for if they were there would be no difference between being ill and being well. yet the subject both of health and of sickness - whether it is humour or blood - is one and the same. ) we can distinguish, then, between the two - just as, to give another example, ' colour ' and visible ' are different - and clearly it is the fulfilment of what is potential as potential that is motion. so this, precisely, is motion. further it is evident that motion is an attribute of a thing just when it is fully real in this way, and neither before nor after. for each thing of this kind is capable of being at one time actual, at another not. take for instance the buildable as buildable. the actuality of the buildable as buildable is the process of building. for the actuality of the buildable must be either this or the house. but when there is a house, the buildable is no longer buildable. on the other hand, it is the buildable which is being built. the process then of being built must be the kind of actuality required but building is a kind of motion, and the same account will apply to the other kinds also. the soundness of this definition is evident both when we consider the accounts of motion that the others have given, and also from the difficulty of defining it otherwise. one could not easily put motion and change in another genus - this is plain if we consider where some people put it ; they identify motion with or ' inequality ' or ' not being ' ; but such things are not necessarily moved, whether they are ' different ' or ' unequal ' or ' non - existent ' ; nor is change either to or from these rather than to or from their opposites. the reason why they put motion into these genera is that it is thought to be something indefinite, and the principles in the second column are indefinite because they are privative : none of them is either ' this ' or ' such ' or comes under any of the other modes of predication. the reason in turn why motion is thought to be indefinite is that it cannot be classed simply as a potentiality or as an actuality - a thing that is merely capable of having a certain size is not undergoing change, nor yet a thing that is actually of a certain size, and motion is thought to be a sort of actuality, but incomplete, the reason for this view being that the potential whose actuality it is is incomplete. this is why it


change, nor yet a thing that is actually of a certain size, and motion is thought to be a sort of actuality, but incomplete, the reason for this view being that the potential whose actuality it is is incomplete. this is why it is hard to grasp what motion is. it is necessary to class it with privation or with potentiality or with sheer actuality, yet none of these seems possible. there remains then the suggested mode of definition, namely that it is a sort of actuality, or actuality of the kind described, hard to grasp, but not incapable of existing. the mover too is moved, as has been said - every mover, that is, which is capable of motion, and whose immobility is rest - when a thing is subject to motion its immobility is rest. for to act on the movable as such is just to move it. but this it does by contact, so that at the same time it is also acted on. hence we can define motion as the fulfilment of the movable qua movable, the cause of the attribute being contact with what can move so that the mover is also acted on. the mover or agent will always be the vehicle of a form, either a ' this ' or ' such ', which, when it acts, will be the source and cause of the change, e. g. the full - formed man begets man from what is potentially man. the solution of the difficulty that is raised about the motion - whether it is in the movable - is plain. it is the fulfilment of this potentiality, and by the action of that which has the power of causing motion ; and the actuality of that which has the power of causing motion is not other than the actuality of the movable, for it must be the fulfilment of both. a thing is capable of causing motion because it can do this, it is a mover because it actually does it. but it is on the movable that it is capable of acting. hence there is a single actuality of both alike, just as one to two and two to one are the same interval, and the steep ascent and the steep descent are one - for these are one and the same, although they can be described in different ways. so it is with the mover and the moved. this view has a dialectical difficulty. perhaps it is necessary that the actuality of the agent and that of the


operation is performed on some patient - it is not cut adrift from a subject, but is of a on b. ( 2 ) there is nothing to prevent two things having one and the same actualization, provided the actualizations are not described in the same way, but are related as what can act to what is acting. ( 3 ) nor is it necessary that the teacher should learn, even if to act and to be acted on are one and the same, provided they are not the same in definition ( as ' raiment ' and ' dress ' ), but are the same merely in the sense in which the road from thebes to athens and the road from athens to thebes are the same, as has been explained above. for it is not things which are in a way the same that have all their attributes the same, but only such as have the same definition. but indeed it by no means follows from the fact that teaching is the same as learning, that to learn is the same as to teach, any more than it follows from the fact that there is one distance between two things which are at a distance from each other, that the two vectors ab and ba, are one and the same. to generalize, teaching is not the same as learning, or agency as patiency, in the full sense, though they belong to the same subject, the motion ; for the ' actualization of x in y ' and the ' actualization of y through the action of x ' differ in definition. what then motion is, has been stated both generally and particularly. it is not difficult to see how each of its types will be defined - alteration is the fulfillment of the alterable qua alterable ( or, more scientifically, the fulfilment of what can act and what can be acted on, as such ) - generally and again in each particular case, building, healing, & c. a similar definition will apply to each of the other kinds of motion. the science of nature is concerned with spatial magnitudes and motion and time, and each of these at least is necessarily infinite or finite, even if some things dealt with by the science are not, e. g. a quality or a point - it is not necessary perhaps that such things should be put under either head. hence it is incumbent on the person who specializes in physics to discuss the infinite and to inquire whether there is such a thing or not, and, if there is, what it is. the appropriateness to the science


put under either head. hence it is incumbent on the person who specializes in physics to discuss the infinite and to inquire whether there is such a thing or not, and, if there is, what it is. the appropriateness to the science of this problem is clearly indicated. all who have touched on this kind of science in a way worth considering have formulated views about the infinite, and indeed, to a man, make it a principle of things. ( 1 ) some, as the pythagoreans and plato, make the infinite a principle in the sense of a self - subsistent substance, and not as a mere attribute of some other thing. only the pythagoreans place the infinite among the objects of sense ( they do not regard number as separable from these ), and assert that what is outside the heaven is infinite. plato, on the other hand, holds that there is no body outside ( the forms are not outside because they are nowhere ), yet that the infinite is present not only in the objects of sense but in the forms also. further, the pythagoreans identify the infinite with the even. for this, they say, when it is cut off and shut in by the odd, provides things with the element of infinity. an indication of this is what happens with numbers. if the gnomons are placed round the one, and without the one, in the one construction the figure that results is always different, in the other it is always the same. but plato has two infinites, the great and the small. the physicists, on the other hand, all of them, always regard the infinite as an attribute of a substance which is different from it and belongs to the class of the so - called elements - water or air or what is intermediate between them. those who make them limited in number never make them infinite in amount. but those who make the elements infinite in number, as anaxagoras and democritus do, say that the infinite is continuous by contact - compounded of the homogeneous parts according to the one, of the seed - mass of the atomic shapes according to the other. further, anaxagoras held that any part is a mixture in the same way as the all, on the ground of the observed fact that anything comes out of anything. for it is probably for this reason that he maintains that once upon a time all things were together. ( this flesh and this bone were together, and so of any


the all, on the ground of the observed fact that anything comes out of anything. for it is probably for this reason that he maintains that once upon a time all things were together. ( this flesh and this bone were together, and so of any thing : therefore all things : and at the same time too. ) for there is a beginning of separation, not only for each thing, but for all. each thing that comes to be comes from a similar body, and there is a coming to be of all things, though not, it is true, at the same time. hence there must also be an origin of coming to be. one such source there is which he calls mind, and mind begins its work of thinking from some starting - point. so necessarily all things must have been together at a certain time, and must have begun to be moved at a certain time. democritus, for his part, asserts the contrary, namely that no element arises from another element. nevertheless for him the common body is a source of all things, differing from part to part in size and in shape. it is clear then from these considerations that the inquiry concerns the physicist. nor is it without reason that they all make it a principle or source. we cannot say that the infinite has no effect, and the only effectiveness which we can ascribe to it is that of a principle. everything is either a source or derived from a source. but there cannot be a source of the infinite or limitless, for that would be a limit of it. further, as it is a beginning, it is both uncreatable and indestructible. for there must be a point at which what has come to be reaches completion, and also a termination of all passing away. that is why, as we say, there is no principle of this, but it is this which is held to be the principle of other things, and to encompass all and to steer all, as those assert who do not recognize, alongside the infinite, other causes, such as mind or friendship. further they identify it with the divine, for it is ' deathless and imperishable ' as anaximander says, with the majority of the physicists. belief in the existence of the infinite comes mainly from five considerations : ( 1 ) from the nature of time - for it is infinite. ( 2 ) from the division of magnitudes - for the mathematicians also use the notion of the infinite. ( 3 ) if coming to be and passing away


mainly from five considerations : ( 1 ) from the nature of time - for it is infinite. ( 2 ) from the division of magnitudes - for the mathematicians also use the notion of the infinite. ( 3 ) if coming to be and passing away do not give out, it is only because that from which things come to be is infinite. ( 4 ) because the limited always finds its limit in something, so that there must be no limit, if everything is always limited by something different from itself. ( 5 ) most of all, a reason which is peculiarly appropriate and presents the difficulty that is felt by everybody - not only number but also mathematical magnitudes and what is outside the heaven are supposed to be infinite because they never give out in our thought. the last fact ( that what is outside is infinite ) leads people to suppose that body also is infinite, and that there is an infinite number of worlds. why should there be body in one part of the void rather than in another? grant only that mass is anywhere and it follows that it must be everywhere. also, if void and place are infinite, there must be infinite body too, for in the case of eternal things what may be must be. but the problem of the infinite is difficult : many contradictions result whether we suppose it to exist or not to exist. if it exists, we have still to ask how it exists ; as a substance or as the essential attribute of some entity? or in neither way, yet none the less is there something which is infinite or some things which are infinitely many? the problem, however, which specially belongs to the physicist is to investigate whether there is a sensible magnitude which is infinite. we must begin by distinguishing the various senses in which the term ' infinite ' is used. ( 1 ) what is incapable of being gone through, because it is not in its nature to be gone through ( the sense in which the voice is ' invisible ' ). ( 2 ) what admits of being gone through, the process however having no termination, or what scarcely admits of being gone through. ( 3 ) what naturally admits of being gone through, but is not actually gone through or does not actually reach an end. further, everything that is infinite may be so in respect of addition or division or both. now it is impossible that the infinite should be a thing which is itself infinite, separable from sensible objects. if the infinite is neither a magnitude nor an aggregate, but is itself a substance and not an attribute, it will be indivisible


. this discussion, however, involves the more general question whether the infinite can be present in mathematical objects and things which are intelligible and do not have extension, as well as among sensible objects. our inquiry ( as physicists ) is limited to its special subject - matter, the objects of sense, and we have to ask whether there is or is not among them a body which is infinite in the direction of increase. we may begin with a dialectical argument and show as follows that there is no such thing. if ' bounded by a surface ' is the definition of body there cannot be an infinite body either intelligible or sensible. nor can number taken in abstraction be infinite, for number or that which has number is numerable. if then the numerable can be numbered, it would also be possible to go through the infinite. if, on the other hand, we investigate the question more in accordance with principles appropriate to physics, we are led as follows to the same result. the infinite body must be either ( 1 ) compound, or ( 2 ) simple ; yet neither alternative is possible. ( 1 ) compound the infinite body will not be, if the elements are finite in number. for they must be more than one, and the contraries must always balance, and no one of them can be infinite. if one of the bodies falls in any degree short of the other in potency - suppose fire is finite in amount while air is infinite and a given quantity of fire exceeds in power the same amount of air in any ratio provided it is numerically definite - the infinite body will obviously prevail over and annihilate the finite body. on the other hand, it is impossible that each should be infinite. ' body ' is what has extension in all directions and the infinite is what is boundlessly extended, so that the infinite body would be extended in all directions ad infinitum. nor ( 2 ) can the infinite body be one and simple, whether it is, as some hold, a thing over and above the elements ( from which they generate the elements ) or is not thus qualified. ( a ) we must consider the former alternative ; for there are some people who make this the infinite, and not air or water, in order that the other elements may not be annihilated by the element which is infinite. they have contrariety with each other - air is cold, water moist, fire hot ; if one were infinite, the others by now would have ceased to be. as it


or ' it is the games ' ; and potentially as matter exists, not independently as what is finite does. by addition then, also, there is potentially an infinite, namely, what we have described as being in a sense the same as the infinite in respect of division. for it will always be possible to take something ah extra. yet the sum of the parts taken will not exceed every determinate magnitude, just as in the direction of division every determinate magnitude is surpassed in smallness and there will be a smaller part. but in respect of addition there cannot be an infinite which even potentially exceeds every assignable magnitude, unless it has the attribute of being actually infinite, as the physicists hold to be true of the body which is outside the world, whose essential nature is air or something of the kind. but if there cannot be in this way a sensible body which is infinite in the full sense, evidently there can no more be a body which is potentially infinite in respect of addition, except as the inverse of the infinite by division, as we have said. it is for this reason that plato also made the infinites two in number, because it is supposed to be possible to exceed all limits and to proceed ad infinitum in the direction both of increase and of reduction. yet though he makes the infinites two, he does not use them. for in the numbers the infinite in the direction of reduction is not present, as the monad is the smallest ; nor is the infinite in the direction of increase, for the parts number only up to the decad. the infinite turns out to be the contrary of what it is said to be. it is not what has nothing outside it that is infinite, but what always has something outside it. this is indicated by the fact that rings also that have no bezel are described as ' endless ', because it is always possible to take a part which is outside a given part. the description depends on a certain similarity, but it is not true in the full sense of the word. this condition alone is not sufficient : it is necessary also that the next part which is taken should never be the same. in the circle, the latter condition is not satisfied : it is only the adjacent part from which the new part is different. our definition then is as follows : a quantity is infinite if it is such that we can always take a part outside what has been already taken. on the other hand, what has nothing outside it is complete and whole. for thus we define the whole -


definition then is as follows : a quantity is infinite if it is such that we can always take a part outside what has been already taken. on the other hand, what has nothing outside it is complete and whole. for thus we define the whole - that from which nothing is wanting, as a whole man or a whole box. what is true of each particular is true of the whole as such - the whole is that of which nothing is outside. on the other hand that from which something is absent and outside, however small that may be, is not ' all '. ' whole ' and ' complete ' are either quite identical or closely akin. nothing is complete ( teleion ) which has no end ( telos ) ; and the end is a limit. hence parmenides must be thought to have spoken better than melissus. the latter says that the whole is infinite, but the former describes it as limited, ' equally balanced from the middle '. for to connect the infinite with the all and the whole is not like joining two pieces of string ; for it is from this they get the dignity they ascribe to the infinite - its containing all things and holding the all in itself - from its having a certain similarity to the whole. it is in fact the matter of the completeness which belongs to size, and what is potentially a whole, though not in the full sense. it is divisible both in the direction of reduction and of the inverse addition. it is a whole and limited ; not, however, in virtue of its own nature, but in virtue of what is other than it. it does not contain, but, in so far as it is infinite, is contained. consequently, also, it is unknowable, qua infinite ; for the matter has no form. ( hence it is plain that the infinite stands in the relation of part rather than of whole. for the matter is part of the whole, as the bronze is of the bronze statue. ) if it contains in the case of sensible things, in the case of intelligible things the great and the small ought to contain them. but it is absurd and impossible to suppose that the unknowable and indeterminate should contain and determine. it is reasonable that there should not be held to be an infinite in respect of addition such as to surpass every magnitude, but that there should be thought to be such an infinite in the direction of division. for the matter and the infinite are contained inside what contains them, while


he is the size he is. the thought is an accident. ( a ) time indeed and movement are infinite, and also thinking, in the sense that each part that is taken passes in succession out of existence. ( b ) magnitude is not infinite either in the way of reduction or of magnification in thought. this concludes my account of the way in which the infinite exists, and of the way in which it does not exist, and of what it is. the physicist must have a knowledge of place, too, as well as of the infinite - namely, whether there is such a thing or not, and the manner of its existence and what it is - both because all suppose that things which exist are somewhere ( the non - existent is nowhere - - where is the goat - stag or the sphinx? ), and because ' motion ' in its most general and primary sense is change of place, which we call ' locomotion '. the question, what is place? presents many difficulties. an examination of all the relevant facts seems to lead to divergent conclusions. moreover, we have inherited nothing from previous thinkers, whether in the way of a statement of difficulties or of a solution. the existence of place is held to be obvious from the fact of mutual replacement. where water now is, there in turn, when the water has gone out as from a vessel, air is present. when therefore another body occupies this same place, the place is thought to be different from all the bodies which come to be in it and replace one another. what now contains air formerly contained water, so that clearly the place or space into which and out of which they passed was something different from both. further, the typical locomotions of the elementary natural bodies - namely, fire, earth, and the like - show not only that place is something, but also that it exerts a certain influence. each is carried to its own place, if it is not hindered, the one up, the other down. now these are regions or kinds of place - up and down and the rest of the six directions. nor do such distinctions ( up and down and right and left, & c. ) hold only in relation to us. to us they are not always the same but change with the direction in which we are turned : that is why the same thing may be both right and left, up and down, before and behind. but in nature each is distinct, taken apart by itself. it is not every chance direction which is '


too have surface and the other limits of body ; for the same statement will apply to them : where the bounding planes of the water were, there in turn will be those of the air. but when we come to a point we cannot make a distinction between it and its place. hence if the place of a point is not different from the point, no more will that of any of the others be different, and place will not be something different from each of them. ( 3 ) what in the world then are we to suppose place to be? if it has the sort of nature described, it cannot be an element or composed of elements, whether these be corporeal or incorporeal : for while it has size, it has not body. but the elements of sensible bodies are bodies, while nothing that has size results from a combination of intelligible elements. ( 4 ) also we may ask : of what in things is space the cause? none of the four modes of causation can be ascribed to it. it is neither in the sense of the matter of existents ( for nothing is composed of it ), nor as the form and definition of things, nor as end, nor does it move existents. ( 5 ) further, too, if it is itself an existent, where will it be? zeno ' s difficulty demands an explanation : for if everything that exists has a place, place too will have a place, and so on ad infinitum. ( 6 ) again, just as every body is in place, so, too, every place has a body in it. what then shall we say about growing things? it follows from these premisses that their place must grow with them, if their place is neither less nor greater than they are. by asking these questions, then, we must raise the whole problem about place - not only as to what it is, but even whether there is such a thing. we may distinguish generally between predicating b of a because it ( a ) is itself, and because it is something else ; and particularly between place which is common and in which all bodies are, and the special place occupied primarily by each. i mean, for instance, that you are now in the heavens because you are in the air and it is in the heavens ; and you are in the air because you are on the earth ; and similarly on the earth because you are in this place which contains no more than you. now if place is what primarily contains each body


in the air and it is in the heavens ; and you are in the air because you are on the earth ; and similarly on the earth because you are in this place which contains no more than you. now if place is what primarily contains each body, it would be a limit, so that the place would be the form or shape of each body by which the magnitude or the matter of the magnitude is defined : for this is the limit of each body. if, then, we look at the question in this way the place of a thing is its form. but, if we regard the place as the extension of the magnitude, it is the matter. for this is different from the magnitude : it is what is contained and defined by the form, as by a bounding plane. matter or the indeterminate is of this nature ; when the boundary and attributes of a sphere are taken away, nothing but the matter is left. this is why plato in the timaeus says that matter and space are the same ; for the ' participant ' and space are identical. ( it is true, indeed, that the account he gives there of the ' participant ' is different from what he says in his so - called ' unwritten teaching '. nevertheless, he did identify place and space. ) i mention plato because, while all hold place to be something, he alone tried to say what it is. in view of these facts we should naturally expect to find difficulty in determining what place is, if indeed it is one of these two things, matter or form. they demand a very close scrutiny, especially as it is not easy to recognize them apart. but it is at any rate not difficult to see that place cannot be either of them. the form and the matter are not separate from the thing, whereas the place can be separated. as we pointed out, where air was, water in turn comes to be, the one replacing the other ; and similarly with other bodies. hence the place of a thing is neither a part nor a state of it, but is separable from it. for place is supposed to be something like a vessel - the vessel being a transportable place. but the vessel is no part of the thing. in so far then as it is separable from the thing, it is not the form : qua containing, it is different from the matter. also it is held that what is anywhere is both itself something and that there is a different thing outside it. ( plato of course, if we may digress


the physicist - namely whether it exists or not, and how it exists or what it is - just as about place. the views taken of it involve arguments both for and against, in much the same sort of way. for those who hold that the void exists regard it as a sort of place or vessel which is supposed to be ' full ' when it holds the bulk which it is capable of containing, ' void ' when it is deprived of that - as if ' void ' and ' full ' and ' place ' denoted the same thing, though the essence of the three is different. we must begin the inquiry by putting down the account given by those who say that it exists, then the account of those who say that it does not exist, and third the current view on these questions. those who try to show that the void does not exist do not disprove what people really mean by it, but only their erroneous way of speaking ; this is true of anaxagoras and of those who refute the existence of the void in this way. they merely give an ingenious demonstration that air is something - - by straining wine - skins and showing the resistance of the air, and by cutting it off in clepsydras. but people really mean that there is an empty interval in which there is no sensible body. they hold that everything which is in body is body and say that what has nothing in it at all is void ( so what is full of air is void ). it is not then the existence of air that needs to be proved, but the non - existence of an interval, different from the bodies, either separable or actual - an interval which divides the whole body so as to break its continuity, as democritus and leucippus hold, and many other physicists - or even perhaps as something which is outside the whole body, which remains continuous. these people, then, have not reached even the threshold of the problem, but rather those who say that the void exists. ( 1 ) they argue, for one thing, that change in place ( i. e. locomotion and increase ) would not be. for it is maintained that motion would seem not to exist, if there were no void, since what is full cannot contain anything more. if it could, and there were two bodies in the same place, it would also be true that any number of bodies could be together ; for it is impossible to draw a line of division beyond which the statement would become


, there must be void. every body, again, they suppose to be tangible ; and of this nature is whatever has weight or lightness. hence, by a syllogism, what has nothing heavy or light in it, is void. this result, then, as i have said, is reached by syllogism. it would be absurd to suppose that the point is void ; for the void must be place which has in it an interval in tangible body. but at all events we observe then that in one way the void is described as what is not full of body perceptible to touch ; and what has heaviness and lightness is perceptible to touch. so we would raise the question : what would they say of an interval that has colour or sound - is it void or not? clearly they would reply that if it could receive what is tangible it was void, and if not, not. in another way void is that in which there is no ' this ' or corporeal substance. so some say that the void is the matter of the body ( they identify the place, too, with this ), and in this they speak incorrectly ; for the matter is not separable from the things, but they are inquiring about the void as about something separable. since we have determined the nature of place, and void must, if it exists, be place deprived of body, and we have stated both in what sense place exists and in what sense it does not, it is plain that on this showing void does not exist, either unseparated or separated ; the void is meant to be, not body but rather an interval in body. this is why the void is thought to be something, viz. because place is, and for the same reasons. for the fact of motion in respect of place comes to the aid both of those who maintain that place is something over and above the bodies that come to occupy it, and of those who maintain that the void is something. they state that the void is the condition of movement in the sense of that in which movement takes place ; and this would be the kind of thing that some say place is. but there is no necessity for there being a void if there is movement. it is not in the least needed as a condition of movement in general, for a reason which, incidentally, escaped melissus ; viz. that the full can suffer qualitative change. but not even movement in respect of place involves a void


time. to sum the matter up, the cause of this result is obvious, viz. that between any two movements there is a ratio ( for they occupy time, and there is a ratio between any two times, so long as both are finite ), but there is no ratio of void to full. these are the consequences that result from a difference in the media ; the following depend upon an excess of one moving body over another. we see that bodies which have a greater impulse either of weight or of lightness, if they are alike in other respects, move faster over an equal space, and in the ratio which their magnitudes bear to each other. therefore they will also move through the void with this ratio of speed. but that is impossible ; for why should one move faster? ( in moving through plena it must be so ; for the greater divides them faster by its force. for a moving thing cleaves the medium either by its shape, or by the impulse which the body that is carried along or is projected possesses. ) therefore all will possess equal velocity. but this is impossible. it is evident from what has been said, then, that, if there is a void, a result follows which is the very opposite of the reason for which those who believe in a void set it up. they think that if movement in respect of place is to exist, the void cannot exist, separated all by itself ; but this is the same as to say that place is a separate cavity ; and this has already been stated to be impossible. but even if we consider it on its own merits the so - called vacuum will be found to be really vacuous. for as, if one puts a cube in water, an amount of water equal to the cube will be displaced ; so too in air ; but the effect is imperceptible to sense. and indeed always in the case of any body that can be displaced, must, if it is not compressed, be displaced in the direction in which it is its nature to be displaced - always either down, if its locomotion is downwards as in the case of earth, or up, if it is fire, or in both directions - whatever be the nature of the inserted body. now in the void this is impossible ; for it is not body ; the void must have penetrated the cube to a distance equal to that which this portion of void formerly occupied in the void, just as if the water or air had not been displaced by the wooden cube, but had penetrated right


for it is not body ; the void must have penetrated the cube to a distance equal to that which this portion of void formerly occupied in the void, just as if the water or air had not been displaced by the wooden cube, but had penetrated right through it. but the cube also has a magnitude equal to that occupied by the void ; a magnitude which, if it is also hot or cold, or heavy or light, is none the less different in essence from all its attributes, even if it is not separable from them ; i mean the volume of the wooden cube. so that even if it were separated from everything else and were neither heavy nor light, it will occupy an equal amount of void, and fill the same place, as the part of place or of the void equal to itself. how then will the body of the cube differ from the void or place that is equal to it? and if there can be two such things, why cannot there be any number coinciding? this, then, is one absurd and impossible implication of the theory. it is also evident that the cube will have this same volume even if it is displaced, which is an attribute possessed by all other bodies also. therefore if this differs in no respect from its place, why need we assume a place for bodies over and above the volume of each, if their volume be conceived of as free from attributes? it contributes nothing to the situation if there is an equal interval attached to it as well. [ further it ought to be clear by the study of moving things what sort of thing void is. but in fact it is found nowhere in the world. for air is something, though it does not seem to be so - nor, for that matter, would water, if fishes were made of iron ; for the discrimination of the tangible is by touch. ] it is clear, then, from these considerations that there is no separate void. there are some who think that the existence of rarity and density shows that there is a void. if rarity and density do not exist, they say, neither can things contract and be compressed. but if this were not to take place, either there would be no movement at all, or the universe would bulge, as xuthus said, or air and water must always change into equal amounts ( e. g. if air has been made out of a cupful of water, at the same time out of an equal amount of air a cupful of water must have been made ), or void must necessarily exist ;


change into equal amounts ( e. g. if air has been made out of a cupful of water, at the same time out of an equal amount of air a cupful of water must have been made ), or void must necessarily exist ; for compression and expansion cannot take place otherwise. now, if they mean by the rare that which has many voids existing separately, it is plain that if void cannot exist separate any more than a place can exist with an extension all to itself, neither can the rare exist in this sense. but if they mean that there is void, not separately existent, but still present in the rare, this is less impossible, yet, first, the void turns out not to be a condition of all movement, but only of movement upwards ( for the rare is light, which is the reason why they say fire is rare ) ; second, the void turns out to be a condition of movement not as that in which it takes place, but in that the void carries things up as skins by being carried up themselves carry up what is continuous with them. yet how can void have a local movement or a place? for thus that into which void moves is till then void of a void. again, how will they explain, in the case of what is heavy, its movement downwards? and it is plain that if the rarer and more void a thing is the quicker it will move upwards, if it were completely void it would move with a maximum speed! but perhaps even this is impossible, that it should move at all ; the same reason which showed that in the void all things are incapable of moving shows that the void cannot move, viz. the fact that the speeds are incomparable. since we deny that a void exists, but for the rest the problem has been truly stated, that either there will be no movement, if there is not to be condensation and rarefaction, or the universe will bulge, or a transformation of water into air will always be balanced by an equal transformation of air into water ( for it is clear that the air produced from water is bulkier than the water ) : it is necessary therefore, if compression does not exist, either that the next portion will be pushed outwards and make the outermost part bulge, or that somewhere else there must be an equal amount of water produced out of air, so that the entire bulk of the whole may be equal, or that nothing moves. for when anything is displaced this will always happen, unless it comes round in a circle


potentially matter for both states ; so that the same thing is dense and rare, and the two qualities have one matter. the dense is heavy, and the rare is light. [ again, as the arc of a circle when contracted into a smaller space does not acquire a new part which is convex, but what was there has been contracted ; and as any part of fire that one takes will be hot ; so, too, it is all a question of contraction and expansion of the same matter. ] there are two types in each case, both in the dense and in the rare ; for both the heavy and the hard are thought to be dense, and contrariwise both the light and the soft are rare ; and weight and hardness fail to coincide in the case of lead and iron. from what has been said it is evident, then, that void does not exist either separate ( either absolutely separate or as a separate element in the rare ) or potentially, unless one is willing to call the condition of movement void, whatever it may be. at that rate the matter of the heavy and the light, qua matter of them, would be the void ; for the dense and the rare are productive of locomotion in virtue of this contrariety, and in virtue of their hardness and softness productive of passivity and impassivity, i. e. not of locomotion but rather of qualitative change. so much, then, for the discussion of the void, and of the sense in which it exists and the sense in which it does not exist. next for discussion after the subjects mentioned is time. the best plan will be to begin by working out the difficulties connected with it, making use of the current arguments. first, does it belong to the class of things that exist or to that of things that do not exist? then secondly, what is its nature? to start, then : the following considerations would make one suspect that it either does not exist at all or barely, and in an obscure way. one part of it has been and is not, while the other is going to be and is not yet. yet time - both infinite time and any time you like to take - is made up of these. one would naturally suppose that what is made up of things which do not exist could have no share in reality. further, if a divisible thing is to exist, it is necessary that, when it exists, all or some of its parts must exist. but of time some parts have been, while others have


is thought to be the movement of the sphere, viz. because the other movements are measured by this, and time by this movement. this also explains the common saying that human affairs form a circle, and that there is a circle in all other things that have a natural movement and coming into being and passing away. this is because all other things are discriminated by time, and end and begin as though conforming to a cycle ; for even time itself is thought to be a circle. and this opinion again is held because time is the measure of this kind of locomotion and is itself measured by such. so that to say that the things that come into being form a circle is to say that there is a circle of time ; and this is to say that it is measured by the circular movement ; for apart from the measure nothing else to be measured is observed ; the whole is just a plurality of measures. it is said rightly, too, that the number of the sheep and of the dogs is the same number if the two numbers are equal, but not the same decad or the same ten ; just as the equilateral and the scalene are not the same triangle, yet they are the same figure, because they are both triangles. for things are called the same so - and - so if they do not differ by a differentia of that thing, but not if they do ; e. g. triangle differs from triangle by a differentia of triangle, therefore they are different triangles ; but they do not differ by a differentia of figure, but are in one and the same division of it. for a figure of the one kind is a circle and a figure of another kind of triangle, and a triangle of one kind is equilateral and a triangle of another kind scalene. they are the same figure, then, that, triangle, but not the same triangle. therefore the number of two groups also - is the same number ( for their number does not differ by a differentia of number ), but it is not the same decad ; for the things of which it is asserted differ ; one group are dogs, and the other horses. we have now discussed time - both time itself and the matters appropriate to the consideration of it. everything which changes does so in one of three senses. it may change ( 1 ) accidentally, as for instance when we say that something musical walks, that which walks being something in which aptitude for music is an accident. again ( 2 ) a thing


neither caused nor experienced by the form or the place or the quantity. so we are left with a mover, a moved, and a goal of motion. i do not include the starting - point of motion : for it is the goal rather than the starting - point of motion that gives its name to a particular process of change. thus ' perishing ' is change to not - being, though it is also true that that that which perishes changes from being : and ' becoming ' is change to being, though it is also change from not - being. now a definition of motion has been given above, from which it will be seen that every goal of motion, whether it be a form, an affection, or a place, is immovable, as, for instance, knowledge and heat. here, however, a difficulty may be raised. affections, it may be said, are motions, and whiteness is an affection : thus there may be change to a motion. to this we may reply that it is not whiteness but whitening that is a motion. here also the same distinctions are to be observed : a goal of motion may be so accidentally, or partially and with reference to something other than itself, or directly and with no reference to anything else : for instance, a thing which is becoming white changes accidentally to an object of thought, the colour being only accidentally the object of thought ; it changes to colour, because white is a part of colour, or to europe, because athens is a part of europe ; but it changes essentially to white colour. it is now clear in what sense a thing is in motion essentially, accidentally, or in respect of something other than itself, and in what sense the phrase ' itself directly ' is used in the case both of the mover and of the moved : and it is also clear that the motion is not in the form but in that which is in motion, that is to say ' the movable in activity '. now accidental change we may leave out of account : for it is to be found in everything, at any time, and in any respect. change which is not accidental on the other hand is not to be found in everything, but only in contraries, in things intermediate contraries, and in contradictories, as may be proved by induction. an intermediate may be a starting - point of change, since for the purposes of the change it serves as contrary to either of two contraries : for the intermediate is in a sense the extremes. hence we


of becoming be becoming, nor can the becoming of any process be that process. finally, since there are three kinds of motion, the substratum and the goal of motion must be one or other of these, e. g. locomotion will have to be altered or to be locally moved. to sum up, then, since everything that is moved is moved in one of three ways, either accidentally, or partially, or essentially, change can change only accidentally, as e. g. when a man who is being restored to health runs or learns : and accidental change we have long ago decided to leave out of account. since, then, motion can belong neither to being nor to relation nor to agent and patient, it remains that there can be motion only in respect of quality, quantity, and place : for with each of these we have a pair of contraries. motion in respect of quality let us call alteration, a general designation that is used to include both contraries : and by quality i do not here mean a property of substance ( in that sense that which constitutes a specific distinction is a quality ) but a passive quality in virtue of which a thing is said to be acted on or to be incapable of being acted on. motion in respect of quantity has no name that includes both contraries, but it is called increase or decrease according as one or the other is designated : that is to say motion in the direction of complete magnitude is increase, motion in the contrary direction is decrease. motion in respect of place has no name either general or particular : but we may designate it by the general name of locomotion, though strictly the term ' locomotion ' is applicable to things that change their place only when they have not the power to come to a stand, and to things that do not move themselves locally. change within the same kind from a lesser to a greater or from a greater to a lesser degree is alteration : for it is motion either from a contrary or to a contrary, whether in an unqualified or in a qualified sense : for change to a lesser degree of a quality will be called change to the contrary of that quality, and change to a greater degree of a quality will be regarded as change from the contrary of that quality to the quality itself. it makes no difference whether the change be qualified or unqualified, except that in the former case the contraries will have to be contrary to one another only in a qualified sense : and a thing ' s possessing a quality in a greater or in a


makes no difference whether the change be qualified or unqualified, except that in the former case the contraries will have to be contrary to one another only in a qualified sense : and a thing ' s possessing a quality in a greater or in a lesser degree means the presence or absence in it of more or less of the opposite quality. it is now clear, then, that there are only these three kinds of motion. the term ' immovable ' we apply in the first place to that which is absolutely incapable of being moved ( just as we correspondingly apply the term invisible to sound ) ; in the second place to that which is moved with difficulty after a long time or whose movement is slow at the start - in fact, what we describe as hard to move ; and in the third place to that which is naturally designed for and capable of motion, but is not in motion when, where, and as it naturally would be so. this last is the only kind of immovable thing of which i use the term ' being at rest ' : for rest is contrary to motion, so that rest will be negation of motion in that which is capable of admitting motion. the foregoing remarks are sufficient to explain the essential nature of motion and rest, the number of kinds of change, and the different varieties of motion. let us now proceed to define the terms ' together ' and ' apart ', ' in contact ', ' between ', ' in succession ', ' contiguous ', and ' continuous ', and to show in what circumstances each of these terms is naturally applicable. things are said to be together in place when they are in one place ( in the strictest sense of the word ' place ' ) and to be apart when they are in different places. things are said to be in contact when their extremities are together. that which a changing thing, if it changes continuously in a natural manner, naturally reaches before it reaches that to which it changes last, is between. thus ' between ' implies the presence of at least three things : for in a process of change it is the contrary that is ' last ' : and a thing is moved continuously if it leaves no gap or only the smallest possible gap in the material - not in the time ( for a gap in the time does not prevent things having a ' between ', while, on the other hand, there is nothing to prevent the highest note sounding immediately after the lowest ) but in the material in which the motion takes place. this is manifestly true not


things prior in definition, e. g. numbers, while contact is not. and if there is continuity there is necessarily contact, but if there is contact, that alone does not imply continuity : for the extremities of things may be ' together ' without necessarily being one : but they cannot be one without being necessarily together. so natural junction is last in coming to be : for the extremities must necessarily come into contact if they are to be naturally joined : but things that are in contact are not all naturally joined, while there is no contact clearly there is no natural junction either. hence, if as some say ' point ' and ' unit ' have an independent existence of their own, it is impossible for the two to be identical : for points can touch while units can only be in succession. moreover, there can always be something between points ( for all lines are intermediate between points ), whereas it is not necessary that there should possibly be anything between units : for there can be nothing between the numbers one and two. we have now defined what is meant by ' together ' and ' apart ', ' contact ', ' between ' and ' in succession ', ' contiguous ' and ' continuous ' : and we have shown in what circumstances each of these terms is applicable. there are many senses in which motion is said to be ' one ' : for we use the term ' one ' in many senses. motion is one generically according to the different categories to which it may be assigned : thus any locomotion is one generically with any other locomotion, whereas alteration is different generically from locomotion. motion is one specifically when besides being one generically it also takes place in a species incapable of subdivision : e. g. colour has specific differences : therefore blackening and whitening differ specifically ; but at all events every whitening will be specifically the same with every other whitening and every blackening with every other blackening. but white is not further subdivided by specific differences : hence any whitening is specifically one with any other whitening. where it happens that the genus is at the same time a species, it is clear that the motion will then in a sense be one specifically though not in an unqualified sense : learning is an example of this, knowledge being on the one hand a species of apprehension and on the other hand a genus including the various knowledges. a difficulty, however, may be raised as to whether a motion is specifically one when the same thing changes from the same to


there might be a case of two men being restored to health at the same time in the same way, e. g. from inflammation of the eye, yet this motion is not really one, but only specifically one ). suppose, however, that socrates undergoes an alteration specifically the same but at one time and again at another : in this case if it is possible for that which ceased to be again to come into being and remain numerically the same, then this motion too will be one : otherwise it will be the same but not one. and akin to this difficulty there is another ; viz. is health one? and generally are the states and affections in bodies severally one in essence although ( as is clear ) the things that contain them are obviously in motion and in flux? thus if a person ' s health at daybreak and at the present moment is one and the same, why should not this health be numerically one with that which he recovers after an interval? the same argument applies in each case. there is, however, we may answer, this difference : that if the states are two then it follows simply from this fact that the activities must also in point of number be two ( for only that which is numerically one can give rise to an activity that is numerically one ), but if the state is one, this is not in itself enough to make us regard the activity also as one : for when a man ceases walking, the walking no longer is, but it will again be if he begins to walk again. but, be this as it may, if in the above instance the health is one and the same, then it must be possible for that which is one and the same to come to be and to cease to be many times. however, these difficulties lie outside our present inquiry. since every motion is continuous, a motion that is one in an unqualified sense must ( since every motion is divisible ) be continuous, and a continuous motion must be one. there will not be continuity between any motion and any other indiscriminately any more than there is between any two things chosen at random in any other sphere : there can be continuity only when the extremities of the two things are one. now some things have no extremities at all : and the extremities of others differ specifically although we give them the same name of ' end ' : how should e. g. the ' end ' of a line and the ' end ' of


will not be simultaneous : for the time may be divided at many points. if, therefore, the present cannot possibly have these characteristics, it follows that it must be the same present that belongs to each of the two times. but if this is so it is evident that the present is also indivisible : for if it is divisible it will be involved in the same implications as before. it is clear, then, from what has been said that time contains something indivisible, and this is what we call a present. we will now show that nothing can be in motion in a present. for if this is possible, there can be both quicker and slower motion in the present. suppose then that in the present n the quicker has traversed the distance ab. that being so, the slower will in the same present traverse a distance less than ab, say ag. but since the slower will have occupied the whole present in traversing ag, the quicker will occupy less than this in traversing it. thus we shall have a division of the present, whereas we found it to be indivisible. it is impossible, therefore, for anything to be in motion in a present. nor can anything be at rest in a present : for, as we were saying, only can be at rest which is naturally designed to be in motion but is not in motion when, where, or as it would naturally be so : since, therefore, nothing is naturally designed to be in motion in a present, it is clear that nothing can be at rest in a present either. moreover, inasmuch as it is the same present that belongs to both the times, and it is possible for a thing to be in motion throughout one time and to be at rest throughout the other, and that which is in motion or at rest for the whole of a time will be in motion or at rest as the case may be in any part of it in which it is naturally designed to be in motion or at rest : this being so, the assumption that there can be motion or rest in a present will carry with it the implication that the same thing can at the same time be at rest and in motion : for both the times have the same extremity, viz. the present. again, when we say that a thing is at rest, we imply that its condition in whole and in part is at the time of speaking uniform with what it was previously : but the present contains no ' previously ' : consequently, there can be no rest in


of the motion must all be susceptible of the same divisions ( though spheres of motion are not all divisible in a like manner : thus quantity is essentially, quality accidentally divisible ). for suppose that a is the time occupied by the motion b. then if all the time has been occupied by the whole motion, it will take less of the motion to occupy half the time, less again to occupy a further subdivision of the time, and so on to infinity. again, the time will be divisible similarly to the motion : for if the whole motion occupies all the time half the motion will occupy half the time, and less of the motion again will occupy less of the time. in the same way the being - in - motion will also be divisible. for let g be the whole being - in - motion. then the being - in - motion that corresponds to half the motion will be less than the whole being - in - motion, that which corresponds to a quarter of the motion will be less again, and so on to infinity. moreover by setting out successively the being - in - motion corresponding to each of the two motions dg ( say ) and ge, we may argue that the whole being - in - motion will correspond to the whole motion ( for if it were some other being - in - motion that corresponded to the whole motion, there would be more than one being - in motion corresponding to the same motion ), the argument being the same as that whereby we showed that the motion of a thing is divisible into the motions of the parts of the thing : for if we take separately the being - in motion corresponding to each of the two motions, we shall see that the whole being - in motion is continuous. the same reasoning will show the divisibility of the length, and in fact of everything that forms a sphere of change ( though some of these are only accidentally divisible because that which changes is so ) : for the division of one term will involve the division of all. so, too, in the matter of their being finite or infinite, they will all alike be either the one or the other. and we now see that in most cases the fact that all the terms are divisible or infinite is a direct consequence of the fact that the thing that changes is divisible or infinite : for the attributes ' divisible ' and ' infinite ' belong in the first instance to the thing that changes. that divisibility does so we have already shown : that infinity does


motion. and so it is impossible for one and the same thing to be infinitely in process of becoming or of perishing. the reasoning he will prove that in a finite time there cannot be an infinite extent of motion or of coming to rest, whether the motion is regular or irregular. for if we take a part which shall be a measure of the whole time, in this part a certain fraction, not the whole, of the magnitude will be traversed, because we assume that the traversing of the whole occupies all the time. again, in another equal part of the time another part of the magnitude will be traversed : and similarly in each part of the time that we take, whether equal or unequal to the part originally taken. it makes no difference whether the parts are equal or not, if only each is finite : for it is clear that while the time is exhausted by the subtraction of its parts, the infinite magnitude will not be thus exhausted, since the process of subtraction is finite both in respect of the quantity subtracted and of the number of times a subtraction is made. consequently the infinite magnitude will not be traversed in finite time : and it makes no difference whether the magnitude is infinite in only one direction or in both : for the same reasoning will hold good. this having been proved, it is evident that neither can a finite magnitude traverse an infinite magnitude in a finite time, the reason being the same as that given above : in part of the time it will traverse a finite magnitude and in each several part likewise, so that in the whole time it will traverse a finite magnitude. and since a finite magnitude will not traverse an infinite in a finite time, it is clear that neither will an infinite traverse a finite in a finite time. for if the infinite could traverse the finite, the finite could traverse the infinite ; for it makes no difference which of the two is the thing in motion ; either case involves the traversing of the infinite by the finite. for when the infinite magnitude a is in motion a part of it, say gd, will occupy the finite and then another, and then another, and so on to infinity. thus the two results will coincide : the infinite will have completed a motion over the finite and the finite will have traversed the infinite : for it would seem to be impossible for the motion of the infinite over the finite to occur in any way other than by the finite traversing the infinite either by locomotion over it or by measuring it. therefore, since this is impossible, the


the aforesaid fallacious assumption. nor in reference to contradictory change shall we find anything unanswerable in the argument that if a thing is changing from not - white, say, to white, and is in neither condition, then it will be neither white nor not - white : for the fact that it is not wholly in either condition will not preclude us from calling it white or not - white. we call a thing white or not - white not necessarily because it is be one or the other, but cause most of its parts or the most essential parts of it are so : not being in a certain condition is different from not being wholly in that condition. so, too, in the case of being and not - being and all other conditions which stand in a contradictory relation : while the changing thing must of necessity be in one of the two opposites, it is never wholly in either. again, in the case of circles and spheres and everything whose motion is confined within the space that it occupies, it is not true to say the motion can be nothing but rest, on the ground that such things in motion, themselves and their parts, will occupy the same position for a period of time, and that therefore they will be at once at rest and in motion. for in the first place the parts do not occupy the same position for any period of time : and in the second place the whole also is always changing to a different position : for if we take the orbit as described from a point a on a circumference, it will not be the same as the orbit as described from b or g or any other point on the same circumference except in an accidental sense, the sense that is to say in which a musical man is the same as a man. thus one orbit is always changing into another, and the thing will never be at rest. and it is the same with the sphere and everything else whose motion is confined within the space that it occupies. our next point is that that which is without parts cannot be in motion except accidentally : i. e. it can be in motion only in so far as the body or the magnitude is in motion and the partless is in motion by inclusion therein, just as that which is in a boat may be in motion in consequence of the locomotion of the boat, or a part may be in motion in virtue of the motion of the whole. ( it must be remembered, however, that by ' that which is without parts ' i mean that


be in motion in consequence of the locomotion of the boat, or a part may be in motion in virtue of the motion of the whole. ( it must be remembered, however, that by ' that which is without parts ' i mean that which is quantitatively indivisible ( and that the case of the motion of a part is not exactly parallel ) : for parts have motions belonging essentially and severally to themselves distinct from the motion of the whole. the distinction may be seen most clearly in the case of a revolving sphere, in which the velocities of the parts near the centre and of those on the surface are different from one another and from that of the whole ; this implies that there is not one motion but many ). as we have said, then, that which is without parts can be in motion in the sense in which a man sitting in a boat is in motion when the boat is travelling, but it cannot be in motion of itself. for suppose that it is changing from ab to bg - either from one magnitude to another, or from one form to another, or from some state to its contradictory - and let d be the primary time in which it undergoes the change. then in the time in which it is changing it must be either in ab or in bg or partly in one and partly in the other : for this, as we saw, is true of everything that is changing. now it cannot be partly in each of the two : for then it would be divisible into parts. nor again can it be in bg : for then it will have completed the change, whereas the assumption is that the change is in process. it remains, then, that in the time in which it is changing, it is in ab. that being so, it will be at rest : for, as we saw, to be in the same condition for a period of time is to be at rest. so it is not possible for that which has no parts to be in motion or to change in any way : for only one condition could have made it possible for it to have motion, viz. that time should be composed of moments, in which case at any moment it would have completed a motion or a change, so that it would never be in motion, but would always have been in motion. but this we have already shown above to be impossible : time is not composed of moments, just as a line is not composed of points, and motion is not composed of starts : for this theory


in motion, but would always have been in motion. but this we have already shown above to be impossible : time is not composed of moments, just as a line is not composed of points, and motion is not composed of starts : for this theory simply makes motion consist of indivisibles in exactly the same way as time is made to consist of moments or a length of points. again, it may be shown in the following way that there can be no motion of a point or of any other indivisible. that which is in motion can never traverse a space greater than itself without first traversing a space equal to or less than itself. that being so, it is evident that the point also must first traverse a space equal to or less than itself. but since it is indivisible, there can be no space less than itself for it to traverse first : so it will have to traverse a distance equal to itself. thus the line will be composed of points, for the point, as it continually traverses a distance equal to itself, will be a measure of the whole line. but since this is impossible, it is likewise impossible for the indivisible to be in motion. again, since motion is always in a period of time and never in a moment, and all time is divisible, for everything that is in motion there must be a time less than that in which it traverses a distance as great as itself. for that in which it is in motion will be a time, because all motion is in a period of time ; and all time has been shown above to be divisible. therefore, if a point is in motion, there must be a time less than that in which it has itself traversed any distance. but this is impossible, for in less time it must traverse less distance, and thus the indivisible will be divisible into something less than itself, just as the time is so divisible : the fact being that the only condition under which that which is without parts and indivisible could be in motion would have been the possibility of the infinitely small being in motion in a moment : for in the two questions - that of motion in a moment and that of motion of something indivisible - the same principle is involved. our next point is that no process of change is infinite : for every change, whether between contradictories or between contraries, is a change from something to something. thus in contradictory changes the positive or the negative,


the same principle is involved. our next point is that no process of change is infinite : for every change, whether between contradictories or between contraries, is a change from something to something. thus in contradictory changes the positive or the negative, as the case may be, is the limit, e. g. being is the limit of coming to be and not - being is the limit of ceasing to be : and in contrary changes the particular contraries are the limits, since these are the extreme points of any such process of change, and consequently of every process of alteration : for alteration is always dependent upon some contraries. similarly contraries are the extreme points of processes of increase and decrease : the limit of increase is to be found in the complete magnitude proper to the peculiar nature of the thing that is increasing, while the limit of decrease is the complete loss of such magnitude. locomotion, it is true, we cannot show to be finite in this way, since it is not always between contraries. but since that which cannot be cut ( in the sense that it is inconceivable that it should be cut, the term ' cannot ' being used in several senses ) - since it is inconceivable that that which in this sense cannot be cut should be in process of being cut, and generally that that which cannot come to be should be in process of coming to be, it follows that it is inconceivable that that which cannot complete a change should be in process of changing to that to which it cannot complete a change. if, then, it is to be assumed that that which is in locomotion is in process of changing, it must be capable of completing the change. consequently its motion is not infinite, and it will not be in locomotion over an infinite distance, for it cannot traverse such a distance. it is evident, then, that a process of change cannot be infinite in the sense that it is not defined by limits. but it remains to be considered whether it is possible in the sense that one and the same process of change may be infinite in respect of the time which it occupies. if it is not one process, it would seem that there is nothing to prevent its being infinite in this sense ; e. g. if a process of locomotion be succeeded by a process of alteration and that by a process of increase and that again by a process of coming to be : in this way there may be motion for ever so far as the time


, e. g. from white to black or from good to bad, which is not of a kind specifically distinct : it is numerically the same if it proceeds from something numerically one to something numerically one in the same period of time, e. g. from a particular white to a particular black, or from a particular place to a particular place, in a particular period of time : for if the period of time were not one and the same, the motion would no longer be numerically one though it would still be specifically one. we have dealt with this question above. now let us further take the time in which a has completed its motion, and let it be represented by k. then since the motion of a is finite the time will also be finite. but since the movents and the things moved are infinite, the motion ezho, i. e. the motion that is composed of all the individual motions, must be infinite. for the motions of a, b, and the others may be equal, or the motions of the others may be greater : but assuming what is conceivable, we find that whether they are equal or some are greater, in both cases the whole motion is infinite. and since the motion of a and that of each of the others are simultaneous, the whole motion must occupy the same time as the motion of a : but the time occupied by the motion of a is finite : consequently the motion will be infinite in a finite time, which is impossible. it might be thought that what we set out to prove has thus been shown, but our argument so far does not prove it, because it does not yet prove that anything impossible results from the contrary supposition : for in a finite time there may be an infinite motion, though not of one thing, but of many : and in the case that we are considering this is so : for each thing accomplishes its own motion, and there is no impossibility in many things being in motion simultaneously. but if ( as we see to be universally the case ) that which primarily is moved locally and corporeally must be either in contact with or continuous with that which moves it, the things moved and the movents must be continuous or in contact with one another, so that together they all form a single unity : whether this unity is finite or infinite makes no difference to our present argument ; for in any case since the things in motion are infinite in number the whole motion will be infinite, if, as is theoretically possible, each


they all form a single unity : whether this unity is finite or infinite makes no difference to our present argument ; for in any case since the things in motion are infinite in number the whole motion will be infinite, if, as is theoretically possible, each motion is either equal to or greater than that which follows it in the series : for we shall take as actual that which is theoretically possible. if, then, a, b, g, d form an infinite magnitude that passes through the motion ezho in the finite time k, this involves the conclusion that an infinite motion is passed through in a finite time : and whether the magnitude in question is finite or infinite this is in either case impossible. therefore the series must come to an end, and there must be a first movent and a first moved : for the fact that this impossibility results only from the assumption of a particular case is immaterial, since the case assumed is theoretically possible, and the assumption of a theoretically possible case ought not to give rise to any impossible result. that which is the first movement of a thing - in the sense that it supplies not ' that for the sake of which ' but the source of the motion - is always together with that which is moved by it by ' together ' i mean that there is nothing intermediate between them ). this is universally true wherever one thing is moved by another. and since there are three kinds of motion, local, qualitative, and quantitative, there must also be three kinds of movent, that which causes locomotion, that which causes alteration, and that which causes increase or decrease. let us begin with locomotion, for this is the primary motion. everything that is in locomotion is moved either by itself or by something else. in the case of things that are moved by themselves it is evident that the moved and the movent are together : for they contain within themselves their first movent, so that there is nothing in between. the motion of things that are moved by something else must proceed in one of four ways : for there are four kinds of locomotion caused by something other than that which is in motion, viz. pulling, pushing, carrying, and twirling. all forms of locomotion are reducible to these. thus pushing on is a form of pushing in which that which is causing motion away from itself follows up that which it pushes and continues to push it : pushing off occurs when the movent does not follow up the thing that it has moved : throwing


with that which suffers increase and that which suffers decrease respectively : and if two things are continuous with one another there can be nothing intermediate between them. it is evident, therefore, that between the extremities of the moved and the movent that are respectively first and last in reference to the moved there is nothing intermediate. everything, we say, that undergoes alteration is altered by sensible causes, and there is alteration only in things that are said to be essentially affected by sensible things. the truth of this is to be seen from the following considerations. of all other things it would be most natural to suppose that there is alteration in figures and shapes, and in acquired states and in the processes of acquiring and losing these : but as a matter of fact in neither of these two classes of things is there alteration. in the first place, when a particular formation of a thing is completed, we do not call it by the name of its material : e. g. we do not call the statue ' bronze ' or the pyramid ' wax ' or the bed ' wood ', but we use a derived expression and call them ' of bronze ', ' waxen ', and ' wooden ' respectively. but when a thing has been affected and altered in any way we still call it by the original name : thus we speak of the bronze or the wax being dry or fluid or hard or hot. and not only so : we also speak of the particular fluid or hot substance as being bronze, giving the material the same name as that which we use to describe the affection. since, therefore, having regard to the figure or shape of a thing we no longer call that which has become of a certain figure by the name of the material that exhibits the figure, whereas having regard to a thing ' s affections or alterations we still call it by the name of its material, it is evident that becomings of the former kind cannot be alterations. moreover it would seem absurd even to speak in this way, to speak, that is to say, of a man or house or anything else that has come into existence as having been altered. though it may be true that every such becoming is necessarily the result of something ' s being altered, the result, e. g. of the material ' s being condensed or rarefied or heated or cooled, nevertheless it is not the things that are coming into existence that are altered, and their becoming is not an alteration. again, acquired states, whether of the body or of the soul, are not alterations. for some are


. that there may be a straight line equal to a circle. but these are not commensurable : and so the corresponding motions are not commensurable either. but may we say that things are always commensurable if the same terms are applied to them without equivocation? e. g. a pen, a wine, and the highest note in a scale are not commensurable : we cannot say whether any one of them is sharper than any other : and why is this? they are incommensurable because it is only equivocally that the same term ' sharp ' is applied to them : whereas the highest note in a scale is commensurable with the leading - note, because the term ' sharp ' has the same meaning as applied to both. can it be, then, that the term ' quick ' has not the same meaning as applied to straight motion and to circular motion respectively? if so, far less will it have the same meaning as applied to alteration and to locomotion. or shall we in the first place deny that things are always commensurable if the same terms are applied to them without equivocation? for the term ' much ' has the same meaning whether applied to water or to air, yet water and air are not commensurable in respect of it : or, if this illustration is not considered satisfactory, ' double ' at any rate would seem to have the same meaning as applied to each ( denoting in each case the proportion of two to one ), yet water and air are not commensurable in respect of it. but here again may we not take up the same position and say that the term ' much ' is equivocal? in fact there are some terms of which even the definitions are equivocal ; e. g. if ' much ' were defined as ' so much and more ', ' so much ' would mean something different in different cases : ' equal ' is similarly equivocal ; and ' one ' again is perhaps inevitably an equivocal term ; and if ' one ' is equivocal, so is ' two '. otherwise why is it that some things are commensurable while others are not, if the nature of the attribute in the two cases is really one and the same? can it be that the incommensurability of two things in respect of any attribute is due to a difference in that which is primarily capable of carrying the


a definite thing that undergoes alteration, and a certain amount, or rather degree, of alteration is completed in a certain amount of time : thus in twice as much time twice as much alteration will be completed and conversely twice as much alteration will occupy twice as much time : and the alteration of half of its object will occupy half as much time and in half as much time half of the object will be altered : or again, in the same amount of time it will be altered twice as much. on the other hand if that which causes alteration or increase causes a certain amount of increase or alteration respectively in a certain amount of time, it does not necessarily follow that half the force will occupy twice the time in altering or increasing the object, or that in twice the time the alteration or increase will be completed by it : it may happen that there will be no alteration or increase at all, the case being the same as with the weight. it remains to consider the following question. was there ever a becoming of motion before which it had no being, and is it perishing again so as to leave nothing in motion? or are we to say that it never had any becoming and is not perishing, but always was and always will be? is it in fact an immortal never - failing property of things that are, a sort of life as it were to all naturally constituted things? now the existence of motion is asserted by all who have anything to say about nature, because they all concern themselves with the construction of the world and study the question of becoming and perishing, which processes could not come about without the existence of motion. but those who say that there is an infinite number of worlds, some of which are in process of becoming while others are in process of perishing, assert that there is always motion ( for these processes of becoming and perishing of the worlds necessarily involve motion ), whereas those who hold that there is only one world, whether everlasting or not, make corresponding assumptions in regard to motion. if then it is possible that at any time nothing should be in motion, this must come about in one of two ways : either in the manner described by anaxagoras, who says that all things were together and at rest for an infinite period of time, and that then mind introduced motion and separated them ; or in the manner described by empedocles, according to whom the universe is alternately in motion and at rest - in motion, when love is making the one out of many, or strife is making many out of one


motion and separated them ; or in the manner described by empedocles, according to whom the universe is alternately in motion and at rest - in motion, when love is making the one out of many, or strife is making many out of one, and at rest in the intermediate periods of time - his account being as follows : ' since one hath learned to spring from manifold, and one disjoined makes manifold arise, thus they become, nor stable is their life : but since their motion must alternate be, thus have they ever rest upon their round ' : for we must suppose that he means by this that they alternate from the one motion to the other. we must consider, then, how this matter stands, for the discovery of the truth about it is of importance, not only for the study of nature, but also for the investigation of the first principle. let us take our start from what we have already laid down in our course on physics. motion, we say, is the fulfilment of the movable in so far as it is movable. each kind of motion, therefore, necessarily involves the presence of the things that are capable of that motion. in fact, even apart from the definition of motion, every one would admit that in each kind of motion it is that which is capable of that motion that is in motion : thus it is that which is capable of alteration that is altered, and that which is capable of local change that is in locomotion : and so there must be something capable of being burned before there can be a process of being burned, and something capable of burning before there can be a process of burning. moreover, these things also must either have a beginning before which they had no being, or they must be eternal. now if there was a becoming of every movable thing, it follows that before the motion in question another change or motion must have taken place in which that which was capable of being moved or of causing motion had its becoming. to suppose, on the other hand, that these things were in being throughout all previous time without there being any motion appears unreasonable on a moment ' s thought, and still more unreasonable, we shall find, on further consideration. for if we are to say that, while there are on the one hand things that are movable, and on the other hand things that are motive, there is a time when there is a first movent and a first moved, and another time when there is no such thing but only something that is at


. if, then, view which we are criticizing involves these impossible consequences, it is clear that motion is eternal and cannot have existed at one time and not at another : in fact such a view can hardly be described as anythling else than fantastic. and much the same may be said of the view that such is the ordinance of nature and that this must be regarded as a principle, as would seem to be the view of empedocles when he says that the constitution of the world is of necessity such that love and strife alternately predominate and cause motion, while in the intermediate period of time there is a state of rest. probably also those who like like anaxagoras, assert a single principle ( of motion ) would hold this view. but that which is produced or directed by nature can never be anything disorderly : for nature is everywhere the cause of order. moreover, there is no ratio in the relation of the infinite to the infinite, whereas order always means ratio. but if we say that there is first a state of rest for an infinite time, and then motion is started at some moment, and that the fact that it is this rather than a previous moment is of no importance, and involves no order, then we can no longer say that it is nature ' s work : for if anything is of a certain character naturally, it either is so invariably and is not sometimes of this and sometimes of another character ( e. g. fire, which travels upwards naturally, does not sometimes do so and sometimes not ) or there is a ratio in the variation. it would be better, therefore, to say with empedocles and any one else who may have maintained such a theory as his that the universe is alternately at rest and in motion : for in a system of this kind we have at once a certain order. but even here the holder of the theory ought not only to assert the fact : he ought to explain the cause of it : i. e. he should not make any mere assumption or lay down any gratuitous axiom, but should employ either inductive or demonstrative reasoning. the love and strife postulated by empedocles are not in themselves causes of the fact in question, nor is it of the essence of either that it should be so, the essential function of the former being to unite, of the latter to separate. if he is to go on to explain this alternate predominance, he should adduce cases where such a state of things exists,


without being at one time present, and at another absent. nevertheless, how this can be so remains matter for inquiry ; how it comes about, i mean, that the same motive force at one time causes a thing to be in motion, and at another does not do so : for the difficulty raised by our objector really amounts to this - why is it that some things are not always at rest, and the rest always in motion? the third objection may be thought to present more difficulty than the others, namely, that which alleges that motion arises in things in which it did not exist before, and adduces in proof the case of animate things : thus an animal is first at rest and afterwards walks, not having been set in motion apparently by anything from without. this, however, is false : for we observe that there is always some part of the animal ' s organism in motion, and the cause of the motion of this part is not the animal itself, but, it may be, its environment. moreover, we say that the animal itself originates not all of its motions but its locomotion. so it may well be the case - or rather we may perhaps say that it must necessarily be the case - that many motions are produced in the body by its environment, and some of these set in motion the intellect or the appetite, and this again then sets the whole animal in motion : this is what happens when animals are asleep : though there is then no perceptive motion in them, there is some motion that causes them to wake up again. but we will leave this point also to be elucidated at a later stage in our discussion. our enquiry will resolve itself at the outset into a consideration of the above - mentioned problem - what can be the reason why some things in the world at one time are in motion and at another are at rest again? now one of three things must be true : either all things are always at rest, or all things are always in motion, or some things are in motion and others at rest : and in this last case again either the things that are in motion are always in motion and the things that are at rest are always at rest, or they are all constituted so as to be capable alike of motion and of rest ; or there is yet a third possibility remaining - it may be that some things in the world are always motionless, others always in motion, while others again admit of both conditions. this last is the account of the matter that we must give


of rest ; or there is yet a third possibility remaining - it may be that some things in the world are always motionless, others always in motion, while others again admit of both conditions. this last is the account of the matter that we must give : for herein lies the solution of all the difficulties raised and the conclusion of the investigation upon which we are engaged. to maintain that all things are at rest, and to disregard sense - perception in an attempt to show the theory to be reasonable, would be an instance of intellectual weakness : it would call in question a whole system, not a particular detail : moreover, it would be an attack not only on the physicist but on almost all sciences and all received opinions, since motion plays a part in all of them. further, just as in arguments about mathematics objections that involve first principles do not affect the mathematician - and the other sciences are in similar case - so, too, objections involving the point that we have just raised do not affect the physicist : for it is a fundamental assumption with him that motion is ultimately referable to nature herself. the assertion that all things are in motion we may fairly regard as equally false, though it is less subversive of physical science : for though in our course on physics it was laid down that rest no less than motion is ultimately referable to nature herself, nevertheless motion is the characteristic fact of nature : moreover, the view is actually held by some that not merely some things but all things in the world are in motion and always in motion, though we cannot apprehend the fact by sense - perception. although the supporters of this theory do not state clearly what kind of motion they mean, or whether they mean all kinds, it is no hard matter to reply to them : thus we may point out that there cannot be a continuous process either of increase or of decrease : that which comes between the two has to be included. the theory resembles that about the stone being worn away by the drop of water or split by plants growing out of it : if so much has been extruded or removed by the drop, it does not follow that half the amount has previously been extruded or removed in half the time : the case of the hauled ship is exactly comparable : here we have so many drops setting so much in motion, but a part of them will not set as much in motion in any period of time. the amount removed is, it is true, divisible into a number of parts, but no one of these was set in motion separately


much in motion, but a part of them will not set as much in motion in any period of time. the amount removed is, it is true, divisible into a number of parts, but no one of these was set in motion separately : they were all set in motion together. it is evident, then, that from the fact that the decrease is divisible into an infinite number of parts it does not follow that some part must always be passing away : it all passes away at a particular moment. similarly, too, in the case of any alteration whatever if that which suffers alteration is infinitely divisible it does not follow from this that the same is true of the alteration itself, which often occurs all at once, as in freezing. again, when any one has fallen ill, there must follow a period of time in which his restoration to health is in the future : the process of change cannot take place in an instant : yet the change cannot be a change to anything else but health. the assertion. therefore, that alteration is continuous is an extravagant calling into question of the obvious : for alteration is a change from one contrary to another. moreover, we notice that a stone becomes neither harder nor softer. again, in the matter of locomotion, it would be a strange thing if a stone could be falling or resting on the ground without our being able to perceive the fact. further, it is a law of nature that earth and all other bodies should remain in their proper places and be moved from them only by violence : from the fact then that some of them are in their proper places it follows that in respect of place also all things cannot be in motion. these and other similar arguments, then, should convince us that it is impossible either that all things are always in motion or that all things are always at rest. nor again can it be that some things are always at rest, others always in motion, and nothing sometimes at rest and sometimes in motion. this theory must be pronounced impossible on the same grounds as those previously mentioned : viz. that we see the above - mentioned changes occurring in the case of the same things. we may further point out that the defender of this position is fighting against the obvious, for on this theory there can be no such thing as increase : nor can there be any such thing as compulsory motion, if it is impossible that a thing can be at rest before being set in motion unnaturally. this theory, then, does away with becoming and perishing. moreover, motion, it would


science potentially in a sense, though not in the same sense as he knew it potentially before he learnt it. and when he is in this condition, if something does not prevent him, he actively exercises his knowledge : otherwise he would be in the contradictory state of not knowing. in regard to natural bodies also the case is similar. thus what is cold is potentially hot : then a change takes place and it is fire, and it burns, unless something prevents and hinders it. so, too, with heavy and light : light is generated from heavy, e. g. air from water ( for water is the first thing that is potentially light ), and air is actually light, and will at once realize its proper activity as such unless something prevents it. the activity of lightness consists in the light thing being in a certain situation, namely high up : when it is in the contrary situation, it is being prevented from rising. the case is similar also in regard to quantity and quality. but, be it noted, this is the question we are trying to answer - how can we account for the motion of light things and heavy things to their proper situations? the reason for it is that they have a natural tendency respectively towards a certain position : and this constitutes the essence of lightness and heaviness, the former being determined by an upward, the latter by a downward, tendency. as we have said, a thing may be potentially light or heavy in more senses than one. thus not only when a thing is water is it in a sense potentially light, but when it has become air it may be still potentially light : for it may be that through some hindrance it does not occupy an upper position, whereas, if what hinders it is removed, it realizes its activity and continues to rise higher. the process whereby what is of a certain quality changes to a condition of active existence is similar : thus the exercise of knowledge follows at once upon the possession of it unless something prevents it. so, too, what is of a certain quantity extends itself over a certain space unless something prevents it. the thing in a sense is and in a sense is not moved by one who moves what is obstructing and preventing its motion ( e. g. one who pulls away a pillar from under a roof or one who removes a stone from a wineskin in the water is the accidental cause of motion ) : and in the same way the real cause of the motion of a ball rebounding from a wall is not the wall but the thrower