as regards a simpler type of theorem than the number-theoretic theorems, viz. those of form " _θ_ ( _x_ ) vanishes identically", where _θ_ ( _x_ ) is primitive recursive. The proof will have to be much abbreviated since we do not wish to go into the formal details of the system _P_. Also there is a certain lack of definiteness in the problem as at present stated, owing to the fact that the formulae _G_ , _E_ , _M P_ were not completely defined. Our attitude here is that it is open to the sceptical reader to give detailed definitions for these formulae and then verify that the remaining details of the proof can be filled in, using his definition. It is not asserted that these details can be filled in whatever be the definitions of _G_ , _E_ , _M P_ consistent with the properties already required of them, only that they can be filled in with the more natural definitions.
I shall prove the completeness theorem in the following form. If [ _x_ 0] is a recursion formula and if [0], [ _f_ 0],... are all provable in _P_ , then there is a **C** -K ordinal formula **A** such that ( _x_ 0) [ _x_ 0] is provable in the system _P_ **A** of logic obtained from _P_ by adjoining as axioms all formulae whose G.R.'s are of the form
(provided they represent propositions).
First let us define the formula **A**. Suppose that **D** is a W.F.F. with the property that **D** ( **n** ) conv 2 if [ _f_ ( _n_ − 1)0] is provable in _P_ , but **D** ( **n** ) conv 1 if ∼ [ _f_ ( _n_ − 1)0] is provable in _P_ ( _P_ is being assumed consistent). Let **Θ** be defined by
**Θ** → { **λ** _vu_. _u_ ( _v_ ( _v_ , _u_ )}( **λ** _vu_. _u_ ( _v_ ( _v_ , _u_ ))),
and let Vi be a formula with the properties
Vi(2) conv **λ** _u_. _u_ (Suc, _U_ ),
Vi(1) conv **λ** _u_. _u_ ( _I_ , **Θ** (Suc)).
The existence of such a formula is established in Kleene [ **1** ], corollary on p. 220. Now put
**A** * → **λ** _ufx_. _u_ ( **λ** _y_. Vi( **D** ( _y_ ), _y_ , _u_ , _f_ , _x_ )),
**A** → Suc ( **A** *).
I assert that **A** *, **A** are C-K ordinal formulae whenever it is true that [0], [ _f_ 0],... are all provable in _P_. For in this case **A** * is **λ** _ufx_. _u_ ( **R** ), where
**R** → **λ** _y_. Vi( **D** ( _y_ ), _y_ , _u_ , _f_ , _x_ ),
and then
and
λ _ufx_. _S_ (n, Suc, _U_ , _u_ , _f_ , _x_ ) conv Suc (λ _ufx_. n(Suc, _U_ , _u_ , _f_ , _x_ )).
These relations hold for an arbitrary positive integer _n_ and therefore **A** * is a C-K ordinal formula [condition (9) p. [163]]: it follows immediately that **A** is also a C-K ordinal formula. It remains to prove that ( _x_ 0) [ _x_ 0] is provable in _P_ A. To do this it is necessary to examine the structure of **A** * in the case in which ( _x_ 0) [ _x_ 0] is false. Let us suppose that ˜ [ _f_ ( _a_ −1)0] is true, so that D(a) conv 1, and let us and consider **B** where
B → λ _ufx_. Vi (D(a), a, _u_ , _f_ , _x_ ).
If **A** * was a C-K ordinal formula, then **B** would be a member of its fundamental sequence; but
This, of course, implies that **B** < **B** and therefore that **B** is no C-K ordinal formula. This, although fundamental in the possibility of proving our completeness theorem, does not form an actual step in the argument. Roughly speaking, our argument amounts to this. The relation 9.3 implies that the system _P_ **B** is inconsistent and therefore that _P_ **A*** is inconsistent and indeed we can prove in _P_ (and _a fortiori_ in _P_ **A** ) that ˜( _x_ 0) [ _x_ 0] implies the inconsistency of _P_ **A***. On the other hand in _P_ **A** we can prove the consistency of _P_ **A***. The inconsistency of _P_ **B** is proved by the Gödel argument. Let us return to the details.
The axioms in _P_ **B** are those whose G.R.'s are of the form
B (λ _mn_. _m_ (ϖ(2, _n_ ), ϖ(3, _n_ )), _K_ , _M P_, r).
When we replace **B** , by Suc (B), this becomes
Suc ( **B** , λ _mn_. _m_ (ϖ(2, _n_ ), ϖ(3, _n_ )), _K_ , _M P_, r)
conv _K_ ( **B** (λ _mn_. _m_ (ϖ(2, _n_ ), ϖ(3, _n_ )), _K_ , _M P_, r)
conv **B** (λ _mn_. _m_ (ϖ(2, _n_ ), ϖ(3, _n_ )), _K_ , _M P_, **p** )
if r conv 2p + 1,
conv _E_ ( **B** (λ _mn_. _m_ (ϖ(2, _n_ ), ϖ(3, _n_ )), _K_ , _M_ _P_ ), **p** )
if r conv 2p.
When we remember the essential property of the formula _E_ , we see that the axioms of _P_ **B** include all formulae of the form
where _q_ is the G.R. of the formula .
Let _b_ be the G.R. of the formula .
Sb[ _x_ 0, _y_ 0, _z_ 0] is a particular recursion formula such that Sb[ _f_ ( _l_ )0, _f_ ( _m_ )0, _f_ ( _n_ )0] holds if and only if _n_ is the G.R. of the result of substituting _f_ ( _m_ )0 for _z_ 0 in the formula whose G.R. is _l_ at all points where _z_ 0 is free. Let _p_ be the G.R. of the formula .
Then we have as an axiom in _P_
( _x_ 0) Proof _P_ B [ _x_ 0, f(p)0] ⊃ ,
and we can prove in _P_ A
since is the result of substituting _f_ ( _b_ )0 for _z_ 0 in ; hence
is provable in _P_. Using 9.4 again, we see that can be proved in _P_ **B**. But, if we can prove in _P_ **B** , then we can prove its provability in _P_ **B** , the proof being in _P_ ; _i.e_. we can prove
( _x_ 0) Proof _P_ B [ _x_ 0, _f_ ( _p_ )0]
in _P_ (since _p_ is the G.R. of ). But this contradicts 9.5, so that, if
˜ [ _f_ ( _a_ − 1)0]
is true, we can prove a contradiction in _P_ **B** or in _P_ **A***. Now I assert that the whole argument up to this point can be carried through formally in the system _P_ , in fact, that, if _c_ is the G.R. of ˜(0 = 0), then
is provable in _P_. I shall not attempt to give any more detailed proof of this assertion.
The formula
is an axiom in _P_ **A**. Combining 9.6, 9.7 we obtain ( _x_ 0) [ _x_ 0]in _P_ **A**.
This completeness theorem as usual is of no value. Although it shows, for instance, that it is possible to prove Fermat's last theorem with **Λ** _P_ (if it is true) yet the truth of the theorem would really be assumed by taking a certain formula as an ordinal formula.
That **Λ** _P_ is not invariant may be proved easily by our general theorem; alternatively it follows from the fact that, in proving our partial completeness theorem, we never used ordinals higher than ω + 1. This fact can also be used to prove that **Λ** _P_ is not C-K invariant up to ω + 2.
#### 10. The continuum hypothesis. A digression
The methods of §9 may be applied to problems which are constructive analogues of the continuum hypothesis problem. The continuum hypothesis asserts that , in other words that, if ω1 is the smallest ordinal α greater than ω such that a series with order type α cannot be put into one–one correspondence with the positive integers, then the ordinals less than ω1 can be put into one–one correspondence with the subsets of the positive integers. To obtain a constructive analogue of this proposition we may replace the ordinals less than ω1 either by the ordinal formulae, or by the ordinals represented by them; we may replace the subsets of the positive integers either by the computable sequences of figures 0, 1, or by the description numbers of the machines which compute these sequences. In the manner in which the correspondence is to be set up there is also more than one possibility. Thus, even when we use only one kind of ordinal formula, there is still great ambiguity concerning what the constructive analogue of the continuum hypothesis should be. I shall prove a single result in this connection. A number of others may be proved in the same way.
We ask "Is it possible to find a computable function of ordinal formulae determining a one-one correspondence between the ordinals represented by ordinal formulae and the computable sequences of figures 0, 1?" More accurately, "Is there a formula **F** such that if **Ω** is an ordinal formula and _n_ a positive integer then **F** ( **Ω** , n) is convertible to 1 or to 2, and such that **F** ( **Ω** , n) conv **F** ( **Ω** ′, n) for each positive integer _n_ , if and only if **Ω** and **Ω** ′ represent the same ordinal?" The answer is "No", as will be seen to be a consequence of the following argument: there is no formula **F** such that **F** ( **Ω** ) enumerates one sequence of integers (each being 1 or 2) when **Ω** represents ω and enumerates another sequence when **Ω** represents 0. If there is such an F, then there is an _a_ such that **F** ( **Ω** , a) conv (Dt, a) if **Ω** represents ω but **F** ( **Ω** , a) and **F** (Dt, a) are convertible to different integers (1 or 2) if **Ω** represents 0. To obtain a contradiction from this we introduce a W.F.F. Gm not unlike Mg. If the machine whose D.N. is _n_ has printed 0 by the time the _m_ -th complete configuration is reached then
Gm( **n, m** ) conv λ _mn_. _m_ ( _n_ , _I_ , 4);
otherwise Gm( **n, m** ) conv λ _pq_. Al 4( _P_ , 2 _p_ \+ 2 _q_ ), 3, 4. Now consider **F** (Dt, a) and **F** Lim Gm(n), a. If never prints 0, Lim Gm(n) represents the ordinal ω. Otherwise it represents 0. Consequently these two formulae are convertible to one another if and only if never prints 0. This gives us a means of determining about any machine whether it ever prints 0, which is impossible.
Results of this kind have of course no real relevance for the classical continuum hypothesis.
#### 11. The purpose of ordinal logics
Mathematical reasoning may be regarded rather schematically as the exercise of a combination of two faculties, which we may call _intuition_ and _ingenuity_. The activity of the intuition consists in making spontaneous judgments which are not the result of conscious trains of reasoning. These judgments are often but by no means invariably correct (leaving aside the question what is meant by "correct"). Often it is possible to find some other way of verifying the correctness of an intuitive judgment. We may, for instance, judge that all positive integers are uniquely factorizable into primes; a detailed mathematical argument leads to the same result. This argument will also involve intuitive judgments, but they will be less open to criticism than the original judgment about factorization. I shall not attempt to explain this idea of "intuition" any more explicitly.
The exercise of ingenuity in mathematics consists in aiding the intuition through suitable arrangements of propositions, and perhaps geometrical figures or drawings. It is intended that when these are really well arranged the validity of the intuitive steps which are required cannot seriously be doubted.
The parts played by these two faculties differ of course from occasion to occasion, and from mathematician to mathematician. This arbitrariness can be removed by the introduction of a formal logic. The necessity for using the intuition is then greatly reduced by setting down formal rules for carrying out inferences which are always intuitively valid. When working with a formal logic, the idea of ingenuity takes a more definite shape. In general a formal logic, will be framed so as to admit a considerable variety of possible steps in any stage in a proof. Ingenuity will then determine which steps are the more profitable for the purpose of proving a particular proposition. In pre-Gödel times it was thought by some that it would probably be possible to carry this programme to such a point that all the intuitive judgments of mathematics could be replaced by a finite number of these rules. The necessity for intuition would then be entirely eliminated.
In our discussions, however, we have gone to the opposite extreme and eliminated not intuition but ingenuity, and this in spite of the fact that our aim has been in much the same direction. We have been trying to see how far it is possible to eliminate intuition, and leave only ingenuity. We do not mind how much ingenuity is required, and therefore assume it to be available in unlimited supply. In our metamathematical discussions we actually express this assumption rather differently. We are always able to obtain from the rules of a formal logic a method of enumerating the propositions proved by its means. We then imagine that all proofs take the form of a search through this enumeration for the theorem for which a proof is desired. In this way ingenuity is replaced by patience. In these heuristic discussions, however, it is better not to make this reduction.
In consequence of the impossibility of finding a formal logic which wholly eliminates the necessity of using intuition, we naturally turn to "non-constructive" systems of logic with which not all the steps in a proof are mechanical, some being intuitive. An example of a non-constructive logic is afforded by any ordinal logic. When we have an ordinal logic, we are in a position to prove number-theoretic theorems by the intuitive steps of recognizing formulae as ordinal formulae, and the mechanical steps of carrying out conversions. What properties do we desire a non-constructive logic to have if we are to make use of it for the expression of mathematical proofs? We want it to show quite clearly when a step makes use of intuition, and when it is purely formal. The strain put on the intuition should be a minimum. Most important of all, it must be beyond all reasonable doubt that the logic leads to correct results whenever the intuitive steps are correct. It is also desirable that the logic shall be adequate for the expression of number-theoretic theorems, in order that it may be used in metamathematical discussions (cf. §5).
Of the particular ordinal logics that we have discussed, **Λ** _H_ and **Λ** _P_ certainly will not satisfy us. In the case of **Λ** _H_ we are in no better position than with a constructive logic. In the case of **Λ** _P_ (and for that matter also **Λ** _H_ ) we are by no means certain that we shall never obtain any but true results, because we do not know whether all the number-theoretic theorems provable in the system _P_ are true. To take **Λ** _P_ as a fundamental non-constructive logic for metamathematical arguments would be most unsound. There remains the system of Church which is free from these objections. It is probably complete (although this would not necessarily mean much) and it is beyond reasonable doubt that it always leads to correct results. In the next section I propose to describe another ordinal logic, of a very different type, which is suggested by the work of Gentzen and which should also be adequate for the formalization of number-theoretic theorems. In particular it should be suitable for proofs of metamathematical theorems (cf. §5).
#### 12. Gentzen type ordinal logics
In proving the consistency of a certain system of formal logic Gentzen (Gentzen [1]) has made use of the principle of transfinite induction for ordinals less than e0, and has suggested that it is to be expected that transfinite induction carried sufficiently far would suffice to solve all problems of consistency. Another suggestion of basing systems of logic on transfinite induction has been made by Zermelo (Zermelo [1]). In this section I propose to show how this method of proof may be put into the form of a formal (non-constructive) logic, and afterwards to obtain from it an ordinal logic.
We can express the Gentzen method of proof formally in this way. Let us take the system _P_ and adjoin to it an axiom **Ω** with the intuitive meaning that the W.F.F. **Ω** is an ordinal formula, whenever we feel certain that **Ω** _is_ an ordinal formula. This is a non-constructive system of logic which may easily be put into the form of an ordinal logic. By the method of §6 we make correspond to the system of logic consisting of _P_ with the axiom **Ω** adjoined a logic formula _L_ **Ω** : _L_ **Ω** is an effectively calculable function of **Ω** , and there is therefore a formula **Λ** _G_ **1** such that **Λ** _G_ **1** ( **Ω** ) conv _L_ **Ω** for each formula **Ω**. **Λ** _G_ 1 is certainly not an ordinal logic unless _P_ is valid, and therefore consistent. This formalization of Gentzen's idea would therefore not be applicable for the problem with which Gentzen himself was concerned, for he was proving the consistency of a system weaker than _P_. However, there are other ways in which the Gentzen method of proof can be formalized. I shall explain one, beginning by describing a certain logical calculus.
The symbols of the calculus are _f_ , _x_ ,1, 1,0, _S_ , _R_ , Γ, Δ, _E_ , |, , !, (,), =, and the comma ",". For clarity we shall use various sizes of brackets (,) in the following. We use capital German letters to stand for variable or undetermined sequences of these symbols.
It is to be understood that the relations that we are about to define hold only when compelled to do so by the conditions that we lay down. The conditions should be taken together as a simultaneous inductive definition of all the relations involved.
##### _Suffixes_
1 is a suffix. If is a suffix then 1 is a suffix.
##### _Indices_
1 is an index. If is an index then is an index.
##### _Numerical variables_
If is a suffix then is a numerical variable.
##### _Functional variables_
If is a suffix and is an index, then is a functional variable of index .
##### _Arguments_
(,) is an argument of index1. If( ) is an argument of index and is a term, then is an argument of index .
##### _Numerals_
0 is a numeral.
If is a numeral, then _S_ (, ,) is a numeral.
In metamathematical statements we shall denote the numeral in which _S_ occurs _r_ times by _S_ ( _r_ )(, 0,).
##### _Expressions of a given index_
A functional variable of index is an expression of index .
_R_ , _S_ are expressions of index111, 11 respectively.
If is a numeral, then it is also an expression of index1.
Suppose that is an expression of index , one of index and one of index ; then (Γ ) and (Δ ) are expressions of index , while ( _E_ ) and and and are expressions of index .
##### _Function constants_
An expression of index in which no functional variable occurs is a function constant of index . If in addition _R_ does not occur, the expression is called a _primitive function constant_.
##### _Terms_
0 is a term.
Every numerical variable is a term.
If is an expression of index and ( ) is an argument of index , then is a term.
##### _Equations_
If and ′ are terms, then = ′ is an equation.
##### _Provable equations_
We define what is meant by the provable equations relative to a given set of equations as axioms.
( _a_ ) The provable equations include all the axioms. The axioms are of the form of equations in which the symbols Γ, **Δ** , _E_ , |, , ! do not appear.
( _b_ ) If is an expression of index and ( ) is an argument of index , then
is a provable equation.
( _c_ ) If is an expression of index , and ( ) is an argument of index , then
is a provable equation.
( _d_ ) If is an expression of index , and ( ) is an argument of index , then
is a provable equation.
( _e_ ) If is an expression of index and is one of index , and ( ) is an argument of index , then
is a provable equation.
( _f_ ) If is an expression of index1, then (,) = is a provable equation.
( _g_ ) If is an expression of index and one of index , and ( ) an argument of index , then
are provable equations. If in addition is an expression of index and
is provable, then
and
are provable.
( _h_ ) If = ′ and = ′ are provable, where , ′, and ′ are terms, then ′ = and the result of substituting ′ for at any particular occurrence in = ′ are provable equations.
( _i_ ) The result of substituting any term for a particular numerical variable throughout a provable equation is provable.
( _j_ ) Suppose that , ′ are expressions of index , that ( ) is an argument of index not containing the numerical variable and that is provable. Also suppose that, if we add
to the axioms and restrict ( _i_ ) so that it can never be applied to the numerical variable , then
becomes a provable equation; in the hypothetical proof of this equation this rule ( _j_ ) itself may be used provided that a different variable is chosen to take the part of .
Under these conditions is a provable equation.
( _k_ ) Suppose that , ′, are expressions of index , that ( ) is an argument of index not containing the numerical variable and that
are provable equations. Suppose also that, if we add
to the axioms, and again restrict ( _i_ ) so that it does not apply to , then
becomes a provable equation; in the hypothetical proof of 12.1 the rule ( _k_ ) may be used if a different variable takes the part of .
Under these conditions 12.1 is a provable equation.
We have now completed the definition of a provable equation relative to a given set of axioms. Next we shall show how to obtain an ordinal logic from this calculus. The first step is to set up a correspondence between some of the equations and number-theoretic theorems, in other words to show how they can be interpreted as number-theoretic theorems. Let be a primitive function constant of index111. describes a certain primitive recursive function _ϕ_ ( _m_ , _n_ ), determined by the condition that, for all natural numbers _m, n_ , the equation
is provable without using the axioms ( _a_ ). Suppose also that is an expression of index . Then to the equation
we make correspond the number-theoretic theorem which asserts that for each natural number _m_ there is a natural number _n_ such that _ϕ_ ( _m_ , _n_ ) = 0. (The circumstance that there is more than one equation to represent each number-theoretic theorem could be avoided by a trivial but inconvenient modification of the calculus.)
Now let us suppose that some definite method is chosen for describing the sets of axioms by means of positive integers, the null set of axioms being described by the integer 1. By an argument used in §6 there is a W.F.F. **Σ** such that, if _r_ is the integer describing a set _A_ of axioms, then **Σ** (r) is a logic formula enabling us to prove just those number-theoretic theorems which are associated with equations provable with the above described calculus, the axioms being those described by the number _r_.
I explain two ways in which the construction of the ordinal logic may be completed.
In the first method we make use of the theory of general recursive functions (Kleene [2]). Let us consider all equations of the form
which are obtainable from the axioms by the use of rules ( _h_ ), ( _i_ ). It is a consequence of the theorem of equivalence of λ-definable and general recursive functions (Kleene [3]) that, if _r_ ( _m, n_ ) is any λ-definable function of two variables, then we can choose the axioms so that 12.2 with _p_ = _r_ ( _m, n_ ) is obtainable in this way for each pair of natural numbers _m, n_ , and no equation of the form
is obtainable. In particular, this is the case if _r_ ( _m, n_ ) is defined by the condition that
**Ω** ( **m** , **n** ) conv _S_ (p) implies _p_ = _r_ ( _m_ , _n_ ),
_r_ (0, _n_ ) = 1, all _n_ > 0, _r_ (0, 0) = 2,
where **Ω** is an ordinal formula. There is a method for obtaining the axioms given the ordinal formula, and consequently a formula Rec such that, for any ordinal formula **Ω** , Rec ( **Ω** ) conv **m** , where _m_ is the integer describing the set of axioms corresponding to **Ω**. Then the formula
is an ordinal logic. Let us leave the proof of this aside for the present.
Our second ordinal logic is to be constructed by a method not unlike the one which we used in constructing **Λ** _P_. We begin by assigning ordinal formulae to all sets of axioms satisfying certain conditions. For this purpose we again consider that part of the calculus which is obtained by restricting "expressions" to be functional variables or _R_ or _S_ and restricting the meaning of "term" accordingly; the new provable equations are given by conditions ( _a_ ), ( _h_ ), ( _i_ ), together with an extra condition ( _l_ ).
( _l_ ) The equation
is provable.
We could design a machine which would obtain all equations of the form 12.2, with _m_ ≠ _n_ , provable in this sense, and all of the form 12.3, except that it would cease to obtain any more equations when it had once obtained one of the latter "contradictory" equations. From the description of the machine we obtain a formula **Ω** such that
is obtained by the machine,
is obtained by the machine, and
**Ω** ( **m, m** ) conv 3 always.
The formula **Ω** is an effectively calculable function of the set of axioms, and therefore also of _m_ : consequently there is a formula _M_ such that _M_ ( **m** ) conv **Ω** when _m_ describes the set of axioms. Now let Cm be a formula such that, if _b_ is the G.R. of a formula _M_ ( **m** ), then Cm( _b_ ) conv m, but otherwise Cm( _b_ ) conv 1. Let
Then **Λ** _G_ 3 ( **Ω** , **A** ) conv 2 if and only if **Ω** conv _M_ ( **m** ), where _m_ describes a set of axioms which, taken with our calculus, suffices to prove the equation which is, roughly speaking, equivalent to "A is dual". To prove that **Λ** _G_ 3 is an ordinal logic, it is sufficient to prove that the calculus with the axioms described by _m_ proves only true number-theoretic theorems when **Ω** is an ordinal formula. This condition on _m_ may also be expressed in this way. Let us put _m_ _n_ if we can prove with ( _a_ ), ( _h_ ), ( _i_ ), ( _l_ ): the condition is that _m_ _n_ is a well ordering of the natural numbers and that no contradictory equation 12.3 is provable with the same rules ( _a_ ), ( _h_ ), ( _i_ ), ( _l_ ). Let us say that such a set of axioms is _admissible_. **Λ** _G_ 3 is an ordinal logic if the calculus leads to none but true number-theoretic theorems when an admissible set of axioms is used.
In the case of **Λ** _G_ 2, Rec ( **Ω** ) describes an admissible set of axioms whenever **Ω** is an ordinal formula. **Λ** _G_ 2 therefore is an ordinal logic if the calculus leads to correct results when admissible axioms are used.
To prove that admissible axioms have the required property, I do not attempt to do more than show how interpretations can be given to the equations of the calculus so that the rules of inference ( _a_ )–( _k_ ) become intuitively valid methods of deduction, and so that the interpretation agrees with our convention regarding number-theoretic theorems.
Each expression is the name of a function, which may be only partially defined. The expression _S_ corresponds simply to the successor function. If is either _R_ or a functional variable and has _p_ \+ 1 symbols in its index, then it corresponds to a function _g_ of _p_ natural numbers defined as follows. If
is provable by the use of ( _a_ ), ( _h_ ), ( _i_ ), ( _l_ ) only, then _g_ ( _r_ 1, _r_ 2,..., _r p_) has the value _p_. It may not be defined for all arguments, but its value is always unique, for otherwise we could prove a "contradictory" equation and _M_ ( **m** ) would then not be an ordinal formula. The functions corresponding to the other expressions are essentially defined by ( _b_ )–( _f_ ). For example, if _g_ is the function corresponding to and _g_ ′ that corresponding to ( **Γ** ), then
_g_ ′( _r_ 1, _r_ 2,..., _r p_, _l_ , _m_ ) = _g_ ( _r_ 1, _r_ 2,..., _r p_, _m_ , _l_ ).
The values of the functions are clearly unique (when defined at all) if given by one of ( _b_ )–( _e_ ). The case ( _f_ ) is less obvious since the function defined appears also in the definiens. I do not treat the case of , since this is the well-known definition by primitive recursion, but I shall show that the values of the function corresponding to ( ! ! ) are unique. Without loss of generality we may suppose that ( ) in ( _f_ ) is of index1. We have then to show that, if _h_ ( _m_ ) is the function corresponding to and _r_ ( _m_ , _n_ ) that corresponding to _R_ , and _k_ ( _u_ , _v_ , _w_ ) is a given function and _a_ a given natural number, then the equations
do not ever assign two different values for the function _l_ ( _m_ ). Consider those values of _r_ for which we obtain more than one value of _l_ ( _r_ ), and suppose that there is at least one such. Clearly 0 is not one, for _l_ (0) can be defined only by (α). Since the relation is a well ordering, there is an integer _r_ 0 such that _r_ 0 > 0, _l_ ( _r_ 0) is not unique, and if _s_ ≠ _r_ 0 and _l_ ( _s_ ) is not unique then _r_ 0 _s_. We may put _s_ = _h_ ( _r_ 0), for, if _l_ ( _h_ ( _r_ 0)) were unique, then _l_ ( _r_ 0), defined by (β), would be unique. But . There is, therefore, no integer _r_ for which we obtain more than one value for the function _l_ ( _r_ ).
Our interpretation of expressions as functions gives us an immediate interpretation for equations with no numerical variables. In general we interpret an equation with numerical variables as the (infinite) conjunction of all equations obtainable by replacing the variables by numerals. With this interpretation ( _h_ ), ( _i_ ) are seen to be valid methods of proof. In ( _j_ ) the provability of
when is assumed to be interpreted as meaning that the implication between these equations holds for all substitutions of numerals for _x_ 1. To justify this, one should satisfy oneself that these implications always hold when the hypothetical proof can be carried out. The rule of procedure ( _j_ ) is now seen to be simply mathematical induction. The rule ( _k_ ) is a form of transfinite induction. In proving the validity of ( _k_ ) we may again suppose ( ) is of index1. Let _r_ ( _m_ , _n_ ), _g_ ( _m_ ), _g_ 1( _m_ ), _h_ ( _n_ ) be the functions corresponding respectively to _R_ , , ′, . We shall prove that, if _g_ (0) = _g_ ′(0) and for each positive integer _n_ and if _g_ ( _n_ \+ 1) = _g_ ′( _n_ \+ 1) whenever = , then _g_ ( _n_ ) = _g_ ′( _n_ ) for each natural number _n_. We consider the class of natural numbers for which _g_ ( _n_ ) = _g_ ′( _n_ ) is not true. If the class is not void it has a positive member _n_ 0 which precedes all other members in the well ordering . But _h_ ( _n_ 0) is another member of the class, for otherwise we should have
and therefore _g_ ( _n_ 0) = _g_ ′( _n_ 0), _i.e_. _n_ 0 would not be in the class. This implies _n_ 0 _h_ ( _n_ 0) contrary to . The class is therefore void.
It should be noticed that we do not really need to make use of the fact that **Ω** is an ordinal formula. It suffices that **Ω** should satisfy conditions ( _a_ )–( _e_ ) (p. [162]) for ordinal formulae, and in place of ( _f_ ) satisfy ( _f_ ′).
( _f_ ′) There is no formula T such that T(n) is convertible to a formula representing a positive integer for each positive integer _n_ , and such that **Ω** (T(n), **n** ) conv 2, for each positive integer _n_ for which **Ω** (n, n) conv 3.
The problem whether a formula satisfies conditions ( _a_ )–( _e_ ), ( _f_ ′) is number-theoretic. If we use formulae satisfying these conditions instead of ordinal formulae with **Λ** _G_ 2 or **Λ** _G_ 3, we have a non-constructive logic with certain advantages over ordinal logics. The intuitive judgments that must be made are all judgments of the truth of number-theoretic theorems. We have seen in §9 that the connection of ordinal logics with the classical theory of ordinals is quite superficial. There seem to be good reasons, therefore, for giving attention to ordinal formulae in this modified sense.
The ordinal logic **Λ** _G_ 3 appears to be adequate for most purposes. It should, for instance, be possible to carry out Gentzen's proof of consistency of number theory, or the proof of the uniqueness of the normal form of a well-formed formula (Church and Rosser [1]) with our calculus and a fairly simple set of axioms. How far this is the case can, of course, only be determined by experiment.
One would prefer a non-constructive system of logic based on transfinite induction rather simpler than the system which we have described. In particular, it would seem that it should be possible to eliminate the necessity of stating explicitly the validity of definitions by primitive recursions, since this principle itself can be shown to be valid by transfinite induction. It is possible to make such modifications in the system, even in such a way that the resulting system is still complete, but no real advantage is gained by doing so. The effect is always, so far as I know, to restrict the class of formulae provable with a given set of axioms, so that we obtain no theorems but trivial restatements of the axioms. We have therefore to compromise between simplicity and comprehensiveness.
#### Index of definitions
No attempt is being made to list heavy type formulae since their meanings are not always constant throughout the paper. Abbreviations for definite well-formed formulae are listed alphabetically. | _Page_
---|---
Ai | 179
Al | 169
Bd | 169
Ck | 173
Cm | 199
Comp | 180
Dt | 151
_E_ | 176
form | 150
_G_ | 177
Gm | 192
Gr | 150
_H_ | 165, 168
_H_ 1 | 168
Hf | 169
Hg | 185
_I_ | 148
Inf | 169
Jh | 183
_K_ | 177
Lim | 169
Ls | 168
_M_ | 199
_M P_ | 177
Mg | 184
Nm | 160
Od | 180
_P_ | 169
Prod | 186
_Q_ | 159
Rec | 198
Rt | 184
S | 149
Sum | 169
Sq | 185
Tn | 173
Ug | 169
_V_ | 160
Vi | 188
_W_ | 159
_W_ ′ | 159
_X_ | 158
_Z_ | 183
**Γ** | 160
δ | 147
Θ | 188
**Λ** _G_ 1 | 194
**Λ** _G_ 2 | 199
**Λ** _G_ 3 | 199
**Λ** _H_ | 178
**Λ** _P_ | 177
ϖ | 151
Σ | 198
1, 2, 3, | 148
| 154
##### **_Bibliography_**
Alonzo Church, [1]. "A proof of freedom from contradiction", _Proc. Nat. Acad. Sci_. 21 (1935), 275–281.
—— [2]. _Mathematical logic_ , Lectures at Princeton University (1935–6), mimeographed, 113 pp.
—— [3]. "An unsolvable problem of elementary number theory", _American J. of Math_. 58 (1936), 345–363.
—— [4]. "The constructive second number class", _Bull. American Math. Soc_. 44 (1938), 224–238.
G. Gentzen, [1]. "Die Widerspruchsfreiheit der reinen Zahlentheorie", _Math. Annalen_ , 112 (1936), 493–565.
K. Gödel, [1]. "Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme, I", _Monatshefte für Math. und Phys_. 38 (1931), 173–189.
—— [2]. _On undecidable propositions of formal mathematical systems_ , Lectures at the Institute for Advanced Study, Princeton, N.J., 1934, mimeographed, 30 pp.
D. Hilbert, [1]. "Über das Unendliche", _Math. Annalen_ , 95 (1926), 161–190.
S. C. Kleene, [1]. "A theory of positive integers in formal logic", _American J. of Math_. 57 (1935), 153–173 and 219–244.
—— [2]. "General recursive functions of natural numbers", _Math. Annalen_ , 112 (1935–6), 727–742.
—— [3]. "λ-definability and recursiveness", _Duke Math. Jour_. 2 (1936), 340–353.
E. L. Post, [1]. "Finite combinatory processes—formulation 1", _Journal Symbolic Logic_ , 1 (1936), 103–105.
J. B. Rosser, [1]. "Gödel theorems for non-constructive logics", _Journal Symbolic Logic_ , 2 (1937), 129–137.
A. Tarski, [1]. "Der Wahrheitsbegriff in den formalisierten Sprachen", _Studia Philosophica_ , 1 (1936), 261–405 (translation from the original paper in Polish dated 1933).
A. M. Turing, [1]. "On computable numbers, with an application to the Entscheidungs-problem". [Chapter 1].
—— [2]. "Computability and λ-definability", _Journal Symbolic Logic_ , 2 (1937), 153–163.
E. Zermelo, [1]. "Grundlagen einer allgemeiner Theorie der mathematischen Satzsysteme, I", _Fund. Math_. 25 (1935), 136–146.
Alonzo Church and S. C. Kleene, [1]. "Formal definitions in the theory of ordinal numbers", _Fund. Math_. 28 (1936), 11–21.
Alonzo Church and J. B. Rosser, [1]. "Some properties of conversion", _Trans. American Math. Soc_. 39 (1936), 472–482.
D. Hilbert and W. Ackermann, [1]. _Grundzüge der theoretischen Logik_ (2nd edition revised, Berlin, 1938), 130 pp.
A. N. Whitehead and Bertrand Russell, [1]. _Principia Mathematica_ (2nd edition, Cambridge, 1925–1927), 3 vols.
[Received 31 May, 1938.—Read 16 June, 1938.]
### **CHAPTER 4
Letters on Logic to Max Newman ( _c. 1940_ )**
_Alan Turing_
#### **Introduction**
##### _Jack Copeland_
At the outbreak of war with Germany in September 1939, Turing left Cambridge to take up work as a codebreaker at Bletchley Park, the wartime headquarters of the Government Code and Cypher School (see 'Enigma', below). In the early months of 1940, Turing received a letter from the Cambridge mathematician M. H. A. Newman, his teacher, colleague, and friend. Turing replied on 23 March, writing from his lodgings at the Crown Inn (situated in the small village of Shenley Brook End): 'Dear Newman, Very glad to get your letter, as I needed some stimulus to make me start thinking about logic.' This was to be the first of five letters that Turing wrote to Newman during the seventeen months before Newman too left Cambridge for Bletchley Park.
In his first letter Turing agreed (presumably at Newman's request—Newman's letters seem not to have been preserved) to 'let Newman] in on... the tricks of the conversion calculus'. The conversion calculus, or 'λ-calculus', is due to Alonzo Church, with whom Turing studied in Princeton from 1936 to 1938 (see the introduction to [Chapter 3). Turing's letters consist for the most part of detailed remarks on the conversion calculus, often elucidating material from what Turing calls 'Church's notes'—a substantial typescript entitled 'Mathematical Logic' which was in circulation at Princeton and elsewhere and which Newman was evidently reading.
Their correspondence on Church's work issued in their joint paper 'A Formal Theorem in Church's Theory of Types', which was submitted to Church's _Journal_ _of Symbolic Logic_ in May 1941 and published in March 1942. The paper was written while Turing played a leading role in the battle to break Naval Enigma (see 'Enigma' and Chapters 5–). Turing would spend his occasional nights off duty 'coming in as usual..., doing his own mathematical research at night, in the warmth and light of the office, without interrupting the routine of daytime sleep'.
The two most interesting items of the correspondence, which are printed here, contain substantial passages in which Turing departs from his commentary on Church's work and expounds his own views. These elegant passages provide information about Turing's thoughts on the logical foundations of mathematics which is not to be found elsewhere in his writings.
Of particular importance are the sections headed 'Intuition. Inspiration. Ingenuity', in which he discusses the unsolvability and incompleteness results in logic and explains the basic idea underlying his ordinal logics (Chapter 3); 'Ingenuity and Intuition', discussing the extent to which provability by Turing machine approximates mathematical truth; 'The Completeness Theorem', concerning the completeness theorem established in Chapter 3; and 'Consequences', in which two notions of logical consequence are compared. These sections contain occasional formulae of the conversion calculus, but the formulae are not necessary to Turing's points, and readers unfamiliar with the notation of the calculus should not be deterred.
##### M. H. A. Newman: Mathematician, Codebreaker, and Computer Pioneer
Max Newman played an important part in Turing's intellectual life over many years. It was Newman who, in a lecture in Cambridge in 1935, launched Turing on the research that led to the universal Turing machine:
I believe it all started because he attended a lecture of mine on foundations of mathematics and logic... I think I said in the course of this lecture that what is meant by saying that [a] process is constructive is that it's a purely mechanical machine—and I may even have said, a machine can do it.
And this of course led [Turing] to the next challenge, what sort of machine, and this inspired him to try and say what one would mean by a perfectly general computing machine.
In April 1936, Turing presented Newman with the draft typescript of 'On Computable Numbers'. Not long after, an offprint of Church's paper proving the undecidability of first-order predicate calculus arrived in Cambridge. Newman proved a staunch ally at what must have been a painful time for Turing. On 29 May 1936 Turing wrote in a letter to his mother:
Meanwhile a paper has appeared in America, written by Alonzo Church, doing the same things in a different way. Mr Newman and I have decided however that the method is sufficiently different to warrant the publication of my paper too.
It was clear to Newman that 'Turing's "machine" had a significance going far beyond this particular application [the _Entscheidungsproblem_ ]'. Turing's paper contained (in Newman's words) 'this extraordinary definition of a perfectly general... computable function, thus giving the first idea... of a perfectly general computing machine.' Newman advised Turing during the final stages of preparation of 'On Computable Numbers', and he wrote to the Secretary of the London Mathematical Society saying that Church's prior publication should not stand in the way of Turing's paper appearing in the _Proceedings_.
In 1942 Newman received a letter from Frank Adcock, another Cambridge man and a veteran of Room 40 (the forerunner of the Government Code and Cypher School): 'Dear Newman, There is some work going at a government institution which would I think interest you and which is certainly important for the War...'. Newman wrote to the Master of St John's to request leave of absence and at the end of August 1942 he joined the Research Section at Bletchley Park.
The Research Section was attempting to break the German cipher machine they nicknamed 'Tunny'. Used mainly by the German Army, Tunny was one of three types of German machine—collectively referred to as 'Fish' by the British—for enciphering the binary teleprinter alphabet (the other two were 'Sturgeon', used mainly by the German Air Force, and 'Thrasher'). From the autumn of 1942 Tunny was used in preference to Enigma for the encryption of messages between the German High Command and the various Army Group commanders in the field—intelligence of the highest grade.
In November 1942 William Tutte found a way of breaking Tunny messages known as the 'Statistical Method'. The rub was that the method seemed impractical, involving a very large amount of time-consuming work—basically, the comparing of two streams of 0s and 1s, counting the number of times that each had 0 in the same position. If the comparing and counting were done by hand, the intelligence in the message would be stale before the work was completed. Tutte explained his method to Newman and Newman suggested using electronic counters. It was a brilliant idea. In December 1942 Newman was given the job of developing the necessary machinery. The electronic counters were designed by C. E. Wynn-Williams at the Telecommunications Research Establishment (TRE) in Malvern. Construction of the new machine was carried out at the Post Office Research Station at Dollis Hill in London and at TRE. In June 1943 the completed machine began work in the 'Newmanry', a newly created section at Bletchley Park headed by Newman.
This first machine—known as 'Heath Robinson', after a popular cartoonist who drew bizarre contraptions—was relay-based, with some electronic circuits for counting and for performing simple logical (i.e. boolean) operations. Heath Robinson was unreliable and slow, and its high-speed paper tapes tended to stretch and tear, but it proved the worth of Newman's approach. Newman ordered a dozen more Robinsons from the Post Office.
During the design phase of Heath Robinson there had been difficulties with the logic unit—the 'combining unit' in the terminology of 1942. At Turing's suggestion Newman had approached the Post Office engineer Thomas H. Flowers for help (Flowers had previously assisted Turing with the design of a machine for use against Enigma). Flowers and his switching group at Dollis Hill successfully redesigned the combining unit; but Flowers did not think much of the overall design of the Robinson, and in February 1943 presented Newman with the alternative of a fully electronic machine. This idea received little encouragement from Bletchley Park, however, where opinion was that a machine containing as many electronic valves (vacuum tubes) as Flowers was proposing—about 2,000—would not work reliably. Flowers, with over ten years' experience of electronic valves, knew better, and on his own initiative began building the machine he could see was necessary, working independently at the Post Office Research Station. Flowers has said that he was probably the only person in Britain who understood at this time that electronic valves could be used in large numbers for high-speed digital computing.
Flowers' 'Colossus', the first large-scale electronic digital computing machine, was installed in the Newmanry on 8 December 1943 (see the introduction to Chapter 9). By the end of the war, there were nine more Colossi working in the Newmanry. The Colossi gave the Allies access to the most secret German radio communications, including messages from Hitler to his front-line generals. Intelligence obtained via Colossus was vital to the planning of the D-day landings and played a major role in the subsequent defeat of Hitler.
In September 1945 Newman took up the Fielden Chair of Mathematics at the University of Manchester. Five months later he wrote the following to the Princeton mathematician and computer pioneer John von Neumann:
I am... hoping to embark on a computing machine section here, having got very interested in electronic devices of this kind during the last two or three years. By about eighteen months ago I had decided to try my hand at starting up a machine unit when I got out.... I am of course in close touch with Turing.
Newman lost no time in establishing the Royal Society Computing Machine Laboratory at the University. He introduced the engineers Frederick Williams and Thomas Kilburn—newly recruited to Manchester University from the Telecommunications Research Establishment, where they had worked on radar (they knew nothing of the top-secret Colossus)—to Turing's idea of a stored-programme computer and explained to them what facilities were necessary in a computer (see the introduction to Chapter 9). It was in Newman's Computing Machine Laboratory that Kilburn and Williams built the world's first electronic stored-programme digital computer. Their prototype ran its first programme on 21 June 1948 (see further the introduction to Chapter 9).
That same year Newman recruited Turing to Manchester from the National Physical Laboratory, appointing him Deputy Director of the Computing Machine Laboratory (see the introduction to Chapter 10). Turing remained at Manchester until his death in 1954.
##### **_Further reading_**
Barendregt, H. P., _The Lambda-Calculus, its Syntax and Semantics_ (Amsterdam: North-Holland, 1984).
Church, A., 'A Set of Postulates for the Foundation of Logic', _Annals of Mathematics_ , 33 (1932), 346–66.
—— 'An Unsolvable Problem of Elementary Number Theory', _American Journal of Mathematics_ , 58 (1936), 345–63.
—— _The Calculi of Lambda-Conversion_ (Princeton: Princeton University Press, 1941).
Copeland, B. J., 'Colossus and the Dawning of the Computer Age', in R. Erskine and M. Smith (eds.), _Action This Day_ (London: Bantam, 2001).
Lalement, R., _Computation as Logic_ (Hemel Hempstead: Prentice Hall, 1993).
##### **_Provenance_**
What follows are transcriptions of Turing's letters. The original letters are among the Turing Papers in the Modern Archive Centre, King's College Library, Cambridge.
#### **Letter from The Crown, Shenley Brook End**
April 21
The Crown
Shenley Brook End
Bletchley
Dear Newman,
The **δ** -function One certainly can manage without **δ** for defining computable functions. The purpose for which it is really brought in in Church's notes is to enable one to 'describe the syntax of the system within itself' i.e. at any rate to define the formula Gr (or something like it) such that Gr (A) conv G.R of a certain normal form of A (if there is one & if A has no free variables, otherwise Gr(A) has no normal form)
Gr → **λ** a. (1, **λ** u. **δ** (form(u), a) & Norm(u)) ****
where Norm(u) conv 2 if u is G.R of a formula in normal form
conv 1 otherwise
(form can be defined without **δ** )
2&1conv1
1&2conv1
1&1conv1
2&2conv2
I do not know that it has been proved that a Gr cannot be obtained without **δ** , but at any rate defining Gr without **δ** would be equivalent to defining a formula without **δ** which would have the properties of **δ**. I haven't got Church's notes with me, but I think most of his bracket technique was in connection with his 'metads' (sort of G. Rs).
There can be no very general picking out function, even using **δ**. The formulae to be picked out must certainly have no free variables (if **Δ** 1 picks out first term of two then **Δ** 1(x, y) conv x and 1.h.s has different free variables from right) but also they must have normal forms, for a formula without normal form will poison any formula in which it enters. If all the formulae involved have normal forms one can pick out with **δ** e.g. in this way –
Gr({A, B}) is G.R of {A, B}
there is a **λ** -definable function (defined by D say) which gives G.R of A as function of G.R of { _A_ , _B_ }
Then A conv form(D(Gr({A, B})))
Intuition. Inspiration. Ingenuity
I am not sure whether my use of the word 'intuition' is right or whether your 'inspiration' would be better. I rather think that you are using 'inspiration' to cover what I had called 'ingenuity'. To give a concrete example of ingenuity, suppose I want a formula **Θ** with the property
**Θ** (x) conv x( **Θ** (x))
I can of course search through an enumeration of all formulae **Θ** and perform conversions on **Θ** (x) (saving time over the possibly infinite conversion processes, by the 'diagonal process'), but if while I am doing this some bystander writes down
**Θ** → { **λ** w.w(w)} ( **λ** vu.u(v(v, u)))
and says 'try that', I should say he had found a formula by 'ingenuity'. In such cases there is no need to worry about how the formula is arrived at. That it is right is verified by a simple conversion, or something equally uncontroversial. Isn't this what you would call inspiration?
The straightforward unsolvability or incompleteness results about systems of logic amount to this
**α** ) One cannot expect to be able to solve the Entscheidungsproblem for a system
**β** ) One cannot expect that a system will cover all possible methods of proof (does not apply to 'restricted function calculus')
It seemed to me that in your account of what we want a system of logic to do you had **α** ) in mind but not **β** ). I should agree with your point of view, in so far as we can shut our eyes to **β** ) i.e. we do not really want to make proofs by hunting through enumerations for them, but by hitting on one and then checking up to see that it is right. However this method is always theoretically, though not practically, replacable by the longer method if one has got a method of checking up. The enumeration of proofs is for instance obtained from an enumeration of all possible sequences of symbols by striking out those which do not pass the test. When one takes **β** ) into account one has to admit that not one but many methods of checking up are needed. In writing about ordinal logics I had this kind of idea in mind. In proofs there is actually an enormous amount of sheer slogging, a certain amount of ingenuity, while in most cases the actual 'methods of proof' are quite well known. Cannot we make it clearer where the slogging comes in, where there is ingenuity involved, and what are the methods of proof? In fact can we not express quite shortly what is the status of each proof? The ordinals were meant to give concise notations for the status of proofs.
The Completeness Theorem
The proof of my completeness theorem (PA etc) is of course completely useless for the purpose of actually producing proofs. PA will only be a convincing logic if A is rather simple, and easily recognized as an ordinal formula. The completeness theorem was written from a rather different point of view from most of the rest, and therefore tends to lead to confusion. I think that all this proof does is to provide an insurance against certain sorts of 'Gödel incompleteness theorems' being proved about the ordinal logic.
As soon as any question arises of having to prove that the formulae one is using are ordinal formulae one is returning to the single logic point of view, unless the kind of proof to be used is something different, being a kind of propaganda rather than formal proof.
The exercise
I have no complaints at all about this. You have evidently got the tools necessary for barging through anywhere where one can get through. I don't remember, even if I ever knew, what the standard way of doing this job is. I have toyed for half an hour or so with trying to do it with things of form
{A,{B,{C,... }}...} instead of {A, B,... }
One might in that way avoid the trouble of looking after the number of variables. One will need a 'picking out function' L which will satisfy
L({m, {n,... }}) conv m
but I cannot find one independent of the number of variables.
Church tells me he is going to publish his form of Principia involving the use of **λ** , and simple theory of types. I am very glad of it, as the system makes things much clearer than any other I know and is not too cumbrous to be used.
Gödel's paper has reached me at last. I am very suspicious of it now but will have to swot up the Zermelo–v. Neumann system a bit before I can put objections down in black & white.
Yours sincerely
A. M. Turing
#### **Letter from King's College, Cambridge**
Sunday
King's College
Cambridge
Dear Newman,
Church's notes certainly are rather a mouthful. I have never worked steadily through them myself, but have taken them in much the same spirit as you are doing. Fortunately I was able to go to the fountainhead for information.
i) Metads certainly are a form of 'Gödel representation' which Ch. finds it convenient to use in his system.
ii) I think the point of using the peculiar form of negation is that one wants Th 1, 2,3p48 to hold in the form in which they stand. If one has (i.e. (p, x)), ˜A will have a normal form sometimes when A is not convertible to 1 or 2 e.g. if A conv 3 (also incidentally, as there is no 0, 3 x has the value 1 in this case and this would be bad apart from the normal form difficulty, but this is more easily corrected).
iii) Consequences. I think one wants here to distinguish two ideas a) 'consequences of an assumption' (p. 82, 14), b) consequences of an assumption relative to a set of rules of procedure. The first of these is an 'intuitive' idea which one tries to approximate by the second with suitable sets of rules of procedure. To get the idea of 'consequences of an assumption' imagine that the underlined letters are admitted as parts of formulae in a new system. I will use only letters for variable underlined letters, and variable formulae involving underlined letters. Then if , are such formulae involving underlined letters we say that is a consequence of if, for all substitutions of formulae (in original sense, with d) for it happens that conv 2 whenever conv 2. This of course implies two different uses of underlined letters from Chap X onwards, but I think Church is really doing this. The idea of consequences of an assumption relative to given rules of procedure I think explains itself. One tries of course to make the rules of procedure such that the consequences will be consequences in the sense a), but also to get as many consequences as one can consistent with this. Of course one cannot get all such with one set of rules.
iv) The Πm's. These certainly are much the same as my ordinal logics, that is to say that the rule by which the Πm's are formed can easily be used to help one construct an ordinal logic. They are better and better approximations by consequences of type b) above to the consequences of type a). The meaning of the Πm's is this. One has defined the rules 1m... 7m and 1r... 7r with r < m. Taking all these and 1... 63 we have a set of rules of procedure Procm say. From them we get Πm which is such that Πm(F, G) conv 2 if and only if F and G are metads of formulae , such that is a consequence of relative to Procm. (There is some confusion between underlined letters and ordinary variables in this definition of the Πm's, as metads are names of formulae without underlined letters. Probably you have to regard all the free variables in the formulae described by the metads as replaced by underlined letters if we are to follow my description under iii).)
v) Ingenuity and Intuition. I think you take a much more radically Hilbertian attitude about mathematics than I do. You say 'If all this whole formal outfit is not about finding proofs which can be checked on a machine it's difficult to know what it is about.' When you say 'on a machine' do you have in mind that there is (or should be or could be, but has not been actually described anywhere) some fixed machine on which proofs are to be checked, and that the formal outfit is, as it were, about this machine. If you take this attitude (and it is this one that seems to me so extreme Hilbertian) there is little more to be said: we simply have to get used to the technique of this machine and resign ourselves to the fact that there are some problems to which we can never get the answer. On these lines my ordinal logics would make no sense. However I don't think you really hold quite this attitude because you admit that in the case of the Gödel example one can decide that the formula is true i.e. you admit that there is a fairly definite idea of a true formula which is quite different from the idea of a provable one. Throughout my paper on ordinal logics I have been assuming this too. It mostly takes the form of talking about such things as a formula A such that A(n) conv 2 for all pos. integers n.
If you think of various machines I don't see your difficulty. One imagines different machines allowing different sets of proofs, and by choosing a suitable machine one can approximate 'truth' by 'provability' better than with a less suitable machine, and can in a sense approximate it as well as you please. The choice of a proof checking machine involves intuition, which is interchangeable with the intuition required for finding an **Ω** if one has an ordinal logic Λ, or as a third alternative one may go straight for the proof and this again requires intuition: or one may go for a proof finding machine. I am rather puzzled why you draw this distinction between proof finders and proof checkers. It seems to me rather unimportant as one can always get a proof finder from a proof checker, and the converse is almost true: the converse fails if for instance one allows the proof finder to go through a proof in the ordinary way, and then, rejecting the steps, to write down the final formula as a 'proof' of itself. One can easily think up suitable restrictions on the idea of proof which will make this converse true and which agree well with our ideas of what a proof should be like.
I am afraid this may be more confusing to you than enlightening. If so I will try again.
Yours sincerely
A. M. Turing
## **Enigma
_Jack Copeland_**
1. Turing Joins the Government Code and Cypher School _217_
2. The Enigma Machine _220_
3. The Polish Contribution, 1932–1940 _231_
4. The Polish Bomba _235_
5. The Bombe and the Spider _246_
6. Naval Enigma _257_
7. Turing Leaves Enigma _262_
### 1. Turing Joins the Government Code and Cypher School
Turing's personal battle with the Enigma machine began some months before the outbreak of the Second World War. At this time there was no more than a handful of people in Britain tackling the problem of Enigma. Turing worked largely in isolation, paying occasional visits to the London office of the Government Code and Cypher School (GC & CS) for discussions with Dillwyn Knox. In 1937, during the Spanish Civil War, Knox had broken the type of Enigma machine used by the Italian Navy. However, the more complicated form of Enigma used by the German military, containing the _Steckerbrett_ or plug-board, was not so easily defeated.
On 4 September 1939, the day following Chamberlain's announcement of war with Germany, Turing took up residence at the new headquarters of the Government Code and Cypher School, Bletchley Park. GC & CS was a tiny organization ill prepared for war. By 1942, however, Bletchley Park had become a veritable factory, and with the help of the codebreaking machines called 'bombes'— designed by Turing, Gordon Welchman, and, on the engineering side, Harold Keen—GC & CS was deciphering about 39,000 Enigma messages each month. By 1945 almost 9,000 people were employed at Bletchley Park. It is estimated that the breaking of Enigma—and in particular the breaking of Home Waters Naval Enigma, in which Turing played the crucial role—may have shortened the war in Europe by some two years.
**Figure 1.** The Mansion, Bletchley Park.
_Source_ : Bletchley Park Trust.
The Government Code and Cypher School had developed from the old 'Room 40', established by the Admiralty during the First World War for the purpose of reading enemy ciphers. A branch of the Foreign Office, GC & CS was located in Whitehall until the summer of 1939. By the beginning of 1938 the Director of Naval Intelligence, Admiral Hugh Sinclair, was looking for premises outside London to which GC & CS could move in the event of war. Bletchley Park—a large Victorian mansion with ample grounds situated in the town of Bletchley, a major railway junction linking London, Oxford, and Cambridge—was purchased in the spring of 1938 (out of Sinclair's own pocket, it is said).
In the course of 1937 and 1938 Commander Alastair Denniston, Head of GC & CS and a veteran of Room 40, supervised a clandestine programme of recruitment, centred largely on Oxford and Cambridge. Denniston's aim was to build up what he described as an 'emergency list [of] men of the Professor type'.
At certain universities... there were men now in senior positions who had worked in our ranks during 1914–18. These men knew the type required. Thus it fell out that our most successful recruiting occurred from these universities. During 1937 and 1938 we were able to arrange a series of courses to which we invited our recruits to give them even a dim idea of what would be required of them... These men joined up in September 1939.
(Frank Adcock and Frank Birch, the two veterans of Room 40 who were most active in recruitment as the new war approached, were both from the same college as Turing, King's.) In the days following the outbreak of war in September 1939 a group of about thirty people assembled at Bletchley Park, many of them—including Turing—drawn from Denniston's 'emergency list'.
An organizational structure rapidly began to emerge at Bletchley, newly formed sections being known simply as 'Hut 4', 'Hut 6', and so on. The 'huts' were single-storey wooden structures hastily constructed in the grounds of the mansion. Here dons worked among uniformed Naval and Army personnel. Military discipline never took root among the 'men of the Professor type' and parts of Bletchley Park had something of the atmosphere of an Oxbridge college. There were some notable eccentrics among the codebreakers. Dilly Knox, another fellow of King's and veteran of Room 40, liked to work in a hot bath. Once, at his lodgings, Knox
stayed so long in the bathroom that his fellow-lodgers at last forced the door. They found him standing by the bath, a faint smile on his face, his gaze fixed on abstractions, both taps full on and the plug out. What then was passing in his mind could possibly have solved a problem that was to win a battle.
It was Knox's Research Section that Turing joined upon his arrival at Bletchley Park.
### 2. The Enigma Machine
The Enigma machine had something of the appearance of an old-fashioned typewriter. Designed by the Berlin engineer Arthur Scherbius, Enigma was marketed commercially from 1923. In 1926 the German Navy adopted Enigma, followed by the German Army in 1928 and the German Air Force in 1935. At the outbreak of war with Britain, Enigma was the Germans' principal method for protecting their military communications. In 1930, the German military had considerably enhanced the security of the machine by adding the _Steckerbrett_ or _plug-board_ (see Figure 4). It is this form of Enigma—German military, or _Wehrmacht_ , Enigma—that is dealt with here. Successive modifications were made to the operating procedures of the military machine, resulting in substantial variation both over time and from one branch of the armed services to another.
Battery powered and highly portable, the _Wehrmacht_ Enigma machine could be used from a general's office in Berlin, an armoured vehicle, a submarine, or a trench. The machine's keyboard had twenty-six keys, each marked with a letter (Figure 4). Instead of an arrangement for typing letters onto paper, the machine had a lampboard consisting of twenty-six bulbs, each of which shone through a stencil on which a letter of the alphabet was marked. The operator of the Enigma machine would be handed a message in plain text. His job was to type the message at the keyboard of the machine. Each time he pressed a key, a letter on the lamp-board would light up. The operator's assistant kept a note of which letters lit up on the lampboard. This enciphered form of the message was then sent to its recipient, if by radio then in Morse code. The sending radio operator would preface the message with his radio call-sign, followed by that of the intended receiver. The Germans also sent Enigma messages by land-lines; for these messages, Morse was not used. (Land-lines are not mentioned further in this introduction, since German message traffic sent in this way was not intercepted in Britain.)
Each time the operator pressed a key, one or more wheels turned inside the machine, and each time a wheel moved it altered the wiring between the keyboard and the lampboard. So if, for example, the operator repeatedly depressed the O-key, the connections between the key and the lampboard would change with each key press, resulting in a succession of different letters lighting up, for example Q M P W A J Y R.
**Figure 2.** A three-wheel Enigma with the plug-board (at the front of the machine) exposed. The lampboard is behind the keyboard. The three wheel-slots are visible behind the lampboard. Beside each wheel-slot is a window through which letters marked on the wheels are visible to the operator.
_Source_ : Science and Society Picture Library, National Museum of Science and Industry.
**Figure 3.** Enigma machine with the three wheels exposed.
_Source_ : Science and Society Picture Library, National Museum of Science and Industry.
The letter O itself would never appear in this succession of letters, however. Because of the action of the reflector, a letter was never enciphered as itself (see Figure 4). This rule was very useful to the codebreakers at Bletchley Park.
At the receiving end of the radio link, the message would be converted from Morse into ordinary letters. This cipher text was then typed at the keyboard of the recipient's Enigma machine. The letters that lit up on the lampboard would be the very same letters that the sender had keyed in—the plain text with which the process had begun. The design of the Enigma machines was such that if a key was pressed on one machine, say O, and the letter that lit up on the machine's
**Figure 4.** Path of electric current through the Enigma. Pressing a key at the keyboard causes a letter to light up at the lampboard. The core of each wheel contains a maze of 26 insulated wires, with each wire joining one of 26 contacts on the right hand side of the wheel to one of 26 contacts on the left-hand side. The wiring is different in each wheel.
Diagram by Dustin A. Barrett.
lampboard was keyed into a second machine, then—provided the two machines had been set up in exactly the same way by their respective operators—the second machine would light up O on its lampboard.
**Figure 5.** View of the wheels with the case closed. The three wheel-adjusters protrude through slots in the case. The windows allow the operator to see one letter from the ring of each wheel. The 'message setting' is the triple of letters visible at the start of typing a message.
Diagram by Dustin A. Barrett.
In a word, the letter-substitutions were _reversible_ : if O produced Q (for example) then, at the same machine-settings, Q produced O. This was the basic principle of the Enigma system, hard-wired into the machine. Figure 4 indicates how this was achieved. If Q were pressed at the keyboard, current would flow along a wire leading to Q at the plug-board, then across the plug-board to Y and through the wheels in the reverse direction to that shown, exiting the wheels at N, crossing the plug-board to O, and lighting O at the lampboard.
#### _The Plug-Board (Steckerbrett) and Wheels_
The operator could make various changes to the settings of his machine before he began typing a message at the keyboard. The recipient would set up his own machine in the same way in order to decode the message. How the recipient knew which settings to use is explained in what follows.
The settings of the machine could be changed in the following ways. (See Figure 4.)
1. The operator could make alterations to the plug-board ( _Steckerbrett_ ) on the front of the machine, pulling electrical leads out of sockets and plugging them back into different sockets. This altered some of the connections between the keyboard and the lampboard. (The plug-board was absent from the commercial version of the machine.)
2. The operator could alter the positions of the rotating wheels inside the machine (sometimes also called 'rotors') by turning them manually. Part of the circumference of each wheel protruded through the case of the machine enabling the operator to click the wheels round with his thumb or finger (Figure 5). In the early years of the war there were three rotatable wheels inside the machine; in 1941, the first Naval machines with a fourth rotatable wheel came into use (see the introduction to Chapter 8). (Another two components of the Enigma are sometimes referred to as wheels or rotors, the _Umkehrwalze_ (described by Mahon on p. 269 of Chapter 5) and the _Eintrittwalze_. In the forms of German military Enigma discussed here, both these components were stationary, and they will be referred to as the _reflector_ and the _entry plate_ respectively (Figure 4).)
3. The operator could open the case of the machine, lift out two or more of the wheels, and replace them in a different order. For example, he might switch the left- and right-hand wheels, leaving the centre wheel untouched. Each wheel was wired differently inside. Since the electrical pathways from the keyboard to the lampboard passed through the wheels, changing the order of the wheels altered the pathways. Alternatively, rather than simply switching the order of the wheels in the machine, the operator might replace one or more of them with different wheels from a box that accompanied the machine. From December 1938 until about the beginning of the war, there were a total of five wheels, numbered I–V, and any three of the five might be inside the machine at any one time. For example, the wheels in use might be I, II, and IV, in the order IV/I/II. From 1940 (or possibly as early as 1939) Enigma machines used by the German Navy were equipped with additional wheels and the operator would select three from a total of eight (numbered I–VIII).
The wheels were somewhat analogous to the wheels of a combination lock, turning through a number of discrete positions. Each wheel had a total of twenty-six possible rotational positions, A–Z. The wheel on the right, the first on the path from keyboard to lampboard, would always turn on one 'click' each time a key was pressed. Hence the term 'fast wheel' (Figure 4). After a certain number of clicks, this wheel would cause the centre wheel to turn one click. Likewise, the centre wheel would at some point cause the wheel on the left— the 'slow wheel'—to move one click. (An extra complication: when this happened, the centre wheel would itself turn forward one click also.)
**Figure 6.** A dismantled wheel.
_Source_ : Science and Society Picture Library, National Museum of Science and Industry.
Precisely when a wheel would cause its neighbour to turn was determined by the position of a notch cut into the ring of the wheel. Since wheels I–V all had their notches in different places, changing or rearranging wheels could affect the 'turnovers' (Bletchley's term for the points at which wheels would cause their neighbours to turn). The Naval wheels VI–VIII were slightly different. These had their notches in the same places as one another, and moreover each had _two_ notches (see pp. 268, 285 below). The extra notch meant that in the course of one revolution, the doubly notched wheel would cause its neighbour to move twice.
Which letter lit up on the lampboard depended, therefore, not only on which key was depressed, but also on how the plug-board was connected up, which of the possible wheels were inside the machine, what order these wheels were arranged in, and which of its twenty-six rotational positions each wheel occupied at the time the key was pressed. In fact, by altering these variables, the operator was able to set up a machine with a total of three wheels in excess of a thousand million million different ways. The message remained protected even if the enemy captured an Enigma machine of the type that the sender was using. In order for a recipient to decipher the message, he or she needed to know which of the astronomically many possible settings the sender had used to encipher the text.
#### _Enigma Keys_
The sender and the (authorized) recipient were issued with printed tables of settings so that they could set up their machines in the same way. A group of Enigma-users operating with the same tables is called a _network_. A set of tables covered a period of one month and specified how, on any given day, the members of the network should set up their machines. Different networks used different tables.
GC & CS referred to a network of Enigma-users as a 'key'. Each key was given a name—Yellow, Red, Green, Light Blue, Shark, Dolphin, Porpoise, Kestrel, Phoenix, Locust, Snowdrop, etc. At the beginning of the war, the number of known keys was small enough for GC & CS to be able to represent them on a chart by means of coloured pencils, the colour used becoming the name of the key. As the war progressed, the number of keys became much larger.
The term 'network' is perhaps clearer than 'key', especially since at Bletchley, 'key' was used ambiguously for a network of Enigma-users and in the term 'daily key' (whose meaning is explained below). Some writers prefer 'crypto-net' to 'network', since the former term makes it clear that it is an Enigma network and not a radio network that is being described. One and the same radio network could carry the message traffic of several crypto-nets.
#### _Wheel Order, Stecker, and Ringstellung_
The wheel order for a particular day for a certain network or key might be III/I/ II, for example.
_Stecker_ is short for _Steckerverbindungen_ , meaning 'plug connections'. The _Stecker_ , or plug-board configuration, for a particular day might be A/C, D/V, F/M, H/W, L/X, R/I. Corresponding to each letter on the plug-board is a pair of sockets, one for a cable leading _to_ another letter, and one for a cable leading _from_ another letter (Figure 4). The operator would set up the plug-board by connecting together the pair of sockets labelled 'A' and the pair of sockets labelled 'C' by means of a short cable with a double plug at each end. Likewise for the 'D' sockets and the 'V' sockets, and so on. The Germans' use of double plugs meant that if A is steckered to C, then C is steckered to A—a fatal simplification, as we shall see.
_Ringstellung_ means 'ring position'. The ring is like a tyre mounted round the core of each wheel. It is marked with the letters of the alphabet, one for each of the twenty-six rotational positions of the wheel (Figure 4). (Sometimes the numerals '01' to '26' were used instead of letters.) The ring could be moved around the wheel core to a selected position and then fixed in position with a clip. The day's ring position for a given wheel was specified by a single letter, say X. The operator would turn the ring until the letter X was aligned against a fixed index mark embossed on the wheel and then would fix the ring in this position. The complete _Ringstellung_ for the day would consist of a trigram, say XYZ, one letter for each wheel in the machine.
#### _The Daily Key_
The daily wheel order, _Stecker_ , and _Ringstellung_ for the machine were specified in the tables issued to each Enigma network. _Stecker_ , wheel order, and _Ringstellung_ were elements of the _daily key_ , or basic settings for the day for a given network of Enigma users.
The reason for changing the basic settings daily was to minimize the number of messages encoded at the same settings. The Germans knew that security could be compromised if too many messages were encoded at the same basic settings. During the later years of the war, some networks changed the _Stecker_ , wheel order, and _Ringstellung_ not daily but every eight hours.
#### _The Message Setting_
Setting up the sender's and recipient's machines in accordance with the specified _Stecker_ , wheel order, and _Ringstellung_ did not suffice to place the two machines completely in register. There was also the question of the rotational positions of the three wheels at the start of the message.
Once the ring position was set, the rotational position of a wheel could be described by saying which of the letters on the ring was uppermost when the wheel was in place inside the machine. The machine's case was fitted with three small windows, one above each wheel, so that the operator could see the uppermost letter (Figure 5).
The positions occupied by the wheels at the start of typing a message were specified by a trigram, for example QVZ, meaning that Q is visible in the window over the left-hand wheel, V in the window over the middle wheel, and Z in the window over the right-hand wheel. QVZ was known as the _message setting_.
Notice that knowing the message setting does not reveal the rotational positions of the wheels at the start of the message unless the _Ringstellung_ is also known—QVZ may specify any one of the 26 × 26 × 26 possible positions, depending on which ring positions have been selected.
#### _Operating Procedures_
In order to decode the message, a recipient needs the wheel order, the _Stecker_ , the _Ringstellung_ , and the message setting. The most direct way to make the message setting available to the authorized recipient would be to make it an element of the daily key printed in the monthly tables. The operator would then simply look up the specified trigram for the day in question, and ensure that it was visible in the windows at the start of each message. This was the procedure used with the commercial form of Enigma. But this method provided very weak security, reducing the problem of breaking a day's messages to that of solving a number of _substitution_ ciphers.
The substitution cipher is an ancient and simple form of cipher in which the alphabet is paired with a 'scrambled' alphabet. For example:
The great Polish cryptanalyst Marian Rejewski explained the weakness of enciphering a day's Enigma traffic at the same message setting:
the first letters of all the messages... constituted an ordinary substitution cipher, a very primitive cipher easily soluable given sufficient material, and all the second letters of the messages... constituted another substitution cipher, and so on. These are not merely theoretical deliberations. It was in that very way that in France in 1940 we solved the Swiss Enigma cipher machine.
The German armed forces employed more secure methods for making the message setting known to the intended recipient. The method adopted varied from service to service and from time to time, generally speaking with increasingly secure methods being used as time went on. From 1937 the German Navy used a particularly complicated method—although Turing did manage to break it. This method is described by Patrick Mahon in Chapter 5, which is an extract from Mahon's previously unpublished 'The History of Hut 8'. (Written in 1945, Mahon's 'History' was kept secret by the British and American governments until 1996.)
From the autumn of 1938 until May 1940 the German Army and Air Force used the following—as it turned out, highly insecure—method for sending the message setting to the recipient. The sender would select two trigrams at random, say RBG and VAK. RBG is the message setting. VAK specifies the starting positions of the wheels that will be used not when encoding the message itself but when encoding the message setting prior to broadcasting it to the recipient. VAK would be broadcast to the recipient as part of an unencoded _preamble_ to the encoded message. (The preamble could also include, for example, the time of origin of the message, the number of letters in the encoded message, and a group of letters called a _discriminant_ , identifying the Enigma network to which the message belonged (e.g. Red). The preamble might also contain an indication that the message was the second (or later) part of a two-part or multi-part message; see Mahon's discussion of 'forts' on pp. 278–9 below.)
#### _The Indicator and Indicator Setting_
Having selected the two trigrams, the sender would first set up VAK in the windows of his machine. He would then type RBGRBG. The group of six letters that lit up, say PRUKAC, is called the _indicator_. VAK is called the _indicator setting_ (or ' _Grundstellung_ '). The indicator would be broadcast immediately before the enciphered message. The reason for sending the encipherment of RBGRBG, rather than simply of RBG, was to provide the recipient with a check that the message setting had been correctly received, radio reception sometimes being poor.
Once the sender had enciphered the message setting to form the indicator, he would set up RBG in the windows of his machine and type the plain text. Then the whole thing would be sent off to the recipient—preamble, indicator, and enciphered text.
The authorized recipient of the message would first rotate the wheels of his machine (already set up in accordance with the daily key) until VAK appeared in the windows. He would then type the indicator PRUKAC and the letters RBGRBG would light up at the lampboard. Now equipped with the message setting, he would set his wheels to RBG and retrieve the plain text by typing the encoded message.
The method just described of selecting and making known the message setting is an example of what is called an _indicator system_.
#### 3. The Polish Contribution, 1932–1940
Unknown to GC & CS, the Biuro Szyfrów—the Polish Cipher Bureau—had already broken _Wehrmacht_ Enigma, with assistance from the French secret service. The Biuro read the message traffic of the German Army regularly from 1933 to the end of 1938, and at other times during this period read the message traffic of other branches of the military, including the Air Force. Statistics gathered by the Biuro early in 1938 showed that, at that time, about 75 per cent of all intercepted Enigma material was being successfully decoded by the Biuro Szyfrów.
Towards the end of 1932 Rejewski had devised a method for reconstructing a day's message settings from the indicators, given about sixty messages sent on the day. He was helped by the fact that, in this early period, the indicator system was simpler than the later system just described. The daily key included an indicator setting _for the day_ , e.g. VAK. The sender would choose his own message setting for each message, e.g. RBG. With the wheels in the positions specified in the daily key (VAK), he would type RBGRBG to produce the indicator. Then he would set the wheels to RBG and type the plain text of the message. The encoded message was sent prefaced by the preamble and the indicator—but, of course, there was no need to send the indicator setting.
Using information obtained from his attack on the indicators, Rejewski devised a method that enabled him to determine the internal wiring of wheels I–III (in those early days there were no additional wheels). This was one of the most far-reaching achievements in the history of cryptanalysis. Rejewski was assisted by the French secret service, whose agent Hans-Thilo Schmidt, a German employed in the cipher branch of the German Army, supplied photographs of two tables setting out the daily keys— _Stecker_ , wheel order, _Ringstellung_ , and the daily indicator setting—for September and October 1932. Rejewski describes this material as the 'decisive factor in breaking the machine's secrets'.
In 1931 the French had attempted to interest the British in documents obtained by Schmidt, including operating manuals for German military Enigma. It is said that the British showed little interest, however, and declined to help the French meet the costs of obtaining them. It was not until 1936 that GC & CS began to study Enigma seriously. By the middle of 1939, Knox had discovered something like the Polish method for obtaining the message settings from the indicators (for German Army traffic). However, he was unable to determine the internal wiring of the wheels. Without the wiring, it was impossible to use the method to decode the messages. GC & CS probably discovered a version of the same method that Rejewski had used to determine the wiring of the wheels, calling the method a 'Saga' (Mahon mentions it briefly on p. 278 of the next chapter). Knox is said to have outlined a 'more complicated version' of the Rejewski method at a meeting in Paris in January 1939. However, he was never able to use this method to find the wiring of the wheels. This was because he was never able to discover the pattern of fixed wiring leading from the plugboard to the right-hand wheel via the entry plate (see Figure 4)—the 'QWERTZU', as he liked to call this unknown pattern, after the letters along the top row of the Enigma keyboard. This entirely humdrum feature of the military machine was what defeated Knox. Rejewski himself discovered the pattern by a lucky guess.
Once Rejewski had worked out the internal wiring of the wheels, he attacked the problem of how to determine the daily keys. This he solved early in 1933. At this stage, Rejewski was joined by Henryk Zygalski and Jerzy Różycki. Zygalski, Różycki, and Rejewski had graduated together from a course in cryptology that the Biuro Szyfrów had given in 1928–9. (Rejewski said later that it could have been the Biuro's fruitless efforts to break Enigma during 1928—the year in which the first messages were intercepted—that prompted the organization of the course at which the three were recruited.)
Now that the Polish cryptanalysts were able to find the daily keys on a regular basis, they needed access to Enigma machines in order to decipher the daily traffic. Using what Rejewski had found out concerning the wiring of the wheels, copies of the Wehrmacht Enigma were built by a Warsaw factory. Initially about half a dozen clerical staff were employed by the Biuro Szyfrów to operate the replica Enigmas. The clerical staff were 'put into a separate room, with the sole assignment of deciphering the stream of messages, the daily keys to which we soon began supplying'. The number of replica Enigmas in use at the Biuro increased to about a dozen by mid-1934.
This state of affairs persisted until September 1938, when the German Army and Air Force abandoned the indicator system that Rejewski had broken in 1932. They switched to the indicator system described above: the indicator setting was no longer supplied in the tables giving the daily key, but was made up by the sender himself. Overnight the Poles' methods for determining the daily keys and message settings became useless. (In German Naval Enigma, the system broken by Rejewski had been abandoned in May 1937, when the complicated indicator system described by Mahon in the next chapter was adopted. Mahon outlines the Polish work on Naval Enigma to 1937.)
Within a few weeks of the September change, however, the Poles had devised two new methods of attack. One involved the use of perforated sheets of paper to determine the daily key, starting from a sufficient number of messages whose indicators displayed certain patterns of repeated letters. (Knox devised a similar method and was planning to use marks on photographic film rather than perforations, but was unable to put the method into practice without knowing the internal wiring of the wheels.) The Poles' other method involved an electromechanical apparatus, designed by Rejewski and (on the engineering side) Antoni Palluth. This was the _bomba_ (plural 'bomby'), forerunner of the Bletchley Park bombe. How the bomba worked is explained in the next section. Six bomby were in operation by mid-November 1938.
The bomby and the perforated sheets depended on the fact that the indicator was formed by enciphering the message setting _twice_ (e.g. enciphering RGBRGB rather than simply RGB). If the indicator system were changed so that the message setting was enciphered only once, the bomby and the perforated sheets would become unusable. This is precisely what was to happen in May 1940. Well before this, however, the bomby became overwhelmed by other changes designed to make Enigma more secure.
In December 1938 the Germans introduced the two extra wheels, IV and V. The Poles were able to determine the internal wiring of the new wheels by the method used in 1932 (thanks to the fact that one Enigma network—the intelligence service of the Nazi party—had not adopted the indicator system that came into force on other networks in September 1938 and was still using the system that the Poles could break by their earlier methods). But the material resources of the Biuro Szyfrów were insufficient to enable the Poles to cope with the increase in the number of wheel orders that the two new wheels produced. Where previously there had been only six possible wheel orders, there were now sixty. In order to investigate the new wheel orders, at least thirty-six replicas of each new wheel were required. The factory could not produce replicas fast enough.
Work with the perforated sheets was affected in the same way. The drawback of the sheet method had always been that the manufacture of a single sheet required the cutting of about 1,000 tiny perforations in exactly the right positions, with twenty-six sheets being required for each possible wheel order. Suddenly a huge number of additional sheets was required.
The result of the addition of the new wheels was that the Poles were able to read German Army and Air Force messages on only those days when it happened that wheels I, II, and III were in the machine—on average one day in ten.
#### _Pyry and After_
In July 1939 the Poles invited members of the British and French intelligence services to a meeting at Pyry near Warsaw. Denniston and Knox represented GC & CS. At this meeting, Rejewski relates, 'we told everything that we knew and showed everything that we had'—a replica Enigma, the bomba, the perforated sheets, and of course the all-important internal wiring of the wheels, which Knox still had not been able to work out. Without the Poles, Knox and Turing might not have found out the wiring of the wheels until May 1940, when the British captured several intact Enigma machines from the German Army in Norway.
Knox's first question to the Poles was 'What is the QWERTZU?' The answer was almost a joke—the connections were in _alphabetical_ order, with the A-socket of the plug-board connected to the first terminal inside the entry plate, the B-socket to the second, and so on. Knox was ecstatic to know the answer at last, chanting in a shared taxi 'Nous avons le QWERTZU, nous marchons ensemble' ('We have the QWERTZU, we march along together').
At Pyry the Poles also undertook to supply their British and French allies with two replica Enigma machines. The replica destined for GC & CS was couriered from Paris to London on 16 August 1939 by two men, Gustave Bertrand, head of the codebreaking section of the French Intelligence Service, and 'Uncle Tom', a diplomatic courier for the British Embassy in Paris. On the platform of Victoria Station they handed the machine over to Admiral Sinclair's deputy, Colonel Stewart Menzies. Menzies, on his way to an evening engagement, was dressed in a dinner jacket and he sported the rosette of the Légion d'Honneur in his buttonhole. _Accueil triomphal_ —a triumphant welcome, Bertrand declared.
Following the invasion of Poland, Rejewski and his colleagues moved to France. By January 1940 GC & CS, with its superior resources, had produced two complete sets of perforated sheets. The Poles received one of the sets in instalments. Turing delivered some of the sheets himself.
Rejewski recollected: 'We treated [Turing] as a younger colleague who had specialized in mathematical logic and was just starting out in cryptology. Our discussions, if I remember correctly, pertained to the commutator [plug-board] and plug connections (Steckerverbindungen) that were Enigma's strong point.' Little did Rejewski know that Turing had already devised the brilliant method of dealing with the _Steckerverbindungen_ on which the British bombe was based.
For several months the British and the Poles worked in cooperation. The first break of wartime traffic since September 1939 was achieved by the Poles in mid-January 1940, followed a few days later by further breaks at GC & CS. During the period of fruitful collaboration that ensued, the Poles with their lesser resources were responsible for about 17 per cent of the daily keys broken.
Then, in May 1940, everything changed. The new indicator system introduced by the German Army and Air Force on 1 May made the perforated sheets useless for all networks except one, Yellow, which continued to employ the old system. Even Yellow, an inter-services key in use during the Norway campaign, went out of service on 14 May. The change of indicator system and the German occupation of France effectively ended the attack on Enigma by the exiled Biuro Szyfrów.
The British were able to continue reading German Air Force messages (from 20 May) by means of methods developed at GC & CS which exploited the bad habits of some German Enigma operators. One was the habit of enciphering the message setting at the position that the wheels happened to be in at the end of the previous message, or at a closely neighbouring position (obtained e.g. by lazily turning only one wheel some small number of clicks).
From the summer of 1940 the codebreakers at GC & CS began to receive assistance from Turing's radically redesigned version of the Polish bomba.
### 4. The Polish Bomba
#### _Origin of the Name 'Bomba'_
In Chapter 5, Mahon says that the British bombe 'was so called because of the ticking noise it made, supposedly similar to that made by an infernal machine regulated by a clock' (p. 291). This story was well entrenched among Bletchleyites. The need-to-know principle meant that few were aware of the Polish bomba. Similarly, the explanation that circulated at Bletchley Park of why certain patterns, involving repetitions of letters at the same places, were known as 'females' took no account of the fact that the terminology had been borrowed from the Poles. The equivalent Polish term 'samiczki', meaning 'females', was quite likely the result of a play on words, 'samiczki' being used as short for a Polish phrase meaning 'the same places.'
Why the Poles chose the name 'bomba' seems not to have been recorded. Rejewski's only comment was that the name was used 'for lack of a better idea'. As well as meaning 'bomb', 'bomba' is the Polish word for a type of ice-cream dessert— _bombe_ in French. Tadeusz Lisicki, who corresponded with Rejewski during the years before the latter's death in 1980, is quoted as saying: 'The name "bomba" was given by Różycki... [T]here was in Warsaw [an] icecream called [a] bomba... [T]he idea [for] the machine came while they were eating it.'
A different story is told in recently declassified American documents. As explained later in this section, the bomba is required to stop immediately it detects a certain feature. How this was achieved by the Polish engineers is not known for sure. The American documents suggest that the stopping mechanism involved the dropping of weights, and the claim is made that this is how the name arose.
[A] bank of Enigma Machines now has the name 'bombe'. This term was used by the Poles and has its origin in the fact that on their device when the correct position was reached a weight was dropped to give the indication.
When a possible solution was reached a part would fall off the machine onto the floor with a loud noise. Hence the name 'bombe'.
It is not implausible that falling weights were used to disengage the bomba's drive mechanism (a printer designed by Babbage as part of his Difference Engine used a similar idea). However, the two American documents in question were written some years after Rejewski and his colleagues destroyed all six bomby in 1939 and neither cites a source for the claim quoted (the documents are dated 1943 and 1944). Moreover, both documents contain inaccurate claims concerning the Polish attack on Enigma (for example, that the bomba was 'hand operated', and that the military Enigma machine had no plug-board until 'about 1938'). The sketch of the bomba that accompanies Rejewski's 'The Mathematical Solution of the Enigma Cipher' shows no system of falling weights—although nor is an alternative system for stopping the bomba depicted.
#### _Simple Enigma and a Mini Bomba_
Let us suppose, for purposes of illustration, that we are dealing with an imaginary, highly simplified, version of the Enigma machine called Simple Enigma. Simple Enigma has one wheel rather than three and no plug-board; in other respects it is the same as a full-scale Enigma.
Suppose that we have a message to decode beginning NYPN... Suppose further that we have a _crib_. A crib is a series of letters or words that are thought likely to occur in the plain language message that the cipher text encrypts. Say we have good reason to believe that the first and fourth letters of the plain text are both E (perhaps a prisoner gasped out the first four letters of the plain text before he died, but his second and third gasps were inaudible). We will use a machine to help us find the message setting—i.e. the rotational position of the wheel at which the sender began typing the message.
Our code-breaking machine consists of two replicas of the Simple Enigma machine plus some additional devices. There is a mechanism for holding down any selected key at the keyboards of the replicas, thereby keeping the current flowing from key to wheel. The wheel of each replica can be locked in step with the other, and there is an electric motor that will click the wheels round in unison through their twenty-six rotational positions, one position at a time. Additional circuitry bridging the two lampboards detects whether a selected letter—E, for example—lights up simultaneously at each lampboard. A switch or relay is wired in such a way that if the selected letter does light simultaneously, the electric motor is turned off, with the result that the wheels stop turning at exactly the position that caused the simultaneous lighting of the letter. This is called a 'stop'.
Assuming that the crib is correct, we know that if the intended recipient of the message sets the wheel of Simple Enigma to the message setting and types the first letter of the cipher text, N, the letter E will light up at the lampboard. The recipient will then type the next two letters of the cipher text, YP, causing unknown letters to light, followed by the fourth letter of the cipher text, N, which will cause E to light up again. Each time the recipient presses a key at the keyboard, the wheel advances one click. So the position of the wheel at which the fourth letter of the cipher text decodes as E is three clicks on from the position at which the first letter of the cipher text decodes as E. This is expressed by saying that these two positions are at a _distance of three_ from each other. What we want our codebreaking machine to do is to search through the twenty-six possible positions of the wheel, looking for a position _p_ that satisfies these two conditions:
1. At position _p_ , keying N causes E to light;
2. At position _p_ \+ 3 (i.e. the position three on from _p_ ), keying N again causes E to light.
We set up the codebreaking machine to perform this search by turning one of the two identical wheels so that it is three positions ahead of the other. For example, we might turn the wheel on the right so that, of the twenty-six letters marked around its ring, Z is uppermost, and then position the wheel on the left three clicks further on, i.e. with C uppermost. The two wheels are then locked together so that they will maintain their position relative to one another while the motor rotates them. The locked wheels are described as being at an _offset_ of three clicks.
Next we set up the additional circuitry at the lampboards so that the simultaneous lighting of the letter E at each board will produce a stop. Finally, we clamp down the N-key at each of the two keyboards and start the electric motor.
The motor turns the wheels from position to position. If all goes well, a point is reached where E lights at both boards and the machine stops. If at that stage the wheels have not yet completed a full revolution, we note the position at which the stop occurred and then start the motor again, since there might be more than one position at which conditions 1 and 2 are jointly satisfied. (If, after a complete revolution, there are no stops, our crib was incorrect.)
If a complete revolution brings only one stop, then the position of the right-hand wheel of the pair must be the position at which the sender began encoding the message. We pass this setting to a clerk sitting at another replica of the Simple Enigma, who turns the wheel to that position and keys in the cipher text, producing the plain text at the lampboard. If there were several stops, then the clerk has to try each of the possible settings in turn until one is found that yields German at the lampboard.
Notice that we have not discovered the actual message setting—the letter visible in the window of the sender's machine at the start of typing the message (and enciphered to form the indicator). Which letter is visible in the window depends on how the sender has positioned the ring around the 'core' of the wheel. Leaving the core in one position, the operator could make any one of the twenty-six letters appear in the window by twisting the ring around the core. What we have found is the position _of the wheel core_ at the start of the message. At GC & CS this was called the 'rod-position' of the wheel. The rod-position is all we need to be able to decipher the message.
Of course, with only twenty-six positions to search through, there is hardly any need for the electric motor, the detector circuitry at the lampboards, and so forth, because one could quite quickly conduct the search simply by turning the wheels of two replica machines manually. However, the additional equipment is certainly necessary when it is the full-scale Enigma machine that is being attacked, since the existence of three wheels and six possible wheel orders means that one must search through not 26 but 6 × 26 ×26 × 26 = 105, 456 possible positions. (This figure ignores the small complications introduced by double-notching and by the extra movement of the middle wheel described above.)
#### _The Actual Bomba_
The Polish bomba was a more complicated version of the mini bomba just described. It consisted in effect of six replica Enigma machines, with six sets of duplicates of wheels I, II, and III—eighteen wheels in all. Each of the six replica Enigmas in a single bomba was usually set up with the same wheel order, for example III/I/II. The wheels used in a bomba had no rings (and so no notches for producing a 'turnover' of the adjacent wheel).
The six replica Enigmas were linked in pairs to form three double-Enigmas— just as in the example of the mini bomba, where two Simple Enigmas are linked to form a double Simple Enigma. Each of these double-Enigmas included three pairs of wheels and equipment equivalent to two keyboards and two lampboards. The complete bomba consisted of the three double-Enigmas plus the electric motor, a mechanism for detecting simultaneities and producing stops, and arrangements for holding constant the letter going into each double-Enigma.
At this point it may be helpful to repeat that the first, or outermost, of the three wheels in an Enigma machine—the wheel linked directly to the keyboard and plug-board and which moved once with every key-stroke—was always the right-hand member of the trio. For example, if the wheel order is I/II/III, it is wheel III that is the outermost of the three wheels.
As in the mini bomba, the identical wheels of a double-Enigma were locked in step, sometimes with one member of a pair a number of positions ahead of the other member. For example, the two IIIs might be locked in step at an offset of three clicks (as above), while the two IIs are locked in step with no offset, and likewise the two Is.
The corresponding wheels of different double-Enigmas in the same bomba were also locked in step with one another. For example, the locked pair of III wheels of one double-Enigma might be locked in step (at an offset of twelve clicks, say) with the locked pair of III wheels of another double-Enigma.
Once all the wheels were appropriately linked, the electric motor would be started and the bomba's six replica Enigmas would move in synchronization, each passing through 26 × 26 × 26 positions. This took about two hours, each outer wheel moving through 676 revolutions, each middle wheel through 26 revolutions, and each left-hand wheel through one revolution. In the space of roughly two hours, the bomba could do the same work that would occupy a human computer for about 200 hours.
#### _The Indicator Method_
In the previous example, we imagined using a mini bomba to discover wheel positions consistent with a crib concerning the first and fourth letters of the cipher text. The method employed by the Poles was different and did not involve text-cribbing (although the method that Turing would later devise for the British bombe did). The Poles focused on the _indicator_ (to recapitulate: the six-letter group preceding the cipher text and produced by enciphering the message setting twice, at an indicator setting that the sender broadcast 'in clear' as part of the preamble to the message).
In a proportion of the intercepted messages, the first and fourth letters of the indicator would be the same, as for example in the indicator WAHWIK. Since an indicator is produced by typing a three-letter message setting twice, the first and fourth letters of any indicator both encode the same letter as each other. This is true also of the second and fifth letters of any indicator, and the third and sixth. So both the occurrences of W in WAHWIK encode the same letter; and moreover three clicks of the right-hand wheel separate the two positions at which W encodes this unknown letter.
Let me use ' _p R_' when referring to a position of the Enigma's right-hand wheel, and similarly ' _p M_' in the case of the middle wheel and ' _p L_' in the case of the left-hand wheel. We could attempt to use the bomba to search for rod-positions _p L_, _p M_, and _p R_ such that at position _p R_ and position _p R_ \+ 3, W encodes the same letter. As I will explain, this is not in fact an effective way to proceed, but in order to get the feel of the bomba, let's briefly consider how to carry out this search.
We select one of the double-Enigmas, pick a wheel order, say I/II/III, and put the three pairs of wheels into this order. We then lock the right-hand pair, the IIIs, in step at an offset of three (just as in the example of the mini bomba). The wheels in the middle pair (the IIs) are locked in step at the same position as one another, and likewise the wheels in the left-hand pair (the Is). Finally, we set the detector circuits to produce a stop whenever the same letter—any letter— lights simultaneously in both Enigmas. (The remaining two double-Enigmas are not needed for this search.) The motor is switched on and each replica Enigma moves through its 26 × 26 × 26 positions. Any stops give pairs of positions, three clicks of the right-hand wheel apart, at which typing W produces the same letter at the lampboard. Another five runs of the bomba are required to explore all six wheel orders. (Alternatively we might use all three double-Enigmas, each with a different wheel order, so enabling the bomba to explore three wheel orders simultaneously. In this case only two runs of the bomba are necessary to cover all the possible wheel orders.)
Notice that an assumption is being made here concerning 'turnovers'. As previously explained, the movement of the right-hand wheel of the Enigma machine at some point causes the centre wheel to turn forward one click; and the movement of the centre wheel at some point causes the left-hand wheel to advance one click. The positions at which these turnovers occur are determined by the _Ringstellung_. In locking the pair of II wheels (the middle wheels) of the double-Enigma together in the _same_ position as one another, we are assuming that, as the sender's machine lights up the letters WAHWIK, no movement of the middle wheel occurs during the three clicks forward of the right-hand wheel that lie between the production of the first and second occurrences of W. And in locking the left-hand wheels of the double-Enigma together in the same position, we are making the same assumption about the left-hand wheel of the sender's machine.
Of course, these assumptions might be wrong, in which case the search will fail. This is no less true in the case of the full-blooded search described below involving three indicators. However, the assumption that only the right-hand wheel moves in the course of typing a group of six letters is true much more often than not, and so searches based on this assumption will, other things being equal, succeed much more often than not.
The problem with the method of searching just described is that it would typically produce excessively many stops—many triples of positions _p L_, _p M_, _p R_ are liable to satisfy the rather mild constraint that W encodes the same letter at both _p R_ and _p R_ \+ 3. It would take the clerk who tries out each stop by hand on a further replica Enigma far, far too long to winnow out the correct wheel positions. It is necessary to find additional indicators from the same day's traffic that can be used to narrow the focus of the bomba's search. Here is what the Poles actually did.
In order to put a bomba to work effectively, it is necessary to find in a single day's traffic (i.e. traffic encoded with the same wheel order and _Stecker_ ) three messages whose indicators exhibit the following patterns of repetitions. One indicator must display the pattern just discussed—the same letter repeated at the first and fourth positions, as in the example
WAHWIK.
A second indicator must have the selfsame letter that is at positions 1 and 4 in the first indicator at its second and fifth positions, as in
DWJMWR.
A third indicator must have that same letter at its third and sixth positions, as in
RAWKTW.
The Poles called these patterns 'females' (see above). At Bletchley Park the three patterns were referred to as a 1–4 female, a 2–5 female, and a 3–6 female respectively. It is because this indicator system admits three types of female that the bomba contains three double-Enigmas, each one utilizing the information contained in one of the three females.
Let the position of the right-hand wheel when the first letter of the first indicator was produced be _p R_ and the position of the right-hand wheel when the first letter of the second indicator was produced be _q R_, and likewise _r R_ in the case of the third indicator. We know from the patterns of repeated letters in the indicators that:
Keying W produces a simultaneity at _p R_ and _p R_ \+ 3 (i.e. at _p R_ and _p R_ \+ 3 the same letter lights). Keying W produces another simultaneity at _q R_ \+ 1 and _q R_ \+ 4 (possibly involving a different letter at the lampboard). Keying W produces a third simultaneity at _r R_ \+ 2 and _r R_ \+ 5.
In fact we know more than this. A rich source of information has not yet been used—the indicator settings which appear in clear in the preambles to the messages. Suppose these are as follows.
Without the wheel order and the _Ringstellung_ for the day in question, which of course we do not yet possess, the indicator setting cannot be used straightforwardly to decode the indicator. Nevertheless, the indicator settings are far from useless to us, because they contain information about the _relative_ positions of the wheels when the indicators were produced; and using this information, we can deduce the relationship between _p R_, _q R_, and _r R_.
The right-hand letter of each indicator setting specifies the position of the right-hand wheel when the encryption—or equivalently the decryption—of each message setting begins. Similarly, the middle letter specifies the position of the middle wheel when the encryption of the message setting begins, and the left-hand letter the position of the left-hand wheel. Picture the letters of the alphabet arranged evenly around the circumference of a circle, as on the ring of a wheel. The right-hand letter of the second indicator setting, Y, is fifteen letters further on than the right-hand letter of the first indicator setting, J. Therefore the position of the right-hand wheel at which the first letter of the second indicator was produced, _q_ _R_ , is fifteen clicks on from the position at which the first letter of the first indicator was produced, _p_ _R_ :
_q_ _R_ = _p_ _R_ \+ 15
The right-hand letter of the third indicator setting, B, is eighteen letters on from J (JKLMNOPQRSTUVWXYZAB). Therefore the position of the right-hand wheel at which the first letter of the third indicator was produced, _r_ _R_ , is eighteen clicks on from _p R_:
_r R_ = _p R_ \+ 18
Inserting this additional information into the above statement about simultaneities gives:
Keying W produces a simultaneity at _p R_ and _p R_ \+ 3; another simultaneity at ( _p R_ \+ 15)+1 and ( _p R_ \+ 15)+ 4; and a third simultaneity at ( _p R_ +18)+2 and ( _p R_ +18) + 5.
Or more simply:
Keying W produces a simultaneity at _p R_ and _p R_ \+ 3; another simultaneity at _p R_ \+ 16 and _p R_ \+ 19; and a third simultaneity at _p R_ \+ 20 and _p R_ \+ 23.
Now we have a much stronger constraint on _p R_ and can use the bomba to search for _p R_ and the accompanying positions of the other wheels in the expectation that the number of stops will be small enough to be manageable.
#### _Using the Bomba_
The bomba is set up for the search as follows. The stopping mechanism is arranged to produce a stop whenever the eighteen wheels move into a configuration that causes a simultaneity at each of the three double-Enigmas at once. The three simultaneities need not involve the same lampboard letter as each other. W is input continuously into the Enigmas.
One double-Enigma is set up as above: the wheel order is I/II/III, the III wheels are locked together at an offset of three, and the other pairs of wheels are locked with no offset (the assumption being, as before, that neither the middle nor the left-hand wheel of the sender's machine moved during the production of WAHWIK). Call this double-Enigma's III wheels l1 and r1 (for the left and right members of the pair); r1 is three clicks ahead of l1.
The second double-Enigma is set up with the same wheel order. Call its III wheels l2 and r2. l2 is locked in step with l1 at an offset of 16, and r2 is locked in step with l2 at an offset of 3 (so r2 is nineteen clicks ahead of l1). As with the first double-Enigma, the II wheels are locked in step with no offset, and likewise the Is. The third double-Enigma is also set up with wheel order I/II/III. Its III wheels are l3 and r3. l3 is locked in step with l1 at an offset of 20, and r3 is locked in step with l3 at an offset of 3 (so r3 is twenty-three clicks ahead of l1). Again, the II wheels are locked in step with no offset, and the same for the Is.
Next, each double-Enigma must have its pair of II wheels suitably synchronized with those of its neighbours, and similarly its I wheels. This is achieved as in the case of the III wheels by making use of the information contained in the indicator settings about the relative positions of the wheels of the sender's machine when the indicators were produced.
The middle letter of the second indicator setting, Q, is twenty-three places ahead of the middle letter of the first indicator setting, T. So the middle wheels of the second double-Enigma—the IIs—are locked in step with the middle wheels of the first at an offset of 23. The middle letter of the third indicator setting, P, is twenty-five places ahead of the middle letter of the second indicator setting, Q, so the middle wheels of the third double-Enigma are locked in step with the middle wheels of the second at an offset of 25. The left-hand letter of the second indicator setting, D, is twelve places ahead of the left hand letter of the first indicator setting, R, so the left-hand wheels of the second double-Enigma—the Is—are locked in step with the left-hand wheels of the first double-Enigma at an offset of 12. Finally, the left-hand letter of the third indicator setting, H, is four places ahead of the left-hand letter of the second indicator setting, D, so the left-hand wheels of the third double-Enigma are locked in step with the left-hand wheels of the second at an offset of 4.
The motor is switched on. As before, the stops that are produced during a run through all 26 × 26 × 26 positions are noted and then tested by a clerk. If none works, it is necessary to set up the bomba again with a different wheel order. Six runs are required to search through all the wheel orders—approximately twelve hours of bomba time in total. By running six bomby simultaneously, one for each wheel order, the Poles reduced the search time to no more than two hours.
The clerk at the replica Enigma tests the various positions at which the stops occurred. He or she eventually finds one that deciphers each indicator into something of the form XYZXYZ. The cryptanalysts now know the message settings and the rod-positions of the wheels at which the message settings were enciphered.
To use the message settings to decode the messages it is necessary to know the _Ringstellung_ (since a message setting XYZ could specify any one of the 26 × 26 × 26 positions, depending on the position of the ring). However, the _Ringstellung_ lies only a step away. It can be deduced by comparing the rod-positions of the wheels at which the first letter of any of the indicators was produced with the corresponding indicator setting.
For example, if the _Ringstellung_ is set correctly, then what should appear in the windows when the wheel cores lie in the positions at which the first W of WAHWIK was produced is RTJ. Since these rod-positions are know, it is a simple matter to take replicas of the wheels and to twist the rings until the letters R, T and J are uppermost at these rod-positions. Once the rings are correctly positioned, a wheel's ring setting is given by the position of the ring against the embossed index mark on the wheel core: whatever letter lies against the index mark is the ring setting for that wheel. The complete _Ringstellung_ is the trigram consisting of the letter for each wheel arranged in the wheel order for the day.
Now the messages can be decoded on a replica Enigma, as can other intercepted messages with the same wheel order and _Ringstellung_.
#### _The Plug-Board Problem_
It remains to explain how the permutations introduced by the plug-board were dealt with. In the military Enigma machine, the plug-board or stecker-board lay in the path both of current flowing from the keyboard to the wheels and of current flowing from the wheels to the lampboard (see Figure 4). Not every keyboard key was affected by the plug-board. When the bomba first came into operation, the Germans were using the plug-board to scramble between ten and sixteen of the twenty-six keys (in effect by swapping the output wires of pairs of keys). The remaining keys were unaffected, being 'self-steckered'.
It was specified in the daily key which (keyboard) keys were to be affected on any given day and how the affected (keyboard) keys were to be paired up. For example, suppose the daily key says that T and K are to be 'steckered'. The operator connects together the plug-board sockets labelled T and K (by means of a cord with a plug at each end). The result of this extra twist is that pressing the T-key at the keyboard produces the effect at the wheels which pressing the K-key would have produced had there been no scrambling of the letters at the plug-board. Likewise pressing the K-key produces the effect which pressing the T-key would have produced in the unsteckered case.
The plug-board comes into play a second time, in between the wheels and the lampboard. If K lights up in the steckered case, then the selfsame output from the wheels would have caused T to light up had T been one of the letters unaffected by the plug-board. Likewise if T lights up, the output would have caused K to light up had K been unaffected by the plug-board.
The bomba took no account at all of _Stecker_. If the females in the chosen indicators had been produced without interference from the plug-board (i.e. if all the letters in the indicators were self-steckered), then the bomba could produce the correct message setting. But if stecker-substitutions were involved, the bomba would be looking for the wrong thing. Returning to the above example, it would not be W that produces simultaneities at _p R_ and _p R_ \+ 3, and so on, but the letter to which W happened to be steckered; and so the bomba's search would fail.
The success of the bomby depended on the fact that, with between ten and sixteen letters unaffected by the plug-board, there was a reasonable chance of the day's traffic containing three indicators unpolluted by _Stecker_ and displaying the requisite females.
Once the wheel order and _Ringstellung_ had been discovered, messages could be deciphered using a replica Enigma on which all letters were self-steckered. The result would be German words peppered with incorrect letters produced by plug-board substitutions. These incorrect letters gave away the plug-board connections of the sender's machine.
On 1 January 1939 the Germans increased the number of letters affected by _Stecker_ (from between five and eight pairs of letters to between seven and ten pairs). The effectiveness of the bomba—already severely compromised by the introduction of wheels IV and V in December 1938—diminished still further.
### 5. The Bombe and the Spider
At Pyry, Knox observed that the indicator system exploited by the bomba might 'at any moment be cancelled'—as did indeed happen in May 1940 (see above). It was clear to Knox that even if the problems engendered by the increases in the number of wheels and the number of steckered letters could be solved, the modified bomba might become unusable overnight. After the Warsaw meeting Knox and Turing considered the possibility of using a bomba-like machine to attack not the indicators but the message text itself, via cribs. The decision was taken to build a flexible machine that could be used both in the Polish manner against the indicators and also with cribs.
Turing was responsible for the logical design of the machine—the 'bombe'. He passed his design to Harold 'Doc' Keen at the factory of the British Tabulating Machine Company in Letchworth. Keen handled the engineering side of the design. Notes dated 1 November 1939 signed by Knox, Turing, Twinn, and Welchman refer to 'the machine now being made at Letchworth, resembling but far larger than the Bombe of the Poles (superbombe machine)' and state: 'A large 30 enigma bomb [ _sic_ ] machine, adapted to use for cribs, is on order and parts are being made at the British Tabulating Company.'
Knox himself appears to have made little or no contribution to the design and development of the bombe. His greatest achievements during the war were breaking the versions of Enigma used by the Italian Navy and by the _Abwehr_ , the secret intelligence service of the German High Command. He died in February 1943.
In its mature form the bombe contained thirty-six replica Enigmas. (The replicas were made at Letchworth and in Chapter 6 Turing refers to them as 'Letchworth Enigmas'.) The intricate bombe contained some ten miles of wire and one million soldered connections. Enclosed in a cabinet, the bombe stood 6 feet 6½ inches tall (5 feet 10 inches without its 8½ inch castors), 7 feet 3¾34; inches long, and 2 feet 7 inches deep. From the front, nine rows of rotating drums were visible. Each drum mimicked a single Enigma wheel. The drums (which were almost 5 inches in diameter and inches deep) were removable and could be arranged to correspond to different wheel orders. Colour-coding was used to indicate which wheel, e.g. IV, a particular drum mimicked. The drums were interconnected by means of a large panel at the rear of the bombe (a panel that 'almost defies description—a mass of dangling plugs on rows of letters and numbers', according to one WRN operator; Mahon says that when viewed from the rear, the bombe appeared to consist 'of coils of coloured wire, reminiscent of a Fair Isle sweater' (p. 291, below)). The replica Enigmas in the bombe could be connected together arbitrarily, according to the demands of whatever crib was being run.
**Figure 7.** A Bletchley bombe.
_Source_ : Science and Society Picture Library, National Museum of Science and Industry.
**Figure 8.** Rear panel of a bombe.
_Source_ : Science and Society Picture Library, National Museum of Science and Industry.
#### _Cribs_
Cribs resulted both from the stereotyped nature of the messages sent by the Germans and from the thoughtlessly insecure habits of some operators. For example, weather stations regularly sent messages beginning in stereotyped ways, such as 'WETTER FUER DIE NACHT' ('Weather for the night') and 'ZUSTAND OST WAERTIGER KANAL' ('Situation Eastern Channel'). In Chapter 5 Mahon relates how a certain station transmitted the confirmation 'FEUER BRANNTEN WIE BEFOHLEN' each evening ('Beacons lit as ordered').
The position of the cribbed phrase within the cipher text could often be found by making use of the fact that the Enigma never encoded a letter as itself. The cryptanalyst would slide a suspected fragment of plain text (e.g. ZUSTAND) along the cipher text, looking for positions at which there were no matches.
In order to uncover cribs, a 'cribster' often had to read through large quantities of decrypts, keeping meticulous records. As the war progressed, 'cribbing' developed to a fine art. The discovering of cribs presupposes that the message traffic is already being read: the period of work from January 1940 with the perforated sheets and other hand methods was an essential preliminary to the success of the bombe.
In the earlier fictitious example, a mini bomba was used in conjunction with a two-letter crib. One replica of the Simple Enigma was dedicated to the first letter of the crib and another to the second (with the two replicas being set in step at a distance of three, as dictated by the crib). Setting all complications to one side— and in particular _Stecker_ —the bombe functions in its bare essentials like the mini bomba in that example.
Suppose we have a message whose first seven letters are
and the one-word crib
Z U S T A N D
In a world without _Stecker_ , we can exploit the crib by connecting seven replica Enigmas together in such a way that the right-hand wheel (or drum) of the second is one position further on than the right-hand wheel of the first, the right-hand wheel of the third is one position further on than the right-hand wheel of the second, and so on. The seven middle wheels are locked in step in the same position as one another, and likewise the left-hand wheels. As with the set-up procedure for the bomba, this assumes that the middle and left-hand wheels of the sender's machine do not turn over during the first seven letters of the message.
During each run, B is input continuously into the first replica Enigma, I into the second, and so on. The electric motor moves the wheels of each replica Enigma through all their possible positions, one by one. The bombe is set up to stop whenever the letters Z U S T A N D light simultaneously at the seven replica Enigmas. When this happens, the positions of the wheels of the first of the seven are noted. These are candidates for the rod-positions of the wheels at the start of the message.
Each stop is tested by hand, using either a replica Enigma or a British Typex cipher machine set up to emulate an Enigma. (The Typex—also written 'Type X'—was in effect an improved form of the Enigma.) If the rest of the message decodes—or at any rate that part of it up to the point where a turnover of the middle or left wheel occurred—then the correct rod-positions have been found.
The seven replica Enigmas all have the same wheel order. By using more replicas, set up in the same way but with different wheel orders, several wheel orders can be tested simultaneously. Several runs of the bombe are required to test all the possible wheel orders.
If there is no success under the assumption that there were no turn-overs of the middle and left-hand wheels during the enciphering of ZUSTAND, then it is necessary to carry out more runs of the bombe, testing the various possibilities for when a turnover occurred. Turing describes this procedure in Chapter 6, p. 316.
Once the correct rod-positions are discovered, the wheel order is known and with some trial and error the _Ringstellung_ can be worked out. In a steckerless world, the codebreakers now have the daily key and all the intercepted messages encoded on that key can be deciphered. This was done by Typex operators. The messages were decoded by following exactly the same steps that the intended recipient would: the indicator setting, transmitted in clear in the message preamble, was used to decrypt the three-letter indicator, producing the message setting.
#### _Turing's Method for Finding the Plug-Board Settings_
Turing employed a simple but brilliant idea in order to deal with the substitutions brought about by the plug-board. He describes this in Chapter 6, which is an extract from his 'Treatise on the Enigma'. (Released in 1996, this material has not previously been published.) 'Treatise on the Enigma' was written in the summer or autumn of 1940 and seems to have been intended for use as a form of training manual. It was known affectionately at Bletchley Park as 'Prof's Book' ('Prof' being Turing's nickname among his colleagues).
Turing's method for finding the plug-board settings dates from 1939. In the example just given, the replica Enigmas are connected 'in parallel'. Turing's idea was to make provision for replica Enigmas (without plug-boards) to be connected nose to tail, with the letter that exits from the wheels of the first being fed into the next in the chain as if it were unsteckered keyboard input. These chains of replica Enigmas could be of varying length, as demanded by the crib.
Each chain exploited a feature of the cribbed message that Turing called a 'closure', but which might equally well be called a 'loop'. There are no closures in the ZUSTAND example. The following, longer, crib (discussed by Turing in Chapter 6, pp. 315ff) contains several examples of closures. (The meaning of the crib is 'No additions to preliminary report'.)
One closure or loop occurs at positions 2 and 5 and is shown in Figure 9. At position 2, E _encodes_ as A and at position 5, A _decodes_ as E. Using an upward-pointing arrow to mean 'encodes' and a downward-pointing arrow to mean 'decodes', the loop is as shown in Figure 9.
(Notice that it is equally true that at position 2, A _decodes_ as E, and at position 5, E _encodes_ as A. It is also true—because the letter substitutions produced by the Enigma are _reversible_ (see p. 224)—that at position 2, A encodes as E, and at position 5, E decodes as A. Any of these equivalent ways of describing the loop will do.)
Another closure, this time involving three letters, occurs at positions 5, 10, and 23 (Figure 10). At position 5, E encodes as A, at position 23, A decodes as I, and at position 10, I decodes as E.
E is called the _central_ letter of these two closures. The crib contains a number of other closures with central letter E (see Turing's Figure 59 on p. 317).
The point about closures is that they are, as Turing says, 'characteristics of the crib which are independent of the Stecker' (p. 316). Figure 9 tells us that there is _some letter_ which, when fed into the wheels at position 2, produces a letter which, if fed into the wheels at position 5, gives the original letter again. At the present stage, we have no idea which letter this is, since unless the central letter E happens to be self-steckered, the letter that goes into the wheels at position 2 is not E itself but whichever letter it is that E is connected to at the plug-board. Turing calls E's mate at the plug-board the 'stecker value' of E. Equally, we have no idea which letter it is that comes out of the wheels at position 2—unless A is self-steckered, the letter that emerges will not be A but A's stecker value.
**Figure 9.** A loop or 'closure'
**Figure 10.** A closure involving three letters.
Figure 10 also represents an assertion about the wheels that is true independently of how the plug-board is set up. There is some letter, _x_ , which when fed into the wheels at position 5—that is to say, with the right-hand wheel four clicks further on than at the start of the message—produces some letter, _y_ , which when fed into the wheels at position 23, produces some letter, _z_ , which when fed into the wheels at position 10, produces _x_ again.
As explained below, these closures are used in determining the stecker value of E. Once E's stecker-mate has been found, then the stecker values of the other letters in the loops are easily found out. For example, A's stecker value is whatever letter emerges from the wheels at position 2 when E's stecker-mate is fed in.
#### _Using the Turing Bombe_
In Turing's bombe, replica Enigmas without plug-boards are connected into chains that mimic the loops in the crib. In general, a crib containing three or more loops was necessary for Turing's bombe to work successfully.
In the case of the loop in Figure 9, two replica Enigmas are connected nose to tail. The right-hand wheel of the second machine is three clicks further on than the right-hand wheel of the first (because three clicks separate positions 2 and 5). As usual, the wheels are locked in step. To deal with the loop in Figure 10, three replica Enigmas are connected nose to tail. The right-hand wheel of the first machine in the chain is set three clicks ahead of the right-hand wheel of the first machine in the chain that corresponds to Figure 9 (three clicks separating positions 2 and 5). The right-hand wheel of the second machine in the chain of three is eighteen clicks ahead of the right-hand wheel of the first machine in that chain (since eighteen clicks separate positions 5 and 23). The right-hand wheel of the third machine in the chain is five clicks ahead of the right-hand wheel of the first (since five clicks separate positions 5 and 10). Other chains are set up for other closures in the crib also having E as central letter (see Turing's Figure 59 on p. 317).
The bombe works like this. We are going to input the same letter into each of the chains. What we are looking for is the stecker value of the central letter, E. We are going to set about finding it by trying out each of the twenty-six possibilities in turn. First we try the hypothesis that E's stecker-mate is A. So we input A into each of the chains.
The bombe is set up to stop whenever the wheels move into a configuration that produces the input letter—A, during the first run—as the output letter of each of the chains. At any stops during the run, we note not only the positions of the wheels, but also the output letter of each of the replica Enigmas in each chain. If the input letter is indeed E's stecker-mate, and the wheel positions are correct, then these 'interior' letters are the stecker-mates of the intermediate letters of the various closures.
If, in searching through all the possible wheel positions, we find no case in which the last machine in every chain produces A as output, then the hypothesis that E is steckered to A must be incorrect. If, however, we do manage to get A lighting up at the end of every chain, the hypothesis that A is E's stecker-mate remains in the running, and is passed on to someone else to investigate further by hand.
Once the first run is complete, we proceed to the hypothesis that E's stecker-mate is B, and again the wheels are moved through all their positions. And so on, taking each of the twenty-six stecker hypotheses in turn.
Additional runs may be required to test various hypotheses concerning the turnover of the middle and left wheels (as mentioned above). There is also the question of the wheel orders. Typically several different wheel orders will be tested simultaneously. (A thirty-six-Enigma bombe could usually test three wheel orders simultaneously, assuming that no more than twelve Enigmas were required for the loops in the crib.) In the case of an 'all wheel order crib', where no information is available to rule out some of the wheel orders, a number of successive runs, or simultaneous runs on several bombes, will be required in order to examine each possible wheel order.
Unless the data provided by a crib is especially scanty, in which case there might be many stops, this procedure would usually produce a manageably small number of stops. These were tried out manually in another building on a replica Enigma or Typex. Usually the stops were tested more or less as they occurred. As soon as one was found that turned part of the remaining ciphertext into German—albeit German peppered with incorrect letters—the instruction would be telephoned back to the bombe operators to strip the bombe and ready it for the next cribbed message in the queue.
The prototype Turing bombe, named 'Victory', was installed at Bletchley Park on 18 March 1940. It seems to have been used exclusively by Turing and other members of Hut 8 in their attempt to break Naval Enigma.
#### _Simultaneous Scanning_
The efficiency of the bombe could be increased greatly by—instead of, as just described, trying out one stecker hypothesis at all positions of the wheels before moving on to the next hypothesis—allowing all the twenty-six possible stecker hypotheses for the central letter (E to A, E to B, etc.) to be tried out together in the short interval before the wheels (drums) shifted from one position to the next. This is 'simultaneous scanning'. Turing's original intention was to include additional electrical apparatus in the prototype bombe to implement simultaneous scanning and he outlines a way of doing this in Chapter 6 (see the section 'Pye simultaneous scanning'—Pye was an electronics company located in Cambridge). However, the problem proved difficult for the engineers and the additional apparatus was not ready in time to be incorporated in Victory. Turing explains in Chapter 6 that the method the engineers were proposing would 'probably have worked if they had had a few more months experimenting', but that their work was in the end overtaken by the discovery of a solution 'which was more along mathematical than along electrical engineering lines' (p. 319).
Turing presents this mathematical solution in two stages in Chapter 6. First he explains (what will in this introduction be called) his _feedback method_ (see his section 'The Spider'). This Turing describes as 'a way of getting simultaneous scanning on the Bombe' (p. 323). Then he goes on to explain the role of Welchman's dazzlingly ingenious invention, the _diagonal board_ (see Turing's section 'The Spider. A Second Description. Actual Form'). Welchman's diagonal board brought about a dramatic increase in the effectiveness of the bombe.
#### _Turing's Feedback Method_
Let us reconsider the previous search for the stecker value of the central letter E. We first tried the hypothesis that E's stecker-mate is A. Inputting A, we rotated the wheels looking for a position at which the letter to emerge is again A. The feedback method is this. Before the wheels are shifted from the current position to the next, whichever letter emerges from the suitably interconnected Enigmas—which will in all probability not be A—is fed back in as the new input letter. (This is done automatically via a braid of twenty-six wires.) This step is then repeated: whichever letter emerges is fed back in, and so on. Unless the first attempt produced A, the effect of these cycles of feedback is that different stecker hypotheses are tested at the current position of the wheels.
If the wheel position is not the starting position for the message then, given a crib with sufficient loops, all twenty-six letters will usually be produced as output during the cycles of feedback. So if the emerging letters are imagined as appearing at a lampboard, all twenty-six lamps will light. At some positions, however—the interesting positions—not all the lamps light. At these positions it is usually true (again given a crib with sufficient loops) either that only one lamp lights, or that only one remains unlit (a reflection of the fact that the Enigma's letter-substitutions are reversible). Either way, the letter on the odd lamp out is a candidate for the stecker value of the central letter, and the position of the wheels is a candidate for the starting position. Letters produced by other Enigmas within the chains are candidates for the stecker values of other letters of the loops.
#### _The Diagonal Board_
Welchman conceived the diagonal board as a way of increasing the effectiveness of the bombe by further exploiting the reciprocal character of the stecker-substitutions. (The substitutions are reciprocal in the sense that if letter L1 is steckered to L2 then—owing to the design of the plug-board—L2 is inevitably steckered to L1.) With the diagonal board in operation, the bombe could work cribs containing fewer than three closures and even cribs containing no closures at all (as in the ZUSTAND example) provided the length of the crib was sufficient. (If Welchman's diagonal board had never been conceived, bombes of the earlier type could have been used successfully against Enigma networks producing enough cribs with at least three closures—although at the expense of greater amounts of bombe time.)
Once Welchman had thought of the diagonal board, Turing quickly saw that it could be used to implement simultaneous scanning. Joan Clarke, who worked alongside Turing in Hut 8, said: 'Turing soon jumped up, saying that Welchman's diagonal board would provide simultaneous scanning.' (Clarke was one of Welchman's mathematics students at Cambridge. For a short period in 1941, she and Turing were engaged to be married.)
The new form of bombe with the diagonal board was initially called the 'Spider' to distinguish it from Turing's earlier form, but soon simply 'bombe' prevailed. (Possibly the name 'Spider' arose in virtue of the practice of using 'web' as a term to refer to the connected parts of a diagram depicting the loops in a crib; see Chapter 6, pp. 325, 329.) The first Spider was installed on 8 August 1940. It was known as 'Agnus', short for 'Agnus Dei' (the name later became corrupted to 'Agnes' and 'Aggie'). Agnus contained thirty replica Enigmas, six fewer than in later models. Both Hut 8 (Naval Enigma) and Hut 6 (Army and Air Force Enigma) were given access to the new machine.
**Figure 11.** Working in a bombe room at Out Station Eastcote. 'Menus' for the outstation bombes were received from Bletchley Park over a teleprinter line.
_Source_ : Photograph from 'Operations of the 6312th Signal Security Detachment, ETOUSA', 1 October 1944 (NARA, document reference: RG 457, Historic Cryptographic Collection, Box 970, NR 2943).
#### _Subsequent Developments_
At first the number of bombes increased relatively slowly, and much of the code-breakers' energy went into the use of hand methods—such as Turing's method of _Banburismus_ —designed to reduce the amount of bombe time required to break a crib. By June 1941 there were only five bombes in operation, rising to fifteen by November. The picture changed markedly when a new factory dedicated to the production of bombes came into operation at Letchworth. The output of Enigma decrypts produced by GC & CS more than doubled during 1942 and 1943, rising to some 84,000 per month by the autumn of 1943. Groups of bombes were housed in 'outstations' in the district surrounding Bletchley Park, and then subsequently at three large satellite sites in the suburbs of London, with dedicated teletype and telephone links to Bletchley Park. By the end of the war there were around 200 bombes in continuous operation at these various sites. From August 1943, US Navy bombes began to go into operation in Washington, DC. About 125 were in operation by the time Germany fell. Good cable communications enabled Bletchley to use the Washington bombes 'almost as conveniently as if they had been at one of our outstations 20 or 30 miles away'.
### 6. Naval Enigma
#### _Turing's Break_
Between 1934 and 1937 the Poles had enjoyed some success against German Naval Enigma. However, on 1 May 1937 a major change of indicator procedure rendered Naval Enigma impenetrable.
During much of 1940 German Air Force traffic was being read in large quantities by GC & CS, but Naval traffic—including the all-important messages to and from the wolf-packs of U-boats in the North Atlantic—remained cloaked. The German strategy was to push Britain toward defeat by sinking the convoys of merchant ships that were Britain's lifeline, bringing food, raw materials, and other supplies across the Atlantic from North America. From the outbreak of war to December 1940 a devastating total of 585 merchant ships were sunk by U-boats, compared to 202 merchant vessels sunk by aircraft during the same period. If Home Waters Naval Enigma ( _Heimische Gewässe_ )—called 'Dolphin' at Bletchley Park—could be broken, the positions of the wolf-packs in the North Atlantic would be known and convoys could be routed around them.
When Turing took up residence at Bletchley Park in September 1939 no work was being done on Naval Enigma, which some thought unbreakable. As late as the summer of 1940 Denniston declared to Birch, the head of the Naval Section at GC & CS (Hut 4): 'You know, the Germans don't mean you to read their stuff, and I don't suppose you ever will.' This was never the opinion of Birch and Turing. Alexander's history of the attack on Naval Enigma (written at the end of the war and kept secret by the British government until very recently) recounted:
Birch thought it could be broken because it had to be broken and Turing thought it could be broken because it would be so interesting to break it... Turing first got interested in the problem for the typical reason that 'no one else was doing anything about it and I could have it to myself'.
The chief reason why Dolphin was so difficult to break was that the indicator system required the sender to encipher the message setting by two different methods before broadcasting it—once by means of the Enigma machine, as was usual, and once by hand. Mahon describes the procedure in detail in Chapter 5. The hand encipherment was performed by means of a set of _bigram tables_. These tables specified substitutions for pairs of letters, such as 'DS' for 'HG' and 'YO' for 'NB'. Enigma operators were issued with a set of nine complete tables, each table giving substitutions for all the 676 possible bigrams. Which table was to be used on any given day was set out in a calendar issued with the tables. New sets of tables came into force periodically. Crews were under strict instructions to destroy the tables before abandoning ship or if the enemy was about to board.
Turing started his attack exactly where the Poles had left off over two years before, studying 100 or so messages from the period 1–8 May 1937 whose message settings were known. Before the end of 1939 he had fathomed out exactly how the complicated indicator system worked. Chapter 5 contains an extract from Turing's 'Treatise on the Enigma' (published here for the first time) in which Turing explains how he performed this remarkable piece of cryptanalysis.
#### _Hut 8_
In 1940 Turing established Hut 8, the section devoted to breaking Naval Enigma. Initially the Naval Enigma group consisted of Turing, Twinn, and 'two girls'. Early in 1940 they were joined by Tony Kendrick, followed by Joan Clarke in June of that year, and then in 1941 by Shaun Wylie, Hugh Alexander, Jack Good, Rolf Nosk-with, Patrick Mahon, and others. Turing was 'rightly recognized by all of us as the authority on any theoretical matter connected with the machine', said Alexander (himself later head of Hut 8). In Chapter 5 Mahon recounts how, under Turing's leadership, Hut 8 slowly gained control of Dolphin during 1940 and 1941.
Unlike _Heimische Gewässe_ (Dolphin), _Ausserheimische Gewässe_ —meaning 'Distant Waters'—would never be broken by Hut 8, and several other Naval Enigma networks also resisted attack. _Süd_ , on the other hand, used in the Mediterranean from mid-1941, was a much easier proposition than _Heimische Gewässe_. As Mahon mentions in the next chapter (p. 273), _Süd_ employed a version of the indicator system broken by the Poles. _Süd_ 's procedure of enciphering the message setting twice (on which the bomby had depended) meant that Hut 8 was able to read _Süd_ traffic without any need for cribs.
#### _Pinches_
Turing's discovery of how the indicator system worked could not be used to read the German traffic until the bigram tables were known. Materials obtained by the Royal Navy from enemy vessels enabled the codebreakers to reconstruct the all-important tables. (Many of the captures are described in Hugh Sebag-Monte-fiore's fast-paced book _Enigma: The Battle for the Code_ ; see the section of further reading.)
The first capture, or 'pinch', of Home Waters daily keys—which Alexander described as 'long awaited'—was on 26 April 1940. A party from the British destroyer HMS _Griffin_ boarded an armed German trawler disguised as a Dutch civilian vessel (bearing the false name 'Polares'). The trawler was bound for the Norwegian port of Narvik to deliver munitions. The 'Narvik Pinch', as it became known, yielded various documents, including notes containing letter-for-letter cribs for 25 and 26 April (see Mahon's account in the next chapter). Among the documents was a loose scrap of paper (overlooked at first) on which were scribbled the _Stecker_ and the indicator setting for 23 and 24 April. Also captured were exact details of the indicator system, confirming Turing's deductions.
The crib for 26 April was tried on the recently arrived Victory, and according to Alexander 'after a series of misadventures and a fortnight's work the machine triumphantly produced the answer' (see also p. 286, below). Alexander reports that 27 April could then also be broken, the 26th and 27th being 'paired days'— days with the same wheel order and _Ringstellung_. Thanks to the Narvik Pinch, the days 22–5 April were also broken (not on the bombe but by hand methods).
Another pinch was needed if Dolphin was to be broken for any substantial period. Various plans were discussed. One, code-named 'Operation Ruthless', was masterminded by Lieutenant Commander Ian Fleming of Naval Intelligence, who later created the character James Bond. Mahon describes the plan, which he credits to Birch, in the next chapter. In the event, Operation Ruthless was not carried out. Turing's reaction is described in a letter by Birch dated 20 October 1940:
Turing and Twinn came to me like undertakers cheated of a nice corpse two days ago, all in a stew about the cancellation of operation Ruthless. The burden of their song was the importance of a pinch. Did the authorities realise that... there was very little hope, if any, of their deciphering current, or even approximately current, enigma for months and months and months—if ever? Contrariwise, if they got a pinch... they could be pretty sure, after an initial delay, of keeping going from day to day from then on... because the level of traffic now is so much higher and because the machinery has been so much improved.
Turing did not get what he wanted until the 'Lofoten Pinch' of March 1941, which Mahon describes as 'one of the landmarks in the history of the Section' (p. 290). On 4 March, during a commando raid on the Norwegian coast— planned with a pinch in mind—the Royal Navy destroyer HMS _Somali_ opened fire on the German armed trawler _Krebs_ near the Lofoten Islands. _Krebs_ was boarded and tables giving the daily keys for the complete month of February 1941 were captured. Short of obtaining the bigram tables as well, this was exactly what was needed. A month's daily keys were sufficient to enable Hut 8 to reconstruct the tables. Suddenly Hut 8 was properly open for business and by the beginning of April was looking forward to breaking the Naval traffic 'as nearly currently as possible'.
Eager to follow up on the _Krebs_ success, Harry Hinsley in Hut 4 put forward a plan to capture a German weather ship, _München_ , operating north-east of Iceland. On 7 May 1941 _München_ was duly boarded by a party from the _Somali_. The booty included the daily keys for the month of June. The July keys soon followed, captured from the weather ship _Lauenburg_ in another raid planned by Hinsley. The capture of the June and July keys helped Hut 8 reconstruct the new bigram tables issued on 15 June (see the next chapter). The new tables were current until November 1941.
During June and July Hut 8 was producing decrypts of Enigma messages within one hour of their being received. Mahon says on pp. 290–291, 'There can be no doubt that at this stage the battle was won and the problem was simply one of perfecting methods, of gaining experience, and of obtaining and above all of training staff.'
Probably the most dramatic pinch of all occurred to the south of Iceland on 9 May 1941, during the pursuit of the submarine _U-110_ by several Royal Naval vessels acting as convoy escorts. Sub-Lieutenant David Balme, of the destroyer HMS _Bulldog_ , led the party that boarded the stricken submarine. In an interview Balme described the depth-charging of the _U-110_ :
Suddenly two ships were torpedoed one after the other. It was obvious where the attack had come from and the corvette _Aubretia_ made a very accurate attack on the U-boat. Must have got the depth-charges just at the right depth. It was a classic attack: depth-charges underneath the U-boat blew it to the surface. It was the dream of every escort vessel to see a U-boat blown to the surface. Usually they just sink when you have a successful attack.
The German crew abandoned ship shortly before Balme boarded the U-boat. He continued:
I couldn't imagine that the Germans would have abandoned this U-boat floating in the Atlantic without someone down below trying to sink her. But at any rate I got on and got my revolver out. Secondary lighting, dim blue lighting, was on and I couldn't see anybody, just a nasty hissing noise that I didn't like the sound of.
But the U-boat was deserted and, inexplicably, the Germans had made no attempt to destroy the Enigma materials on board. Balme and his men carried off the Enigma machine and the bigram tables. However, the tables had already been reconstructed laboriously by Turing and co. (see p. 290). Balme's pinch was not of major significance to Hut 8 and does not even rate a mention by Mahon or Alexander.
#### _Banburismus_
Another of Turing's pivotal contributions to the breaking of Naval Enigma was his invention of the hand method called _Banburismus_. The name arose because the method involved the use of specially made sheets bearing the alphabet which, being printed in the nearby town of Banbury, came to be called 'Banburies'. Mahon records that Turing invented the method the same night in 1939 that he worked out the indicator system (see Chapter 5).
The aim of Banburismus was to identify the day's right-hand and middle wheels. This meant that fewer wheel orders had to be tried on the bombe, thereby saving large amounts of bombe time. During the years when so few bombes were available, it was Banburismus which made it possible to read Dolphin. As Mahon says, for two or three years Banburismus was 'the fundamental process which Hut 8 performed' (p. 281). Banburismus was discontinued in September 1943, bombes being plentiful enough by that stage.
#### _The Battle of the Atlantic_
Hut 8's ability to decode the U-boat messages had an immediate effect on the course of the war.
At the beginning of June 1941 Churchill had been informed by his planners that, as a result of the attacks on convoys, Britain's predicted imports amounted to substantially less than the minimum quantity of food necessary to keep the population fed during the remainder of 1941. Oil and other imports would also arrive in insufficient quantities for war production to be maintained. The U-boats were crippling Britain. However, during June 1941—when Dolphin was read currently for the first time—reroutings based on Hut 8 decrypts were so successful that for the first twenty-three days of the month, the North Atlantic U-boats made not a single sighting of a convoy.
The pattern continued in subsequent months. The Admiralty's Operational Intelligence Centre (OIC) became increasingly skilled at evasive routing based on Bletchley's Ultra intelligence, and the wolf-packs spent more and more time searching fruitlessly. Although Hut 8's battle with the U-boats was to seesaw—for eleven long months of 1942, Hut 8 was blacked out of the North Atlantic U-boat traffic by the new fourth wheel inside the Enigma—the intelligence from Naval Enigma decrypts played a crucial role in the struggle for supremacy in the North Atlantic.
### 7. Turing Leaves Enigma
Mahon records that towards the end of 1941 Turing was running out of theoretical problems to solve concerning Naval Enigma (p. 312). Soon Turing was taking little part in Hut 8's activities. His talent for groundbreaking work was needed elsewhere.
For a period during 1942 Turing rejoined the Research Section to work on the new problem of 'Tunny'. From June 1941 GC & CS had begun to receive enciphered messages that were very different from the Enigma traffic. These were carried by an experimental radio link between Berlin and Greece. Numerous other links soon came into existence, connecting Berlin to German Army Group commands throughout Europe. Unlike Enigma radio transmissions, which were in Morse code, the messages on these links were broadcast in binary teleprinter code. The British code-named the machine encrypting the new traffic 'Tunny'. Tunny was one of three different types of non-Morse 'Fish' traffic known to Bletchley (the others were codenamed 'Sturgeon' and 'Thrasher').
It was not until July 1942 that up-to-date Tunny traffic was read for the first time, by means of a paper-and-pencil method invented by Turing and known simply as 'Turingery'. The Germans used Tunny for high-level Army communications and sometimes messages signed by Hitler himself would be deciphered. With the arrival of the 'Heath Robinson' in June 1943, followed a few months later by the first of the electronic Colossus computers, the Tunny traffic, like Enigma before it, succumbed to the Bletchley machines (see further the introductions to Chapters 4 and ).
Alexander gradually took over the running of Hut 8. In November 1942, Turing departed for the United States, where he liased with the US Navy's codebreakers and bombe-builders. He was never to do any more work in Hut 8. Following his return to Bletchley, in March 1943, he held a wider brief, acting as scientific policy adviser. Turing eventually left Bletchley Park at the end of 1943, moving to Hanslope Park to work on the problem of automatically enciphering speech. He remained at Hanslope until the end of the war.
In his history of Bletchley's attack on Naval Enigma, Alexander included the following appreciation of Turing's 'great contribution':
There should be no question in anyone's mind that Turing's work was the biggest factor in Hut 8's success. In the early days he was the only cryptographer who thought the problem worth tackling and not only was he primarily responsible for the main theoretical work within the Hut (particularly the developing of a satisfactory scoring technique for dealing with Banburismus) but he also shared with Welchman and Keen the chief credit for the invention of the Bombe. It is always difficult to say that anyone is absolutely indispensable but if anyone was indispensable to Hut 8 it was Turing. The pioneer work always tends to be forgotten when experience and routine later make everything seem easy and many of us in Hut 8 felt that the magnitude of Turing's contribution was never fully realized by the outside world.
In July 1941 Turing, Alexander, and Welchman were summoned to the Foreign Office in London to be thanked for what they had done. Each was given £200 (a sizeable sum in those days—Turing's Fellowship at King's paid him less than twice this amount per annum). At the end of the war, Turing received the Order of the British Empire for the role he had played in defeating Hitler—a role that, after more than half a century of secrecy, has only now come fully into the light of day.
#### **_Further reading_**
Bauer, F. L., _Decrypted Secrets: Methods and Maxims of Cryptology_ (Berlin: Springer-Verlag, 2nd edn. 2000).
Budiansky, S., _Battle of Wits: The Complete Story of Codebreaking in World War II_ (New York: Free Press, 2000).
Erskine, R., and Smith, M. (eds.), _Action This Day_ (London: Bantam, 2001).
Hinsley, F. H., and Stripp, A. (eds), _Codebreakers: The Inside Story of Bletchley Park_ (Oxford: Oxford University Press, 1993).
Kahn, D., _Seizing the Enigma: The Race to Break the German U-Boat Codes, 1939–1943_ (Boston: Houghton Mifflin, 1991).
Sebag-Montefiore, H., _Enigma: The Battle for the Code_ (London: Weidenfeld and Nicolson, 2000).
Smith, M., _Station X: The Codebreakers of Bletchley Park_ (London: Channel 4 Books, 1998).
Welchman, G., _The Hut Six Story: Breaking the Enigma Codes_ (Cleobury Mortimer: M&M Baldwin, 2nd edn. 2000).
### **CHAPTER 5
History of Hut 8 to December 1941 ( _1945_ )**
_Patrick Mahon_
#### **Introduction**
##### _Jack Copeland_
Patrick Mahon (A. P. Mahon) was born on 18 April 1921, the son of C. P. Mahon, Chief Cashier of the Bank of England from 1925 to 1930 and Comptroller from 1929 to 1932. From 1934 to 1939 he attended Marlborough College before going up to Clare College, Cambridge, in October 1939 to read Modern Languages. In July 1941, having achieved a First in both German and French in the Modern Languages Part II, he joined the Army, serving as a private (acting lance-corporal) in the Essex Regiment for several months before being sent to Bletchley. He joined Hut 8 in October 1941, and was its head from the autumn of 1944 until the end of the war. On his release from Bletchley in early 1946 he decided not to return to Cambridge to obtain his degree but instead joined the John Lewis Partnership group of department stores. John Spedan Lewis, founder of the company, was a friend of Hut 8 veteran Hugh Alexander, who effected the introduction. At John Lewis, where he spent his entire subsequent career, Mahon rapidly achieved promotion to director level, but his health deteriorated over a long period. He died on 13 April 1972.
This chapter consists of approximately the first half of Mahon's 'The History of Hut Eight, 1939–1945'. Mahon's typescript is dated June 1945 and was written at Hut 8. It remained secret until 1996, when a copy was released by the US government into the National Archives and Records Administration (NARA) in Washington, DC. Subsequently another copy was released by the British government into the Public Record Office at Kew. Mahon's 'History' is published here for the first time.
Mahon's account is first-hand from October 1941. Mahon says, 'for the early history I am indebted primarily to Turing, the first Head of Hut 8, and most of the early information is based on conversations I have had with him'.
#### **History of Hut 8 to December 1941**
The king hath note of all that they intend,
By interception which they dream not of.
King Henry V
#### Introduction
Before writing this history I have not had the unpleasant task of reading voluminous records and scanning innumerable documents. We have never been enthusiastic keepers of diaries and log books and have habitually destroyed records when their period of utility was over, and it is the merest chance that has preserved a few documents of interest; hardly any of these are dated. Since March 1943, the Weekly Report to the Director provides a valuable record of our activities, but it is naturally this more recent period which human memory most easily recalls, and it is the lack of documentary evidence about early days which is the most serious. A very large portion of this history is simply an effort of memory confirmed by referring to other members of the Section. I joined the Section myself in October 1941 and have fairly clear personal recollections from that time; for the early history I am indebted primarily to Turing, the first Head of Hut 8, and most of the early information is based on conversations I have had with him. I also owe a considerable debt of gratitude to Mr. Birch who lent me the surviving 1939–1940 Naval Section documents which yielded several valuable pieces of information and afforded an interesting opportunity of seeing Hut 8 as others saw us. Many past members of the Section and many people from elsewhere in B. P. have been kind enough to answer questions....
With the exception of Turing, whose position as founder of the Section is a very special one, I have adopted the policy of not mentioning individuals by name. Attributing this or that accomplishment to an individual would be an invidious process contrary to the traditional attitude of the Section towards its work and it would inevitably give a misleading impression of the relative contributions of the different members of the Section.
It is impossible to write a truly objective history of a Section which has been one's principal interest in life for the last years and so this account is written on a comparatively personal note—mostly in the first person plural with occasionally a purely personal recollection or opinion included: to the best of my ability I have only introduced by 'we' opinions with which the Section as a whole would have agreed. We have always prided ourselves on not trying to conceal our failures, and on admitting where we might have done better, and I have attempted to avoid any tendency to 'whitewash' our efforts for the benefit of posterity.
This history is intended for the layman. Our work has been traditionally incomprehensible—the last distinguished visitor I remember had barely sat down before he announced that he was not a mathematician and did not expect to understand anything. (Anyone wishing to probe the more abstruse mathematical aspect of it should turn to the technical volume which is being compiled in collaboration with Hut 6.) In fact, there is nothing very difficult to understand in the work we did, although it was confusing at first sight. I have attempted to explain only the basic principles involved in the methods we used. As a result of this I hope that anyone interested in Hut 8 and willing to read the semi-technical passages with some care will get a fairly clear idea of our work and I make no excuse for having deliberately avoided mentioning many of the complications which arose—thus when describing the machine I say that after pressing the keys 26 × 26 × 26 times the machine has returned to its starting place, but the mathematician will realize that the introduction of wheels 6, 7, and 8 with 2 turnovers each is liable to split this cycle into several smaller cycles.
The account starts with a description of the machine and the methods of sending messages. [This] is followed by some information as to where the machine was used and the volume of traffic carried. After these tedious but rather necessary pages of background [the account] follows the course of events more or less chronologically, starting some time before the war. Certain subjects—like Banburismus...—required whole sections to themselves outside the historical narrative and these are the subjects of a series of digressions....
#### The Machine and the Traffic
If the history of Hut 8 is to be understood, it is essential to understand roughly how the [Enigma] machine works and thus obtain some idea of the problem which had to be tackled. Developments which have taken place during the war have complicated the problem but have left the machine fundamentally the same.
The process of cyphering is simple and quick. The message is 'typed' on a normal keyboard and as each letter is pressed, another letter is illuminated on a lampboard containing the 26 letters of the alphabet. The series of letters illuminated on the lampboard form the cypher text and the recipient of the cypher message, in possession of an identical machine, types out the cypher text and the decoded message appears on the lampboard.
##### _Wheels_
The main scrambler unit consists of 3 (later 4) wheels and an Umkehrwalze which I shall refer to henceforth as the Reflector—an admirable American translation. These wheels have on each side 26 contacts which we will for convenience label A to Z. The contacts on the one side are wired in an arbitrary and haphazard fashion to the contacts on the other. Each wheel is, of course, wired differently. The reflector has 26 contacts which are wired together arbitrarily in pairs. What happens when one of the letters of the keyboard is pressed may be seen from the following diagram.
The current in this example enters the right hand wheel at A and leaves it at M, A being wired to M in this wheel: it enters the middle wheel at M and leaves it at Q, and so on until it reaches the reflector, where it turns around and returns through the wheels in a similar fashion, eventually leaving the right hand wheel at position N and lighting the appropriate lamp in the lampboard. Pressing a key may light up any bulb except that which is the same as the key pressed—for a letter to light up itself it would be necessary for the current to return through the wheels by the same route as it entered and, from the nature of the reflector, this is clearly impossible. This inability of the machine to encypher a letter as itself is a vital factor in the breaking of Enigma. It should also be noted at this point that the machine is reciprocal, that is to say that, if at a given position of the machine N lights up A, then A will light up N.
Each time a key is pressed the right-hand wheel moves on one so that if, in the position immediately following our example above, the same key is pressed, the current will enter the right-hand wheel at B and not A, and will pursue an entirely different course. Once in every 26 positions, the right hand wheel moves the middle wheel over one so that when the right hand wheel returns to position A, the middle wheel is in a new position. Similarly, the middle wheel turns over the left hand wheel once for every complete revolution it makes. Thus it will be seen that 26 × 26 × 26 (about 17,000) letters have to be encyphered before the machine returns to the position at which it started.
For most of the period with which we are concerned, there have been 8 wheels. Wheels 1 to 5 turn over the wheel next to them once per revolution, wheels 6, 7, 8, twice per revolution, this somewhat complicating the cycle of the machine as described in the previous paragraph. The turnovers (by which I mean the position of the wheel at which it turns over its next door neighbour) on wheels 1 to 5 are all in different places; in 6, 7, and 8 they are always at M and Z. As we shall see later, this was an important development.
##### _Ringstellung_
On each wheel is a tyre, marked with the letters of the alphabet. One of these letters can be seen through a window on top of the machine and the position of the wheel is referred to by the letter shown in the window. The tyre is completely independent of the core of the wheel, which contains the wiring, the relative position being fixed by the Ringstellung or clip which connects the tyre with the core. Thus even if the starting position of the message is known, it still cannot be decoded unless the clips, which fix the relative position of tyre and core, are known also.
##### _Stecker_
The Enigma machine would be a comparatively simple affair if it were not for the Stecker. This is a substitution process affecting 20 of the 26 letters before and after the current travels through the wheels. Let us return to our original example and assume for the moment that A is steckered to F and N to T. In our example we pressed key A and entered the right-hand wheel at a position we called A, but if we now press A the current will be sidetracked before entering the wheel, and will in fact enter at F and pursue a quite different course. If, on the other hand, we press F, the current will enter at A and proceed as before, coming out at N. N will not, however, light up on the lampboard, but rather T, because N has been steckered to T. This steckering process affects 20 letters; the remaining 6 are referred to as self-steckered and, when they are involved, the current proceeds directly to or from the wheels.
##### _Set-Up_
In order to decode a message one has, then, to know wheel order, clips, starting position of message, and Stecker. Any 3 of the 8 wheels may be chosen—336 possibilities. There are 17,000 possible clip combinations and 17,000 possible starting positions—in the 4-wheeled machine half a million. The number of possible Stecker combinations is in the region of half a billion. In fact, the number of ways the machine may be set up is astronomical, and it is out of the question to attempt to get messages out by a process of trial and error. I mention this as it is a hypothetical solution to the problem often put forward by the uninitiated, when in fact all the coolies in China could experiment for months without reading a single message. On most keys the wheel order and clips were changed every two days—this is the so-called 'innere Einstellung' which was supposed to be changed only by officers and which was printed on a separate sheet of paper. The Stecker and Grundstellung (of which more later) normally changed every 24 hours. In a 31-day month, the odd day was coped with by having 3 days on 1 wheel order—a triplet. This sometimes came at the end of the month and sometimes in the middle. Triplets happened also in 30-day months, with the result that the last day was a 'singleton' with a wheel order of its own. These rules were not obeyed by all keys, but we shall meet the exceptions as we proceed with the historical survey. The later 4-wheeled keys had a choice of 2 reflectors and 2 reflector wheels; these were only changed once a month.
##### _Indicating System_
The next essential is to understand the indicating system. Various indicating systems were used simultaneously, but if we examine the most complicated fairly carefully the others will be simple to explain.
[The keys] Dolphin, Plaice, Shark, Narwhal, and Sucker all built up their indicators with the help of bigram tables and K book.
##### _K Book_
One half of the K book consists of a Spaltenliste containing all 17,576 existing trigrams, divided into 733 numbered columns of 24 trigrams chosen at random. The second half consists of a Gruppenliste where the trigrams are sorted into alphabetical order; after each trigram are 2 numbers, the first giving the number of the column in the Spaltenliste in which the trigram occurs, the second giving the position of the trigram in the column.
By means of a Zuteilungsliste the columns of the K book are divided amongst the various keys, the large keys being given several blocks of columns, small keys as few as 10. The K book is a large document which has probably changed only once—the current edition having come into force in 1941—but the Zuteilungsliste was changed fairly frequently.
##### _Bigram Tables_
A set of bigram tables consisted of 9 tables, each giving a series of equivalents for the 676 existing bigrams. These tables were reciprocal, i.e. if AN=OD then OD=AN, a useful property as we shall see later. Which bigram table was in force on any given day was determined by means of a calendar which was issued with the tables. New sets of bigram tables were introduced in June 1941, November 1941, March 1943, July 1944.
Now that we are familiar with all the necessary documents—key sheets, K book, bigram tables, etc.—it will be profitable to follow in detail the steps taken by a German operator wishing to send a message.
He is on board a U-boat and has Shark keys and, after consulting his Zuteilungsliste, goes to columns 272–81, which have been allotted to Shark. Here he selects the trigram HNH to serve as his Schluesselkenngruppe. He then selects from anywhere in the book another trigram (PGB) and writes them down like this:
He then fills in the 2 blanks with dummy letters of his own choice:
Q H N H
P G B L
Taking the bigram table which is in force at the time, [he] substitutes for each vertical pair of letters QP, HG, etc.:
I D Y B
N S O I
The indicator groups of his message will then be IDYB NSOI.
The trigram which he chose at random (Verfahrenkenngruppe) now provides him with the starting position of his message. To obtain this he sets up the Grundstellung in the window of his machine and taps out PGB. The three encyphered letters which result are the set-up for his message.
The first step in decyphering the message is, of course, to decypher the indicator groups by means of the bigram tables. At this stage the Schluessel-kenngruppe can be looked up in the K book, and it can then be established by each station whether the message is on a key which it possesses. The Grundstellung is then set up, the trigram tapped out, and the message decoded.
Some form of Grundstellung procedure was common to all keys, but there were a number of keys not using bigram tables and K book. In the Mediterranean area Kenngruppenverfahren Sued was used and discrimination between keys was dependent on the first letters of the first and second groups—the resulting bigram indicating the key. In this case, the operator chose any trigram he wished and encyphered it twice at the Grundstellung and the resulting 6 letters formed the last three letters of the first 2 groups.... Of the other keys, Bonito and Bounce relied for recognition on the fact that in external appearance their traffic was unlike any other and so used no discriminating procedure, simply sending as indicators a trigram encyphered at the Grundstellung and filling the groups up with dummy letters as required.
This machine was in general use by the German Navy in all parts of the world; it was used alike for communication between ship and shore and shore and shore and by all vessels from mine sweepers and MTBs up to U-boats and major units.
##### _Traffic_
The history of Hut 8 is conditioned very largely by the rapid growth of the German Naval communications system and the resulting increases of traffic and increasing number of keys.... It is not necessary to go into details here, but a few figures will illustrate very clearly that the gradual contraction of German occupied territory in no way signified a decrease of traffic and a simplification of the problem. Our traffic figures do not go back to 1940 but the following are the daily averages of messages for March 1941–5:
1941 | 465
---|---
1942 | 458
1943 | 981
1944 | 1,560
1945 | 1,790
The largest number of messages ever registered in Hut 8 on one day was on March 13, 1945, when 2,133 were registered.
As a general rule German Naval traffic was sent out in 4 letter groups with the indicator groups at the beginning and repeated at the end. Both from the German and our own point of view this made Naval traffic easily recognizable, and gave a check on the correct interception of the indicator groups. Messages were normally broadcast on fixed frequencies which changed comparatively rarely, so that it was possible for the cryptographer without W/T knowledge to keep the different frequencies and the areas to which they belonged in his head. The principal exception to this was the U-boats, which used a complicated W/T programme, but from our point of view identification of services was made easy by the use of an independent set of serial numbers for each service. We should have had some difficulty also with the Mediterranean area if the intercept stations had not given a group letter to each W/T service and appended it to the frequency when teleprinting the traffic.
Except for a short period in the early days of Bonito, fixed call signs were always used, though it was unfortunately by no means always possible to tell from whom a message originated. 'Addressee' call signs—a very great help to the cryptographer— were little used except in the Mediterranean where, if we were fortunate, it might be possible to tell both the originator and the destination of a message. Like the Mediterranean keys, Bonito gave a lot away by its call signs, but other keys stuck to the old procedure. The only exception of any note to this rule was the emergency W/T links which replaced teleprinter communications if the latter broke down.
From early days Scarborough was our chief intercept station and was responsible for picking up most of the traffic. Other stations were brought in if interception at Scarborough was unsatisfactory, as was often the case in the Mediterranean and North Norwegian areas. For a considerable time North Norwegian traffic was being sent back from Murmansk, while the W/T station at Alexandria played an important part in covering many Mediterranean frequencies. The principal disadvantage of traffic from distant intercept stations was the length of time it took to reach us. Traffic came from Alexandria by cable and an average delay of 6 hours between time of interception and time of receipt at B/P was considered good.
Unlike Hut 6 we never controlled the disposition of the various receiving sets at our disposal but made our requests, which were considerable, through Naval Section. As I do not ever recollect an urgent request having been refused, this system worked very satisfactorily from our point of view.
The reason for our numerous demands for double, treble, and even quadruple banking of certain frequencies was that, for cryptographic reasons which I will explain later, it was absolutely essential to have a 100% accurate text of any message that might be used for crib purposes. In our experience it was most unwise to believe in the accuracy of single text, even if it was transmitted with Q.S.A. 5, and so we asked automatically for double banking on frequencies likely to be used for cribbing. Especially on the Mediterranean keys and Bonito, interception was extremely unreliable and quadruple banking on crib frequencies was often necessary. On Bonito the assistance of R.S.S., who did not normally work on Naval traffic, was enlisted, and for some Bounce traffic originating from weak transmitters in Northern Italy, we relied on R.A.F. stations in Italy.
Requests for double banking for cryptographic reasons became extremely numerous by the end of the war. In 1941, with only one key, special cover on a small group of frequencies was sufficient, but by late 1944 we were normally breaking some 9 or 10 keys, for each of which special cover on some frequencies was required. On the whole the policy was to ask for double banking if it was at all likely that it might be useful and accept the fact that a certain amount of unnecessary work was being done by intercept stations. There were at any rate sufficient hazards involved in breaking keys and it was felt that it would be foolhardy policy to take risks in the matter of interception when these could be obviated by double banking.
Most traffic arrived from the intercept stations by teleprinter, being duplicated by carbon. Retransmissions, dupes as we called them, were also teleprinted in full after 1942; before this, German preamble and differing groups only of dupes had been teleprinted, but we found that we were unable to rely on the intercept stations to notice all differences.
Shortage of teleprinters was a perennial problem, as the traffic constantly increased while those responsible for teleprinters persisted in believing that it would decrease. The effect of this shortage was that traffic got delayed at Scarborough for considerable lengths of time before teleprinting and was not in fact cleared until there was a lull in the traffic. It was usually true that there were sufficient teleprinters to cope over a period of 24 hours with the traffic sent in that period, but they were quite insufficient for the rush hours on the evening and early night shift. In the spring of 1944 the teleprinter situation was reviewed and considerably improved, in anticipation of the Second Front and possible heavy increases of traffic. Experiments were carried out with a priority teleprinting system for certain frequencies but the list tended to be so large (having to cover cryptographic and intelligence needs) and so fluid that it afforded no more than a theoretical solution to the problem. It was our experience that it was possible to 'rush' very small groups of traffic at very high speed—some very remarkable results were achieved with the frequency which carried Flying Bomb information—but that rushing a large quantity was comparatively ineffective.
As a result of the increased number of teleprinters, the average time elapsing between interception and teleprinting was reduced to about 30 minutes, which was thought to be satisfactory. For the opening of the Second Front a small W/T station was opened at B/P, most appropriately in the old Hut 8. This covered certain frequencies of special operational urgency or crib importance and produced very satisfactory results. A new record was established when a signal reached Admiralty in translation 12 minutes after being intercepted here. As the excitement over the success of the Second Front died down and the sense of urgency disappeared, the time lag became somewhat worse, but the situation remained under control, with one or two brief exceptions, even during the final peak period in March 1945.
As was to be expected, stations other than Scarborough which had less interest in Naval traffic and less facilities for teleprinting were appreciably slower in passing us the traffic; a time check late in 1944 revealed an average delay of 103 minutes at Flowerdown. It should perhaps also be gratefully recorded that Scarborough's standard of teleprinting, accuracy, and neatness remained right through the war a model which other stations were far from rivalling.
In early days traffic was teleprinted to the Main Building whence it was carried every half hour, later every = hour, to the old Hut 8 Registration Room. This was inevitably a slow process, but it mattered comparatively little as in those days keys were not often being read currently. The move into Block D (February 1943) and the introduction of conveyor belts greatly improved the situation and traffic now came to the teleprinter room a few yards away whence it was conveyed to the Registration Room by belt.
Once a message had arrived in Hut 8, a considerable number of things had to be done before it could arrive decoded in Naval Section. The Registration Room had to sort the traffic—partly by frequency and partly with the help of the K book—into the various keys and, if the key in question was current, the message was then handed to the Decoding Room. For a long time decoded traffic was carried to Naval Section, a considerable walk either from the old or the new Hut 8; this wasteful method of conveyance was only superseded when the pneumatic tube system was introduced some time after we arrived in Block D.
These tubes were violently opposed on various grounds at first but, when the permanent two-way tube system had been introduced, they carried a terrific load and much toing and froing between Naval Section and ourselves was cut out. There were undoubted disadvantages in having to screw the messages up to put them into the containers but there can be no doubt that they saved us both time and trouble and that messages reached Naval Section much more quickly than before. A conveyor belt would certainly have been more satisfactory but, given the distance separating us from Naval Section (A15), was presumably out of the question.
The time taken for decodeable traffic to pass through the Section varied appreciably with the degree of excitement caused by the war news. In early Second Front days some messages were arriving in Naval Section 20 minutes after being intercepted, but speeds of this sort could not be kept up indefinitely. The introduction of time stamping with the help of Stromberg time-clocks enabled us to make a regular check of the time it took for traffic to pass through the Hut and we were normally able to keep the average in the region of half an hour. This reflects, I think, great credit on all concerned as the work was tiring and, especially in the Decoding Room, noisy, and at peak periods everyone had to work very fast indeed. In the week ending March 16th 1945, the record total of 19,902 messages were decoded, a remarkable feat for an average of perhaps 10 typists per shift.
This introduction has, I hope, supplied the necessary background on how the machine works and on how the traffic was received and dealt with. We can now turn to a more interesting subject and examine the history of the breaking of Enigma from the earliest days.
#### Early work on Enigma
Nearly all the early work on German Naval Enigma was done by Polish cryptographers, who handed over the details of their very considerable achievements just before the outbreak of war. Most of the information I have collected about prewar days comes from them through Turing, who joined G.C.C.S. in 1939 and began to interest himself in Naval cyphers, which so far had received scant attention.
The Heimsoeth & Rinke machine which was in use throughout the war and which I described [earlier] was not the first machine to be used by the German Navy. In the 1920s, the so-called O Bar machine had been in use. This had 3 wheels and no Stecker, and the curious characteristic of 29 keys—the modified vowels O, U, and A being included. Of these 29 symbols, X always encyphered as X without the current entering the machine, and the remaining 28 letters were encyphered in the normal way. The tyres of the wheels necessarily had 28 letters printed on them and it was decided that the letter which had been omitted was the modified O: hence the name of the machine, which was broken by the Poles and the traffic read.
The O Bar machine went out of force for fleet units in 1931, when the present machine was introduced, and gradually disappeared altogether. The new machine had originally been sold commercially by the Swedes: as sold by them it had no Stecker and it was they who recommended the boxing indicator system which enabled so many Navy cyphers to be read.
When the German Navy first started to use the machine there were only 3 wheels in existence instead of the later 8 and only 6 Stecker were used. The reflector in force was reflector A and boxing or throw-on indicators were used....
Having obtained photographs of the keys for 3 months, during which period the wheel order obligingly remained unchanged, the Poles broke the wiring of wheels 1, 2, and 3 by a 'Saga', a long and complicated hand process which I shall not attempt to explain. Having obtained all the details of the machine, they were able to read the traffic more or less currently with the help of the indicating system and catalogues of 'box shapes'. These catalogues listed positions of the machine which would satisfy certain conditions which were implied by the indicator groups, and from them the machine set-up for the day could be worked out.
On May 1st 1937 a new indicating system was introduced. The first 2 groups of the message were repeated at the end, thus showing clearly that they formed the indicator, but it was immediately apparent that throwing-on had been given up. This was a sad blow, but the Poles succeeded in breaking May 8th and they discovered the Grundstellung with the help of messages to and from a torpedo boat with call-sign 'AFÄ', which had not got the instructions for the new indicating system. On breaking May 8th, the Poles discovered that it had the same wheel order as April 30th, and the intervening days were soon broken. AFÄ's lack of instructions and the continuation of the wheel order are typical examples of good fortune such as we have often experienced, and also of the German failure to appreciate that for a cypher innovation to be successful it must be absolutely complete.
May 8th and the preceding days could not, of course, be broken with the catalogue of box shapes, as the indicating system had changed, and for them the Poles devised a new method which is of considerable interest. Their account of this system, written in stilted German, still exists and makes amusing reading for anyone who has dealt with machines. The process was fundamentally a form of cribbing, the earliest known form. On the basis of external evidence, one message was assumed to be a continuation (a FORT) of another—apparently identification was easy and FORTS numerous in those days. The second message was then assume to start FORT followed by the time of origin of the first message, repeated twice between Ys. German security must have been non-existent in those days, as these cribs appear to have been good and the Poles quote an example which, by its pronounceability, gave its name to the method of attack they had evolved—FORTYWEEPYYWEEPY = continuation of message 2330, numerals at this time being read off the top row of the keyboard and inserted between Ys. I will not attempt to explain the details of the method: it involved a series of assumptions to the effect that certain pairs of letters were both self-steckered. If the assumption was correct, the method worked. At this time, it will be remembered, 14 of the 26 letters were still unsteckered so that the assumption was not a very rash one and also the number of wheel orders was very small.
Even when they had found the Grundstellung with the help of the AFÄ messages, the Poles still could not read the traffic as they did not know how the indicating system worked. They set to work, therefore, to break individual messages on cribs—largely of the FORTY WEEPY TYPE—not a very difficult process when Stecker and wheels are known. By this method they broke out about 15 messages a day and came to the conclusion that the indicating system involved a bigram substitution, but they got little further than this.
All witnesses agree that Naval Enigma was generally considered in 1939 to be unbreakable; indeed pessimism about cryptographic prospects in all fields appears to have been fairly prevalent. This attitude is constantly referred to in such letters of the period as still survive; Mr. Birch records that he was told when war broke out that 'all German codes were unbreakable. I was told it wasn't worth while putting pundits onto them' (letter to Commander Travis, August 1940) and, writing to Commander Denniston in December of the same year, he expresses the view that 'Defeatism at the beginning of the war, to my mind, played a large part in delaying the breaking of codes.' When Turing joined the organization in 1939 no work was being done on Naval Enigma and he himself became interested in it 'because no one else was doing anything about it and I could have it to myself'. Machine cryptographers were on the whole working on the Army and Air Force cyphers with which considerable success had been obtained.
Turing started work where the Poles had given up; he set out to discover from the traffic of May 1937 how the new indicating system worked. 'Prof's Book', the write-up he made in 1940 of the work he had done and of the theory of Banburismus, describes the successful conclusion of this work.
#### Excerpt from Turing's **_Treatise on the Enigma_**
[T]he Poles found the keys for the 8th of May 1937, and as they found that the wheel order and the turnovers were the same as for the end of April they rightly assumed that the wheel order and Ringstellung had remained the same during the end of April and the beginning of May. This made it easier for them to find the keys for other days at the beginning of May and they actually found the Stecker for the 2nd, 3rd, 4th, 5th, and 8th, and read about 100 messages. The indicators and window positions of four (selected) messages for the 5th were
The repetition of the EW combined with the repetition of V suggests that the fifth and sixth letters describe the third letter of the window position, and similarly one is led to believe that the first two letters of the indicator represent the first letter of the window position, and that the third and fourth represent the second. Presumably this effect is somehow produced by means of a table of bigramme equivalents of letters, but it cannot be done simply by replacing the letters of the window position with one of their bigramme equivalents, and then putting in a dummy bigramme, for in this case the window position corresponding to JMFE FEVC would have to be say MYY instead of MYK. Probably some encipherment is involved somewhere. The two most natural alternatives are i) The letters of the window position are replaced by some bigramme equivalents and then the whole enciphered at some 'Grundstellung', or ii) The window position is enciphered at the Grundstellung, and the resulting letters replaced by bigramme equivalents. The second of these alternatives was made far more probable by the following indicators occurring on the 2nd May
With this second alternative we can deduce from the first two indicators that the bigrammes EX and XX have the same value, and this is confirmed from the second and third, where XX and EX occur in the second position instead of the first.
It so happened that the change of indicating system had not been very well made, and a certain torpedo boat, with the call sign AFÄ, had not been provided with the bigramme tables. This boat sent a message in another cipher explaining this on the 1st May, and it was arranged that traffic with AFÄ was to take [place] according to the old system until May 4, when the bigramme tables would be supplied. Sufficient traffic passed on May 2, 3 to and from AFÄ for the Grundstellung used to be found, the Stecker having already been found from the FORTYWEEPY messages. It was natural to assume that the Grundstellung used by AFÄ was the Grundstellung to be used with the correct method of indication, and as soon as we noticed the two indicators mentioned above we tried this out and found it to be the case.
There actually turned out to be some more complications. There were two Grundstellungen at least instead of one. One of them was called the Allgemeine and the other the Offiziere Grundstellung. This made it extremely difficult to find either Grundstellung. The Poles pointed out another possibility, viz that the trigrammes were still probably not chosen at random. They suggested that probably the window positions enciphered at the Grundstellung, rather than the window positions themselves, were taken off the restricted list.
In Nov. 1939 a prisoner told us that the German Navy had now given up writing numbers with Y... YY... Y and that the digits of the numbers were spelt out in full. When we heard this we examined the messages toward the end of 1937 which were expected to be continuations and wrote the expected beginnings under them. The proportion of 'crashes' i.e. of letters apparently left unaltered by encipherment, then shews how nearly correct our guesses were. Assuming that the change mentioned by the prisoner had already taken place we found that about 70% of these cribs must have been right.
[ _End of Excerpt_ ]
[Turing's] theory was further confirmed when the Grundstellung which AFÄ had been using was discovered to encypher these trigrams in such a way that the Vs and Us all came out as the same letter. Turing had in fact solved the essential part of the indicator problem and that same night he conceived the idea of Banburismus 'though I was not sure that it would work in practice, and was not in fact sure until some days had actually broken'.
As Banburismus was the fundamental process which Hut 8 performed for the next 2 or 3 years, it is essential to understand roughly the principle on which it works.
#### Banburismus
Banburismus is not possible unless you have the bigram tables.
The idea behind Banburismus is based on the fact that if two rows of letters of the alphabet, selected at random, are placed on top of each other, the repeat rate between them will be 1 in 26, while if two stretches of German Naval plain language are compared in the same way the repeat rate will be 1 in 17. Cypher texts of Enigma signals are in effect a selection of random letters and if compared in this way the repeat rate will be 1 in 26 but if, by any chance, both cypher texts were encyphered at the same position of the machine and [are] then written level under each other, the repeat rate will be 1 in 17—because, wherever there was a plain language repeat, there will be a cypher repeat also. Two messages thus aligned are said to be set _in depth_ : their correct relative position has been found. If by any chance the two messages have identical content for 4 or 6 or 8 or more letters then the cypher texts will be the same for the number of letters concerned—such a coincidence between cypher texts is known as a 'fit'. Banburismus aims first of all at setting messages in depth with the help of fits and of a repeat rate much higher than the random expectation.
Long before the day is broken, a certain amount can be done to the indicators of the messages. The bigram substitution can be performed and the trigrams obtained: these trigrams, when encyphered or 'transposed' at the Grundstellung, will give us the starting positions of the messages. Once the day has been broken, the Grundstellung alphabets, i.e. the effect of encyphering each of the 26 letters of the alphabet at positions one, two, and three of the trigrams, can be produced. The alphabets will look something like this:
The aim of Banburismus is to obtain, with the help of the trigrams and fits between messages, alphabets 2 and 3, the middle and right hand wheel alphabets.
The chance a priori of 2 messages with completely different trigrams ZLE and OUX being correctly set in depth is 1 in 17,000, but if the trigrams are NPE and NLO, the factor against them being in depth is only 1 in 676 as, although we do not know the transposed value of the trigrams, N will in each case transpose to the same letter, and therefore both messages were encyphered with the left hand wheel in the same position. These messages are said to be 'at 676'. Messages with such trigrams as PDP and PDB are said to be 'at 26': they are known to have had starting positions close together on the machine. On our alphabets:
PDP = KWH
PDB = KWG
therefore PDB started one place earlier than PDP. This is expressed as B + 1 = P: in the right hand wheel alphabet, P will be seen one place ahead of B.
The first stage in attacking a day by Banburismus is to discover the fits. This was done largely by Freeborn who sorted all messages against all other messages and listed fits of 4 letters or more. At the same time messages were punched by hand onto Banburies, long strips of paper with alphabets printed vertically, so that any 2 messages could be compared together and the number of repeats be recorded by counting the number of holes showing through both Banburies. A scoring system by 'decibans' recorded the value of fits. All messages at 26 were compared with 25 positions to the right and left of the level position and the scores recorded.
The completion of a Banburismus can best be explained by a simple example, the type of Banburismus that would take the expert 10 minutes rather than many hours or even days of work. This is our list of fits. Note 3.7 = 3 alphabets + 7 letters.
The fits of better than even chance concern the letters of the right hand wheel.
C—E
F—Q
C—L
X—C
S—Y
R—X
One chain X R C L E can be expressed in this form:
R.... X.... C. E.......... L
We know that those letters must appear in those relative positions in the right hand wheel alphabet. We now 'scritch' the 26 possible positions (R under A, then under B etc.) and cross out those which imply contradictions: the first position, with the reciprocals (R = A, etc) written in and ringed, looks like this.
Here there is the contradiction that L appears under X, but X which is on our chain also is under F, giving two values for one letter. Similar reasons reject most other positions. Those left in are:
Of these, 2 contradicts the S + 3 = Y fit, 1 and 9 contradict the good tetra I + 3 = D: they are, therefore, unlikely to be right. 6, however, is extremely interesting.
We scritched L R X C E and got the position:
In this alphabet F + 7 = Q and we have a good fit at this distance which was not used in our scritching. We have in fact 'picked up' this extra fit and thereby obtained a considerable factor in favour of this alphabet. We now fit in S in such a position that S + 3 = Y in accordance with our hexagram:
This makes S equal U and we notice at once that we pick up the tetragram D + 16 = U, not a very good tetra but valuable as a further contribution.
We now have to see for which wheels this alphabet is valid. The turnovers on the wheels are in the positions marked on the alphabet above. From the fit RWC + 13 = RWL we know there is no turnover between C and L; if there were a turnover, the position of the middle wheel would have changed, but both trigrams have W in the middle. This fit knocks out wheels 4, 6, 7, and 8. BBC + 2 = BBE knocks out wheel 2. Further, we have the fit PNX + 5 = PIC: as the two trigrams have different letters in the middle, there must be a turnover between X and C. The wheel must therefore be 5, 6, 7, or 8; 6, 7, and 8 are already denied by C + 13 = L, so the wheel is 5.
The next stage is to count up the score for the alphabet, assuming it to be correct. For instance, if there are two messages BDL and BDS, the score for BDL + 4 = BDS will be recorded and should be better than random if the alphabet is right. The final score should be a handsome plus total. An attempt, usually not difficult, is then made to complete the alphabet with the help of any further fits or good scores which may exist. It will be noticed that the fit QQG + 2.7 = QDU is contradicted: it had, however, only an even chance of being right and we do not let this worry us unduly.
A similar process finds the alphabet for the middle wheel and sufficient material is then available to break the day on the bombe. The number of wheel orders which have to be run will probably have been reduced from 276 to something between 3 and 90.
This example shows clearly the fallacy of the system of having all the wheels turning over in different places. It was this characteristic alone which made it possible to distinguish the wheels by Banburismus and reduce the number of wheel orders to be tried. Wheels 6, 7, and 8 were indistinguishable from one another and a great nuisance to the Banburist.
Like depth cribbing, which was closely allied to it and which will be described in due course, Banburismus was a delightful intellectual game. It was eventually killed in 1943 by the rapidly increasing number of bombes, which made it unnecessary to spend much time and labour in reducing the number of wheel orders to be run: it was simpler and quicker to run all wheel orders.
#### January 1940 to July 1941
Turing's solution of the indicating system came at the end of 1939 but it was well over a year before Banburismus was established as a practical proposition and used as a successful method of attack. The reason for this was primarily a lack of bigram tables.
The next interesting development was the interrogation of Funkmaat Meyer, who revealed valuable information about Short Signals and also the fact that the German Navy now spelt out numerals in full instead of using the top row of the keyboard. This encouraged Turing to look again at the FORTY WEEPY cribs which in 1937 had begun inexplicably to crash, and he came to the conclusion that the cribs remained fundamentally correct provided that the numerals were spelt out.
In early 1940, now joined by Twinn and 2 girls, he started an attack on November 1938 by the FORTY WEEPY method and the new-style crib. The reasons for choosing a period so long ago were various but were primarily based on the knowledge that modern keys were more complicated and would require more work. Two new wheels (4 and 5) had been introduced in December 1938, and from the beginning of the war they were unable to trace the FORTY WEEPY messages owing to call-signs being no longer used.
After about a fortnight's work they broke November 28th, and 4 further days were broken on the same wheel order. Only the Spanish Waters came out; the rest of the traffic was on a different key. There were still only 6 Stecker and there was a powerful and extremely helpful rule by which a letter was never steckered 2 days running: if continuity was preserved, 12 self-stecker were known in advance. No Grundstellungs and no bigrams were broken, messages being broken individually or on the EINS catalogue which was invented at this time and was to play an important part in the exploitation of Enigma.
EINS was the commonest tetragram in German Naval traffic: something in the region of 90% of the genuine messages contained at least one EINS. An EINS catalogue consisted of the results of encyphering EINS at all the 17,000 positions of the machine on the keys of the day in question. These 17,000 tetragrams were then compared with the messages of the day for repeats. When a repeat was found, it meant that [at] a certain position of the machine the messages could be made to say EINS, and further letters were then decoded to see if the answer was a genuine one. If it was, the starting position of the message was known and it could be decoded. In fact about one answer in 4 was right, so that messages were broken fairly rapidly. In later days, the whole process of preparation and comparison was done rapidly and efficiently by Hollerith machinery, but at first slow and laborious hand methods were used.
The plan was to read as many messages as possible, to gain some knowledge of cribs, and then to make rapid progress with the help of the Stecker rule. 'There seemed', says Turing in his book, 'to be some doubt as to the feasibility of this plan', and in fact it proved over-optimistic. Work was fizzling out when Norway was invaded and the cryptographic forces of Hut 8 were transferred en bloc to assist with Army and Air Force cyphers.
By the time work on Naval could be started again, the 'Narvik Pinch' of April 19th had taken place. This pinch revealed the precise form of the indicating system, supplied the Stecker and Grundstellung for April 23rd and 24th (though the scrap of paper on which they were written was for some time ignored) and the operators' log, which gave long letter for letter cribs for the 25th and 26th. Wheels 6 and 7 had been introduced by this time and were already in our possession.
In all 6 days were broken, April 22nd to 27th. The 23rd and 24th presented no difficulty and the days paired with them, the 22nd and 25th, were also broken (by this time the wheel order was only lasting 2 days and there were ten Stecker). The 26th gave much more trouble, being on a new wheel order with unknown Stecker. At first a hand method, the Stecker Knock Out, was unsuccessfully tried and then the Bombe, which had arrived in April and which will be discussed later, was put onto the problem. After about a fortnight of failure, due largely to running unsuitable menus, the day was broken on a freak menu, to be known later as a Wylie menu and tried unsuccessfully on Shark of February 28th 1943. The paired day, the 27th, was also broken.
All hands now turned to EINSing out messages and building up the bigram tables. Provided that the Grundstellung was known, the starting position of the message when broken by EINSing could be transposed and the trigram discovered. This then gave a value for 3 bigrams. A message with 2 of the 3 operative bigrams known could be 'twiddled' out: the 2 known bigrams fixed the positions of 2 of the wheels, and only 26 positions for the remaining wheel had to be tried. This was quickly done and a further bigram was added to the store. April 27th was on the same bigram table as the 24th and this table came near enough to completion to make Banburismus feasible on another day using the same table. May 8th was identified as using this table and a Banburismus was started, but no results were obtained for many months. Turing wrongly deduced that June was using different bigram tables.
The next 6 months produced depressingly few results. Such Banburismus as was tried was unsuccessful, and there was little bombe time for running cribs. Such cribs as there were were supplied by Naval Section and failed to come out; 'Hinsley's certain cribs' became a standing joke. After consulting many people I have come to the conclusion that it is impossible to get an impartial and moderately accurate picture of cribbing attempts at this period: Hut 8 and Naval Section each remain convinced that cribbing failures were due to the other section's shortcomings.
1940 was clearly a very trying period for those outside Hut 8 whose hopes had been raised by the April Pinch and the results obtained from it. On August 21st Mr. Birch wrote to Commander Travis:
I'm worried about Naval Enigma. I've been worried for a long time, but haven't liked to say as much...Turing and Twinn are like people waiting for a miracle, without believing in miracles...
I'm not concerned with the cryptographic problem of Enigma. Pinches are beyond my control, but the cribs are ours. We supply them, we know the degree of reliability, the alternative letterings, etc. and I am confident if they were tried out systematically, they would work.
Turing and Twinn are brilliant, but like many brilliant people, they are not practical. They are untidy, they lose things, they can't copy out right, and they dither between theory and cribbing. Nor have they the determination of practical men...
Of the cribs we supply, some are tried out partially, some not at all, and one, at least, was copied out wrong before being put on the machine...
Sometimes we produce a crib of 90% certainty. Turing and Twinn insist on adding another word of less than 50% probability, because that reduces the number of answers and makes the result quicker. Quicker, my foot→ It hasn't produced any result at all so far. The 'slower' method might have won the war by now.
Presumably the number of answers possible on a given crib is mathematically ascertainable. Suppose the one we back 90% has 100,000 possible answers: is that a superhuman labour?...
When a crib, with or without unauthorised and very doubtful additions, has been tried once unsuccessfully we are not usually consulted as to what should be tried next, but, generally speaking, instead of exhausting the possibilities of the best crib, a new one is pottered with under similar handicaps. No crib has been tried systematically and failed; and a few have been tried partially and the partial trial has been unsuccessful...
Turing has stated categorically that with 10 machines [bombes] he could be sure of breaking Enigma and keeping it broken. Well can't we have 10 machines?...
At one end, we're responsible for cribs; at the other end we're responsible to Admiralty. We know the cribs and the odds on them and we believe in them and it's horrible to have no hold, no say, no nothing, on the use that is made of them or the way they are worked...
This letter has great value as a reflection of the relationship of Hut 8 with the outside world and of the cryptographic organization for breaking Enigma; the fact that the letter misinterprets the true position tends to show Mr. Birch at a disadvantage, but it would be most unfair to look at it in this light, as anyone must know who has read the early Naval Section documents and has seen the efforts which Mr. Birch extended to further any work which might in any way assist Hut 8.
First of all the letter demonstrates clearly Turing's almost total inability to make himself understood. Nearly all Mr. Birch's suggestions, as is immediately obvious to anyone with actual experience of Hut 8 work, are impossible and are simply the result of not understanding the problem—his 100,000 answers (had the bombe been able to run the job, which it couldn't have done) would have taken 5 men about 8 months to test. Such problems as this and the disadvantages of the other suggestions should clearly have been explained but Turing was a lamentable explainer and, as Mr. Birch rightly says, not a good practical man: it was for these reasons that he left Hut 8 when the research work was done and the back of the problem broken. The lack of satisfactory liaison was a great disadvantage in early days, but was fortunately most completely overcome later; in a letter to myself of May 16th, 1945, Mr. Birch speaks truly of 'two independent entities so closely, continuously, and cordially united as our two Sections'.
The second point of interest in the letter is the assumption that cribbing and all to do with it was the business of Naval Section and something quite separate from the mathematical work, classed as cryptography and belonging to Hut 8. This concept prevailed until 1941 when Hut 8 set up a Crib Room of its own. No one now would maintain that it would be feasible to separate cribbing from cryptography in this way: to be a good cribster it was essential to understand fully the working of the machine and the problems of Banburismus, bombe management, etc. On the other hand, it remained highly valuable to us that Naval Section were always crib conscious and would send over suggestions for us to explore.
The view frequently expressed by Hut 8 was that a successful pinch of a month's keys with all appurtenances (such as bigram tables) offered the best chance of our being able to get into a position where regular breaking would be possible, as in the course of that month crib records and modern statistics could be built up. Naval Section papers of the barren days of the autumn of 1940 discuss various plans for obtaining keys in this way.
On September 7th Mr. Birch distributed the following document to his subsections, requesting their comments:
When talking to Lt. Cd. Fleming the other day, Mr. Knox put forward the following suggestion: The Enigma Key for one day might be obtained by asking for it in a bogus signal. Lt. Cdr. Fleming suggested that the possibilities should be examined and something got ready and kept ready for use in emergency. Four groups of questions need answering:
1. In the light of our knowledge of German codes and cyphers, W/T routine, and coding [and] cyphering instructions, what signals could be made for the purpose,
(1) in what code,
(2) on what frequency,
(3) at what hour(s),
(4) from what geographical position accessible to us?
2. Of the various alternative possibilities, in what circumstances would which be most likely to fox the enemy?
This scheme found little favour and was soon rejected as impracticable but a week later Mr. Birch produced his own plan in a letter to D.N.I.:
Operation Ruthless:
I suggest we obtain the loot by the following means
1. Obtain from Air Ministry an air-worthy German bomber (they have some).
2. Pick a tough crew of five, including a pilot, W/T operator and word-perfect German speaker. Dress them in German Air Force uniform, add blood and bandages to suit.
3. Crash plane in Channel after making S.O.S. to rescue service in P/L.
4. Once aboard rescue boat, shoot German crew, dump overboard, bring rescue boat to English port.
In order to increase the chances of capturing an R. or M. with its richer booty, the crash might be staged in mid-Channel. The Germans would presumably employ one of this type for the longer and more hazardous journey.
This somewhat ungentlemanly scheme was never put into practice although detailed plans for it were made and it is discussed several times in Naval Section papers. In fact, the only valuable acquisition during this period was the finding of wheel 8 in August 1940, the last new wheel to be introduced during the war.
The next event in the cryptographic world was the breaking in November of May 8th, known to history as Foss's day. Foss had joined temporarily to assist in exploiting the Banburismus idea, and after a labour of many months broke the first day on Banburismus. The moral effect of this triumph was considerable and about a fortnight later another Banburismus, April 14th, was broken at what was considered lightning speed. June 26th was also broken (June having by now been established to be on the same bigram tables) and contained the information that new bigram tables would come into force on July 1st, so Banburismus after that date was out of the question.
A second sensational event was the breaking of April 28th on a crib, the first all wheel order crib success. Hut 8 at this time contained no linguists and no cribster by profession and the crib was produced as a result of the labour of two mathematicians, who take great delight in recalling that the correct form of the crib had been rejected by the rival Naval Section cribster.
This last break was obtained in February 1941 and was followed shortly afterwards by the first Lofoten pinch which is one of the landmarks in the history of the Section. This pinch gave us the complete keys for February—but no bigram tables or K book.
The immediate problem was to build up the bigram tables by EINSing and twiddling, the methods for which had now been much improved. With a whole month's traffic to deal with, there was a vast amount of work to be done and the staff position was acute. Rapid expansion and training of new people had to take place and greatly slowed up the work, but by late in March the bigram tables were more or less complete.
Much of the theory of the Banburismus scoring system had been worked out at the end of 1940 and now statistics were brought up to date and satisfactory charts produced. Much work was done on the identification and utilization of dummy messages which at this time formed about half the traffic. It was most important to know and allow for the chance of a message being dummy. The end of a dummy message consisted of a string of consonants and yielded a totally different repeat rate. If dummy was not allowed for, Banburismus could become difficult and even insoluble.
In March also shift work was started and Hut 8 was manned 24 hours a day for the rest of the war. In April, teleprinting of traffic from Scarborough was begun and the Registration Room was started, all traffic being registered currently. Banburismus was started on some March days and there was a rather depressing period of inexplicable failure before the first break and then the system began to get under way. Sooner or later a large part of April and May were broken. There can be no doubt that at this stage the battle was won and the problem was simply one of perfecting methods, of gaining experience, and of obtaining and above all of training staff. These last stages were made much simpler by the pinch of June and July keys.
These last two pinches were a great stroke of good fortune, for the bigram tables changed on June 15th and had once again to be reconstructed. The methods of reconstruction were, however, by now efficient and between June 15th and the end of July this task was easily accomplished. Had we not had the keys, all days subsequent to June would have had to be broken on all wheel order cribs and the messages subsequently broken by EINSing. With the lack of bombes and comparatively crude knowledge of cribs which existed at that stage, this would have been a slow process, and the beginning of what I have called the operational period of Hut 8 would have been delayed by perhaps 2 or 3 months.
#### Bombes
Throughout these early [sections] I have avoided as far as possible all mention of bombes. Bombes are a complicated subject and their workings are to a large extent incomprehensible to the layman, but without them Hut 8 and Hut 6 could not have existed and it is essential to attempt to describe briefly the part they played.
The bombe was so called because of the ticking noise it made, supposedly similar to that made by an infernal machine regulated by a clock. From one side, a bombe appears to consist of 9 rows of revolving drums; from the other, of coils of coloured wire, reminiscent of a Fairisle sweater.
Put briefly, the function of a bombe was to take one wheel order at a time and discover which of the 17,000 possible positions of the machine combined with which of the half billion possible Stecker combinations would satisfy the conditions of the problem presented to it. This problem was called a menu and was in fact a crib in diagram form. If the crib and cypher text were
the bombe would be asked to find a position on the machine where V would encypher as E, followed at the next position by V encyphering as T, etc. To perform this function for one wheel order the bombe would take about 20 minutes. The wheel order would then be changed and the process repeated.
The bombe was a highly complicated electrical apparatus, involving some 10 miles of wire and about 1 million soldered connections. Its intricate and delicate apparatus had to be kept in perfect condition or the right answer was likely to be missed. An embryonic bombe was evolved by the Poles and could be used on the comparatively simple pre-war Enigma problems. The invention of the bombe as we have known it was largely the work of Turing, Welchman, and, on the technical side, Keen.
Unfortunately, the bombe was an expensive apparatus and it was far from certain that it would work or, even if the bombe itself worked, that it would enable us to break Enigma. Its original production, and above all the acceptance of a scheme for large scale production, was the subject of long and bitter battles. Hut 8, and, of course, Hut 6, owe very much to Commander Travis, and to a lesser extent to Mr. Birch, for the energy and courage with which they sponsored its production.
The first bombe arrived in April 1940. In August, the first bombe to incorporate the vital development of the diagonal board arrived.
On the 21st of December, 1940, Mr. Birch wrote:
The chances of reading current Enigma depend ultimately on the number of bombes available. The pundits promise that given 35 bombes they guarantee to break Enigma continuously at an average delay of 48 hours.
At present they get the part time use of one machine. The reason that they don't get more is that there are only two bombes available and that, owing to increased complications of Air Enigma, Hut 6 requires the use of both machines. It is true that more bombes are on their way, up to a limit of 12, but the situation may well be as bad when they have all arrived, owing to the introduction to Air Enigma of further complications and owing to the further success with other Enigma colours, Air or Army.
The long and the short of it is Navy is not getting fair does. Nor is it likely to.
It has been argued that a large number of bombes would cost a lot of money, a lot of skilled labour to make and a lot of labour to run, as well as more electric power than is at present available here. Well, the issue is a simple one. Tot up the difficulties and balance them against the value to the Nation of being able to read current Enigma.
By August 1941, when Hut 8 really started work on an operational basis, 6 bombes were available. By this time it appears that they were considered to have proved their worth and production went ahead steadily. Some idea of the increasing bombe capacity may be obtained from the following figures of the number of jobs run in each year:
1940 | 273
---|---
1941 | 1,344
1942 | 4,655
1943 | 9,193
1944 | 15,303
The running and maintenance of the machinery was in no respect the responsibility of Huts 6 and 8 but was under the control of Squadron Leader Jones, who had working for him a team of technical experts from the R.A.F. and large numbers of Wrens (about 2,000 in all) to operate the machines. The bombe organisation started in one Hut at B.P. and finished at four Out Stations organized and fed with menus from a central Station at B.P. The final organisation was complex and highly efficient, and we owe much to Squadron Leader Jones and his Section. At all times they gave every possible assistance with our problem and no labour was too much to ask of them: certainly no one in Hut 8 worked for lengths of time comparable to those worked frequently by the Bombe Hut mechanics.
The fact that the German Army, Navy, and Air Force used the same cypher machine had the fortunate result that the bombes could be used by both Hut 6 and ourselves and it was a universally accepted principle that they were to be used in the most profitable way possible, irrespective of the Service or Section concerned. As Hut 6 had more keys to run than ourselves, the bombes were normally left in their hands and we applied for them as required. The relative priority of Hut 6 and Hut 8 keys in their claim for bombe time was decided at a weekly meeting between Hut 6, Hut 8, Naval Section, and Hut 3; in very early days Commander Travis decided what use should be made of the few bombes then available.
To have sufficient material to break a day on the bombe a crib of 30 letters or more was normally needed; when cribs are referred to, the phrase should be taken to mean a guess at the plain text for not less than 30 letters.
The bombe was rather like the traditional German soldier, highly efficient but totally unintelligent; it could spot the perfectly correct answer but would ignore an immensely promising position involving one contradiction. The effect of this was that if one letter of the cypher text had been incorrectly intercepted, the menu would fail although both the crib and the text were elsewhere absolutely correct. This was the [reason for] the extensive double-banking programme, which has already been described.
Cribs were sent to the Bombe Hut in the form of menus with directions as to the wheel orders on which they were to be run. No more was heard of them until the possible positions, known as 'stops', started to come from the bombes as they worked through wheel orders. The strength of menus was calculated with a view to the bombe giving one stop on each wheel order, thus supplying a check that the machine was working correctly. The identification of the right stop, the stop giving the correct Stecker, and the rejection of the wrong ones was done in Hut 8.
In order to get the maximum use from the bombes they had, of course, to be kept fed with menus for 24 hours a day, and the art of bombe management required a certain amount of skill and experience. The plugging up of a new menu was a comparatively complicated and lengthy process, so that it was desirable to give a bombe as long a run on a menu as possible: on the other hand an urgent job would justify plugging up a large number of bombes for only a few runs, because of the importance of saving time. Efficient bombe management was largely a matter of striking the happy medium between speed and economy, of making sure that, with a limited amount of bombe time, everything of importance got run and that, as far as possible, the urgent jobs were run first. Bombe management was interesting because the situation a few hours ahead was to a large extent incalculable: allowance had to be made on the one hand for jobs which one expected to have to run in 12 hours time (and which when the time came did not always materialise) and for the fact that sometimes 2 or 3 jobs would come out in quick succession on one of the first wheel orders to be run, thus releasing large numbers of bombes: because of this possibility it was necessary to keep a reserve of fairly unimportant jobs to fill the gaps.
Pressure on the bombes varied greatly with the immediate cryptographic situation and the period of the month, it being generally true to say that towards the end of the month the number of wheel orders was restricted by wheel order rules. It was, however, extremely rare for us to be unable to keep all the machinery busy even in latter days when there were very large numbers of bombes both here and in America.
#### Cribbing
##### _The Beginning of the Crib Room_
Autumn of 1941 found us at last approaching a position where there was some hope of breaking Naval Enigma with regularity. For the first time we had read, in June and July, a fairly long series of days and the traffic was heavy enough—in the region of 400 [messages] a day—to make Banburismus practicable.
Hut 8 immediately began to increase in numbers so as to be able to staff 3 shifts for an attack on current traffic. Before the end of the year our senior staff numbered 16. Rather curiously, this was the highest total it ever reached. As methods improved, and as we ourselves became quicker and more skilled, we found ever increasing difficulty in keeping busy and by the end of 1942 our numbers were already on the decline. Although the number of keys to be broken and the volume of traffic rose steadily, we reduced ourselves by March 1944 to a staff of 4, with which we were able to keep the situation under control for the rest of the war.
Autumn of 1941 saw the birth of the Crib Room as an independent body from the Banburists, an important date as the Cribsters were to outlive the Banburists by 18 months and cribbing was to become the only means of breaking after the introduction of the 4-wheeled machine.
It is interesting to note here that by this time cribbing was accepted as a natural part of Hut 8 work—we have seen that earlier it was considered separate from 'cryptography' and the function of Naval Section. No one now would dispute that the only possible arrangement was to have the cribbing done by people who understood the whole problem of breaking Enigma, though the more 'crib conscious' people there were in Naval Section and the more suggestions they sent over the better....
Earlier in the year, as we have seen, days had been broken on cribs obtained from the operator's log of the Lofoten pinch, but little had been done in the way of analysing the traffic for routine messages. Cribbing is essentially merely a matter of guessing what a message says and then presenting the result to the bombe in the form of a menu on which the bombe has to find the correct answer. Any fool, as has recently been shown, can find an occasional right crib, though some skill and judgment is required to avoid wasting time on wrong ones, or rather to waste as little time as possible.
Cribs may be divided into 3 basic groups:
1. Depth cribs
2. Straight cribs
3. Re-encodements
As the history of the Hut is from this point to a large extent the history of cribbing, we must digress considerably at this point and study the 3 basic groups with some care. This process will take us far beyond August 1941....
##### _Depth Cribbing_
The concept of depth is very simple to understand. If two operators choose the same trigram they will, after transposing it at the Grundstellung, get the same starting position for their messages. Now suppose that one encyphers
W E T T E R F U E R D I E N A C H T
and the other
M I T M M M D R E I S I E B E N E I N S
It is clear that their cypher texts must have the 3rd, 9th, 12th, and 13th letters in common, as both hit the same letter at the same position of the machine. Let us write them under each other with the encyphered text:
Here we get the repeats—'clicks' we called them—as expected and also 'reciprocals' where the cypher text in one message equals the clear text in the other. Now let us assume that we are cribsters possessing only the cypher text but suspecting that the top message is a weather message which says 'Wetter fuer die Nacht und Morgen'—quite a likely state of affairs as, from time of origin, frequency, call signs, and length, we consider that this message is a plausible candidate for a weather message which occurs every day.
Here we see the state of our knowledge about the lower message after assuming we know the clear text of the upper message. It may well be that, from our knowledge of the traffic on this frequency and of the minesweepers known to be operating, we can guess the text of the lower message. We have now done a depth crib, and one that is certainly right. For each 'click' successfully cribbed we receive a factor of 17 (the language repeat rate) and for each reciprocal one of 26, so we have in our favour a factor of 174 × 262—an astronomical number in 8 figures, which completely lulls any lingering doubts we may have had about the a priori improbability of a given message starting Mit MMM 371... In fact, no experienced cribster would have troubled to do the calculation, but I include it as an example of a method which may very usefully be applied when assessing a crib.
Such a depth crib as that illustrated was rare in the early days of cribbing, largely because we had not on the whole accumulated enough general knowledge or enough experience to have guessed correctly the beginning of the second message. Much of the early depth was 'dummy' depth. We have already noticed that the traffic contained very large quantities of dummies ending in a series of consonants. These began with a few dummy words and then said such things as
HATKEINENSINNVONVONMNOOOBOULOGNE
FUELLFUNKVVVHANSJOTAAERGER
(HJÄ, the call sign of Brest.)
The large number of messages saying VONVON naturally led to hexagram repeats being discovered by Freeborn when he analysed the traffic. One would be presented with something of this sort: two messages, trigrams HBN and HDS, have a hexa repeat if HDS is written out 17 places in front of HBN. The overlapping part of the texts look like this:
This was the simplest form of dummy depth and, as will be realized, the solution of it was extremely easy, and could in fact be found by consultation of a chart listing all known dummy expressions one under the other: all that was necessary was to find two phrases having 2 letters in common the appropriate distance in front of the 'vonvon'. To use this chart was thought by some cribsters to be unsporting and unaesthetic, but it was none the less a very useful document. The value of having cribbed the above messages was not, of course, only that we now possessed a crib which would break the day but also that the hexagram had been proved beyond doubt to be a correct fit—a valuable contribution to the Banburismus.
By the early months of 1942, depth cribbing had become a fairly highly developed art. The amount of dummy traffic was steadily decreasing and much of the cribbing had to be between genuine messages; this, of course, was much more varied, much wider in its scope, and required a far greater knowledge of what messages were likely to say. This knowledge was obtained by extensive reading of traffic and by the keeping of such useful records as lists of ships likely to be addressed on the various frequencies.
The various applications of depth cribbing were also developed and the amount of assistance cribsters were able to give to Banburists steadily increased. I will give one example of the growth of a 'monster depth' such as warmed the cockles of a cribster's heart.
We start by noting that there are 2 messages about which we think we know something and which have trigrams HEX and HEN, that is to say that they started within 26 places of each other on the machine and it is therefore possible that we may be able to set them in depth.
HEX we think says one of 2 things:
VORHERSAGEBEREICHDREITEILEINS
or
WETTERBEREICHDREITEILEINS
HEN we think says
ZUSTANDOSTWAERTIGERKANALXX
We now proceed to examine all the possible relative positions for confirmations or contradictions.
Position HEX = HEN + 1 looks like this:
Now in this position there are clicks between the 7th letter of HEX and the 8th of HEN and the 10th of HEX and 11th of HEN, but in neither case have the cribs for HEX letters in common with the crib for HEN, so the position is impossible. Continuing we get to the position HEX = HEN + 4, which is obviously correct, having 4 confirmations and no contradictions.
We now take message HEK about which we know nothing except that it probably comes from Boulogne and may be a dummy. Here again we examine position by position to see what consequences are implied.
Position HEK + 1 = HEN looks like this:
Both UW and CN are more or less impossible bigrams, so we reject the position. Position HEK + 3 = HEN, however, looks like this:
Which we guess correctly as OHNESINNVVVMNOOOBOULOGNE, the first N being just a stray letter in the dummy words. We are now well under way, and have supplied the Banburists with 2 certain distances: with the distances obtained from the Freeborn catalogue in addition they will probably soon produce the correct alphabets.
The next move is to examine a tetragram fit between HIP and HEX—not a very good tetragram, estimated to have about 1 chance in 3 of being right. With HEX, HEN, and HEK cribbed we can easily prove or disprove this tetra, and great is our delight on finding that it is the cypher text of the word DREI; this almost certainly a correct fit and the matter is proved beyond doubt when we crib the whole beginning of the message. With the amount of crib we now possess, attaching HIJ is not very difficult and we have a solid piece of depth on five messages.
At this stage it is certain that the Banburists will get the alphabet, but we can still assist by confirming and by finding new distances on the middle wheel. If we have two messages HOD and HIJ and know that on the right hand wheel D + 1 = J, we can say that HOD starts in one of 50 positions in a known relationship to HIJ—1 alphabet plus 1 letter, or 2 plus 1 letter, etc. Inasmuch as HOD is long enough, we can try these possibilities also. Although the whole message cannot be cribbed, the position illustrated where I = O + 3 on the middle wheel is clearly right, all the consequences are good letters, and at one place much of EINSNULYYEINSNUL is thrown up. HOS can then be attached.
It would be pointless to pursue this depth further. More might well be added and, once started, the messages might be read for some distance. The delight of depth lay in its great variety—no two depths were ever quite the same—and it were a blasé cryptographer indeed who experienced no thrill at discovering a right position and correctly guessing large chunks of message about the contents of which he at first knew nothing. Needless to say things rarely went as smoothly as in our example, and it was possible to work for a very long time without even getting a piece of depth started.
There were other applications of depth cribbing of which only one need be mentioned here—the slide. The slide is similar to the process by which we started our last example, but in this case we only have a crib for one message—a fairly long and probably a right crib. If SWI is our cribbed message, we examine the consequences of trying SWL for 25 places to either side of it and, although we cannot attempt to crib [the] result, we look to see whether the letters thrown up are good or bad. For this a simple scoring system was used, E being worth something in the region of 57 and Q about 100. On a really long crib the right position usually showed up clearly and a large number of distances were established by this method.
Depth cribbing died with Banburismus in autumn 1943.
##### _Straight Cribs_
I should perhaps have dealt with straight cribs before depth. The theory of straight cribs is simple: it is merely a matter of guessing the contents of a message without the assistance of depth and without the contents having already been passed in another cypher which has been broken.
Finding straight cribs was largely a matter of analysing the traffic. At first, with the traffic fairly small, this was comparatively easy and an organisation was created for writing down any message which looked as if it might occur regularly. Having found a message of a routine type, details about it were recorded. Significant facts were normally frequency and frequencies on which it was retransmitted, call-signs, German time of origin, and length. When a few examples had been written down, it became possible to assess:
(a) whether, given an unbroken day's traffic, it would be possible to identify this particular message;
(b) whether, once identified, it had few enough forms to be used as a crib.
These two factors—identification and forms—were the essential factors in all cribbing. In our experience identification, though often tricky, could usually be cleared up by careful examination of the evidence and there have been comparatively few cribs that have been unusable because they had been unfindable. In very difficult cases we tried to enlist the assistance of R. F. P. but, though help was most readily given, the experiments were never very successful. The German Security Service appears to have considered that retransmitting of messages of one area in another area was dangerous and did something towards stopping curious linkages by recyphering messages and adding dummy at the end before retransmission. This sort of thing was a nuisance to us, but never became sufficiently widespread to cause serious difficulty.
The ideal crib is not shorter than 35 letters and uses the identical wording every day. Such cribs never existed, though occasionally we possessed for a time what seemed to be the crib to end all cribbing. For some 3 months in the summer of 1942, Boulogne sent a weather message which began ZUSTANDOSTWAERTIGERKANAL and, if I remember rightly, it only failed twice during that period. On the whole, a crib that had more than 2 or 3 basic forms was little use except for getting out paired days on known wheel orders or for depth cribbing: most cribs produced 'horrors' from time to time which we classed as 'other forms' and made no attempt to allow for them as a possible form of a crib, but when assessing a crib it was of course necessary to take into consideration the frequency with which 'other forms' were tending to appear.
The crashing property of the machine was an essential element of all cribbing and most especially of straight cribbing. When writing a wrong crib under a portion of cypher text, each letter of crib had 1 chance in 25 of being the same as the letter of cypher text above it and of thus proving by a 'crash' that the crib was wrong. If therefore a crib had 2 good forms, each about 30 letters long, there was an odds on chance that the wrong form would crash out; in this case the remaining form is, of course, left with a heavily odds on—instead of an approximately evens—chance of coming out. If the good forms crashed out and only some rather poor form went in, it proved to be bad policy to believe the poor form to be correct, although on the basis of mathematical calculation it might appear to have a reasonably good chance. It was the ability to assess this type of problem which distinguished the good cribster. It was impossible to become a good cribster until one had got beyond the stage of believing all one's own cribs were right—a very common form of optimism which died hard.
The pleasure of straight cribbing lay in the fact that no crib ever lasted for very long; it was always necessary to be looking for new cribs and to preserve an open mind as to which cribs offered the best chance of breaking a day. A crib which lasted for 2 months was a rarity; most cribs gradually deteriorated and never recovered until eventually we only recorded them every 2 or 3 days. To give up recording cribs because they were bad was a fool's policy; it unquestionably paid to keep a record of anything that might one day assist in breaking. Time and again, when a good crib died, we were thrown back onto a reserve at which we would have turned up our noses a week before, only to find that the reserve was quite good enough to enable us to break regularly.
The perennial mortality of cribs was undoubtedly to a considerable extent the result of the work of the German Security Service to whose work as crib hunters we must in all fairness pay tribute. The information recently received that they kept an expert permanently at work analysing the weaknesses of the machine does on the other hand little credit to their technical ability. The crib chasers gave us a fairly bad time, especially in the areas nearer home where the cyphers were better organised. By March 1945 Dolphin—a large key of some 400 messages a day—had been rendered almost cribless and we might have failed to read some of the last days had we not captured the Hackle keys which kept us supplied with re-encodements.
Cribs in the early days were largely weather messages. A very large amount of weather was sent in Enigma and it was obvious that it was regarded as of first importance. In 1941 weather cribs from the Channel ports were our principal stand-by and WEWA BOULOGNE and WEWA CHERBOURG were trusty friends. One day during the late autumn, the security officers pounced on this habit of announcing internally from whom the weather originated, but first class cribs of a rather shorter variety continued to come regularly at the beginnings and ends of messages. Rather curiously, this habit of signing weather messages at the end never caught on elsewhere and after the death of the Channel weather cribs in spring 1942 we never again had cribs at the ends of messages. This was, on the whole, convenient as a message always finished with a complete 4 letter group (irrespective of the number of letters in the plain text, dummy letters being added at the end), so that an end crib could be written in 4 different places, and one had to be fortunate with the crashing out for a single really good shot to be available.
The reprimand to the Channel weather stations for insecurity in April 1942 is something of a landmark as the Channel cribs never recovered, except for the remarkable run of ZUSTANDOSTWAERTIGERKANAL during the summer of the same year. Henceforth they omitted such lengthy statements as WETTER-ZUSTANDEINSACHTNULNULUHR and satisfied themselves with a terse NANTESBISBIARRITZ buried somewhere in the middle of the message. This habit of burying sign-offs was an almost completely effective anti-crib measure and became more and more widespread as time went on.
Weather cribs in Norway and the Baltic were useful for a long time, and in the Mediterranean for even longer, but there can be little doubt that the security services were weather conscious and one after another cribs of this type disappeared. They were replaced by other cribs of a type which generally required more finding—some were situation reports of a fairly obviously routine nature, others were much more elusive. For instance, when looking through a day's traffic one was not likely to be struck immediately by a message from Alderney to Seekommandant Kanalinseln which said FEUER BRANNTEN WIE BEFOHLEN but it was in fact a daily confirmation that various lights had been shown as ordered and was a very excellent crib.
To spot cribs of this type it was necessary to read through large quantities of traffic, covering perhaps a week or two, and to have a good short-term memory which would react to seeing two similar messages. Work of this type was naturally more difficult and, as the years went by, the finding of possible cribs (as distinct from their exploitation) began to require more and more high-grade labour. For a long time, junior members of the Crib Room were relied upon for discovering new cribs by reading traffic in Naval Section but, as cribs became scarcer, it became obvious that this system was inadequate, and we started having traffic redecoded on the carbon copies of messages so that they were available for scrutiny by senior cribsters. The desirability of this system was further stressed by the increasing numbers of re-encodements which had to be recognized and preserved, so that by 1945 nearly all the traffic was being decoded twice. In some respects this was an extravagant system, as it required a large decoding staff, but it meant that Naval Section received their decodes more quickly, as all available typists set to work first of all on their copies and subsequently typed those for the Crib Room. The alternative scheme would have been to have an increased number of high-grade cribsters so that someone was always available to examine decodes before they went to Naval Section.
The crib chosen as an example in the paragraph before last is interesting also for being a crib for a complete message. Cribs on the whole were only 'beginners', but we had much success with very short messages for in these we got additional confirmation that our crib was right, in that it finished up exactly in the last group of the message. An interesting example of this type was a little harbour report from the Mediterranean. It said: HANSMAXVVVLECHXXAAAYYDDDFEHLANZEIGE. This was a 13 group message. If the message was 14 groups, VVV had been changed to VONVON, one of the normal alternatives for which the cribster had to allow—others were FUNF or FUENF, SIBEN or SIEBEN, VIR or VIER, etc. Most of these little messages that could be cribbed in toto were situation reports which said, in some form or other, 'Nothing to report'; some of them were security conscious and filled up the message with dummy words, which had the effect of degrading them to the level of a normal crib.
... One further crib must be mentioned here for fear it be forgotten altogether as, though often useful, it was never one of the cribs which formed our daily bread and butter. This was the POPTI crib, so christened by the decoders who were amused by the curious selection of letters it contained. We have seen that the German Navy had ceased in general using QWERTZUIOP numerals, but these numerals continued to be used for certain types of figure—weather and notably for certain observations affecting gunnery. As a result we received messages of this sort:
NUEMBERG VVV WEWA SWINEMUENDE LUFTBALTA 05 UHR:
ERDBALTA 05 UHR:
The letters enclosed in the boxes are constant. The bigrams in columns 1 and 2 are the station indices and the 13 letters which follow are the observations from each station: the Ys are commas and the second half of the observation always has a one as its first digit—hence the Q. The stations were always listed in the same order and so we had right through a long message a series of known letters in fixed relative positions. The crib would look something like this: PP?????YQ?????YPT?????YQ etc. These cribs were good and were successfully used, but unfortunately usually lived for a short time only. The example given, for instance, was dependent on the _Nuemberg_ being out exercising. Occasionally, similar messages occurred with the numerals written out in full, but were less useful in most cases owing to the varying lengths of numerals.
In the spring of 1942 the first shift system for a crib was discovered. This was a very important discovery which affected straight cribbing for the rest of the war. The reason for failing to make this discovery earlier was doubtless the same [as the] reason for our failing to notice in early days cribs of a type which were in regular use later; the obvious cribs were very good and we simply did not look very hard for the less obvious, a reprehensible but very understandable state of mind into which we seem frequently to have lapsed. The fact was that, when a key was breaking regularly and satisfactorily, there was very little incentive to do energetic research work.
The incentive which led to the discovery of the shift system was a deterioration of the Channel weather cribs, soon to be finally killed. The position was that they were developing too many different forms and it was at this stage that it was noticed that certain forms occurred at regular intervals. The discovery was made on Cherbourg weather and was rapidly exploited elsewhere. This system involved 4 operators and I very much regret that all records of this original shift system have long since been destroyed. The interesting thing is that all later shift systems, some of which have been proved beyond a shadow of doubt, have worked on a 3-day cycle and have involved 3 operators. It would be interesting to re-examine the original shifts in the light of this evidence to see if our conclusions were only in part correct. The most common form of 3 day cycle, and the only one to be proved in detail, divides the day into 3 shifts (approx. 0–9, 9–16, 16–24) and works as follows:/BAC/ACB/CBA/BAC/ACB/ etc. As an example of the uses of these shifts we may take the twice daily weather report sent out to Arctic U-boats. This message stated the day and the month for which it was a forecast and the month appeared as a jumble of names and numbers—in fact, 2 forms of the crib had to be run each time, one with April and one with VIER. On being divided into shifts on the above principle, however, it was discovered that one man said April consistently and the other two VIER so that on any day we knew which would occur. Also one man was security conscious and was responsible for nearly all the 'horrors' and it was best to leave the crib severely alone when he was on duty.
The practical use of crib records, that is to say, the identification of cribs and the decisions as to what was and was not worth running, always remained a job for high-grade labour, rather surprisingly, as in theory it is simple enough. An attempt to 'mechanize' cribbing with the help of mathematical formulae was a lamentable failure and disappeared amidst howls of derision, though in justice to the inventor of the system it must be said that this was not really due to the essential faultiness of the system as he proposed to use it, but rather the fact that it was misapplied by the inexpert and ignored by the expert, who felt rightly that it was no real assistance. Mathematical computation of the probability of cribs was a system which could not be ignored, but results needed to be modified and analysed by the judgement of experience. Nothing really could replace the knowledge which was gained by experience. A cribster had to come to realize that a crib was not right because he had thought of it himself, that he should not be discouraged because a series of apparently good cribs had failed and start making wild assumptions about new wheels and new keys. He had constantly to decide between two or more cribs as to which was the best and in doing so had to rely as much on a very wide experience as on written records. He had to know which risks could be wisely, which unwisely, taken and had constantly to make decisions on the policy to adopt in breaking a key: would it be better to run 1 crib with a 60% chance of coming out and, if this failed, change to another message, or start on a programme of a 3 form crib on one message which would finish by giving a 95% chance of success? Problems like this would, of course, have been easy had it not been necessary to consider such other factors as the bombe time available, the intelligence and cryptographic advantages of a quick break, the probable pressure on bombes in 12 hours time, and so on. On the face of it, straight cribbing appears to be impossibly tedious when compared with depth but, though depth had its great moments which straight cribbing could not touch, the problem did not become less interesting when Banburismus had died. With perhaps 6 to 10 keys to break regularly, the cribster was a busy man faced with an interesting problem in tackling which he had to consider not only the total number of keys eventually broken but also economy of bombe time and the demands of Intelligence.
##### _Re-encodements_
Re-encodements are repetitions in a cypher of messages which have already been transmitted on other cyphers, or indeed in plain language. The great advantage of re-encodements over straight cribs is the factor they receive in favour of their being right, owing to the length of crib which has been written in without its crashing against the cypher text. We have already seen that for each letter of a wrong crib there is a 1 in 25 chance of a crash, so that a 50 letter crib which does not crash gets a factor of 7 in its favour, a 100 letter crib a factor of 50, a 200 letter crib a factor of 2,500, a 300 letter crib a factor of 40,000—in fact there is really no chance of a long re-encodement being wrong. We have met re-encodements in a variety of fairly distinct forms which I will deal with separately.
##### _Re-encodements from hand cyphers_
The two main sources of re-encodements from hand cyphers were Werftschluessel and R.H.V. Werftschluessel was used by small ships in the German Home Waters area, mostly in the Baltic: it was read continuously from early 1941 to February 1945, when it was abandoned as being of little further value. R.H.V., the reserve hand cypher of the German Navy, was being used, apart from its function as an emergency cypher, by a number of small ships in the Norwegian area when it was first captured in December 1941. Presumably owing to the completion of the distribution of the [Enigma] machine, its use gradually declined and there has been little traffic since the end of 1943.
Re-encodements from these cyphers were of the type one would expect— messages of significance to great and small alike: weather messages, gale warnings, aircraft reports, mine warnings, wreck warnings, etc. Given the R.H.V. or Werft version of the signal, it was not normally difficult to find its Enigma pair. Habits about relative time of origin varied somewhat from area to area but were fairly consistent in any one area, while length, and the frequencies the Enigma version was likely to be passed on, were normally fairly accurately predictable.
The most famous of all these re-encodements was Bereich 7, which broke Dolphin consistently from late 1942 to late 1943. This was a twice daily reencodement of weather from Trondhjem, and it was normally possible to produce a right crib from it. The contents were rarely hatted and the only hazard was an addition at the end of the Dolphin message giving the information that a certain R.H.V. message need not be decoded as it had identical content. Another famous weather crib was Bereich 5, which broke most of the first 6 months of Plaice but which subsequently became security conscious and hatted, the Werft version to such an extent that we could do nothing with it. By early 1945 it was so bad that we did not feel justified in asking that Werft should continue to be broken to assist us with Plaice. It was generally true of all re-encodements that, if much hatting took place, we could make little use of them.
The other most prolific source of re-encodements of this type was the mine laying, especially in the Baltic. Mine warnings and 'all clears' were sent out in Werft and then repeated for the Baltic U-boats. So good were these cribs that, if we were under pressure to obtain an early break, mines were laid deliberately with a view to producing cribs and most profitable results were obtained. We even put up suggestions as to the best places to lay mines: obviously WEG FUNF (or FUENF) ZIFFER (or ZIFFX or ZIFF) SIEBEN (or SIBEN) was a rotten place but KRIEGSANSTEUERUNGSTONNE SWINEMUENDE was likely to go unaltered into the Enigma version. The golden age of these cribs was the first half of 1942. By 1943 they were already falling off and eventually, by using dummy words and inversions, they ceased to be of any interest to us.
##### _Re-encodements from Army and Air cyphers_
There was surprisingly little re-encodement between our traffic and that of Hut 6. The only re-encodements to have occurred consistently over a long period were Mediterranean reconnaissance reports. These were usually easy to tie up and fairly difficult and amusing to attack; they had their heyday in early 1944 when Porpoise ceased to be breakable on its indicating system. These differed from all other re-encodements in that they presented a problem of 'translation': the German Air Force used very different cypher conventions from the Navy, so each message had to be turned into Naval language before it could be used. With a little practice this game could be played very successfully.
##### _Re-encodements between Naval machine cyphers_
Apart from those already mentioned, all re-encodements have been between keys we ourselves have broken. Hence they only began fairly late in our history as the number of our keys increased. The most famous series of these re-encodements were those originating in Shark and being repeated in Dolphin (later Narwhal) for the Arctic U-boats, Plaice for the Baltic U-boats, [and] Turtle for the Mediterranean U-boats. These re-encodements were messages of general interest to all U-boats—corrections to existing documents, descriptions of new Allied weapons, significant experiences of other boats, etc.—and they would start their career by being sent on all existing Shark services, with often a note at the end ordering repetitions on other keys. Re-encodements of this type were usually dead easy and, as they were usually long, one knew definitely when one had a right crib. A considerable time-lag before re-encoding—on Narwhal sometimes 3 or 4 days—often made identification difficult, but had compensating advantages. Shark of the 2nd of a month might break Narwhal of the 4th and Narwhal of the 4th might also contain a re-encodement of a Shark message of the 3rd. The 3rd Shark would break and its paired day the 4th would follow, providing perhaps a store of messages likely to appear in Narwhal of the 5th or 6th. Re-encodements were fairly frequent, an overall average of perhaps 1 or 2 a day, and, with several keys involved, it was often possible to break all keys concerned for considerable periods on re-encodements without having to use a single straight crib—a very economical process.
Another remarkable series of re-encodements linked Dolphin and Sucker from November 1944 until the end of the war. In this case it was a weather message from the Hook of Holland which was repeated in almost identical form in the 2 cyphers. This crib was responsible for our consistently good results in Dolphin during the last months of the war.
Another weather re-encodement of considerable interest was that of BEREICHDORA—an area in the Skagerrak and West Baltic of interest to Plaice and Dolphin. This message originated from Swinemunde on Plaice and was repeated on Dolphin by one of a variety of stations, in accordance with a complicated programme. The repeating station was responsible for hatting the original and for sending it out on the Dolphin frequencies; normally the repeating station encyphered it twice with the text in a different order and sent it out as 2 independent messages. Security precautions were in fact extremely strictly enforced. However, the original Plaice message always set out its observations in the same order—wind—cloud—visibility—sea—and it was possible to take the Dolphin message and reshuffle it into its original order. BEREICHDORA remained far too tricky to be a high-grade crib, but it was used regularly and quite successfully when nothing better was available: there were several periods when it was in fact our only crib into Plaice. The final refinement of discovering the principle on which the Plaice was hatted and using it for breaking Dolphin remained beyond our powers.
The remaining and most curious type of re-encodement was between one day's traffic and another on the same cypher. These re-encodements were found largely in the Baltic where a series of BELEHRUNGSFUKSPRUECHE were sent to the training U-boats and were repeated frequently for some weeks. There was no law as to when these might be retransmitted but, presumably because their content was [of] didactic value and the decyphering of them good practice, they occurred at one period fairly frequently, and could often be identified by their exceptional length. Dolphin also produced one crib of this type, perhaps the longest lived of all our cribs and one which would have broken Dolphin daily, had it been possible to intercept the message satisfactorily. It was a statement of areas off North Norway in which anti-submarine activities were to be allowed during the next 24 hours and, while the areas did indeed sometimes change, much more often they remained the same and the message of the day before supplied the crib for the message of the new day.
Commander Denniston wrote: 'The German Signal Service will do its best to prevent compromise of Enigma by inferior low-grade cyphers... the Germans do not intend their cyphers to be read.' The German Signals Service failed, however, lamentably in its functions (though Werft and R.H.V. are not, of course, low-grade cyphers), and its cyphers were most extensively read. In later years, we should never have been able to continue regular breaking on a large number of keys if it had not been for the steady flow of re-encodements, and their complete failure to master the re-encodement problem was without doubt one of the biggest blunders the German Security Service made.
#### The Beginning of the Operational Period
During the Autumn of 1941, the outlook became steadily brighter. In August there were 6 bombes in action and we had the bigram tables without which Banburismus was impossible.
The results speak for themselves. All August was broken except 1–4, 24–25, all September, and all October except 3–4 and 12–13. From October 14th, 1941, Dolphin was broken consistently until March 7th, 1945.
During these months, methods of Banburismus and cribbing improved considerably. The situation gradually developed that stability which can only arise from a long period of regular breaking.
Early in October, on the 3rd, 4th, or 5th, we were mystified by the failure of the U-boat traffic to decode when Dolphin for the day had been broken. The situation was saved by a Werft crib, one of Werft's earliest triumphs, and it transpired that the key was the same as for the rest of the traffic except for the Grundstellung, which was entirely different. This was a further development of the innovation of April when the U-boats started to use a Grundstellung which was the reverse of that used by surface craft. This new development made very little difference except that it delayed slightly the reading of the U-boat traffic.
November 29th brought the first real crisis in the form of a change of bigram tables and K book. 6 months earlier this would have beaten us, but we now knew enough about cribs to be able to break the traffic without Banburismus if sufficient bombes were available, and there were now 12 bombes in action. Rather curiously, we thought it worth while to indulge in the luxury of a Freeborn catalogue of fits in the hope of being able to do a piece of depth— dummy depth—a remarkable reflection on our comparative ignorance of straight cribs and of their value. The policy was, however, to some extent justified, as on November 30th we got a right dummy depth crib and Commander Travis, then the arbiter of bombe policy, decreed that we might have all 12 bombes to run it.
This crib and many others duly came out in the course of December, and the process of EINSing, twiddling, and bigram table building, described in an earlier [section], proceeded merrily enough and considerably more smoothly than before. By the end of December, the tables were near enough to completion to begin to think of restarting Banburismus and Turing was just starting to reconstruct the K book—almost the last theoretical problem he tackled in the Section, although he remained with us for some time to come. On December 30th a pinch of keys, bigram tables, and K book made further work unnecessary and we were able to restart Banburismus at once.
### **CHAPTER 6
Bombe and Spider ( _1940_ )**
_Alan Turing_
#### **Introduction**
##### _Jack Copeland_
This material forms chapter 6 of Turing's _Treatise on the Enigma_ (known at Bletchley Park as 'Prof's Book'). The text has been prepared by Ralph Erskine, Philip Marks and Frode Weierud from the two known surviving copies of Turing's original typescript. The pagination of Turing's typescript indicates that the chapter may possibly have continued for a further four pages; however, these pages are not to be found in either of the archived copies.
#### **The steckered Enigma. Bombe and Spider.**
When one has as teckered Enigma to deal with one's problems naturally divide themselves into what is to be done to find the Stecker, and what is to be done afterwards. Unless the indicating system is very well designed there will be no problem at all when the Stecker have been found, and even with a good indicating system we shall be able to apply the methods of the last two chapters [of Turing's _Treatise_ ] to the individual messages. The obvious example of a good indicating system is the German Naval Enigma cipher, which is dealt with in Chapter VII [of the _Treatise_ ]. This chapter is devoted to methods of finding the Stecker. Naturally enough we never find the Stecker without at the same time finding much other information.
#### Cribs
The most obvious kind of data for finding the keys is a 'crib', i.e. a message of which a part of the decode is known. We shall mostly assume that our data is a crib, although actually it may be a number of constatations arising from another source, e.g. a number of CILLIs or a Naval Banburismus.
#### FORTYWEEPYYWEEPY methods
It is sometimes possible to find the keys by pencil and paper methods when the number of Stecker is not very great, e.g. 5 to 7. One would have to hope that several of the constatations of the crib were 'unsteckered'. The best chance would be if the same pair of letters occurred twice in the crib (a 'half-bombe'). In this case, assuming 6 or 7 Stecker there would be a 25% chance of both constatations being unsteckered. The positions at which these constatations occurred could be found by means of the Turing sheets (if there were three wheels) or the Jeffreys sheets. The positions at which this occurred could be separately tested. Another possibility is to set up the inverse rods for the crib and to look for clicks. There is quite a good chance of any apparent click being a real click arising because all four letters involved are unsteckered. The position on the right hand wheel is given by the column of the inverse rod set-up, and we can find all possible positions where the click coupling occurs from the Turing sheets or the Jeffreys sheets. In some cases there will be other constatations which are made up from letters supposed to be unsteckered because they occur in the click, and these will further reduce the number of places to be tested.
These methods have both of them given successful results, but they are not practicable for cases where there are many Stecker, or even where there are few Stecker and many wheel orders.
#### A mechanical method. The Bombe
Now let us turn to the case where there is a large number of Stecker, so many that any attempt to make use of the unsteckered letters is not likely to succeed. To fix our ideas let us take a particular crib.
Presumably the method of solution will depend on taking hypotheses about parts of the keys and drawing what conclusions one can, hoping to get either a confirmation or a contradiction. The parts of the keys involved are the wheel order, the rod start of the crib, whether there are any turnovers in the crib and if so where, and the Stecker. As regards the wheel order one is almost bound to consider all of these separately. If the crib were of very great length one might make no assumption about what wheels were in the L.H.W. position and M.W. position, and apply a method we have called 'Stecker knock-out' (an attempt of this kind was made with the 'Feindseligkeiten' crib in Nov. –39), or one might sometimes make assumptions about the L.H.W. and M.W. but none, until a late stage, about the R.H.W. In this case we have to work entirely with constatations where the R.H.W. has the same position. This method was used for the crib from the Schluesselzettel of the Vorpostenboot, with success; however I shall assume that all wheel orders are being treated separately. As regards the turnover one will normally take several different hypotheses, e.g.:
1) turnover between positions 1 and 5
2) turnover between positions 5 and 10
3) turnover between positions 10 and 15
4) turnover between positions 15 and 20
5) turnover between positions 20 and 25
With the first of these hypotheses one would have to leave out the constatations in positions 2 to 4, and similarly in all the other hypotheses four constatations would have to be omitted. One could of course manage without leaving out any constatations at all if one took 25 different hypotheses, and there will always be a problem as to what constatations can best be dispensed with. In what follows I shall assume we are working the T.O. hypothesis numbered 5) above. We have not yet made sufficiently many hypotheses to be able to draw any immediate conclusions, and must therefore either assume something about the Stecker or about the rod start. If we were to assume something about the Stecker our best chance would be to assume the Stecker values of A and E, or of E and I, as we should then have two constatations corrected for Stecker, with only two Stecker assumptions. With Turing sheets one could find all possible places where these constatations occurred, of which we should, on the average, find about 28.1. As there would be 626 hypotheses of this kind to be worked we should gain very little in comparison with separate examination of all rod starts. If there had not been any half-bombes in the crib we should have fared even worse. We therefore work all possible hypotheses as to the rod start, and to simplify this we try to find characteristics of the crib which are independent of the Stecker. Such characteristics can be seen most easily if the crib is put in to the form of a picture like Fig. 59. From this picture we see that one characteristic which is independent of the Stecker is that there must be a letter which enciphered at either position 2 or position 5 of the crib gives the same result. This may also be expressed by saying that there must be a letter such that, if it is enciphered at position 2, and the result re-enciphered at position 5, the final result will be the original letter. Another such condition is that the same letter enciphered successively at the positions 3, 10 must lead back to the original letter. Three other conditions of this kind are that the successive encipherments at positions 2, 23, 3 or at 2, 9, 8, 6, 24, 3 or at 13, 12, 8, 9, 5 starting from the same letter as before must lead back to it. There are other such series, e.g. 13, 12, 6, 24, 3, but these do not give conditions independent of the others. The letter to which all these multiple encipherments are applied is, of course, the Stecker value of E. We shall call E the 'central letter'. Any letter can of course be chosen as the] 'central letter', but the choice affects the series of positions or 'chains' for the multiple encipherments. There are other conditions, as well as these that involve the multiple encipherments. For instance the Stecker values of the letters in [Fig. [59] must all be different. The Stecker values for E, I, M, Z, Q, S, A are the letters that arise at the various stages in the multiple encipherments and the values for W, T, V, N, D, K can be found similarly. There is also the condition that the Stecker must be self-reciprocal, and the other parts or 'webs' of Fig. 59, P-B-U-O and R-H, will also restrict the possibilities somewhat. Of these conditions the multiple encipherment one is obviously the easiest to apply, and with a crib as long as the one above this condition will be quite sufficient to reduce the possible positions to a number which can be tested by hand methods. It is actually possible to make use of some of the other conditions mechanically also; this will be explained later.
**Figure 59.** Picture from KEINE ZUSAETZE crib. Constatations 16 to 19 omitted to allow for turnover.
In order to apply the multiple encipherment condition one naturally wants to be able to perform the multiple encipherments without Stecker in one operation. To do this we make a new kind of machine which we call a 'Letchworth Enigma'. There are two rows of contacts in a Letchworth Enigma each labelled A to Z and called the input and output rows: there are also moveable wheels. For each position of an ordinary Enigma there is a corresponding position of the Letch-worth Enigma, and if the result of enciphering F at this position is R, then F on the input row of the Letchworth Enigma is connected to R on the output row, and of course R on the input row to F on the output row. Such a 'Letchworth Enigma' can be made like an ordinary Enigma, but with all the wiring of the moveable wheels in duplicate, one set of wires being used for the journey towards the Umkehrwalze, and the others for the return journey. The Umkehrwalze has two sets of contacts, one in contact with the inward-journey wiring of the L.H.W. and one in contact with the outward-journey wiring. The Umkehrwalze wiring is from the one set of contacts across to the other. In the actual design used there were some other differences; the wheels did not actually come into contact with one another, but each came into contact with a 'commutator' bearing 104 fixed contacts. These contacts would be connected by fixed wiring to contacts of other commutators. These contacts of the commutators can be regarded as physical counterparts of the 'rod points' and 'output points' for the wheels.
If one has two of these 'Letchworth Enigmas' one can connect the output points of the one to the input points of the other and then the connections through the two Enigmas between the two sets of contacts left over will give the effect of successive encipherments at the positions occupied by the two Enigmas. Naturally this can be extended to the case of longer series of Enigmas, the output of each being connected to the input of the next.
Now let us return to our crib and see how we could use these Letchworth Enigmas. For each of our 'chains' we could set up a series of Enigmas. We should in fact use 18 Enigmas which we will name as follows.
By 'position 8' we here mean 'the position at which the constatation numbered 8 in the crib, is, under the hypothesis we are testing, supposed to be enciphered'. The Enigmas are connected up in this way: output of A1 to input of A2: output of B1 to input of B2: output of C1 to input of C2, output of C2 to input of C3: etc. This gives us five 'chains of Enigmas' which we may call A, B, C, D, E, and there must be some letter, which enciphered with each chain gives itself. We could easily arrange to have all five chains controlled by one keyboard, and to have five lampboards shewing the results of the five multiple encipherments of the letter on the depressed key. After one hypothesis as to the rod start had been tested one would go on to the next, and this would usually involve simply moving the R.H.W. of each Enigma forward one place. When 26 positions of the R.H.W. have been tested the M.W. must be made to move forward too. This movement of the wheels in step can be very easily done mechanically, the right hand wheels all being driven continuously from one shaft, and the motion of the other wheels being controlled by a carry mechanism.
It now only remains to find a mechanical method of registering whether the multiple encipherment condition is fulfilled. This can be done most simply if we are willing to test each Stecker value of the central letter throughout all rod starts before trying the next Stecker value. Suppose we are investigating the case where the Stecker value of the central letter E is K. We let a current enter all of the chains of Enigmas at their K input points, and at the K output points of the chains we put relays. The 'on' points of the five relays are put in series with a battery (say), and another relay. A current flows through this last relay if and only if a current flows through all the other five relays, i.e. if the five multiple encipherments applied to K all give K. When this happens the effect is, essentially, to stop the machine, and such an occurrence is known at Letchworth as a 'straight'. An alternative possibility is to have a quickly rotating 'scanner' which, during a revolution, would first connect the input points A of the chains to the current supply, and the output points A to the relays, and then would connect the input and output points B to the supply and relays. In a revolution of the scanner the output and input points A to Z would all have their turn, and the right hand wheels would then move on. This last possible solution was called 'serial scanning' and led to all the possible forms of registration being known as different kinds of 'scanning'. The simple possibility that we first mentioned was called 'single line scanning'. Naturally there was much research into possible alternatives to these two kinds of scanning, which would enable all 26 possible Stecker values of the central letter to be tested simultaneously without any parts of the machine moving. Any device to do this was described as 'simultaneous scanning'.
The solution which was eventually found for this problem was more along mathematical than along electrical engineering lines, and would really not have been a solution of the problem as it was put to the electricians, to whom we gave, as we thought, just the essentials of the problem. It turned out in the end that we had given them rather less than the essentials, and they therefore cannot be blamed for not having found the best solution. They did find a solution of the problem as it was put to them, which would probably have worked if they had had a few more months experimenting. As it was the mathematical solution was found before they had finished.
#### Pye simultaneous scanning
The problem as given to the electricians was this. There are 52 contacts labelled A... Z, A′,..., Z′. At any moment each one of A,..., Z is connected to one and only one of A′,..., Z′: the connections are changing all the time very quickly. For each letter of the alphabet there is a relay, and we want to arrange that the relay for the letter R will only close if contact R is connected to contact R′.
The latest solution proposed for this problem depended on having current at 26 equidistant phases corresponding to the 26 different letters. There is also a thyratron valve for each letter. The filaments of the thyratrons are given potentials corresponding to their letters, and the grids are connected to the corresponding points A′,..., Z′. The points A,..., Z are also given potentials with the phase of the letter concerned. The result is that the difference of potential of the filament and the grid of thyratron A oscillates with an amplitude of at least , , E being the amplitude of the original 26 phase supply, unless A and A′ are connected through the chain, in which case the potentials remain the same or differ only by whatever grid bias has been put into the grid circuit. The thyratrons are so adjusted that an oscillation of amplitude will bring the potential of the grid to the critical value and the valve will 'fire'. The valve is coupled with a relay which only trips if the thyratron fails to fire. This relay is actually a 'differential relay', with two sets of windings, one carrying a constant current and the other carrying the current from the anode circuit of the thyratron. Fig. 60 shews a possible form of circuit. It is probably not the exact form of circuit used in the Pye experiments, but is given to illustrate the theoretical possibility.
#### The Spider
We can look at the Bombe in a slightly different way as a machine for making deductions about Stecker when the rod start is assumed. Suppose we were to put lamp-boards in between the Enigmas of the chains, and label the lamp-boards with the appropriate letters off figure [ _number missing but presumably 59_ ]. For example in chain C the lamp-board between C1 and C2 would be labelled A. The keyboard, if we were using one, could be labelled with the 'central letter'. Now when we depress a letter of the key-board we can read off from the lamp-boards some of the Stecker consequences of the hypothesis that the depressed letter is steckered to the central letter: for one such consequence could be read off each lamp-board, namely that the letter lighting is steckered to the name of the lamp-board.
**Figure 60.** Circuit for Pye simultaneous scanning.
Editors' note: The figure shows a thyratron valve (the smaller of the two circles) in a circuit connected to a 26-phase supply. A thyratron is a gas or mercury vapour filled valve with three connections, called the anode, the cathode, and, between them, the grid. Current begins to flow through the valve once the grid voltage reaches a certain level. Once current has started flowing through the valve, it can be turned off only by dropping the anode voltage to zero. The component depicted at the upper left of the diagram (a breakable contact with two parallel coils shown beneath it) is a relay controlled by the thyratron. The two vertical lines of dots indicate identical repetitions of the same circuit. In total there are 26 thyratrons and 26 relays. The cathode of each thyratron is connected to one of the twenty-six input points of the first Letchworth Enigma in a chain, and the grid of each thyratron is connected to the corresponding output point of the last Letchworth Enigma in the chain. Each input point is supplied with one phase of the 26-phase supply.
The relay is of the differential type described by Turing on pp. 325, 327 and used extensively in the bombe. A differential relay has two coils; in Fig. 60 one of the relay's coils is permanently powered from the cathode supply via a resistor, and the second coil is powered via the thyratron. The idea is that if, for instance, point R is connected to point R′ (as in Turing's description on p. 320) then there will be no grid supply signal to this thyratron and so it will not 'fire' (i.e. current will not start to flow through the valve), while all the other thyratrons will 'fire', thereby energizing the second coil of their respective differential relays. This will result in the relay attached to R—but none of the other relays—being switched on by the constant current in its first coil. At the start of each round of scanning, the anode supply would be momentarily interrupted, switching all the thyratrons off and resetting all the differential relays. This interruption can itself be performed electronically.
The 26-phase supply is not a rotary arrangement but a fixed supply generating sinusoidal signals with 26 equidistant phases. Such supplies can be made in various ways, usually using a number of transformers with different taps; the taps sum together to produce the desired phase relationships. Turing does not provide details of the construction of the 26 phase supply. In correspondence concerning Fig. 60, Donald Davies has suggested that the supply might consist of an 'arrangement using two transformers in quadrature with signals 3, 2, 1, 0, 1, 2, 3 times some unit value (E/3) from each transformer. This gives 49 signals and 26 of these can be chosen so that the maximum individual amplitude E corresponds to a minimum difference signal of E/3, better than their value 0.24E. This would be easier to produce than 26 equidistant phases. There are better distributions still, but a bit more difficult to produce.' Turing's sketch shows the cathode, anode, and grid of the thyratron as being powered by batteries. In an engineered form of the circuit, the power would be more likely to come from a mains supply via suitable transformers. The grid bias supply might be regulated electronically.
(Thanks to Martin Slack for information.)
When we look at the Bombe in this way we see that it would be natural to modify it so as to make this idea fit even better. We have not so far allowed for lengthy chains of deductions; the possible deductions stop as soon as one comes back to the central letter. There is however no reason why, when from one hypothesis about the Stecker value of the central letter we have deduced that the central letter must have another Stecker value, we should not go on and draw further conclusions from this second Stecker value. At first sight this seems quite useless, but, as all the deductions are reversible, it is actually very useful, for all the conclusions that can be drawn will then be false, and those that remain will stand out clearly as possible correct hypotheses. In order that all these deductions may be made mechanically we shall have to connect the 26 contacts at the end of each chain to the common beginning of all the chains. With this arrangement we can think of each output or input point of an Enigma as representing a possible Stecker, and if two of these points are connected together through the Enigmas then the corresponding Stecker imply one another. At this point we might see how it all works out in the case of the crib given above. This crib was actually enciphered with alphabets which, when corrected for their Stecker, are those given below, the numbers over the crib constatations giving the column headings. The alphabets most used below are 2, 3, 5, 10, 23, and these are reproduced here for reference.
In Fig. 61 at the top are the chains, with the positions and the letters of the chain. In each column are written some of the letters which can be inferred to be Stecker values of the letters at the heads of their columns from the hypothesis that X is a Stecker value of the central letter E. By no means all possible inferences of this kind are made in the figure, but among those that are made are all possible Stecker values for E except the right one, L. If we had taken a rod start that was wrong we should almost certainly have found that all of the Stecker values of E could be deduced from any one of them, and this will hold for any cribs with two or more chains. Remembering now that with our Bombe one Stecker is deducible from another if the corresponding points on the lamp-boards are connected through the Enigmas, a correct rod start can only be one for which not all the input points of the chains are connected together; the positions at which this happens are almost exactly those at which a Bombe with simultaneous scanning would have stopped.
This is roughly the idea of the 'spider'. It has been described in this section as a way of getting simultaneous scanning on the Bombe, and has been made to look as much like the Bombe as possible. In the next section another description of the spider is given.
#### The Spider. A Second Description. Actual Form
In our original description of the Bombe we thought of it as a method of looking for characteristics of a crib which are independent of Stecker, but in the last section we thought of it more as a machine for making Stecker deductions. This last way of looking at it has obviously great possibilities, and so we will start afresh with this idea.
In the last section various points of the circuit were regarded as having certain Stecker corresponding to them. We are now going to carry this idea further and have a metal point for each possible Stecker. These we can imagine arranged in a rectangle. Each point has a name such as Pv: here the capital letters refer to 'outside' points and the small letters to 'inside letters'; an outside letter is the name of a key or bulb, and so can be a letter of a crib, while an inside letter is the name of a contact of the Eintrittwalze, so that all constatations obtained from the Enigma without Stecker give information about inside letters rather than outside. Our statements will usually be put in rather illogical form: statements like 'J is an outside letter' will usually mean 'J is occurring in so and so as the name of a key rather than of a contact of the Eintrittwalze'. The rectangle is called the 'diagonal board' and the rows are named after the outside letters, the columns after the inside letters. Now let us take any constatation of our crib e.g. at 24. For the position we are supposed to be testing we will have an Enigma set up at the right position for encoding this constatation, but of course without any Stecker. Let us suppose it set up for the correct position, then one of the pairs in the alphabet in position 24 is OC: consequently if Qo then Ic (i.e. if outside letter Q is associated with inside o then outside I is associated with inside c). Now if we connect the input of the (Letchworth) Enigma to the corresponding points of the diagonal board on line Q and the output to line I then since the 'o' input point is connected to the 'c' output point we shall have Qo on the diagonal board connected to Ic through the Letchworth Enigma. We can of course put in a Letchworth Enigma for every constatation of the crib, and then we shall have all the possible deductions that can be made about the association of inside and outside letters paralleled in the connections between the points of the diagonal board. We can also bring in the reciprocal property of the Stecker by connecting together diagonally opposite points of the diagonal board, e.g. connecting Pv to Vp. One can also bring in other conditions about the Stecker, e.g. if one knows that the letters which were unsteckered on one day are invariably steckered on the next then, having found the keys for one day's traffic, one could when looking for the keys for the next day connect together all points of the diagonal board which correspond to non-steckers which had occurred on the previous day. This would of course not entirely eliminate the inadmissible solutions, but would enormously reduce their number, the only solutions which would not be eliminated being those which were inadmissible on every count.
**Figure 61.** Stecker deductions with crib on p. [315], with correct rod start and correct alphabets, but starting from an incorrect Stecker hypothesis E/X. All other incorrect Stecker values of E are deduced.
**Figure 62.** Stecker deductions with same alphabets as Fig. 61, but from correct Stecker hypothesis E/L.
One difference between this arrangement and the Bombe, or the spider as we described it in the last section, is that we need only one Enigma for each constatation.
Our machine is still not complete, as we have not put in any mechanism for distinguishing correct from incorrect positions. In the case of a crib giving a picture like Fig. 59 where most of the letters are connected together in one 'web' it is sufficient to let current into the diagonal board at some point on some line named after a letter on the main web, e.g. at the Ea point in the case of the crib we have been considering. In this case the only possible positions will be ones in which the current fails to reach all the other points of the E line of the diagonal board. We can detect whether this happens by connecting the points of the E line through differential relays to the other pole of our current supply, and putting the 'on' points of the relays in parallel with one another and in series with the stop mech[ _anism_ ]. Normally current will flow through all the differential relays, and they will not move. When one reaches a position that might be correct the current fails to reach one of these relays, and the current permanently flowing in the other coil of the relay causes it to close, and bring the stopping mechanism into play. Mostly what will happen is that there will be just one relay which closes, and this will be one connected to a point of the diagonal board which corresponds to a Stecker which is possibly correct: more accurately, if this Stecker is not correct the position is not correct. Another possibility is that all relays close except the one connected to the point at which current enters the diagonal board, and this point then corresponds to the only possible Stecker. In cases where the data is rather scanty, and the stops therefore very frequent, other things may happen, e.g. we might find four relays closing simultaneously, all of them connected together through the Enigmas and the cross connections of the diagonal board, and therefore none of them corresponding to possible Stecker.
**Figure 62a.**
Editors' note: This figure, which has no caption or other explanation, follows Fig. 62. The label 'Figure 62a' is our own: in Turing's typescript, neither the figure nor the page containing it is numbered. The appearance is that of a worksheet rather than of a figure that belongs to the text.
In order for it to be possible to make the necessary connections between the Enigmas, the diagonal board and the relays there has to be a good deal of additional gear. The input and output rows of the Enigmas are brought to rows of 26 contacts called 'female jacks'. The rows of the diagonal board are also brought to female jacks. The 26 relays and the current supply are also brought to a jack. Any two female jacks can be connected with 'plaited jacks' consisting of 26 wires plaited together and ending in male jacks which can be plugged into the female jacks. In order to make it possible to connect three or more rows of contacts together one is also provided with 'commons' consisting of four female jacks with corresponding points connected together. There is also a device for connecting together the output jack of one Enigma and the input of the next, both being connected to another female jack, which can be used for connecting them to anywhere else one wishes.
On the first spider made there were 30 Enigmas, and three diagonal boards and 'inputs' i.e. sets of relays and stopping devices. There were also 15 sets of commons.
Figs 63, 64 shew the connections of Enigmas and diagonal board in a particular case. The case of a six-letter alphabet has been taken to reduce the size of the figure.
The actual origin of the spider was not an attempt to find simultaneous scanning for the Bombe, but to make use of the reciprocal character of the Stecker. This occurred at a time when it was clear that very much shorter cribs would have to be worked than could be managed on the Bombe. Welchman then discovered that by using a diagonal board one could get the complete set of consequences of a hypothesis. The ideal machine that Welchman was aiming at was to reject any position in which a certain fixed-for-the-time Stecker hypothesis led to any direct contradiction: by a direct contradiction I do not mean to include any contradictions which can only be obtained by considering all Stecker values of some letter independently and shewing each one inconsistent with the original hypothesis. Actually the spider does more than this in one way and less in another. It is not restricted to dealing with one Stecker hypothesis at a time, and it does not find all direct contradictions.
**Figure 63.** Spider connections with Enigma for 6 letter alphabet and crib
Names of contact are given in purple ink, contacts to which they are connected in green ** _in the original_** ]. Connections of diagonal board to Enigmas [Fig. 64.
**Figure 64.** Connections of diagonal board. See Fig. 63. 'Input' is at E. Correct hypothesis E/A. The squares in this figure represent contacts. As in Fig. [63] the purple letters are names and the green letters shew the contacts to which they are connected [ _in the original_ ].
Naturally enough Welchman and Keen set to work to find some way of adapting the spider so as to detect all direct contradictions. The result of this research is described in the next section. Before we can leave the spider however we should see what sort of contradictions it will detect, and about how many stops one will get with given data.
First of all let us simplify the problem and consider only 'normal' stops, i.e. positions at which by altering the point at which the current enters the diagonal board to another pt _point_ ] on the same line one can make 25 relays close [ _illegible; may be_ 'and there is only one such pt']. The current will then be entering at a correct Stecker if the position is correct. Let us further simplify the problem by supposing that there is only one 'web', i.e. that the 'picture' formed from the part of the crib that is being used forms one connected piece, e.g. with the crib on p [ _missing, presumably a reference to[Fig. 59_] we should have one web if we omit the constatations
Some of the constatations of the web could still be omitted without any of the letters becoming disconnected from the rest. Let us choose some set of such constatations, in such a way that we cannot omit any more constatations without the web breaking up. When the constatations are omitted there will of course be no 'chains' or 'closures'. This set of constatations may be called the 'chain-closing constatations' and the others will be called the 'web-forming constatations'. At any position we may imagine that the web-forming constatations are brought into play first, and only if the position is possible for these are the chain-closing constatations used. Now the Stecker value of the input letter and the web-forming constatations will completely determine the Stecker values of the letters occurring in the web. When the chain closing constatations are brought in it will already be completely determined what are the corresponding 'unsteckered' constatations, so that if there are c chain-closing constatations the final number of stops will be a proportion 26 c of the stops which occur if they are omitted. Our problem reduces therefore to the case in which there are no closures. It is, I hope, also fairly clear that the number of stops will not be appreciably affected by the branch arrangement of the web, but only by the number of letters occurring in it. These facts enable us to make a table for the calculation of the number of stops in any case where there is only one web. The method of construction of the table is very tedious and uninteresting. The table is reproduced below.
A similar table has also been made to allow for two webs, with up to five letters on the second. To the case of three webs it is not worth while and hardly possible to go. One can often get a sufficiently good estimate in such cases by using common-sense inequalities, e.g. if we denote the H-M factor for the case of webs with m, n, and p letters by H(m, n, p) we shall have the common-sense inequalities
To see what kind of contradictions are detected by the machine we can take the picture, Fig. 59, and on it write against each letter any Stecker values of that letter which can be deduced from the Stecker hypothesis which is read off the spider when it stops. This has been done in Fig. 65 for a case where the input was on letter E of the diagonal board, and the relay R closed when the machine stopped; if the position of the stop were correct at all the correct Stecker would be given by the points of the diagonal board which were connected to Er, and they will also be the direct consequences of the Stecker hypothesis E/R. As we are assuming that R was the only relay to close this relay cannot have been connected to any of the others, or it would have behaved similarly. We cannot therefore deduce any other Stecker value for E than R, and this explains why on the 'main web' in Fig. 65 there is only one pencil letter against each ink letter. Wherever any pencil letter is the same as an ink letter we are able to write down another pencil letter corresponding to the reciprocal Stecker or to the diagonal connections of the board. In one or two cases we find that the letter we might write down is there already. In others the new letter is written against a letter of one of the minor webs; in such a case we clearly have a contradiction, but as it does not result in a second set of pencil letters on the main web the machine is not prevented from stopping. There are other contradictions; e.g. we have Z/L, W/L, but as L does not occur in the crib this has no effect.
#### The machine gun
When using the spider there is a great deal of work in taking down data about stops from the machine and in testing these out afterwards, making it hardly feasible to run cribs which give more than 5 stops per wheel order. As the complete data about the direct consequences of any Stecker hypothesis at any position are already contained in the connections of the points of the diagonal board it seems that it should be possible to make the machine do the testing itself. It would not be necessary to improve on the stopping arrangement of the spider itself, as one could use the spider as already described, and have an arrangement by which, whenever it stopped, a further mechanism is brought into play, which looks more closely into the Stecker. Such a mechanism will be described as a machine gun, regardless of what its construction may be.
**Figure 65.** Illustrating the kind of position at which the spider will stop. Here the input letter may be supposed to be E and the relay which closed R. The Stecker values of the letters, which are consequences of the hypothesis E/R are written against the letters. There are contradictions such as Z/L, W/L: P/D, P/F, P/M which are not observed by the spider.
1 Editors' note. Originally ND, but ND are the menu letters. Should be AU.
2 Editors' note. Orginally QL, should be TL.
3 Editors' note. Originally ME, but ME are the menu letters. Should be PR.
4 Editors' note. The table in the _Treatise_ has column 20 positioned under column 19 and all the subsequent columns shifted one place to the left with a column 26 added at the end. This is clearly an error. There is no column 19 in this menu as the four constatations 16 to 19 have been left out (as Turing has already explained) due to the turnover hypothesis selected.
With almost any crib the proportion of spider stops that could be passed by a machine gun as possible would be higher than the ratio of spider stops to total possible hypotheses. Consequently the amount of time that can economically be allowed to the machine gun for examining a position is vastly greater than can be allowed to the spider. We might for instance run a crib which gives 100 spider stops per wheel order, and the time for running, apart from time spent during stops, might be 25 minutes. If the machine gun were allowed 5 seconds per position, as compared with the spider's 1/10 second, only 8 minutes would be added to the time for the run.
When the spider stops, normally the points of the diagonal board which are energised are those corresponding to supposedly false Stecker. Naturally it would be easier for the machine gun if the points energised corresponded to supposedly correct Stecker. It is therefore necessary to have some arrangement by which immediately after the spider stops the point of entry of the current is altered to the point to which the relay which closed was connected, or is left unaltered in the case that 25 relays closed. Mr. Keen has invented some device for doing this, depending entirely on relay wiring. I do not know the details at present, but apparently the effect is that the machine does not stop at all except in cases in which either just one relay closes or 25 relays close. In the case that 25 relays close the current is allowed to continue to enter at the same point, but if just one relay closes the point of entry is changed over to this relay. This method has the possible disadvantage that a certain number of possible solutions may be missed through not being of normal type. This will only be serious in cases where the frequency of spider stops is very high indeed, e.g. 20%, and some other method, such as 'Ringstellung cut-out' is being used for further reducing the stops. An alternative method is to have some kind of a 'scanner' which will look for relays which are not connected to any others. Which method is to be used is not yet decided.
At the next stage in the process we have to see whether there are any contradictions in the Stecker; in order to reduce the number of relays involved this is done in stages. In the first stage we see whether or not there are two different Stecker values for A, in the second whether there are two different values for B, and so on. To do this testing we have 26 relays which are wired up in such a way that we can distinguish whether or not two or more of them are energised. When we are testing the Stecker values of A we have the 26 contacts of the A line of the diagonal board connected to the corresponding relays in this set. What is principally lacking is some device for connecting the rows of the diagonal board successively to the set of relays. This fortunately was found in post-office standard equipment; the clicking noise that this gadget makes when in operation gives the whole apparatus its name. If we find no contradictions in the Steckers of any letter the whole position is passed as good. The machine is designed to print the position and the Stecker in such a case. Here again I do not know the exact method used, but the following simple arrangement seems to give much the same effect, although perhaps it could not be made to work quite fast enough. The Stecker are given by typing one letter in a column headed by the other. When any letter is being tested for Stecker contradictions the relays corresponding to the Stecker values of the letter close. We can arrange that these relays operate corresponding keys of the typewriter, but that in the case that there is a contradiction this is prevented and some special symbol is typed instead shewing that the whole is wrong. When no relay closes nothing is typed. The carriage of the typewriter is not operated by the keys but only by the space bar, and this is moved whenever there is a change of the letter whose Stecker are being examined.
#### Additional gadgets
Besides the spider and machine gun a number of other improvements of the Bombe are now being planned. We have already mentioned that it is possible to use additional data about Stecker by connecting up points of the diagonal board. It is planned to make this more straightforward by leading the points of the diagonal board to 325 points of a plug board; the plug board also has a great many points all connected together, and any Stecker which one believes to be false one simply connects to this set.
Another gadget is designed to deal with cases such as that in which there are two 'webs' with six letters and no chains on each. A little experiment will show that in the great majority of cases with such data, when the solution is found, the Stecker value of a letter on either web will imply the whole set of Steckers for the letters of both webs: in the current terminology, 'In the right place we can nearly always get from one web onto the other.' If however we try to run such data on the spider, even with the machine gun attachment, there will be an enormous number of stops, and the vast majority of these will be cases in which 'we have not got onto the second web'. If we are prepared to reject these possibilities without testing them we shall not very greatly decrease the probability of our finding the right solution, but very greatly reduce the amount of testing to be done. If in addition the spider can be persuaded not to stop in these positions, the spider time saved will be enormous. Some arrangement of this kind is being made but I will not attempt to describe how it works.
With some of the ciphers there is information about the Ringstellung (Herivelismus) which makes certain stopping places wrong in virtue of their position, and not of the alphabets produced at those positions. There is an arrangement, known as a 'Ringstellung cut-out', which will prevent the machine from stopping in such positions. The design of such a cut-out clearly presents no difficulties of principle.
There are also plans for 'majority vote' gadgets which will enable one to make use of data which is not very reliable. A hypothesis will only be regarded as rejected if it contradicts three (say) of the unreliable pieces of data. This method may be applied to the case of unreliable data about Stecker.
### **CHAPTER 7
Letter to Winston Churchill ( _1941_ )**
_Alan Turing, Gordon Welchman, Hugh Alexander, Stuart Milner-Barry_
#### **Introduction**
##### _Jack Copeland_
During 1941, codebreaking at Bletchley Park was hindered by shortages of typists and unskilled staff. These shortages could have been easily rectified, but the codebreakers' urgent requests were ignored by officials in Whitehall. Going over the heads of those in command at GC & CS, Turing and his co-signatories wrote directly to the Prime Minister, Winston Churchill. On receiving the letter Churchill minuted his Chief of Staff, General Ismay: 'action this day Make sure they have all they want on extreme priority and report to me that this had been done.'
It fell to Stuart Milner-Barry of Hut 6 to deliver the letter by hand to 10 Downing Street. In 1986, Milner-Barry recalled his trip to Whitehall:
Why I was deputed to carry the letter to No. 10 I do not remember—at a guess, because I was the most readily expendable from the scene of action. What I do recall is arriving at Euston Station, hailing a taxi, and with a sense of total incredulity (can this really be happening?) inviting the driver to take me to 10 Downing Street. The taxi-driver never blinked an eyelid: without comment he directed himself to Whitehall. Arrived at the entrance to Downing Street, I was again surprised at the lack of formality: there was just a wooden barrier across the road, and one uniformed policeman who waved my driver on. At the door to No. 10 I paid off the taxi, rang the bell, was courteously ushered in, explained that I had an urgent letter which I was anxious to deliver to the Prime Minister personally, and was invited to wait. Of course I did not see the Prime Minister himself; but very shortly there appeared a dapper dark-suited figure of shortish stature whom I subsequently identified as Brigadier Harvie-Watt, Mr. Churchill's PPS from 1941 to 1945. To him I again explained my errand; and while obviously and understandably puzzled as to who I might be and what this was all about, he took me sufficiently seriously to promise that he would without fail deliver the letter to the Prime Minister and stress its urgency. That accomplished, I took my leave and took myself back to Bletchley by the next train. It was some forty years later before I found out what happened to that letter, when I saw the Prime Minister's minute dated the following day... All that we did notice was that almost from that day the rough ways began miraculously to be made smooth. The flow of bombes was speeded up, the staff bottlenecks were relieved, and we were able to devote ourselves uninterruptedly to the business in hand.
Milner-Barry added, 'I by chance met Commander Denniston in the corridors some days later, and he made some rather wry remark about our unorthodox behaviour; but he was much too nice a man to bear malice.'
#### **Letter to Winston Churchill**
_Secret and Confidential
Prime Minister only_
Hut 6 and Hut 8
21st October 1941
Dear Prime Minister,
Some weeks ago you paid us the honour of a visit, and we believe that you regard our work as important. You will have seen that, thanks largely to the energy and foresight of Commander Travis, we have been well supplied with the 'bombes' for the breaking of the German Enigma codes. We think, however, that you ought to know that this work is being held up, and in some cases is not being done at all, principally because we cannot get sufficient staff to deal with it. Our reason for writing to you direct is that for months we have done everything that we possibly can through the normal channels, and that we despair of any early improvement without your intervention. No doubt in the long run these particular requirements will be met, but meanwhile still more precious months will have been wasted, and as our needs are continually expanding we see little hope of ever being adequately staffed.
We realise that there is a tremendous demand for labour of all kinds and that its allocation is a matter of priorities. The trouble to our mind is that as we are a very small section with numerically trivial requirements it is very difficult to bring home to the authorities finally responsible either the importance of what is done here or the urgent necessity of dealing promptly with our requests. At the same time we find it hard to believe that it is really impossible to produce quickly the additional staff that we need, even if this meant interfering with the normal machinery of allocations.
We do not wish to burden you with a detailed list of our difficulties, but the following are the bottlenecks which are causing us the most acute anxiety.
#### 1. Breaking of Naval Enigma (Hut 8)
Owing to shortage of staff and the overworking of his present team the Hollerith section here under Mr Freeborn has had to stop working night shifts. The effect of this is that the finding of the naval keys is being delayed at least twelve hours every day. In order to enable him to start night shifts again Freeborn needs immediately about twenty more untrained Grade III women clerks. To put himself in a really adequate position to deal with any likely demands he will want a good many more.
A further serious danger now threatening us is that some of the skilled male staff, both with the British Tabulating Company at Letchworth and in Freeborn's section here, who have so far been exempt from military service, are now liable to be called up.
#### 2. Military and Air Force Enigma (Hut 6)
We are intercepting quite a substantial proportion of wireless traffic in the Middle East which cannot be picked up by our intercepting stations here. This contains among other things a good deal of new 'Light Blue' intelligence. Owing to shortage of trained typists, however, and the fatigue of our present decoding staff, we cannot get all this traffic decoded. This has been the state of affairs since May. Yet all that we need to put matters right is about twenty trained typists.
#### 3. Bombe testing, Hut 6 and Hut 8
In July we were promised that the testing of the 'stories' produced by the bombes would be taken over by the WRNS in the bombe hut and that sufficient WRNS would be provided for this purpose. It is now late in October and nothing has been done. We do not wish to stress this so strongly as the two preceding points, because it has not actually delayed us in delivering the goods. It has, however, meant that staff in Huts 6 and 8 who are needed for other jobs have had to do the testing themselves. We cannot help feeling that with a Service matter of this kind it should have been possible to detail a body of WRNS for this purpose, if sufficiently urgent instructions had been sent to the right quarters.
**4.** Apart altogether from staff matters, there are a number of other directions in which it seems to us that we have met with unnecessary impediments. It would take too long to set these out in full, and we realise that some of the matters involved are controversial. The cumulative effect, however, has been to drive us to the conviction that the importance of the work is not being impressed with sufficient force upon those outside authorities with whom we have to deal.
We have written this letter entirely on our own initiative. We do not know who or what is responsible for our difficulties, and most emphatically we do not want to be taken as criticising Commander Travis who has all along done his utmost to help us in every possible way. But if we are to do our job as well as it could and should be done it is absolutely vital that our wants, small as they are, should be promptly attended to. We have felt that we should be failing in our duty if we did not draw your attention to the facts and to the effects which they are having and must continue to have on our work, unless immediate action is taken.
We are, Sir, Your obedient servants,
A M Turing
W G Welchman
C H O'D Alexander
P S Milner-Barry
### **CHAPTER 8
Memorandum to OP-20-G on Naval Enigma ( _c.1941_ )**
_Alan Turing_
#### **Introduction: Turing Questions OP-20-G's Attempts to Break Naval Enigma in 1941**
##### _Ralph Erskine, Colin Burke, and Philip Marks_
The document that forms this chapter was written by Alan Turing and sent to the US Navy codebreaking unit, OP-20-G, in Washington, DC, probably to the celebrated cryptanalyst Mrs Agnes Driscoll. It is undated, but was probably dispatched in the autumn of 1941.
#### Background
Turing solved the indicating system of the principal Naval Enigma cipher, _Heimisch_ (codenamed 'Dolphin' by the Government Code and Cypher School (GC & CS)), at Bletchley Park by the end of 1939. Typically, he thought Dolphin 'could be broken because it would be so interesting to break it' (see 'Enigma', p. 257). Hut 8 at Bletchley Park solved some wartime Naval Enigma signals in May and June 1940. Internal evidence shows that Turing wrote his outstanding _Treatise on the Enigma_ around autumn 1940. GC & CS read Dolphin traffic currently, using captured keys, in June and July 1941. The resulting data provided enough cribs and other information to break Dolphin signals cryptanalytically from August onwards, within 24 to 36 hours of their transmission.
Mrs Driscoll was assigned to attack Naval Enigma, with two assistants, around October 1940. However, the US Navy was then intercepting only a small proportion of the Naval Enigma signals being transmitted, and was unable to make any progress against Dolphin, especially since it could not reconstruct the wiring of Enigma's wheels. It did not even fully understand the wheels' noncyclometric motion, which considerably complicated any solution of Naval Enigma, in particular, since each of the special _Kriegsmarine_ wheels VI to VIII had two notches. Notching made the wheel motion irregular, especially when a doubly notched wheel was in the middle or right-hand position. Using two doubly notched wheels could reduce Enigma's period from its normal 16,900 (26 × 25 × 26) to 4,056 (24 × 13 × 13).
In February 1941, following lengthy negotiations between the US Army and Navy, a four-man team led by Abraham Sinkov visited GC & CS. (Sinkov was accompanied by Leo Rosen, also from the US Army's Signal Intelligence Service, and Lt. Robert Weeks and Lt. Prescott Currier, both from OP-20-G.) They brought with them various items of codebreaking material, including one example of 'Purple', the American clone of the Japanese Foreign Ministry's high-level cipher machine. GC & CS staff briefed them fully about Bletchley's work and, with the blessing of Winston Churchill, showed them the top secret bombes used to break Enigma, although they had to undertake to tell only their superior officers about what they had learned about Enigma. Neither Weeks nor Currier was an Enigma cryptanalyst, so that they are unlikely to have followed all the explanations they heard about breaking Enigma. They were given only a 'paper' copy of Enigma's wiring to bring home, since GC & CS did not have an Enigma machine to spare.
Alastair Denniston, the operational head of GC & CS, wanted further cooperation with the US Army and Navy codebreaking units. During a visit he made to Washington in August 1941, Mrs Driscoll told him about her attack on Naval Enigma. She also gave him some questions on which she needed help from GC & CS. Her questions reveal just how little OP-20-G then knew about some important details of Naval Enigma. GC & CS responded to most of her questions and requests for data in early October 1941, although the packet unfortunately went astray, and did not turn up until mid-December.
Turing's memorandum can only have been written at some time after Denniston returned from the United States in late August 1941. It does not refer to the four-wheel Enigma, M4, and was therefore almost certainly prepared before 1 February 1942, when the Atlantic U-boats started to use M4 for their _Triton_ cipher (codenamed 'Shark' by GC & CS). It was probably sent between mid-October and the end of November. There is no record of any reply by OP-20-G: it is quite possible that none was made.
Turing writes concerning Mrs Driscoll's hand attack on Naval Enigma, and therefore does not mention the bombes. However, breaking Dolphin manually was seldom anything other than an extremely slow process, because of its enormous key-space (6.014 × 1023 possible combinations, assuming that the wheel wiring was known). Turing begins by trying to solve only a single wheel order. But he adopts a more rigorous approach in his penultimate paragraph, by requiring all 336 wheel orders to be tested, as would sometimes have been necessary in 1941 when using Mrs Driscoll's method.
#### Aftermath
Turing's memorandum should have been a turning point for OP-20-G's management, since it demonstrated that Mrs Driscoll's methods had no chance whatsoever of producing operationally useful intelligence. It also implied that she did not fully appreciate some of the subtleties of Naval Enigma. But it may not have reached OP-20-G before the Japanese attack on Pearl Harbor, and there is no evidence that it caused OP-20-G to change its approach to Naval Enigma, or to Mrs Driscoll's work, since she continued to receive support on Enigma until well into 1942. Lt.-Col. John Tiltman, GC & CS's most senior cryptanalyst, considered that she 'was making no original contribution at all', but she remained part of the attack on Naval Enigma throughout 1942. However, it is not known whether Tiltman passed his views on to OP-20-G's management, who treated her with 'exaggerated respect' on account of her pre-war achievements.
A new generation of young, mathematically trained cryptanalysts, many of whom had been naval reservists, was recruited for OP-20-G after Pearl Harbor. They developed an outline bombe design for OP-20-G in April 1942, although it would have been twenty-six times less efficient than GC & CS's bombe, since it did not employ 'simultaneous scanning' (i.e. the simultaneous testing of twenty-six possible Stecker values at each position of the bombe's wheels; see 'Enigma', pp. 254–5, and Chapter 6, pp. 319–20), about which OP-20-G seems to have known nothing at that time. OP-20-G therefore appears not to have seen Turing's _Treatise_ in early 1942. This is not wholly surprising. Until May 1942, GC & CS did not want OP-20-G to attack Naval Enigma, and may deliberately have withheld a copy of the _Treatise_. A copy was sent to the United States at some point, but probably only to the US Army. OP-20-G only learned the full details of the GC & CS bombe, and about simultaneous scanning, after Lt. Joseph Eachus and Lt. Robert Ely arrived at Bletchley Park in July 1942, to learn how GC & CS attacked Naval Enigma.
M4 Enigma proved devastating when it entered service on Shark in February 1942, since it took effect shortly after the introduction of a new edition of a weather short signal book (the _Wetterkurzschlüssel_ ), which deprived Hut 8 of cribs for Shark. Without cribs, and lacking four-wheel bombes, GC & CS became blind against Shark. M4 was not a true four-wheel machine. The new wheel, beta (which was stationary during encipherment), and its associated reflector, thin B, gave M4 the equivalent of twenty-six reflectors, but beta was not interchangeable with wheels I to VIII. Fortunately, GC & CS had solved beta and thin B in December 1941, when M4 was used inadvertently.
The new _Wetterkurzschlüssel_ was captured from _U-559_ on 30 October 1942, and reached GC & CS in late November. Hut 8 found that M4 emulated M3 when enciphering weather short signals, allowing Shark to be solved using relatively few three-wheel bombes. On 13 December, Hut 8 solved Shark keys for 5 to 7 December. It continued to do so for most days using three-wheel bombes, albeit with delays, until British four-wheel bombes entered service in June 1943.
GC & CS's failure to develop a four-wheel bombe quickly had led the US Navy to decide, on 4 September 1942, to embark on a massive four-wheel bombe-building programme of its own. In a considerable tour de force, Joseph Desch, of the National Cash Register Co., in Dayton, Ohio, submitted a detailed design proposal to OP-20-G on 15 September. In October 1942, OP-20-G and GC & CS entered into the Holden Agreement, which provided for 'full collaboration upon the German submarine [Shark] and naval cryptanalysis problems'.
After a difficult testing period for two bombe prototypes from May to late July 1943, OP-20-G four-wheel bombes came into service in August 1943. They performed so well that GC & CS came to rely on them extensively for work against Shark and other _Kriegsmarine_ four-wheel ciphers, not least because as late as March 1944 the performance of the British four-wheel bombes was, in the words of Hugh Alexander, Turing's successor as head of Hut 8, 'poor and likely to remain so'. The OP-20-G bombes also carried out a vast amount of work for Hut 6 on German Army and Air Force Enigma ciphers: for much of 1944 OP-20-G devoted around 60 per cent of its bombe time to those ciphers.
Agnes Driscoll gradually faded out of the Naval Enigma scene in 1942, as indeed, for different reasons, did Alan Turing. She is mentioned only once or twice in the extant records for 1942, and did not write any of the extensive series of papers on Naval Enigma which have been preserved in the American archives. She was transferred to the attack on JNA-20 (codenamed 'Coral' by OP-20-G), the cipher machine used by Japanese naval attachés, on 31 January 1943.
##### _Provenance_
Turing's typescript and figures were released only in late 1998. Ralph Erskine found the figures at College Park in October 1999. He recognized them as being in Turing's handwriting but, not being attached to Turing's unsigned typescript (which he had received earlier from Stephen Budiansky), they completely lacked context, except that they clearly dealt with Naval Enigma. A typescript by Colin Burke 'Agnes Meyer Driscoll v the Enigma and the Bombe' described the typescript's author as being 'perhaps Alan Turing'. A careful study of the type-script showed that it was indeed by Turing: two words are in Turing's hand, and the typewriter used is identical with that employed for Turing's _Treatise_. When Erskine remembered the figures and re-examined them, the connection with Turing became patent. The figures tie in perfectly with references to them in the typescript, and the crib (VVVBDUU) is the same in both documents. Turing's authorship of the typescript is therefore beyond question.
#### **Memorandum to OP-20-G on Naval Enigma**
We are rather surprised to hear that you are able to find the keys, given that a message when deciphered says VVVBDUUU. Our experience shows that with a 'crib' as short as 8 letters there are always far too many sets of keys consistent with the data, so that whatever method may be used for discovering the keys the time required to test these solutions out further becomes prohibitive. To illustrate this I have enciphered VVVBDUUU with a random chosen key viz wheel order 457, English Ringstellung RWH, pre-start window position SZK and Stecker A/P, B/Y, C/L, E/Q, F/X, K/R, M/W, N/T, O/V, S/Z, giving YFZONMTY. I then imagined that
was a crib that I had to solve, but that I knew the wheel order and Ringstellung: I tried out the hypothesis that the pre-start window position was the right one (SZK) and also the five which follow it (allowing correctly for turnovers) viz TAL, TAM, TBN, TBO, TBP, and found that with pre-start TBP there is a solution with V/J, F/G, Z/H, Y/E, U/X, M/L, T/K, D/P and either B/S and O/W or B/W and O/S. The 'unsteckered alphabets' for the relevant positions of the machine are shown in Fig [8.1], and the working in Fig [8.2]. I hope that this working is self-explanatory. Each column of letters consists of steckers of the letters VFZYUMT which imply one another on account of the crib.
A continuation of this process would probably give about 3000 essentially different solutions per wheel order. Of course these solutions are not all equally likely: e.g. the solution with pre-start TBP is unlikely as the complete set of 10 Stecker has to be assumed to account for the whole crib, whereas with the right solution we can only deduce O/V, F/X, S/Z, Y/B, W/M, N/T, and D and U self-steckered. But there will still be a great many that look as good as this. A fairly simple calculation tells us the probability of this solution being the right one, under the assumption that the wheel order and Ringstellung are right. The total number of ways of setting up 10 Stecker is about 1.5 × 1014, and the number of essentially different window positions is 16,224, so that the total number of sets of keys in question is about 2.4 × 1018. From this we can obtain the expected number of sets of keys consistent with the data by multiplying by 26−8. We get 1.15 × 107. Now the solution in question can be made into a complete set of keys, by completing the Stecker in 51,975 ways, i.e. it corresponds to 51,975 of the 1.15 × 107 solutions and therefore has a probability of 51,975/1.15 × 107 or 0.0045. We may therefore expect to have anything from say 50 to 1,000 solutions to test further on each wheel order, even if we assume the Ringstellung, or, what comes to the same, the position of the turnover in the message. The examination of these solutions is not very easy, especially in the case of likely looking solutions, as in such cases we necessarily know comparatively few Stecker, and so can get very little of the plain text of the message.
**Figure 8.1.** Alphabets for
**Figure 8.2.** Workings to solve
Editors' note: Turing continues with similar arrays to the one set out above, for one to five places after the right one. In the interests of space, those arrays are not reproduced here, but may be downloaded from www.AlanTuring.net/OP-20-G_figures.
The working shown in Fig [8.2] is not given as a suggested method for solving these cribs. It is part of an _a fortiori_ argument to the effect that even if all solutions had been found by this method or some other the remaining work to be done would still be too much.
Leaving aside this general aspect of the problem I should be interested to be sure that I understand your method correctly: the argument given above depends essentially on the length of the crib, and it may well be that you have a method which will deal with rather longer cribs.
As I understand it your method is to assume Stecker for certain letters thereby obtaining certain 'unsteckered' constatations. One then takes 26 separate hypotheses concerning the position of the R.H.W. [right-hand wheel] and deduces, for each hypothesis, the 'output' of the two left hand wheels and U.K.W. Assuming the wheel order one then looks up in a catalogue and finds the possible positions of the two wheels on the left. The whole effect of the process so far is to find the positions of the wheels consistent with the unsteckered constatations. Each position must be examined more closely afterwards with a machine.
The process may be explained by means of an example.
This is a favourable one as the same constatation VD occurs twice over. Suppose now that we wish to try out the hypothesis that V and D are both self-steckered on wheel order 457. Assume that there is no turnover between the two occurrences of VD. We lay down the 'Inverse rods' for V and D and wheel 7; the effect of this is shewn in Fig [8.3]. The information we get from them is for instance that if VD were enciphered with no Stecker and with the R.H.W. in absolute position (window position less English Ringstellung) 17, then, if the R.H.W. were replaced by a 'straight through' wheel, and the other wheels kept the same and in the same positions, then the effect of enciphering G would be V. We wish therefore to find where, with wheel order '4 5 straight', we can get the pairings EH and JF, also where we can get JF and PO, and so on. We have catalogues in which we can look these pairs up. We find for instance that PO, AZ occurs in position 8 of the L.H.W. and 22 of the M.W. [middle wheel] and therefore that without Stecker we get VD in absolute positions 8, 22, 3 and 8, 22, 4. The complete set of solutions is shewn in Fig [8.4]. These solutions have now to be tested out on the remainder of the crib. Take the case of the solution 8, 22, 3 and suppose we are assuming there is no turnover in the whole crib. Then the DU will have been enciphered at 8, 22, 9 at which position D enciphers without Stecker to Z. Since we are assuming that D is self-steckered, we must have Z/U. Now the UY constatation was enciphered at 8, 22, 6 where Z without Stecker enciphers to V. We therefore have V/Y contrary to the hypothesis that V was self-steckered.
**Figure 8.3.** Inverse rods of wheel VII, for solving
Editors' note: Figure 8.3 shows the V and D inverse rods for wheel 7. These show that V and D map to G and V, respectively, when taken through wheel 7 in position 17. Since V and D are paired in the crib, this means that the G and V contacts are connected together through the unsteckered 'two-wheel Enigma' formed by the other two wheels (4 and 5) and the reflector. The catalogue (which is not reproduced here) shows the results produced by such a two-wheel Engima for all possible wheel orders and wheel positions.
The full examination of the possibilities of turnover takes some considerable time. Of course it is only worth while considering rather longer cribs than VVVBDUUU: with cribs of length 20 it would be possible to deal with a wheel order on one assumption of Stecker for the letters taking the place of V and D in about five hours, of which about half an hour or less would be the looking up in catalogues. Suppose that we have a very large supply of cribs, 100 a day say, each with probability [½] of being right. The chance of the two letters being self-steckered is 3/65, and therefore working on 336 wheel orders we should have on average 22⅓ × 336 × 5 × 2 i.e. 72,800 hours work to obtain a solution.
**Figure 8.4.** Position where we get on wheel order 4 5 7.
Editors' note: Since the two V/D pairings are adjacent in the crib, solutions can occur only at positions where the two-wheel Enigma produces two adjacent pairings on the inverse rods. Figure 8.4 shows the complete set of such positions. In Turing's example (wheel positions 8, 22, 3), with V and D self-steckered, the successive pairings on the rods are P/O and A/Z, and the catalogue tells us that the two-wheel Enigma with wheel order 4 5 produces both of these pairings when wheel 4 is at position 8 and wheel 5 is at position 22. The inverse rods show pairing PO at position 3, thus the first of the V/D pairings in the crib must occur when wheel 7 is at position 3. The solution would be much more laborious if the very favourable V/D constatation had not been repeated, since more Stecker assumptions would have to be made in order to constrain the number of possible solutions.
Would you mind telling us your method
1) Does it give the keys starting from scratch, or does one need to start with the Stecker?
2) Is the above account substantially correct?
3) Do you work with cribs as short as VVVBDUUU? Have you any longer ones?
4) About how many hours work do you estimate would be necessary to obtain a solution on 336 wheel orders?
## **Artificial Intelligence**
_Jack Copeland_
Turing was the first to carry out substantial research in the field now known as Artificial Intelligence or AI. The term 'Artificial Intelligence' itself did not come into use until after Turing's death, making a prominent appearance in 1956, in the title of a conference held at Dartmouth College, New Hampshire: _The Dartmouth Summer Research Project on Artificial Intelligence_. Turing's original term, 'machine intelligence', remains in use, especially in Britain.
#### 1. AI at Bletchley Park
Turing was thinking about machine intelligence at least as early as 1941. During the war he circulated a typewritten paper on machine intelligence among some of his colleagues at the Government Code and Cypher School (GC & CS). Now lost, this was undoubtedly the earliest paper in the field of AI. It probably concerned the mechanization of problem-solving and the idea of machines learning from experience; both were topics that Turing discussed extensively during the war years at GC & CS.
Turing enthusiastically discussed the mechanization of chess with Donald Michie and others at Bletchley Park (see the introduction to Chapter 16). Michie (a leading codebreaker at GC & CS) recalls Turing's speaking often during the war about the possibility of computing machines solving problems by means of searching through the space of possible solutions, guided by rules of thumb. (Convinced by Turing that AI was worth pursuing, Michie himself went on to found the influential Department of Machine Intelligence and Perception at the University of Edinburgh.)
Turing's thinking on AI was probably influenced by his work on the bombe (see 'Enigma' and Chapter 6). Central to the bombe was the idea of solving a problem by means of a guided mechanical search through the space of possible solutions. The space of possible solutions searched by the bombe consisted of configurations of the Enigma machine, but the space searched by a different form of machine might consist of configurations of a chess board, for example. The bombe's search could be guided in various ways; one involved the 'multiple encipherment condition' associated with a crib (see Chapter 6, p. 317). The search would 'reduce the possible positions to a number which can be tested by hand methods' (ibid.).
Modern AI researchers speak of the method of 'generate-and-test'. Potential solutions to a given problem are generated by means of a guided search. These potential solutions are then tested by an auxiliary method in order to find out if any actually is a solution. The bombe mechanized the first process. The testing of the stops, or potential solutions, was then carried out manually (by setting up a replica Enigma accordingly, typing in the cipher text, and seeing whether or not German came out). Nowadays in AI both processes, generate and test, are typically carried out by the same programme.
In 1948 Turing boldly hypothesized that 'intellectual activity consists mainly of various kinds of search' (Chapter 10, p. 431). His readers would no doubt have been astonished to learn of his wartime experience with mechanized search (still secret at that time). Some eight years later the same hypothesis was put forward independently by Herbert Simon and Allen Newell in the USA; through their influential work, it became one of the central tenets of AI.
Turing's work on the bombe involved him in the design of the _Ringstellung_ 'cutout' (Chapter 6, p. 335). The _Ringstellung_ cut-out is an early example of constraining search by means of what modern AI researchers call _heuristics_. A heuristic is a rule that cuts down the amount of searching required in order to find potential solutions. Unlike a decision method (see 'Computable Numbers: A Guide'), a heuristic is not necessarily _guaranteed_ to produce the correct solution, but works often enough to be useful. For example, one heuristic that everybody uses from time to time is: if you have lost something, search in the vicinity of the place where you think you dropped it. Heuristic search is one of modern AI's central techniques. The Poles in effect used heuristic search in the bomba: the number of possibilities to be examined was reduced by the assumption, true for more messages than not, that the letters of the indicator were unsteckered (see 'Enigma').
The _Ringstellung_ cut-out would be set so as to prevent the bombe from stopping at certain positions at which it would otherwise have stopped—positions ruled out by conjectured information about the ring settings. The device thereby reduced the number of stops to be tested by hand. Early in 1940 John Herivel had discovered the heuristic concerning ring settings that helped Hut 6 break 'Red' Enigma daily. When a German operator sent his first message of the day, he would be liable to use for the indicator setting either the three letters visible in the windows when he had finished setting the rings, or the three letters visible at a closely neighbouring position of the wheels, for example the position produced by his having lazily turned a single wheel some small number of clicks (see Chapter 6, n. 26). By assuming that this was so in the case of any given first message, the codebreakers could (Herivel said) 'narrow down the 17,576 possible ring settings to a manageable number, say twenty or thirty'. Of course, the assumption might be wrong in any particular case, but was correct often enough to be useful.
In _Treatise on the Enigma_ Turing also described a 'majority vote gadget', mechanizing the process of evaluating certain hypotheses on the basis of unreliable data, e.g. unreliable data about _Stecker_ (see Chapter 6, p. 335). Although the task mechanized by the gadget is a relatively undemanding one, it is nevertheless of a type that might be described as requiring or evidencing intelligence when carried out by human beings. The fact that such a task can be carried out by a machine is suggestive. Leaving aside tasks that are provably uncomputable (concerning which see Turing's discussion, in Chapters 9, , , and , of what he called the 'Mathematical Objection'), might it be the case that all the tasks we normally describe as demanding or showing intelligence can be reduced to rules that a computing machine can follow? In 1940 nothing was known to falsify the daring hypothesis that this is so, and the same remains true today.
#### 2. AI in Post-war Britain
The birth of Artificial Intelligence as a field of research is usually placed at 1956, the year of the Dartmouth Summer Research Project on Artificial Intelligence and also the year in which a programme written by Newell, Simon, and Shaw—later named 'the Logic Theorist'—proved theorems from Whitehead and Russell's famous work on mathematical logic _Principia Mathematica_. However, this received view of the matter is hardly accurate. By 1956 computer intelligence had been actively pursued for more than a decade in Britain; the earliest AI programmes to run were written there in 1951–2. That the earliest work in the field was done in Britain is a consequence of the fact that the first electronic stored-programme digital computers to function were built at Manchester and Cambridge universities (see the introduction to Chapter 9), and another important factor was the influence of Turing on the first generation of computer programmers.
In London in 1947 Turing gave what was, so far as is known, the earliest public lecture to mention computer intelligence, providing a breathtaking glimpse of a new field (Chapter 9). In 1948 he wrote his National Physical Laboratory report 'Intelligent Machinery' (Chapter 10). This, the first manifesto of Artificial Intelligence, adumbrated the methods of the new field and included Turing's proposals for connectionist-style neural simulation. In 1950 Turing published 'Computing Machinery and Intelligence' (Chapter 11), probably the best known of all his papers. In it Turing addressed mainly philosophical and logical issues, introducing his now famous imitation game or 'Turing test'. There then followed his three radio broadcasts on AI: the lectures 'Intelligent Machinery, A Heretical Theory' (Chapter 12) and 'Can Digital Computers Think?' (Chapter 13), and the panel discussion 'Can Automatic Calculating Machines Be Said to Think?' (Chapter 14). In 1953 Turing's last work on AI was published, a groundbreaking essay on computer chess (Chapter 16).
#### 3. The First AI Programmes
Both during and after the war Turing experimented with machine routines for playing chess (see Chapter 16). In the absence of an electronic computer, the machine's behaviour was simulated by hand, using paper and pencil. The first chess programme to run electronically was written for the Manchester University computer by Dietrich Prinz in 1951 (see Chapter 16).
When Turing delivered his lecture 'Can Digital Computers Think?' on British radio one of his listeners was Christopher Strachey. Strachey's draughts (or checkers) programme first ran successfully in Turing's Computing Machine Laboratory at Manchester University. The programme used simple heuristics and looked ahead three to four turns of play. The state of the board was represented on the face of a cathode ray tube—one of the earliest uses of computer graphics. (Strachey was at this time a schoolmaster at Harrow; he later became Director of the Programming Research Group at Oxford University, where with the mathematical logician Dana Scott he did the work on the semantics of programming languages for which he is best known.)
Strachey initially coded his draughts programme in May 1951 for the Pilot Model of Turing's Automatic Computing Engine at the National Physical Laboratory (see Chapter 9). This version of the programme never ran successfully. An attempt to run it in July 1951 foundered due to programming errors. When Strachey was ready to try the programme again, he discovered that the computer's hardware had been modified; his programme could not be run without extensive changes. Strachey switched his attention to the Manchester University Computing Machine Laboratory, where the first Ferranti Mark I computer was installed in February 1952. With Turing's encouragement, and using Turing's recently completed _Programmers' Handbook_ for the Ferranti machine, Strachey finally got his programme working. By the summer of 1952 the programme could play a complete game of draughts at a reasonable speed. (Built by the Manchester firm of Ferranti in close collaboration with the Computing Machine Laboratory, the Mark I was the world's first commercially available electronic stored-programme computer.)
**Figure 1.** Turing standing at the console of the Manchester Ferranti computer.
_Source_ : Reprinted with permission of the Department of Computer Science, University of Manchester.
In 1952 Strachey described his draughts programme at a computing conference in North America. Arthur Samuel of IBM took over the essentials of Strachey's programme and wrote a checkers player for the IBM 701 (IBM's first mass-produced electronic stored-programme computer). Samuel's checkers programme first ran at the end of 1952 and appears to have been the earliest AI programme in the USA. In 1955 Samuel added learning to the programme and over a period of years made successive improvements to the learning apparatus. In 1962 his programme won a game against a former Connecticut checkers champion, R. W. Nealey. Nealey, who immediately turned the tables and beat the programme in six consecutive games, is reported to have said: 'In the matter of the end game, I have not had such competition from any human being since 1954, when I lost my last game.'
Strachey was thinking about mechanized learning at the time of writing his draughts player. He devised a simple rote-learning scheme which he envisaged being implemented in a NIM-playing programme. Strachey wrote at length concerning learning in a letter to Turing, composed on the evening of Turing's lecture 'Can Digital Computers Think?' He said:
I have just been listening to your talk on the Third Programme. Most stimulating... [i]n particular your remark... that the programme for making a machine think would probably have great similarities with the process of teaching; this seems to me absolutely fundamental.... I am convinced that the crux of the problem of learning is recognizing relationships and being able to use them.... There are, I think, three main stages in learning from a teacher. The first is the exhibition of a few special cases of the rule to be learned. The second is the process of generalisation—i.e. the underlining of the important features that these cases have in common. The third is that of verifying the rule in further special cases and asking questions about it. I have omitted any mention of 'understanding' the rule, because this is not appropriate at the moment to the action of a machine. I think, as a matter of fact, that the process of understanding a rule is connected with finding relationships between it and other rules—i.e. second (or higher) order relations between relations and this might well become important for a machine later.... I think it might well be possible to programme the Manchester machine to do all of these stages, though how much it would be able to learn in this way before the storage became inadequate remains to be seen.
Strachey missed the opportunity to be the first to achieve a functioning programme incorporating learning, however. The earliest programmes to do so were written by Anthony Oettinger for the EDSAC computer at the University of Cambridge Mathematical Laboratory. Oettinger was considerably influenced by Turing's 'Computing Machinery and Intelligence' (Chapter 11).
Oettinger's 'response-learning programme', dating from 1951, could be taught to respond appropriately to given stimuli by means of expressions of 'approval' or 'disapproval' by the teacher. As training proceeded errors became less frequent, and the learned response would be initiated by a progressively weaker stimulus. Oettinger described the response-learning programme as 'operating at a level roughly corresponding to that of conditioned reflexes', and he noted that the 'behaviour pattern of the response-learning... machine is sufficiently complex to provide a difficult task for an observer required to discover the mechanism by which the behaviour of the... machine is determined.'
Oettinger's 'shopping machine', also dating from 1951, incorporated rote-learning. Adopting Turing's terminology, Oettinger described this programme as a 'child machine' (see Chapter 11, p. 460). Shopper's simulated world was a mall of eight shops. When sent out to purchase an item whose location was unknown Shopper would search for it, visiting shops at random until the item was found. While searching, Shopper would memorize a few of the items stocked in each shop that it visited. Next time Shopper was sent out for the same item, or for some other item that it had already located, it would go to the right shop straight away. (Oettinger was the first of many programmers to claim a programme capable of passing a restricted form of the Turing test. The shopping machine could, he remarked, successfully play a version of Turing's imitation game in which the 'questions are restricted to... the form "In what shop may article _j_ be found?"')
#### 4. Subsequent Developments
Within a decade of this early work, Artificial Intelligence had become an established and burgeoning area of research. Some landmarks in the development of the field were:
• During the latter part of the 1950s and the early 1960s, Artificial Intelligence laboratories were set up at a number of US and British universities, notably at Carnegie Mellon University (under Newell and Simon), Edinburgh University (under Michie), Massachusetts Institute of Technology (under Marvin Minsky), and Stanford University (under John McCarthy, the organizer of the Dartmouth Summer Research Project on Artificial Intelligence).
• The 1950s saw the development of a number of programmes able to carry out tasks of a sort usually said to require intelligence when carried out by human beings. The most famous of these early programmes was the General Problem Solver or GPS (written by Newell, Simon, and Shaw). GPS could solve a variety of puzzles. One example is the 'missionaries and cannibals' problem: how can a party of three missionaries and three cannibals cross a river by means of a boat holding at most two people, without the missionaries on either bank ever becoming outnumbered by cannibals?
• Work on neuron-like computation got under way in Britain and the USA during the 1950s. (Those involved included J. T. Allanson, R. L. Beurle, W. A. Clark, B. G. Farley, F. Rosenblatt, W. Ross Ashby, W. K. Taylor, and A. M. Uttley). Ross Ashby's influential book _Design for a Brain_ was published in 1952 and Rosenblatt's _Principles of Neurodynamics_ in 1962. Rosenblatt was widely influential and numerous research groups in the USA pursued his approach. He called this approach 'connectionist', emphasizing—as Turing had in 1948 (Chapter 10)—the role in learning of the creation and modification of connections between (real or simulated) neurons.
• At the close of the 1950s John McCarthy developed the computer language that he called LISP (from 'list processor'). Designed specifically for AI programming, LISP remains today one of the principal languages for AI work. McCarthy took some of the ideas used in LISP from Church's lambda calculus (see the introduction to Chapter 4).
• In 1965 AI researcher Edward Feigenbaum and geneticist Joshua Lederberg (both of Stanford University) began work on their programme Heuristic Dendral (subsequently shortened to dendral). The programme's task was chemical analysis. The substance to be analysed might be a complicated compound of carbon, hydrogen, and nitrogen, for example. Starting from spectrographic data obtained from the substance, dendral would hypothesize the substance's molecular structure. dendral's performance rivalled that of human chemists expert at this task, and the programme was used in industry and in universities. This high-performance programme was the model for much of the ensuing work in the important area of _expert systems_ (see the introduction to Chapter 10).
##### _Further reading_
Boden, M. A., _Mind as Machine: A History of Cognitive Science_ (Oxford: Oxford University Press, 2005).
Copeland, B. J., _Artificial Intelligence: A Philosophical Introduction_ (Oxford: Blackwell, 1993).
Copeland, B. J., 'Artificial Intelligence', _Encyclopaedia Britannica_ (15th ed. 2001).
Haugeland, J., _Artificial Intelligence: The Very Idea_ (Cambridge, Mass.: MIT Press, 1985).
McCorduck, P., _Machines Who Think: A Personal Enquiry into the History and Prospects of Artificial Intelligence_ (New York: W. H. Freeman, 1979).
Michie, D., _On Machine Intelligence_ (2nd ed. Chichester: Ellis Horwood, 1986).
### **CHAPTER 9
Lecture on the Automatic Computing Engine ( _1947_ )**
_Alan Turing_
#### **Introduction**
##### _Jack Copeland_
#### Electronics at Bletchley Park
On 8 December 1943 the world's first large-scale special-purpose electronic digital computer—'Colossus', as it became known—went into operation at the Government Code and Cypher School (see 'Computable Numbers: A Guide', 'Enigma', and the introduction to Chapter 4). Colossus was built by Thomas H. Flowers and his team of engineers at the Post Office Research Station in Dollis Hill, London. Until relatively recently, few had any idea that electronic digital computation was used successfully during the Second World War, since those who built and worked with Colossus were prohibited by the Official Secrets Act from sharing their knowledge.
Colossus contained approximately the same number of electronic valves (vacuum tubes) as von Neumann's IAS computer, built at the Princeton Institute of Advanced Study and dedicated in 1952. The IAS computer was forerunner of the IBM 701, the company's first mass-produced stored-programme electronic computer (1953). The first Colossus had 1,600 electronic valves and Colossus II, installed in mid-1944, 2,400, while the IAS computer had 2,600.
Colossus lacked two important features of modern computers. First, it had no internally stored programmes (see 'Computable Numbers: A Guide'). To set up Colossus for a new task, the operators had to alter the machine's physical wiring, using plugs and switches. Second, Colossus was not a general-purpose machine, being designed for a specific cryptanalytic task (involving only logical operations and counting). Nevertheless, Flowers had established decisively and for the first time that large-scale electronic computing machinery was practicable.
The implication of Flowers's racks of electronic equipment would have been obvious to Turing. Once Turing had seen Colossus it was, Flowers said, just a matter of Turing's waiting to see what opportunity might arise to put the idea of his universal computing machine into practice.
#### Turing Joins the National Physical Laboratory
Precisely such an opportunity fell into Turing's lap in 1945, when John Womersley invited him to join the Mathematics Division of the National Physical Laboratory (NPL) at Teddington in London, in order to design and develop an electronic stored-programme digital computer—a concrete form of the universal Turing machine of 1936. Womersley named the proposed computer the Automatic Computing Engine, or ACE, in homage to Babbage and his planned calculating machines, the Difference Engine and the Analytical Engine (see 'Computable Numbers: A Guide').
The formal date of Turing's appointment was 1 October 1945. Womersley reviewed the events leading up to the appointment in an NPL document entitled 'A.C.E. Project – Origin and Early History' (26 November 1946).
_1936–37_ Publication of paper by A.M. Turing 'On Computable Numbers, with an Application to the Entscheidungsproblem'....
_1937–38_ Paper seen by J.R.W. [J. R. Womersley] and read. J.R.W. met C. L. Norfolk, a telephone engineer who had specialised in totalisator design and discussed with him the planning of a 'Turing machine' using automatic telephone equipment. Rough schematics prepared, and possibility of submitting a proposal to N.P.L. discussed. It was decided that machine would be too slow to be effective.
_June 1938_ J.R.W. purchased a uniselector and some relays on Petty Cash at R.D. Woolwich for spare-time experiments. Experiments abandoned owing to pressure of work on ballistics,....
_Late 1943_ J.R.W. first heard of [the] American machines. [Editor's note: Aiken's Sequence-Controlled Calculator at Harvard University and Stibitz's Relay Computer at Bell Telephone Laboratories—these machines were neither electronic nor stored-programme.]
_1944_ Interdepartmental Committee on a Central Mathematical Station. D. R. Hartree mentioned at one meeting the possible use of automatic telephone equipment in the design of large calculating machines. J.R.W. submitted suggestions for a research programme to be included in Committee's Report.
_1944 Sept_. J.R.W. chosen for Maths. Division.
_1944 Oct_. J.R.W. prepares research programme for Maths. Division which includes an item covering the A.C.E.
_1944 Nov_. J.R.W. addresses Executive Committee of N.P.L. Quotation from M/S (delivered verbatim)...'
'Are we to have a mixed team developing gadgets of many kinds... Or are we, following Comrie... to rely on sheer virtuosity in the handling of the ordinary types of calculating machines? I think either attitude would be disastrous... We can gain the advantages of both methods by adopting electronic counting and by making the instructions to the machine automatic...'
_1945 Feb–May_ J.R.W. sent to the U.S.A. by Director. Sees Harvard machine and calls it 'Turing in hardware'. (Can be confirmed by reference to letters to wife during visit). J.R.W. sees ENIAC and is given information about EDVAC by Von Neumann and Goldstine.
_1945 June_ J.R.W. meets Professor M. H. A. Newman. Tells Newman he wishes to meet Turing. Meets Turing same day and invites him home. J.R.W. shows Turing the first report on the EDVAC and persuades him to join N.P.L. staff, arranges interview and convinces Director and Secretary.
#### The Automatic Computing Engine
During the remainder of 1945 Turing drafted his technical report 'Proposed Electronic Calculator'. According to Michael Woodger—Turing's assistant at the NPL from 1946—an NPL file gave the date of Turing's completed report as 1945 (unfortunately, this file was destroyed in 1952). Woodger believes that Turing probably wrote the report between October and December 1945. The report was submitted to the Executive Committee of the NPL in February 1946, under the title 'Proposals for the Development of an Automatic Computing Engine (ACE)', and on 19 March 1946 the 'Committee resolved unanimously to support with enthusiasm the proposal that Mathematics Division should undertake the development and construction of an automatic computing engine of the type proposed by Dr. A. M. Turing'.
**Figure 1.** The Pilot ACE in December 1950. On the left are the control table and the modified Hollerith punched card unit (on the table with contoured legs). The tray slung below the main frame contains the short delay lines used for temporary storage.
_Source_ : Crown copyright. Reproduced by permission of the National Physical Laboratory.
Turing's 'Proposed Electronic Calculator' gave the first relatively complete specification of an electronic stored-programme digital computer. The earlier 'First Draft of a Report on the EDVAC', written by von Neumann in the USA in about May 1945 (see 'Computable Numbers: A Guide'), contained little engineering detail, in particular concerning electronic hardware. Harry Huskey, the electronic engineer who subsequently drew up the first detailed specifications of possible hardware configurations for the EDVAC, has stated that the 'information in the "First Draft" was of no help in this' and that 'von Neumann's "First Draft" provided no technical contribution to the development of computers'. Turing's proposal, on the other hand, supplied detailed specifications of hardware units, including circuit designs, and specimen programmes in machine code. Turing even gave an exact estimate of the cost of building the machine (£11,200).
Turing's ACE and the EDVAC (which was not fully working until 1952) differed fundamentally in design. The EDVAC had what is now called a central processing unit or cpu, whereas in the ACE different memory locations had specific logical or numerical functions associated with them. For example, if two numbers were transferred to a certain destination in memory their sum would be formed there, ready to be transferred elsewhere by a subsequent instruction. Programmes for the ACE were made up entirely of instructions such as 'Transfer the contents of Temporary Store 27 to Temporary Store 6'. Instead of writing mathematically significant instructions like
MULTIPLY _x_ BY _y_ AND STORE THE RESULT IN _z_
the programmer composed a series of transfer instructions producing that effect.
Turing saw that size of memory and speed were the keys to computing. (His assistant at the NPL, James Wilkinson, observed that Turing 'was obsessed with the idea of speed on the machine'.) Turing's design specified a high-speed memory of roughly the same capacity as an early Macintosh computer—enormous by the standards of his day. In order to increase the speed of a programme's execution, he proposed that instructions be stored, not consecutively, but at carefully chosen positions in memory, with each instruction containing a reference to the position of the next. Also with a view to speed, he included a small fast-access memory for the temporary storage of whichever numbers were used most frequently at a given stage of a computation. According to Wilkinson in 1955, Turing 'was the first to realise that it was possible to overcome access time difficulties with... mercury lines... or drum stores by providing a comparatively small amount of fast access store. Many of the commercial machines in the USA and... in this country make great use of this principle.'
Turing's philosophy—very different from that embraced in the EDVAC—was to dispense with additional hardware in favour of software: in his design, complex behaviour was to be achieved by complex programming rather than by complex equipment (such as a hardware multiplier and divider, and special hardware for floating-point arithmetic). The ACE therefore had much in common with today's RISC (Reduced Instruction Set Computing) architectures. Turing spoke disparagingly of the contrary 'American tradition of solving one's difficulties by means of much equipment rather than thought'.
Had Turing's ACE been built as he planned, it would have been in a different league from the other early computers. However, from early 1947—for reasons that Turing might have regarded as amounting to 'cowardly and irrational doubts'—his colleagues in the ACE Section were in favour of devoting their efforts to building a much scaled-down and simplified form of Turing's design, which they called the 'Test Assembly', rather than pressing on ahead immediately with the full-scale ACE. It was not in Turing's nature to direct them otherwise. He 'tended to ignore the Test Assembly', simply 'standing to one side' (Wilkinson said). Finally, in 1948, a disenchanted Turing left the NPL for Newman's Computing Machine Laboratory at Manchester University. The introduction to the next chapter tells the full story of the frustrating delays at the NPL and of Turing's move to Manchester (and see also the introduction to Chapter 4).
It was not until May 1950 that a small 'pilot model' of the Automatic Computing Engine, built by Wilkinson, David Clayden, Donald Davies, Edward Newman, Michael Woodger, and others, executed its first programme. With an operating speed of 1 MHz, the Pilot Model ACE was for some time the fastest computer in the world. However, years of delays beyond Turing's control had cost the NPL the race to build the world's first stored-programme electronic digital computer—an honour that went to the University of Manchester, where, in Newman's Computing Machine Laboratory, the 'Manchester Baby' ran its first programme on 21 June 1948. (As its name implies, the Baby was a very small computer, and the news that it had run what was only a tiny programme—just seventeen instructions long—for a mathematically trivial task was 'greeted with hilarity' by the NPL team developing the much more sophisticated Pilot Model ACE.) Turing's influence on the Manchester computer is described below.
The EDSAC computer, built by Maurice Wilkes at Cambridge University, became operational in 1949 and was the second stored-programme electronic computer to run. The Pilot Model ACE was also preceded by the BINAC (1949), built by Eckert and Mauchly at their Electronic Control Company, Philadelphia (although opinions differ over whether the BINAC ever actually worked), the CSIR Mark 1 (1949), built by Trevor Pearcey at the Commonwealth Scientific and Industrial Research Organization, Division of Radiophysics, Sydney, Australia, the Whirlwind I (1949), built by Jay Forrester at the Digital Computer Laboratory, Massachusetts Institute of Technology, and the SEAC (1950), built by Samuel Alexander and Ralph Slutz at the US Bureau of Standards Eastern Division, Washington, DC. Huskey's SWAC, built at the US Bureau of Standards Western Division, Los Angeles, ran a few months after the Pilot Model ACE (in August 1950).
#### Derivatives of Turing's ACE Design
The DEUCE, the production version of the Pilot Model ACE, was built by the English Electric Company. The first was delivered in March 1955 (to the NPL) and the last went out of service around 1970. Sales of this large and expensive machine exceeded thirty, confounding the suggestion, made in 1946 by Sir Charles Darwin, Director of the NPL, that 'it is very possible that... one machine would suffice to solve all the problems that are demanded of it from the whole country'. (Douglas Hartee, a leading British expert on automatic computation, thought that a total of three digital computers would probably be adequate for the country's computing needs.) The NPL's DEUCE replaced the Pilot Model ACE, and in 1956 much of the Pilot Model was transferred to the London Science Museum (where it is on permanent display).
Work began on a large-scale ACE in the autumn of 1954. Built and housed at the NPL, the 'Big ACE' was in operation by late 1958. Wilkinson, Clayden, Davies, (Ted) Newman, and Woodger all contributed to the final design. The Big ACE filled a room the size of an auditorium. It remained in service until 1967.
At the Press Day held in 1958 to announce the inauguration of the Big ACE, A. M. Uttley—Superintendent of the NPL's Control Mechanisms and Electronics Division—announced: 'Today, Turing's dream has come true.' If so, it was a dream whose time had passed. Technology had moved ahead in the thirteen years since Turing wrote 'Proposed Electronic Calculator'. The Big ACE was not the revolutionary machine that it would have been if completed six or seven years earlier. Not only did the Big ACE use valves in the era of the transistor; the designers also retained the by then outmoded mercury delay line memory proposed by Turing in 1945 (see below).
Nevertheless, the Big ACE was a fast machine with a large memory, and the decision to stick with the principles used in the Pilot ACE and the DEUCE was reasonable in the circumstances. In 1953 Francis Colebrook (Head of the Electronics Section responsible for building Pilot ACE) urged that the proposed large-scale ACE should 'be based on well proved components and techniques, even when revolutionary developments seem to be just around the corner. Otherwise the [Mathematics] Division will get nothing but a succession of pilot models.'
The Big ACE ran at 1.5 MHz, 50 per cent faster than the Pilot ACE and the DEUCE, both 1 MHz machines. A 1957 report stated that the Big ACE 'appears in fact to be about as fast as present-day parallel core-store computers'. 'Core-store' or magnetic core memory was the most advanced high-speed storage medium at that time.
The basic principles of Turing's ACE design were used in the G15 computer, built and marketed by the Detroit-based Bendix Corporation. The G15 was designed by Huskey, who had spent the year 1947 at the NPL working in the ACE Section. The first G15 ran in 1954. It was arguably the first personal computer. By following Turing's philosophy of minimizing hardware in favour of software, Huskey was able to make the G15 small enough (it was the size of a large domestic refrigerator) and cheap enough to be marketed as a single-user computer. Yet thanks to the design of the ACE-like memory (implemented in a magnetic drum), the G15 was as fast as computers many times its size. Over 400 were sold worldwide and the G15 remained in use until about 1970.
Another computer deriving from Turing's ACE design, the MOSAIC or Ministry of Supply Automatic Integrator and Computer, played a role in Britain's air defences during the Cold War period. In 1946 Flowers established a small team at the Post Office Research Station to build a computer to Turing's logical design (see the introduction to the next chapter). The team consisted of two engineers, William Chandler and Allen Coombs, both of whom had assisted Flowers in the construction of Colossus. Working alone, Coombs and Chandler carried out the engineering design of the MOSAIC, a large computer based on Turing's Version VII of the ACE design (Version VII dated from 1946). Of the various ACE-type computers that were built, the MOSAIC was the closest to Turing's original conception.
The MOSAIC consisted of some 70 mercury delay lines, 2,000 semi-conductors (germanium diodes), and 7,000 thermionic valves. It first ran a programme in 1952 or early 1953. Once completed, the MOSAIC was installed at the Radar Research and Development Establishment (RRDE) in Malvern. It was used to calculate aircraft trajectories from radar data in connection with anti-aircraft measures (the details appear still to be classified). The data was generated by two mobile data-recorders working in conjunction with a radar tracking system.
Coombs has emphasized: 'it was just Chandler and I—we designed every scrap of that machine.' Given, therefore, that two engineers working alone succeeded in completing the large MOSAIC, there seems little doubt that, had the NPL possessed the organizational capability and sufficient manpower, a computer not too distant from Turing's original conception could have been up and running by the early 1950s. Thanks to their experience with Colossus, Chandler and Coombs had a substantial start on everyone else in the field. Turing was unable to share his knowledge of their wartime work with Darwin. Had he been able to do so, the NPL might have acted to boost the resources available to Chandler and Coombs, and so made Turing's dream a reality much sooner.
Other derivatives of the ACE included the EMI Business Machine and the Packard-Bell PB250. Designed for business applications involving the shallow processing of large quantities of data, the EMI Business Machine was a relatively slow electronic computer with a large memory. The PB250 was a low-cost transistorized computer.
#### Turing, Newman, and the Manchester Computer
At the time of the Manchester Baby and its successor, the Manchester Mark I, the electronic engineers Frederic Williams and Thomas Kilburn, who had translated the logico-mathematical idea of the stored-programme computer into hardware, were given too little credit by the mathematicians at Manchester—Williams and Kilburn were regarded as excellent engineers but not as 'ideas men'. Nowadays the tables have turned too far and the triumph at Manchester is usually credited to Williams and Kilburn alone. Fortunately the words of the late Williams survive to set the record straight:
Now let's be clear before we go any further that neither Tom Kilburn nor I knew the first thing about computers when we arrived in Manchester University... Newman explained the whole business of how a computer works to us.
Tom Kilburn and I knew nothing about computers... Professor Newman and Mr A. M. Turing... knew a lot about computers... They took us by the hand and explained how numbers could live in houses with addresses...
In an address to the Royal Society on 4 March 1948, Newman presented this very explanation:
In modern times the idea of a universal calculating machine was independently [of Babbage] introduced by Turing... There is provision for storing numbers, say in the scale of 2, so that each number appears as a row of, say, forty 0's and 1's in certain places or 'houses' in the machine.... Certain of these numbers, or 'words' are read, one after another, as orders. In one possible type of machine an order consists of four numbers, for example 11, 13, 27, 4. The number 4 signifies 'add', and when control shifts to this word the 'houses' _H_ 11 and _H_ 13 will be connected to the adder as inputs, and _H_ 27 as output. The numbers stored in _H_ 11 and _H_ 13 pass through the adder, are added, and the sum is passed on to _H_ 27. The control then shifts to the next order. In most real machines the process just described would be done by three separate orders, the first bringing < _H_ 11> (= content of _H_ 11) to a central accumulator, the second adding < _H_ 13> into the accumulator, and the third sending the result to _H_ 27; thus only one address would be required in each order.... A machine with storage, with this automatic-telephone-exchange arrangement and with the necessary adders, subtractors and so on, is, in a sense, already a universal machine.
Following this explanation of Turing's three-address concept (source 1, source 2, destination, function) Newman went on to describe programme storage ('the orders shall be in a series of houses _X_ 1, _X_ 2,...') and conditional branching. He then summed up:
From this highly simplified account it emerges that the essential internal parts of the machine are, first, a storage for numbers (which may also be orders).... Secondly, adders, multipliers, etc. Thirdly, an 'automatic telephone exchange' for selecting 'houses', connecting them to the arithmetic organ, and writing the answers in other prescribed houses. Finally, means of moving control at any stage to any chosen order, if a certain condition is satisfied, otherwise passing to the next order in the normal sequence. Besides these there must be ways of setting up the machine at the outset, and extracting the final answer in useable form.
In a letter written in 1972 Williams described in some detail what he and Kilburn were told by Newman:
About the middle of the year [1946] the possibility of an appointment at Manchester University arose and I had a talk with Professor Newman who was already interested in the possibility of developing computers and had acquired a grant from the Royal Society of £30,000 for this purpose. Since he understood computers and I understood electronics the possibilities of fruitful collaboration were obvious. I remember Newman giving us a few lectures in which he outlined the organisation of a computer in terms of numbers being identified by the address of the house in which they were placed and in terms of numbers being transferred from this address, one at a time, to an accumulator where each entering number was added to what was already there. At any time the number in the accumulator could be transferred back to an assigned address in the store and the accumulator cleared for further use. The transfers were to be effected by a stored program in which a list of instructions was obeyed sequentially. Ordered progress through the list could be interrupted by a test instruction which examined the sign of the number in the accumulator. Thereafter operation started from a new point in the list of instructions. This was the first information I received about the organisation of computers.... Our first computer was the simplest embodiment of these principles, with the sole difference that it used a subtracting rather than an adding accumulator.
Turing's early input to the developments at Manchester, hinted at by Williams in his above-quoted reference to Turing, may have been via the lectures on computer design that Turing and Wilkinson gave in London during the period December 1946 to February 1947. The lectures were attended by representatives of various organizations planning to use or build an electronic computer. Kilburn was in the audience. (Kilburn usually said, when asked where he obtained his basic knowledge of the computer from, that he could not remember; for example, in a 1992 interview he said: 'Between early 1945 and early 1947, in that period, somehow or other I knew what a digital computer was... Where I got this knowledge from I've no idea.')
Whatever role Turing's lectures may have played in informing Kilburn, there is little doubt that credit for the Manchester computer—called the 'Newman– Williams machine' by Huskey in a report written shortly after a visit in 1947 to the Manchester project—belongs not only to Williams and Kilburn but also to Newman, and that the influence on Newman of Turing's 'On Computable Numbers' was crucial, as was the influence of Flowers's Colossus (see the introduction to Chapter 4).
There is more information concerning Turing and the Manchester computer in the chapter 'Artificial Life'.
##### _The Manchester computer and the EDVAC_
The Baby and the Manchester Mark I are sometimes said to have descended from the EDVAC. Newman was well aware of von Neumann's 'First Draft of a Report on the EDVAC'. In the summer of 1946 he sent David Rees, a lecturer in his department at Manchester and an ex-member of the Newmanry, to a series of lectures at the Moore School, where Eckert, Mauchly, and other members of the ENIAC-EDVAC group publicized their ideas on computer design. In the autumn of 1946 Newman himself went to Princeton for three months.
Newman's advocacy of 'a central accumulator'—a characteristic feature of the EDVAC but not of the ACE—was probably influenced by his knowledge of the American proposals. However, von Neumann's ideas seem to have had little influence on other members of the Manchester project. Kilburn spoke scathingly of the von Neumann 'dictat'. Geoffrey Tootill said:
Williams, Kilburn and I (the three designers of the first Manchester machine) had all spent the 1939–1945 war at the Telecommunications Research Establishment doing R & D on radiolocation equipments. The main U.S. ideas that we accepted in return for our initiatives on these and later on computers were the terms 'radar' and 'memory'... We disliked the latter term, incidentally, as encouraging the anthropomorphic concept of 'machines that think'.
To the best of my recollection FC [Williams], Tom [Kilburn] and I never discussed... von Neumann's... ideas during the development of the Small-Scale Experimental Machine [the Baby], nor did I have any knowledge of them when I designed the Ferranti Mk I. I don't think FC was influenced at all by von Neumann, because I think he was in general quite punctilious in acknowledging other people's ideas.
Tootill added:
As well as our own ideas, we incorporated functions suggested by Turing and Newman in the improvement and extension of the first machine. When I did the logic design of the Ferranti Mark 1, I got them to approve the list of functions.
#### The ACE and Artificial Intelligence
In designing the ACE, Artificial Intelligence was not far from Turing's thoughts —he described himself as building 'a brain'. The otherwise austere 'Proposed Electronic Calculator' contains a cameo discussion of computer intelligence and chess (Turing's earliest surviving remarks concerning AI):
'Can the machine play chess?' It could fairly easily be made to play a rather bad game. It would be bad because chess requires intelligence. We stated at the beginning of this section that the machine should be treated as entirely without intelligence. There are indications however that it is possible to make the machine display intelligence at the risk of its making occasional serious mistakes. By following up this aspect the machine could probably be made to play very good chess.
What is probably Turing's earliest mention to survive of his interest in neural simulation (see further Chapter 10) occurs in a letter to the cyberneticist W. Ross Ashby:
In working on the ACE I am more interested in the possibility of producing models of the action of the brain than in the practical applications to computing.... The ACE will be used, as you suggest, in the first instance in an entirely disciplined manner, similar to the action of the lower centres, although the reflexes will be extremely complicated. The disciplined action carries with it the disagreeable feature, which you mentioned, that it will be entirely uncritical when anything goes wrong. It will also be necessarily devoid of anything that could be called originality. There is, however, no reason why the machine should always be used in such a manner: there is nothing in its construction which obliges us to do so. It would be quite possible for the machine to try out variations of behaviour and accept or reject them in the manner you describe and I have been hoping to make the machine do this. This is possible because, without altering the design of the machine itself, it can, in theory at any rate, be used as a model of any other machine, by making it remember a suitable set of instructions. The ACE is in fact analogous to the 'universal machine' described in my paper on computable numbers. This theoretical possibility is attainable in practice, in all reasonable cases, at worst at the expense of operating slightly slower than a machine specially designed for the purpose in question. Thus, although the brain may in fact operate by changing its neuron circuits by the growth of axons and dendrites, we could nevertheless make a model, within the ACE, in which this possibility was allowed for, but in which the actual construction of the ACE did not alter, but only the remembered data, describing the mode of behaviour applicable at any time. I feel that you would be well advised to take advantage of this principle, and do your experiments on the ACE, instead of building a special machine. I should be very glad to help you over this.
#### The Lecture
On 20 February 1947 Turing lectured on the ACE to the London Mathematical Society. So far as is known, this was the earliest public lecture to mention computer intelligence, providing a breathtaking glimpe of a new field. Turing discussed the prospect of machines acting intelligently, learning, and beating human opponents at chess, remarking that '[w]hat we want is a machine that can learn from experience' and that '[t]he possibility of letting the machine alter its own instructions provides the mechanism for this'.
The lecture is also of note for its early discussion of computer programming. By the time of the lecture, Turing had been developing programmes—then called 'instruction tables' (see 'Computable Numbers: A Guide')—for the notyet-existent ACE for well over a year. As Womersley was to remark in 1948, 'The planning of the ACE is far ahead of the hardware.' During this period, Turing and the other members of the ACE Section brought the nascent science of computer programming to a state of considerable sophistication.
Among Turing's many technical innovations was the use of what are now called _subroutines_ , or in Turing's term, 'subsidiary tables' (see 'Computable Numbers: A Guide').
Turing devoted about a quarter of his lecture to a discussion of computer memory, saying (p. 383):
I have spent a considerable time in this lecture on this question of memory, because I believe that the provision of proper storage is the key to the problem of the digital computer, and certainly if they are to be persuaded to show any sort of genuine intelligence much larger capacities than are yet available must be provided.
It was precisely Turing's desire to conduct experiments in AI that led him to propose such a large memory in the ACE.
#### Delay Line Memory and Optimum Programming
The fundamental feature of the ACE design, which influenced virtually all other aspects of it, was Turing's adoption of acoustic delay lines to form the high-speed memory. He remarked in the lecture that the chief advantage of delay lines as a memory medium—they were far from ideal—was that they were 'already a going concern' (p. 380). The acoustic delay line was pioneered in 1942 by W. B. Shockley, later one of the co-inventors of the transistor, and in 1943 Presper Eckert and others at the Moore School (home of the ENIAC) independently investigated a different type of acoustic delay line filled with mercury. Subsequently, acoustic delay lines were widely employed in radar. It was Eckert who first proposed their use in digital computers (see 'Computable Numbers: A Guide').
The following description of a mercury delay line is from Turing's 'Proposed Electronic Calculator':
It is proposed to build 'delay line' units consisting of mercury... tubes about 5' long and 1" in diameter in contact with a quartz crystal at each end. The velocity of sound in... mercury... is such that the delay will be 1.024 ms. The information to be stored may be considered to be a sequence of 1024 'digits' (0 or 1)... These digits will be represented by a corresponding sequence of pulses. The digit 0... will be represented by the absence of a pulse at the appropriate time, the digit 1... by its presence. This series of pulses is impressed on the end of the line by one piezo-crystal, it is transmitted down the line in the form of supersonic waves, and is reconverted into a varying voltage by the crystal at the far end. This voltage is amplified sufficiently to give an output of the order of 10 volts peak to peak and is used to gate a standard pulse generated by the clock. This pulse may be again fed into the line by means of the transmitting crystal, or we may feed in some altogether different signal. We also have the possibility of leading the gated pulse to some other part of the calculator, if we have need of that information at the time. Making use of the information does not of course preclude keeping it also.
As Turing noted in his 1947 lecture, the ACE's memory was to consist of approximately 200 such delay lines. In the event, the Big ACE contained only a few dozen delay lines, supplemented by four magnetic drums.
Having decided to use delay lines, Turing was determined to maximize their effectiveness. The time taken for an instruction, or number, to emerge from a delay line will depend on where in the delay line it happens to be. In order to minimize waiting time, Turing arranged for instructions to be stored not at regular intervals in a delay line, but in irregular positions. These positions were selected by the programmer in such a way that each instruction would emerge from its delay line at precisely the time it was required. This system became known at the NPL as 'optimum coding' (and later 'optimum programming').
Optimum coding made for difficult and untidy programming but the advantage in terms of speed was considerable. Thanks to optimum coding, the Pilot Model ACE was able to do a floating point multiplication in 3 milliseconds. The EDSAC computer at Cambridge University—a slower delay line machine lacking optimum coding—required 4.5 milliseconds to perform a single fixed point multiplication.
##### _Further reading_
Campbell-Kelly, M., 'Programming the Pilot ACE: Early Programming Activity at the National Physical Laboratory', _Annals of the History of Computing_ , 3 (1981), 133–62.
Carpenter, B. E., and Doran, R. W. (eds.), _A. M. Turing's ACE Report of 1946 and Other Papers_ (Cambridge, Mass.: MIT Press, 1986).
Copeland, B. J. (ed.), _Alan Turing's Automatic Computing Engine_ (Oxford: Oxford University Press, 2004).
Yates, D. M., _Turing's Legacy: A History of Computing at the National Physical Laboratory 1945–1995_ (London: Science Museum, 1997).
##### _Provenance_
What follows is the text of the lecture taken from Turing's own typescript (which is headed 'Lecture to L.M.S. Feb. 20 1947').
#### **Lecture on the Automatic Computing Engine**
The automatic computing engine now being designed at N.P.L. is a typical large scale electronic digital computing machine. In a single lecture it will not be possible to give much technical detail of this machine, and most of what I shall say will apply equally to any other machine of this type now being planned.
From the point of view of the mathematician the property of being digital should be of greater interest than that of being electronic. That it is electronic is certainly important because these machines owe their high speed to this, and without the speed it is doubtful if financial support for their construction would be forthcoming. But this is virtually all that there is to be said on that subject. That the machine is digital however has more subtle significance. It means firstly that numbers are represented by sequences of digits which can be as long as one wishes. One can therefore work to any desired degree of accuracy. This accuracy is not obtained by more careful machining of parts, control of temperature variations, and such means, but by a slight increase in the amount of equipment in the machine. To double the number of significant figures used would involve increasing the equipment by a factor definitely less than two, and would also have some effect in increasing the time taken over each job. This is in sharp contrast with analogue machines, and continuous variable machines such as the differential analyser, where each additional decimal digit required necessitates a complete redesign of the machine, and an increase in the cost by perhaps as much as a factor of 10. A second advantage of digital computing machines is that they are not restricted in their applications to any particular type of problem. The differential analyser is by far the most general type of analogue machine yet produced, but even it is comparatively limited in its scope. It can be made to deal with almost any kind of ordinary differential equation, but it is hardly able to deal with partial differential equations at all, and certainly cannot manage large numbers of linear simultaneous equations, or the zeros of polynomials. With digital machines however it is almost literally true that they are able to tackle any computing problem. A good working rule is that the ACE can be made to do any job that could be done by a human computer, and will do it in one ten-thousandth of the time. This time estimate is fairly reliable, except in cases where the job is too trivial to be worth while giving to the ACE.
Some years ago I was researching on what might now be described as an investigation of the theoretical possibilities and limitations of digital computing machines. I considered a type of machine which had a central mechanism, and an infinite memory which was contained on an infinite tape. This type of machine appeared to be sufficiently general. One of my conclusions was that the idea of a 'rule of thumb' process and a 'machine process' were synonymous. The expression 'machine process' of course means one which could be carried out by the type of machine I was considering. It was essential in these theoretical arguments that the memory should be infinite. It can easily be shown that otherwise the machine can only execute periodic operations. Machines such as the ACE may be regarded as practical versions of this same type of machine. There is at least a very close analogy. Digital computing machines all have a central mechanism or control and some very extensive form of memory. The memory does not have to be infinite, but it certainly needs to be very large. In general the arrangement of the memory on an infinite tape is unsatisfactory in a practical machine, because of the large amount of time which is liable to be spent in shifting up and down the tape to reach the point at which a particular piece of information required at the moment is stored. Thus a problem might easily need a storage of three million entries, and if each entry was equally likely to be the next required the average journey up the tape would be through a million entries, and this would be intolerable. One needs some form of memory with which any required entry can be reached at short notice. This difficulty presumably used to worry the Egyptians when their books were written on papyrus scrolls. It must have been slow work looking up references in them, and the present arrangement of written matter in books which can be opened at any point is greatly to be preferred. We may say that storage on tape and papyrus scrolls is somewhat _inaccessible_. It takes a considerable time to find a given entry. Memory in book form is a good deal better, and is certainly highly suitable when it is to be read by the human eye. We could even imagine a computing machine that was made to work with a memory based on books. It would not be very easy but would be immensely preferable to the single long tape. Let us for the sake of argument suppose that the difficulties involved in using books as memory were overcome, that is to say that mechanical devices for finding the right book and opening it at the right page, etc. etc. had been developed, imitating the use of human hands and eyes. The information contained in the books would still be rather inaccessible because of the time occupied in the mechanical motions. One cannot turn a page over very quickly without tearing it, and if one were to do much book transportation, and do it fast the energy involved would be very great. Thus if we moved one book every millisecond and each was moved ten metres and weighed 200 grams, and if the kinetic energy were wasted each time we should consume 1010 watts, about half the country's power consumption. If we are to have a really fast machine then, we must have our information, or at any rate a part of it, in a more accessible form than can be obtained with books. It seems that this can only be done at the expense of compactness and economy, e.g. by cutting the pages out of the books, and putting each one into a separate reading mechanism. Some of the methods of storage which are being developed at the present time are not unlike this.
If one wishes to go to the extreme of accessibility in storage mechanisms one is liable to find that it is gained at the price of an intolerable loss of compactness and economy. For instance the most accessible known form of storage is that provided by the valve flip-flop or Jordan Eccles trigger circuit. This enables us to store one digit, capable of two values, and uses two thermionic valves. To store the content of an ordinary novel by such means would cost many millions of pounds. We clearly need some compromise method of storage which is more accessible than paper, film etc, but more economical in space and money than the straightforward use of valves. Another desirable feature is that it should be possible to record into the memory from within the computing machine, and this should be possible whether or not the storage already contains something, i.e. the storage should be _erasible_.
There are three main types of storage which have been developed recently and have these properties in greater or less degree. Magnetic wire is very compact, is erasible, can be recorded on from within the machine, and is moderately accessible. There is storage in the form of charge patterns on the screen of a cathode ray tube. This is probably the ultimate solution. It could eventually be nearly as accessible as the Jordan Eccles circuit. A third possibility is provided by acoustic delay lines. They give greater accessibility than the magnetic wire, though less than the C.R.T type. The accessibility is adequate for most purposes. Their chief advantage is that they are already a going concern. It is intended that the main memory of the ACE shall be provided by acoustic delay lines, consisting of mercury tanks.
**Figure 9.1.**
The idea of using acoustic delay lines as memory units is due I believe to Eckert of Philadelphia University, who was the engineer chiefly responsible for the Eniac. The idea is to store the information in the form of compression waves travelling along a column of mercury. Liquids and solids will transmit sound of surprisingly high frequency, and it is quite feasible to put as many as 1,000 pulses into a single 5′ tube. The signals may be conveyed into the mercury by a piezoelectric crystal; and also detected at the far end by another quartz crystal. A train of pulses or the information which they represent may be regarded as stored in the mercury whilst it is travelling through it. If the information is not required when the train emerges it can be fed back into the column again and again until such time as it is required. This requires a 'recirculating circuit' to read the signal as it emerges from the tank and amplify it and feed it in again. If this were done with a simple amplifier it is clear that the characteristics of both the tank and the amplifier would have to be extremely good to permit the signal to pass through even as many as ten times. Actually the recirculating circuit does something slightly different. What it does may perhaps be best expressed in terms of point set topology. Let the plane of the diagram represent the space of all possible signals. I do not of course wish to imply that this is two dimensional. Let the function _f_ be defined for arguments in this signal space and have values in it. In fact let _f_ ( _s_ ) represent the effect on the signal _s_ when it is passed through the tank and the recirculating mechanism. We assume however that owing to thermal agitation the effect of recirculation may be to give any pt [point] within a circle of radius δ of _f_ ( _s_ ). Then a necessary and sufficient condition that the tank can be used as a storage which will distinguish between _N_ different signals is that there must be _N_ sets _E_ 1... _E N_ such that if _F r_ is the set of pts [points] within distance ε of _E r_
and the sets _F r_ are disjoint. It is clearly sufficient for we have only then to ensure that the signals initially fed in belong to one or other of the sets _F r_, and it will remain in the set after any number of recirculations, without any danger of confusion. It is necessary for suppose _s_ 1... _s N_ are signals which have different meanings and which can be fed into the machine at any time and read out later without fear of confusion.
Let _E r_ be the set of signals which _could_ be obtained for _s r_ by successive applications of _f_ and shifts of distance not more than ε. Then the sets _E r_ are disjoint, [ _five or six illegible words_ ], and by applying a shift of distance ε or less to pts [points] of _E r_ we obtain [ _line missing from bottom of page_ ]. In the case of a mercury delay line used for _N_ = 16 the set would consist of all continuous signals within the shaded area.
**Figure 9.2.**
One of the sets would consist of all continuous signals lying in the region below. It would represent the signal 1001.
**Figure 9.3.**
In order to put such a recirculation system into effect it is essential that a clock signal be supplied to the memory system so that it will be able to distinguish the times when a pulse if any should be present. It would for instance be natural to supply a timing sine wave as shown above to the recirculator.
The idea of a process _f_ with the properties we have described is a very common one in connection with storage devices. It is known as 'regeneration' of storage. It is always present in some form, but sometimes the regeneration is as it were naturally occurring and no precautions have to be taken. In other cases special precautions have to be taken to improve such an _f_ process or else the impression will fade.
The importance of a clock to the regeneration process in delay lines may be illustrated by an interesting little theorem. Suppose that instead of the condition _s_ ∈ _F r_ ⊃ _f_ ( _s_ ) ∈ _E r_ we impose a stronger one, viz _f_ n( _s_ ) → _c r_ if _s_ ∈ _E r_, i.e. there are ideal forms of the distinguishable signals, and each admissible signal converges towards the ideal form after recirculating. Then we can show that unless there is a clock the ideal signals are all constants. For let _U_ α represent a shift of origin, i.e. _U_ α _s_ ( _t_ ) = _s_ ( _t_ \+ α). Then since there is no clock the properties of the recirculator are the same at all times and _f_ therefore commutes with _U_ α. Then _fU_ α( _c r_) = _U_ α _f_ ( _c r_) = _U_ α _c r_, for _f_ ( _c r_) = _c r_ since _c r_ is an ideal signal. But this means that _U_ α( _c r_) is an ideal signal, and therefore for sufficiently small a must be _c r_, since the ideal signals are discrete. Then for any β and sufficiently large _n_ , β/ _n_ will be sufficiently small and _U_ β/n( _c_ ) = _c_. But then by iteration . This means that the ideal signal _c_ is a constant.
We might say that the clock enables us to introduce a discreteness into time, so that time can for some purposes be regarded as a succession of instants instead of as a continuous flow. A digital machine must essentially deal with discrete objects, and in the case of the ACE this is made possible by the use of a clock. All other digital computing machines that I know of except for human and other brains do the same. One can think up ways of avoiding it, but they are very awkward. I should mention that the use of the clock in the ACE is not confined to the recirculation process, but is used in almost every part.
It may be as well to mention some figures connected with the mercury delay line as we shall use it. We shall use five foot tubes, with a[n] inside diameter of half an inch. Each of these will enable us to store 1024 binary digits. The unit I have used here to describe storage capacity is self explanatory. A storage mechanism has a capacity of _m_ binary digits if it can remember any sequence of _m_ digits each being a 0 or a 1. The storage capacity is also the logarithm to the base 2 of the number of different signals which can be remembered, i.e. log2 _N_. The digits will be placed at a time interval of one microsecond, so that the time taken for the waves to travel down the tube is just over a millisecond. The velocity is about one and a half kilometres per second. The delay in accessibility time or average waiting for a given piece of information is about half a millisecond. In practice this is reduced to an effective 150 μs. The full storage capacity of the ACE available on Hg delay lines will be about 200,000 binary digits. This is probably comparable with the memory capacity of a minnow.
I have spent a considerable time in this lecture on this question of memory, because I believe that the provision of proper storage is the key to the problem of the digital computer, and certainly if they are to be persuaded to show any sort of genuine intelligence much larger capacities than are yet available must be provided. In my opinion this problem of making a large memory available at reasonably short notice is much more important than that of doing operations such as multiplication at high speed. Speed is necessary if the machine is to work fast enough for the machine to be commercially valuable, but a large storage capacity is necessary if it is to be capable of anything more than rather trivial operations. The storage capacity is therefore the more fundamental requirement.
Let us now return to the analogy of the theoretical computing machines with an infinite tape. It can be shown that a single special machine of that type can be made to do the work of all. It could in fact be made to work as a model of any other machine. The special machine may be called the universal machine; it works in the following quite simple manner. When we have decided what machine we wish to imitate we punch a description of it on the tape of the universal machine. This description explains what the machine would do in every configuration in which it might find itself. The universal machine has only to keep looking at this description in order to find out what it should do at each stage. Thus the complexity of the machine to be imitated is concentrated in the tape and does not appear in the universal machine proper in any way.
If we take the properties of the universal machine in combination with the fact that machine processes and rule of thumb processes are synonymous we may say that the universal machine is one which, when supplied with the appropriate instructions, can be made to do any rule of thumb process. This feature is paralleled in digital computing machines such as the ACE. They are in fact practical versions of the universal machine. There is a certain central pool of electronic equipment, and a large memory. When any particular problem has to be handled the appropriate instructions for the computing process involved are stored in the memory of the ACE and it is then 'set up' for carrying out that process.
I have indicated the main strategic ideas behind digital computing machinery, and will now follow this account up with the very briefest description of the ACE. It may be divided for the sake of argument into the following parts
Memory
Control
Arithmetic part
Input and Output
I have already said enough about the memory and will only repeat that in the ACE the memory will consist mainly of 200 mercury delay lines each holding 1024 binary digits. The purpose of the control is to take the right instructions from the memory, see what they mean, and arrange for them to be carried out. It is understood that a certain 'code of instructions' has been laid down, whereby each 'word' or combination of say 32 binary digits describes some particular operation. The circuit of the control is made in accordance with the code, so that the right effect is produced. To a large extent we have also allowed the circuit to determine the code, i.e. we have not just thought up an imaginary 'best code' and then found a circuit to put it into effect, but have often simplified the circuit at the expense of the code. It is also quite difficult to think about the code entirely in abstracto without any kind of circuit.
The arithmetic part of the machine is the part concerned with addition, multiplication and any other operations which it seems worth while to do by means of special circuits rather than through the simple facilities provided by the control. The distinction between control and arithmetic part is a rather hazy one, but at any rate it is clear that the machine should at least have an adder and a multiplier, even if they turn out in the end to be part of the control. This is the point at which I should mention that the machine is operated in the binary scale, with two qualifications. Inputs from externally provided data are in decimal, and so are outputs intended for human eyes rather than for later reconsumption by the ACE. This is the first qualification. The second is that, in spite of the intention of binary working there can be no bar on decimal working of a kind, because of the relation of the ACE to the universal machine. Binary working is the most natural thing to do with any large scale computer. It is much easier to work in the scale of two than any other, because it is so easy to produce mechanisms which have two positions of stability: the two positions may then be regarded as representing 0 and 1. Examples are lever as diagram, Jordan Eccles circuit, thyratron. If one is concerned with a small scale calculating machine then there is at least one serious objection to binary working. For practical use it will be necessary to build a converter to transform numbers from the binary form to the decimal and back. This may well be a larger undertaking than the binary calculator. With the large scale machines this argument carries no weight. In the first place a converter would become a relatively small piece of apparatus, and in the second it would not really be necessary. This last statement sounds quite paradoxical, but it is a simple consequence of the fact that these machines can be made to do any rule of thumb process by remembering suitable instructions. In particular it can be made to do binary decimal conversion. For example in the case of the ACE the provision of the converter involves no more than adding two extra delay lines to the memory. This situation is very typical of what happens with the ACE. There are many fussy little details which have to be taken care of, and which, according to normal engineering practice, would require special circuits. We are able to deal with these points without modification of the machine itself, by pure paper work, eventually resulting in feeding in appropriate instructions.
**Figure 9.4.**
**Figure 9.5.**
Editor's note. I am grateful to John Andreae and David Clayden for assistance with redrawing Turing's original figure. The figure shows an Eccles–Jordan trigger circuit, also known as a bistable circuit or flipflop. The circuit has two stable states. In one, current flows through the left-hand triode valve but not through the right; in the other, current flows through the right-hand triode but not the left. The circuit can be switched from one state to the other by applying a pulse to one of the valves.
To return to the various parts of the machine. I was saying that it will work in the scale of two. It is not unnatural to use the convention that an electrical pulse shall represent the digit 1 and that absence of a pulse shall represent a digit 0. Thus a sequence of digits 0010110 would be represented by a signal like
**Figure 9.6.**
where the time interval might be one microsecond. Let us now look at what the process of binary addition is like. In ordinary decimal addition we always begin from the right, and the same naturally applies to binary. We have to do this because we cannot tell whether to carry unless we have already dealt with the less significant columns. The same applies with electronic addition, and therefore it is convenient to use the convention that if a sequence of pulses is coming down a line, then the least significant pulse always comes first. This has the unfortunate result that we must either write the least significant digit on the left in our binary numbers or else make time flow from right to left in our diagrams. As the latter alternative would involve writing from right to left as well as adding in that way, we have decided to put the least significant digit on the left. Now let us do a typical addition. Let us write the carry digits above the addends.
Note that I can do the addition only looking at a small part of the data. To do the addition electronically we need to produce a circuit with three inputs and two outputs.
This circuit must be such that
If no. of 1s on inputs α, β, γ is
It is very easy to produce a voltage proportional to the number of pulses on the inputs, and one then merely has to provide a circuit which will discriminate between four different levels and put out the appropriate sum and carry digits. I will not attempt to describe such a circuit; it can be quite simple. When we are given the circuit we merely have to connect it up with feedback and it is an adder. Thus:
**Figure 9.7.**
It will be seen that we have made use of the fact that the same process is used in addition with each digit, and also the fact that the properties of the electrical circuit are invariant under time shifts, at any rate if these are multiples of the clock period. It might be said that we have made use of the isomorphism between the group of these time shifts and the multiplicative group of real numbers to simplify our apparatus, though I doubt if many other applications of this principle could be found.
It will be seen that with such an adder the addition is broken down into the most elementary steps possible, such as adding one and one. Each of these occupies a microsecond. Our numbers will normally consist of 32 binary digits, so that two of them can be added in 32 microseconds. Likewise we shall do multiplications in the form of a number of consecutive additions of one and one or one and zero etc. There are 1024 such additions or thereabouts to be done in a multiplication of one 32 digit number by another, so that one might expect a multiplication to take about a millisecond. Actually the multiplier to be used on ACE will take rather over two milliseconds. This may sound rather long, when the unit operation is only a microsecond, but it actually seems that the machine is fairly well balanced in this respect, i.e. the multiplication time is not a serious bottleneck. Computers always spend just as long in writing numbers down and deciding what to do next as they do in actual multiplications, and it is just the same with the ACE. A great deal of time is spent in getting numbers in and out of storage and deciding what to do next. To complete the four elementary processes, subtraction is done by complementation and addition, and division is done by the use of the iteration formula
_u n_ converges to _a_ −1 provided |1 − _au_ 0| < 1. The error is squared at each step, so that the convergence is very rapid. This process is of course programmed, i.e. the only extra apparatus required is the delay lines required for storing the relevant instructions.
Passing on from the arithmetic part there remains the input and output. For this purpose we have chosen Hollerith card equipment. We are able to obtain this without having to do any special development work. The speeds obtainable are not very impressive compared with the speeds at which the electronic equipment works, but they are quite sufficient in all cases where the calculation is long and the result concise: the interesting cases in fact. It might appear that there would be a difficulty in converting the information provided at the slow speeds appropriate to the Hollerith equipment to the high speeds required with the ACE, but it is really quite easy. The Hollerith speeds are so slow as to be counted zero or stop for many purposes, and the problem reduces to the simple one of converting a number of statically given digits into a stream of pulses. This can be done by means of a form of electronic commutator.
Before leaving the outline of the description of the machine I should mention some of the tactical situations that are met with in programming. I can illustrate two of them in connection with the calculation of the reciprocal described above. One of these is the idea of the iterative cycle. Each time that we go from _u_ _r_ to _u_ _r_ +1 we apply the same sequence of operations, and it will therefore be economical in storage space if we use the same instructions. Thus we go round and round a cycle of instructions:
**Figure 9.8.**
Editor's note. I am grateful to Mike Woodger for assistance with redrawing Turing's original diagram.
It looks however as if we were in danger of getting stuck in this cycle, and unable to get out. The solution of this difficulty involves another tactical idea, 'discrimination' i.e. of deciding what to do next partly according to the results of the machine itself, instead of according to data available to the programmer. In that of this case we include a discrimination in each cycle, which takes us out of the cycle when the value of |1 − _au_ | is sufficiently small. It is like an aeroplane circling over an aerodrome, and asking permission to land after each circle. This is a very simple idea, but is of the utmost importance. The idea of the iterative cycle of instructions will also be seen to be rather fundamental when it is realised that the majority of the instructions in the memory must be obeyed a great number of times. If the whole memory were occupied by instructions, none of it being used for numbers or other data, and if each instruction were obeyed once only, but took the longest possible time, the machine could only remain working for sixteen seconds.
Another important idea is that of constructing an instruction and then obeying it. This can be used amongst other things for discrimination. In the example I have just taken for instance we could calculate a quantity which was 1 if |1 − _au_ | was less than 2−31 and 0 otherwise. By adding this quantity to the instruction that is obeyed at the forking point that instruction can be completely altered in its effect when finally 1 − _au_ is reduced to sufficiently small dimensions.
Probably the most important idea involved in instruction tables is that of standard _subsidiary tables_. Certain processes are used repeatedly in all sorts of different connections, and we wish to use the same instructions, from the same part of the memory every time. Thus we may use interpolation for the calculation of a great number of different functions, but we shall always use the same instruction table for interpolation. We have only to think out how this is to be done once, and forget then how it is done. Each time we want to do an interpolation we have only to remember the memory position where this table is kept, and make the appropriate reference in the instruction table which is using the interpolation. We might for instance be making up an instruction table for finding values of _J_ 0( _x_ ) and use the interpolation table in this way. We should then say that the interpolation table was a subsidiary to the table for calculating _J_ 0( _x_ ). There is thus a sort of hierarchy of tables. The interpolation table might be regarded as taking its orders from the _J_ 0 table, and reporting its answers back to it. The master servant analogy is however not a very good one, as there are many more masters than servants, and many masters have to share the same servants.
Now let me give a picture of the operation of the machine. Let us begin with some problem which has been brought in by a customer. It will first go to the problems preparation section where it is examined to see whether it is in a suitable form and self-consistent, and a very rough computing procedure made out. It then goes to the tables preparation section. Let us suppose for example that the problem was to tabulate solutions of the equation
with initial conditions _x_ = _y_ = 0, _y_ ′ = _a_. This would be regarded as a particular case of solving the equation
for which one would have instruction tables already prepared. One would need also a table to produce the function _F_ ( _x, y, z_ ) (in this case _F_ ( _x, y, z_ ) = _J_ 0( _x_ ) – _xz_ which would mainly involve a table to produce _J_ 0( _x_ ), and this we might expect to get off the shelf). A few additional details about the boundary conditions and the length of the arc would have to be dealt with, but much of this detail would also be found on the shelf, just like the table for obtaining _J_ 0( _x_ ). The instructions for the job would therefore consist of a considerable number taken off the shelf together with a few made up specially for the job in question. The instruction cards for the standard processes would have already been punched, but the new ones would have to be done separately. When these had all been assembled and checked they would be taken to the input mechanism, which is simply a Hollerith card feed. They would be put into the card hopper and a button pressed to start the cards moving through. It must be remembered that initially there are no instructions in the machine, and one's normal facilities are therefore not available. The first few cards that pass in have therefore to be carefully thought out to deal with this situation. They are the initial input cards and are always the same. When they have passed in a few rather fundamental instruction tables will have been set up in the machine, including sufficient to enable the machine to read the special pack of cards that has been prepared for the job we are doing. When this has been done there are various possibilities as to what happens next, depending on the way the job has been programmed. The machine might have been made to go straight on through, and carry out the job, punching or printing all the answers required, and stopping when all of this has been done. But more probably it will have been arranged that the machine stops as soon as the instruction tables have been put in. This allows for the possibility of checking that the content of the memories is correct, and for a number of variations of procedure. It is clearly a suitable moment for a break. We might also make a number of other breaks. For instance we might be interested in certain particular values of the parameter _a_ , which were experimentally obtained figures, and it would then be convenient to pause after each parameter value, and feed the next parameter value in from another card. Or one might prefer to have the cards all ready in the hopper and let the ACE take them in as it wanted them. One can do as one wishes, but one must make up one's mind. Each time the machine pauses in this way a 'word' or sequence of 32 binary digits is displayed on neon bulbs. This word indicates the reason for stopping. I have already mentioned two possible reasons. A large class of further possible reasons is provided by the checks. The programming should be done in such a way that the ACE is frequently investigating identities which should be satisfied if all is as it should be. Whenever one of these checks fails the machine stops and displays a word which describes what check has failed.
It will be seen that the possibilities as to what one may do are immense. One of our difficulties will be the maintenance of an appropriate discipline, so that we do not lose track of what we are doing. We shall need a number of efficient librarian types to keep us in order.
Finally I should like to make a few conjectures as to the repercussions that electronic digital computing machinery will have on mathematics. I have already mentioned that the ACE will do the work of about 10,000 computers. It is to be expected therefore that large scale hand-computing will die out. Computers will still be employed on small calculations, such as the substitution of values in formulae, but whenever a single calculation may be expected to take a human computer days of work, it will presumably be done by an electronic computer instead. This will not necessitate everyone interested in such work having an electronic computer. It would be quite possible to arrange to control a distant computer by means of a telephone line. Special input and output machinery would be developed for use at these out stations, and would cost a few hundred pounds at most. The main bulk of the work done by these computers will however consist of problems which could not have been tackled by hand computing because of the scale of the undertaking. In order to supply the machine with these problems we shall need a great number of mathematicians of ability. These mathematicians will be needed in order to do the preliminary research on the problems, putting them into a form for computation. There will be considerable scope for analysts. When a human computer is working on a problem he can usually apply some common sense to give him an idea of how accurate his answers are. With a digital computer we can no longer rely on common sense, and the bounds of error must be based on some proved inequalities. We need analysts to find the appropriate inequalities for us. The inequalities need not always be explicit, i.e. one need not have them in such a form that we can tell, before the calculation starts, and using only pencil and paper, how big the error will be. The error calculation may be a serious part of the ACE's duties. To an extent it may be possible to replace the estimates of error by statistical estimates obtained by repeating the job several times, and doing the rounding off differently each time, controlling it by some random element, some electronic roulette wheel. Such statistical estimates however leave much in doubt, are wasteful in machine time, and give no indication of what can be done if it turns out that the errors are intolerably large. The statistical method can only help the analyst, not replace him.
Analysis is just one of the purposes for which we shall need good mathematicians. Roughly speaking those who work in connection with the ACE will be divided into its masters and its servants. Its masters will plan out instruction tables for it, thinking up deeper and deeper ways of using it. Its servants will feed it with cards as it calls for them. They will put right any parts that go wrong. They will assemble data that it requires. In fact the servants will take the place of limbs. As time goes on the calculator itself will take over the functions both of masters and of servants. The servants will be replaced by mechanical and electrical limbs and sense organs. One might for instance provide curve followers to enable data to be taken direct from curves instead of having girls read off values and punch them on cards. The masters are liable to get replaced because as soon as any technique becomes at all stereotyped it becomes possible to devise a system of instruction tables which will enable the electronic computer to do it for itself. It may happen however that the masters will refuse to do this. They may be unwilling to let their jobs be stolen from them in this way. In that case they would surround the whole of their work with mystery and make excuses, couched in well chosen gibberish, whenever any dangerous suggestions were made. I think that a reaction of this kind is a very real danger. This topic naturally leads to the question as to how far it is possible in principle for a computing machine to simulate human activities. I will return to this later, when I have discussed the effects of these machines on mathematics a little further.
I expect that digital computing machines will eventually stimulate a considerable interest in symbolic logic and mathematical philosophy. The language in which one communicates with these machines, i.e. the language of instruction tables, forms a sort of symbolic logic. The machine interprets whatever it is told in a quite definite manner without any sense of humour or sense of proportion. Unless in communicating with it one says exactly what one means, trouble is bound to result. Actually one could communicate with these machines in any language provided it was an exact language, i.e. in principle one should be able to communicate in any symbolic logic, provided that the machine were given instruction tables which would enable it to interpret that logical system. This should mean that there will be much more practical scope for logical systems than there has been in the past. Some attempts will probably be made to get the machines to do actual manipulations of mathematical formulae. To do so will require the development of a special logical system for the purpose. This system should resemble normal mathematical procedure closely, but at the same time should be as unambiguous as possible. As regards mathematical philosophy, since the machines will be doing more and more mathematics themselves, the centre of gravity of the human interest will be driven further and further into philosophical questions of what can in principle be done etc.
It has been said that computing machines can only carry out the processes that they are instructed to do. This is certainly true in the sense that if they do something other than what they were instructed then they have just made some mistake. It is also true that the intention in constructing these machines in the first instance is to treat them as slaves, giving them only jobs which have been thought out in detail, jobs such that the user of the machine fully understands what in principle is going on all the time. Up till the present machines have only been used in this way. But is it necessary that they should always be used in such a manner? Let us suppose we have set up a machine with certain initial instruction tables, so constructed that these tables might on occasion, if good reason arose, modify those tables. One can imagine that after the machine had been operating for some time, the instructions would have altered out of all recognition, but nevertheless still be such that one would have to admit that the machine was still doing very worthwhile calculations. Possibly it might still be getting results of the type desired when the machine was first set up, but in a much more efficient manner. In such a case one would have to admit that the progress of the machine had not been foreseen when its original instructions were put in. It would be like a pupil who had learnt much from his master, but had added much more by his own work. When this happens I feel that one is obliged to regard the machine as showing intelligence. As soon as one can provide a reasonably large memory capacity it should be possible to begin to experiment on these lines. The memory capacity of the human brain is probably of the order of ten thousand million binary digits. But most of this is probably used in remembering visual impressions, and other comparatively wasteful ways. One might reasonably hope to be able to make some real progress with a few million digits, especially if one confined one's investigations to some rather limited field such as the game of chess. It would probably be quite easy to find instruction tables which would enable the ACE to win against an average player. Indeed Shannon of Bell Telephone laboratories tells me that he has won games playing by rule of thumb: the skill of his opponents is not stated. But I would not consider such a victory very significant. What we want is a machine that can learn from experience. The possibility of letting the machine alter its own instructions provides the mechanism for this, but this of course does not get us very far.
It might be argued that there is a fundamental contradiction in the idea of a machine with intelligence. It is certainly true that 'acting like a machine' has become synonymous with lack of adaptability. But the reason for this is obvious. Machines in the past have had very little storage, and there has been no question of the machine having any discretion. The argument might however be put into a more aggressive form. It has for instance been shown that with certain logical systems there can be no machine which will distinguish provable formulae of the system from unprovable, i.e. that there is no test that the machine can apply which will divide propositions with certainty into these two classes. Thus if a machine is made for this purpose it must in some cases fail to give an answer. On the other hand if a mathematician is confronted with such a problem he would search around a[nd] find new methods of proof, so that he ought eventually to be able to reach a decision about any given formula. This would be the argument. Against it I would say that fair play must be given to the machine. Instead of it sometimes giving no answer we could arrange that it gives occasional wrong answers. But the human mathematician would likewise make blunders when trying out new techniques. It is easy for us to regard these blunders as not counting and give him another chance, but the machine would probably be allowed no mercy. In other words then, if a machine is expected to be infallible, it cannot also be intelligent. There are several mathematical theorems which say almost exactly that. But these theorems say nothing about how much intelligence may be displayed if a machine makes no pretence at infallibility. To continue my plea for 'fair play for the machines' when testing their I.Q. A human mathematician has always undergone an extensive training. This training may be regarded as not unlike putting instruction tables into a machine. One must therefore not expect a machine to do a very great deal of building up of instruction tables on its own. No man adds very much to the body of knowledge, why should we expect more of a machine? Putting the same point differently, the machine must be allowed to have contact with human beings in order that it may adapt itself to their standards. The game of chess may perhaps be rather suitable for this purpose, as the moves of the machine's opponent will automatically provide this contact.
### **CHAPTER 10
Intelligent Machinery ( _1948_ )**
_Alan Turing_
#### Introduction
##### _Jack Copeland_
#### Slow Progress on the ACE
By the beginning of 1947 much effort had gone into writing programmes or 'instruction tables' for the ACE. In 'Proposed Electronic Calculator', Turing had said that work on instruction tables should start immediately, since the 'earlier stages of the making of instruction tables will have serious repercussions on the design [of the machine]'; moreover, the programming work should 'go on whilst the machine is being built, in order to avoid some of the delay between the delivery of the machine and the production of results'.
However, little progress had been made on the physical construction of the ACE. The actual engineering work was being carried out not at the National Physical Laboratory but at the Post Office Research Station, under the supervision of Turing's wartime associate Flowers (see the introductions to Chapters 4 and ). Flowers was asked by the NPL early in 1946 to assist with the engineering design of, and to build, the ACE. Ominously, the letter of agreement from Flowers's superior to Turing's superior spoke of 'very considerable arrears of work' and warned that 'the manpower position is difficult'.
Initial progress was nevertheless promising, with some early successes in the experimental work on the delay line memory units. According to an NPL document dated March 1946, 'Mr. Flowers states that they can have ready for N.P.L. a minimal ACE by August or September.' Unfortunately it proved impossible to keep to Flowers's timetable. Dollis Hill was occupied with a backlog of urgent work on the national telephone system (at that time managed by the Post Office). Flowers's Section was 'too busy to do other people's work' (he said in 1998). Only two men could be spared to work on the ACE, Chandler and Coombs (both of whom had played leading roles in the wartime Colossus project).
Sir Charles Darwin, the Director of the NPL, noted in August 1946 that the Post Office was 'not in a position to plunge very deep', and by November was expressing concern to Post Office staff about the slow rate of progress on the ACE. The fault was not all with the Post Office, however. In November 1946, the NPL proposed a radical change in the hardware design, with cathode ray tube memory units taking the place of mercury delay lines. Such a change would have meant that most of the work done by Chandler and Coombs up to that point was wasted. (In the end CRT memory was used at Manchester but not in the ACE.) The logical design kept changing, too—by the end of 1946 Turing had reached Version VII of the ACE design. Coombs said:
One of the problems was, I remember, that NPL kept on changing its ideas, and every time we went down there and said 'Right now! What do you want us to make?', we'd find that the last idea, that they gave us last week, was old hat and they'd got a quite different one, and we couldn't get a consolidated idea at all until eventually we dug our toes in and said 'Stop! Tell us what to make.'
Some eight years later, Chandler and Coombs finally completed a computer based on Version VII of Turing's ACE design, the MOSAIC (see the introduction to Chapter 9).
#### Turing Proposes an ACE Electronics Section
In January 1947 Turing had gone to the United States, visiting several of the groups there that were attempting to build an electronic stored-programme computer. In his report on his visit he wrote:
One point concerning the form of organisation struck me very strongly. The engineering development work was in every case being done in the same building with the more mathematical work. I am convinced that this is the right approach. It is not possible for the two parts of the organisation to keep in sufficiently close touch otherwise. They are too deeply interdependent. We are frequently finding that we are held up due to ignorance of some point which could be cleared up by a conversation with the engineers, and the Post Office find similar difficulty; a telephone conversation is seldom effective because we cannot use diagrams. Probably more important are the points which are misunderstood, but which would be cleared up if closer contact were maintained, because they would come to light in casual discussion. It is clear that we must have an engineering section at the ACE site eventually, the sooner the better, I would say.
Darwin suggested to Dr Horace Thomas, a member of the NPL's Radio Division, that he 'be put in charge of the work of making by a suitable firm a prototype model' of the ACE. A 'pre-prototype model' would be started in-house at the NPL 'in Radio and Metrology workshops before approaching an outside firm', Darwin said, and 'Metrology and Radio workshops could get on with the hardware part of the job straight away'. (The outside firm eventually approached was English Electric, who subsequently built and marketed the DEUCE (see the introduction to Chapter 9). In 1949 wiremen and engineers from English Electric would join the NPL team to assist with the completion of Pilot Model ACE.)
Darwin's directive that an in-house electronics section begin work on the hardware took several months to implement, however—the wheels of administration turned slowly. At the end of April an NPL minute spoke of the need for an electronics group working 'together in one place as a whole in close contact with the planning staff in the Mathematics Division'. The minute emphasized that the 'various parts' of the ACE project were 'so interwoven' that it was 'not practicable at present to farm out portions of the work to isolated groups', adding 'Our experience with the Post Office confirms this.' Yet it was not until August that suitable staff were seconded from elsewhere in the NPL and Thomas's electronics group finally came into existence.
Womersley proposed that the new group should complete the 'Test Assembly', a pilot version of the ACE begun by Huskey in Mathematics Division in the spring of 1947 (see the introduction to Chapter 9). Womersley's proposal was disregarded, however. Thomas was an empire-builder. The week before the official launch of Thomas's electronics group on 18 August 1947, E. S. Hiscocks, Secretary to the NPL, had written:
Thomas has apparently shown some signs of behaving as if he is starting up a new Division, and so as to allay certain qualms which both Smith-Rose and Womersley have, I think it would be better for it to be explained to the whole team that Mathematics Division is the parent Division, and the one which is to justify the financial outlay on this work; that the work is being put out on contract, as it were, to Radio Division, and that Thomas's team is a part of Radio Division. I think, even if only for our own peace of mind, this is desirable, because Thomas has already shown some signs of wanting to set up a separate office, etc.
Unfortunately Thomas 'didn't like... the idea of this group in Mathematics Division... working independently', Wilkinson recalled. Rivalry quickly sprang up between Thomas's group in Radio Division and the ACE Section in Mathematics Division, and soon after Thomas's appointment the work on the Test Assembly was summarily stopped by Darwin. This was a dire turn of events—in Huskey's words, 'morale in the Mathematics Division collapsed'.
The Test Assembly—based on Version V of Turing's design for the ACE—was in fact a highly promising project and considerable progress had been made during the course of 1947. By August 1947 the NPL workshops were fabricating a mercury delay line to Huskey's specifications, valve types had been chosen and circuit block diagrams made, source and destination decisions had been taken, and programmes were being written to check these decisions. In October, Womersley and E. C. Fieller (who became Superintendent of Mathematics Division when Womersley left in 1950) expected, optimistically, that the Test Assembly would 'be ready by the end of November'.
Work on building the ACE drew nearly to a standstill after Thomas persuaded Darwin to shut down the Test Assembly. Only two members of Thomas's group were skilled in digital electronics and they knew little about computers. Thomas's own background was not in digital electronics at all but in radio and industrial electronics. The group 'began to develop their knowledge of pulse techniques', said Wilkinson, and 'for a while they just did basic things and became more familiar with the electronics they needed to learn to build a computer'. Then, in February 1948, Thomas resigned from the NPL to join Unilever Ltd. (the manufacturers of 'Sunlight' soap). As Womersley summed up the situation in April 1948, hardware development was 'probably as far advanced 18 months ago'.
Given only slightly better management on the NPL's part, a minimal computer based on Turing's Version V of the ACE design could probably have been working at the NPL during 1948. By September 1947 the NPL had an electronics group which, by joining forces with Huskey and the ACE Section, would have been capable of carrying Huskey's Test Assembly to completion. (Huskey said: 'I never hoped to have the Test Assembly working before I left [the NPL for the USA] in December. I certainly hoped the group would have it working in 1948.') Womersley had advocated this course of action but Thomas threw a spanner in the works. The Test Assembly could easily have been the world's first electronic stored-programme digital computer to run a trial calculation (a title that in the event went to the Manchester 'Baby' in June 1948).
Following Thomas's replacement by Francis Colebrook in March 1948 the fortunes of the ACE did at last begin to improve. Colebrook drew the ACE Section and the Electronics Section together to work harmoniously on what became the Pilot Model ACE (see the introduction to Chapter 9). But Turing did not wait.
#### Turing Leaves the National Physical Laboratory
In the middle of 1947 a thoroughly disheartened Turing applied for a twelve-month period of sabbatical leave to be spent in Cambridge. The purpose of the leave, as described by Darwin in July 1947, was to enable Turing
to extend his work on the machine [the ACE] still further towards the biological side. I can best describe it by saying that hitherto the machine has been planned for work equivalent to that of the lower parts of the brain, and he [Turing] wants to see how much a machine can do for the higher ones; for example, could a machine be made that could learn by experience? This will be theoretical work, and better done away from here.
Turing left the NPL for Cambridge in the autumn of 1947. Then in May 1948 he gave up his position at the NPL altogether, breaking what Darwin referred to as 'a gentleman's agreement to return here for at least two years after the year's absence'. Newman's offer of a job lured a 'very fed up' Turing—Robin Gandy's description—to Manchester University, where in May 1948 he was appointed Deputy Director of the Computing Machine Laboratory (there being no director).
At Manchester, Turing designed the input mechanism and programming system for an expanded version of Kilburn and William's 'Baby' (see the introduction to Chapter 9) and wrote a programming manual for the new machine. At last Turing had his hands on a functioning stored-programme computer. He was soon using it to model biological growth— pioneering work in the Weld now known as Artificial Life (see Chapter 15 and 'Artifical Life').
#### The First Manifesto of Artificial Intelligence
In the summer of 1948 Turing completed a report for Darwin describing the outcomes of his research into 'how much a machine can do for the higher' parts of the brain. It was entitled 'Intelligent Machinery'. Donald Michie recalls that Turing 'was in a state of some agitation about its reception by his superiors at NPL: "A bit thin for a year's time off!"' The headmasterly Darwin—who once complained about the 'smudgy' appearance of Turing's work—was, as Turing predicted, displeased with the report, describing it as a 'schoolboy's essay' and 'not suitable for publication'. In reality this far-sighted paper was the first manifesto of AI; sadly Turing never published it.
'Intelligent Machinery' is a wide-ranging and strikingly original survey of the prospects for Artificial Intelligence. In it Turing brilliantly introduced many of the concepts that were later to become central in the Weld, in some cases after reinvention by others. These included the logic-based approach to problem-solving, and the idea, subsequently made popular by Newell and Simon, that (as Turing put it) 'intellectual activity consists mainly of various kinds of search' (p. 431). Turing anticipated the concept of a genetic algorithm (GA), in a brief passage concerning what he calls 'genetical or evolutionary search' (see further Chapter 11, p. 463, and 'Artificial Life'). 'Intelligent Machinery' also contains the earliest description of (a restricted form of) what Turing was later to call the 'imitation game' and is now known simply as the Turing test (see further Chapter 11). The major part of 'Intelligent Machinery', however, consists of an exquisite discussion of machine learning, in which Turing anticipated the modern approach to AI known as connectionism.
#### Expert Systems
Search and the logic-based approach are both used in modern 'expert systems'. An expert system is an AI programme for solving problems and giving advice within a specialized Weld of knowledge, such as medical diagnosis or corporate planning. The area of expert systems is one of the most successful in modern AI.
The basic components of an expert system are a knowledge base (KB), an inference engine, and a search engine for searching the KB. The KB is built up by interviewing (human) experts in the area in question. A 'knowledge engineer' organizes the information elicited from the experts into a system of propositions, typically of 'if-then' form.
The inference engine enables the expert system to draw deductions from propositions in the KB. For example, from two propositions 'if _x_ then _y_ ' and 'if _y_ then _z_ ', the inference engine is able to deduce 'if _x_ then _z_ '. The expert system might then query its user 'Is _x_ true in the situation that we are considering?' (e.g. 'Does the patient have a rash?'), and if the answer is affirmative, the system will proceed to infer _z_.
Modern systems using search and inference can produce impressive results. For example, Douglas Lenat's common-sense reasoning system CYC is able to conclude 'Garcia is wet' from the statement 'Garcia is finishing a marathon run', by means of searching for and using such items of common-sense knowledge as that running a marathon entails high exertion, that people sweat at high levels of exertion, and that when something sweats it is wet.
#### Connectionism
Connectionism—still in its infancy—is the science of computing with networks of artificial neurons. This approach came to widespread attention in the mid-1980s when a group based at the University of California at San Diego reported some striking experiments. In one, an artificial neural network learned to form the past tenses of English verbs, responding correctly to irregular verbs not previously encountered (such as 'weep' and 'cling').
Modern connectionists regard Donald Hebb and Frank Rosenblatt as the founding figures of their approach and it is not widely realized that Turing wrote a blueprint for much of the connectionist project as early as 1948. In 'Intelligent Machinery' he introduced what he called 'unorganised machines', giving as examples networks of neuron-like elements connected together in a largely random manner. He described a certain type of network as 'the simplest model of a nervous system' (p. 418). From a historical point of view, his idea that an initially unorganized neural network can be organized by means of 'interfering training' is of considerable significance, since it did not appear in the earlier work of McCulloch and Pitts (see below). In Turing's model, the training process renders certain neural pathways effective and others ineffective. So far as is known, he was the first person to consider building computing machines out of trainable networks of randomly arranged neuron-like elements.
#### B-Type Unorganized Machines
In 'Intelligent Machinery' Turing introduced the type of neural network that he called a 'B-type unorganised machine'. A B-type consists of interconnected artificial neurons, depicted in Figure 10.1 as circles, and connection-modifiers, depicted as squares. A B-type may contain any number of neurons connected together in any pattern, but subject always to the restriction that each neuron-to-neuron connection passes through a connection-modifier (Figure 10.2). The connection-modifiers are used in training the network.
Turing's model neurons work as follows. Each neuron has two input fibres, and the output of a neuron is a simple logical function of its two inputs. Every neuron in the network performs the same logical operation, 'nand' (Table 1).
**Figure 10.1**. Two neurons from a B-type network. The two fibres on the connection-modifier between the neurons enable training by an external agent.
**Figure 10.2**. A fragment of a large, intially random B-type neural network
**Table 1** Behaviour of a nand-neuron
Turing chose nand as the basic operation of his model neurons because every other boolean operation can be carried out by groups of nand-neurons. Turing showed that even the connection-modifier itself can be built out of nand-neurons. So each B-type network consists of nothing more than nand-neurons and their connecting fibres. This is about the simplest possible model of the cortex.
#### Training a B-Type
Each connection-modifier has two training fibres (Figure 10.1). Applying a pulse to one fibre sets the modifier to _interchange mode_. When the modifier is in interchange mode and the input into the modifier is 1, the modifier's output is 0; and when the input is 0, the output is 1.
The effect of a pulse on the other training fibre is to place the modifier in _interrupt mode_. In this mode, the output of the modifier is always 1, no matter what its input. When in interrupt mode, the modifier destroys all information attempting to pass along the connection to which the modifier is attached. Once set, a modifier will maintain its function unless it receives a pulse on the other training fibre.
The presence of these connection-modifiers enables a B-type unorganized machine to be trained, by means of what Turing called 'appropriate interference, mimicking education' (p. 422). Turing theorized that 'the cortex of an infant is an unorganised machine, which can be organised by suitable interfering training' (p. 424).
Initially a network that is to be trained contains random inter-neural connections, and the modifiers on these connections are also set randomly (Figure 10.2). Unwanted connections are destroyed by switching their attached modifiers to interrupt mode. The output of the neuron immediately upstream of the modifier no longer finds its way along the connection to the neuron on the downstream end (see Table 1). Conversely, switching the setting of the modifier on an initially interrupted connection to the other mode is in effect to create a new connection. This selective culling and enlivening of connections hones the initially random network into one organized for a given task.
#### Neural Simulation
Turing wished to investigate other types of 'unorganised machine', and he envisaged the procedure—now a days used extensively by connectionists—of simulating a neural network and its training regimen using an ordinary digital computer (just as an engineer may use a computer to simulate an aircraft wing or a weather analyst to simulate a storm system). He would, he said, 'allow the whole system to run for an appreciable period, and then break in as a kind of "inspector of schools" and see what progress had been made' (p. 428).
However, Turing's research on neural networks was carried out shortly before the first general-purpose electronic computers were up and running, and he used only paper and pencil. Once he did have access to a computer in the Manchester Computing Machine Laboratory, Turing turned his attention to research in Artificial Life, and he seems never to have used computer simulation to explore the behaviour of neural networks.
In the year of Turing's death (1954), two researchers at MIT, Wesley Clark and Belmont Farley, succeeded in running the first computer simulations of neural networks. Clark and Farley were unaware of Turing's earlier work and their neural architecture was quite different from his, using inter-neural connections of variable 'weight' (or 'strength'). Clark and Farley were able to train their networks—which contained a maximum of 128 neurons—to recognize simple patterns. (In addition, they discovered that the random destruction of up to 10 per cent of the neurons in a trained network does not affect the network's performance at its task.) The work begun by Clark and Farley was considerably developed by Rosenblatt, in whose theory of 'perceptrons' modern connectionism took shape. Meanwhile, Turing's pioneering work on a distinctively different type of connectionist architecture was forgotten.
#### B-Types Redefined
There appears to be an inconsistency in Turing's presentation of B-types. He claimed that B-types are able, with appropriate modifier settings, to 'do any required job, given sufficient time and provided the number of units i.e. neurons] is sufficient' (p. 422). This claim is false. The problem lies with the connection-modifiers. If each modifier must either interrupt or interchange, then there are simple logical operations that no B-type can perform. For example, no B-type can produce the _exclusive disjunction_ of a pair of inputs ([Table 2). Nor can a B-type compute the identity function (i.e. produce outputs identical to its inputs). In effect the difficulty is one that Turing mentions in a different connection. In Chapter 17 he points out that 'an odd number of... interchanges... can never bring one back to where one started' (p. 589).
A simple remedy is to use connection-modifiers that, when not in interrupt mode, are in _pass mode_ rather than interchange mode. In pass mode the modifier's output is identical to its input. Turing's claim that B-types can 'do any required job...' is true if the two available modes of each modifier are interrupt and pass.
The required interrupt/pass modifiers can in fact be built out of pairs of interrupt/interchange modifiers. A pair of modifiers both in interchange mode amounts to a single modifier in pass mode. Similarly, a pair of modifiers whose downstream member is in interrupt mode functions indistinguishably from a single modifier in interrupt mode.
**Table 2** Exclusive disjunction (XOR)
#### B-Types and the Universal Turing Machine
Turing claimed a proof (not given in 'Intelligent Machinery' and now lost) of the proposition that an initially unorganized B-type network with sufficient neurons can be organized (via its connection-modifiers) to become 'a universal Turing machine with a given storage capacity' (p. 422)—i.e. to become equivalent to a universal Turing machine with a truncated tape.
This feature of neural networks may shed light on one of the most fundamental problems concerning human cognition. Examining cognition from a top-down perspective, we find complex sequential processes, often involving language or other forms of symbolic representation—for example, logical reasoning, the planning out of activities, and mathematical calculation. Yet from a bottom-up perspective, cognition is nothing but the simple firings of neurons. How is the view from the top to be reconciled with the prima facie very different view from the bottom?
Turing's proof first opened up the possibility, noted in 'Intelligent Machinery'(p. 424), that the brain is in part a universal computing machine (of a given storage capacity) implemented in a neural network. This, then, is a possible solution to the fundamental problem: it is by virtue of being a neural network acting as a universal computing machine (of a given storage capacity) that the cortex is able to carry out the sequential, symbol-rich processing discerned in the view from the top. In 1948 this hypothesis was well ahead of its time, and today it remains one of our best guesses concerning the fundamental problem of cognitive science.
#### McCulloch–Pitts Neural Nets
It is interesting that Turing makes no reference in 'Intelligent Machinery' to the now famous work of McCulloch and Pitts, itself influenced by his own 'On Computable Numbers'. Their 1943 article represents the first attempt to apply what they call 'the Turing definition of computability' to the study of neuronal function. Like Turing, McCulloch and Pitts considered networks of simple two-state boolean 'neurons', although there were important differences between McCulloch–Pitts nets and Turing nets. For example, inhibitory synapses are a fundamental feature of McCulloch–Pitts nets but not of Turing nets. (An input of 1 at an inhibitory synapse at moment _m_ unconditionally sets the output of the unit to 0 at _m_ \+ 1.)
McCulloch stressed the extent to which his and Pitts's work is indebted to Turing in the course of some autobiographical remarks made during the public discussion of a lecture given by von Neumann in 1948:
I started at entirely the wrong angle... and it was not until I saw Turing's paper ['On Computable Numbers'] that I began to get going the right way around, and with Pitts' help formulated the required logical calculus. What we thought we were doing (and I think we succeeded fairly well) was treating the brain as a Turing machine.
Turing had undoubtedly heard something of the work of McCulloch and Pitts. Wiener—with McCulloch a founding member of the cybernetics movement— would almost certainly have mentioned McCulloch in the course of his 'talk over the fundamental ideas of cybernetics with Mr Turing' at the NPL in the spring of 1947. Moreover, von Neumann mentioned the McCulloch–Pitts article of 1943—albeit very briefly—in the 'First Draft of a Report on the EDVAC', which Turing read in 1945. In order to depict the EDVAC's logic gates, von Neumann employed a modified version of the diagrammatic notation for neural nets used by McCulloch and Pitts. (In his 'Proposed Electronic Calculator' (see Chapter 9) Turing considerably extended the notation that he found in the 'First Draft'.) Turing and McCulloch seem not to have met until 1949. After their meeting Turing spoke dismissively of McCulloch, referring to him as a 'charlatan'.
It is an open question whether the work of McCulloch and Pitts had any influence at all on the development of the ideas presented in 'Intelligent Machinery'. Probably not. As Newman once remarked of Turing: 'It was, perhaps, a defect of his qualities that he found it hard to use the work of others, preferring to work things out for himself.'
Whatever the influences were on Turing at that time, there is no doubt that his work on neural nets goes importantly beyond the earlier work of McCulloch and Pitts. The latter give only a perfunctory discussion of learning, saying no more than that the mechanisms supposedly underlying learning in the brain—they specifically mention the formation of new connections and neuronal threshold change—can be mimicked by means of nets whose connections and thresholds are fixed. Turing's idea of using supervised interference to train an initially random arrangement of neurons to compute a specified function is nowhere prefigured.
#### _Further reading_
Copeland, B. J., and Proudfoot, D., 'On Alan Turing's Anticipation of Connectionism', _Synthese_ , 108 (1996), 361–77. Reprinted in R. Chrisley (ed.), _Artificial Intelligence: Critical Concepts in Cognitive Science, ii: Symbolic AI_ (London: Routledge, 2000).
Jackson, P., _Introduction to Expert Systems_ (Wokingham: Addison-Wesley, 1986).
Rumelhart, D. E., McClelland, J. L., and the PDP Research Group, _Parallel Distributed Processing: Explorations in the Microstructure of Cognition, i: Foundations, (Cambridge_ , Mass.: MIT Press, 1986).
#### _Provenance_
What follows is the text of the National Physical Laboratory document 'Intelligent Machinery: A Report by A. M. Turing' (dated 1948). Symbols missing from the Report itself but present in Turing's draft typescript have been restored.
#### Intelligent Machinery
I propose to investigate the question as to whether it is possible for machinery to show intelligent behaviour. It is usually assumed without argument that it is not possible. Common catch phrases such as 'acting like a machine', 'purely mechanical behaviour' reveal this common attitude. It is not difficult to see why such an attitude should have arisen. Some of the reasons are:
(a) An unwillingness to admit the possibility that mankind can have any rivals in intellectual power. This occurs as much amongst intellectual people as amongst others: they have more to lose. Those who admit the possibility all agree that its realization would be very disagreeable. The same situation arises in connection with the possibility of our being superseded by some other animal species. This is almost as disagreeable and its theoretical possibility is indisputable.
(b) A religious belief that any attempt to construct such machines is a sort of Promethean irreverence.
(c) The very limited character of the machinery which has been used until recent times (e.g. up to 1940). This encouraged the belief that machinery was necessarily limited to extremely straightforward, possibly even to repetitive, jobs. This attitude is very well expressed by Dorothy Sayers _(The Mind of the Maker, p. 46)_ '... which imagines that God, having created his Universe, has now screwed the cap on His pen, put His feet on the mantel-piece and left the work to get on with itself.' This, however, rather comes into St. Augustine's category of figures of speech or enigmatic sayings framed from things which do not exist at all. We simply do not know of any creation which goes on creating itself in variety when the creator has withdrawn from it. The idea is that God simply created a vast machine and has left it working until it runs down from lack of fuel. This is another of those obscure analogies, since we have no experience of machines that produce variety of their own accord; the nature of a machine is to do the same thing over and over again so long as it keeps going.
(d) Recently the theorem of Gödel and related results (Gödel, Church, Turing) have shown that if one tries to use machines for such purposes as determining the truth or falsity of mathematical theorems and one is not willing to tolerate an occasional wrong result, then any given machine will in some cases be unable to give an answer at all. On the other hand the human intelligence seems to be able to find methods of ever increasing power for dealing with such problems 'transcending' the methods available to machines.
(e) In so far as a machine can show intelligence this is to be regarded as nothing but a reflection of the intelligence of its creator.
#### 2. Refutation of some objections
In this section I propose to outline reasons why we do not need to be influenced by the above described objections. The objections (a) and (b), being purely emotional, do not really need to be refuted. If one feels it necessary to refute them there is little to be said that could hope to prevail, though the actual production of the machines would probably have some effect. In so far then as we are influenced by such arguments we are bound to be left feeling rather uneasy about the whole project, at any rate for the present. These arguments cannot be wholly ignored, because the idea of 'intelligence' is itself emotional rather than mathematical.
The objection (c) in its crudest form is refuted at once by the actual existence of machinery (ENIAC etc.) which can go on through immense numbers (e.g. 1060,000 about for ACE) of operations without repetition, assuming no breakdown. The more subtle forms of this objection will be considered at length in § 11 and 12.
The argument from Gödel's and other theorems (objection (d)) rests essentially on the condition that the machine must not make mistakes. But this is not a requirement for intelligence. It is related that the infant Gauss was asked at school to do the addition 15 + 18 + 21 +... + 54 (or something of the kind) and that he immediately wrote down 483, presumably having calculated it as (15 + 54)(54 − 12)/2.3. One can imagine circumstances where a foolish master told the child that he ought instead to have added 18 to 15 obtaining 33, then added 21 etc. From some points of view this would be a 'mistake', in spite of the obvious intelligence involved. One can also imagine a situation where the children were given a number of additions to do, of which the first 5 were all arithmetic progressions, but the 6th was say 23 + 34 + 45 +... + 100 + 112 + 122 +... + 199. Gauss might have given the answer to this as if it were an arithmetic progression, not having noticed that the 9th term was 112 instead of 111. This would be a definite mistake, which the less intelligent children would not have been likely to make.
The view (e) that intelligence in machinery is merely a reflection of that of its creator is rather similar to the view that the credit for the discoveries of a pupil should be given to his teacher. In such a case the teacher would be pleased with the success of his methods of education, but would not claim the results themselves unless he had actually communicated them to his pupil. He would certainly have envisaged in very broad outline the sort of thing his pupil might be expected to do, but would not expect to foresee any sort of detail. It is already possible to produce machines where this sort of situation arises in a small degree. One can produce 'paper machines' for playing chess. Playing against such a machine gives a definite feeling that one is pitting one's wits against something alive.
These views will all be developed more completely below.
#### 3. Varieties of machinery
It will not be possible to discuss possible means of producing intelligent machinery without introducing a number of technical terms to describe different kinds of existent machinery.
_'Discrete' and 'Continuous' machinery_. We may call a machine 'discrete' when it is natural to describe its possible states as a discrete set, the motion of the machine occurring by jumping from one state to another. The states of 'continuous' machinery on the other hand form a continuous manifold, and the behaviour of the machine is described by a curve on this manifold. All machinery can be regarded as continuous, but when it is possible to regard it as discrete it is usually best to do so. The states of discrete machinery will be described as 'configurations'.
_'Controlling' and 'Active' machinery_. Machinery may be described as 'controlling' if it only deals with information. In practice this condition is much the same as saying that the magnitude of the machine's effects may be as small as we please, so long as we do not introduce confusion through Brownian movement etc. 'Active' machinery is intended to produce some definite physical effect.
We shall mainly be concerned with discrete controlling machinery. As we have mentioned, brains very nearly fall into this class, and there seems every reason to believe that they could have been made to fall genuinely into it without any change in their essential properties. However, the property of being 'discrete' is only an advantage for the theoretical investigator, and serves no evolutionary purpose, so we could not expect Nature to assist us by producing truly 'discrete' brains.
Given any discrete machine the first thing we wish to find out about it is the number of states (configurations) it can have. This number may be infinite (but enumerable) in which case we say that the machine has infinite memory (or storage) capacity. If the machine has a finite number N of possible states then we say that it has a memory capacity of (or equivalent to) log2 N binary digits. According to this definition we have the following table of capacities, very roughly
Brunsviga | 90
---|---
ENIAC without cards and with fixed programme | 600
ENIAC with cards | ∞
ACE as proposed | 60,000
Manchester machine (as actually working 8/7/48) | 1,100
The memory capacity of a machine more than anything else determines the complexity of its possible behaviour.
The behaviour of a discrete machine is completely described when we are given the state (configuration) of the machine as a function of the immediately preceding state and the relevant external data.
#### _Logical Computing Machines (L.C.M.s)_
In [Chapter 1] a certain type of discrete machine was described. It had an infinite memory capacity obtained in the form of an infinite tape marked out into squares on each of which a symbol could be printed. At any moment there is one symbol in the machine; it is called the scanned symbol. The machine can alter the scanned symbol and its behaviour is in part determined by that symbol, but the symbols on the tape elsewhere do not affect the behaviour of the machine. However the tape can be moved back and forth through the machine, this being one of the elementary operations of the machine. Any symbol on the tape may therefore eventually have an innings.
These machines will here be called 'Logical Computing Machines'. They are chiefly of interest when we wish to consider what a machine could in principle be designed to do, when we are willing to allow it both unlimited time and unlimited storage capacity.
_Universal Logical Computing Machines_. It is possible to describe L.C.M.s in a very standard way, and to put the description into a form which can be 'understood' (i.e. applied by) a special machine. In particular it is possible to design a 'universal machine' which is an L.C.M. such that if the standard description of some other L.C.M. is imposed on the otherwise blank tape from outside, and the (universal) machine then set going it will carry out the operations of the particular machine whose description it was given. For details the reader must refer to [Chapter 1].
The importance of the universal machine is clear. We do not need to have an infinity of different machines doing different jobs. A single one will suffice. The engineering problem of producing various machines for various jobs is replaced by the office work of 'programming' the universal machine to do these jobs.
It is found in practice that L.C.M.s can do anything that could be described as 'rule of thumb' or 'purely mechanical'. This is sufficiently well established that it is now agreed amongst logicians that 'calculable by means of an L.C.M.' is the correct accurate rendering of such phrases. There are several mathematically equivalent but superficially very different renderings.
##### _Practical Computing Machines (P.C.M.s)_
Although the operations which can be performed by L.C.M.s include every rule of thumb process, the number of steps involved tends to be enormous. This is mainly due to the arrangement of the memory along the tape. Two facts which need to be used together may be stored very far apart on the tape. There is also rather little encouragement, when dealing with these machines, to condense the stored expressions at all. For instance the number of symbols required in order to express a number in Arabic form (e.g. 149056) cannot be given any definite bound, any more than if the numbers are expressed in the 'simplified Roman' form (IIIII... I, with 149056 occurrences of I). As the simplified Roman system obeys very much simpler laws one uses it instead of the Arabic system.
In practice however one _can_ assign finite bounds to the numbers that one will deal with. For instance we can assign a bound to the number of steps that we will admit in a calculation performed with a real machine in the following sort of way. Suppose that the storage system depends on charging condensers of capacity _C_ = 1 μf, and that we use two states of charging, _E_ = 100 volts and − _E_ = −100 volts. When we wish to use the information carried by the condenser we have to observe its voltage. Owing to thermal agitation the voltage observed will always be slightly wrong, and the probability of an error between _V_ and _V_ − d _V_ volts is
where _k_ is Boltzmann's constant. Taking the values suggested we find that the probability of reading the sign of the voltage wrong is about 10−1.2×1016. If then a job took more than 101017 steps we should be virtually certain of getting the wrong answer, and we may therefore restrict ourselves to jobs with fewer steps. Even a bound of this order might have useful simplifying effects. More practical bounds are obtained by assuming that a light wave must travel at least 1 cm between steps (this would only be false with a very small machine) and that we could not wait more than 100 years for an answer. This would give a limit of 1020 steps. The storage capacity will probably have a rather similar bound, so that we could use sequences of 20 decimal digits for describing the position in which a given piece of data was to be found, and this would be a really valuable possibility.
Machines of the type generally known as 'Automatic Digital Computing Machines' often make great use of this possibility. They also usually put a great deal of their stored information in a form very different from the tape form. By means of a system rather reminiscent of a telephone exchange it is made possible to obtain a piece of information almost immediately by 'dialling' the position of this information in the store. The delay may be only a few microseconds with some systems. Such machines will be described as 'Practical Computing Machines'.
##### _Universal Practical Computing Machines_
Nearly all of the P.C.M.s now under construction have the essential properties of the 'Universal Logical Computing' machines mentioned earlier. In practice, given any job which could have been done on an L.C.M. one can also do it on one of these digital computers. I do not mean that we can design a digital computer to do it, but that we stick to one, say the ACE, and that we can do any required job of the type mentioned on it, by suitable programming. The programming is pure paper work. It naturally occurs to one to ask whether e.g. the ACE would be truly universal if its memory capacity were infinitely extended. I have investigated this question, and the answer appears to be as follows, though I have not proved any formal mathematical theorem about it. As has been explained, the ACE at present uses finite sequences of digits to describe positions in its memory: they are (Sept 1947) actually sequences of 9 binary digits. The ACE also works largely for other purposes with sequences of 32 binary digits. If the memory were extended e.g. to 1000 times its present capacity it would be natural to arrange the memory in blocks of nearly the maximum capacity which can be handled with the 9 digits, and from time to time to switch from block to block. A relatively small part would never be switched. This would contain some of the more fundamental instruction tables and those concerned with switching. This part might be called the 'central part'. One would then need to have a number which described which block was in action at any moment. This number might however be as large as one pleased. Eventually the point would be reached where it could not be stored in a word (32 digits), or even in the central part. One would then have to set aside a block for storing the number, or even a sequence of blocks, say blocks 1, 2,... _n_. We should then have to store n, and in theory it would be of indefinite size. This sort of process can be extended in all sorts of ways, but we shall always be left with a positive integer which is of indefinite size and which needs to be stored somewhere, and there seems to be no way out of the difficulty but to introduce a 'tape'. But once this has been done, and since we are only trying to prove a theoretical result, one might as well, whilst proving the theorem, ignore all the other forms of storage. One will in fact have a U.L.C.M. with some complications. This in effect means that one will not be able to prove any result of the required kind which gives any intellectual satisfaction.
##### _Paper machines_
It is possible to produce the effect of a computing machine by writing down a set of rules of procedure and asking a man to carry them out. Such a combination of a man with written instructions will be called a 'Paper Machine'. A man provided with paper, pencil, and rubber, and subject to strict discipline, is in effect a universal machine. The expression 'paper machine' will often be used below.
##### _Partially random and apparently partially random machines_
It is possible to modify the above described types of discrete machines by allowing several alternative operations to be applied at some points, the alternatives to be chosen by a random process. Such a machine will be described as 'partially random'. If we wish to say definitely that a machine is not of this kind we will describe it as 'determined'. Sometimes a machine may be strictly speaking determined but appear superficially as if it were partially random. This would occur if for instance the digits of the number π were used to determine the choices of a partially random machine, where previously a dice thrower or electronic equivalent had been used. These machines are known as apparently partially random.
#### 4. Unorganised machines
So far we have been considering machines which are designed for a definite purpose (though the universal machines are in a sense an exception). We might instead consider what happens when we make up a machine in a comparatively unsystematic way from some kind of standard components. We could consider some particular machine of this nature and find out what sort of things it is likely to do. Machines which are largely random in their construction in this way will be called 'unorganised machines'. This does not pretend to be an accurate term. It is conceivable that the same machine might be regarded by one man as organised and by another as unorganised.
A typical example of an unorganised machine would be as follows. The machine is made up from a rather large number N of similar units. Each unit has two input terminals, and has an output terminal which can be connected to the input terminals of (0 or more) other units. We may imagine that for each integer _r_ , 1 ≤ _r_ ≤ _N_ , two numbers _i_ ( _r_ ) and _j_ ( _r_ ) are chosen at random from 1... _N_ and that we connect the inputs of unit r to the outputs of units _i_ ( _r_ ) and _j_ ( _r_ ). All of the units are connected to a central synchronising unit from which synchronising pulses are emitted at more or less equal intervals of time. The times when these pulses arrive will be called 'moments'. Each unit is capable of having two states at each moment. These states may be called 0 and 1. The state is determined by the rule that the states of the units from which the input leads come are to be taken at the previous moment, multiplied together and the result subtracted from 1. An unorganised machine of this character is shown in the diagram below.
**Figure 10.1.**
A sequence of six possible consecutive conditions for the whole machine is:
The behaviour of a machine with so few units is naturally very trivial. However, machines of this character can behave in a very complicated manner when the number of units is large. We may call these A-type unorganised machines. Thus the machine in the diagram is an A-type unorganised machine of 5 units. The motion of an A-type machine with N units is of course eventually periodic, as is any determined machine with finite memory capacity. The period cannot exceed 2N moments, nor can the length of time before the periodic motion begins. In the example above the period is 2 moments and there are 3 moments before the periodic motion begins. 2N is 32.
The A-type unorganised machines are of interest as being about the simplest model of a nervous system with a random arrangement of neurons. It would therefore be of very great interest to find out something about their behaviour. A second type of unorganised machine will now be described, not because it is of any great intrinsic importance, but because it will be useful later for illustrative purposes. Let us denote the circuit as an abbreviation. Then for each A-type unorganised machine we can construct another machine by replacing each connection in it by The resulting machines will be called B-type unorganised machines. It may be said that the B-type machines are all A-type. To this I would reply that the above definitions if correctly (but drily!) set out would take the form of describing the probability of an A- (or B-) type machine belonging to a given set; it is not merely a definition of which are the A-type machines and which are the B-type machines. If one chooses an A-type machine, with a given number of units, at random, it will be extremely unlikely that one will get a B-type machine.
**Figure 10.2.**
It is easily seen that the connection can have three conditions. It may i) pass all signals through with interchange of 0 and 1, or ii) it may convert all signals into 1, or again iii) it may act as in i) and ii) in alternate moments. (Alternative iii) has two sub-cases.) Which of these cases applies depends on the initial conditions. There is a delay of two moments in going through .
#### 5. Interference with machinery. Modifiable and self-modifying machinery
The types of machine that we have considered so far are mainly ones that are allowed to continue in their own way for indefinite periods without interference from outside. The universal machines were an exception to this, in that from time to time one might change the description of the machine which is being imitated. We shall now consider machines in which such interference is the rule rather than the exception.
We may distinguish two kinds of interference. There is the extreme form in which parts of the machine are removed and replaced by others. This may be described as 'screwdriver interference'. At the other end of the scale is 'paper interference', which consists in the mere communication of information to the machine, which alters its behaviour. In view of the properties of the universal machine we do not need to consider the difference between these two kinds of machine as being so very radical after all. Paper interference when applied to the universal machine can be as useful as screwdriver interference.
We shall mainly be interested in paper interference. Since screwdriver interference can produce a completely new machine without difficulty there is rather little to be said about it. In future 'interference' will normally mean 'paper interference'.
When it is possible to alter the behaviour of a machine very radically we may speak of the machine as being 'modifiable'. This is a relative term. One machine may be spoken of as being more modifiable than another.
One may also sometimes speak of a machine modifying itself, or of a machine changing its own instructions. This is really a nonsensical form of phraseology, but is convenient. Of course, according to our conventions the 'machine' is completely described by the relation between its possible configurations at consecutive moments. It is an abstraction which by the form of its definition cannot change in time. If we consider the machine as starting in a particular configuration, however, we may be tempted to ignore those configurations which cannot be reached without interference from it. If we do this we should get a 'successor relation' for the configurations with different properties from the original one and so a different 'machine'.
If we now consider interference, we should say that each time interference occurs the machine is probably changed. It is in this sense that interference 'modifies' a machine. The sense in which a machine can modify itself is even more remote. We may if we wish divide the operations of the machine into two classes, normal and self-modifying operations. So long as only normal operations are performed we regard the machine as unaltered. Clearly the idea of 'self-modification' will not be of much interest except where the division of operations into the two classes is very carefully made. The sort of case I have in mind is a computing machine like the ACE where large parts of the storage are normally occupied in holding instruction tables. (Instruction tables are the equivalent in U.P.C.M.s of descriptions of machines in U.L.C.M.s). Whenever the content of this storage was altered by the internal operations of the machine, one would naturally speak of the machine 'modifying itself'.
#### 6. Man as a machine
A great positive reason for believing in the possibility of making thinking machinery is the fact that it is possible to make machinery to imitate any small part of a man. That the microphone does this for the ear, and the television camera for the eye, are commonplaces. One can also produce remote controlled Robots whose limbs balance the body with the aid of servo-mechanisms. Here we are chiefly interested in the nervous system. We could produce fairly accurate electrical models to copy the behaviour of nerves, but there seems very little point in doing so. It would be rather like putting a lot of work into cars which walked on legs instead of continuing to use wheels. The electrical circuits which are used in electronic computing machinery seem to have the essential properties of nerves. They are able to transmit information from place to place, and also to store it. Certainly the nerve has many advantages. It is extremely compact, does not wear out (probably for hundreds of years if kept in a suitable medium!) and has a very low energy consumption. Against these advantages the electronic circuits have only one counter attraction, that of speed. This advantage is however on such a scale that it may possibly outweigh the advantages of the nerve.
One way of setting about our task of building a 'thinking machine' would be to take a man as a whole and to try to replace all the parts of him by machinery. He would include television cameras, microphones, loudspeakers, wheels and 'handling servo-mechanisms' as well as some sort of 'electronic brain'. This would of course be a tremendous undertaking. The object if produced by present techniques would be of immense size, even if the 'brain' part were stationary and controlled the body from a distance. In order that the machine should have a chance of finding things out for itself it should be allowed to roam the countryside, and the danger to the ordinary citizen would be serious. Moreover even when the facilities mentioned above were provided, the creature would still have no contact with food, sex, sport and many other things of interest to the human being. Thus although this method is probably the 'sure' way of producing a thinking machine it seems to be altogether too slow and impracticable.
Instead we propose to try and see what can be done with a 'brain' which is more or less without a body, providing at most organs of sight, speech and hearing. We are then faced with the problem of finding suitable branches of thought for the machine to exercise its powers in. The following Welds appear to me to have advantages:
(i) Various games e.g. chess, noughts and crosses, bridge, poker
(ii) The learning of languages
(iii) Translation of languages
(iv) Cryptography
(v) Mathematics.
Of these (i), (iv), and to a lesser extent (iii) and (v) are good in that they require little contact with the outside world. For instance in order that the machine should be able to play chess its only organs need be 'eyes' capable of distinguishing the various positions on a specially made board, and means for announcing its own moves. Mathematics should preferably be restricted to branches where diagrams are not much used. Of the above possible Welds the learning of languages would be the most impressive, since it is the most human of these activities. This field seems however to depend rather too much on sense organs and locomotion to be feasible.
The field of cryptography will perhaps be the most rewarding. There is a remarkably close parallel between the problems of the physicist and those of the cryptographer. The system on which a message is enciphered corresponds to the laws of the universe, the intercepted messages to the evidence available, the keys for a day or a message to important constants which have to be determined. The correspondence is very close, but the subject matter of cryptography is very easily dealt with by discrete machinery, physics not so easily.
#### 7. Education of machinery
Although we have abandoned the plan to make a 'whole man', we should be wise to sometimes compare the circumstances of our machine with those of a man. It would be quite unfair to expect a machine straight from the factory to compete on equal terms with a university graduate. The graduate has had contact with human beings for twenty years or more. This contact has throughout that period been modifying his behaviour pattern. His teachers have been intentionally trying to modify it. At the end of the period a large number of standard routines will have been superimposed on the original pattern of his brain. These routines will be known to the community as a whole. He is then in a position to try out new combinations of these routines, to make slight variations on them, and to apply them in new ways.
We may say then that in so far as a man is a machine he is one that is subject to very much interference. In fact interference will be the rule rather than the exception. He is in frequent communication with other men, and is continually receiving visual and other stimuli which themselves constitute a form of interference. It will only be when the man is 'concentrating' with a view to eliminating these stimuli or 'distractions' that he approximates a machine without interference.
We are chiefly interested in machines with comparatively little interference, for reasons given in the last section, but it is important to remember that although a man when concentrating may behave like a machine without interference, his behaviour when concentrating is largely determined by the way he has been conditioned by previous interference.
If we are trying to produce an intelligent machine, and are following the human model as closely as we can, we should begin with a machine with very little capacity to carry out elaborate operations or to react in a disciplined manner to orders (taking the form of interference). Then by applying appropriate interference, mimicking education, we should hope to modify the machine until it could be relied on to produce definite reactions to certain commands. This would be the beginning of the process. I will not attempt to follow it further now.
#### 8. Organising unorganised machinery
Many unorganised machines have configurations such that if once that con-figuration is reached, and if the interference thereafter is appropriately restricted, the machine behaves as one organised for some definite purpose. For instance the B-type machine shown below was chosen at random.
**Figure 10.3.**
If the connections numbered 1, 3, 6, 4 are in condition ii) initially and connections 2, 5, 7 are in condition i), then the machine may be considered to be one for the purpose of passing on signals with a delay of 4 moments. This is a particular case of a very general property of B-type machines (and many other types), viz. that with suitable initial conditions they will do any required job, given sufficient time and provided the number of units is sufficient. In particular with a B-type unorganised machine with sufficient units one can find initial conditions which will make it into a universal machine with a given storage capacity. (A formal proof to this effect might be of some interest, or even a demonstration of it starting with a particular unorganised B-type machine, but I am not giving it as it lies rather too far outside the main argument.)
With these B-type machines the possibility of interference which could set in appropriate initial conditions has not been arranged for. It is however not difficult to think of appropriate methods by which this could be done. For instance instead of the connection Here _A, B_ are interfering inputs, normally giving the signal '1'. By supplying appropriate other signals at _A, B_ we can get the connection into condition i) or ii) or either form of iii), as desired. However, this requires two special interfering inputs for each connection.
.
**Figure 10.4.**
one might use
.
**Figure 10.5.**
We shall be mainly interested in cases where there are only quite few independent inputs altogether, so that all the interference which sets up the 'initial conditions' of the machine has to be provided through one or two inputs. The process of setting up these initial conditions so that the machine will carry out some particular useful task may be called 'organising the machine'. 'Organising' is thus a form of 'modification'.
#### 9. The cortex as an unorganised machine
Many parts of a man's brain are definite nerve circuits required for quite definite purposes. Examples of these are the 'centres' which control respiration, sneezing, following moving objects with the eyes, etc.: all the reflexes proper (not 'conditioned') are due to the activities of these definite structures in the brain. Likewise the apparatus for the more elementary analysis of shapes and sounds probably comes into this category. But the more intellectual activities of the brain are too varied to be managed on this basis. The difference between the languages spoken on the two sides of the Channel is not due to differences in development of the French-speaking and English-speaking parts of the brain. It is due to the linguistic parts having been subjected to different training. We believe then that there are large parts of the brain, chiefly in the cortex, whose function is largely indeterminate. In the infant these parts do not have much effect: the effect they have is uncoordinated. In the adult they have great and purposive effect: the form of this effect depends on the training in childhood. A large remnant of the random behaviour of infancy remains in the adult.
All of this suggests that the cortex of the infant is an unorganised machine, which can be organised by suitable interfering training. The organising might result in the modification of the machine into a universal machine or something like it. This would mean that the adult will obey orders given in appropriate language, even if they were very complicated; he would have no common sense, and would obey the most ridiculous orders unflinchingly. When all his orders had been fulfilled he would sink into a comatose state or perhaps obey some standing order, such as eating. Creatures not unlike this can really be found, but most people behave quite differently under many circumstances. However the resemblance to a universal machine is still very great, and suggests to us that the step from the unorganised infant to a universal machine is one which should be understood. When this has been mastered we shall be in a far better position to consider how the organising process might have been modified to produce a more normal type of mind.
This picture of the cortex as an unorganised machine is very satisfactory from the point of view of evolution and genetics. It clearly would not require any very complex system of genes to produce something like the A- or B-type unorganised machine. In fact this should be much easier than the production of such things as the respiratory centre. This might suggest that intelligent races could be produced comparatively easily. I think this is wrong because the possession of a human cortex (say) would be virtually useless if no attempt was made to organise it. Thus if a wolf by a mutation acquired a human cortex there is little reason to believe that he would have any selective advantage. If however the mutation occurred in a milieu where speech had developed (parrot-like wolves), and if the mutation by chance had well permeated a small community, then some selective advantage might be felt. It would then be possible to pass information on from generation to generation. However this is all rather speculative.
#### 10. Experiments in organising. Pleasure–pain systems
It is interesting to experiment with unorganised machines admitting definite types of interference and trying to organize them, e.g. to modify them into universal machines.
The organisation of a machine into a universal machine would be most impressive if the arrangements of interference involve very few inputs. The training of the human child depends largely on a system of rewards and punishments, and this suggests that it ought to be possible to carry through the organising with only two interfering inputs, one for 'pleasure' or 'reward' (R) and the other for 'pain' or 'punishment' (P). One can devise a large number of such 'pleasure–pain' systems. I will use this term to mean an unorganised machine of the following general character:
The configurations of the machine are described by two expressions, which we may call the character-expression and the situation-expression. The character and situation at any moment, together with the input signals, determine the character and situation at the next moment. The character may be subject to some random variation. Pleasure interference has a tendency to fix the character i.e. towards preventing it changing, whereas pain stimuli tend to disrupt the character, causing features which had become fixed to change, or to become again subject to random variation.
This definition is probably too vague and general to be very helpful. The idea is that when the 'character' changes we like to think of it as a change in the machine, but the 'situation' is merely the configuration of the machine described by the character. It is intended that pain stimuli occur when the machine's behaviour is wrong, pleasure stimuli when it is particularly right. With appropriate stimuli on these lines, judiciously operated by the 'teacher', one may hope that the 'character' will converge towards the one desired, i.e. that wrong behaviour will tend to become rare.
I have investigated a particular type of pleasure–pain system, which I will now describe.
#### 11. The P-type unorganised machine
The P-type machine may be regarded as an L.C.M. without a tape, and whose description is largely incomplete. When a configuration is reached for which the action is undetermined, a random choice for the missing data is made and the appropriate entry is made in the description, tentatively, and is applied. When a pain stimulus occurs all tentative entries are cancelled, and when a pleasure stimulus occurs they are all made permanent.
Specifically. The situation is a number _s_ = 1, 2,..., _N_ and corresponds to the configuration of the incomplete machine. The character is a table of _N_ entries showing the behaviour of the machine in each situation. Each entry has to say something both about the next situation and about what action the machine has to take. The action part may be either
(i) To do some externally visible act A1 or A2... A _K_
(ii) To set one of the memory units M1... M _R_ either into the '1' condition or into the '0' condition.
The next situation is always the remainder either of 2 _s_ or of 2 _s_ \+ 1 on division by _N_. These may be called alternatives 0 and 1. Which alternative applies may be determined by either
(a) One of the memory units
(b) A sense stimulus
(c) The pleasure–pain arrangements.
In each situation it is determined which of these applies when the machine is made, i.e. interference cannot alter which of the three cases applies. Also in cases (a) and (b) interference can have no effect. In case (c) the entry in the character table may be either U ('uncertain'), or T0 (tentative 0), T1, D0 (definite 0) or D1. When the entry in the character for the current situation is U then the alternative is chosen at random, and the entry in the character is changed to T0 or T1 according as 0 or 1 was chosen. If the character entry was T0 or D0 then the alternative is 0 and if it is T1 or D1 then the alternative is 1. The changes in character include the above mentioned change from U to T0 or T1, and a change of every T to D when a pleasure stimulus occurs, changes of T0 and T1 to U when a pain stimulus occurs.
We may imagine the memory units essentially as 'trigger circuits' or switches. The sense stimuli are means by which the teacher communicates 'unemotionally' to the machine, i.e. otherwise than by pleasure and pain stimuli. There are a finite number S of sense stimulus lines, and each always carries either the signal 0 or 1.
A small P-type machine is described in the table below
In this machine there is only one memory unit M1 and one sense line S1. Its behaviour can be described by giving the successive situations together with the actions of the teacher: the latter consist of the values of S1 and the rewards and punishments. At any moment the 'character' consists of the above table with each 'P' replaced by either U, T0, T1, D0 or D1. In working out the behaviour of the machine it is convenient first of all to make up a sequence of random digits for use when the U cases occur. Underneath these we may write the sequence of situations, and have other rows for the corresponding entries from the character, and for the actions of the teacher. The character and the values stored in the memory units may be kept on another sheet. The T entries may be made in pencil and the D entries in ink. A bit of the behaviour of the machine is given below:
It will be noticed that the machine very soon got into a repetitive cycle. This became externally visible through the repetitive BABAB... By means of a pain stimulus this cycle was broken.
It is probably possible to organise these P-type machines into universal machines, but it is not easy because of the form of memory available. It would be necessary to organise the randomly distributed 'memory units' to provide a systematic form of memory, and this would not be easy. If, however, we supply the P-type machine with a systematic external memory this organising becomes quite feasible. Such a memory could be provided in the form of a tape, and the externally visible operations could include movement to right and left along the tape, and altering the symbol on the tape to 0 or to 1. The sense lines could include one from the symbol on the tape. Alternatively, if the memory were to be finite, e.g. not more than 232 binary digits, we could use a dialling system. (Dialling systems can also be used with an infinite memory, but this is not of much practical interest.) I have succeeded in organising such a (paper) machine into a universal machine.
The details of the machine involved were as follows. There was a circular memory consisting of 64 squares of which at any moment one was in the machine ('scanned') and motion to right or left were among the 'visible actions'. Changing the symbol on the square was another 'visible action', and the symbol was connected to one of the sense lines S1. The even-numbered squares also had another function, they controlled the dialling of information to or from the main memory. This main memory consisted of 232 binary digits. At any moment one of these digits was connected to the sense line S2. The digit of the main memory concerned was that indicated by the 32 even positioned digits of the circular memory. Another two of the 'visible actions' were printing 0 or 1 in this square of the main memory. There were also three ordinary memory units and three sense units S3, S4, S5. Also six other externally visible actions A, B, C, D, E, F.
This P-type machine with external memory has, it must be admitted, considerably more 'organisation' than say the A-type unorganised machine. Nevertheless the fact that it can be organised into a universal machine still remains interesting.
The actual technique by which the 'organising' of the P-type machine was carried through is perhaps a little disappointing. It is not sufficiently analogous to the kind of process by which a child would really be taught. The process actually adopted was first to let the machine run for a long time with continuous application of pain, and with various changes of the sense data S3, S4, S5. Observation of the sequence of externally visible actions for some thousands of moments made it possible to set up a scheme for identifying the situations, i.e. by which one could at any moment find out what the situation was, except that the situations as a whole had been renamed. A similar investigation, with less use of punishment, enables one to find the situations which are affected by the sense lines; the data about the situations involving the memory units can also be found but with more difficulty. At this stage the character has been reconstructed. There are no occurrences of T0, T1, D0, D1. The next stage is to think up some way of replacing the Us of the character by D0, D1 in such a way as to give the desired modification. This will normally be possible with the suggested number of situations (1000), memory units etc. The final stage is the conversion of the character into the chosen one. This may be done simply by allowing the machine to wander at random through a sequence of situations, and applying pain stimuli when the wrong choice is made, pleasure stimuli when the right one is made. It is best also to apply pain stimuli when irrelevant choices are made. This is to prevent getting isolated in a ring of irrelevant situations. The machine is now 'ready for use'.
The form of universal machine actually produced in this process was as follows. Each instruction consisted of 128 digits, which we may regard as forming four sets of 32 each of which describes one place in the main memory. These places may be called P, Q, R, S. The meaning of the instruction is that if _p_ is the digit at P and _q_ that at Q then 1 _pq_ is to be transferred to position R and that the next instruction will be found in the 128 digits beginning at S. This gives a U.P.C.M., though with rather less facilities than are available say on the ACE.
I feel that more should be done on these lines. I would like to investigate other types of unorganised machine, and also to try out organising methods that would be more nearly analogous to our 'methods of education'. I made a start on the latter but found the work altogether too laborious at present. When some electronic machines are in actual operation I hope that they will make this more feasible. It should be easy to make a model of any particular machine that one wishes to work on within such a U.P.C.M. instead of having to work with a paper machine as at present. If also one decided on quite definite 'teaching policies' these could also be programmed into the machine. One would then allow the whole system to run for an appreciable period, and then break in as a kind of 'inspector of schools' and see what progress had been made. One might also be able to make some progress with unorganised machines more like the A- and B-types. The work involved with these is altogether too great for pure paper-machine work.
One particular kind of phenomenon I had been hoping to find in connection with the P-type machines. This was the incorporation of old routines into new. One might have 'taught' (i.e. modified or organised) a machine to add (say). Later one might teach it to multiply by small numbers by repeated addition and so arrange matters that the same set of situations which formed the addition routine, as originally taught, was also used in the additions involved in the multiplication. Although I was able to obtain a fairly detailed picture of how this might happen I was not able to do experiments on a sufficient scale for such phenomena to be seen as part of a larger context.
I also hoped to find something rather similar to the 'irregular verbs' which add variety to language. We seem to be quite content that things should not obey too mathematically regular rules. By long experience we can pick up and apply the most complicated rules without being able to enunciate them at all. I rather suspect that a P-type machine without the systematic memory would behave in a rather similar manner because of the randomly distributed memory units. Clearly this could only be verified by very painstaking work; by the very nature of the problem 'mass production' methods like built-in teaching procedures could not help.
#### 12. Discipline and initiative
If the untrained infant's mind is to become an intelligent one, it must acquire both discipline and initiative. So far we have been considering only discipline. To convert a brain or machine into a universal machine is the extremest form of discipline. Without something of this kind one cannot set up proper communication. But discipline is certainly not enough in itself to produce intelligence. That which is required in addition we call initiative. This statement will have to serve as a definition. Our task is to discover the nature of this residue as it occurs in man, and to try and copy it in machines.
Two possible methods of setting about this present themselves. On the one hand we have fully disciplined machines immediately available, or in a matter of months or years, in the form of various U.P.C.M.s. We might try to graft some initiative onto these. This would probably take the form of programming the machine to do every kind of job that could be done, as a matter of principle, whether it were economical to do it by machine or not. Bit by bit one would be able to allow the machine to make more and more 'choices' or 'decisions'. One would eventually find it possible to programme it so as to make its behaviour be the logical result of a comparatively small number of general principles. When these became sufficiently general, interference would no longer be necessary, and the machine would have 'grown up'. This may be called the 'direct method'.
The other method is to start with an unorganised machine and to try to bring both discipline and initiative into it at once, i.e. instead of trying to organise the machine to become a universal machine, to organise it for initiative as well. Both methods should, I think, be attempted.
##### _Intellectual, Genetical, and Cultural Searches_
A very typical sort of problem requiring some sort of initiative consists of those of the form 'Find a number _n_ such that...'. This form covers a very great variety of problems. For instance problems of the form 'See if you can find a way of calculating the function... which will enable us to obtain the values for arguments... to accuracy... within a time... using the U.P.C.M....' are reducible to this form, for the problem is clearly equivalent to that of finding a programme to put on the machine in question, and it is easy to put the programmes into correspondence with the positive integers in such a way that given either the number or the programme the other can easily be found. We should not go far wrong for the time being if we assumed that all problems were reducible to this form. It will be time to think again when something turns up which is obviously not of this form.
The crudest way of dealing with such a problem is to take the integers in order and to test each one to see whether it has the required property, and to go on until one is found which has it. Such a method will only be successful in the simplest cases. For instance in the case of problems of the kind mentioned above, where one is really searching for a programme, the number required will normally be somewhere between 21000 and 21,000,000. For practical work therefore some more expeditious method is necessary. In a number of cases the following method would be successful. Starting with a U.P.C.M. we first put a programme into it which corresponds to building in a logical system (like Russell's Principia Mathematica). This would not determine the behaviour of the machine completely: at various stages more than one choice as to the next step would be possible. We might however arrange to take all possible arrangements of choices in order, and go on until the machine proved a theorem which, by its form, could be verified to give a solution of the problem. This may be seen to be a conversion of the original problem into another of the same form. Instead of searching through values of the original variable _n_ one searches through values of something else. In practice when solving problems of the above kind one will probably apply some very complex 'transformation' of the original problem, involving searching through various variables, some more analogous to the original one, some more like a 'search through all proofs'. Further research into intelligence of machinery will probably be very greatly concerned with 'searches' of this kind. We may perhaps call such searches 'intellectual searches'. They might very briefly be defined as 'searches carried out by brains for combinations with particular properties'.
It may be of interest to mention two other kinds of search in this connection. There is the genetical or evolutionary search by which a combination of genes is looked for, the criterion being survival value. The remarkable success of this search confirms to some extent the idea that intellectual activity consists mainly of various kinds of search.
The remaining form of search is what I should like to call the 'cultural search'. As I have mentioned, the isolated man does not develop any intellectual power. It is necessary for him to be immersed in an environment of other men, whose techniques he absorbs during the first 20 years of his life. He may then perhaps do a little research of his own and make a very few discoveries which are passed on to other men. From this point of view the search for new techniques must be regarded as carried out by the human community as a whole, rather than by individuals.
#### 13. Intelligence as an emotional concept
The extent to which we regard something as behaving in an intelligent manner is determined as much by our own state of mind and training as by the properties of the object under consideration. If we are able to explain and predict its behaviour or if there seems to be little underlying plan, we have little temptation to imagine intelligence. With the same object therefore it is possible that one man would consider it as intelligent and another would not; the second man would have found out the rules of its behaviour.
It is possible to do a little experiment on these lines, even at the present stage of knowledge. It is not difficult to devise a paper machine which will play a not very bad game of chess. Now get three men as subjects for the experiment A, B, C. A and C are to be rather poor chess players, B is the operator who works the paper machine. (In order that he should be able to work it fairly fast it is advisable that he be both mathematician and chess player.) Two rooms are used with some arrangement for communicating moves, and a game is played between C and either A or the paper machine. C may find it quite difficult to tell which he is playing.
(This is a rather idealized form of an experiment I have actually done.)
#### Summary
The possible ways in which machinery might be made to show intelligent behaviour are discussed. The analogy with the human brain is used as a guiding principle. It is pointed out that the potentialities of the human intelligence can only be realised if suitable education is provided. The investigation mainly centres round an analogous teaching process applied to machines. The idea of an unorganised machine is defined, and it is suggested that the infant human cortex is of this nature. Simple examples of such machines are given, and their education by means of rewards and punishments is discussed. In one case the education process is carried through until the organisation is similar to that of an ACE.
### **CHAPTER 11
Computing Machinery and Intelligence ( _1950_ )**
_Alan Turing_
#### **Introduction**
##### _Jack Copeland_
Together with 'On Computable Numbers' (Chapter 1), 'Computing Machinery and Intelligence' forms Turing's best-known work. This elegant and sometimes amusing essay was originally published in 1950 in the leading philosophy journal _Mind_. Turing's friend Robin Gandy (like Turing a mathematical logician) said that 'Computing Machinery and Intelligence'
was intended not so much as a penetrating contribution to philosophy but as propaganda. Turing thought the time had come for philosophers and mathematicians and scientists to take seriously the fact that computers were not merely calculating engines but were capable of behaviour which must be accounted as intelligent; he sought to persuade people that this was so. He wrote this paper—unlike his mathematical papers—quickly and with enjoyment. I can remember him reading aloud to me some of the passages— always with a smile, sometimes with a giggle.
The quality and originality of 'Computing Machinery and Intelligence' have earned it a place among the classics of philosophy of mind.
#### The Turing Test
'Computing Machinery and Intelligence' contains Turing's principal exposition of the famous 'imitation game' or Turing test. The test first appeared, in a restricted form, in the closing paragraphs of 'Intelligent Machinery' (Chapter 10). Chapters 13 and , dating from 1951 and 1952 respectively, contain further discussion and amplification; unpublished until 1999, this important additional material throws new light on how the Turing test is to be understood.
The imitation game involves three participants: a computer, a human interrogator, and a human 'foil'. The interrogator attempts to determine, by asking questions of the other two participants, which of them is the computer. All communication is via keyboard and screen, or an equivalent arrangement (Turing suggested a teleprinter link). The interrogator may ask questions as penetrating and wide-ranging as he or she likes, and the computer is permitted to do everything possible to force a wrong identification. (So the computer might answer 'No' in response to 'Are you a computer?' and might follow a request to multiply one large number by another with a long pause and a plausibly incorrect answer.) The foil must help the interrogator to make a correct identification.
The ability to play the imitation game successfully is Turing's proposed 'criterion for "thinking"' (pp. 442, 443). He gives two examples of the sort of exchange that might occur between an interrogator and a machine that plays successfully. The following is from p. 452.
Interrogator: In the first line of your sonnet which reads 'Shall I compare thee to a summer's day', would not 'a spring day' do as well or better?
Machine: It wouldn't scan.
Interrogator: How about 'a winter's day'? That would scan all right.
Machine: Yes, but nobody wants to be compared to a winter's day.
Interrogator: Would you say Mr Pickwick reminded you of Christmas?
Machine: In a way.
Interrogator: Yet Christmas is a winter's day, and I do not think Mr Pickwick would mind the comparison.
Machine: I don't think you're serious. By a winter's day one means a typical winter's day, rather than a special one like Christmas.
#### Did Turing Propose a Definition?
Turing is sometimes said to have proposed a definition of 'thinking' or 'intelligence'; and sometimes his supposed definition is said to be an 'operational' or 'behaviourist' definition. For example:
An especially influential behaviorist definition of intelligence was put forward by Turing. (Ned Block)
[Turing] introduced...an operational definition of 'thinking' or 'intelligence'... by means of a sexual guessing game. (Andrew Hodges)
The Turing Test [was] originally proposed as a simple operational definition of intelligence. (Robert French)
There is no textual evidence to support this interpretation of Turing, however. In 'Computing Machinery and Intelligence' Turing claimed to be offering only a ' _criterion_ for "thinking"' (emphasis added). Moreover, in his discussion of the Turing test in Chapter 14, Turing says quite specifically that his aim is not 'to give a definition of thinking' (p. 494).
In fact, Turing made it plain in 'Computing Machinery and Intelligence' that his intention was not to offer a definition, for he said:
The game may perhaps be criticised on the ground that the odds are weighted too heavily against the machine. If the man were to try and pretend to be the machine he would clearly make a very poor showing. He would be given away at once by slowness and inaccuracy in arithmetic. May not machines carry out something which ought to be described as thinking but which is very different from what a man does? (p. 442)
A computer carrying out something that 'ought to be described as thinking' would nevertheless fail the Turing test if for any reason it stood out in conversation as very different from a man. It follows that 'thinking' cannot be defined in terms of success in the imitation game. Success in the game is arguably a sufficient condition for thinking; but success in the imitation game is not also a _necessary_ condition for thinking. (Someone's breathing spontaneously is a sufficient condition for their being alive, but it is not also a necessary condition, for someone may be alive without breathing spontaneously.)
#### The Male–Female Imitation Game
Turing introduced his criterion for 'thinking' by first describing an imitation game involving a human interrogator and two _human_ subjects, one male (A) and one female (B). The interrogator must determine, by question and answer, which of A and B is the man. A's object in the game is to try to cause the interrogator to make the wrong identification. Having introduced the imitation game in this way, Turing said:
We now ask the question, 'What will happen when a machine takes the part of A in this game?' Will the interrogator decide wrongly as often when the game is played like this as he does when the game is played between a man and a woman? These questions replace our original, 'Can machines think?' (p. 441)
Some commentators have suggested, on the basis of this passage, that Turing's criterion for thinking is that the computer in the Turing test be able to impersonate a woman. Later in the article, however, Turing described matters differently, saying that the part of A is taken by a machine and 'the part of B... by a man' (p. 448). This runs contrary to the suggestion that the computer is supposed to imitate a woman (rather than a man or a woman). Moreover in Chapter 14 Turing says that 't]he idea of the test is that the machine has to try and pretend to be a man... and it will pass only if the pretence is reasonably convincing' (p. 495). In [Chapter 13 Turing presents the test in a starkly ungendered form: here the point of the test is to determine whether or not a computer can 'imitate a brain' (p. 485). On balance, then, it seems rather unlikely that Turing's intention in 'Computing Machinery and Intelligence' was to put forward a test in which the computer must impersonate a woman.
The role of the man-imitates-woman game is frequently misunderstood. For example, Hodges claims that this game is irrelevant as an introduction to the Turing test—indeed, it is a 'red herring'. However, the man-imitates-woman game forms part of the protocol for scoring the test. Will interrogators decide wrongly as often in man-imitates-woman imitation games as they do in computer-imitates-human games? This question, Turing said, replaces 'Can machines think?'
#### The Current Status of the Turing Test
Section 6 of 'Computing Machinery and Intelligence', entitled 'Contrary Views on the Main Question', occupies nearly half of the article. It contains no fewer than nine objections to Turing's position, together with Turing's rebuttal of each. One of them, the 'Mathematical Objection', is also discussed in Chapters 10 and (the introduction to Chapter 12 gives some further information about this important and controversial objection).
Since 'Computing Machinery and Intelligence' first appeared, Turing's test has received considerable attention from philosophers, computer scientists, psychologists, and others, and numerous additional objections have been raised to the test, some of them ingenious indeed. Nevertheless, it seems to me that none of these objections is successful (see my chapter in Moor's _The Turing Test_ in the list of further reading). A discussion of one such objection, called here the Shannon– McCarthy objection, will give something of the flavour of the debate that still rages over the Turing test. Another form of objection—the 'Fiendish Expert' objection—is discussed in the introduction to Chapter 14.
#### The Shannon–McCarthy Objection
This objection envisages a hypothetical computer that is able to play the imitation game successfully, for any set length of time, in virtue of incorporating a very large—but nevertheless finite—'look-up' table. The table contains _all_ the exchanges that could possibly occur between the computer and the interrogator during the length of time for which the test is run. The number of these is astronomical—but finite. For example, the exchange displayed earlier concerning sonnets and Mr Pickwick forms part of this (imaginary) table.
Clearly an interrogator would have no means by which to distinguish a computer using this table from a human respondent. Yet presumably the computer—which does nothing but search the table provided by its (hypothetical) programmers—does not think. In principle, therefore, an unthinking, unintelligent computer can pass the test.
Claude Shannon and John McCarthy put the objection forward in 1956:
The problem of giving a precise definition to the concept of 'thinking' and of deciding whether or not a given machine is capable of thinking has aroused a great deal of heated discussion. One interesting definition has been proposed by A. M. Turing: a machine is termed capable of thinking if it can, under certain prescribed conditions, imitate a human being by answering questions sufficiently well to deceive a human questioner for a reasonable period of time. A definition of this type has the advantages of being operational, or, in the psychologists' term, behavioristic.... A disadvantage of the Turing definition of thinking is that it is possible, in principle, to design a machine with a complete set of arbitrarily chosen responses to all possible input stimuli... Such a machine, in a sense, for any given input situation (including past history) merely looks up in a 'dictionary' the appropriate response. With a suitable dictionary such a machine would surely satisfy Turing's definition but does not reflect our usual intuitive concept of thinking.
This objection has been rediscovered by a number of philosophers, and it is in fact usually credited to Block, who published a version of it in 1981. (It is sometimes referred to as the 'blockhead' objection to the Turing test.)
What might Turing have said in response to the objection? A hint is perhaps provided by the following exchange between Turing and Newman (Chapter 14, p. 503):
Newman: It is all very well to say that a machine could... be made to do this or that, but, to take only one practical point, what about the time it would take to do it? It would only take an hour or two to make up a routine to make our Manchester machine analyse all possible variations of the game of chess right out, and find the best move that way— _if_ you didn't mind its taking thousands of millions of years to run through the routine. Solving a problem on the machine doesn't mean finding a way to do it between now and eternity, but within a reasonable time....
Turing: To my mind this time factor is the one question which will involve all the real technical difficulty.
The Shannon–McCarthy objection establishes only that the
_Turing Test Principle_ If _x_ plays Turing's imitation game satisfactorily, then _x_ thinks
is false in _some possible world_. The objection directs our imagination toward a possible world that is very different from the actual world—a world in which an astronomically large look-up table can be stored in a computer's memory and searched in a reasonable time—and points out that the Turing test principle is false in _that_ world. However, there is no textual evidence to indicate that Turing was claiming anything more than that the Turing test principle is _actually_ true, i.e. true in the actual world. Nor did he need to claim more than this in order to advocate the imitation game as a satisfactory real-world test.
Had Turing been proposing a definition of 'thinking', then he would indeed have had to say, consistently, that the Turing test principle is true in _all_ possible worlds. (To take a more obvious case, if 'bachelor' is defined as 'unmarried male of marriageable age', then it is true not only in the actual world but in every possible world that if _x_ is an unmarried male of marriageable age, then _x_ is a bachelor.) At bottom, then, the Shannon–McCarthy objection depends on the interpretational mistake of taking Turing to be proposing a definition.
There is further discussion of the Turing test in Chapters 13, , and .
#### Learning Machines
The discussion of learning begun in Chapter 10 is continued in the iconoclastic Section 7 of 'Computing Machinery and Intelligence', entitled 'Learning Machines'. Turing poses the rhetorical question: 'Instead of trying to produce a programme to simulate the adult mind, why not rather try to produce one which simulates the child's?' (p. 460). The child's mind may contain 'so little mechanism' that 'something like it can be easily programmed'. If this child-machine 'were then subjected to an appropriate course of education one would obtain the adult brain'. These remarks are of a piece with Turing's suggestion in Chapter 10 that 'the cortex of an infant is an unorganised machine, which can be organised by suitable interfering training' (p. 424).
Turing mentions in Section 7 that he has 'done some experiments with one such child-machine, and succeeded in teaching it a few things, but the teaching method was too unorthodox for the experiment to be considered really successful' (p. 461). Here he is probably referring to the experiments with an unorganized machine that are described in Chapter 10, where he says that he has 'succeeded in organising such a (paper) machine into a universal machine', but that the technique used 'is not sufficiently analogous to the kind of process by which a child would really be taught' (pp. 427-8).
#### Situated AI
AI traditionally has attempted to build disembodied intelligences carrying out abstract activities—e.g. chess-playing—and whose only way of interacting with the world is by means of a screen or printer. An alternative approach now called 'situated AI' aims at building embodied intelligences situated in the real world. 'Computing Machinery and Intelligence' ends with a characteristically farsighted statement in which Turing sketches each of these two approaches to AI. He contrasts research that focuses on 'abstract activity, like the playing of chess' with research aiming 'to provide the machine with the best sense organs that money can buy, and then teach it to understand and speak English' (p. 463). Turing recommended that 'both approaches should be tried' (ibid.; compare Chapter 10, pp. 420–1).
Rodney Brooks, a modern pioneer of situated AI and Director of the MIT Artificial Intelligence Laboratory, pointed out that although Turing proposed both these 'paths toward his goal of a thinking machine', Artificial Intelligence for a long time 'all but ignored' the situated approach. Now the tables have turned and there is huge interest in situated AI.
One of Brooks's experimental robots, Herbert—named after Herbert Simon— searched the offices and work-spaces of the MIT AI Lab for empty soda cans, picking them up and carrying them to the trash. Herbert, unlike previous generations of experimental robots, operated in real time in a busy, cluttered, and unpredictably changing real-world environment. Brooks's humanoid learning robot Cog—from 'cognizer'—has four microphone-type 'ears' and saccading foveated vision provided by cameras mounted on its 'head'. Cog's legless torso is able to lean and twist. Strain gauges on the spine give Cog information about posture, while heat and current sensors on the robot's motors provide feedback concerning exertion. Cog's arm and manipulating hand are coated with electrically conducting rubber membranes providing tactile information. Those working in situated AI regard Cog as a milestone on the road toward the realization of Turing's dream.
##### _Further reading_
Block, N., 'Psychologism and Behaviorism', _Philosophical Review_ , 90 (1981), 5–43.
Dennett, D. C., 'Can Machines Think?', in his _Brainchildren: Essays on Designing Minds_ (Cambridge, Mass.: MIT Press, 1998).
French, R., 'The Turing Test: The First 50 Years', _Trends in Cognitive Sciences_ , 4 (2000), 115–22.
Michie, D., 'Turing's Test and Conscious Thought', _Artificial Intelligence_ , 60 (1993), 1–22. Reprinted in P. Millican and A. Clark (eds.), _Machines and Thought: The Legacy of Alan Turing_ (Oxford: Clarendon Press, 1996).
Moor, J. H. (ed.), _The Turing Test_ (Dordrecht: Kluwer, 2003).
—— 'An Analysis of the Turing Test', _Philosophical Studies_ , 30 (1976), 249–57.
##### _Provenance_
What follows is the text of the original printing of 'Computing Machinery and Intelligence' in _Mind_. (Unfortunately Turing's typescript has been lost.)
#### **Computing Machinery and Intelligence**
#### 1. The Imitation Game
I propose to consider the question, 'Can machines think?' This should begin with definitions of the meaning of the terms 'machine' and 'think'. The definitions might be framed so as to reflect so far as possible the normal use of the words, but this attitude is dangerous. If the meaning of the words 'machine' and 'think' are to be found by examining how they are commonly used it is difficult to escape the conclusion that the meaning and the answer to the question, 'Can machines think?' is to be sought in a statistical survey such as a Gallup poll. But this is absurd. Instead of attempting such a definition I shall replace the question by another, which is closely related to it and is expressed in relatively unambiguous words.
The new form of the problem can be described in terms of a game which we call the 'imitation game'. It is played with three people, a man (A), a woman (B), and an interrogator (C) who may be of either sex. The interrogator stays in a room apart from the other two. The object of the game for the interrogator is to determine which of the other two is the man and which is the woman. He knows them by labels X and Y, and at the end of the game he says either 'X is A and Y is B' or 'X is B and Y is A'. The interrogator is allowed to put questions to A and B thus:
C: Will X please tell me the length of his or her hair? Now suppose X is actually A, then A must answer. It is A's object in the game to try and cause C to make the wrong identification. His answer might therefore be
'My hair is shingled, and the longest strands are about nine inches long.'
In order that tones of voice may not help the interrogator the answers should be written, or better still, typewritten. The ideal arrangement is to have a teleprinter communicating between the two rooms. Alternatively the question and answers can be repeated by an intermediary. The object of the game for the third player (B) is to help the interrogator. The best strategy for her is probably to give truthful answers. She can add such things as 'I am the woman, don't listen to him!' to her answers, but it will avail nothing as the man can make similar remarks.
We now ask the question, 'What will happen when a machine takes the part of A in this game?' Will the interrogator decide wrongly as often when the game is played like this as he does when the game is played between a man and a woman? These questions replace our original, 'Can machines think?'
#### 2. Critique of the New Problem
As well as asking, 'What is the answer to this new form of the question', one may ask, 'Is this new question a worthy one to investigate?' This latter question we investigate without further ado, thereby cutting short an infinite regress.
The new problem has the advantage of drawing a fairly sharp line between the physical and the intellectual capacities of a man. No engineer or chemist claims to be able to produce a material which is indistinguishable from the human skin. It is possible that at some time this might be done, but even supposing this invention available we should feel there was little point in trying to make a 'thinking machine' more human by dressing it up in such artificial flesh. The form in which we have set the problem reflects this fact in the condition which prevents the interrogator from seeing or touching the other competitors, or hearing their voices. Some other advantages of the proposed criterion may be shown up by specimen questions and answers. Thus:
Q: Please write me a sonnet on the subject of the Forth Bridge.
A: Count me out on this one. I never could write poetry.
Q: Add 34957 to 70764.
A: (Pause about 30 seconds and then give as answer) 105621.
Q: Do you play chess?
A: Yes.
Q: I have K at my K1, and no other pieces. You have only K at K6 and R at R1. It is your move. What do you play?
A: (After a pause of 15 seconds) R-R8 mate.
The question and answer method seems to be suitable for introducing almost any one of the fields of human endeavour that we wish to include. We do not wish to penalise the machine for its inability to shine in beauty competitions, nor to penalise a man for losing in a race against an aeroplane. The conditions of our game make these disabilities irrelevant. The 'witnesses' can brag, if they consider it advisable, as much as they please about their charms, strength or heroism, but the interrogator cannot demand practical demonstrations.
The game may perhaps be criticised on the ground that the odds are weighted too heavily against the machine. If the man were to try and pretend to be the machine he would clearly make a very poor showing. He would be given away at once by slowness and inaccuracy in arithmetic. May not machines carry out something which ought to be described as thinking but which is very different from what a man does? This objection is a very strong one, but at least we can say that if, nevertheless, a machine can be constructed to play the imitation game satisfactorily, we need not be troubled by this objection.
It might be urged that when playing the 'imitation game' the best strategy for the machine may possibly be something other than imitation of the behaviour of a man. This may be, but I think it is unlikely that there is any great effect of this kind. In any case there is no intention to investigate here the theory of the game, and it will be assumed that the best strategy is to try to provide answers that would naturally be given by a man.
#### 3. The Machines concerned in the Game
The question which we put in §1 will not be quite definite until we have specified what we mean by the word 'machine'. It is natural that we should wish to permit every kind of engineering technique to be used in our machines. We also wish to allow the possibility that an engineer or team of engineers may construct a machine which works, but whose manner of operation cannot be satisfactorily described by its constructors because they have applied a method which is largely experimental. Finally, we wish to exclude from the machines men born in the usual manner. It is difficult to frame the definitions so as to satisfy these three conditions. One might for instance insist that the team of engineers should be all of one sex, but this would not really be satisfactory, for it is probably possible to rear a complete individual from a single cell of the skin (say) of a man. To do so would be a feat of biological technique deserving of the very highest praise, but we would not be inclined to regard it as a case of 'constructing a thinking machine'. This prompts us to abandon the requirement that every kind of technique should be permitted. We are the more ready to do so in view of the fact that the present interest in 'thinking machines' has been aroused by a particular kind of machine, usually called an 'electronic computer' or 'digital computer'. Following this suggestion we only permit digital computers to take part in our game.
This restriction appears at first sight to be a very drastic one. I shall attempt to show that it is not so in reality. To do this necessitates a short account of the nature and properties of these computers.
It may also be said that this identification of machines with digital computers, like our criterion for 'thinking', will only be unsatisfactory if (contrary to my belief), it turns out that digital computers are unable to give a good showing in the game.
There are already a number of digital computers in working order, and it may be asked, 'Why not try the experiment straight away? It would be easy to satisfy the conditions of the game. A number of interrogators could be used, and statistics compiled to show how often the right identification was given.' The short answer is that we are not asking whether all digital computers would do well in the game nor whether the computers at present available would do well, but whether there are imaginable computers which would do well. But this is only the short answer. We shall see this question in a different light later.
#### 4. Digital Computers
The idea behind digital computers may be explained by saying that these machines are intended to carry out any operations which could be done by a human computer. The human computer is supposed to be following fixed rules; he has no authority to deviate from them in any detail. We may suppose that these rules are supplied in a book, which is altered whenever he is put on to a new job. He has also an unlimited supply of paper on which he does his calculations. He may also do his multiplications and additions on a 'desk machine', but this is not important.
If we use the above explanation as a definition we shall be in danger of circularity of argument. We avoid this by giving an outline of the means by which the desired effect is achieved. A digital computer can usually be regarded as consisting of three parts:
(i) Store.
(ii) Executive unit.
(iii) Control.
The store is a store of information, and corresponds to the human computer's paper, whether this is the paper on which he does his calculations or that on which his book of rules is printed. In so far as the human computer does calculations in his head a part of the store will correspond to his memory.
The executive unit is the part which carries out the various individual operations involved in a calculation. What these individual operations are will vary from machine to machine. Usually fairly lengthy operations can be done such as 'Multiply 3540675445 by 7076345687' but in some machines only very simple ones such as 'Write down 0' are possible.
We have mentioned that the 'book of rules' supplied to the computer is replaced in the machine by a part of the store. It is then called the 'table of instructions'. It is the duty of the control to see that these instructions are obeyed correctly and in the right order. The control is so constructed that this necessarily happens.
The information in the store is usually broken up into packets of moderately small size. In one machine, for instance, a packet might consist of ten decimal digits. Numbers are assigned to the parts of the store in which the various packets of information are stored, in some systematic manner. A typical instruction might say—
'Add the number stored in position 6809 to that in 4302 and put the result back into the latter storage position'.
Needless to say it would not occur in the machine expressed in English. It would more likely be coded in a form such as 6809430217. Here 17 says which of various possible operations is to be performed on the two numbers. In this case the operation is that described above, _viz_. 'Add the number...' It will be noticed that the instruction takes up 10 digits and so forms one packet of information, very conveniently. The control will normally take the instructions to be obeyed in the order of the positions in which they are stored, but occasionally an instruction such as
'Now obey the instruction stored in position 5606, and continue from there' may be encountered, or again
'If position 4505 contains 0 obey next the instruction stored in 6707, otherwise continue straight on.'
Instructions of these latter types are very important because they make it possible for a sequence of operations to be repeated over and over again until some condition is fulfilled, but in doing so to obey, not fresh instructions on each repetition, but the same ones over and over again. To take a domestic analogy. Suppose Mother wants Tommy to call at the cobbler's every morning on his way to school to see if her shoes are done, she can ask him afresh every morning. Alternatively she can stick up a notice once and for all in the hall which he will see when he leaves for school and which tells him to call for the shoes, and also to destroy the notice when he comes back if he has the shoes with him.
The reader must accept it as a fact that digital computers can be constructed, and indeed have been constructed, according to the principles we have described, and that they can in fact mimic the actions of a human computer very closely.
The book of rules which we have described our human computer as using is of course a convenient fiction. Actual human computers really remember what they have got to do. If one wants to make a machine mimic the behaviour of the human computer in some complex operation one has to ask him how it is done, and then translate the answer into the form of an instruction table. Constructing instruction tables is usually described as 'programming'. To 'programme a machine to carry out the operation A' means to put the appropriate instruction table into the machine so that it will do A.
An interesting variant on the idea of a digital computer is a 'digital computer with a random element'. These have instructions involving the throwing of a die or some equivalent electronic process; one such instruction might for instance be, 'Throw the die and put the resulting number into store 1000'. Sometimes such a machine is described as having free will (though I would not use this phrase myself). It is not normally possible to determine from observing a machine whether it has a random element, for a similar effect can be produced by such devices as making the choices depend on the digits of the decimal for _π_.
Most actual digital computers have only a finite store. There is no theoretical difficulty in the idea of a computer with an unlimited store. Of course only a finite part can have been used at any one time. Likewise only a finite amount can have been constructed, but we can imagine more and more being added as required. Such computers have special theoretical interest and will be called infinitive capacity computers.
The idea of a digital computer is an old one. Charles Babbage, Lucasian Professor of Mathematics at Cambridge from 1828 to 1839, planned such a machine, called the Analytical Engine, but it was never completed. Although Babbage had all the essential ideas, his machine was not at that time such a very attractive prospect. The speed which would have been available would be definitely faster than a human computer but something like 100 times slower than the Manchester machine, itself one of the slower of the modern machines. The storage was to be purely mechanical, using wheels and cards.
The fact that Babbage's Analytical Engine was to be entirely mechanical will help us to rid ourselves of a superstition. Importance is often attached to the fact that modern digital computers are electrical, and that the nervous system also is electrical. Since Babbage's machine was not electrical, and since all digital computers are in a sense equivalent, we see that this use of electricity cannot be of theoretical importance. Of course electricity usually comes in where fast signalling is concerned, so that it is not surprising that we find it in both these connections. In the nervous system chemical phenomena are at least as important as electrical. In certain computers the storage system is mainly acoustic. The feature of using electricity is thus seen to be only a very superficial similarity. If we wish to find such similarities we should look rather for mathematical analogies of function.
#### 5. Universality of Digital Computers
The digital computers considered in the last section may be classified amongst the 'discrete state machines'. These are the machines which move by sudden jumps or clicks from one quite definite state to another. These states are sufficiently different for the possibility of confusion between them to be ignored. Strictly speaking there are no such machines. Everything really moves continuously. But there are many kinds of machine which can profitably be _thought of_ as being discrete state machines. For instance in considering the switches for a lighting system it is a convenient fiction that each switch must be definitely on or definitely off. There must be intermediate positions, but for most purposes we can forget about them. As an example of a discrete state machine we might consider a wheel which clicks round through 120° once a second, but may be stopped by a lever which can be operated from outside; in addition a lamp is to light in one of the positions of the wheel. This machine could be described abstractly as follows. The internal state of the machine (which is described by the position of the wheel) may be _q_ 1, _q_ 2 or _q_ 3. There is an input signal _i_ 0 or _i_ 1 (position of lever). The internal state at any moment is determined by the last state and input signal according to the table
The output signals, the only externally visible indication of the internal state (the light) are described by the table
This example is typical of discrete state machines. They can be described by such tables provided they have only a finite number of possible states.
It will seem that given the initial state of the machine and the input signals it is always possible to predict all future states. This is reminiscent of Laplace's view that from the complete state of the universe at one moment of time, as described by the positions and velocities of all particles, it should be possible to predict all future states. The prediction which we are considering is, however, rather nearer to practicability than that considered by Laplace. The system of the 'universe as a whole' is such that quite small errors in the initial conditions can have an overwhelming effect at a later time. The displacement of a single electron by a billionth of a centimetre at one moment might make the difference between a man being killed by an avalanche a year later, or escaping. It is an essential property of the mechanical systems which we have called 'discrete state machines' that this phenomenon does not occur. Even when we consider the actual physical machines instead of the idealised machines, reasonably accurate knowledge of the state at one moment yields reasonably accurate knowledge any number of steps later.
As we have mentioned, digital computers fall within the class of discrete state machines. But the number of states of which such a machine is capable is usually enormously large. For instance, the number for the machine now working at Manchester is about 2165,000, _i.e_. about 1050,000. Compare this with our example of the clicking wheel described above, which had three states. It is not difficult to see why the number of states should be so immense. The computer includes a store corresponding to the paper used by a human computer. It must be possible to write into the store any one of the combinations of symbols which might have been written on the paper. For simplicity suppose that only digits from 0 to 9 are used as symbols. Variations in handwriting are ignored. Suppose the computer is allowed 100 sheets of paper each containing 50 lines each with room for 30 digits. Then the number of states is 10100 × 50 × 30, _i.e_.10150,000. This is about the number of states of three Manchester machines put together. The logarithm to the base two of the number of states is usually called the 'storage capacity' of the machine. Thus the Manchester machine has a storage capacity of about 165,000 and the wheel machine of our example about 1.6. If two machines are put together their capacities must be added to obtain the capacity of the resultant machine. This leads to the possibility of statements such as 'The Manchester machine contains 64 magnetic tracks each with a capacity of 2560, eight electronic tubes with a capacity of 1280. Miscellaneous storage amounts to about 300 making a total of 174,380.'
Given the table corresponding to a discrete state machine it is possible to predict what it will do. There is no reason why this calculation should not be carried out by means of a digital computer. Provided it could be carried out sufficiently quickly the digital computer could mimic the behaviour of any discrete state machine. The imitation game could then be played with the machine in question (as B) and the mimicking digital computer (as A) and the interrogator would be unable to distinguish them. Of course the digital computer must have an adequate storage capacity as well as working sufficiently fast. Moreover, it must be programmed afresh for each new machine which it is desired to mimic.
This special property of digital computers, that they can mimic any discrete state machine, is described by saying that they are _universal_ machines. The existence of machines with this property has the important consequence that, considerations of speed apart, it is unnecessary to design various new machines to do various computing processes. They can all be done with one digital computer, suitably programmed for each case. It will be seen that as a consequence of this all digital computers are in a sense equivalent.
We may now consider again the point raised at the end of §3. It was suggested tentatively that the question, 'Can machines think?' should be replaced by 'Are there imaginable digital computers which would do well in the imitation game?' If we wish we can make this superficially more general and ask 'Are there discrete state machines which would do well?' But in view of the universality property we see that either of these questions is equivalent to this, 'Let us fix our attention on one particular digital computer _C_. Is it true that by modifying this computer to have an adequate storage, suitably increasing its speed of action, and providing it with an appropriate programme, _C_ can be made to play satisfactorily the part of A in the imitation game, the part of B being taken by a man?'
#### 6. Contrary Views on the Main Question
We may now consider the ground to have been cleared and we are ready to proceed to the debate on our question, 'Can machines think?' and the variant of it quoted at the end of the last section. We cannot altogether abandon the original form of the problem, for opinions will differ as to the appropriateness of the substitution and we must at least listen to what has to be said in this connexion.
It will simplify matters for the reader if I explain first my own beliefs in the matter. Consider first the more accurate form of the question. I believe that in about fifty years' time it will be possible to programme computers, with a storage capacity of about 109, to make them play the imitation game so well that an average interrogator will not have more than 70 per cent. chance of making the right identification after five minutes of questioning. The original question, 'Can machines think?' I believe to be too meaningless to deserve discussion. Nevertheless I believe that at the end of the century the use of words and general educated opinion will have altered so much that one will be able to speak of machines thinking without expecting to be contradicted. I believe further that no useful purpose is served by concealing these beliefs. The popular view that scientists proceed inexorably from well-established fact to well-established fact, never being influenced by any unproved conjecture, is quite mistaken. Provided it is made clear which are proved facts and which are conjectures, no harm can result. Conjectures are of great importance since they suggest useful lines of research.
I now proceed to consider opinions opposed to my own.
(1) _The Theological Objection_. Thinking is a function of man's immortal soul. God has given an immortal soul to every man and woman, but not to any other animal or to machines. Hence no animal or machine can think.
I am unable to accept any part of this, but will attempt to reply in theological terms. I should find the argument more convincing if animals were classed with men, for there is a greater difference, to my mind, between the typical animate and the inanimate than there is between man and the other animals. The arbitrary character of the orthodox view becomes clearer if we consider how it might appear to a member of some other religious community. How do Christians regard the Moslem view that women have no souls? But let us leave this point aside and return to the main argument. It appears to me that the argument quoted above implies a serious restriction of the omnipotence of the Almighty. It is admitted that there are certain things that He cannot do such as making one equal to two, but should we not believe that He has freedom to confer a soul on an elephant if He sees fit? We might expect that He would only exercise this power in conjunction with a mutation which provided the elephant with an appropriately improved brain to minister to the needs of this soul. An argument of exactly similar form may be made for the case of machines. It may seem different because it is more difficult to "swallow". But this really only means that we think it would be less likely that He would consider the circumstances suitable for conferring a soul. The circumstances in question are discussed in the rest of this paper. In attempting to construct such machines we should not be irreverently usurping His power of creating souls, any more than we are in the procreation of children: rather we are, in either case, instruments of His will providing mansions for the souls that He creates.
However, this is mere speculation. I am not very impressed with theological arguments whatever they may be used to support. Such arguments have often been found unsatisfactory in the past. In the time of Galileo it was argued that the texts, "And the sun stood still... and hasted not to go down about a whole day" (Joshua x. 13) and "He laid the foundations of the earth, that it should not move at any time" (Psalm cv. 5) were an adequate refutation of the Copernican theory. With our present knowledge such an argument appears futile. When that knowledge was not available it made a quite different impression.
(2) _The 'Heads in the Sand' Objection_. "The consequences of machines thinking would be too dreadful. Let us hope and believe that they cannot do so."
This argument is seldom expressed quite so openly as in the form above. But it affects most of us who think about it at all. We like to believe that Man is in some subtle way superior to the rest of creation. It is best if he can be shown to be _necessarily_ superior, for then there is no danger of him losing his commanding position. The popularity of the theological argument is clearly connected with this feeling. It is likely to be quite strong in intellectual people, since they value the power of thinking more highly than others, and are more inclined to base their belief in the superiority of Man on this power.
I do not think that this argument is sufficiently substantial to require refutation. Consolation would be more appropriate: perhaps this should be sought in the transmigration of souls.
(3) _The Mathematical Objection_. There are a number of results of mathematical logic which can be used to show that there are limitations to the powers of discrete-state machines. The best known of these results is known as _Gödel_ ' _s_ theorem, and shows that in any sufficiently powerful logical system statements can be formulated which can neither be proved nor disproved within the system, unless possibly the system itself is inconsistent. There are other, in some respects similar, results due to _Church_ , _Kleene_ , _Rosser_ , and _Turing_. The latter result is the most convenient to consider, since it refers directly to machines, whereas the others can only be used in a comparatively indirect argument: for instance if Gödel's theorem is to be used we need in addition to have some means of describing logical systems in terms of machines, and machines in terms of logical systems. The result in question refers to a type of machine which is essentially a digital computer with an infinite capacity. It states that there are certain things that such a machine cannot do. If it is rigged up to give answers to questions as in the imitation game, there will be some questions to which it will either give a wrong answer, or fail to give an answer at all however much time is allowed for a reply. There may, of course, be many such questions, and questions which cannot be answered by one machine may be satisfactorily answered by another. We are of course supposing for the present that the questions are of the kind to which an answer 'Yes' or 'No' is appropriate, rather than questions such as 'What do you think of Picasso?' The questions that we know the machines must fail on are of this type, "Consider the machine specified as follows... Will this machine ever answer 'Yes' to any question?" The dots are to be replaced by a description of some machine in a standard form, which could be something like that used in §5. When the machine described bears a certain comparatively simple relation to the machine which is under interrogation, it can be shown that the answer is either wrong or not forthcoming. This is the mathematical result: it is argued that it proves a disability of machines to which the human intellect is not subject.
The short answer to this argument is that although it is established that there are limitations to the powers of any particular machine, it has only been stated, without any sort of proof, that no such limitations apply to the human intellect. But I do not think this view can be dismissed quite so lightly. Whenever one of these machines is asked the appropriate critical question, and gives a definite answer, we know that this answer must be wrong, and this gives us a certain feeling of superiority. Is this feeling illusory? It is no doubt quite genuine, but I do not think too much importance should be attached to it. We too often give wrong answers to questions ourselves to be justified in being very pleased at such evidence of fallibility on the part of the machines. Further, our superiority can only be felt on such an occasion in relation to the one machine over which we have scored our petty triumph. There would be no question of triumphing simultaneously over _all_ machines. In short, then, there might be men cleverer than any given machine, but then again there might be other machines cleverer again, and so on.
Those who hold to the mathematical argument would, I think, mostly be willing to accept the imitation game as a basis for discussion. Those who believe in the two previous objections would probably not be interested in any criteria.
(4) _The Argument from Consciousness_. This argument is very well expressed in _Professor Jefferson's_ Lister Oration for 1949, from which I quote. "Not until a machine can write a sonnet or compose a concerto because of thoughts and emotions felt, and not by the chance fall of symbols, could we agree that machine equals brain—that is, not only write it but know that it had written it. No mechanism could feel (and not merely artificially signal, an easy contrivance) pleasure at its successes, grief when its valves fuse, be warmed by flattery, be made miserable by its mistakes, be charmed by sex, be angry or depressed when it cannot get what it wants."
This argument appears to be a denial of the validity of our test. According to the most extreme form of this view the only way by which one could be sure that a machine thinks is to _be_ the machine and to feel oneself thinking. One could then describe these feelings to the world, but of course no one would be justified in taking any notice. Likewise according to this view the only way to know that a _man_ thinks is to be that particular man. It is in fact the solipsist point of view. It may be the most logical view to hold but it makes communication of ideas difficult. A is liable to believe 'A thinks but B does not' whilst B believes 'B thinks but A does not'. Instead of arguing continually over this point it is usual to have the polite convention that everyone thinks.
I am sure that Professor Jefferson does not wish to adopt the extreme and solipsist point of view. Probably he would be quite willing to accept the imitation game as a test. The game (with the player B omitted) is frequently used in practice under the name of _viva voce_ to discover whether some one really understands something or has 'learnt it parrot fashion'. Let us listen in to a part of such a _viva voce_ :
Interrogator: In the first line of your sonnet which reads 'Shall I compare thee to a summer's day', would not 'a spring day' do as well or better?
Witness: It wouldn't scan.
Interrogator: How about 'a winter's day'. That would scan all right.
Witness: Yes, but nobody wants to be compared to a winter's day.
Interrogator: Would you say Mr. Pickwick reminded you of Christmas?
Witness: In a way.
Interrogator: Yet Christmas is a winter's day, and I do not think Mr. Pickwick would mind the comparison.
Witness: I don't think you're serious. By a winter's day one means a typical winter's day, rather than a special one like Christmas.
And so on. What would Professor Jefferson say if the sonnet-writing machine was able to answer like this in the _viva voce_? I do not know whether he would regard the machine as 'merely artificially signalling' these answers, but if the answers were as satisfactory and sustained as in the above passage I do not think he would describe it as 'an easy contrivance'. This phrase is, I think, intended to cover such devices as the inclusion in the machine of a record of someone reading a sonnet, with appropriate switching to turn it on from time to time.
In short then, I think that most of those who support the argument from consciousness could be persuaded to abandon it rather than be forced into the solipsist position. They will then probably be willing to accept our test.
I do not wish to give the impression that I think there is no mystery about consciousness. There is, for instance, something of a paradox connected with any attempt to localise it. But I do not think these mysteries necessarily need to be solved before we can answer the question with which we are concerned in this paper.
(5) _Arguments from Various Disabilities_. These arguments take the form, "I grant you that you can make machines do all the things you have mentioned but you will never be able to make one to do X". Numerous features X are suggested in this connexion. I offer a selection:
Be kind, resourceful, beautiful, friendly (p. [454]), have initiative, have a sense of humour, tell right from wrong, make mistakes (p. [454]), fall in love, enjoy strawberries and cream (p. [453]), make some one fall in love with it, learn from experience (pp. [460]f.), use words properly, be the subject of its own thought (pp. [454–5]), have as much diversity of behaviour as a man, do something really new (pp. [455–6]). (Some of these disabilities are given special consideration as indicated by the page numbers.)
No support is usually offered for these statements. I believe they are mostly founded on the principle of scientific induction. A man has seen thousands of machines in his lifetime. From what he sees of them he draws a number of general conclusions. They are ugly, each is designed for a very limited purpose, when required for a minutely different purpose they are useless, the variety of behaviour of any one of them is very small, etc., etc. Naturally he concludes that these are necessary properties of machines in general. Many of these limitations are associated with the very small storage capacity of most machines. (I am assuming that the idea of storage capacity is extended in some way to cover machines other than discrete-state machines. The exact definition does not matter as no mathematical accuracy is claimed in the present discussion.) A few years ago, when very little had been heard of digital computers, it was possible to elicit much incredulity concerning them, if one mentioned their properties without describing their construction. That was presumably due to a similar application of the principle of scientific induction. These applications of the principle are of course largely unconscious. When a burnt child fears the fire and shows that he fears it by avoiding it, I should say that he was applying scientific induction. (I could of course also describe his behaviour in many other ways.) The works and customs of mankind do not seem to be very suitable material to which to apply scientific induction. A very large part of space-time must be investigated, if reliable results are to be obtained. Otherwise we may (as most English children do) decide that everybody speaks English, and that it is silly to learn French.
There are, however, special remarks to be made about many of the disabilities that have been mentioned. The inability to enjoy strawberries and cream may have struck the reader as frivolous. Possibly a machine might be made to enjoy this delicious dish, but any attempt to make one do so would be idiotic. What is important about this disability is that it contributes to some of the other disabilities, _e.g_. to the difficulty of the same kind of friendliness occurring between man and machine as between white man and white man, or between black man and black man.
The claim that "machines cannot make mistakes" seems a curious one. One is tempted to retort, "Are they any the worse for that?" But let us adopt a more sympathetic attitude, and try to see what is really meant. I think this criticism can be explained in terms of the imitation game. It is claimed that the interrogator could distinguish the machine from the man simply by setting them a number of problems in arithmetic. The machine would be unmasked because of its deadly accuracy. The reply to this is simple. The machine (programmed for playing the game) would not attempt to give the _right_ answers to the arithmetic problems. It would deliberately introduce mistakes in a manner calculated to confuse the interrogator. A mechanical fault would probably show itself through an unsuitable decision as to what sort of a mistake to make in the arithmetic. Even this interpretation of the criticism is not sufficiently sympathetic. But we cannot afford the space to go into it much further. It seems to me that this criticism depends on a confusion between two kinds of mistake. We may call them 'errors of functioning' and 'errors of conclusion'. Errors of functioning are due to some mechanical or electrical fault which causes the machine to behave otherwise than it was designed to do. In philosophical discussions one likes to ignore the possibility of such errors; one is therefore discussing 'abstract machines'. These abstract machines are mathematical fictions rather than physical objects. By definition they are incapable of errors of functioning. In this sense we can truly say that 'machines can never make mistakes'. Errors of conclusion can only arise when some meaning is attached to the output signals from the machine. The machine might, for instance, type out mathematical equations, or sentences in English. When a false proposition is typed we say that the machine has committed an error of conclusion. There is clearly no reason at all for saying that a machine cannot make this kind of mistake. It might do nothing but type out repeatedly '0 = 1'. To take a less perverse example, it might have some method for drawing conclusions by scientific induction. We must expect such a method to lead occasionally to erroneous results.
The claim that a machine cannot be the subject of its own thought can of course only be answered if it can be shown that the machine has _some_ thought with _some_ subject matter. Nevertheless, 'the subject matter of a machine's operations' does seem to mean something, at least to the people who deal with it. If, for instance, the machine was trying to find a solution of the equation _x_ 2 − 40 _x_ − 11 = 0 one would be tempted to describe this equation as part of the machine's subject matter at that moment. In this sort of sense a machine undoubtedly can be its own subject matter. It may be used to help in making up its own programmes, or to predict the effect of alterations in its own structure. By observing the results of its own behaviour it can modify its own programmes so as to achieve some purpose more effectively. These are possibilities of the near future, rather than Utopian dreams.
The criticism that a machine cannot have much diversity of behaviour is just a way of saying that it cannot have much storage capacity. Until fairly recently a storage capacity of even a thousand digits was very rare.
The criticisms that we are considering here are often disguised forms of the argument from consciousness. Usually if one maintains that a machine _can_ do one of these things, and describes the kind of method that the machine could use, one will not make much of an impression. It is thought that the method (whatever it may be, for it must be mechanical) is really rather base. Compare the parenthesis in Jefferson's statement quoted on p. [451].
(6) _Lady Lovelace's Objection_. Our most detailed information of Babbage's Analytical Engine comes from a memoir by _Lady Lovelace_. In it she states, "The Analytical Engine has no pretensions to _originate_ anything. It can do _whatever we know how to order it_ to perform" (her italics). This statement is quoted by _Hartree_ (p. 70) who adds: "This does not imply that it may not be possible to construct electronic equipment which will 'think for itself', or in which, in biological terms, one could set up a conditioned reflex, which would serve as a basis for 'learning'. Whether this is possible in principle or not is a stimulating and exciting question, suggested by some of these recent developments. But it did not seem that the machines constructed or projected at the time had this property."
I am in thorough agreement with Hartree over this. It will be noticed that he does not assert that the machines in question had not got the property, but rather that the evidence available to Lady Lovelace did not encourage her to believe that they had it. It is quite possible that the machines in question had in a sense got this property. For suppose that some discrete-state machine has the property. The Analytical Engine was a universal digital computer, so that, if its storage capacity and speed were adequate, it could by suitable programming be made to mimic the machine in question. Probably this argument did not occur to the Countess or to Babbage. In any case there was no obligation on them to claim all that could be claimed.
This whole question will be considered again under the heading of learning machines.
A variant of Lady Lovelace's objection states that a machine can 'never do anything really new'. This may be parried for a moment with the saw, 'There is nothing new under the sun'. Who can be certain that 'original work' that he has done was not simply the growth of the seed planted in him by teaching, or the effect of following well-known general principles. A better variant of the objection says that a machine can never 'take us by surprise'. This statement is a more direct challenge and can be met directly. Machines take me by surprise with great frequency. This is largely because I do not do sufficient calculation to decide what to expect them to do, or rather because, although I do a calculation, I do it in a hurried, slipshod fashion, taking risks. Perhaps I say to myself, 'I suppose the voltage here ought to be the same as there: anyway let's assume it is'. Naturally I am often wrong, and the result is a surprise for me for by the time the experiment is done these assumptions have been forgotten. These admissions lay me open to lectures on the subject of my vicious ways, but do not throw any doubt on my credibility when I testify to the surprises I experience.
I do not expect this reply to silence my critic. He will probably say that such surprises are due to some creative mental act on my part, and reflect no credit on the machine. This leads us back to the argument from consciousness, and far from the idea of surprise. It is a line of argument we must consider closed, but it is perhaps worth remarking that the appreciation of something as surprising requires as much of a 'creative mental act' whether the surprising event originates from a man, a book, a machine or anything else.
The view that machines cannot give rise to surprises is due, I believe, to a fallacy to which philosophers and mathematicians are particularly subject. This is the assumption that as soon as a fact is presented to a mind all consequences of that fact spring into the mind simultaneously with it. It is a very useful assumption under many circumstances, but one too easily forgets that it is false. A natural consequence of doing so is that one then assumes that there is no virtue in the mere working out of consequences from data and general principles.
(7) _Argument from Continuity in the Nervous System_. The nervous system is certainly not a discrete-state machine. A small error in the information about the size of a nervous impulse impinging on a neuron, may make a large difference to the size of the outgoing impulse. It may be argued that, this being so, one cannot expect to be able to mimic the behaviour of the nervous system with a discrete-state system.
It is true that a discrete-state machine must be different from a continuous machine. But if we adhere to the conditions of the imitation game, the interrogator will not be able to take any advantage of this difference. The situation can be made clearer if we consider some other simpler continuous machine. A differential analyser will do very well. (A differential analyser is a certain kind of machine not of the discrete-state type used for some kinds of calculation.) Some of these provide their answers in a typed form, and so are suitable for taking part in the game. It would not be possible for a digital computer to predict exactly what answers the differential analyser would give to a problem, but it would be quite capable of giving the right sort of answer. For instance, if asked to give the value of π (actually about 3.1416) it would be reasonable to choose at random between the values 3.12, 3.13, 3.14, 3.15, 3.16 with the probabilities of 0.05, 0.15, 0.55, 0.19, 0.06 (say). Under these circumstances it would be very difficult for the interrogator to distinguish the differential analyser from the digital computer.
(8) _The Argument from Informality of Behaviour_. It is not possible to produce a set of rules purporting to describe what a man should do in every conceivable set of circumstances. One might for instance have a rule that one is to stop when one sees a red traffic light, and to go if one sees a green one, but what if by some fault both appear together? One may perhaps decide that it is safest to stop. But some further difficulty may well arise from this decision later. To attempt to provide rules of conduct to cover every eventuality, even those arising from traffic lights, appears to be impossible. With all this I agree.
From this it is argued that we cannot be machines. I shall try to reproduce the argument, but I fear I shall hardly do it justice. It seems to run something like this. 'If each man had a definite set of rules of conduct by which he regulated his life he would be no better than a machine. But there are no such rules, so men cannot be machines.' The undistributed middle is glaring. I do not think the argument is ever put quite like this, but I believe this is the argument used nevertheless. There may however be a certain confusion between 'rules of conduct' and 'laws of behaviour' to cloud the issue. By 'rules of conduct' I mean precepts such as 'Stop if you see red lights', on which one can act, and of which one can be conscious. By 'laws of behaviour' I mean laws of nature as applied to a man's body such as 'if you pinch him he will squeak'. If we substitute 'laws of behaviour which regulate his life' for 'laws of conduct by which he regulates his life' in the argument quoted the undistributed middle is no longer insuperable. For we believe that it is not only true that being regulated by laws of behaviour implies being some sort of machine (though not necessarily a discrete-state machine), but that conversely being such a machine implies being regulated by such laws. However, we cannot so easily convince ourselves of the absence of complete laws of behaviour as of complete rules of conduct. The only way we know of for finding such laws is scientific observation, and we certainly know of no circumstances under which we could say, 'We have searched enough. There are no such laws.'
We can demonstrate more forcibly that any such statement would be unjustified. For suppose we could be sure of finding such laws if they existed. Then given a discrete-state machine it should certainly be possible to discover by observation sufficient about it to predict its future behaviour, and this within a reasonable time, say a thousand years. But this does not seem to be the case. I have set up on the Manchester computer a small programme using only 1000 units of storage, whereby the machine supplied with one sixteen figure number replies with another within two seconds. I would defy anyone to learn from these replies sufficient about the programme to be able to predict any replies to untried values.
(9) _The Argument from Extra-Sensory Perception_. I assume that the reader is familiar with the idea of extra-sensory perception, and the meaning of the four items of it, _viz_. telepathy, clairvoyance, precognition and psycho-kinesis. These disturbing phenomena seem to deny all our usual scientific ideas. How we should like to discredit them! Unfortunately the statistical evidence, at least for telepathy, is overwhelming. It is very difficult to rearrange one's ideas so as to fit these new facts in. Once one has accepted them it does not seem a very big step to believe in ghosts and bogies. The idea that our bodies move simply according to the known laws of physics, together with some others not yet discovered but somewhat similar, would be one of the first to go.
This argument is to my mind quite a strong one. One can say in reply that many scientific theories seem to remain workable in practice, in spite of clashing with E.S.P.; that in fact one can get along very nicely if one forgets about it. This is rather cold comfort, and one fears that thinking is just the kind of phenomenon where E.S.P. may be especially relevant.
A more specific argument based on E.S.P. might run as follows: "Let us play the imitation game, using as witnesses a man who is good as a telepathic receiver, and a digital computer. The interrogator can ask such questions as 'What suit does the card in my right hand belong to?' The man by telepathy or clairvoyance gives the right answer 130 times out of 400 cards. The machine can only guess at random, and perhaps gets 104 right, so the interrogator makes the right identification." There is an interesting possibility which opens here. Suppose the digital computer contains a random number generator. Then it will be natural to use this to decide what answer to give. But then the random number generator will be subject to the psycho-kinetic powers of the interrogator. Perhaps this psycho-kinesis might cause the machine to guess right more often than would be expected on a probability calculation, so that the interrogator might still be unable to make the right identification. On the other hand, he might be able to guess right without any questioning, by clairvoyance. With E.S.P. anything may happen.
If telepathy is admitted it will be necessary to tighten our test up. The situation could be regarded as analogous to that which would occur if the interrogator were talking to himself and one of the competitors was listening with his ear to the wall. To put the competitors into a 'telepathy-proof room' would satisfy all requirements.
#### 7. Learning Machines
The reader will have anticipated that I have no very convincing arguments of a positive nature to support my views. If I had I should not have taken such pains to point out the fallacies in contrary views. Such evidence as I have I shall now give.
Let us return for a moment to Lady Lovelace's objection, which stated that the machine can only do what we tell it to do. One could say that a man can 'inject' an idea into the machine, and that it will respond to a certain extent and then drop into quiescence, like a piano string struck by a hammer. Another simile would be an atomic pile of less than critical size: an injected idea is to correspond to a neutron entering the pile from without. Each such neutron will cause a certain disturbance which eventually dies away. If, however, the size of the pile is sufficiently increased, the disturbance caused by such an incoming neutron will very likely go on and on increasing until the whole pile is destroyed. Is there a corresponding phenomenon for minds, and is there one for machines? There does seem to be one for the human mind. The majority of them seem to be 'sub-critical', _i.e_. to correspond in this analogy to piles of sub-critical size. An idea presented to such a mind will on average give rise to less than one idea in reply. A smallish proportion are super-critical. An idea presented to such a mind may give rise to a whole 'theory' consisting of secondary, tertiary and more remote ideas. Animals minds seem to be very definitely sub-critical. Adhering to this analogy we ask, 'Can a machine be made to be super-critical?'
The 'skin of an onion' analogy is also helpful. In considering the functions of the mind or the brain we find certain operations which we can explain in purely mechanical terms. This we say does not correspond to the real mind: it is a sort of skin which we must strip off if we are to find the real mind. But then in what remains we find a further skin to be stripped off, and so on. Proceeding in this way do we ever come to the 'real' mind, or do we eventually come to the skin which has nothing in it? In the latter case the whole mind is mechanical. (It would not be a discrete-state machine however. We have discussed this.)
These last two paragraphs do not claim to be convincing arguments. They should rather be described as 'recitations tending to produce belief'.
The only really satisfactory support that can be given for the view expressed at the beginning of §6, will be that provided by waiting for the end of the century and then doing the experiment described. But what can we say in the meantime? What steps should be taken now if the experiment is to be successful?
As I have explained, the problem is mainly one of programming. Advances in engineering will have to be made too, but it seems unlikely that these will not be adequate for the requirements. Estimates of the storage capacity of the brain vary from 1010 to 1015 binary digits. I incline to the lower values and believe that only a very small fraction is used for the higher types of thinking. Most of it is probably used for the retention of visual impressions. I should be surprised if more than 109 was required for satisfactory playing of the imitation game, at any rate against a blind man. (Note—The capacity of the _Encyclopaedia Britannica_ , 11th edition, is 2 × 109.) A storage capacity of 107 would be a very practicable possibility even by present techniques. It is probably not necessary to increase the speed of operations of the machines at all. Parts of modern machines which can be regarded as analogues of nerve cells work about a thousand times faster than the latter. This should provide a 'margin of safety' which could cover losses of speed arising in many ways. Our problem then is to find out how to programme these machines to play the game. At my present rate of working I produce about a thousand digits of programme a day, so that about sixty workers, working steadily through the fifty years might accomplish the job, if nothing went into the waste-paper basket. Some more expeditious method seems desirable.
In the process of trying to imitate an adult human mind we are bound to think a good deal about the process which has brought it to the state that it is in. We may notice three components,
( _a_ ) The initial state of the mind, say at birth,
( _b_ ) The education to which it has been subjected,
( _c_ ) Other experience, not to be described as education, to which it has been subjected.
Instead of trying to produce a programme to simulate the adult mind, why not rather try to produce one which simulates the child's? If this were then subjected to an appropriate course of education one would obtain the adult brain. Presumably the child-brain is something like a note-book as one buys it from the stationers. Rather little mechanism, and lots of blank sheets. (Mechanism and writing are from our point of view almost synonymous.) Our hope is that there is so little mechanism in the child-brain that something like it can be easily programmed. The amount of work in the education we can assume, as a first approximation, to be much the same as for the human child.
We have thus divided our problem into two parts. The child-programme and the education process. These two remain very closely connected. We cannot expect to find a good child-machine at the first attempt. One must experiment with teaching one such machine and see how well it learns. One can then try another and see if it is better or worse. There is an obvious connection between this process and evolution, by the identifications
One may hope, however, that this process will be more expeditious than evolution. The survival of the fittest is a slow method for measuring advantages. The experimenter, by the exercise of intelligence, should be able to speed it up. Equally important is the fact that he is not restricted to random mutations. If he can trace a cause for some weakness he can probably think of the kind of mutation which will improve it.
It will not be possible to apply exactly the same teaching process to the machine as to a normal child. It will not, for instance, be provided with legs, so that it could not be asked to go out and fill the coal scuttle. Possibly it might not have eyes. But however well these deficiencies might be overcome by clever engineering, one could not send the creature to school without the other children making excessive fun of it. It must be given some tuition. We need not be too concerned about the legs, eyes, etc. The example of Miss Helen Keller shows that education can take place provided that communication in both directions between teacher and pupil can take place by some means or other.
We normally associate punishments and rewards with the teaching process. Some simple child-machines can be constructed or programmed on this sort of principle. The machine has to be so constructed that events which shortly preceded the occurrence of a punishment-signal are unlikely to be repeated, whereas a reward-signal increased the probability of repetition of the events which led up to it. These definitions do not presuppose any feelings on the part of the machine. I have done some experiments with one such child-machine, and succeeded in teaching it a few things, but the teaching method was too unorthodox for the experiment to be considered really successful.
The use of punishments and rewards can at best be a part of the teaching process. Roughly speaking, if the teacher has no other means of communicating to the pupil, the amount of information which can reach him does not exceed the total number of rewards and punishments applied. By the time a child has learnt to repeat 'Casabianca' he would probably feel very sore indeed, if the text could only be discovered by a 'Twenty Questions' technique, every 'no' taking the form of a blow. It is necessary therefore to have some other 'unemotional' channels of communication. If these are available it is possible to teach a machine by punishments and rewards to obey orders given in some language, _e.g_. a symbolic language. These orders are to be transmitted through the 'unemotional' channels. The use of this language will diminish greatly the number of punishments and rewards required.
Opinions may vary as to the complexity which is suitable in the child machine. One might try to make it as simple as possible consistently with the general principles. Alternatively one might have a complete system of logical inference 'built in'. In the latter case the store would be largely occupied with definitions and propositions. The propositions would have various kinds of status, _e.g_. well-established facts, conjectures, mathematically proved theorems, statements given by an authority, expressions having the logical form of proposition but not belief-value. Certain propositions may be described as 'imperatives.' The machine should be so constructed that as soon as an imperative is classed as 'well-established' the appropriate action automatically takes place. To illustrate this, suppose the teacher says to the machine, 'Do your homework now'. This may cause "Teacher says 'Do your homework now'" to be included amongst the well-established facts. Another such fact might be, "Everything that teacher says is true". Combining these may eventually lead to the imperative, 'Do your homework now', being included amongst the well-established facts, and this, by the construction of the machine, will mean that the homework actually gets started, but the effect is very satisfactory. The processes of inference used by the machine need not be such as would satisfy the most exacting logicians. There might for instance be no hierarchy of types. But this need not mean that type fallacies will occur, any more than we are bound to fall over unfenced cliffs. Suitable imperatives (expressed _within_ the systems, not forming part of the rules _of_ the system) such as 'Do not use a class unless it is a subclass of one which has been mentioned by teacher' can have a similar effect to 'Do not go too near the edge'.
The imperatives that can be obeyed by a machine that has no limbs are bound to be of a rather intellectual character, as in the example (doing homework) given above. Important amongst such imperatives will be ones which regulate the order in which the rules of the logical system concerned are to be applied. For at each stage when one is using a logical system, there is a very large number of alternative steps, any of which one is permitted to apply, so far as obedience to the rules of the logical system is concerned. These choices make the difference between a brilliant and a footling reasoner, not the difference between a sound and a fallacious one. Propositions leading to imperatives of this kind might be "When Socrates is mentioned, use the syllogism in Barbara" or "If one method has been proved to be quicker than another, do not use the slower method". Some of these may be 'given by authority', but others may be produced by the machine itself, _e.g_. by scientific induction.
The idea of a learning machine may appear paradoxical to some readers. How can the rules of operation of the machine change? They should describe completely how the machine will react whatever its history might be, whatever changes it might undergo. The rules are thus quite time-invariant. This is quite true. The explanation of the paradox is that the rules which get changed in the learning process are of a rather less pretentious kind, claiming only an ephemeral validity. The reader may draw a parallel with the Constitution of the United States.
An important feature of a learning machine is that its teacher will often be very largely ignorant of quite what is going on inside, although he may still be able to some extent to predict his pupil's behaviour. This should apply most strongly to the later education of a machine arising from a child-machine of well-tried design (or programme). This is in clear contrast with normal procedure when using a machine to do computations: one's object is then to have a clear mental picture of the state of the machine at each moment in the computation. This object can only be achieved with a struggle. The view that 'the machine can only do what we know how to order it to do' appears strange in face of this. Most of the programmes which we can put into the machine will result in its doing something that we cannot make sense of at all, or which we regard as completely random behaviour. Intelligent behaviour presumably consists in a departure from the completely disciplined behaviour involved in computation, but a rather slight one, which does not give rise to random behaviour, or to pointless repetitive loops. Another important result of preparing our machine for its part in the imitation game by a process of teaching and learning is that 'human fallibility' is likely to be omitted in a rather natural way, _i.e_. without special 'coaching'. (The reader should reconcile this with the point of view on p. [454].) Processes that are learnt do not produce a hundred per cent. certainty of result; if they did they could not be unlearnt.
It is probably wise to include a random element in a learning machine (see p. [445]). A random element is rather useful when we are searching for a solution of some problem. Suppose for instance we wanted to find a number between 50 and 200 which was equal to the square of the sum of its digits, we might start at 51 then try 52 and go on until we got a number that worked. Alternatively we might choose numbers at random until we got a good one. This method has the advantage that it is unnecessary to keep track of the values that have been tried, but the disadvantage that one may try the same one twice, but this is not very important if there are several solutions. The systematic method has the disadvantage that there may be an enormous block without any solutions in the region which has to be investigated first. Now the learning process may be regarded as a search for a form of behaviour which will satisfy the teacher (or some other criterion). Since there is probably a very large number of satisfactory solutions the random method seems to be better than the systematic. It should be noticed that it is used in the analogous process of evolution. But there the systematic method is not possible. How could one keep track of the different genetical combinations that had been tried, so as to avoid trying them again?
We may hope that machines will eventually compete with men in all purely intellectual fields. But which are the best ones to start with? Even this is a difficult decision. Many people think that a very abstract activity, like the playing of chess, would be best. It can also be maintained that it is best to provide the machine with the best sense organs that money can buy, and then teach it to understand and speak English. This process could follow the normal teaching of a child. Things would be pointed out and named, etc. Again I do not know what the right answer is, but I think both approaches should be tried.
We can only see a short distance ahead, but we can see plenty there that needs to be done.
##### _Bibliography_
Samuel Butler, Erewhon, London, 1865. Chapters 23, 24, 25, _The Book of the Machines_.
Alonzo Church, "An Unsolvable Problem of Elementary Number Theory", _American Journal of Mathematics_ , 58 (1936), 345–363.
K. Gödel, "Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I", _Monatshefte für Mathematik und Physik_ (1931), 173–189.
D. R. Hartree, _Calculating Instruments and Machines_ , New York, 1949.
S. C. Kleene, "General Recursive Functions of Natural Numbers", _American Journal of Mathematics_ , 57 (1935), 153–173 and 219–244.
G. Jefferson, "The Mind of Mechanical Man". Lister Oration for 1949. _British Medical Journal_ , vol. i (1949), 1105–1121.
Countess of Lovelace, 'Translator's notes to an article on Babbage's Analytical Engine', _Scientific Memoirs_ (ed. by R. Taylor), vol. 3 (1842), 691–731.
Bertrand Russell, _History of Western Philosophy_ , London, 1940.
A. M. Turing, "On Computable Numbers, with an Application to the Entscheidungsproblem" [Chapter 1].
### **CHAPTER 12
Intelligent Machinery, A Heretical Theory ( _c. 1951_ )**
_Alan Turing_
#### Introduction
##### _Jack Copeland_
#### The '51 Society
Turing gave the presentation 'Intelligent Machinery, A Heretical Theory' on a radio discussion programme called _The '51 Society_. Named after the year in which the programme first went to air, _The '51 Society_ was produced by the BBC Home Service at their Manchester studio and ran for several years. A presentation by the week's guest would be followed by a panel discussion. Regulars on the panel included Max Newman, Professor of Mathematics at Manchester, the philosopher Michael Polanyi, then Professor of Social Studies at Manchester, and the mathematician Peter Hilton, a younger member of Newman's department at Manchester who had worked with Turing and Newman at Bletchley Park.
#### Machine Learning
Turing's target in 'Intelligent Machinery, A Heretical Theory' is the claim that 'You cannot make a machine to think for you' (p. 472). A common theme in his writing is that if a machine is to be intelligent, then it will need to 'learn by experience' (probably with some pre-selection, by an external educator, of the experiences to which the machine will be subjected). The present article continues the discussion of machine learning begun in Chapters 10 and . Turing remarks that the 'human analogy alone' suggests that a process of education 'would in practice be an essential to the production of a reasonably intelligent machine within a reasonably short space of time' (p. 473). He emphasizes the point, also made in Chapter 11, that one might 'start from a comparatively simple machine, and, by subjecting it to a suitable range of "experience" transform it into one which was more elaborate, and was able to deal with a far greater range of contingencies' (p. 473).
Turing goes on to give some indication of how learning might be accomplished, introducing the idea of a machine's building up what he calls 'indexes of experiences' (p. 474). (This idea is not mentioned elsewhere in his writings.) An example of an index of experiences is a list (ordered in some way) of situations in which the machine has found itself, coupled with the action that was taken, and the outcome, good or bad. The situations are described in terms of features. Faced with a choice as to what to do next, the machine looks up features of its present situation in whatever indexes it has. If this procedure affords more than one candidate action, the machine selects between them by means of some rule, possibly itself learned through experience. Turing very reasonably grounds his belief that comparatively crude selection-rules will lead to satisfactory behaviour in the fact that engineering problems are regularly solved by 'the crudest rule of thumb procedure... e.g. whether a function increases or decreases with one of its variables' (p. 474).
In response to the problem of how the educator is to indicate to the machine whether a situation or outcome is a 'favourable' one or not, Turing returns to the possibility of incorporating two 'keys' in the machine, which can be manipulated by the educator, and which represent 'pleasure' and 'pain' (p. 474). This is an idea that Turing discusses more fully in Chapter 10, where he considers adding two input lines to a (modified) Turing machine, the pleasure (or reward) line and the pain (or punishment) line. He calls the result a 'P-type machine' ('P' standing for 'pleasure–pain').
#### Random Elements
Turing ends his discussion of machine learning with the suggestion that a 'random element' be incorporated in the machine (p. 475). This would, as he says, result in the behaviour of the machine being by no means completely determined by the experiences to which it was subjected (p. 475). The idea that a random element be included in a learning machine appears elsewhere in Turing's discussions of machine intelligence. In Chapter 11 he says: 'A random element is rather useful when... searching for a solution of some problem' (p. 463). He gives this example:
Suppose for instance we wanted to find a number between 50 and 200 which was equal to the square of the sum of its digits, we might start at 51 then try 52 and so on until we got a number that worked. Alternatively we might choose numbers at random until we got a good one.
Turing continues (p. 463):
The systematic method has the disadvantage that there may be an enormous block without any solutions in the region which has to be investigated first. Now the learning process may be regarded as a search for a form of behaviour which will satisfy the teacher (or some other criterion). Since there is probably a very large number of satisfactory solutions the random method seems to be better than the systematic. It should be noticed that it is used in the analogous process of evolution.
Turing's discussion of 'pleasure–pain systems' in Chapter 10 also mentions randomness (p. 425):
I will use this term ['pleasure–pain' system] to mean an unorganised machine of the following general character: The configurations of the machine are described by two expressions, which we may call the character-expression and the situation-expression. The character and situation at any moment, together with the input signals, determine the character and situation at the next moment. The character may be subject to some random variation. Pleasure interference has a tendency to fix the character i.e. towards preventing it changing, whereas pain stimuli tend to disrupt the character, causing features which had become fixed to change, or to become again subject to random variation.
#### The Mathematical Objection
In what are some of the most interesting remarks in 'Intelligent Machinery, A Heretical Theory', Turing sketches and rebuts an argument against the possibility of computing machines emulating the full intelligence of human beings. The objection is stated as follows in Chapter 10 (pp. 410–11):
Recently the theorem of Gödel and related results... have shown that if one tries to use machines for such purposes as determining the truth or falsity of mathematical theorems and one is not willing to tolerate an occasional wrong result, then any given machine will in some cases be unable to give an answer at all. On the other hand the human intelligence seems to be able to find methods of ever-increasing power for dealing with such problems 'transcending' the methods available to machines.
In Chapter 11 he terms this the 'Mathematical Objection' (p. 450).
As Turing notes, the 'related results' include what he himself proved in 'On Computable Numbers'. The import of the satisfactoriness problem (explained in 'Computable Numbers: A Guide') is that no Turing machine can correctly determine the truth or falsity of each statement of the form 'such-and-such Turing machine is circle-free'. Whichever Turing machine one chooses to ask, there will be statements of this form for which the chosen machine either gives no answer or gives the wrong answer (compare Chapter 11, pp. 450–1). (In Chapter 3, Turing extends this result to his oracle machines: no oracle machine can correctly determine the truth or falsity of each statement of the form 'such-and-such oracle machine is circle-free' (pp. 156–7).)
Post formulated a version of the Mathematical Objection as early as 1921. However, the objection has become known over the years as the 'Gödel argument'. In 1961, in a famous article, the philosopher John Lucas claimed the Gödel argument establishes that 'mechanism'—which Lucas characterizes as the view that 'minds [can] be explained as machines'—is false. More recently, the mathematical physicist Roger Penrose has endorsed a version of the Gödel argument.
Lucas was happy to assert, on the basis of the Mathematical Objection, that 'no scientific enquiry can ever exhaust the... human mind'. Not many who admire the explanatory power of science would be happy to endorse this conclusion. Penrose himself appears to hold that the mind can be explained in ultimately physical terms. However, it is difficult to say what scientific conception of the mind could be available to someone who endorses the Mathematical Objection. This is because the objection, if sound, could be used equally well to support the conclusion, not only that the mind is not a Turing machine, but also that it is not any one of a very broad range of machines (which includes the oracle machines). Given the enormous diversity of types of machine in this range, it is an open question whether there is any scientific conception of the mind that the Mathematical Objection (if sound) would not rule out.
Penrose acknowledges that the objection applies not only to the view that the mind is equivalent to a Turing machine but 'much more generally', saying: 'No doubt there are readers who believe that the last vestige of credibility of my [version of the Gödel] argument has disappeared at this stage! I certainly should not blame any reader for feeling this way.'
So far, however, Penrose has not made it clear what scientific conception of the mind can remain for one who endorses the argument, remarking only that, since the argument 'can be applied in very general circumstances indeed', the mind is 'something very mysterious'.
#### Turing's Answer to the Mathematical Objection
In Chapter 11 Turing says (p. 451): 'The short answer to this argument is that although it is established that there are limitations to the powers of any particular machine, it has only been stated, without any sort of proof, that no such limitations apply to the human intellect.' This remark might appear to cut to the heart of the matter. However, Turing expresses dissatisfaction with it, saying that the Mathematical Objection cannot 'be dismissed so lightly'. He goes on to broach a further line of attack on the argument, pointing out that humans 'often give wrong answers to questions', and it is this line of attack that he pursues in 'Intelligent Machinery, A Heretical Theory'.
In the quotation from Chapter 10 given above, Turing notes that the Mathematical Objection rests on a proviso that the machine is not allowed to make mistakes, and as he goes on to point out, 'the condition that the machine must not make mistakes... is not a requirement for intelligence' (p. 411). In 'Intelligent Machinery, A Heretical Theory' he suggests that the 'danger of the mathematician making mistakes is an unavoidable corollary of his power of sometimes hitting upon an entirely new method' (p. 472). Turing envisages machines also able to hit upon new methods: 'My contention is that machines can be constructed which will simulate the behaviour of the human mind very closely. They will make mistakes at times, and at times they may make new and very interesting statements.'
Turing makes a similar point in Chapter 9 (pp. 393–4):
[I]f a mathematician is confronted with such a problem [e.g. determining the truth or falsity of statements of the form ' _p_ is provable in such-and-such system'—Ed.] he would search around and find new methods of proof, so that he ought eventually to be able to reach a decision about any given formula.... I would say that fair play must be given to the machine. Instead of it sometimes giving no answer we could arrange that it gives occasional wrong answers. But the human mathematician would likewise make blunders when trying out new techniques. It is easy for us to regard these blunders as not counting and give him another chance, but the machine would probably be allowed no mercy.
The use of heuristic search carries with it the risk of the computer producing a proportion of incorrect answers (see 'Artificial Intelligence'). This fact would have been very familiar to Turing from his experience with the bombe. Probably Turing was thinking of heuristic search when he wrote this, the earliest surviving statement of his views concerning machine intelligence, in 'Proposed Electronic Calculator': 'There are indications however that it is possible to make the machine display intelligence at the risk of its making occasional serious mistakes. By following up this aspect the machine could probably be made to play very good chess.'
In 'Intelligent Machinery, A Heretical Theory' Turing passes immediately from his remarks on the Mathematical Objection to a discussion of machine learning. This juxtaposition perhaps indicates that Turing's view was this: it is the possibility of a machine's learning _new_ methods and techniques that ultimately defeats the Mathematical Objection. In the simplest possible case, the machine's tutor— a human mathematician—can just present the machine with a better method whenever the machine produces an incorrect answer to a problem. This new input in effect alters the machine's standard description, transforming it into a different Turing machine (see 'Computable Numbers: A Guide'). Alternatively a machine may itself be able to search around (albeit fallibly) for better methods. The search might involve the use of a random element. As in the preceding case, the standard description of the machine alters in consequence of the learning process, as the machine overwrites its previous algorithm with a successor. (As Turing says in Chapter 9: 'What we want is a machine that can learn from experience. The possibility of letting the machine alter its own instructions provides the mechanism for this.') Thus the learning machine may traverse the space of what in one of his letters to Newman (Chapter 4, p. 215) Turing calls 'proof finding' machines. In the same letter Turing says:
One imagines different machines allowing different sets of proofs, and by choosing a suitable machine one can approximate 'truth' by 'provability' better than with a less suitable machine, and can in a sense approximate it as well as you please.
The learning machine successively mutates from one proof-finding Turing machine into another, becoming capable of wider sets of proofs as new, more powerful methods of proof are acquired.
#### The Future
Turing ends 'Intelligent Machinery, A Heretical Theory' with a vision of the future, now hackneyed, in which intelligent computers 'outstrip our feeble powers' and 'take control'. There is more of the same in Chapter 13. No doubt this is comic-strip stuff. Nevertheless, these images of Turing's reveal his profound grasp of the potential of the universal Turing machine at a time when the only computers in existence were minuscule, and none but the most straightforward of tasks had been successfully programmed.
##### _Further reading_
Benacerraf, P., 'God, the Devil, and Gödel', _Monist_ , 51 (1967), 9–32.
Copeland, B. J., 'Turing's O-machines, Penrose, Searle, and the Brain', _Analysis_ , 58 (1998), 128–38.
Gandy, R., 'Human versus Mechanical Intelligence', in P. Millican and A. Clark (eds.), _Machines and Thought: The Legacy of Alan Turing_ (Oxford: Clarendon Press, 1996).
Lucas, J. R., 'Minds, Machines and Gödel', _Philosophy_ , 36 (1961), 112–27.
—— 'Minds, Machines and Gödel: A Retrospect', in P. Millican and A. Clark (eds.), _Machines and Thought: The Legacy of Alan Turing_ (Oxford: Clarendon Press, 1996).
Penrose, R., _Shadows of the Mind: A Search for the Missing Science of Consciousness_ (Oxford: Oxford University Press, 1994).
Piccinini, G., 'Alan Turing and the Mathematical Objection', _Minds and Machines_ , 13 (2003), 23–48.
##### _Provenance_
The text that follows is from a typescript entitled 'Intelligent Machinery, A Heretical Theory' and marked 'Typist's Typescript'.
#### Intelligent Machinery, A Heretical Theory
'You cannot make a machine to think for you.' This is a commonplace that is usually accepted without question. It will be the purpose of this paper to question it.
Most machinery developed for commercial purposes is intended to carry out some very specific job, and to carry it out with certainty and considerable speed. Very often it does the same series of operations over and over again without any variety. This fact about the actual machinery available is a powerful argument to many in favour of the slogan quoted above. To a mathematical logician this argument is not available, for it has been shown that there are machines theoretically possible which will do something very close to thinking. They will, for instance, test the validity of a formal proof in the system of Principia Mathematica, or even tell of a formula of that system whether it is provable or disprovable. In the case that the formula is neither provable nor disprovable such a machine certainly does not behave in a very satisfactory manner, for it continues to work indefinitely without producing any result at all, but this cannot be regarded as very different from the reaction of the mathematicians, who have for instance worked for hundreds of years on the question as to whether Fermat's last theorem is true or not. For the case of machines of this kind a more subtle argument is necessary. By Gödel's famous theorem, or some similar argument, one can show that however the machine is constructed there are bound to be cases where the machine fails to give an answer, but a mathematician would be able to. On the other hand, the machine has certain advantages over the mathematician. Whatever it does can be relied upon, assuming no mechanical 'breakdown', whereas the mathematician makes a certain proportion of mistakes. I believe that this danger of the mathematician making mistakes is an unavoidable corollary of his power of sometimes hitting upon an entirely new method. This seems to be confirmed by the well known fact that the most reliable people will not usually hit upon really new methods.
My contention is that machines can be constructed which will simulate the behaviour of the human mind very closely. They will make mistakes at times, and at times they may make new and very interesting statements, and on the whole the output of them will be worth attention to the same sort of extent as the output of a human mind. The content of this statement lies in the greater frequency expected for the true statements, and it cannot, I think, be given an exact statement. It would not, for instance, be sufficient to say simply that the machine will make any true statement sooner or later, for an example of such a machine would be one which makes all possible statements sooner or later. We know how to construct these, and as they would (probably) produce true and false statements about equally frequently, their verdicts would be quite worthless. It would be the actual reaction of the machine to circumstances that would prove my contention, if indeed it can be proved at all.
Let us go rather more carefully into the nature of this 'proof'. It is clearly possible to produce a machine which would give a very good account of itself for any range of tests, if the machine were made sufficiently elaborate. However, this again would hardly be considered an adequate proof. Such a machine would give itself away by making the same sort of mistake over and over again, and being quite unable to correct itself, or to be corrected by argument from outside. If the machine were able in some way to 'learn by experience' it would be much more impressive. If this were the case there seems to be no real reason why one should not start from a comparatively simple machine, and, by subjecting it to a suitable range of 'experience' transform it into one which was more elaborate, and was able to deal with a far greater range of contingencies. This process could probably be hastened by a suitable selection of the experiences to which it was subjected. This might be called 'education'. But here we have to be careful. It would be quite easy to arrange the experiences in such a way that they automatically caused the structure of the machine to build up into a previously intended form, and this would obviously be a gross form of cheating, almost on a par with having a man inside the machine. Here again the criterion as to what would be considered reasonable in the way of 'education' cannot be put into mathematical terms, but I suggest that the following would be adequate in practice. Let us suppose that it is intended that the machine shall understand English, and that owing to its having no hands or feet, and not needing to eat, nor desiring to smoke, it will occupy its time mostly in playing games such as Chess and GO, and possibly Bridge. The machine is provided with a typewriter keyboard on which any remarks to it are typed, and it also types out any remarks that it wishes to make. I suggest that the education of the machine should be entrusted to some highly competent schoolmaster who is interested in the project but who is forbidden any detailed knowledge of the inner workings of the machine. The mechanic who has constructed the machine, however, is permitted to keep the machine in running order, and if he suspects that the machine has been operating incorrectly may put it back to one of its previous positions and ask the schoolmaster to repeat his lessons from that point on, but he may not take any part in the teaching. Since this procedure would only serve to test the bona fides of the mechanic, I need hardly say that it would not be adopted in the experimental stages. As I see it, this education process would in practice be an essential to the production of a reasonably intelligent machine within a reasonably short space of time. The human analogy alone suggests this.
I may now give some indication of the way in which such a machine might be expected to function. The machine would incorporate a memory. This does not need very much explanation. It would simply be a list of all the statements that had been made to it or by it, and all the moves it had made and the cards it had played in its games. This would be listed in chronological order. Besides this straightforward memory there would be a number of 'indexes of experiences'. To explain this idea I will suggest the form which one such index might possibly take. It might be an alphabetical index of the words that had been used giving the 'times' at which they had been used, so that they could be looked up in the memory. Another such index might contain patterns of men on parts of a GO board that had occurred. At comparatively late stages of education the memory might be extended to include important parts of the configuration of the machine at each moment, or in other words it would begin to remember what its thoughts had been. This would give rise to fruitful new forms of indexing. New forms of index might be introduced on account of special features observed in the indexes already used. The indexes would be used in this sort of way. Whenever a choice has to be made as to what to do next, features of the present situation are looked up in the indexes available, and the previous choice in the similar situations, and the outcome, good or bad, is discovered. The new choice is made accordingly. This raises a number of problems. If some of the indications are favourable and some are unfavourable what is one to do? The answer to this will probably differ from machine to machine and will also vary with its degree of education. At first probably some quite crude rule will suffice, e.g. to do whichever has the greatest number of votes in its favour. At a very late stage of education the whole question of procedure in such cases will probably have been investigated by the machine itself, by means of some kind of index, and this may result in some highly sophisticated, and, one hopes, highly satisfactory, form of rule. It seems probable however that the comparatively crude forms of rule will themselves be reasonably satisfactory, so that progress can on the whole be made in spite of the crudeness of the choice [of] rules. This seems to be verified by the fact that engineering problems are sometimes solved by the crudest rule of thumb procedure which only deals with the most superficial aspects of the problem, e.g. whether a function increases or decreases with one of its variables. Another problem raised by this picture of the way behaviour is determined is the idea of 'favourable outcome'. Without some such idea, corresponding to the 'pleasure principle' of the psychologists, it is very difficult to see how to proceed. Certainly it would be most natural to introduce some such thing into the machine. I suggest that there should be two keys which can be manipulated by the schoolmaster, and which represent the ideas of pleasure and pain. At later stages in education the machine would recognise certain other conditions as desirable owing to their having been constantly associated in the past with pleasure, and likewise certain others as undesirable. Certain expressions of anger on the part of the schoolmaster might, for instance, be recognised as so ominous that they could never be overlooked, so that the schoolmaster would find that it became unnecessary to 'apply the cane' any more.
To make further suggestions along these lines would perhaps be unfruitful at this stage, as they are likely to consist of nothing more than an analysis of actual methods of education applied to human children. There is, however, one feature that I would like to suggest should be incorporated in the machines, and that is a 'random element'. Each machine should be supplied with a tape bearing a random series of figures, e.g. 0 and 1 in equal quantities, and this series of figures should be used in the choices made by the machine. This would result in the behaviour of the machine not being by any means completely determined by the experiences to which it was subjected, and would have some valuable uses when one was experimenting with it. By faking the choices made one would be able to control the development of the machine to some extent. One might, for instance, insist on the choice made being a particular one at, say, 10 particular places, and this would mean that about one machine in 1024 or more would develop to as high a degree as the one which had been faked. This cannot very well be given an accurate statement because of the subjective nature of the idea of 'degree of development' to say nothing of the fact that the machine that had been faked might have been also fortunate in its unfaked choices.
Let us now assume, for the sake of argument, that these machines are a genuine possibility, and look at the consequences of constructing them. To do so would of course meet with great opposition, unless we have advanced greatly in religious toleration from the days of Galileo. There would be great opposition from the intellectuals who were afraid of being put out of a job. It is probable though that the intellectuals would be mistaken about this. There would be plenty to do, [trying to understand what the machines were trying to say,] i.e. in trying to keep one's intelligence up to the standard set by the machines, for it seems probable that once the machine thinking method had started, it would not take long to outstrip our feeble powers. There would be no question of the machines dying, and they would be able to converse with each other to sharpen their wits. At some stage therefore we should have to expect the machines to take control, in the way that is mentioned in Samuel Butler's 'Erewhon'.
### **CHAPTER 13
Can Digital Computers Think? ( _1951_ )**
_Alan Turing_
#### Introduction
##### _Jack Copeland_
The lecture 'Can Digital Computers Think?' was broadcast on BBC Radio on 15 May 1951, and was repeated on 3 July of that year. (Sara Turing relates that Turing did not listen to the first broadcast but did 'pluck up courage' to listen to the repeat.) Turing's was the second lecture in a series with the general title 'Automatic Calculating Machines'. Other speakers in the series included Newman, D. R. Hartree, M. V. Wilkes, and F. C. Williams.
#### Imitating the Brain
Turing's principal aim in this lecture is to defend his view that 'it is not altogether unreasonable to describe digital computers as brains', and he argues for the proposition that 'If any machine can appropriately be described as a brain, then any digital computer can be so described'.
The lecture casts light upon Turing's attitude towards talk of machines thinking. In Chapter 11 he says that in his view the question 'Can machines think?' is 'too meaningless to deserve discussion' (p. 449). However, in the present chapter he makes liberal use of such phrases as 'programming] a machine... to think' and 'the attempt to make a thinking machine'. In one passage, Turing says (p. 485): 'our main problem [is] how to programme a machine to imitate a brain, or as we might say more briefly, if less accurately, to think.' He shows the same willingness to discuss the question 'Can machines think?' in [Chapter 14.
Turing's view is that a machine which imitates the intellectual behaviour of a human brain can itself appropriately be described as a brain or as thinking. In Chapter 14, Turing emphasizes that it is only the _intellectual_ behaviour of the brain that need be considered (pp. 494–5): 'To take an extreme case, we are not interested in the fact that the brain has the consistency of cold porridge. We don't want to say "This machine's quite hard, so it isn't a brain, and so it can't think."'
It is, of course, the ability of the machine to imitate the intellectual behaviour of a human brain that is examined in the Turing test (Chapter 11). Thus: any machine that plays the imitation game successfully can appropriately be described as a brain or as thinking.
#### Freedom of the Will
This chapter contains one of the two discussions of free will occurring in Turing's mature writings, both of which are tantalizingly brief. (The early essay entitled 'Nature of Spirit, which possibly dates from Turing's undergraduate days, also contains a discussion of free will. There Turing wrote: 'the theory which held that as eclipses etc. are predestined so were all our actions breaks down... We have a will which is able to determine the action of the atoms probably in a small portion of the brain, or possibly all over it.') The other discussion occurs in Chapter 11, where Turing says (p. 445):
An interesting variant on the idea of a digital computer is a 'digital computer with a random element'. These have instructions involving the throwing of a die or some equivalent electronic process... Sometimes such a machine is described as having free will (though I would not use this phrase myself). It is not normally possible to determine from observing a machine whether it has a random element, for a similar effect can be produced by such devices as making the choices depend on the digits of the decimal for π.
Unfortunately, Turing does not expand on the remark 'I would not use this phrase myself.' Possibly he means simply that the addition of a random element to a computer is not in itself sufficient to warrant the attribution of free will. Presumably one would at least need to add cognition and initiative as well before the machine could reasonably be described as having free will (compare Chapter 10, pp. 424, 429–30). Alternatively, it is possible that Turing is objecting to the term 'free will' itself, much as he objects elsewhere in Chapter 11 to the word 'think' ('too meaningless to deserve discussion').
Turing introduced this idea of a 'digital computer with a random element' more fully in Chapter 10 (p. 416):
It is possible to modify the above described types of discrete machines by allowing several alternative operations to be applied at some points, the alternatives to be chosen by a random process. Such a machine will be described as 'partially random'. If we wish to say definitely that a machine is not of this kind we will describe it as 'determined'. Sometimes a machine may be strictly speaking determined but appear superficially as if it were partially random. This would occur if for instance the digits of the number π were used to determine the choices of a partially random machine where previously a dice thrower or electronic equivalent had been used. These machines are known as apparently partially random.
Turing discusses partially random machines further in Chapter 12 and in Chapter 9, where he mentions the possibility of including in the ACE a 'random element, some electronic roulette wheel' (p. 391).
In 'Can Digital Computers Think?' Turing raises both the possibility that 'the feeling of free will which we all have is an illusion' and the possibility that 'we really have got free will but yet there is no way of telling from our behaviour that this is so'. In parallel, he raises the question whether the behaviour of the brain 'is in principle predictable by calculation' (i.e. by Turing machine). In discussing this possibility, he observes that we 'certainly do not know how any such calculation should be done'. He points out that, furthermore, some physicists argue that no such prediction is even theoretically possible, 'on account of the indeterminacy principle in quantum mechanics'. Turing does not state his own position on these issues in the course of the lecture. However, in an interview given long after Turing's death, Max Newman stated that Turing 'had a deep-seated conviction that the real brain has a "roulette wheel" somewhere in it.' This seems to indicate that Turing's view was that the brain is a partially random machine. Whether or not Turing would have asserted, on that basis, that we 'really have got free will' is not known.
#### Can Computers Think?
Turing's overarching aim in the lecture is to answer the question posed by his title. His strategy is to argue for the proposition mentioned above:
If any machine can appropriately be described as a brain, then any digital computer can be so described.
His initial bald statement of his argument is (p. 483):
If now some particular machine can be described as a brain we have only to programme our digital computer to imitate it and it will also be a brain. If it is accepted that real brains, as found in animals, and in particular in men, are a sort of machine it will follow that our digital computer suitably programmed, will behave like a brain.
Turing goes on to flesh out his argument in various ways, turning eventually to the problem of free will (p. 484): 'There are still some difficulties. To behave like a brain seems to involve free will, but the behaviour of a digital computer, when it has been programmed, is completely determined.' Turing argues resourcefully that, even if it is true that brains have free will, this in fact presents no difficulty for his claim that a suitably programmed computer can imitate the brain (pp. 484–5):
a machine which is to imitate a brain must appear to behave as if it had free will, and it may well be asked how this is to be achieved. One possibility is to make its behaviour depend on something like a roulette wheel or a supply of radium.... It is, however, not really even necessary to do this. It is not difficult to design machines whose behaviour appears quite random to anyone who does not know the details of their construction.
Such machines are 'apparently partially random' (p. 416). Examples of apparently partially random machines are the German Enigma machine and the Lorenz SZ 40 cipher machine ('Tunny'). Since both these machines can be simulated by a digital computer, an appropriately programmed digital computer is apparently partially random. (In Chapter 11 Turing mentions having written a programme for the Manchester computer that produced apparently partially random behaviour. When given a number, the programme would reply with a number. Turing said 'I would defy anyone to learn from these replies sufficient about the programme to be able to predict any replies to untried values' (p. 457).)
Apparently partially random machines imitate partially random machines. If the brain is a partially random machine, an appropriately programmed digital computer may nevertheless give a convincing imitation of a brain. The appearance that this deterministic machine gives of possessing free will may be said to be mere sham, but this will not affect the machine's ability to play the imitation game successfully. And by Turing's principle, above, any machine that plays the imitation game successfully can appropriately be described as a brain.
#### The Church–Turing Thesis and Calculating Machines
In 'Can Digital Computers Think?' Turing puts foward a thesis that, while not the same as the Church–Turing thesis (see Chapter 1 and 'Computable Numbers: A Guide'), is in effect the result of replacing the term 'human computer' in the Church–Turing thesis by 'calculating machine', and replacing 'universal Turing machine' by 'digital computer of sufficient speed and storage capacity'.
The Church–Turing thesis states that any work that can be carried out by a human computer (i.e. by an obedient clerk working with pencil on paper in accordance with an effective procedure) can equally well be carried out by the universal Turing machine. The present thesis (pp. 482–3) states that any work that can be carried out by any calculating machine can equally well be carried out by a digital computer of sufficient speed and storage capacity:
A digital computer is a _universal_ machine in the sense that it can be made to replace any machine of a certain very wide class. It will not replace a bulldozer or a steam-engine or a telescope, but it will replace any rival design of calculating machine.
Newman wrote in the same vein a few years previously:
A universal machine is a single machine which, when provided with suitable instructions, will perform any calculation that could be done by a specially constructed machine. No real machine can be truly universal because its size is limited... but subject to this limitation of size, the machines now being made in America and in this country will be 'universal'—if they work at all; that is, they will do every kind of job that can be done by special machines.
If Turing were requested to clarify the notion of a 'calculating machine', he would perhaps offer paradigm examples such as the Brunsviga (a popular desk calculating machine), a differential analyser (an analogue computing device), special-purpose electronic machines like Colossus and ENIAC, and so on (compare Chapter 10, pp. 412–13). Or perhaps he would say, with greater generality, that a calculating machine is any machine that duplicates the abilities of a human mathematician working mechanically with paper and pencil, i.e. in accordance with an effective ('rule of thumb') procedure. It was in this manner that he explained the idea of an electronic computing machine in the opening paragraph of his _Programmers' Handbook_ : 'Electronic computers are intended to carry out any definite rule of thumb process which could have been done by a human operator working in a disciplined but unintelligent manner.'
Turing's remarks in Chapter 17 on the status of the Church–Turing thesis are also relevant here (and see also the section 'Normal Forms and the Church-Turing Thesis' in the introduction to Chapter 17).
#### Other Notable Features
Other features of note in the lecture include the continuation of the discussion of 'Lady Lovelace's dictum', begun in Chapter 11, and Turing's glorious analogy comparing trying to programme a computer to behave like a brain with trying to write a treatise about family life on Mars—and moreover with insufficient paper. (Newman once remarked on the 'comical but brilliantly apt analogies with which he [Turing] explained his ideas'.)
##### _Further reading_
Copeland, B. J., _Artificial Intelligence: A Philosophical Introduction_ (Oxford: Blackwell, 1993). Chapter 3: 'Can a Machine Think?'; Chapter 7: 'Freedom'.
—— 'Narrow versus Wide Mechanism: Including a Re-examination of Turing's Views on the Mind–Machine Issue', _Journal of Philosophy_ , 97 (2000), 5–32. Reprinted in M. Scheutz, _Computationalism: New Directions_ (Cambridge, Mass.: MIT Press, 2002).
Dennett, D. C., _Elbow Room: The Varieties of Freewill Worth Wanting_ (Oxford: Clarendon Press, 1984).
Simons, G., _The Biology of Computer Life_ (Brighton: Harvester, 1985).
##### _Provenance_
The text that follows is from Turing's typescript and incorporates corrections made in his hand.
#### Can Digital Computers Think?
Digital computers have often been described as mechanical brains. Most scientists probably regard this description as a mere newspaper stunt, but some do not. One mathematician has expressed the opposite point of view to me rather forcefully in the words 'It is commonly said that these machines are not brains, but you and I know that they are.' In this talk I shall try to explain the ideas behind the various possible points of view, though not altogether impartially. I shall give most attention to the view which I hold myself, that it is not altogether unreasonable to describe digital computers as brains. A different point of view has already been put by Professor Hartree.
First we may consider the naive point of view of the man in the street. He hears amazing accounts of what these machines can do: most of them apparently involve intellectual feats of which he would be quite incapable. He can only explain it by supposing that the machine is a sort of brain, though he may prefer simply to disbelieve what he has heard.
The majority of scientists are contemptuous of this almost superstitious attitude. They know something of the principles on which the machines are constructed and of the way in which they are used. Their outlook was well summed up by Lady Lovelace over a hundred years ago, speaking of Babbage's Analytical Engine. She said, as Hartree has already quoted, 'The Analytical Engine has no pretensions whatever to _originate_ anything. It can do whatever _we know how to order it_ to perform.' This very well describes the way in which digital computers are actually used at the present time, and in which they will probably mainly be used for many years to come. For any one calculation the whole procedure that the machine is to go through is planned out in advance by a mathematician. The less doubt there is about what is going to happen the better the mathematician is pleased. It is like planning a military operation. Under these circumstances it is fair to say that the machine doesn't originate anything.
There is however a third point of view, which I hold myself. I agree with Lady Lovelace's dictum as far as it goes, but I believe that its validity depends on considering how digital computers _are_ used rather than how they _could be_ used. In fact I believe that they could be used in such a manner that they could appropriately be described as brains. I should also say that 'If any machine can appropriately be described as a brain, then any digital computer can be so described.'
This last statement needs some explanation. It may appear rather startling, but with some reservations it appears to be an inescapable fact. It can be shown to follow from a characteristic property of digital computers, which I will call their _universality_. A digital computer is a _universal_ machine in the sense that it can be made to replace any machine of a certain very wide class. It will not replace a bulldozer or a steam-engine or a telescope, but it will replace any rival design of calculating machine, that is to say any machine into which one can feed data and which will later print out results. In order to arrange for our computer to imitate a given machine it is only necessary to programme the computer to calculate what the machine in question would do under given circumstances, and in particular what answers it would print out. The computer can then be made to print out the same answers.
If now some particular machine can be described as a brain we have only to programme our digital computer to imitate it and it will also be a brain. If it is accepted that real brains, as found in animals, and in particular in men, are a sort of machine it will follow that our digital computer, suitably programmed, will behave like a brain.
This argument involves several assumptions which can quite reasonably be challenged. I have already explained that the machine to be imitated must be more like a calculator than a bulldozer. This is merely a reflection of the fact that we are speaking of mechanical analogues of brains, rather than of feet or jaws. It was also necessary that this machine should be of the sort whose behaviour is in principle predictable by calculation. We certainly do not know how any such calculation should be done, and it was even argued by Sir Arthur Eddington that on account of the indeterminacy principle in quantum mechanics no such prediction is even theoretically possible.
Another assumption was that the storage capacity of the computer used should be sufficient to carry out the prediction of the behaviour of the machine to be imitated. It should also have sufficient speed. Our present computers probably have not got the necessary storage capacity, though they may well have the speed. This means in effect that if we wish to imitate anything so complicated as the human brain we need a very much larger machine than any of the computers at present available. We probably need something at least a hundred times as large as the Manchester Computer. Alternatively of course a machine of equal size or smaller would do if sufficient progress were made in the technique of storing information.
It should be noticed that there is no need for there to be any increase in the complexity of the computers used. If we try to imitate ever more complicated machines or brains we must use larger and larger computers to do it. We do not need to use successively more complicated ones. This may appear paradoxical, but the explanation is not difficult. The imitation of a machine by a computer requires not only that we should have made the computer, but that we should have programmed it appropriately. The more complicated the machine to be imitated the more complicated must the programme be.
This may perhaps be made clearer by an analogy. Suppose two men both wanted to write their autobiographies, and that one had had an eventful life, but very little had happened to the other. There would be two difficulties troubling the man with the more eventful life more seriously than the other. He would have to spend more on paper and he would have to take more trouble over thinking what to say. The supply of paper would not be likely to be a serious difficulty, unless for instance he were on a desert island, and in any case it could only be a technical or a financial problem. The other difficulty would be more fundamental and would become more serious still if he were not writing his life but a work on something he knew nothing about, let us say about family life on Mars. Our problem of programming a computer to behave like a brain is something like trying to write this treatise on a desert island. We cannot get the storage capacity we need: in other words we cannot get enough paper to write the treatise on, and in any case we don't know what we should write down if we had it. This is a poor state of affairs, but, to continue the analogy, it is something to know how to write, and to appreciate the fact that most knowledge can be embodied in books.
In view of this it seems that the wisest ground on which to criticise the description of digital computers as 'mechanical brains' or 'electronic brains' is that, although they might be programmed to behave like brains, we do not at present know how this should be done. With this outlook I am in full agreement. It leaves open the question as to whether we will or will not eventually succeed in finding such a programme. I, personally, am inclined to believe that such a programme will be found. I think it is probable for instance that at the end of the century it will be possible to programme a machine to answer questions in such a way that it will be extremely difficult to guess whether the answers are being given by a man or by the machine. I am imagining something like a viva-voce examination, but with the questions and answers all typewritten in order that we need not consider such irrelevant matters as the faithfulness with which the human voice can be imitated. This only represents my opinion; there is plenty of room for others.
There are still some difficulties. To behave like a brain seems to involve free will, but the behaviour of a digital computer, when it has been programmed, is completely determined. These two facts must somehow be reconciled, but to do so seems to involve us in an age-old controversy, that of 'free will and determinism'. There are two ways out. It may be that the feeling of free will which we all have is an illusion. Or it may be that we really have got free will, but yet there is no way of telling from our behaviour that this is so. In the latter case, however well a machine imitates a man's behaviour it is to be regarded as a mere sham. I do not know how we can ever decide between these alternatives but whichever is the correct one it is certain that a machine which is to imitate a brain must appear to behave as if it had free will, and it may well be asked how this is to be achieved. One possibility is to make its behaviour depend on something like a roulette wheel or a supply of radium. The behaviour of these may perhaps be predictable, but if so, we do not know how to do the prediction.
It is, however, not really even necessary to do this. It is not difficult to design machines whose behaviour appears quite random to anyone who does not know the details of their construction. Naturally enough the inclusion of this random element, whichever technique is used, does not solve our main problem, how to programme a machine to imitate a brain, or as we might say more briefly, if less accurately, to think. But it gives us some indication of what the process will be like. We must not always expect to know what the computer is going to do. We should be pleased when the machine surprises us, in rather the same way as one is pleased when a pupil does something which he had not been explicitly taught to do.
Let us now reconsider Lady Lovelace's dictum. 'The machine can do whatever _we know how to order it_ to perform.' The sense of the rest of the passage is such that one is tempted to say that the machine can _only_ do what we know how to order it to perform. But I think this would not be true. Certainly the machine can only do what we _do_ order it to perform, anything else would be a mechanical fault. But there is no need to suppose that, when we give it its orders we know what we are doing, what the consequences of these orders are going to be. One does not need to be able to understand how these orders lead to the machine's subsequent behaviour, any more than one needs to understand the mechanism of germination when one puts a seed in the ground. The plant comes up whether one understands or not. If we give the machine a programme which results in its doing something interesting which we had not anticipated I should be inclined to say that the machine _had_ originated something, rather than to claim that its behaviour was implicit in the programme, and therefore that the originality lies entirely with us.
I will not attempt to say much about how this process of 'programming a machine to think' is to be done. The fact is that we know very little about it, and very little research has yet been done. There are plentiful ideas, but we do not yet know which of them are of importance. As in the detective stories, at the beginning of the investigation any trifle may be of importance to the investigator. When the problem has been solved, only the essential facts need to be told to the jury. But at present we have nothing worth putting before a jury. I will only say this, that I believe the process should bear a close relation of that of teaching.
I have tried to explain what are the main rational arguments for and against the theory that machines could be made to think, but something should also be said about the irrational arguments. Many people are extremely opposed to the idea of machine that thinks, but I do not believe that it is for any of the reasons that I have given, or any other rational reason, but simply because they do not like the idea. One can see many features which make it unpleasant. If a machine can think, it might think more intelligently than we do, and then where should we be? Even if we could keep the machines in a subservient position, for instance by turning off the power at strategic moments, we should, as a species, feel greatly humbled. A similar danger and humiliation threatens us from the possibility that we might be superseded by the pig or the rat. This is a theoretical possibility which is hardly controversial, but we have lived with pigs and rats for so long without their intelligence much increasing, that we no longer trouble ourselves about this possibility. We feel that if it is to happen at all it will not be for several million years to come. But this new danger is much closer. If it comes at all it will almost certainly be within the next millennium. It is remote but not astronomically remote, and is certainly something which can give us anxiety.
It is customary, in a talk or article on this subject, to offer a grain of comfort, in the form of a statement that some particularly human characteristic could never be imitated by a machine. It might for instance be said that no machine could write good English, or that it could not be influenced by sex-appeal or smoke a pipe. I cannot offer any such comfort, for I believe that no such bounds can be set. But I certainly hope and believe that no great efforts will be put into making machines with the most distinctively human, but non-intellectual characteristics such as the shape of the human body; it appears to me to be quite futile to make such attempts and their results would have something like the unpleasant quality of artificial flowers. Attempts to produce a thinking machine seem to me to be in a different category. The whole thinking process is still rather mysterious to us, but I believe that the attempt to make a thinking machine will help us greatly in finding out how we think ourselves.
### **CHAPTER 14
Can Automatic Calculating Machines Be Said To Think? ( _1952_ )**
_Alan Turing, Richard Braithwaite, Geoffrey Jefferson, Max Newman_
#### Introduction
##### _Jack Copeland_
This discussion between Turing, Newman, R. B. Braithwaite, and G. Jefferson was recorded by the BBC on 10 January 1952 and broadcast on BBC Radio on the 14th, and again on the 23rd, of that month. This is the earliest known recorded discussion of artificial intelligence.
#### The Participants
The anchor man of the discussion is Richard Braithwaite (1900–90). Braithwaite was at the time Sidgwick Lecturer in Moral Science at the University of Cambridge, where the following year he was appointed Knightsbridge Professor of Moral Philosophy. Like Turing, he was a Fellow of King's College. Braithwaite's main work lay in the philosophy of science and in decision and games theory (which he applied in moral philosophy).
Geoffrey Jefferson (1886–1961) retired from the Chair of Neurosurgery at Manchester University in 1951. In his Lister Oration, delivered at the Royal College of Surgeons of England on 9 June 1949, he had declared: 'When we hear it said that wireless valves think, we may despair of language.'
Turing gave a substantial discussion of Jefferson's views in 'Computing Machinery and Intelligence' (pp. 451–2), rebutting the 'argument from consciousness' that he found in the Lister Oration. In the present chapter, Jefferson takes numerous pot shots at the notion of a machine thinking, which for the most part Turing and Newman are easily able to turn aside.
Jefferson may have thought little of the idea of machine intelligence, but he held Turing in considerable regard, saying after Turing's death that he 'had real genius, it shone from him'.
#### The Turing Test Revisited
From the point of view of Turing scholarship, the most important parts of 'Can Automatic Calculating Machines Be Said to Think' are the passages containing Turing's exposition of the imitation game or Turing test. The description of the test that Turing gave in 'Computing Machinery and Intelligence' is here modified in a number of significant ways.
The lone interrogator of the original version is replaced by a 'jury' (p. 495). Each jury must judge 'quite a number of times' and 'sometimes they really are dealing with a man and not a machine'. For a machine to pass the test, a 'considerable proportion' of the jury 'must be taken in by the pretence'. The members of the jury interrogate the contestants, but their contributions 'don't really have to be questions, any more than questions in a law court are really questions'; for example, 'I put it to you that you are only pretending to be a man' is 'quite in order'.
In its original presentation (Chapter 11), the Turing test is a three-party game involving the parallel interrogation by a human of a computer and a human foil. According to the 1952 formulation, however, members of a jury interview a series of contestants one at a time, some of the contestants being machines and some humans. In the original form of the test, each interrogator knows that one of each pair of interviewees is a human and one a machine, but in the single-interviewee version of the test, this condition is necessarily absent. It appears that the earlier formulation is in fact superior, since the single-interviewee version is open to a biasing effect which disfavours the machine. Results of the Loebner series of single-interviewee Turing tests reveal a strong propensity among jurors to classify human respondents as machines. (New York businessman Hugh Loebner started the annual Loebner Prize competition in 1991, offering the sum of $100,000 to the programmer(s) of the first programme to pass the Turing test. $2,000 is awarded each year for the best effort. So far the grand prize remains unclaimed.) In the Loebner competition held at Dartmouth College, New Hampshire, in January 2000, human respondents were mistaken for computers on ten occasions, a computer for a human on none. The same effect was present in a series of single-interviewee tests performed with Kenneth Colby's historic programme Parry. In a total of ten interviews, there were five misidentifications; in four of these a human respondent was mistaken for a computer. Presumably this phenomenon is the result of a determination on the part of the jurors not to be fooled by a programme. This lengthening of the odds against the machine cannot occur in the three-player form of the test.
#### Turing's Predictions
Turing is often misquoted as having predicted that, by the turn of the twentieth century, artificial intelligence indistinguishable from human intelligence would be in existence. What he in fact wrote in 1950 was that (p. 449):
in about fifty years' time it will be possible to programme computers...to make them play the imitation game so well that an average interrogator will not have more than 70 per cent chance of making the right identification after five minutes of questioning.
Some commentators have reported this prediction the wrong way about, as the claim that, by the end of the twentieth century, computers would succeed in _deceiving_ the interrogator 70 per cent of the time.
In 'Can Automatic Calculating Machines Be Said to Think', Turing offered a prediction that is interestingly different from the above, and which seems to concern success of a more substantial nature (p. 495):
_Newman_ : I should like to be there when your match between a man and a machine takes place, and perhaps to try my hand at making up some of the questions. But that will be a long time from now, if the machine is to stand any chance with no questions barred?
_Turing_ : Oh yes, at least 100 years, I should say.
In Chapter 11, Turing describes a form of the imitation game in which the interrogator must attempt to distinguish between a woman (the foil) and a man pretending to be a woman. This form of the game serves as a reference point for evaluating the computer's performance in the computer-human imitation game (p. 441):
We now ask the question, 'What will happen when a machine takes the part of A in this [man-imitates-woman] game?' Will the interrogator decide wrongly as often when the game is played like this as he does when the game is played between a man and a woman?
Reformulating Turing's 1952 prediction in these terms produces: It will be at least 100 years (2052) before a computer is able to play the imitation game sufficiently well so that jurors will decide wrongly as often in man-imitates-woman imitation games as in computer-imitates-human imitation games, in each case no questions being barred.
#### 'Fiendish Expert' Objections to the Turing Test
'Fiendish expert' objections, of which there are many, are of the form: 'An expert could unmask the computer by asking it...'. All can be dealt with in a similar fashion. (Some other objections to the test are discussed in the introduction to Chapter 11.)
An interesting example is put forward by Robert French. French's objection involves the phenomenon of associative priming, observed in what psychologists call the word/non-word recognition task. A subject seated in front of a screen is presented for a brief time with what may or may not be a word (e.g. 'dog', 'DOK'). If the subject recognizes the letters as a word he or she must press a button. The experimenter measures the time that the subject takes to respond. It is found that, on average, subjects require less time to recognize a word if the word is preceded by a brief presentation of an associated word (e.g. 'fish' may facilitate the recognition of 'chips', 'bread' of 'butter'). French's claim is that this priming effect may be used to unmask the computer in the Turing test:
The Turing Test interrogator makes use of this phenomenon as follows. The day before the Test, she selects a set of words (and non-words), runs the lexical decision task on the interviewees and records average recognition times. She then comes to the Test armed with the results... [and] identifies as the human being the candidate whose results more closely resemble the average results produced by her sample population of interviewees. The machine would invariably fail this type of test because there is no a priori way of determining associative strengths... Virtually the only way a machine could determine, even on average, all of the associative strengths between human concepts is to have experienced the world as the human candidate and the interviewers had.
Turing, however, was happy to rule out expert jurors. In 'Can Automatic Calculating Machines Be Said to Think' he said that the interrogator 'should not be expert about machines' (p. 495), and in Chapter 10, describing the chessplayer version of the test, he said that the discriminator should be a 'rather poor' chess player (p. 431). Turing did not mention other kinds of expert, but the reasons for excluding experts about machines apply equally well to experts about minds.
In any case, French's proposal is illegitimate. The specifications of the Turing test are clear: the interrogator is allowed only to put questions. There is no provision for the use of the equipment necessary for administering the lexical decision task and for measuring the contestants' reaction times. One might as well allow the interrogator to bring along equipment for measuring the contestants' magnetic fields or energy dissipation.
#### Can Machines Think?
In Chapter 11, Turing said that the question 'Can machines think?' is 'too meaningless to deserve discussion' (p. 449). He certainly did not allow this view to prevent him from indulging rather often in such discussion. In the present chapter, Turing records a considerably milder attitude to the question (p. 495):
You might call it a test to see whether the machine thinks, but it would be better to avoid begging the question, and say that the machines that pass are (let's say) 'Grade A' machines.... [The question whether] machines really could pass the test [is] not the same as 'Do machines think?', but it seems near enough for our present purpose, and raises much the same difficulties.
In Chapter 10 Turing wrote (p. 431):
The extent to which we regard something as behaving in an intelligent manner is determined as much by our own state of mind and training as by the properties of the object under consideration. If we are able to explain or predict its behaviour... we have little temptation to imagine intelligence. With the same object therefore it is possible that one man would consider it as intelligent and another would not; the second man would have found out the rules of its behaviour.
Turing develops this point in the present discussion (p. 500):
As soon as one can see the cause and effect working themselves out in the brain, one regards it as not being thinking, but a sort of unimaginative donkey-work. From this point of view one might be tempted to define thinking as consisting of 'those mental processes that we don't understand'. If this is right then to make a thinking machine is to make one which does interesting things without our really understanding quite how it is done.
Many years later Marvin Minsky put forward this same view, saying that 'intelligence' is simply our name for whichever problem-solving mental processes we do not yet understand. Minsky likens intelligence to the concept 'unexplored regions of Africa': it disappears as soon as we discover it.
#### Analogy and Creativity
Turing sketches an interesting mechanical explanation of how analogy works in the human brain, and suggests that a digital computer can be made to do the same (p. 499). Another proposal which does not appear elsewhere in Turing's writings concerns the vexed issue of creativity. A machine that combines words more or less at random and then scores the combinations 'for various merits' would, he says, be able to find useful new concepts. His example is 'lumping together rain, hail, snow and sleet, under the word "precipitation"' (p. 499). He agrees with Newman that this process would be 'shockingly slow'. Nevertheless, this brief suggestion of Turing's is an economical illustration of the possibility of a machine's acquiring new concepts for itself.
#### Machine Learning
'Can Automatic Calculating Machines Be Said to Think' contains a lengthy discussion of machine learning, in which Turing alludes to the P-type machines of Chapter 10 (pp. 425–9). He says in response to Jefferson's point that real learning involves intervention by teachers, 'I have made some experiments in teaching a machine to do some simple operation, and a very great deal of such intervention was needed' (p. 497). Turing emphasizes several times that a learning machine, if it is to be effective, must not only learn first-order facts but must also be able to learn to improve its learning methods (compare the discussion of the Mathematical Objection in Chapter 12 and elsewhere).
Newman gives a simple illustration (on p. 496) of how a computer can learn to do better with practice. His example makes use of the idea of the computer modifying its own programme, mentioned by Turing in Chapter 9 (p. 393). Faced with a two-move chess problem, the machine initially follows an instruction to choose a move at random. If the move, say B-Q5, is found to lead to forced mate in two moves, then the machine changes the instruction to 'Try BQ5'. When presented with the same problem again, the machine immediately gives the right answer.
#### Newman's Test
Towards the end of the discussion Newman suggested a test—and the term 'Newman's test' seems appropriate—for when machines have 'begun to think' (pp. 504–5). That stage of development has been reached, he said, when a machine can solve a mathematical problem for which no effective method exists.
##### _Provenance_
The discussion is taken from a BBC script (which is marked 'Not checked in Talks Department with "as broadcast" script').
#### Can Automatic Calculating Machines Be Said To Think?
_Braithwaite_ : We're here today to discuss whether calculating machines can be said to think in any proper sense of the word. Thinking is ordinarily regarded as so much a speciality of man, and perhaps of other higher animals, that the question may seem too absurd to be discussed. But, of course, it all depends on what is to be included in thinking. The word is used to cover a multitude of different activities. What would you, Jefferson, as a physiologist, say were the most important elements involved in thinking?
_Jefferson_ : I don't think that we need waste too much time on [a] definition of thinking since it will be hard to get beyond phrases in common usage, such as having ideas in the mind, cogitating, meditating, deliberating, solving problems or imagining. Philologists say that the word 'Man' is derived from a Sanskrit word that means 'to think', probably in the sense of judging between one idea and another. I agree that we could no longer use the word 'thinking' in a sense that restricted it to man. No one would deny that many animals think, though in a very limited way. They lack insight. For example, a dog learns that it is wrong to get on cushions or chairs with muddy paws, but he only learns it as a venture that doesn't pay. He has no conception of the real reason, that he damages fabrics by doing that.
The average person would perhaps be content to define thinking in very general terms such as revolving ideas in the mind, of having notions in one's head, of having one's mind occupied by a problem, and so on. But it is only right to add that our minds are occupied much of the time with trivialities. One might say in the end that thinking was the general result of having a sufficiently complex nervous system. Very simple ones do not provide the creature with any problems that are not answered by simple reflex mechanisms. Thinking then becomes all the things that go on in one's brain, things that often end in an action but don't necessarily do so. I should say that it was the sum total of what the brain of man or animal does. Turing, what do you think about it? Have you a mechanical definition?
_Turing_ : I don't want to give a definition of thinking, but if I had to I should probably be unable to say anything more about it than that it was a sort of buzzing that went on inside my head. But I don't really see that we need to agree on a definition at all. The important thing is to try to draw a line between the properties of a brain, or of a man, that we want to discuss, and those that we don't. To take an extreme case, we are not interested in the fact that the brain has the consistency of cold porridge. We don't want to say 'This machine's quite hard, so it isn't a brain, and so it can't think.' I would like to suggest a particular kind of _test_ that one might apply to a machine. You might call it a test to see whether the machine thinks, but it would be better to avoid begging the question, and say that the machines that pass are (let's say) 'Grade A' machines. The idea of the test is that the machine has to try and pretend to be a man, by answering questions put to it, and it will only pass if the pretence is reasonably convincing. A considerable proportion of a jury, who should not be expert about machines, must be taken in by the pretence. They aren't allowed to see the machine itself—that would make it too easy. So the machine is kept in a far away room and the jury are allowed to ask it questions, which are transmitted through to it: it sends back a typewritten answer.
_Braithwaite_ : Would the questions have to be sums, or could I ask it what it had had for breakfast?
_Turing_ : Oh yes, anything. And the questions don't really have to be questions, any more than questions in a law court are really questions. You know the sort of thing. 'I put it to you that you are only pretending to be a man' would be quite in order. Likewise the machine would be permitted all sorts of tricks so as to appear more man-like, such as waiting a bit before giving the answer, or making spelling mistakes, but it can't make smudges on the paper, any more than one can send smudges by telegraph. We had better suppose that each jury has to judge quite a number of times, and that sometimes they really are dealing with a man and not a machine. That will prevent them saying 'It must be a machine' every time without proper consideration.
Well, that's my test. Of course I am not saying at present either that machines really could pass the test, or that they couldn't. My suggestion is just that this is the question we should discuss. It's not the same as 'Do machines think,' but it seems near enough for our present purpose, and raises much the same difficulties.
_Newman_ : I should like to be there when your match between a man and a machine takes place, and perhaps to try my hand at making up some of the questions. But that will be a long time from now, if the machine is to stand any chance with no questions barred?
_Turing_ : Oh yes, at least 100 years, I should say.
_Jefferson_ : Newman, how well would existing machines stand up to this test? What kind of things can they do now?
_Newman_ : Of course, their strongest line is mathematical computing, which they were designed to do, but they would also do well at some questions that don't look numerical, but can easily be made so, like solving a chess problem or looking you up a train in the time-table.
_Braithwaite_ : Could they do that?
_Newman_ : Yes. Both these jobs can be done by trying all the possibilities, one after another. The whole of the information in an ordinary time-table would have to be written in as part of the programme, and the simplest possible routine would be one that found the trains from London to Manchester by testing every train in the time-table to see if it calls at both places, and printing out those that do. Of course, this is a dull, plodding method, and you could improve on it by using a more complicated routine, but if I have understood Turing's test properly, you are not allowed to go behind the scenes and criticise the method, but must abide by the scoring on correct answers, found reasonably quickly.
_Jefferson_ : Yes, but all the same a man who has to look up trains frequently gets better at it, as he learns his way about the time-table. Suppose I give a machine the same problem again, can it learn to do better without going through the whole rigmarole of trying everything over every time? I'd like to have your answer to that because it's such an important point. Can machines learn to do better with practice?
_Newman_ : Yes, it could. Perhaps the chess problem provides a better illustration of this. First I should mention that _all_ the information required in any job—the numbers, times of trains, positions of pieces, or whatever it is, and also the instructions saying what is to be done with them—all this material is stored in the same way. (In the Manchester machine it is stored as a pattern on something resembling a television screen.) As the work goes on the pattern is changed. Usually it is the part of the pattern that contains the data that changes, while the instructions stay fixed. But it is just as simple to arrange that the instructions themselves shall be changed now and then. Well, now a programme could be composed that would cause the machine to do this: a 2-move chess problem is recorded into the machine in some suitable coding, and whenever the machine is started, a white move is chosen at random (there is a device for making random choices in our machine). All the consequences of this move are now analysed, and if it does _not_ lead to forced mate in two moves, the machine prints, say, 'P-Q3, wrong move', and stops. But the analysis shows that when the right move is chosen the machine not only prints, say, 'B-Q5, solution', but it changes the instruction calling for a random choice to one that says 'Try B-Q5.' The result is that whenever the machine is started again it will immediately print out the right solution—and this without the man who made up the routine knowing beforehand what it was. Such a routine could certainly be made now, and I think this can fairly be called learning.
_Jefferson_ : Yes, I suppose it is. Human beings learn by repeating the same exercises until they have perfected them. Of course it goes further, and at the same time —we learn generally to shift the knowledge gained about one thing to another set of problems, seeing relevances and relationships. Learning means remembering. How long can a machine store information for?
_Newman_ : Oh, at least as long as a man's lifetime, if it is refreshed occasionally.
_Jefferson_ : Another difference would be that in the learning process there is much more frequent intervention by teachers, parental or otherwise, guiding the arts of learning. You mathematicians put the programme once into the machine and leave it to it. You wouldn't get any distance at all with human beings if that is what you did. In fact, the only time you do that in the learning period is at examinations.
_Turing_ : It's quite true that when a child is being taught, his parents and teachers are repeatedly intervening to stop him doing this or encourage him to do that. But this will not be any the less so when one is trying to teach a machine. I have made some experiments in teaching a machine to do some simple operation, and a very great deal of such intervention was needed before I could get any results at all. In other words the machine learnt so slowly that it needed a great deal of teaching.
_Jefferson_ : But who was learning, you or the machine?
_Turing_ : Well, I suppose we both were. One will have to find out how to make machines that will learn more quickly if there is to be any real success. One hopes too that there will be a sort of snowball effect. The more things the machine has learnt the easier it ought to be for it to learn others. In learning to do any particular thing it will probably also be learning to learn more efficiently. I am inclined to believe that when one has taught it to do certain things one will find that some other things which one had planned to teach it are happening without any special teaching being required. This certainly happens with an intelligent human mind, and if it doesn't happen when one is teaching a machine there is something lacking in the machine. What do you think about learning possibilities, Braithwaite?
_Braithwaite_ : No-one has mentioned what seems to me the great difficulty about learning, since we've only discussed learning to solve a particular problem. But the most important part of human learning is learning from experience—not learning from one particular kind of experience, but being able to learn from experience in general. A machine can easily be constructed with a feed-back device so that the programming of the machine is controlled by the relation of its output to some feature in its external environment—so that the working of the machine in relation to the environment is self-corrective. But this requires that it should be some particular feature of the environment to which the machine has to adjust itself. The peculiarity of men and animals is that they have the power of adjusting themselves to almost all the features. The feature to which adjustment is made on a particular occasion is the one the man is attending to and he attends to what he is _interested in_. His interests are determined, by and large, by his appetites, desires, drives, instincts—all the things that together make up his 'springs of action'. If we want to construct a machine which will vary its attention to things in its environment so that it will sometimes adjust itself to one and sometimes to another, it would seem to be necessary to equip the machine with something corresponding to a set of appetites. If the machine is built to be treated only as a domestic pet, and is spoon-fed with particular problems, it will not be able to learn in the varying way in which human beings learn. This arises from the necessity of adapting behaviour suitably to environment if human appetites are to be satisfied.
_Jefferson_ : Turing, you spoke with great confidence about what you are going to be able to do. You make it sound as if it would be fairly easy to modify construction so that the machine reacted more like a man. But I recollect that from the time of Descartes and Borelli on people have said that it would be only a matter of a few years, perhaps 3 or 4 or maybe 50, and a replica of man would have been artificially created. We shall be wrong, I am sure, if we give the impression that these things would be easy to do.
_Newman_ : I agree that we are getting rather far away from computing machines as they exist at present. These machines have rather restricted appetites, and they can't blush when they're embarrassed, but it's quite hard enough, and I think a very interesting problem, to discover how near these actually existing machines can get to thinking. Even if we stick to the reasoning side of thinking, it is a long way from solving chess problems to the invention of new mathematical concepts or making a generalisation that takes in ideas that were current before, but had never been brought together as instances of a single general notion.
_Braithwaite_ : For example?
_Newman_ : The different kinds of number. There are the integers, 0, 1, −2, and so on; there are the real numbers used in comparing lengths, for example the circumference of a circle and its diameter; and the complex numbers involving √ −1; and so on. It is not at all obvious that these are instances of one thing, 'number'. The Greek mathematicians used entirely different words for the integers and the real numbers, and had no single idea to cover both. It is really only recently that the general notion of kinds of number has been abstracted from these instances and accurately defined. To make this sort of generalisation you need to have the power of recognising similarities, seeing analogies between things that had not been put together before. It is not just a matter of testing things for a specified property and classifying them accordingly. The concept itself has to be framed, something has to be created, say the idea of a number-field. Can we even guess at the way a machine could make such an invention from a programme composed by a man who had not the concept in his own mind?
_Turing_ : It seems to me, Newman, that what you said about 'trying out possibilities' as a method applies to quite an extent, even when a machine is required to do something as advanced as finding a useful new concept. I wouldn't like to have to define the meaning of the word 'concept', nor to give rules for rating their usefulness, but whatever they are they've got outward and visible forms, which are words and combinations of words. A machine could make up such combinations of words more or less at random, and then give them marks for various merits.
_Newman_ : Wouldn't that take a prohibitively long time?
_Turing_ : It would certainly be shockingly slow, but it could start on easy things, such as lumping together rain, hail, snow and sleet, under the word 'precipitation.' Perhaps it might do more difficult things later on if it was learning all the time how to improve its methods.
_Braithwaite_ : I don't think there's much difficulty about seeing analogies that can be formally analysed and explicitly stated. It is then only a question of designing the machine so that it can recognise similarities of mathematical structure. The difficulty arises if the analogy is a vague one about which little more can be said than that one has a feeling that there is some sort of similarity between two cases but one hasn't any idea as to the respect in which the two cases are similar. A machine can't recognise similarities when there is nothing in its programme to say what are the similarities it is expected to recognise.
_Turing_ : I think you could make a machine spot an analogy, in fact it's quite a good instance of how a machine could be made to do some of those things that one usually regards as essentially a human monopoly. Suppose that someone was trying to explain the double negative to me, for instance, that when something isn't not green it must be green, and he couldn't quite get it across. He might say 'Well, it's like crossing the road. You cross it, and then you cross it again, and you're back where you started.' This remark might just clinch it. This is one of the things one would like to work with machines, and I think it would be likely to happen with them. I imagine that the way analogy works in our brains is something like this. When two or more sets of ideas have the same pattern of logical connections, the brain may very likely economise parts by using some of them twice over, to remember the logical connections both in the one case and in the other. One must suppose that some part of my brain was used twice over in this way, once for the idea of double negation and once for crossing the road, there and back. I am really supposed to know about both these things but can't get what it is the man is driving at, so long as he is talking about all those dreary nots and not-nots. Somehow it doesn't get through to the right part of the brain. But as soon as he says his piece about crossing the road it gets through to the right part, but by a different route. If there is some such purely mechanical explanation of how this argument by analogy goes on in the brain, one could make a digital computer do the same.
_Jefferson_ : Well, there isn't a mechanical explanation in terms of cells and connecting fibres in the brain.
_Braithwaite_ : But could a machine really do this? How would it do it?
_Turing_ : I've certainly left a great deal to the imagination. If I had given a longer explanation I might have made it seem more certain that what I was describing was feasible, but you would probably feel rather uneasy about it all, and you'd probably exclaim impatiently, 'Well, yes, I see that a machine could do all that, but I wouldn't call it thinking.' As soon as one can see the cause and effect working themselves out in the brain, one regards it as not being thinking, but a sort of unimaginative donkey-work. From this point of view one might be tempted to define thinking as consisting of 'those mental processes that we don't understand'. If this is right then to make a thinking machine is to make one which does interesting things without our really understanding quite how it is done.
_Jefferson_ : If you mean that we don't know the wiring in men, as it were, that is quite true.
_Turing_ : No, that isn't at all what I mean. We know the wiring of our machine, but it already happens there in a limited sort of way. Sometimes a computing machine does do something rather weird that we hadn't expected. In principle one could have predicted it, but in practice it's usually too much trouble. Obviously if one were to predict everything a computer was going to do one might just as well do without it.
_Newman_ : It is quite true that people are disappointed when they discover what the big computing machines actually do, which is just to add and multiply, and use the results to decide what further additions and multiplications to do. ' _That's_ not thinking', is the natural comment, but this is rather begging the question. If you go into one of the ancient churches in Ravenna you see some most beautiful pictures round the walls, but if you peer at them through binoculars you might say, 'Why, they aren't really pictures at all, but just a lot of little coloured stones with cement in between.' The machine's processes are mosaics of very simple standard parts, but the designs can be of great complexity, and it is not obvious where the limit is to the patterns of thought they could imitate.
_Braithwaite_ : But how many stones are there in your mosaic? Jefferson, is there a sufficient multiplicity of the cells in the brain for them to behave like a computing machine?
_Jefferson_ : Yes, there are thousands, tens of thousands more cells in the brain than there are in a computing machine, because the present machine contains— how many did you say?
_Turing_ : Half a million digits. I think we can assume that is the equivalent of half a million nerve cells.
_Braithwaite_ : If the brain works like a computing machine then the present computing machine cannot do all the things the brain does. Agreed; but if a computing machine were made that could do all the things the brain does, wouldn't it require more digits than there is room for in the brain?
_Jefferson_ : Well, I don't know. Suppose that it is right to equate digits in a machine with nerve cells in a brain. There are various estimates, somewhere between ten thousand million and fifteen thousand million cells are supposed to be there. Nobody knows for certain, you see. It is a colossal number. You would need 20,000 or more of your machines to equate digits with nerve cells. But it is not, surely, just a question of size. There would be too much logic in your huge machine. It wouldn't be really like a human output of thought. To make it more like, a lot of the machine parts would have to be designed quite differently to give greater flexibility and more diverse possibilities of use. It's a very tall order indeed.
_Turing_ : It really is the size that matters in this case. It is the amount of information that can be stored up. If you think of something very complicated that you want one of these machines to do, you may find the particular machine you have got won't do, but if any machine can do it at all, then it can be done by your first computer, simply increased in its storage capacity.
_Jefferson_ : If we are really to get near to anything that can be truly called 'thinking' the effects of external stimuli cannot be missed out; the intervention of all sorts of extraneous factors, like the worries of having to make one's living, or pay one's taxes, or get food that one likes. These are not in any sense minor factors, they are very important indeed, and worries concerned with them may greatly interfere with good thinking, especially with creative thinking. You see a machine has no environment, and man is in constant relation to his environment, which, as it were punches him whilst he punches back. There is a vast background of memories in a man's brain that each new idea or experience has to fit in with. I wonder if you could tell me how far a calculating machine meets that situation. Most people agree that man's first reaction to a new idea (such as the one we are discussing today) is one of rejection, often immediate and horrified denial of it. I don't see how a machine could as it were say 'Now Professor Newman or Mr. Turing, I don't like this programme at all that you've just put into me, in fact I'm not going to have anything to do with it.'
_Newman_ : One difficulty about answering that is one that Turing has already mentioned. If someone says, 'Could a machine do this, e.g. could it say "I don't like the programme you have just put into me"', and a programme for doing that very thing is duly produced, it is apt to have an artificial and ad hoc air, and appear to be more of a trick than a serious answer to the question. It is like those passages in the Bible, which worried me as a small boy, that say that such and such was done 'that the prophecy might be fulfilled which says' so and so. This always seemed to me a most unfair way of making sure that the prophecy came true. If I answer your question, Jefferson, by making a routine which simply caused the machine to say just the words 'Newman and Turing, I don't like your programme', you would certainly feel this was a rather childish trick, and not the answer to what you really wanted to know. But yet it's hard to pin down what you want.
_Jefferson_ : I want the machine to reject the problem because it offends it in some way. That leads me to enquire what the ingredients are of ideas that we reject because we instinctively don't care for them. I don't know why I like some pictures and some music and am bored by other sorts. But I'm not going to carry that line on because we are all different, our dislikes are based on our personal histories and probably too on small differences of construction in all of us, I mean by heredity. Your machines have no genes, no pedigrees. Mendelian inheritance means nothing to wireless valves. But I don't want to score debating points! We ought to make it clear that not even Turing thinks that all that he has to do is to put a skin on the machine and that it is alive! We've been trying for a more limited objective whether the sort of thing that machines do can be considered as thinking. But is not your machine more certain than any human being of getting its problem right at once, and infallibly?
_Newman_ : Oh!
_Turing_ : Computing machines aren't really infallible at all. Making up checks on their accuracy is quite an important part of the art of using them. Besides making mistakes they sometimes haven't done quite the calculation one had expected, and one gets something that might be called a 'misunderstanding'.
_Jefferson_ : At any rate, they are not influenced by the emotions. You have only to upset a person enough and he becomes confused, he can't think of the answers and may make a fool of himself. It is high emotional content of mental processes in the human being that makes him quite different from a machine. It seems to me to come from the great complexity of his nervous system with its 1010 cells and also from his endocrine system which imports all sorts of emotions and instincts, such as those to do with sex. Man is essentially a chemical machine, he is much affected by hunger and fatigue, by being 'out of sorts' as we say, also by innate judgements, and by sexual urges. This chemical side is tremendously important, not the least so because the brain does exercise a remote control over the most important chemical processes that go on in our bodies. Your machines don't have to bother with that, with being tired or cold or happy or satisfied. They show no delight at having done something never done before. No, they are 'mentally' simple things. I mean that however complicated their structure is (and I know it _is_ very complicated), compared with man they are very simple and perform their tasks with an absence of distracting thoughts which is quite _inhuman_.
_Braithwaite_ : I'm not sure that I agree. I believe that it will be necessary to provide the machine with something corresponding to appetites, or other 'springs of action', in order that it will pay enough attention to relevant features in its environment to be able to learn from experience. Many psychologists have held that the emotions in men are by-products of their appetites and that they serve a biological function in calling higher levels of mental activity into play when the lower levels are incapable of coping with an external situation. For example, one does not feel afraid when there is no danger, or a danger which can be avoided more or less automatically: fear is a symptom showing that the danger has to be met by conscious thought. Perhaps it will be impossible to build a machine capable of learning in general from experience without incorporating in it an emotional apparatus, the function of which will be to switch over to a different part of the machine when the external environment differs too much from what would satisfy the machine's appetites by more than a certain amount. I don't want to suggest that it will be necessary for the machine to be able to throw a fit of tantrums. But in humans tantrums frequently ful fil a definite function—that of escaping from responsibility; and to protect a machine against a too hostile environment it may be essential to allow it, as it were, to go to bed with a neurosis, or psychogenic illness—just as, in a simpler way, it is provided with a fuse to blow, if the electric power working it threatens its continued existence.
_Turing_ : Well, I don't envisage teaching the machine to throw temperamental scenes. I think some such effects are likely to occur as a sort of by-product of genuine teaching, and that one will be more interested in curbing such displays than in encouraging them. Such effects would probably be distinctly different from the corresponding human ones, but recognisable as variations on them. This means that if the machine was being put through one of my imitation tests, it would have to do quite a bit of acting, but if one was comparing it with a man in a less strict sort of way the resemblance might be quite impressive.
_Newman_ : I still feel that too much of our argument is about what hypothetical future machines will do. It is all very well to say that a machine could easily be made to do this or that, but, to take only one practical point, what about the time it would take to do it? It would only take an hour or two to make up a routine to make our Manchester machine analyse all possible variations of the game of chess right out, and find the best move that way— _if_ you didn't mind its taking thousands of millions of years to run through the routine. Solving a problem on the machine doesn't mean finding a way to do it between now and eternity, but within a reasonable time. This is not just a technical detail that will be taken care of by future improvements. It's most unlikely that the engineers can ever give us a factor of more than a thousand or two times our present speeds. To assume that runs that would take thousands of millions of years on our present machines will be done in a flash on machines of the future, is to move into the realms of science fiction.
_Turing_ : To my mind this time factor is the one question which will involve all the real technical difficulty. If one didn't know already that these things can be done by brains within a reasonable time one might think it hopeless to try with a machine. The fact that a brain _can_ do it seems to suggest that the difficulties may not really be so bad as they now seem.
_Braithwaite_ : I agree that we ought not to extend our discussion to cover whether calculating machines could be made which would do everything that a man can do. The point is, surely, whether they can do all that it is proper to call thinking. Appreciation of a picture contains elements of thinking, but it also contains elements of feeling; and we're not concerned with whether a machine can be made that will feel. Similarly with moral questions: we're only concerned with them so far as they are also intellectual ones. We haven't got to give the machine a sense of duty or anything corresponding to a will: still less need it be given temptations which it would then have to have an apparatus for resisting. All that it has got to do in order to think is to be able to solve, or to make a good attempt at solving, all the intellectual problems with which it might be confronted by the environment in which it finds itself. This environment, of course, must include Turing asking it awkward questions as well as natural events such as being rained upon, or being shaken up by an earthquake.
_Newman_ : But I thought it was you who said that a machine wouldn't be able to learn to adjust to its environment if it hadn't been provided with a set of appetites and all that went with them?
_Braithwaite_ : Yes, certainly. But the problems raised by a machine having appetites are not properly our concern today. It may be the case that it wouldn't be able to learn from experience without them; but we're only required to consider whether it would be able to learn at all—since I agree that being able to learn is an essential part of thinking. So oughtn't we to get back to something centred on thinking? Can a machine make up new concepts, for example?
_Newman_ : There are really two questions that can be asked about machines and thinking, first, what do we require before we agree that the machine does _everything_ that we call thinking? This is really what we have been talking about for most of the time; but there is also another interesting and important question: Where does the doubtful territory begin? What is the _nearest_ thing to straight computing that the present machines perhaps can't do?
_Braithwaite_ : And what would your own answer be?
_Newman_ : I think perhaps to solve mathematical problems for which no method is known, in the way that men do; to find new methods. This is a much more modest aim than inventing new mathematical concepts. What happens when you try to solve a new problem in the ordinary way is that you think about it for a few seconds, or a few years, trying out all the analogies you can think of with problems that have been solved, and then you have an idea. You try it out in detail. If it is no good you must wait for another idea. This is a little like the chess-problem routine, where one move after another is tried, but with one very important difference, that if I am even a moderately good mathematician the ideas that I get are not just random ones, but are pre-selected so that there is an appreciable chance that after a few trials one of them will be successful. Henry Moore says about the studies he does for his sculpture, 'When the work is more than an exercise, inexplicable jumps occur. This is where the imagination comes in.' If a machine could really be got to imitate this sudden pounce on an idea, I believe that everyone would agree that it had begun to think, even though it didn't have appetites or worry about the income tax. And suppose that we also stuck to what we know about the physiology of human thinking, how much would that amount to, Jefferson?
_Jefferson_ : We know a great deal about the end-product, thinking itself. Are not the contents of our libraries and museums the total up to date? Experimental psychology has taught us a lot about the way that we use memory and association of ideas, how we fill in gaps in knowledge and improvise from a few given facts. But exactly how we do it in terms of nerve cell actions we don't know. We are particularly ignorant of the very point that you mentioned just now, Newman, the actual physiology of the pounce on an idea, of the sudden inspiration. Thinking is clearly a motor activity of the brain's cells, a suggestion supported by the common experience that so many people think better with a pen in their hand than viva voce or by reverie and reflection. But you can't so far produce ideas in a man's mind by stimulating his exposed brain here or there electrically. It would have been really exciting if one could have done that—if one could have perhaps excited original thoughts by local stimulation. It can't be done. Nor does the electro-encephalograph show us how the process of thinking is carried out. It can't tell you what a man is thinking about. We can trace the course, say, of a page of print or of a stream of words into the brain, but we eventually lose them. If we could follow them to their storage places we still couldn't see how they are reassembled later as ideas. You have the great advantage of knowing how your machine was made. We only know that we have in the human nervous system a concern compact in size and in its way perfect for its job. We know a great deal about its microscopical structure and its connections. If fact, we know everything except how these myriads of cells allow us to think. But, Newman, before we say 'not only does this machine think but also here in this machine we have an exact counterpart of the wiring and circuits of human nervous systems,' I ought to ask whether machines have been built or could be built which are as it were anatomically different, and yet produce the same work.
_Newman_ : The logical plan of all of them is rather similar, but certainly their anatomy, and I suppose you could say their physiology, varies a lot.
_Jefferson_ : Yes, that's what I imagined—we cannot then assume that any one of these electronic machines is a replica of part of a man's brain even though the result of its actions has to be conceded as thought. The real value of the machine to you is its end results, its performance, rather than that its plan reveals to us a model of our brains and nerves. Its usefulness lies in the fact that electricity travels along wires 2 or 3 million times faster than nerve impulses pass along nerves. You can set it to do things that man would need thousands of lives to complete. But that old slow coach, man, is the one with the ideas—or so I think. It would be fun some day, Turing, to listen to a discussion, say on the Fourth Programme, between two machines on why human beings think that they think!
## **Artificial Life**
_Jack Copeland_
#### 1. What is 'Artificial Life'?
The highly interdisciplinary field of Artificial Life was so named by Christopher Langton, a physicist working at the Los Alamos National Laboratory. In 1987 Langton organized at Los Alamos what he described as an 'interdisciplinary workshop on the synthesis and simulation of living systems'. The workshop was a rallying point, bringing together researchers with shared interests and diverse backgrounds.
Artificial Life ('A-Life') aims to achieve theoretical understanding of naturally occurring biological life, in particular of the most conspicuous feature of living matter, its ability to _self-organize_ , i.e. to develop form and structure spontaneously. Langton defines Artificial Life as 'the study of man-made systems that exhibit behaviors characteristic of natural living systems'. A-Life, he says, 'complements the traditional biological sciences concerned with the _analysis_ of living organisms by attempting to _synthesise_ life-like behaviors within computers and other artificial media'.
The use of computers to simulate living and life-like systems is central to A-Life. Langton says:
Computers should be thought of as an important laboratory tool for the study of life, substituting for the array of incubators, culture dishes, microscopes, electrophoretic gels, pipettes, centrifuges, and other assorted wet-lab paraphernalia, one simple-to-master piece of experimental equipment.
Langton even suggests that the 'ultimate goal of the study of artificial life would be to create "life" in some other medium, ideally a _virtual_ medium where the essence of life has been abstracted from the details of its implementation in _any_ particular hardware'.
#### 2. Morphogenesis
Turing was the earliest pioneer of computer-based A-Life: he was the first to use computer simulation to investigate a theory of the development of organization and pattern in living things. Early in 1951, the world's first commercially manufactured general-purpose electronic digital computer, the Manchester-built Ferranti Mark I, was installed in the Computing Machine Laboratory at Manchester University. Turing immediately set about using the machine to model biological growth. In February 1951 he wrote to a colleague at the National Physical Laboratory:
Our new machine [the Ferranti Mark I] is to start arriving on Monday. I am hoping as one of the first jobs to do something about 'chemical embryology'. In particular I think one can account for the appearance of Fibonacci numbers in connection with fir-cones.
In a Fibonacci number-series, each number, except for the first two, is the sum of the two previous numbers: for example 1, 3, 4, 7, 11...
In brief, Turing's 'chemical embryology' is the hypothesis that the development of anatomical structure in the animal or plant embryo is a result of the fact that diffusing chemicals reacting with one another can form spatial patterns. This is a thoroughly reductionist view: forms found in living matter are accounted for by the fact that, under appropriate conditions, sheer chemical reaction produces pattern and form.
Turing took his cue from the zoologist D'Arcy Thompson (to whose work Turing refers at the end of Chapter 15). D'Arcy Thompson held that the forms of living things, no less than naturally occurring forms in inorganic matter, are to be explained in terms of 'the operation of physical forces or mathematical laws'.
Cell and tissue, shell and bone, leaf and flower, are so many portions of matter, and it is in obedience to the laws of physics that their particles have been moved, moulded and conformed.... The form... of any portion of matter, whether it be living or dead, and the changes of form which are apparent in its... growth, may in all cases alike be described as due to the action of force.
Concerning the puzzling fact that the scales of a fir-cone, or the florets of a sunflower, are grouped together in numbers that form a Fibonacci series, D'Arcy Thompson boldly declared:
while the Fibonacci series stares us in the face in the fir-cone, it does so for mathematical reasons; and its supposed usefulness, and the hypothesis of its introduction into plant-structure through natural selection, are matters which deserve no place in the plain study of botanical phenomena.
Turing summarized his own theory as the suggestion that 'certain well-known physical laws are sufficient to account for many of the facts' of morphogenesis (p. 519). ('Morphogenesis' means 'generation of form'.) Turing described an idealized chemical mechanism, now called the _reaction-diffusion model_. He showed that this mechanism could lead to a number of simple but life-like patterns and forms. The reaction-diffusion model is the topic of Chapter 15, 'The Chemical Basis of Morphogenesis'. 'The Chemical Basis of Morphogenesis' has been widely cited in the biological literature and today reaction-diffusion remains a possible, although still unconfirmed, explanation of aspects of the generation of biological pattern and form.
The geneticist C. H. Waddington commented in a letter to Turing in 1952 that the most clear-cut application of Turing's theory appeared to be 'in the arising of spots, streaks, and flecks of various kinds in apparently uniform areas such as the wings of butterflies, the shells of molluscs, the skin of tigers, leopards, etc'. Modern computer simulations of Turing's reaction-diffusion mechanism have indeed produced leopard-like spots, cheetah-like spots, and giraffe-like stripes, as well as textures reminiscent of reptile skin, corals, and the surfaces of some fungi. Turing himself, however, envisaged diverse applications of his theory (as he indicated in the letter to biologist J. Z. Young quoted below), including leaf arrangements and the appearance of Fibonacci sequences, and phenomena such as gastrulation. (Gastrulation is a process of rearrangement of cells in the spherical embryo, involving the folding inwards of part of the surface, producing a depression akin to the dent formed by poking a balloon with a finger.) Furthermore, Turing described his research into morphogenesis as 'not altogether unconnected' to his work on neural networks (see Chapter 10).
Turing simulated the reaction-diffusion mechanism using the Ferranti computer. A mathematician using paper and pencil to analyse the behaviour of a reaction-diffusion system risks becoming overwhelmed by formidable mathematical complexity, and must make (what Turing calls) 'simplifying assumptions'. Turing explains that use of the computer enables such assumptions to be dispensed with to some extent. This freedom enables Turing to employ _nonlinear_ differential equations to describe the chemical interactions hypothesized by his theory. Non-linear differential equations are mathematically intractable. Turing used the computer to explore in detail particular cases of interactions governed by equations of this type. He may well have been the first researcher to engage in the computer-assisted exploration of non-linear systems. (It was not until Benoit Mandelbrot's discovery of the 'Mandelbrot set' in 1979 that the computer-assisted investigation of non-linear systems gained widespead attention.)
Turing's work on morphogenesis was in every respect ahead of its time. He died while in the midst of this groundbreaking work, leaving a large pile of handwritten notes and various programmes. This material is still not fully understood.
The computer programme shown in Figure 1, which is in Turing's own hand, formed part of his study of the development of the fir-cone. He also investigated the development of the sunflower. The photograph and diagram shown in Figure 2 are from his notes.
#### 3. The Reaction-Diffusion Model
In reaction-diffusion, two or more chemicals diffuse through the embryo reacting with each other. Turing showed that under certain conditions a stable pattern of chemical concentrations will be reached. For example, in the case of an artificially simple embryo consisting of nothing but a ring of twenty cells, reaction-diffusion will produce a stable, regular pattern of concentration and rarefaction around the circumference of the ring. The points of highest concentration occur equidistantly from one another. Turing picturesquely describes this pattern as a stationary 'chemical wave'. His suggestion is that at the points of high concentration around the ring, the chemical acts as a trigger to stimulate growth. For example, a ring of cells might sprout leaves at these points, or tentacles, producing a structure reminiscent of _Hydra_ ('something like a sea-anenome but liv[ing] in fresh water and hav[ing] from about five to ten tentacles' (p. 556)). Regular but non-stationary patterns of concentration—travelling waves—are also possible. Turing suggested that the movements of the tail of a spermatozoon may provide an example of these travelling waves.
**Figure 1.**
It might be wondered how genes fit into this picture. Given our current knowledge of the role played by genes in the determination of anatomical structure, is Turing's theory archaic? Not at all. On Turing's account, reaction-diffusion is the _mechanism by which_ genes determine the anatomical structure of the resulting organism. The function of genes, he suggested, is catalytic. The genes are presumed to catalyse the production of appropriate chemicals, so setting reaction-diffusion in train.
Turing calls the chemicals that diffuse and react 'morphogens', suggesting hormones as an example. However, his aim is not to give a taxonomy of morphogens or to describe specific morphogens, but rather to demonstrate in the abstract that, given certain realistic assumptions about _unspecified_ morphogens, reaction-diffusion will produce pattern. These assumptions include the rates at which the morphogens diffuse between the cells, the rates at which the various chemical reactions between the morphogens take place, and the ways in which these rates change (due, for example, to increases in temperature in the tissue, or increases in concentration of morphogens that act as catalysts). The initial concentrations of the morphogens must be specified, and the number, dimensions, and positions of the cells making up the mass of tissue through which the morphogens diffuse. This mass—the embryo—is assumed to be initially homogeneous. (Other features of the situation, such as the motions and elasticities of the cells, are ignored in order to simplify the model.)
Turing represents the reactions between morphogens purely schematically, making no assumptions concerning the actual chemical compositions of the substances involved. For example, two reactions might be specified like this: morphogens _X_ and _Y_ react to produce _Z_ ; _Z_ and morphogen _A_ react to produce 2 _Y_. The first reaction therefore depletes the supply of morphogen _Y_ , while the second tends to build up the concentration of _Y_.
**Figure 2.** Turing's numbering of the individual florets of a sunflower.
Uniform diffusion and uniform reaction in a uniform mass of tissue can produce only uniformity, not differentiation and pattern. Some sort of symmetry-breaker must be thrown into the mix. As Turing put it (p. 525):
There appears superficially to be a difficulty confronting this theory of morphogenesis, or, indeed, almost any other theory of it. An embryo in its spherical blastula stage has spherical symmetry, or if there are any deviations from perfect symmetry, they cannot be regarded as of any particular importance, for the deviations vary greatly from embryo to embryo within a species, though the organisms developed from them are barely distinguishable. One may take it therefore that there is perfect spherical symmetry. But a system which has spherical symmetry, and whose state is changing because of chemical reactions and diffusion, will remain spherically symmetrical forever.... It certainly cannot result in an organism such as a horse, which is not spherically symmetrical.
Turing suggested various possible symmetry-breakers, including small disturbances caused by the presence of anatomical structures neighbouring the embryo, and purely statistical fluctuations at the molecular level. An example of the latter is statistical fluctuation in the number of molecules of a given morphogen passing through the wall of a cell, so producing small fluctuations in the concentration of that morphogen within the cell. Turing's point is that in certain circumstances, small fluctuations such as this can bring about large effects, just as a small nudge that under normal circumstances would have no effect on a person's balance could be enough to topple someone balancing on one foot.
Turing demonstrated that under appropriate conditions small departures from uniformity can indeed lead to the formation of chemical waves.
#### 4. Genetic Algorithms
An important concept both in Artificial Life and in Artificial Intelligence is that of a _genetic algorithm_ (GA). GAs employ methods analogous to the processes of natural evolution in order to produce successive generations of software entities that are increasingly fit for their intended purpose. Turing anticipated the concept of a genetic algorithm in a brief passage of his 'Intelligent Machinery', where he described what he called a 'genetical or evolutionary search' (Chapter 10, p. 431; see also Chapter 11, p. 463, and the introduction to Chapter 16, p. 565). The actual term 'genetic algorithm' was introduced circa 1975 by John Holland and his research group at the University of Michigan. Holland's work is responsible for the current intense interest in GAs. (Holland, a student of Arthur Burks, was influenced by von Neumann's ideas—see below.)
Turing described an early example of a GA in connection with his chess-player in Chapter 16.
One of the first GAs to be implemented (in the 1950s) formed part of the learning mechanism of Samuel's checkers or draughts programme mentioned above in 'Artificial Intelligence'. Samuel's programme used heuristics to rank moves and board positions (the programme 'looked ahead' as many as ten turns of play). To speed up learning, Samuel would set up two copies of the programme, Alpha and Beta, on the same computer and leave them to play game after game with each other. The learning procedure consisted in the computer making small numerical changes to Alpha's ranking procedure, leaving Beta's unchanged, and then comparing Alpha's and Beta's performance over a few games. If Alpha played worse than Beta, these changes to the ranking procedure were discarded, but if Alpha played better than Beta then Beta's ranking procedure was replaced with Alpha's. As in biological evolution, the fitter survived. Over many such cycles of mutation and selection, the programme's quality of play increased markedly.
The use of GAs is burgeoning, in AI and elsewhere. In one application a GA-based system and a witness to a crime cooperate to generate on-screen faces that become closer and closer to the recollected face of the criminal. In A-Life, researchers study GAs as a means of studying the process of evolution itself.
#### 5. John von Neumann and A-Life
John von Neumann was another important early pioneer of Artificial Life. In his 1948 Hixon Symposium, entitled 'The General and Logical Theory of Automata', he said:
Natural organisms are, as a rule, much more complicated and subtle, and therefore much less well understood in detail, than are artificial automata. Nevertheless... a good deal of our experiences and difficulties with our artificial automata can be to some extent projected on our interpretations of natural organisms.
Arthur Burks (who edited and completed von Neumann's posthumously published volume _Theory of Self-Reproducing Automata_ , listed in the section of Further Reading) wrote this concerning von Neumann's research on the problem of self-reproduction:
Von Neumann had the familiar natural phenomenon of self-reproduction in mind... but he was not trying to simulate the self-reproduction of a natural system at the levels of genetics and biochemistry. He wished to abstract from the natural self-reproduction problem its logical form.
This passage is quoted approvingly by Langton, who italicizes the final sentence and comments:
This approach is the first to capture the essence of Artificial Life. To understand the field of Artificial Life, one need only replace references to 'self-reproduction' in the above with references to any other biological phenomenon.
Von Neumann was thinking about issues relevant to A-Life at least as early as 1946. In a letter to the cyberneticist Norbert Wiener (dated 29 November 1946), von Neumann wrote: 'I did think a good deal about self-reproductive mechanisms. I can formulate the problem rigorously, in about the style in which Turing did it for his mechanisms.' There is no doubt that von Neumann's theorizing about self-reproduction was strongly influenced by Turing's discovery of the universal computing machine (or 'universal automaton', as von Neumann called it). Von Neumann's colleague Herman Goldstine wrote:
von Neumann had a profound concern for automata. In particular, he always had a deep interest in Turing's work.... Turing proved a most remarkable and unexpected result.... In essence what he showed is that any particular automaton can be described by a finite set of instructions, and that when this is fed to his universal automaton it in turn imitates the special one.... Von Neumann was enormously intrigued with these ideas, and he started in 1947 working on... how complex a device or construct needed to be in order to be self-reproductive.
In his Hixon Symposium, von Neumann said:
For the question which concerns me here, that of 'self-reproduction' of automata, Turing's procedure is too narrow in one respect only. His automata are purely computing machines. Their output is a piece of tape with zeros and ones on it. What is needed... is an automaton whose output is other automata. There is, however, no difficulty in principle in dealing with this broader concept and in deriving from it the equivalent of Turing's result.... The problem of self-reproduction can... be stated like this: Can one build an aggregate out of... elements in such a manner that if it is put into a reservoir, in which there float all these elements in large numbers, it will then begin to construct other aggregates, each of which will at the end turn out to be another automaton exactly like the original one? This is feasible, and the principle on which it can be based is closely related to Turing's [universal automaton] outlined earlier.
In lectures von Neumann described a 'universal constructor' (remarking, 'You see, I'm coming quite close to Turing's trick with universal automata'). Just as complete descriptions of Turing machines can be fed into the universal automaton in the form of programmes, complete descriptions of automata can be inserted into the universal constructor. The universal constructor floats in a medium—or sea—in which also float, in practically unlimited supply, the components from which the universal constructor is made. Given a complete description of an automaton, the universal constructor will assemble that automaton.
If what is inserted into the universal constructor is a description _I_ of the universal constructor itself, the constructor assembles a duplicate of itself—self-reproduction. Since the duplicate is to be an exact copy, it must contain the description _I_. This is done by a copying mechanism in the universal constructor and the copy of _I_ is inserted into the offspring. Thus the offspring, too, is capable of self-reproduction. Von Neumann likened the description _I_ to the gene, saying that the copying mechanism 'performs the fundamental act of reproduction, the duplication of the genetic material, which is clearly the fundamental operation in the multiplication of living cells'. Allowing the copying mechanism to make occasional random errors affords the possibility of random mutation in the genes of the offspring, thus opening the door to Darwinian evolution.
In his letter to Wiener (29 November 1946), von Neumann voiced some concerns that might equally well be raised concerning the focus of modern research in A-Life and Artificial Intelligence. Von Neumann pointed out that when automata theorists choose the human nervous system as their model, they are unrealistically selecting 'the most complicated object under the sun— literally'. Moreover, he said, there is little advantage in choosing instead simpler organisms with fewer neurons, for example, the ant: _any_ nervous system exhibits 'exceptional complexity'. Von Neumann suggested that automata theorists 'turn to simpler systems', and he recommended attention to 'organisms of the virus or bacteriophage type'. These, he pointed out, are 'self-reproductive and... are able to orient themselves in an unorganized milieu, to move towards food, to appropriate it and to use it'. He estimated that a typical bacteriophage might consist of 6 million atoms grouped into a few hundred thousand 'mechanical elements', saying that this represents 'a degree of complexity which is not necessarily beyond human endurance'. By following this path, he said, the 'decisive break' might be achieved.
#### 6. Letter from Turing to Young
Shortly before the Ferranti computer arrived in 1951, Turing wrote about his work on morphogenesis in a letter to the biologist J. Z. Young. The letter connects Turing's work on morphogenesis with his interest in neural networks (Chapter 10), and moreover to some extent explains why he did not follow up his earlier suggestion (Chapter 10, p. 428) and use the Ferranti computer to simulate his 'unorganised machines'.
I am afraid I am very far from the stage where I feel inclined to start asking any anatomical questions [about the brain]. According to my notions of how to set about it that will not occur until quite a late stage when I have a fairly definite theory about how things are done.
At present I am not working on the problem at all, but on my mathematical theory of embryology... This is yielding to treatment, and it will so far as I can see, give satisfactory explanations of
i) Gastrulation.
ii) Polyogonally symmetrical structures, e.g., starfish, flowers.
iii) Leaf arrangement, in particular the way the Fibonacci series (0, 1, 1, 2, 3, 5, 8, 13,...) comes to be involved.
iv) Colour patterns on animals, e.g., stripes, spots and dappling.
v) Patterns on nearly spherical structures such as some Radiolaria, but this is more difficult and doubtful.
I am really doing this now because it is yielding more easily to treatment. I think it is not altogether unconnected with the other problem. The brain structure has to be one which can be achieved by the genetical embryological mechanism, and I hope that this theory that I am now working on may make clearer what restrictions this really implies. What you tell me about growth of neurons under stimulation is very interesting in this connection. It suggests means by which the neurons might be made to grow so as to form a particular circuit, rather than to reach a particular place.
##### _Further reading_
Boden, M. A., _Mind as Machine: A History of Cognitive Science_ (Oxford: Oxford University Press, 2005).
—— (ed.), _The Philosophy of Artificial Life_ (Oxford: Oxford University Press, 1996).
Holland, J. H., 'Genetic Algorithms', _Scientific American_ , 267 (July 1992), 44–50.
Swinton, J., 'Watching the Daisies Grow: Turing and Fibonacci Phyllotaxis', in C. Teuscher (ed.), _Alan Turing: Life and Legacy of a Great Thinker_ (Berlin: Springer-Verlag, 2004).
Turing, A. M., _Morphogenesis: Collected Works of A. M. Turing_ , ed. P. T. Saunders (Amsterdam: North-Holland, 1992).
Turk, G., 'Generating Textures on Arbitrary Surfaces Using Reaction-Diffusion', _Computer Graphics_ , 25 (1991), 289–98.
von Neumann, J., _Theory of Self-Reproducing Automata_ , ed. and completed by A. W. Burks (Urbana: University of Illinois Press, 1966).
### **CHAPTER 15
The Chemical Basis of Morphogenesis ( _1952_ )**
_Alan Turing_
It is suggested that a system of chemical substances, called morphogens, reacting together and diffusing through a tissue, is adequate to account for the main phenomena of morphogenesis. Such a system, although it may originally be quite homogenous, may later develop a pattern or structure due to an instability of the homogeneous equilibrium, which is triggered off by random disturbances. Such reaction-diffusion systems are considered in some detail in the case of an isolated ring of cells, a mathematically convenient, though biologically unusual system. The investigation is chiefly concerned with the onset of instability. It is found that there are six essentially different forms which this may take. In the most interesting form stationary waves appear on the ring. It is suggested that this might account, for instance, for the tentacle patterns on _Hydra_ and for whorled leaves. A system of reactions and diffusion on a sphere is also considered. Such a system appears to account for gastrulation. Another reaction system in two dimensions gives rise to patterns reminiscent of dappling. It is also suggested that stationary waves in two dimensions could account for the phenomena of phyllotaxis.
The purpose of this paper is to discuss a possible mechanism by which the genes of a zygote may determine the anatomical structure of the resulting organism. The theory does not make any new hypotheses; it merely suggests that certain well-known physical laws are sufficient to account for many of the facts. The full understanding of the paper requires a good knowledge of mathematics, some biology, and some elementary chemistry. Since readers cannot be expected to be experts in all of these subjects, a number of elementary facts are explained, which can be found in text-books, but whose omission would make the paper difficult reading.
#### 1. A model of the embryo. Morphogens
In this section a mathematical model of the growing embryo will be described. This model will be a simplification and an idealization, and consequently a falsification. It is to be hoped that the features retained for discussion are those of greatest importance in the present state of knowledge.
The model takes two slightly different forms. In one of them the cell theory is recognized but the cells are idealized into geometrical points. In the other the matter of the organism is imagined as continuously distributed. The cells are not, however, completely ignored, for various physical and physico-chemical characteristics of the matter as a whole are assumed to have values appropriate to the cellular matter.
With either of the models one proceeds as with a physical theory and defines an entity called 'the state of the system'. One then describes how that state is to be determined from the state at a moment very shortly before. With either model the description of the state consists of two parts, the mechanical and the chemical. The mechanical part of the state describes the positions, masses, velocities and elastic properties of the cells, and the forces between them. In the continuous form of the theory essentially the same information is given in the form of the stress, velocity, density and elasticity of the matter. The chemical part of the state is given (in the cell form of theory) as the chemical composition of each separate cell; the diffusibility of each substance between each two adjacent cells must also be given. In the continuous form of the theory the concentrations and diffusibilities of each substance have to be given at each point. In determining the changes of state one should take into account
(i) The changes of position and velocity as given by Newton's laws of motion.
(ii) The stresses as given by the elasticities and motions, also taking into account the osmotic pressures as given from the chemical data.
(iii) The chemical reactions.
(iv) The diffusion of the chemical substances. The region in which this diffusion is possible is given from the mechanical data.
This account of the problem omits many features, e.g. electrical properties and the internal structure of the cell. But even so it is a problem of formidable mathematical complexity. One cannot at present hope to make any progress with the understanding of such systems except in very simplified cases. The interdependence of the chemical and mechanical data adds enormously to the difficulty, and attention will therefore be confined, so far as is possible, to cases where these can be separated. The mathematics of elastic solids is a well-developed subject, and has often been applied to biological systems. In this paper it is proposed to give attention rather to cases where the mechanical aspect can be ignored and the chemical aspect is the most significant. These cases promise greater interest, for the characteristic action of the genes themselves is presumably chemical. The systems actually to be considered consist therefore of masses of tissues which are not growing, but within which certain substances are reacting chemically, and through which they are diffusing. These substances will be called morphogens, the word being intended to convey the idea of a form producer. It is not intended to have any very exact meaning, but is simply the kind of substance concerned in this theory. The evocators of Waddington provide a good example of morphogens (Waddington 1940). These evocators diffusing into a tissue somehow persuade it to develop along different lines from those which would have been followed in its absence. The genes themselves may also be considered to be morphogens. But they certainly form rather a special class. They are quite indiffusible. Moreover, it is only by courtesy that genes can be regarded as separate molecules. It would be more accurate (at any rate at mitosis) to regard them as radicals of the giant molecules known as chromosomes. But presumably these radicals act almost independently, so that it is unlikely that serious errors will arise through regarding the genes as molecules. Hormones may also be regarded as quite typical morphogens. Skin pigments may be regarded as morphogens if desired. But those whose action is to be considered here do not come squarely within any of these categories.
The function of genes is presumed to be purely catalytic. They catalyze the production of other morphogens, which in turn may only be catalysts. Eventually, presumably, the chain leads to some morphogens whose duties are not purely catalytic. For instance, a substance might break down into a number of smaller molecules, thereby increasing the osmotic pressure in a cell and promoting its growth. The genes might thus be said to influence the anatomical form of the organism by determining the rates of those reactions which they catalyze. If the rates are assumed to be those determined by the genes, and if a comparison of organisms is not in question, the genes themselves may be eliminated from the discussion. Likewise any other catalysts obtained secondarily through the agency of the genes may equally be ignored, if there is no question of their concentrations varying. There may, however, be some other morphogens, of the nature of evocators, which cannot be altogether forgotten, but whose role may nevertheless be subsidiary, from the point of view of the formation of a particular organ. Suppose, for instance, that a 'leg-evocator' morphogen were being produced in a certain region of an embryo, or perhaps diffusing into it, and that an attempt was being made to explain the mechanism by which the leg was formed in the presence of the evocator. It would then be reasonable to take the distribution of the evocator in space and time as given in advance and to consider the chemical reactions set in train by it. That at any rate is the procedure adopted in the few examples considered here.
#### 2. Mathematical background required
The greater part of this present paper requires only a very moderate knowledge of mathematics. What is chiefly required is an understanding of the solution of linear differential equations with constant coefficients. (This is also what is chiefly required for an understanding of mechanical and electrical oscillations.) The solution of such an equation takes the form of a sum Σ _A_ e _bt_ , where the quantities _A, b_ may be complex, i.e. of the form α + iβ, where α and β are ordinary (real) numbers and . It is of great importance that the physical significance of the various possible solutions of this kind should be appreciated, for instance, that
( _a_ ) Since the solutions will normally be real one can also write them in the form or ( means 'real part of').
( _b_ ) That if _A_ = _A_ ′ei _ϕ_ and _b_ = α + iβ, where _A_ ′, α, β, _ϕ_ are real, then
Thus each such term represents a sinusoidal oscillation if α = 0, a damped oscillation if α < 0, and an oscillation of ever-increasing amplitude if α > 0.
( _c_ ) If any one of the numbers _b_ has a positive real part the system in question is unstable.
( _d_ ) After a sufficiently great lapse of time all the terms _A_ e _bt_ will be negligible in comparison with those for which _b_ has the greatest real part, but unless this greatest real part is itself zero these dominant terms will eventually either tend to zero or to infinite values.
( _e_ ) That the indefinite growth mentioned in ( _b_ ) and ( _d_ ) will in any physical or biological situation eventually be arrested due to a breakdown of the assumptions under which the solution was valid. Thus, for example, the growth of a colony of bacteria will normally be taken to satisfy the equation d _y_ /d _t_ = α _y_ (α > 0), _y_ being the number of organisms at time _t_ , and this has the solution _y_ = _A_ eα _t_. When, however, the factor e _αt_ has reached some billions the food supply can no longer be regarded as unlimited and the equation d _y_ /d _t_ = α _y_ will no longer apply.
The following relatively elementary result will be needed, but may not be known to all readers:
but
The first case can easily be proved when it is noticed that the left-hand side is a geometric progression. In the second case all the terms are equal to 1.
The relative degrees of difficulty of the various sections are believed to be as follows. Those who are unable to follow the points made in this section should only attempt §§3, 4, 11, 12, 14 and part of §13. Those who can just understand this section should profit also from §§7, 8, 9. The remainder, §§5, 10, 13, will probably only be understood by those definitely trained as mathematicians.
#### 3. Chemical reactions
It has been explained in a preceding section that the system to be considered consists of a number of chemical substances (morphogens) diffusing through a mass of tissue of given geometrical form and reacting together within it. What laws are to control the development of this situation? They are quite simple. The diffusion follows the ordinary laws of diffusion, i.e. each morphogen moves from regions of greater to regions of less concentration, at a rate proportional to the gradient of the concentration, and also proportional to the 'diffusibility' of the substance. This is very like the conduction of heat, diffusibility taking the place of conductivity. If it were not for the walls of the cells the diffusibilities would be inversely proportional to the square roots of the molecular weights. The pores of the cell walls put a further handicap on the movement of the larger molecules in addition to that imposed by their inertia, and most of them are not able to pass through the walls at all.
The reaction rates will be assumed to obey the 'law of mass action'. This states that the rate at which a reaction takes place is proportional to the concentrations of the reacting substances. Thus, for instance, the rate at which silver chloride will be formed and precipitated from a solution of silver nitrate and sodium chloride by the reaction
Ag+ \+ Cl− → AgCl
will be proportional to the product of the concentrations of the silver ion Ag+ and the chloride ion Cl−. It should be noticed that the equation
AgNO3 \+ NaCl → AgCl + NaNO3
is not used because it does not correspond to an actual reaction but to the final outcome of a number of reactions. The law of mass action must only be applied to the _actual_ reactions. Very often certain substances appear in the individual reactions of a group, but not in the final outcome. For instance, a reaction _A_ → _B_ may really take the form of two steps _A_ \+ _G_ → _C_ and _C_ → _B_ \+ _G_. In such a case the substance _G_ is described as a catalyst, and as catalyzing the reaction _A_ → _B_. (Catalysis according to this plan has been considered in detail by Michaelis & Menten (1913).) The effect of the genes is presumably achieved almost entirely by catalysis. They are certainly not permanently used up in the reactions.
Sometimes one can regard the effect of a catalyst as merely altering a reaction rate. Consider, for example, the case mentioned above, but suppose also that _A_ can become detached from _G_ , i.e. that the reaction _C_ → _A_ \+ _G_ is taken into account. Also suppose that the reactions _A_ \+ _G_ _C_ both proceed much faster than _C_ → _B_ \+ _G_. Then the concentrations of _A, G, C_ will be related by the condition that there is equilibrium between the reactions _A_ \+ _G_ → _C_ and _C_ → _A_ \+ _G_ , so that (denoting concentrations by square brackets) [ _A_ ][ _G_ ] = _k_ [ _C_ ] for some constant _k_. The reaction _C_ → _B_ \+ _G_ will of course proceed at a rate proportional to [ _C_ ], i.e. to [ _A_ ][ _G_ ]. If the amount of _C_ is always small compared with the amount of _G_ one can say that the presence of the catalyst and its amount merely alter the mass action constant for the reaction _A_ → _B_ , for the whole proceeds at a rate proportional to [ _A_ ]. This situation does not, however, hold invariably. It may well happen that nearly all of _G_ takes the combined form _C_ so long as any of _A_ is left. In this case the reaction proceeds at a rate independent of the concentration of _A_ until _A_ is entirely consumed. In either of these cases the rate of the complete group of reactions depends only on the concentrations of the reagents, although usually not according to the law of mass action applied crudely to the chemical equation for the whole group. The same applies in any case where all reactions of the group with one exception proceed at speeds much greater than that of the exceptional one. In these cases the rate of the reaction is a function of the concentrations of the reagents. More generally again, no such approximation is applicable. One simply has to take all the actual reactions into account.
According to the cell model then, the number and positions of the cells are given in advance, and so are the rates at which the various morphogens diffuse between the cells. Suppose that there are _N_ cells and _M_ morphogens. The state of the whole system is then given by _MN_ numbers, the quantities of the _M_ morphogens in each of _N_ cells. These numbers change with time, partly because of the reactions, partly because of the diffusion. To determine the part of the rate of change of one of these numbers due to diffusion, at any one moment, one only needs to know the amounts of the same morphogen in the cell and its neighbours, and the diffusion coefficient for that morphogen. To find the rate of change due to chemical reaction one only needs to know the concentrations of all morphogens at that moment in the one cell concerned.
This description of the system in terms of the concentrations in the various cells is, of course, only an approximation. It would be justified if, for instance, the contents were perfectly stirred. Alternatively, it may often be justified on the understanding that the 'concentration in the cell' is the concentration at a certain representative point, although the idea of 'concentration at a point' clearly itself raises difficulties. The author believes that the approximation is a good one, whatever argument is used to justify it, and it is certainly a convenient one.
It would be possible to extend much of the theory to the case of organisms immersed in a fluid, considering the diffusion within the fluid as well as from cell to cell. Such problems are not, however, considered here.
#### 4. The breakdown of symmetry and homogeneity
There appears superficially to be a difficulty confronting this theory of morphogenesis, or, indeed, almost any other theory of it. An embryo in its spherical blastula stage has spherical symmetry, or if there are any deviations from perfect symmetry, they cannot be regarded as of any particular importance, for the deviations vary greatly from embryo to embryo within a species, though the organisms developed from them are barely distinguishable. One may take it therefore that there is perfect spherical symmetry. But a system which has spherical symmetry, and whose state is changing because of chemical reactions and diffusion, will remain spherically symmetrical for ever. (The same would hold true if the state were changing according to the laws of electricity and magnetism, or of quantum mechanics.) It certainly cannot result in an organism such as a horse, which is not spherically symmetrical.
There is a fallacy in this argument. It was assumed that the deviations from spherical symmetry in the blastula could be ignored because it makes no particular difference what form of asymmetry there is. It is, however, important that there are _some_ deviations, for the system may reach a state of instability in which these irregularities, or certain components of them, tend to grow. If this happens a new and stable equilibrium is usually reached, with the symmetry entirely gone. The variety of such new equilibria will normally not be so great as the variety of irregularities giving rise to them. In the case, for instance, of the gastrulating sphere, discussed at the end of this paper, the direction of the axis of the gastrula can vary, but nothing else.
The situation is very similar to that which arises in connexion with electrical oscillators. It is usually easy to understand how an oscillator keeps going when once it has started, but on a first acquaintance it is not obvious how the oscillation begins. The explanation is that there are random disturbances always present in the circuit. Any disturbance whose frequency is the natural frequency of the oscillator will tend to set it going. The ultimate fate of the system will be a state of oscillation at its appropriate frequency, and with an amplitude (and a wave form) which are also determined by the circuit. The phase of the oscillation alone is determined by the disturbance.
If chemical reactions and diffusion are the only forms of physical change which are taken into account the argument above can take a slightly different form. For if the system originally has no sort of geometrical symmetry but is a perfectly homogeneous and possibly irregularly shaped mass of tissue, it will continue indefinitely to be homogeneous. In practice, however, the presence of irregularities, including statistical fluctuations in the numbers of molecules undergoing the various reactions, will, if the system has an appropriate kind of instability, result in this homogeneity disappearing.
This breakdown of symmetry or homogeneity may be illustrated by the case of a pair of cells originally having the same, or very nearly the same, contents. The system is homogeneous: it is also symmetrical with respect to the operation of interchanging the cells. The contents of either cell will be supposed describable by giving the concentrations _X_ and _Y_ of two morphogens. The chemical reactions will be supposed such that, on balance, the first morphogen ( _X_ ) is produced at the rate 5 _X_ − 6 _Y_ \+ 1 and the second ( _Y_ ) at the rate 6 _X_ − 7 _Y_ \+ 1. When, however, the strict application of these formulae would involve the concentration of a morphogen in a cell becoming negative, it is understood that it is instead destroyed only at the rate at which it is reaching that cell by diffusion. The first morphogen will be supposed to diffuse at the rate 0 · 5 for unit difference of concentration between the cells, the second, for the same difference, at the rate 4 · 5. Now if both morphogens have unit concentration in both cells there is equilibrium. There is no resultant passage of either morphogen across the cell walls, since there is no concentration difference, and there is no resultant production (or destruction) of either morphogen in either cell since 5 _X_ − 6 _Y_ \+ 1 and 6 _X_ − 7 _Y_ \+ 1 both have the value zero for _X_ = 1, _Y_ = 1. But suppose the values are _X_ 1 = 1 · 06, _Y_ 1 = 1 · 02 for the first cell and _X_ 2 = 0 · 94, _Y_ 2 = 0 · 98 for the second. Then the two morphogens will be being produced by chemical action at the rates 0 · 18, 0 · 22 respectively in the first cell and destroyed at the same rates in the second. At the same time there is a flow due to diffusion from the first cell to the second at the rate 0 · 06 for the first morphogen and 0 · 18 for the second. In sum the effect is a flow from the second cell to the first at the rates 0 · 12, 0 · 04 for the two morphogens respectively. This flow tends to accentuate the already existing differences between the two cells. More generally, if at some moment the four concentrations continue afterwards to be expressible in this form, and increases at the rate . Thus there is an exponential drift away from the equilibrium condition. It will be appreciated that a drift away from the equilibrium occurs with almost any small displacement from the equilibrium condition, though not normally according to an exact exponential curve. A particular choice was made in the above argument in order to exhibit the drift with only very simple mathematics.
Before it can be said to follow that a two-cell system can be unstable, with inhomogeneity succeeding homogeneity, it is necessary to show that the reaction rate functions postulated really can occur. To specify actual substances, concentrations and temperatures giving rise to these functions would settle the matter finally, but would be difficult and somewhat out of the spirit of the present inquiry. Instead, it is proposed merely to mention imaginary reactions which give rise to the required functions by the law of mass action, if suitable reaction constants are assumed. It will be sufficient to describe
(i) A set of reactions producing the first morphogen at the constant rate 1, and a similar set forming the second morphogen at the same rate.
(ii) A set destroying the second morphogen ( _Y_ ) at the rate 7 _Y_.
(iii) A set converting the first morphogen ( _X_ ) into the second ( _Y_ ) at the rate 6 _X_.
(iv) A set producing the first morphogen ( _X_ ) at the rate 11 _X_.
(v) A set destroying the first morphogen ( _X_ ) at the rate 6 _Y_ , so long as any of it is present.
The conditions of (i) can be fulfilled by reactions of the type _A_ → _X_ , _B_ → _Y_ , where _A_ and _B_ are substances continually present in large and invariable concentrations. The conditions of (ii) are satisfied by a reaction of the form _Y_ → _D_ , _D_ being an inert substance, and (iii) by the reaction _X_ → _Y_ or _X_ → _Y_ \+ _E_. The remaining two sets are rather more difficult. To satisfy the conditions of (iv) one may suppose that _X_ is a catalyst for its own formation from _A_. The actual reactions could be the formation of an unstable compound _U_ by the reaction _A_ \+ _X_ → _U_ , and the subsequent almost instantaneous breakdown _U_ →2 _X_. To destroy _X_ at a rate proportional to _Y_ as required in (v) one may suppose that a catalyst _C_ is present in small but constant concentration and immediately combines with _X_ , _X_ \+ _C_ → _V_. The modified catalyst reacting with _Y_ , at a rate proportional to _Y_ , restores the catalyst but not the morphogen _X_ , by the reactions _V_ \+ _Y_ → _W_ , _W_ → _C_ \+ _H_ , of which the latter is assumed instantaneous.
It should be emphasized that the reactions here described are by no means those which are most likely to give rise to instability in nature. The choice of the reactions to be discussed was dictated entirely by the fact that it was desirable that the argument be easy to follow. More plausible reaction systems are described in §10.
Unstable equilibrium is not, of course, a condition which occurs very naturally. It usually requires some rather artificial interference, such as placing a marble on the top of a dome. Since systems tend to leave unstable equilibria they cannot often be in them. Such equilibria can, however, occur naturally through a stable equilibrium changing into an unstable one. For example, if a rod is hanging from a point a little above its centre of gravity it will be in stable equilibrium. If, however, a mouse climbs up the rod the equilibrium eventually becomes unstable and the rod starts to swing. A chemical analogue of this mouse-and-pendulum system would be that described above with the same diffusibilities but with the two morphogens produced at the rates
(3 + _I_ ) _X_ −6 _Y_ \+ _I_ −1 and 6 _X_ − (9 + _I_ ) _Y_ − _I_ \+ 1.
This system is stable if _I_ < 0 but unstable if _I_ > 0. If _I_ is allowed to increase, corresponding to the mouse running up the pendulum, it will eventually become positive and the equilibrium will collapse. The system which was originally discussed was the case _I_ = 2, and might be supposed to correspond to the mouse somehow reaching the top of the pendulum without disaster, perhaps by falling vertically on to it.
#### 5. Left-handed and right-handed organisms
The object of this section is to discuss a certain difficulty which might be thought to show that the morphogen theory of morphogenesis cannot be right. The difficulty is mainly concerned with organisms which have not got bilateral symmetry. The argument, although carried through here without the use of mathematical formulae, may be found difficult by non-mathematicians, and these are therefore recommended to ignore it unless they are already troubled by such a difficulty.
An organism is said to have 'bilateral symmetry' if it is identical with its own reflexion in some plane. This plane of course always has to pass through some part of the organism, in particular through its centre of gravity. For the purpose of this argument it is more general to consider what may be called 'left–right symmetry'. An organism has left-right symmetry if its description in any right-handed set of rectangular Cartesian co-ordinates is identical with its description in some set of left-handed axes. An example of a body with left–right symmetry, but not bilateral symmetry, is a cylinder with the letter _P_ printed on one end, and with the mirror image of a _P_ on the other end, but with the two upright strokes of the two letters not parallel. The distinction may possibly be without a difference so far as the biological world is concerned, but mathematically it should not be ignored.
If the organisms of a species are sufficiently alike, and the absence of left–right symmetry sufficiently pronounced, it is possible to describe each individual as either right-handed or left-handed without there being difficulty in classifying any particular specimen. In man, for instance, one could take the _X_ -axis in the forward direction, the _Y_ -axis at right angles to it in the direction towards the side on which the heart is felt, and the _Z_ -axis upwards. The specimen is classed as left-handed or right-handed according as the axes so chosen are left-handed or right-handed. A new classification has of course to be defined for each species.
The fact that there exist organisms which do not have left–right symmetry does not in itself cause any difficulty. It has already been explained how various kinds of symmetry can be lost in the development of the embryo, due to the particular disturbances (or 'noise') influencing the particular specimen not having that kind of symmetry, taken in conjunction with appropriate kinds of instability. The difficulty lies in the fact that there are species in which the proportions of left-handed and right-handed types are very unequal. It will be as well to describe first an argument which appears to show that this should not happen. The argument is very general, and might be applied to a very wide class of theories of morphogenesis.
An entity may be described as ' _P_ -symmetrical' if its description in terms of one set of right-handed axes is identical with its description in terms of any other set of right-handed axes with the same origin. Thus, for instance, the totality of positions that a corkscrew would take up when rotated in all possible ways about the origin has _P_ -symmetry. The entity will be said to be ' _F_ -symmetrical' when changes from right-handed axes to left-handed may also be made. This would apply if the corkscrew were replaced by a bilaterally symmetrical object such as a coal scuttle, or a left–right symmetrical object. In these terms one may say that there are species such that the totality of specimens from that species, together with the rotated specimens, is _P_ -symmetrical, but very far from _F_ -symmetrical. On the other hand, it is reasonable to suppose that
(i) The laws of physics are _F_ -symmetrical.
(ii) The initial totality of zygotes for the species is _F_ -symmetrical.
(iii) The statistical distribution of disturbances is _F_ -symmetrical. The individual disturbances of course will in general have neither _F_ -symmetry nor _P_ -symmetry.
It should be noticed that the ideas of _P_ -symmetry and _F_ -symmetry as defined above apply even to so elaborate an entity as 'the laws of physics'. It should also be understood that the laws are to be the laws taken into account in the theory in question rather than some ideal as yet undiscovered laws.
Now it follows from these assumptions that the statistical distribution of resulting organisms will have _F_ -symmetry, or more strictly that the distribution deduced as the result of working out such a theory will have such symmetry. The distribution of observed mature organisms, however, has no such symmetry. In the first place, for instance, men are more often found standing on their feet than their heads. This may be corrected by taking gravity into account in the laws, together with an appropriate change of definition of the two kinds of symmetry. But it will be more convenient if, for the sake of argument, it is imagined that some species has been reared in the absence of gravity, and that the resulting distribution of mature organisms is found to be _P_ -symmetrical but to yield more right-handed specimens than left-handed and so not to have _F_ -symmetry. It remains therefore to explain this absence of _F_ -symmetry.
Evidently one or other of the assumptions (i) to (iii) must be wrong, i.e. in a correct theory one of them would not apply. In the morphogen theory already described these three assumptions do all apply, and it must therefore be regarded as defective to some extent. The theory may be corrected by taking into account the fact that the morphogens do not always have an equal number of left-and right-handed molecules. According to one's point of view one may regard this as invalidating either (i), (ii) or even (iii). Simplest perhaps is to say that the totality of zygotes just is not _F_ -symmetrical, and that this could be seen if one looked at the molecules. This is, however, not very satisfactory from the point of view of this paper, as it would not be consistent with describing states in terms of concentrations only. It would be preferable if it was found possible to find more accurate laws concerning reactions and diffusion. For the purpose of accounting for unequal numbers of left- and right-handed organisms it is unnecessary to do more than show that there are corrections which would not be _F_ -symmetrical when there are laevo- or dextrorotatory morphogens, and which would be large enough to account for the effects observed. It is not very difficult to think of such effects. They do not have to be very large, but must, of course, be larger than the purely statistical effects, such as thermal noise or Brownian movement.
There may also be other reasons why the totality of zygotes is not _F_ -symmetrical, e.g. an asymmetry of the chromosomes themselves. If these also produce a sufficiently large effect, so much the better.
Though these effects may be large compared with the statistical disturbances they are almost certainly small compared with the ordinary diffusion and reaction effects. This will mean that they only have an appreciable effect during a short period in which the breakdown of left–right symmetry is occurring. Once their existence is admitted, whether on a theoretical or experimental basis, it is probably most convenient to give them mathematical expression by regarding them as _P_ -symmetrically (but not _F_ -symmetrically) distributed disturbances. However, they will not be considered further in this paper.
#### 6. Reactions and diffusion in a ring of cells
The original reason for considering the breakdown of homogeneity was an apparent difficulty in the diffusion-reaction theory of morphogenesis. Now that the difficulty is resolved it might be supposed that there is no reason for pursuing this aspect of the problem further, and that it would be best to proceed to consider what occurs when the system is very far from homogeneous. A great deal more attention will nevertheless be given to the breakdown of homogeneity. This is largely because the assumption that the system is still nearly homogeneous brings the problem within the range of what is capable of being treated mathematically. Even so many further simplifying assumptions have to be made. Another reason for giving this phase such attention is that it is in a sense the most critical period. That is to say, that if there is any doubt as to how the organism is going to develop it is conceivable that a minute examination of it just after instability has set in might settle the matter, but an examination of it at any earlier time could never do so.
There is a great variety of geometrical arrangement of cells which might be considered, but one particular type of configuration stands out as being particularly simple in its theory, and also illustrates the general principles very well. This configuration is a ring of similar cells. One may suppose that there are _N_ such cells. It must be admitted that there is no biological example to which the theory of the ring can be immediately applied, though it is not difficult to find ones in which the principles illustrated by the ring apply.
It will be assumed at first that there are only two morphogens, or rather only two interesting morphogens. There may be others whose concentration does not vary either in space or time, or which can be eliminated from the discussion for one reason or another. These other morphogens may, for instance, be catalysts involved in the reactions between the interesting morphogens. An example of a complete system of reactions is given in §10. Some consideration will also be given in §§8, 9 to the case of three morphogens. The reader should have no difficulty in extending the results to any number of morphogens, but no essentially new features appear when the number is increased beyond three.
The two morphogens will be called _X_ and _Y_. These letters will also be used to denote their concentrations. This need not lead to any real confusion. The concentration of _X_ in cell _r_ may be written _X r_, and _Y r_ has a similar meaning. It is convenient to regard 'cell _N_ ' and 'cell _O_ ' as synonymous, and likewise 'cell 1' and cell ' _N_ \+ 1'. One can then say that for each _r_ satisfying cell _r_ exchanges material by diffusion with cells _r_ − 1 and _r_ \+ 1. The cell-to-cell diffusion constant for _X_ will be called _μ_ , and that for _Y_ will be called _ν_. This means that for unit concentration difference of _X_ , this morphogen passes at the rate _μ_ from the cell with the higher concentration to the (neighbouring) cell with the lower concentration. It is also necessary to make assumptions about the rates of chemical reaction. The most general assumption that can be made is that for concentrations _X_ and _Y_ chemical reactions are tending to increase _X_ at the rate _f_ ( _X, Y_ ) and _Y_ at the rate _g_ ( _X, Y_ ). When the changes in _X_ and _Y_ due to diffusion are also taken into account the behaviour of the system may be described by the 2 _N_ differential equations
If _f_ ( _h_ , _k_ ): _g_ ( _h_ , _k_ ) = 0, then an isolated cell has an equilibrium with concentrations _X_ = _h_ , _Y_ = _k_. The ring system also has an equilibrium, stable or unstable, with each _X r_ equal to _h_ and each _Y r_ equal to _k_. Assuming that the system is not very far from this equilibrium it is convenient to put _X r_ = _h_ \+ _x r_, _Y r_ = _k_ \+ _y r_. One may also write _ax_ \+ _by_ for _f_ ( _h_ \+ _x_ , _y_ \+ _k_ ) and _cx_ \+ _dy_ for _g_ ( _h_ \+ _x_ , _y_ \+ _k_ ). Since _f_ ( _h_ , _k_ ) = _g_ ( _h_ , _k_ ) = 0 no constant terms are required, and since _x_ and _y_ are supposed small the terms in higher powers of _x_ and _y_ will have relatively little effect and one is justified in ignoring them. The four quantities _a, b, c, d_ may be called the 'marginal reaction rates'. Collectively they may be described as the 'marginal reaction rate matrix'. When there are _M_ morphogens this matrix consists of _M_ 2 numbers. A marginal reaction rate has the dimensions of the reciprocal of a time, like a radioactive decay rate, which is in fact an example of a marginal (nuclear) reaction rate.
With these assumptions the equations can be rewritten as
To solve the equations one introduces new co-ordinates ,..., and _ɳ_ 0,..., _ɳ N − 1_ by putting
These relations can also be written as
as may be shown by using the equations
(referred to in §2). Making this substitution one obtains
Likewise
The equations have now been converted into a quite manageable form, with the variables separated. There are now two equations concerned with 1 and _ɳ_ 1, two concerned with _ξ_ 2 and _η_ 2, etc. The equations themselves are also of a well-known standard form, being linear with constant coefficients. Let _p s_ and be the roots of the equation
(with for definiteness), then the solution of the equations is of the form
where, however, the coefficients _A s_, _B s_, _C s_, _D s_ are not independent but are restricted to satisfy
If it should happen that the equations (6.9) have to be replaced by
and (6.10) remains true. Substituting back into (6.3) and replacing the variables _x r_, _y r_ by _X r_, _Y r_ (the actual concentrations) the solution can be written
Here _A s_, _B s_, _C s_, _D s_ are still related by (6.10), but otherwise are arbitrary complex numbers; _p s_ and are the roots of (6.8).
The expression (6.11) gives the general solution of the equations (6.1) when one assumes that departures from homogeneity are sufficiently small that the functions _f_ ( _X, Y_ ) and _g_ ( _X, Y_ ) can safely be taken as linear. The form (6.11) given is not very informative. It will be considerably simplified in §8. Another implicit assumption concerns random disturbing influences. Strictly speaking one should consider such influences to be continuously at work. This would make the mathematical treatment considerably more difficult without substantially altering the conclusions. The assumption which is implicit in the analysis, here and in §8, is that the state of the system at _t_ = 0 is not one of homogeneity, since it has been displaced from such a state by the disturbances; but after _t_ = 0 further disturbances are ignored. In §9 the theory is reconsidered without this latter assumption.
#### 7. Continuous ring of tissue
As an alternative to a ring of separate cells one might prefer to consider a continuous ring of tissue. In this case one can describe the position of a point of the ring by the angle _θ_ which a radius to the point makes with a fixed reference radius. Let the diffusibilities of the two substances be _μ_ ′ and _ν_ ′. These are not quite the same as μ and _v_ of the last section, since _μ_ and _ν_ are in effect referred to a cell diameter as unit of length, whereas _μ_ ′ and _ν_ ′ are referred to a conventional unit, the same unit in which the radius _ρ_ of the ring is measured.
Then
The equations are
which will be seen to be the limiting case of (6.2). The marginal reaction rates _a, b, c, d_ are, as before, the values at the equilibrium position of ∂ _f_ /∂ _X_ , ∂ _f_ /∂ _Y_ , ∂ _g_ /∂ _X_ , ∂ _g_ /∂ _Y_. The general solution of the equations is
where _p s_, are now roots of
and
This solution may be justified by considering the limiting case of the solution (6.11). Alternatively, one may observe that the formula proposed is a solution, so that it only remains to prove that it is the most general one. This will follow if values of _A s_, _B s_, _C s_, _D s_ can be found to fit any given initial conditions. It is well known that any function of an angle (such as _X_ ) can be expanded as a 'Fourier series'
Provided, for instance, that its first derivative is continuous. If also
then the required initial conditions are satisfied provided _A s_ \+ _B s_ = _G s_ and _C s_ \+ _D s_ = _H s_. Values _A s_, _B s_, _C s_, _D s_ to satisfy these conditions can be found unless . This is an exceptional case and its solution if required may be found as the limit of the normal case.
#### 8. Types of asymptotic behaviour in the ring after a lapse of time
As the reader was reminded in § 2, after a lapse of time the behaviour of an expression of the form of (6.11) is eventually dominated by the terms for which the corresponding _p s_ has the largest real part. There may, however, be several terms for which this real part has the same value, and these terms will together dominate the situation, the other terms being ignored by comparison. There will, in fact, normally be either two or four such 'leading' terms. For if _p s0_ is one of them then _p N−s0_ = _p s0_, since
so that _p s0_ and _p N−s0_ are roots of the same equation (6.8). If also _p s0_ is complex then , and so in all
One need not, however, normally anticipate that any further terms will have to be included. If _p s0_ and _p s1_ are to have the same real part, then, unless _s_ 1 = _s_ 0 or _s_ 0 \+ _s_ 1 = _N_ the quantities _a_ , _b_ , _c_ , _d_ , _μ_ , _ν_ will be restricted to satisfy some special condition, which they would be unlikely to satisfy by chance. It is possible to find circumstances in which as many as ten terms have to be included if such special conditions _are_ satisfied, but these have no particular physical or biological importance. It is assumed below that none of these chance relations hold.
It has already been seen that it is necessary to distinguish the cases where the value of _p s0_ for one of the dominant terms is real from those where it is complex. These may be called respectively the _stationary_ and the _oscillatory_ cases.
_Stationary case_. After a sufficient lapse of time _X r_ − _h_ and _Y r_ − _k_ approach asymptotically to the forms
_Oscillatory case_. After a sufficient lapse of time _X r_ − _h_ and _Y r_ − _k_ approach the forms
The real part of _p s0_ has been represented by _I_ , standing for 'instability', and in the oscillatory case its imaginary part is ω. By the use of the operation (real part of), two terms have in each case been combined in one.
The meaning of these formulae may be conveniently described in terms of waves. In the stationary case there are stationary waves on the ring having _s_ 0lobes or crests. The coefficients _A s0_ and _C s0_ are in a definite ratio given by (6 · 10), so that the pattern for one morphogen determines that for the other. With the lapse of time the waves become more pronounced provided there is genuine instability, i.e. if _I_ is positive. The wave-length of the waves may be obtained by dividing the number of lobes into the circumference of the ring. In the oscillatory case the interpretation is similar, but the waves are now not stationary but travelling. As well as having a wave-length they have a velocity and a frequency. The frequency is ω/2π, and the velocity is obtained by multiplying the wavelength by the frequency. There are two wave trains moving round the ring in opposite directions.
The wave-lengths of the patterns on the ring do not depend only on the chemical data _a_ , _b_ , _c_ , _d_ , _μ_ ′, _ν_ ′ but on the circumference of the ring, since they must be submultiples of the latter. There is a sense, however, in which there is a 'chemical wave-length' which does not depend on the dimensions of the ring. This may be described as the limit to which the wave-lengths tend when the rings are made successively larger. Alternatively (at any rate in the case of continuous tissue), it may be described as the wave-length when the radius is chosen to give the largest possible instability _I_. One may picture the situation by supposing that the chemical wave-length is true wave-length which is achieved whenever possible, but that on a ring it is necessary to 'make do' with an approximation which divides exactly into the circumference.
Although all the possibilities are covered by the stationary and oscillatory alternatives there are special cases of them which deserve to be treated separately. One of these occurs when _s_ 0 = 0, and may be described as the 'case of extreme long wave-length', though this term may perhaps preferably be reserved to describe the chemical data when they are such that _s_ 0 is zero whatever the dimensions of the ring. There is also the case of 'extreme short wave-length'. This means that sin2 is as large as possible, which is achieved by _s_ 0 being either , or . If the remaining possibilities are regarded as forming the 'case of finite wave-length', there are six subcases altogether. It will be shown that each of these really can occur, although two of them require three or more morphogens for their realization.
( _a_ ) _Stationary case with extreme long wave-length_. This occurs for instance if , _b_ = _c_ = 1, _a_ = _d_. Then . This is certainly real and is greatest when _s_ = 0. In this case the contents of all the cells are the same; there is no resultant flow from cell to cell due to diffusion, so that each is behaving as if it were isolated. Each is in unstable equilibrium, and slips out of it in synchronism with the others.
( _b_ ) _Oscillatory case with extreme long wave-length_. This occurs, for instance, if , _b_ = − _c_ = 1, _a_ = _d_. Then . This is complex and its real part is greatest when _s_ = 0. As in case ( _a_ ) each cell behaves as if it were isolated. The difference from case ( _a_ ) is that the departure from the equilibrium is oscillatory.
( _c_ ) _Stationary waves of extreme short wave-length_. This occurs, for instance, if _v_ = 0, μ = 1, _d_ = _I_ , _a_ = _I_ − 1, _b_ = − _c_ = 1. _p s_ is
and is greatest when sin2(π _s_ / _N_ ) is greatest. If _N_ is even the contents of each cell are similar to those of the next but one, but distinctly different from those of its immediate neighbours. If, however, the number of cells is odd this arrangement is impossible, and the magnitude of the difference between neighbouring cells varies round the ring, from zero at one point to a maximum at a point diametrically opposite.
( _d_ ) _Stationary waves of finite wave-length_. This is the case which is of greatest interest, and has most biological application. It occurs, for instance, if _a_ = _I_ − 2, _b_ = 2·5, _c_ = − 1·25, _d_ = _I_ \+ 1·5, μ′ = 1, , and . As before _ρ_ is the radius of the ring, and _N_ the number of cells in it. If one writes _U_ for , then equation (6.8) can, with these special values, be written
This has a solution _p_ = _I_ if . One the other hand, it will be shown that if _U_ has any other (positive) value then both roots for _p_ − _I_ have negative real parts. Their product is positive being , so that if they are real they both have the same sign. Their sum in this case is which is negative. Their common sign is therefore negative. If, however, the roots are complex their real parts are both equal to , which is negative.
If the radius _ρ_ of the ring be chosen so that for some integer _s_ 0, , there will be stationary waves with _s_ 0 lobes and a wave-length which is also equal to the chemical wave-length, for _p s0_ will be equal to _I_ , whereas every other _p s_ will have a real part smaller than _I_. If, however, the radius is chosen so that cannot hold with an integral _s_ , then (in this example) the actual number of lobes will be one of the two integers nearest to the (non-integral) solutions of this equation, and usually _the_ nearest. Examples can, however, be constructed where this simple rule does not apply.
Figure 1 shows the relation (8.3) in graphical form. The curved portions of the graphs are hyperbolae.
The two remaining possibilities can only occur with three or more morphogens. With one morphogen the only possibility is ( _a_ ).
( _e_ ) _Oscillatory case with a finite wave-length_. This means that there are genuine travelling waves. Since the example to be given involves three morphogens it is not possible to use the formulae of § 6. Instead, one must use the corresponding three morphogen formulae. That which corresponds to (6·8) or (7·3) is most conveniently written as
**Figure 1.** Values of (instability or growth rate), and (radian frequency of oscillation), related to wave-length as in the relation (8.3) with **_I_** = 0. This is a case of stationary waves with finite wave-length. Full line, ; broken line, − (zero for **_U_** > 0·071); dotted line, . The full circles on the curve for indicate the values of **_U_** , **_p_** actually achievable on the finite ring considered in §10, with **_s_** = 0 on the extreme left, **_s_** = 5 on the right.
where again _U_ has been written for . (This means essentially that , where λ is the wave-length.) The four marginal reactivities are superseded by nine _a_ 11,..., _a_ 33, and the three diffusibilities are _μ_ 1, _μ_ 2, _μ_ 3. Special values leading to travelling waves are
and with them (8.4) reduces to
If _U_ = 1 the roots are _±_ i and −2. If _U_ is near to _I_ they are approximately −1 − _U_ and , and all have negative real parts. If the greatest real part is not the value zero, achieved with _U_ = 1, then the value zero must be reached again at some intermediate value of _U_. Since _P_ is then pure imaginary the even terms of (8.6) must vanish, i.e. ( _p_ 2 \+ 1)( _U_ \+ 1) = 0. But this can only happen if _p_ = _±_ i, and the vanishing of the odd terms then shows that _U_ = 1. Hence zero is the largest real part for any root _p_ of (8.6). The corresponding _p_ is _±_ i and _U_ is 1. This means that there are travelling waves with unit (chemical) radian frequency and unit (chemical) velocity. If _I_ is added to _a_ 11, _a_ 22 and _a_ 33, the instability will become _I_ in place of zero.
( _f_ ) Oscillatory case with extreme short wave-length. This means that there is metabolic oscillation with neighbouring cells nearly 180° out of phase. It can be achieved with three morphogens and the following chemical data:
With these values (8.4) reduces to
This may be shown to have all the real parts of its roots negative if _U_ ≥ 0, for if _U_ = 0 the roots are near to −0·6, 0·2 _±_ 1·3i, and if _U_ be continuously increased the values of _p_ will alter continuously. If they ever attain values with a positive real part they must pass through pure imaginary values (or zero). But if _p_ is pure imaginary _p_ 3 \+ 2 _p_ and ( _p_ 2 \+ 1)( _U_ \+ 1) must both vanish, which is impossible if _U_ ≥ 0. As _U_ approaches infinity, however, one of the roots approaches i. Thus can be approached as closely as desired by large values of _U_ , but not attained.
#### 9. Further consideration of the mathematics of the ring
In this section some of the finer points concerning the development of wave patterns are considered. These will be of interest mainly to those who wish to do further research on the subject, and can well be omitted on a first reading.
(1) _General formulae for the two morphogen case_. Taking the limiting case of a ring of large radius (or a filament), one may write in (6.11) or in (7.3) and obtain
which has the solution
One may put _I_ ( _U_ ) for the real part of this, representing the instability for waves of wave-length . The dominant waves correspond to the maximum of _I_ ( _U_ ). This maximum may either be at _U_ = 0 or _U_ = ∞ or at a stationary point on the part of the curve which is hyperbolic (rather than straight). When this last case occurs the values of _p_ (or _I_ ) and _U_ at the maximum are
The conditions which lead to the four cases ( _a_ ), ( _b_ ), ( _c_ ), ( _d_ ) described in the last section are
( _a_ ) (Stationary waves of extreme long wave-length.) This occurs if either (i) _bc_ > 0, (ii) _bc_ < 0 and (iii) _bc_ < 0 and .
The condition for instability in either case is that either _bc_ > _ad_ or _a_ \+ _d_ > 0. ( _b_ ) (Oscillating case with extreme long wave-length, i.e. synchronized oscillations.)
This occurs if
There is instability if in addition _a_ \+ _d_ > 0.
( _c_ ) (Stationary waves of extreme short wave-length.) This occurs if _bc_ < 0, _μ_ ′ > _ν_ ′ = 0.
There is instability if, in addition, _a_ \+ _d_ > 0.
( _d_ ) (Stationary waves of finite wave-length.) This occurs if
and there is instability if also
It has been assumed that . The case where can be obtained by interchanging the two morphogens. In the case _μ_ ′ = _ν_ ′ = 0 there is no cooperation between the cells whatever.
Some additional formulae will be given for the case of stationary waves of finite wave-length. The marginal reaction rates may be expressed parametrically in terms of the diffusibilities, the wave-length, the instability, and two other parameters _α_ and _χ_. Of these _α_ may be described as the ratio of _X_ − _h_ to _Y_ − _k_ in the waves. The expressions for the marginal reaction rates in terms of these parameters are
and when these are substituted into (9.2) it becomes
Here is the chemical wave-length and the wave-length of the Fourier component under consideration. _χ_ must be positive for case ( _d_ ) to apply.
If _s_ be regarded as a continuous variable one can consider (9.2) or (9.6) as relating _s_ to _p_ , and d _p_ /d _s_ and d2 _p_ /d _s_ 2 have meaning. The value of d2 _p_ /d _s_ 2at the maximum is of some interest, and will be used below in this section. Its value is
(2) In §§6, 7, 8 it was supposed that the disturbances were not continuously operative, and that the marginal reaction rates did not change with the passage of time. These assumptions will now be dropped, though it will be necessary to make some other, less drastic, approximations to replace them. The (statistical) amplitude of the 'noise' disturbances will be assumed constant in time. Instead of (6.6), (6.7), one then has
where _ξ_ , _η_ have been written for _ξ s_, _η s_ since _s_ may now be supposed fixed. For the same reason has been replaced by _a_ ′ and by _d_ ′. The noise disturbances may be supposed to constitute white noise, i.e. if ( _t_ 1, _t_ 2) and ( _t_ 3, _t_ 4) are two non-overlapping intervals then and are statistically independent and each is normally distributed with variances β1( _t_ 2 − _t_ 1) and β1( _t_ 4 − _t_ 3) respectively, β1 being a constant describing the amplitude of the noise. Likewise for _R_ 2( _t_ ), the constant β1 being replaced by β2. If _p_ and _p_ ′ are the roots of ( _p_ − _d_ ′)( _p_ − _d_ ′) = _bc_ and _p_ is the greater (both being real), one can make the substitution
which transforms (9.8) into
with a similar equation for _v_ , of which the leading terms are . This indicates that _v_ will be small, or at least small in comparison with _u_ after a lapse of time. If it is assumed that _v_ = 0 holds (9.11) may be written
where
The solution of this equation is
One is, however, not so much interested in such a solution in terms of the statistical disturbances as in the consequent statistical distribution of values of _u_ , _ξ_ and _η_ at various times after instability has set in. In view of the properties of 'white noise' assumed above, the values of _u_ at time _t_ will be distributed according to the normal error law, with the variance
There are two commonly occurring cases in which one can simplify this expression considerably without great loss of accuracy. If the system is in a distinctly stable state, then _q_ ( _t_ ), which is near to _p_ ( _t_ ), will be distinctly negative, and exp will be small unless _w_ is near to _t_. But then _L_ 1( _w_ ) and _L_ 2( _w_ ) be replaced by _L_ 1( _t_ ) and _L_ 2( _t_ ) in the integral, and also _q_ ( _z_ ) may be replaced by _q_ ( _t_ ). With these approximations the variance is
A second case where there is a convenient approximation concerns times when the system is unstable, so that _q_ ( _t_ ) > 0. For the approximation concerned to apply must have its maximum at the last moment _w_ (= _t_ 0) when _q_ ( _t_ 0) = 0, and it must be the maximum by a considerable margin (e.g. at least 5) over all other local maxima. These conditions would apply for instance if _q_ ( _z_ ) were always increasing and had negative values at a sufficiently early time. One also requires _q_ ′( _t_ 0) (the rate of increase of _q_ at time _t_ 0) to be reasonably large; it must at least be so large that over a period of time of length near to _t_ 0the changes in _L_ 1( _t_ ) and _L_ 2( _t_ ) are small, and _q_ ′( _t_ ) itself must not appreciably alter in this period. Under these circumstances the integrand is negligible when _w_ is considerably different from _t_ 0, in comparison with its values at that time, and therefore one may replace _L_ 1( _w_ ) and _L_ 2( _w_ ) by _L_ 1( _t_ 0) and _L_ 2( _t_ 0), and _q_ ′( _w_ ) by _q_ ′( _t_ 0). This gives the value
for the variance of _u_.
The physical significance of this latter approximation is that the disturbances near the time when the instability is zero are the only ones which have any appreciable ultimate effect. Those which occur earlier are damped out by the subsequent period of stability. Those which occur later have a shorter period of instability within which to develop to greater amplitude. This principle is familiar in radio, and is fundamental to the theory of the superregenerative receiver.
Naturally one does not often wish to calculate the expression (9.17), but it is valuable as justifying a common-sense point of view of the matter. The factor exp is essentially the integrated instability and describes the extent to which one would expect disturbances of appropriate wave-length to grow between times _t_ 0 and _t_. Taking the terms in β1, β2 into consideration separately, the factor indicates that the disturbances on the first morphogen should be regarded as lasting for a time . The dimensionless quantities _bL_ 1( _t_ 0), _bL_ 2( _t_ 0) will not usually be sufficiently large or small to justify their detailed calculation.
(3) The extent to which the component for which _p s_ is greatest may be expected to outdistance the others will now be considered in case ( _d_ ). The greatest of the _p s_ will be called _p s0_. The two closest competitors to _s_ 0 will be _s_ 0 − 1 and _s_ 0 \+ 1; it is required to determine how close the competition is. If the variation in the chemical data is sufficiently small it may be assumed that, although the exponents _p s0 − 1_, _p s0_, _p s0+1_ may themselves vary appreciably in time, the differences _p s0_ − _p s0 − 1_ and _p s0_ − _p s0+1_ are constant. It certainly can happen that one of these differences is zero or nearly zero, and there is then 'neck and neck' competition. The weakest competition occurs when _p s0 − 1_ = _p s0+1_. In this case
But if _s_ 0 is reasonably large _p s0+1_ − 2 _p s0_ \+ _p s0 − 1_ can be set equal to (d2 _p_ /d _s_ 2) _s=s 0_. It may be concluded that the rate at which the most quickly growing component grows cannot exceed the rate for its closest competitor by more than about . The formula (9.7), by which d2 _p_ /d _s_ 2 can be estimated, may be regarded as the product of two factors. The dimensionless factor never exceeds 4. The factor may be described in very rough terms as 'the reciprocal of the time for the morphogens to diffuse a length equal to a radius'. In equally rough terms one may say that a time of this order of magnitude is required for the most quickly growing component to get a lead, amounting to a factor whose logarithm is of the order of unity, over its closest competitors, in the favourable case where _p s0−1_ = _p s0+1_.
(4) Very little has yet been said about the effect of considering non-linear reaction rate functions when far from homogeneity. Any treatment so systematic as that given for the linear case seems to be out of the question. It is possible, however, to reach some qualitative conclusions about the effects of non-linear terms. Suppose that _z_ 1 is the amplitude of the Fourier component which is most unstable (on a basis of the linear terms), and which may be supposed to have wave-length λ. The non-linear terms will cause components with wave-lengths , , ,... to appear as well as a space-independent component. If only quadratic terms are taken into account and if these are somewhat small, then the component of wave-length and the space-independent component will be the strongest. Suppose these have amplitudes _z_ 2 and _z_ 1. The state of the system is thus being described by the numbers _z_ 0, _z_ 1, _z_ 2. In the absence of non-linear terms they would satisfy equations
and if there is slight instability _p_ 1 would be a small positive number, but _p_ 0 and _p_ 2 distinctly negative. The effect of the non-linear terms is to replace these equations by ones of the form
As a first approximation one may put d _z_ 0/d _t_ = d _z_ 2/d _t_ = 0 and ignore and higher powers; _z_ 0 and _z_ 1 are then found to be proportional to , and the equation for _z_ 1 can be written . The sign of _k_ in this differential equation is of great importance. If it is positive, then the effect of the term is to arrest the exponential growth of _z_ 1 at the value . The 'instability' is then very confined in its effect, for the waves can only reach a finite amplitude, and this amplitude tends to zero as the instability ( _p_ 1) tends to zero. If, however, _k_ is negative the growth becomes something even faster than exponential, and, if the equation held universally, it would result in the amplitude becoming infinite in a finite time. This phenomenon may be called 'catastrophic instability'. In the case of two-dimensional systems catastrophic instability is almost universal, and the corresponding equation takes the form . Naturally enough in the case of catastrophic instability the amplitude does not really reach infinity, but when it is sufficiently large some effect previously ignored becomes large enough to halt the growth.
(5) Case ( _a_ ) as described in §8 represents a most extremely featureless form of pattern development. This may be remedied quite simply by making less drastic simplifying assumptions, so that a less gross account of the pattern can be given by the theory. It was assumed in § 9 that only the most unstable Fourier components would contribute appreciably to the pattern, though it was seen above (heading (3) of this section) that (in case ( _d_ )) this will only apply if the period of time involved is adequate to permit the morphogens, supposed for this purpose to be chemically inactive, to diffuse over the whole ring or organ concerned. The same may be shown to apply for case ( _a_ ). If this assumption is dropped a much more interesting form of pattern can be accounted for. To do this it is necessary to consider not merely the components with _U_ = 0 but some others with small positive values of _U_. One may assume the form _At_ − _BU_ for _p_. Linearity in _U_ is assumed because only small values of _U_ are concerned, and the term _At_ is included to represent the steady increase in instability. By measuring time from the moment of zero instability the necessity for a constant term is avoided. The formula (9.17) may be applied to estimate the statistical distribution of the amplitudes of the components. Only the factor exp will depend very much on _U_ , and taking _q_ ( _t_ ) = _p_ ( _t_ ) = _At_ − _BU_ , _t_ 0 must be _BU_ / _A_ and the factor is
The term in _U_ 2 can be ignored if _At_ 2 is fairly large, for then either _B_ 2 _U_ 2/ _A_ 2 is small or the factor e− _BUt_ is. But _At_ 2 certainly is large if the factor , applying when _U_ = 0, is large. With this approximation the variance takes the form , with only the two parameters _C_ , _k_ to distinguish the pattern populations. By choosing appropriate units of concentration and length these pattern populations may all be reduced to a standard one, e.g. with _C_ = _k_ = 1. Random members of this population may be produced by considering any one of the type ( _a_ ) systems to which the approximations used above apply. They are also produced, but with only a very small amplitude scale, if a homogeneous one-morphogen system undergoes random disturbances without diffusion for a period, and then diffusion without disturbance. This process is very convenient for computation, and can also be applied to two dimensions. Figure 2 shows such a pattern, obtained in a few hours by a manual computation.
**Figure 2.** An example of a 'dappled' pattern as resulting from a type ( ** _a_** ) morphogen system. A marker of unit length is shown. See text, §9, 11.
To be more definite a set of numbers _u r_, _s_ was chosen, each being _±_ 1, and taking the two values with equal probability. A function _f_ ( _x_ , _y_ ) is related to these numbers by the formula
In the actual computation a somewhat crude approximation to the function
was used and _h_ was about 0·7. In the figure the set of points where _f_ ( _x, y_ ) is positive is shown black. The outlines of the black patches are somewhat less irregular than they should be due to an inadequacy in the computation procedure.
#### 10. A numerical example
The numerous approximations and assumptions that have been made in the foregoing analysis may be rather confusing to many readers. In the present section it is proposed to consider in detail a single example of the case of most interest, ( _d_ ). This will be made as specific as possible. It is unfortunately not possible to specify actual chemical reactions with the required properties, but it is thought that the reaction rates associated with the imagined reactions are not unreasonable.
The detail to be specified includes
(i) The number and dimensions of the cells of the ring.
(ii) The diffusibilities of the morphogens.
(iii) The reactions concerned.
(iv) The rates at which the reactions occur.
(v) Information about random disturbances.
(vi) Information about the distribution, in space and time, of those morphogens which are of the nature of evocators.
These will be taken in order.
(i) It will be assumed that there are twenty cells in the ring, and that they have a diameter of 0·1 mm each. These cells are certainly on the large rather than the small side, but by no means impossibly so. The number of cells in the ring has been chosen rather small in order that it should not be necessary to make the approximation of continuous tissue.
(ii) Two morphogens are considered. They will be called _X_ and _Y_ , and the same letters will be used for their concentrations. This will not lead to any real confusion. The diffusion constant for _X_ will be assumed to be 5×10−8 cm2s−1 and that for _Y_ to be 2·5× 10−8 cm2s−1. With cells of diameter 0·01 cm this means that _X_ flows between neighbouring cells at the rate 5 ×10−4 of the difference of _X_ -content of the two cells per second. In other words, if there is nothing altering the concentrations but diffusion the difference of concentrations suffers an exponential decay with time constant 1000 s, or 'half-period' of 700 s. These times are doubled for _Y_.
If the cell membrane is regarded as the only obstacle to diffusion the permeability of the membranes to the morphogen is 5 ×10−6 cm/s or 0·018 cm/h. Values as large as 0·1 cm/h have been observed (Davson & Danielli 1943, figure 28).
(iii) The reactions are the most important part of the assumptions. Four substances _A_ , _X_ , _Y_ , _B_ are involved; these are isomeric, i.e. the molecules of the four substances are all rearrangements of the same atoms. Substances _C_ , _C_ ′, _W_ will also be concerned. The thermodynamics of the problem will not be discussed except to say that it is contemplated that of the substances _A_ , _X_ , _Y_ , _B_ the one with the greatest free energy is _A_ , and that with the least is _B_. Energy for the whole process is obtained by the degradation of _A_ into _B_. The substance _C_ is in effect a catalyst for the reaction _Y_ → _X_ , and may also be regarded as an evocator, the system being unstable if there is a sufficient concentration of _C_.
The reactions postulated are
(iv) For the purpose of stating the reaction rates special units will be introduced (for the purpose of this section only). They will be based on a period of 1000 s as units of time, and 10−11 mole/cm3 as concentration unit. There will be little occasion to use any but these special units (S.U.). The concentration of _A_ will be assumed to have the large value of 1000 S.U. and the catalyst _C_ , together with its combined form _C_ ′ the concentration 10−3(1 + γ) S.U., the dimensionless quantity γ being often supposed somewhat small, though values over as large a range as from −0·5 to 0·5 may be considered. The rates assumed will be
With the values assumed for _A_ and _C_ ′ the net effect of these reactions is to convert _X_ into _Y_ at the rate at the same time producing _X_ at the constant rate , and destroying _Y_ at the rate _Y/16_. If, however, the concentration of _Y_ is zero and the rate of increase of _Y_ required by these formulae is negative, the rate of conversion of _Y_ into _X_ is reduced sufficiently to permit _Y_ to remain zero.
In the special units , .
(v) Statistical theory describes in detail what irregularities arise from the molecular nature of matter. In a period in which, on the average, one should expect a reaction to occur between _n_ pairs (or other combinations) of molecules, the actual number will differ from the mean by an amount whose mean square is also _n_ , and is distributed according to the normal error law. Applying this to a reaction proceeding at a rate _F_ (S.U.) and taking the volume of the cell as 10−8 cm3 (assuming some elongation tangentially to the ring) it will be found that the root mean square irregularity of the quantity reacting in a period _τ_ of time (S.U.) is .
The diffusion of a morphogen from a cell to a neighbour may be treated as if the passage of a molecule from one cell to another were a monomolecular reaction; a molecule must be imagined to change its form slightly as it passes the cell wall. If the diffusion constant for a wall is _μ_ , and quantities _M_ 1, _M_ 2 of the relevant morphogen lie on the two sides of it, the root-mean-square irregularity in the amount passing the wall in a period _τ_ is
These two sources of irregularity are the most significant of those which arise from truly statistical cause, and are the only ones which are taken into account in the calculations whose results are given below. There may also be disturbances due to the presence of neighbouring anatomical structures, and other similar causes. These are of great importance, but of too great variety and complexity to be suitable for consideration here.
(vi) The only morphogen which is being treated as an evocator is _C_. Changes in the concentration of _A_ might have similar effects, but the change would have to be rather great. It is preferable to assume that _A_ is a 'fuel substance' (e.g. glucose) whose concentration does not change. The concentration of _C_ , together with its combined form _C_ ′, will be supposed the same in all cells, but it changes with the passage of time. Two different varieties of the problem will be considered, with slightly different assumptions.
The results are shown in table 1. There are eight columns, each of which gives the concentration of a morphogen in each of the twenty cells; the circumstances to which these concentrations refer differ from column to column. The first five columns all refer to the same 'variety' of the imaginary organism, but there are two specimens shown. The specimens differ merely in the chance factors which were involved. With this variety the value of γ was allowed to increase at the rate of 2−7 S.U. from the value to . At this point a pattern had definitely begun to appear, and was recorded. The parameter γ was then allowed to decrease at the same rate to zero and then remained there until there was no more appreciable change. The pattern was then recorded again. The concentrations of _Y_ in these two recordings are shown in figure 3 as well as in table 1. For the second specimen only one column of figures is given, viz. those for the _Y_ morphogen in the incipient pattern. At this stage the _X_ values are closely related to the _Y_ values, as may be seen from the first specimen (or from theory). The final values can be made almost indistinguishable from those for the first specimen by renumbering the cells and have therefore not been given. These two specimens may be said to belong to the 'variety with quick cooking', because the instability is allowed to increase so quickly that the pattern appears relatively soon. The effect of this haste might be regarded as rather unsatisfactory, as the incipient pattern is very irregular. In both specimens the four-lobed component is present in considerable strength in the incipient pattern. It 'beats' with the three-lobed component producing considerable irregularity. The relative magnitudes of the three- and four-lobed components depend on chance and vary from specimen to specimen. The four-lobed component may often be the stronger, and may occasionally be so strong that the final pattern is four-lobed. How often this happens is not known, but the pattern, when it occurs, is shown in the last two columns of the table. In this case the disturbances were supposed removed for some time before recording, so as to give a perfectly regular pattern.
**Table 1.** Some stationary-wave patterns
**Figure 3.** Concentrations of **_Y_** in the development of the first specimen (taken from table 1). ------- original homogeneous equilibrium; incipient pattern; ———— final equilibrium.
The remaining column refers to a second variety, one with 'slow cooking'. In this the value of γ was allowed to increase only at the rate 10−5. Its initial value was −0·010, but is of no significance. The final value was 0·003. With this pattern, when shown graphically, the irregularities are definitely perceptible, but are altogether overshadowed by the three-lobed component. The possibility of the ultimate pattern being four-lobed is not to be taken seriously with this variety.
The set of reactions chosen is such that the instability becomes 'catastrophic' when the second-order terms are taken into account, i.e. the growth of the waves tends to make the whole system more unstable than ever. This effect is finally halted when (in some cells) the concentration of _Y_ has become zero. The constant conversion of _Y_ into _X_ through the agency of the catalyst _C_ can then no longer continue in these cells, and the continued growth of the amplitude of the waves is arrested. When γ = 0 there is of course an equilibrium with _X_ = _Y_ = 1 in all cells, which is very slightly stable. There are, however, also other stable equilibria with γ = 0, two of which are shown in the table. These final equilibria may, with some trouble but little difficulty, be verified to be solutions of the equations (6.1) with
and
The morphogen concentrations recorded at the earlier times connect more directly with the theory given in §§6 to 9. The amplitude of the waves was then still sufficiently small for the approximation of linearity to be still appropriate, and consequently the 'catastrophic' growth had not yet set in.
The functions _f_ ( _X, Y_ ) and _g_ ( _X, Y_ ) of §6 depend also on γ and are
In applying the theory it will be as well to consider principally the behaviour of the system when γ remains permanently zero. Then for equilibrium _f_ ( _X_ , _Y_ ) = _g_ ( _X_ , _Y_ ) = 0 which means that _X_ = _Y_ = 1, i.e. _h_ = _k_ = 1. One also finds the following values for various quantities mentioned in §§ 6 to 9:
(The relation between _p_ and _U_ for these chemical data, and the values _p n_, can be seen in figure 1, the values being so related as to make the curves apply to this example as well as that in §8.) The value _s_ = 3 333 leads one to expect a three-lobed pattern as the commonest, and this is confirmed by the values _p n_. The four-lobed pattern is evidently the closest competitor. The closeness of the competition may be judged from the difference _p_ 3 − _p_ 4 = 0·0084, which suggests that the three-lobed component takes about 120 S.U. or about 33 h to gain an advantage of a neper (i.e. about 2 7:1) over the four-lobed one. However, the fact that γ is different from 0 and is changing invalidates this calculation to some extent.
The figures in table 1 were mainly obtained with the aid of the Manchester University Computer.
Although the above example is quite adequate to illustrate the mathematical principles involved it may be thought that the chemical reaction system is somewhat artificial. The following example is perhaps less so. The same 'special units' are used. The reactions assumed are
The effect of the reactions _X_ \+ _Y_ _C_ is that _C_ = 10−3 _XY_. The reaction _C_ → _D_ destroys _C_ , and therefore in effect both _X_ and _Y_ , at the rate _XY_. The reaction _A_ → _X_ forms _X_ at the constant rate 1, and the pair _Y_ \+ _V_ → _V_ ′→ _E_ \+ _V_ destroys _Y_ at the constant rate β. The pair _B_ \+ _C_ → _W_ → _Y_ \+ _C_ forms _Y_ at the rate _XY_ , and _Y_ → _E_ destroys it at the rate _Y_. The total effect therefore is that _X_ is produced at the rate _f_ ( _X_ , _Y_ ) = (16 − _XY_ ), and _Y_ at the rate _g_ ( _X_ , _Y_ ) = ( _XY_ − _Y_ − β). However, _g_ ( _X_ , _Y_ ) = 0 if _Y_ ≤ 0. The diffusion constants will be supposed to be _μ_ = , _ν_ = . The homogeneity condition gives _hk_ = 16, _k_ = 16 − β.
It will be seen from conditions (9.4 _a_) that case ( _d_ ) applies if and only if , i.e. if _k_ lies between 1·725 and 9·257. Condition (9.4 _b_) shows that there will be instability if in addition , i.e. if _k_ does not lie between 4·98 and 12·8. It will also be found that the wave-length corresponding to _k_ = 4·98 is 4·86 cell diameters.
In the case of a ring of six cells with β = 12 there is a stable equilibrium, as shown in table 2.
**Table 2.**
It should be recognized that these equilibria are only dynamic equilibria. The molecules which together make up the chemical waves are continually changing, though their concentrations in any particular cell are only undergoing small statistical fluctuations. Moreover, in order to maintain the wave pattern a continual supply of free energy is required. It is clear that this must be so since there is a continual degradation of energy through diffusion. This energy is supplied through the 'fuel substances' ( _A, B_ in the last example), which are degraded into 'waste products' ( _D, E_ ).
#### 11. Restatement and biological interpretation of the results
Certain readers may have preferred to omit the detailed mathematical treatment of §§ 6 to 10. For their benefit the assumptions and results will be briefly summarized, with some change of emphasis.
The system considered was either a ring of cells each in contact with its neighbours, or a continuous ring of tissue. The effects are extremely similar in the two cases. For the purposes of this summary it is not necessary to distinguish between them. A system with two or three morphogens only was considered, but the results apply quite generally. The system was supposed to be initially in a stable homogeneous condition, but disturbed slightly from this state by some influences unspecified, such as Brownian movement or the effects of neighbouring structures or slight irregularities of form. It was supposed also that slow changes are taking place in the reaction rates (or, possibly, the diffusibilities) of the two or three morphogens under consideration. These might, for instance, be due to changes of concentration of other morphogens acting in the role of catalyst or of fuel supply, or to a concurrent growth of the cells, or a change of temperature. Such changes are supposed ultimately to bring the system out of the stable state. The phenomena when the system is just unstable were the particular subject of the inquiry. In order to make the problem mathematically tractable it was necessary to assume that the system never deviated very far from the original homogeneous condition. This assumption was called the 'linearity assumption' because it permitted the replacement of the general reaction rate functions by linear ones. This linearity assumption is a serious one. Its justification lies in the fact that the patterns produced in the early stages when it is valid may be expected to have strong qualitative similarity to those prevailing in the later stages when it is not. Other, less important, assumptions were also made at the beginning of the mathematical theory, but the detailed effects of these were mostly considered in § 9, and were qualitatively unimportant.
The conclusions reached were as follows. After the lapse of a certain period of time from the beginning of instability, a pattern of morphogen concentrations appears which can best be described in terms of 'waves'. There are six types of possibility which may arise.
( _a_ ) The equilibrium concentrations and reaction rates may become such that there would be instability for an isolated cell with the same content as any one of the cells of the ring. If that cell drifts away from the equilibrium position, like an upright stick falling over, then, in the ring, each cell may be expected to do likewise. In neighbouring cells the drift may be expected to be in the same direction, but in distant cells, e.g. at opposite ends of a diameter there is no reason to expect this to be so.
This is the least interesting of the cases. It is possible, however, that it might account for 'dappled' colour patterns, and an example of a pattern in two dimensions produced by this type of process is shown in figure 2 for comparison with 'dappling'. If dappled patterns are to be explained in this way they must be laid down in a latent form when the foetus is only a few inches long. Later the distances would be greater than the morphogens could travel by diffusion.
( _b_ ) This case is similar to ( _a_ ), except that the departure from equilibrium is not a unidirectional drift, but is oscillatory. As in case ( _a_ ) there may not be agreement between the contents of cells at great distances.
There are probably many biological examples of this metabolic oscillation, but no really satisfactory one is known to the author.
( _c_ ) There may be a drift from equilibrium, which is in opposite directions in contiguous cells.
No biological examples of this are known.
( _d_ ) There is a stationary wave pattern on the ring, with no time variation, apart from a slow increase in amplitude, i.e. the pattern is slowly becoming more marked. In the case of a ring of continuous tissue the pattern is sinusoidal, i.e. the concentration of one of the morphogens plotted against position on the ring is a sine curve. The peaks of the waves will be uniformly spaced round the ring. The number of such peaks can be obtained approximately by dividing the so-called 'chemical wave-length' of the system into the circumference of the ring. The chemical wave-length is given for the case of two morphogens by the formula (9.3). This formula for the number of peaks of course does not give a whole number, but the actual number of peaks will always be one of the two whole numbers nearest to it, and will usually be _the_ nearest. The degree of instability is also shown in (9.3).
The mathematical conditions under which this case applies are given in equations (9.4 _a_), (9.4 _b_).
Biological examples of this case are discussed at some length below.
( _e_ ) For a two-morphogen system only the alternatives ( _a_ ) to ( _d_ ) are possible, but with three or more morphogens it is possible to have travelling waves. With a ring there would be two sets of waves, one travelling clockwise and the other anticlockwise. There is a natural chemical wave-length and wave frequency in this case as well as a wave-length; no attempt was made to develop formulae for these.
In looking for biological examples of this there is no need to consider only rings. The waves could arise in a tissue of any anatomical form. It is important to know what wave-lengths, velocities and frequencies would be consistent with the theory. These quantities are determined by the rates at which the reactions occur (more accurately by the 'marginal reaction rates', which have the dimensions of the reciprocal of a time), and the diffusibilities of the morphogens. The possible range of values of the reaction rates is so immensely wide that they do not even give an indication of orders of magnitude. The diffusibilities are more helpful. If one were to assume that all the _dimensionless_ parameters in a system of travelling waves were the same as in the example given in § 8, one could say that the product of the velocity and wave-length of the waves was 3π times the diffusibility of the most diffusible morphogen. But this assumption is certainly false, and it is by no means obvious what is the true range of possible values for the numerical constant (here 3π). The movements of the tail of a spermatozoon suggest themselves as an example of these travelling waves. That the waves are within one cell is no real difficulty. However, the speed of propagation seems to be somewhat greater than can be accounted for except with a rather large numerical constant.
( _f_ ) Metabolic oscillation with neighbouring cells in opposite phases. No biological examples of this are known to the author.
It is difficult also to find cases to which case ( _d_ ) applies directly, but this is simply because isolated rings of tissue are very rare. On the other hand, systems that have the same kind of symmetry as a ring are extremely common, and it is to be expected that under appropriate chemical conditions, stationary waves may develop on these bodies, and that their circular symmetry will be replaced by a polygonal symmetry. Thus, for instance, a plant shoot may at one time have circular symmetry, i.e. appear essentially the same when rotated through any angle about a certain axis; this shoot may later develop a whorl of leaves, and then it will only suffer rotation through the angle which separates the leaves, or any multiple of it. This same example demonstrates the complexity of the situation when more than one dimension is involved. The leaves on the shoots may not appear in whorls, but be imbricated. This possibility is also capable of mathematical analysis, and will be considered in detail in a later paper. The cases which appear to the writer to come closest biologically to the 'isolated ring of cells' are the tentacles of (e.g.) _Hydra_ , and the whorls of leaves of certain plants such as Woodruff ( _Asperula odorata_ ).
_Hydra_ is something like a sea-anemone but lives in fresh water and has from about five to ten tentacles. A part of a _Hydra_ cut off from the rest will rearrange itself so as to form a complete new organism. At one stage of this proceeding the organism has reached the form of a tube open at the head end and closed at the other end. The external diameter is somewhat greater at the head end than over the rest of the tube. The whole still has circular symmetry. At a somewhat later stage the symmetry has gone to the extent that an appropriate stain will bring out a number of patches on the widened head end. These patches arise at the points where the tentacles are subsequently to appear (Child 1941, p. 101 and figure 30). According to morphogen theory it is natural to suppose that reactions, similar to those which were considered in connection with the ring of tissue, take place in the widened head end, leading to a similar breakdown of symmetry. The situation is more complicated than the case of the thin isolated ring, for the portion of the _Hydra_ concerned is neither isolated nor very thin. It is not unreasonable to suppose that this head region is the only one in which the chemical conditions are such as to give instability. But substances produced in this region are still free to diffuse through the surrounding region of lesser activity. There is no great difficulty in extending the mathematics to cover this point in particular cases. But if the active region is too wide the system no longer approximates the behaviour of a thin ring and one can no longer expect the tentacles to form a single whorl. This also cannot be considered in detail in the present paper.
In the case of woodruff the leaves appear in whorls on the stem, the number of leaves in a whorl varying considerably, sometimes being as few as five or as many as nine. The numbers in consecutive whorls on the same stem are often equal, but by no means invariably. It is to be presumed that the whorls originate in rings of active tissue in the meristematic area, and that the rings arise at sufficiently great distance to have little influence on one another. The number of leaves in the whorl will presumably be obtainable by the rule given above, viz. by dividing the chemical wave-length into the circumference, though both these quantities will have to be given some new interpretation more appropriate to woodruff than to the ring. Another important example of a structure with polygonal symmetry is provided by young root fibres just breaking out from the parent root. Initially these are almost homogeneous in cross-section, but eventually a ring of fairly evenly spaced spots appear, and these later develop into vascular strands. In this case again the full explanation must be in terms of a two-dimensional or even a three-dimensional problem, although the analysis for the ring is still illuminating. When the cross-section is very large the strands may be in more than one ring, or more or less randomly or hexagonally arranged. The two-dimensional theory (not expounded here) also goes a long way to explain this.
Flowers might appear superficially to provide the most obvious examples of polygonal symmetry, and it is probable that there are many species for which this 'waves round a ring' theory is essentially correct. But it is certain that it does not apply for all species. If it did it would follow that, taking flowers as a whole, i.e. mixing up all species, there would be no very markedly preferred petal (or corolla, segment, stamen, etc.) numbers. For when all species are taken into account one must expect that the diameters of the rings concerned will take on nearly all values within a considerable range, and that neighbouring diameters will be almost equally common. There may also be some variation in chemical wave-length. Neighbouring values of the ratio circumferences to wave-length should therefore be more or less equally frequent, and this must mean that neighbouring petal numbers will have much the same frequency. But this is not borne out by the facts. The number five is extremely common, and the number seven rather rare. Such facts are, in the author's opinion, capable of explanation on the basis of morphogen theory, and are closely connected with the theory of phyllotaxis. They cannot be considered in detail here.
The case of a filament of tissue calls for some comment. The equilibrium patterns on such a filament will be the same as on a ring, which has been cut at a point where the concentrations of the morphogens are a maximum or a minimum. This could account for the segmentation of such filaments. It should be noticed, however, that the theory will not apply unmodified for filaments immersed in water.
#### 12. Chemical waves on spheres. Gastrulation
The treatment of homogeneity breakdown on the surface of a sphere is not much more difficult than in the case of the ring. The theory of spherical harmonics, on which it is based, is not, however, known to many that are not mathematical specialists. Although the essential properties of spherical harmonics that are used are stated below, many readers will prefer to proceed directly to the last paragraph of this section.
The anatomical structure concerned in this problem is a hollow sphere of continuous tissue such as a blastula. It is supposed sufficiently thin that one can treat it as a 'spherical shell'. This latter assumption is merely for the purpose of mathematical simplification; the results are almost exactly similar if it is omitted. As in § 7 there are to be two morphogens, and _a_ , _b_ , _c_ , _d_ , _μ_ ′, _ν_ ′, _h_ , _k_ are also to have the same meaning as they did there. The operator will be used here to mean the superficial part of the Laplacian, i.e. will be an abbreviation of
where _θ_ and _ϕ_ are spherical polar co-ordinates on the surface of the sphere and _ρ_ is its radius. The equations corresponding to (7.1) may then be written
It is well known (e.g. Jeans 1927, chapter 8) that any function on the surface of the sphere, or at least any that is likely to arise in a physical problem, can be 'expanded in spherical surface harmonics'. This means that it can be expressed in the form
The expression in the square bracket is described as a 'surface harmonic of degree _n_ '. Its nearest analogue in the ring theory is a Fourier component. The essential property of a spherical harmonic of degree _n_ is that when the operator is applied to it the effect is the same as multiplication by − _n_ ( _n_ \+ 1)/ _ρ_ 2. In view of this fact it is evident that a solution of (12.1) is
where _q n_ and are the two roots of
and
This is the most general solution, since the coefficients and can be chosen to give any required values of _X, Y_ when _t_ = 0, except when (12.3) has two equal roots, in which case a treatment is required which is similar to that applied in similar circumstances in § 7. The analogy with § 7 throughout will indeed be obvious, though the summation with respect to _m_ does not appear there. The meaning of this summation is that there are a number of different patterns with the same wave-length, which can be superposed with various amplitude factors. Then supposing that, as in § 8, one particular wave-length predominates, (12.2) reduces to In other words, the concentrations of the two morphogens are proportional, and both of them are surface harmonics of the same degree _n_ 0, viz. that which makes the greater of the roots _q n0_, have the greatest value.
It is probable that the forms of various nearly spherical structures, such as radiolarian skeletons, are closely related to these spherical harmonic patterns. The most important application of the theory seems, however, to be to the gastrulation of a blastula. Suppose that the chemical data, including the chemical wave-length, remain constant as the radius of the blastula increases. To be quite specific suppose that
With these values the system is quite stable so long as the radius is less than about 2. Near this point, however, the harmonics of degree 1 begin to develop and a pattern of form (12.5) with _n_ 0 = 1 makes its appearance. Making use of the facts that
it is seen that _X_ − _h_ is of the form
which may also be interpreted as
where _θ_ ′ is the angle which the radius _θ_ , _ϕ_ makes with the fixed direction having direction cosines proportional to _B_ , _C_ , _A_ and .
The outcome of the analysis therefore is quite simply this. Under certain not very restrictive conditions (which include a requirement that the sphere be relatively small but increasing in size) the pattern of the breakdown of homogeneity is axially symmetrical, not about the original axis of spherical polar coordinates, but about some new axis determined by the disturbing influences. The concentrations of the first morphogen are given by (12.7), where _θ_ ′ is measured from this new axis; and _Y_ – _k_ is proportional to _X_ – _h_. Supposing that the first morphogen is, or encourages the production of, a growth hormone, one must expect the blastula to grow in an axially symmetric manner, but at a greater rate at one end of the axis than at the other. This might under many circumstances lead to gastrulation, though the effects of such growth are not very easily determinable. They depend on the elastic properties of the tissue as well as on the growth rate at each point. This growth will certainly lead to a solid of revolution with a marked difference between the two poles, unless, in addition to the chemical instability, there is a mechanical instability causing the breakdown of axial symmetry. The direction of the axis of gastrulation will be quite random according to this theory. It may be that it is found experimentally that the axis is normally in some definite direction such as that of the animal pole. This is not essentially contradictory to the theory, for any small asymmetry of the zygote may be sufficient to provide the 'disturbance' which determines the axis.
#### 13. Non-linear theory. Use of digital computers
The 'wave' theory which has been developed here depends essentially on the assumption that the reaction rates are linear functions of the concentrations, an assumption which is justifiable in the case of a system just beginning to leave a homogeneous condition. Such systems certainly have a special interest as giving the first appearance of a pattern, but they are the exception rather than the rule. Most of an organism, most of the time, is developing from one pattern into another, rather than from homogeneity into a pattern. One would like to be able to follow this more general process mathematically also. The difficulties are, however, such that one cannot hope to have any very embracing _theory_ of such processes, beyond the statement of the equations. It might be possible, however, to treat a few particular cases in detail with the aid of a digital computer. This method has the advantage that it is not so necessary to make simplifying assumptions as it is when doing a more theoretical type of analysis. It might even be possible to take the mechanical aspects of the problem into account as well as the chemical, when applying this type of method. The essential disadvantage of the method is that one only gets results for particular cases. But this disadvantage is probably of comparatively little importance. Even with the ring problem, considered in this paper, for which a reasonably complete mathematical analysis was possible, the computational treatment of a particular case was most illuminating. The morphogen theory of phyllotaxis, to be described, as already mentioned, in a later paper, will be covered by this computational method. Non-linear equations will be used.
It must be admitted that the biological examples which it has been possible to give in the present paper are very limited. This can be ascribed quite simply to the fact that biological phenomena are usually very complicated. Taking this in combination with the relatively elementary mathematics used in this paper one could hardly expect to find that many observed biological phenomena would be covered. It is thought, however, that the imaginary biological systems which have been treated, and the principles which have been discussed, should be of some help in interpreting real biological forms.
##### _References_
Child, C. M. 1941. _Patterns and problems of development_. University of Chicago Press.
Davson, H. & Danielli, J. F. 1943. _The permeability of natural membranes_. Cambridge University Press.
Jeans, J. H. 1927. _The mathematical theory of elasticity and magnetism_ , 5th ed. Cambridge University Press.
Michaelis, L. & Menten, M. L. 1913. Die Kinetik der Invertinwirkung. _Biochemische Zeitschrift_ 49: 333–369.
Thompson, Sir D'Arcy 1942. _On growth and form_ , 2nd ed. Cambridge University Press.
Waddington, C. H. 1940. _Organisers and genes_. Cambridge University Press.
### **CHAPTER 16
Chess ( _1953_ )**
_Alan Turing_
#### Introduction
##### _Jack Copeland_
#### Chess and AI
Chess and some other board games are a test-bed for ideas in Artificial Intelligence. Donald Michie—Turing's wartime colleague and subsequently founder of the Department of Machine Intelligence and Perception at the University of Edinburgh—explains the relevance of chess to AI:
Computer chess has been described as the _Drosophila melanogaster_ of machine intelligence. Just as Thomas Hunt Morgan and his colleagues were able to exploit the special limitations and conveniences of the _Drosophila_ fruit fly to develop a methodology of genetic mapping, so the game of chess holds special interest for the study of the representation of human knowledge in machines. Its chief advantages are: (1) chess constitutes a fully defined and well-formalized domain; (2) the game challenges the highest levels of human intellectual capacity; (3) the challenge extends over the full range of cognitive functions such as logical calculation, rote learning, concept-formation, analogical thinking, imagination, deductive and inductive reasoning; (4) a massive and detailed corpus of chess knowledge has accumulated over the centuries in the form of chess instructional works and commentaries; (5) a generally accepted numerical scale of performance is available in the form of the U.S. Chess Federation and International ELO rating system.
#### History of Computer Chess
In 1945, in his paper 'Proposed Electronic Calculator', Turing predicted that computers would probably play 'very good chess', an opinion echoed in 1949 by Claude Shannon of Bell Telephone Laboratories, another leading early theoretician of computer chess. By 1958, Herbert Simon and Allen Newell were predicting that within ten years the world chess champion would be a computer, unless barred by the rules. Just under forty years later, on 11 May 1997, IBM's Deep Blue beat the reigning world champion, Gary Kasparov, in a six-game match.
Turing was theorizing about the mechanization of chess as early as 1941. Fellow codebreakers at GC & CS remember him experimenting with two heuristics now commonly used in computer chess, _minimax_ and _best-first_. The minimax heuristic involves assuming that one's opponent will move in such a way as to maximize their gains; one then makes one's own move in such a way as to minimize the losses caused by the opponent's expected move. The best-first heuristic involves ranking the moves available to a player by means of a rule of thumb scoring system and examining the consequences of the highest-scoring move first.
#### Turochamp
In 1948 Turing and the mathematical economist David Champernowne designed a chess-playing routine known as 'Turochamp'.
Champernowne later gave this description of Turochamp:
It was in the late summer of 1948 that Turing and I did try out a loose system of rules for deciding on the next move in a chess game which we thought could be fairly easily programmed for a computer. My long-suffering wife, a beginner at chess, took on the system and lost.... Here is what I think I remember about the system but I may have been influenced by what I have since read about other people's systems. There was a system for estimating the effects of any move on White's estimated net advantage over Black. This allowed for:
(1) Captures, using the conventional scale of 10 for pawn, 30 for knight or bishop, 50 for rook, 100 for queen and something huge, say 5000, for king.
(2) Change in mobility; i.e., change in the number of squares to which any piece or pawn could immediately move legitimately (1 each).
(3) Special incentives for: (a) Castling (3 points). (b) Advancing a passed pawn (1 or 2 points). (c) Getting a rook onto the seventh rank (4 points perhaps).
(I don't think occupation of one of the 4 central squares gained any special bonus. We did not cater to the end-game.) Most of our attention went to deciding which moves were to be followed up. My memory about this is infuriatingly weak. Captures had to be followed up at least to the point where no further capture was immediately possible. Checks and forcing moves had to be followed further. We were particularly keen on the idea that whereas certain moves would be scorned as pointless and pursued no further others would be followed quite a long way down certain paths. In the actual experiment I suspect we were a bit slapdash about all this and must have made a number of slips since the arithmetic was extremely tedious with pencil and paper. Our general conclusion was that a computer should be fairly easy to programme to play a game of chess against a beginner and stand a fair chance of winning or least reaching a winning position.
Turing started to code the Turochamp for the Ferranti Mark I computer at Manchester University but he never completed the task.
#### The First Working Chess Programme
Dietrich Prinz, who worked for the engineering firm Ferranti Ltd., wrote the first chess programme to be fully implemented. It ran in November 1951 on the University of Manchester Ferranti Mark I. Prinz, like Michie, Strachey, and others, was influenced by an important article published in 1950 by Donald Davies, 'A Theory of Chess and Noughts and Crosses' (see Further Reading). Prinz 'learned all about programming the [Mark I] computer at seminars given by Alan Turing and Cecily Popplewell'. (Like Turing, Prinz wrote a programming manual for the Mark I.)
Prinz's programme was for solving simple problems of the mate-in-two variety. The programme would examine every possible move until a solution was found. On average, several thousand moves had to be examined in the course of solving a problem, and the programme was considerably slower than a human player. Unlike Prinz's programme, Turochamp could in principle play a complete game and operated not by exhaustive search but under the guidance of heuristics.
Prinz saw in chess programming an 'indication of the methods that might be used to treat structural or logistic problems occurring in other fields by means of electronic computers'. Prinz used the Ferranti Mark I to solve logical problems (as also did Audrey Bates), and in 1949, 1950, and 1951 Ferranti built three small experimental special-purpose computers for theorem-proving and other logical work. (This was several years before the Logic Theorist of Newell, Simon, and Shaw—often incorrectly said to be the first AI programme—made its debut at the Dartmouth conference (see 'Artificial Intelligence', above).)
#### Turing's Approach in 'Chess'
Turing says that the system of rules set out in 'Chess' is based on an 'introspective analysis' of his own thought processes when playing (but with 'considerable simplifications'). His system anticipates much that has become standard in chess programming: the use of heuristics to guide the search through the 'tree' of possible moves and counter-moves; the use of evaluation rules which assign numerical values, indicative of strength or weakness, to board configurations; the minimax strategy; and variable look-ahead whereby, instead of the consequences of every possible move being followed equally far, the 'more profitable moves [are] considered in greater detail than the less' (p. 571). Turing also realized the necessity of using 'an entirely different system for the end-game' (p. 574).
The learning procedure that Turing proposes in 'Chess' involves the machine trying out variations in its method of play—e.g. varying the numerical values that are assigned to the various pieces. The machine adopts any variation that leads to more satisfactory results. This procedure is an early example of a genetic algorithm or GA (see 'Artificial Life', above).
#### Chess Bulldozers
Turing likened the claim that no chess programme can outplay its programmer to the claim that no animal can swallow an animal heavier than itself. Both claims, he said, 'are, so far as I know, untrue'. He pointed out that a programme might outplay its programmer simply in virtue of 'the speed of the machine, which might make it feasible to carry the analysis... farther than a man could do in the same time' (p. 575).
Critics question the worth of research into computer chess. Noam Chomsky has said famously that a computer beating a grandmaster at chess is as interesting as a bulldozer winning a weight-lifting competition. Deep Blue did indeed bulldoze its way to victory—256 parallel processors enabled it to examine 200 million possible moves per second and to look ahead as many as fourteen turns of play.
The huge improvement in computer chess since Turing's day owes much more to advances in hardware engineering than to advances in AI. Massive increases in cpu speed and memory have meant that successive generations of machines have been able to examine increasingly many possible moves. Turing's expectation was that chess-programming would contribute to the study of how human beings think. In fact, little or nothing about human thought processes appears to have been learned from the series of projects that culminated in Deep Blue.
#### The Turing Test Again
In the introductory paragraphs of his essay, Turing touches on some philosophical issues 'unconnected with chess'. In a reference to the imitation game (Chapter 11), he again states his belief that one could 'make a machine which would answer questions put to it, in such a way that it would not be possible to distinguish its answers from those of a man' (p. 569). On the question of what justification there is for this, he says that he knows of 'no really convincing argument to support this belief and certainly of none to disprove it.'
#### Consciousness
In 'Chess' Turing is careful to distinguish the question whether one could make a machine that would play the imitation game successfully from the question 'Could one make a machine which would have feelings like you and I do?' (p. 569). Examples of 'feelings' are pleasure, grief, misery, and anger (Chapter 11, p. 451). The assertion 'No mechanism could feel' forms the basis of the objection to the Turing test that in Chapter 11 Turing dubs the 'Argument from Consciousness'. His remarks in the present chapter clarify the discussion that he gives of this objection in Chapter 11.
The answer that Turing gives in 'Chess' to his question 'Could one make a machine which would have feelings' is this (p. 569): 'I shall never know, any more than I shall ever be quite certain that _you_ feel as I do.' Thus his view appears to be that the question 'Can machines think?' is independent of the question whether machines can feel, and that an affirmative answer may be given to the former in the absence of our having any answer at all to the latter. If he is right—and arguably he is—then the assertion 'No mechanism could feel', even if true, goes wide of the mark as an objection to the Turing test. As Turing sensibly says in Chapter 11 (pp. 452–3):
I do not wish to give the impression that I think there is no mystery about consciousness.... But I do not think these mysteries necessarily need to be solved before we can answer the question with which we are concerned in this paper [i.e. 'Can machines think?'].
#### Consciousness and Computer Simulation
Turing asserts (p. 569) that in the case of the questions 'Could one make a machine to play chess, and to improve its play?', and 'Could one make a machine which would answer questions put to it, in such a way that it would not be possible to distinguish its answers from those of a man?', the phrase 'Could one make a machine to...' might equally well be replaced by 'Could one programme an electronic computer to...'. On pp. 569–70 Turing supports this claim with an argument that also appears, in somewhat different forms, in Chapters 11 and :
if some other machine had been constructed to do the job we could use an electronic computer (of sufficient storage capacity), suitably programmed, to calculate what this machine would do, and in particular what answer it would give.
(If no superior method presented itself, then the computer could be supplied with a 'look-up table' setting out the behaviour of the other machine (see Chapter 11).)
In the case, however, of the question 'Could one make a machine which would have feelings like you and I do?', Turing indicates (on p. 569) that replacing 'Could one make a machine to...' by 'Could one programme an electronic computer to...' might possibly result in a question that is not equivalent. Turing says in Chapter 13 that 'any machine of a certain very wide class' can be replaced equivalently by a digital computer, but notes that not _every_ machine can be so replaced, giving as examples of those that cannot bulldozers, steam-engines, and telescopes (p. 483). Presumably, therefore, Turing considered it an open question whether a machine capable of feeling belongs to the first or second of these two classes.
#### The Church–Turing Thesis
'Chess' contains a formulation of the Church–Turing thesis that does not appear elsewhere in Turing's writings (p. 570):
If one can explain quite unambiguously in English, with the aid of mathematical symbols if required, how a calculation is to be done, then it is always possible to programme any digital computer to do that calculation, provided the storage capacity is adequate.
As elsewhere (e.g. in Chapter 10 (p. 414) and Chapter 17 (pp. 588–9)), Turing emphasizes here that this thesis 'is not the sort of thing that admits of clear cut proof', but nevertheless 'amongst workers in the field it is regarded as being clear as day' (p. 570). (The phrase 'how a calculation is to be done' should be understood as being elliptical for some such phrase as 'how a calculation is to be done by an obedient clerk working in accordance with a systematic method'.)
Turing applies this thesis to the question 'Could one programme an electronic computer to play a reasonably good game of chess?' He points out that, in view of the thesis, there is no need to set out an actual programme in order to show that the answer to the question is affirmative. It suffices to explain, 'unambiguously in English', the rules by which the machine is to choose its move in each position. This Turing proceeds to do.
##### _Further reading_
Davies, D. W., 'A Theory of Chess and Noughts and Crosses', _Science News_ , 16 (1950), 40–64.
Newborn, M., _Kasparov Versus Deep Blue: Computer Chess Comes of Age_ , (Springer: New York, 1997).
Shannon, C. E. 'Programming a Computer for Playing Chess', _Philosophical Magazine_ , 41 (1950), 256–75.
—— 'A Chess-Playing Machine', _Scientific American_ , 182 (1950), 48–51.
##### _Provenance_
Turing's essay 'Chess' was published in the 1953 collection _Faster Than Thought_ , where it formed a section of a chapter entitled 'Digital Computers Applied To Games'. (Edited by Vivian Bowden, _Faster Than Thought_ is a fascinating survey of the then state of the art in digital computing.) The chapter was co-authored by Audrey Bates, Bowden, and Christopher Strachey. Bowden's editorial remarks do not make it clear which parts of 'Digital Computers Applied To Games' were written by which author. Fortunately, Turing's typescript of his contribution has survived. It is this which is printed here.
#### Chess
When one is asked 'Could one make a machine to play chess?', there are several possible meanings which might be given to the words. Here are a few:
i) Could one make a machine which would obey the rules of chess, i.e. one which would play random legal moves, or which could tell one whether a given move is a legal one?
ii) Could one make a machine which would solve chess problems, e.g. tell one whether, in a given position, white has a forced mate in three?
iii) Could one make a machine which would play a reasonably good game of chess, i.e. which, confronted with an ordinary (that is, not particularly unusual) chess position, would after two or three minutes of calculation, indicate a passably good legal move?
iv) Could one make a machine to play chess, and to improve its play, game by game, profiting from its experience?
To these we may add two further questions, unconnected with chess, which are likely to be on the tip of the reader's tongue.
v) Could one make a machine which would answer questions put to it, in such a way that it would not be possible to distinguish its answers from those of a man?
vi) Could one make a machine which would have feelings like you and I do?
The problem to be considered here is iii), but to put this problem into perspective with the others I shall give the very briefest of answers to each of them.
To i) and ii) I should say 'This certainly can be done. If it has not been done already it is merely because there is something better to do.'
Question iii) we are to consider in greater detail, but the short answer is 'Yes, but the better the standard of play required, the more complex will the machine be, and the more ingenious perhaps the designer.'
To iv) and v) I should answer 'I believe so. I know of no really convincing argument to support this belief and certainly of none to disprove it.'
To vi) I should say 'I shall never know, any more than I shall ever be quite certain that _you_ feel as I do.'
In each of these problems except possibly the last, the phrase 'Could one make a machine to...' might equally well be replaced by 'Could one programme an electronic computer to...'. Clearly the electronic computer so programmed would itself constitute a machine. And on the other hand if some other machine had been constructed to do the job we could use an electronic computer (of sufficient storage capacity), suitably programmed, to calculate what this machine would do, and in particular what answer it would give.
After these preliminaries let us give our minds to the problem of making a machine, or of programming a computer, to play a tolerable game of chess. In this short discussion it is of course out of the question to provide actual programmes, but this does not really matter on account of the following principle:
If one can explain quite unambiguously in English, with the aid of mathematical symbols if required, how a calculation is to be done, then it is always possible to programme any digital computer to do that calculation, provided the storage capacity is adequate.
This is not the sort of thing that admits of clear cut proof, but amongst workers in the field it is regarded as being clear as day. Accepting this principle, our problem is reduced to explaining 'unambiguously in English' the rules by which the machine is to choose its move in each position. For definiteness we will suppose the machine is playing white.
If the machine could calculate at an infinite speed, and also had unlimited storage capacity, a comparatively simple rule would suffice, and would give a result that in a sense could not be improved on. This rule could be stated:
Consider every possible continuation of the game from the given position. There is only a finite number of them (at any rate if the fifty-move rule makes a draw obligatory, not merely permissive). Work back from the end of these continuations, marking a position with white to play as 'win' if there is a move which turns it into a position previously marked as 'win'. If this does not occur, but there is a move which leads to a position marked 'draw', then mark the position 'draw'. Failing this, mark it 'lose'. Mark a position with black to play by a similar rule with 'win' and 'lose' interchanged. If after this process has been completed it is found that there are moves which lead to a position marked 'win', one of these should be chosen. If there is none marked 'win' choose one marked 'draw' if such exists. If all moves lead to a position marked 'lose', any move may be chosen.
Such a rule is practically applicable in the game of noughts and crosses, but in chess is of merely academic interest.
Even when the rule can be applied it is not very appropriate for use against a weak opponent, who may make mistakes which ought to be exploited.
In spite of the impracticability of this rule it bears some resemblance to what one really does when playing chess. One does not follow all the continuations of play, but one follows some of them. One does not follow them until the end of the game, but one follows them a move or two, perhaps more. Eventually a position seems, rightly or wrongly, too bad to be worth further consideration, or (less frequently) too good to hesitate longer over. The further a position is from the one on the board the less likely is it to occur, and therefore the shorter is the time which can be assigned for its consideration. Following this idea we might have a rule something like this:
Consider all continuations of the game consisting of a move by white, a reply by black, and another move and reply. The value of the position at the end of each of these sequences of moves is estimated according to some suitable rule. The values at earlier positions are then calculated by working backwards move by move as in the theoretical rule given before. The move to be chosen is that which leads to the position with the greatest value.
It is possible to arrange that no two positions have the same value. The rule is then unambiguous. A very simple form of values, but one not having this property, is an 'evaluation of material', e.g. on the basis
If _B_ is black's total and _W_ is white's, then _W_ / _B_ is quite a good measure of value. This is better than _W_ − _B_ as the latter does not encourage exchanges when one has the advantage. Some small extra arbitrary function of position may be added to ensure definiteness in the result.
The weakness of this rule is that it follows all combinations equally far. It would be much better if the more profitable moves were considered in greater detail than the less. It would also be desirable to take into account more than mere 'value of material'.
After this introduction I shall describe a particular set of rules, which could without difficulty be made into a machine programme. It is understood that the machine is white and white is next to play. The current position is called the 'position of the board', and the positions arising from it by later moves 'positions in the analysis'.
##### _'Considerable' Moves, i.e. Moves to be considered in the analysis by the machine_
Every possibility for white's next move and for black's reply is 'considerable'. If a capture is considerable then any recapture is considerable. The capture of an undefended piece or the capture of a piece of higher value by one of lower value is always considerable. A move giving checkmate is considerable.
##### _Dead position_
A position in the analysis is dead if there are no considerable moves in that position, i.e. if it is more than two moves ahead of the present position, and no capture or recapture or mate can be made in the next move.
##### _Value of position_
The value of a dead position is obtained by adding up the piece values as above, and forming the ratio _W_ / _B_ of white's total to black's. In other positions with white to play the value is the greatest value of: ( _a_ ) the positions obtained by considerable moves, or ( _b_ ) the position itself evaluated as if a dead position, the latter alternative to be omitted if all moves are considerable. The same process is to be undertaken for one of black's moves, but the machine will then choose the _least_ value.
##### _Position-play value_
Each white piece has a certain position-play contribution and so has the black king. These must all be added up to give the position-play value.
For a Q, R, B or Kt, count
i) The square root of the number of moves the piece can make from the position, counting a capture as two moves, and not forgetting that the king must not be left in check.
ii) (If not a Q) 1·0 if it is defended, and an additional 0·5 if twice defended.
For a K, count
iii) For moves other than castling as i) above.
iv) It is then necessary to make some allowance for the vulnerability of the K. This can be done by assuming it to be replaced by a friendly Q on the same square, estimating as in i), but subtracting instead of adding.
v) Count 1·0 for the possibility of castling later not being lost by moves of K or rooks, a further 1·0 if castling could take place on the next move, and yet another 1·0 for the actual performance of castling.
For a P, count
vi) 0·2 for each rank advanced.
vii) 0·3 for being defended by at least one piece (not P).
For the black K, count
viii) 1·0 for the threat of checkmate.
ix) 0·5 for check.
We can now state the rule for play as follows.
The move chosen must have the greatest possible value, and, consistent with this, the greatest possible position-play value. If this condition admits of several solutions a choice may be made at random, or according to an arbitrary additional condition.
Note that no 'analysis' is involved in position-play evaluation. This is in order to reduce the amount of work done on deciding the move.
The game below was played between this machine and a weak player who did not know the system. To simplify the calculations the square roots were rounded off to one decimal place (i.e. the table below was used). No 'random choices' actually arose in this game. The increase of position-play value is given after white's move if relevant. An asterisk indicates that every other move had a lower position-play value. 0—0 indicates castling.
Numerous criticisms of the machine's play may be made. It is quite defenceless against 'forks', although it may be able to see certain other kinds of combination. It is of course not difficult to devise improvements of the programme [so] that these simple forks are foreseen. The reader may be able to think of some such improvements for himself. Since no claim is made that the above rule is particularly good, I have been content to leave this flaw without remedy; clearly a line has to be drawn between the flaws which one will attempt to eliminate and those which must be accepted as a risk. Another criticism is that the scheme proposed, although reasonable in the middle game, is futile in the end game. The changeover from the middle game to the end-game is usually sufficiently clean cut for it to be possible to have an entirely different system for the end-game. This should of course include quite definite programmes for the standard situations, such as mate with rook and king, or king and pawn against king. There is no intention to discuss the end-game further here.
If I were to sum up the weakness of the above system in a few words I would describe it as a caricature of my own play. It was in fact based on an introspective analysis of my thought processes when playing, with considerable simplifications. It makes oversights which are very similar to those which I make myself, and which may in both cases be ascribed to the considerable moves being rather inappropriately chosen. This fact might be regarded as supporting the rather glib view which is often expressed, to the effect that 'one cannot programme a machine to play a better game than one plays oneself'. This statement should I think be compared with another of rather similar form. 'No animal can swallow an animal heavier than himself.' Both statements are, so far as I know, untrue. They are also both of a kind that one is rather easily bluffed into accepting, partly because one thinks that there ought to be some rather slick way of demonstrating them, and one does not like to admit that one does not see what this argument is. They are also both supported by normal experience, and need rather exceptional cases to falsify them. The statement about chess programming may be falsified quite simply by the speed of the machine, which might make it feasible to carry the analysis a move farther than a man could do in the same time. This effect is rather less than might be supposed. Although electronic computers are very fast where conventional computing is concerned, their advantage is much reduced where enumeration of cases, etc., is involved on a large scale. Take for instance the problem of counting the possible moves from a given position in chess. If the number is 30 a man might do it in 45 seconds and the machine in 1 second. The machine has still an advantage, but it is much less overwhelming than it would be for instance where calculating cosines.
In connection with the question, numbered iv) above, as to the ability of a chess-machine to profit from experience, one can see that it would be quite possible to programme the machine to try out variations in its method of play (e.g. variations in piece value) and adopt the one giving the most satisfactory results. This could certainly be described as 'learning', though it is not quite representative of learning as we know it. It might also be possible to programme the machine to search for new types of combination in chess. If this product produced results which were quite new, and also interesting to the programmer, who should have the credit? Compare this with the situation where the Defence Minister has given orders for research to be done to find a counter to the bow and arrow. Should the inventor of the shield have the credit, or should the Defence Minister?
### **CHAPTER 17
Solvable and Unsolvable Problems ( _1954_ )**
_Alan Turing_
#### Introduction
##### _Jack Copeland_
#### Unsolvable Problems
In Chapter 1 Turing proves the existence of mathematical problems that cannot be solved by the universal Turing machine. There he also advances the thesis, now called the Church–Turing thesis, that any systematic method for solving mathematical problems can be carried out by the universal Turing machine. Combining these two propositions yields the result that there are mathematical problems which cannot be solved by any systematic method—cannot, in other words, be solved by any algorithm.
#### Substitution Puzzles
In 'Solvable and Unsolvable Problems' Turing sets out to explain this result to a lay audience. The article first appeared in _Science News_ , a popular science journal of the time. Starting from concrete examples of problems that do admit of algorithmic solution, Turing works his way towards an example of a problem that is not solvable by any systematic method. Loosely put, this is the problem of sorting puzzles into those that will 'come out' and those that will not. Turing gives an elegant argument showing that a sharpened form of this problem is not solvable by means of a systematic method (pp. 591–2).
The sharpened form of the problem involves what Turing calls 'the substitution type of puzzle'. An typical example of a substitution puzzle is this. Starting with the word BOB, is it possible to produce BOOOB by replacing selected occurrences of the pair OB by BOOB and selected occurences of the triple BOB by O? The answer is yes:
Turing suggests that any puzzle can be re-expressed as a substitution puzzle. Some row of letters can always be used to represent the 'starting position' envisaged in a particular puzzle, e.g. in the case of a chess problem, the pieces on the board and their positions. Desired outcomes, for example board positions that count as wins, can be described by further rows of letters, and the rules of the puzzle, whatever they are, are to be represented in terms of permissible substitutions of groups of letters for other groups of letters.
As Turing points out, it is not only 'toy' puzzles that can be re-expressed as substitution puzzles, but also mathematical problems, for instance the problem of finding a proof of a given mathematical theorem within an axiom system (which Turing describes as 'a very good example of a puzzle'). The axioms— which are simply strings of mathematical symbols—form the starting position. The theorem—another string of symbols—is the winning position. The rules of the puzzle are substitutions that enable strings of mathematical symbols to be transformed into other strings, much as in the case of the transition from BOB to BBOOB in the earlier example.
Turing calls the substitution formulation of any puzzle its 'normal form' and states the following _normal form principle_ (p. 588):
Given any puzzle, we can find a corresponding substitution puzzle which is equivalent to it in the sense that given a solution of the one we can easily use it to find a solution of the other.
#### Normal Forms and the Church–Turing Thesis
The normal form principle for puzzles closely parallels the Church–Turing thesis, which says that given any systematic method, we can find a corresponding Turing machine that is equivalent to it.
Neither the normal form principle for puzzles nor the Church–Turing thesis is susceptible to definite proof (see 'Computable Numbers: A Guide'). While few doubt that the Church–Turing thesis is in fact true, the very nature of the thesis has always been a matter for debate. Church, for example, described the thesis as a _definition_. Post, on the other hand, described it as a 'working hypothesis' that is in need of 'continual verification', and he criticized Church for masking this hypothesis as a definition. Turing's remarks in 'Solvable and Unsolvable Problems' about the status of the normal form principle for puzzles are of outstanding interest for the light that they may cast on his view concerning the status of the Church–Turing thesis. In this connection, see also the material from Turing's draft typescript quoted in n. 9 on p. 590.
Turing says of the normal form principle (pp. 588–9):
The statement is... one which one does not attempt to prove. Propaganda is more appropriate to it than proof, for its status is something between a theorem and a definition. In so far as we know a _priori_ what is a puzzle and what is not, the statement is a theorem. In so far as we do not know what puzzles are, the statement is a definition which tells us something about what they are. One can of course define a puzzle by some phrase beginning, for instance, 'A set of definite rules...', but this just throws us back on the definition of 'definite rules'. Equally one can reduce it to the definition of 'computable function' or 'systematic procedure'. A definition of any one of these would define all the rest.
Turing would perhaps have said much the same concerning not only the Church–Turing thesis but also the thesis introduced in Chapter 13:
A digital computer will replace any rival design of calculating machine.
In so far as we do not know what calculating machines are, the statement is a definition which tells us something about what they are.
#### Proof of Unsolvability
Having introduced the normal form principle for puzzles, Turing turns to his central project of establishing that 'there cannot be any systematic procedure for determining whether a puzzle be solvable or not' (p. 590). In particular, there cannot be a systematic procedure for determining whether substitution puzzles are or are not solvable. Turing argues by _reductio ad absurdum_. He shows that the supposition that there is a systematic procedure for determining whether substitution puzzles are or are not solvable leads to an outright contradiction, and on that basis concludes that there can be no such procedure. The argument turns on the impossibility of applying a certain procedure to itself.
Any systematic procedure is in effect a puzzle, since in following the procedure one applies rules to some 'starting position' until one or another result is achieved. So if there were a systematic procedure for determining whether each puzzle is or is not solvable, then by the normal form principle, there is a substitution puzzle—call it K—that is equivalent to this procedure. When applied to any substitution puzzle, K—if it exists—must 'come out' either with the result SOLVABLE or with the result NOT SOLVABLE. Since K is applicable to any substitution puzzle, K can be applied to itself in order to determine whether it itself is or is not solvable. Turing shows (p. 592) that this supposed ability of K to pronounce on its own solvability leads to outright contradiction, and so concludes that K cannot exist.
#### The Meaning of 'Unsolvable'
Turing points out that the result he has established, namely that there is no systematic method for deciding whether or not substitution puzzles come out, is often expressed by saying that there is no _decision procedure_ for puzzles of this type, and that the _decision problem_ for this type of puzzle is _unsolvable_. He continues (p. 592): 'so one comes to speak (as in the title of this article) about "unsolvable problems" meaning in effect puzzles for which there is no decision procedure. This is the technical meaning which the words are now given by mathematical logicians.'
As Turing says, this terminology is potentially confusing. It is natural to use the words 'unsolvable problem' to mean a problem for which no solution can possibly be found. It would be a confusion to think that Turing has shown that the problem of deciding whether or not substitution puzzles come out is an unsolvable problem in this natural sense. Indeed, with sufficient time, inventiveness, and patience, mathematicians may always be able to establish whether or not any given substitution puzzle comes out. If that is so, then the problem of deciding whether or not substitution puzzles come out is solvable, in the natural sense of the word.
What Turing has shown is that there is no systematic method for deciding whether or not substitution puzzles come out, i.e. there is no general procedure, applicable by rote, that one can employ in order to decide whether or not each substitution puzzle comes out. The 'decision problem' for substitution puzzles is the problem of finding such a rote procedure (a 'decision procedure'); in showing that there is no such procedure, Turing has shown that the decision problem for substitution puzzles is unsolvable in the natural sense.
Turing therefore recommends that, in order to 'minimize confusion', one should 'always speak of "unsolvable decision problems", rather than just "unsolvable problems"' (p. 592).
#### Significance of Turing's Result
Turing ends the chapter with a comment on the significance of what he has shown. His result concerning the decision problem for substitution puzzles 'may be regarded as going some way towards a demonstration, within mathematics itself, of the inadequacy of "reason" unsupported by common sense'. For he has, he says, set 'certain bounds to what we can hope to achieve purely by reasoning'.
The phrase 'purely by reasoning' here presumably means 'purely by algorithmic methods'. Some mathematical problems require for their solution not only 'reason', in this sense, but also what Turing refers to in Chapter 3 as 'intuition' (see also Chapter 4). There he says (pp. 192–3):
The activity of the intuition consists in making spontaneous judgements which are not the result of conscious trains of reasoning.... Often it is possible to find some other way of verifying the correctness of an intuitive judgement. We may, for instance, judge that all positive integers are uniquely factorizable into primes; a detailed mathematical argument leads to the same result. This argument will also involve intuitive judgements, but they will be less open to criticism than the original judgement about factorization.... The necessity for using the intuition is... greatly reduced by setting down formal rules for carrying out inferences which are always intuitively valid.... In pre-Gödel times it was thought by some that it would probably be possible to carry this programme to such a point that all the intuitive judgements of mathematics could be replaced by a finite number of these rules. The necessity for intuition would then be entirely eliminated.
The argument of 'Solvable and Unsolvable Problems' illustrates why it is that the need for intuition cannot always be eliminated in favour of formal rules.
#### Gödel's Theorem
Turing notes that the unsolvability of the decision problem for substitution puzzles affords an elegant proof of the following rather general statement (p. 593):
no systematic method of proving mathematical theorems is sufficiently complete to settle every mathematical question, yes or no.
The proof Turing gives is as follows. Each statement of the form 'such-and-such substitution puzzle comes out' can be expressed in the form of a mathematical statement. So if there were a systematic method of settling every question that can be posed in mathematical form, this method would serve as a decision procedure for substitution puzzles. Given that there is no such decision procedure, it follows that no systematic method is able to settle every mathematical question.
Turing remarks that the above statement follows 'by a famous theorem of Gödel' and describes himself as providing 'an independent proof' of the statement (p. 593). Turing might also have pointed out that his own 'On Computable Numbers' yields a proof of this statement.
Gödel's famous incompleteness theorem of 1931 is, however, importantly less general than the above statement, since it concerns only one particular systematic method of proving mathematical theorems, the system set out by Whitehead and Russell in _Principia Mathematica_ (as explained in 'Computable Numbers: A Guide'). Gödel did later generalize his result of 1931 to all formal systems (containing a certain amount of arithmetic), but emphasized the importance that Turing's work played in this generalization. Gödel said in 1964:
[D]ue to A. M. Turing's work, a precise and unquestionably adequate definition of the general concept of formal system can now be given... Turing's work gives an analysis of the concept of 'mechanical procedure' (alias 'algorithm' or 'computation procedure' or 'finite combinatorial procedure').... A formal system can simply be defined to be any mechanical procedure for producing formulas, called provable formulas.
In his references to Gödel's work, Turing hides his own light under a bushel.
##### _Further reading_
Boone, W. W., review of Turing's 'The Word Problem in Semi-Groups with Cancellation', _Journal of Symbolic Logic_ , 17 (1952), 74–6.
Turing, A. M., 'The Word Problem in Semi-Groups with Cancellation', _Annals of Mathematics_ , 52 (1950), 491–505. Reprinted in _Pure Mathematics: Collected Works of A. M. Turing_ , ed. J. L. Britton (Amsterdam: North-Holland, 1992).
##### _Provenance_
What follows is the text of the original printing of 'Solvable and Unsolvable Problems' in _Science News._ Unfortunately Turing's own typescript appears to have been lost. However, a sizeable fragment of a draft typescript, with additions in Turing's handwriting, has been preserved. (Turing recycled the draft pages, covering the reverse sides with handwritten notes concerning morphogenesis.) The fragment corresponds to pp. 584–9. For the most part the published version follows the draft pages closely (except for punctuation and occasional changes of word and word-order). Significant differences between the draft and the published version are mentioned in footnotes.
#### Solvable and Unsolvable Problems
If one is given a puzzle to solve one will usually, if it proves to be difficult, ask the owner whether it can be done. Such a question should have a quite definite answer, yes or no, at any rate provided the rules describing what you are allowed to do are perfectly clear. Of course the owner of the puzzle may not know the answer. One might equally ask, 'How can one tell whether a puzzle is solvable?', but this cannot be answered so straightforwardly. The fact of the matter is that there is no systematic method of testing puzzles to see whether they are solvable or not. If by this one meant merely that nobody had ever yet found a test which could be applied to any puzzle, there would be nothing at all remarkable in the statement. It would have been a great achievement to have invented such a test, so we can hardly be surprised that it has never been done. But it is not merely that the test has never been found. It has been proved that no such test ever can be found.
Let us get away from generalities a little and consider a particular puzzle. One which has been on sale during the last few years and has probably been seen by most of the readers of this article illustrates a number of the points involved quite well. The puzzle consists of a large square within which are some smaller movable squares numbered 1 to 15, and one empty space, into which any of the neighbouring squares can be slid leaving a new empty space behind it. One may be asked to transform a given arrangement of the squares into another by a succession of such movements of a square into an empty space. For this puzzle there is a fairly simple and quite practicable rule by which one can tell whether the transformation required is possible or not. One first imagines the transformation carried out according to a different set of rules. As well as sliding the squares into the empty space one is allowed to make moves each consisting of two interchanges, each of one pair of squares. One would, for instance, be allowed as one move to interchange the squares numbered 4 and 7, and also the squares numbered 3 and 5. One is permitted to use the same number in both pairs. Thus one may replace 1 by 2, 2 by 3, and 3 by 1 as a move because this is the same as interchanging first (1, 2) and then (1, 3). The original puzzle is solvable by sliding if it is solvable according to the new rules. It is not solvable by sliding if the required position can be reached by the new rules, together with a 'cheat' consisting of _one single_ interchange of a pair of squares. Suppose, for instance, that one is asked to get back to the standard position—
One may, according to the modified rules, first get the empty square into the correct position by moving the squares 15 and 12, and then get the squares 1, 2, 3,... successively into their correct positions by the interchanges (1, 10), (2, 10), (3, 4), (4, 5), (5, 9), (6, 10), (7, 10), (9, 11), (10, 11), (11, 15). The squares 8, 12, 13, 14, 15 are found to be already in their correct positions when their turns are reached. Since the number of interchanges required is even, this transformation is possible by sliding. If one were required after this to interchange say square 14 and 15 it could not be done.
This explanation of the theory of the puzzle can be regarded as entirely satisfactory. It gives one a simple rule for determining for any two positions whether one can get from one to the other or not. That the rule is so satisfactory depends very largely on the fact that it does not take very long to apply. No mathematical method can be useful for any problem if it involves much calculation. It is nevertheless sometimes interesting to consider whether something is possible at all or not, without worrying whether, in case it _is_ possible, the amount of labour or calculation is economically prohibitive. These investigations that are not concerned with the amount of work involved are in some ways easier to carry out, and they certainly have a greater aesthetic appeal. The results are not altogether without value, for if one has proved that there is no method of doing something it follows _a fortiori_ that there is no practicable method. On the other hand, if one method has been proved to exist by which the decision can be made, it gives some encouragement to anyone who wishes to find a workable method.
From this point of view, in which one is only interested in the question, 'Is there a systematic way of deciding whether puzzles of this kind are solvable?', the rules which have been described for the sliding-squares puzzle are much more special and detailed than is really necessary. It would be quite enough to say: 'Certainly one can find out whether one position can be reached from another by a systematic procedure. There are only a finite number of positions in which the numbered squares can be arranged (viz. 20922789888000) and only a finite number (2, 3, or 4) of moves in each position. By making a list of all the positions and working through all the moves, one can divide the positions into classes, such that sliding the squares allows one to get to any position which is in the same class as the one started from. By looking up which classes the two positions belong to one can tell whether one can get from one to the other or not.' This is all, of course, perfectly true, but one would hardly find such remarks helpful if they were made in reply to a request for an explanation of how the puzzle should be done. In fact they are so obvious that under such circumstances one might find them some how rather insulting. But the fact of the matter is, that if one is interested in the question as put, 'Can one tell by a systematic method in which cases the puzzle is solvable?', this answer is entirely appropriate, because one wants to know if there is a systematic method, rather than to know of a good one.
The same kind of argument will apply for any puzzle where one is allowed to move certain 'pieces' around in a specified manner, provided that the total number of essentially different positions which the pieces can take up is finite. A slight variation on the argument is necessary in general to allow for the fact that in many puzzles some moves are allowed which one is not permitted to reverse. But one can still make a list of the positions, and list against these first the positions which can be reached from them in one move. One then adds the positions which are reached by two moves and so on until an increase in the number of moves does not give rise to any further entries. For instance, we can say at once that there is a method of deciding whether a patience can be got out with a given order of the cards in the pack: it is to be understood that there is only a finite number of places in which a card is ever to be placed on the table. It may be argued that one is permitted to put the cards down in a manner which is not perfectly regular, but one can still say that there is only a finite number of 'essentially different' positions. A more interesting example is provided by those puzzles made (apparently at least) of two or more pieces of very thick twisted wire which one is required to separate. It is understood that one is not allowed to bend the wires at all, and when one makes the right movement there is always plenty of room to get the pieces apart without them ever touching, if one wishes to do so. One may describe the positions of the pieces by saying where some three definite points of each piece are. Because of the spare space it is not necessary to give these positions quite exactly. It would be enough to give them to, say, a tenth of a millimetre. One does not need to take any notice of movements of the puzzle as a whole: in fact one could suppose one of the pieces quite fixed. The second piece can be supposed to be not very far away, for, if it is, the puzzle is already solved. These considerations enable us to reduce the number of 'essentially different' positions to a finite number, probably a few hundred millions, and the usual argument will then apply. There are some further complications, which we will not consider in detail, if we do not know how much clearance to allow for. It is necessary to repeat the process again and again allowing successively smaller and smaller clearances. Eventually one will find that either it can be solved, allowing a small clearance margin, or else it cannot be solved even allowing a small margin of 'cheating' (i.e. of 'forcing', or having the pieces slightly overlapping in space). It will, of course, be understood that this process of trying out the possible positions is not to be done with the physical puzzle itself, but on paper, with mathematical descriptions of the positions, and mathematical criteria for deciding whether in a given position the pieces overlap, etc.
These puzzles where one is asked to separate rigid bodies are in a way like the 'puzzle' of trying to undo a tangle, or more generally of trying to turn one knot into another without cutting the string. The difference is that one is allowed to bend the string, but not the wire forming the rigid bodies. In either case, if one wants to treat the problem seriously and systematically one has to replace the physical puzzle by a mathematical equivalent. The knot puzzle lends itself quite conveniently to this. A knot is just a closed curve in three dimensions nowhere crossing itself; but, for the purpose we are interested in, any knot can be given accurately enough as a series of segments in the directions of the three coordinate axes. Thus, for instance, the trefoil knot (Figure 1 _a_) may be regarded as consisting of a number of segments joining the points given, in the usual ( _x, y, z_ ) system of coordinates, as (1, 1, 1), (4, 1, 1,), (4, 2, 1), (4, 2, −1), (2, 2, −1), (2, 2, 2), (2, 0, 2), (3, 0, 2), (3, 0, 0), (3, 3, 0), (1, 3, 0), (1, 3, 1), and returning again with a twelfth segment to the starting point (1, 1, 1). This representation of the knot is shown in perspective in Figure 1 _b_. There is no special virtue in the representation which has been chosen. If it is desired to follow the original curve more closely a greater number of segments must be used. Now let _a_ and _d_ represent unit steps in the positive and negative X-directions respectively, _b_ and _e_ in the Y-directions, and _c_ and _f_ in the Z-directions: then this knot may be described as _aaabffddccceeaffbbbddcee_. One can then, if one wishes, deal entirely with such sequences of letters. In order that such a sequence should represent a knot it is necessary and sufficient that the numbers of a's and d's should be equal, and likewise the number of _b_ 's equal to the number of _e_ 's and the number of _c_ 's equal to the number of _f_ 's, and it must not be possible to obtain another sequence of letters with these properties by omitting a number of consecutive letters at the beginning or the end or both. One can turn a knot into an equivalent one by operations of the following kinds—
**Figure 1.** ( _a_ ) The trefoil knot ( _b_ ) a possible representation of this knot as a number of segments joining points.
(i) One may move a letter from one end of the row to the other.
(ii) One may interchange two consecutive letters provided this still gives a knot.
(iii) One may introduce a letter _a_ in one place in the row, and _d_ somewhere else, or _b_ and _e_ , or _c_ and _f_ , or take such pairs out, provided it still gives a knot.
(iv) One may replace _a_ everywhere by _aa_ and _d_ by _dd_ or replace each _b_ and _e_ by _bb_ and _ee_ or each _c_ and _f_ by _cc_ and _ff_. One may also reverse any such operation.
—and these are all the moves that are necessary.
It is also possible to give a similar symbolic equivalent for the problem of separating rigid bodies, but it is less straightforward than in the case of knots.
These knots provide an example of a puzzle where one cannot tell in advance how many arrangements of pieces may be involved (in this case the pieces are the letters _a, b, c, d, e, f_ ), so that the usual method of determining whether the puzzle is solvable cannot be applied. Because of rules (iii) and (iv) the lengths of the sequences describing the knots may become indefinitely great. No systematic method is yet known by which one can tell whether two knots are the same.
Another type of puzzle which we shall find very important is the 'substitution puzzle'. In such a puzzle one is supposed to be supplied with a finite number of different kinds of counters, perhaps just black _(B)_ and white _(W)_. Each kind is in unlimited supply. Initially a number of counters are arranged in a row and one is asked to transform it into another pattern by substitutions. A finite list of the substitutions allowed is given. Thus, for instance, one might be allowed the substitutions
and be asked to transform _WBW_ into _WBBBW_ , which could be done as follows
Here the substitutions used are indicated by the numbers below the arrows, and their effects by underlinings. On the other hand if one were asked to transform _WBB_ into _BW_ it could not be done, for there are no admissible steps which reduce the number of _B_ 's.
It will be seen that with this puzzle, and with the majority of substitution puzzles, one cannot set any bound to the number of positions that the original position might give rise to.
It will have been realized by now that a puzzle can be something rather more important than just a toy. For instance the task of proving a given mathematical theorem within an axiomatic system is a very good example of a puzzle.
It would be helpful if one had some kind of 'normal form' or 'standard form' for describing puzzles. There is, in fact, quite a reasonably simple one which I shall attempt to describe. It will be necessary for reasons of space to take a good deal for granted, but this need not obscure the main ideas. First of all we may suppose that the puzzle is somehow reduced to a mathematical form in the sort of way that was used in the case of the knots. The position of the puzzle may be described, as was done in that case, by sequences of symbols in a row. There is usually very little difficulty in reducing other arrangements of symbols (e.g. the squares in the sliding squares puzzle) to this form. The question which remains to be answered is, 'What sort of rules should one be allowed to have for rearranging the symbols or counters?' In order to answer this one needs to think about what kinds of processes ever do occur in such rules, and, in order to reduce their number, to break them up into simpler processes. Typical of such processes are counting, copying, comparing, substituting. When one is doing such processes, it is necessary, especially if there are many symbols involved, and if one wishes to avoid carrying too much information in one's head, either to make a number of jottings elsewhere or to use a number of marker objects as well as the pieces of the puzzle itself. For instance, if one were making a copy of a row of counters concerned in the puzzle it would be as well to have a marker which divided the pieces which have been copied from those which have not and another showing the end of the portion to be copied. Now there is no reason why the rules of the puzzle itself should not be expressed in such a way as to take account of these markers. If one does express the rules in this way they can be made to be just substitutions. This means to say that the _normal form for puzzles is the substitution type of puzzle_. More definitely we can say:
_Given any puzzle we can find a corresponding_ substitution puzzle _which is equivalent to it in the sense that given a solution of the one we can easily use it to find a solution of the other. If the original puzzle is concerned with rows of pieces of a finite number of different kinds, then the substitutions may be applied as an alternative set of rules to the pieces of the original puzzle. A transformation can be carried out by the rules of the original puzzle if and only if it can be carried out by the substitutions and leads to a final position from which all marker symbols have disappeared_.
This statement is still somewhat lacking in definiteness, and will remain so. I do not propose, for instance, to enter here into the question as to what I mean by the word 'easily'. The statement is moreover one which one does not attempt to prove. Propaganda is more appropriate to it than proof, for its status is something between a theorem and a definition. In so far as we know a priori what is a puzzle and what is not, the statement is a theorem. In so far as we do not know what puzzles are, the statement is a definition which tells us something about what they are. One can of course define a puzzle by some phrase beginning, for instance, 'A set of definite rules...', but this just throws us back on the definition of 'definite rules'. Equally one can reduce it to the definition of 'computable function' or 'systematic procedure'. A definition of any one of these would define all the rest. Since 1935 a number of definitions have been given, explaining in detail the meaning of one or other of these terms, and these have all been proved equivalent to one another and also equivalent to the above statement. In effect there is no opposition to the view that every puzzle is equivalent to a substitution puzzle.
After these preliminaries let us think again about puzzles as a whole. First let us recapitulate. There are a number of questions to which a puzzle may give rise. When given a particular task one may ask quite simply
(a) _Can this be done?_
Such a straightforward question admits only the straightforward answers, 'Yes' or 'No', or perhaps 'I don't know'. In the case that the answer is 'Yes' the answerer need only have done the puzzle himself beforehand to be sure. If the answer is to be 'No', some rather more subtle kind of argument, more or less mathematical, is necessary. For instance, in the case of the sliding squares one can state that the impossible cases _are_ impossible because of the mathematical fact that an odd number of simple interchanges of a number of objects can never bring one back to where one started. One may also be asked
(b) _What is the best way of doing this?_
Such a question does not admit of a straightforward answer. It depends partly on individual differences in people's ideas as to what they find easy. If it is put in the form, 'What is the solution which involves the smallest number of steps?', we again have a straightforward question, but now it is one which is somehow of remarkably little interest. In any particular case where the answer to (a) is 'Yes' one can find the smallest possible number of steps by a tedious and usually impracticable process of enumeration, but the result hardly justifies the labour.
When one has been asked a number of times whether a number of different puzzles of similar nature can be solved one is naturally led to ask oneself
(c) _Is there a systematic procedure by which I can answer these questions, for puzzles of this type?_
If one were feeling rather more ambitious one might even ask
(d) _Is there a systematic procedure by which one can tell whether a puzzle is solvable?_
I hope to show that the answer to this last question is 'No'.
There are in fact certain types of puzzle for which the answer to (c) is 'No'.
Before we can consider this question properly we shall need to be quite clear what we mean by a 'systematic procedure' for deciding a question. But this need not now give us any particular difficulty. A 'systematic procedure' was one of the phrases which we mentioned as being equivalent to the idea of a puzzle, because either could be reduced to the other. If we are now clear as to what a puzzle is, then we should be equally clear about 'systematic procedures'. In fact a systematic procedure is just a puzzle _in which there is never more than one possible move in any of the positions which arise and in which some significance is attached to the final result_.
Now that we have explained the meaning both of the term 'puzzle' and of 'systematic procedure', we are in a position to prove the assertion made in the first paragraph of this article, that there cannot be any systematic procedure for determining whether a puzzle be solvable or not. The proof does not really require the detailed definition of either of the terms, but only the relation between them which we have just explained. Any systematic procedure for deciding whether a puzzle were solvable could certainly be put in the form of a puzzle, with unambiguous moves (i.e. only one move from any one position), and having for its starting position a combination of the rules, the starting position and the final position of the puzzle under investigation.
The puzzle under investigation is also to be described by its rules and starting position. Each of these is to be just a row of symbols. As we are only considering substitution puzzles, the rules need only be a list of all the substitution pairs appropriately punctuated. One possible form of punctuation would be to separate the first member of a pair from the second by an arrow, and to separate the different substitution pairs with colons. In this case the rules
_B_ may be replaced by _BC_
_WBW_ may be deleted
would be represented by ': B→ BC : WBW→ :'. For the purposes of the argument which follows, however, these arrows and colons are an embarrassment. We shall need the rules to be expressed without the use of any symbols which are barred from appearing in the starting positions. This can be achieved by the following simple, though slightly artificial trick. We first double all the symbols other than the punctuation symbols, thus ': _BB_ → _BBCC_ : _WWBBWW_ →:'. We then replace each arrow by a single symbol, which must be different from those on either side of it, and each colon by three similar symbols, also chosen to avoid clashes. This can always be done if we have at least three symbols available, and the rules above could then be represented as, for instance, ' _CCCBBWBBCC BBBWWBBWWBWWW'_. Of course according to these conventions a great variety of different rows of symbols will describe essentially the same puzzle. Quite apart from the arbitrary choice of the punctuating symbols the substitution pairs can be given in any order, and the same pair can be repeated again and again.
Now let _P (R, S)_ stand for 'the puzzle whose rules are described by the row of symbols _R_ and whose starting position is described by _S_ '. Owing to the special form in which we have chosen to describe the rules of puzzles, there is no reason why we should not consider _P (R, R)_ for which the 'rules' also serve as starting position: in fact the success of the argument which follows depends on our doing so. The argument will also be mainly concerned with puzzles in which there is at most one possible move in any position; these may be called 'puzzles with unambiguous moves'. Such a puzzle may be said to have 'come out' if one reaches either the position _B_ or the position _W_ , and the rules do not permit any further moves. Clearly if a puzzle has unambiguous moves it cannot both come out with the end result _B_ and with the end result _W_.
We now consider the problem of classifying rules _R_ of puzzles into two classes, I and II, as follows:
_Class I_ is to consist of sets _R_ of rules, which represent puzzles with unambiguous moves, and such that _P(R,R)_ comes out with the end result _W_.
_Class II_ is to include all other cases, i.e. either _P(R,R)_ does not come out, or comes out with the end result _B_ , or else _R_ does not represent a puzzle with unambiguous moves. We may also, if we wish, include in this class sequences of symbols such as _BBBBB_ which do not represent a set of rules at all.
Now suppose that, contrary to the theorem that we wish to prove, we have a systematic procedure for deciding whether puzzles come out or not. Then with the aid of this procedure we shall be able to distinguish rules of class I from those of class II. There is no difficulty in deciding whether _R_ really represents a set of rules, and whether they are unambiguous. If there is any difficulty it lies in finding the end result in the cases where the puzzle is known to come out: but this can be decided by actually working the puzzle through. By a principle which has already been explained, this systematic procedure for distinguishing the two classes can itself be put into the form of a substitution puzzle (with rules _K_ , say). When applying these rules _K_ , the rules _R_ of the puzzle under investigation form the starting position, and the end result of the puzzle gives the result of the test. Since the procedure always gives an answer, the puzzle _P(K,R)_ always comes out. The puzzle _K_ might be made to announce its results in a variety of ways, and we may be permitted to suppose that the end result is _B_ for rules _R_ of class I, and W for rules of class II. The opposite choice would be equally possible, and would hold for a slightly different set of rules _K_ ′, which however we do not choose to favour with our attention. The puzzle with rules _K_ may without difficulty be made to have unambiguous moves. Its essential properties are therefore:
_K_ has unambiguous moves.
_P(K,R)_ always comes out whatever _R_.
If _R_ is in class I, then _P(K,R)_ has end result _B_.
If _R_ is in class II, then _P(K,R)_ has end result _W_.
These properties are however inconsistent with the definitions of the two classes. If we ask ourselves which class _K_ belongs to, we find that neither will do. The puzzle _P(K,K)_ is bound to come out, but the properties of _K_ tell us that we must get end result _B_ if _K_ is in class I and _W_ if it is in class II, whereas the definitions of the classes tell us that the end results must be the other way round. The assumption that there was a systematic procedure for telling whether puzzles come out has thus been reduced to an absurdity.
Thus in connexion with question (c) above we can say that there are some types of puzzle for which no systematic method of deciding the question exists. This is often expressed in the form, 'There is no _decision procedure_ for this type of puzzle', or again, 'The decision problem for this type of puzzle is unsolvable', and so one comes to speak (as in the title of this article) about 'unsolvable problems' meaning in effect puzzles for which there is no decision procedure. This is the technical meaning which the words are now given by mathematical logicians. It would seem more natural to use the phrase 'unsolvable problem' to mean just an unsolvable puzzle, as for example 'to transform 1, 2, 3 into 2, 1, 3 by cyclic permutation of the symbols', but this is not the meaning it now has. However, to minimize confusion I shall here always speak of 'unsolvable decision problems', rather than just 'unsolvable problems', and also speak of puzzles rather than problems where it is puzzles and not decision problems that are concerned.
It should be noticed that a decision problem only arises when one has an infinity of questions to ask. If you ask, 'Is this apple good to eat?', or 'Is this number prime?', or 'Is this puzzle solvable?' the question can be settled with a single 'Yes' or 'No'. A finite number of answers will deal with a question about a finite number of objects, such as the apples in a basket. When the number is infinite, or in some way not yet completed concerning say all the apples one may ever be offered, or all whole numbers or puzzles, a list of answers will not suffice. Some kind of rule or systematic procedure must be given. Even if the number concerned is finite one may still prefer to have a rule rather than a list: it may be easier to remember. But there certainly cannot be an unsolvable decision problem in such cases, because of the possibility of using finite list.
Regarding decision problems as being concerned with classes of puzzles, we see that if we have a decision method for one class it will apply also for any subclass. Likewise, if we have proved that there is no decision procedure for the subclass, it follows that there is none for the whole class. The most interesting and valuable results about unsolvable decision problems concern the smaller classes of puzzle.
Another point which is worth noticing is quite well illustrated by the puzzle which we considered first of all in which the pieces were sliding squares. If one wants to know whether the puzzle is solvable with a given starting position, one can try moving the pieces about in the hope of reaching the required end-position. If one succeeds, then one will have solved the puzzle and consequently will be able to answer the question, 'Is it solvable?' In the case that the puzzle is solvable one will eventually come on the right set of moves. If one has also a procedure by which, if the puzzle is unsolvable, one would eventually establish the fact that it was so, then one would have a solution of the decision problem for the puzzle. For it is only necessary to apply both processes, a bit of one alternating with a bit of the other, in order eventually to reach a conclusion by one or the other. Actually, in the case of the sliding squares problem, we have got such a procedure, for we know that if, by sliding, one ever reaches the required final position, with squares 14 and 15 interchanged, then the puzzle is impossible.
It is clear then that the difficulty in finding decision procedures for types of puzzle lies in establishing that the puzzle is unsolvable in those cases where it is unsolvable. This, as was mentioned on page [589], requires some sort of mathematical argument. This suggests that we might try expressing the statement that the puzzle comes out in a mathematical form and then try and prove it by some systematic process. There is no particular difficulty in the first part of this project, the mathematical expression of the statement about the puzzle. But the second half of the project is bound to fail, because by a famous theorem of Gödel no systematic method of proving mathematical theorems is sufficiently complete to settle every mathematical question, yes or no. In any case we are now in a position to give an independent proof of this. If there were such a systematic method of proving mathematical theorems we could apply it to our puzzles and for each one eventually either prove that it was solvable or unsolvable; this would provide a systematic method of determining whether the puzzle was solvable or not, contrary to what we have already proved.
This result about the decision problem for puzzles, or, more accurately speaking, a number of others very similar to it, was proved in 1936–7. Since then a considerable number of further decision problems have been shown to be unsolvable. They are all proved to be unsolvable by showing that if they were solvable one could use the solution to provide a solution of the original one. They could all without difficulty be reduced to the same unsolvable problem. A number of these results are mentioned very shortly below. No attempt is made to explain the technical terms used, as most readers will be familiar with some of them, and the space required for the explanation would be quite out of proportion to its usefulness in this context.
(1) It is not possible to solve the decision problem even for substitution processes applied to rows of black and white counters only.
(2) There are certain particular puzzles for which there is no decision procedure, the rules being fixed and the only variable element being the starting position.
(3) There is no procedure for deciding whether a given set of axioms leads to a contradiction or not.
(4) The 'word problem in semi-groups with cancellation' is not solvable.
(5) It has recently been announced from Russia that the 'word problem in groups' is not solvable. This is a decision problem not unlike the 'word problem in semi-groups', but very much more important, having applications in topology: attempts were being made to solve this decision problem before any such problems had been proved unsolvable. No adequately complete proof is yet available, but if it is correct this is a considerable step forward.
(6) There is a set of 102 matrices of order 4, with integral coefficients such that there is no decision method for determining whether another given matrix is or is not expressible as a product of matrices from the given set.
These are, of course, only a selection from the results. Although quite a number of decision problems are now known to be unsolvable, we are still very far from being in a position to say of a given decision problem, whether it is solvable or not. Indeed, we shall never be quite in that position, for the question whether a given decision problem is solvable is itself one of the undecidable decision problems. The results which have been found are on the whole ones which have fallen into our laps rather than ones which have positively been searched for. Considerable efforts have however been made over the word problem in groups (see (5) above). Another problem which mathematicians are very anxious to settle is known as 'the decision problem of the equivalence of manifolds'. This is something like one of the problems we have already mentioned, that concerning the twisted wire puzzles. But whereas with the twisted wire puzzles the pieces are quite rigid, the 'equivalence of manifolds' problem concerns pieces which one is allowed to bend, stretch, twist, or compress as much as one likes, without ever actually breaking them or making new junctions or filling in holes. Given a number of interlacing pieces of plasticine one may be asked to transform them in this way into another given form. The decision problem for this class of problem is the 'decision problem for the equivalence of manifolds'. It is probably unsolvable, but has never been proved to be so. A similar decision problem which might well be unsolvable is the one concerning knots which has already been mentioned.
The results which have been described in this article are mainly of a negative character, setting certain bounds to what we can hope to achieve purely by reasoning. These, and some other results of mathematical logic may be regarded as going some way towards a demonstration, within mathematics itself, of the inadequacy of 'reason' unsupported by common sense.
##### _Further reading_
Kleene, S. C. _Introduction to Metamathematics, Amsterdam_ , 1952.
## **Index**
Ackermann, W. 49, 84
ADA 28
Adcock, F. 207, 219
Agnus Dei (2nd bombe) 255
Aiken, H. H. 29, 363
Alexander, C. H. O'D 257, 258, 259, 261, 263, 264, 265, 340, 345
Alexander, S. 368
Alexandria 'Y' (intercept) station 274
all inclusive (logic formula) 179ff
Allanson, J. T. 360
_a_ -machine 60, 156
_see also_ choice machine
analogy 492, 498–9
Analytical Engine 28–30, 363, 446, 455, 482
Andreae, J. H. 385
Apple Macintosh 366
Aquinas, St Thomas 450
ARPANET 92
Artificial Intelligence
bacteriophage and 516
discussion concerning 494–506
draughts (checkers) and 356–8, 514
genetic algorithms and 430–1, 513–14, 565, 575
history of 1, 2, 3, 353–60, 401–32, 469–70, 487–506, 562–6
intelligence as emotional concept 431, 491–2, 500, 501
origin of term 353
situated 439
Turing pioneers 2, 3, 353–9, 374–6, 392–4, 401–9, 410–32, 433–9, 441–63, 465–71, 472–5, 476–8, 480, 482–6, 487–92, 494–5, 563–4, 565, 566–7, 569–75
_see also_ brain, chess, connectionism, expert system, heuristic, learning, Mathematical Objection, mechanical theorem proving, neural simulation, neuron-like computation, thinking machine, Turing test, search
Artificial Life
AI and 439, 508
computer simulation and 507, 508, 510, 560–1
genetic algorithms and 513–14
history of 507–17
meaning of term 507
self-reproduction and 515–16
Turing pioneers 1, 3, 401, 405, 508–14, 517, 519–61
von Neumann and 22, 513, 514–17
_see also_ connectionism, Fibonacci number, gastrulation, genetic algorithm, morphogenesis, neural simulation, neuron-like computation, non-linear equations, phyllotaxis, reaction-diffusion model, ring of cells or tissue
_Asperula odorata_ (woodruff) 556, 557
atomic bomb 22
A-type unorganized machine 417, 418, 427, 429
Automatic Computing Engine (ACE)
ACE Test Assembly 367, 398–400
AI and 356, 374–5, 392–4
as example of discrete controlling machine 412
as example of universal practical computing machine 415
Big ACE 368–9, 377
chess and 393
compared with trained P-type 428, 432
derivatives of 368–71
description of 365–6, 383–8
draughts (checkers) and 356
efforts to build 395–400
letter from Turing to Ross Ashby about 374–5
memory in 365, 366, 369, 376–7, 380–4
number of operations without repetition 411
optimum programming and 377
Pilot Model of 16, 92, 365, 367, 368, 369, 377, 397, 399, 400
preparation of problems for 389–91
random element in 391, 478
self-modification and 374–5, 392–3, 419, 462, 470
start-up procedure 390
storage capacity of 382–3, 413
Turing designs 2, 12, 27, 30, 31, 363–7, 369, 376–7, 378–94
universal Turing machine and 16, 378–9, 383
universality and 415–6
Version H (Huskey) 32
Version V (of Turing's design for) 399
Version VII (of Turing's design for) 370, 396
_see also_ DEUCE
axiomatic (class or property) 151ff
_see also_ choice machine
Babbage, C. 27–30, 236, 363, 371, 446, 455, 482
Babbage, D. 29
Baer, R. 128
Balme, D. 261
ban (unit) _see_ deciban
Banburismus
as method for reducing bombe time 256, 261–2
Banburies 261, 282
explanation of 281–5, 297, 299
origin of name 261
Turing invents 256, 261–2, 263, 279, 281, 285
use in Hut 8 261–2, 285ff, 311–12, 314
Bates, M. A. 565, 568
Baudot-Murray code _see_ teleprinter code
Bayley, D. 374
BBC 465, 476, 487, 493
Bell Telephone Laboratories 363, 393, 562
Bendix Corp. 369
Bernays, P. 48–9, 82, 127, 132, 191
Bernstein, F. 191
Bertrand, G. 234
best-first 563
Beurle, R. L. 360
Bigelow, J. 23
bigram tables (in Naval Enigma) 257, 258–61, 271–3, 280–1, 285–91, 311–12
BINAC 367
Birch, F. 219, 257, 259–60, 267, 279, 287–9, 292
BirkhoV, G. D. 130
Biuro Szyfrów 231ff
Bletchley Park
Babbage discussed at 29
establishment and development as codebreaking centre 217–20
history of AI and 353–5, 563
home of Colossus, first electronic computer 208–9, 362–3
Turing comes to 1, 205, 217, 220, 257, 279
work on Enigma at 217–31, 232, 234–5, 235–6, 238, 246–64, 265–312, 313–35, 336–40, 341–52, 353–5, 465
work on Fish at 207–9, 262–3, 362–3, 465
_see also_ Government Code and Cypher School
Block, N. 434, 437
bomba
demise of 233–4, 236, 246
explanation of name 235–7, 291, 314
function of 237–45
history of AI and 354
Poles invent 233, 292
Poles reveal to British and French 234
_Stecker_ and 245–6
bombe
appearance of 246–7, 248, 256, 291–2
Banburismus and 261–2, 285, 287, 311
bomba and 233, 234, 246
designers of 218, 246, 254–5, 263, 292, 327, 329
diagonal board of 254, 255, 323–34
explanation of name 235–7, 291, 314
first installed 253, 292
four-wheel 344 –5
function of 246–53, 291–4, 315–35
history of AI and 353–5, 469
importance of 2, 218, 256–7, 262
numbers of 256–7, 292, 311, 337, 338
simultaneous scanning and 254–5, 319–20, 321, 323, 327, 343– 4
Spider and 255, 320 –31
_Stecker_ and 235, 250 –5, 291, 293, 314ff
Turing and 218, 235, 246, 250–5, 314–35
Turing's feedback method and 254–5, 322–3
US Navy bombes 256–7, 342–5
use against Naval Enigma 218, 253, 285ff, 315ff
use of cribs with 240, 246, 248–55, 259, 287–8, 291, 293– 4, 307, 315–35, 339, 344
_see also_ Agnus Dei, menu, Victory
Borelli, G. A. 498
Bowden, B. V. 568
brain
analogy and 499
as machine 2, 374, 382, 403, 405, 407, 412–13, 418, 423–4, 425, 429–30, 431–2, 451, 456–7, 459–61, 478, 482, 483, 499, 500–1, 503–5
continuity and 412–13, 456 –7, 459
digital computer as 2, 374, 375, 476 –7, 478, 479, 480, 482–6, 500ff
electronic 374, 420, 484
free will and 479, 484
growth of 375, 517
higher parts of 400–1
imitation by computer 463, 456–7, 476, 477, 478, 479, 494–5, 483–5
intellectual search and 401, 430–1
learning and 408–9, 421, 423–4, 438
mechanical 482, 483, 484
of child 424, 429, 432, 438, 460
random element and 424, 478–9
storage capacity of 383, 393, 459, 483, 500–1
Turing machine and 407, 408, 424, 429
Turing test and 477, 479
_see also_ connectionism, human being as machine, learning, neural simulation, neuron-like computing
Braithwaite, R. B. 131, 487
British Tabulating Machine Co. 246, 330, 339
Brooks, R. 439
Brouwer, L. E. J. 96
Brunsviga 412, 413, 480
B-type unorganized machine 403–9, 418, 422, 429
universal Turing machine and 407, 422
Burali-Forti paradox 170
Burks, A. W. 24, 27, 32, 513, 514–15
Bush, V. 29
Butler, S. 475
C 12
calculating machine 479–80, 483, 487ff, 578, 591–2
Cambridge, University of 1, 15, 17, 27, 125, 127, 131, 133, 205, 219, 265, 355, 377, 400, 446, 487
Mathematical Laboratory 358, 367
_see also_ King's College, St John's College
Cantor theory of ordinals 161–70
Carnegie Mellon University 359
central letter (of Enigma crib) 251–5, 317ff
Chamberlain, N. 217
Champernowne, D. G. 130, 563–4
Chandler, W. W. 369–70, 396
checkers _see_ draughts
chess
exhaustive search and 503
genetic algorithm and 514, 565, 575
heuristic search and 353–4, 374, 470
history of computer chess 353, 356, 374, 375, 393, 562–6, 569–75
importance in AI 393, 394, 420, 439, 463, 473, 562, 566
learning and 375, 393, 492, 496, 498
Max Newman on 492, 495, 496, 498, 503, 504
Turing test and 431, 442
Turing's chess programmes 3, 353, 356, 412, 431, 563–4, 565, 570–5
_see also_ Turochamp
child-machine 460–3; _cf_ 424ff
choice machine 60, 77, 88
Chomsky, N. 565
Church, A.
Church–Turing thesis and 44–5, 577
comments on Turing as graduate student 126
corresponds with Turing 205
founder of _Journal of Symbolic Logic_ 205–6
introduces term 'Turing machine' 6
lambda calculus and 44, 52, 88, 126, 147ff, 205–7, 211, 214–15, 360
letter from Turing to concerning Post critique 92, 102
mentioned by Turing in correspondence 126, 127, 128, 134, 205–6, 207, 211, 213, 214
on effective calculability 44–5, 59, 125–6, 150
ordinal logics and 125–6, 134, 137, 146, 163, 177, 194, 205, 206
theory of types and 205–6, 213
work on _Entscheidungsproblem_ 45, 48, 49, 52, 59, 99, 125, 126, 207, 410, 450
Church's thesis 44–5;
_see also_ Church–Turing thesis
Church–Turing thesis
ACE and 378, 383
application of 43, 52, 53, 84–7
arguments in favour of 42–3, 45, 74–9
calculating machines and 479–80, 482–3, 578
chess and 567–8, 570
Church and 44–5, 577
converse of 43
Gödel and 45, 48, 581
statement of 40–5, 58, 74, 414, 567, 570, 576, 577
status of 42–3, 414, 568, 570, 577–8, 588–9, 590
Churchill, W. L. S. 262, 336–40, 342
CILLI 315
Clark, W. A. 360, 405–6
Clarke, J. 255, 258, 259, 330
class-subclass rule 462
Clayden, D. O. 31, 367, 368, 385
closure ('chain' in crib) 250ff, 317ff, 330
Cog (robot) 439
Colby, K. 489
Colebrook, F. M. 369, 400
Colossus 8, 208–9, 263, 362–3, 370, 373, 396, 480
colour (in Enigma) 227, 292
Commonwealth Scientific and Industrial Research Organisation (CSIRO) 367
computable function
Church–Turing thesis and 44–5, 150–1, 578, 589
computable number and 44, 58
lambda calculus and 151–2, 211
Max Newman on significance of 207
meaning of term 44, 58, 79–80
of integral variable 79–81, 151–2
ordinal logics and 152–4, 158–9, 162, 163, 191
_see also_ computable number, computable sequence
computable number
as opposed to definable number 58, 78–9
Church–Turing thesis and 41, 43, 58, 60
enumerability of 58, 72–4
examples 58, 79–83, 95
extent of 58, 74–9
meaning of term 36, 41, 58–61, 95–6
_see also_ axiomatic, Church–Turing thesis, computable function, computable sequence, effective calculability, effective method, general recursive, human computer, primitive recursive, systematic method, Turing machine, uncomputable number, uncomputable sequence
computable sequence
as opposed to definable 78–9
Church–Turing thesis and 43
computable function and 79ff
continuum hypothesis and 191–2
definition of computable number and 61, 95–6
diagonal argument and 34–5, 37–9, 72–4
effective calculability and 88–90
enumeration of 66–8, 72–4
meaning of term 33, 61
of logical systems 171ff
universal machine and 68
_see also_ computable number, uncomputable sequence
computer, history of
at Bell Telephone Laboratories 363
at Bletchley Park 8, 29, 208–9, 362–3, 373, 396
at Cambridge 17, 355, 358, 367, 377
at Commonwealth Scientific and Industrial Research Organization, Sydney 367
at EMI 370–1
at English Electric Co. 368, 369, 397
at Harvard 29, 363, 364
at IBM 17, 29, 357, 362
at Manchester 2–3, 16, 17, 30, 209, 355, 356 –7, 367, 369, 371–4, 396, 400, 401, 457, 480, 496, 508, 564, 565
at MIT 29, 367
at Moore School and Philadelphia 8, 16, 17, 21–7, 32, 364, 365, 366, 367, 373–4, 376, 380, 408
at National Physical Laboratory 2, 12, 16, 27, 30–2, 92, 209, 356, 363–70, 372–3, 374–7, 378–94, 395–400
at Packard-Bell 370–1
at Post Office Research Station, Dollis Hill 208–9, 263, 362–3, 369–70, 373, 395–6, 397, 398
at Princeton Institute for Advanced Study 16, 21–7, 32, 362, 373–4
at Radar Research and Development Establishment, Malvern 370
at Telecommunications Research Establishment, Malvern 208, 209, 373
at US Bureau of Standards 367–8
Babbage and 27–30, 236, 363, 446, 455
first 'personal' computer 369
Turing and 1, 2, 6, 9, 12, 15–17, 21–7, 30–1, 55, 58–87, 206, 207, 209, 363, 371, 375, 378–9, 383, 414–5
concept creation 492, 498–9
_see also_ learning
connectionism
early work on 360, 403, 405–6, 507–8
irregular verbs and 402, 429
meaning of term 360, 402
Turing anticipates 356, 403–5, 406–7, 408–9, 416–24, 429–30, 431–2, 510, 517
_see also_ A-type, B-type unorganized machine, learning, neuron-like computation
consciousness 451–3, 455, 456, 488, 566–7, 569
constatation 314ff, 349ff
continuum hypothesis 191–2
convertible _see_ lambda calculus
Coombs, A. W. M. 370, 396
Copernican theory 450
coral (Japanese cipher) 345
Courant, R. 127
crib
all wheel order crib 253, 290, 291
Banburismus and 256
cribbing, art of 294–311
Hut 8 Crib Room 294–5, 304
in early days 278–81, 285ff, 294, 297, 306, 311
meaning of term 237
mine-laying and 308
Poles' use of 278–9
types of 295–311
use with bombe 240, 246, 248–55, 259, 287–8, 291, 293–4, 307, 315–35, 339, 344
use with 'mini bomba' 237–8
W/T interception and 274, 275
worked examples 248–50, 295–300, 315ff, 347ff
_see also_ EINS catalogue, depth
CSIR Mark 1 (CSIRAC) 367
cultural search 430–1
Currier, P. 342
CYC 402
daily key 228ff, 421
D'Arcy Thompson, W. 508–9
Dartmouth College 353, 489, 565
Dartmouth Summer Research Project on Artificial Intelligence 353, 355, 359, 565
Darwin, C. G. 368, 370, 396, 397, 399, 400, 401
Davies, D. W. 21, 92–3, 367, 368, 564
Davis, M. 40, 41
D-Day 209
deciban 283ff
decision problem 6, 45ff, 143–4, 207, 393–4, 469, 472, 579, 592–5
_see also Entscheidungsproblem_ , unsolvable problem
Dedekind, J. W. R. 81
Deep Blue 563, 565–6
degree of unsolvability 99, 143–4
DENDRAL 360
Denniston, A. G. 217, 219, 234, 257, 279, 310, 337, 342–3
depth (in Enigma breaking) 281ff, 295–301, 302, 307, 311
Descartes, R. 498
Desch, J. 344
description number
diagonal argument and 34–5, 37–9, 72–4
halting problem and 39
meaning of term 10–12, 67–8, 69
of complete configuration 69, 89
of oracle machine 142, 156–7
ordinal logics and 184
Post critique and 98ff
printing problem and 39, 73–4
satisfactoriness problem and 36–7, 68, 72–4
determinism 416, 447, 466, 475, 477–9, 483, 484–5
_see also_ free will, partially random machine, prediction, random element
DEUCE 368, 369, 397
diagonal argument 34–5, 37–9, 72–4, 142, 157, 578–9, 591–2
diagonal board (of bombe) 254–5, 323–34
Difference Engine 28, 236, 363
differential analyser 29, 378, 412, 456–7, 480
discrete state machine
brain and 412–13, 456, 459
characterization of 412, 446–7
compared with continuous machinery 412–13, 446, 456–7, 459
complete description of behaviour of 413, 447, 448
cryptography and 421
digital computer as 446, 447, 450
Logical Computing Machine as 413
Mathematical Objection and 450–1
numbers of states of 413, 447–8, 453
partially random 416, 477–8
prediction of 447, 448, 455–6, 457, 475, 485, 500
simulation of by digital computer 448
thinking and 455
Turing test and 448, 456–7
universality and 448, 455
discriminant (in Enigma) 230, 273
Dollis Hill _see_ Post Office Research Station
draughts (in history of AI) 356–8, 514
Driscoll, A. M. 341–3, 345
dual (formula) 154ff
Eachus, J. J. 344
Eastcote (bombe outstation) 256
Eckert, J. P. 22, 25–7, 32, 367, 373, 376, 380
Eckert-Mauchly Computer Corp. 17
Eckert-Mauchly Electronic Control Co. 367
Eddington, A. S. 483
Edinburgh, University of 353, 359, 562
EDSAC 17, 358, 367, 377
education _see_ learning
EDVAC 25–7, 364, 365, 366, 373–4, 408
Edward VIII 129–30
effective calculability
abbreviation of treatment 150–2
Church on 44–5, 59, 125–6, 150
Church–Turing thesis and 44–5
computability and 44–5, 59, 88–90
Gentzen type ordinal logics and 194, 199
_see also_ effective method
effective method
Church–Turing thesis and 42, 45, 125–6, 137, 479, 480
duality and 158
meaning of term 42
Newman's test and 493
ordinal formula and 139, 170
rules of procedure and 171
_see also_ Church–Turing thesis, human computer, effective calculability, systematic method
EINS catalogue 286–7, 290, 291, 311
Einstein, A. 127
Eisenhart, L. 21, 131, 132
Ely, R. B. 344
EMI 370–1
EMI Business Machine 370–1
ENIAC 8, 22–3, 24, 26–7, 32, 364, 373, 376, 411, 412, 413, 480
Enigma
_Abwehr_ Enigma 246, 274
appearance of machine 220, 221, 222, 223, 224, 226
as example of apparently partially random machine 479
Battle of the Atlantic and 2, 218, 257–61, 262
breaking 231–64, 273–312, 314–35, 336–40, 341–52
design of machine 220–8
diagrams of machine 223, 224, 269
four-wheel 225, 262, 270, 271, 295, 343–5
German Air Force Enigma 220, 229–31, 233, 235, 255, 257, 279, 286, 291, 292, 293, 309, 339, 345
German Army Enigma 220, 229–31, 232, 233, 235, 255, 257, 279, 286, 291, 292, 293, 309, 339, 345
German Naval Enigma 2, 218, 220, 225, 226, 229, 233, 271–312, 338–9, 341–52
German Railway Enigma 322
history of AI and 353–5
indicator system for German Naval Enigma 257–8, 278–81
Italian Naval Enigma 217, 246
Knox's early work on Enigma 217, 232
O Bar machine 277
OP-20-G and 341–52
operating procedures for 227–31, 269ff
Polish work on Enigma 231–46, 257–8, 277–9, 292
Tunny compared with 207
Turing breaks German Naval Enigma 2, 206, 218, 253, 257–62, 279–82, 285–9, 314ff
Turing's first work on Enigma 217, 277, 279–81, 285–9
US attempts to break 342–5, 347–52
_see also_ Banburismus, Bletchley Park, bomba, bombe, crib, daily key, discriminant, Government Code and Cipher School, Hut 8, indicator, indicator setting, indicator system, key, message setting, Ringstellung, Stecker, turnover, wheel order
English Electric Co. 368, 397
_Entscheidungsproblem_ 6, 43, 45–53, 84–7, 125, 126, 207, 212, 393–4
_see also_ decision problem, unsolvable problem
Erskine, R. 29
expert system 360, 402
extra-sensory perception 457–8
Farley, B. G. 360, 405
Feferman, S. 140, 141
Feigenbaum, E. A. 360
females (in Enigma indicators) 236, 242, 245
Fermat's last theorem 141, 155, 191, 472
Ferranti Ltd. 17, 356–7, 564–5
Ferranti Mark I computer 3, 17, 356–7, 374, 437, 483, 496, 503, 508, 510, 517, 552, 564–5
_see also_ Manchester computers
Fibonacci number 508, 509, 517
Fieller, E. C. 399
Fish 207, 263
_see also_ Tunny, Sturgeon, Thrasher
Fleming, I. 259, 289
flip-flop _see_ Jordan Eccles trigger circuit
Flowerdown 'Y' (intercept) station 276
Flowers, T. H. 29, 208–9, 362–3, 369–70, 373, 395–6
flying bomb 275
formally definable (l-definable) 88ff, 149ff
_see also_ lambda calculus
Forrester, J. W. 367
fort (continuation) 230, 278–9
Foss, H. 290
Frankel, S. 22
Freeborn catalogue 282, 299, 311
Freeborn, F. 282, 297, 338–9
Freebornery (Hollerith section at Bletchley Park) 282, 338–9
free will 445, 449, 477–9, 484–5
French, R. M. 435, 490–1
G15 computer 369
Galilei, Galileo 450, 475
Gandy, R. O. 30, 42, 126, 400, 408, 433
gastrulation 509, 517, 519, 525, 558–60
Gauss, J. C. F. 411
General Problem Solver 359–60
general recursive function 150ff, 198ff
generalized recursion theory 143–4
generate-and-test 354
genetic algorithm (GA) 401, 430–1, 460, 463, 513–14, 565, 575
genetical search 430–1
_see also_ genetic algorithm
Gentzen, G. 49, 51, 135, 137, 139, 141, 194, 202
GO 473, 474
Gödel argument 468
_see also_ Mathematical Objection
Gödel, K.
general recursive functions and 150, 153
mentioned by Turing in correspondence 127, 213, 214
ordinal logics and 1, 126, 136, 137, 138–9, 140, 146, 177, 180, 192
remarks concerning Turing 45, 48, 581
Turing's influence on 45, 48, 581
work on incompleteness 1, 47–8, 59, 84, 126, 136, 138, 139, 140, 146, 160, 173, 189, 410, 411, 450, 467, 472, 580–1, 593
Gödel representation 74, 147ff
Gödel's incompleteness theorems
Hilbert programme and 47–9, 84, 126, 135–9
Mathematical Objection and 410, 411, 450, 467, 472
ordinal logics and 1, 126, 136, 137, 138–9, 140, 141, 146, 160, 178, 180, 189, 192–3, 206, 212, 213, 215
statement of 47–8, 84, 580–1
substitution puzzles and 580–1, 593
_see also_ Gödel
Goldstine, H. H. 22, 24, 25, 27, 32, 364, 515
Good, I. J. 2, 258
Goodwin, E. T. (Charles) 32
Government Code and Cypher School (GC & CS)
early history of 217–20
history of AI and 353–5, 563
significance of work of 2, 217, 262
Turing joins 1, 205, 217, 220, 257, 279
work on Enigma by 217–31, 232, 234–5, 235–6, 238, 246–64, 265–312, 313–35, 336–40, 341–52, 353–5, 465
work on Fish by 207–9, 262–3, 362–3, 465
_see also_ Bletchley Park, Room 40
Grey Walter, W. 508
growth _see_ Artificial Life, morphogenesis
Grundstellung 230, 271, 312ff
_see also_ indicator setting
Hall, P. 131
halting problem 39–40, 41
_see also_ satisfactoriness problem
Hanslope Park 263
Hardy, G. H. 53, 127, 128
Harper, J. 247
Hartree, D. R. 363, 368, 455, 476, 482
Harvard Automatic Sequence Controlled Calculator 29, 363, 364
Harvard University 363, 364
Harvie-Watt, Brigadier 337
hat book 308, 310
Hayes, J. G. 31, 400
Heath Robinson (codebreaking machine) 208, 263
Hebb, D. O. 403
Heimsoeth & Rinke Co. 277
Herbert (robot) 439
Herbrand, J. 45, 150, 153
Herivel, J. 335, 354–5
Herivelismus 335
heuristic 353–5, 356, 360, 514, 563, 564
_see also_ search
Hilbert programme 46–9, 52–3, 84–7, 126, 136–8, 142, 143, 215
Hilbert, D. 46–9, 53, 75, 77, 82, 84, 136, 139, 143, 177
_see also_ Hilbert programme
Hilton, P. 263, 371, 465
Hinsley, F. H. 218, 260, 287
Hiscocks, E. S. 398
Hitler, A. 209, 263, 264
Hodges, A. 263, 435, 436
Holden Agreement 344
Holland, J. 513
Hollerith, H. 29, 31
Hollerith punched card
description of punched card plug-board equipment 30–1
invention of 31
relation to Babbage's Analytical Engine 29
use at Bletchley Park 282, 286, 338–9
use in Automatic Computing Engine 31, 365, 388, 390
_see also_ Freebornery
human being as machine 2, 3, 354, 355, 358, 374–5, 382, 394–5, 401, 403, 405, 407, 408, 420, 421, 422, 423–4, 425, 429–30, 431–2, 438–9, 450–1, 456–7, 459–60, 478, 482, 483, 499, 500, 502, 508–9
_see also_ Artificial Intelligence, brain, connectionism, consciousness, free will, human computer, neuron-like computation, thinking machine
human computer
ACE and 378, 387, 391–2
Analytical Engine and 446
calculating machine and 479–80
characterization of 40
characterization of digital computer and 444–5, 447, 480
Church–Turing thesis and 41, 479–80
computable number and 41
history of computer and 40–1
systematic method and 42, 43
Turing machine as idealisation of 41, 42, 59, 75–7, 79
Huskey, H. D. 26, 32, 55, 365, 368, 369, 373, 398, 399
Hut 3 293
Hut 4 219, 257, 260
Hut 6 29, 219, 255, 274, 291, 292, 293, 309, 336, 338, 339, 345, 354
Hut 8
Alexander takes over 263
early personnel of 258
first breaks of into wartime traffic 259, 260, 273, 286–7, 289–91, 341
four wheel bombes and 256–7, 345
impact of on Battle of Atlantic 2, 217, 262
indispensability of Turing to 263
letter to Churchill concerning 338–40
loses and regains Shark 344
Mahon's history of, based on conversations with Turing 267–312
pinches and 259–61
Turing establishes 258
Turing leaves 262–3
uses Wrst bombes 253, 255, 259
Hydra 511, 519, 556
hydrogen bomb 22
IAS computer 362
_see also_ computer, history of at Princeton Institute for Advanced Study
IBM 17, 29, 362
IBM 701 computer 17, 357, 362
Illinois, University of 24
imitation game _see_ Turing test
indeterminacy principle 478, 483
index of experiences 466, 474
indicator
Banburismus and 282ff
explanation of 230–1
in Naval Enigma 257–8, 272, 273, 280–1
Narvik Pinch and 258–9, 286
Poles' method of attacking 240–6, 278, 279
Turing attacks 257–8, 279–81, 285
indicator setting 230ff, 354–5
_see also Grundstellung_ , indicator, indicator system
indicator system
boxing (or throw-on) 273, 277–8
change of 233, 235, 246
for German Naval Enigma 257–8, 259, 261, 271–3, 278–81, 286ff, 314
involving daily setting 231, 233
involving enciphering message setting twice 229–30, 231–3, 240–3, 258
meaning of term 231
induction 453, 454, 462
informality of behaviour 457
ingenuity (in mathematical proof) 135–8, 140, 192–3, 212–13, 215–16
initiative 429–31, 477
Institute for Advanced Study, Princeton 21, 23, 24, 125, 362
_see also_ IAS computer
intellectual search 430–1
interception of Naval Enigma traffic 273–6, 293
Internet 92
intuition (in mathematics) 126, 135–8, 140, 142–3, 192–3, 202, 206, 212–13, 215–16, 579–80
irregular verbs 402, 429
Ismay, H. 336
Jacquard loom 28
James, W. 134
Jefferson, G. 451–2, 455, 487–8, 492
Jeffreys sheets 315
Jones, Squadron Leader 292–3
Jordan Eccles trigger circuit 380, 385, 426
Kalmár, L. 43
Kasparov, G. 563
K Book (in Naval Enigma) 271–3, 276, 277, 290, 311, 312
Key (Enigma)
interception of 275
meaning of term 227
_Ausserheimische Gewässe_ 258
Bonito 273, 274
Bounce 273
Dolphin 257, 258, 259, 261, 271, 303, 308, 309, 310, 311, 341–3, 351
Hackle 303
_Heimische Gewässe_ 257, 258, 259, 341
Narwhal 271, 309
Plaice 271, 308, 309, 310
Porpoise 309
Red 230, 335, 354
Shark 271, 272, 286, 295, 309, 343, 344, 345
Sucker 271, 309
Süd 258, 273, 277
Triton 343
Turtle 309
Yellow 235
_see also_ daily key
Keen, H. 218, 246, 263, 292, 329, 333
Keller, H. 461
Kendrick, F. A. 258
Kilburn, T. 209, 371–4, 401, 471
King's College, Cambridge 1, 131, 134, 214, 219, 264, 487
Kleene, S. C. 43, 44, 45, 88, 92, 102, 126, 127, 150, 153, 163, 211, 450
knots 585–7, 595
Knox, D. 217, 219, 220, 232, 233, 234, 246, 289
lambda calculus 44, 52, 88–90, 126, 139, 147ff, 205–6, 211–16, 360
Langton, C. G. 507, 508, 515
Laplace, P. 447
laws of behaviour 457
learning (by machine)
chess and 565, 569, 575
child-machine and 438–9, 460–3
connectionist 360, 402–3, 405–6, 406–7, 422–3, 424
Darwin on Turing on 400
discussed at Bletchley Park 353
draughts (checkers) and 357–8, 514
education of machinery 421–2, 460–3, 465–6, 473–5, 485, 492, 497, 503
genetic algorithm and 514, 565, 575
in B-type unorganized machines 403–5, 406–7, 422–3, 424
in P-type unorganized machines 425–9
index of experiences and 466, 474
initiative and 430
learning to improve learning methods 492, 497
Lovelace objection and 455, 458–9, 485
Mathematical Objection and 470, 504–5
McCulloch–Pitts and 408–9
NIM and 358
Oettinger and 358–9
pleasure-pain system and 424–5, 432, 461, 466, 474–5
programme-modification and 374–5, 392–3, 419, 462, 470, 492, 496
random element and 463, 466–7, 475
Strachey on 358
training 394, 403, 404ff, 424ff
Lederberg, J. 360
Lefschetz, S. 127
Lenat, D. 402
LEO 17
Letchworth bombe factory 256
_see also_ British Tabulating Machine Co.
Letchworth Enigma 246, 318–19, 321, 325
Lewis, J. S. 265
limit system 171ff
Lisicki, T. 236
LISP 360
Loebner, H. 488–9
Lofoten Pinch 260, 290, 295
logic formula (in lambda calculus) 158ff
Logic Theorist 355, 565
logical computing machine see Turing machine
London Mathematical Society 5–6, 91, 92, 125, 130, 131, 132, 133, 207, 375
London Science Museum 28, 368
look-ahead 356, 565, 566
Lorenz SZ 40/42 cipher machine 479
_see also Tunny_
Los Alamos 22, 507, 510
Lovelace, A. A. 28–9, 455, 458, 480, 482, 485
Lucas, J. R. 468
machine intelligence _see_ Artificial Intelligence
Macrae, N. 25
Mahon, A. P. 229, 258, 259, 260–1, 265–6
Manchester computers
AI and 356–7
'Baby' machine 2, 16, 209, 367, 371, 373, 374, 401, 413
EDVAC and 373–4
Ferranti Mark I 3, 17, 356–7, 374, 437, 483, 496, 503, 508, 510, 517, 552, 564–5
first transistorized computer 369
history of computer chess and 356, 564
Manchester Mark I 16–17, 371, 373, 446, 447–8, 457, 479
theorem-proving and 565
Turing and 2–3, 209, 367, 371–4, 401, 564
Manchester University 3, 16–17, 209, 355, 367, 371–4, 400–1, 465, 487, 552, 564
Computing Machine Laboratory 2, 3, 16–17, 209, 356, 357, 367, 371–4, 396, 400–1, 405, 508
Mandelbrot set 510
Mandelbrot, B. 510
Manhattan Project 22
Massachusetts Institute of Technology 359, 367, 405, 439
Mathematical Objection 355, 393–4, 410, 411, 436, 450–1, 467–70, 472–3
Mauchly, J. W. 22, 25–7, 32, 367, 373
McCarthy, J. 359, 360, 436–8
McCulloch, W. S. 403, 407–9
mechanic 471, 473
mechanical theorem proving 206, 215–16, 355, 401–2, 430–1, 564–5
_see also_ Entscheidungsproblem, expert system, Logic Theorist, Mathematical Objection, Newman's test, ordinal logics, Turing machine
memory
cathode ray tube 380, 396, 496
delay line 26–7, 365, 366, 369, 376–7, 380–4, 385, 387, 395, 396, 399, 446
drum 366, 369
magnetic core 369
menu (for bombe) 256, 291, 293, 295, 332
Menzies, S. 234
message setting 224, 228ff, 247
Meyer, _Funkmaat_ 285
_m_ -function _see_ subroutine
Michie, D. 353, 359, 401, 562, 564
Michigan, University of 513
Milner-Barry, P. S. 336–7, 340
minimax 563, 565
Ministry of Supply 369
Minsky, M. L. 359, 491–2
Moore School of Electrical Engineering 22–3, 25–7, 32, 373, 376
_see also_ University of Pennsylvania
Moore, H. 505
morphogenesis
brain structure and 517
genes as catalysts in 512, 523
letter from Turing to Young about 517
symmetry-breakers in 513, 519, 524ff
Turing's theory of 3, 508–13, 519–61, 581
_see also_ Artificial Life, non-linear equations, reaction-diffusion model
Morse code 220, 222, 263, 273, 301
MOSAIC 369–70, 396
Napper, B. 373
Narvik Pinch 259, 286
National Cash Register Corp. 344
National Physical Laboratory 2, 16, 31–2, 55, 92, 103, 209, 355, 356, 363–70, 377, 378, 395–401, 408, 409, 508
Naval Section (at Bletchley Park) 267, 274, 276, 287–9, 290, 293, 295, 304
Nealey, R. W. 358
neural simulation 356, 374–5, 402–9, 418, 420, 423–4, 432
_see also_ connectionism, neuron-like computation
neuron-like computation 356, 360, 402–9, 416–19, 422–4, 517
_see also_ connectionism, neural simulation
New York 125, 127, 129
Newell, A. 354, 355, 359, 401, 563, 565
Newman, E. A. 367, 368
Newman, M. H. A.
addresses Royal Society 371–2, 480
arranges for Womersley to meet Turing 364
attacks Tunny by machine 208–9
attracts Turing to Manchester 3, 209, 367, 400
biography and obituaries of Turing quoted 3, 15, 48, 207, 408, 480
interviewed concerning Turing 15, 206, 207
involvement with 'On Computable Numbers' 15, 206–7
joins GC & CS 205, 207
letter to von Neumann mentioning Turing 209
mentioned by Turing in correspondence 130, 133, 207
on chess 492, 495, 496, 498, 503, 504
on universality 207, 371, 480
pioneers electronic computing 1, 2, 16, 208–9, 371–4
radio broadcasts 2, 437–8, 465, 476, 487–506
Turing's correspondence with 135, 139, 140, 205–6, 211–16, 470
Turing's influence on 1, 2, 16, 205–6, 209, 372, 373
_see also_ Newman's test, Newmanry
Newmanry (section at Bletchley Park) 208–9, 373
Newman's test 492–3, 504–5
NIM 358
non-linear equations (in morphogenesis) 510, 544–5, 554, 560–1
Norfolk, C. L. 363
normal form 147, 149ff, 577–8, 587ff
Noskwith, R. 258, 267
noughts and crosses 420, 570
number-theoretic (theorem or problem) 152ff validity of logical system and 171ff
Oettinger, A. G. 358–9
OP-20-G 341–5, 347
Operational Intelligence Centre (OIC) 262
oracle machine ( _o_ -machine)
circle-free 142, 157, 468
completeness of ordinal logics and 143, 179–80
degrees of unsolvability and 143–4
description of 141–2, 156–7
diagonal argument and 142, 157
generalized recursion theory and 143–4
intuition and 142–3
relative computability and 126, 144
ordinal formula 162ff
C-K ordinal formula 163ff
representation of ordinals by 162ff
ordinal logic
Church and 125
completeness of 139, 140–1, 159–60, 178–91, 206, 213
definition of 170
Gentzen type 194–202
Gödel's theorem and 1, 126, 136, 138–9, 146, 178ff, 192–3
Hilbert programme and 126, 136, 138–9, 146, 178ff, 192–3
invariance of 180ff
proof-finding machines and 136, 139–40, 206, 212, 215–16
purpose of 1, 135–8, 146, 192–4
Turing writes to Newman concerning 206, 212–13, 214–16
Turing's work on supervised by Church 125–6, 134
_see also_ Church, intuition, oracle machine
Oxford, University of 219, 356
_P_ (logical system) 139, 173ff, 177ff, 194ff
Packard-Bell PB250 370–1
Paired day (in Enigma) 259, 302, 309
Palluth, A. 233
paper machine 412, 416, 429, 431
Parry 489
partially random machine 416, 477–9
_see also_ random element
Pascal 12
Pearcey, T. 367
Pearl Harbor 343
Penrose, R. 468–9
Péter, R. 43
Philadelphia 32
_see also_ Moore School
phyllotaxis 519, 557, 561
pinch (of Enigma materials) 258–61, 288–9, 291
_see also_ Narvik Pinch, Lofoten Pinch
Pitts, W. 403, 407–9
pleasure-pain system 424–9, 466, 477, 474–5
_see also_ punishment, P-type unorganized machine
plug-board (of Enigma) 220ff
_see also Stecker_
plug-board calculators 30–2, 282
_see also_ Hollerith card, programme-controlled
Polanyi, M. 465, 487
Popplewell, C. 564
Post Office Research Station (Dollis Hill) 208–9, 362, 369–70, 395–6, 397, 398
Post, E. L. 91–2, 102, 143–4, 150, 468, 577
preamble (of Enigma message) 230, 275
prediction
in Laplacian universe 447
of brain 477, 478, 483
of continuous machine by discrete-state machine 456–7
of discrete-state machine 447–8, 455–6, 457, 475, 485, 500
of learning machine 462
quantum mechanics and 478, 483
Price, F. 128–9, 130
Price, R. 128
primitive recursive (function or relation) 152ff, 174ff, 188, 200, 202
Princeton University 1, 21, 26, 125, 126, 130, 147, 150, 205, 373
_see also_ Institute for Advanced Study
Principia Mathematica (logical system) 47–8, 52, 84, 138, 139, 173, 213, 355, 430, 472, 580
printing problem
application to _Entscheidungsproblem_ 52–3, 84–7
characterization of 39, 73–4
continuum hypothesis and 192
for oracle machines 156–7
ordinal logics and 185
Post critique and 98ff
satisfactoriness problem and 39, 79
strengthened form of Church-Turing thesis and 41
Prinz, D. G. 356, 564–5
programme-controlled (opposed to _stored-programme_ ) 8, 22–3, 26, 28, 29–30, 31, 32, 362–3
programming, history of 2, 8, 10ff, 24–7, 28, 30–2, 55, 355, 365, 366, 375, 377, 388–92, 395, 399, 445, 460, 511
_see also_ computer, history of, menu, stored-programme concept, subroutine, Turing machine, Turing pioneers computer programming
Prolog 12
Pryce, M. 128, 129, 130, 131, 132
PSILLI 315
P-type unorganized machine 425–9, 466, 492
compared with ACE 428
universal Turing machine and 427–8
_see also_ pleasure-pain system, punishment
punishment (and reward) 425, 426, 427, 428, 432, 461, 466, 474–5
_see also_ learning, pleasure-pain system, P-type unorganized machine
Purple (Japanese cipher) 342
Pye Ltd. 254, 320, 321
Pyry 234, 246
Radar Research and Development Establishment (RRDE) 370
_Radiolaria_ 517, 559
Radley, W. G. 395
Randell, B. 16, 22
random element
brain and 424, 478–9
determinism and 466, 475
evolution and 463, 516
free will and 445, 477–9, 484–5
in ACE 391, 478
in digital computer 391, 445, 466, 475, 477–8
in learning 425, 426, 427, 429, 463, 466–7, 475, 496
in partially random machine 416, 478
in pleasure-pain system 425, 426, 427, 428, 429, 467
in Turing test 458
mathematical problem solving and 470, 505
search and 463, 466, 467, 470
Ratio Club 508
reaction-diffusion model 509, 510–13, 519ff
recursion formula 193ff
re-encodements (in Enigma) 307–11
Rees, D. 373
Rejewski, M. 229, 231–46
comments about Turing 235
R.H.V. (reserve hand cipher) 307–8, 310, 311
relative computability 126, 143–4
Research Section (at Bletchley Park) 207–8, 220, 262
Riemann hypothesis 155
ring (of cells or continuous tissue) 510–11, 519, 530ff
_Ringstellung_ 227ff, 270ff, 335, 347ff, 354–5
_Ringstellung_ cut-out (in bombe) 333, 335, 354–5
RISC 366
robot 392, 420–1, 439, 460–1, 463, 486, 508
rod-position (in Enigma) 238ff, 249ff, 315ff
Room 40 207, 218, 219
_see also_ Government Code and Cypher School
Rosenblatt, F. 360, 403, 406
Ross Ashby, W. 360, 374–5
Rosser, J. B. 127, 137, 156, 450
Royal Astronomical Society 375
Royal Navy 259–61
Royal Society of London 3, 209, 371, 372
Różycki, J. 232, 236
Russell, B. A. W. 47, 131, 138, 139, 355, 430, 580
saga (in Enigma wheel-breaking) 232, 278
St John's College, Cambridge 207
Samuel, A. L. 357–8, 514
satisfactoriness problem
characterization of 36–7, 68
diagonal argument and 37–9, 72–4
for oracle machines 142, 156–7
Mathematical Objection and 467–8
Post critique and 98ff
printing problem and 39, 73–4, 79
strengthened form of Church–Turing thesis and 41
Sayers, D. 410
Scarborough 'Y' (intercept) station 274–6, 290
Scherbius, A. 220, 277
Scheutz, E. 28
Scheutz, G. 28
Schmidt, H.-T. 231–2
Scholz, H. 131, 133
Scott, D. S. 356
SEAC 367
search (heuristic)
bomba and 354
bombe and 353–5
chess and 353, 563–4, 565, 570ff
draughts (checkers) and 356, 514
intellectual activity as 354, 430–1
Mathematical Objection and 469–70
meaning of 'heuristic' 354
random element and 463, 466, 467, 470
theorem-proving and 401–2, 430–1
self-reproduction 514–16
_see also_ universal constructor
sense organs 392, 420–1, 426, 439, 459, 460–1, 463
Shannon, C. E. 22, 393, 436–8, 562
Shaw, J. C. 355, 359, 565
Sherborne School 1
Shockley, W. B. 376
Shoenfield, J. R. 141, 143
Shopper 359
Simon, H. A. 354, 355, 359, 401, 439, 563, 565
simultaneous scanning (in bombe) 254–5, 319–20, 321, 323, 327, 343–4
Sinclair, H. 219, 234
Sinkov Mission 342
Sinkov, A. 342
sliding squares puzzle 582–4, 588, 589, 593
Slutz, R. J. 368
Smith-Rose, R. L. 398
Soare, R. I. 5
solipsism 452
Spanish Civil War 217, 224
spermatozoon 511, 556
Spider _see_ bombe
standard description
as first programming language 25
halting problem and 39
Mathematical Objection and 470
meaning of term 10–12, 67–8
Post critique and 98–101
printing problem and 39
satisfactoriness problem and 36–7
universal Turing machine and 15, 17–20, 68–72, 105ff, 413–14
standardized logic 158ff
Stanford University 359, 360
statistical method (against Tunny) 208
_Stecker_ 221, 222, 227ff, 270ff, 347ff
bomba and 245
explanation of 223, 224–6, 227, 270
majority vote gadget and 335, 355
self-steckered 245, 270, 279, 347, 349
stecker hypothesis 253–5, 316ff
Stecker Knock Out 286, 316
stecker value of letter 252ff, 316ff
Turing's method for discovering using bombe 235, 250–5, 314ff
_see also_ plug-board
Stibitz Relay Computer 363
Stibitz, G. R. 363
stop (in bomba or bombe) 237ff, 293, 319, 325, 327, 329–34, 354
stored-programme concept 1, 2, 3, 6, 8, 9, 12, 15–21, 21–7, 29–30, 30–2, 68–72, 105–117, 209, 362–8, 371–4, 375, 378–9, 389, 393
Strachey, C. S. 356–8, 564, 568
Sturgeon 207, 263
subroutine (subsidiary table)
examples of 13–14, 54–7, 63–6, 108–12
explanation of term 12
history of programming and 12, 55, 375
in ACE 12, 389–90
_m_ -functions and 54–7, 63ff
universal Turing machine and 69–72, 112–15
substitution cipher 229
substitution puzzle 576–80, 587–92, 594
SWAC 368
systematic method
Church–Turing thesis and 41–5, 568, 576, 577–8, 589
compared with search involving random element 463, 467
decidability and 47, 592
_Entscheidungsproblem_ and 52–3
Gödel's theorem and 580–1, 593
halting problem and 39
meaning of term 42, 590
meaning of 'unsolvable' and 579, 592
solvability of puzzles and 578, 582, 583–4, 587, 589, 590, 591–2, 593
_see also_ Church–Turing thesis, human computer, effective calculability, effective method
Tarski, A. 177
Taylor, W. K. 360
Telecommunications Research Establishment(TRE) 208, 209, 373
teleprinter code 207–8, 263, 273
The '51 Society 465
thinking machine 2, 356, 358, 420, 434–9, 441–3, 448–52, 454–5, 459, 465, 472, 474, 475, 476–7, 478–9, 482, 485–6, 487, 488, 491, 492–3, 494–5, 498, 500, 501, 502, 504, 505, 566; _see also_ Artificial Intelligence, brain, child-machine
Thomas, H. A. 397–400
Thrasher 207, 263
Thue problem (after Axel Thue) 99ff
_see also_ substitution puzzle, word problem
Tiltman, J. H. 343
Tootill, G. C. 373–4
totalisator 363
Travis, E. 279, 287, 292, 293, 311, 338, 339
Tunny 207–9, 262–3, 479
_see also_ Lorenz SZ 40/42
Turing, A. M.
born (1912) 1
educated at Sherborne School and King's College, Cambridge 1
elected Fellow of King's College (1935) 1
while proving _Entscheidungsproblem_ unsolvable, invents universal Turing machine and fundamental stored-programme principle of modern computer 1, 2, 5–57, 58–87, 206, 207, 209, 363, 371, 375, 378–9, 383, 414–5
discovers and explores the uncomputable 1, 3, 6, 32–53, 72–9, 84–7, 125–44, 146–202, 206, 212–13, 214–16, 355, 393–4, 410, 411, 450–1, 467–70, 472–3, 576–81, 582–95
learns of Church's work on _Entscheidungsproblem_ 125, 207
studies at Princeton University (1936–8) 125–34
writes to Sara Turing from Princeton (1936–8) 126–34
explores ordinal logics and place of intuition in mathematics 1, 125–6, 135–44, 146–202, 206, 211–16
returns from US to Fellowship at King's (1938) 1, 21, 134
with outbreak of war transfers to Government Code and Cypher School at Bletchley Park (1939) 1–2, 205, 217–20, 257, 279
breaks Naval Enigma indicator system and invents 'Banburismus' 1–2, 218, 257–8, 279–81, 256, 261–2, 281, 285
designs bombe 218, 235, 240, 246, 250–5, 263, 314–35
visits exiled Polish codebreakers in France 235
leads attack on Naval Enigma 2, 218, 253, 257–62, 279–81, 285–9, 314ff, 341ff
writes to Newman concerning logic (c.1940) 205–6, 211–16
writes to Churchill (1941) 336–40
advises US codebreakers 341–52
leaves Enigma (1942) 262–4, 288, 312
works on Tunny and invents 'Turingery'(1942) 262–3
visits US (Nov 1942–Mar 1943) 263
works on speech encipherment at Hanslope Park(1943–5) 263
OBE for war work 2, 264
joins National Physical Laboratory and designs electronic stored-programme digital computer (1945) 2, 12, 27, 30, 31, 363–4, 364–71, 376–7, 378–94
pioneers computer programming before hardware in existence (1945–7) 2, 12, 25, 27, 30–2, 55, 356, 365, 366, 372, 375, 377, 378ff, 388–92, 395, 563–4
pioneers Artificial Intelligence and cognitive science 2, 3, 353–9, 374–6, 392–4, 401–9, 410–32, 433–9, 441–63, 465–71, 472–5, 476–8, 480, 482–6, 487–92, 494–5, 563–4, 565, 566–7, 569–75
writes to Ross Ashby concerning computer and brain 374–5
pioneers computer chess 3, 353, 356, 374, 375, 393, 394, 412, 420, 431, 439, 463, 470, 473, 514, 562–4, 565–6, 569–75
lectures on computer design in London (1946–7) 2, 355–6, 372–3, 375–7, 378–94
visits computer projects in US (Jan 1947) 397
proposes electronics section at NPL (1947) 397–8
spends sabbatical in Cambridge (1947–8) 400–1
leaves NPL for Manchester University (1948) 2–3, 209, 367, 400–1
influence on computer developments at Manchester 2–3, 209, 371–4, 401, 564
anticipates connectionism 356, 402–9, 416–24, 429–30, 431–2, 510, 517
invents Turing test 2, 356, 359, 401, 431, 433–9, 441–63, 477, 479, 484, 485, 488, 489–90, 494–5, 496, 503
pioneers Artificial Life with study of morphogenesis 1, 3, 401, 405, 508–14, 517, 519–61
writes to Young concerning morphogenesis and neuron growth (1951) 517
broadcasts on BBC radio (1951–2) 356, 358, 465–71, 472–5, 476–80, 482–6, 487–93, 494–506
uses computer to explore non-linear systems empirically 510, 561
dies (1954) 1, 510
Turing degree 144
Turing, E. S. (Sara) 21, 125–34, 471, 476
Turing machine
circle-free 32–3, 34–5, 36, 37ff, 60–1, 72–4, 79, 98ff, 142–3, 144, 153, 154, 468
circular 32–3, 36, 37, 60–1, 72–4
continuum hypothesis and 191–2
diagonal argument and 34–5, 37–9, 72–4
effective computability and 88–90, 150
_Entscheidungsproblem_ and 52–3, 84–7
introduction to 6–14, 32–41, 104–5
intuition and 137–8, 139–40, 215–16
learning and 407, 422, 424–30, 438–9, 470
Mathematical Objection and 450–1, 468, 470
oracle machine and 141–3, 144, 156–7
ordinal logics and 137–8, 184–5
provability by 52–3, 72–9, 84–7, 136, 140–1, 193, 206, 215–16, 430–1, 470, 472
Turing's exposition of 59ff, 413–14
_see also_ universal Turing machine
Turing sheets (in Enigma breaking) 314–15, 316
Turing test
1948 presentation of 401, 431, 433
1950 presentation of 2, 356, 433–6, 441–448–58, 488, 489–90
1951 presentation of 433–4, 477, 479, 484, 485
1952 presentation of 433–4, 488–9, 494–5, 496, 503
consciousness and 451–3, 455, 456, 566–7, 569
imitating brain and 477, 479, 494–5
learning and 460ff
Loebner Prize for 488–9
objections to 436–8, 448–58, 490–1
predictions concerning success in 449, 459, 460, 484, 489–90, 495, 566, 569
Shopper and 359
_see also_ thinking machine
Turingery (in Tunny breaking) 263
Turing-reducible 143–4
Turing's feedback method (in bombe) 254–5, 322–3
turnover (of Enigma wheel) 225–6, 239, 241, 249, 253, 268, 270, 279, 284–5, 315ff, 342, 347ff
Turochamp 563–4
Tutte, W. T. 208
twiddle (in Enigma breaking) 287, 290, 311
Twinn, P. 217, 258, 260, 286, 287
twisted-wire puzzle 584–5, 594
type fallacy 462
Typex machine 249, 250, 253
U-boat 257, 261, 262, 272, 273–4, 291, 306, 308, 309, 310, 311, 343, 347
_U_ -110 41, 261
_U_ -559 344
Ulam, S. M. 21
uncomputable number 36, 58, 72–4, 79
_see also_ computable number
uncomputable sequence 33ff, 72–4, 79, 83
_see also_ computable sequence
Unilever Ltd. 399
UNIVAC 17
universal constructor 516
universal Turing machine
Artificial Life and 515–16
B- and P-type machines and 407, 422, 424 –5, 427–8
brain and 407, 424, 429–30, 478
Church-Turing thesis and 41, 43, 479–80
compared with ACE 375, 383, 384
compared with Analytical Engine 29–30, 455
corrections to 91–101, 115–24
history of computer and 1, 2, 6, 9, 12, 15–17, 21–7, 30–1, 55, 206, 207, 209, 363, 371, 375, 378–9, 383, 414–15
introduction to 15–21, 105–15
lambda-definability and 44, 88–90
'paper interference' and 418–19
Turing's exposition of 68–72, 383, 413–14
unsolvable problems and 576
_see also Entscheidungsproblem_ , halting problem, printing problem, satisfactoriness problem, Turing machine
universality 415–16, 446–8, 455, 480, 482–3, 501, 516, 569–70
_see also_ universal Turing machine
University of California at San Diego 402
University of Pennsylvania 22, 380
_see also_ Moore School
unorganized machine 402–9, 416–19, 422–9, 430, 432, 467, 517
_see also_ A-type, B-type, connectionism, neural simulation, neuron-like computing, pleasure-pain system, P-type
unsolvable problem 6, 33, 98ff, 576–81, 582–95
meaning of term 144, 579, 592
_see also_ decision problem, degree of unsolvability, _Entscheidungsproblem_ , halting problem, printing problem, satisfactoriness problem, systematic method, Thue problem
US Army 342, 344
US Bureau of Standards 368
US Navy 256–7, 342, 344
Uttley, A. M. 360, 368–9
Vassar College 129, 132
Victory (1st bombe) 253, 254, 259
von Neumann, J.
as pioneer of Artificial Life 513, 514–17
as pioneer of stored-programme computer 1, 16, 21–7, 32, 362, 364, 365, 408
letter to Wiener mentioning Turing 23, 515, 516–17
Manchester computer and 373–4
Max Newman writes to 209
mentioned by Turing in correspondence 127, 134, 213
minimax and 563
on undecidability 53
Turing offered job as assistant to 21, 134
Turing's influence on 1, 16, 21–2, 23–5, 515–16
W/T (wireless telegraphy) 273–5
Waddington, C. H. 509, 521
Wang, H. 45
Weeks, R. 342
Welchman, W. G. 218, 254, 255, 263, 264, 292, 327, 329, 340
_Werft_ (dockyard cipher) 307–8, 310, 311
Weyl, H. 48, 127, 131
wheel core (in Enigma) 223, 226, 227, 238, 244, 270
_see also_ rod-position
wheel order (in Enigma) 222, 223, 225, 227ff, 270ff, 315ff, 343ff, 347ff
Whirlwind I computer 367
Whitehead, A. N. 47, 138, 139, 355, 580
Wiener, N. 23, 408, 515, 516
Wilkes, M. V. 367, 476
Wilkinson, J. H. 366, 367, 368, 398, 399
Williams, F. C. 16, 209, 367, 371–4, 401, 471, 476
Wilmslow 1
Wittgenstein, L. 41, 130, 487
Womersley, J. R. 31, 363–4, 375, 395, 398, 399
Woodger, M. 31, 32, 356, 364, 367, 368, 376, 388, 508
word problem 594
_see also_ substitution puzzle, Thue problem
Wylie, S. 258, 286
Wynn-Williams, C. E. 208
Young, J. Z. 487, 509, 517
Zermelo, E. 194, 213
Zygalski, H. 232
At 2 Warrington Crescent, London W9, where now there is a commemorative plaque.
Jack Good in an interview with Pamela McCorduck, on p. 53 of her _Machines Who Think_ (New York: W. H. Freeman, 1979).
This estimate is given by Sir Harry Hinsley, official historian of the British Secret Service, writing on p. 12 of his and Alan Stripp's edited volume _Codebreakers: The Inside Story of Bletchley Park_ (Oxford: Oxford University Press, 1993).
M. H. A. Newman, 'Alan Mathison Turing, 1912–1954', _Biographical Memoirs of Fellows of the Royal Society_ , 1 (1955), 253–63 (256).
_Proceedings of the London Mathematical Society_ , 42 (1936–7), 230–65. The publication date of 'On Computable Numbers' is sometimes cited, incorrectly, as 1937. The article was published in two parts, both parts appearing in 1936. The break between the two parts occurred, rather inelegantly, in the middle of Section 5, at the bottom of p. 240 (p. 67 in the present volume). Pages 230–40 appeared in part 3 of volume 42, issued on 30 Nov. 1936, and the remainder of the article appeared in part 4, issued on 23 Dec. 1936. This information is given on the title pages of parts 3 and 4 of volume 42, which show the contents of each part and their dates of issue. (I am grateful to Robert Soare for sending me these pages. See R. I. Soare, 'Computability and Recursion', _Bulletin of Symbolic Logic, 2_ (1996), 284–321.)
The article was published bearing the information 'Received 28 May, 1936.—Read 12 November, 1936.' However, Turing was in the United States on 12 November, having left England in September 1936 for what was to be a stay of almost two years (see the introductions to Chapters 3 and 4). Although papers were read at the meetings of the London Mathematical Society, many of those published in the _Proceedings_ were 'taken as read', the author not necessarily being present at the meeting in question. Mysteriously, the minutes of the meeting held on 18 June 1936 list 'On Computable Numbers, with an Application to the Entscheidungs-problem' as one of 22 papers taken as read at that meeting. The minutes of an Annual General Meeting held on 12 Nov. 1936 contain no reference to the paper. (I am grateful to Janet Foster, Archives Consultant to the London Mathematical Society, for information.)
The first was 'Equivalence of Left and Right Almost Periodicity', _Journal of the London Mathematical Society_ , 10 (1935), 284–5.
Church introduced the term 'Turing machine' in a review of Turing's paper in the _Journal of Symbolic Logic_ , 2 (1937), 42–3.
A. Church, 'An Unsolvable Problem of Elementary Number Theory', _American Journal of Mathematics_ , 58 (1936), 345–63, and 'A Note on the Entscheidungsproblem', _Journal of Symbolic Logic_ , 1 (1936), 40–1.
There is a subtle issue concerning the placement of the semicolons. See Davies's 'Corrections to Turing's Universal Computing Machine', Sections 3, 7, 10.
Properly speaking, the description number is not the string '313325311731133531117311133225 31111731111335317', but is the number denoted by this string of numerals.
The quotation is from p. 389 below.
Newman in interview with Christopher Evans ('The Pioneers of Computing: An Oral History of Computing', London, Science Museum).
Ibid.
'Dr. A. M. Turing', _The Times_ , 16 June 1954, p. 10.
F. C. Williams described some of Turing's contributions to the Manchester machine in a letter written in 1972 to Brian Randell (parts of which are quoted in B. Randell, 'On Alan Turing and the Origins of Digital Computers', in B. Meltzer and D. Michie (eds.), _Machine Intelligence 7_ (Edinburgh: Edinburgh University Press, 1972)); see the introduction to Chapter 9 below. A digital facsimile of Turing's _Programmers' Handbook for Manchester Electronic Computer_ (University of Manchester Computing Machine Laboratory, 1950) is in The Turing Archive for the History of Computing <www.AlanTuring.net/programmers_handbook>.
S. Lavington, 'Computer Development at Manchester University', in N. Metropolis, J. Howlett, and G. C. Rota (eds.), _A History of Computing in the Twentieth Century_ (New York: Academic Press, 1980).
See M. V. Wilkes, _Memoirs of a Computer Pioneer_ (Cambridge, Mass.: MIT Press, 1985).
Turing's letters to his mother are among the Turing Papers in the Modern Archive Centre, King's College Library, Cambridge (catalogue reference K 1).
S. Turing, _Alan M. Turing_ (Cambridge: Heffer, 1959), 55.
'I know that von Neumann was influenced by Turing... during his Princeton stay before the war,' said von Neumann's friend and colleague Stanislaw Ulam (in an interview with Christopher Evans in 1976; 'The Pioneers of Computing: An Oral History of Computing', Science Museum, London). When Ulam and von Neumann were touring in Europe during the summer of 1938, von Neumann devised a mathematical game involving Turing-machine-like descriptions of numbers (Ulam reported by W. Aspray on pp. 178, 313 of his _John von Neumann and the Origins of Modern Computing_ (Cambridge, Mass.: MIT Press, 1990)). The word 'intrigued' is used in this connection by von Neumann's colleague Herman Goldstine on p. 275 of his _The Computer from Pascal to von Neumann_ (Princeton: Princeton University Press, 1972).)
Turing's universal machine was crucial to von Neumann's construction of a self-reproducing automaton; see the chapter 'Artificial Life', below.
J. von Neumann, _The Computer and the Brain_ (New Haven: Yale University Press, 1958).
Letter from Frankel to Brain Randell, 1972 (first published in B. Randell, 'On Alan Turing and the Origins of Digital Computers', in Meltzer and Michie (eds.), _Machine Intelligence 7_. I am grateful to Randell for giving me a copy of this letter.
John Mauchly recalled that 7 September 1944 'was the first day that von Neumann had security clearance to see the ENIAC and talk with Eckert and me' (J. Mauchly, 'Amending the ENIAC Story', _Datamation_ , 25/11 (1979), 217–20 (217)). Goldstine ( _The Computer from Pascal to von Neumann_ , 185) suggests that the date of von Neumann's first visit may have been a month earlier: 'I probably took von Neumann for a first visit to the ENIAC on or about 7 August'.
Goldstine, _The Computer from Pascal to von Neumann_ , 182.
Letter from Julian Bigelow to Copeland (12 Apr. 2002). See also Aspray, _John von Neumann_ , 178.
Bigelow in a tape-recorded interview made in 1971 by the Smithsonian Institution and released in 2002. I am grateful to Bigelow for sending me a transcript of excerpts from the interview.
The letter, dated 29 Nov. 1946, is in the von Neumann Archive at the Library of Congress, Washington, DC. In the letter von Neumann also remarked that Turing had 'demonstrated in absolute... generality that anything and everything Brouwerian can be done by an appropriate mechanism' (a Turing machine). He made a related remark in a lecture: 'It has been pointed out by A. M. Turing [in "On Computable Numbers"]... that effectively constructive logics, that is, intuitionistic logics, can be best studied in terms of automata' ('Probabilistic Logics and the Synthesis of Reliable Organisms from Unreliable Components', in vol. v of von Neumann's _Collected Works_ , ed. A. H. Taub (Oxford: Pergamon Press, 1963), 329).
The text of 'The General and Logical Theory of Automata' is in vol. v of von Neumann, _Collected Works_ ; see pp. 313–14.
The text of 'Rigorous Theories of Control and Information' is printed in J. von Neumann, _Theory of Self-Reproducing Automata_ , ed. A. W. Burks (Urbana: University of Illinois Press, 1966); see p. 50.
The first papers in the series were the 'First Draft of a Report on the EDVAC' (1945, von Neumann; see n. 31), and 'Preliminary Discussion of the Logical Design of an Electronic Computing Instrument' (1946, Burks, Goldstine, von Neumann; see n. 28).
A. W. Burks, H. H. Goldstine, and J. von Neumann, 'Preliminary Discussion of the Logical Design of an Electronic Computing Instrument', 28 June 1946, Institute for Advanced Study, Princeton University, Section 3.1 (p. 37); reprinted in vol. v of von Neumann, _Collected Works_.
Letter from Burks to Copeland (22 Apr. 1998). See also Goldstine, _The Computer from Pascal to von Neumann_ , 258.
N. Macrae, _John von Neumann_ (New York: Pantheon Books, 1992), 23.
J. von Neumann, 'First Draft of a Report on the EDVAC', Moore School of Electrical Engineering, University of Pennsylvania, 1945; reprinted in full in N. Stern, _From ENIAC to UNIVAC: An Appraisal of the Eckert-Mauchly Computers_ (Bedford, Mass.: Digital Press, 1981).
See N. Stern, 'John von Neumann's Influence on Electronic Digital Computing, 1944–1946', _Annals of the History of Computing_ , 2 (1980), 349–62.
Huskey in interview with Copeland (Feb. 1998). (Huskey was offered the directorship of the EDVAC project in 1946 but other commitments prevented him from accepting.)
Mauchly, 'Amending the ENIAC Story'; J. P. Eckert, 'The ENIAC', in Metropolis, Howlett, and Rota, _A History of Computing in the Twentieth Century_ ; letter from Burks to Copeland (16 Aug. 2003): 'before von Neumann came' to the Moore School, Eckert and Mauchly were 'saying that they would build a mercury memory large enough to store the program for a problem as well as the arithmetic data'. Burks points out that von Neumann was however the first of the Moore School group to note the possibility, implict in the stored-programme concept, of allowing the computer to modify the addresses of selected instructions in a programme while it runs (A. W. Burks, 'From ENIAC to the Stored-Program Computer: Two Revolutions in Computers', in Metropolis, Howlett, and Rota, _A History of Computing in the Twentieth Century_ , 340–1). Turing employed a more general form of the idea of instruction modification in his 1945 technical report 'Proposed Electronic Calculator' (in order to carry out conditional branching), and the idea of instruction modification lay at the foundation of his theory of machine learning (see Chapter 9).
H. D. Huskey, 'The Early Days', _Annals of the History of Computing_ , 13 (1991), 290–306 (292–3). The date of the conversation was 'perhaps the spring of 1945' (letter from Huskey to Copeland (5 Aug. 2003)).
Goldstine, _The Computer from Pascal to von Neumann_ , 186.
Ibid.
J. P. Eckert and J. W. Mauchly, 'Automatic High Speed Computing: A Progress Report on the EDVAC', Moore School of Electrical Engineering, University of Pennsylvania (Sept. 1945), Section 1; this section of the report is reproduced on pp. 184–6 of L. R. Johnson, _System Structure in Data, Programs, and Computers_ (Englewood Cliffs, NJ: Prentice-Hall, 1970).
Burks, 'From ENIAC to the Stored-Program Computer: Two Revolutions in Computers', 312.
C. Babbage, _Passages from the Life of a Philosopher_ , vol. xi of _The Works of Charles Babbage_ , ed. M. Campbell-Kelly (London: William Pickering, 1989); see also B. Randell (ed.), _The Origins of Digital Computers: Selected Papers_ (Berlin: Springer-Verlag, 3rd edn. 1982), ch. 1.
See Babbage, _Passages from the Life of a Philosopher;_ A. A. Lovelace and L. F. Menabrea, 'Sketch of the Analytical Engine Invented by Charles Babbage, Esq.' (1843), in B. V. Bowden (ed.), _Faster than Thought_ (London: Pitman, 1953); Randell, _The Origins of Digital Computers: Selected Papers_ , ch. 2; A. Bromley, 'Charles Babbage's Analytical Engine, 1838', _Annals of the History of Computing_ , 4 (1982), 196–217.
Lovelace and Menabrea, 'Sketch of the Analytical Engine', 365.
V. Bush, 'Instrumental Analysis', _Bulletin of the American Mathematical Society_ , 42 (1936), 649–69 (654) (the text of Bush's 1936 Josiah Willard Gibbs Lecture).
H. Aiken, 'Proposed Automatic Calculating Machine' (1937), in Randell, _The Origins of Digital Computers: Selected Papers_ , 196.
Thomas H. Flowers in interview with Copeland (July 1996).
Dennis Babbage, chief cryptanalyst in Hut 6, the section at Bletchley Park responsible for Army, Airforce, and Railway Enigma, is sometimes said to have been a descendant of Charles Babbage. This was not in fact so. (Dennis Babbage in interview with Ralph Erskine.)
Babbage, _Passages from the Life of a Philosopher_ , 97.
R. Gandy, 'The Confluence of Ideas in 1936', in R. Herken (ed.), _The Universal Turing Machine: A Half-Century Survey_ (Oxford: Oxford University Press, 1998), 90. Emphasis added.
'Program' is the original English spelling, in conformity with 'anagram', 'diagram', etc. The spelling 'programme' was introduced into Britain from France in approximately 1800 _(Oxford English Dictionary)_. The earlier spelling persisted in the United States. Turing's spelling is followed in this volume (except in quotations from other authors and in the section by Davies).
See also 'The Turing-Wilkinson Lecture Series on the Automatic Computing Engine' (ed. Copeland), in K. Furukawa, D. Michie, and S. Muggleton (eds.), _Machine Intelligence 15_ (Oxford: Oxford University Press, 1999).
Letter from Clayden to Copeland (3 Oct. 2000).
Letter from Woodger to Copeland (6 Oct. 2000).
'Draft Report of the Executive Committee for the Year 1946', National Physical Laboratory, paper E.910, section Ma. 1, anon., but probably by Womersley (NPL Library; a digital facsimile is in The Turing Archive for the History of Computing <www.AlanTuring.net/annual_report_1946>).
J. G. Hayes, 'The Place of Pilot Programming', MS, 2000.
M. Woodger, 'A Program for Version H', handwritten MS, 1947 (in the Woodger Papers, National Museum of Science and Industry, Kensington, London (catalogue reference N30/37)).
Letter from Woodger to Copeland (6 Oct. 2000).
J. W. Mauchly, 'The Use of High Speed Vacuum Tube Devices for Calculating' (1942), in Randell, _The Origins of Digital Computers: Selected Papers_.
J. P. Eckert, 'A Preview of a Digital Computing Machine' (15 July 1946), in M. Campbell-Kelly and M. R. Williams (eds.), _The Moore School Lectures_ (Cambridge, Mass.: MIT Press, 1985), 114.
Sections 1.2, 5.3 of Burks, Goldstine, and von Neumann, 'Preliminary Discussion of the Logical Design of an Electronic Computing Instrument' (von Neumann, _Collected Works_ , vol. v, 15, 43).
Letter from Huskey to Copeland (3 Feb. 2002).
See M. Davis, _Computability and Unsolvability_ (New York: McGraw-Hill, 1958), 70. Davis thinks it likely that he first used the term 'halting problem' in a series of lectures that he gave at the Control Systems Laboratory at the University of Illinois in 1952 (letter from Davis to Copeland, 12 Dec. 2001).
It is interesting that if one lifts the restriction that the determination must be carried out in a _finite_ number of steps, then Turing machines _are_ able to solve the halting and printing problems, and moreover in a finite time. See B. J. Copeland, 'Super Turing-Machines', _Complexity_ , 4 (1998), 30–2, and 'Accelerating Turing Machines', _Minds and Machines_ , 12 (2002), 281–301.
Turing's _Programmers' Handbook for Manchester Electronic Computer_ , 1.
L. Wittgenstein, _Remarks on the Philosophy of Psychology_ , vol. i (Oxford: Blackwell, 1980), § 1096.
As Martin Davis emphasized long ago in his _Computability and Unsolvability_ , p. vii.
See R. Gandy, 'Church's Thesis and Principles for Mechanisms', in J. Barwise, H. J. Keisler, and K. Kunen (eds.), _The Kleene Symposium_ (Amsterdam: North-Holland, 1980).
There is a survey of the evidence in chapters 12 and 13 of S. C. Kleene, _Introduction to Metamathematics_ (Amsterdam: North-Holland, 1952).
L. Kalmár, 'An Argument against the Plausibility of Church's Thesis', R. Péter, 'Rekursivität und Konstruktivität'; both in A. Heyting (ed.), _Constructivity in Mathematics_ (Amsterdam: North-Holland, 1959).
Church, 'An Unsolvable Problem of Elementary Number Theory'.
Equivalent, that is, if the computable functions are restricted to functions of positive integers. Turing's concerns were rather more general than Church's, in that whereas Church considered only functions of positive integers, Turing described his work as encompassing 'computable functions of an integral variable or a real or computable variable, computable predicates, and so forth' (p. 58, below). Turing intended to pursue the theory of computable functions of a real variable in a subsequent paper, but in fact did not do so.
S. C. Kleene, _Mathematical Logic_ (New York: Wiley, 1967), 232.
Turing, 'Computability and λ-Definability', _Journal of Symbolic Logic_ , 2 (1937), 153–63 (153).
Church's review of 'On Computable Numbers' in _Journal of Symbolic Logic_ , 43.
S. C. Kleene, 'Origins of Recursive Function Theory', _Annals of the History of Computing_ , 3 (1981), 52–67 (59, 61).
H. Wang, _From Mathematics to Philosophy_ (New York: Humanities Press, 1974), 85.
K. Gödel, _Collected Works_ , ed. S. Feferman et al., vol. iii (Oxford: Oxford University Press, 1995), 304, 168.
Ibid., vol. ii. (Oxford: Oxford University Press, 1990), 150.
Church, 'A Note on the Entscheidungsproblem', 41.
D. Hilbert and W. Ackermann, _Grundzüge der Theoretischen Logik_ [Principles of Mathematical Logic] (Berlin: Julius Springer, 1928), 73, 77.
D. Hilbert, 'The Foundations of Mathematics' (English translation of a lecture given in Hamburg in 1927, entitled 'Die Grundlagen der Mathematik'), in J. van Heijenoort (ed.), _From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931_ (Cambridge, Mass.: Harvard University Press, 1967), 475.
D. Hilbert, 'Über das Unendliche' [On the Infinite], _Mathematische Annalen_ , 95 (1926), 161–90 (180); English translation by E. Putnam and G. Massey in R. L. Epstein and W. A. Carnielli, _Computability: Computable Functions, Logic, and the Foundations of Mathematics_ (2nd edn. Belmont, Calif.: Wadsworth, 2000).
D. Hilbert, 'Mathematical Problems: Lecture Delivered before the International Congress of Mathematicians at Paris in 1900', _Bulletin of the American Mathematical Society_ , 8 (1902), 437–79 (445).
To prove an arbitrary statement from a contradiction _P_ & _not P_ , one may use the following rules of inference (see further pp. 49–52, below):
(a) not _p_ _not_ ( _P_ & _X_ )
(b) _P & not_ ( _P & X_) _not X_.
Rule ( _a_ ) says: from the statement that it is not the case that _P_ , it can be inferred that not _both P_ and _X_ are the case—i.e. inferred that one at least of _P_ and _X_ is not the case—where _X_ is any statement that you please. Rule ( _b_ ) says: given that _P_ is the case and that not _both P_ and _X_ are the case, it can be inferred that _X_ is not the case. Via ( _a_ ), the contradiction ' _P_ & _not P_ ' leads to ' _not_ ( _P_ & _X_ )'; and since the contradiction also offers us _P_ , we may then move, via ( _b_ ), to ' _not X_ '. So we have deduced an arbitrary statement, ' _not X_ ', from the contradiction. (To deduce simply _X_ , replace _X_ in ( _a_ ) and ( _b_ ) by ' _not X_ ', and at the last step use the rule saying that two negations 'cancel out': _not not X_ _X_.)
D. Hilbert, 'Probleme der Grundlegung der Mathematik' [Problems Concerning the Foundation of Mathematics], _Mathematische Annalen_ , 102 (1930), 1–9 (3, 9). Translation by Elisabeth Norcliffe.
K. Gödel, 'Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I.' [On Formally Undecidable Propositions of Principia Mathematica and Related Systems I], _Monatshefte für Mathematik und Physik_ , 38 (1931), 173–98. English translation in M. Davis (ed.), _The Undecidable: Basic Papers on Undecidable Propositions, Unsolvable Problems and Computable Functions_ (New York: Raven, 1965), 5–38.
A. N. Whitehead and B. Russell, _Principia Mathematica_ , vols. i–iii (Cambridge: Cambridge University Press, 1910–13).
Gödel, 'Postscriptum', in Davis, _The Undecidable: Basic Papers on Undecidable Propositions, Unsolvable Problems and Computable Functions_ , 71–3 (71); the Postscriptum, dated 1964, is to Gödel's 1934 paper 'On Undecidable Propositions of Formal Mathematical Systems' (ibid. 41–71).
Ibid. 72.
H. Weyl, 'David Hilbert and his Mathematical Work', _Bulletin of the American Mathematical Society_ , 50 (1944), 612–54 (644).
M. H. A. Newman, 'Alan Mathison Turing, 1912–1954', _Biographical Memoirs of Fellows of the Royal Society_ , 1 (1955), 253–63 (256).
P. Bernays, 'Die Philosophie der Mathematik und die Hilbertsche Beweistheorie' [The Philosophy of Mathematics and Hilbert's Proof Theory], _Blätter für Deutsche Philosophie_ , 4 (1930/1931), 326–67. See also H. Wang, _Reflections on Kurt Gödel_ (Cambridge, Mass.: MIT Press, 1987), 87–8.
G. Gentzen, 'Investigations into Logical Deduction' (1934), in _The Collected Papers of Gerhard Gentzen_ , ed. M. E. Szabo (Amsterdam: North-Holland, 1969).
In Gentzen's system this rule can itself be derived from the basic rules. It should be mentioned that in the full system it is permissible to write any finite number of statements (including zero) on the right hand side of ' '.
In a lecture given in April 1935—the text of which was printed the following year as 'An Unsolvable Problem of Elementary Number Theory' (a short 'Preliminary report' dated 22 Mar. 1935 having appeared in the _Bulletin of the American Mathematical Society_ (41 (1935), 332–3))—Church proved the undecidability of a system that includes FOPC as a part. This system is known as _Principia Mathematica_ , or PM, after the treatise in which it was first set out (see n. 86). PM is obtained by adding mathematical axioms to FOPC. Church established the conditional result that if PM is _omega-consistent_ , then PM is undecidable. Omega-consistency (first defined by Gödel) is a stronger property than consistency, in the sense that a consistent system is not necessarily omega-consistent. As explained above, a system is consistent when there is no statement _S_ such that both _S_ and _not-S_ are provable in the system. A system is omega-consistent when there is no predicate _F_ of integers such that all the following are provable in the system: (∃ _x_ ) _F_ ( _x_ ), _not-F_ (1), _not-F_ (2), _not-F_ (3), and so on, for every integer. In his later paper 'A Note on the Entscheidungsproblem' (completed in April 1936) Church improved on this earlier result, showing unconditionally that FOPC is undecidable.
J. von Neumann, 'Zur Hilbertschen Beweistheorie' [On Hilbert's Proof Theory], _Mathematische Zeitschrift_ , 26 (1927), 1–46 (12); reprinted in vol. i of von Neumann's _Collected Works_ , ed. A. H. Taub (Oxford: Pergamon Press, 1961).
G. H. Hardy, 'Mathematical Proof', _Mind_ , 38 (1929), 1–25 (16) (the text of Hardy's 1928 Rouse Ball Lecture).
By Andrés Sicard and Jack Copeland.
H. D. Huskey, untitled typescript, National Physical Laboratory, n.d. but _c_. Mar. 1947 (in the Woodger Papers, National Museum of Science and Industry, Kensington, London (catalogue reference M12/105); a digital facsimile is in The Turing Archive for the History of Computing <www.AlanTuring.net/huskey_1947>).
A. Sicard, 'Máquinas de Turing dinámicas: historia y desarrollo de una idea' [Dynamic Turing Machines: Story and Development of an Idea], appendix 3 (Master's thesis, Universidad EAFIT, 1998); 'Máquina universal de Turing: algunas indicaciones para su construcción' [The Universal Turing Machine: Some Directions for its Construction], _Revista Universidad EAFIT_ , vol. 108 (1998), pp. 61–106.
Crown copyright. Reproduced with permission of the Controller of HMSO, the National Physical Laboratory, and the Estate of Alan Turing.
Gödel, "Über formal unentscheidbare Sӓtze der Principia Mathematica und verwandter Systeme, I", _Monatshefte Math. Phys_., 38 (1931), 173–198.
[Received 28 May, 1936.—Read 12 November, 1936.]
This article first appeared in _Proceedings of the London Mathematical Society_ , Series 2, 42 (1936–7). It is reprinted with the permission of the London Mathematical Society and the Estate of Alan Turing.
Alonzo Church, "An unsolvable problem of elementary number theory", _American J. of Math_., 58 (1936), 345–363.
Alonzo Church, "A note on the Entscheidungsproblem", _J. of Symbolic Logic_ , 1 (1936), 40–41.
_Cf_. Hobson, _Theory of functions of a real variable_ (2nd ed., 1921), 87, 88.
If we regard a symbol as literally printed on a square we may suppose that the square is 0≤ _x_ ≤1, 0≤ _y_ ≤1. The symbol is defined as a set of points in this square, viz. the set occupied by printer's ink. If these sets are restricted to be measurable, we can define the "distance" between two symbols as the cost of transforming one symbol into the other if the cost of moving unit area of printer's ink unit distance is unity, and there is an infinite supply of ink at _x_ = 2, _y_ = 0. With this topology the symbols form a conditionally compact space.
The expression "the functional calculus" is used throughout to mean the _restricted_ Hilbert functional calculus.
It is most natural to construct first a choice machine ( 2) to do this. But it is then easy to construct the required automatic machine. We can suppose that the choices are always choices between two possibilities 0 and 1. Each proof will then be determined by a sequence of choices _i_ 1, _i_ 2,..., _i n_ ( _i_ 1 = 0 or 1, _i_ 2 = 0 or 1,..., _i n_ = 0 or 1), and hence the number 2n \+ _i_ 12n− 1 \+ _i_ 22n − 2 \+... + _i n_ completely determines the proof. The automatic machine carries out successively proof 1, proof 2, proof 3,....
The author has found a description of such a machine.
The negation sign is written before an expression and not over it.
A sequence of _r_ primes is denoted by( _r_ ).
If computes **γ** , then the problem whether prints 0 infinitely often is of the same character as the problem whether is circle-free.
A function **α** _n_ may be defined in many other ways so as to run through the computable numbers.
Although it is not possible to find a general process for determining whether a given number is satisfactory, it is often possible to show that certain classes of numbers are satisfactory.
_Loc. cit_.
The Graduate College, Princeton University, New Jersey, USA.
In a complete proof of the **λ** -definability of computable sequences it would be best to modify this method by replacing the numerical description of the complete configurations by a description which can be handled more easily with our apparatus. Let us choose certain integers to represent the symbols and the _m_ -configurations of the machine. Suppose that in a certain complete configuration the numbers representing the successive symbols on the tape are _s_ 1 _s_ 2... _s n_, that the _m_ -th symbol is scanned, and that the _m_ -configuration has the number _t_ ; then we may represent this complete configuration by the formula
[[ _N s1_, _N s2_,..., _N_ _sm_ −1], [ _N_ _t_ , _N_ _sm_ ], [ _N_ _sm+1_ ,..., _N_ _sn_ ]],
where [ _a_ , _b_ ] stands for **λ** _u_ [{{ _u_ }( _a_ )}( _b_ )],
[ _a_ , _b_ , _c_ ] stands for **λ** _u_ [{{{ _u_ }( _a_ )}( _b_ )}( _c_ )],
etc.
_Journal of Symbolic Logic_ , 12 (1947), 1–11.
E. L. Post, 'Finite Combinatory Processes—Formulation 1', _Journal of Symbolic Logic_ , 1 (1936), 103–4.
The draft is among the Turing Papers in the Modern Archive Centre, King's College Library, Cambridge; catalogue reference D 2.
Davies in interview with Christopher Evans ('The Pioneers of Computing: An Oral History of Computing' (London: Science Museum, 1975)).
I am grateful to Diane Davies for her permission to publish this article.
The author is indebted to P. Bernays for pointing out these errors.
This article first appeared in _Proceedings of the London Mathematical Society_ , Series 2, 43 (1937), 544–6. It is reprinted with the permission of the London Mathematical Society and the Estate of Alan Turing.
This use of overlapping intervals for the definition of real numbers is due originally to Brouwer.
Editor's note. This paragraph originally formed a footnote (the first) to Post's appendix.
Post's critique originally formed an untitled appendix occupying pp. 7–11 of 'Recursive Unsolvability of a Problem of Thue', _Journal of Symbolic Logic_ , 12 (1947), 1–11. The critique is reprinted here by permission of the Association for Symbolic Logic. All rights reserved. This reproduction is by special permission for this publication only.
Our quadruplets are quintuplets in the Turing development. That is, where our standard instruction orders either a printing (overprinting) or motion, left or right, Turing's standard instruction always orders a printing and a motion, right, left, or none. Turing's method has certain technical advantages, but complicates theory by introducing an irrelevant "printing" of a symbol each time that symbol is merely passed over.
"Genuinely prints", that is, a genuine printing being a printing in an empty square. See the previous footnote.
Turing in each case refers to the S.D of a machine being supplied. But the proof of the first theorem, and the second theorem depends on the first, shows that it is really a positive integer _n_ that is supplied. Turing's proof of the second theorem is unusual in that while it uses the unsolvability result of the first theorem, it does not "reduce" [Post (1944)] the problem of the first theorem to that of the second. In fact, the first problem is almost surely of "higher degree of unsolvability" [Post (1944)] than the second, in which case it could not be "reduced" to the second. Despite appearances, that second unsolvability proof, like the first, is a _reductio ad absurdum_ proof based on the definition of unsolvability, at the conclusion of which, the first result is used.
Our experience with proving that "normal unsolvability" in a sense implicit in [Post (1943)] is equivalent to unsolvability in the sense of Church [(1936)], at least when the set of questions is recursive, suggests that a fair amount of additional labor would here be involved. That is probably our chief reason for making our proof of the recursive unsolvability of the problem of Thue independent of Turing's development.
Editor's note. Thue's problem is that of determining, for arbitrary strings of symbols _A_ , _B_ from a given finite alphabet, whether or not _A_ and _B_ are interderivable by means of a succession of certain simple substitutions. (See further Chapter 17.)
Granted the corrections [detailed above].
This modification of the concept of motion left is assumed throughout the rest of the discussion, with the exception of the last paragraph.
So far as recursiveness is concerned, the distinction between the Turing convention and the (0, 1) convention is that the former concerns the history of the machine in action, the latter only the instructions defining the machine. Likewise, despite appearances, the later ( _q_ , _q_ ′) convention.
It is here assumed that the suggested extension of [Turing (1937)] includes a proof to the effect that the existence of an arbitrary Turing machine for solving a given problem is equivalent to the existence of a Turing convention-machine for solving that problem.
Editor's note. See the reference at the foot of p. 97.
S. Turing, _Alan M. Turing_ (Cambridge: Heffer, 1959), 51.
Letter from Turing to Sara Turing, his mother, 24 May 1935 (in the Turing Papers, Modern Archive Centre, King's College Library, Cambridge (catalogue reference K 1)).
Letter from Turing to Sara Turing, 29 May 1936 (Turing Papers, catalogue reference K 1).
Letter from Turing to Sara Turing, 7 May 1938 (see below); Turing, _Alan M. Turing_ , 54. The thesis is held in the Seeley G. Mudd Manuscript Library at Princeton University (catalogue reference P685.1938.47).
See the subsection 'Church's contribution' of 'Computable Numbers: A Guide'.
Church in interview with William Aspray (17 May 1984); transcript no. 5 in the series 'The Princeton Mathematics Community in the 1930s', Princeton University.
Ibid.
See below.
S. C. Kleene, 'Origins of Recursive Function Theory', _Annals of the History of Computing_ , 3 (1981), 52–67 (62).
Letter from Gandy to Max Newman, n.d. (Turing Papers, catalogue reference A 8).
All the letters are in the Turing Papers (catalogue reference K 1).
Editor's note. Sara Turing's dating of the letters is followed where dates are absent or incomplete.
Editor's note. Turing's previous letter to Sara on 28 Sept. 1936 was written on board the vessel _Berengaria_ bound for New York.
Editor's note. Presumably 'On Computable Numbers, with an Application to the Entscheidungsproblem'.
Editor's note. This may refer to the 'development of the theory of functions of a real variable' mentioned by Turing on p. 58 of Chapter 1. No such paper ever appeared, nor a book.
Editor's note. Here the letter is marked '"On Computable Numbers"' in Sara's hand.
Editor's note. The author's copies of 'On Computable Numbers'.
Editor's note. Trinity College, Cambridge.
Editor's note. Sara has written 'probably' against the date.
Editor's note. In 1938 Turing published two papers in group theory: 'Finite Approximations to Lie Groups' ( _Annals of Mathematics_ , 39: 105–11), which developed a method due to R. Baer, and 'The Extensions of a Group' ( _Compositio Mathematica_ , 5: 357–67).
Editor's note. Presumably 'Finite Approximations to Lie Groups' (see above).
Editor's note. London Mathematical Society.
Editor's note. _New Statesman and Nation_.
Editor's note. See the introduction to Chapter 16.
Editor's note. Luther Eisenhart, Dean of the Graduate College.
Editor's note. The Jane Eliza Procter Visiting Fellowship.
Editor's note. Reprints of 'On Computable Numbers'.
Editor's note. See Chapter 14.
Editor's note. H. Scholz. Scholz's postcard is in the Turing Papers (catalogue reference D 5).
Editor's note. Turing wrote '15 Feb '37'; 'Feb' has been corrected on the letter to 'Mar'.
Editor's note. Provost of King's College, Cambridge.
Editor's note. Hall was a Fellow of King's. In 1938 Hall became Secretary of the London Mathematical Society.
Editor's note. Turing did not get the lectureship.
Editor's note. Presumably the ideas that formed 'Systems of Logic Based on Ordinals'.
Editor's note. To England.
Editor's note. From England to the United States.
Editor's note. 'On Computable Numbers'.
Editor's note. 'Lectured'.
Editor's note. The letter is undated and marked 'Recd. Jan 14. 38'.
Editor's note. 'Computability and λ-Definability' ( _Journal of Symbolic Logic_ , 2 (1937), 153–63), and 'The ' '-Function in λ- _K_ -Conversion' ( _Journal of Symbolic Logic_ , 2 (1937), 164).
Editor's note. The Cambridge long vacation.
D. Hilbert, 'Über das Unendliche' [On the Infinite], _Mathematische Annalen_ , 95 (1926), 161–90 (180); English translation by E. Putnam and G. Massey in R. L. Epstein and W. A. Carnielli, _Computability: Computable Functions, Logic, and the Foundations of Mathematics_ (2nd edn. Belmont, Calif.: Wadsworth, 2000).
D. Hilbert, 'Probleme der Grundlegung der Mathematik' [Problems Concerning the Foundation of Mathematics], _Mathematische Annalen_ , 102 (1930), 1–9 (9).
This post-Gödelian programme was also investigated by Church's student Barkley Rosser in his 'Gödel Theorems for Non-Constructive Logics', _Journal of Symbolic Logic_ , 2 (1937), 129–37.
The 'Gentzen-type' logics of Section 12 are different from the Gentzen-style formulation of first-order predicate calculus discussed above (although, as the names imply, both are suggested by work of Gentzen).
S. Feferman, 'Turing in the Land of O(z)', in R. Herken (ed.), _The Universal Turing Machine: A Half-Century Survey_ (Oxford: Oxford University Press, 1988), 122–3.
S. Feferman, 'Transfinite Recursive Progressions of Axiomatic Theories', _Journal of Symbolic Logic_ , 27 (1962), 259–316.
Ibid. 261; J. R. Shoenfield, 'On a Restricted v-Rule', _Bulletin de l'Académie Polonaise des Sciences_ , 7 (1959), 405–7.
J. R. Shoenfield, _Degrees of Unsolvability_ (Amsterdam: North-Holland, 1971), 15.
E. L. Post, 'Recursively Enumerable Sets of Positive Integers and their Decision Problems', _Bulletin of the American Mathematical Society_ , 50 (1944), 284–316; E. L. Post, 'Recursive Unsolvability of a Problem of Thue', _Journal of Symbolic Logic_ , 12 (1947), 1–11; E. L. Post, 'Degrees of Recursive Unsolvability-Preliminary Report', _Bulletin of the American Mathematical Society_ , 54 (1948), 641–2.
'Recursively Enumerable Sets of Positive Integers and their Decision Problems', 311.
Ibid. 289.
Ibid. 289–90.
I am grateful to Peter Hilton and Gualtiero Piccinini for their comments on a draft of this introduction.
This paper represents work done while a Jane Eliza Procter Visiting Fellow at Princeton University, where the author received most valuable advice and assistance from Prof. Alonzo Church.
This article first appeared in _Proceedings of the London Mathematical Society_ , Series 2, 45 (1939), 161–228. It is reprinted with the permission of the London Mathematical Society and the Estate of Alan Turing.
The situation is not quite so simple as is suggested by this crude argument. See pages [162–73], [181–3].
This follows from (A) below.
We shall use the expression "computable function" to mean a function calculable by a machine, and we let "effectively calculable" refer to the intuitive idea without particular identification with any one of these definitions. We do not restrict the values taken by a computable function to be natural numbers; we may for instance have computable propositional functions.
I believe that there is no generally accepted meaning for this term, but it should be noticed that we are using it in a rather restricted sense. The most generally accepted meaning is probably this: suppose that we take an arbitrary formula of the functional calculus of the first order and replace the function variables by primitive recursive relations. The resulting formula represents a typical number-theoretic theorem in this (more general) sense.
Primitive recursive functions of natural numbers are defined inductively as follows. Suppose that φ( _x_ 1,..., _x_ _n–1_ ), _g_ ( _x_ 1,..., _x_ _n_ ), _h_ ( _x_ 1,..., _x_ _n_ \+ 1) are primitive recursive, then φ( _x_ 1,..., _x_ _n_ ) is primitive recursive if it is defined by one of the sets of equations (a) to (e).
( _a_ ) _φ_ ( _x_ 1,..., _x_ _n_ ) = _h_ ( _x_ 1,..., _x_ _m_ – 1, _g_ ( _x_ 1,..., _x_ n), _x_ _m_ +1,..., _x_ _n_ –1, _x_ _m_ ) (1 ≥ _m_ ≥ _n_ );
(b) _φ_ ( _x_ 1,..., _x_ _n_ ) = _f_ ( _x_ 2,..., _x_ _n_ );
(c) _φ_ ( _x_ 1) = _a_ , where _n_ = 1 and _a_ is some particular natural number;
(d) _φ_ ( _x_ 1) = _x_ 1 \+ 1 ( _n_ = 1);
(e) _φ_ ( _x_ 1,..., _x_ _n_ – 1, 0) = _φ_ ( _x_ 1,..., _x_ _n_ – 1);
_φ_ ( _x_ 1,..., _x_ _n_ – 1, _x_ _n_ \+ 1) = _h_ ( _x_ 1,..., _x_ _n_ , _φ_ ( _x_ 1,..., _x_ _n_ )):
The class of primitive recursive functions is more restricted than the class of computable functions, but it has the advantage that there is a process whereby it can be said of a set of equations whether it defines a primitive recursive function in the manner described above.
Kleene [3], 727. This result is really superfluous for our purpose, since the proof that every computable function is general recursive proceeds by showing that these functions are of the form (3.2). (Turing [2], 161).
Compare Rosser [1].
Without real loss of generality we may suppose that A is always well formed.
Compare Turing [1], §6, 7.
Compare Turing [1], [p. 77, n. 7], [2], 156.
To require _G_ ( _x, y_ ) to be axiomatic amounts to requiring _G_ ( _x, y_ ) to be computable on account of (7.1)
On the other hand, if _D_ ( _x_ ) is axiomatic and _G_ ( _x, y_ ) is computable in the modified sense that there is a rule for determining whether _G_ ( _x, y_ ) is true which leads to a definite result in all cases where _D_ ( _x_ ) and _D_ ( _y_ ) are true, the corresponding definition of ordinal formula is equivalent to our definition. To give the proof would be too much of a digression. Probably other equivalences of this kind hold.
If we also allow λ _ufx_. _u_ (R) to be a C-K ordinal formula when
λ _ufx_. **n** ( **R** ) conv λ _ufx_. _S_ ( **n** , **R** )
for all _n_ , then the formulae for sum, product and exponentiation of C-K ordinal formulae can be much simplified. For instance, if **A** and **B** represent α and β, then
λ _ufx_ : **B** ( _u_ , _f_ , **A** ( _u_ , _f_ , _x_ ))
represents α + β. Property (6) remains true.
The order type is β, where 1 + β = Ξ **A** ; but β = Ξ **A** , since Ξ **A** is infinite.
Whitehead and Russell [1]. The axioms and rules of procedure of a similar system _P_ will be found in a convenient form in Gödel [1], and I follow Gödel. The symbols for the natural numbers in _P_ are 0, _f_ 0, _ff_ 0,..., _f_ ( _n_ )0... Variables with the suffix "0" stand for natural numbers.
It is sometimes regarded as necessary that the set of axioms used should be computable, the intention being that it should be possible to verify of a formula reputed to be an axiom whether it really is so. We can obtain the same effect with axiomatic sets of axioms in this way. In the rules of procedure describing which are the axioms, we incorporate a method of enumerating them, and we also introduce a rule that in the main part of the deduction, whenever we write down an axiom as such, we must also write down its position in the enumeration. It is possible to verify whether this has been done correctly.
A relation _F_ ( _m_ 1,..., _m r_) is primitive recursive if it is a necessary and sufficient condition for the vanishing of a primitive recursive function φ( _m_ 1,..., _m r_).
Capital German letters will be used to stand for variable or undetermined formulae in _P_. An expression such as stands for the result of substituting and for _x_ 0 and _y_ 0 in .
In outline Church [1], 279–280. In greater detail Church [2], Chap. X.
In the case of _P_ we adjoined all of the axioms (∃ _x_ 0) Proof [ _x_ 0, _f_ ( _m_ )0] , where _m_ is the G.R. of ; the Gödel theorem shows that _some_ of them are unprovable in _P_.
This part of the argument can equally well be based on the impossibility of determining about two W.F.F. whether they are interconvertible. (Church [3], 363.)
The subset _U_ wholly supersedes _V_ in what follows. The introduction of _V_ serves to emphasise the fact that the set of ordinals represented by members of _U_ may have gaps.
A suggestion to consider this problem came to me indirectly from F. Bernstein. A related problem was suggested by P. Bernays.
We are leaving out of account that most important faculty which distinguishes topics of interest from others; in fact, we are regarding the function of the mathematician as simply to determine the truth or falsity of propositions.
This requirement is very vague. It is not of course intended that the criterion of the correctness of the intuitive steps be the correctness of the final result. The meaning becomes clearer if each intuitive step is regarded as a judgment that a particular proposition is true. In the case of an ordinal logic it is always a judgment that a formula is an ordinal formula, and this is equivalent to judging that a number-theoretic proposition is true. In this case then the requirement is that the reputed ordinal logic _is_ an ordinal logic.
This ordinal logic arises from a certain system _C 0_ in essentially the same way as L _P_ arose from _P_. By an argument similar to one occurring in §8 we can show that the ordinal logic leads to correct results if and only if _C 0_ is valid; the validity of _C 0_ is proved in Church [1], making use of the results of Church and Rosser [1].
Editor's note. The remainder of Turing's index has been incorporated into the general index at the rear of the book.
Turing and Church also corresponded at this time. A letter from Church addressed to Turing at the Crown is dated 15 May 1940 and replies to Turing's of 15 April. I am grateful to Alonzo Church's son, Alonzo Church Jnr, for sending me a copy of Church's letter.
The title page reads: 'MATHEMATICAL LOGIC Lectures by Alonzo Church Princeton University, October 1935–January 1936. (Notes by F. A. Ficken, H. G. Landau, H. Ruja, R. R. Singleton, N. E. Steenrod, J. H. Sweer, F. J. Weyl).' I am grateful to Alonzo Church Jnr for information concerning this typescript.
M. H. A. Newman and A. M. Turing, 'A Formal Theorem in Church's Theory of Types', _Journal of Symbolic Logic_ , 7 (1942), 28–33.
J. Murray, 'Hut 8 and Naval Enigma, Part I', in H. Hinsley and A. Stripp (eds.), _Codebreakers: The Inside Story of Bletchley Park_ (Oxford: Oxford University Press, 1993), 117.
Newman in interview with Christopher Evans ('The Pioneers of Computing: An Oral History of Computing' (London: Science Museum)).
A. Hodges, _Alan Turing: The Enigma_ (London: Vintage, 1992), 109.
M. H. A. Newman, 'Alan Mathison Turing, 1912–1954', _Biographical Memoirs of Fellows of the Royal Society_ , 1 (1955), 253–63 (258). On the paper by Church, 'A Note on the Entscheidungsproblem', see 'Computable Numbers: A Guide'.
The letter is among the Turing Papers in the Modern Archive Centre, King's College Library, Cambridge (catalogue reference K 1).
Newman, 'Alan Mathison Turing, 1912–1954', 258.
Newman in interview with Christopher Evans (see n. 5).
See further 'Max Newman: Mathematician, Codebreaker and Computer Pioneer', by William Newman (Max's son), to appear in B. J. Copeland (ed.), _Colossus: The First Electronic Computer_ (Oxford University Press).
Quoted in W. Newman, 'Max Newman: Mathematician, Codebreaker and Computer Pioneer'.
W. T. Tutte, 'At Bletchley Park', to appear in Copeland (ed.), _Colossus: The First Electronic Computer_.
Part 1 of 'General Report on Tunny'. 'General Report on Tunny' was written in 1945 by Jack Good, Donald Michie, and Geoffrey Timms, all members of Newman's section at GC & CS. This document was released by the British government in 2000 to the Public Record Office at Kew (document reference HW 25/4, HW 25/5). A digital facsimile of the document is available in The Turing Archive for the History of Computing <www. AlanTuring.net/tunny_report>.
Flowers in interview with Copeland (July 1996, July 1998).
Flowers in interview with Copeland (July 1996).
F. H. Hinsley et al., _British Intelligence in the Second World War_ , vol. iii, part 2 (London: Her Majesty's Stationery Office, 1988), 53, 799.
Letter from Newman to von Neumann, 8 Feb. 1946 (in the von Neumann Archive at the Library of Congress, Washington, DC; a digital facsimile is in The Turing Archive for the History of Computing <www. AlanTuring.net/newman_vonneumann_8feb46>).
Williams in interview with Christopher Evans in 1976 ('The Pioneers of Computing: An Oral History of Computing' (London: Science Museum)).
Catalogue reference D 2. The letters are published with the permission of the Estate of Alan Turing.
Editor's note. See S. C. Kleene, 'A Theory of Positive Integers in Formal Logic, Part II', _American Journal of Mathematics_ , 57 (1935), 219–44 (231–2).
Editor's note. 'Metad' is defined in Kleene, 'A Theory of Positive Integers in Formal Logic', 233.
Editor's note. Presumably 'left-hand side'.
Editor's note. Presumably this should read: Θ(x) conv λx.x(Θ( _x_ )).
Editor's note. Turing's example concerns a formula that he himself found and published in his 'The -Function in λ- _K_ -Conversion' ( _Journal of Symbolic Logic_ , 2 (1937), 164). The published formula is Θ → {λ _vu.u_ ( _v_ ( _v_ , _u_ ))} (λ _vu.u_ ( _v_ ( _v_ , _u_ ))).
Editor's note. Turing is referring to 'Systems of Logic Based on Ordinals' (Chapter 3).
Editor's note. A reference to Church's 'Mathematical Logic' (see p. 205 n. 2).
Editor's note. A reference to Church's 'Mathematical Logic'.
Editor's note. Chapter X of Church's 'Mathematical Logic' is entitled 'The Universal Quantifier'.
Editor's note. Turing is referring to 'Systems of Logic Based on Ordinals' (Chapter 3).
Letters from Peter Twinn to Copeland (28 Jan. 2001, 21 Feb. 2001). Twinn himself joined the attack on Enigma in February 1939. Turing was placed on Denniston's 'emergency list' (see below) in March 1939, according to 'Staff and Establishment of G.C.C.S.' (undated), held in the Public Record Office: National Archives (PRO), Kew, Richmond, Surrey (document reference HW 3/82). (I am grateful to Ralph Erskine for drawing my attention to this document.)
Letters from Twinn to Copeland (see n. 1).
M. Batey, 'Breaking Italian Naval Enigma', in R. Erskine and M. Smith (eds.), _Action This Day_ (London: Bantam, 2001), 98.
Letter from A. G. Denniston to T. J. Wilson of the Foreign Office (7 Sept. 1939). PRO document reference FO 366/1059.
F. H. Hinsley et al., _British Intelligence in the Second World War_ , vol. ii (London: Her Majesty's Stationery Office, 1981), 29.
F. H. Hinsley et al., _British Intelligence in the Second World War_ , vol. iii, part 1 (London: Her Majesty's Stationery Office, 1984), 461.
This estimate was given by Hinsley, official historian of the British Secret Service, on p. 12 of his and Alan Stripp's edited volume _Codebreakers: The Inside Story of Bletchley Park_ (Oxford: Oxford University Press, 1993). If, wrote Hinsley, the achievements of GC & CS 'had not prevented the U-boats from dominating the Atlantic... it is not unreasonable to believe that... Overlord [the invasion of Normandy, 1944] would have had to be deferred till 1946'.
The older spelling 'cypher' and the newer 'cipher' were both in use at GC & CS during 1939–45. Mahon used 'cypher' in a 1945 document, part of which forms Chapter 5, and Turing used 'cipher' in a 1940 document, parts of which appear in Chapters 5 and 6.
Probably in August (R. Erskine, 'GC and CS Mobilizes "Men of the Professor Type"', _Cryptologia_ , 10 (1986), 50–9 (50)).
Letter from Denniston to Wilson (3 Sept. 1939). PRO document reference FO 366/1059.
A. G. Denniston, 'The Government Code and Cypher School between the Wars', in C. W. Andrew (ed.), _Codebreaking and Signals Intelligence_ (London: Cass, 1986), 52.
Andrew, _Codebreaking and Signals Intelligence_ , 4.
S. Milner-Barry, 'Hut 6: Early Days', in Hinsley and Stripp (eds.), _Codebreakers_ , 90; 'Staff and Establishment of G.C.C.S.'; Erskine, 'GC and CS Mobilizes "Men of the Professor Type"', 50.
E. R. Vincent, Unpublished Memoirs, Corpus Christi College Archives, Cambridge; quoted in C. W. Andrew, _Secret Service: The Making of the British Intelligence Community_ (London: Guild, 1985), 94.
F. L. Bauer, _Decrypted Secrets: Methods and Maxims of Cryptology_ (Berlin: Springer-Verlag, 2nd edn. 2000), 107.
F. H. Hinsley et al. _British Intelligence in the Second World War_ , vol. iii, part 2 (London: Her Majesty's Stationery Office, 1988), 946.
M. Rejewski, 'Remarks on Appendix 1 to British Intelligence in the Second World War by F. H. Hinsley', _Cryptologia_ , 6 (1982), 75–83 (76).
The commercial model remained on sale after the German military adopted Enigma. The Germans knew how to break the commercial model and from 1938 several hundred were sold to neutral Switzerland by the German manufacturers. The commercial model was also sold by Germany to Hungary during the war. Commercial model Enigmas sold to Spain were used during the Spanish Civil War. (I am grateful to Frode Weierud for this information (personal communication).)
The fourth wheel differed from the other three in that once the operator had set it to one of its twenty-six positions, it remained stationary during the encipherment of the message. (That the fourth wheel came into Naval use in 1941 is documented in R. Erskine, 'Breaking German Naval Enigma on Both Sides of the Atlantic', in Erskine and Smith (eds.), _Action this Day_ , 181.
H. Alexander, 'Cryptographic History of Work on the German Naval Enigma' (no date ( _c_.1945), PRO document reference HW 25/1), 3; a digital facsimile of Alexander's typescript is available in The Turing Archive for the History of Computing <www.AlanTuring.net/alexander_naval_enigma>.
See, for example, G. Welchman, _The Hut Six Story: Breaking the Enigma Codes_ (Kidderminster: M. & M. Baldwin, 2nd edn. 1997), 205.
M. Rejewski, 'Summary of our Methods for Reconstructing Enigma and Reconstructing Daily Keys, and of German Efforts to Frustrate Those Methods', in W. Kozaczuk, _Enigma: How the German Machine Cipher Was Broken, and How It Was Read by the Allies in World War Two_ , trans. C. Kasparek (London: Arms and Armour Press, 1984), 243.
Rejewski's accounts of the work of the Polish cryptanalysts use 'message key' instead of the Bletchley term 'message setting'. See, for example, M. Rejewski, 'Jak Matematycy polscy rozszyfrowali Enigme' [How the Polish Mathematicians Broke Enigma], _Annals of the Polish Mathematical Society, Series II: Mathematical News_ , 23 (1980), 1–28. (This article appears in an English translation by C. Kasparek as appendix D of Kozaczuk, _Enigma_ ; another translation, by J. Stepenske, appears in _Annals of the History of Computing_ , 3 (1981), 213–34, under the title 'How Polish Mathematicians Deciphered the Enigma'.)
Rejewski, 'Remarks on Appendix 1 to British Intelligence in the Second World War by F. H. Hinsley',79.
Rejewski, 'How the Polish Mathematicians Broke Enigma', trans. Kasparek, 251.
Mahon's 'The History of Hut 8' is in the US National Archives and Records Administration (NARA) in Washington, DC (document reference: RG 457, Historic Cryptographic Collection, Box 1424, NR 4685) and in the UK Public Record Office (document reference HW 25/2). A digital facsimile of the original typescript is available in The Turing Archive for the History of Computing <www.AlanTuring.net/mahon_hut_8>.
Rejewski, 'How the Polish Mathematicians Broke Enigma', trans. Kasparek, 265–6; Hinsley, _British Intelligence in the Second World War_ , vol. iii, part 2, 949, 953.
G. Bloch and R. Erskine, 'Enigma: The Dropping of the Double Encipherment', _Cryptologia_ , 10 (1986), 134–41.
The term 'indicator' is used by Mahon and Turing in the next chapter and is listed in 'A Cryptographic Dictionary', GC & CS (1944). ('A Cryptographic Dictionary' was declassified in 1996 (NARA document reference: RG 457, Historic Cryptographic Collection, Box 1413, NR 4559); a digital facsimile is available in The Turing Archive for the History of Computing <www.AlanTuring.net/crypt_dic_1944>.) However, the term 'indicator setting', which is from Welchman ( _The Hut Six Story_ , 36, 46) may not have been in use at Bletchley Park, where the German term _Grundstellung_ (or 'Grund') was used (see e.g. pp. 272–3, below), as it was by the Poles (letter from Rejewski to Woytak, quoted on p. 237 of Kozaczuk, _Enigma_ ).
The sources for this section are: 'A Conversation with Marian Rejewski' (in Kozaczuk, _Enigma_ ), Rejewski's articles 'How the Polish Mathematicians Broke Enigma', 'Summary of our Methods for Reconstructing Enigma and Reconstructing Daily Keys, and of German Efforts to Frustrate Those Methods', 'The Mathematical Solution of the Enigma Cipher' (in Kozaczuk, _Enigma_ ), and 'Remarks on Appendix 1 to British Intelligence in the Second World War by F. H. Hinsley', together with Hinsley, vol. iii, part 2, appendix 30 'The Polish, French and British Contributions to the Breaking of the Enigma: A Revised Account'. (Appendix 30 replaces the sometimes very inaccurate appendix 1, 'The Polish, French and British Contributions to the Breaking of the Enigma', of Hinsley et al., _British Intelligence in the Second World War_ , vol. i (London: Her Majesty's Stationery Office, 1979).
Rejewski, 'How Polish Mathematicians Deciphered the Enigma', trans. Stepenske, 221.
Hinsley, _British Intelligence in the Second World War_ , vol. iii, part 2, appendix 30, 951.
Ibid.
Letter from Rejewski to Richard Woytak, 15 Apr. 1979; the letter is printed in Kozaczuk, _Enigma_ , 237–8.
Rejewski, 'How the Polish Mathematicians Broke Enigma', trans. Kasparek, 261.
Hinsley, _British Intelligence in the Second World War_ , vol. iii, part 2, appendix 30, 951.
Rejewski, 'How the Polish Mathematicians Broke Enigma', 267.
Ibid. 269.
Ibid. 257; P. Twinn, 'The _Abwehr_ Enigma', in Hinsley and Stripp (eds.), _Codebreakers_ , 126.
Twinn, 'The _Abwehr_ Enigma', 126–7.
G. Bertrand, _Enigma, ou la plus grande énigme de la guerre 1939–1945_ (Paris: Plon, 1973), 60–1.
Quoted in Kozaczuk, _Enigma_ , 97. On Turing's visit to the Poles, see ibid 96–7; Welchman, _The Hut Six Story_ , 220; and R. Erskine, 'Breaking Air Force and Army Enigma', in Erskine and Smith, _Action This Day_ , 54.
Erskine, 'Breaking Air Force and Army Enigma', 55.
Kozaczuk, _Enigma_ , 63.
Rejewski, 'How the Polish Mathematicians Broke Enigma', 267.
Tadeusz Lisicki quoted in Kozaczuk, _Enigma_ , 63.
Untitled typescript dated 11 Oct. 1943 (NARA, document reference RG 457, Historic Cryptographic Collection, Box 705, NR 4584), 1.
'Operations of the 6312th Signal Security Detachment, ETOUSA', 1 Oct. 1944 (NARA, document reference: RG 457, Historic Cryptographic Collection, Box 970, NR 2943), 5. (Thanks to Ralph Erskine for drawing my attention to this quotation and to Frode Weierud for sending me a copy of the document.)
Rejewski, 'Remarks on Appendix 1 to British Intelligence in the Second World War by F. H. Hinsley', 81.
Untitled typescript dated 11 Oct. 1943, 2; 'Operations of the 6312th Signal Security Detachment, ETOUSA', 5.
Kozaczuk, _Enigma_ , figure E-8, 289.
See p. 40.
The indicators and indicator settings used in this example are adapted from p. 266 of Kasparek's translation of Rejewski's 'Jak Matematycy polscy rozszyfrowali Enigme' in Kozaczuk, _Enigma_. The present description of the bomby has been reconstructed from Rejewski's rather compressed account appearing on that page. Unfortunately, Stepenske's translation of these same paragraphs in the _Annals of the History of Computing_ is marred by an error that seriously affects the sense. The phrase that Stepenske translates 'by striking key W three times in a row, the same lamp would light' (p. 226) should be translated 'if key W is struck the same lamp will light again after three more strokes'.
Hinsley, _British Intelligence in the Second World War_ , vol. iii, part 2, appendix 30, p. 954.
Ibid.
'Enigma—Position' and 'Naval Enigma Situation', notes dated 1 Nov. 1939 and signed by Knox, Twinn, Welchman, and Turing. Both notes are in the Public Record Office (document reference HW 14/2).
Batey, 'Breaking Italian Naval Enigma'; Twinn, 'The _Abwehr_ Enigma'.
'Operations of the 6312th Signal Security Detachment, ETOUSA', 60. (Thanks to John Harper for additional information.)
'Operations of the 6312th Signal Security Detachment, ETOUSA', 67.
D. Payne, 'The Bombes', in Hinsley and Stripp (eds.), _Codebreakers_ , 134. The coils of wire described by Mahon were probably red in colour. Red wire and very rarely black wire were used by the Letchworth bombe factory (letter from John Harper to Copeland (25 Feb. 2003), reporting interviews with engineers who worked on the bombes at the Letchworth factory).
Bauer, _Decrypted Secrets_ , 112, 135.
The title 'Treatise on the Enigma' was probably added to Turing's document by a third party outside GC & CS and quite probably in the United States. The copy of the otherwise untitled document held in the US National Archives and Records Administration (document reference RG 457, Historic Cryptographic Collection, Box 201, NR 964) is prefaced by a page typed some years later than the document itself. It is this page that bears the title 'Turing's Treatise on the Enigma'. Another copy of the document held in the British Public Record Office (document reference HW 25/3) carries the title 'Mathematical theory of ENIGMA machine by A M Turing'; this, too, was possibly added at a later date. Mahon refers to the document simply as 'Prof's Book'. The PRO copy is complete, and much more legible than the incomplete NARA copy, which lacks many figures. A digital facsimile of the PRO typescript is available in The Turing Archive for the History of Computing <www.AlanTuring.net/profs_book>. A retyped version of the complete work, prepared by Ralph Erskine, Philip Marks, and Frode Weierud, is available at <<http://home.cern.ch/frode/crypto>>.
See J. Murray, 'Hut 8 and Naval Enigma, Part I', in Hinsley and Stripp (eds.), _Codebreakers_ , 116. The date of composition of the document, summer 1940, is given by Hinsley, _British Intelligence in the Second World War_ , vol. iii, part 2, appendix 30, 955.
Hinsley, _British Intelligence in the Second World War_ , vol. iii, part 2, appendix 30, 954. When Mahon says in the next chapter that the 'first bombe arrived in April 1940' (p. 292), he is probably referring to the time at which the bombe became available to the codebreakers.
'Squadron Leader Jones, Section' (Public Record Office, document reference HW 3/164). (Thanks to Ralph Erskine for sending me a copy of this document.)
Hinsley, _British Intelligence in the Second World War_ , vol. iii, part 2, appendix 30, p. 954.
Welchman gives an account of the method, _The Hut Six Story_ , 237–41.
C. A. Deavours and L. Kruh, 'The Turing Bombe: Was It Enough?', _Cryptologia_ , 14 (1990), 331–49 (346–8).
Murray (née Clarke), 'Hut 8 and Naval Enigma, Part I', 115.
I am indebted to Frank Carter for this suggestion.
Hinsley, _British Intelligence in the Second World War_ , vol. iii, part 2, appendix 30, 955.
'Squadron Leader Jones, Section' (see n. 65); R. Erskine, 'Breaking Air Force and Army Enigma', in Erskine and Smith, _Action this Day_ , 56.
'Squadron Leader Jones, Section'.
Alexander 'Cryptographic History of Work on the German Naval Enigma', 31, 35.
Hinsley, _British Intelligence in the Second World War_ , ii. 29.
Welchman, _The Hut Six Story_ , 139–41, 147.
Ibid. 147.
Erskine, 'Breaking German Naval Enigma on Both Sides of the Atlantic', 192–3.
Alexander, 'Cryptographic History of Work on the German Naval Enigma', 90.
S. W. Roskill, _The War at Sea 1939–1945_ (London: HMSO, 1954), 615–16.
C. Morris, 'Navy Ultra's Poor Relations', in Hinsley and Stripp (eds.), _Codebreakers_ , 237.
Alexander, 'Cryptographic History of Work on the German Naval Enigma', 19–20.
Alexander, 'Cryptographic History of Work on the German Naval Enigma', 7.
Chapter 5, p. 285.
Alexander, 'Cryptographic History of Work on the German Naval Enigma', 26, 28, 30; Murray, 'Hut 8 and Naval Enigma, Part I', 112.
Alexander, 'Cryptographic History of Work on the German Naval Enigma', 33.
Erskine, 'Breaking German Naval Enigma on Both Sides of the Atlantic', and 'Naval Enigma: The Breaking of Heimisch and Triton', _Intelligence and National Security_ , 3 (1988), 162–83.
_Süd_ is discussed in Erskine, 'Naval Enigma: An Astonishing Blunder', _Intelligence and National Security_ , 11 (1996), 468–73, and 'Breaking German Naval Enigma on Both Sides of the Atlantic', 186–9.
Alexander, 'Cryptographic History of Work on the German Naval Enigma', 24.
The report of the engagement, 'Second and Last War Cruise', is in PRO (document reference ADM 186/805). See also R. Erskine, 'The First Naval Enigma Decrypts of World War II', _Cryptologia_ , 21 (1997), 42–6.
Alexander, 'Cryptographic History of Work on the German Naval Enigma', 24; Ralph Erskine (personal communication).
Alexander, 'Cryptographic History of Work on the German Naval Enigma', 24; Murray, 'Hut 8 and Naval Enigma, Part I', 113.
Alexander, 'Cryptographic History of Work on the German Naval Enigma', 24.
Ibid. 25.
Ibid. 5, 25.
Alexander's statements on pp. 24–5 (or possibly Mahon's on p. 286, below) have been interpreted, probably incorrectly, by the authors of _British Intelligence in the Second World War_ as implying that materials obtained from the Narvik Pinch enabled Hut 8 to read Naval Enigma traffic for the six days 22–7 April _during May_ (Hinsley et al., _British Intelligence in the Second World War_ , i. 163, 336). In 1993 Joan Clarke stated that some of these days were not broken until June (Murray, 'Hut 8 and Naval Enigma, Part I', 113; see also Erskine, 'The First Naval Enigma Decrypts of World War II', 43).
Birch's letter is included in a contemporary report entitled 'Operation Ruthless' by C. Morgan (PRO document reference ADM 223/463).
Alexander, 'Cryptographic History of Work on the German Naval Enigma', 27. An official report of the operation is in PRO (document reference DEFE 2/142). The operation is described in Erskine 'Breaking German Naval Enigma on Both Sides of the Atlantic', 178.
Chapter 5, p. 290; Alexander, 'Cryptographic History of Work on the German Naval Enigma', 27.
Chapter 5, p. 290.
Alexander, 'Cryptographic History of Work on the German Naval Enigma', 28.
Hinsley, _British Intelligence in the Second World War_ , i. 337.
A report of the capture is in PRO (document reference ADM 199/447).
Hinsley, _British Intelligence in the Second World War_ , i. 337.
Alexander, 'Cryptographic History of Work on the German Naval Enigma', 31.
Ibid. 7.
The official account of the pursuit is in PRO (document reference ADM 1/11133). See also R. Erskine, 'Naval Enigma: A Missing Link', _International Journal of Intelligence and Counterintelligence_ , 3 (1989), 493–508.
Balme interviewed on British Channel 4 TV, 1998.
Hinsley, _British Intelligence in the Second World War_ , ii. 168–71.
Ibid. 171.
Ibid. 169–70, 172–5.
W. Tutte, 'Bletchley Park Days', in B. J. Copeland (ed.), _Colossus: The First Electronic Computer_ (Oxford: Oxford University Press, 2005).
I. J. Good, D. Michie, and G. Timms, 'General Report on Tunny' (1945), 458. 'General Report on Tunny' was released by the British government in 2000 to the Public Record Office (document reference HW 25/4, HW 25/5). A digital facsimile is in The Turing Archive for the History of Computing <www.AlanTuring.net/tunny_report>.
Peter Hilton in interview with Copeland (July 2001).
S. Turing, _Alan M. Turing_ (Cambridge: Heffer, 1959), 71. Turing's report 'Visit to National Cash Register Corporation of Dayton, Ohio' (n.d.; _c_. Dec. 1942) is now declassified (document reference: NARA, RG 38, CNSG Library, 5750/441). A digital facsimile of the report is in The Turing Archive for the History of Computing <www.AlanTuring.net/turing_ncr>.
Alexander, 'Cryptographic History of Work on the German Naval Enigma', 42.
S. Turing, _Alan M. Turing_ , 72; Don Horwood in interview with Copeland (Oct. 2001).
There is an account of Turing's Hanslope period on pp. 269–90 of Hodges's biography (see the section of further reading in 'Alan Turing 1912–1954', above).
Alexander, 'Cryptographic History of Work on the German Naval Enigma', 42–3.
Diary of Sir Alexander Cadogan, Permanent Under-Secretary at the Foreign Office, 15 July 1941: Andrew, _Codebreaking and Signals Intelligence_ , 3.
I am grateful to Friedrich Bauer, Frank Carter, Ralph Erskine, John Harper, Diane Proudfoot, and Frode Weierud for their comments on a draft of this chapter.
With thanks to Elizabeth Stratton (Edgar Bowring Archivist at Clare College) for information.
This paragraph by Elizabeth Mahon.
Document reference RG 457, Historic Cryptographic Collection, Box 1424, NR 4685.
Document reference HW 25/2.
I have made as few editorial changes as possible. Mahon's chapter headings have been replaced by section headings. Errors in typing have been corrected and some punctuation marks have been added. Sometimes sentences which Mahon linked by 'and', or a punctuation mark, have been separated by a full stop. Mahon preferred the lower-case 'enigma', which has been altered to 'Enigma' for the sake of consistency with other chapters. In Mahon's introductory remarks 'account' has been substituted for 'book' (and the second word of that section has been changed from 'reading' to 'writing'). Occasionally a word or sentence has been omitted (indicated '...') and sometimes a word or phrase has been added (indicated '[]'). In every case, the omitted material consists of a reference to later parts of the 'History' not reproduced here.
Mahon's 'The History of Hut Eight, 1939–1945' is Crown copyright. This extract is published with the permission of the Public Record Office and Elizabeth Mahon. The extract from Turing's 'Treatise on the Enigma' is Crown copyright and is published with the permission of the Public Record Office and the Estate of Alan Turing. A digital facsimile of the complete original typescript of Mahon's 'The History of Hut Eight, 1939–1945' is available in The Turing Archive for the History of Computing <www. AlanTuring.net/ mahon_hut_8>.
Editor's note. Bletchley Park. Also B/P.
Editor's note. _Innere Einstellung_ = inner settings.
Editor's note. K book = _Kenngruppenbuch_ or _Kennbuch_ = Identification Group Book. 'Group' refers to groups of letters.
Editor's note. _Spaltenliste_ = column list.
Editor's note. A trigram is any trio of letters, e.g. ABC, XYZ.
Editor's note. _Gruppenliste_ = group list.
Editor's note. _Zuteilungsliste_ = assignation list.
Editor's note. _Schluesselkenngruppe_ = key identification group.
Editor's note. _Verfahrenkenngruppe_ = procedure identification group.
Editor's note. _Kenngruppenverfahren Sued_ = identification group procedure South.
Editor's note. W/T = wireless telegraphy. R/T or radio-telephony involves the transmission of speech, W/T the transmission of e.g. Morse code or Baudot-Murray (teleprinter) code.
Editor's note. Where two independent operators are assigned to monitor the same radio frequency, the frequency is described as 'double-banked'.
Editor's note. In International Q-Code (radio operators' code) the strength of the received signal is represented on a scale of 1 to 5. 'Q.S.A. 5' means 'signal strength excellent'. (I am grateful to Stephen Blunt for this information.)
Editor's note. Radio Security Service: a branch of the SIS (Secret Intelligence Service) which intercepted Enigma and other enciphered traffic.
Editor's note. The Heimsoeth & Rinke company manufactured Enigma machines following Scherbius' death. See F. L. Bauer, _Decrypted Secrets: Methods and Maxims of Cryptology_ (2nd edn. Berlin: Springer-Verlag, 2000), 107.
Editor's note. The 'boxing' or 'throw-on' indicator system was an earlier and much less secure method. K book and bigram tables were not used. Mahon explains the system later in his 'History' (on p. 56, which is not printed here): 'By this system a trigram of the operator's own choosing was encyphered twice at the Grundstellung and the resulting 6 letters became letters 2, 3, 4, 6, 7, 8, of the indicator groups.' Letters 1 and 5 were either dummies or, in Mediterranean traffic, identified which particular key was being used in accordance with _Kenngruppenverfahren Sued_. (See p. 273 and also H. Alexander, 'Cryptographic History of Work on the German Naval Enigma' (n.d. ( _c_.1945), Public Record Office (document reference HW 25/1), 8–9; a digital facsimile of Alexander's typescript is available in The Turing Archive for the History of Computing <www. AlanTuring.net/alexander_naval_enigma>).
Editor's note. Alexander (ibid. 17–18) gives the following explanation. With the boxing indicator system, 'the letters of the encyphered indicators can be associated together so as to produce various patterns—known as "box shapes"—independent of the stecker and determined only by the wheel order and Grundstellung at which the indicators are encyphered. With only 6 possible wheel orders catalogues could be made of all possible box shapes with the machine positions at which they occurred and thus the daily key could be worked out.'
Editor's note. The top line of the German keyboard is QWERTZUIO. Q was used for 1, W for 2, E for three, and so on. The time of origin of the original message, 23.30, becomes weep, 0 being represented by P, the first key of the bottom line of the keyboard. (See ibid. 18.)
Editor's note. Edward Travis was Denniston's deputy.
Editor's note. Freeborn's department, known as the 'Freebornery', used Hollerith equipment to sort and analyse cipher material. 'Freeborn' was commonly used as an adjective, e.g. attached to 'catalogue'. 'Freeborn, a[dj]: Performed, produced, or obtained by means of the (Hollerith) electrical calculating, sorting, collating, reproducing, and tabulating machines in Mr. Freeborn's department' (Cryptographic Dictionary, 39).
Editor's note. The name arose because the printed strips of paper were produced in the nearby town of Banbury.
Editor's note. 'Ban: Fundamental scoring unit for the odds on, or probability factor of, one of a series of hypotheses which, in order that multiplication may be replaced by addition, are expressed in logarithms. One ban thus represents an odds of 10 to 1 in favour, and as this is too large a unit for most practical purposes _decibans_ and _centibans_ are normally employed instead' (Cryptographic Dictionary, 4).
Editor's note. 'Scritch: To test (a hypothesis or possible solution) by examining its implications in conjunction with each of a set of (usually 26) further assumptions in turn, eliminating those cases which yield contradictions and scoring the others' (Cryptographic Dictionary, 72).
Editor's note. _Funkmaat_ = radio operator.
Editor's note. 'Twiddle: To turn round the wheels of an Enigma machine in hand-testing' (Cryptographic Dictionary, 90).
Editor's note. Director of Naval Intelligence.
Editor's note. Plain language.
Editor's note. See p. 253 for an explanation of 'all wheel order crib'.
Editor's note. Hut 6 dealt with German Army and Air Force Enigma.
Editor's note. This belief seems to have been widespread at Bletchley Park. See pp. 235–7, for remarks concerning the possible origin of the Polish term 'bomba', from which the British term 'bombe' was derived.
32 Editor's note. FROM VON DER GROEBEN (probably a U-boat commander). 'VVV' was standard radio spelling for 'from' (von).
Editor's note. See p. 227 for an explanation of 'Enigma colours'.
Editor's note. Alexander, 'Cryptographic History of Work on the German Naval Enigma', 6: 'There were a number of ways in which the German key maker quite unnecessarily restricted his choice of wheel orders e.g. the W.O. always contained a [wheel] 6, 7 or 8 in it. Restrictions of this kind were known as Wheel Order Rules and could on occasions narrow the choice from 336 to as little as 10 or 20 which was of enormous value to us.'
Editor's note. Concerning the four-wheeled machine, Mahon says later in his 'History' (on p. 55, which is not printed here): 'January 1942 passed peacefully enough but on February 1st Shark the U-boat key] changed over to the 4-wheeled machine and was not broken again, with one exception, until December. This was a depressing period for us as clearly we had lost the most valuable part of the traffic and no form of cryptographic attack was available to us.' Further information concerning the four-wheeled machine is given in the introduction to [Chapter 8, pp. 343–5.
Editor's note. 'Depth-cribbing = Fun and Games: The interesting process of simultaneously fitting two cribs (especially for the beginnings) to two messages on the same setting' (Cryptographic Dictionary, 27, 40). Alexander expands ('Cryptographic History of Work on the German Naval Enigma', 11): 'when the relative positions of a number of messages have been discovered it is frequently possible by examining them in conjunction with each other to work out the contents of some or all of the messages. This process is known as DEPTH CRIBBING and cribs produced in this way give cross checks from one message to the other of such a kind that one can be virtually certain of their correctness.'
Editor's note. WEATHER FOR THE NIGHT.
Editor's note. WITH M 371. 'M' refers to a class of vessel, minesweeper.
Editor's note. 'Click: A repeat or repetition of one or more cipher units usually in two or more messages, especially a repeat which, by its position in the messages or from the fact that it is one of a significant series, suggests that the messages are in depth' (Cryptographic Dictionary, 16).
Editor's note. WEATHER FOR THE NIGHT AND MORNING.
Editor's note. HAS NO MEANING FROM MNO BOULOGNE. 'MNO' stands for _Marine Nachrichten Offizier_ = Naval Communications Officer. 'OOO' was standard radio spelling for 'O'.
Editor's note. FILLER RADIO FROM HJÄ.
Editor's note. FORECAST AREA 3 PART 1.
Editor's note. WEATHER AREA 3 PART 1.
Editor's note. SITUATION EASTERN CHANNEL. (The weather situation.)
Editor's note. WITHOUT MEANING FROM MNO BOULOGNE.
Editor's note. Drei = three.
48 Editor's note. BEE BSO FROM M 1913. I am grateful to Frode Weierud and Ralph Erskine for the following information: 'BSO' stands for _Befehlshaber der Seestreitkraefte Ostsee_ = Commanding Officer Naval Forces Baltic Sea (see M. van der Meulen, 'Werftschluessel: A German Navy Hand Cipher System–Part I', _Cryptologia_ , 19 (1995), 349–64). 'Bee' translates 'Bine', a code word for the urgency indicator 'SSD': 'Sehr, Sehr Dringend' (Very, Very Urgent). The German Navy used three alternative code words for SSD: 'Bine', short for 'Biene' (bee), 'Wespe' (wasp), and 'Mucke' or 'Muke', short for 'Muecke' (midge).
49 Editor's note. NIGHT TO 13.4. OCCUPY.
Editor's note. TEN TEN.
51 Editor's note. 00 HOURS LEFT ROYAN.
Editor's note. 'Radio Finger Print: Enlarged or elongated film-record of morse transmission by means of which the type of transmitter used and the peculiarities of the individual sets of any type can be distinguished, serving to identify stations' (Cryptographic Dictionary, 63).
Editor's note. SITUATION EASTERN CHANNEL. (Weather situation.)
Editor's note. 'Crash: The occurrence of a plain letter opposite the same letter in the cipher text in one of the positions or versions in which a crib is tried, normally involving rejection of that position or version' (Cryptographic Dictionary, 22).
Editor's note. 'WEWA' abbreviates 'Wetter Warte': weather station.
Editor's note. WEATHER SITUATION 1800 HOURS.
Editor's note. NANTES TO BIARRITZ.
Editor's note. Naval Commander.
Editor's note. BEACONS LIT AS ORDERED.
Editor's note. HM FROM LECH A-D ERROR MESSAGE.
Editor's note. Alternative spellings of the German words for five, seven, and four.
Editor's note. NUREMBERG FROM WEATHER STATION SWINEMUENDA AIR [BALTA] 05 HOURS. 'Balta' is clearly a meteorological term but the meaning is obscure.
Editor's note. GROUND [BALTA] 05 HOURS.
Editor's note. Dockyard Key.
Editor's note. RHV = _Reservehandverfahren_ : Reserve Hand Cipher, used should an Enigma machine break down (see F. H. Hinsley and A. Stripp (eds.), _Codebreakers: The Inside Story of Bletchley Park_ (Oxford: Oxford University Press, 1993), 238–9).
Editor's note. Area 7.
Editor's note. 'Hat book: A code-book characterized by the fact that when the plain-language terms are arranged in alphabetical order the code groups are not in numerical (or alphabetical) order... Hatted: Arranged in other than numerical (or alphabetical) order.' (Cryptographic Dictionary, 43.)
Editor's note. WARTIME MARKER BUOY SWINEMUENDE. (I am grateful to Ralph Erskine and Frode Weierud for assistance with the translation of this crib.)
Editor's note. See p. 259 for an explanation of 'paired day'.
Editor's note. Area D.
Editor's note. Training radio messages.
Editor's note. 'Low-grade (of a code or cipher system): Not expected to resist attempts to break it for long, esp. if used to any great extent' (Cryptographic Dictionary, 52).
This material is Crown copyright and is published with the permission of the Public Record Office and the Estate of Alan Turing.
One copy is held in the Public Record Office in Kew, Richmond, Surrey (document reference HW 25/3), and the other in the National Archives and Records Administration, College Park, Maryland (document reference RG 457, Historic Cryptographic Collection, Box 201, NR 964). A digital facsimile of the typescript HW 25/3 is available in The Turing Archive for the History of Computing <www.AlanTuring.net/profs_book>. A retyped version of the _Treatise on the Enigma_ prepared by Erskine, Marks, and Weierud is available at <<http://home.cern.ch/frode/crypto>>. See also p. 250, n. 62.
A small number of typographical errors in the original typescript have been corrected. Occasionally punctuation has been added and sometimes a superfluous punctuation mark has been removed. Material in the text appearing within square brackets has been added by the editors (e.g. mech[ _anism_ ]). Footnotes beginning 'Editors' note' have been added by Copeland, Erskine, Marks, and Weierud; other footnotes are Turing's own.
Editors' note. This is the original title of chapter 6 of Turing's _Treatise_.
Editors' note. 'Constatation: The association of a cipher letter and its assumed plain equivalent for a particular position' (Cryptographic Dictionary, 20; see p. 269 of Chapter 5 for details of the Dictionary: a digital facsimile of the Dictionary is available in The Turing Archive for the History of Computing <www.AlanTuring.net/crypt_dic_1944>).
Editors' note. 'CILLI: The employment or occurrence of the finishing position of one Enigma message as the setting for enciphering the message setting of a second,... thus enabling the possible settings of the first message to be calculated for the various permissible wheel-orders and... the setting and probable wheel-order to be determined' (Cryptographic Dictionary, 14). A psychological CILLI, PSILLI for short, is 'any setting which can be guessed from a knowledge of the idiosyncrasies of the operator concerned' (Cryptographic Dictionary, 63).
Editors' note. In an earlier chapter of the _Treatise_ Turing says, '... repetitions of constatations (half-bombes as they are rather absurdly called)' (p. 32 of the original typescript).
Editors' note. A catalogue of wheel positions of a type described elsewhere in the _Treatise_ (pp. 87 ff. of Turing's original typescript).
Editors' note. A type of perforated sheet; see also pp. 233–4.
Editors' note. 'Rod: Strip of wood or other suitable material ruled off in equal compartments' (Cryptographic Dictionary, 70). 'Direct rod: A rod showing the letters on the right side of a wheel of an Enigma machine [i.e. the side of the wheel farther from the reflector] that are consecutively connected to a fixed point in space at the left side, for the twenty-six different positions of that wheel which occur in one revolution. Twenty-six such rods can be constructed for each wheel, one for each of twenty-six fixed points on its left side (corresponding to the contacts on a non-turning second wheel)' (ibid. 30). 'Inverse rod: A rod showing the letters on the left side of a wheel of an Enigma machine that are consecutively connected to a fixed point in space at the right side for the twenty-six different positions which occur in one revolution of that wheel' (ibid. 46).
Editors' note. See p. 296 and n. 39.
Editors' note. The twenty-six rods formed what was a called a 'rod-square' with columns and rows. 'Rod-square: Square of 26 letters by 26 (or rectangle of 26 by 52) formed by the direct or inverse rods of a particular Enigma wheel arranged in order' (Cryptographic Dictionary, 70).
Editors' note. The crib means: NO ADDITIONS TO PRELIMINARY REPORT.
Editors' note. The rod-position (see p. 238) of the wheels at the start of the cribbed ciphertext.
Editors' note. See p. 226 for the explanation of turnovers.
Editors' note. L.H.W. = left-hand wheel; similarly for M.W. and R.H.W.
Editors' note. _Feindseligkeiten_ = hostilities.
Editors' note. _Schluesselzettel_ = key sheet; _Vorpostenboot_ = patrol boat. German records show that the boat was the _Schiff 26_ (erroneously identified as the _VP 2623_ in some accounts). See R. Erskine, 'The First Naval Enigma Decrypts of World War II', _Cryptologia_ , 21 (1997), 42–6.
Editors' note. Turnover.
Editors' note. Turing says hypothesis number 5 but the menu in Fig. 59 shows that the turnover hypothesis is number 4. (Turing's original numbering of the figures has been retained.)
Editors' note. See p. 269.
Editors' note. Pye Limited was an electronics company located in Cambridge. Prior to the war Pye marketed radio and TV receivers. From the end of 1939 the company devoted itself to war work, chiefly the development and production of R.D.F. (radar) equipment and radio-based proximity fuses for munitions. (Thanks to David Clayden and M. Cosgrove for this information.)
A thyratron valve has the property that no current flows in the anode circuit until the grid potential becomes more positive] than a certain critical amount, after which the current continues to flow, regardless of the grid potential, until the anode potential is switched off. [See note to [Fig. 60.]
Editors' note. Turing is using an approximation to describe the amplitude. We have inserted the real expression in square brackets. This expression has been independently derived by Donald Davies and Martin Slack and we are grateful for their assistance.
Editors' note. The alphabets have been generated by a one-wheel Enigma machine consisting of the _Umkehrwalze_ (UKW) and wheel III (Green) from the Railway Enigma.
Editors' note. See p. 225.
24 Editors' note. In the PRO copy there is a handwritten note by Joan Murray, formerly Joan Clarke of Hut 8, explaining that H-M stands for Holland-Martin of the British Tabulating Machine Company.
Now has been decided to use scanner.
Editors' note. 'Herivelismus' is probably an alternative name for the procedure devised by John Herivel known as the 'Herivel tip'. The 'Herivel tip' helped the codebreakers find the day's ring settings from the indicators of the first messages of the day to be sent by the various German stations making up a given network. Herivel describes the 'Herivel tip' in M. Smith, _Station X: The Codebreakers of Bletchley Park_ (London: Channel 4 Books, 1998), 43: 'I thought of this imaginary German fellow with his wheels and his book of keys. He would open the book and find what wheels and settings he was supposed to use that day. He would set the rings on the wheels, put them into the machine and the next thing he would have to do would be to choose a three-letter indicator for his first message of the day. So I began to think, how would he choose that indicator. He might just take it out of a book, or he might pluck it out of the air like ABC or whatever. Then I had the thought, suppose he was a lazy fellow, or in a tearing hurry, or had the wind up, or something or other and he were to leave the wheels untouched in the machine and bang the top down and look at the windows, see what letters were showing and just use them. Then another thought struck me. What about the rings? Would he set them for each of the three given wheels before he put them into the machine or would he set them afterwards? Then I had a flash of illumination. If he set them afterwards and, at the same time, simply chose the letters in the windows as the indicator for his first message, then the indicator would tend to be close to the ring setting of the day. He would, as it were, be sending it almost in clear. If the intercept sites could send us the indicators of all the Red messages they judged to be the first messages of the day for the individual German operators there was a sporting chance that they would cluster around the ring settings for the day and we might be able to narrow down the 17,576 possible ring settings to a manageable number, say twenty or thirty, and simply test these one after the other in the hope of hitting on the right answer.'
Editors' note. 'Ringstellung cut out: A running of a bombe with a restriction on the range of possible Ringstellungen' (Cryptographic Dictionary, 69).
Both the letter and Churchill's minute appear in F. H. Hinsley et al., _British Intelligence in the Second World War_ , vol. ii (London: Her Majesty's Stationery Office, 1981), appendix 3, pp. 655–7. A facsimile of Churchill's minute appears on p. xiii of R. Erskine and M. Smith (eds.), _Action This Day_ (London: Bantam, 2001). The letter and minute are in the Public Record Office in Kew, Richmond, Surrey (document reference HW 1/155). They are Crown copyright and are reproduced with the permission of the Controller of Her Majesty's Stationery Office.
P. S. Milner-Barry, '"Action This Day": The Letter from Bletchley Park Cryptanalysts to the Prime Minister, 21 October 1941', _Intelligence and National Security_ , 1 (1986), 272–3. Reproduced with the permission of Frank Cass Publishers.
RG 38, CNSG Library, Box 117, 5750/205 (National Archives and Records Administration (NARA), College Park, Maryland).
For an extensive treatment of the full background, see J. Debrosse and C. Burke, _The Secret in Building 26_ (New York: Random House, 2004), ch. 3; also R. Erskine, 'What Did the Sinkov Mission Receive from Bletchley Park?', _Cryptologia_ , 24 (2000), 97–109.
R. Erskine, 'The First Naval Enigma Decrypts of World War II', _Cryptologia_ , 21 (1997), 42–6.
'Naval Security Group History to World War II' 400: NARA, RG 457, SRH 355.
D. Hamer, 'ENIGMA: Actions Involved in the "Double Stepping" of the Middle Rotor', _Cryptologia_ , 21 (1997), 47–50.
On the Sinkov mission, see B. Smith, _The Ultra-Magic Deals and the Most Secret Relationship_ (Novato, Calif.: Presidio, 1993), chapter 3, pp. 54, 56, 58.
R. Erskine, 'Churchill and the Start of the Ultra-Magic Deals', _International Journal of Intelligence and Counterintelligence_ , 10 (1997), 57–74.
Most of the questions and requests for data are set out in Erskine, 'What Did the Sinkov Mission Receive from Bletchley Park?'
Donald Davies, 'The Bombe: A Remarkable Logic Machine', _Cryptologia_ , 23 (1999), 108–138.
'Report by Lieut-Colonel J. H. Tiltman on his visit to North America during March and April 1942', para. 14, 20 May 1942: PRO HW 14/46.
See e.g. Turing's report, 'Visit to National Cash Register Corporation of Dayton, Ohio' (n.d., but _c_. Dec. 1942): NARA RG 38, CNSG Library, Box 183, 5750/441. We are indebted to Stephen Budiansky for a copy of this document.
'Report by Lieut-Colonel J. H. Tiltman'.
Memorandum (no title or author), 25 Apr 1942: NARA RG 38, Inactive Stations, Box 54, 3200/1; R. B. Ely, 'Easy Research to Date', 24 July 1942: NARA RG 38, CNSG Library, Box 117, 5750/205; J. J. Eachus, 'Cold Spot Method: Short Row Test: New Bombe' (n.d., but _c_. July 1942): ibid., Box 113, 5750/177; cf. 'American Hot-Point Method', Aug. 1942: NARA RG 38, Radio Intelligence Publications, RIP 602, 1.
R. Erskine, and F. Weierud, 'Naval Enigma: M4 and its Rotors', _Cryptologia_ , 11 (1987), 235–44.
R. Erskine, 'Naval Enigma: The Breaking of Heimisch and Triton', _Intelligence and National Security_ , 3(1) (1988), 162–83.
See R. Erskine, 'The Holden Agreement on Naval Sigint: The First BRUSA?', _Intelligence and National Security_ , 14(2) (1999), 187–97.
See 'Listing of Incoming/Outgoing Messages OP-20-G-4 from NCML': NARA RG 38, CNSG Library, Box 184, 5830/116.
On the development of the US Navy bombes, see S. Budiansky, _Battle of Wits: The Complete Story of Codebreaking in World War II_ (New York: The Free Press; London: Viking, 2000), 238–9, 241–2, 294–5; C. Burke, _Information and Secrecy: Vannevar Bush, Ultra and the Other Memex_ (Metuchen, NJ: Scarecrow Press, 1994), _passim_.
Alexander to Church, signal, 24 Mar. 1944: NARA RG 38, Inactive Stations, Box 55, 3200/2.
See OP-20-GM-1's monthly 'Summary of Attack on Naval Enigma Traffic' during 1944: NARA RG 38, CNSG Library, Box 117, 5750/205.
OP-20-GM-6 war diary, 31 Jan. 1943: ibid. Box 113, 5750/176.
NARA, RG 38, CNSG Library, Box 117, 5750/205.
The typescript and figures were found independently in May 2002 by Lee Gladwin, who published them, with a commentary, in 'Alan M. Turing's "Critique of Running Short Cribs on the U.S. Navy Bombe"', _Cryptologia_ , 27 (2003), 44–9 and 50–4. However, Gladwin's commentary contains several errors: e.g. Turing's memorandum questioned the use of short cribs in solving Enigma manually, and not running them on the US Navy bombe, which had not even been designed when Turing wrote the memorandum. (The present chapter was completed and accepted for publication in this volume in January 2001.—Ed.)
This title has been added by the editors, as have all footnotes.
Editors' note. This crib was short for 'Von Befehlshaber der U-Boote' (From BdU—Admiral Commanding U-boats). 'BDUUU' was an abbreviation used in signals to and from the U-boats.
Editors' note. Following recoveries of wheel wiring by the Poles before the war, the _Ringstellung_ on the wheels used by GC & CS was displaced by one for wheels I to III and VI to VIII, by two for wheel IV, and by three for wheel V (all with the neutral position at Z). Thus the English _Ringstellung_ RWH given in Turing's example corresponds to German _Ringstellung_ TZI. The different ring settings were of no consequence when solving Enigma or using the bombes.
Editors' note. The pre-start window position was the message setting—the starting position of the wheels for a specific message.
Editors' note. An Enigma simulator confirms that the middle wheel does indeed move after only three keystrokes, when set as indicated by Turing. This turnover would have complicated a solution here.
Editors' note. Turing's number is incorrect. 16,224 = 17,576 − 1352: the reduction (1,352 = 2 × 26 × 26) attempts to allow for the stepping pattern of wheel order 457. However, Philip Marks and Frode Weierud have independently calculated that 457 gives 16,952 (= 17,576 − 24 × 26) essentially different starting positions.
Editors' note. 'Constatation: The association of a cipher letter and its assumed plain equivalent for a particular position' ('Cryptographic Dictionary', 20; see the notes to the introduction to Chapter 5 for details of the Dictionary).
Editors' note. U.K.W. = _Umkehrwalze_ (reflector).
Editors' note. An inverse rod was a cardboard strip giving the letters on the left-hand side of the wheel that were consecutively connected to a fixed point at the right-hand side for all 26 positions of a full wheel revolution. See further Chapter 6, n. 7.
Editors' note. A 'straight through' wheel was a notional wheel whose wiring was an identity (A wired to A, B to B and so on). It could therefore be temporarily disregarded in any solution.
Editors' note. Although Turing also suggested that a message could be solved with a 70-letter crib, or two depths of 26 ('Mathematical Theory of ENIGMA Machine by A M Turing', 60; cf. p. 96: PRO HW 25/3), in practice GC & CS seldom, if ever, tried to solve messages by hand with cribs that were shorter than 250 letters.
Editors' note. This fraction is illegible. Turing's subsequent calculation, and the fact that it is typed with a single keystroke, show that it must be ½.
Editors' note. (8 × 7 × 6): Dolphin used a three-wheel machine, M3, in 1941.
Editors' note. 22⅓ is a typo for 21⅔ (65/3 — see n. 15).
Editors' note. 8.3 years! Dolphin keys had 10 Stecker pairs, leaving 6 letters unsteckered. The probability of two letters randomly selected being unsteckered is 6/26 × 5/25 = 3/65. Taking the crib probability correctness factor as 1/2, on average it would have been necessary to test 2 × 65/3 cribs to obtain a 'hit', giving Turing's 21⅔ [65/3] × 336 [wheel orders] × 2 × 5 [hours per test].
Donald Michie in interview with Copeland (Oct. 1995, Feb. 1998).
Ibid.
Ibid.
See, for example, A. Newell, J. C. Shaw, and H. A. Simon, 'Empirical Explorations with the Logic Theory Machine: A Case Study in Heuristics', _Proceedings of the Western Joint Computer Conference_ , 15 (1957), 218–39 (reprinted in E. A. Feigenbaum and J. Feldman (eds.), _Computers and Thought_ (New York: McGraw-Hill, 1963); A. Newell and H. A. Simon, 'Computer Science as Empirical Inquiry: Symbols and Search', _Communications of the Association for Computing Machinery_ , 19 (1976), 113–26.
M. Smith, _Station X: The Codebreakers of Bletchley Park_ (London: Channel 4 Books, 1998), 43.
Newell, Shaw, and Simon, 'Empirical Explorations with the Logic Theory Machine'; A. N. Whitehead and B. Russell, _Principia Mathematica_ , vols. i–iii (Cambridge: Cambridge University Press, 1910–13).
D. S. Scott and C. S. Strachey, 'Towards a Mathematical Semantics for Computer Languages', _Proceedings of a Symposium on Computers and Automata_ , Polytechnic Institute of Brooklyn, and Technical Monograph 6, Programming Research Group, Oxford University (1971).
Letter from Strachey to Michael Woodger, 13 May 1951 (in the Woodger Papers, National Museum of Science and Industry, Kensington, London).
Letters from Woodger to Copeland (15 July 1999, 15 Sept. 1999).
M. Campbell-Kelly, 'Christopher Strachey, 1916–1975: A Biographical Note', _Annals of the History of Computing_ , 7 (1985), 19–42(24); A. M. Turing, 'Programmers' Handbook for Manchester Electronic Computer', Computing Machine Laboratory, University of Manchester (n.d., _c_. 1950); a digital facsimile is available in The Turing Archive for the History of Computing <www.AlanTuring.net/programmers_handbook>.
C. S. Strachey, 'Logical or Non-Mathematical Programmes', _Proceedings of the Association for Computing Machinery_ , Toronto (Sept. 1952), 46–9 (47).
Letter from Samuel to Copeland (6 Dec. 1988).
A. L. Samuel, 'Some Studies in Machine Learning Using the Game of Checkers', _IBM Journal of Research and Development_ , 3 (1959), 211–29; reprinted in E. A. Feigenbaum and J. Feldman (eds.), _Computers and Thought_ (New York: McGraw-Hill, 1963).
Ibid. (Feigenbaum and Feldman (eds.), _Computers and Thought_ , 104).
Letter from Strachey to Turing, 15 May 1951 (in the Turing Papers, Modern Archive Centre, King's College, Cambridge (catalogue reference D 5)).
Ibid. (This extract is published by permission of Henry Strachey and the Strachey family.)
Oettinger in interview with Copeland (Jan. 2000).
Letter from Oettinger to Copeland (19 June 2000).
A. G. Oettinger, 'Programming a Digital Computer to Learn', _Philosophical Magazine_ , 43 (1952), 1243–63.
Ibid. 1251, 1257.
Ibid. 1247–51.
'Shopper' is my term; Oettinger uses 'shopping programme' and 'shopping machine'.
Oettinger, 'Programming a Digital Computer to Learn', 1250.
A. Newell and H. Simon, 'GPS, a Program that Simulates Human Thought', in Feigenbaum and Feldman (eds.), _Computers and Thought_ ; G. W. Ernst and A. Newell, _GPS: A Case Study in Generality and Problem Solving_ (New York: Academic Press, 1969).
For a synopsis see B. J. Copeland and D. Proudfoot, 'On Alan Turing's Anticipation of Connectionism', _Synthese_ , 108 (1996), 361–77; reprinted in R. Chrisley (ed.), _Artificial Intelligence: Critical Concepts in Cognitive Science_ , ii: _Symbolic AI_ (London: Routledge, 2000).
W. R. Ashby, _Design for a Brain_ (London: Chapman and Hall, 1952); F. Rosenblatt, _Principles of Neurodynamics_ (Washington, DC: Spartan, 1962).
J. McCarthy, 'Recursive Functions of Symbolic Expressions and their Computation by Machine, Part I', _Communications of the Association for Computing Machinery_ , 3 (1960), 184–95.
E. A. Feigenbaum, B. G. Buchanan, and J. Lederberg, 'On Generality and Problem Solving: A Case Study Using the dendral Program', in B. Meltzer and D. Michie (eds.), _Machine Intelligence 6_ (Edinburgh: Edinburgh University Press, 1971).
C. C. Hurd, 'Computer Development at IBM', in N. Metropolis, J. Howlett, and G. C. Rota (eds.), _A History of Computing in the Twentieth Century_ (New York: Academic Press, 1980).
J. Bigelow, 'Computer Development at the Institute for Advanced Study', ibid.
Flowers in interview with Copeland (July 1996).
Minutes of the Executive Committee of the National Physical Laboratory for 23 Oct. 1945 (National Physical Laboratory library; a digital facsimile is in The Turing Archive for the History of Computing <www.AlanTuring.net/npl_minutes_oct1945>).
J. R. Womersley, 'A.C.E. Project – Origin and Early History', National Physical Laboratory, 26 Nov. 1946 (Public Record Office, Kew, Richmond, Surrey (document reference DSIR 10/385); a digital facsimile is in The Turing Archive for the History of Computing <www.AlanTuring.net/ace_early_history>).
Turing's technical report was reprinted by the NPL in April 1972 as Computer Science Division Report No. 57. The report is reprinted in full in B. J. Copeland (ed.), _Alan Turing's Automatic Computing Engine_ (Oxford: Oxford University Press, 2004) under the title 'Proposed Electronic Calculator'; and in B. E. Carpenter and R. W. Doran (eds.), _A.M.Turing's ACE Report of 1946 and Other Papers_ (Cambridge, Mass.: MIT Press, 1986) under the title 'Proposal for Development in the Mathematics Division of an Automatic Computing Engine (ACE)'. A copy of the original typewritten report is in the Woodger Papers (National Museum of Science and Industry, Kensington, London (catalogue reference M15/83)); a digital facsimile is in The Turing Archive for the History of Computing <www.AlanTuring.net/proposed_electronic_calculator>. Page references in what follows are to the original typescript.
M. Woodger, handwritten note, undated (Woodger Papers (catalogue reference M15/78)); letter from Woodger to Copeland (27 Nov. 1999).
Minutes of the Executive Committee of the National Physical Laboratory for 19 Mar. 1946 (National Physical Laboratory library; a digital facsimile is in The Turing Archive for the History of Computing <www.AlanTuring.net/npl_minutes_mar1946>).
Letter from Huskey to Copeland (4 Feb. 2002).
H. Huskey, 'The Development of Automatic Computing', in _Proceedings of the First USA–JAPAN Computer Conference_ , Tokyo (1972), 698–704 (702).
Wilkinson in interview with Christopher Evans in 1976 ('The Pioneers of Computing: An Oral History of Computing' (London: Science Museum)).
Letter from Wilkinson to Newman, 10 June 1955 (Turing Papers, Modern Archive Centre, King's College, Cambridge (catalogue reference A 7)). Quoted by permission of Heather Wilkinson.
Memo from Turing to Womersley, n.d. but _c_. Dec. 1946 (Woodger Papers (catalogue reference M15/77); a digital facsimile is in The Turing Archive for the History of Computing <www.AlanTuring.net/turing_womersley_cdec46>).
Turing used this phrase in a different connection on p. 42 of his 'Proposed Electronic Calculator'.
Wilkinson in interview with Evans (see n. 11).
Williams described the Computing Machine Laboratory on p. 328 of his 'Early Computers at Manchester University' ( _Radio and Electronic Engineer_ , 45 (1975), 327–31): 'It was one room in a Victorian building whose architectural features are best described as "late lavatorial". The walls were of brown glazed brick and the door was labelled "Magnetism Room'."
Woodger in interview with Copeland (June 1998).
Letter from J. Illingworth to Fryer, 6 Nov. 1956 (Woodger Papers (catalogue reference M15/87); a digital facsimile is in The Turing Archive for the History of Computing <www.AlanTuring.net/illingworth_fryer_6nov56>).
C. Darwin, 'Automatic Computing Engine (ACE)', National Physical Laboratory, 17 Apr. 1946 (Public Record Office (document reference DSIR 10/385); a digital facsimile is in The Turing Archive for the History of Computing <www.AlanTuring.net/darwin_ace>).
Hartree's opinion is quoted in V. Bowden, 'The 25th Anniversary of the Stored Program Computer', _Radio and Electronic Engineer_ , 45 (1975), 326.
Memorandum from Hiscocks to the DSIR, 30 Jan. 1956 (Public Record Office (document reference DSIR 10/275); a digital facsimile is in The Turing Archive for the History of Computing <www.AlanTuring.net/hiscocks_dsir_30jan1956>); letter from D. Brunt to Sir Charles Darwin (20 July 1956) (Public Record Office (document reference DSIR 10/275); <www.AlanTuring.net/brunt_darwin_20july56>); 'Disposal of Pilot ACE', memorandum from Hiscocks to the DSIR, 26 July 1956 (Public Record Office (document reference DSIR 10/275); <www.AlanTuring.net/hiscocks_disposal_pilot_ace>).
Letter from Illingworth to Fryer (see n. 18).
Letter from A. M. Uttley to Sara Turing, 19 Dec. 1958 (in the Turing Papers, the Modern Archive Centre, King's College, Cambridge (catalogue reference A 11)).
F. M. Blake, D. O. Clayden, D. W. Davies, L. J. Page, and J. B. Stringer, 'Some Features of the ACE Computer', National Physical Laboratory, 8 May 1957 (Woodger Papers (catalogue reference N12/102); a digital facsimile is in The Turing Archive for the History of Computing <www.AlanTuring.net/ace_ features>).
Letter from Uttley to Sara Turing (see n. 23).
An experimental transistorized machine went into operation at Manchester University in 1953; see S. H. Lavington, _Early British Computers_ (Manchester: Manchester University Press, 1980).
F. M. Colebrook, 4 May 1953; quoted in D. M. Yates, _Turing's Legacy: A History of Computing at the National Physical Laboratory 1945–1995_ (London: Science Museum, 1997), 67.
Blake, Clayden, Davies, Page, and Stringer, 'Some Features of the ACE Computer', 3.
Huskey in interview with Copeland (Feb. 1998).
Letter from Huskey to Copeland (20 Dec. 2001).
Coombs in interview with Evans in 1976 ('The Pioneers of Computing: An Oral History of Computing' (London: Science Museum)); A. W. M. Coombs, 'MOSAIC', in _Automatic Digital Computation: Proceedings of a Symposium Held at the National Physical Laboratory_ (London: Her Majesty's Stationery Office, 1954); B. J. Copeland (ed.), 'The Turing–Wilkinson Lectures on the Automatic Computing Engine', in K. Furukawa, D. Michie, and S. Muggleton (eds.), _Machine Intelligence 15_ (Oxford: Oxford University Press, 1999). Digital facsimiles of a series of technical reports concerning the MOSAIC by Coombs, Chandler, and others are available in The Turing Archive for the History of Computing <www.AlanTuring.net/mosaic>.
'Engineer-in-Chief's Report on the Work of the Engineering Department for the Year 1 April 1954 to 31 March 1955', Post Office Engineering Department (The Post Office Archive, London); Coombs, 'MOSAIC'.
'Engineer-in-Chief's Report on the Work of the Engineering Department for the Year 1 April 1952 to 31 March 1953', Post Office Engineering Department (The Post Office Archive, London).
'Engineer-in-Chief's Report on the Work of the Engineering Department for the Year 1 April 1951 to 31 March 1952', Post Office Engineering Department (The Post Office Archive, London).
Coombs in interview with Evans (see n. 31).
C. G. Bell, and A. Newell, _Computer Structures: Readings and Examples_ (New York: McGraw-Hill, 1971), 44, 74; R. J. Froggatt, 'Logical Design of a Computer for Business Use', _Journal of the British Institution of Radio Engineers_ , 17 (1957), 681–96; Yates, _Turing's Legacy: A History of Computing at the National Physical Laboratory 1945–1995_ , 43–4.
Peter Hilton in interview with Copeland (June 2001).
Williams in interview with Evans in 1976 ('The Pioneers of Computing: An Oral History of Computing' (London: Science Museum)).
Williams, 'Early Computers at Manchester University', 328.
M. H. A. Newman, 'General Principles of the Design of All-Purpose Computing Machines', _Proceedings of the Royal Society of London_ , Series A, 195 (1948), 271–74 (271–2).
M. H. A. Newman, 'General Principles of the Design of All-Purpose Computing Machines', _Proceedings of the Royal Society of London_ , Series A, 195 (1948), 271–74 (271–2).
Letter from Williams to Randell, 1972 (quoted in B. Randell, 'On Alan Turing and the Origins of Digital Computers', in B. Meltzer and D. Michie (eds.), _Machine Intelligence 7_ (Edinburgh: Edinburgh University Press, 1972), 9).
'The Turing–Wilkinson Lectures on the Automatic Computing Engine'.
G. Bowker and R. Giordano, 'Interview with Tom Kilburn', _Annals of the History of Computing_ , 15 (1993), 17–32.
Letter from Brian Napper to Copeland (16 June 2002).
Bowker and Giordano, 'Interview with Tom Kilburn', 19.
H. D. Huskey, untitled typescript, National Physical Laboratory, n.d. but _c_. Mar. 1947 (Woodger Papers (catalogue reference M12/105); a digital facsimile is in The Turing Archive for the History of Computing <www.AlanTuring.net/huskey_1947>).
Letter from Rees to Copeland (2 Apr. 2001).
W. Newman, 'Max Newman: Mathematician, Codebreaker and Computer Pioneer', to appear in B. J. Copeland (ed.) _Colossus: The First Electronic Computer_ (Oxford: Oxford University Press).
Kilburn in interview with Copeland (July 1997).
Letter from Tootill to Copeland (18 Apr. 2001).
Letter from Tootill to Copeland (16 May 2001).
Letter from Tootill to Copeland (18 Apr. 2001).
As explained in the chapter 'Artificial Intelligence', the term 'Artificial Intelligence' did not come into use until after Turing's death.
Don Bayley in interview with Copeland (Dec. 1997).
'Proposed Electronic Calculator', 16.
Letter from Turing to W. Ross Ashby, no date (Woodger Papers (catalogue reference M11/99); a digital facsimile is in The Turing Archive for the History of Computing <www.AlanTuring.net/turing_ashby>). The letter was probably written in 1946 and certainly prior to October 1947.
The lecture was held at 5 p.m. in the rooms of the Royal Astronomical Society at Burlington House in London (entry in Woodger's diary for 20 Feb. 1947 (Copeland is grateful to Woodger for this information)).
Minutes of the NPL Executive Committee, 20 Apr. 1948 (National Physical Laboratory library; a digital facsimile is in The Turing Archive for the History of Computing <www.AlanTuring.net/ npl_minutes_apr1948>).
Woodger in interview with Copeland (June 1998).
A. G. Emslie, H. B. Huntington, H. Shapiro, and A. E. Benfield, 'Ultrasonic Delay Lines II', _Journal of the Franklin Institute_ , 245 (1948), 101–15 (101–2). In Shockley's delay line, the transmitting medium was not mercury but ethylene glycol.
'Proposed Electronic Calculator', 5.
Wilkinson in interview with Evans (see n. 11).
M. Campbell-Kelly, 'Programming the EDSAC: Early Programming Activity at the University of Cambridge', _Annals of the History of Computing_ , 2 (1980), 7–36.
Turing's typescript is among the Turing Papers in the Modern Archive Centre, King's College, Cambridge (catalogue reference B 1). It was first published in 1986 in Carpenter and Doran, _A. M. Turing's ACE Report of 1946 and Other Papers_. The present edition differs from the 1986 edition in various small respects. A few missing words have been restored (on pp. 379 and 381) and some minor departures in the earlier edition from Turing's original text have been rectified. Many of Turing's sketch diagrams (reproduced in the 1986 edition) have been redrawn. Obvious typing and spelling errors in Turing's typescript have been corrected without comment. Words or letters enclosed in square brackets have been added by the editor.
Reproduced with permission of the Estate of Alan Turing.
Editor's note. Turing means the University of Pennsylvania in Philadelphia.
Editor's note. _μ_ S = microseconds.
Editor's note. Hg = mercury.
'Proposed Electronic Calculator', 18 (see n. 6 of the introduction to Chapter 9). A digital facsimile of the original typescript of 'Proposed Electronic Calculator' is in The Turing Archive for the History of Computing <www.AlanTuring.net/proposed_electronic_calculator>. Page references are to this typescript.
Flowers in interview with Copeland (July 1996).
Letter from W. G. Radley to Womersley, 25 Feb. 1946 (Public Record Office, Kew, Richmond, Surrey (document reference DSIR 10/385); a digital facsimile is in The Turing Archive for the History of Computing <www.AlanTuring.net/radley_womersley_25feb46>).
'Status of the Delay Line Computing Machine at the P.O. Research Station', National Physical Laboratory, 7 Mar. 1946, anon. (Woodger Papers, National Museum of Science and Industry, Kensington, London (catalogue reference M12/105); a digital facsimile is in The Turing Archive for the History of Computing <www.AlanTuring.net/delay_line_status>).
Flowers in interview with Copeland (July 1998).
Letter from Darwin to Sir Edward Appleton, 13 Aug. 1946 (Public Record Office (document reference DSIR 10/275); a digital facsimile is in The Turing Archive for the History of Computing <www.AlanTuring.net/darwin_appleton_13aug46>); letter from Radley to Darwin, 1 Nov. 1946 (Public Record Office (document reference DSIR 10/385); <www.AlanTuring.net/radley_darwin_1nov46>).
Letter from Darwin to Radley, 26 Nov. 1946 (Public Record Office (document reference DSIR 10/385); a digital facsimile is in The Turing Archive for the History of Computing <www.AlanTuring.net/darwin_radley_26nov46>).
See B. J. Copeland (ed.), 'The Turing–Wilkinson Lectures on the Automatic Computing Engine', in K. Furukawa, D. Michie, and S. Muggleton (eds.), _Machine Intelligence 15_ (Oxford University Press, 1999).
Coombs in interview with Christopher Evans in 1976 ('The Pioneers of Computing: an Oral History of Computing' (London: Science Museum)).
A. M. Turing, 'Report on visit to U.S.A., January 1st–20th, 1947', National Physical Laboratory, 3 Feb. 1947 (Public Record Office (document reference DSIR 10/385); a digital facsimile is in The Turing Archive for the History of Computing<www.AlanTuring.net/turing_usa_visit>).
Report by Darwin to the NPL Executive Committee in the Minutes of the Executive Committee for 18 Mar. 1947 (NPL library; a digital facsimile is in The Turing Archive for the History of Computing <www.AlanTuring.net/npl_minutes_mar1947>).
Ibid.
The placing of a contract with English Electric 'i]n order to expedite the construction of [the] pilot assembly and to make possible the construction of the final machine' was proposed by Mathematics Division in February 1949 and approved by Treasury in May of that year (J. R. Womersley, 'A.C.E. Project: Present Position, and request for financial provision for a Study Contract to be placed with the English Electric Co. Ltd.', 1 Feb. 1949 (Woodger Papers; a digital facsimile is in The Turing Archive for the History of Computing <[www.AlanTuring.net/ace_project_position>); letter from Evans to Darwin, 28 May 1949 (Public Record Office (document reference DSIR 10/275); a digital facsimile is in The Turing Archive for the History of Computing <www.AlanTuring.net/evans_darwin_28may49>)).
J. R. Womersley and R. L. Smith-Rose, 'A.C.E. Pilot Test Assembly and later Development', National Physical Laboratory, 30 Apr. 1947 (Public Record Office (document reference DSIR 10/385); a digital facsimile is in The Turing Archive for the History of Computing <www.AlanTuring.net/pilot_test_assembly>).
Memorandum from Hiscocks to Womersley, National Physical Laboratory, 6 Aug. 1947 (Public Record Office (document reference DSIR 10/385); a digital facsimile is in The Turing Archive for the History of Computing <www.AlanTuring.net/hiscocks_womersley_6aug47>).
'A.C.E. Project', National Physical Laboratory, 21 Aug. 1947, initialled 'JWC/JG' (Public Record Office (document reference DSIR 10/385); a digital facsimile is in The Turing Archive for the History of Computing <www.AlanTuring.net/ace_project_meeting>).
Letter from Hiscocks to Darwin, 12 Aug. 1947 (Public Record Office (document reference DSIR 10/ 385); a digital facsimile is in The Turing Archive for the History of Computing <www.AlanTuring.net/hiscocks_darwin_12aug47>).
Wilkinson in interview with Evans in 1976 ('The Pioneers of Computing: An Oral History of Computing' (London: Science Museum)).
H. D. Huskey, 'From ACE to the G-15', _Annals of the History of Computing_ , 6 (1984), 350–71 (361).
M. Woodger, 'ACE Test Assembly, Sept./Oct. 1947', National Physical Laboratory, n.d. (Woodger Papers (catalogue reference M15/84); a digital facsimile is in The Turing Archive for the History of Computing <www.AlanTuring.net/test_assembly>).
Letter from Huskey to Copeland (3 June 2003).
E. C. Fieller, 'Hollerith Equipment for A.C.E. Work-Immediate Requirements', National Physical Laboratory, 16 Oct. 1947 (Public Record Office (document reference DSIR 10/385); a digital facsimile is in The Turing Archive for the History of Computing <www.AlanTuring.net/hollerith_equipment>).
Minutes of the Executive Committee of the National Physical Laboratory for 20 Apr. 1948 (NPL library; a digital facsimile is in The Turing Archive for the History of Computing <www.AlanTuring.net/npl_minutes_apr1948>).
J. R. Womersley, 'A.C.E. Project', National Physical Laboratory, n.d., attached to a letter from Womersley to the Secretary of the NPL dated 21 Aug. 1947 (Public Record Office (document reference DSIR 10/385); a digital facsimile is in The Turing Archive for the History of Computing <www.AlanTuring.net/womersley_ace_project>).
The Thomas era and the ACE's change in fortunes under Colebrook are described in B. J. Copeland, 'The Origins and Development of the ACE Project', in B. J. Copeland (ed.), _Alan Turing's Automatic Computing Engine_ (Oxford: Oxford University Press, 2004).
Letter from Darwin to Appleton, 23 July 1947 (Public Record Office (document reference DSIR 10/ 385); a digital facsimile is in The Turing Archive for the History of Computing <www.AlanTuring.net/darwin_appleton_23jul47>).
Probably at the end of September. Turing was still at the NPL when Geoff Hayes arrived in Maths Division on 23 Sept. 1947 (communication from Hayes to Woodger, Nov. 1979). Turing was on half-pay during his sabbatical (Minutes of the Executive Committee of the National Physical Laboratory for 28 Sept. 1948 (NPL library; a digital facsimile is in The Turing Archive for the History of Computing <www.AlanTuring.net/npl_minutes_sept1948>)). Darwin was in favour of paying Turing his full salary, but Turing preferred not, 'because if he were earning full pay, he would feel that "I ought not to play tennis in the morning, when I want to"' (Darwin to Appleton, 23 July 1947 (see n. 26)).
Darwin to Appleton, 23 July 1947 (see n. 26).
Gandy in interview with Copeland (Oct. 1995).
Turing's salary was paid wholly from a Royal Society grant to Newman for the purpose of developing a stored-programme electronic computer (letter from Newman to D. Brunt at the Royal Society, 22 Dec. 1948; a digital facsimile is in The Turing Archive for the History of Computing <www.AlanTuring.net/newman_brunt_22dec48>).
Letter from Williams to Brain Randell (1972); printed on p. 9 of B. Randell, 'On Alan Turing and the Origins of Digital Computers', in B. Meltzer and D. Michie (eds.), _Machine Intelligence 5_ (Edinburgh: Edinburgh University Press, 1972); Turing, _Programmers' Handbook for Manchester Electronic Computer_ (University of Manchester Computing Machine Laboratory, 1950; a digital facsimile is in The Turing Archive for the History of Computing <www.AlanTuring.net/programmers_handbook>).
During his sabbatical year Turing also proved that the word problem for semi-groups with cancellation is unsolvable (A. M. Turing, 'The Word Problem in Semi-Groups with Cancellation', _Annals of Mathematics_ , 52 (1950), 491–505).
Michie, unpublished note (in the Woodger papers).
Letter from Darwin to Turing, 11 Nov. 1947 (in the Modern Archive Centre, King's College, Cambridge (catalogue reference D 5); a digital facsimile is in The Turing Archive for the History of Computing <www.AlanTuring.net/darwin_turing_11nov47>).
Gandy in interview with Copeland (Oct. 1995); Minutes of the Executive Committee of the National Physical Laboratory for 28 Sept. 1948.
D. E. Rumelhart and J. L. McClelland, 'On Learning the Past Tenses of English Verbs', in J. L. McClelland, D. E. Rumelhart, and the PDP Research Group, _Parallel Distributed Processing: Explorations in the Microstructure of Cognition_ , ii: _Psychological and Biological Models_ (Cambridge, Mass.: MIT Press, 1986).
D. O. Hebb, _The Organization of Behavior: A Neuropsychological Theory_ (New York: John Wiley, 1949).
F. Rosenblatt, _Principles of Neurodynamics_ (Washington, DC: Spartan, 1962).
See further B. J. Copeland and D. Proudfoot, 'On Alan Turing's Anticipation of Connectionism' ( _Synthese_ , 108 (1996), 361–77) and 'Alan Turing's Forgotten Ideas in Computer Science' ( _Scientific American_ , 280 (1999), 99–103).
W. S. McCulloch and W. Pitts, 'A Logical Calculus of the Ideas Immanent in Nervous Activity', _Bulletin of Mathematical Biophysics_ , 5 (1943), 115–33.
B. G. Farley and W. A. Clark, 'Simulation of Self-Organising Systems by Digital Computer', _Institute of Radio Engineers Transactions on Information Theory_ , 4 (1954), 76–84; W. A. Clark and B. G. Farley, 'Generalisation of Pattern Recognition in a Self-Organising System', in _Proceedings of the Western Joint Computer Conference_ (1955).
F. Rosenblatt, 'The Perceptron, a Perceiving and Recognizing Automaton', Cornell Aeronautical Laboratory Report No. 85-460-1 (1957); 'The Perceptron: a Theory of Statistical Separability in Cognitive Systems', Cornell Aeronautical Laboratory Report No. VG-1196-G-1 (1958); 'The Perception: a Probabilistic Model for Information Storage and Organisation in the Brain', _Psychological Review_ , 65 (1958), 386–408.
Copeland and Proudfoot, 'On Alan Turing's Anticipation of Connectionism', 367.
Ibid. 367–8.
McCulloch and Pitts, 'A Logical Calculus of the Ideas Immanent in Nervous Activity', 129.
J. von Neumann, _Collected Works_ , vol. v, ed. A. H. Taub (Oxford: Pergamon Press, 1963), 319.
N. Wiener, _Cybernetics_ (New York: John Wiley, 1948), 32.
'Proposed Electronic Calculator'; see also D. R. Hartree, _Calculating Instruments and Machines_ (Illinois: University of Illinois Press, 1949), 97, 102, and B. E. Carpenter and R. W. Doran (eds.), _A. M. Turing's ACE Report of 1946 and Other Papers_ (Cambridge, Mass.: MIT Press, 1986), 277.
Gandy in interview with Copeland (Oct. 1995).
_Manchester Guardian_ , 11 June 1954.
McCulloch and Pitts, 'A Logical Calculus of the Ideas Immanent in Nervous Activity', 117, 124.
'Intelligent Machinery: A Report by A. M. Turing' is in the Woodger Papers; a digital facsimile is in The Turing Archive for the History of Computing <www.AlanTuring.net/intelligent_machinery>. Turing's draft typescript of the Report is among the Turing Papers in the Modern Archive Centre, King's College, Cambridge (catalogue reference C 11); there is a digital facsimile at <www.turingarchive.org>.
'Intelligent Machinery' appeared in 1968 in a collection of essays entitled _Key Papers: Cybernetics_ (London: Butterworths), ed. C. R. Evans and A. D. J. Robertson, two members of the Autonomics Division of the National Physical Laboratory. The following year another edition of 'Intelligent Machinery' appeared in the volume _Machine Intelligence 5_ (ed. B. Meltzer and D. Michie, Edinburgh University Press). Unlike the 1968 and 1969 editions, the present edition follows Turing's layout and his numbering of headings. Unfortunately the 1969 edition (which is reproduced in _Collected Works of A. M. Turing: Mechanical Intelligence_ , ed. D. C. Ince (Amsterdam: Elsevier, 1992)) contained numerous departures from Turing's own wording, as well as typographical errors, and outright mistakes. To mention only the most significant of these: (1) Turing's words 'determined by that symbol' on p. 413, below, were replaced by 'described by that symbol'; (2) Turing's words 'we can design a digital computer to do it, but that we stick to one, say the ACE, and that' on p. 415 were omitted, making nonsense of Turing's statement; (3) the words 'or either form of iii)' on p. 423 were omitted; (4) 'T1' was omitted from Turing' list 'U, T0, T1, D0 or D1' on p. 426; (5) the phrase 'replacing the Us of the character by D0' on p. 428 was incorrectly rendered'replacing the 0s of the character by D0'. In their introduction to the 1969 edition, the editors state that 'Intelligent Machinery' was 'written in September 1947'. This statement has caused some confusion in the literature. In fact 'Intelligent Machinery' was written in the summer of 1948. The phrase 'Manchester machine (as actually working 8/7/48)' (p. 413 below) appears in both the finished NPL Report and Turing's draft typescript. In the 1969 edition Turing' date '8/7/48' (8 July 1948) has been rewritten '8 August 1947'. The Manchester machine first operated on 21 June 1948.
K. Gödel, 'Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I', _Monatshefte für Mathematik und Physik_ , 38 (1931), 173–98.
A. Church, 'An Unsolvable Problem of Elementary Number Theory', _American Journal of Mathematics_ , 58 (1936), 345–63.
'On Computable Numbers, with an Application to the Entscheidungsproblem' (Chapter 1).
4 Editor's note. The Brunsviga was a popular desk calculating machine.
Editor's note. The symbol '1' is omitted from the original 1948 NPL Report and from the 1968 and 1969 editions. It is present in Turing's draft typescript.
Editor's note. μf = micro-farad.
7 Editor's note. This formula is taken from Turing's draft typescript.
R. Gandy, 'Human versus Mechanical Intelligence', in P. Millican and A. Clark (eds.), _Machines and Thought: The Legacy of Alan Turing_ , vol. i (Oxford: Clarendon Press, 1996), 125.
This additional material was first published in B. J. Copeland (ed.), 'A Lecture and Two Radio Broadcasts on Machine Intelligence by Alan Turing', in K. Furukawa, D. Michie and S. Muggleton (eds.), _Machine Intelligence 15_ (Oxford University Press, 1999). See also B. J. Copeland, 'The Turing Test', _Minds and Machines_ , 10 (2000), 519–39 (reprinted in J. H. Moor (ed.), _The Turing Test_ (Dordrecht: Kluwer, 2003)).
The term 'foil' is from p. 40 of B. J. Copeland, _Artificial Intelligence: A Philosophical Introduction_ (Oxford: Blackwell, 1993).
N. Block, 'The Computer Model of the Mind', in D. N. Osherson and H. Lasnik (eds.), _An Invitation to Cognitive Science_ , vol. iii (Cambridge, Mass.: MIT Press, 1990), 248.
A. Hodges, _Alan Turing: The Enigma_ (London: Vintage, 1992), 415.
R. French, 'The Turing Test: The First 50 Years', _Trends in Cognitive Sciences_ , 4 (2000), 115–22 (115).
See, for example, S. G. Sterrett, 'Turing's Two Tests for Intelligence', _Minds and Machines_ , 10 (2000), 541–59; S. Traiger, 'Making the Right Identification in the Turing Test', _Minds and Machines_ , 10 (2000), 561–72 (both reprinted in J. H. Moor (ed.), _The Turing Test_ (Dordrecht: Kluwer, 2003)).
Hodges, _Alan Turing_ , 415.
C. E. Shannon and J. McCarthy (eds.), _Automata Studies_ (Princeton: Princeton University Press, 1956), pp. v–vi.
N. Block, 'Psychologism and Behaviorism', _Philosophical Review_ , 90 (1981), 5–43.
R. Brooks, 'Intelligence without Reason', in L. Steels and R. Brooks (eds.), _The Artificial Life Route to Artificial Intelligence_ (Hillsdale, NJ: Erlbaum, 1995), 34. See also R. Brooks, _Cambrian Intelligence: The History of the New AI_ (Cambridge, Mass.: MIT Press, 1999).
R. Brooks, 'Elephants Don't Play Chess', _Robotics and Autonomous Systems_ , 6 (1990), 3–15.
R. A. Brooks and L. A. Stein, 'Building Brains for Bodies', _Autonomous Robots_ , 1 (1994), 7–25.
Footnotes have been renumbered consecutively. All footnotes not marked 'Editor's note' appeared in _Mind_. Where the text contains numbers referring to pages of _Mind_ these have been replaced by the numbers of the corresponding pages of the present edition, enclosed in square brackets. Not all cross-references in Turing's article were dealt with correctly by the editor of _Mind_ —some of the numbers appearing in _Mind_ presumably refer to pages of Turing's original typescript. These also have been replaced by the numbers of the corresponding pages of this volume.
This article first appeared in _Mind_ , 59 (1950), 433–60. It is reprinted with the permission of the Mind Association and the Estate of Alan Turing.
Editor's note. Perhaps 'infinitive' is a mis-printing in _Mind_ of 'infinite'.
2 Possibly this view is heretical. St Thomas Aquinas ( _Summa Theologica_ , quoted by Bertrand Russell, p. 480) states that God cannot make a man to have no soul. But this may not be a real restriction on His powers, but only a result of the fact that men's souls are immortal, and therefore indestructible. (Editor's note: the text in Mind contains no reference-marker for this footnote.)
Author's names in italics refer to the Bibliography.
Or rather 'programmed in' for our child-machine will be programmed in a digital computer. But the logical system will not have to be learnt.
Compare Lady Lovelace's statement (p. [455]), which does not contain the word 'only'.
Editor's note. Presumably 'omitted' is a typographical error in _Mind_.
Editor's note. The cross-reference in _Mind_ is to 'pp. 24, 25'. These are presumably pages of Turing's original typescript. The approximate position of the material is indicated by the fact that another uncorrected cross-reference in _Mind_ places Turing's quotation from Jefferson on p. 21 of the original typescript.
Peter Hilton in interview with Copeland (June 2001).
A detailed description of Turing's P-type machines is given in B. J. Copeland and D. Proudfoot, 'On Alan Turing's Anticipation of Connectionism', _Synthese_ , 108 (1996), 361–77 (reprinted in R. Chrisley (ed.), _Artificial Intelligence: Critical Concepts in Cognitive Science_ , ii: _Symbolic AI_ (London: Routledge, 2000)).
E. L. Post, 'Absolutely Unsolvable Problems and Relatively Undecidable Propositions: Account of an Anticipation', in M. Davis (ed.), _The Undecidable: Basic Papers on Undecidable Propositions, Unsolvable Problems and Computable Functions_ (New York: Raven, 1965), 417; see also 423.
J. R. Lucas, 'Minds, Machines and Gödel', _Philosophy_ , 36 (1961), 112–27 (112).
See his _The Emperor's New Mind: Concerning Computers, Minds, and the Laws of Physics_ (Oxford: Oxford University Press, 1989); 'Précis of _The Emperor's New Mind: Concerning Computers, Minds, and the Laws of Physics_ ', _Behavioral and Brain Sciences_ , 13 (1990), 643–55 and 692–705; _Shadows of the Mind: A Search for the Missing Science of Consciousness_ (Oxford: Oxford University Press, 1994); 'Beyond the Doubting of a Shadow', _Psyche_ , 2/23 (1996).
Lucas, 'Minds, Machines and Gödel', 127.
See B. J. Copeland, 'Turing's O-machines, Penrose, Searle, and the Brain', _Analysis_ , 58 (1998), 128–38.
Penrose, 'Beyond the Doubting of a Shadow', section 3.10, and _Shadows of the Mind_ , 381.
'Beyond the Doubting of a Shadow', section 13.2.
'Proposed Electronic Calculator', National Physical Laboratory, 1945, 16 (National Physical Laboratory library; a digital facsimile of the original typescript is in The Turing Archive for the History of Computing <www.AlanTuring.net/proposed_electronic_calculator> (page reference is to the original typescript)).
In this chapter Turing speaks of the 'mechanic who has constructed the machine'. This is perhaps a glimpse of Turing's attitude toward Kilburn, Williams, and the other engineers who built the Manchester computer. Kilburn himself was hardly less dismissive of the logicians' contributions (for example in an interview with Christopher Evans in 1976, 'The Pioneers of Computing: An Oral History of Computing' (London: Science Museum)).
The typescript is among the Turing Papers in the Modern Archive Centre, King's College, Cambridge (catalogue reference B 4). Turing's mother Sara included the text of 'Intelligent Machinery, A Heretical Theory' in her biography _Alan M. Turing_ but unfortunately incorporated some errors (S. Turing, _Alan M. Turing_ (Cambridge: Heffer, 1959), 128–34.) The present edition first appeared in B. J. Copeland (ed.), 'A Lecture and Two Radio Broadcasts on Machine Intelligence by Alan Turing', in K. Furukawa, D. Michie, and S. Muggleton (eds.), _Machine Intelligence 15_ (Oxford: Oxford University Press, 1999).
Editor's note. Words enclosed in square brackets do not appear in the typescript.
Editor's note. The words 'trying to understand what the machines were trying to say,' are handwritten and are marked in the margin 'Inserted from Turing's Typescript'.
_Alan M. Turing_ (Cambridge: Heffer, 1959), 102.
Ibid.
Letter from Maurice Wilkes to Copeland (9 July 1997).
A copy of 'Nature of Spirit' is among the Turing Papers in the Modern Archive Centre, King's College, Cambridge.
Newman in interview with Christopher Evans ('The Pioneers of Computing: An Oral History of Computing' (London: Science Museum)).
M. H. A. Newman, 'General Principles of the Design of All-Purpose Computing Machines', _Proceedings of the Royal Society of London_ , Series A, 195 (1948), 271–4 (271–2).
A. M. Turing, _Programmers' Handbook for Manchester Electronic Computer_ , University of Manchester Computing Laboratory (1950), 1. A digital facsimile of the _Programmers' Handbook_ is available in The Turing Archive for the History of Computing <www.AlanTuring.net/programmers_handbook>.
Newman writing in the _Manchester Guardian_ , 11 June 1954.
The typescript is in the Modern Archive Centre, King's College, Cambridge (catalogue reference B 5). The present edition first appeared in B. J. Copeland (ed.), 'A Lecture and Two Radio Broadcasts on Machine Intelligence by Alan Turing', in K. Furukawa, D. Michie and S. Muggleton (eds.), _Machine Intelligence 15_ (Oxford: Oxford University Press, 1999).
Printed with the permission of the BBC and the Estate of Alan Turing.
The present material together with the dialogues recorded in _Wittgenstein's Lectures on the Foundations of Mathematics_ (ed. C. Diamond, Ithaca, NY: Cornell University Press, 1976) are the only known transcriptions of discussions involving Turing. (Some rather compressed notes of a discussion between Turing, Newman, Young, Polanyi, and others, entitled 'Rough Draft of the Discussion on the Mind and the Computing Machine, held on Thursday, 27th October, 1949, in the Philosophy Seminar' (anon., n.d., University of Manchester Philosophy Department), are in The Turing Archive for the History of Computing <www.AlanTuring.net/philosophy_seminar_oct1949>. I am grateful to Wolfe Mays for making these notes available.)
G. Jefferson, 'The Mind of Mechanical Man', _British Medical Journal_ , 25 June 1949, 1105–10 (1110).
Letter from Jefferson to Sara Turing, 18 Oct. 1954. The letter is among the Turing Papers in the Modern Archive Centre, King's College, Cambridge (catalogue reference A 16).
J. F. Heiser, K. M. Colby, W. S. Faught and R. C. Parkison, 'Can Psychiatrists Distinguish a Computer Simulation of Paranoia from the Real Thing?', _Journal of Psychiatric Research_ , 15 (1980), 149–62.
See, for example, R. Brooks, 'Intelligence without Reason', in L. Steels and R. Brooks (eds.), _The Artificial Life Route to Artificial Intelligence_ (Mahwah, NJ: Erlbaum, 1995), n. 8, and P. Millican and A. Clark (eds.), _Machines and Thought: The Legacy of Alan Turing_ (Oxford: Oxford University Press, 1996), 61.
R. French, 'Subcognition and the Limits of the Turing Test', _Mind_ , 99 (1990), 53–65.
Ibid. 17.
There is additional discussion of French's objections to the Turing test in my 'The Turing Test', _Minds and Machines_ , 10 (2000), 519–39 (reprinted in J. H. Moor (ed.), _The Turing Test_ (Dordrecht: Kluwer, 2003)).
M. Minsky, _The Society of Mind_ (London: Pan Books, 1988), 71.
The script is in the Modern Archive Centre, King's College, Cambridge (catalogue reference B 6). It was first published in B. J. Copeland (ed.), 'A Lecture and Two Radio Broadcasts on Machine Intelligence by Alan Turing', in K. Furukawa, D. Michie, and S. Muggleton (eds.), _Machine Intelligence 15_ (Oxford: Oxford University Press, 1999).
Printed with the permission of the BBC, Lewis C. Braithwaite, Antony A. Jefferson, Edward Newman, and the Estate of Alan Turing.
C. G. Langton, 'Studying Artificial Life with Cellular Automata', _Physica D_ , 22 (1986), 120–49.
C. G. Langton (ed.), _Artificial Life: The Proceedings of an Interdisciplinary Workshop on the Synthesis and Simulation of Living Systems_ (Redwood City, Calif.: Addison-Wesley, 1989).
C. G. Langton, 'Artificial Life', in Langton, _Artificial Life_ , 1.
Ibid.
Ibid. 32.
Langton, 'Studying Artificial Life with Cellular Automata', 147.
Another early pioneer of A-Life was the neurophysiologist W. Grey Walter (a founder of the Ratio Club, of which Turing was also a member). His famous 'tortoises'—built from about 1949 onwards—were mobile, battery-powered devices exhibiting life-like properties. He constructed them in order to show that seemingly complex behaviour can result from simple mechanisms (W. Grey Walter, _The Living Brain_ (London: Gerald Duckworth, 1953), 82–7 and appendix B). Although digital computers were not involved, Grey Walter's work certainly conforms to Langton's description given above: 'Artificial Life... attempt[s] to synthesise life-like behaviors within computers _and other artificial media_ ' (italics added).
A digital facsimile of Turing's _Programmers' Handbook for Manchester Electronic Computer_ (University of Manchester Computing Machine Laboratory, 1950) is in The Turing Archive for the History of Computing<www.AlanTuring.net/programmers_handbook>.
Letter from Turing to Michael Woodger, undated, received 12 Feb. 1951 (in the Woodger Papers, National Museum of Science and Industry, Kensington, London; a digital facsimile is in the Turing Archive for the History of Computing<www.AlanTuring.net/turing_woodger_feb51>).
W. D'Arcy Thompson, _On Growth and Form_ (2nd edn. Cambridge: Cambridge University Press, 1942), 3.
Ibid. 10, 16.
Ibid. 933.
Letter from Waddington to Turing (11 Sept. 1952). The letter is among the Turing Papers in the Modern Archive Centre, King's College, Cambridge (catalogue reference D 5).
J. D. Murray, 'A Pre-pattern Formation Mechanism for Animal Coat Markings', _Journal of Theoretical Biology_ , 88 (1981), 161–99; G. Turk, 'Generating Textures on Arbitrary Surfaces Using Reaction-Diffusion', _Computer Graphics_ , 25 (1991), 289–98; A. Witkin and M. Kass, 'Reaction-Diffusion Textures', _Computer Graphics_ , 25 (1991), 299–308.
See p. 561 of Chapter 15; also section 2 of Turing's 'Morphogen Theory of Phyllotaxis', in _Morphogenesis: Collected Works of A. M. Turing_ , ed. P. T. Saunders (Amsterdam: North-Holland, 1992).
Unless perhaps the computer-assisted modelling of non-linear systems was first undertaken a little earlier, in secret, by members of the Los Alamos group in connection with nuclear explosions.
B. B. Mandelbrot, _The Fractal Geometry of Nature_ (New York: Freeman, 1977; revised and expanded 1982, 1983). For a popular exposition see J. Gleick, _Chaos: Making a New Science_ (London: Cardinal, 1988).
Some—but by no means all—of this material appears in _Morphogenesis_ , ed. Saunders.
The notes and programme sheets are among the Turing Papers in the Modern Archive Centre, King's College, Cambridge (catalogue references C 25, C 27).
Editor's note. A blastula is a hollow sphere of cells, one cell in thickness.
J. H. Holland, _Adaptation in Natural and Artificial Systems_ (Cambridge, Mass.: MIT Press, 1992), p. x.
A. L. Samuel, 'Some Studies in Machine Learning Using the Game of Checkers', _IBM Journal of Research and Development_ , 3 (1959), 211–29; reprinted in E. A. Feigenbaum and J. Feldman (eds.), _Computers and Thought_ (New York: McGraw-Hill, 1963).
D. E. Goldberg, 'Genetic and Evolutionary Algorithms Come of Age', _Communications of the Association for Computing Machinery_ , 37 (1994), 113–19.
J. von Neumann, 'The General and Logical Theory of Automata', in vol. v of von Neumann's _Collected Works_ , ed. A. H. Taub (Oxford: Pergamon Press, 1963), 288–9.
A. W. Burks (ed.), _Essays on Cellular Automata_ (Urbana: University of Illinois Press, 1970), p. xv.
C. G. Langton, 'Artificial Life', in M. A. Boden (ed.), _The Philosophy of Artificial Life_ (Oxford: Oxford University Press, 1996), 47.
Letter from von Neumann to Wiener, 29 Nov. 1946 (in the von Neumann Archive at the Library of Congress, Washington, DC).
H. H. Goldstine, _The Computer from Pascal to von Neumann_ (Princeton: Princeton University Press, 1972), 271, 274–5.
von Neumann, 'The General and Logical Theory of Automata', 315.
J. von Neumann, _Theory of Self-Reproducing Automata_ , ed. and completed by A. W. Burks (Urbana: University of Illinois Press, 1966), 83.
von Neumann 'The General and Logical Theory of Automata', 316–17.
Ibid., 317.
Ibid. 317–18.
Letter from Turing to Young, 8 Feb. 1951 (a copy of the letter is in the Modern Archive Centre, King's College, Cambridge (catalogue reference K 78)).
A somewhat larger value of concentration unit (e.g. 10−9 mole/cm3) is probably more suitable. The choice of unit only affects the calculations through the amplitude of the random disturbances.
This article first appeared in _Philosophical Transactions of the Royal Society of London_ , Series B, 237 (1952–54), 37–72. It is reprinted with the permission of the Royal Society and the Estate of Alan Turing.
D. Michie, _On Machine Intelligence_ (2nd edn. Chichester: Ellis Horwood, 1986), 78–9.
C. E. Shannon, 'Programming a Computer for Playing Chess', _Philosophical Magazine_ , 41 (1950), 256–75.
H. A. Simon and A. Newell, 'Heuristic Problem Solving: The Next Advance in Operations Research', _Operations Research_ , 6 (1958), 1–10.
Michie in interview with Copeland (Oct. 1995).
The foundations of the minimax approach were laid in 1928 by von Neumann in his fundamental 'minimax theorem' (J. von Neumann, 'Zur Theorie der Gesellschaftsspiele', _Mathematische Annalen_ , 100 (1928), 295–320).
Champernowne's account is from a letter that he wrote to _Computer Chess_ (4 (Jan. 1980), 80–1); it is published here by permission of W. and R. P. Champernowne.
D. Michie, 'Game-Playing and Game-Learning Automata', in L. Fox (ed.), _Advances in Programming and Non-numerical Computation_ (New York: Pergamon, 1966), 189.
B. V. Bowden (ed.), _Faster Than Thought_ (London: Pitman, 1953), 295.
C. Gradwell, 'Early Days', reminiscences in a Newsletter 'For those who worked on the Manchester Mk I computers', Apr. 1994. (I am grateful to Prinz's daughter, Daniela Derbyshire, for sending me a copy of Gradwell's article.)
D. G. Prinz, 'Introduction to Programming on the Manchester Electronic Digital Computer', n.d., Ferranti Ltd. (a digital facsimile is in The Turing Archive for the History of Computing <www.Alan Turing.net/prinz>).
Prinz's programme is described in Bowden (ed.), _Faster Than Thought_ , 295–7 and in D. G. Prinz, 'Robot Chess', _Research_ , 5 (1952), 261–6.
Prinz, 'Robot Chess', 266.
D. G. Prinz and J. B. Smith, 'Machines for the Solution of Logical Problems', in Bowden (ed.), _Faster Than Thought_ , ch. 15; W. Mays and D. G. Prinz, 'A Relay Machine for the Demonstration of Symbolic Logic', _Nature_ , 165/4188 (4 Feb. 1950), 197–8; M. A. Bates, 'On the Mechanical Solution of a Problem in Church's Lambda Calculus', M.Sc. thesis, University of Manchester.
N. Chomsky, _Language and Thought_ (London: Moyer Bell, 1993), 93.
See section 3.1 ('Is Consciousness Necessary for Thought?') of B. J. Copeland, _Artificial Intelligence: A Philosophical Introduction_ (Oxford: Blackwell, 1993).
The whole article is mistakenly attributed to Turing alone in _Mechanical Intelligence: Collected Works of A. M. Turing_ , ed. D. C. Ince (Amsterdam: North-Holland, 1992).
Turing's typescript is itself entitled 'Digital Computers applied to Games'. In _Faster Than Thought_ Turing's essay was published under the narrower and more accurate title 'Chess'. The typescript is among the Turing Papers in the Modern Archive Centre, King's College, Cambridge (catalogue reference B 7).
Bowden's edition of the essay differs from Turing's typescript in numerous respects, many of them minor. The present edition follows Turing's typescript. Some obvious typing errors have been corrected and in one case a conjectured missing word has been added in square brackets. Some significant differences between the typescript and Bowden's edition are mentioned in footnotes. (Bowden's edition is reprinted in Turing, _Mechanical Intelligence_ , ed. Ince, 288–95.)
Printed with the permission of the BBC and the Estate of Alan Turing.
1 Editor's note. In the Bowden edition. Turing's '±' has been replaced with '='.
Printed with the permission of Financial Times Management and the Estate of Alan Turing.
Editor's note. In the Bowden edition, Turing's 'position of the board' has been replaced by 'position on the board'.
Editor's note. In the Bowden edition a different game is to shown, diverging from the present game at the 21st move.
A. Church, 'An Unsolvable Problem of Elementary Number Theory', _American Journal of Mathematics_ , 58 (1936), 345–63 (356).
E. L. Post, 'Finite Combinatory Processes - Formulation 1', _Journal of Symbolic Logic_ , 1 (1936), 103–5 (105).
A. N. Whitehead and B. Russell, _Principia Mathematica_ , vols. i–iii (Cambridge: Cambridge University Press, 1910–13).
K. Gödel, 'Postscriptum', in M. Davis (ed.), _The Undecidable_ (New York: Raven, 1965), 71–3 (71–2); the Postscriptum, dated 1964, is to Gödel's 1934 paper 'On Undecidable Propositions of Formal Mathematical Systems' (ibid. 41–71).
Footnotes have been renumbered consecutively. Footnotes not marked 'Editor's note' appeared in _Science News_. A page reference to _Science News_ been replaced by the number (in square brackets) of the corresponding page of this volume.
The fragment is among the Turing Papers in the Modern Archive Centre, King's College, Cambridge; at the time of writing it is uncatalogued.
It would take us too far from our main purpose to give the proof of this rule: the reader should have little difficulty in proving it by making use of the fact that an odd number of interchanges can never bring a set of objects back to the position it started from.
This article first appeared in _Science News_ , 31 (1954), 7–23, published by the Penguin Press, and is printed by permission of the Estate of Alan Turing.
It can in fact be done by sliding successively the squares numbered 7, 14, 13, 11, 9, 10, 1, 2, 3, 7, 15, 8, 5, 4, 6, 3, 10, 1, 2, 6, 3, 10, 6, 2, 1, 6, 7, 15, 8, 5, 10, 8, 5, 10, 8, 7, 6, 9, 15, 5, 10, 8, 7, 6, 5, 15, 9, 5, 6, 7, 8, 12, 14, 13, 15, 10, 13, 15, 11, 9, 10, 11, 15, 13, 12, 14, 13, 15, 9, 10, 11, 12, 14, 13, 15, 14, 13, 15, 14, 13, 12, 11, 10, 9, 13, 14, 15, 12, 11, 10, 9, 13, 14, 15.
Editor's note. In place of this sentence Turing's draft has: 'Thus for instance the trefoil knot may be regarded as consisting of a number of segments joining the points (0, 0, 0), (0, 2, 0), (1, 2, 0), (1, 2, 2), (1, −1, 2), (1, −1, 1), (−1, −1, 1), (−1, 1, 1), (2, 1, 1), (2, 0, 1), (2, 0, 3), (0, 0, 3), (0, 0, 0).'
Editor's note. Turing's draft has 'bbacceeefddbbaaaeccddfff'.
Editor's note. Turing's draft has: 'Thus for instance one might be allowed the substitutions
and be asked to transform WBWWBWBBB into WBB, and this one could do first by substituting W for WWB and getting WBWBBBB and then successively WWWWBBBB, WWBBB, WBB.'
Editor's note. Turing's draft has 'positions'.
Editor's note. At this point Turing's draft contains the following: 'Some of these other definitions will be found in Refs (1), (3), (11), (13), and (16) vol II. Some equivalence theorems are proved in (4) and (14), and some propaganda on the matter will be found in (13). A very satisfactory account of all these problems will be found in (5).' Tantalizingly, the list of references is omitted.
Editor's note. Turing's draft has 'systematic method'.
Editor's note. At this point Turing's draft contains the following material, which is crossed out. 'It is a phrase which, like many others e.g. "vegetable" one understands well enough in the ordinary way. But one can have difficulties when speaking to greengrocers or microbiologists or when playing "twenty questions". Are rhubarb and tomatoes vegetables or fruits? Is coal vegetable or mineral? What about coal gas, marrow, fossilised trees, streptococci, viruses? Has the lettuce I ate at lunch yet become animal? The fact of the matter is that when one is applying a word, say an adjective, to something definite, one chooses the word itself so that it describes what one wants to describe fairly and squarely. If it doesn't one had better look for another word. But if one is playing twenty questions this just can't be done. The questions are about "the object", and one doesn't know what it is. The same sort of difficulty arises about question c) above. An ordinary sort of acquaintance with the meaning of the phrase "systematic method" won't do, because one has got to be able to say quite clearly about any kind of method that might be proposed whether it is allowable or not. Fortunately a number of satisfactory definitions were found in the late thirties, and they have...' [the fragment ends at this point].
| {
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} | 590 |
Billerud AB is a Swedish pulp and paper manufacturer with headquarters in Solna, Sweden. The company simplified its name from BillerudKorsnäs to Billerud after the acquisition of Verso 2022, an American producer of coated paper. Billerud has nine production facilities in Sweden, Finland and the USA with around 5,800 employees in over 13 countries.
Its production units are located in Grums, Skärblacka, Frövi/Rockhammar, Gävle and Karlsborg in Sweden, Jakobstad in Finland, Escanaba, Quinnesec and Wisconsin Rapids in the US.
History
The company was formed in November 2012 through the merger of Billerud AB and Korsnäs AB.
In the 2012 merger, Billerud was the formal buyer. Korsnäs' owner, Investment Kinnevik AB, received a sum of SEK 3.2 billion as well as 25.1% of the votes in the new company and consequently became the largest owner in the merged BillerudKorsnäs. Behind the merger, was an ambition to become an internationally leading packaging manufacturer.
During the same year, BillerudKorsnäs acquired paper production facilities both in Jakobstad and Tervasaari, Valkeakoski, in Finland from UPM Kymmene. Valkeakoski was closed down and the paper machine for MG paper was moved to Skärblacka mill in 2017.
In 2013, Kinnevik sold its shares in BillerudKorsnäs to pension firms AMF, Alecta and the Fourth Swedish National Pension Fund.
In March 2022, BillerudKorsnäs completed its acquisition of the US-based coated paper company, Verso Corporation for $825m in cash. As a result of the acquisition, according to a press release in October 2022, the company changed its name from BillerudKorsnäs to Billerud "to cater to its international customers".
Production units
In Sweden
Gruvön Mill – produces cup stock, fluting, formable paper, kraft paper, liners, sack paper and liquid packaging board
Skärblacka Mill – produces white machine glazed kraft paper, brown sack paper and fluting
Karlsborg Mill – produces kraft paper, sack paper, formable paper (FibreForm®) and pulp
Frövi Mill – produces cartonboard and liquid packaging board
Gävle Mill – produces liquid carton and liners.
In the US
Escanaba Mill – produces graphic papers used in commercial printing, media and marketing
Quinnesec Mill – produces graphic papers used in commercial printing, media and marketing
Wisconsin Rapids Converting Facility – converts rolls of paper into folio and digital sheet.
In Finland
Jakobstad Mill – produces kraft- and sack paper for food packaging and carrier bags.
Product areas
In 2021, Board corresponds to 57% of the group's net sales and paper corresponds to 31%.
Corporate leadership
Chairmen of the Board
Hannu Ryöppönen, 2012–2014
Lennart Holm, 2014–2019
Jan Åström 2019-2021
Jan Svensson 2021-
Chief Executive Officer
Per Lindberg, 2012–2017
Petra Einarsson, 2018–2019
Lennart Holm, 2019-2020 (acting CEO)
Christoph Michalski, 2020-
Ownership Structure
The largest shareholders are AMF Pension & Funds with 14.2 percent of the votes and Frapag Beteiligungholding AG with 12.1 percent (as of April 30, 2022).
See also
Holmen
Stora Enso
Svenska Cellulosa Aktiebolaget
UPM (company)
References
External links
Solna Municipality
Companies based in Solna Municipality
Pulp and paper companies of Sweden
Manufacturing companies established in 2012
Swedish companies established in 2012
Companies listed on Nasdaq Stockholm | {
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} | 9,521 |
{"url":"https:\/\/nspyrison.github.io\/spinifex\/","text":"Data visualization tour is a class of linear projections that are animate small changes in a projection basis over time. Shadows are an example of a 3D object casting a 2D linear projection. Imagine a bar stool cast a circular shadow of its seat. Watching the shadow alone, a rotation of the objection would quickly give structural information about the object; when the legs are rotated into frame you can be more certain the object is a bar stool. In the same way, a tour watches a lower-dimensional projection of the data object. The object permanence of the observations between frames holds structural information that is revealed as the object is rotated.\n\n## Scope\n\nSuch visualizations are applicable to numeric matrices which could be approximations of component spaces or feature decomposition of non-tabular data. Continuous quantitative variables are preferred, although ordinal discrete variables can also be used. Mapping color and\/or shape to a (supervised) class tends to be helpful.\n\n## Description\n\nThe tourr R package implements grand tours (constrained random walks in the basis), projection pursuit (basis anneals to some objective function), and several other variants and options of visualization tours.\n\nGrand tour\n\nThe work and contribution of spinifex are primarily two-fold. The addition of manual tours which allow for user defined (and interactive!) control of the basis. Secondly, a layered API to create tour elements that mirrors the additive creation of ggplot2 visuals. The extension to gganimate and plotly graphics packages, which allow for more control over display and exporting .gif, .mp4, and .html widgets of tours. The later can also consume tours produced in tourr!\n\nWe see that bill length is important for separating the orange and green species. Let\u2019s change its contribution with a manual tour:\n\nManual tour\n\n## Getting started\n\n#install.packages(\"spinifex\") ## Install from CRAN\nvignette(\"getting started with spinifex\") ## Introduction to tours and spinifex\nvignette(\"ggproto api\") ## Introduction to the new ggproto api\n?ggtour ## Examples and links to ggproto functions\nhelp(package = \"spinifex\") ## Review the package contents and documentation\n\n## ggproto API\n\nThe ggproto API is a layered approach to the composition of animated tours, it should feel similar to the composition of ggplots.\n\nFamily Function Related to Description\nprocessing scale_01\/sd - scale each column to [01]\/std dev away from the mean\nprocessing basis_pca\/olda\/\u2026 Rdimtools::do.* basis of orthogonal component spaces\nprocessing basis_half_circle - basis with uniform contribution across half of a circle\nprocessing basis_guided tourr::guided_tour silently return the basis from a guided tour\ntour path manual_tour - basis and interpolation information for a manual tour\ntour path save_history tourr::save_history silent extended wrapper returning other tour arrays\ndisplay ggtour ggplot2::ggplot canvas and initialization for a tour animation\ndisplay proto_point\/text geom_point\/text adds observation points\/text\ndisplay proto_density\/2d geom_density\/2d adds density curve\/2d contours\ndisplay proto_hex geom_hex adds hexagonal heatmap of observations","date":"2022-07-01 05:43:26","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.21445590257644653, \"perplexity\": 7475.8403790146895}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-27\/segments\/1656103920118.49\/warc\/CC-MAIN-20220701034437-20220701064437-00174.warc.gz\"}"} | null | null |
I briefly mentioned this Feature request in another thread, but here it is formally.
How about when people post to the forums, their message is scanned for phrases, like "Amanda Tapping", "RDA", "Goa'uld", "G.D.O", "M.A.L.P" and when these phrases are found, they are hyperlinked to the appropriate Actor/Character/Technology/Whatever page in the omnipedia. For this to work, it may also have to look for alternate/incorrect spellings of these phrases, like Goa'uld/Gou'ald/Gould/Gao'uld/Goauld, etc.
Of course, if the phrase is already hyperlinked to something else, it would not change it.
Maybe a checkbox in the editor or near the "submit reply" button to allow the user to not have their message hyperlinked? | {
"redpajama_set_name": "RedPajamaC4"
} | 1,453 |
Q: tensorflow model predict runs out of memory We have a tensorflow keras model which we would like to evaluate after training but the predict call after the training runs into out of memory errors even though the fit call works just fine.
The dataset is loaded like this:
options = tf.data.Options()
options.experimental_distribute.auto_shard_policy = tf.data.experimental.AutoShardPolicy.OFF
dataset = (
tf.data.Dataset.list_files(f'{dataset_location}/*.csv')
.flat_map(tf.data.TextLineDataset)
.skip(1)
.map(decode_line) # custom function how to read the csv files
)
dataset = dataset.apply(tf.data.experimental.ignore_errors())
dataset = dataset.batch(batch_size).with_options(options)
Using model.fit on the dataset just works fine with batch size of 128
model.fit(dataset, epochs=10)
But using predict after the model is trained gives me an out of memory error even when I used batch size 1.
model.predict(dataset)
We are using google AI-platform with a custom image which is using the following base image:
gcr.io/deeplearning-platform-release/tf-gpu.2-8
Why would the fit call work but the predict call for the same model with the same dataset runs out of memory?
I would expect that the fit call is more memory intensive than the predict call.
This happens on tensorflow 2.8
Update:
I found a way how to solve the problem although I still think there is a memory leak in the predict function in tensorflow 2.8.
The solution which worked for me is not to load all csv files in the dataset but load one by one and do the prediction on the smaller datasets. Even though for training loading all csv files in one dataset works.
predictions = []
for file_name in tf.io.gfile.listdir(dataset_location):
file_path = f'{dataset_location}/{file_name}'
dataset = tf.data.experimental.make_csv_dataset(
file_pattern=file_path,
batch_size=10,
shuffle=False,
label_name="label",
field_delim=';',
column_defaults=[[""], [""], [0.0]],
num_epochs=1
)
pred = model.predict(dataset)
predictions.append(pred)
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,568 |
package org.knowm.xchange.btc38.service;
import java.io.IOException;
import java.util.HashMap;
import java.util.Map;
import org.knowm.xchange.Exchange;
import org.knowm.xchange.btc38.Btc38;
import org.knowm.xchange.btc38.dto.marketdata.Btc38Ticker;
import org.knowm.xchange.btc38.dto.marketdata.Btc38TickerReturn;
import org.knowm.xchange.btc38.dto.marketdata.Btc38Trade;
import org.knowm.xchange.currency.CurrencyPair;
/**
* Created by Yingzhe on 12/19/2014.
*/
public class Btc38MarketDataServiceRaw extends Btc38BaseService<Btc38> {
/**
* Constructor for Btc38MarketDataServiceRaw
*
* @param exchange The {@link org.knowm.xchange.Exchange}
*/
public Btc38MarketDataServiceRaw(Exchange exchange) {
super(Btc38.class, exchange);
}
/**
* Gets ticker from Btc38
*
* @param baseCurrency Base currency
* @param targetCurrency Target currency
* @return Btc38Ticker object
* @throws IOException
*/
public Btc38Ticker getBtc38Ticker(String baseCurrency, String targetCurrency) throws IOException {
if (!this.getCurrencyPairMap().containsKey(baseCurrency.toUpperCase() + "_" + targetCurrency.toUpperCase())) {
return null;
}
Map<String, Btc38TickerReturn> allTickers = this.btc38.getMarketTicker(targetCurrency);
Btc38Ticker ticker = allTickers != null && allTickers.containsKey(baseCurrency.toLowerCase())
? allTickers.get(baseCurrency.toLowerCase()).getTicker() : null;
return ticker.getBuy() != null || ticker.getHigh() != null || ticker.getLast() != null || ticker.getLow() != null || ticker.getSell() != null
|| ticker.getVol() != null ? ticker : null;
}
public HashMap<String, CurrencyPair> getCurrencyPairMap() throws IOException {
HashMap<String, CurrencyPair> currencyPairMap = new HashMap<>();
Map<String, Btc38TickerReturn> btcTickers = this.btc38.getMarketTicker("BTC");
Map<String, Btc38TickerReturn> cnyTickers = this.btc38.getMarketTicker("CNY");
if (btcTickers != null) {
for (String key : btcTickers.keySet()) {
String base = key.toUpperCase();
String target = "BTC";
currencyPairMap.put(base + "_" + target, new CurrencyPair(base, target));
}
}
if (cnyTickers != null) {
for (String key : cnyTickers.keySet()) {
String base = key.toUpperCase();
String target = "CNY";
currencyPairMap.put(base + "_" + target, new CurrencyPair(base, target));
}
}
return currencyPairMap;
}
public Btc38Trade[] getBtc38Trades(CurrencyPair currencyPair, Object[] args) throws IOException {
return (args != null && args.length > 0 && args[0] != null && args[0] instanceof Long)
? btc38.getTradesFrom(currencyPair.base.getCurrencyCode().toUpperCase(), currencyPair.counter.getCurrencyCode().toUpperCase(), (Long) args[0])
: btc38.getTrades(currencyPair.base.getCurrencyCode().toUpperCase(), currencyPair.counter.getCurrencyCode().toUpperCase());
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,619 |
This section of our website is dedicated to providing visitors with information about the musical landscape here in Luxembourg.
It includes a list of venues, festivals, orchestras, organisations, information services, record labels, publishers as well as some information regarding music:LX's publications.
Supported by the Ministry of Culture. | {
"redpajama_set_name": "RedPajamaC4"
} | 9,476 |
Antonín Maděra (22. října 1846) Veselá – 17. října 1918 Praha), byl rakouský a český právník a politik, na počátku 20. století poslanec Českého zemského sněmu.
Biografie
Studoval na gymnáziu v Mladé Boleslavi a v Litoměřicích, pak absolvoval práva na Karlo-Ferdinandově univerzitě v Praze. Nastoupil jako notářský koncipient v Pacově, později působil ve Vysokém Mýtě, Pelhřimově, Březnici a v Roudnici nad Labem.
Roku 1883 se přestěhoval do Lišova, kde si otevřel notářskou kancelář. V Lišově se postupně zapojil i do veřejného života jako člen obecního výboru, okresního zastupitelstva a místní školní rady. Stál u zrodu místní organizace Sokola a roku 1895 krátce působil jako její starosta. Roku 1899 byl zvolen starostou Lišova. V této době se uvádí jako zemský důvěrník mladočeské strany.
Počátkem 20. století se zapojil i do zemské politiky. Ve volbách v roce 1901 byl zvolen do Českého zemského sněmu v kurii městské (volební obvod Lišov, Třeboň a Týn nad Vltavou). Na sněmu prosadil zákon o právních a hmotných poměrech úředníků samosprávy.
Antonín Maděra se roku 1904 se přestěhoval do Jindřichova Hradce. Z tohoto důvodu rezignoval 22. října 1904 na funkci starosty Lišova. V dubnu 1911 se stal starostou okresního zastupitelstva v J. Hradci, v červenci byl zvolen také do městského zastupitelstva a městské rady. Zemřel roku 17.10.1918 v Praze, ale jeho přáním bylo být pohřben v Lišově.
Odkazy
Externí odkazy
Reference
Absolventi Univerzity Karlovy
Rakouští právníci
Čeští notáři
Čeští starostové
Členové Sokola
Poslanci českého zemského sněmu
Mladočeši
Narození v roce 1845
Narození 18. října
Úmrtí v roce 1918
Úmrtí 17. října
Muži
Úmrtí v Praze
Starostové Lišova | {
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A Ghost Waits (2020)
Posted on 9 August 2020 9 August 2020 By AntBit
A Ghost Waits first published by SciFiNow
"I don't know why I do what I do."
The speaker, Jack (MacLeod Andrews, They Look Like People, The Siren) is going through what might be called an existential crisis. Displaced from his own home by a neighbour's cockroach infestation, he is having to make his bed wherever he can – and right now, given that neither his boss, his friends or his clients seem willing to offer their couch or even answer his calls, the thirtysomething handyman is having to sleep where he works: a rental home whose previous occupants left in a real hurry. As Jack drifts about the premises, trying to do minor repairs and report on any greater property damage while working around a family's abandoned furniture and belongings, he is like a restless ghost – alone, unsettled, practically invisible. He even, to stave off his loneliness, talks with the toilet (voicing both sides of this one-way conversation), or – in a dream – with his own doppelgänger.
As it turns out, this ordinary-looking suburban unit is also accommodating a ghost of a more straightforwardly literal kind. Muriel (Natalie Walker) has been moving furniture about, making noises, entering tenants' dreams and ultimately frightening people away from her house for very many years. Yet perhaps there is something metaphorical to this 'spectral agent' after all. For as someone also starting to question the purpose of her 'job' and her 'life', and also longing for companionship, Muriel is a perfect double for Jack – a personified expression of his inner monologue, like that flapping toilet lid.
As its title implies, A Ghost Waits is a ghost story – but with strong psychological/philosophical preoccupations, it is as much about a haunted man as a haunted house. As Jack struggles to find a way out of his personal and professional rut, Muriel's failure to get rid of him reflects his own sense of failure. They say that like attracts like, and although Jack and Muriel occupy the same space in different dimensions, before you can say "The Ghost and Mrs Muir" this well-matched pair of disgruntled working folk find themselves wanting just to hang out together and discuss matters theological, teleological and eschatological.
If the feature debut of writer/director Adam Stovall certainly includes the kinds of scares associated with its genre, they are shown to be part of a ghost's performative bag of tricks, little different from the tools and expertise that Jack brings to his jobs. And just as Jack answers to his boss Neal (voiced by Stovall), ghosts like Muriel and Rosie (Sydney Vollmer) – with whom Muriel is temporarily forced to partner up on her job – are overseen by an office-bound supervisor (Amanda Miller), in a low-key version of the infernal bureaucracy from Tim Burton's Beetlejuice (1988).
Indeed everything here is low-key, as Stovall compensates for his small budget with big ideas, deconstructing and reconceptualising the ghost story from the ground up. Likewise the decision to present everything in old-fashioned black and white may be a cinematic way of merging Jack's contemporary present with Muriel's past, and to show the colourless quality that the world has assumed in both Jack's and Muriel's eyes (in what is, in the final analysis, a film about Jack's depression) – but this monochrome also helps cover over some very simple (and cheap) makeup and visual effects. Give more money to Stovall, and it will be interesting to see what he does next – but meanwhile this uncanny romance, melancholic if generous-spirited, makes an interesting companion piece to David Lowery's similarly lo-fi, location-fixed A Ghost Story (2017). For here too, haunting is a business that is kept strictly in-house, and where the work, though able to be combined with pleasure, never ends.
Strap: Adam Stovall's debut feature uses ghostly romance to fix up a damaged psyche.
A Night of Horror: Nightmare Radio (2019)
Unseeing Evil (2020) | {
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