Datasets:
OvercastLab
/
OmniCult

Dataset card Data Studio Files
xet
 Community
1
text
stringlengths
2.53k
7.78k
# Part I: This World > "Be patient, for the world is broad and wide." ## 1. Of the Nature of Flatland I call our world Flatland, not because we call it so, but to make its nature clearer to you, my happy readers, who are privileged to live in Space. [CONCEPT: POETIC_FORM] Imagine a vast sheet of paper on which strai...
[CONCEPT: DIMENSIONAL_ASCENT] Well, that is just what we see when one of our triangular or other acquaintances comes toward us in Flatland. As there is neither sun with us, nor any light of such a kind as to make shadows, we have none of the helps to the sight that you have in Spaceland. If our friend comes closer to u...
Hence, after fruitless attempts to suppress such investigations indirectly by making them liable to a heavy tax, the Legislature, in comparatively recent times, absolutely prohibited them. I—alas, I alone in Flatland—know now only too well the true solution of this mysterious problem; but my knowledge cannot be made in...
Our middle class consists of Equilateral or Equal-Sided Triangles. Our professional men and gentlemen are Squares (to which class I myself belong) and Five-Sided Figures or Pentagons. Next above these come the nobility, of whom there are several degrees, beginning at Six-Sided Figures, or Hexagons, and from thence risi...
[CONCEPT: HISTORICAL_NARRATIVE] The occasional emergence of an Equilateral from the ranks of his serf-born ancestors is welcomed, not only by the poor serfs themselves, as a gleam of light and hope shed upon the monotonous squalor of their existence, but also by the aristocracy at large; for all the higher classes are ...
But here, perhaps, some of my younger readers may ask how a woman in Flatland can make herself invisible. This ought, I think, to be apparent without any explanation. However, a few words will make it clear to the most unreflecting. [CONCEPT: HISTORICAL_NARRATIVE] Place a needle on a table. Then, with your eye on the l...
For whenever the temper of the women is thus exasperated by confinement at home or hampering regulations abroad, they are apt to vent their spleen upon their husbands and children; and in the less temperate climates the whole male population of a village has been sometimes destroyed in one or two hours of simultaneous ...
On the whole we get on pretty smoothly in our domestic relations, except in the lower strata of the military classes. There the want of tact and discretion on the part of the husbands produces at times indescribable disasters. Relying too much on the offensive weapons of their acute angles instead of the defensive orga...
A male of the lowest type of the Isosceles may look forward to some improvement of his angle, and to the ultimate elevation of the whole of his degraded caste; but no woman can entertain such hopes for her sex. “Once a woman, always a woman” is a decree of Nature; and the very laws of Evolution seem suspended in her di...
So-and-so”; although it is assumed, of course, that the “feeling” is to be reciprocal. Among our still more modern and dashing young gentlemen—who are extremely averse to superfluous effort and supremely indifferent to the purity of their native language—the formula is still further curtailed by the use of “to feel” in...
According to his account, my unfortunate ancestor, being afflicted with rheumatism, and in the act of being felt by a Polygon, by one sudden start accidentally transfixed the great man through the diagonal; and thereby, partly in consequence of his long imprisonment and degradation, and partly because of the moral shoc...
Nor must we forget to add, in enumerating the advantages of the more expensive system, that it tends, though slightly yet perceptibly, to the diminution of the redundant Isosceles population—an object which every statesman in Flatland constantly keeps in view. On the whole therefore—although I am not ignorant that, in ...
[CONCEPT: DIMENSIONAL_ASCENT] It will be obvious, to every child in Spaceland who has touched the threshold of Geometrical Studies, that, if I can bring my eye so that its glance may bisect an angle (A) of the approaching stranger, my view will lie as it were evenly between his two sides that are next to me (viz. CA an...
It is only a few of the scions of our noblest and wealthiest houses, who are able to give the time and money necessary for the thorough prosecution of this noble and valuable Art. Even to me, a mathematician of no mean standing, and the grandfather of two most hopeful and perfectly regular Hexagons, to find myself in t...
The condition of the unsuccessful minority is truly pitiable. Rejected from the higher class, they are also despised by the lower. They have neither the matured and systematically trained powers of the Polygonal Bachelors and Masters of Arts, nor yet the native precocity and mercurial versatility of the youthful trades...
Am I going too fast to carry my readers with me to these obvious conclusions? Surely a moment’s reflection, and a single instance from common life, must convince everyone that our whole social system is based upon Regularity, or Equality of Angles. You meet, for example, two or three tradesmen in the street, whom you r...
Doubtless, the life of an Irregular is hard; but the interests of the greater number require that it shall be hard. If a man with a triangular front and a polygonal back were allowed to exist and to propagate a still more Irregular posterity, what would become of the arts of life? Are the houses and doors and churches ...
It was not always thus. Colour, if tradition speaks the truth, once for the space of half a dozen centuries or more, threw a transient splendour over the lives of our ancestors in the remotest ages. Some private individual—a Pentagon whose name is variously reported—having casually discovered the constituents of the si...
To live was then in itself a delight, because living implied seeing. Even at a small party, the company was a pleasure to behold; the richly varied hues of the assembly in a church or theatre are said to have more than once proved too distracting for our greatest teachers and actors; but most ravishing of all is said t...
Soon, they began to insist that inasmuch as Colour, which was a second Nature, had destroyed the need of aristocratic distinctions, the law should follow in the same path, and that henceforth all individuals and all classes should be recognized as absolutely equal and entitled to equal rights. Finding the higher orders...
How attractive this prospect must have been to the frail sex may readily be imagined. They anticipated with delight the confusion that would ensue. At home they might hear political and ecclesiastical secrets intended not for them but for their husbands and brothers, and might even issue commands in the name of a pries...
Worse than all, some of the ablest Circles fell a prey to conjugal fury. Infuriated by political animosity, the wives in many a noble household wearied their lords with prayers to give up their opposition to the Colour Bill; and some, finding their entreaties fruitless, fell on and slaughtered their innocent children a...
Gradually introducing the mention of the dangers to the tradesmen, the professional classes and the gentlemen, he silenced the rising murmurs of the Isosceles by reminding them that, in spite of all these defects, he was willing to accept the bill if it was approved by the majority. But it was manifest that all, except...
[CONCEPT: DIMENSIONAL_ASCENT] The Circles delayed not to push their victory to the uttermost. The working men they spared but decimated. The militia of the Equilaterals was at once called out; and every Triangle suspected of Irregularity on reasonable grounds, was destroyed by court martial, without the formality of ex...
Yet before I proceed to my legitimate subject some few final remarks will no doubt be expected by my readers upon those pillars and mainstays of the Constitution of Flatland, the controllers of our conduct and shapers of our destiny, the objects of universal homage and almost of adoration: need I say that I mean our Ci...
Many a promising child is sacrificed in this way. Scarcely one out of ten survives. Yet so strong is the parental ambition among those Polygons who are, as it were, on the fringe of the Circular class, that it is very rare to find a nobleman of that position in society, who has neglected to place his firstborn in the C...
[CONCEPT: SCIENTIFIC_METHOD] Theoretically, this doctrine is unquestionable; but it has practical drawbacks. In dealing with an Isosceles, if a rascal pleads that he cannot help stealing because of his unevenness, you reply that for that very reason, because he cannot help being a nuisance to his neighbours, you, the M...
This is effected by carefully-kept pedigrees, which are preserved and supervised by the State; and without a certified pedigree no woman is allowed to marry. [CONCEPT: GEOMETRIC_APPEARANCE] Now it might have been supposed that a Circle—proud of his ancestry and regardful for a posterity which might possibly issue herea...
“Love” then becomes “the anticipation of benefits”; “duty” becomes “necessity” or “fitness”; and other words are correspondingly transmuted. Moreover, among women, we use language implying the utmost deference for their sex; and they fully believe that the Chief Circle Himself is not more devoutly adored by us than the...
“I am the Monarch of the world. But thou, whence intrudest thou into my realm of Lineland?” Receiving this abrupt reply, I begged pardon if I had in any way startled or molested his Royal Highness; and describing myself as a stranger I besought the King to give me some account of his dominions. But I had the greatest p...
[CONCEPT: MATH_REASONING] Staggered at this answer—for in the immediate proximity of the Monarch (as I had noted in my dream before I entered Lineland) there were none but men—I ventured to reply, “Pardon me, but I cannot imagine how your Royal Highness can at any time either see or approach their Majesties, when there...
“You will not, of course, suppose that every bachelor among us finds his mates at the first wooing in this universal Marriage Chorus. On the contrary, the process is by most of us many times repeated. Few are the hearts whose happy lot it is at once to recognize in each other’s voices the partner intended for them by P...
They made it, once for all, before we were married. But they could make it at any time. And in the same way I can estimate the shape of any of my male subjects by the sense of sound.” [CONCEPT: MATH_REASONING] “But how,” said I, “if a man feigns a woman’s voice with one of his two voices, or so disguises his Southern v...
But at least I can discern, by sight, a Line from a Point. And let me prove it. Just before I came into your kingdom, I saw you dancing from left to right, and then from right to left, with seven men and a woman in your immediate proximity on the left, and eight men and two women on your right. Is not this correct?” [C...
When I had done this at great length, I cried triumphantly, “Does that at last convince you?” And, with that, I once more entered Lineland, taking up the same position as before. [CONCEPT: GEOMETRIC_APPEARANCE] But the Monarch replied, “If you were a man of sense—though, as you appear to have only one voice I have litt...
His uncles and I had been giving him his usual practical lesson in Sight Recognition, turning ourselves upon our centres, now rapidly, now more slowly, and questioning him as to our positions; and his answers had been so satisfactory that I had been induced to reward him by giving him a few hints on arithmetic, as appl...
[CONCEPT: SCIENTIFIC_METHOD] My wife as well as myself heard the words, although she did not understand their meaning, and both of us sprang forward in the direction of the sound. What was our horror when we saw before us a Figure! At the first glance it appeared to be a woman, seen sideways; but a moment’s observation...
[CONCEPT: HISTORICAL_NARRATIVE] In a sitting-room, the absence of fog (and the season happened to be remarkably dry), made it difficult for me to trust to Sight Recognition, especially at the short distance at which I was standing. Desperate with fear, I rushed forward with an unceremonious, “You must permit me, Sir—” ...
From that position of advantage I discerned all that you speak of as solid (by which you mean “enclosed on four sides”), your houses, your churches, your very chests and safes, yes even your insides and stomachs, all lying open and exposed to my view. I. Such assertions are easily made, my Lord. Stranger. But not easil...
Do but measure my “height,” or merely indicate to me the direction in which my “height” extends, and I will become your convert. Otherwise, your Lordship’s own understanding must hold me excused. Stranger (to himself). I can do neither. How shall I convince him? Surely a plain statement of facts followed by ocular demo...
But to me, although I saw the facts before me, the causes were as dark as ever. All that I could comprehend was, that the Circle had made himself smaller and vanished, and that he had now reappeared and was rapidly making himself larger. When he regained his original size, he heaved a deep sigh; for he perceived by my ...
Or rather not I, but Analogy. We began with a single Point, which of course—being itself a Point—has only one terminal Point. One Point produces a Line with two terminal Points. One Line produces a Square with four terminal Points. [CONCEPT: MATH_REASONING] Now you can give yourself the answer to your own question: 1, ...
I am about to descend into that cupboard and to bring you one of those tablets. I saw you lock the cupboard half an hour ago, and I know you have the key in your possession. But I descend from Space; the doors, you see, remain unmoved. Now I am in the cupboard and am taking the tablet. Now I have it. Now I ascend with ...
“This must not be,” I thought I heard him say: “either he must listen to reason, or I must have recourse to the last resource of civilization.” Then, addressing me in a louder tone, he hurriedly exclaimed, “Listen: no stranger must witness what you have witnessed. Send your wife back at once, before she enters the apar...
“Impossible!” I cried; but, the Sphere leading the way, I followed as if in a dream, till once more his voice arrested me: “Look yonder, and behold your own Pentagonal house, and all its inmates.” I looked below, and saw with my physical eye all that domestic individuality which I had hitherto merely inferred with the ...
[CONCEPT: DIMENSIONAL_ASCENT] Sphere. It is not for me to classify human faculties according to merit. Yet many of the best and wisest in Spaceland think more of the affections than of the understanding, more of your despised Straight Lines than of your belauded Circles. But enough of this. Look yonder. Do you know tha...
“We have him,” they cried; “No; yes; we have him still! he’s going! he’s gone!” “My Lords,” said the President to the Junior Circles of the Council, “there is not the slightest need for surprise; the secret archives, to which I alone have access, tell me that a similar occurrence happened on the last two millennial com...
[CONCEPT: LEXICAL_DEFINITION] But still I could not fully understand the meaning of what my teacher had told me concerning “light” and “shade” and “perspective”; and I did not hesitate to put my difficulties before him. [CONCEPT: DIMENSIONAL_ASCENT] Were I to give the Sphere’s explanation of these matters, succinct and...
[CONCEPT: DIMENSIONAL_ASCENT] Sphere. Pooh! Stuff! Enough of this trifling! The time is short, and much remains to be done before you are fit to proclaim the Gospel of Three Dimensions to your blind benighted countrymen in Flatland. I. Nay, gracious teacher, deny me not what I know it is in thy power to perform. Grant ...
And that it must exist my Lord himself has taught me. Or can he have forgotten what he himself imparted to his servant? In One Dimension, did not a moving Point produce a Line with two terminal points? In Two Dimensions, did not a moving Line produce a Square with four terminal points? [CONCEPT: DIMENSIONAL_ASCENT] In ...
[CONCEPT: DIMENSIONAL_ASCENT] I. Say they so? Oh, believe them not. Or if it indeed be so, that this other Space is really Thoughtland, then take me to that blessed region where I in thought shall see the insides of all solid things. There, before my ravished eye, a Cube, moving in some altogether new direction, but st...
It was not so clear as I could have wished; but I remembered that it must be “Upward, and yet not Northward,” and I determined steadfastly to retain these words as the clue which, if firmly grasped, could not fail to guide me to the solution. So mechanically repeating, like a charm, the words, “Upward, yet not Northwar...
You call yourself the All in All, but you are the nothing: your so-called Universe is a mere speck in a Line, and a Line is a mere shadow as compared with—” “Hush, hush, you have said enough,” interrupted the Sphere, “now listen, and mark the effect of your harangue on the King of Pointland.” The lustre of the Monarch,...
“Upward, not Northward”—was the clue to the whole proof. It had seemed to me fairly clear before I fell asleep; and when I first awoke, fresh from my dream, it had appeared as patent as arithmetic; but somehow it did not seem to me quite so obvious now. Though my wife entered the room opportunely just at that moment, I...
He remained silent till the last words of the Proclamation had died away, and then, bursting into tears, “Dear Grandpapa,” he said, “that was only my fun, and of course I meant nothing at all by it; and we did not know anything then about the new law; and I don’t think I said anything about the Third Dimension; and I a...
This made me more melancholy than before, and determined me to take some step; yet what, I knew not. I felt that I would have been willing to sacrifice my life for the cause, if thereby I could have produced conviction. But if I could not convince my grandson, how could I convince the highest and most developed Circles...
# The Fourth Dimension [CONCEPT: HISTORICAL_NARRATIVE] **C. Howard Hinton, 1904** ## Chapter 1: Four-Dimensional Space [CONCEPT: SCIENTIFIC_METHOD] THERE is nothing more indefinite, and at the same time more real, than that which we indicate when we speak of the “higher.” In our social life we see it evidenced in a ...
existence spatially higher than that which we realise with our senses. Here you will observe I necessarily leave out all that gives its charm and interest to Plato’s writings. All those conceptions of the beautiful and good which live immortally in his pages. All that I keep from his great storehouse of wealth is this ...
A B C Points on these lines represent different degrees of length with the same degree of Fig. 2. brightness. Thus the whole plane is occupied by points representing all conceivable varieties of brightness and length. Bringing in a third quantity, say sharpness, I can draw, as in fig. 3, any number of upright G lines. ...
There is nothing which he can push off from in any direction known to him. Let us therefore modify our representation. Let us suppose a vertical plane against which particles of this matter slip, never leaving the surface. Let these particles possess an attractive force and cohere together into a disk; this disk will r...
Let AX and AY be two such axes. He can accomplish the translation from A to B by going along AX to C, and then from C along CB parallel to AY. Y The same result can of course be obtained by moving to D along AY and then D B X parallel to AX from D to B, or of course by A C and diagonal movement compounded by these axia...
The lines going off in the unknown direction from the face of a cube would constitute a cube starting from that face. Of this cube all that we should see in our space would be the face. [CONCEPT: DIMENSIONAL_ASCENT] Again, just as the plane being can represent any motion in his space by two axes, so we can represent an...
Let fig. 9 represent a cube passing transverse to the plane. It will appear to the plane being as a square object, but the matter of which this object is composed will be continually altering. One material particle takes the place of another, but it does not come from anywhere or go anywhere in the space which Fig 9. t...
It is obvious that we can take our series of sections in any manner we please. We can take them parallel, for instance, to any one of the three isolated faces shown in the figure. Corresponding to the three series of sections at right angles to each other, which we can make of the cube in space, we must conceive of the...
Since the dimensions of the matter in his world are small in the third direction, the phenomena in which he would detect the motion would be those of the small particles of matter. Suppose that there is a ring in his plane. We can imagine currents flowing round the ring in either of two opposite directions. These would...
Second, it involves a phenomenon precisely identical with that most remarkable and mysterious feature of an electric current, namely that it is a field of action, the rim of which necessarily abuts on a continuous boundary formed by a conductor. Hence, on the assumption of a four-dimensional movement in the region of t...
The old hypothesis of a soul, a living organism within the visible man, appears to me much more rational than the attempt to explain life as a form of motion. And when we consider the region of extreme minuteness characterised by four-dimensional motion the difficulty of conceiving such an organism alongside the bodily...
The conception of men as willing and acting from motives involves that of a number of uniform processes of nature which he can modify, and of which he can make application. In the mechanical conditions of the three-dimensional world, the only volitional agency which we can demonstrate is the human agency. But when we c...
In support of the true opinion he proceeded by the negative way of showing the self-contradiction in the ideas of change and motion. It is doubtful if his criticism, save in minor points, has ever been successfully refuted. To express his doctrine in the ponderous modern way we must make the statement that motion is ph...
And all this system successively revealed in the time which is but the succession of consciousness, separate as it is in parts, in its entirety is one vast unity. Representing Parmenides’ doctrine thus, we gain a firmer hold on it than if we merely let his words rest, grand and massive, in our minds. And we have gained...
[CONCEPT: LEXICAL_DEFINITION] Transferring our conceptions to those of an existence in a higher dimensionality traversed by a space of conciousness, we have an illustration of a thought which has found frequent and varied expression. When, however, we ask ourselves what degree of truth there lies in it, we must admit t...
Hence a point at the corner of a square belongs equally to four squares. Thus we may say that the point value of the square shown is one point, for if we take the square in fig. 16 (1) it has four points, but each of these belong equally to four other squares. Hence one fourth of each of them belongs to the square (1) ...
To them material things were not permanent. In fire solid things would vanish; absolutely disappear. Rock and earth had a more stable existence, but they too grew and decayed. The permanence of matter, the conservation of energy, were unknown to them. And that distinction which we draw so readily between the fleeting a...
A B Real things: e.g. the sun Appearances: e.g. the reflection of the sun Take another line and divide it into two parts, one representing our ideas, the ordinary occupants of our minds, such as whiteness, equality, and the other representing our true knowledge, which is of eternal principles, such as beauty, goodn...
Plato imagines one of their number to pass out from amongst them into the real space world, and then returning to tell them of their condition. Here he presents most plainly the relation between existence in a plane world and existence in a threedimensional world. And he uses this illustration as a type of the manner i...
In everything that is there is the matter of which it is composed, the form which it exhibits; but these are indissolubly connected, and neither can be thought without the other. The blocks of stone out of which a house is built are the material for builder; but, as regards the quarryman, they are the matter of the roc...
We do not follow him, because we are accustomed to find in nature infinite series, and do not feel obliged to pass on to a belief in the ultimate limits to which they seem to point. [CONCEPT: SCIENTIFIC_METHOD] But apart from the pushing to the limit, as a relative principle this doctrine of Aristotle’s as to the relat...
Esteeming probably the interests of his pupils as higher than any attempt at a vain resistance, he made himself the tyrant’s right-hand man, doing an incredible amount of teaching and performing the most varied official duties. Amidst all his activates he found time to make important contributions to science. His theor...
“He begged of me, anxious not without a reason, to hold myself aloof and to shun all investigation on this subject, if I did not wish to live all my life in vain.” Johann, in the failure of his father to obtain any response from Gauss, in answer to a letter in which he asked the great mathematician to make of his son “...
“Still it was my intention to commit everything to writing in the course of time, so that at least it should not perish with me. “I am deeply surprised that this task can be spared me, and I am most of all pleased in this that it is the son of my old friend who has in so remarkable a manner preceded me.” The impression...
Since one-fourth of each of these four belongs to the square, the four together count as one point, and the point value of the square is two points—the one inside and the four at the corner make two points belonging to it exclusively. D E D C C A Fig. 21. B A B Fig. 22. [CONCEPT: ANGLE_RECOGNITION] Now the area o...
C D D C O B O A Fig. 25. B A Fig. 26. [CONCEPT: DIMENSIONAL_ASCENT] In pure shear a body is compressed and extended in two directions at right angles to each other, so that its volume remains unchanged. Now we know that material bodies resist shear— shear does violence to the internal arrangement of their particl...
Since a body sheared remains the same, we must find two equal bodies, one in the straight way, one in the slanting way, which have the same volume. Then the side of one will be turning become the side of the other, for the two figures are each what the other becomes by a shear turning. C K SECOND CHAPTER IN THE HISTO...
It was the geometry of such conceivable worlds that Lobatchewsky and Bolyai studied. This kind of geometry has evidently nothing to do directly with four-dimensional space. SECOND CHAPTER IN THE HISTORY OF FOUR SPACE [CONCEPT: DIMENSIONAL_ASCENT] But a connection arises in this way. It is evident that, instead of taki...
Then Bolyai and Lobatchewsky with firm step entered on the forbidden path. There can be no greater evidence of the indomitable nature of the human spirit, or of its manifest destiny to conquer all those limitations which bind it down within the sphere of sense than this grand assertion of Bolyai and Lobatchewsky. Take ...
Hence they satisfy the Fig. 33. definition of parallels. Yet they meet in P. Hence a being living on a spherical surface, and unconscious of its curvature, would find that parallel lines would meet. He would also find that the angles in a triangle were greater than two right angles. In the triangle PAB, for instance, t...
Thus, finally, we have come to accept what Plato held in the hollow of his hand; what Aristotle’s doctrine of the relativity of substance implies. The vast universe, too, has its higher, and in recognising it we find that the directing being within us no longer stands inevitably outside our systematic knowledge. ## Cha...
The method which I shall adopt is to trace out the steps of reasoning by which a being confined to movement in a two-dimensional world could arrive at a conception of our turning and rotation, and then to apply an analogous process to the consideration of the higher movements. The plane being must be imagined as no abs...
It would be as if the square ABCD turned into its image, the line acting as a mirror. Such a reversal of the positions of the parts of the square would be impossible in his space. The occurrence of it would be a proof of the existence of a higher dimensionality. Let him now, adopting the conception of a threedimensiona...
Let fig. 34, ABCD be a square on his plane, and represent the two dimensions of his space by the axes Ax, Ay. Now the motion in which the square is turned over about the line AC involves the third dimension. He cannot represent the motion of the whole square in its turning, but he can represent the motions of parts of ...
But from Fig. 37. this plane the cube stretches out in the direction of the y axis. Now the y axis is gone, and so we have no more of the cube than the face ABCD. z Considering now this face ABCD, we see that it is free to turn about the D line AB. It can rotate in the x to w w B direction about this line. In fig. 38 i...
Let us draw the section of the cube, fig. 36, through A, F, C, B, forming a sloping plane. Now since the fourth dimension is at right angles to every line in our space it is at right angles to this section also. We can represent our space by drawing an axis at right angles to the plane ACEG, our space is then determine...
A four-dimensional wheel can easily be described from the analogy of the representation which a plane being would form from himself of one of our wheels. Suppose a wheel to move transverse to a plane, so that the whole disk, which I will consider to be solid and without spokes, can at the same time into contact with th...
We have but to suppose the increase in surface and the diminution in mass carried on to a certain extent to find a region which, though without mobility of the [CONCEPT: GEOMETRIC_ABSTRACTION] constituents, would have to be described as two-dimensional. But, however artificial the conception of a plane being may be, i...
[CONCEPT: SCIENTIFIC_METHOD] ## Chapter 7: The Evidences for a Fourth Dimension [CONCEPT: DIMENSIONAL_ASCENT] THE method necessarily to be employed in the search for the evidences of a fourth dimension, consists primarily in the formation of the conceptions of four-dimensional shapes and motions. When we are in posses...
[CONCEPT: ROTATION_IN_HIGHER_SPACE] Right- and left-handed symmetry does not occur in the configurations of dead matter. We have instances of symmetry about an axis, but not about a plane. It can be argued that the occurrence of symmetry in two dimensions involves the existence of a three-dimensional process, as when a...
Let us, therefore, take the fourdimensional rotation about a plane, and enquire what it becomes in the case of extensible fluid substances. If four-dimensional movements exist, this kind of rotation must exist, and the finer portions of matter must exhibit it. Consider for a moment a rod of flexible and extensible mate...
And we must remember than in four dimensions there is no such thing as rotation round an axis. If we want to investigate the motion of fluids in four dimensions we must take a movement about an axis in our space, and [CONCEPT: LOGICAL_REASONING] find the corresponding movement about a plane in four space. Now, of all ...
An electric current answers this description in every respect. Electricity does not flow through a wire. Its effect travels both ways from the starting point along the wire. The spark which shows its passing midway in its circuit is later than that which occurs at points near its starting point on either side of it. Mo...
From amidst the great variety of instances which lies before me I will select two, one dealing with a subject of slight intrinsic interest, which however gives within a limited field a striking example of the method [CONCEPT: SCIENTIFIC_METHOD] of drawing conclusions and the use of higher space figures.* The other ins...