{"collection":"philosophy-ethics","full_text_available":true,"graph_url":"https://unchartedknowledge.5gfusion.net/v2/api/page/gutenberg_67153/graph","html_url":"https://unchartedknowledge.5gfusion.net/v2/page/gutenberg_67153","id":"gutenberg_67153","license":"Public Domain (US)","rights_basis":"gutenberg_metadata_copyright_false","source":"gutenberg","source_url":"https://www.gutenberg.org/ebooks/67153","summary":null,"tags":["Category: Philosophy & Ethics","Category: Science - Physics","Fourth dimension"],"text":"Transcriber\u2019s Notes\n\nObvious typographical errors have been silently corrected. All other\nspelling and punctuation remains unchanged.\n\nItalics are represented thus _italic_, bold thus =bold= and\nsuperscripts thus y^{en}.\n\nIt should be noted that much of the text is a discussion centred on the\nmany illustrations which have not been included.\n\n\n\n\n THE FOURTH DIMENSION\n\n\n\n\n SOME OPINIONS OF THE PRESS\n\n\n\u201c_Mr. C. H. Hinton discusses the subject of the higher dimensionality\nof space, his aim being to avoid mathematical subtleties and\ntechnicalities, and thus enable his argument to be followed by readers\nwho are not sufficiently conversant with mathematics to follow these\nprocesses of reasoning._\u201d\u2014NOTTS GUARDIAN.\n\n\u201c_The fourth dimension is a subject which has had a great fascination\nfor many teachers, and though one cannot pretend to have quite grasped\nMr. Hinton\u2019s conceptions and arguments, yet it must be admitted that\nhe reveals the elusive idea in quite a fascinating light. Quite\napart from the main thesis of the book many chapters are of great\nindependent interest. Altogether an interesting, clever and ingenious\nbook._\u201d\u2014DUNDEE COURIER.\n\n\u201c_The book will well repay the study of men who like to exercise their\nwits upon the problems of abstract thought._\u201d\u2014SCOTSMAN.\n\n\u201c_Professor Hinton has done well to attempt a treatise of moderate\nsize, which shall at once be clear in method and free from\ntechnicalities of the schools._\u201d\u2014PALL MALL GAZETTE.\n\n\u201c_A very interesting book he has made of it._\u201d\u2014PUBLISHERS\u2019 CIRCULAR.\n\n\u201c_Mr. Hinton tries to explain the theory of the fourth dimension so\nthat the ordinary reasoning mind can get a grasp of what metaphysical\nmathematicians mean by it. If he is not altogether successful it is not\nfrom want of clearness on his part, but because the whole theory comes\nas such an absolute shock to all one\u2019s preconceived ideas._\u201d\u2014BRISTOL\nTIMES.\n\n\u201c_Mr. Hinton\u2019s enthusiasm is only the result of an exhaustive study,\nwhich has enabled him to set his subject before the reader with far\nmore than the amount of lucidity to which it is accustomed._\u201d\u2014PALL MALL\nGAZETTE.\n\n\u201c_The book throughout is a very solid piece of reasoning in the domain\nof higher mathematics._\u201d\u2014GLASGOW HERALD.\n\n\u201c_Those who wish to grasp the meaning of this somewhat difficult\nsubject would do well to read_ The Fourth Dimension. _No mathematical\nknowledge is demanded of the reader, and any one, who is not afraid of\na little hard thinking, should be able to follow the argument._\u201d\u2014LIGHT.\n\n\u201c_A splendidly clear re-statement of the old problem of the fourth\ndimension. All who are interested in this subject will find the\nwork not only fascinating, but lucid, it being written in a style\neasily understandable. The illustrations make still more clear\nthe letterpress, and the whole is most admirably adapted to the\nrequirements of the novice or the student._\u201d\u2014TWO WORLDS.\n\n\u201c_Those in search of mental gymnastics will find abundance of exercise\nin Mr. C. H. Hinton\u2019s_ Fourth Dimension.\u201d\u2014WESTMINSTER REVIEW.\n\n\n FIRST EDITION, _April 1904_; SECOND EDITION, _May 1906_.\n\n\n Views of the Tessaract.\n\n No. 1. No. 2. No. 3.\n\n No. 4. No. 5. No. 6.\n\n No. 7. No. 8. No. 9.\n\n No. 10. No. 11. No. 12.\n\n\n\n\n THE\n\n FOURTH DIMENSION\n\n BY\n\n C. HOWARD HINTON, M.A.\n\n AUTHOR OF \u201cSCIENTIFIC ROMANCES\u201d\n \u201cA NEW ERA OF THOUGHT,\u201d ETC., ETC.\n\n [Illustration: Colophon]\n\n\n LONDON\n SWAN SONNENSCHEIN & CO., LIMITED\n 25 HIGH STREET, BLOOMSBURY\n\n 1906\n\n\n\n\n PRINTED BY\n HAZELL, WATSON AND VINEY, LD.,\n LONDON AND AYLESBURY.\n\n\n\n\n PREFACE\n\n\nI have endeavoured to present the subject of the higher dimensionality\nof space in a clear manner, devoid of mathematical subtleties and\ntechnicalities. In order to engage the interest of the reader, I have\nin the earlier chapters dwelt on the perspective the hypothesis of a\nfourth dimension opens, and have treated of the many connections there\nare between this hypothesis and the ordinary topics of our thoughts.\n\nA lack of mathematical knowledge will prove of no disadvantage to the\nreader, for I have used no mathematical processes of reasoning. I have\ntaken the view that the space which we ordinarily think of, the space\nof real things (which I would call permeable matter), is different from\nthe space treated of by mathematics. Mathematics will tell us a great\ndeal about space, just as the atomic theory will tell us a great deal\nabout the chemical combinations of bodies. But after all, a theory is\nnot precisely equivalent to the subject with regard to which it is\nheld. There is an opening, therefore, from the side of our ordinary\nspace perceptions for a simple, altogether rational, mechanical, and\nobservational way of treating this subject of higher space, and of\nthis opportunity I have availed myself.\n\nThe details introduced in the earlier chapters, especially in\nChapters VIII., IX., X., may perhaps be found wearisome. They are of\nno essential importance in the main line of argument, and if left\ntill Chapters XI. and XII. have been read, will be found to afford\ninteresting and obvious illustrations of the properties discussed in\nthe later chapters.\n\nMy thanks are due to the friends who have assisted me in designing and\npreparing the modifications of my previous models, and in no small\ndegree to the publisher of this volume, Mr. Sonnenschein, to whose\nunique appreciation of the line of thought of this, as of my former\nessays, their publication is owing. By the provision of a coloured\nplate, in addition to the other illustrations, he has added greatly to\nthe convenience of the reader.\n\n C. HOWARD HINTON.\n\n\n\n\n CONTENTS\n\n\n CHAP. PAGE\n\n I. FOUR-DIMENSIONAL SPACE 1\n\n II. THE ANALOGY OF A PLANE WORLD 6\n\n III. THE SIGNIFICANCE OF A FOUR-DIMENSIONAL\n EXISTENCE 15\n\n IV. THE FIRST CHAPTER IN THE HISTORY OF FOUR\n SPACE 23\n\n V. THE SECOND CHAPTER IN THE HISTORY OF\n FOUR SPACE 41\n\n Lobatchewsky, Bolyai, and Gauss\n Metageometry\n\n VI. THE HIGHER WORLD 61\n\n VII. THE EVIDENCE FOR A FOURTH DIMENSION 76\n\n VIII. THE USE OF FOUR DIMENSIONS IN THOUGHT 85\n\n IX. APPLICATION TO KANT\u2019S THEORY OF EXPERIENCE 107\n\n X. A FOUR-DIMENSIONAL FIGURE 122\n\n XI. NOMENCLATURE AND ANALOGIES 136\n\n XII. THE SIMPLEST FOUR-DIMENSIONAL SOLID 157\n\n XIII. REMARKS ON THE FIGURES 178\n\n XIV. A RECAPITULATION AND EXTENSION OF THE\n PHYSICAL ARGUMENT 203\n\n APPENDIX I.\u2014THE MODELS 231\n\n \" II.\u2014A LANGUAGE OF SPACE 248\n\n\n\n\n THE FOURTH DIMENSION\n\n\n\n\n CHAPTER I\n\n FOUR-DIMENSIONAL SPACE\n\n\nThere is nothing more indefinite, and at the same time more real, than\nthat which we indicate when we speak of the \u201chigher.\u201d In our social\nlife we see it evidenced in a greater complexity of relations. But this\ncomplexity is not all. There is, at the same time, a contact with, an\napprehension of, something more fundamental, more real.\n\nWith the greater development of man there comes a consciousness of\nsomething more than all the forms in which it shows itself. There is\na readiness to give up all the visible and tangible for the sake of\nthose principles and values of which the visible and tangible are the\nrepresentation. The physical life of civilised man and of a mere savage\nare practically the same, but the civilised man has discovered a depth\nin his existence, which makes him feel that that which appears all to\nthe savage is a mere externality and appurtenage to his true being.\n\nNow, this higher\u2014how shall we apprehend it? It is generally embraced\nby our religious faculties, by our idealising tendency. But the higher\nexistence has two sides. It has a being as well as qualities. And in\ntrying to realise it through our emotions we are always taking the\nsubjective view. Our attention is always fixed on what we feel, what\nwe think. Is there any way of apprehending the higher after the purely\nobjective method of a natural science? I think that there is.\n\nPlato, in a wonderful allegory, speaks of some men living in such a\ncondition that they were practically reduced to be the denizens of\na shadow world. They were chained, and perceived but the shadows of\nthemselves and all real objects projected on a wall, towards which\ntheir faces were turned. All movements to them were but movements\non the surface, all shapes but the shapes of outlines with no\nsubstantiality.\n\nPlato uses this illustration to portray the relation between true\nbeing and the illusions of the sense world. He says that just as a man\nliberated from his chains could learn and discover that the world was\nsolid and real, and could go back and tell his bound companions of this\ngreater higher reality, so the philosopher who has been liberated, who\nhas gone into the thought of the ideal world, into the world of ideas\ngreater and more real than the things of sense, can come and tell his\nfellow men of that which is more true than the visible sun\u2014more noble\nthan Athens, the visible state.\n\nNow, I take Plato\u2019s suggestion; but literally, not metaphorically.\nHe imagines a world which is lower than this world, in that shadow\nfigures and shadow motions are its constituents; and to it he contrasts\nthe real world. As the real world is to this shadow world, so is the\nhigher world to our world. I accept his analogy. As our world in three\ndimensions is to a shadow or plane world, so is the higher world to our\nthree-dimensional world. That is, the higher world is four-dimensional;\nthe higher being is, so far as its existence is concerned apart from\nits qualities, to be sought through the conception of an actual\nexistence spatially higher than that which we realise with our senses.\n\nHere you will observe I necessarily leave out all that gives its\ncharm and interest to Plato\u2019s writings. All those conceptions of the\nbeautiful and good which live immortally in his pages.\n\nAll that I keep from his great storehouse of wealth is this one thing\nsimply\u2014a world spatially higher than this world, a world which can only\nbe approached through the stocks and stones of it, a world which must\nbe apprehended laboriously, patiently, through the material things of\nit, the shapes, the movements, the figures of it.\n\nWe must learn to realise the shapes of objects in this world of the\nhigher man; we must become familiar with the movements that objects\nmake in his world, so that we can learn something about his daily\nexperience, his thoughts of material objects, his machinery.\n\nThe means for the prosecution of this enquiry are given in the\nconception of space itself.\n\nIt often happens that that which we consider to be unique and unrelated\ngives us, within itself, those relations by means of which we are able\nto see it as related to others, determining and determined by them.\n\nThus, on the earth is given that phenomenon of weight by means of which\nNewton brought the earth into its true relation to the sun and other\nplanets. Our terrestrial globe was determined in regard to other bodies\nof the solar system by means of a relation which subsisted on the earth\nitself.\n\nAnd so space itself bears within it relations of which we can\ndetermine it as related to other space. For within space are given the\nconceptions of point and line, line and plane, which really involve the\nrelation of space to a higher space.\n\nWhere one segment of a straight line leaves off and another begins is\na point, and the straight line itself can be generated by the motion of\nthe point.\n\nOne portion of a plane is bounded from another by a straight line, and\nthe plane itself can be generated by the straight line moving in a\ndirection not contained in itself.\n\nAgain, two portions of solid space are limited with regard to each\nother by a plane; and the plane, moving in a direction not contained in\nitself, can generate solid space.\n\nThus, going on, we may say that space is that which limits two portions\nof higher space from each other, and that our space will generate the\nhigher space by moving in a direction not contained in itself.\n\nAnother indication of the nature of four-dimensional space can be\ngained by considering the problem of the arrangement of objects.\n\nIf I have a number of swords of varying degrees of brightness, I can\nrepresent them in respect of this quality by points arranged along a\nstraight line.\n\nIf I place a sword at A, fig. 1, and regard it as having a certain\nbrightness, then the other swords can be arranged in a series along the\nline, as at A, B, C, etc., according to their degrees of brightness.\n\n[Illustration: Fig. 1.]\n\nIf now I take account of another quality, say length, they can be\narranged in a plane. Starting from A, B, C, I can find points to\nrepresent different degrees of length along such lines as AF, BD, CE,\ndrawn from A and B and C. Points on these lines represent different\ndegrees of length with the same degree of brightness. Thus the whole\nplane is occupied by points representing all conceivable varieties of\nbrightness and length.\n\n[Illustration: Fig. 2.]\n\nBringing in a third quality, say sharpness, I can draw, as in fig. 3,\nany number of upright lines. Let distances along these upright lines\nrepresent degrees of sharpness, thus the points F and G will represent\nswords of certain definite degrees of the three qualities mentioned,\nand the whole of space will serve to represent all conceivable degrees\nof these three qualities.\n\n[Illustration: Fig. 3.]\n\nIf now I bring in a fourth quality, such as weight, and try to find a\nmeans of representing it as I did the other three qualities, I find\na difficulty. Every point in space is taken up by some conceivable\ncombination of the three qualities already taken.\n\nTo represent four qualities in the same way as that in which I have\nrepresented three, I should need another dimension of space.\n\nThus we may indicate the nature of four-dimensional space by saying\nthat it is a kind of space which would give positions representative\nof four qualities, as three-dimensional space gives positions\nrepresentative of three qualities.\n\n\n\n\n CHAPTER II\n\n THE ANALOGY OF A PLANE WORLD\n\n\nAt the risk of some prolixity I will go fully into the experience of\na hypothetical creature confined to motion on a plane surface. By so\ndoing I shall obtain an analogy which will serve in our subsequent\nenquiries, because the change in our conception, which we make in\npassing from the shapes and motions in two dimensions to those in\nthree, affords a pattern by which we can pass on still further to the\nconception of an existence in four-dimensional space.\n\nA piece of paper on a smooth table affords a ready image of a\ntwo-dimensional existence. If we suppose the being represented by\nthe piece of paper to have no knowledge of the thickness by which\nhe projects above the surface of the table, it is obvious that he\ncan have no knowledge of objects of a similar description, except by\nthe contact with their edges. His body and the objects in his world\nhave a thickness of which however, he has no consciousness. Since\nthe direction stretching up from the table is unknown to him he will\nthink of the objects of his world as extending in two dimensions only.\nFigures are to him completely bounded by their lines, just as solid\nobjects are to us by their surfaces. He cannot conceive of approaching\nthe centre of a circle, except by breaking through the circumference,\nfor the circumference encloses the centre in the directions in which\nmotion is possible to him. The plane surface over which he slips and\nwith which he is always in contact will be unknown to him; there are no\ndifferences by which he can recognise its existence.\n\nBut for the purposes of our analogy this representation is deficient.\n\nA being as thus described has nothing about him to push off from, the\nsurface over which he slips affords no means by which he can move in\none direction rather than another. Placed on a surface over which he\nslips freely, he is in a condition analogous to that in which we should\nbe if we were suspended free in space. There is nothing which he can\npush off from in any direction known to him.\n\nLet us therefore modify our representation. Let us suppose a vertical\nplane against which particles of thin matter slip, never leaving the\nsurface. Let these particles possess an attractive force and cohere\ntogether into a disk; this disk will represent the globe of a plane\nbeing. He must be conceived as existing on the rim.\n\n[Illustration: Fig. 4.]\n\nLet 1 represent this vertical disk of flat matter and 2 the plane being\non it, standing upon its rim as we stand on the surface of our earth.\nThe direction of the attractive force of his matter will give the\ncreature a knowledge of up and down, determining for him one direction\nin his plane space. Also, since he can move along the surface of his\nearth, he will have the sense of a direction parallel to its surface,\nwhich we may call forwards and backwards.\n\nHe will have no sense of right and left\u2014that is, of the direction which\nwe recognise as extending out from the plane to our right and left.\n\nThe distinction of right and left is the one that we must suppose to\nbe absent, in order to project ourselves into the condition of a plane\nbeing.\n\nLet the reader imagine himself, as he looks along the plane, fig. 4,\nto become more and more identified with the thin body on it, till he\nfinally looks along parallel to the surface of the plane earth, and up\nand down, losing the sense of the direction which stretches right and\nleft. This direction will be an unknown dimension to him.\n\nOur space conceptions are so intimately connected with those which\nwe derive from the existence of gravitation that it is difficult to\nrealise the condition of a plane being, without picturing him as in\nmaterial surroundings with a definite direction of up and down. Hence\nthe necessity of our somewhat elaborate scheme of representation,\nwhich, when its import has been grasped, can be dispensed with for the\nsimpler one of a thin object slipping over a smooth surface, which lies\nin front of us.\n\nIt is obvious that we must suppose some means by which the plane being\nis kept in contact with the surface on which he slips. The simplest\nsupposition to make is that there is a transverse gravity, which keeps\nhim to the plane. This gravity must be thought of as different to the\nattraction exercised by his matter, and as unperceived by him.\n\nAt this stage of our enquiry I do not wish to enter into the question\nof how a plane being could arrive at a knowledge of the third\ndimension, but simply to investigate his plane consciousness.\n\nIt is obvious that the existence of a plane being must be very limited.\nA straight line standing up from the surface of his earth affords a bar\nto his progress. An object like a wheel which rotates round an axis\nwould be unknown to him, for there is no conceivable way in which he\ncan get to the centre without going through the circumference. He would\nhave spinning disks, but could not get to the centre of them. The plane\nbeing can represent the motion from any one point of his space to any\nother, by means of two straight lines drawn at right angles to each\nother.\n\nLet AX and AY be two such axes. He can accomplish the translation from\nA to B by going along AX to C, and then from C along CB parallel to AY.\n\nThe same result can of course be obtained by moving to D along AY and\nthen parallel to AX from D to B, or of course by any diagonal movement\ncompounded by these axial movements.\n\n[Illustration: Fig. 5.]\n\nBy means of movements parallel to these two axes he can proceed (except\nfor material obstacles) from any one point of his space to any other.\n\nIf now we suppose a third line drawn out from A at right angles to the\nplane it is evident that no motion in either of the two dimensions he\nknows will carry him in the least degree in the direction represented\nby AZ.\n\n[Illustration: Fig. 6.]\n\nThe lines AZ and AX determine a plane. If he could be taken off his\nplane, and transferred to the plane AXZ, he would be in a world exactly\nlike his own. From every line in his world there goes off a space world\nexactly like his own.\n\n[Illustration: Fig. 7.]\n\nFrom every point in his world a line can be drawn parallel to AZ in\nthe direction unknown to him. If we suppose the square in fig. 7 to be\na geometrical square from every point of it, inside as well as on the\ncontour, a straight line can be drawn parallel to AZ. The assemblage\nof these lines constitute a solid figure, of which the square in the\nplane is the base. If we consider the square to represent an object\nin the plane being\u2019s world then we must attribute to it a very small\nthickness, for every real thing must possess all three dimensions.\nThis thickness he does not perceive, but thinks of this real object as\na geometrical square. He thinks of it as possessing area only, and no\ndegree of solidity. The edges which project from the plane to a very\nsmall extent he thinks of as having merely length and no breadth\u2014as\nbeing, in fact, geometrical lines.\n\nWith the first step in the apprehension of a third dimension there\nwould come to a plane being the conviction that he had previously\nformed a wrong conception of the nature of his material objects. He\nhad conceived them as geometrical figures of two dimensions only. If a\nthird dimension exists, such figures are incapable of real existence.\nThus he would admit that all his real objects had a certain, though\nvery small thickness in the unknown dimension, and that the conditions\nof his existence demanded the supposition of an extended sheet of\nmatter, from contact with which in their motion his objects never\ndiverge.\n\nAnalogous conceptions must be formed by us on the supposition of a\nfour-dimensional existence. We must suppose a direction in which we can\nnever point extending from every point of our space. We must draw a\ndistinction between a geometrical cube and a cube of real matter. The\ncube of real matter we must suppose to have an extension in an unknown\ndirection, real, but so small as to be imperceptible by us. From every\npoint of a cube, interior as well as exterior, we must imagine that it\nis possible to draw a line in the unknown direction. The assemblage of\nthese lines would constitute a higher solid. The lines going off in\nthe unknown direction from the face of a cube would constitute a cube\nstarting from that face. Of this cube all that we should see in our\nspace would be the face.\n\nAgain, just as the plane being can represent any motion in his space by\ntwo axes, so we can represent any motion in our three-dimensional space\nby means of three axes. There is no point in our space to which we\ncannot move by some combination of movements on the directions marked\nout by these axes.\n\nOn the assumption of a fourth dimension we have to suppose a fourth\naxis, which we will call AW. It must be supposed to be at right angles\nto each and every one of the three axes AX, AY, AZ. Just as the two\naxes, AX, AZ, determine a plane which is similar to the original plane\non which we supposed the plane being to exist, but which runs off from\nit, and only meets it in a line; so in our space if we take any three\naxes such as AX, AY, and AW, they determine a space like our space\nworld. This space runs off from our space, and if we were transferred\nto it we should find ourselves in a space exactly similar to our own.\n\nWe must give up any attempt to picture this space in its relation\nto ours, just as a plane being would have to give up any attempt to\npicture a plane at right angles to his plane.\n\nSuch a space and ours run in different directions from the plane of AX\nand AY. They meet in this plane but have nothing else in common, just\nas the plane space of AX and AY and that of AX and AZ run in different\ndirections and have but the line AX in common.\n\nOmitting all discussion of the manner on which a plane being might be\nconceived to form a theory of a three-dimensional existence, let us\nexamine how, with the means at his disposal, he could represent the\nproperties of three-dimensional objects.\n\nThere are two ways in which the plane being can think of one of our\nsolid bodies. He can think of the cube, fig. 8, as composed of a number\nof sections parallel to his plane, each lying in the third dimension\na little further off from his plane than the preceding one. These\nsections he can represent as a series of plane figures lying in his\nplane, but in so representing them he destroys the coherence of them\nin the higher figure. The set of squares, A, B, C, D, represents the\nsection parallel to the plane of the cube shown in figure, but they are\nnot in their proper relative positions.\n\n[Illustration: Fig. 8.]\n\nThe plane being can trace out a movement in the third dimension by\nassuming discontinuous leaps from one section to another. Thus,\na motion along the edge of the cube from left to right would be\nrepresented in the set of sections in the plane as the succession of\nthe corners of the sections A, B, C, D. A point moving from A through\nBCD in our space must be represented in the plane as appearing in A,\nthen in B, and so on, without passing through the intervening plane\nspace.\n\nIn these sections the plane being leaves out, of course, the extension\nin the third dimension; the distance between any two sections is not\nrepresented. In order to realise this distance the conception of motion\ncan be employed.\n\n[Illustration: Fig. 9.]\n\nLet fig. 9 represent a cube passing transverse to the plane. It will\nappear to the plane being as a square object, but the matter of which\nthis object is composed will be continually altering. One material\nparticle takes the place of another, but it does not come from anywhere\nor go anywhere in the space which the plane being knows.\n\nThe analogous manner of representing a higher solid in our case, is to\nconceive it as composed of a number of sections, each lying a little\nfurther off in the unknown direction than the preceding.\n\n[Illustration: Fig. 10.]\n\nWe can represent these sections as a number of solids. Thus the cubes\nA, B, C, D, may be considered as the sections at different intervals in\nthe unknown dimension of a higher cube. Arranged thus their coherence\nin the higher figure is destroyed, they are mere representations.\n\nA motion in the fourth dimension from A through B, C, etc., would be\ncontinuous, but we can only represent it as the occupation of the\npositions A, B, C, etc., in succession. We can exhibit the results of\nthe motion at different stages, but no more.\n\nIn this representation we have left out the distance between one\nsection and another; we have considered the higher body merely as a\nseries of sections, and so left out its contents. The only way to\nexhibit its contents is to call in the aid of the conception of motion.\n\n[Illustration: Fig. 11.]\n\nIf a higher cube passes transverse to our space, it will appear as\na cube isolated in space, the part that has not come into our space\nand the part that has passed through will not be visible. The gradual\npassing through our space would appear as the change of the matter\nof the cube before us. One material particle in it is succeeded by\nanother, neither coming nor going in any direction we can point to. In\nthis manner, by the duration of the figure, we can exhibit the higher\ndimensionality of it; a cube of our matter, under the circumstances\nsupposed, namely, that it has a motion transverse to our space, would\ninstantly disappear. A higher cube would last till it had passed\ntransverse to our space by its whole distance of extension in the\nfourth dimension.\n\nAs the plane being can think of the cube as consisting of sections,\neach like a figure he knows, extending away from his plane, so we can\nthink of a higher solid as composed of sections, each like a solid\nwhich we know, but extending away from our space.\n\nThus, taking a higher cube, we can look on it as starting from a cube\nin our space and extending in the unknown dimension.\n\n[Illustration: Fig. 12.]\n\nTake the face A and conceive it to exist as simply a face, a square\nwith no thickness. From this face the cube in our space extends by the\noccupation of space which we can see.\n\nBut from this face there extends equally a cube in the unknown\ndimension. We can think of the higher cube, then, by taking the set\nof sections A, B, C, D, etc., and considering that from each of them\nthere runs a cube. These cubes have nothing in common with each other,\nand of each of them in its actual position all that we can have in our\nspace is an isolated square. It is obvious that we can take our series\nof sections in any manner we please. We can take them parallel, for\ninstance, to any one of the three isolated faces shown in the figure.\nCorresponding to the three series of sections at right angles to each\nother, which we can make of the cube in space, we must conceive of the\nhigher cube, as composed of cubes starting from squares parallel to the\nfaces of the cube, and of these cubes all that exist in our space are\nthe isolated squares from which they start.\n\n\n\n\n CHAPTER III\n\n THE SIGNIFICANCE OF A FOUR-DIMENSIONAL EXISTENCE\n\n\nHaving now obtained the conception of a four-dimensional space, and\nhaving formed the analogy which, without any further geometrical\ndifficulties, enables us to enquire into its properties, I will refer\nthe reader, whose interest is principally in the mechanical aspect,\nto Chapters VI. and VII. In the present chapter I will deal with\nthe general significance of the enquiry, and in the next with the\nhistorical origin of the idea.\n\nFirst, with regard to the question of whether there is any evidence\nthat we are really in four-dimensional space, I will go back to the\nanalogy of the plane world.\n\nA being in a plane world could not have any experience of\nthree-dimensional shapes, but he could have an experience of\nthree-dimensional movements.\n\nWe have seen that his matter must be supposed to have an extension,\nthough a very small one, in the third dimension. And thus, in the\nsmall particles of his matter, three-dimensional movements may well\nbe conceived to take place. Of these movements he would only perceive\nthe resultants. Since all movements of an observable size in the plane\nworld are two-dimensional, he would only perceive the resultants in\ntwo dimensions of the small three-dimensional movements. Thus, there\nwould be phenomena which he could not explain by his theory of\nmechanics\u2014motions would take place which he could not explain by his\ntheory of motion. Hence, to determine if we are in a four-dimensional\nworld, we must examine the phenomena of motion in our space. If\nmovements occur which are not explicable on the suppositions of our\nthree-dimensional mechanics, we should have an indication of a possible\nfour-dimensional motion, and if, moreover, it could be shown that such\nmovements would be a consequence of a four-dimensional motion in the\nminute particles of bodies or of the ether, we should have a strong\npresumption in favour of the reality of the fourth dimension.\n\nBy proceeding in the direction of finer and finer subdivision, we come\nto forms of matter possessing properties different to those of the\nlarger masses. It is probable that at some stage in this process we\nshould come to a form of matter of such minute subdivision that its\nparticles possess a freedom of movement in four dimensions. This form\nof matter I speak of as four-dimensional ether, and attribute to it\nproperties approximating to those of a perfect liquid.\n\nDeferring the detailed discussion of this form of matter to Chapter\nVI., we will now examine the means by which a plane being would come to\nthe conclusion that three-dimensional movements existed in his world,\nand point out the analogy by which we can conclude the existence of\nfour-dimensional movements in our world. Since the dimensions of the\nmatter in his world are small in the third direction, the phenomena in\nwhich he would detect the motion would be those of the small particles\nof matter.\n\nSuppose that there is a ring in his plane. We can imagine currents\nflowing round the ring in either of two opposite directions. These\nwould produce unlike effects, and give rise to two different fields\nof influence. If the ring with a current in it in one direction be\ntaken up and turned over, and put down again on the plane, it would be\nidentical with the ring having a current in the opposite direction. An\noperation of this kind would be impossible to the plane being. Hence\nhe would have in his space two irreconcilable objects, namely, the\ntwo fields of influence due to the two rings with currents in them in\nopposite directions. By irreconcilable objects in the plane I mean\nobjects which cannot be thought of as transformed one into the other by\nany movement in the plane.\n\nInstead of currents flowing in the rings we can imagine a different\nkind of current. Imagine a number of small rings strung on the original\nring. A current round these secondary rings would give two varieties\nof effect, or two different fields of influence, according to its\ndirection. These two varieties of current could be turned one into\nthe other by taking one of the rings up, turning it over, and putting\nit down again in the plane. This operation is impossible to the plane\nbeing, hence in this case also there would be two irreconcilable fields\nin the plane. Now, if the plane being found two such irreconcilable\nfields and could prove that they could not be accounted for by currents\nin the rings, he would have to admit the existence of currents round\nthe rings\u2014that is, in rings strung on the primary ring. Thus he would\ncome to admit the existence of a three-dimensional motion, for such a\ndisposition of currents is in three dimensions.\n\nNow in our space there are two fields of different properties, which\ncan be produced by an electric current flowing in a closed circuit or\nring. These two fields can be changed one into the other by reversing\nthe currents, but they cannot be changed one into the other by any\nturning about of the rings in our space; for the disposition of the\nfield with regard to the ring itself is different when we turn the\nring, over and when we reverse the direction of the current in the ring.\n\nAs hypotheses to explain the differences of these two fields and their\neffects we can suppose the following kinds of space motions:\u2014First, a\ncurrent along the conductor; second, a current round the conductor\u2014that\nis, of rings of currents strung on the conductor as an axis. Neither of\nthese suppositions accounts for facts of observation.\n\nHence we have to make the supposition of a four-dimensional motion.\nWe find that a four-dimensional rotation of the nature explained in a\nsubsequent chapter, has the following characteristics:\u2014First, it would\ngive us two fields of influence, the one of which could be turned into\nthe other by taking the circuit up into the fourth dimension, turning\nit over, and putting it down in our space again, precisely as the two\nkinds of fields in the plane could be turned one into the other by a\nreversal of the current in our space. Second, it involves a phenomenon\nprecisely identical with that most remarkable and mysterious feature of\nan electric current, namely that it is a field of action, the rim of\nwhich necessarily abuts on a continuous boundary formed by a conductor.\nHence, on the assumption of a four-dimensional movement in the region\nof the minute particles of matter, we should expect to find a motion\nanalogous to electricity.\n\nNow, a phenomenon of such universal occurrence as electricity cannot be\ndue to matter and motion in any very complex relation, but ought to be\nseen as a simple and natural consequence of their properties. I infer\nthat the difficulty in its theory is due to the attempt to explain a\nfour-dimensional phenomenon by a three-dimensional geometry.\n\nIn view of this piece of evidence we cannot disregard that afforded\nby the existence of symmetry. In this connection I will allude to the\nsimple way of producing the images of insects, sometimes practised by\nchildren. They put a few blots of ink in a straight line on a piece of\npaper, fold the paper along the blots, and on opening it the lifelike\npresentment of an insect is obtained. If we were to find a multitude\nof these figures, we should conclude that they had originated from a\nprocess of folding over; the chances against this kind of reduplication\nof parts is too great to admit of the assumption that they had been\nformed in any other way.\n\nThe production of the symmetrical forms of organised beings, though not\nof course due to a turning over of bodies of any appreciable size in\nfour-dimensional space, can well be imagined as due to a disposition in\nthat manner of the smallest living particles from which they are built\nup. Thus, not only electricity, but life, and the processes by which we\nthink and feel, must be attributed to that region of magnitude in which\nfour-dimensional movements take place.\n\nI do not mean, however, that life can be explained as a\nfour-dimensional movement. It seems to me that the whole bias of\nthought, which tends to explain the phenomena of life and volition, as\ndue to matter and motion in some peculiar relation, is adopted rather\nin the interests of the explicability of things than with any regard to\nprobability.\n\nOf course, if we could show that life were a phenomenon of motion, we\nshould be able to explain a great deal that is at present obscure. But\nthere are two great difficulties in the way. It would be necessary to\nshow that in a germ capable of developing into a living being, there\nwere modifications of structure capable of determining in the developed\ngerm all the characteristics of its form, and not only this, but of\ndetermining those of all the descendants of such a form in an infinite\nseries. Such a complexity of mechanical relations, undeniable though\nit be, cannot surely be the best way of grouping the phenomena and\ngiving a practical account of them. And another difficulty is this,\nthat no amount of mechanical adaptation would give that element of\nconsciousness which we possess, and which is shared in to a modified\ndegree by the animal world.\n\nIn those complex structures which men build up and direct, such as a\nship or a railway train (and which, if seen by an observer of such a\nsize that the men guiding them were invisible, would seem to present\nsome of the phenomena of life) the appearance of animation is not due\nto any diffusion of life in the material parts of the structure, but to\nthe presence of a living being.\n\nThe old hypothesis of a soul, a living organism within the visible one,\nappears to me much more rational than the attempt to explain life as a\nform of motion. And when we consider the region of extreme minuteness\ncharacterised by four-dimensional motion the difficulty of conceiving\nsuch an organism alongside the bodily one disappears. Lord Kelvin\nsupposes that matter is formed from the ether. We may very well suppose\nthat the living organisms directing the material ones are co-ordinate\nwith them, not composed of matter, but consisting of etherial bodies,\nand as such capable of motion through the ether, and able to originate\nmaterial living bodies throughout the mineral.\n\nHypotheses such as these find no immediate ground for proof or disproof\nin the physical world. Let us, therefore, turn to a different field,\nand, assuming that the human soul is a four-dimensional being, capable\nin itself of four dimensional movements, but in its experiences through\nthe senses limited to three dimensions, ask if the history of thought,\nof these productivities which characterise man, correspond to our\nassumption. Let us pass in review those steps by which man, presumably\na four-dimensional being, despite his bodily environment, has come to\nrecognise the fact of four-dimensional existence.\n\nDeferring this enquiry to another chapter, I will here recapitulate the\nargument in order to show that our purpose is entirely practical and\nindependent of any philosophical or metaphysical considerations.\n\nIf two shots are fired at a target, and the second bullet hits it\nat a different place to the first, we suppose that there was some\ndifference in the conditions under which the second shot was fired\nfrom those affecting the first shot. The force of the powder, the\ndirection of aim, the strength of the wind, or some condition must\nhave been different in the second case, if the course of the bullet\nwas not exactly the same as in the first case. Corresponding to every\ndifference in a result there must be some difference in the antecedent\nmaterial conditions. By tracing out this chain of relations we explain\nnature.\n\nBut there is also another mode of explanation which we apply. If we ask\nwhat was the cause that a certain ship was built, or that a certain\nstructure was erected, we might proceed to investigate the changes in\nthe brain cells of the men who designed the works. Every variation in\none ship or building from another ship or building is accompanied by\na variation in the processes that go on in the brain matter of the\ndesigners. But practically this would be a very long task.\n\nA more effective mode of explaining the production of the ship or\nbuilding would be to enquire into the motives, plans, and aims of the\nmen who constructed them. We obtain a cumulative and consistent body of\nknowledge much more easily and effectively in the latter way.\n\nSometimes we apply the one, sometimes the other mode of explanation.\n\nBut it must be observed that the method of explanation founded on\naim, purpose, volition, always presupposes a mechanical system on\nwhich the volition and aim works. The conception of man as willing and\nacting from motives involves that of a number of uniform processes of\nnature which he can modify, and of which he can make application. In\nthe mechanical conditions of the three-dimensional world, the only\nvolitional agency which we can demonstrate is the human agency. But\nwhen we consider the four-dimensional world the conclusion remains\nperfectly open.\n\nThe method of explanation founded on purpose and aim does not, surely,\nsuddenly begin with man and end with him. There is as much behind the\nexhibition of will and motive which we see in man as there is behind\nthe phenomena of movement; they are co-ordinate, neither to be resolved\ninto the other. And the commencement of the investigation of that will\nand motive which lies behind the will and motive manifested in the\nthree-dimensional mechanical field is in the conception of a soul\u2014a\nfour-dimensional organism, which expresses its higher physical being\nin the symmetry of the body, and gives the aims and motives of human\nexistence.\n\nOur primary task is to form a systematic knowledge of the phenomena\nof a four-dimensional world and find those points in which this\nknowledge must be called in to complete our mechanical explanation of\nthe universe. But a subsidiary contribution towards the verification\nof the hypothesis may be made by passing in review the history of\nhuman thought, and enquiring if it presents such features as would be\nnaturally expected on this assumption.\n\n\n\n\n CHAPTER IV\n\n THE FIRST CHAPTER IN THE HISTORY OF FOUR SPACE\n\n\nParmenides, and the Asiatic thinkers with whom he is in close\naffinity, propound a theory of existence which is in close accord\nwith a conception of a possible relation between a higher and a lower\ndimensional space. This theory, prior and in marked contrast to the\nmain stream of thought, which we shall afterwards describe, forms a\nclosed circle by itself. It is one which in all ages has had a strong\nattraction for pure intellect, and is the natural mode of thought for\nthose who refrain from projecting their own volition into nature under\nthe guise of causality.\n\nAccording to Parmenides of the school of Elea the all is one, unmoving\nand unchanging. The permanent amid the transient\u2014that foothold for\nthought, that solid ground for feeling on the discovery of which\ndepends all our life\u2014is no phantom; it is the image amidst deception of\ntrue being, the eternal, the unmoved, the one. Thus says Parmenides.\n\nBut how explain the shifting scene, these mutations of things!\n\n\u201cIllusion,\u201d answers Parmenides. Distinguishing between truth and\nerror, he tells of the true doctrine of the one\u2014the false opinion of a\nchanging world. He is no less memorable for the manner of his advocacy\nthan for the cause he advocates. It is as if from his firm foothold\nof being he could play with the thoughts under the burden of which\nothers laboured, for from him springs that fluency of supposition and\nhypothesis which forms the texture of Plato\u2019s dialectic.\n\nCan the mind conceive a more delightful intellectual picture than that\nof Parmenides, pointing to the one, the true, the unchanging, and yet\non the other hand ready to discuss all manner of false opinion, forming\na cosmogony too, false \u201cbut mine own\u201d after the fashion of the time?\n\nIn support of the true opinion he proceeded by the negative way of\nshowing the self-contradictions in the ideas of change and motion.\nIt is doubtful if his criticism, save in minor points, has ever been\nsuccessfully refuted. To express his doctrine in the ponderous modern\nway we must make the statement that motion is phenomenal, not real.\n\nLet us represent his doctrine.\n\n[Illustration: Fig. 13.]\n\nImagine a sheet of still water into which a slanting stick is being\nlowered with a motion vertically downwards. Let 1, 2, 3 (Fig. 13),\nbe three consecutive positions of the stick. A, B, C, will be three\nconsecutive positions of the meeting of the stick, with the surface of\nthe water. As the stick passes down, the meeting will move from A on to\nB and C.\n\nSuppose now all the water to be removed except a film. At the meeting\nof the film and the stick there will be an interruption of the film.\nIf we suppose the film to have a property, like that of a soap bubble,\nof closing up round any penetrating object, then as the stick goes\nvertically downwards the interruption in the film will move on.\n\n[Illustration: Fig. 14.]\n\nIf we pass a spiral through the film the intersection will give a point\nmoving in a circle shown by the dotted lines in the figure. Suppose\nnow the spiral to be still and the film to move vertically upwards,\nthe whole spiral will be represented in the film of the consecutive\npositions of the point of intersection. In the film the permanent\nexistence of the spiral is experienced as a time series\u2014the record\nof traversing the spiral is a point moving in a circle. If now we\nsuppose a consciousness connected with the film in such a way that the\nintersection of the spiral with the film gives rise to a conscious\nexperience, we see that we shall have in the film a point moving in a\ncircle, conscious of its motion, knowing nothing of that real spiral\nthe record of the successive intersections of which by the film is the\nmotion of the point.\n\nIt is easy to imagine complicated structures of the nature of the\nspiral, structures consisting of filaments, and to suppose also that\nthese structures are distinguishable from each other at every section.\nIf we consider the intersections of these filaments with the film as it\npasses to be the atoms constituting a filmar universe, we shall have in\nthe film a world of apparent motion; we shall have bodies corresponding\nto the filamentary structure, and the positions of these structures\nwith regard to one another will give rise to bodies in the film moving\namongst one another. This mutual motion is apparent merely. The reality\nis of permanent structures stationary, and all the relative motions\naccounted for by one steady movement of the film as a whole.\n\nThus we can imagine a plane world, in which all the variety of motion\nis the phenomenon of structures consisting of filamentary atoms\ntraversed by a plane of consciousness. Passing to four dimensions and\nour space, we can conceive that all things and movements in our world\nare the reading off of a permanent reality by a space of consciousness.\nEach atom at every moment is not what it was, but a new part of that\nendless line which is itself. And all this system successively revealed\nin the time which is but the succession of consciousness, separate\nas it is in parts, in its entirety is one vast unity. Representing\nParmenides\u2019 doctrine thus, we gain a firmer hold on it than if we\nmerely let his words rest, grand and massive, in our minds. And we have\ngained the means also of representing phases of that Eastern thought\nto which Parmenides was no stranger. Modifying his uncompromising\ndoctrine, let us suppose, to go back to the plane of consciousness\nand the structure of filamentary atoms, that these structures are\nthemselves moving\u2014are acting, living. Then, in the transverse motion\nof the film, there would be two phenomena of motion, one due to the\nreading off in the film of the permanent existences as they are in\nthemselves, and another phenomenon of motion due to the modification of\nthe record of the things themselves, by their proper motion during the\nprocess of traversing them.\n\nThus a conscious being in the plane would have, as it were, a\ntwo-fold experience. In the complete traversing of the structure, the\nintersection of which with the film gives his conscious all, the main\nand principal movements and actions which he went through would be the\nrecord of his higher self as it existed unmoved and unacting. Slight\nmodifications and deviations from these movements and actions would\nrepresent the activity and self-determination of the complete being, of\nhis higher self.\n\nIt is admissible to suppose that the consciousness in the plane has\na share in that volition by which the complete existence determines\nitself. Thus the motive and will, the initiative and life, of the\nhigher being, would be represented in the case of the being in the\nfilm by an initiative and a will capable, not of determining any great\nthings or important movements in his existence, but only of small and\nrelatively insignificant activities. In all the main features of his\nlife his experience would be representative of one state of the higher\nbeing whose existence determines his as the film passes on. But in his\nminute and apparently unimportant actions he would share in that will\nand determination by which the whole of the being he really is acts and\nlives.\n\nAn alteration of the higher being would correspond to a different life\nhistory for him. Let us now make the supposition that film after film\ntraverses these higher structures, that the life of the real being is\nread off again and again in successive waves of consciousness. There\nwould be a succession of lives in the different advancing planes of\nconsciousness, each differing from the preceding, and differing in\nvirtue of that will and activity which in the preceding had not been\ndevoted to the greater and apparently most significant things in life,\nbut the minute and apparently unimportant. In all great things the\nbeing of the film shares in the existence of his higher self as it is\nat any one time. In the small things he shares in that volition by\nwhich the higher being alters and changes, acts and lives.\n\nThus we gain the conception of a life changing and developing as a\nwhole, a life in which our separation and cessation and fugitiveness\nare merely apparent, but which in its events and course alters,\nchanges, develops; and the power of altering and changing this whole\nlies in the will and power the limited being has of directing, guiding,\naltering himself in the minute things of his existence.\n\nTransferring our conceptions to those of an existence in a higher\ndimensionality traversed by a space of consciousness, we have an\nillustration of a thought which has found frequent and varied\nexpression. When, however, we ask ourselves what degree of truth\nthere lies in it, we must admit that, as far as we can see, it is\nmerely symbolical. The true path in the investigation of a higher\ndimensionality lies in another direction.\n\nThe significance of the Parmenidean doctrine lies in this that here, as\nagain and again, we find that those conceptions which man introduces of\nhimself, which he does not derive from the mere record of his outward\nexperience, have a striking and significant correspondence to the\nconception of a physical existence in a world of a higher space. How\nclose we come to Parmenides\u2019 thought by this manner of representation\nit is impossible to say. What I want to point out is the adequateness\nof the illustration, not only to give a static model of his doctrine,\nbut one capable as it were, of a plastic modification into a\ncorrespondence into kindred forms of thought. Either one of two things\nmust be true\u2014that four-dimensional conceptions give a wonderful power\nof representing the thought of the East, or that the thinkers of the\nEast must have been looking at and regarding four-dimensional existence.\n\nComing now to the main stream of thought we must dwell in some detail\non Pythagoras, not because of his direct relation to the subject, but\nbecause of his relation to investigators who came later.\n\nPythagoras invented the two-way counting. Let us represent the\nsingle-way counting by the posits _aa_, _ab_, _ac_, _ad_, using these\npairs of letters instead of the numbers 1, 2, 3, 4. I put an _a_ in\neach case first for a reason which will immediately appear.\n\nWe have a sequence and order. There is no conception of distance\nnecessarily involved. The difference between the posits is one of\norder not of distance\u2014only when identified with a number of equal\nmaterial things in juxtaposition does the notion of distance arise.\n\nNow, besides the simple series I can have, starting from _aa_, _ba_,\n_ca_, _da_, from _ab_, _bb_, _cb_, _db_, and so on, and forming a\nscheme:\n\n _da_ _db_ _dc_ _dd_\n _ca_ _cb_ _cc_ _cd_\n _ba_ _bb_ _bc_ _bd_\n _aa_ _ab_ _ac_ _ad_\n\nThis complex or manifold gives a two-way order. I can represent it by\na set of points, if I am on my guard against assuming any relation of\ndistance.\n\n[Illustration: Fig. 15.]\n\nPythagoras studied this two-fold way of counting in reference to\nmaterial bodies, and discovered that most remarkable property of the\ncombination of number and matter that bears his name.\n\nThe Pythagorean property of an extended material system can be\nexhibited in a manner which will be of use to us afterwards, and which\ntherefore I will employ now instead of using the kind of figure which\nhe himself employed.\n\nConsider a two-fold field of points arranged in regular rows. Such a\nfield will be presupposed in the following argument.\n\n[Illustration: Fig. 16. 1 and 2]\n\nIt is evident that in fig. 16 four of the points determine a square,\nwhich square we may take as the unit of measurement for areas. But we\ncan also measure areas in another way.\n\nFig. 16 (1) shows four points determining a square.\n\nBut four squares also meet in a point, fig. 16 (2).\n\nHence a point at the corner of a square belongs equally to four\nsquares.\n\nThus we may say that the point value of the square shown is one point,\nfor if we take the square in fig. 16 (1) it has four points, but each\nof these belong equally to four other squares. Hence one fourth of each\nof them belongs to the square (1) in fig. 16. Thus the point value of\nthe square is one point.\n\nThe result of counting the points is the same as that arrived at by\nreckoning the square units enclosed.\n\nHence, if we wish to measure the area of any square we can take the\nnumber of points it encloses, count these as one each, and take\none-fourth of the number of points at its corners.\n\n[Illustration: Fig. 17.]\n\nNow draw a diagonal square as shown in fig. 17. It contains one point\nand the four corners count for one point more; hence its point value is\n2. The value is the measure of its area\u2014the size of this square is two\nof the unit squares.\n\nLooking now at the sides of this figure we see that there is a unit\nsquare on each of them\u2014the two squares contain no points, but have four\ncorner points each, which gives the point value of each as one point.\n\nHence we see that the square on the diagonal is equal to the squares\non the two sides; or as it is generally expressed, the square on the\nhypothenuse is equal to the sum of the squares on the sides.\n\n[Illustration: Fig. 18.]\n\nNoticing this fact we can proceed to ask if it is always true. Drawing\nthe square shown in fig. 18, we can count the number of its points.\nThere are five altogether. There are four points inside the square on\nthe diagonal, and hence, with the four points at its corners the point\nvalue is 5\u2014that is, the area is 5. Now the squares on the sides are\nrespectively of the area 4 and 1. Hence in this case also the square\non the diagonal is equal to the sum of the square on the sides. This\nproperty of matter is one of the first great discoveries of applied\nmathematics. We shall prove afterwards that it is not a property of\nspace. For the present it is enough to remark that the positions in\nwhich the points are arranged is entirely experimental. It is by means\nof equal pieces of some material, or the same piece of material moved\nfrom one place to another, that the points are arranged.\n\nPythagoras next enquired what the relation must be so that a square\ndrawn slanting-wise should be equal to one straight-wise. He found that\na square whose side is five can be placed either rectangularly along\nthe lines of points, or in a slanting position. And this square is\nequivalent to two squares of sides 4 and 3.\n\nHere he came upon a numerical relation embodied in a property of\nmatter. Numbers immanent in the objects produced the equality so\nsatisfactory for intellectual apprehension. And he found that numbers\nwhen immanent in sound\u2014when the strings of a musical instrument were\ngiven certain definite proportions of length\u2014were no less captivating\nto the ear than the equality of squares was to the reason. What wonder\nthen that he ascribed an active power to number!\n\nWe must remember that, sharing like ourselves the search for the\npermanent in changing phenomena, the Greeks had not that conception of\nthe permanent in matter that we have. To them material things were not\npermanent. In fire solid things would vanish; absolutely disappear.\nRock and earth had a more stable existence, but they too grew and\ndecayed. The permanence of matter, the conservation of energy, were\nunknown to them. And that distinction which we draw so readily between\nthe fleeting and permanent causes of sensation, between a sound and\na material object, for instance, had not the same meaning to them\nwhich it has for us. Let us but imagine for a moment that material\nthings are fleeting, disappearing, and we shall enter with a far better\nappreciation into that search for the permanent which, with the Greeks,\nas with us, is the primary intellectual demand.\n\nWhat is that which amid a thousand forms is ever the same, which we can\nrecognise under all its vicissitudes, of which the diverse phenomena\nare the appearances?\n\nTo think that this is number is not so very wide of the mark. With\nan intellectual apprehension which far outran the evidences for its\napplication, the atomists asserted that there were everlasting material\nparticles, which, by their union, produced all the varying forms and\nstates of bodies. But in view of the observed facts of nature as\nthen known, Aristotle, with perfect reason, refused to accept this\nhypothesis.\n\nHe expressly states that there is a change of quality, and that the\nchange due to motion is only one of the possible modes of change.\n\nWith no permanent material world about us, with the fleeting, the\nunpermanent, all around we should, I think, be ready to follow\nPythagoras in his identification of number with that principle which\nsubsists amidst all changes, which in multitudinous forms we apprehend\nimmanent in the changing and disappearing substance of things.\n\nAnd from the numerical idealism of Pythagoras there is but a step to\nthe more rich and full idealism of Plato. That which is apprehended by\nthe sense of touch we put as primary and real, and the other senses we\nsay are merely concerned with appearances. But Plato took them all as\nvalid, as giving qualities of existence. That the qualities were not\npermanent in the world as given to the senses forced him to attribute\nto them a different kind of permanence. He formed the conception of a\nworld of ideas, in which all that really is, all that affects us and\ngives the rich and wonderful wealth of our experience, is not fleeting\nand transitory, but eternal. And of this real and eternal we see in the\nthings about us the fleeting and transient images.\n\nAnd this world of ideas was no exclusive one, wherein was no place\nfor the innermost convictions of the soul and its most authoritative\nassertions. Therein existed justice, beauty\u2014the one, the good, all\nthat the soul demanded to be. The world of ideas, Plato\u2019s wonderful\ncreation preserved for man, for his deliberate investigation and their\nsure development, all that the rude incomprehensible changes of a harsh\nexperience scatters and destroys.\n\nPlato believed in the reality of ideas. He meets us fairly and\nsquarely. Divide a line into two parts, he says; one to represent\nthe real objects in the world, the other to represent the transitory\nappearances, such as the image in still water, the glitter of the sun\non a bright surface, the shadows on the clouds.\n\n A B\n \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014|\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014-\n Real things: Appearances:\n _e.g._, the sun. _e.g._, the reflection of the sun.\n\nTake another line and divide it into two parts, one representing\nour ideas, the ordinary occupants of our minds, such as whiteness,\nequality, and the other representing our true knowledge, which is of\neternal principles, such as beauty, goodness.\n\n A^1 B^1\n \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014|\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014-\n Eternal principles, Appearances in the mind,\n as beauty as whiteness, equality\n\nThen as A is to B, so is A^1 to B^1\n\nThat is, the soul can proceed, going away from real things to a region\nof perfect certainty, where it beholds what is, not the scattered\nreflections; beholds the sun, not the glitter on the sands; true being,\nnot chance opinion.\n\nNow, this is to us, as it was to Aristotle, absolutely inconceivable\nfrom a scientific point of view. We can understand that a being is\nknown in the fulness of his relations; it is in his relations to his\ncircumstances that a man\u2019s character is known; it is in his acts under\nhis conditions that his character exists. We cannot grasp or conceive\nany principle of individuation apart from the fulness of the relations\nto the surroundings.\n\nBut suppose now that Plato is talking about the higher man\u2014the\nfour-dimensional being that is limited in our external experience to a\nthree-dimensional world. Do not his words begin to have a meaning? Such\na being would have a consciousness of motion which is not as the motion\nhe can see with the eyes of the body. He, in his own being, knows a\nreality to which the outward matter of this too solid earth is flimsy\nsuperficiality. He too knows a mode of being, the fulness of relations,\nin which can only be represented in the limited world of sense, as the\npainter unsubstantially portrays the depths of woodland, plains, and\nair. Thinking of such a being in man, was not Plato\u2019s line well divided?\n\nIt is noteworthy that, if Plato omitted his doctrine of the independent\norigin of ideas, he would present exactly the four-dimensional\nargument; a real thing as we think it is an idea. A plane being\u2019s idea\nof a square object is the idea of an abstraction, namely, a geometrical\nsquare. Similarly our idea of a solid thing is an abstraction, for\nin our idea there is not the four-dimensional thickness which is\nnecessary, however slight, to give reality. The argument would then\nrun, as a shadow is to a solid object, so is the solid object to the\nreality. Thus A and B\u00b4 would be identified.\n\nIn the allegory which I have already alluded to, Plato in almost as\nmany words shows forth the relation between existence in a superficies\nand in solid space. And he uses this relation to point to the\nconditions of a higher being.\n\nHe imagines a number of men prisoners, chained so that they look at\nthe wall of a cavern in which they are confined, with their backs to\nthe road and the light. Over the road pass men and women, figures and\nprocessions, but of all this pageant all that the prisoners behold\nis the shadow of it on the wall whereon they gaze. Their own shadows\nand the shadows of the things in the world are all that they see, and\nidentifying themselves with their shadows related as shadows to a world\nof shadows, they live in a kind of dream.\n\nPlato imagines one of their number to pass out from amongst them\ninto the real space world, and then returning to tell them of their\ncondition.\n\nHere he presents most plainly the relation between existence in a plane\nworld and existence in a three-dimensional world. And he uses this\nillustration as a type of the manner in which we are to proceed to a\nhigher state from the three-dimensional life we know.\n\nIt must have hung upon the weight of a shadow which path he\ntook!\u2014whether the one we shall follow toward the higher solid and the\nfour-dimensional existence, or the one which makes ideas the higher\nrealities, and the direct perception of them the contact with the truer\nworld.\n\nPassing on to Aristotle, we will touch on the points which most\nimmediately concern our enquiry.\n\nJust as a scientific man of the present day in reviewing the\nspeculations of the ancient world would treat them with a curiosity\nhalf amused but wholly respectful, asking of each and all wherein lay\ntheir relation to fact, so Aristotle, in discussing the philosophy\nof Greece as he found it, asks, above all other things: \u201cDoes this\nrepresent the world? In this system is there an adequate presentation\nof what is?\u201d\n\nHe finds them all defective, some for the very reasons which we esteem\nthem most highly, as when he criticises the Atomic theory for its\nreduction of all change to motion. But in the lofty march of his reason\nhe never loses sight of the whole; and that wherein our views differ\nfrom his lies not so much in a superiority of our point of view, as\nin the fact which he himself enunciates\u2014that it is impossible for one\nprinciple to be valid in all branches of enquiry. The conceptions\nof one method of investigation are not those of another; and our\ndivergence lies in our exclusive attention to the conceptions useful\nin one way of apprehending nature rather than in any possibility we\nfind in our theories of giving a view of the whole transcending that of\nAristotle.\n\nHe takes account of everything; he does not separate matter and the\nmanifestation of matter; he fires all together in a conception of a\nvast world process in which everything takes part\u2014the motion of a grain\nof dust, the unfolding of a leaf, the ordered motion of the spheres in\nheaven\u2014all are parts of one whole which he will not separate into dead\nmatter and adventitious modifications.\n\nAnd just as our theories, as representative of actuality, fall before\nhis unequalled grasp of fact, so the doctrine of ideas fell. It is\nnot an adequate account of existence, as Plato himself shows in his\n\u201cParmenides\u201d; it only explains things by putting their doubles beside\nthem.\n\nFor his own part Aristotle invented a great marching definition which,\nwith a kind of power of its own, cleaves its way through phenomena\nto limiting conceptions on either hand, towards whose existence all\nexperience points.\n\nIn Aristotle\u2019s definition of matter and form as the constituent of\nreality, as in Plato\u2019s mystical vision of the kingdom of ideas, the\nexistence of the higher dimensionality is implicitly involved.\n\nSubstance according to Aristotle is relative, not absolute. In\neverything that is there is the matter of which it is composed, the\nform which it exhibits; but these are indissolubly connected, and\nneither can be thought without the other.\n\nThe blocks of stone out of which a house is built are the material for\nthe builder; but, as regards the quarrymen, they are the matter of the\nrocks with the form he has imposed on them. Words are the final product\nof the grammarian, but the mere matter of the orator or poet. The atom\nis, with us, that out of which chemical substances are built up, but\nlooked at from another point of view is the result of complex processes.\n\nNowhere do we find finality. The matter in one sphere is the matter,\nplus form, of another sphere of thought. Making an obvious application\nto geometry, plane figures exist as the limitation of different\nportions of the plane by one another. In the bounding lines the\nseparated matter of the plane shows its determination into form. And\nas the plane is the matter relatively to determinations in the plane,\nso the plane itself exists in virtue of the determination of space. A\nplane is that wherein formless space has form superimposed on it, and\ngives an actuality of real relations. We cannot refuse to carry this\nprocess of reasoning a step farther back, and say that space itself is\nthat which gives form to higher space. As a line is the determination\nof a plane, and a plane of a solid, so solid space itself is the\ndetermination of a higher space.\n\nAs a line by itself is inconceivable without that plane which it\nseparates, so the plane is inconceivable without the solids which\nit limits on either hand. And so space itself cannot be positively\ndefined. It is the negation of the possibility of movement in more than\nthree dimensions. The conception of space demands that of a higher\nspace. As a surface is thin and unsubstantial without the substance of\nwhich it is the surface, so matter itself is thin without the higher\nmatter.\n\nJust as Aristotle invented that algebraical method of representing\nunknown quantities by mere symbols, not by lines necessarily\ndeterminate in length as was the habit of the Greek geometers, and so\nstruck out the path towards those objectifications of thought which,\nlike independent machines for reasoning, supply the mathematician\nwith his analytical weapons, so in the formulation of the doctrine\nof matter and form, of potentiality and actuality, of the relativity\nof substance, he produced another kind of objectification of mind\u2014a\ndefinition which had a vital force and an activity of its own.\n\nIn none of his writings, as far as we know, did he carry it to its\nlegitimate conclusion on the side of matter, but in the direction of\nthe formal qualities he was led to his limiting conception of that\nexistence of pure form which lies beyond all known determination\nof matter. The unmoved mover of all things is Aristotle\u2019s highest\nprinciple. Towards it, to partake of its perfection all things move.\nThe universe, according to Aristotle, is an active process\u2014he does\nnot adopt the illogical conception that it was once set in motion\nand has kept on ever since. There is room for activity, will,\nself-determination, in Aristotle\u2019s system, and for the contingent and\naccidental as well. We do not follow him, because we are accustomed to\nfind in nature infinite series, and do not feel obliged to pass on to a\nbelief in the ultimate limits to which they seem to point.\n\nBut apart from the pushing to the limit, as a relative principle\nthis doctrine of Aristotle\u2019s as to the relativity of substance is\nirrefragible in its logic. He was the first to show the necessity\nof that path of thought which when followed leads to a belief in a\nfour-dimensional space.\n\nAntagonistic as he was to Plato in his conception of the practical\nrelation of reason to the world of phenomena, yet in one point he\ncoincided with him. And in this he showed the candour of his intellect.\nHe was more anxious to lose nothing than to explain everything. And\nthat wherein so many have detected an inconsistency, an inability to\nfree himself from the school of Plato, appears to us in connection with\nour enquiry as an instance of the acuteness of his observation. For\nbeyond all knowledge given by the senses Aristotle held that there is\nan active intelligence, a mind not the passive recipient of impressions\nfrom without, but an active and originative being, capable of grasping\nknowledge at first hand. In the active soul Aristotle recognised\nsomething in man not produced by his physical surroundings, something\nwhich creates, whose activity is a knowledge underived from sense.\nThis, he says, is the immortal and undying being in man.\n\nThus we see that Aristotle was not far from the recognition of the\nfour-dimensional existence, both without and within man, and the\nprocess of adequately realising the higher dimensional figures to which\nwe shall come subsequently is a simple reduction to practice of his\nhypothesis of a soul.\n\nThe next step in the unfolding of the drama of the recognition of\nthe soul as connected with our scientific conception of the world,\nand, at the same time, the recognition of that higher of which a\nthree-dimensional world presents the superficial appearance, took place\nmany centuries later. If we pass over the intervening time without a\nword it is because the soul was occupied with the assertion of itself\nin other ways than that of knowledge. When it took up the task in\nearnest of knowing this material world in which it found itself, and of\ndirecting the course of inanimate nature, from that most objective aim\ncame, reflected back as from a mirror, its knowledge of itself.\n\n\n\n\n CHAPTER V\n\n THE SECOND CHAPTER IN THE HISTORY OF FOUR SPACE\n\n\n LOBATCHEWSKY, BOLYAI, AND GAUSS\n\nBefore entering on a description of the work of Lobatchewsky and Bolyai\nit will not be out of place to give a brief account of them, the\nmaterials for which are to be found in an article by Franz Schmidt in\nthe forty-second volume of the _Mathematische Annalen_, and in Engel\u2019s\nedition of Lobatchewsky.\n\nLobatchewsky was a man of the most complete and wonderful talents.\nAs a youth he was full of vivacity, carrying his exuberance so far\nas to fall into serious trouble for hazing a professor, and other\nfreaks. Saved by the good offices of the mathematician Bartels, who\nappreciated his ability, he managed to restrain himself within the\nbounds of prudence. Appointed professor at his own University, Kasan,\nhe entered on his duties under the regime of a pietistic reactionary,\nwho surrounded himself with sycophants and hypocrites. Esteeming\nprobably the interests of his pupils as higher than any attempt at a\nvain resistance, he made himself the tyrant\u2019s right-hand man, doing an\nincredible amount of teaching and performing the most varied official\nduties. Amidst all his activities he found time to make important\ncontributions to science. His theory of parallels is most closely\nconnected with his name, but a study of his writings shows that he was\na man capable of carrying on mathematics in its main lines of advance,\nand of a judgment equal to discerning what these lines were. Appointed\nrector of his University, he died at an advanced age, surrounded by\nfriends, honoured, with the results of his beneficent activity all\naround him. To him no subject came amiss, from the foundations of\ngeometry to the improvement of the stoves by which the peasants warmed\ntheir houses.\n\nHe was born in 1793. His scientific work was unnoticed till, in 1867,\nHouel, the French mathematician, drew attention to its importance.\n\nJohann Bolyai de Bolyai was born in Klausenburg, a town in\nTransylvania, December 15th, 1802.\n\nHis father, Wolfgang Bolyai, a professor in the Reformed College of\nMaros Vasarhely, retained the ardour in mathematical studies which had\nmade him a chosen companion of Gauss in their early student days at\nG\u00f6ttingen.\n\nHe found an eager pupil in Johann. He relates that the boy sprang\nbefore him like a devil. As soon as he had enunciated a problem the\nchild would give the solution and command him to go on further. As a\nthirteen-year-old boy his father sometimes sent him to fill his place\nwhen incapacitated from taking his classes. The pupils listened to him\nwith more attention than to his father for they found him clearer to\nunderstand.\n\nIn a letter to Gauss Wolfgang Bolyai writes:\u2014\n\n \u201cMy boy is strongly built. He has learned to recognise many\n constellations, and the ordinary figures of geometry. He makes apt\n applications of his notions, drawing for instance the positions of the\n stars with their constellations. Last winter in the country, seeing\n Jupiter he asked: \u2018How is it that we can see him from here as well as\n from the town? He must be far off.\u2019 And as to three different places\n to which he had been he asked me to tell him about them in one word. I\n did not know what he meant, and then he asked me if one was in a line\n with the other and all in a row, or if they were in a triangle.\n\n \u201cHe enjoys cutting paper figures with a pair of scissors, and without\n my ever having told him about triangles remarked that a right-angled\n triangle which he had cut out was half of an oblong. I exercise his\n body with care, he can dig well in the earth with his little hands.\n The blossom can fall and no fruit left. When he is fifteen I want to\n send him to you to be your pupil.\u201d\n\nIn Johann\u2019s autobiography he says:\u2014\n\n \u201cMy father called my attention to the imperfections and gaps in the\n theory of parallels. He told me he had gained more satisfactory\n results than his predecessors, but had obtained no perfect and\n satisfying conclusion. None of his assumptions had the necessary\n degree of geometrical certainty, although they sufficed to prove the\n eleventh axiom and appeared acceptable on first sight.\n\n \u201cHe begged of me, anxious not without a reason, to hold myself aloof\n and to shun all investigation on this subject, if I did not wish to\n live all my life in vain.\u201d\n\nJohann, in the failure of his father to obtain any response from Gauss,\nin answer to a letter in which he asked the great mathematician to make\nof his son \u201can apostle of truth in a far land,\u201d entered the Engineering\nSchool at Vienna. He writes from Temesvar, where he was appointed\nsub-lieutenant September, 1823:\u2014\n\n\n \u201cTemesvar, November 3rd, 1823.\n\n \u201cDEAR GOOD FATHER,\n\n \u201cI have so overwhelmingly much to write about my discovery that I know\n no other way of checking myself than taking a quarter of a sheet only\n to write on. I want an answer to my four-sheet letter.\n\n \u201cI am unbroken in my determination to publish a work on Parallels, as\n soon as I have put my material in order and have the means.\n\n \u201cAt present I have not made any discovery, but the way I have followed\n almost certainly promises me the attainment of my object if any\n possibility of it exists.\n\n \u201cI have not got my object yet, but I have produced such stupendous\n things that I was overwhelmed myself, and it would be an eternal shame\n if they were lost. When you see them you will find that it is so. Now\n I can only say that I have made a new world out of nothing. Everything\n that I have sent you before is a house of cards in comparison with a\n tower. I am convinced that it will be no less to my honour than if I\n had already discovered it.\u201d\n\nThe discovery of which Johann here speaks was published as an appendix\nto Wolfgang Bolyai\u2019s _Tentamen_.\n\nSending the book to Gauss, Wolfgang writes, after an interruption of\neighteen years in his correspondence:\u2014\n\n \u201cMy son is first lieutenant of Engineers and will soon be captain.\n He is a fine youth, a good violin player, a skilful fencer, and\n brave, but has had many duels, and is wild even for a soldier. Yet he\n is distinguished\u2014light in darkness and darkness in light. He is an\n impassioned mathematician with extraordinary capacities.... He will\n think more of your judgment on his work than that of all Europe.\u201d\n\nWolfgang received no answer from Gauss to this letter, but sending a\nsecond copy of the book received the following reply:\u2014\n\n \u201cYou have rejoiced me, my unforgotten friend, by your letters. I\n delayed answering the first because I wanted to wait for the arrival\n of the promised little book.\n\n \u201cNow something about your son\u2019s work.\n\n \u201cIf I begin with saying that \u2018I ought not to praise it,\u2019 you will be\n staggered for a moment. But I cannot say anything else. To praise it\n is to praise myself, for the path your son has broken in upon and the\n results to which he has been led are almost exactly the same as my own\n reflections, some of which date from thirty to thirty-five years ago.\n\n \u201cIn fact I am astonished to the uttermost. My intention was to let\n nothing be known in my lifetime about my own work, of which, for the\n rest, but little is committed to writing. Most people have but little\n perception of the problem, and I have found very few who took any\n interest in the views I expressed to them. To be able to do that one\n must first of all have had a real live feeling of what is wanting, and\n as to that most men are completely in the dark.\n\n \u201cStill it was my intention to commit everything to writing in the\n course of time, so that at least it should not perish with me.\n\n \u201cI am deeply surprised that this task can be spared me, and I am most\n of all pleased in this that it is the son of my old friend who has in\n so remarkable a manner preceded me.\u201d\n\nThe impression which we receive from Gauss\u2019s inexplicable silence\ntowards his old friend is swept away by this letter. Hence we breathe\nthe clear air of the mountain tops. Gauss would not have failed to\nperceive the vast significance of his thoughts, sure to be all the\ngreater in their effect on future ages from the want of comprehension\nof the present. Yet there is not a word or a sign in his writing to\nclaim the thought for himself. He published no single line on the\nsubject. By the measure of what he thus silently relinquishes, by\nsuch a measure of a world-transforming thought, we can appreciate his\ngreatness.\n\nIt is a long step from Gauss\u2019s serenity to the disturbed and passionate\nlife of Johann Bolyai\u2014he and Galois, the two most interesting figures\nin the history of mathematics. For Bolyai, the wild soldier, the\nduellist, fell at odds with the world. It is related of him that he was\nchallenged by thirteen officers of his garrison, a thing not unlikely\nto happen considering how differently he thought from every one else.\nHe fought them all in succession\u2014making it his only condition that he\nshould be allowed to play on his violin for an interval between meeting\neach opponent. He disarmed or wounded all his antagonists. It can be\neasily imagined that a temperament such as his was one not congenial to\nhis military superiors. He was retired in 1833.\n\nHis epoch-making discovery awoke no attention. He seems to have\nconceived the idea that his father had betrayed him in some\ninexplicable way by his communications with Gauss, and he challenged\nthe excellent Wolfgang to a duel. He passed his life in poverty, many a\ntime, says his biographer, seeking to snatch himself from dissipation\nand apply himself again to mathematics. But his efforts had no result.\nHe died January 27th, 1860, fallen out with the world and with himself.\n\n\n METAGEOMETRY\n\nThe theories which are generally connected with the names of\nLobatchewsky and Bolyai bear a singular and curious relation to the\nsubject of higher space.\n\nIn order to show what this relation is, I must ask the reader to be\nat the pains to count carefully the sets of points by which I shall\nestimate the volumes of certain figures.\n\nNo mathematical processes beyond this simple one of counting will be\nnecessary.\n\n[Illustration: Fig. 19.]\n\nLet us suppose we have before us in fig. 19 a plane covered with points\nat regular intervals, so placed that every four determine a square.\n\nNow it is evident that as four points determine a square, so four\nsquares meet in a point.\n\n[Illustration: Fig. 20.]\n\nThus, considering a point inside a square as belonging to it, we may\nsay that a point on the corner of a square belongs to it and to three\nothers equally: belongs a quarter of it to each square.\n\n[Illustration: Fig. 21.]\n\n[Illustration: Fig. 22.]\n\nThus the square ACDE (fig. 21) contains one point, and has four points\nat the four corners. Since one-fourth of each of these four belongs to\nthe square, the four together count as one point, and the point value\nof the square is two points\u2014the one inside and the four at the corner\nmake two points belonging to it exclusively.\n\nNow the area of this square is two unit squares, as can be seen by\ndrawing two diagonals in fig. 22.\n\nWe also notice that the square in question is equal to the sum of the\nsquares on the sides AB, BC, of the right-angled triangle ABC. Thus we\nrecognise the proposition that the square on the hypothenuse is equal\nto the sum of the squares on the two sides of a right-angled triangle.\n\nNow suppose we set ourselves the question of determining the\nwhereabouts in the ordered system of points, the end of a line would\ncome when it turned about a point keeping one extremity fixed at the\npoint.\n\nWe can solve this problem in a particular case. If we can find a square\nlying slantwise amongst the dots which is equal to one which goes\nregularly, we shall know that the two sides are equal, and that the\nslanting side is equal to the straight-way side. Thus the volume and\nshape of a figure remaining unchanged will be the test of its having\nrotated about the point, so that we can say that its side in its first\nposition would turn into its side in the second position.\n\nNow, such a square can be found in the one whose side is five units in\nlength.\n\n[Illustration: Fig. 23.]\n\nIn fig. 23, in the square on AB, there are\u2014\n\n 9 points interior 9\n 4 at the corners 1\n 4 sides with 3 on each side, considered as\n 1\u00bd on each side, because belonging\n equally to two squares 6\n\nThe total is 16. There are 9 points in the square on BC.\n\nIn the square on AC there are\u2014\n\n 24 points inside 24\n 4 at the corners 1\n\nor 25 altogether.\n\nHence we see again that the square on the hypothenuse is equal to the\nsquares on the sides.\n\nNow take the square AFHG, which is larger than the square on AB. It\ncontains 25 points.\n\n 16 inside 16\n 16 on the sides, counting as 8\n 4 on the corners 1\n\nmaking 25 altogether.\n\nIf two squares are equal we conclude the sides are equal. Hence, the\nline AF turning round A would move so that it would after a certain\nturning coincide with AC.\n\nThis is preliminary, but it involves all the mathematical difficulties\nthat will present themselves.\n\nThere are two alterations of a body by which its volume is not changed.\n\nOne is the one we have just considered, rotation, the other is what is\ncalled shear.\n\nConsider a book, or heap of loose pages. They can be slid so that each\none slips over the preceding one, and the whole assumes the shape _b_\nin fig. 24.\n\n[Illustration: Fig. 24.]\n\nThis deformation is not shear alone, but shear accompanied by rotation.\n\nShear can be considered as produced in another way.\n\nTake the square ABCD (fig. 25), and suppose that it is pulled out from\nalong one of its diagonals both ways, and proportionately compressed\nalong the other diagonal. It will assume the shape in fig. 26.\n\nThis compression and expansion along two lines at right angles is what\nis called shear; it is equivalent to the sliding illustrated above,\ncombined with a turning round.\n\n[Illustration: Fig. 25.] [Illustration: Fig. 26.]\n\nIn pure shear a body is compressed and extended in two directions at\nright angles to each other, so that its volume remains unchanged.\n\nNow we know that our material bodies resist shear\u2014shear does violence\nto the internal arrangement of their particles, but they turn as wholes\nwithout such internal resistance.\n\nBut there is an exception. In a liquid shear and rotation take place\nequally easily, there is no more resistance against a shear than there\nis against a rotation.\n\nNow, suppose all bodies were to be reduced to the liquid state, in\nwhich they yield to shear and to rotation equally easily, and then\nwere to be reconstructed as solids, but in such a way that shear and\nrotation had interchanged places.\n\nThat is to say, let us suppose that when they had become solids again\nthey would shear without offering any internal resistance, but a\nrotation would do violence to their internal arrangement.\n\nThat is, we should have a world in which shear would have taken the\nplace of rotation.\n\nA shear does not alter the volume of a body: thus an inhabitant living\nin such a world would look on a body sheared as we look on a body\nrotated. He would say that it was of the same shape, but had turned a\nbit round.\n\nLet us imagine a Pythagoras in this world going to work to investigate,\nas is his wont.\n\n[Illustration: Fig. 27.] [Illustration: Fig. 28.]\n\nFig. 27 represents a square unsheared. Fig. 28 represents a square\nsheared. It is not the figure into which the square in fig. 27 would\nturn, but the result of shear on some square not drawn. It is a simple\nslanting placed figure, taken now as we took a simple slanting placed\nsquare before. Now, since bodies in this world of shear offer no\ninternal resistance to shearing, and keep their volume when sheared,\nan inhabitant accustomed to them would not consider that they altered\ntheir shape under shear. He would call ACDE as much a square as the\nsquare in fig. 27. We will call such figures shear squares. Counting\nthe dots in ACDE, we find\u2014\n\n 2 inside = 2\n 4 at corners = 1\n\nor a total of 3.\n\nNow, the square on the side AB has 4 points, that on BC has 1 point.\nHere the shear square on the hypothenuse has not 5 points but 3; it is\nnot the sum of the squares on the sides, but the difference.\n\nThis relation always holds. Look at fig. 29.\n\n[Illustration: Fig. 29.]\n\nShear square on hypothenuse\u2014\n\n 7 internal 7\n 4 at corners 1\n \u2014\n 8\n\n\n[Illustration: Fig. 29 _bis_.]\n\nSquare on one side\u2014which the reader can draw for himself\u2014\n\n 4 internal 4\n 8 on sides 4\n 4 at corners 1\n \u2014\n 9\n\nand the square on the other side is 1. Hence in this case again the\ndifference is equal to the shear square on the hypothenuse, 9 - 1 = 8.\n\nThus in a world of shear the square on the hypothenuse would be equal\nto the difference of the squares on the sides of a right-angled\ntriangle.\n\nIn fig. 29 _bis_ another shear square is drawn on which the above\nrelation can be tested.\n\nWhat now would be the position a line on turning by shear would take up?\n\nWe must settle this in the same way as previously with our turning.\n\nSince a body sheared remains the same, we must find two equal bodies,\none in the straight way, one in the slanting way, which have the same\nvolume. Then the side of one will by turning become the side of the\nother, for the two figures are each what the other becomes by a shear\nturning.\n\nWe can solve the problem in a particular case\u2014\n\n[Illustration: Fig. 30.]\n\nIn the figure ACDE (fig. 30) there are\u2014\n\n 15 inside 15\n 4 at corners 1\n\na total of 16.\n\nNow in the square ABGF, there are 16\u2014\n\n 9 inside 9\n 12 on sides 6\n 4 at corners 1\n \u2014\n 16\n\nHence the square on AB would, by the shear turning, become the shear\nsquare ACDE.\n\nAnd hence the inhabitant of this world would say that the line AB\nturned into the line AC. These two lines would be to him two lines of\nequal length, one turned a little way round from the other.\n\nThat is, putting shear in place of rotation, we get a different kind\nof figure, as the result of the shear rotation, from what we got with\nour ordinary rotation. And as a consequence we get a position for the\nend of a line of invariable length when it turns by the shear rotation,\ndifferent from the position which it would assume on turning by our\nrotation.\n\nA real material rod in the shear world would, on turning about A, pass\nfrom the position AB to the position AC. We say that its length alters\nwhen it becomes AC, but this transformation of AB would seem to an\ninhabitant of the shear world like a turning of AB without altering in\nlength.\n\nIf now we suppose a communication of ideas that takes place between\none of ourselves and an inhabitant of the shear world, there would\nevidently be a difference between his views of distance and ours.\n\nWe should say that his line AB increased in length in turning to AC. He\nwould say that our line AF (fig. 23) decreased in length in turning to\nAC. He would think that what we called an equal line was in reality a\nshorter one.\n\nWe should say that a rod turning round would have its extremities in\nthe positions we call at equal distances. So would he\u2014but the positions\nwould be different. He could, like us, appeal to the properties of\nmatter. His rod to him alters as little as ours does to us.\n\nNow, is there any standard to which we could appeal, to say which of\nthe two is right in this argument? There is no standard.\n\nWe should say that, with a change of position, the configuration and\nshape of his objects altered. He would say that the configuration and\nshape of our objects altered in what we called merely a change of\nposition. Hence distance independent of position is inconceivable, or\npractically distance is solely a property of matter.\n\nThere is no principle to which either party in this controversy could\nappeal. There is nothing to connect the definition of distance with our\nideas rather than with his, except the behaviour of an actual piece of\nmatter.\n\nFor the study of the processes which go on in our world the definition\nof distance given by taking the sum of the squares is of paramount\nimportance to us. But as a question of pure space without making any\nunnecessary assumptions the shear world is just as possible and just as\ninteresting as our world.\n\nIt was the geometry of such conceivable worlds that Lobatchewsky and\nBolyai studied.\n\nThis kind of geometry has evidently nothing to do directly with\nfour-dimensional space.\n\nBut a connection arises in this way. It is evident that, instead of\ntaking a simple shear as I have done, and defining it as that change\nof the arrangement of the particles of a solid which they will undergo\nwithout offering any resistance due to their mutual action, I might\ntake a complex motion, composed of a shear and a rotation together, or\nsome other kind of deformation.\n\nLet us suppose such an alteration picked out and defined as the one\nwhich means simple rotation, then the type, according to which all\nbodies will alter by this rotation, is fixed.\n\nLooking at the movements of this kind, we should say that the objects\nwere altering their shape as well as rotating. But to the inhabitants\nof that world they would seem to be unaltered, and our figures in their\nmotions would seem to them to alter.\n\nIn such a world the features of geometry are different. We have seen\none such difference in the case of our illustration of the world of\nshear, where the square on the hypothenuse was equal to the difference,\nnot the sum, of the squares on the sides.\n\nIn our illustration we have the same laws of parallel lines as in our\nordinary rotation world, but in general the laws of parallel lines are\ndifferent.\n\nIn one of these worlds of a different constitution of matter through\none point there can be two parallels to a given line, in another of\nthem there can be none, that is, although a line be drawn parallel to\nanother it will meet it after a time.\n\nNow it was precisely in this respect of parallels that Lobatchewsky and\nBolyai discovered these different worlds. They did not think of them as\nworlds of matter, but they discovered that space did not necessarily\nmean that our law of parallels is true. They made the distinction\nbetween laws of space and laws of matter, although that is not the\nform in which they stated their results.\n\nThe way in which they were led to these results was the\nfollowing. Euclid had stated the existence of parallel lines as a\npostulate\u2014putting frankly this unproved proposition\u2014that one line and\nonly one parallel to a given straight line can be drawn, as a demand,\nas something that must be assumed. The words of his ninth postulate are\nthese: \u201cIf a straight line meeting two other straight lines makes the\ninterior angles on the same side of it equal to two right angles, the\ntwo straight lines will never meet.\u201d\n\nThe mathematicians of later ages did not like this bald assumption, and\nnot being able to prove the proposition they called it an axiom\u2014the\neleventh axiom.\n\nMany attempts were made to prove the axiom; no one doubted of its\ntruth, but no means could be found to demonstrate it. At last an\nItalian, Sacchieri, unable to find a proof, said: \u201cLet us suppose it\nnot true.\u201d He deduced the results of there being possibly two parallels\nto one given line through a given point, but feeling the waters too\ndeep for the human reason, he devoted the latter half of his book to\ndisproving what he had assumed in the first part.\n\nThen Bolyai and Lobatchewsky with firm step entered on the forbidden\npath. There can be no greater evidence of the indomitable nature of\nthe human spirit, or of its manifest destiny to conquer all those\nlimitations which bind it down within the sphere of sense than this\ngrand assertion of Bolyai and Lobatchewsky.\n\n \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n C D\n \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n A B\nTake a line AB and a point C. We say and see and know that through C\ncan only be drawn one line parallel to AB.\n\nBut Bolyai said: \u201cI will draw two.\u201d Let CD be parallel to AB, that\nis, not meet AB however far produced, and let lines beyond CD also not\nmeet AB; let there be a certain region between CD and CE, in which no\nline drawn meets AB. CE and CD produced backwards through C will give a\nsimilar region on the other side of C.\n\n[Illustration: Fig. 32.]\n\nNothing so triumphantly, one may almost say so insolently, ignoring\nof sense had ever been written before. Men had struggled against the\nlimitations of the body, fought them, despised them, conquered them.\nBut no one had ever thought simply as if the body, the bodily eyes,\nthe organs of vision, all this vast experience of space, had never\nexisted. The age-long contest of the soul with the body, the struggle\nfor mastery, had come to a culmination. Bolyai and Lobatchewsky simply\nthought as if the body was not. The struggle for dominion, the strife\nand combat of the soul were over; they had mastered, and the Hungarian\ndrew his line.\n\nCan we point out any connection, as in the case of Parmenides, between\nthese speculations and higher space? Can we suppose it was any inner\nperception by the soul of a motion not known to the senses, which\nresulted in this theory so free from the bonds of sense? No such\nsupposition appears to be possible.\n\nPractically, however, metageometry had a great influence in bringing\nthe higher space to the front as a working hypothesis. This can\nbe traced to the tendency the mind has to move in the direction\nof least resistance. The results of the new geometry could not be\nneglected, the problem of parallels had occupied a place too prominent\nin the development of mathematical thought for its final solution\nto be neglected. But this utter independence of all mechanical\nconsiderations, this perfect cutting loose from the familiar\nintuitions, was so difficult that almost any other hypothesis was\nmore easy of acceptance, and when Beltrami showed that the geometry\nof Lobatchewsky and Bolyai was the geometry of shortest lines drawn\non certain curved surfaces, the ordinary definitions of measurement\nbeing retained, attention was drawn to the theory of a higher space.\nAn illustration of Beltrami\u2019s theory is furnished by the simple\nconsideration of hypothetical beings living on a spherical surface.\n\n[Illustration: Fig. 33.]\n\nLet ABCD be the equator of a globe, and AP, BP, meridian lines drawn to\nthe pole, P. The lines AB, AP, BP would seem to be perfectly straight\nto a person moving on the surface of the sphere, and unconscious of its\ncurvature. Now AP and BP both make right angles with AB. Hence they\nsatisfy the definition of parallels. Yet they meet in P. Hence a being\nliving on a spherical surface, and unconscious of its curvature, would\nfind that parallel lines would meet. He would also find that the angles\nin a triangle were greater than two right angles. In the triangle PAB,\nfor instance, the angles at A and B are right angles, so the three\nangles of the triangle PAB are greater than two right angles.\n\nNow in one of the systems of metageometry (for after Lobatchewsky had\nshown the way it was found that other systems were possible besides\nhis) the angles of a triangle are greater than two right angles.\n\nThus a being on a sphere would form conclusions about his space which\nare the same as he would form if he lived on a plane, the matter in\nwhich had such properties as are presupposed by one of these systems\nof geometry. Beltrami also discovered a certain surface on which\nthere could be drawn more than one \u201cstraight\u201d line through a point\nwhich would not meet another given line. I use the word straight as\nequivalent to the line having the property of giving the shortest path\nbetween any two points on it. Hence, without giving up the ordinary\nmethods of measurement, it was possible to find conditions in which\na plane being would necessarily have an experience corresponding to\nLobatchewsky\u2019s geometry. And by the consideration of a higher space,\nand a solid curved in such a higher space, it was possible to account\nfor a similar experience in a space of three dimensions.\n\nNow, it is far more easy to conceive of a higher dimensionality to\nspace than to imagine that a rod in rotating does not move so that\nits end describes a circle. Hence, a logical conception having been\nfound harder than that of a four dimensional space, thought turned\nto the latter as a simple explanation of the possibilities to which\nLobatchewsky had awakened it. Thinkers became accustomed to deal with\nthe geometry of higher space\u2014it was Kant, says Veronese, who first\nused the expression of \u201cdifferent spaces\u201d\u2014and with familiarity the\ninevitableness of the conception made itself felt.\n\nFrom this point it is but a small step to adapt the ordinary mechanical\nconceptions to a higher spatial existence, and then the recognition of\nits objective existence could be delayed no longer. Here, too, as in so\nmany cases, it turns out that the order and connection of our ideas is\nthe order and connection of things.\n\nWhat is the significance of Lobatchewsky\u2019s and Bolyai\u2019s work?\n\nIt must be recognised as something totally different from the\nconception of a higher space; it is applicable to spaces of any number\nof dimensions. By immersing the conception of distance in matter to\nwhich it properly belongs, it promises to be of the greatest aid in\nanalysis for the effective distance of any two particles is the\nproduct of complex material conditions and cannot be measured by hard\nand fast rules. Its ultimate significance is altogether unknown. It\nis a cutting loose from the bonds of sense, not coincident with the\nrecognition of a higher dimensionality, but indirectly contributory\nthereto.\n\nThus, finally, we have come to accept what Plato held in the hollow\nof his hand; what Aristotle\u2019s doctrine of the relativity of substance\nimplies. The vast universe, too, has its higher, and in recognising it\nwe find that the directing being within us no longer stands inevitably\noutside our systematic knowledge.\n\n\n\n\n CHAPTER VI\n\n THE HIGHER WORLD\n\n\nIt is indeed strange, the manner in which we must begin to think about\nthe higher world.\n\nThose simplest objects analogous to those which are about us on every\nside in our daily experience such as a door, a table, a wheel are\nremote and uncognisable in the world of four dimensions, while the\nabstract ideas of rotation, stress and strain, elasticity into which\nanalysis resolves the familiar elements of our daily experience are\ntransferable and applicable with no difficulty whatever. Thus we are\nin the unwonted position of being obliged to construct the daily and\nhabitual experience of a four-dimensional being, from a knowledge of\nthe abstract theories of the space, the matter, the motion of it;\ninstead of, as in our case, passing to the abstract theories from the\nrichness of sensible things.\n\nWhat would a wheel be in four dimensions? What the shafting for the\ntransmission of power which a four-dimensional being would use.\n\nThe four-dimensional wheel, and the four-dimensional shafting are\nwhat will occupy us for these few pages. And it is no futile or\ninsignificant enquiry. For in the attempt to penetrate into the nature\nof the higher, to grasp within our ken that which transcends all\nanalogies, because what we know are merely partial views of it, the\npurely material and physical path affords a means of approach pursuing\nwhich we are in less likelihood of error than if we use the more\nfrequently trodden path of framing conceptions which in their elevation\nand beauty seem to us ideally perfect.\n\nFor where we are concerned with our own thoughts, the development of\nour own ideals, we are as it were on a curve, moving at any moment\nin a direction of tangency. Whither we go, what we set up and exalt\nas perfect, represents not the true trend of the curve, but our own\ndirection at the present\u2014a tendency conditioned by the past, and by\na vital energy of motion essential but only true when perpetually\nmodified. That eternal corrector of our aspirations and ideals, the\nmaterial universe draws sublimely away from the simplest things we can\ntouch or handle to the infinite depths of starry space, in one and\nall uninfluenced by what we think or feel, presenting unmoved fact\nto which, think it good or think it evil, we can but conform, yet\nout of all that impassivity with a reference to something beyond our\nindividual hopes and fears supporting us and giving us our being.\n\nAnd to this great being we come with the question: \u201cYou, too, what is\nyour higher?\u201d\n\nOr to put it in a form which will leave our conclusions in the shape\nof no barren formula, and attacking the problem on its most assailable\nside: \u201cWhat is the wheel and the shafting of the four-dimensional\nmechanic?\u201d\n\nIn entering on this enquiry we must make a plan of procedure. The\nmethod which I shall adopt is to trace out the steps of reasoning by\nwhich a being confined to movement in a two-dimensional world could\narrive at a conception of our turning and rotation, and then to apply\nan analogous process to the consideration of the higher movements. The\nplane being must be imagined as no abstract figure, but as a real body\npossessing all three dimensions. His limitation to a plane must be the\nresult of physical conditions.\n\nWe will therefore think of him as of a figure cut out of paper placed\non a smooth plane. Sliding over this plane, and coming into contact\nwith other figures equally thin as he in the third dimension, he will\napprehend them only by their edges. To him they will be completely\nbounded by lines. A \u201csolid\u201d body will be to him a two-dimensional\nextent, the interior of which can only be reached by penetrating\nthrough the bounding lines.\n\nNow such a plane being can think of our three-dimensional existence in\ntwo ways.\n\nFirst, he can think of it as a series of sections, each like the solid\nhe knows of extending in a direction unknown to him, which stretches\ntransverse to his tangible universe, which lies in a direction at right\nangles to every motion which he made.\n\nSecondly, relinquishing the attempt to think of the three-dimensional\nsolid body in its entirety he can regard it as consisting of a\nnumber of plane sections, each of them in itself exactly like\nthe two-dimensional bodies he knows, but extending away from his\ntwo-dimensional space.\n\nA square lying in his space he regards as a solid bounded by four\nlines, each of which lies in his space.\n\nA square standing at right angles to his plane appears to him as simply\na line in his plane, for all of it except the line stretches in the\nthird dimension.\n\nHe can think of a three-dimensional body as consisting of a number of\nsuch sections, each of which starts from a line in his space.\n\nNow, since in his world he can make any drawing or model which involves\nonly two dimensions, he can represent each such upright section as it\nactually is, and can represent a turning from a known into the unknown\ndimension as a turning from one to another of his known dimensions.\n\nTo see the whole he must relinquish part of that which he has, and take\nthe whole portion by portion.\n\nConsider now a plane being in front of a square, fig. 34. The square\ncan turn about any point in the plane\u2014say the point A. But it cannot\nturn about a line, as AB. For, in order to turn about the line AB,\nthe square must leave the plane and move in the third dimension. This\nmotion is out of his range of observation, and is therefore, except for\na process of reasoning, inconceivable to him.\n\n[Illustration: Fig. 34.]\n\nRotation will therefore be to him rotation about a point. Rotation\nabout a line will be inconceivable to him.\n\nThe result of rotation about a line he can apprehend. He can see the\nfirst and last positions occupied in a half-revolution about the line\nAC. The result of such a half revolution is to place the square ABCD\non the left hand instead of on the right hand of the line AC. It would\ncorrespond to a pulling of the whole body ABCD through the line AC,\nor to the production of a solid body which was the exact reflection\nof it in the line AC. It would be as if the square ABCD turned into\nits image, the line AB acting as a mirror. Such a reversal of the\npositions of the parts of the square would be impossible in his space.\nThe occurrence of it would be a proof of the existence of a higher\ndimensionality.\n\nLet him now, adopting the conception of a three-dimensional body as\na series of sections lying, each removed a little farther than the\npreceding one, in direction at right angles to his plane, regard a\ncube, fig. 36, as a series of sections, each like the square which\nforms its base, all rigidly connected together.\n\n[Illustration: Fig. 35.]\n\nIf now he turns the square about the point A in the plane of _xy_,\neach parallel section turns with the square he moves. In each of the\nsections there is a point at rest, that vertically over A. Hence he\nwould conclude that in the turning of a three-dimensional body there\nis one line which is at rest. That is a three-dimensional turning in a\nturning about a line.\n\n * * * * *\n\nIn a similar way let us regard ourselves as limited to a\nthree-dimensional world by a physical condition. Let us imagine that\nthere is a direction at right angles to every direction in which we can\nmove, and that we are prevented from passing in this direction by a\nvast solid, that against which in every movement we make we slip as the\nplane being slips against his plane sheet.\n\nWe can then consider a four-dimensional body as consisting of a series\nof sections, each parallel to our space, and each a little farther off\nthan the preceding on the unknown dimension.\n\nTake the simplest four-dimensional body\u2014one which begins as a cube,\nfig. 36, in our space, and consists of sections, each a cube like fig.\n36, lying away from our space. If we turn the cube which is its base in\nour space about a line, if, _e.g._, in fig. 36 we turn the cube about\nthe line AB, not only it but each of the parallel cubes moves about a\nline. The cube we see moves about the line AB, the cube beyond it about\na line parallel to AB and so on. Hence the whole four-dimensional body\nmoves about a plane, for the assemblage of these lines is our way of\nthinking about the plane which, starting from the line AB in our space,\nruns off in the unknown direction.\n\n[Illustration: Fig. 36.]\n\nIn this case all that we see of the plane about which the turning takes\nplace is the line AB.\n\nBut it is obvious that the axis plane may lie in our space. A point\nnear the plane determines with it a three-dimensional space. When it\nbegins to rotate round the plane it does not move anywhere in this\nthree-dimensional space, but moves out of it. A point can no more\nrotate round a plane in three-dimensional space than a point can move\nround a line in two-dimensional space.\n\nWe will now apply the second of the modes of representation to this\ncase of turning about a plane, building up our analogy step by step\nfrom the turning in a plane about a point and that in space about a\nline, and so on.\n\nIn order to reduce our considerations to those of the greatest\nsimplicity possible, let us realise how the plane being would think of\nthe motion by which a square is turned round a line.\n\nLet, fig. 34, ABCD be a square on his plane, and represent the two\ndimensions of his space by the axes A_x_ A_y_.\n\nNow the motion by which the square is turned over about the line AC\ninvolves the third dimension.\n\nHe cannot represent the motion of the whole square in its turning,\nbut he can represent the motions of parts of it. Let the third axis\nperpendicular to the plane of the paper be called the axis of _z_. Of\nthe three axes _x_, _y_, _z_, the plane being can represent any two in\nhis space. Let him then draw, in fig. 35, two axes, _x_ and _z_. Here\nhe has in his plane a representation of what exists in the plane which\ngoes off perpendicularly to his space.\n\nIn this representation the square would not be shown, for in the plane\nof _xz_ simply the line AB of the square is contained.\n\nThe plane being then would have before him, in fig. 35, the\nrepresentation of one line AB of his square and two axes, _x_ and _z_,\nat right angles. Now it would be obvious to him that, by a turning\nsuch as he knows, by a rotation about a point, the line AB can turn\nround A, and occupying all the intermediate positions, such as AB_{1},\ncome after half a revolution to lie as A_x_ produced through A.\n\nAgain, just as he can represent the vertical plane through AB, so he\ncan represent the vertical plane through A\u00b4B\u00b4, fig. 34, and in a like\nmanner can see that the line A\u00b4B\u00b4 can turn about the point A\u00b4 till it\nlies in the opposite direction from that which it ran in at first.\n\nNow these two turnings are not inconsistent. In his plane, if AB\nturned about A, and A\u00b4B\u00b4 about A\u00b4, the consistency of the square would\nbe destroyed, it would be an impossible motion for a rigid body to\nperform. But in the turning which he studies portion by portion there\nis nothing inconsistent. Each line in the square can turn in this way,\nhence he would realise the turning of the whole square as the sum of\na number of turnings of isolated parts. Such turnings, if they took\nplace in his plane, would be inconsistent, but by virtue of a third\ndimension they are consistent, and the result of them all is that the\nsquare turns about the line AC and lies in a position in which it is\nthe mirror image of what it was in its first position. Thus he can\nrealise a turning about a line by relinquishing one of his axes, and\nrepresenting his body part by part.\n\nLet us apply this method to the turning of a cube so as to become the\nmirror image of itself. In our space we can construct three independent\naxes, _x_, _y_, _z_, shown in fig. 36. Suppose that there is a fourth\naxis, _w_, at right angles to each and every one of them. We cannot,\nkeeping all three axes, _x_, _y_, _z_, represent _w_ in our space; but\nif we relinquish one of our three axes we can let the fourth axis take\nits place, and we can represent what lies in the space, determined by\nthe two axes we retain and the fourth axis.\n\n[Illustration: Fig. 37.]\n\nLet us suppose that we let the _y_ axis drop, and that we represent\nthe _w_ axis as occupying its direction. We have in fig. 37 a drawing\nof what we should then see of the cube. The square ABCD, remains\nunchanged, for that is in the plane of _xz_, and we still have that\nplane. But from this plane the cube stretches out in the direction of\nthe _y_ axis. Now the _y_ axis is gone, and so we have no more of the\ncube than the face ABCD. Considering now this face ABCD, we see that\nit is free to turn about the line AB. It can rotate in the _x_ to _w_\ndirection about this line. In fig. 38 it is shown on its way, and it\ncan evidently continue this rotation till it lies on the other side of\nthe _z_ axis in the plane of _xz_.\n\nWe can also take a section parallel to the face ABCD, and then letting\ndrop all of our space except the plane of that section, introduce\nthe _w_ axis, running in the old _y_ direction. This section can be\nrepresented by the same drawing, fig. 38, and we see that it can rotate\nabout the line on its left until it swings half way round and runs in\nthe opposite direction to that which it ran in before. These turnings\nof the different sections are not inconsistent, and taken all together\nthey will bring the cube from the position shown in fig. 36 to that\nshown in fig. 41.\n\n[Illustration: Fig. 38.]\n\nSince we have three axes at our disposal in our space, we are not\nobliged to represent the _w_ axis by any particular one. We may let any\naxis we like disappear, and let the fourth axis take its place.\n\n[Illustration: Fig. 39.]\n\n[Illustration: Fig. 40.]\n\n[Illustration: Fig. 41.]\n\nIn fig. 36 suppose the _z_ axis to go. We have then simply the plane of\n_xy_ and the square base of the cube ACEG, fig. 39, is all that could\nbe seen of it. Let now the _w_ axis take the place of the _z_ axis and\nwe have, in fig. 39 again, a representation of the space of _xyw_, in\nwhich all that exists of the cube is its square base. Now, by a turning\nof _x_ to _w_, this base can rotate around the line AE, it is shown\non its way in fig. 40, and finally it will, after half a revolution,\nlie on the other side of the _y_ axis. In a similar way we may rotate\nsections parallel to the base of the _xw_ rotation, and each of them\ncomes to run in the opposite direction from that which they occupied at\nfirst.\n\nThus again the cube comes from the position of fig. 36. to that of\nfig. 41. In this _x_ to _w_ turning, we see that it takes place by\nthe rotations of sections parallel to the front face about lines\nparallel to AB, or else we may consider it as consisting of the\nrotation of sections parallel to the base about lines parallel to AE.\nIt is a rotation of the whole cube about the plane ABEF. Two separate\nsections could not rotate about two separate lines in our space without\nconflicting, but their motion is consistent when we consider another\ndimension. Just, then, as a plane being can think of rotation about\na line as a rotation about a number of points, these rotations not\ninterfering as they would if they took place in his two-dimensional\nspace, so we can think of a rotation about a plane as the rotation\nof a number of sections of a body about a number of lines in a plane,\nthese rotations not being inconsistent in a four-dimensional space as\nthey are in three-dimensional space.\n\nWe are not limited to any particular direction for the lines in the\nplane about which we suppose the rotation of the particular sections to\ntake place. Let us draw the section of the cube, fig. 36, through A,\nF, C, H, forming a sloping plane. Now since the fourth dimension is at\nright angles to every line in our space it is at right angles to this\nsection also. We can represent our space by drawing an axis at right\nangles to the plane ACEG, our space is then determined by the plane\nACEG, and the perpendicular axis. If we let this axis drop and suppose\nthe fourth axis, _w_, to take its place, we have a representation of\nthe space which runs off in the fourth dimension from the plane ACEG.\nIn this space we shall see simply the section ACEG of the cube, and\nnothing else, for one cube does not extend to any distance in the\nfourth dimension.\n\nIf, keeping this plane, we bring in the fourth dimension, we shall have\na space in which simply this section of the cube exists and nothing\nelse. The section can turn about the line AF, and parallel sections can\nturn about parallel lines. Thus in considering the rotation about a\nplane we can draw any lines we like and consider the rotation as taking\nplace in sections about them.\n\n[Illustration: Fig. 42.]\n\nTo bring out this point more clearly let us take two parallel lines,\nA and B, in the space of _xyz_, and let CD and EF be two rods running\nabove and below the plane of _xy_, from these lines. If we turn these\nrods in our space about the lines A and B, as the upper end of one,\nF, is going down, the lower end of the other, C, will be coming up.\nThey will meet and conflict. But it is quite possible for these two\nrods each of them to turn about the two lines without altering their\nrelative distances.\n\nTo see this suppose the _y_ axis to go, and let the _w_ axis take its\nplace. We shall see the lines A and B no longer, for they run in the\n_y_ direction from the points G and H.\n\n[Illustration: Fig. 43.]\n\nFig. 43 is a picture of the two rods seen in the space of _xzw_. If\nthey rotate in the direction shown by the arrows\u2014in the _z_ to _w_\ndirection\u2014they move parallel to one another, keeping their relative\ndistances. Each will rotate about its own line, but their rotation will\nnot be inconsistent with their forming part of a rigid body.\n\nNow we have but to suppose a central plane with rods crossing it\nat every point, like CD and EF cross the plane of _xy_, to have an\nimage of a mass of matter extending equal distances on each side of a\ndiametral plane. As two of these rods can rotate round, so can all, and\nthe whole mass of matter can rotate round its diametral plane.\n\nThis rotation round a plane corresponds, in four dimensions, to the\nrotation round an axis in three dimensions. Rotation of a body round a\nplane is the analogue of rotation of a rod round an axis.\n\nIn a plane we have rotation round a point, in three-space rotation\nround an axis line, in four-space rotation round an axis plane.\n\nThe four-dimensional being\u2019s shaft by which he transmits power is a\ndisk rotating round its central plane\u2014the whole contour corresponds\nto the ends of an axis of rotation in our space. He can impart the\nrotation at any point and take it off at any other point on the\ncontour, just as rotation round a line can in three-space be imparted\nat one end of a rod and taken off at the other end.\n\nA four-dimensional wheel can easily be described from the analogy of\nthe representation which a plane being would form for himself of one of\nour wheels.\n\nSuppose a wheel to move transverse to a plane, so that the whole disk,\nwhich I will consider to be solid and without spokes, came at the same\ntime into contact with the plane. It would appear as a circular portion\nof plane matter completely enclosing another and smaller portion\u2014the\naxle.\n\nThis appearance would last, supposing the motion of the wheel to\ncontinue until it had traversed the plane by the extent of its\nthickness, when there would remain in the plane only the small disk\nwhich is the section of the axle. There would be no means obvious in\nthe plane at first by which the axle could be reached, except by going\nthrough the substance of the wheel. But the possibility of reaching it\nwithout destroying the substance of the wheel would be shown by the\ncontinued existence of the axle section after that of the wheel had\ndisappeared.\n\nIn a similar way a four-dimensional wheel moving transverse to our\nspace would appear first as a solid sphere, completely surrounding\na smaller solid sphere. The outer sphere would represent the wheel,\nand would last until the wheel has traversed our space by a distance\nequal to its thickness. Then the small sphere alone would remain,\nrepresenting the section of the axle. The large sphere could move\nround the small one quite freely. Any line in space could be taken as\nan axis, and round this line the outer sphere could rotate, while the\ninner sphere remained still. But in all these directions of revolution\nthere would be in reality one line which remained unaltered, that is\nthe line which stretches away in the fourth direction, forming the\naxis of the axle. The four-dimensional wheel can rotate in any number\nof planes, but all these planes are such that there is a line at right\nangles to them all unaffected by rotation in them.\n\nAn objection is sometimes experienced as to this mode of reasoning from\na plane world to a higher dimensionality. How artificial, it is argued,\nthis conception of a plane world is. If any real existence confined to\na superficies could be shown to exist, there would be an argument for\none relative to which our three-dimensional existence is superficial.\nBut, both on the one side and the other of the space we are familiar\nwith, spaces either with less or more than three dimensions are merely\narbitrary conceptions.\n\nIn reply to this I would remark that a plane being having one less\ndimension than our three would have one-third of our possibilities of\nmotion, while we have only one-fourth less than those of the higher\nspace. It may very well be that there may be a certain amount of\nfreedom of motion which is demanded as a condition of an organised\nexistence, and that no material existence is possible with a more\nlimited dimensionality than ours. This is well seen if we try to\nconstruct the mechanics of a two-dimensional world. No tube could\nexist, for unless joined together completely at one end two parallel\nlines would be completely separate. The possibility of an organic\nstructure, subject to conditions such as this, is highly problematical;\nyet, possibly in the convolutions of the brain there may be a mode of\nexistence to be described as two-dimensional.\n\nWe have but to suppose the increase in surface and the diminution in\nmass carried on to a certain extent to find a region which, though\nwithout mobility of the constituents, would have to be described as\ntwo-dimensional.\n\nBut, however artificial the conception of a plane being may be, it is\nnone the less to be used in passing to the conception of a greater\ndimensionality than ours, and hence the validity of the first part of\nthis objection altogether disappears directly we find evidence for such\na state of being.\n\nThe second part of the objection has more weight. How is it possible\nto conceive that in a four-dimensional space any creatures should be\nconfined to a three-dimensional existence?\n\nIn reply I would say that we know as a matter of fact that life is\nessentially a phenomenon of surface. The amplitude of the movements\nwhich we can make is much greater along the surface of the earth than\nit is up or down.\n\nNow we have but to conceive the extent of a solid surface increased,\nwhile the motions possible tranverse to it are diminished in the\nsame proportion, to obtain the image of a three-dimensional world in\nfour-dimensional space.\n\nAnd as our habitat is the meeting of air and earth on the world, so\nwe must think of the meeting place of two as affording the condition\nfor our universe. The meeting of what two? What can that vastness be\nin the higher space which stretches in such a perfect level that our\nastronomical observations fail to detect the slightest curvature?\n\nThe perfection of the level suggests a liquid\u2014a lake amidst what vast\nscenery!\u2014whereon the matter of the universe floats speck-like.\n\nBut this aspect of the problem is like what are called in mathematics\nboundary conditions.\n\nWe can trace out all the consequences of four-dimensional movements\ndown to their last detail. Then, knowing the mode of action which\nwould be characteristic of the minutest particles, if they were\nfree, we can draw conclusions from what they actually do of what the\nconstraint on them is. Of the two things, the material conditions and\nthe motion, one is known, and the other can be inferred. If the place\nof this universe is a meeting of two, there would be a one-sideness\nto space. If it lies so that what stretches away in one direction in\nthe unknown is unlike what stretches away in the other, then, as far\nas the movements which participate in that dimension are concerned,\nthere would be a difference as to which way the motion took place. This\nwould be shown in the dissimilarity of phenomena, which, so far as\nall three-space movements are concerned, were perfectly symmetrical.\nTo take an instance, merely, for the sake of precising our ideas,\nnot for any inherent probability in it; if it could be shown that\nthe electric current in the positive direction were exactly like the\nelectric current in the negative direction, except for a reversal of\nthe components of the motion in three-dimensional space, then the\ndissimilarity of the discharge from the positive and negative poles\nwould be an indication of a one-sideness to our space. The only cause\nof difference in the two discharges would be due to a component in\nthe fourth dimension, which directed in one direction transverse to\nour space, met with a different resistance to that which it met when\ndirected in the opposite direction.\n\n\n\n\n CHAPTER VII\n\n THE EVIDENCES FOR A FOURTH DIMENSION\n\n\nThe method necessarily to be employed in the search for the evidences\nof a fourth dimension, consists primarily in the formation of the\nconceptions of four-dimensional shapes and motions. When we are in\npossession of these it is possible to call in the aid of observation,\nwithout them we may have been all our lives in the familiar presence of\na four-dimensional phenomenon without ever recognising its nature.\n\nTo take one of the conceptions we have already formed, the turning of a\nreal thing into its mirror image would be an occurrence which it would\nbe hard to explain, except on the assumption of a fourth dimension.\n\nWe know of no such turning. But there exist a multitude of forms which\nshow a certain relation to a plane, a relation of symmetry, which\nindicates more than an accidental juxtaposition of parts. In organic\nlife the universal type is of right- and left-handed symmetry, there\nis a plane on each side of which the parts correspond. Now we have\nseen that in four dimensions a plane takes the place of a line in\nthree dimensions. In our space, rotation about an axis is the type of\nrotation, and the origin of bodies symmetrical about a line as the\nearth is symmetrical about an axis can easily be explained. But where\nthere is symmetry about a plane no simple physical motion, such as we\nare accustomed to, suffices to explain it. In our space a symmetrical\nobject must be built up by equal additions on each side of a central\nplane. Such additions about such a plane are as little likely as any\nother increments. The probability against the existence of symmetrical\nform in inorganic nature is overwhelming in our space, and in organic\nforms they would be as difficult of production as any other variety\nof configuration. To illustrate this point we may take the child\u2019s\namusement of making from dots of ink on a piece of paper a lifelike\nrepresentation of an insect by simply folding the paper over. The\ndots spread out on a symmetrical line, and give the impression of a\nsegmented form with antenn\u00e6 and legs.\n\nNow seeing a number of such figures we should naturally infer a folding\nover. Can, then, a folding over in four-dimensional space account for\nthe symmetry of organic forms? The folding cannot of course be of the\nbodies we see, but it may be of those minute constituents, the ultimate\nelements of living matter which, turned in one way or the other, become\nright- or left-handed, and so produce a corresponding structure.\n\nThere is something in life not included in our conceptions of\nmechanical movement. Is this something a four-dimensional movement?\n\nIf we look at it from the broadest point of view, there is something\nstriking in the fact that where life comes in there arises an entirely\ndifferent set of phenomena to those of the inorganic world.\n\nThe interest and values of life as we know it in ourselves, as we\nknow it existing around us in subordinate forms, is entirely and\ncompletely different to anything which inorganic nature shows. And in\nliving beings we have a kind of form, a disposition of matter which\nis entirely different from that shown in inorganic matter. Right-\nand left-handed symmetry does not occur in the configurations of dead\nmatter. We have instances of symmetry about an axis, but not about\na plane. It can be argued that the occurrence of symmetry in two\ndimensions involves the existence of a three-dimensional process, as\nwhen a stone falls into water and makes rings of ripples, or as when\na mass of soft material rotates about an axis. It can be argued that\nsymmetry in any number of dimensions is the evidence of an action in\na higher dimensionality. Thus considering living beings, there is an\nevidence both in their structure, and their different mode of activity,\nof a something coming in from without into the inorganic world.\n\nAnd the objections which will readily occur, such as those derived from\nthe forms of twin crystals and the theoretical structure of chemical\nmolecules, do not invalidate the argument; for in these forms too the\npresumable seat of the activity producing them lies in that very minute\nregion in which we necessarily place the seat of a four-dimensional\nmobility.\n\nIn another respect also the existence of symmetrical forms is\nnoteworthy. It is puzzling to conceive how two shapes exactly equal can\nexist which are not superposible. Such a pair of symmetrical figures\nas the two hands, right and left, show either a limitation in our\npower of movement, by which we cannot superpose the one on the other,\nor a definite influence and compulsion of space on matter, inflicting\nlimitations which are additional to those of the proportions of the\nparts.\n\nWe will, however, put aside the arguments to be drawn from the\nconsideration of symmetry as inconclusive, retaining one valuable\nindication which they afford. If it is in virtue of a four-dimensional\nmotion that symmetry exists, it is only in the very minute particles\nof bodies that that motion is to be found, for there is no such thing\nas a bending over in four dimensions of any object of a size which we\ncan observe. The region of the extremely minute is the one, then, which\nwe shall have to investigate. We must look for some phenomenon which,\noccasioning movements of the kind we know, still is itself inexplicable\nas any form of motion which we know.\n\nNow in the theories of the actions of the minute particles of bodies\non one another, and in the motions of the ether, mathematicians\nhave tacitly assumed that the mechanical principles are the same as\nthose which prevail in the case of bodies which can be observed, it\nhas been assumed without proof that the conception of motion being\nthree-dimensional, holds beyond the region from observations in which\nit was formed.\n\nHence it is not from any phenomenon explained by mathematics that we\ncan derive a proof of four dimensions. Every phenomenon that has been\nexplained is explained as three-dimensional. And, moreover, since in\nthe region of the very minute we do not find rigid bodies acting on\neach other at a distance, but elastic substances and continuous fluids\nsuch as ether, we shall have a double task.\n\nWe must form the conceptions of the possible movements of elastic and\nliquid four-dimensional matter, before we can begin to observe. Let\nus, therefore, take the four-dimensional rotation about a plane, and\nenquire what it becomes in the case of extensible fluid substances. If\nfour-dimensional movements exist, this kind of rotation must exist, and\nthe finer portions of matter must exhibit it.\n\nConsider for a moment a rod of flexible and extensible material. It can\nturn about an axis, even if not straight; a ring of india rubber can\nturn inside out.\n\nWhat would this be in the case of four dimensions?\n\nLet us consider a sphere of our three-dimensional matter having a\ndefinite thickness. To represent this thickness let us suppose that\nfrom every point of the sphere in fig. 44 rods project both ways, in\nand out, like D and F. We can only see the external portion, because\nthe internal parts are hidden by the sphere.\n\n[Illustration: Fig. 44.\n\n_Axis of x running towards the observer._]\n\nIn this sphere the axis of _x_ is supposed to come towards the\nobserver, the axis of _z_ to run up, the axis of _y_ to go to the right.\n\n[Illustration: Fig. 45.]\n\nNow take the section determined by the _zy_ plane. This will be a\ncircle as shown in fig. 45. If we let drop the _x_ axis, this circle\nis all we have of the sphere. Letting the _w_ axis now run in the\nplace of the old _x_ axis we have the space _yzw_, and in this space\nall that we have of the sphere is the circle. Fig. 45 then represents\nall that there is of the sphere in the space of _yzw_. In this space\nit is evident that the rods CD and EF can turn round the circumference\nas an axis. If the matter of the spherical shell is sufficiently\nextensible to allow the particles C and E to become as widely separated\nas they would be in the positions D and F, then the strip of matter\nrepresented by CD and EF and a multitude of rods like them can turn\nround the circular circumference.\n\nThus this particular section of the sphere can turn inside out, and\nwhat holds for any one section holds for all. Hence in four dimensions\nthe whole sphere can, if extensible turn inside out. Moreover, any part\nof it\u2014a bowl-shaped portion, for instance\u2014can turn inside out, and so\non round and round.\n\nThis is really no more than we had before in the rotation about a\nplane, except that we see that the plane can, in the case of extensible\nmatter, be curved, and still play the part of an axis.\n\nIf we suppose the spherical shell to be of four-dimensional matter, our\nrepresentation will be a little different. Let us suppose there to be a\nsmall thickness to the matter in the fourth dimension. This would make\nno difference in fig. 44, for that merely shows the view in the _xyz_\nspace. But when the _x_ axis is let drop, and the _w_ axis comes in,\nthen the rods CD and EF which represent the matter of the shell, will\nhave a certain thickness perpendicular to the plane of the paper on\nwhich they are drawn. If they have a thickness in the fourth dimension\nthey will show this thickness when looked at from the direction of the\n_w_ axis.\n\nSupposing these rods, then, to be small slabs strung on the\ncircumference of the circle in fig. 45, we see that there will not\nbe in this case either any obstacle to their turning round the\ncircumference. We can have a shell of extensible material or of fluid\nmaterial turning inside out in four dimensions.\n\nAnd we must remember that in four dimensions there is no such thing as\nrotation round an axis. If we want to investigate the motion of fluids\nin four dimensions we must take a movement about an axis in our space,\nand find the corresponding movement about a plane in four space.\n\nNow, of all the movements which take place in fluids, the most\nimportant from a physical point of view is vortex motion.\n\nA vortex is a whirl or eddy\u2014it is shown in the gyrating wreaths of\ndust seen on a summer day; it is exhibited on a larger scale in the\ndestructive march of a cyclone.\n\nA wheel whirling round will throw off the water on it. But when\nthis circling motion takes place in a liquid itself it is strangely\npersistent. There is, of course, a certain cohesion between the\nparticles of water by which they mutually impede their motions. But\nin a liquid devoid of friction, such that every particle is free from\nlateral cohesion on its path of motion, it can be shown that a vortex\nor eddy separates from the mass of the fluid a certain portion, which\nalways remain in that vortex.\n\nThe shape of the vortex may alter, but it always consists of the same\nparticles of the fluid.\n\nNow, a very remarkable fact about such a vortex is that the ends of the\nvortex cannot remain suspended and isolated in the fluid. They must\nalways run to the boundary of the fluid. An eddy in water that remains\nhalf way down without coming to the top is impossible.\n\nThe ends of a vortex must reach the boundary of a fluid\u2014the boundary\nmay be external or internal\u2014a vortex may exist between two objects\nin the fluid, terminating one end on each object, the objects being\ninternal boundaries of the fluid. Again, a vortex may have its ends\nlinked together, so that it forms a ring. Circular vortex rings of\nthis description are often seen in puffs of smoke, and that the smoke\ntravels on in the ring is a proof that the vortex always consists of\nthe same particles of air.\n\nLet us now enquire what a vortex would be in a four-dimensional fluid.\n\nWe must replace the line axis by a plane axis. We should have therefore\na portion of fluid rotating round a plane.\n\nWe have seen that the contour of this plane corresponds with the ends\nof the axis line. Hence such a four-dimensional vortex must have its\nrim on a boundary of the fluid. There would be a region of vorticity\nwith a contour. If such a rotation were started at one part of a\ncircular boundary, its edges would run round the boundary in both\ndirections till the whole interior region was filled with the vortex\nsheet.\n\nA vortex in a three-dimensional liquid may consist of a number of\nvortex filaments lying together producing a tube, or rod of vorticity.\n\nIn the same way we can have in four dimensions a number of vortex\nsheets alongside each other, each of which can be thought of as a\nbowl-shaped portion of a spherical shell turning inside out. The\nrotation takes place at any point not in the space occupied by the\nshell, but from that space to the fourth dimension and round back again.\n\nIs there anything analogous to this within the range of our observation?\n\nAn electric current answers this description in every respect.\nElectricity does not flow through a wire. Its effect travels both ways\nfrom the starting point along the wire. The spark which shows its\npassing midway in its circuit is later than that which occurs at points\nnear its starting point on either side of it.\n\nMoreover, it is known that the action of the current is not in the\nwire. It is in the region enclosed by the wire, this is the field of\nforce, the locus of the exhibition of the effects of the current.\n\nAnd the necessity of a conducting circuit for a current is exactly\nthat which we should expect if it were a four-dimensional vortex.\nAccording to Maxwell every current forms a closed circuit, and this,\nfrom the four-dimensional point of view, is the same as saying a vortex\nmust have its ends on a boundary of the fluid.\n\nThus, on the hypothesis of a fourth dimension, the rotation of the\nfluid ether would give the phenomenon of an electric current. We must\nsuppose the ether to be full of movement, for the more we examine into\nthe conditions which prevail in the obscurity of the minute, the more\nwe find that an unceasing and perpetual motion reigns. Thus we may say\nthat the conception of the fourth dimension means that there must be a\nphenomenon which presents the characteristics of electricity.\n\nWe know now that light is an electro-magnetic action, and that so far\nfrom being a special and isolated phenomenon this electric action is\nuniversal in the realm of the minute. Hence, may we not conclude that,\nso far from the fourth dimension being remote and far away, being a\nthing of symbolic import, a term for the explanation of dubious facts\nby a more obscure theory, it is really the most important fact within\nour knowledge. Our three-dimensional world is superficial. These\nprocesses, which really lie at the basis of all phenomena of matter,\nescape our observation by their minuteness, but reveal to our intellect\nan amplitude of motion surpassing any that we can see. In such shapes\nand motions there is a realm of the utmost intellectual beauty, and one\nto which our symbolic methods apply with a better grace than they do to\nthose of three dimensions.\n\n\n\n\n CHAPTER VIII\n\n THE USE OF FOUR DIMENSIONS IN THOUGHT\n\n\nHaving held before ourselves this outline of a conjecture of the world\nas four-dimensional, having roughly thrown together those facts of\nmovement which we can see apply to our actual experience, let us pass\nto another branch of our subject.\n\nThe engineer uses drawings, graphical constructions, in a variety of\nmanners. He has, for instance, diagrams which represent the expansion\nof steam, the efficiency of his valves. These exist alongside the\nactual plans of his machines. They are not the pictures of anything\nreally existing, but enable him to think about the relations which\nexist in his mechanisms.\n\nAnd so, besides showing us the actual existence of that world which\nlies beneath the one of visible movements, four-dimensional space\nenables us to make ideal constructions which serve to represent the\nrelations of things, and throw what would otherwise be obscure into a\ndefinite and suggestive form.\n\nFrom amidst the great variety of instances which lies before me I will\nselect two, one dealing with a subject of slight intrinsic interest,\nwhich however gives within a limited field a striking example of the\nmethod of drawing conclusions and the use of higher space figures.[1]\n\n [1] It is suggestive also in another respect, because it shows very\n clearly that in our processes of thought there are in play faculties\n other than logical; in it the origin of the idea which proves to be\n justified is drawn from the consideration of symmetry, a branch of the\n beautiful.\n\nThe other instance is chosen on account of the bearing it has on our\nfundamental conceptions. In it I try to discover the real meaning of\nKant\u2019s theory of experience.\n\nThe investigation of the properties of numbers is much facilitated\nby the fact that relations between numbers are themselves able to be\nrepresented as numbers\u2014_e.g._, 12, and 3 are both numbers, and the\nrelation between them is 4, another number. The way is thus opened for\na process of constructive theory, without there being any necessity for\na recourse to another class of concepts besides that which is given in\nthe phenomena to be studied.\n\nThe discipline of number thus created is of great and varied\napplicability, but it is not solely as quantitative that we learn to\nunderstand the phenomena of nature. It is not possible to explain the\nproperties of matter by number simply, but all the activities of matter\nare energies in space. They are numerically definite and also, we may\nsay, directedly definite, _i.e._ definite in direction.\n\nIs there, then, a body of doctrine about space which, like that of\nnumber, is available in science? It is needless to answer: Yes;\ngeometry. But there is a method lying alongside the ordinary methods of\ngeometry, which tacitly used and presenting an analogy to the method of\nnumerical thought deserves to be brought into greater prominence than\nit usually occupies.\n\nThe relation of numbers is a number.\n\nCan we say in the same way that the relation of shapes is a shape?\n\nWe can.\n\nTo take an instance chosen on account of its ready availability. Let\nus take two right-angled triangles of a given hypothenuse, but having\nsides of different lengths (fig. 46). These triangles are shapes which\nhave a certain relation to each other. Let us exhibit their relation as\na figure.\n\n[Illustration: Fig. 46.]\n\nDraw two straight lines at right angles to each other, the one HL a\nhorizontal level, the other VL a vertical level (fig. 47). By means\nof these two co-ordinating lines we can represent a double set of\nmagnitudes; one set as distances to the right of the vertical level,\nthe other as distances above the horizontal level, a suitable unit\nbeing chosen.\n\n[Illustration: Fig. 47.]\n\nThus the line marked 7 will pick out the assemblage of points whose\ndistance from the vertical level is 7, and the line marked 1 will pick\nout the points whose distance above the horizontal level is 1. The\nmeeting point of these two lines, 7 and 1, will define a point which\nwith regard to the one set of magnitudes is 7, with regard to the\nother is 1. Let us take the sides of our triangles as the two sets of\nmagnitudes in question.\n\nThen the point 7, 1, will represent the triangle whose sides are 7 and\n1. Similarly the point 5, 5\u20145, that is, to the right of the vertical\nlevel and 5 above the horizontal level\u2014will represent the triangle\nwhose sides are 5 and 5 (fig. 48).\n\n[Illustration: Fig. 48.]\n\nThus we have obtained a figure consisting of the two points 7, 1, and\n5, 5, representative of our two triangles. But we can go further,\nand, drawing an arc of a circle about O, the meeting point of the\nhorizontal and vertical levels, which passes through 7, 1, and 5,\n5, assert that all the triangles which are right-angled and have a\nhypothenuse whose square is 50 are represented by the points on this\narc.\n\nThus, each individual of a class being represented by a point, the\nwhole class is represented by an assemblage of points forming a\nfigure. Accepting this representation we can attach a definite and\ncalculable significance to the expression, resemblance, or similarity\nbetween two individuals of the class represented, the difference being\nmeasured by the length of the line between two representative points.\nIt is needless to multiply examples, or to show how, corresponding to\ndifferent classes of triangles, we obtain different curves.\n\nA representation of this kind in which an object, a thing in space,\nis represented as a point, and all its properties are left out, their\neffect remaining only in the relative position which the representative\npoint bears to the representative points of the other objects, may be\ncalled, after the analogy of Sir William R. Hamilton\u2019s hodograph, a\n\u201cPoiograph.\u201d\n\nRepresentations thus made have the character of natural objects;\nthey have a determinate and definite character of their own. Any\nlack of completeness in them is probably due to a failure in point\nof completeness of those observations which form the ground of their\nconstruction.\n\nEvery system of classification is a poiograph. In Mendel\u00e9eff\u2019s scheme\nof the elements, for instance, each element is represented by a point,\nand the relations between the elements are represented by the relations\nbetween the points.\n\nSo far I have simply brought into prominence processes and\nconsiderations with which we are all familiar. But it is worth while\nto bring into the full light of our attention our habitual assumptions\nand processes. It often happens that we find there are two of them\nwhich have a bearing on each other, which, without this dragging into\nthe light, we should have allowed to remain without mutual influence.\n\nThere is a fact which it concerns us to take into account in discussing\nthe theory of the poiograph.\n\nWith respect to our knowledge of the world we are far from that\ncondition which Laplace imagined when he asserted that an all-knowing\nmind could determine the future condition of every object, if he knew\nthe co-ordinates of its particles in space, and their velocity at any\nparticular moment.\n\nOn the contrary, in the presence of any natural object, we have a great\ncomplexity of conditions before us, which we cannot reduce to position\nin space and date in time.\n\nThere is mass, attraction apparently spontaneous, electrical and\nmagnetic properties which must be superadded to spatial configuration.\nTo cut the list short we must say that practically the phenomena of the\nworld present us problems involving many variables, which we must take\nas independent.\n\nFrom this it follows that in making poiographs we must be prepared\nto use space of more than three dimensions. If the symmetry and\ncompleteness of our representation is to be of use to us we must be\nprepared to appreciate and criticise figures of a complexity greater\nthan of those in three dimensions. It is impossible to give an example\nof such a poiograph which will not be merely trivial, without going\ninto details of some kind irrelevant to our subject. I prefer to\nintroduce the irrelevant details rather than treat this part of the\nsubject perfunctorily.\n\nTo take an instance of a poiograph which does not lead us into the\ncomplexities incident on its application in classificatory science,\nlet us follow Mrs. Alicia Boole Stott in her representation of the\nsyllogism by its means. She will be interested to find that the curious\ngap she detected has a significance.\n\nA syllogism consists of two statements, the major and the minor\npremiss, with the conclusion that can be drawn from them. Thus, to take\nan instance, fig. 49. It is evident, from looking at the successive\nfigures that, if we know that the region M lies altogether within the\nregion P, and also know that the region S lies altogether within the\nregion M, we can conclude that the region S lies altogether within\nthe region P. M is P, major premiss; S is M, minor premiss; S is P,\nconclusion. Given the first two data we must conclude that S lies\nin P. The conclusion S is P involves two terms, S and P, which are\nrespectively called the subject and the predicate, the letters S and\nP being chosen with reference to the parts the notions they designate\nplay in the conclusion. S is the subject of the conclusion, P is the\npredicate of the conclusion. The major premiss we take to be, that\nwhich does not involve S, and here we always write it first.\n\n[Illustration: Fig. 49.]\n\nThere are several varieties of statement possessing different degrees\nof universality and manners of assertiveness. These different forms of\nstatement are called the moods.\n\nWe will take the major premiss as one variable, as a thing capable of\ndifferent modifications of the same kind, the minor premiss as another,\nand the different moods we will consider as defining the variations\nwhich these variables undergo.\n\nThere are four moods:\u2014\n\n 1. The universal affirmative; all M is P, called mood A.\n\n 2. The universal negative; no M is P, mood E.\n\n 3. The particular affirmative; some M is P, mood I.\n\n 4. The particular negative; some M is not P, mood O.\n\n[Illustration: 1. 2. 3. 4. Mood A. Mood E. Mood I. Mood O.\nFig. 50.]\n\nThe dotted lines in 3 and 4, fig. 50, denote that it is not known\nwhether or no any objects exist, corresponding to the space of which\nthe dotted line forms one delimiting boundary; thus, in mood I we do\nnot know if there are any M\u2019S which are not P, we only know some M\u2019S\nare P.\n\n[Illustration: Fig. 51.]\n\nRepresenting the first premiss in its various moods by regions marked\nby vertical lines to the right of PQ, we have in fig. 51, running up\nfrom the four letters AEIO, four columns, each of which indicates that\nthe major premiss is in the mood denoted by the respective letter. In\nthe first column to the right of PQ is the mood A. Now above the line\nRS let there be marked off four regions corresponding to the four moods\nof the minor premiss. Thus, in the first row above RS all the region\nbetween RS and the first horizontal line above it denotes that the\nminor premiss is in the mood A. The letters E, I, O, in the same way\nshow the mood characterising the minor premiss in the rows opposite\nthese letters.\n\nWe have still to exhibit the conclusion. To do this we must consider\nthe conclusion as a third variable, characterised in its different\nvarieties by four moods\u2014this being the syllogistic classification. The\nintroduction of a third variable involves a change in our system of\nrepresentation.\n\nBefore we started with the regions to the right of a certain line as\nrepresenting successively the major premiss in its moods; now we must\nstart with the regions to the right of a certain plane. Let LMNR be\nthe plane face of a cube, fig. 52, and let the cube be divided into\nfour parts by vertical sections parallel to LMNR. The variable, the\nmajor premiss, is represented by the successive regions which occur to\nthe right of the plane LMNR\u2014that region to which A stands opposite,\nthat slice of the cube, is significative of the mood A. This whole\nquarter-part of the cube represents that for every part of it the major\npremiss is in the mood A.\n\n[Illustration: Fig. 52.]\n\nIn a similar manner the next section, the second with the letter E\nopposite it, represents that for every one of the sixteen small cubic\nspaces in it, the major premiss is in the mood E. The third and fourth\ncompartments made by the vertical sections denote the major premiss in\nthe moods I and O. But the cube can be divided in other ways by other\nplanes. Let the divisions, of which four stretch from the front face,\ncorrespond to the minor premiss. The first wall of sixteen cubes,\nfacing the observer, has as its characteristic that in each of the\nsmall cubes, whatever else may be the case, the minor premiss is in the\nmood A. The variable\u2014the minor premiss\u2014varies through the phases A, E,\nI, O, away from the front face of the cube, or the front plane of which\nthe front face is a part.\n\nAnd now we can represent the third variable in a precisely similar way.\nWe can take the conclusion as the third variable, going through its\nfour phases from the ground plane upwards. Each of the small cubes at\nthe base of the whole cube has this true about it, whatever else may\nbe the case, that the conclusion is, in it, in the mood A. Thus, to\nrecapitulate, the first wall of sixteen small cubes, the first of the\nfour walls which, proceeding from left to right, build up the whole\ncube, is characterised in each part of it by this, that the major\npremiss is in the mood A.\n\nThe next wall denotes that the major premiss is in the mood E, and\nso on. Proceeding from the front to the back the first wall presents\na region in every part of which the minor premiss is in the mood A.\nThe second wall is a region throughout which the minor premiss is in\nthe mood E, and so on. In the layers, from the bottom upwards, the\nconclusion goes through its various moods beginning with A in the\nlowest, E in the second, I in the third, O in the fourth.\n\nIn the general case, in which the variables represented in the\npoiograph pass through a wide range of values, the planes from which we\nmeasure their degrees of variation in our representation are taken to\nbe indefinitely extended. In this case, however, all we are concerned\nwith is the finite region.\n\nWe have now to represent, by some limitation of the complex we have\nobtained, the fact that not every combination of premisses justifies\nany kind of conclusion. This can be simply effected by marking the\nregions in which the premisses, being such as are defined by the\npositions, a conclusion which is valid is found.\n\nTaking the conjunction of the major premiss, all M is P, and the minor,\nall S is M, we conclude that all S is P. Hence, that region must be\nmarked in which we have the conjunction of major premiss in mood A;\nminor premiss, mood A; conclusion, mood A. This is the cube occupying\nthe lowest left-hand corner of the large cube.\n\n[Illustration: Fig. 53.]\n\nProceeding in this way, we find that the regions which must be marked\nare those shown in fig. 53. To discuss the case shown in the marked\ncube which appears at the top of fig. 53. Here the major premiss is\nin the second wall to the right\u2014it is in the mood E and is of the\ntype no M is P. The minor premiss is in the mood characterised by the\nthird wall from the front. It is of the type some S is M. From these\npremisses we draw the conclusion that some S is not P, a conclusion in\nthe mood O. Now the mood O of the conclusion is represented in the top\nlayer. Hence we see that the marking is correct in this respect.\n\n[Illustration: Fig. 54.]\n\nIt would, of course, be possible to represent the cube on a plane by\nmeans of four squares, as in fig. 54, if we consider each square to\nrepresent merely the beginning of the region it stands for. Thus the\nwhole cube can be represented by four vertical squares, each standing\nfor a kind of vertical tray, and the markings would be as shown. In No.\n1 the major premiss is in mood A for the whole of the region indicated\nby the vertical square of sixteen divisions; in No. 2 it is in the mood\nE, and so on.\n\nA creature confined to a plane would have to adopt some such\ndisjunctive way of representing the whole cube. He would be obliged to\nrepresent that which we see as a whole in separate parts, and each part\nwould merely represent, would not be, that solid content which we see.\n\nThe view of these four squares which the plane creature would have\nwould not be such as ours. He would not see the interior of the four\nsquares represented above, but each would be entirely contained within\nits outline, the internal boundaries of the separate small squares he\ncould not see except by removing the outer squares.\n\nWe are now ready to introduce the fourth variable involved in the\nsyllogism.\n\nIn assigning letters to denote the terms of the syllogism we have taken\nS and P to represent the subject and predicate in the conclusion, and\nthus in the conclusion their order is invariable. But in the premisses\nwe have taken arbitrarily the order all M is P, and all S is M. There\nis no reason why M instead of P should not be the predicate of the\nmajor premiss, and so on.\n\nAccordingly we take the order of the terms in the premisses as the\nfourth variable. Of this order there are four varieties, and these\nvarieties are called figures.\n\nUsing the order in which the letters are written to denote that the\nletter first written is subject, the one written second is predicate,\nwe have the following possibilities:\u2014\n\n 1st Figure. 2nd Figure. 3rd Figure. 4th Figure.\n Major M P P M M P P M\n Minor S M S M M S M S\n\nThere are therefore four possibilities with regard to this fourth\nvariable as with regard to the premisses.\n\nWe have used up our dimensions of space in representing the phases of\nthe premisses and the conclusion in respect of mood, and to represent\nin an analogous manner the variations in figure we require a fourth\ndimension.\n\nNow in bringing in this fourth dimension we must make a change in our\norigins of measurement analogous to that which we made in passing from\nthe plane to the solid.\n\nThis fourth dimension is supposed to run at right angles to any of the\nthree space dimensions, as the third space dimension runs at right\nangles to the two dimensions of a plane, and thus it gives us the\nopportunity of generating a new kind of volume. If the whole cube moves\nin this dimension, the solid itself traces out a path, each section of\nwhich, made at right angles to the direction in which it moves, is a\nsolid, an exact repetition of the cube itself.\n\nThe cube as we see it is the beginning of a solid of such a kind. It\nrepresents a kind of tray, as the square face of the cube is a kind of\ntray against which the cube rests.\n\nSuppose the cube to move in this fourth dimension in four stages,\nand let the hyper-solid region traced out in the first stage of its\nprogress be characterised by this, that the terms of the syllogism\nare in the first figure, then we can represent in each of the three\nsubsequent stages the remaining three figures. Thus the whole cube\nforms the basis from which we measure the variation in figure. The\nfirst figure holds good for the cube as we see it, and for that\nhyper-solid which lies within the first stage; the second figure holds\ngood in the second stage, and so on.\n\nThus we measure from the whole cube as far as figures are concerned.\n\nBut we saw that when we measured in the cube itself having three\nvariables, namely, the two premisses and the conclusion, we measured\nfrom three planes. The base from which we measured was in every case\nthe same.\n\nHence, in measuring in this higher space we should have bases of the\nsame kind to measure from, we should have solid bases.\n\nThe first solid base is easily seen, it is the cube itself. The other\ncan be found from this consideration.\n\nThat solid from which we measure figure is that in which the remaining\nvariables run through their full range of varieties.\n\nNow, if we want to measure in respect of the moods of the major\npremiss, we must let the minor premiss, the conclusion, run through\ntheir range, and also the order of the terms. That is we must take as\nbasis of measurement in respect to the moods of the major that which\nrepresents the variation of the moods of the minor, the conclusion and\nthe variation of the figures.\n\nNow the variation of the moods of the minor and of the conclusion are\nrepresented in the square face on the left of the cube. Here are all\nvarieties of the minor premiss and the conclusion. The varieties of\nthe figures are represented by stages in a motion proceeding at right\nangles to all space directions, at right angles consequently to the\nface in question, the left-hand face of the cube.\n\nConsequently letting the left-hand face move in this direction we get\na cube, and in this cube all the varieties of the minor premiss, the\nconclusion, and the figure are represented.\n\nThus another cubic base of measurement is given to the cube, generated\nby movement of the left-hand square in the fourth dimension.\n\nWe find the other bases in a similar manner, one is the cube generated\nby the front square moved in the fourth dimension so as to generate a\ncube. From this cube variations in the mood of the minor are measured.\nThe fourth base is that found by moving the bottom square of the cube\nin the fourth dimension. In this cube the variations of the major,\nthe minor, and the figure are given. Considering this as a basis in\nthe four stages proceeding from it, the variation in the moods of the\nconclusion are given.\n\nAny one of these cubic bases can be represented in space, and then the\nhigher solid generated from them lies out of our space. It can only\nbe represented by a device analogous to that by which the plane being\nrepresents a cube.\n\nHe represents the cube shown above, by taking four square sections and\nplacing them arbitrarily at convenient distances the one from the other.\n\nSo we must represent this higher solid by four cubes: each cube\nrepresents only the beginning of the corresponding higher volume.\n\nIt is sufficient for us, then, if we draw four cubes, the first\nrepresenting that region in which the figure is of the first kind,\nthe second that region in which the figure is of the second kind,\nand so on. These cubes are the beginnings merely of the respective\nregions\u2014they are the trays, as it were, against which the real solids\nmust be conceived as resting, from which they start. The first one, as\nit is the beginning of the region of the first figure, is characterised\nby the order of the terms in the premisses being that of the first\nfigure. The second similarly has the terms of the premisses in the\norder of the second figure, and so on.\n\nThese cubes are shown below.\n\nFor the sake of showing the properties of the method of representation,\nnot for the logical problem, I will make a digression. I will represent\nin space the moods of the minor and of the conclusion and the different\nfigures, keeping the major always in mood A. Here we have three\nvariables in different stages, the minor, the conclusion, and the\nfigure. Let the square of the left-hand side of the original cube be\nimagined to be standing by itself, without the solid part of the cube,\nrepresented by (2) fig. 55. The A, E, I, O, which run away represent\nthe moods of the minor, the A, E, I, O, which run up represent the\nmoods of the conclusion. The whole square, since it is the beginning\nof the region in the major premiss, mood A, is to be considered as in\nmajor premiss, mood A.\n\nFrom this square, let it be supposed that that direction in which the\nfigures are represented runs to the left hand. Thus we have a cube (1)\nrunning from the square above, in which the square itself is hidden,\nbut the letters A, E, I, O, of the conclusion are seen. In this cube\nwe have the minor premiss and the conclusion in all their moods, and\nall the figures represented. With regard to the major premiss, since\nthe face (2) belongs to the first wall from the left in the original\narrangement, and in this arrangement was characterised by the major\npremiss in the mood A, we may say that the whole of the cube we now\nhave put up represents the mood A of the major premiss.\n\n[Illustration: Fig. 55.]\n\nHence the small cube at the bottom to the right in 1, nearest to the\nspectator, is major premiss, mood A; minor premiss, mood A; conclusion,\nmood A; and figure the first. The cube next to it, running to the left,\nis major premiss, mood A; minor premiss, mood A; conclusion, mood A;\nfigure 2.\n\nSo in this cube we have the representations of all the combinations\nwhich can occur when the major premiss, remaining in the mood A, the\nminor premiss, the conclusion, and the figures pass through their\nvarieties.\n\nIn this case there is no room in space for a natural representation of\nthe moods of the major premiss. To represent them we must suppose as\nbefore that there is a fourth dimension, and starting from this cube as\nbase in the fourth direction in four equal stages, all the first volume\ncorresponds to major premiss A, the second to major premiss, mood E,\nthe next to the mood I, and the last to mood O.\n\nThe cube we see is as it were merely a tray against which the\nfour-dimensional figure rests. Its section at any stage is a cube. But\na transition in this direction being transverse to the whole of our\nspace is represented by no space motion. We can exhibit successive\nstages of the result of transference of the cube in that direction, but\ncannot exhibit the product of a transference, however small, in that\ndirection.\n\n[Illustration: Fig. 56.]\n\nTo return to the original method of representing our variables,\nconsider fig. 56. These four cubes represent four sections of the\nfigure derived from the first of them by moving it in the fourth\ndimension. The first portion of the motion, which begins with 1, traces\nout a more than solid body, which is all in the first figure. The\nbeginning of this body is shown in 1. The next portion of the motion\ntraces out a more than solid body, all of which is in the second\nfigure; the beginning of this body is shown in 2; 3 and 4 follow on in\nlike manner. Here, then, in one four-dimensional figure we have all\nthe combinations of the four variables, major premiss, minor premiss,\nfigure, conclusion, represented, each variable going through its four\nvarieties. The disconnected cubes drawn are our representation in space\nby means of disconnected sections of this higher body.\n\nNow it is only a limited number of conclusions which are true\u2014their\ntruth depends on the particular combinations of the premisses and\nfigures which they accompany. The total figure thus represented may be\ncalled the universe of thought in respect to these four constituents,\nand out of the universe of possibly existing combinations it is the\nprovince of logic to select those which correspond to the results of\nour reasoning faculties.\n\nWe can go over each of the premisses in each of the moods, and find out\nwhat conclusion logically follows. But this is done in the works on\nlogic; most simply and clearly I believe in \u201cJevon\u2019s Logic.\u201d As we are\nonly concerned with a formal presentation of the results we will make\nuse of the mnemonic lines printed below, in which the words enclosed in\nbrackets refer to the figures, and are not significative:\u2014\n\n Barbara celarent Darii ferio_que_ [prioris].\n Caesare Camestris Festino Baroko [secundae].\n [Tertia] darapti disamis datisi felapton.\n Bokardo ferisson _habet_ [Quarta insuper addit].\n Bramantip camenes dimaris ferapton fresison.\n\nIn these lines each significative word has three vowels, the first\nvowel refers to the major premiss, and gives the mood of that premiss,\n\u201ca\u201d signifying, for instance, that the major mood is in mood _a_. The\nsecond vowel refers to the minor premiss, and gives its mood. The third\nvowel refers to the conclusion, and gives its mood. Thus (prioris)\u2014of\nthe first figure\u2014the first mnemonic word is \u201cbarbara,\u201d and this gives\nmajor premiss, mood A; minor premiss, mood A; conclusion, mood A.\nAccordingly in the first of our four cubes we mark the lowest left-hand\nfront cube. To take another instance in the third figure \u201cTertia,\u201d\nthe word \u201cferisson\u201d gives us major premiss mood E\u2014_e.g._, no M is P,\nminor premiss mood I; some M is S, conclusion, mood O; some S is not P.\nThe region to be marked then in the third representative cube is the\none in the second wall to the right for the major premiss, the third\nwall from the front for the minor premiss, and the top layer for the\nconclusion.\n\nIt is easily seen that in the diagram this cube is marked, and so with\nall the valid conclusions. The regions marked in the total region show\nwhich combinations of the four variables, major premiss, minor premiss,\nfigure, and conclusion exist.\n\nThat is to say, we objectify all possible conclusions, and build up an\nideal manifold, containing all possible combinations of them with the\npremisses, and then out of this we eliminate all that do not satisfy\nthe laws of logic. The residue is the syllogism, considered as a canon\nof reasoning.\n\nLooking at the shape which represents the totality of the valid\nconclusions, it does not present any obvious symmetry, or easily\ncharacterisable nature. A striking configuration, however, is\nobtained, if we project the four-dimensional figure obtained into a\nthree-dimensional one; that is, if we take in the base cube all those\ncubes which have a marked space anywhere in the series of four regions\nwhich start from that cube.\n\nThis corresponds to making abstraction of the figures, giving all the\nconclusions which are valid whatever the figure may be.\n\n[Illustration: Fig. 57.]\n\nProceeding in this way we obtain the arrangement of marked cubes shown\nin fig. 57. We see that the valid conclusions are arranged almost\nsymmetrically round one cube\u2014the one on the top of the column starting\nfrom AAA. There is one breach of continuity however in this scheme.\nOne cube is unmarked, which if marked would give symmetry. It is the\none which would be denoted by the letters I, E, O, in the third\nwall to the right, the second wall away, the topmost layer. Now this\ncombination of premisses in the mood IE, with a conclusion in the mood\nO, is not noticed in any book on logic with which I am familiar. Let\nus look at it for ourselves, as it seems that there must be something\ncurious in connection with this break of continuity in the poiograph.\n\n[Illustration: Fig. 58.]\n\nThe propositions I, E, in the various figures are the following, as\nshown in the accompanying scheme, fig. 58:\u2014First figure: some M is P;\nno S is M. Second figure: some P is M; no S is M. Third figure: some M\nis P; no M is S. Fourth figure: some P is M; no M is S.\n\nExamining these figures, we see, taking the first, that if some M is P\nand no S is M, we have no conclusion of the form S is P in the various\nmoods. It is quite indeterminate how the circle representing S lies\nwith regard to the circle representing P. It may lie inside, outside,\nor partly inside P. The same is true in the other figures 2 and 3.\nBut when we come to the fourth figure, since M and S lie completely\noutside each other, there cannot lie inside S that part of P which lies\ninside M. Now we know by the major premiss that some of P does lie in\nM. Hence S cannot contain the whole of P. In words, some P is M, no\nM is S, therefore S does not contain the whole of P. If we take P as\nthe subject, this gives us a conclusion in the mood O about P. Some\nP is not S. But it does not give us conclusion about S in any one of\nthe four forms recognised in the syllogism and called its moods. Hence\nthe breach of the continuity in the poiograph has enabled us to detect\na lack of completeness in the relations which are considered in the\nsyllogism.\n\nTo take an instance:\u2014Some Americans (P) are of African stock (M); No\nAryans (S) are of African stock (M); Aryans (S) do not include all of\nAmericans (P).\n\nIn order to draw a conclusion about S we have to admit the statement,\n\u201cS does not contain the whole of P,\u201d as a valid logical form\u2014it is a\nstatement about S which can be made. The logic which gives us the form,\n\u201csome P is not S,\u201d and which does not allow us to give the exactly\nequivalent and equally primary form, \u201cS does not contain the whole of\nP,\u201d is artificial.\n\nAnd I wish to point out that this artificiality leads to an error.\n\nIf one trusted to the mnemonic lines given above, one would conclude\nthat no logical conclusion about S can be drawn from the statement,\n\u201csome P are M, no M are S.\u201d\n\nBut a conclusion can be drawn: S does not contain the whole of P.\n\nIt is not that the result is given expressed in another form. The\nmnemonic lines deny that any conclusion can be drawn from premisses in\nthe moods I, E, respectively.\n\nThus a simple four-dimensional poiograph has enabled us to detect a\nmistake in the mnemonic lines which have been handed down unchallenged\nfrom medi\u00e6val times. To discuss the subject of these lines more fully a\nlogician defending them would probably say that a particular statement\ncannot be a major premiss; and so deny the existence of the fourth\nfigure in the combination of moods.\n\nTo take our instance: some Americans are of African stock; no Aryans\nare of African stock. He would say that the conclusion is some\nAmericans are not Aryans; and that the second statement is the major.\nHe would refuse to say anything about Aryans, condemning us to an\neternal silence about them, as far as these premisses are concerned!\nBut, if there is a statement involving the relation of two classes, it\nmust be expressible as a statement about either of them.\n\nTo bar the conclusion, \u201cAryans do not include the whole of Americans,\u201d\nis purely a makeshift in favour of a false classification.\n\nAnd the argument drawn from the universality of the major premiss\ncannot be consistently maintained. It would preclude such combinations\nas major O, minor A, conclusion O\u2014_i.e._, such as some mountains (M)\nare not permanent (P); all mountains (M) are scenery (S); some scenery\n(S) is not permanent (P).\n\nThis is allowed in \u201cJevon\u2019s Logic,\u201d and his omission to discuss I, E,\nO, in the fourth figure, is inexplicable. A satisfactory poiograph\nof the logical scheme can be made by admitting the use of the words\nsome, none, or all, about the predicate as well as about the subject.\nThen we can express the statement, \u201cAryans do not include the whole of\nAmericans,\u201d clumsily, but, when its obscurity is fathomed, correctly,\nas \u201cSome Aryans are not all Americans.\u201d And this method is what is\ncalled the \u201cquantification of the predicate.\u201d\n\nThe laws of formal logic are coincident with the conclusions which\ncan be drawn about regions of space, which overlap one another in the\nvarious possible ways. It is not difficult so to state the relations\nor to obtain a symmetrical poiograph. But to enter into this branch\nof geometry is beside our present purpose, which is to show the\napplication of the poiograph in a finite and limited region, without\nany of those complexities which attend its use in regard to natural\nobjects.\n\nIf we take the latter\u2014plants, for instance\u2014and, without assuming\nfixed directions in space as representative of definite variations,\narrange the representative points in such a manner as to correspond to\nthe similarities of the objects, we obtain configuration of singular\ninterest; and perhaps in this way, in the making of shapes of shapes,\nbodies with bodies omitted, some insight into the structure of the\nspecies and genera might be obtained.\n\n\n\n\n CHAPTER IX\n\n APPLICATION TO KANT\u2019S THEORY OF EXPERIENCE\n\n\nWhen we observe the heavenly bodies we become aware that they all\nparticipate in one universal motion\u2014a diurnal revolution round the\npolar axis.\n\nIn the case of fixed stars this is most unqualifiedly true, but in the\ncase of the sun, and the planets also, the single motion of revolution\ncan be discerned, modified, and slightly altered by other and secondary\nmotions.\n\nHence the universal characteristic of the celestial bodies is that they\nmove in a diurnal circle.\n\nBut we know that this one great fact which is true of them all has in\nreality nothing to do with them. The diurnal revolution which they\nvisibly perform is the result of the condition of the observer. It is\nbecause the observer is on a rotating earth that a universal statement\ncan be made about all the celestial bodies.\n\nThe universal statement which is valid about every one of the celestial\nbodies is that which does not concern them at all, and is but a\nstatement of the condition of the observer.\n\nNow there are universal statements of other kinds which we can make. We\ncan say that all objects of experience are in space and subject to the\nlaws of geometry.\n\nDoes this mean that space and all that it means is due to a condition\nof the observer?\n\nIf a universal law in one case means nothing affecting the objects\nthemselves, but only a condition of observation, is this true in every\ncase? There is shown us in astronomy a _vera causa_ for the assertion\nof a universal. Is the same cause to be traced everywhere?\n\nSuch is a first approximation to the doctrine of Kant\u2019s critique.\n\nIt is the apprehension of a relation into which, on the one side and\nthe other, perfectly definite constituents enter\u2014the human observer and\nthe stars\u2014and a transference of this relation to a region in which the\nconstituents on either side are perfectly unknown.\n\nIf spatiality is due to a condition of the observer, the observer\ncannot be this bodily self of ours\u2014the body, like the objects around\nit, are equally in space.\n\nThis conception Kant applied, not only to the intuitions of sense, but\nto the concepts of reason\u2014wherever a universal statement is made there\nis afforded him an opportunity for the application of his principle.\nHe constructed a system in which one hardly knows which the most to\nadmire, the architectonic skill, or the reticence with regard to things\nin themselves, and the observer in himself.\n\nHis system can be compared to a garden, somewhat formal perhaps, but\nwith the charm of a quality more than intellectual, a _besonnenheit_,\nan exquisite moderation over all. And from the ground he so carefully\nprepared with that buried in obscurity, which it is fitting should be\nobscure, science blossoms and the tree of real knowledge grows.\n\nThe critique is a storehouse of ideas of profound interest. The one\nof which I have given a partial statement leads, as we shall see\non studying it in detail, to a theory of mathematics suggestive of\nenquiries in many directions.\n\nThe justification for my treatment will be found amongst other passages\nin that part of the transcendental analytic, in which Kant speaks of\nobjects of experience subject to the forms of sensibility, not subject\nto the concepts of reason.\n\nKant asserts that whenever we think we think of objects in space and\ntime, but he denies that the space and time exist as independent\nentities. He goes about to explain them, and their universality, not by\nassuming them, as most other philosophers do, but by postulating their\nabsence. How then does it come to pass that the world is in space and\ntime to us?\n\nKant takes the same position with regard to what we call nature\u2014a great\nsystem subject to law and order. \u201cHow do you explain the law and order\nin nature?\u201d we ask the philosophers. All except Kant reply by assuming\nlaw and order somewhere, and then showing how we can recognise it.\n\nIn explaining our notions, philosophers from other than the Kantian\nstandpoint, assume the notions as existing outside us, and then it is\nno difficult task to show how they come to us, either by inspiration or\nby observation.\n\nWe ask \u201cWhy do we have an idea of law in nature?\u201d \u201cBecause natural\nprocesses go according to law,\u201d we are answered, \u201cand experience\ninherited or acquired, gives us this notion.\u201d\n\nBut when we speak about the law in nature we are speaking about a\nnotion of our own. So all that these expositors do is to explain our\nnotion by an assumption of it.\n\nKant is very different. He supposes nothing. An experience such as ours\nis very different from experience in the abstract. Imagine just simply\nexperience, succession of states, of consciousness! Why, there would be\nno connecting any two together, there would be no personal identity,\nno memory. It is out of a general experience such as this, which, in\nrespect to anything we call real, is less than a dream, that Kant shows\nthe genesis of an experience such as ours.\n\nKant takes up the problem of the explanation of space, time, order, and\nso quite logically does not presuppose them.\n\nBut how, when every act of thought is of things in space, and time,\nand ordered, shall we represent to ourselves that perfectly indefinite\nsomewhat which is Kant\u2019s necessary hypothesis\u2014that which is not in\nspace or time and is not ordered. That is our problem, to represent\nthat which Kant assumes not subject to any of our forms of thought, and\nthen show some function which working on that makes it into a \u201cnature\u201d\nsubject to law and order, in space and time. Such a function Kant\ncalls the \u201cUnity of Apperception\u201d; _i.e._, that which makes our state\nof consciousness capable of being woven into a system with a self, an\nouter world, memory, law, cause, and order.\n\nThe difficulty that meets us in discussing Kant\u2019s hypothesis is that\neverything we think of is in space and time\u2014how then shall we represent\nin space an existence not in space, and in time an existence not in\ntime? This difficulty is still more evident when we come to construct\na poiograph, for a poiograph is essentially a space structure. But\nbecause more evident the difficulty is nearer a solution. If we always\nthink in space, _i.e._ using space concepts, the first condition\nrequisite for adapting them to the representation of non-spatial\nexistence, is to be aware of the limitation of our thought, and so be\nable to take the proper steps to overcome it. The problem before us,\nthen, is to represent in space an existence not in space.\n\nThe solution is an easy one. It is provided by the conception of\nalternativity.\n\nTo get our ideas clear let us go right back behind the distinctions of\nan inner and an outer world. Both of these, Kant says, are products.\nLet us take merely states of consciousness, and not ask the question\nwhether they are produced or superinduced\u2014to ask such a question is to\nhave got too far on, to have assumed something of which we have not\ntraced the origin. Of these states let us simply say that they occur.\nLet us now use the word a \u201cposit\u201d for a phase of consciousness reduced\nto its last possible stage of evanescence; let a posit be that phase of\nconsciousness of which all that can be said is that it occurs.\n\nLet _a_, _b_, _c_, be three such posits. We cannot represent them in\nspace without placing them in a certain order, as _a_, _b_, _c_. But\nKant distinguishes between the forms of sensibility and the concepts\nof reason. A dream in which everything happens at haphazard would be\nan experience subject to the form of sensibility and only partially\nsubject to the concepts of reason. It is partially subject to the\nconcepts of reason because, although there is no order of sequence,\nstill at any given time there is order. Perception of a thing as in\nspace is a form of sensibility, the perception of an order is a concept\nof reason.\n\nWe must, therefore, in order to get at that process which Kant supposes\nto be constitutive of an ordered experience imagine the posits as in\nspace without order.\n\nAs we know them they must be in some order, _abc_, _bca_, _cab_, _acb_,\n_cba_, _bac_, one or another.\n\nTo represent them as having no order conceive all these different\norders as equally existing. Introduce the conception of\nalternativity\u2014let us suppose that the order _abc_, and _bac_, for\nexample, exist equally, so that we cannot say about _a_ that it comes\nbefore or after _b_. This would correspond to a sudden and arbitrary\nchange of _a_ into _b_ and _b_ into _a_, so that, to use Kant\u2019s words,\nit would be possible to call one thing by one name at one time and at\nanother time by another name.\n\nIn an experience of this kind we have a kind of chaos, in which no\norder exists; it is a manifold not subject to the concepts of reason.\n\nNow is there any process by which order can be introduced into such a\nmanifold\u2014is there any function of consciousness in virtue of which an\nordered experience could arise?\n\nIn the precise condition in which the posits are, as described above,\nit does not seem to be possible. But if we imagine a duality to exist\nin the manifold, a function of consciousness can be easily discovered\nwhich will produce order out of no order.\n\nLet us imagine each posit, then, as having, a dual aspect. Let _a_ be\n1_a_ in which the dual aspect is represented by the combination of\nsymbols. And similarly let _b_ be 2_b_, _c_ be 3_c_, in which 2 and _b_\nrepresent the dual aspects of _b_, 3 and _c_ those of _c_.\n\nSince _a_ can arbitrarily change into _b_, or into _c_, and so on, the\nparticular combinations written above cannot be kept. We have to assume\nthe equally possible occurrence of form such as 2_a_, 2_b_, and so on;\nand in order to get a representation of all those combinations out of\nwhich any set is alternatively possible, we must take every aspect with\nevery aspect. We must, that is, have every letter with every number.\n\nLet us now apply the method of space representation.\n\n _Note._\u2014At the beginning of the next chapter the same structures as\n those which follow are exhibited in more detail and a reference to\n them will remove any obscurity which may be found in the immediately\n following passages. They are there carried on to a greater\n multiplicity of dimensions, and the significance of the process here\n briefly explained becomes more apparent.\n\n[Illustration: Fig. 59.]\n\nTake three mutually rectangular axes in space 1, 2, 3 (fig. 59), and\non each mark three points, the common meeting point being the first on\neach axis. Then by means of these three points on each axis we define\n27 positions, 27 points in a cubical cluster, shown in fig. 60, the\nsame method of co-ordination being used as has been described before.\nEach of these positions can be named by means of the axes and the\npoints combined.\n\n[Illustration: Fig. 60.]\n\nThus, for instance, the one marked by an asterisk can be called 1_c_,\n2_b_, 3_c_, because it is opposite to _c_ on 1, to _b_ on 2, to _c_ on\n3.\n\nLet us now treat of the states of consciousness corresponding to\nthese positions. Each point represents a composite of posits, and\nthe manifold of consciousness corresponding to them is of a certain\ncomplexity.\n\nSuppose now the constituents, the points on the axes, to interchange\narbitrarily, any one to become any other, and also the axes 1, 2, and\n3, to interchange amongst themselves, any one to become any other, and\nto be subject to no system or law, that is to say, that order does not\nexist, and that the points which run _abc_ on each axis may run _bac_,\nand so on.\n\nThen any one of the states of consciousness represented by the points\nin the cluster can become any other. We have a representation of a\nrandom consciousness of a certain degree of complexity.\n\nNow let us examine carefully one particular case of arbitrary\ninterchange of the points, _a_, _b_, _c_; as one such case, carefully\nconsidered, makes the whole clear.\n\n[Illustration: Fig. 61.]\n\nConsider the points named in the figure 1_c_, 2_a_, 3_c_; 1_c_, 2_c_,\n3_a_; 1_a_, 2_c_, 3_c_, and examine the effect on them when a change of\norder takes place. Let us suppose, for instance, that _a_ changes into\n_b_, and let us call the two sets of points we get, the one before and\nthe one after, their change conjugates.\n\n Before the change 1_c_ 2_a_ 3_c_ 1_c_ 2_c_ 3_a_ 1_a_ 2_c_ 3_c_}Conjug-\n After the change 1_c_ 2_b_ 3_c_ 1_c_ 2_c_ 3_b_ 1_b_ 2_c_ 3_c_} ates.\n\nThe points surrounded by rings represent the conjugate points.\n\nIt is evident that as consciousness, represented first by the first\nset of points and afterwards by the second set of points, would have\nnothing in common in its two phases. It would not be capable of giving\nan account of itself. There would be no identity.\n\nIf, however, we can find any set of points in the cubical cluster,\nwhich, when any arbitrary change takes place in the points on the\naxes, or in the axes themselves, repeats itself, is reproduced, then a\nconsciousness represented by those points would have a permanence. It\nwould have a principle of identity. Despite the no law, the no order,\nof the ultimate constituents, it would have an order, it would form a\nsystem, the condition of a personal identity would be fulfilled.\n\nThe question comes to this, then. Can we find a system of points\nwhich is self-conjugate which is such that when any posit on the axes\nbecomes any other, or when any axis becomes any other, such a set\nis transformed into itself, its identity is not submerged, but rises\nsuperior to the chaos of its constituents?\n\n[Illustration: Fig. 62.]\n\nSuch a set can be found. Consider the set represented in fig. 62, and\nwritten down in the first of the two lines\u2014\n\n Self- {1_a_ 2_b_ 3_c_ 1_b_ 2_a_ 3_c_ 1_c_ 2_a_ 3_b_\n conjugate. {1_c_ 2_b_ 3_a_ 1_b_ 2_c_ 3_a_ 1_a_ 2_c_ 3_b_\n\n Self- {1_c_ 2_b_ 3_a_ 1_b_ 2_c_ 3_a_ 1_a_ 2_c_ 3_b_\n conjugate. {1_a_ 2_b_ 3_c_ 1_b_ 2_a_ 3_c_ 1_c_ 2_a_ 3_b_\n\nIf now _a_ change into _c_ and _c_ into _a_, we get the set in the\nsecond line, which has the same members as are in the upper line.\nLooking at the diagram we see that it would correspond simply to the\nturning of the figures as a whole.[2] Any arbitrary change of the\npoints on the axes, or of the axes themselves, reproduces the same set.\n\n [2] These figures are described more fully, and extended, in the next\n chapter.\n\nThus, a function, by which a random, an unordered, consciousness\ncould give an ordered and systematic one, can be represented. It\nis noteworthy that it is a system of selection. If out of all the\nalternative forms that only is attended to which is self-conjugate,\nan ordered consciousness is formed. A selection gives a feature of\npermanence.\n\nCan we say that the permanent consciousness is this selection?\n\nAn analogy between Kant and Darwin comes into light. That which is\nswings clear of the fleeting, in virtue of its presenting a feature of\npermanence. There is no need to suppose any function of \u201cattending to.\u201d\nA consciousness capable of giving an account of itself is one which is\ncharacterised by this combination. All combinations exist\u2014of this kind\nis the consciousness which can give an account of itself. And the very\nduality which we have presupposed may be regarded as originated by a\nprocess of selection.\n\nDarwin set himself to explain the origin of the fauna and flora of\nthe world. He denied specific tendencies. He assumed an indefinite\nvariability\u2014that is, chance\u2014but a chance confined within narrow limits\nas regards the magnitude of any consecutive variations. He showed that\norganisms possessing features of permanence, if they occurred would be\npreserved. So his account of any structure or organised being was that\nit possessed features of permanence.\n\nKant, undertaking not the explanation of any particular phenomena but\nof that which we call nature as a whole, had an origin of species\nof his own, an account of the flora and fauna of consciousness. He\ndenied any specific tendency of the elements of consciousness, but\ntaking our own consciousness, pointed out that in which it resembled\nany consciousness which could survive, which could give an account of\nitself.\n\nHe assumes a chance or random world, and as great and small were not\nto him any given notions of which he could make use, he did not limit\nthe chance, the randomness, in any way. But any consciousness which\nis permanent must possess certain features\u2014those attributes namely\nwhich give it permanence. Any consciousness like our own is simply a\nconsciousness which possesses those attributes. The main thing is that\nwhich he calls the unity of apperception, which we have seen above is\nsimply the statement that a particular set of phases of consciousness\non the basis of complete randomness will be self-conjugate, and so\npermanent.\n\nAs with Darwin so with Kant, the reason for existence of any feature\ncomes to this\u2014show that it tends to the permanence of that which\npossesses it.\n\nWe can thus regard Kant as the creator of the first of the modern\nevolution theories. And, as is so often the case, the first effort was\nthe most stupendous in its scope. Kant does not investigate the origin\nof any special part of the world, such as its organisms, its chemical\nelements, its social communities of men. He simply investigates the\norigin of the whole\u2014of all that is included in consciousness, the\norigin of that \u201cthought thing\u201d whose progressive realisation is the\nknowable universe.\n\nThis point of view is very different from the ordinary one, in which a\nman is supposed to be placed in a world like that which he has come to\nthink of it, and then to learn what he has found out from this model\nwhich he himself has placed on the scene.\n\nWe all know that there are a number of questions in attempting an\nanswer to which such an assumption is not allowable.\n\nMill, for instance, explains our notion of \u201claw\u201d by an invariable\nsequence in nature. But what we call nature is something given in\nthought. So he explains a thought of law and order by a thought of an\ninvariable sequence. He leaves the problem where he found it.\n\nKant\u2019s theory is not unique and alone. It is one of a number of\nevolution theories. A notion of its import and significance can be\nobtained by a comparison of it with other theories.\n\nThus in Darwin\u2019s theoretical world of natural selection a certain\nassumption is made, the assumption of indefinite variability\u2014slight\nvariability it is true, over any appreciable lapse of time, but\nindefinite in the postulated epochs of transformation\u2014and a whole chain\nof results is shown to follow.\n\nThis element of chance variation is not, however, an ultimate resting\nplace. It is a preliminary stage. This supposing the all is a\npreliminary step towards finding out what is. If every kind of organism\ncan come into being, those that do survive will present such and such\ncharacteristics. This is the necessary beginning for ascertaining what\nkinds of organisms do come into existence. And so Kant\u2019s hypothesis\nof a random consciousness is the necessary beginning for the rational\ninvestigation of consciousness as it is. His assumption supplies, as\nit were, the space in which we can observe the phenomena. It gives the\ngeneral laws constitutive of any experience. If, on the assumption\nof absolute randomness in the constituents, such and such would be\ncharacteristic of the experience, then, whatever the constituents,\nthese characteristics must be universally valid.\n\nWe will now proceed to examine more carefully the poiograph,\nconstructed for the purpose of exhibiting an illustration of Kant\u2019s\nunity of apperception.\n\nIn order to show the derivation order out of non-order it has been\nnecessary to assume a principle of duality\u2014we have had the axes and the\nposits on the axes\u2014there are two sets of elements, each non-ordered,\nand it is in the reciprocal relation of them that the order, the\ndefinite system, originates.\n\nIs there anything in our experience of the nature of a duality?\n\nThere certainly are objects in our experience which have order and\nthose which are incapable of order. The two roots of a quadratic\nequation have no order. No one can tell which comes first. If a body\nrises vertically and then goes at right angles to its former course,\nno one can assign any priority to the direction of the north or to\nthe east. There is no priority in directions of turning. We associate\nturnings with no order progressions in a line with order. But in the\naxes and points we have assumed above there is no such distinction.\nIt is the same, whether we assume an order among the turnings, and no\norder among the points on the axes, or, _vice versa_, an order in the\npoints and no order in the turnings. A being with an infinite number of\naxes mutually at right angles, with a definite sequence between them\nand no sequence between the points on the axes, would be in a condition\nformally indistinguishable from that of a creature who, according to an\nassumption more natural to us, had on each axis an infinite number of\nordered points and no order of priority amongst the axes. A being in\nsuch a constituted world would not be able to tell which was turning\nand which was length along an axis, in order to distinguish between\nthem. Thus to take a pertinent illustration, we may be in a world\nof an infinite number of dimensions, with three arbitrary points on\neach\u2014three points whose order is indifferent, or in a world of three\naxes of arbitrary sequence with an infinite number of ordered points on\neach. We can\u2019t tell which is which, to distinguish it from the other.\n\nThus it appears the mode of illustration which we have used is not an\nartificial one. There really exists in nature a duality of the kind\nwhich is necessary to explain the origin of order out of no order\u2014the\nduality, namely, of dimension and position. Let us use the term group\nfor that system of points which remains unchanged, whatever arbitrary\nchange of its constituents takes place. We notice that a group involves\na duality, is inconceivable without a duality.\n\nThus, according to Kant, the primary element of experience is the\ngroup, and the theory of groups would be the most fundamental branch\nof science. Owing to an expression in the critique the authority of\nKant is sometimes adduced against the assumption of more than three\ndimensions to space. It seems to me, however, that the whole tendency\nof his theory lies in the opposite direction, and points to a perfect\nduality between dimension and position in a dimension.\n\nIf the order and the law we see is due to the conditions of conscious\nexperience, we must conceive nature as spontaneous, free, subject to no\npredication that we can devise, but, however apprehended, subject to\nour logic.\n\nAnd our logic is simply spatiality in the general sense\u2014that resultant\nof a selection of the permanent from the unpermanent, the ordered from\nthe unordered, by the means of the group and its underlying duality.\n\nWe can predicate nothing about nature, only about the way in which\nwe can apprehend nature. All that we can say is that all that which\nexperience gives us will be conditioned as spatial, subject to our\nlogic. Thus, in exploring the facts of geometry from the simplest\nlogical relations to the properties of space of any number of\ndimensions, we are merely observing ourselves, becoming aware of the\nconditions under which we must perceive. Do any phenomena present\nthemselves incapable of explanation under the assumption of the space\nwe are dealing with, then we must habituate ourselves to the conception\nof a higher space, in order that our logic may be equal to the task\nbefore us.\n\nWe gain a repetition of the thought that came before, experimentally\nsuggested. If the laws of the intellectual comprehension of nature are\nthose derived from considering her as absolute chance, subject to no\nlaw save that derived from a process of selection, then, perhaps, the\norder of nature requires different faculties from the intellectual to\napprehend it. The source and origin of ideas may have to be sought\nelsewhere than in reasoning.\n\nThe total outcome of the critique is to leave the ordinary man just\nwhere he is, justified in his practical attitude towards nature,\nliberated from the fetters of his own mental representations.\n\nThe truth of a picture lies in its total effect. It is vain to seek\ninformation about the landscape from an examination of the pigments.\nAnd in any method of thought it is the complexity of the whole that\nbrings us to a knowledge of nature. Dimensions are artificial enough,\nbut in the multiplicity of them we catch some breath of nature.\n\nWe must therefore, and this seems to me the practical conclusion of the\nwhole matter, proceed to form means of intellectual apprehension of a\ngreater and greater degree of complexity, both dimensionally and in\nextent in any dimension. Such means of representation must always be\nartificial, but in the multiplicity of the elements with which we deal,\nhowever incipiently arbitrary, lies our chance of apprehending nature.\n\nAnd as a concluding chapter to this part of the book, I will extend\nthe figures, which have been used to represent Kant\u2019s theory, two\nsteps, so that the reader may have the opportunity of looking at a\nfour-dimensional figure which can be delineated without any of the\nspecial apparatus, to the consideration of which I shall subsequently\npass on.\n\n\n\n\n CHAPTER X\n\n A FOUR-DIMENSIONAL FIGURE\n\n\nThe method used in the preceding chapter to illustrate the problem\nof Kant\u2019s critique, gives a singularly easy and direct mode of\nconstructing a series of important figures in any number of dimensions.\n\nWe have seen that to represent our space a plane being must give up one\nof his axes, and similarly to represent the higher shapes we must give\nup one amongst our three axes.\n\nBut there is another kind of giving up which reduces the construction\nof higher shapes to a matter of the utmost simplicity.\n\nOrdinarily we have on a straight line any number of positions. The\nwealth of space in position is illimitable, while there are only three\ndimensions.\n\nI propose to give up this wealth of positions, and to consider the\nfigures obtained by taking just as many positions as dimensions.\n\nIn this way I consider dimensions and positions as two \u201ckinds,\u201d and\napplying the simple rule of selecting every one of one kind with every\nother of every other kind, get a series of figures which are noteworthy\nbecause they exactly fill space of any number of dimensions (as the\nhexagon fills a plane) by equal repetitions of themselves.\n\nThe rule will be made more evident by a simple application.\n\nLet us consider one dimension and one position. I will call the axis\n_i_, and the position _o_.\n\n \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014-_i_\n _o_\n\nHere the figure is the position _o_ on the line _i_. Take now two\ndimensions and two positions on each.\n\n[Illustration: Fig. 63.]\n\nWe have the two positions _o_; 1 on _i_, and the two positions _o_, 1\non _j_, fig. 63. These give rise to a certain complexity. I will let\nthe two lines _i_ and _j_ meet in the position I call _o_ on each, and\nI will consider _i_ as a direction starting equally from every position\non _j_, and _j_ as starting equally from every position on _i_. We thus\nobtain the following figure:\u2014A is both _oi_ and _oj_, B is 1_i_ and\n_oj_, and so on as shown in fig. 63_b_. The positions on AC are all\n_oi_ positions. They are, if we like to consider it in that way, points\nat no distance in the _i_ direction from the line AC. We can call the\nline AC the _oi_ line. Similarly the points on AB are those no distance\nfrom AB in the _j_ direction, and we can call them _oj_ points and the\nline AB the _oj_ line. Again, the line CD can be called the 1_j_ line\nbecause the points on it are at a distance, 1 in the _j_ direction.\n\n[Illustration: Fig. 63_b_.]\n\nWe have then four positions or points named as shown, and, considering\ndirections and positions as \u201ckinds,\u201d we have the combination of two\nkinds with two kinds. Now, selecting every one of one kind with every\nother of every other kind will mean that we take 1 of the kind _i_ and\nwith it _o_ of the kind _j_; and then, that we take _o_ of the kind _i_\nand with it 1 of the kind _j_.\n\nThus we get a pair of positions lying in the straight line BC, fig.\n64. We can call this pair 10 and 01 if we adopt the plan of mentally,\nadding an _i_ to the first and a _j_ to the second of the symbols\nwritten thus\u201401 is a short expression for O_i_, 1_j_.\n\n[Illustration: Fig. 64.]\n\nComing now to our space, we have three dimensions, so we take three\npositions on each. These positions I will suppose to be at equal\ndistances along each axis. The three axes and the three positions on\neach are shown in the accompanying diagrams, fig. 65, of which the\nfirst represents a cube with the front faces visible, the second the\nrear faces of the same cube; the positions I will call 0, 1, 2; the\naxes, _i_, _j_, _k_. I take the base ABC as the starting place, from\nwhich to determine distances in the _k_ direction, and hence every\npoint in the base ABC will be an _ok_ position, and the base ABC can be\ncalled an _ok_ plane.\n\n[Illustration: Fig. 65.]\n\nIn the same way, measuring the distances from the face ADC, we see\nthat every position in the face ADC is an _oi_ position, and the whole\nplane of the face may be called an _oi_ plane. Thus we see that with\nthe introduction of a new dimension the signification of a compound\nsymbol, such as \u201c_oi_,\u201d alters. In the plane it meant the line AC. In\nspace it means the whole plane ACD.\n\nNow, it is evident that we have twenty-seven positions, each of them\nnamed. If the reader will follow this nomenclature in respect of the\npositions marked in the figures he will have no difficulty in assigning\nnames to each one of the twenty-seven positions. A is _oi_, _oj_, _ok_.\nIt is at the distance 0 along _i_, 0 along _j_, 0 along _k_, and _io_\ncan be written in short 000, where the _ijk_ symbols are omitted.\n\nThe point immediately above is 001, for it is no distance in the _i_\ndirection, and a distance of 1 in the _k_ direction. Again, looking at\nB, it is at a distance of 2 from A, or from the plane ADC, in the _i_\ndirection, 0 in the _j_ direction from the plane ABD, and 0 in the _k_\ndirection, measured from the plane ABC. Hence it is 200 written for\n2_i_, 0_j_, 0_k_.\n\nNow, out of these twenty-seven \u201cthings\u201d or compounds of position and\ndimension, select those which are given by the rule, every one of one\nkind with every other of every other kind.\n\nTake 2 of the _i_ kind. With this we must have a 1 of the _j_ kind, and\nthen by the rule we can only have a 0 of the _k_ kind, for if we had\nany other of the _k_ kind we should repeat one of the kinds we already\nhad. In 2_i_, 1_j_, 1_k_, for instance, 1 is repeated. The point we\nobtain is that marked 210, fig. 66.\n\n[Illustration: Fig. 66.]\n\nProceeding in this way, we pick out the following cluster of points,\nfig. 67. They are joined by lines, dotted where they are hidden by the\nbody of the cube, and we see that they form a figure\u2014a hexagon which\ncould be taken out of the cube and placed on a plane. It is a figure\nwhich will fill a plane by equal repetitions of itself. The plane being\nrepresenting this construction in his plane would take three squares to\nrepresent the cube. Let us suppose that he takes the _ij_ axes in his\nspace and _k_ represents the axis running out of his space, fig. 68.\nIn each of the three squares shown here as drawn separately he could\nselect the points given by the rule, and he would then have to try to\ndiscover the figure determined by the three lines drawn. The line from\n210 to 120 is given in the figure, but the line from 201 to 102 or GK\nis not given. He can determine GK by making another set of drawings and\ndiscovering in them what the relation between these two extremities is.\n\n[Illustration: Fig. 67.]\n\n[Illustration: Fig. 68.]\n\n[Illustration: Fig. 69.]\n\nLet him draw the _i_ and _k_ axes in his plane, fig. 69. The _j_ axis\nthen runs out and he has the accompanying figure. In the first of these\nthree squares, fig. 69, he can pick out by the rule the two points\n201, 102\u2014G, and K. Here they occur in one plane and he can measure the\ndistance between them. In his first representation they occur at G and\nK in separate figures.\n\nThus the plane being would find that the ends of each of the lines was\ndistant by the diagonal of a unit square from the corresponding end\nof the last and he could then place the three lines in their right\nrelative position. Joining them he would have the figure of a hexagon.\n\n[Illustration: Fig. 70.]\n\nWe may also notice that the plane being could make a representation of\nthe whole cube simultaneously. The three squares, shown in perspective\nin fig. 70, all lie in one plane, and on these the plane being could\npick out any selection of points just as well as on three separate\nsquares. He would obtain a hexagon by joining the points marked. This\nhexagon, as drawn, is of the right shape, but it would not be so if\nactual squares were used instead of perspective, because the relation\nbetween the separate squares as they lie in the plane figure is not\ntheir real relation. The figure, however, as thus constructed, would\ngive him an idea of the correct figure, and he could determine it\naccurately by remembering that distances in each square were correct,\nbut in passing from one square to another their distance in the third\ndimension had to be taken into account.\n\nComing now to the figure made by selecting according to our rule from\nthe whole mass of points given by four axes and four positions in each,\nwe must first draw a catalogue figure in which the whole assemblage is\nshown.\n\nWe can represent this assemblage of points by four solid figures. The\nfirst giving all those positions which are at a distance O from our\nspace in the fourth dimension, the second showing all those that are at\na distance 1, and so on.\n\nThese figures will each be cubes. The first two are drawn showing the\nfront faces, the second two the rear faces. We will mark the points 0,\n1, 2, 3, putting points at those distances along each of these axes,\nand suppose all the points thus determined to be contained in solid\nmodels of which our drawings in fig. 71 are representatives. Here we\nnotice that as on the plane 0_i_ meant the whole line from which the\ndistances in the _i_ direction was measured, and as in space 0_i_\nmeans the whole plane from which distances in the _i_ direction are\nmeasured, so now 0_h_ means the whole space in which the first cube\nstands\u2014measuring away from that space by a distance of one we come to\nthe second cube represented.\n\n[Illustration: Fig. 71.]\n\nNow selecting according to the rule every one of one kind with every\nother of every other kind, we must take, for instance, 3_i_, 2_j_,\n1_k_, 0_h_. This point is marked 3210 at the lower star in the figure.\nIt is 3 in the _i_ direction, 2 in the _j_ direction, 1 in the _k_\ndirection, 0 in the _h_ direction.\n\nWith 3_i_ we must also take 1_j_, 2_k_, 0_h_. This point is shown by\nthe second star in the cube 0_h_.\n\n[Illustration: Fig. 72.]\n\nIn the first cube, since all the points are 0_h_ points, we can only\nhave varieties in which _i_, _j_, _k_, are accompanied by 3, 2, 1.\n\nThe points determined are marked off in the diagram fig. 72, and lines\nare drawn joining the adjacent pairs in each figure, the lines being\ndotted when they pass within the substance of the cube in the first two\ndiagrams.\n\nOpposite each point, on one side or the other of each cube, is written\nits name. It will be noticed that the figures are symmetrical right and\nleft; and right and left the first two numbers are simply interchanged.\n\nNow this being our selection of points, what figure do they make when\nall are put together in their proper relative positions?\n\nTo determine this we must find the distance between corresponding\ncorners of the separate hexagons.\n\n[Illustration: Fig. 73.]\n\nTo do this let us keep the axes _i_, _j_, in our space, and draw _h_\ninstead of _k_, letting _k_ run out in the fourth dimension, fig. 73.\n\nHere we have four cubes again, in the first of which all the points are\n0_k_ points; that is, points at a distance zero in the _k_ direction\nfrom the space of the three dimensions _ijh_. We have all the points\nselected before, and some of the distances, which in the last diagram\nled from figure to figure are shown here in the same figure, and so\ncapable of measurement. Take for instance the points 3120 to 3021,\nwhich in the first diagram (fig. 72) lie in the first and second\nfigures. Their actual relation is shown in fig. 73 in the cube marked\n2K, where the points in question are marked with a *. We see that the\ndistance in question is the diagonal of a unit square. In like manner\nwe find that the distance between corresponding points of any two\nhexagonal figures is the diagonal of a unit square. The total figure\nis now easily constructed. An idea of it may be gained by drawing all\nthe four cubes in the catalogue figure in one (fig. 74). These cubes\nare exact repetitions of one another, so one drawing will serve as a\nrepresentation of the whole series, if we take care to remember where\nwe are, whether in a 0_h_, a 1_h_, a 2_h_, or a 3_h_ figure, when we\npick out the points required. Fig. 74 is a representation of all the\ncatalogue cubes put in one. For the sake of clearness the front faces\nand the back faces of this cube are represented separately.\n\n[Illustration: Fig. 74.]\n\nThe figure determined by the selected points is shown below.\n\nIn putting the sections together some of the outlines in them\ndisappear. The line TW for instance is not wanted.\n\nWe notice that PQTW and TWRS are each the half of a hexagon. Now QV and\nVR lie in one straight line. Hence these two hexagons fit together,\nforming one hexagon, and the line TW is only wanted when we consider a\nsection of the whole figure, we thus obtain the solid represented in\nthe lower part of fig. 74. Equal repetitions of this figure, called a\ntetrakaidecagon, will fill up three-dimensional space.\n\nTo make the corresponding four-dimensional figure we have to take five\naxes mutually at right angles with five points on each. A catalogue of\nthe positions determined in five-dimensional space can be found thus.\n\nTake a cube with five points on each of its axes, the fifth point is\nat a distance of four units of length from the first on any one of\nthe axes. And since the fourth dimension also stretches to a distance\nof four we shall need to represent the successive sets of points at\ndistances 0, 1, 2, 3, 4, in the fourth dimensions, five cubes. Now\nall of these extend to no distance at all in the fifth dimension. To\nrepresent what lies in the fifth dimension we shall have to draw,\nstarting from each of our cubes, five similar cubes to represent the\nfour steps on in the fifth dimension. By this assemblage we get a\ncatalogue of all the points shown in fig. 75, in which _L_ represents\nthe fifth dimension.\n\n[Illustration: Fig. 75.]\n\nNow, as we saw before, there is nothing to prevent us from putting all\nthe cubes representing the different stages in the fourth dimension in\none figure, if we take note when we look at it, whether we consider\nit as a 0_h_, a 1_h_, a 2_h_, etc., cube. Putting then the 0_h_, 1_h_,\n2_h_, 3_h_, 4_h_ cubes of each row in one, we have five cubes with the\nsides of each containing five positions, the first of these five cubes\nrepresents the 0_l_ points, and has in it the _i_ points from 0 to 4,\nthe _j_ points from 0 to 4, the _k_ points from 0 to 4, while we have\nto specify with regard to any selection we make from it, whether we\nregard it as a 0_h_, a 1_h_, a 2_h_, a 3_h_, or a 4_h_ figure. In fig.\n76 each cube is represented by two drawings, one of the front part, the\nother of the rear part.\n\nLet then our five cubes be arranged before us and our selection be made\naccording to the rule. Take the first figure in which all points are\n0_l_ points. We cannot have 0 with any other letter. Then, keeping in\nthe first figure, which is that of the 0_l_ positions, take first of\nall that selection which always contains 1_h_. We suppose, therefore,\nthat the cube is a 1_h_ cube, and in it we take _i_, _j_, _k_ in\ncombination with 4, 3, 2 according to the rule.\n\nThe figure we obtain is a hexagon, as shown, the one in front. The\npoints on the right hand have the same figures as those on the left,\nwith the first two numerals interchanged. Next keeping still to the\n0_l_ figure let us suppose that the cube before us represents a section\nat a distance of 2 in the _h_ direction. Let all the points in it be\nconsidered as 2_h_ points. We then have a 0_l_, 2_h_ region, and have\nthe sets _ijk_ and 431 left over. We must then pick out in accordance\nwith our rule all such points as 4_i_, 3_j_, 1_k_.\n\nThese are shown in the figure and we find that we can draw them without\nconfusion, forming the second hexagon from the front. Going on in this\nway it will be seen that in each of the five figures a set of hexagons\nis picked out, which put together form a three-space figure something\nlike the tetrakaidecagon.\n\n[Illustration: Fig. 76.]\n\nThese separate figures are the successive stages in which the whole\nfour-dimensional figure in which they cohere can be apprehended.\n\nThe first figure and the last are tetrakaidecagons. These are two\nof the solid boundaries of the figure. The other solid boundaries\ncan be traced easily. Some of them are complete from one face in the\nfigure to the corresponding face in the next, as for instance the\nsolid which extends from the hexagonal base of the first figure to the\nequal hexagonal base of the second figure. This kind of boundary is a\nhexagonal prism. The hexagonal prism also occurs in another sectional\nseries, as for instance, in the square at the bottom of the first\nfigure, the oblong at the base of the second and the square at the base\nof the third figure.\n\nOther solid boundaries can be traced through four of the five sectional\nfigures. Thus taking the hexagon at the top of the first figure we\nfind in the next a hexagon also, of which some alternate sides are\nelongated. The top of the third figure is also a hexagon with the other\nset of alternate rules elongated, and finally we come in the fourth\nfigure to a regular hexagon.\n\nThese four sections are the sections of a tetrakaidecagon as can\nbe recognised from the sections of this figure which we have had\npreviously. Hence the boundaries are of two kinds, hexagonal prisms and\ntetrakaidecagons.\n\nThese four-dimensional figures exactly fill four-dimensional space by\nequal repetitions of themselves.\n\n\n\n\n CHAPTER XI\n\nNOMENCLATURE AND ANALOGIES PRELIMINARY TO THE STUDY OF FOUR-DIMENSIONAL\n FIGURES\n\n\nIn the following pages a method of designating different regions of\nspace by a systematic colour scheme has been adopted. The explanations\nhave been given in such a manner as to involve no reference to models,\nthe diagrams will be found sufficient. But to facilitate the study a\ndescription of a set of models is given in an appendix which the reader\ncan either make for himself or obtain. If models are used the diagrams\nin Chapters XI. and XII. will form a guide sufficient to indicate their\nuse. Cubes of the colours designated by the diagrams should be picked\nout and used to reinforce the diagrams. The reader, in the following\ndescription, should suppose that a board or wall stretches away from\nhim, against which the figures are placed.\n\n[Illustration: Fig. 77.]\n\nTake a square, one of those shown in Fig. 77 and give it a neutral\ncolour, let this colour be called \u201cnull,\u201d and be such that it makes no\nappreciable difference to any colour with which it mixed. If there is\nno such real colour let us imagine such a colour, and assign to it the\nproperties of the number zero, which makes no difference in any number\nto which it is added.\n\nAbove this square place a red square. Thus we symbolise the going up by\nadding red to null.\n\nAway from this null square place a yellow square, and represent going\naway by adding yellow to null.\n\nTo complete the figure we need a fourth square. Colour this orange,\nwhich is a mixture of red and yellow, and so appropriately represents a\ngoing in a direction compounded of up and away. We have thus a colour\nscheme which will serve to name the set of squares drawn. We have two\naxes of colours\u2014red and yellow\u2014and they may occupy as in the figure\nthe direction up and away, or they may be turned about; in any case\nthey enable us to name the four squares drawn in their relation to one\nanother.\n\nNow take, in Fig. 78, nine squares, and suppose that at the end of the\ngoing in any direction the colour started with repeats itself.\n\n[Illustration: Fig. 78.]\n\nWe obtain a square named as shown.\n\nLet us now, in fig. 79, suppose the number of squares to be increased,\nkeeping still to the principle of colouring already used.\n\nHere the nulls remain four in number. There are three reds between the\nfirst null and the null above it, three yellows between the first null\nand the null beyond it, while the oranges increase in a double way.\n\n[Illustration: Fig. 79.]\n\nSuppose this process of enlarging the number of the squares to be\nindefinitely pursued and the total figure obtained to be reduced in\nsize, we should obtain a square of which the interior was all orange,\nwhile the lines round it were red and yellow, and merely the points\nnull colour, as in fig. 80. Thus all the points, lines, and the area\nwould have a colour.\n\n[Illustration: Fig. 80.]\n\nWe can consider this scheme to originate thus:\u2014Let a null point move\nin a yellow direction and trace out a yellow line and end in a null\npoint. Then let the whole line thus traced move in a red direction. The\nnull points at the ends of the line will produce red lines, and end in\nnull points. The yellow line will trace out a yellow and red, or orange\nsquare.\n\nNow, turning back to fig. 78, we see that these two ways of naming, the\none we started with and the one we arrived at, can be combined.\n\nBy its position in the group of four squares, in fig. 77, the null\nsquare has a relation to the yellow and to the red directions. We can\nspeak therefore of the red line of the null square without confusion,\nmeaning thereby the line AB, fig. 81, which runs up from the initial\nnull point A in the figure as drawn. The yellow line of the null square\nis its lower horizontal line AC as it is situated in the figure.\n\n[Illustration: Fig. 81.]\n\nIf we wish to denote the upper yellow line BD, fig. 81, we can speak\nof it as the yellow \u03b3 line, meaning the yellow line which is separated\nfrom the primary yellow line by the red movement.\n\nIn a similar way each of the other squares has null points, red and\nyellow lines. Although the yellow square is all yellow, its line CD,\nfor instance, can be referred to as its red line.\n\nThis nomenclature can be extended.\n\nIf the eight cubes drawn, in fig. 82, are put close together, as on\nthe right hand of the diagram, they form a cube, and in them, as thus\narranged, a going up is represented by adding red to the zero, or\nnull colour, a going away by adding yellow, a going to the right by\nadding white. White is used as a colour, as a pigment, which produces\na colour change in the pigments with which it is mixed. From whatever\ncube of the lower set we start, a motion up brings us to a cube showing\na change to red, thus light yellow becomes light yellow red, or light\norange, which is called ochre. And going to the right from the null on\nthe left we have a change involving the introduction of white, while\nthe yellow change runs from front to back. There are three colour\naxes\u2014the red, the white, the yellow\u2014and these run in the position the\ncubes occupy in the drawing\u2014up, to the right, away\u2014but they could be\nturned about to occupy any positions in space.\n\n[Illustration: Fig. 82.]\n\n[Illustration: Fig. 83. The three layers.]\n\nWe can conveniently represent a block of cubes by three sets of\nsquares, representing each the base of a cube.\n\nThus the block, fig. 83, can be represented by the layers on the\nright. Here, as in the case of the plane, the initial colours repeat\nthemselves at the end of the series.\n\nProceeding now to increase the number of the cubes we obtain fig. 84,\nin which the initial letters of the colours are given instead of their\nfull names.\n\nHere we see that there are four null cubes as before, but the series\nwhich spring from the initial corner will tend to become lines of\ncubes, as also the sets of cubes parallel to them, starting from other\ncorners. Thus, from the initial null springs a line of red cubes, a\nline of white cubes, and a line of yellow cubes.\n\nIf the number of the cubes is largely increased, and the size of the\nwhole cube is diminished, we get a cube with null points, and the edges\ncoloured with these three colours.\n\n[Illustration: Fig. 84.]\n\nThe light yellow cubes increase in two ways, forming ultimately a sheet\nof cubes, and the same is true of the orange and pink sets. Hence,\nultimately the cube thus formed would have red, white, and yellow\nlines surrounding pink, orange, and light yellow faces. The ochre cubes\nincrease in three ways, and hence ultimately the whole interior of the\ncube would be coloured ochre.\n\nWe have thus a nomenclature for the points, lines, faces, and solid\ncontent of a cube, and it can be named as exhibited in fig. 85.\n\n[Illustration: Fig. 85.]\n\nWe can consider the cube to be produced in the following way. A null\npoint moves in a direction to which we attach the colour indication\nyellow; it generates a yellow line and ends in a null point. The yellow\nline thus generated moves in a direction to which we give the colour\nindication red. This lies up in the figure. The yellow line traces out\na yellow, red, or orange square, and each of its null points trace out\na red line, and ends in a null point.\n\nThis orange square moves in a direction to which we attribute the\ncolour indication white, in this case the direction is the right. The\nsquare traces out a cube coloured orange, red, or ochre, the red lines\ntrace out red to white or pink squares, and the yellow lines trace out\nlight yellow squares, each line ending in a line of its own colour.\nWhile the points each trace out a null + white, or white line to end in\na null point.\n\nNow returning to the first block of eight cubes we can name each point,\nline, and square in them by reference to the colour scheme, which they\ndetermine by their relation to each other.\n\nThus, in fig. 86, the null cube touches the red cube by a light yellow\nsquare; it touches the yellow cube by a pink square, and touches the\nwhite cube by an orange square.\n\nThere are three axes to which the colours red, yellow, and white are\nassigned, the faces of each cube are designated by taking these colours\nin pairs. Taking all the colours together we get a colour name for the\nsolidity of a cube.\n\n[Illustration: Fig. 86.]\n\nLet us now ask ourselves how the cube could be presented to the plane\nbeing. Without going into the question of how he could have a real\nexperience of it, let us see how, if we could turn it about and show it\nto him, he, under his limitations, could get information about it. If\nthe cube were placed with its red and yellow axes against a plane, that\nis resting against it by its orange face, the plane being would observe\na square surrounded by red and yellow lines, and having null points.\nSee the dotted square, fig. 87.\n\n[Illustration: Fig. 87.]\n\nWe could turn the cube about the red line so that a different face\ncomes into juxtaposition with the plane.\n\nSuppose the cube turned about the red line. As it is turning from its\nfirst position all of it except the red line leaves the plane\u2014goes\nabsolutely out of the range of the plane being\u2019s apprehension. But when\nthe yellow line points straight out from the plane then the pink face\ncomes into contact with it. Thus the same red line remaining as he saw\nit at first, now towards him comes a face surrounded by white and red\nlines.\n\nIf we call the direction to the right the unknown direction, then\nthe line he saw before, the yellow line, goes out into this unknown\ndirection, and the line which before went into the unknown direction,\ncomes in. It comes in in the opposite direction to that in which the\nyellow line ran before; the interior of the face now against the plane\nis pink. It is a property of two lines at right angles that, if one\nturns out of a given direction and stands at right angles to it, then\nthe other of the two lines comes in, but runs the opposite way in that\ngiven direction, as in fig. 88.\n\n[Illustration: Fig. 88.]\n\nNow these two presentations of the cube would seem, to the plane\ncreature like perfectly different material bodies, with only that line\nin common in which they both meet.\n\nAgain our cube can be turned about the yellow line. In this case the\nyellow square would disappear as before, but a new square would come\ninto the plane after the cube had rotated by an angle of 90\u00b0 about this\nline. The bottom square of the cube would come in thus in figure 89.\nThe cube supposed in contact with the plane is rotated about the lower\nyellow line and then the bottom face is in contact with the plane.\n\nHere, as before, the red line going out into the unknown dimension,\nthe white line which before ran in the unknown dimension would come\nin downwards in the opposite sense to that in which the red line ran\nbefore.\n\n[Illustration: Fig. 89.]\n\nNow if we use _i_, _j_, _k_, for the three space directions, _i_ left\nto right, _j_ from near away, _k_ from below up; then, using the colour\nnames for the axes, we have that first of all white runs _i_, yellow\nruns _j_, red runs _k_; then after the first turning round the _k_\naxis, white runs negative _j_, yellow runs _i_, red runs _k_; thus we\nhave the table:\u2014\n\n _i_ _j_ _k_\n 1st position white yellow red\n 2nd position yellow white\u2014 red\n 3rd position red yellow white\u2014\n\nHere white with a negative sign after it in the column under _j_ means\nthat white runs in the negative sense of the _j_ direction.\n\nWe may express the fact in the following way:\u2014 In the plane there is\nroom for two axes while the body has three. Therefore in the plane we\ncan represent any two. If we want to keep the axis that goes in the\nunknown dimension always running in the positive sense, then the axis\nwhich originally ran in the unknown dimension (the white axis) must\ncome in in the negative sense of that axis which goes out of the plane\ninto the unknown dimension.\n\nIt is obvious that the unknown direction, the direction in which the\nwhite line runs at first, is quite distinct from any direction which\nthe plane creature knows. The white line may come in towards him, or\nrunning down. If he is looking at a square, which is the face of a cube\n(looking at it by a line), then any one of the bounding lines remaining\nunmoved, another face of the cube may come in, any one of the faces,\nnamely, which have the white line in them. And the white line comes\nsometimes in one of the space directions he knows, sometimes in another.\n\nNow this turning which leaves a line unchanged is something quite\nunlike any turning he knows in the plane. In the plane a figure turns\nround a point. The square can turn round the null point in his plane,\nand the red and yellow lines change places, only of course, as with\nevery rotation of lines at right angles, if red goes where yellow went,\nyellow comes in negative of red\u2019s old direction.\n\nThis turning, as the plane creature conceives it, we should call\nturning about an axis perpendicular to the plane. What he calls turning\nabout the null point we call turning about the white line as it stands\nout from his plane. There is no such thing as turning about a point,\nthere is always an axis, and really much more turns than the plane\nbeing is aware of.\n\nTaking now a different point of view, let us suppose the cubes to be\npresented to the plane being by being passed transverse to his plane.\nLet us suppose the sheet of matter over which the plane being and all\nobjects in his world slide, to be of such a nature that objects can\npass through it without breaking it. Let us suppose it to be of the\nsame nature as the film of a soap bubble, so that it closes around\nobjects pushed through it, and, however the object alters its shape as\nit passes through it, let us suppose this film to run up to the contour\nof the object in every part, maintaining its plane surface unbroken.\n\nThen we can push a cube or any object through the film and the plane\nbeing who slips about in the film will know the contour of the cube\njust and exactly where the film meets it.\n\n[Illustration: Fig. 90.]\n\nFig. 90 represents a cube passing through a plane film. The plane being\nnow comes into contact with a very thin slice of the cube somewhere\nbetween the left and right hand faces. This very thin slice he thinks\nof as having no thickness, and consequently his idea of it is what we\ncall a section. It is bounded by him by pink lines front and back,\ncoming from the part of the pink face he is in contact with, and above\nand below, by light yellow lines. Its corners are not null-coloured\npoints, but white points, and its interior is ochre, the colour of the\ninterior of the cube.\n\nIf now we suppose the cube to be an inch in each dimension, and to pass\nacross, from right to left, through the plane, then we should explain\nthe appearances presented to the plane being by saying: First of all\nyou have the face of a cube, this lasts only a moment; then you have a\nfigure of the same shape but differently coloured. This, which appears\nnot to move to you in any direction which you know of, is really moving\ntransverse to your plane world. Its appearance is unaltered, but each\nmoment it is something different\u2014a section further on, in the white,\nthe unknown dimension. Finally, at the end of the minute, a face comes\nin exactly like the face you first saw. This finishes up the cube\u2014it is\nthe further face in the unknown dimension.\n\nThe white line, which extends in length just like the red or the\nyellow, you do not see as extensive; you apprehend it simply as an\nenduring white point. The null point, under the condition of movement\nof the cube, vanishes in a moment, the lasting white point is really\nyour apprehension of a white line, running in the unknown dimension.\nIn the same way the red line of the face by which the cube is first in\ncontact with the plane lasts only a moment, it is succeeded by the pink\nline, and this pink line lasts for the inside of a minute. This lasting\npink line in your apprehension of a surface, which extends in two\ndimensions just like the orange surface extends, as you know it, when\nthe cube is at rest.\n\nBut the plane creature might answer, \u201cThis orange object is substance,\nsolid substance, bounded completely and on every side.\u201d\n\nHere, of course, the difficulty comes in. His solid is our surface\u2014his\nnotion of a solid is our notion of an abstract surface with no\nthickness at all.\n\nWe should have to explain to him that, from every point of what he\ncalled a solid, a new dimension runs away. From every point a line\ncan be drawn in a direction unknown to him, and there is a solidity\nof a kind greater than that which he knows. This solidity can only\nbe realised by him by his supposing an unknown direction, by motion\nin which what he conceives to be solid matter instantly disappears.\nThe higher solid, however, which extends in this dimension as well\nas in those which he knows, lasts when a motion of that kind takes\nplace, different sections of it come consecutively in the plane\nof his apprehension, and take the place of the solid which he at\nfirst conceives to be all. Thus, the higher solid\u2014our solid in\ncontradistinction to his area solid, his two-dimensional solid, must\nbe conceived by him as something which has duration in it, under\ncircumstances in which his matter disappears out of his world.\n\nWe may put the matter thus, using the conception of motion.\n\nA null point moving in a direction away generates a yellow line, and\nthe yellow line ends in a null point. We suppose, that is, a point\nto move and mark out the products of this motion in such a manner.\nNow suppose this whole line as thus produced to move in an upward\ndirection; it traces out the two-dimensional solid, and the plane being\ngets an orange square. The null point moves in a red line and ends in\na null point, the yellow line moves and generates an orange square and\nends in a yellow line, the farther null point generates a red line and\nends in a null point. Thus, by movement in two successive directions\nknown to him, he can imagine his two-dimensional solid produced with\nall its boundaries.\n\nNow we tell him: \u201cThis whole two-dimensional solid can move in a third\nor unknown dimension to you. The null point moving in this dimension\nout of your world generates a white line and ends in a null point. The\nyellow line moving generates a light yellow two-dimensional solid and\nends in a yellow line, and this two-dimensional solid, lying end on to\nyour plane world, is bounded on the far side by the other yellow line.\nIn the same way each of the lines surrounding your square traces out an\narea, just like the orange area you know. But there is something new\nproduced, something which you had no idea of before; it is that which\nis produced by the movement of the orange square. That, than which you\ncan imagine nothing more solid, itself moves in a direction open to it\nand produces a three-dimensional solid. Using the addition of white\nto symbolise the products of this motion this new kind of solid will\nbe light orange or ochre, and it will be bounded on the far side by\nthe final position of the orange square which traced it out, and this\nfinal position we suppose to be coloured like the square in its first\nposition, orange with yellow and red boundaries and null corners.\u201d\n\nThis product of movement, which it is so easy for us to describe, would\nbe difficult for him to conceive. But this difficulty is connected\nrather with its totality than with any particular part of it.\n\nAny line, or plane of this, to him higher, solid we could show to him,\nand put in his sensible world.\n\nWe have already seen how the pink square could be put in his world by\na turning of the cube about the red line. And any section which we can\nconceive made of the cube could be exhibited to him. You have simply to\nturn the cube and push it through, so that the plane of his existence\nis the plane which cuts out the given section of the cube, then the\nsection would appear to him as a solid. In his world he would see the\ncontour, get to any part of it by digging down into it.\n\n\n THE PROCESS BY WHICH A PLANE BEING WOULD GAIN A NOTION OF A SOLID.\n\nIf we suppose the plane being to have a general idea of the existence\nof a higher solid\u2014our solid\u2014we must next trace out in detail the\nmethod, the discipline, by which he would acquire a working familiarity\nwith our space existence. The process begins with an adequate\nrealisation of a simple solid figure. For this purpose we will suppose\neight cubes forming a larger cube, and first we will suppose each cube\nto be coloured throughout uniformly. Let the cubes in fig. 91 be the\neight making a larger cube.\n\n[Illustration: Fig. 91.]\n\nNow, although each cube is supposed to be coloured entirely through\nwith the colour, the name of which is written on it, still we can\nspeak of the faces, edges, and corners of each cube as if the colour\nscheme we have investigated held for it. Thus, on the null cube we can\nspeak of a null point, a red line, a white line, a pink face, and so\non. These colour designations are shown on No. 1 of the views of the\ntesseract in the plate. Here these colour names are used simply in\ntheir geometrical significance. They denote what the particular line,\netc., referred to would have as its colour, if in reference to the\nparticular cube the colour scheme described previously were carried out.\n\nIf such a block of cubes were put against the plane and then passed\nthrough it from right to left, at the rate of an inch a minute, each\ncube being an inch each way, the plane being would have the following\nappearances:\u2014\n\nFirst of all, four squares null, yellow, red, orange, lasting each a\nminute; and secondly, taking the exact places of these four squares,\nfour others, coloured white, light yellow, pink, ochre. Thus, to make\na catalogue of the solid body, he would have to put side by side in\nhis world two sets of four squares each, as in fig. 92. The first are\nsupposed to last a minute, and then the others to come in in place of\nthem, and also last a minute.\n\n[Illustration: Fig. 92.]\n\nIn speaking of them he would have to denote what part of the respective\ncube each square represents. Thus, at the beginning he would have null\ncube orange face, and after the motion had begun he would have null\ncube ochre section. As he could get the same coloured section whichever\nway the cube passed through, it would be best for him to call this\nsection white section, meaning that it is transverse to the white axis.\nThese colour-names, of course, are merely used as names, and do not\nimply in this case that the object is really coloured. Finally, after\na minute, as the first cube was passing beyond his plane he would have\nnull cube orange face again.\n\nThe same names will hold for each of the other cubes, describing what\nface or section of them the plane being has before him; and the second\nwall of cubes will come on, continue, and go out in the same manner. In\nthe area he thus has he can represent any movement which we carry out\nin the cubes, as long as it does not involve a motion in the direction\nof the white axis. The relation of parts that succeed one another in\nthe direction of the white axis is realised by him as a consecution of\nstates.\n\nNow, his means of developing his space apprehension lies in this, that\nthat which is represented as a time sequence in one position of the\ncubes, can become a real co-existence, _if something that has a real\nco-existence becomes a time sequence_.\n\nWe must suppose the cubes turned round each of the axes, the red line,\nand the yellow line, then something, which was given as time before,\nwill now be given as the plane creature\u2019s space; something, which was\ngiven as space before, will now be given as a time series as the cube\nis passed through the plane.\n\nThe three positions in which the cubes must be studied are the one\ngiven above and the two following ones. In each case the original null\npoint which was nearest to us at first is marked by an asterisk. In\nfigs. 93 and 94 the point marked with a star is the same in the cubes\nand in the plane view.\n\n[Illustration: Fig. 93. The cube swung round the red line, so that the\nwhite line points towards us.]\n\nIn fig. 93 the cube is swung round the red line so as to point towards\nus, and consequently the pink face comes next to the plane. As it\npasses through there are two varieties of appearance designated by\nthe figures 1 and 2 in the plane. These appearances are named in the\nfigure, and are determined by the order in which the cubes come in the\nmotion of the whole block through the plane.\n\nWith regard to these squares severally, however, different names must\nbe used, determined by their relations in the block.\n\nThus, in fig. 93, when the cube first rests against the plane the null\ncube is in contact by its pink face; as the block passes through we get\nan ochre section of the null cube, but this is better called a yellow\nsection, as it is made by a plane perpendicular to the yellow line.\nWhen the null cube has passed through the plane, as it is leaving it,\nwe get again a pink face.\n\n[Illustration: Fig. 94. The cube swung round yellow line, with red line\nrunning from left to right, and white line running down.]\n\nThe same series of changes take place with the cube appearances which\nfollow on those of the null cube. In this motion the yellow cube\nfollows on the null cube, and the square marked yellow in 2 in the\nplane will be first \u201cyellow pink face,\u201d then \u201cyellow yellow section,\u201d\nthen \u201cyellow pink face.\u201d\n\nIn fig. 94, in which the cube is turned about the yellow line, we have\na certain difficulty, for the plane being will find that the position\nhis squares are to be placed in will lie below that which they first\noccupied. They will come where the support was on which he stood his\nfirst set of squares. He will get over this difficulty by moving his\nsupport.\n\nThen, since the cubes come upon his plane by the light yellow face, he\nwill have, taking the null cube as before for an example, null, light\nyellow face; null, red section, because the section is perpendicular\nto the red line; and finally, as the null cube leaves the plane, null,\nlight yellow face. Then, in this case red following on null, he will\nhave the same series of views of the red as he had of the null cube.\n\n[Illustration: Fig. 95.]\n\nThere is another set of considerations which we will briefly allude to.\n\nSuppose there is a hollow cube, and a string is stretched across it\nfrom null to null, _r_, _y_, _wh_, as we may call the far diagonal\npoint, how will this string appear to the plane being as the cube moves\ntransverse to his plane?\n\nLet us represent the cube as a number of sections, say 5, corresponding\nto 4 equal divisions made along the white line perpendicular to it.\n\nWe number these sections 0, 1, 2, 3, 4, corresponding to the distances\nalong the white line at which they are taken, and imagine each section\nto come in successively, taking the place of the preceding one.\n\nThese sections appear to the plane being, counting from the first, to\nexactly coincide each with the preceding one. But the section of the\nstring occupies a different place in each to that which it does in the\npreceding section. The section of the string appears in the position\nmarked by the dots. Hence the slant of the string appears as a motion\nin the frame work marked out by the cube sides. If we suppose the\nmotion of the cube not to be recognised, then the string appears to the\nplane being as a moving point. Hence extension on the unknown dimension\nappears as duration. Extension sloping in the unknown direction appears\nas continuous movement.\n\n\n\n\n CHAPTER XII\n\n THE SIMPLEST FOUR-DIMENSIONAL SOLID\n\n\nA plane being, in learning to apprehend solid existence, must first\nof all realise that there is a sense of direction altogether wanting\nto him. That which we call right and left does not exist in his\nperception. He must assume a movement in a direction, and a distinction\nof positive and negative in that direction, which has no reality\ncorresponding to it in the movements he can make. This direction, this\nnew dimension, he can only make sensible to himself by bringing in\ntime, and supposing that changes, which take place in time, are due\nto objects of a definite configuration in three dimensions passing\ntransverse to his plane, and the different sections of it being\napprehended as changes of one and the same plane figure.\n\nHe must also acquire a distinct notion about his plane world, he must\nno longer believe that it is the all of space, but that space extends\non both sides of it. In order, then, to prevent his moving off in this\nunknown direction, he must assume a sheet, an extended solid sheet, in\ntwo dimensions, against which, in contact with which, all his movements\ntake place.\n\nWhen we come to think of a four-dimensional solid, what are the\ncorresponding assumptions which we must make?\n\nWe must suppose a sense which we have not, a sense of direction\nwanting in us, something which a being in a four-dimensional world\nhas, and which we have not. It is a sense corresponding to a new space\ndirection, a direction which extends positively and negatively from\nevery point of our space, and which goes right away from any space\ndirection we know of. The perpendicular to a plane is perpendicular,\nnot only to two lines in it, but to every line, and so we must conceive\nthis fourth dimension as running perpendicularly to each and every line\nwe can draw in our space.\n\nAnd as the plane being had to suppose something which prevented his\nmoving off in the third, the unknown dimension to him, so we have to\nsuppose something which prevents us moving off in the direction unknown\nto us. This something, since we must be in contact with it in every one\nof our movements, must not be a plane surface, but a solid; it must be\na solid, which in every one of our movements we are against, not in.\nIt must be supposed as stretching out in every space dimension that we\nknow; but we are not in it, we are against it, we are next to it, in\nthe fourth dimension.\n\nThat is, as the plane being conceives himself as having a very small\nthickness in the third dimension, of which he is not aware in his\nsense experience, so we must suppose ourselves as having a very small\nthickness in the fourth dimension, and, being thus four-dimensional\nbeings, to be prevented from realising that we are such beings by a\nconstraint which keeps us always in contact with a vast solid sheet,\nwhich stretches on in every direction. We are against that sheet, so\nthat, if we had the power of four-dimensional movement, we should\neither go away from it or through it; all our space movements as we\nknow them being such that, performing them, we keep in contact with\nthis solid sheet.\n\nNow consider the exposition a plane being would make for himself as to\nthe question of the enclosure of a square, and of a cube.\n\nHe would say the square A, in Fig. 96, is completely enclosed by the\nfour squares, A far, A near, A above, A below, or as they are written\nA_n_, A_f_, A_a_, A_b_.\n\n[Illustration: Fig. 96.]\n\nIf now he conceives the square A to move in the, to him, unknown\ndimension it will trace out a cube, and the bounding squares will\nform cubes. Will these completely surround the cube generated by A?\nNo; there will be two faces of the cube made by A left uncovered;\nthe first, that face which coincides with the square A in its first\nposition; the next, that which coincides with the square A in its\nfinal position. Against these two faces cubes must be placed in order\nto completely enclose the cube A. These may be called the cubes left\nand right or A_l_ and A_r_. Thus each of the enclosing squares of the\nsquare A becomes a cube and two more cubes are wanted to enclose the\ncube formed by the movement of A in the third dimension.\n\n[Illustration: Fig. 97.]\n\nThe plane being could not see the square A with the squares A_n_, A_f_,\netc., placed about it, because they completely hide it from view; and\nso we, in the analogous case in our three-dimensional world, cannot\nsee a cube A surrounded by six other cubes. These cubes we will call A\nnear A_n_, A far A_f_, A above A_a_, A below A_b_, A left A_l_, A right\nA_r_, shown in fig. 97. If now the cube A moves in the fourth dimension\nright out of space, it traces out a higher cube\u2014a tesseract, as it may\nbe called. Each of the six surrounding cubes carried on in the same\nmotion will make a tesseract also, and these will be grouped around the\ntesseract formed by A. But will they enclose it completely?\n\nAll the cubes A_n_, A_f_, etc., lie in our space. But there is nothing\nbetween the cube A and that solid sheet in contact with which every\nparticle of matter is. When the cube A moves in the fourth direction\nit starts from its position, say A_k_, and ends in a final position\nA_n_ (using the words \u201cana\u201d and \u201ckata\u201d for up and down in the fourth\ndimension). Now the movement in this fourth dimension is not bounded by\nany of the cubes A_n_, A_f_, nor by what they form when thus moved. The\ntesseract which A becomes is bounded in the positive and negative ways\nin this new direction by the first position of A and the last position\nof A. Or, if we ask how many tesseracts lie around the tesseract which\nA forms, there are eight, of which one meets it by the cube A, and\nanother meets it by a cube like A at the end of its motion.\n\nWe come here to a very curious thing. The whole solid cube A is to be\nlooked on merely as a boundary of the tesseract.\n\nYet this is exactly analogous to what the plane being would come to in\nhis study of the solid world. The square A (fig. 96), which the plane\nbeing looks on as a solid existence in his plane world, is merely the\nboundary of the cube which he supposes generated by its motion.\n\nThe fact is that we have to recognise that, if there is another\ndimension of space, our present idea of a solid body, as one which\nhas three dimensions only, does not correspond to anything real,\nbut is the abstract idea of a three-dimensional boundary limiting a\nfour-dimensional solid, which a four-dimensional being would form. The\nplane being\u2019s thought of a square is not the thought of what we should\ncall a possibly existing real square, but the thought of an abstract\nboundary, the face of a cube.\n\nLet us now take our eight coloured cubes, which form a cube in\nspace, and ask what additions we must make to them to represent\nthe simplest collection of four-dimensional bodies\u2014namely, a group\nof them of the same extent in every direction. In plane space we\nhave four squares. In solid space we have eight cubes. So we should\nexpect in four-dimensional space to have sixteen four-dimensional\nbodies-bodies which in four-dimensional space correspond to cubes in\nthree-dimensional space, and these bodies we call tesseracts.\n\nGiven then the null, white, red, yellow cubes, and those which make up\nthe block, we notice that we represent perfectly well the extension\nin three directions (fig. 98). From the null point of the null cube,\ntravelling one inch, we come to the white cube; travelling one inch\naway we come to the yellow cube; travelling one inch up we come to the\nred cube. Now, if there is a fourth dimension, then travelling from the\nsame null point for one inch in that direction, we must come to the\nbody lying beyond the null region.\n\n[Illustration: Fig. 98.]\n\nI say null region, not cube; for with the introduction of the fourth\ndimension each of our cubes must become something different from cubes.\nIf they are to have existence in the fourth dimension, they must be\n\u201cfilled up from\u201d in this fourth dimension.\n\nNow we will assume that as we get a transference from null to white\ngoing in one way, from null to yellow going in another, so going\nfrom null in the fourth direction we have a transference from null\nto blue, using thus the colours white, yellow, red, blue, to denote\ntransferences in each of the four directions\u2014right, away, up, unknown\nor fourth dimension.\n\n[Illustration: Fig. 99.\n\nA plane being\u2019s representation of a block of eight cubes by two sets of\nfour squares.]\n\nHence, as the plane being must represent the solid regions, he would\ncome to by going right, as four squares lying in some position in his\nplane, arbitrarily chosen, side by side with his original four squares,\nso we must represent those eight four-dimensional regions, which we\nshould come to by going in the fourth dimension from each of our eight\ncubes, by eight cubes placed in some arbitrary position relative to our\nfirst eight cubes.\n\n[Illustration: Fig. 100.]\n\nOur representation of a block of sixteen tesseracts by two blocks of\neight cubes.[3]\n\n [3] The eight cubes used here in 2 can be found in the second of the\n model blocks. They can be taken out and used.\n\nHence, of the two sets of eight cubes, each one will serve us as a\nrepresentation of one of the sixteen tesseracts which form one single\nblock in four-dimensional space. Each cube, as we have it, is a tray,\nas it were, against which the real four-dimensional figure rests\u2014just\nas each of the squares which the plane being has is a tray, so to\nspeak, against which the cube it represents could rest.\n\nIf we suppose the cubes to be one inch each way, then the original\neight cubes will give eight tesseracts of the same colours, or the\ncubes, extending each one inch in the fourth dimension.\n\nBut after these there come, going on in the fourth dimension, eight\nother bodies, eight other tesseracts. These must be there, if we\nsuppose the four-dimensional body we make up to have two divisions, one\ninch each in each of four directions.\n\nThe colour we choose to designate the transference to this second\nregion in the fourth dimension is blue. Thus, starting from the null\ncube and going in the fourth dimension, we first go through one inch of\nthe null tesseract, then we come to a blue cube, which is the beginning\nof a blue tesseract. This blue tesseract stretches one inch farther on\nin the fourth dimension.\n\nThus, beyond each of the eight tesseracts, which are of the same colour\nas the cubes which are their bases, lie eight tesseracts whose colours\nare derived from the colours of the first eight by adding blue. Thus\u2014\n\n Null gives blue\n Yellow \u201d green\n Red \u201d purple\n Orange \u201d brown\n White \u201d light blue\n Pink \u201d light purple\n Light yellow \u201d light green\n Ochre \u201d light brown\n\nThe addition of blue to yellow gives green\u2014this is a natural\nsupposition to make. It is also natural to suppose that blue added to\nred makes purple. Orange and blue can be made to give a brown, by using\ncertain shades and proportions. And ochre and blue can be made to give\na light brown.\n\nBut the scheme of colours is merely used for getting a definite and\nrealisable set of names and distinctions visible to the eye. Their\nnaturalness is apparent to any one in the habit of using colours, and\nmay be assumed to be justifiable, as the sole purpose is to devise a\nset of names which are easy to remember, and which will give us a set\nof colours by which diagrams may be made easy of comprehension. No\nscientific classification of colours has been attempted.\n\nStarting, then, with these sixteen colour names, we have a catalogue of\nthe sixteen tesseracts, which form a four-dimensional block analogous\nto the cubic block. But the cube which we can put in space and look at\nis not one of the constituent tesseracts; it is merely the beginning,\nthe solid face, the side, the aspect, of a tesseract.\n\nWe will now proceed to derive a name for each region, point, edge,\nplane face, solid and a face of the tesseract.\n\nThe system will be clear, if we look at a representation in the plane\nof a tesseract with three, and one with four divisions in its side.\n\nThe tesseract made up of three tesseracts each way corresponds to the\ncube made up of three cubes each way, and will give us a complete\nnomenclature.\n\nIn this diagram, fig. 101, 1 represents a cube of 27 cubes, each of\nwhich is the beginning of a tesseract. These cubes are represented\nsimply by their lowest squares, the solid content must be understood. 2\nrepresents the 27 cubes which are the beginnings of the 27 tesseracts\none inch on in the fourth dimension. These tesseracts are represented\nas a block of cubes put side by side with the first block, but in\ntheir proper positions they could not be in space with the first set. 3\nrepresents 27 cubes (forming a larger cube) which are the beginnings of\nthe tesseracts, which begin two inches in the fourth direction from our\nspace and continue another inch.\n\n[Illustration: Fig. 101.]\n\n\n[Illustration: Fig. 102[4]]\n\n [4] The coloured plate, figs. 1, 2, 3, shows these relations more\n conspicuously.\n\nIn fig. 102, we have the representation of a block of 4 \u00d7 4 \u00d7 4 \u00d7 4\nor 256 tesseracts. They are given in four consecutive sections, each\nsupposed to be taken one inch apart in the fourth dimension, and so\ngiving four blocks of cubes, 64 in each block. Here we see, comparing\nit with the figure of 81 tesseracts, that the number of the different\nregions show a different tendency of increase. By taking five blocks of\nfive divisions each way this would become even more clear.\n\nWe see, fig. 102, that starting from the point at any corner, the white\ncoloured regions only extend out in a line. The same is true for the\nyellow, red, and blue. With regard to the latter it should be noticed\nthat the line of blues does not consist in regions next to each other\nin the drawing, but in portions which come in in different cubes.\nThe portions which lie next to one another in the fourth dimension\nmust always be represented so, when we have a three-dimensional\nrepresentation. Again, those regions such as the pink one, go on\nincreasing in two dimensions. About the pink region this is seen\nwithout going out of the cube itself, the pink regions increase in\nlength and height, but in no other dimension. In examining these\nregions it is sufficient to take one as a sample.\n\nThe purple increases in the same manner, for it comes in in a\nsuccession from below to above in block 2, and in a succession from\nblock to block in 2 and 3. Now, a succession from below to above\nrepresents a continuous extension upwards, and a succession from block\nto block represents a continuous extension in the fourth dimension.\nThus the purple regions increase in two dimensions, the upward and\nthe fourth, so when we take a very great many divisions, and let each\nbecome very small, the purple region forms a two-dimensional extension.\n\nIn the same way, looking at the regions marked l. b. or light blue,\nwhich starts nearest a corner, we see that the tesseracts occupying\nit increase in length from left to right, forming a line, and that\nthere are as many lines of light blue tesseracts as there are sections\nbetween the first and last section. Hence the light blue tesseracts\nincrease in number in two ways\u2014in the right and left, and in the fourth\ndimension. They ultimately form what we may call a plane surface.\n\nNow all those regions which contain a mixture of two simple colours,\nwhite, yellow, red, blue, increase in two ways. On the other hand,\nthose which contain a mixture of three colours increase in three ways.\nTake, for instance, the ochre region; this has three colours, white,\nyellow, red; and in the cube itself it increases in three ways.\n\nNow regard the orange region; if we add blue to this we get a brown.\nThe region of the brown tesseracts extends in two ways on the left of\nthe second block, No. 2 in the figure. It extends also from left to\nright in succession from one section to another, from section 2 to\nsection 3 in our figure.\n\nHence the brown tesseracts increase in number in three dimensions\nupwards, to and fro, fourth dimension. Hence they form a cubic, a\nthree-dimensional region; this region extends up and down, near\nand far, and in the fourth direction, but is thin in the direction\nfrom left to right. It is a cube which, when the complete tesseract\nis represented in our space, appears as a series of faces on the\nsuccessive cubic sections of the tesseract. Compare fig. 103 in which\nthe middle block, 2, stands as representing a great number of sections\nintermediate between 1 and 3.\n\nIn a similar way from the pink region by addition of blue we have\nthe light purple region, which can be seen to increase in three ways\nas the number of divisions becomes greater. The three ways in which\nthis region of tesseracts extends is up and down, right and left,\nfourth dimension. Finally, therefore, it forms a cubic mass of very\nsmall tesseracts, and when the tesseract is given in space sections\nit appears on the faces containing the upward and the right and left\ndimensions.\n\nWe get then altogether, as three-dimensional regions, ochre, brown,\nlight purple, light green.\n\nFinally, there is the region which corresponds to a mixture of all the\ncolours; there is only one region such as this. It is the one that\nsprings from ochre by the addition of blue\u2014this colour we call light\nbrown.\n\nLooking at the light brown region we see that it increases in four\nways. Hence, the tesseracts of which it is composed increase in\nnumber in each of four dimensions, and the shape they form does not\nremain thin in any of the four dimensions. Consequently this region\nbecomes the solid content of the block of tesseracts, itself; it\nis the real four-dimensional solid. All the other regions are then\nboundaries of this light brown region. If we suppose the process\nof increasing the number of tesseracts and diminishing their size\ncarried on indefinitely, then the light brown coloured tesseracts\nbecome the whole interior mass, the three-coloured tesseracts become\nthree-dimensional boundaries, thin in one dimension, and form the\nochre, the brown, the light purple, the light green. The two-coloured\ntesseracts become two-dimensional boundaries, thin in two dimensions,\n_e.g._, the pink, the green, the purple, the orange, the light blue,\nthe light yellow. The one-coloured tesseracts become bounding lines,\nthin in three dimensions, and the null points become bounding corners,\nthin in four dimensions. From these thin real boundaries we can pass in\nthought to the abstractions\u2014points, lines, faces, solids\u2014bounding the\nfour-dimensional solid, which in this case is light brown coloured, and\nunder this supposition the light brown coloured region is the only real\none, is the only one which is not an abstraction.\n\nIt should be observed that, in taking a square as the representation\nof a cube on a plane, we only represent one face, or the section\nbetween two faces. The squares, as drawn by a plane being, are not the\ncubes themselves, but represent the faces or the sections of a cube.\nThus in the plane being\u2019s diagram a cube of twenty-seven cubes \u201cnull\u201d\nrepresents a cube, but is really, in the normal position, the orange\nsquare of a null cube, and may be called null, orange square.\n\nA plane being would save himself confusion if he named his\nrepresentative squares, not by using the names of the cubes simply, but\nby adding to the names of the cubes a word to show what part of a cube\nhis representative square was.\n\nThus a cube null standing against his plane touches it by null orange\nface, passing through his plane it has in the plane a square as trace,\nwhich is null white section, if we use the phrase white section to\nmean a section drawn perpendicular to the white line. In the same way\nthe cubes which we take as representative of the tesseract are not\nthe tesseract itself, but definite faces or sections of it. In the\npreceding figures we should say then, not null, but \u201cnull tesseract\nochre cube,\u201d because the cube we actually have is the one determined by\nthe three axes, white, red, yellow.\n\nThere is another way in which we can regard the colour nomenclature of\nthe boundaries of a tesseract.\n\nConsider a null point to move tracing out a white line one inch in\nlength, and terminating in a null point, see fig. 103 or in the\ncoloured plate.\n\nThen consider this white line with its terminal points itself to move\nin a second dimension, each of the points traces out a line, the line\nitself traces out an area, and gives two lines as well, its initial and\nits final position.\n\nThus, if we call \u201ca region\u201d any element of the figure, such as a point,\nor a line, etc., every \u201cregion\u201d in moving traces out a new kind of\nregion, \u201ca higher region,\u201d and gives two regions of its own kind, an\ninitial and a final position. The \u201chigher region\u201d means a region with\nanother dimension in it.\n\nNow the square can move and generate a cube. The square light yellow\nmoves and traces out the mass of the cube. Letting the addition of\nred denote the region made by the motion in the upward direction we\nget an ochre solid. The light yellow face in its initial and terminal\npositions give the two square boundaries of the cube above and below.\nThen each of the four lines of the light yellow square\u2014white, yellow,\nand the white, yellow opposite them\u2014trace out a bounding square. So\nthere are in all six bounding squares, four of these squares being\ndesignated in colour by adding red to the colour of the generating\nlines. Finally, each point moving in the up direction gives rise to\na line coloured null + red, or red, and then there are the initial\nand terminal positions of the points giving eight points. The number\nof the lines is evidently twelve, for the four lines of this light\nyellow square give four lines in their initial, four lines in their\nfinal position, while the four points trace out four lines, that is\naltogether twelve lines.\n\nNow the squares are each of them separate boundaries of the cube, while\nthe lines belong, each of them, to two squares, thus the red line is\nthat which is common to the orange and pink squares.\n\nNow suppose that there is a direction, the fourth dimension, which is\nperpendicular alike to every one of the space dimensions already used\u2014a\ndimension perpendicular, for instance, to up and to right hand, so that\nthe pink square moving in this direction traces out a cube.\n\nA dimension, moreover, perpendicular to the up and away directions,\nso that the orange square moving in this direction also traces out\na cube, and the light yellow square, too, moving in this direction\ntraces out a cube. Under this supposition, the whole cube moving in\nthe unknown dimension, traces out something new\u2014a new kind of volume,\na higher volume. This higher volume is a four-dimensional volume, and\nwe designate it in colour by adding blue to the colour of that which by\nmoving generates it.\n\nIt is generated by the motion of the ochre solid, and hence it is\nof the colour we call light brown (white, yellow, red, blue, mixed\ntogether). It is represented by a number of sections like 2 in fig. 103.\n\nNow this light brown higher solid has for boundaries: first, the ochre\ncube in its initial position, second, the same cube in its final\nposition, 1 and 3, fig. 103. Each of the squares which bound the cube,\nmoreover, by movement in this new direction traces out a cube, so we\nhave from the front pink faces of the cube, third, a pink blue or\nlight purple cube, shown as a light purple face on cube 2 in fig. 103,\nthis cube standing for any number of intermediate sections; fourth,\na similar cube from the opposite pink face; fifth, a cube traced out\nby the orange face\u2014this is coloured brown and is represented by the\nbrown face of the section cube in fig. 103; sixth, a corresponding\nbrown cube on the right hand; seventh, a cube starting from the light\nyellow square below; the unknown dimension is at right angles to this\nalso. This cube is coloured light yellow and blue or light green; and,\nfinally, eighth, a corresponding cube from the upper light yellow face,\nshown as the light green square at the top of the section cube.\n\nThe tesseract has thus eight cubic boundaries. These completely enclose\nit, so that it would be invisible to a four-dimensional being. Now, as\nto the other boundaries, just as the cube has squares, lines, points,\nas boundaries, so the tesseract has cubes, squares, lines, points, as\nboundaries.\n\nThe number of squares is found thus\u2014round the cube are six squares,\nthese will give six squares in their initial and six in their final\npositions. Then each of the twelve lines of the cube trace out a square\nin the motion in the fourth dimension. Hence there will be altogether\n12 + 12 = 24 squares.\n\nIf we look at any one of these squares we see that it is the meeting\nsurface of two of the cubic sides. Thus, the red line by its movement\nin the fourth dimension, traces out a purple square\u2014this is common\nto two cubes, one of which is traced out by the pink square moving\nin the fourth dimension, and the other is traced out by the orange\nsquare moving in the same way. To take another square, the light yellow\none, this is common to the ochre cube and the light green cube. The\nochre cube comes from the light yellow square by moving it in the up\ndirection, the light green cube is made from the light yellow square by\nmoving it in the fourth dimension. The number of lines is thirty-two,\nfor the twelve lines of the cube give twelve lines of the tesseract\nin their initial position, and twelve in their final position, making\ntwenty-four, while each of the eight points traces out a line, thus\nforming thirty-two lines altogether.\n\nThe lines are each of them common to three cubes, or to three square\nfaces; take, for instance, the red line. This is common to the orange\nface, the pink face, and that face which is formed by moving the red\nline in the sixth dimension, namely, the purple face. It is also common\nto the ochre cube, the pale purple cube, and the brown cube.\n\nThe points are common to six square faces and to four cubes; thus,\nthe null point from which we start is common to the three square\nfaces\u2014pink, light yellow, orange, and to the three square faces made by\nmoving the three lines white, yellow, red, in the fourth dimension,\nnamely, the light blue, the light green, the purple faces\u2014that is, to\nsix faces in all. The four cubes which meet in it are the ochre cube,\nthe light purple cube, the brown cube, and the light green cube.\n\n[Illustration: Fig. 103.\n\nThe tesseract, red, white, yellow axes in space. In the lower line the\nthree rear faces are shown, the interior being removed.]\n\n[Illustration: Fig. 104.\n\nThe tesseract, red, yellow, blue axes in space, the blue axis running\nto the left, opposite faces are coloured identically.]\n\nA complete view of the tesseract in its various space presentations\nis given in the following figures or catalogue cubes, figs. 103-106.\nThe first cube in each figure represents the view of a tesseract\ncoloured as described as it begins to pass transverse to our space.\nThe intermediate figure represents a sectional view when it is partly\nthrough, and the final figure represents the far end as it is just\npassing out. These figures will be explained in detail in the next\nchapter.\n\n[Illustration: Fig. 105.\n\nThe tesseract, with red, white, blue axes in space. Opposite faces are\ncoloured identically.]\n\n[Illustration: Fig. 106.\n\nThe tesseract, with blue, white, yellow axes in space. The blue axis\nruns downward from the base of the ochre cube as it stands originally.\nOpposite faces are coloured identically.]\n\nWe have thus obtained a nomenclature for each of the regions of a\ntesseract; we can speak of any one of the eight bounding cubes, the\ntwenty square faces, the thirty-two lines, the sixteen points.\n\n\n\n\n CHAPTER XIII\n\n REMARKS ON THE FIGURES\n\n\nAn inspection of above figures will give an answer to many questions\nabout the tesseract. If we have a tesseract one inch each way, then it\ncan be represented as a cube\u2014a cube having white, yellow, red axes,\nand from this cube as a beginning, a volume extending into the fourth\ndimension. Now suppose the tesseract to pass transverse to our space,\nthe cube of the red, yellow, white axes disappears at once, it is\nindefinitely thin in the fourth dimension. Its place is occupied by\nthose parts of the tesseract which lie further away from our space in\nthe fourth dimension. Each one of these sections will last only for\none moment, but the whole of them will take up some appreciable time\nin passing. If we take the rate of one inch a minute the sections will\ntake the whole of the minute in their passage across our space, they\nwill take the whole of the minute except the moment which the beginning\ncube and the end cube occupy in their crossing our space. In each one\nof the cubes, the section cubes, we can draw lines in all directions\nexcept in the direction occupied by the blue line, the fourth\ndimension; lines in that direction are represented by the transition\nfrom one section cube to another. Thus to give ourselves an adequate\nrepresentation of the tesseract we ought to have a limitless number of\nsection cubes intermediate between the first bounding cube, the ochre\ncube, and the last bounding cube, the other ochre cube. Practically\nthree intermediate sectional cubes will be found sufficient for most\npurposes. We will take then a series of five figures\u2014two terminal\ncubes, and three intermediate sections\u2014and show how the different\nregions appear in our space when we take each set of three out of the\nfour axes of the tesseract as lying in our space.\n\nIn fig. 107 initial letters are used for the colours. A reference to\nfig. 103 will show the complete nomenclature, which is merely indicated\nhere.\n\n[Illustration: Fig. 107.]\n\nIn this figure the tesseract is shown in five stages distant from our\nspace: first, zero; second, 1/4 in.; third, 2/4 in.; fourth, 3/4 in.;\nfifth, 1 in.; which are called _b_0, _b_1, _b_2, _b_3, _b_4, because\nthey are sections taken at distances 0, 1, 2, 3, 4 quarter inches along\nthe blue line. All the regions can be named from the first cube, the\n_b_0 cube, as before, simply by remembering that transference along\nthe b axis gives the addition of blue to the colour of the region in\nthe ochre, the _b_0 cube. In the final cube _b_4, the colouring of the\noriginal _b_0 cube is repeated. Thus the red line moved along the blue\naxis gives a red and blue or purple square. This purple square appears\nas the three purple lines in the sections _b_1, _b_2, _b_3, taken at\n1/4, 2/4, 3/4 of an inch in the fourth dimension. If the tesseract\nmoves transverse to our space we have then in this particular region,\nfirst of all a red line which lasts for a moment, secondly a purple\nline which takes its place. This purple line lasts for a minute\u2014that\nis, all of a minute, except the moment taken by the crossing our space\nof the initial and final red line. The purple line having lasted for\nthis period is succeeded by a red line, which lasts for a moment; then\nthis goes and the tesseract has passed across our space. The final red\nline we call red bl., because it is separated from the initial red\nline by a distance along the axis for which we use the colour blue.\nThus a line that lasts represents an area duration; is in this mode\nof presentation equivalent to a dimension of space. In the same way\nthe white line, during the crossing our space by the tesseract, is\nsucceeded by a light blue line which lasts for the inside of a minute,\nand as the tesseract leaves our space, having crossed it, the white bl.\nline appears as the final termination.\n\nTake now the pink face. Moved in the blue direction it traces out a\nlight purple cube. This light purple cube is shown in sections in\n_b__{1}, _b__{2}, _b__{3}, and the farther face of this cube in the\nblue direction is shown in _b__{4}\u2014a pink face, called pink _b_ because\nit is distant from the pink face we began with in the blue direction.\nThus the cube which we colour light purple appears as a lasting square.\nThe square face itself, the pink face, vanishes instantly the tesseract\nbegins to move, but the light purple cube appears as a lasting square.\nHere also duration is the equivalent of a dimension of space\u2014a lasting\nsquare is a cube. It is useful to connect these diagrams with the views\ngiven in the coloured plate.\n\nTake again the orange face, that determined by the red and yellow axes;\nfrom it goes a brown cube in the blue direction, for red and yellow\nand blue are supposed to make brown. This brown cube is shown in three\nsections in the faces _b__{1}, _b__{2}, _b__{3}. In _b__{4} is the\nopposite orange face of the brown cube, the face called orange _b_,\nfor it is distant in the blue direction from the orange face. As the\ntesseract passes transverse to our space, we have then in this region\nan instantly vanishing orange square, followed by a lasting brown\nsquare, and finally an orange face which vanishes instantly.\n\nNow, as any three axes will be in our space, let us send the white\naxis out into the unknown, the fourth dimension, and take the blue\naxis into our known space dimension. Since the white and blue axes are\nperpendicular to each other, if the white axis goes out into the fourth\ndimension in the positive sense, the blue axis will come into the\ndirection the white axis occupied, in the negative sense.\n\n[Illustration: Fig. 108.]\n\nHence, not to complicate matters by having to think of two senses in\nthe unknown direction, let us send the white line into the positive\nsense of the fourth dimension, and take the blue one as running in the\nnegative sense of that direction which the white line has left; let the\nblue line, that is, run to the left. We have now the row of figures\nin fig. 108. The dotted cube shows where we had a cube when the white\nline ran in our space\u2014now it has turned out of our space, and another\nsolid boundary, another cubic face of the tesseract comes into our\nspace. This cube has red and yellow axes as before; but now, instead\nof a white axis running to the right, there is a blue axis running to\nthe left. Here we can distinguish the regions by colours in a perfectly\nsystematic way. The red line traces out a purple square in the\ntransference along the blue axis by which this cube is generated from\nthe orange face. This purple square made by the motion of the red line\nis the same purple face that we saw before as a series of lines in the\nsections _b__{1}, _b__{2}, _b__{3}. Here, since both red and blue axes\nare in our space, we have no need of duration to represent the area\nthey determine. In the motion of the tesseract across space this purple\nface would instantly disappear.\n\nFrom the orange face, which is common to the initial cubes in fig. 107\nand fig. 108, there goes in the blue direction a cube coloured brown.\nThis brown cube is now all in our space, because each of its three axes\nrun in space directions, up, away, to the left. It is the same brown\ncube which appeared as the successive faces on the sections _b__{1},\n_b__{2}, _b__{3}. Having all its three axes in our space, it is given\nin extension; no part of it needs to be represented as a succession.\nThe tesseract is now in a new position with regard to our space, and\nwhen it moves across our space the brown cube instantly disappears.\n\nIn order to exhibit the other regions of the tesseract we must remember\nthat now the white line runs in the unknown dimension. Where shall we\nput the sections at distances along the line? Any arbitrary position in\nour space will do: there is no way by which we can represent their real\nposition.\n\nHowever, as the brown cube comes off from the orange face to the left,\nlet us put these successive sections to the left. We can call them\n_wh__{0}, _wh__{1}, _wh__{2}, _wh__{3}, _wh__{4}, because they are\nsections along the white axis, which now runs in the unknown dimension.\n\nRunning from the purple square in the white direction we find the light\npurple cube. This is represented in the sections _wh__{1}, _wh__{2},\n_wh__{3}, _wh__{4}, fig. 108. It is the same cube that is represented\nin the sections _b__{1}, _b__{2}, _b__{3}: in fig. 107 the red and\nwhite axes are in our space, the blue out of it; in the other case, the\nred and blue are in our space, the white out of it. It is evident that\nthe face pink _y_, opposite the pink face in fig. 107, makes a cube\nshown in squares in _b__{1}, _b__{2}, _b__{3}, _b__{4}, on the opposite\nside to the _l_ purple squares. Also the light yellow face at the base\nof the cube _b__{0}, makes a light green cube, shown as a series of\nbase squares.\n\nThe same light green cube can be found in fig. 107. The base square in\n_wh__{0} is a green square, for it is enclosed by blue and yellow axes.\nFrom it goes a cube in the white direction, this is then a light green\ncube and the same as the one just mentioned as existing in the sections\n_b__{0}, _b__{1}, _b__{2}, _b__{3}, _b__{4}.\n\nThe case is, however, a little different with the brown cube. This cube\nwe have altogether in space in the section _wh__{0}, fig. 108, while\nit exists as a series of squares, the left-hand ones, in the sections\n_b__{0}, _b__{1}, _b__{2}, _b__{3}, _b__{4}. The brown cube exists as a\nsolid in our space, as shown in fig. 108. In the mode of representation\nof the tesseract exhibited in fig. 107, the same brown cube appears as\na succession of squares. That is, as the tesseract moves across space,\nthe brown cube would actually be to us a square\u2014it would be merely\nthe lasting boundary of another solid. It would have no thickness at\nall, only extension in two dimensions, and its duration would show its\nsolidity in three dimensions.\n\nIt is obvious that, if there is a four-dimensional space, matter in\nthree dimensions only is a mere abstraction; all material objects\nmust then have a slight four-dimensional thickness. In this case the\nabove statement will undergo modification. The material cube which is\nused as the model of the boundary of a tesseract will have a slight\nthickness in the fourth dimension, and when the cube is presented to\nus in another aspect, it would not be a mere surface. But it is most\nconvenient to regard the cubes we use as having no extension at all in\nthe fourth dimension. This consideration serves to bring out a point\nalluded to before, that, if there is a fourth dimension, our conception\nof a solid is the conception of a mere abstraction, and our talking\nabout real three-dimensional objects would seem to a four-dimensional\nbeing as incorrect as a two-dimensional being\u2019s telling about real\nsquares, real triangles, etc., would seem to us.\n\nThe consideration of the two views of the brown cube shows that any\nsection of a cube can be looked at by a presentation of the cube in\na different position in four-dimensional space. The brown faces in\n_b__{1}, _b__{2}, _b__{3}, are the very same brown sections that would\nbe obtained by cutting the brown cube, _wh__{0}, across at the right\ndistances along the blue line, as shown in fig. 108. But as these\nsections are placed in the brown cube, _wh__{0}, they come behind one\nanother in the blue direction. Now, in the sections _wh__{1}, _wh__{2},\n_wh__{3}, we are looking at these sections from the white direction\u2014the\nblue direction does not exist in these figures. So we see them in\na direction at right angles to that in which they occur behind one\nanother in _wh__{0}. There are intermediate views, which would come in\nthe rotation of a tesseract. These brown squares can be looked at from\ndirections intermediate between the white and blue axes. It must be\nremembered that the fourth dimension is perpendicular equally to all\nthree space axes. Hence we must take the combinations of the blue axis,\nwith each two of our three axes, white, red, yellow, in turn.\n\nIn fig. 109 we take red, white, and blue axes in space, sending yellow\ninto the fourth dimension. If it goes into the positive sense of the\nfourth dimension the blue line will come in the opposite direction to\nthat in which the yellow line ran before. Hence, the cube determined\nby the white, red, blue axes, will start from the pink plane and run\ntowards us. The dotted cube shows where the ochre cube was. When it is\nturned out of space, the cube coming towards from its front face is\nthe one which comes into our space in this turning. Since the yellow\nline now runs in the unknown dimension we call the sections _y__{0},\n_y__{1}, _y__{2}, _y__{3}, _y__{4}, as they are made at distances 0, 1,\n2, 3, 4, quarter inches along the yellow line. We suppose these cubes\narranged in a line coming towards us\u2014not that that is any more natural\nthan any other arbitrary series of positions, but it agrees with the\nplan previously adopted.\n\n[Illustration: Fig. 109.]\n\nThe interior of the first cube, _y__{0}, is that derived from pink by\nadding blue, or, as we call it, light purple. The faces of the cube are\nlight blue, purple, pink. As drawn, we can only see the face nearest to\nus, which is not the one from which the cube starts\u2014but the face on the\nopposite side has the same colour name as the face towards us.\n\nThe successive sections of the series, _y__{0}, _y__{1}, _y__{2}, etc.,\ncan be considered as derived from sections of the _b__{0} cube made at\ndistances along the yellow axis. What is distant a quarter inch from\nthe pink face in the yellow direction? This question is answered by\ntaking a section from a point a quarter inch along the yellow axis in\nthe cube _b__{0}, fig. 107. It is an ochre section with lines orange\nand light yellow. This section will therefore take the place of the\npink face in _y__{1} when we go on in the yellow direction. Thus, the\nfirst section, _y__{1}, will begin from an ochre face with light yellow\nand orange lines. The colour of the axis which lies in space towards\nus is blue, hence the regions of this section-cube are determined in\nnomenclature, they will be found in full in fig. 105.\n\nThere remains only one figure to be drawn, and that is the one in which\nthe red axis is replaced by the blue. Here, as before, if the red axis\ngoes out into the positive sense of the fourth dimension, the blue line\nmust come into our space in the negative sense of the direction which\nthe red line has left. Accordingly, the first cube will come in beneath\nthe position of our ochre cube, the one we have been in the habit of\nstarting with.\n\n[Illustration: Fig. 110.]\n\nTo show these figures we must suppose the ochre cube to be on a movable\nstand. When the red line swings out into the unknown dimension, and the\nblue line comes in downwards, a cube appears below the place occupied\nby the ochre cube. The dotted cube shows where the ochre cube was.\nThat cube has gone and a different cube runs downwards from its base.\nThis cube has white, yellow, and blue axes. Its top is a light yellow\nsquare, and hence its interior is light yellow + blue or light green.\nIts front face is formed by the white line moving along the blue axis,\nand is therefore light blue, the left-hand side is formed by the yellow\nline moving along the blue axis, and therefore green.\n\nAs the red line now runs in the fourth dimension, the successive\nsections can he called _r__{0}, _r__{1}, _r__{2}, _r__{3}, _r__{4},\nthese letters indicating that at distances 0, 1/4, 2/4, 3/4, 1 inch\nalong the red axis we take all of the tesseract that can be found in a\nthree-dimensional space, this three-dimensional space extending not at\nall in the fourth dimension, but up and down, right and left, far and\nnear.\n\nWe can see what should replace the light yellow face of _r__{0}, when\nthe section _r__{1} comes in, by looking at the cube _b__{0}, fig. 107.\nWhat is distant in it one-quarter of an inch from the light yellow face\nin the red direction? It is an ochre section with orange and pink lines\nand red points; see also fig. 103.\n\nThis square then forms the top square of _r__{1}. Now we can determine\nthe nomenclature of all the regions of _r__{1} by considering what\nwould be formed by the motion of this square along a blue axis.\n\nBut we can adopt another plan. Let us take a horizontal section of\n_r__{0}, and finding that section in the figures, of fig. 107 or fig.\n103, from them determine what will replace it, going on in the red\ndirection.\n\nA section of the _r__{0} cube has green, light blue, green, light blue\nsides and blue points.\n\nNow this square occurs on the base of each of the section figures,\n_b__{1}, _b__{2}, etc. In them we see that 1/4 inch in the red\ndirection from it lies a section with brown and light purple lines and\npurple corners, the interior being of light brown. Hence this is the\nnomenclature of the section which in _r__{1} replaces the section of\n_r__{0} made from a point along the blue axis.\n\nHence the colouring as given can be derived.\n\nWe have thus obtained a perfectly named group of tesseracts. We can\ntake a group of eighty-one of them 3 \u00d7 3 \u00d7 3 \u00d7 3, in four dimensions,\nand each tesseract will have its name null, red, white, yellow, blue,\netc., and whatever cubic view we take of them we can say exactly\nwhat sides of the tesseracts we are handling, and how they touch each\nother.[5]\n\n [5] At this point the reader will find it advantageous, if he has the\n models, to go through the manipulations described in the appendix.\n\nThus, for instance, if we have the sixteen tesseracts shown below, we\ncan ask how does null touch blue.\n\n[Illustration: Fig. 111.]\n\nIn the arrangement given in fig. 111 we have the axes white, red,\nyellow, in space, blue running in the fourth dimension. Hence we have\nthe ochre cubes as bases. Imagine now the tesseractic group to pass\ntransverse to our space\u2014we have first of all null ochre cube, white\nochre cube, etc.; these instantly vanish, and we get the section shown\nin the middle cube in fig. 103, and finally, just when the tesseract\nblock has moved one inch transverse to our space, we have null ochre\ncube, and then immediately afterwards the ochre cube of blue comes in.\nHence the tesseract null touches the tesseract blue by its ochre cube,\nwhich is in contact, each and every point of it, with the ochre cube of\nblue.\n\nHow does null touch white, we may ask? Looking at the beginning A, fig.\n111, where we have the ochre cubes, we see that null ochre touches\nwhite ochre by an orange face. Now let us generate the null and white\ntesseracts by a motion in the blue direction of each of these cubes.\nEach of them generates the corresponding tesseract, and the plane of\ncontact of the cubes generates the cube by which the tesseracts are\nin contact. Now an orange plane carried along a blue axis generates a\nbrown cube. Hence null touches white by a brown cube.\n\n[Illustration: Fig. 112.]\n\nIf we ask again how red touches light blue tesseract, let us rearrange\nour group, fig. 112, or rather turn it about so that we have a\ndifferent space view of it; let the red axis and the white axis run\nup and right, and let the blue axis come in space towards us, then\nthe yellow axis runs in the fourth dimension. We have then two blocks\nin which the bounding cubes of the tesseracts are given, differently\narranged with regard to us\u2014the arrangement is really the same, but it\nappears different to us. Starting from the plane of the red and white\naxes we have the four squares of the null, white, red, pink tesseracts\nas shown in A, on the red, white plane, unaltered, only from them now\ncomes out towards us the blue axis. Hence we have null, white, red,\npink tesseracts in contact with our space by their cubes which have\nthe red, white, blue axis in them, that is by the light purple cubes.\nFollowing on these four tesseracts we have that which comes next to\nthem in the blue direction, that is the four blue, light blue, purple,\nlight purple. These are likewise in contact with our space by their\nlight purple cubes, so we see a block as named in the figure, of which\neach cube is the one determined by the red, white, blue, axes.\n\nThe yellow line now runs out of space; accordingly one inch on in the\nfourth dimension we come to the tesseracts which follow on the eight\nnamed in C, fig. 112, in the yellow direction.\n\nThese are shown in C.y_{1}, fig. 112. Between figure C and C.y_{1} is\nthat four-dimensional mass which is formed by moving each of the cubes\nin C one inch in the fourth dimension\u2014that is, along a yellow axis; for\nthe yellow axis now runs in the fourth dimension.\n\nIn the block C we observe that red (light purple cube) touches light\nblue (light purple cube) by a point. Now these two cubes moving\ntogether remain in contact during the period in which they trace out\nthe tesseracts red and light blue. This motion is along the yellow\naxis, consequently red and light blue touch by a yellow line.\n\nWe have seen that the pink face moved in a yellow direction traces out\na cube; moved in the blue direction it also traces out a cube. Let us\nask what the pink face will trace out if it is moved in a direction\nwithin the tesseract lying equally between the yellow and blue\ndirections. What section of the tesseract will it make?\n\nWe will first consider the red line alone. Let us take a cube with the\nred line in it and the yellow and blue axes.\n\nThe cube with the yellow, red, blue axes is shown in fig. 113. If the\nred line is moved equally in the yellow and in the blue direction by\nfour equal motions of \u00bc inch each, it takes the positions 11, 22, 33,\nand ends as a red line.\n\n[Illustration: Fig. 113.]\n\nNow, the whole of this red, yellow, blue, or brown cube appears as a\nseries of faces on the successive sections of the tesseract starting\nfrom the ochre cube and letting the blue axis run in the fourth\ndimension. Hence the plane traced out by the red line appears as a\nseries of lines in the successive sections, in our ordinary way of\nrepresenting the tesseract; these lines are in different places in each\nsuccessive section.\n\n[Illustration: Fig. 114.]\n\nThus drawing our initial cube and the successive sections, calling them\n_b__{0}, _b__{1}, _b__{2}, _b__{3}, _b__{4}, fig. 115, we have the red\nline subject to this movement appearing in the positions indicated.\n\nWe will now investigate what positions in the tesseract another line in\nthe pink face assumes when it is moved in a similar manner.\n\nTake a section of the original cube containing a vertical line, 4,\nin the pink plane, fig. 115. We have, in the section, the yellow\ndirection, but not the blue.\n\nFrom this section a cube goes off in the fourth dimension, which is\nformed by moving each point of the section in the blue direction.\n\n[Illustration: Fig. 115.]\n\n[Illustration: Fig. 116.]\n\nDrawing this cube we have fig. 116.\n\nNow this cube occurs as a series of sections in our original\nrepresentation of the tesseract. Taking four steps as before this cube\nappears as the sections drawn in _b__{0}, _b__{1}, _b__{2}, _b__{3},\n_b__{4}, fig. 117, and if the line 4 is subjected to a movement equal\nin the blue and yellow directions, it will occupy the positions\ndesignated by 4, 4_{1}, 4_{2}, 4_{3}, 4_{4}.\n\n[Illustration: Fig. 117.]\n\nHence, reasoning in a similar manner about every line, it is evident\nthat, moved equally in the blue and yellow directions, the pink plane\nwill trace out a space which is shown by the series of section planes\nrepresented in the diagram.\n\nThus the space traced out by the pink face, if it is moved equally in\nthe yellow and blue directions, is represented by the set of planes\ndelineated in Fig. 118, pink face or 0, then 1, 2, 3, and finally pink\nface or 4. This solid is a diagonal solid of the tesseract, running\nfrom a pink face to a pink face. Its length is the length of the\ndiagonal of a square, its side is a square.\n\nLet us now consider the unlimited space which springs from the pink\nface extended.\n\nThis space, if it goes off in the yellow direction, gives us in it the\nochre cube of the tesseract. Thus, if we have the pink face given and a\npoint in the ochre cube, we have determined this particular space.\n\nSimilarly going off from the pink face in the blue direction is another\nspace, which gives us the light purple cube of the tesseract in it. And\nany point being taken in the light purple cube, this space going off\nfrom the pink face is fixed.\n\n[Illustration: Fig. 118.]\n\nThe space we are speaking of can be conceived as swinging round the\npink face, and in each of its positions it cuts out a solid figure from\nthe tesseract, one of which we have seen represented in fig. 118.\n\nEach of these solid figures is given by one position of the swinging\nspace, and by one only. Hence in each of them, if one point is taken,\nthe particular one of the slanting spaces is fixed. Thus we see that\ngiven a plane and a point out of it a space is determined.\n\nNow, two points determine a line.\n\nAgain, think of a line and a point outside it. Imagine a plane rotating\nround the line. At some time in its rotation it passes through the\npoint. Thus a line and a point, or three points, determine a plane.\nAnd finally four points determine a space. We have seen that a plane\nand a point determine a space, and that three points determine a plane;\nso four points will determine a space.\n\nThese four points may be any points, and we can take, for instance, the\nfour points at the extremities of the red, white, yellow, blue axes, in\nthe tesseract. These will determine a space slanting with regard to the\nsection spaces we have been previously considering. This space will cut\nthe tesseract in a certain figure.\n\nOne of the simplest sections of a cube by a plane is that in which the\nplane passes through the extremities of the three edges which meet in a\npoint. We see at once that this plane would cut the cube in a triangle,\nbut we will go through the process by which a plane being would most\nconveniently treat the problem of the determination of this shape, in\norder that we may apply the method to the determination of the figure\nin which a space cuts a tesseract when it passes through the 4 points\nat unit distance from a corner.\n\nWe know that two points determine a line, three points determine a\nplane, and given any two points in a plane the line between them lies\nwholly in the plane.\n\n[Illustration: Fig. 119.]\n\nLet now the plane being study the section made by a plane passing\nthrough the null _r_, null _wh_, and null _y_ points, fig. 119. Looking\nat the orange square, which, as usual, we suppose to be initially in\nhis plane, he sees that the line from null _r_ to null _y_, which is\na line in the section plane, the plane, namely, through the three\nextremities of the edges meeting in null, cuts the orange face in an\norange line with null points. This then is one of the boundaries of the\nsection figure.\n\nLet now the cube be so turned that the pink face comes in his plane.\nThe points null _r_ and null _wh_ are now visible. The line between\nthem is pink with null points, and since this line is common to the\nsurface of the cube and the cutting plane, it is a boundary of the\nfigure in which the plane cuts the cube.\n\nAgain, suppose the cube turned so that the light yellow face is in\ncontact with the plane being\u2019s plane. He sees two points, the null _wh_\nand the null _y_. The line between these lies in the cutting plane.\nHence, since the three cutting lines meet and enclose a portion of\nthe cube between them, he has determined the figure he sought. It is\na triangle with orange, pink, and light yellow sides, all equal, and\nenclosing an ochre area.\n\nLet us now determine in what figure the space, determined by the four\npoints, null _r_, null _y_, null _wh_, null _b_, cuts the tesseract. We\ncan see three of these points in the primary position of the tesseract\nresting against our solid sheet by the ochre cube. These three points\ndetermine a plane which lies in the space we are considering, and this\nplane cuts the ochre cube in a triangle, the interior of which is\nochre (fig. 119 will serve for this view), with pink, light yellow and\norange sides, and null points. Going in the fourth direction, in one\nsense, from this plane we pass into the tesseract, in the other sense\nwe pass away from it. The whole area inside the triangle is common to\nthe cutting plane we see, and a boundary of the tesseract. Hence we\nconclude that the triangle drawn is common to the tesseract and the\ncutting space.\n\nNow let the ochre cube turn out and the brown cube come in. The dotted\nlines show the position the ochre cube has left (fig. 120).\n\n[Illustration: Fig. 120.]\n\nHere we see three out of the four points through which the cutting\nplane passes, null _r_, null _y_, and null _b_. The plane they\ndetermine lies in the cutting space, and this plane cuts out of the\nbrown cube a triangle with orange, purple and green sides, and null\npoints. The orange line of this figure is the same as the orange line\nin the last figure.\n\nNow let the light purple cube swing into our space, towards us, fig.\n121.\n\n[Illustration: Fig. 121.]\n\nThe cutting space which passes through the four points, null _r_, _y_,\n_wh_, _b_, passes through the null _r_, _wh_, _b_, and therefore the\nplane these determine lies in the cutting space.\n\nThis triangle lies before us. It has a light purple interior and pink,\nlight blue, and purple edges with null points.\n\nThis, since it is all of the plane that is common to it, and this\nbounding of the tesseract, gives us one of the bounding faces of our\nsectional figure. The pink line in it is the same as the pink line we\nfound in the first figure\u2014that of the ochre cube.\n\nFinally, let the tesseract swing about the light yellow plane, so that\nthe light green cube comes into our space. It will point downwards.\n\nThe three points, _n.y_, _n.wh_, _n.b_, are in the cutting space, and\nthe triangle they determine is common to the tesseract and the cutting\nspace. Hence this boundary is a triangle having a light yellow line,\nwhich is the same as the light yellow line of the first figure, a light\nblue line and a green line.\n\n[Illustration: Fig. 122.]\n\nWe have now traced the cutting space between every set of three that\ncan be made out of the four points in which it cuts the tesseract, and\nhave got four faces which all join on to each other by lines.\n\n[Illustration: Fig. 123.]\n\nThe triangles are shown in fig. 123 as they join on to the triangle\nin the ochre cube. But they join on each to the other in an exactly\nsimilar manner; their edges are all identical two and two. They form a\nclosed figure, a tetrahedron, enclosing a light brown portion which is\nthe portion of the cutting space which lies inside the tesseract.\n\nWe cannot expect to see this light brown portion, any more than a plane\nbeing could expect to see the inside of a cube if an angle of it were\npushed through his plane. All he can do is to come upon the boundaries\nof it in a different way to that in which he would if it passed\nstraight through his plane.\n\nThus in this solid section; the whole interior lies perfectly open in\nthe fourth dimension. Go round it as we may we are simply looking at\nthe boundaries of the tesseract which penetrates through our solid\nsheet. If the tesseract were not to pass across so far, the triangle\nwould be smaller; if it were to pass farther, we should have a\ndifferent figure, the outlines of which can be determined in a similar\nmanner.\n\nThe preceding method is open to the objection that it depends rather on\nour inferring what must be, than our seeing what is. Let us therefore\nconsider our sectional space as consisting of a number of planes, each\nvery close to the last, and observe what is to be found in each plane.\n\nThe corresponding method in the case of two dimensions is as\nfollows:\u2014The plane being can see that line of the sectional plane\nthrough null _y_, null _wh_, null _r_, which lies in the orange plane.\nLet him now suppose the cube and the section plane to pass half way\nthrough his plane. Replacing the red and yellow axes are lines parallel\nto them, sections of the pink and light yellow faces.\n\n[Illustration: Fig. 124.]\n\nWhere will the section plane cut these parallels to the red and yellow\naxes?\n\nLet him suppose the cube, in the position of the drawing, fig. 124,\nturned so that the pink face lies against his plane. He can see the\nline from the null _r_ point to the null _wh_ point, and can see\n(compare fig. 119) that it cuts AB a parallel to his red axis, drawn\nat a point half way along the white line, in a point B, half way up. I\nshall speak of the axis as having the length of an edge of the cube.\nSimilarly, by letting the cube turn so that the light yellow square\nswings against his plane, he can see (compare fig. 119) that a parallel\nto his yellow axis drawn from a point half-way along the white axis, is\ncut at half its length by the trace of the section plane in the light\nyellow face.\n\nHence when the cube had passed half-way through he would have\u2014instead\nof the orange line with null points, which he had at first\u2014an ochre\nline of half its length, with pink and light yellow points. Thus, as\nthe cube passed slowly through his plane, he would have a succession\nof lines gradually diminishing in length and forming an equilateral\ntriangle. The whole interior would be ochre, the line from which it\nstarted would be orange. The succession of points at the ends of\nthe succeeding lines would form pink and light yellow lines and the\nfinal point would be null. Thus looking at the successive lines in\nthe section plane as it and the cube passed across his plane he would\ndetermine the figure cut out bit by bit.\n\nComing now to the section of the tesseract, let us imagine that the\ntesseract and its cutting _space_ pass slowly across our space; we can\nexamine portions of it, and their relation to portions of the cutting\nspace. Take the section space which passes through the four points,\nnull _r_, _wh_, _y_, _b_; we can see in the ochre cube (fig. 119) the\nplane belonging to this section space, which passes through the three\nextremities of the red, white, yellow axes.\n\nNow let the tesseract pass half way through our space. Instead of our\noriginal axes we have parallels to them, purple, light blue, and green,\neach of the same length as the first axes, for the section of the\ntesseract is of exactly the same shape as its ochre cube.\n\nBut the sectional space seen at this stage of the transference would\nnot cut the section of the tesseract in a plane disposed as at first.\n\nTo see where the sectional space would cut these parallels to the\noriginal axes let the tesseract swing so that, the orange face\nremaining stationary, the blue line comes in to the left.\n\nHere (fig. 125) we have the null _r_, _y_, _b_ points, and of the\nsectional space all we see is the plane through these three points in\nit.\n\n[Illustration: Fig. 125.]\n\nIn this figure we can draw the parallels to the red and yellow axes and\nsee that, if they started at a point half way along the blue axis, they\nwould each be cut at a point so as to be half of their previous length.\n\nSwinging the tesseract into our space about the pink face of the ochre\ncube we likewise find that the parallel to the white axis is cut at\nhalf its length by the sectional space.\n\nHence in a section made when the tesseract had passed half across our\nspace the parallels to the red, white, yellow axes, which are now in\nour space, are cut by the section space, each of them half way along,\nand for this stage of the traversing motion we should have fig. 126.\nThe section made of this cube by the plane in which the sectional space\ncuts it, is an equilateral triangle with purple, l. blue, green points,\nand l. purple, brown, l. green lines.\n\n[Illustration: Fig. 126.]\n\nThus the original ochre triangle, with null points and pink, orange,\nlight yellow lines, would be succeeded by a triangle coloured in manner\njust described.\n\nThis triangle would initially be only a very little smaller than the\noriginal triangle, it would gradually diminish, until it ended in a\npoint, a null point. Each of its edges would be of the same length.\nThus the successive sections of the successive planes into which we\nanalyse the cutting space would be a tetrahedron of the description\nshown (fig. 123), and the whole interior of the tetrahedron would be\nlight brown.\n\n[Illustration: Fig. 127. Front view. The rear faces.]\n\nIn fig. 127 the tetrahedron is represented by means of its faces as\ntwo triangles which meet in the p. line, and two rear triangles which\njoin on to them, the diagonal of the pink face being supposed to run\nvertically upward.\n\nWe have now reached a natural termination. The reader may pursue\nthe subject in further detail, but will find no essential novelty.\nI conclude with an indication as to the manner in which figures\npreviously given may be used in determining sections by the method\ndeveloped above.\n\nApplying this method to the tesseract, as represented in Chapter IX.,\nsections made by a space cutting the axes equidistantly at any distance\ncan be drawn, and also the sections of tesseracts arranged in a block.\n\nIf we draw a plane, cutting all four axes at a point six units distance\nfrom null, we have a slanting space. This space cuts the red, white,\nyellow axes in the points LMN (fig. 128), and so in the region of our\nspace before we go off into the fourth dimension, we have the plane\nrepresented by LMN extended. This is what is common to the slanting\nspace and our space.\n\n[Illustration: Fig. 128.]\n\nThis plane cuts the ochre cube in the triangle EFG.\n\nComparing this with (fig. 72) _oh_, we see that the hexagon there drawn\nis part of the triangle EFG.\n\nLet us now imagine the tesseract and the slanting space both together\nto pass transverse to our space, a distance of one unit, we have in\n1_h_ a section of the tesseract, whose axes are parallels to the\nprevious axes. The slanting space cuts them at a distance of five units\nalong each. Drawing the plane through these points in 1_h_ it will be\nfound to cut the cubical section of the tesseract in the hexagonal\nfigure drawn. In 2_h_ (fig. 72) the slanting space cuts the parallels\nto the axes at a distance of four along each, and the hexagonal figure\nis the section of this section of the tesseract by it. Finally when\n3_h_ comes in the slanting space cuts the axes at a distance of three\nalong each, and the section is a triangle, of which the hexagon drawn\nis a truncated portion. After this the tesseract, which extends only\nthree units in each of the four dimensions, has completely passed\ntransverse of our space, and there is no more of it to be cut. Hence,\nputting the plane sections together in the right relations, we have\nthe section determined by the particular slanting space: namely an\noctahedron.\n\n\n\n\nCHAPTER XIV.[6]\n\nA RECAPITULATION AND EXTENSION OF THE PHYSICAL ARGUMENT\n\n\nThere are two directions of inquiry in which the research for the\nphysical reality of a fourth dimension can be prosecuted. One is the\ninvestigation of the infinitely great, the other is the investigation\nof the infinitely small.\n\n [6] The contents of this chapter are taken from a paper read before\n the Philosophical Society of Washington. The mathematical portion\n of the paper has appeared in part in the Proceedings of the Royal\n Irish Academy under the title, \u201cCayley\u2019s formul\u00e6 of orthogonal\n transformation,\u201d Nov. 29th, 1903.\n\nBy the measurement of the angles of vast triangles, whose sides are the\ndistances between the stars, astronomers have sought to determine if\nthere is any deviation from the values given by geometrical deduction.\nIf the angles of a celestial triangle do not together equal two right\nangles, there would be an evidence for the physical reality of a fourth\ndimension.\n\nThis conclusion deserves a word of explanation. If space is really\nfour-dimensional, certain conclusions follow which must be brought\nclearly into evidence if we are to frame the questions definitely which\nwe put to Nature. To account for our limitation let us assume a solid\nmaterial sheet against which we move. This sheet must stretch alongside\nevery object in every direction in which it visibly moves. Every\nmaterial body must slip or slide along this sheet, not deviating from\ncontact with it in any motion which we can observe.\n\nThe necessity for this assumption is clearly apparent, if we consider\nthe analogous case of a suppositionary plane world. If there were\nany creatures whose experiences were confined to a plane, we must\naccount for their limitation. If they were free to move in every space\ndirection, they would have a three-dimensional motion; hence they must\nbe physically limited, and the only way in which we can conceive such\na limitation to exist is by means of a material surface against which\nthey slide. The existence of this surface could only be known to them\nindirectly. It does not lie in any direction from them in which the\nkinds of motion they know of leads them. If it were perfectly smooth\nand always in contact with every material object, there would be no\ndifference in their relations to it which would direct their attention\nto it.\n\nBut if this surface were curved\u2014if it were, say, in the form of a vast\nsphere\u2014the triangles they drew would really be triangles of a sphere,\nand when these triangles are large enough the angles diverge from\nthe magnitudes they would have for the same lengths of sides if the\nsurface were plane. Hence by the measurement of triangles of very great\nmagnitude a plane being might detect a difference from the laws of a\nplane world in his physical world, and so be led to the conclusion that\nthere was in reality another dimension to space\u2014a third dimension\u2014as\nwell as the two which his ordinary experience made him familiar with.\n\nNow, astronomers have thought it worth while to examine the\nmeasurements of vast triangles drawn from one celestial body to another\nwith a view to determine if there is anything like a curvature in our\nspace\u2014that is to say, they have tried astronomical measurements to\nfind out if the vast solid sheet against which, on the supposition of\na fourth dimension, everything slides is curved or not. These results\nhave been negative. The solid sheet, if it exists, is not curved or,\nbeing curved, has not a sufficient curvature to cause any observable\ndeviation from the theoretical value of the angles calculated.\n\nHence the examination of the infinitely great leads to no decisive\ncriterion. If it did we should have to decide between the present\ntheory and that of metageometry.\n\nComing now to the prosecution of the inquiry in the direction of\nthe infinitely small, we have to state the question thus: Our laws\nof movement are derived from the examination of bodies which move\nin three-dimensional space. All our conceptions are founded on the\nsupposition of a space which is represented analytically by three\nindependent axes and variations along them\u2014that is, it is a space in\nwhich there are three independent movements. Any motion possible in it\ncan be compounded out of these three movements, which we may call: up,\nright, away.\n\nTo examine the actions of the very small portions of matter with the\nview of ascertaining if there is any evidence in the phenomena for\nthe supposition of a fourth dimension of space, we must commence by\nclearly defining what the laws of mechanics would be on the supposition\nof a fourth dimension. It is of no use asking if the phenomena of the\nsmallest particles of matter are like\u2014we do not know what. We must\nhave a definite conception of what the laws of motion would be on the\nsupposition of the fourth dimension, and then inquire if the phenomena\nof the activity of the smaller particles of matter resemble the\nconceptions which we have elaborated.\n\nNow, the task of forming these conceptions is by no means one to be\nlightly dismissed. Movement in space has many features which differ\nentirely from movement on a plane; and when we set about to form the\nconception of motion in four dimensions, we find that there is at least\nas great a step as from the plane to three-dimensional space.\n\nI do not say that the step is difficult, but I want to point out\nthat it must be taken. When we have formed the conception of\nfour-dimensional motion, we can ask a rational question of Nature.\nBefore we have elaborated our conceptions we are asking if an unknown\nis like an unknown\u2014a futile inquiry.\n\nAs a matter of fact, four-dimensional movements are in every way simple\nand more easy to calculate than three-dimensional movements, for\nfour-dimensional movements are simply two sets of plane movements put\ntogether.\n\nWithout the formation of an experience of four-dimensional bodies,\ntheir shapes and motions, the subject can be but formal\u2014logically\nconclusive, not intuitively evident. It is to this logical apprehension\nthat I must appeal.\n\nIt is perfectly simple to form an experiential familiarity with the\nfacts of four-dimensional movement. The method is analogous to that\nwhich a plane being would have to adopt to form an experiential\nfamiliarity with three-dimensional movements, and may be briefly summed\nup as the formation of a compound sense by means of which duration is\nregarded as equivalent to extension.\n\nConsider a being confined to a plane. A square enclosed by four lines\nwill be to him a solid, the interior of which can only be examined by\nbreaking through the lines. If such a square were to pass transverse to\nhis plane, it would immediately disappear. It would vanish, going in no\ndirection to which he could point.\n\nIf, now, a cube be placed in contact with his plane, its surface of\ncontact would appear like the square which we have just mentioned.\nBut if it were to pass transverse to his plane, breaking through it,\nit would appear as a lasting square. The three-dimensional matter will\ngive a lasting appearance in circumstances under which two-dimensional\nmatter will at once disappear.\n\nSimilarly, a four-dimensional cube, or, as we may call it, a tesseract,\nwhich is generated from a cube by a movement of every part of the cube\nin a fourth direction at right angles to each of the three visible\ndirections in the cube, if it moved transverse to our space, would\nappear as a lasting cube.\n\nA cube of three-dimensional matter, since it extends to no distance at\nall in the fourth dimension, would instantly disappear, if subjected\nto a motion transverse to our space. It would disappear and be gone,\nwithout it being possible to point to any direction in which it had\nmoved.\n\nAll attempts to visualise a fourth dimension are futile. It must be\nconnected with a time experience in three space.\n\nThe most difficult notion for a plane being to acquire would be that of\nrotation about a line. Consider a plane being facing a square. If he\nwere told that rotation about a line were possible, he would move his\nsquare this way and that. A square in a plane can rotate about a point,\nbut to rotate about a line would seem to the plane being perfectly\nimpossible. How could those parts of his square which were on one side\nof an edge come to the other side without the edge moving? He could\nunderstand their reflection in the edge. He could form an idea of the\nlooking-glass image of his square lying on the opposite side of the\nline of an edge, but by no motion that he knows of can he make the\nactual square assume that position. The result of the rotation would be\nlike reflection in the edge, but it would be a physical impossibility\nto produce it in the plane.\n\nThe demonstration of rotation about a line must be to him purely\nformal. If he conceived the notion of a cube stretching out in an\nunknown direction away from his plane, then he can see the base of\nit, his square in the plane, rotating round a point. He can likewise\napprehend that every parallel section taken at successive intervals in\nthe unknown direction rotates in like manner round a point. Thus he\nwould come to conclude that the whole body rotates round a line\u2014the\nline consisting of the succession of points round which the plane\nsections rotate. Thus, given three axes, _x_, _y_, _z_, if _x_ rotates\nto take the place of _y_, and _y_ turns so as to point to negative\n_x_, then the third axis remaining unaffected by this turning is the\naxis about which the rotation takes place. This, then, would have to be\nhis criterion of the axis of a rotation\u2014that which remains unchanged\nwhen a rotation of every plane section of a body takes place.\n\nThere is another way in which a plane being can think about\nthree-dimensional movements; and, as it affords the type by which we\ncan most conveniently think about four-dimensional movements, it will\nbe no loss of time to consider it in detail.\n\n[Illustration: Fig. 1 (129).]\n\nWe can represent the plane being and his object by figures cut out of\npaper, which slip on a smooth surface. The thickness of these bodies\nmust be taken as so minute that their extension in the third dimension\nescapes the observation of the plane being, and he thinks about them\nas if they were mathematical plane figures in a plane instead of being\nmaterial bodies capable of moving on a plane surface. Let A_x_, A_y_\nbe two axes and ABCD a square. As far as movements in the plane are\nconcerned, the square can rotate about a point A, for example. It\ncannot rotate about a side, such as AC.\n\nBut if the plane being is aware of the existence of a third dimension\nhe can study the movements possible in the ample space, taking his\nfigure portion by portion.\n\nHis plane can only hold two axes. But, since it can hold two, he is\nable to represent a turning into the third dimension if he neglects one\nof his axes and represents the third axis as lying in his plane. He can\nmake a drawing in his plane of what stands up perpendicularly from his\nplane. Let A_z_ be the axis, which stands perpendicular to his plane at\nA. He can draw in his plane two lines to represent the two axes, A_x_\nand A_z_. Let Fig. 2 be this drawing. Here the _z_ axis has taken the\nplace of the _y_ axis, and the plane of A_x_ A_z_ is represented in his\nplane. In this figure all that exists of the square ABCD will be the\nline AB.\n\n[Illustration: Fig. 2 (130).]\n\nThe square extends from this line in the _y_ direction, but more of\nthat direction is represented in Fig. 2. The plane being can study the\nturning of the line AB in this diagram. It is simply a case of plane\nturning around the point A. The line AB occupies intermediate portions\nlike AB_{1} and after half a revolution will lie on A_x_ produced\nthrough A.\n\nNow, in the same way, the plane being can take another point, A\u00b4, and\nanother line, A\u00b4B\u00b4, in his square. He can make the drawing of the two\ndirections at A\u00b4, one along A\u00b4B\u00b4, the other perpendicular to his plane.\nHe will obtain a figure precisely similar to Fig. 2, and will see that,\nas AB can turn around A, so A\u00b4C\u00b4 around A.\n\nIn this turning AB and A\u00b4B\u00b4 would not interfere with each other, as\nthey would if they moved in the plane around the separate points A and\nA\u00b4.\n\nHence the plane being would conclude that a rotation round a line was\npossible. He could see his square as it began to make this turning. He\ncould see it half way round when it came to lie on the opposite side of\nthe line AC. But in intermediate portions he could not see it, for it\nruns out of the plane.\n\nComing now to the question of a four-dimensional body, let us conceive\nof it as a series of cubic sections, the first in our space, the rest\nat intervals, stretching away from our space in the unknown direction.\n\nWe must not think of a four-dimensional body as formed by moving a\nthree-dimensional body in any direction which we can see.\n\nRefer for a moment to Fig. 3. The point A, moving to the right, traces\nout the line AC. The line AC, moving away in a new direction, traces\nout the square ACEG at the base of the cube. The square AEGC, moving\nin a new direction, will trace out the cube ACEGBDHF. The vertical\ndirection of this last motion is not identical with any motion possible\nin the plane of the base of the cube. It is an entirely new direction,\nat right angles to every line that can be drawn in the base. To trace\nout a tesseract the cube must move in a new direction\u2014a direction at\nright angles to any and every line that can be drawn in the space of\nthe cube.\n\nThe cubic sections of the tesseract are related to the cube we see, as\nthe square sections of the cube are related to the square of its base\nwhich a plane being sees.\n\nLet us imagine the cube in our space, which is the base of a tesseract,\nto turn about one of its edges. The rotation will carry the whole body\nwith it, and each of the cubic sections will rotate. The axis we see\nin our space will remain unchanged, and likewise the series of axes\nparallel to it about which each of the parallel cubic sections rotates.\nThe assemblage of all of these is a plane.\n\nHence in four dimensions a body rotates about a plane. There is no such\nthing as rotation round an axis.\n\nWe may regard the rotation from a different point of view. Consider\nfour independent axes each at right angles to all the others, drawn in\na four-dimensional body. Of these four axes we can see any three. The\nfourth extends normal to our space.\n\nRotation is the turning of one axis into a second, and the second\nturning to take the place of the negative of the first. It involves\ntwo axes. Thus, in this rotation of a four-dimensional body, two axes\nchange and two remain at rest. Four-dimensional rotation is therefore a\nturning about a plane.\n\nAs in the case of a plane being, the result of rotation about a\nline would appear as the production of a looking-glass image of the\noriginal object on the other side of the line, so to us the result\nof a four-dimensional rotation would appear like the production of a\nlooking-glass image of a body on the other side of a plane. The plane\nwould be the axis of the rotation, and the path of the body between its\ntwo appearances would be unimaginable in three-dimensional space.\n\n[Illustration: Fig. 3 (131).]\n\nLet us now apply the method by which a plane being could examine\nthe nature of rotation about a line in our examination of rotation\nabout a plane. Fig. 3 represents a cube in our space, the three axes\n_x_, _y_, _z_ denoting its three dimensions. Let _w_ represent the\nfourth dimension. Now, since in our space we can represent any three\ndimensions, we can, if we choose, make a representation of what is\nin the space determined by the three axes _x_, _z_, _w_. This is a\nthree-dimensional space determined by two of the axes we have drawn,\n_x_ and _z_, and in place of _y_ the fourth axis, _w_. We cannot,\nkeeping _x_ and _z_, have both _y_ and _w_ in our space; so we will\nlet _y_ go and draw _w_ in its place. What will be our view of the cube?\n\nEvidently we shall have simply the square that is in the plane of _xz_,\nthe square ACDB. The rest of the cube stretches in the _y_ direction,\nand, as we have none of the space so determined, we have only the face\nof the cube. This is represented in fig. 4.\n\n[Illustration: Fig. 4 (132).]\n\nNow, suppose the whole cube to be turned from the _x_ to the _w_\ndirection. Conformably with our method, we will not take the whole of\nthe cube into consideration at once, but will begin with the face ABCD.\n\nLet this face begin to turn. Fig. 5 represents one of the positions it\nwill occupy; the line AB remains on the _z_ axis. The rest of the face\nextends between the _x_ and the _w_ direction.\n\n[Illustration: Fig. 5 (133).]\n\nNow, since we can take any three axes, let us look at what lies in the\nspace of _zyw_, and examine the turning there. We must now let the _z_\naxis disappear and let the _w_ axis run in the direction in which the\n_z_ ran.\n\nMaking this representation, what do we see of the cube? Obviously we\nsee only the lower face. The rest of the cube lies in the space of\n_xyz_. In the space of _xyz_ we have merely the base of the cube lying\nin the plane of _xy_, as shown in fig. 6.\n\n[Illustration: Fig. 6 (134).]\n\nNow let the _x_ to _w_ turning take place. The square ACEG will turn\nabout the line AE. This edge will remain along the _y_ axis and will be\nstationary, however far the square turns.\n\nThus, if the cube be turned by an _x_ to _w_ turning, both the edge AB\nand the edge AC remain stationary; hence the whole face ABEF in the\n_yz_ plane remains fixed. The turning has taken place about the face\nABEF.\n\n[Illustration: Fig. 7 (135).]\n\nSuppose this turning to continue till AC runs to the left from\nA. The cube will occupy the position shown in fig. 8. This is\nthe looking-glass image of the cube in fig. 3. By no rotation in\nthree-dimensional space can the cube be brought from the position in\nfig. 3 to that shown in fig. 8.\n\n[Illustration: Fig. 8 (136).]\n\nWe can think of this turning as a turning of the face ABCD about AB,\nand a turning of each section parallel to ABCD round the vertical line\nin which it intersects the face ABEF, the space in which the turning\ntakes place being a different one from that in which the cube lies.\n\nOne of the conditions, then, of our inquiry in the direction of the\ninfinitely small is that we form the conception of a rotation about\na plane. The production of a body in a state in which it presents\nthe appearance of a looking-glass image of its former state is the\ncriterion for a four-dimensional rotation.\n\nThere is some evidence for the occurrence of such transformations\nof bodies in the change of bodies from those which produce a\nright-handed polarisation of light to those which produce a left-handed\npolarisation; but this is not a point to which any very great\nimportance can be attached.\n\nStill, in this connection, let me quote a remark from Prof. John G.\nMcKendrick\u2019s address on Physiology before the British Association\nat Glasgow. Discussing the possibility of the hereditary production\nof characteristics through the material structure of the ovum, he\nestimates that in it there exist 12,000,000,000 biophors, or ultimate\nparticles of living matter, a sufficient number to account for\nhereditary transmission, and observes: \u201cThus it is conceivable that\nvital activities may also be determined by the kind of motion that\ntakes place in the molecules of that which we speak of as living\nmatter. It may be different in kind from some of the motions known to\nphysicists, and it is conceivable that life may be the transmission\nto dead matter, the molecules of which have already a special kind of\nmotion, of a form of motion _sui generis_.\u201d\n\nNow, in the realm of organic beings symmetrical structures\u2014those with a\nright and left symmetry\u2014are everywhere in evidence. Granted that four\ndimensions exist, the simplest turning produces the image form, and by\na folding-over structures could be produced, duplicated right and left,\njust as is the case of symmetry in a plane.\n\nThus one very general characteristic of the forms of organisms could\nbe accounted for by the supposition that a four-dimensional motion was\ninvolved in the process of life.\n\nBut whether four-dimensional motions correspond in other respects to\nthe physiologist\u2019s demand for a special kind of motion, or not, I\ndo not know. Our business is with the evidence for their existence\nin physics. For this purpose it is necessary to examine into the\nsignificance of rotation round a plane in the case of extensible and of\nfluid matter.\n\nLet us dwell a moment longer on the rotation of a rigid body. Looking\nat the cube in fig. 3, which turns about the face of ABFE, we see that\nany line in the face can take the place of the vertical and horizontal\nlines we have examined. Take the diagonal line AF and the section\nthrough it to GH. The portions of matter which were on one side of AF\nin this section in fig. 3 are on the opposite side of it in fig. 8.\nThey have gone round the line AF. Thus the rotation round a face can be\nconsidered as a number of rotations of sections round parallel lines in\nit.\n\nThe turning about two different lines is impossible in\nthree-dimensional space. To take another illustration, suppose A and\nB are two parallel lines in the _xy_ plane, and let CD and EF be two\nrods crossing them. Now, in the space of _xyz_ if the rods turn round\nthe lines A and B in the same direction they will make two independent\ncircles.\n\nWhen the end F is going down the end C will be coming up. They will\nmeet and conflict.\n\n[Illustration: Fig. 9 (137).]\n\nBut if we rotate the rods about the plane of AB by the _z_ to _w_\nrotation these movements will not conflict. Suppose all the figure\nremoved with the exception of the plane _xz_, and from this plane draw\nthe axis of _w_, so that we are looking at the space of _xzw_.\n\nHere, fig. 10, we cannot see the lines A and B. We see the points G and\nH, in which A and B intercept the _x_ axis, but we cannot see the lines\nthemselves, for they run in the _y_ direction, and that is not in our\ndrawing.\n\nNow, if the rods move with the _z_ to _w_ rotation they will turn in\nparallel planes, keeping their relative positions. The point D, for\ninstance, will describe a circle. At one time it will be above the line\nA, at another time below it. Hence it rotates round A.\n\n[Illustration: Fig. 10 (138).]\n\nNot only two rods but any number of rods crossing the plane will move\nround it harmoniously. We can think of this rotation by supposing the\nrods standing up from one line to move round that line and remembering\nthat it is not inconsistent with this rotation for the rods standing up\nalong another line also to move round it, the relative positions of all\nthe rods being preserved. Now, if the rods are thick together, they may\nrepresent a disk of matter, and we see that a disk of matter can rotate\nround a central plane.\n\nRotation round a plane is exactly analogous to rotation round an axis\nin three dimensions. If we want a rod to turn round, the ends must be\nfree; so if we want a disk of matter to turn round its central plane\nby a four-dimensional turning, all the contour must be free. The whole\ncontour corresponds to the ends of the rod. Each point of the contour\ncan be looked on as the extremity of an axis in the body, round each\npoint of which there is a rotation of the matter in the disk.\n\nIf the one end of a rod be clamped, we can twist the rod, but not turn\nit round; so if any part of the contour of a disk is clamped we can\nimpart a twist to the disk, but not turn it round its central plane. In\nthe case of extensible materials a long, thin rod will twist round its\naxis, even when the axis is curved, as, for instance, in the case of a\nring of India rubber.\n\nIn an analogous manner, in four dimensions we can have rotation round\na curved plane, if I may use the expression. A sphere can be turned\ninside out in four dimensions.\n\n[Illustration: Fig. 11 (139).]\n\nLet fig. 11 represent a spherical surface, on each side of which a\nlayer of matter exists. The thickness of the matter is represented by\nthe rods CD and EF, extending equally without and within.\n\n[Illustration: Fig. 12 (140).]\n\nNow, take the section of the sphere by the _yz_ plane we have a\ncircle\u2014fig. 12. Now, let the _w_ axis be drawn in place of the _x_ axis\nso that we have the space of _yzw_ represented. In this space all that\nthere will be seen of the sphere is the circle drawn.\n\nHere we see that there is no obstacle to prevent the rods turning\nround. If the matter is so elastic that it will give enough for the\nparticles at E and C to be separated as they are at F and D, they\ncan rotate round to the position D and F, and a similar motion is\npossible for all other particles. There is no matter or obstacle to\nprevent them from moving out in the _w_ direction, and then on round\nthe circumference as an axis. Now, what will hold for one section will\nhold for all, as the fourth dimension is at right angles to all the\nsections which can be made of the sphere.\n\nWe have supposed the matter of which the sphere is composed to be\nthree-dimensional. If the matter had a small thickness in the fourth\ndimension, there would be a slight thickness in fig. 12 above the\nplane of the paper\u2014a thickness equal to the thickness of the matter\nin the fourth dimension. The rods would have to be replaced by thin\nslabs. But this would make no difference as to the possibility of the\nrotation. This motion is discussed by Newcomb in the first volume of\nthe _American Journal of Mathematics_.\n\nLet us now consider, not a merely extensible body, but a liquid one. A\nmass of rotating liquid, a whirl, eddy, or vortex, has many remarkable\nproperties. On first consideration we should expect the rotating mass\nof liquid immediately to spread off and lose itself in the surrounding\nliquid. The water flies off a wheel whirled round, and we should expect\nthe rotating liquid to be dispersed. But see the eddies in a river\nstrangely persistent. The rings that occur in puffs of smoke and last\nso long are whirls or vortices curved round so that their opposite ends\njoin together. A cyclone will travel over great distances.\n\nHelmholtz was the first to investigate the properties of vortices.\nHe studied them as they would occur in a perfect fluid\u2014that is, one\nwithout friction of one moving portion or another. In such a medium\nvortices would be indestructible. They would go on for ever, altering\ntheir shape, but consisting always of the same portion of the fluid.\nBut a straight vortex could not exist surrounded entirely by the fluid.\nThe ends of a vortex must reach to some boundary inside or outside the\nfluid.\n\nA vortex which is bent round so that its opposite ends join is capable\nof existing, but no vortex has a free end in the fluid. The fluid\nround the vortex is always in motion, and one produces a definite\nmovement in another.\n\nLord Kelvin has proposed the hypothesis that portions of a fluid\nsegregated in vortices account for the origin of matter. The properties\nof the ether in respect of its capacity of propagating disturbances\ncan be explained by the assumption of vortices in it instead of by a\nproperty of rigidity. It is difficult to conceive, however, of any\narrangement of the vortex rings and endless vortex filaments in the\nether.\n\nNow, the further consideration of four-dimensional rotations shows the\nexistence of a kind of vortex which would make an ether filled with a\nhomogeneous vortex motion easily thinkable.\n\nTo understand the nature of this vortex, we must go on and take a\nstep by which we accept the full significance of the four-dimensional\nhypothesis. Granted four-dimensional axes, we have seen that a rotation\nof one into another leaves two unaltered, and these two form the axial\nplane about which the rotation takes place. But what about these two?\nDo they necessarily remain motionless? There is nothing to prevent a\nrotation of these two, one into the other, taking place concurrently\nwith the first rotation. This possibility of a double rotation deserves\nthe most careful attention, for it is the kind of movement which is\ndistinctly typical of four dimensions.\n\nRotation round a plane is analogous to rotation round an axis. But in\nthree-dimensional space there is no motion analogous to the double\nrotation, in which, while axis 1 changes into axis 2, axis 3 changes\ninto axis 4.\n\nConsider a four-dimensional body, with four independent axes, _x_,\n_y_, _z_, _w_. A point in it can move in only one direction at a given\nmoment. If the body has a velocity of rotation by which the _x_ axis\nchanges into the _y_ axis and all parallel sections move in a similar\nmanner, then the point will describe a circle. If, now, in addition\nto the rotation by which the _x_ axis changes into the _y_ axis the\nbody has a rotation by which the _z_ axis turns into the _w_ axis, the\npoint in question will have a double motion in consequence of the two\nturnings. The motions will compound, and the point will describe a\ncircle, but not the same circle which it would describe in virtue of\neither rotation separately.\n\nWe know that if a body in three-dimensional space is given two\nmovements of rotation they will combine into a single movement of\nrotation round a definite axis. It is in no different condition\nfrom that in which it is subjected to one movement of rotation. The\ndirection of the axis changes; that is all. The same is not true about\na four-dimensional body. The two rotations, _x_ to _y_ and _z_ to _w_,\nare independent. A body subject to the two is in a totally different\ncondition to that which it is in when subject to one only. When subject\nto a rotation such as that of _x_ to _y_, a whole plane in the body,\nas we have seen, is stationary. When subject to the double rotation\nno part of the body is stationary except the point common to the two\nplanes of rotation.\n\nIf the two rotations are equal in velocity, every point in the body\ndescribes a circle. All points equally distant from the stationary\npoint describe circles of equal size.\n\nWe can represent a four-dimensional sphere by means of two diagrams,\nin one of which we take the three axes, _x_, _y_, _z_; in the\nother the axes _x_, _w_, and _z_. In fig. 13 we have the view of a\nfour-dimensional sphere in the space of _xyz_. Fig. 13 shows all that\nwe can see of the four sphere in the space of _xyz_, for it represents\nall the points in that space, which are at an equal distance from the\ncentre.\n\nLet us now take the _xz_ section, and let the axis of _w_ take the\nplace of the _y_ axis. Here, in fig. 14, we have the space of _xzw_.\nIn this space we have to take all the points which are at the same\ndistance from the centre, consequently we have another sphere. If we\nhad a three-dimensional sphere, as has been shown before, we should\nhave merely a circle in the _xzw_ space, the _xz_ circle seen in the\nspace of _xzw_. But now, taking the view in the space of _xzw_, we have\na sphere in that space also. In a similar manner, whichever set of\nthree axes we take, we obtain a sphere.\n\n[Illustration: _Showing axes xyz_\nFig. 13 (141).]\n\n[Illustration: _Showing axes xwz_\nFig. 14 (142).]\n\nIn fig. 13, let us imagine the rotation in the direction _xy_ to be\ntaking place. The point _x_ will turn to _y_, and _p_ to _p\u00b4_. The axis\n_zz\u00b4_ remains stationary, and this axis is all of the plane _zw_ which\nwe can see in the space section exhibited in the figure.\n\nIn fig. 14, imagine the rotation from _z_ to _w_ to be taking place.\nThe _w_ axis now occupies the position previously occupied by the _y_\naxis. This does not mean that the _w_ axis can coincide with the _y_\naxis. It indicates that we are looking at the four-dimensional sphere\nfrom a different point of view. Any three-space view will show us three\naxes, and in fig. 14 we are looking at _xzw_.\n\nThe only part that is identical in the two diagrams is the circle of\nthe _x_ and _z_ axes, which axes are contained in both diagrams. Thus\nthe plane _zxz\u00b4_ is the same in both, and the point _p_ represents the\nsame point in both diagrams. Now, in fig. 14 let the _zw_ rotation\ntake place, the _z_ axis will turn toward the point _w_ of the _w_\naxis, and the point _p_ will move in a circle about the point _x_.\n\nThus in fig. 13 the point _p_ moves in a circle parallel to the _xy_\nplane; in fig. 14 it moves in a circle parallel to the _zw_ plane,\nindicated by the arrow.\n\nNow, suppose both of these independent rotations compounded, the point\n_p_ will move in a circle, but this circle will coincide with neither\nof the circles in which either one of the rotations will take it. The\ncircle the point _p_ will move in will depend on its position on the\nsurface of the four sphere.\n\nIn this double rotation, possible in four-dimensional space, there\nis a kind of movement totally unlike any with which we are familiar\nin three-dimensional space. It is a requisite preliminary to the\ndiscussion of the behaviour of the small particles of matter,\nwith a view to determining whether they show the characteristics\nof four-dimensional movements, to become familiar with the main\ncharacteristics of this double rotation. And here I must rely on a\nformal and logical assent rather than on the intuitive apprehension,\nwhich can only be obtained by a more detailed study.\n\nIn the first place this double rotation consists in two varieties or\nkinds, which we will call the A and B kinds. Consider four axes, _x_,\n_y_, _z_, _w_. The rotation of _x_ to _y_ can be accompanied with the\nrotation of _z_ to _w_. Call this the A kind.\n\nBut also the rotation of _x_ to _y_ can be accompanied by the rotation,\nof not _z_ to _w_, but _w_ to _z_. Call this the B kind.\n\nThey differ in only one of the component rotations. One is not the\nnegative of the other. It is the semi-negative. The opposite of an\n_x_ to _y_, _z_ to _w_ rotation would be _y_ to _x_, _w_ to _z_. The\nsemi-negative is _x_ to _y_ and _w_ to _z_.\n\nIf four dimensions exist and we cannot perceive them, because the\nextension of matter is so small in the fourth dimension that all\nmovements are withheld from direct observation except those which are\nthree-dimensional, we should not observe these double rotations, but\nonly the effects of them in three-dimensional movements of the type\nwith which we are familiar.\n\nIf matter in its small particles is four-dimensional, we should expect\nthis double rotation to be a universal characteristic of the atoms\nand molecules, for no portion of matter is at rest. The consequences\nof this corpuscular motion can be perceived, but only under the form\nof ordinary rotation or displacement. Thus, if the theory of four\ndimensions is true, we have in the corpuscles of matter a whole world\nof movement, which we can never study directly, but only by means of\ninference.\n\nThe rotation A, as I have defined it, consists of two equal\nrotations\u2014one about the plane of _zw_, the other about the plane\nof _xy_. It is evident that these rotations are not necessarily\nequal. A body may be moving with a double rotation, in which these\ntwo independent components are not equal; but in such a case we can\nconsider the body to be moving with a composite rotation\u2014a rotation of\nthe A or B kind and, in addition, a rotation about a plane.\n\nIf we combine an A and a B movement, we obtain a rotation about a\nplane; for, the first being _x_ to _y_ and _z_ to _w_, and the second\nbeing _x_ to _y_ and _w_ to _z_, when they are put together the _z_\nto _w_ and _w_ to _z_ rotations neutralise each other, and we obtain\nan _x_ to _y_ rotation only, which is a rotation about the plane of\n_zw_. Similarly, if we take a B rotation, _y_ to _x_ and _z_ to _w_,\nwe get, on combining this with the A rotation, a rotation of _z_ to\n_w_ about the _xy_ plane. In this case the plane of rotation is in the\nthree-dimensional space of _xyz_, and we have\u2014what has been described\nbefore\u2014a twisting about a plane in our space.\n\nConsider now a portion of a perfect liquid having an A motion. It\ncan be proved that it possesses the properties of a vortex. It\nforms a permanent individuality\u2014a separated-out portion of the\nliquid\u2014accompanied by a motion of the surrounding liquid. It has\nproperties analogous to those of a vortex filament. But it is not\nnecessary for its existence that its ends should reach the boundary of\nthe liquid. It is self-contained and, unless disturbed, is circular in\nevery section.\n\n[Illustration: Fig. 15 (143).]\n\nIf we suppose the ether to have its properties of transmitting\nvibration given it by such vortices, we must inquire how they lie\ntogether in four-dimensional space. Placing a circular disk on a plane\nand surrounding it by six others, we find that if the central one is\ngiven a motion of rotation, it imparts to the others a rotation which\nis antagonistic in every two adjacent ones. If A goes round, as shown\nby the arrow, B and C will be moving in opposite ways, and each tends\nto destroy the motion of the other.\n\nNow, if we suppose spheres to be arranged in a corresponding manner\nin three-dimensional space, they will be grouped in figures which are\nfor three-dimensional space what hexagons are for plane space. If a\nnumber of spheres of soft clay be pressed together, so as to fill up\nthe interstices, each will assume the form of a fourteen-sided figure\ncalled a tetrakaidecagon.\n\nNow, assuming space to be filled with such tetrakaidecagons, and\nplacing a sphere in each, it will be found that one sphere is touched\nby eight others. The remaining six spheres of the fourteen which\nsurround the central one will not touch it, but will touch three of\nthose in contact with it. Hence, if the central sphere rotates, it\nwill not necessarily drive those around it so that their motions will\nbe antagonistic to each other, but the velocities will not arrange\nthemselves in a systematic manner.\n\nIn four-dimensional space the figure which forms the next term of the\nseries hexagon, tetrakaidecagon, is a thirty-sided figure. It has for\nits faces ten solid tetrakaidecagons and twenty hexagonal prisms. Such\nfigures will exactly fill four-dimensional space, five of them meeting\nat every point. If, now, in each of these figures we suppose a solid\nfour-dimensional sphere to be placed, any one sphere is surrounded by\nthirty others. Of these it touches ten, and, if it rotates, it drives\nthe rest by means of these. Now, if we imagine the central sphere to be\ngiven an A or a B rotation, it will turn the whole mass of sphere round\nin a systematic manner. Suppose four-dimensional space to be filled\nwith such spheres, each rotating with a double rotation, the whole mass\nwould form one consistent system of motion, in which each one drove\nevery other one, with no friction or lagging behind.\n\nEvery sphere would have the same kind of rotation. In three-dimensional\nspace, if one body drives another round the second body rotates\nwith the opposite kind of rotation; but in four-dimensional space\nthese four-dimensional spheres would each have the double negative\nof the rotation of the one next it, and we have seen that the\ndouble negative of an A or B rotation is still an A or B rotation.\nThus four-dimensional space could be filled with a system of\nself-preservative living energy. If we imagine the four-dimensional\nspheres to be of liquid and not of solid matter, then, even if the\nliquid were not quite perfect and there were a slight retarding effect\nof one vortex on another, the system would still maintain itself.\n\nIn this hypothesis we must look on the ether as possessing energy,\nand its transmission of vibrations, not as the conveying of a motion\nimparted from without, but as a modification of its own motion.\n\nWe are now in possession of some of the conceptions of four-dimensional\nmechanics, and will turn aside from the line of their development\nto inquire if there is any evidence of their applicability to the\nprocesses of nature.\n\nIs there any mode of motion in the region of the minute which, giving\nthree-dimensional movements for its effect, still in itself escapes the\ngrasp of our mechanical theories? I would point to electricity. Through\nthe labours of Faraday and Maxwell we are convinced that the phenomena\nof electricity are of the nature of the stress and strain of a medium;\nbut there is still a gap to be bridged over in their explanation\u2014the\nlaws of elasticity, which Maxwell assumes, are not those of ordinary\nmatter. And, to take another instance: a magnetic pole in the\nneighbourhood of a current tends to move. Maxwell has shown that the\npressures on it are analogous to the velocities in a liquid which would\nexist if a vortex took the place of the electric current: but we cannot\npoint out the definite mechanical explanation of these pressures. There\nmust be some mode of motion of a body or of the medium in virtue of\nwhich a body is said to be electrified.\n\nTake the ions which convey charges of electricity 500 times greater in\nproportion to their mass than are carried by the molecules of hydrogen\nin electrolysis. In respect of what motion can these ions be said to\nbe electrified? It can be shown that the energy they possess is not\nenergy of rotation. Think of a short rod rotating. If it is turned\nover it is found to be rotating in the opposite direction. Now, if\nrotation in one direction corresponds to positive electricity, rotation\nin the opposite direction corresponds to negative electricity, and the\nsmallest electrified particles would have their charges reversed by\nbeing turned over\u2014an absurd supposition.\n\nIf we fix on a mode of motion as a definition of electricity, we must\nhave two varieties of it, one for positive and one for negative; and a\nbody possessing the one kind must not become possessed of the other by\nany change in its position.\n\nAll three-dimensional motions are compounded of rotations and\ntranslations, and none of them satisfy this first condition for serving\nas a definition of electricity.\n\nBut consider the double rotation of the A and B kinds. A body rotating\nwith the A motion cannot have its motion transformed into the B kind\nby being turned over in any way. Suppose a body has the rotation _x_\nto _y_ and _z_ to _w_. Turning it about the _xy_ plane, we reverse the\ndirection of the motion _x_ to _y_. But we also reverse the _z_ to _w_\nmotion, for the point at the extremity of the positive _z_ axis is\nnow at the extremity of the negative _z_ axis, and since we have not\ninterfered with its motion it goes in the direction of position _w_.\nHence we have _y_ to _x_ and _w_ to _z_, which is the same as _x_ to\n_y_ and _z_ to _w_. Thus both components are reversed, and there is the\nA motion over again. The B kind is the semi-negative, with only one\ncomponent reversed.\n\nHence a system of molecules with the A motion would not destroy it in\none another, and would impart it to a body in contact with them. Thus A\nand B motions possess the first requisite which must be demanded in any\nmode of motion representative of electricity.\n\nLet us trace out the consequences of defining positive electricity as\nan A motion and negative electricity as a B motion. The combination of\npositive and negative electricity produces a current. Imagine a vortex\nin the ether of the A kind and unite with this one of the B kind. An\nA motion and B motion produce rotation round a plane, which is in the\nether a vortex round an axial surface. It is a vortex of the kind we\nrepresent as a part of a sphere turning inside out. Now such a vortex\nmust have its rim on a boundary of the ether\u2014on a body in the ether.\n\nLet us suppose that a conductor is a body which has the property of\nserving as the terminal abutment of such a vortex. Then the conception\nwe must form of a closed current is of a vortex sheet having its edge\nalong the circuit of the conducting wire. The whole wire will then be\nlike the centres on which a spindle turns in three-dimensional space,\nand any interruption of the continuity of the wire will produce a\ntension in place of a continuous revolution.\n\nAs the direction of the rotation of the vortex is from a three-space\ndirection into the fourth dimension and back again, there will be no\ndirection of flow to the current; but it will have two sides, according\nto whether _z_ goes to _w_ or _z_ goes to negative _w_.\n\nWe can draw any line from one part of the circuit to another; then the\nether along that line is rotating round its points.\n\nThis geometric image corresponds to the definition of an electric\ncircuit. It is known that the action does not lie in the wire, but in\nthe medium, and it is known that there is no direction of flow in the\nwire.\n\nNo explanation has been offered in three-dimensional mechanics of how\nan action can be impressed throughout a region and yet necessarily\nrun itself out along a closed boundary, as is the case in an electric\ncurrent. But this phenomenon corresponds exactly to the definition of a\nfour-dimensional vortex.\n\nIf we take a very long magnet, so long that one of its poles is\npractically isolated, and put this pole in the vicinity of an electric\ncircuit, we find that it moves.\n\nNow, assuming for the sake of simplicity that the wire which determines\nthe current is in the form of a circle, if we take a number of small\nmagnets and place them all pointing in the same direction normal to\nthe plane of the circle, so that they fill it and the wire binds them\nround, we find that this sheet of magnets has the same effect on\nthe magnetic pole that the current has. The sheet of magnets may be\ncurved, but the edge of it must coincide with the wire. The collection\nof magnets is then equivalent to the vortex sheet, and an elementary\nmagnet to a part of it. Thus, we must think of a magnet as conditioning\na rotation in the ether round the plane which bisects at right angles\nthe line joining its poles.\n\nIf a current is started in a circuit, we must imagine vortices like\nbowls turning themselves inside out, starting from the contour. In\nreaching a parallel circuit, if the vortex sheet were interrupted and\njoined momentarily to the second circuit by a free rim, the axis plane\nwould lie between the two circuits, and a point on the second circuit\nopposite a point on the first would correspond to a point opposite\nto it on the first; hence we should expect a current in the opposite\ndirection in the second circuit. Thus the phenomena of induction are\nnot inconsistent with the hypothesis of a vortex about an axial plane.\n\nIn four-dimensional space, in which all four dimensions were\ncommensurable, the intensity of the action transmitted by the medium\nwould vary inversely as the cube of the distance. Now, the action of\na current on a magnetic pole varies inversely as the square of the\ndistance; hence, over measurable distances the extension of the ether\nin the fourth dimension cannot be assumed as other than small in\ncomparison with those distances.\n\nIf we suppose the ether to be filled with vortices in the shape of\nfour-dimensional spheres rotating with the A motion, the B motion would\ncorrespond to electricity in the one-fluid theory. There would thus\nbe a possibility of electricity existing in two forms, statically,\nby itself, and, combined with the universal motion, in the form of a\ncurrent.\n\nTo arrive at a definite conclusion it will be necessary to investigate\nthe resultant pressures which accompany the collocation of solid\nvortices with surface ones.\n\nTo recapitulate:\n\nThe movements and mechanics of four-dimensional space are definite and\nintelligible. A vortex with a surface as its axis affords a geometric\nimage of a closed circuit, and there are rotations which by their\npolarity afford a possible definition of statical electricity.[7]\n\n [7] These double rotations of the A and B kinds I should like to call\n Hamiltons and co-Hamiltons, for it is a singular fact that in his\n \u201cQuaternions\u201d Sir Wm. Rowan Hamilton has given the theory of either\n the A or the B kind. They follow the laws of his symbols, I, J, K.\n\nHamiltons and co-Hamiltons seem to be natural units of geometrical\nexpression. In the paper in the \u201cProceedings of the Royal Irish\nAcademy,\u201d Nov. 1903, already alluded to, I have shown something of the\nremarkable facility which is gained in dealing with the composition of\nthree- and four-dimensional rotations by an alteration in Hamilton\u2019s\nnotation, which enables his system to be applied to both the A and B\nkinds of rotations.\n\nThe objection which has been often made to Hamilton\u2019s system, namely,\nthat it is only under special conditions of application that his\nprocesses give geometrically interpretable results, can be removed, if\nwe assume that he was really dealing with a four-dimensional motion,\nand alter his notation to bring this circumstance into explicit\nrecognition.\n\n\n\n\n APPENDIX I\n\n THE MODELS\n\n\nIn Chapter XI. a description has been given which will enable any\none to make a set of models illustrative of the tesseract and its\nproperties. The set here supposed to be employed consists of:\u2014\n\n 1. Three sets of twenty-seven cubes each.\n 2. Twenty-seven slabs.\n 3. Twelve cubes with points, lines, faces, distinguished by colours,\n which will be called the catalogue cubes.\n\nThe preparation of the twelve catalogue cubes involves the expenditure\nof a considerable amount of time. It is advantageous to use them, but\nthey can be replaced by the drawing of the views of the tesseract or by\na reference to figs. 103, 104, 105, 106 of the text.\n\nThe slabs are coloured like the twenty-seven cubes of the first cubic\nblock in fig. 101, the one with red, white, yellow axes.\n\nThe colours of the three sets of twenty-seven cubes are those of the\ncubes shown in fig. 101.\n\nThe slabs are used to form the representation of a cube in a plane, and\ncan well be dispensed with by any one who is accustomed to deal with\nsolid figures. But the whole theory depends on a careful observation of\nhow the cube would be represented by these slabs.\n\nIn the first step, that of forming a clear idea how a plane being\nwould represent three-dimensional space, only one of the catalogue\ncubes and one of the three blocks is needed.\n\n\n APPLICATION TO THE STEP FROM PLANE TO SOLID.\n\nLook at fig. 1 of the views of the tesseract, or, what comes to the\nsame thing, take catalogue cube No. 1 and place it before you with the\nred line running up, the white line running to the right, the yellow\nline running away. The three dimensions of space are then marked out\nby these lines or axes. Now take a piece of cardboard, or a book, and\nplace it so that it forms a wall extending up and down not opposite to\nyou, but running away parallel to the wall of the room on your left\nhand.\n\nPlacing the catalogue cube against this wall we see that it comes into\ncontact with it by the red and yellow lines, and by the included orange\nface.\n\nIn the plane being\u2019s world the aspect he has of the cube would be a\nsquare surrounded by red and yellow lines with grey points.\n\nNow, keeping the red line fixed, turn the cube about it so that the\nyellow line goes out to the right, and the white line comes into\ncontact with the plane.\n\nIn this case a different aspect is presented to the plane being, a\nsquare, namely, surrounded by red and white lines and grey points. You\nshould particularly notice that when the yellow line goes out, at right\nangles to the plane, and the white comes in, the latter does not run in\nthe same sense that the yellow did.\n\nFrom the fixed grey point at the base of the red line the yellow line\nran away from you. The white line now runs towards you. This turning\nat right angles makes the line which was out of the plane before, come\ninto it in an opposite sense to that in which the line ran which has\njust left the plane. If the cube does not break through the plane this\nis always the rule.\n\nAgain turn the cube back to the normal position with red running up,\nwhite to the right, and yellow away, and try another turning.\n\nYou can keep the yellow line fixed, and turn the cube about it. In this\ncase the red line going out to the right the white line will come in\npointing downwards.\n\nYou will be obliged to elevate the cube from the table in order to\ncarry out this turning. It is always necessary when a vertical axis\ngoes out of a space to imagine a movable support which will allow the\nline which ran out before to come in below.\n\nHaving looked at the three ways of turning the cube so as to present\ndifferent faces to the plane, examine what would be the appearance if\na square hole were cut in the piece of cardboard, and the cube were to\npass through it. A hole can be actually cut, and it will be seen that\nin the normal position, with red axis running up, yellow away, and\nwhite to the right, the square first perceived by the plane being\u2014the\none contained by red and yellow lines\u2014would be replaced by another\nsquare of which the line towards you is pink\u2014the section line of the\npink face. The line above is light yellow, below is light yellow and on\nthe opposite side away from you is pink.\n\nIn the same way the cube can be pushed through a square opening in the\nplane from any of the positions which you have already turned it into.\nIn each case the plane being will perceive a different set of contour\nlines.\n\nHaving observed these facts about the catalogue cube, turn now to the\nfirst block of twenty-seven cubes.\n\nYou notice that the colour scheme on the catalogue cube and that of\nthis set of blocks is the same.\n\nPlace them before you, a grey or null cube on the table, above it a\nred cube, and on the top a null cube again. Then away from you place a\nyellow cube, and beyond it a null cube. Then to the right place a white\ncube and beyond it another null. Then complete the block, according to\nthe scheme of the catalogue cube, putting in the centre of all an ochre\ncube.\n\nYou have now a cube like that which is described in the text. For the\nsake of simplicity, in some cases, this cubic block can be reduced to\none of eight cubes, by leaving out the terminations in each direction.\nThus, instead of null, red, null, three cubes, you can take null, red,\ntwo cubes, and so on.\n\nIt is useful, however, to practise the representation in a plane of a\nblock of twenty-seven cubes. For this purpose take the slabs, and build\nthem up against the piece of cardboard, or the book in such a way as to\nrepresent the different aspects of the cube.\n\nProceed as follows:\u2014\n\nFirst, cube in normal position.\n\nPlace nine slabs against the cardboard to represent the nine cubes\nin the wall of the red and yellow axes, facing the cardboard; these\nrepresent the aspect of the cube as it touches the plane.\n\nNow push these along the cardboard and make a different set of nine\nslabs to represent the appearance which the cube would present to a\nplane being, if it were to pass half way through the plane.\n\nThere would be a white slab, above it a pink one, above that another\nwhite one, and six others, representing what would be the nature of a\nsection across the middle of the block of cubes. The section can be\nthought of as a thin slice cut out by two parallel cuts across the\ncube. Having arranged these nine slabs, push them along the plane, and\nmake another set of nine to represent what would be the appearance of\nthe cube when it had almost completely gone through. This set of nine\nwill be the same as the first set of nine.\n\nNow we have in the plane three sets of nine slabs each, which represent\nthree sections of the twenty-seven block.\n\nThey are put alongside one another. We see that it does not matter in\nwhat order the sets of nine are put. As the cube passes through the\nplane they represent appearances which follow the one after the other.\nIf they were what they represented, they could not exist in the same\nplane together.\n\nThis is a rather important point, namely, to notice that they should\nnot co-exist on the plane, and that the order in which they are placed\nis indifferent. When we represent a four-dimensional body our solid\ncubes are to us in the same position that the slabs are to the plane\nbeing. You should also notice that each of these slabs represents only\nthe very thinnest slice of a cube. The set of nine slabs first set up\nrepresents the side surface of the block. It is, as it were, a kind\nof tray\u2014a beginning from which the solid cube goes off. The slabs\nas we use them have thickness, but this thickness is a necessity of\nconstruction. They are to be thought of as merely of the thickness of a\nline.\n\nIf now the block of cubes passed through the plane at the rate of an\ninch a minute the appearance to a plane being would be represented by:\u2014\n\n1. The first set of nine slabs lasting for one minute.\n\n2. The second set of nine slabs lasting for one minute.\n\n3. The third set of nine slabs lasting for one minute.\n\nNow the appearances which the cube would present to the plane being\nin other positions can be shown by means of these slabs. The use of\nsuch slabs would be the means by which a plane being could acquire a\nfamiliarity with our cube. Turn the catalogue cube (or imagine the\ncoloured figure turned) so that the red line runs up, the yellow line\nout to the right, and the white line towards you. Then turn the block\nof cubes to occupy a similar position.\n\nThe block has now a different wall in contact with the plane. Its\nappearance to a plane being will not be the same as before. He has,\nhowever, enough slabs to represent this new set of appearances. But he\nmust remodel his former arrangement of them.\n\nHe must take a null, a red, and a null slab from the first of his sets\nof slabs, then a white, a pink, and a white from the second, and then a\nnull, a red, and a null from the third set of slabs.\n\nHe takes the first column from the first set, the first column from the\nsecond set, and the first column from the third set.\n\nTo represent the half-way-through appearance, which is as if a very\nthin slice were cut out half way through the block, he must take the\nsecond column of each of his sets of slabs, and to represent the final\nappearance, the third column of each set.\n\nNow turn the catalogue cube back to the normal position, and also the\nblock of cubes.\n\nThere is another turning\u2014a turning about the yellow line, in which the\nwhite axis comes below the support.\n\nYou cannot break through the surface of the table, so you must imagine\nthe old support to be raised. Then the top of the block of cubes in its\nnew position is at the level at which the base of it was before.\n\nNow representing the appearance on the plane, we must draw a horizontal\nline to represent the old base. The line should be drawn three inches\nhigh on the cardboard.\n\nBelow this the representative slabs can be arranged.\n\nIt is easy to see what they are. The old arrangements have to be\nbroken up, and the layers taken in order, the first layer of each for\nthe representation of the aspect of the block as it touches the plane.\n\nThen the second layers will represent the appearance half way through,\nand the third layers will represent the final appearance.\n\nIt is evident that the slabs individually do not represent the same\nportion of the cube in these different presentations.\n\nIn the first case each slab represents a section or a face\nperpendicular to the white axis, in the second case a face or a section\nwhich runs perpendicularly to the yellow axis, and in the third case a\nsection or a face perpendicular to the red axis.\n\nBut by means of these nine slabs the plane being can represent the\nwhole of the cubic block. He can touch and handle each portion of the\ncubic block, there is no part of it which he cannot observe. Taking it\nbit by bit, two axes at a time, he can examine the whole of it.\n\n\n OUR REPRESENTATION OF A BLOCK OF TESSERACTS.\n\nLook at the views of the tesseract 1, 2, 3, or take the catalogue cubes\n1, 2, 3, and place them in front of you, in any order, say running from\nleft to right, placing 1 in the normal position, the red axis running\nup, the white to the right, and yellow away.\n\nNow notice that in catalogue cube 2 the colours of each region are\nderived from those of the corresponding region of cube 1 by the\naddition of blue. Thus null + blue = blue, and the corners of number 2\nare blue. Again, red + blue = purple, and the vertical lines of 2 are\npurple. Blue + yellow = green, and the line which runs away is coloured\ngreen.\n\nBy means of these observations you may be sure that catalogue cube 2\nis rightly placed. Catalogue cube 3 is just like number 1.\n\nHaving these cubes in what we may call their normal position, proceed\nto build up the three sets of blocks.\n\nThis is easily done in accordance with the colour scheme on the\ncatalogue cubes.\n\nThe first block we already know. Build up the second block, beginning\nwith a blue corner cube, placing a purple on it, and so on.\n\nHaving these three blocks we have the means of representing the\nappearances of a group of eighty-one tesseracts.\n\nLet us consider a moment what the analogy in the case of the plane\nbeing is.\n\nHe has his three sets of nine slabs each. We have our three sets of\ntwenty-seven cubes each.\n\nOur cubes are like his slabs. As his slabs are not the things which\nthey represent to him, so our cubes are not the things they represent\nto us.\n\nThe plane being\u2019s slabs are to him the faces of cubes.\n\nOur cubes then are the faces of tesseracts, the cubes by which they are\nin contact with our space.\n\nAs each set of slabs in the case of the plane being might be considered\nas a sort of tray from which the solid contents of the cubes came out,\nso our three blocks of cubes may be considered as three-space trays,\neach of which is the beginning of an inch of the solid contents of the\nfour-dimensional solids starting from them.\n\nWe want now to use the names null, red, white, etc., for tesseracts.\nThe cubes we use are only tesseract faces. Let us denote that fact\nby calling the cube of null colour, null face; or, shortly, null f.,\nmeaning that it is the face of a tesseract.\n\nTo determine which face it is let us look at the catalogue cube 1 or\nthe first of the views of the tesseract, which can be used instead of\nthe models. It has three axes, red, white, yellow, in our space. Hence\nthe cube determined by these axes is the face of the tesseract which we\nnow have before us. It is the ochre face. It is enough, however, simply\nto say null f., red f. for the cubes which we use.\n\nTo impress this in your mind, imagine that tesseracts do actually run\nfrom each cube. Then, when you move the cubes about, you move the\ntesseracts about with them. You move the face but the tesseract follows\nwith it, as the cube follows when its face is shifted in a plane.\n\nThe cube null in the normal position is the cube which has in it the\nred, yellow, white axes. It is the face having these, but wanting the\nblue. In this way you can define which face it is you are handling. I\nwill write an \u201cf.\u201d after the name of each tesseract just as the plane\nbeing might call each of his slabs null slab, yellow slab, etc., to\ndenote that they were representations.\n\nWe have then in the first block of twenty-seven cubes, the\nfollowing\u2014null f., red f., null f., going up; white f., null f., lying\nto the right, and so on. Starting from the null point and travelling\nup one inch we are in the null region, the same for the away and the\nright-hand directions. And if we were to travel in the fourth dimension\nfor an inch we should still be in a null region. The tesseract\nstretches equally all four ways. Hence the appearance we have in this\nfirst block would do equally well if the tesseract block were to move\nacross our space for a certain distance. For anything less than an inch\nof their transverse motion we should still have the same appearance.\nYou must notice, however, that we should not have null face after the\nmotion had begun.\n\nWhen the tesseract, null for instance, had moved ever so little we\nshould not have a face of null but a section of null in our space.\nHence, when we think of the motion across our space we must call our\ncubes tesseract sections. Thus on null passing across we should see\nfirst null f., then null s., and then, finally, null f. again.\n\nImagine now the whole first block of twenty-seven tesseracts to have\nmoved tranverse to our space a distance of one inch. Then the second\nset of tesseracts, which originally were an inch distant from our\nspace, would be ready to come in.\n\nTheir colours are shown in the second block of twenty-seven cubes which\nyou have before you. These represent the tesseract faces of the set of\ntesseracts that lay before an inch away from our space. They are ready\nnow to come in, and we can observe their colours. In the place which\nnull f. occupied before we have blue f., in place of red f. we have\npurple f., and so on. Each tesseract is coloured like the one whose\nplace it takes in this motion with the addition of blue.\n\nNow if the tesseract block goes on moving at the rate of an inch a\nminute, this next set of tesseracts will occupy a minute in passing\nacross. We shall see, to take the null one for instance, first of all\nnull face, then null section, then null face again.\n\nAt the end of the second minute the second set of tesseracts has gone\nthrough, and the third set comes in. This, as you see, is coloured just\nlike the first. Altogether, these three sets extend three inches in the\nfourth dimension, making the tesseract block of equal magnitude in all\ndimensions.\n\nWe have now before us a complete catalogue of all the tesseracts in our\ngroup. We have seen them all, and we shall refer to this arrangement\nof the blocks as the \u201cnormal position.\u201d We have seen as much of each\ntesseract at a time as could be done in a three-dimensional space. Each\npart of each tesseract has been in our space, and we could have touched\nit.\n\nThe fourth dimension appeared to us as the duration of the block.\n\nIf a bit of our matter were to be subjected to the same motion it\nwould be instantly removed out of our space. Being thin in the fourth\ndimension it is at once taken out of our space by a motion in the\nfourth dimension.\n\nBut the tesseract block we represent having length in the fourth\ndimension remains steadily before our eyes for three minutes, when it\nis subjected to this transverse motion.\n\nWe have now to form representations of the other views of the same\ntesseract group which are possible in our space.\n\nLet us then turn the block of tesseracts so that another face of it\ncomes into contact with our space, and then by observing what we have,\nand what changes come when the block traverses our space, we shall have\nanother view of it. The dimension which appeared as duration before\nwill become extension in one of our known dimensions, and a dimension\nwhich coincided with one of our space dimensions will appear as\nduration.\n\nLeaving catalogue cube 1 in the normal position, remove the other two,\nor suppose them removed. We have in space the red, the yellow, and the\nwhite axes. Let the white axis go out into the unknown, and occupy the\nposition the blue axis holds. Then the blue axis, which runs in that\ndirection now will come into space. But it will not come in pointing\nin the same way that the white axis does now. It will point in the\nopposite sense. It will come in running to the left instead of running\nto the right as the white axis does now.\n\nWhen this turning takes place every part of the cube 1 will disappear\nexcept the left-hand face\u2014the orange face.\n\nAnd the new cube that appears in our space will run to the left from\nthis orange face, having axes, red, yellow, blue.\n\nTake models 4, 5, 6. Place 4, or suppose No. 4 of the tesseract views\nplaced, with its orange face coincident with the orange face of 1, red\nline to red line, and yellow line to yellow line, with the blue line\npointing to the left. Then remove cube 1 and we have the tesseract face\nwhich comes in when the white axis runs in the positive unknown, and\nthe blue axis comes into our space.\n\nNow place catalogue cube 5 in some position, it does not matter which,\nsay to the left; and place it so that there is a correspondence of\ncolour corresponding to the colour of the line that runs out of space.\nThe line that runs out of space is white, hence, every part of this\ncube 5 should differ from the corresponding part of 4 by an alteration\nin the direction of white.\n\nThus we have white points in 5 corresponding to the null points in\n4. We have a pink line corresponding to a red line, a light yellow\nline corresponding to a yellow line, an ochre face corresponding to\nan orange face. This cube section is completely named in Chapter XI.\nFinally cube 6 is a replica of 1.\n\nThese catalogue cubes will enable us to set up our models of the block\nof tesseracts.\n\nFirst of all for the set of tesseracts, which beginning in our space\nreach out one inch in the unknown, we have the pattern of catalogue\ncube 4.\n\nWe see that we can build up a block of twenty-seven tesseract faces\nafter the colour scheme of cube 4, by taking the left-hand wall of\nblock 1, then the left-hand wall of block 2, and finally that of block\n3. We take, that is, the three first walls of our previous arrangement\nto form the first cubic block of this new one.\n\nThis will represent the cubic faces by which the group of tesseracts in\nits new position touches our space. We have running up, null f., red\nf., null f. In the next vertical line, on the side remote from us, we\nhave yellow f., orange f., yellow f., and then the first colours over\nagain. Then the three following columns are, blue f., purple f., blue\nf.; green f., brown f., green f.; blue f., purple f., blue f. The last\nthree columns are like the first.\n\nThese tesseracts touch our space, and none of them are by any part of\nthem distant more than an inch from it. What lies beyond them in the\nunknown?\n\nThis can be told by looking at catalogue cube 5. According to its\nscheme of colour we see that the second wall of each of our old\narrangements must be taken. Putting them together we have, as the\ncorner, white f. above it, pink f. above it, white f. The column next\nto this remote from us is as follows:\u2014light yellow f., ochre f., light\nyellow f., and beyond this a column like the first. Then for the middle\nof the block, light blue f., above it light purple, then light blue.\nThe centre column has, at the bottom, light green f., light brown f.\nin the centre and at the top light green f. The last wall is like the\nfirst.\n\nThe third block is made by taking the third walls of our previous\narrangement, which we called the normal one.\n\nYou may ask what faces and what sections our cubes represent. To answer\nthis question look at what axes you have in our space. You have red,\nyellow, blue. Now these determine brown. The colours red, yellow, blue\nare supposed by us when mixed to produce a brown colour. And that cube\nwhich is determined by the red, yellow, blue axes we call the brown\ncube.\n\nWhen the tesseract block in its new position begins to move across our\nspace each tesseract in it gives a section in our space. This section\nis transverse to the white axis, which now runs in the unknown.\n\nAs the tesseract in its present position passes across our space, we\nshould see first of all the first of the blocks of cubic faces we have\nput up\u2014these would last for a minute, then would come the second block\nand then the third. At first we should have a cube of tesseract faces,\neach of which would be brown. Directly the movement began, we should\nhave tesseract sections transverse to the white line.\n\nThere are two more analogous positions in which the block of tesseracts\ncan be placed. To find the third position, restore the blocks to the\nnormal arrangement.\n\nLet us make the yellow axis go out into the positive unknown, and let\nthe blue axis, consequently, come in running towards us. The yellow ran\naway, so the blue will come in running towards us.\n\nPut catalogue cube 1 in its normal position. Take catalogue cube 7\nand place it so that its pink face coincides with the pink face of\ncube 1, making also its red axis coincide with the red axis of 1 and\nits white with the white. Moreover, make cube 7 come towards us from\ncube 1. Looking at it we see in our space, red, white, and blue axes.\nThe yellow runs out. Place catalogue cube 8 in the neighbourhood\nof 7\u2014observe that every region in 8 has a change in the direction\nof yellow from the corresponding region in 7. This is because it\nrepresents what you come to now in going in the unknown, when the\nyellow axis runs out of our space. Finally catalogue cube 9, which is\nlike number 7, shows the colours of the third set of tesseracts. Now\nevidently, starting from the normal position, to make up our three\nblocks of tesseract faces we have to take the near wall from the first\nblock, the near wall from the second, and then the near wall from the\nthird block. This gives us the cubic block formed by the faces of the\ntwenty-seven tesseracts which are now immediately touching our space.\n\nFollowing the colour scheme of catalogue cube 8, we make the next set\nof twenty-seven tesseract faces, representing the tesseracts, each of\nwhich begins one inch off from our space, by putting the second walls\nof our previous arrangement together, and the representation of the\nthird set of tesseracts is the cubic block formed of the remaining\nthree walls.\n\nSince we have red, white, blue axes in our space to begin with, the\ncubes we see at first are light purple tesseract faces, and after the\ntransverse motion begins we have cubic sections transverse to the\nyellow line.\n\nRestore the blocks to the normal position, there remains the case in\nwhich the red axis turns out of space. In this case the blue axis will\ncome in downwards, opposite to the sense in which the red axis ran.\n\nIn this case take catalogue cubes 10, 11, 12. Lift up catalogue cube 1\nand put 10 underneath it, imagining that it goes down from the previous\nposition of 1.\n\nWe have to keep in space the white and the yellow axes, and let the red\ngo out, the blue come in.\n\nNow, you will find on cube 10 a light yellow face; this should coincide\nwith the base of 1, and the white and yellow lines on the two cubes\nshould coincide. Then the blue axis running down you have the catalogue\ncube correctly placed, and it forms a guide for putting up the first\nrepresentative block.\n\nCatalogue cube 11 will represent what lies in the fourth dimension\u2014now\nthe red line runs in the fourth dimension. Thus the change from 10 to\n11 should be towards red, corresponding to a null point is a red point,\nto a white line is a pink line, to a yellow line an orange line, and so\non.\n\nCatalogue cube 12 is like 10. Hence we see that to build up our blocks\nof tesseract faces we must take the bottom layer of the first block,\nhold that up in the air, underneath it place the bottom layer of the\nsecond block, and finally underneath this last the bottom layer of the\nlast of our normal blocks.\n\nSimilarly we make the second representative group by taking the middle\ncourses of our three blocks. The last is made by taking the three\ntopmost layers. The three axes in our space before the transverse\nmotion begins are blue, white, yellow, so we have light green tesseract\nfaces, and after the motion begins sections transverse to the red light.\n\nThese three blocks represent the appearances as the tesseract group in\nits new position passes across our space. The cubes of contact in this\ncase are those determinal by the three axes in our space, namely, the\nwhite, the yellow, the blue. Hence they are light green.\n\nIt follows from this that light green is the interior cube of the first\nblock of representative cubic faces.\n\nPractice in the manipulations described, with a realization in each\ncase of the face or section which is in our space, is one of the best\nmeans of a thorough comprehension of the subject.\n\nWe have to learn how to get any part of these four-dimensional figures\ninto space, so that we can look at them. We must first learn to swing a\ntesseract, and a group of tesseracts about in any way.\n\nWhen these operations have been repeated and the method of arrangement\nof the set of blocks has become familiar, it is a good plan to rotate\nthe axes of the normal cube 1 about a diagonal, and then repeat the\nwhole series of turnings.\n\nThus, in the normal position, red goes up, white to the right, yellow\naway. Make white go up, yellow to the right, and red away. Learn the\ncube in this position by putting up the set of blocks of the normal\ncube, over and over again till it becomes as familiar to you as in the\nnormal position. Then when this is learned, and the corresponding\nchanges in the arrangements of the tesseract groups are made, another\nchange should be made: let, in the normal cube, yellow go up, red to\nthe right, and white away.\n\nLearn the normal block of cubes in this new position by arranging them\nand re-arranging them till you know without thought where each one\ngoes. Then carry out all the tesseract arrangements and turnings.\n\nIf you want to understand the subject, but do not see your way clearly,\nif it does not seem natural and easy to you, practise these turnings.\nPractise, first of all, the turning of a block of cubes round, so that\nyou know it in every position as well as in the normal one. Practise by\ngradually putting up the set of cubes in their new arrangements. Then\nput up the tesseract blocks in their arrangements. This will give you\na working conception of higher space, you will gain the feeling of it,\nwhether you take up the mathematical treatment of it or not.\n\n\n\n\n APPENDIX II\n\n A LANGUAGE OF SPACE\n\n\nThe mere naming the parts of the figures we consider involves a certain\namount of time and attention. This time and attention leads to no\nresult, for with each new figure the nomenclature applied is completely\nchanged, every letter or symbol is used in a different significance.\n\nSurely it must be possible in some way to utilise the labour thus at\npresent wasted!\n\nWhy should we not make a language for space itself, so that every\nposition we want to refer to would have its own name? Then every time\nwe named a figure in order to demonstrate its properties we should be\nexercising ourselves in the vocabulary of place.\n\nIf we use a definite system of names, and always refer to the same\nspace position by the same name, we create as it were a multitude of\nlittle hands, each prepared to grasp a special point, position, or\nelement, and hold it for us in its proper relations.\n\nWe make, to use another analogy, a kind of mental paper, which has\nsomewhat of the properties of a sensitive plate, in that it will\nregister, without effort, complex, visual, or tactual impressions.\n\nBut of far more importance than the applications of a space language to\nthe plane and to solid space is the facilitation it brings with it to\nthe study of four-dimensional shapes.\n\nI have delayed introducing a space language because all the systems I\nmade turned out, after giving them a fair trial, to be intolerable. I\nhave now come upon one which seems to present features of permanence,\nand I will here give an outline of it, so that it can be applied to the\nsubject of the text, and in order that it may be subjected to criticism.\n\nThe principle on which the language is constructed is to sacrifice\nevery other consideration for brevity.\n\nIt is indeed curious that we are able to talk and converse on every\nsubject of thought except the fundamental one of space. The only way of\nspeaking about the spatial configurations that underlie every subject\nof discursive thought is a co-ordinate system of numbers. This is so\nawkward and incommodious that it is never used. In thinking also, in\nrealising shapes, we do not use it; we confine ourselves to a direct\nvisualisation.\n\nNow, the use of words corresponds to the storing up of our experience\nin a definite brain structure. A child, in the endless tactual, visual,\nmental manipulations it makes for itself, is best left to itself, but\nin the course of instruction the introduction of space names would\nmake the teachers work more cumulative, and the child\u2019s knowledge more\nsocial.\n\nTheir full use can only be appreciated, if they are introduced early\nin the course of education; but in a minor degree any one can convince\nhimself of their utility, especially in our immediate subject of\nhandling four-dimensional shapes. The sum total of the results obtained\nin the preceding pages can be compendiously and accurately expressed in\nnine words of the Space Language.\n\nIn one of Plato\u2019s dialogues Socrates makes an experiment on a slave boy\nstanding by. He makes certain perceptions of space awake in the mind\nof Meno\u2019s slave by directing his close attention on some simple facts\nof geometry.\n\nBy means of a few words and some simple forms we can repeat Plato\u2019s\nexperiment on new ground.\n\nDo we by directing our close attention on the facts of four dimensions\nawaken a latent faculty in ourselves? The old experiment of Plato\u2019s, it\nseems to me, has come down to us as novel as on the day he incepted it,\nand its significance not better understood through all the discussion\nof which it has been the subject.\n\nImagine a voiceless people living in a region where everything had\na velvety surface, and who were thus deprived of all opportunity of\nexperiencing what sound is. They could observe the slow pulsations\nof the air caused by their movements, and arguing from analogy, they\nwould no doubt infer that more rapid vibrations were possible. From\nthe theoretical side they could determine all about these more rapid\nvibrations. They merely differ, they would say, from slower ones,\nby the number that occur in a given time; there is a merely formal\ndifference.\n\nBut suppose they were to take the trouble, go to the pains of producing\nthese more rapid vibrations, then a totally new sensation would fall\non their rudimentary ears. Probably at first they would only be dimly\nconscious of Sound, but even from the first they would become aware\nthat a merely formal difference, a mere difference in point of number\nin this particular respect, made a great difference practically, as\nrelated to them. And to us the difference between three and four\ndimensions is merely formal, numerical. We can tell formally all about\nfour dimensions, calculate the relations that would exist. But that\nthe difference is merely formal does not prove that it is a futile and\nempty task, to present to ourselves as closely as we can the phenomena\nof four dimensions. In our formal knowledge of it, the whole question\nof its actual relation to us, as we are, is left in abeyance.\n\nPossibly a new apprehension of nature may come to us through the\npractical, as distinguished from the mathematical and formal, study\nof four dimensions. As a child handles and examines the objects with\nwhich he comes in contact, so we can mentally handle and examine\nfour-dimensional objects. The point to be determined is this. Do we\nfind something cognate and natural to our faculties, or are we merely\nbuilding up an artificial presentation of a scheme only formally\npossible, conceivable, but which has no real connection with any\nexisting or possible experience?\n\nThis, it seems to me, is a question which can only be settled by\nactually trying. This practical attempt is the logical and direct\ncontinuation of the experiment Plato devised in the \u201cMeno.\u201d\n\nWhy do we think true? Why, by our processes of thought, can we predict\nwhat will happen, and correctly conjecture the constitution of the\nthings around us? This is a problem which every modern philosopher has\nconsidered, and of which Descartes, Leibnitz, Kant, to name a few,\nhave given memorable solutions. Plato was the first to suggest it.\nAnd as he had the unique position of being the first devisor of the\nproblem, so his solution is the most unique. Later philosophers have\ntalked about consciousness and its laws, sensations, categories. But\nPlato never used such words. Consciousness apart from a conscious being\nmeant nothing to him. His was always an objective search. He made man\u2019s\nintuitions the basis of a new kind of natural history.\n\nIn a few simple words Plato puts us in an attitude with regard to\npsychic phenomena\u2014the mind\u2014the ego\u2014\u201cwhat we are,\u201d which is analogous\nto the attitude scientific men of the present day have with regard\nto the phenomena of outward nature. Behind this first apprehension\nof ours of nature, there is an infinite depth to be learned and\nknown. Plato said that behind the phenomena of mind that Meno\u2019s slave\nboy exhibited, there was a vast, an infinite perspective. And his\nsingularity, his originality, comes out most strongly marked in this,\nthat the perspective, the complex phenomena beyond were, according to\nhim, phenomena of personal experience. A footprint in the sand means a\nman to a being that has the conception of a man. But to a creature that\nhas no such conception, it means a curious mark, somehow resulting from\nthe concatenation of ordinary occurrences. Such a being would attempt\nmerely to explain how causes known to him could so coincide as to\nproduce such a result; he would not recognise its significance.\n\nPlato introduced the conception which made a new kind of natural\nhistory possible. He said that Meno\u2019s slave boy thought true about\nthings he had never learned, because his \u201csoul\u201d had experience. I\nknow this will sound absurd to some people, and it flies straight in\nthe face of the maxim, that explanation consists in showing how an\neffect depends on simple causes. But what a mistaken maxim that is!\nCan any single instance be shown of a simple cause? Take the behaviour\nof spheres for instance; say those ivory spheres, billiard balls,\nfor example. We can explain their behaviour by supposing they are\nhomogeneous elastic solids. We can give formul\u00e6 which will account for\ntheir movements in every variety. But are they homogeneous elastic\nsolids? No, certainly not. They are complex in physical and molecular\nstructure, and atoms and ions beyond open an endless vista. Our simple\nexplanation is false, false as it can be. The balls act as if they\nwere homogeneous elastic spheres. There is a statistical simplicity in\nthe resultant of very complex conditions, which makes that artificial\nconception useful. But its usefulness must not blind us to the fact\nthat it is artificial. If we really look deep into nature, we find a\nmuch greater complexity than we at first suspect. And so behind this\nsimple \u201cI,\u201d this myself, is there not a parallel complexity? Plato\u2019s\n\u201csoul\u201d would be quite acceptable to a large class of thinkers, if by\n\u201csoul\u201d and the complexity he attributes to it, he meant the product of\na long course of evolutionary changes, whereby simple forms of living\nmatter endowed with rudimentary sensation had gradually developed into\nfully conscious beings.\n\nBut Plato does not mean by \u201csoul\u201d a being of such a kind. His soul is\na being whose faculties are clogged by its bodily environment, or at\nleast hampered by the difficulty of directing its bodily frame\u2014a being\nwhich is essentially higher than the account it gives of itself through\nits organs. At the same time Plato\u2019s soul is not incorporeal. It is a\nreal being with a real experience. The question of whether Plato had\nthe conception of non-spatial existence has been much discussed. The\nverdict is, I believe, that even his \u201cideas\u201d were conceived by him as\nbeings in space, or, as we should say, real. Plato\u2019s attitude is that\nof Science, inasmuch as he thinks of a world in Space. But, granting\nthis, it cannot be denied that there is a fundamental divergence\nbetween Plato\u2019s conception and the evolutionary theory, and also an\nabsolute divergence between his conception and the genetic account of\nthe origin of the human faculties. The functions and capacities of\nPlato\u2019s \u201csoul\u201d are not derived by the interaction of the body and its\nenvironment.\n\nPlato was engaged on a variety of problems, and his religious and\nethical thoughts were so keen and fertile that the experimental\ninvestigation of his soul appears involved with many other motives.\nIn one passage Plato will combine matter of thought of all kinds and\nfrom all sources, overlapping, interrunning. And in no case is he more\ninvolved and rich than in this question of the soul. In fact, I wish\nthere were two words, one denoting that being, corporeal and real, but\nwith higher faculties than we manifest in our bodily actions, which is\nto be taken as the subject of experimental investigation; and the other\nword denoting \u201csoul\u201d in the sense in which it is made the recipient and\nthe promise of so much that men desire. It is the soul in the former\nsense that I wish to investigate, and in a limited sphere only. I wish\nto find out, in continuation of the experiment in the Meno, what the\n\u201csoul\u201d in us thinks about extension, experimenting on the grounds laid\ndown by Plato. He made, to state the matter briefly, the hypothesis\nwith regard to the thinking power of a being in us, a \u201csoul.\u201d This\nsoul is not accessible to observation by sight or touch, but it can be\nobserved by its functions; it is the object of a new kind of natural\nhistory, the materials for constructing which lie in what it is natural\nto us to think. With Plato \u201cthought\u201d was a very wide-reaching term, but\nstill I would claim in his general plan of procedure a place for the\nparticular question of extension.\n\nThe problem comes to be, \u201cWhat is it natural to us to think about\nmatter _qua_ extended?\u201d\n\nFirst of all, I find that the ordinary intuition of any simple object\nis extremely imperfect. Take a block of differently marked cubes, for\ninstance, and become acquainted with them in their positions. You may\nthink you know them quite well, but when you turn them round\u2014rotate\nthe block round a diagonal, for instance\u2014you will find that you have\nlost track of the individuals in their new positions. You can mentally\nconstruct the block in its new position, by a rule, by taking the\nremembered sequences, but you don\u2019t know it intuitively. By observation\nof a block of cubes in various positions, and very expeditiously\nby a use of Space names applied to the cubes in their different\npresentations, it is possible to get an intuitive knowledge of the\nblock of cubes, which is not disturbed by any displacement. Now, with\nregard to this intuition, we moderns would say that I had formed it by\nmy tactual visual experiences (aided by hereditary pre-disposition).\nPlato would say that the soul had been stimulated to recognise an\ninstance of shape which it knew. Plato would consider the operation\nof learning merely as a stimulus; we as completely accounting for\nthe result. The latter is the more common-sense view. But, on the\nother hand, it presupposes the generation of experience from physical\nchanges. The world of sentient experience, according to the modern\nview, is closed and limited; only the physical world is ample and large\nand of ever-to-be-discovered complexity. Plato\u2019s world of soul, on the\nother hand, is at least as large and ample as the world of things.\n\nLet us now try a crucial experiment. Can I form an intuition of a\nfour-dimensional object? Such an object is not given in the physical\nrange of my sense contacts. All I can do is to present to myself the\nsequences of solids, which would mean the presentation to me under my\nconditions of a four-dimensional object. All I can do is to visualise\nand tactualise different series of solids which are alternative sets of\nsectional views of a four-dimensional shape.\n\nIf now, on presenting these sequences, I find a power in me of\nintuitively passing from one of these sets of sequences to another, of,\nbeing given one, intuitively constructing another, not using a rule,\nbut directly apprehending it, then I have found a new fact about my\nsoul, that it has a four-dimensional experience; I have observed it by\na function it has.\n\nI do not like to speak positively, for I might occasion a loss of time\non the part of others, if, as may very well be, I am mistaken. But for\nmy own part, I think there are indications of such an intuition; from\nthe results of my experiments, I adopt the hypothesis that that which\nthinks in us has an ample experience, of which the intuitions we use in\ndealing with the world of real objects are a part; of which experience,\nthe intuition of four-dimensional forms and motions is also a part. The\nprocess we are engaged in intellectually is the reading the obscure\nsignals of our nerves into a world of reality, by means of intuitions\nderived from the inner experience.\n\nThe image I form is as follows. Imagine the captain of a modern\nbattle-ship directing its course. He has his charts before him; he\nis in communication with his associates and subordinates; can convey\nhis messages and commands to every part of the ship, and receive\ninformation from the conning-tower and the engine-room. Now suppose the\ncaptain immersed in the problem of the navigation of his ship over the\nocean, to have so absorbed himself in the problem of the direction of\nhis craft over the plane surface of the sea that he forgets himself.\nAll that occupies his attention is the kind of movement that his ship\nmakes. The operations by which that movement is produced have sunk\nbelow the threshold of his consciousness, his own actions, by which\nhe pushes the buttons, gives the orders, are so familiar as to be\nautomatic, his mind is on the motion of the ship as a whole. In such a\ncase we can imagine that he identifies himself with his ship; all that\nenters his conscious thought is the direction of its movement over the\nplane surface of the ocean.\n\nSuch is the relation, as I imagine it, of the soul to the body. A\nrelation which we can imagine as existing momentarily in the case\nof the captain is the normal one in the case of the soul with its\ncraft. As the captain is capable of a kind of movement, an amplitude\nof motion, which does not enter into his thoughts with regard to the\ndirecting the ship over the plane surface of the ocean, so the soul is\ncapable of a kind of movement, has an amplitude of motion, which is\nnot used in its task of directing the body in the three-dimensional\nregion in which the body\u2019s activity lies. If for any reason it became\nnecessary for the captain to consider three-dimensional motions with\nregard to his ship, it would not be difficult for him to gain the\nmaterials for thinking about such motions; all he has to do is to\ncall his own intimate experience into play. As far as the navigation\nof the ship, however, is concerned, he is not obliged to call on\nsuch experience. The ship as a whole simply moves on a surface. The\nproblem of three-dimensional movement does not ordinarily concern its\nsteering. And thus with regard to ourselves all those movements and\nactivities which characterise our bodily organs are three-dimensional;\nwe never need to consider the ampler movements. But we do more than\nuse the movements of our body to effect our aims by direct means; we\nhave now come to the pass when we act indirectly on nature, when we\ncall processes into play which lie beyond the reach of any explanation\nwe can give by the kind of thought which has been sufficient for the\nsteering of our craft as a whole. When we come to the problem of what\ngoes on in the minute, and apply ourselves to the mechanism of the\nminute, we find our habitual conceptions inadequate.\n\nThe captain in us must wake up to his own intimate nature, realise\nthose functions of movement which are his own, and in virtue of his\nknowledge of them apprehend how to deal with the problems he has come\nto.\n\nThink of the history of man. When has there been a time, in which his\nthoughts of form and movement were not exclusively of such varieties as\nwere adapted for his bodily performance? We have never had a demand to\nconceive what our own most intimate powers are. But, just as little as\nby immersing himself in the steering of his ship over the plane surface\nof the ocean, a captain can lose the faculty of thinking about what he\nactually does, so little can the soul lose its own nature. It can be\nroused to an intuition that is not derived from the experience which\nthe senses give. All that is necessary is to present some few of those\nappearances which, while inconsistent with three-dimensional matter,\nare yet consistent with our formal knowledge of four-dimensional\nmatter, in order for the soul to wake up and not begin to learn, but of\nits own intimate feeling fill up the gaps in the presentiment, grasp\nthe full orb of possibilities from the isolated points presented to\nit. In relation to this question of our perceptions, let me suggest\nanother illustration, not taking it too seriously, only propounding it\nto exhibit the possibilities in a broad and general way.\n\nIn the heavens, amongst the multitude of stars, there are some which,\nwhen the telescope is directed on them, seem not to be single stars,\nbut to be split up into two. Regarding these twin stars through a\nspectroscope, an astronomer sees in each a spectrum of bands of colour\nand black lines. Comparing these spectrums with one another, he finds\nthat there is a slight relative shifting of the dark lines, and from\nthat shifting he knows that the stars are rotating round one another,\nand can tell their relative velocity with regard to the earth. By\nmeans of his terrestrial physics he reads this signal of the skies.\nThis shifting of lines, the mere slight variation of a black line in a\nspectrum, is very unlike that which the astronomer knows it means. But\nit is probably much more like what it means than the signals which the\nnerves deliver are like the phenomena of the outer world.\n\nNo picture of an object is conveyed through the nerves. No picture of\nmotion, in the sense in which we postulate its existence, is conveyed\nthrough the nerves. The actual deliverances of which our consciousness\ntakes account are probably identical for eye and ear, sight and touch.\n\nIf for a moment I take the whole earth together and regard it as a\nsentient being, I find that the problem of its apprehension is a very\ncomplex one, and involves a long series of personal and physical\nevents. Similarly the problem of our apprehension is a very complex\none. I only use this illustration to exhibit my meaning. It has this\nespecial merit, that, as the process of conscious apprehension takes\nplace in our case in the minute, so, with regard to this earth being,\nthe corresponding process takes place in what is relatively to it very\nminute.\n\nNow, Plato\u2019s view of a soul leads us to the hypothesis that that\nwhich we designate as an act of apprehension may be a very complex\nevent, both physically and personally. He does not seek to explain\nwhat an intuition is; he makes it a basis from whence he sets out on\na voyage of discovery. Knowledge means knowledge; he puts conscious\nbeing to account for conscious being. He makes an hypothesis of the\nkind that is so fertile in physical science\u2014an hypothesis making no\nclaim to finality, which marks out a vista of possible determination\nbehind determination, like the hypothesis of space itself, the type of\nserviceable hypotheses.\n\nAnd, above all, Plato\u2019s hypothesis is conducive to experiment. He\ngives the perspective in which real objects can be determined; and,\nin our present enquiry, we are making the simplest of all possible\nexperiments\u2014we are enquiring what it is natural to the soul to think of\nmatter as extended.\n\nAristotle says we always use a \u201cphantasm\u201d in thinking, a phantasm of\nour corporeal senses a visualisation or a tactualisation. But we can\nso modify that visualisation or tactualisation that it represents\nsomething not known by the senses. Do we by that representation wake\nup an intuition of the soul? Can we by the presentation of these\nhypothetical forms, that are the subject of our present discussion,\nwake ourselves up to higher intuitions? And can we explain the world\naround by a motion that we only know by our souls?\n\nApart from all speculation, however, it seems to me that the interest\nof these four-dimensional shapes and motions is sufficient reason for\nstudying them, and that they are the way by which we can grow into a\nfuller apprehension of the world as a concrete whole.\n\n\n SPACE NAMES.\n\nIf the words written in the squares drawn in fig. 1 are used as the\nnames of the squares in the positions in which they are placed, it is\nevident that a combination of these names will denote a figure composed\nof the designated squares. It is found to be most convenient to take as\nthe initial square that marked with an asterisk, so that the directions\nof progression are towards the observer and to his right. The\ndirections of progression, however, are arbitrary, and can be chosen at\nwill.\n\n[Illustration: Fig. 1.]\n\nThus _et_, _at_, _it_, _an_, _al_ will denote a figure in the form of a\ncross composed of five squares.\n\nHere, by means of the double sequence, _e_, _a_, _i_ and _n_, _t_, _l_,\nit is possible to name a limited collection of space elements.\n\nThe system can obviously be extended by using letter sequences of more\nmembers.\n\nBut, without introducing such a complexity, the principles of a space\nlanguage can be exhibited, and a nomenclature obtained adequate to all\nthe considerations of the preceding pages.\n\n\n1. _Extension._\n\nCall the large squares in fig. 2 by the name written in them. It is\nevident that each can be divided as shown in fig. 1. Then the small\nsquare marked 1 will be \u201cen\u201d in \u201cEn,\u201d or \u201cEnen.\u201d The square marked 2\nwill be \u201cet\u201d in \u201cEn\u201d or \u201cEnet,\u201d while the square marked 4 will be \u201cen\u201d\nin \u201cEt\u201d or \u201cEten.\u201d Thus the square 5 will be called \u201cIlil.\u201d\n\n[Illustration: Fig. 2.]\n\nThis principle of extension can be applied in any number of dimensions.\n\n\n2. _Application to Three-Dimensional Space._\n\nTo name a three-dimensional collocation of cubes take the upward\ndirection first, secondly the direction towards the observer, thirdly\nthe direction to his right hand.\n\n[Illustration]\n\nThese form a word in which the first letter gives the place of the cube\nupwards, the second letter its place towards the observer, the third\nletter its place to the right.\n\nWe have thus the following scheme, which represents the set of cubes of\ncolumn 1, fig. 101, page 165.\n\nWe begin with the remote lowest cube at the left hand, where the\nasterisk is placed (this proves to be by far the most convenient origin\nto take for the normal system).\n\nThus \u201cnen\u201d is a \u201cnull\u201d cube, \u201cten\u201d a red cube on it, and \u201clen\u201d a \u201cnull\u201d\ncube above \u201cten.\u201d\n\nBy using a more extended sequence of consonants and vowels a larger set\nof cubes can be named.\n\nTo name a four-dimensional block of tesseracts it is simply necessary\nto prefix an \u201ce,\u201d an \u201ca,\u201d or an \u201ci\u201d to the cube names.\n\nThus the tesseract blocks schematically represented on page 165, fig.\n101 are named as follows:\u2014\n\n[Illustration: 1 2 3]\n\n\n2. DERIVATION OF POINT, LINE, FACE, ETC., NAMES.\n\n[Illustration]\n\nThe principle of derivation can be shown as follows: Taking the square\nof squares the number of squares in it can be enlarged and the whole\nkept the same size.\n\n[Illustration]\n\nCompare fig. 79, p. 138, for instance, or the bottom layer of fig. 84.\n\nNow use an initial \u201cs\u201d to denote the result of carrying this process on\nto a great extent, and we obtain the limit names, that is the point,\nline, area names for a square. \u201cSat\u201d is the whole interior. The corners\nare \u201csen,\u201d \u201csel,\u201d \u201csin,\u201d \u201csil,\u201d while the lines are \u201csan,\u201d \u201csal,\u201d\n\u201cset,\u201d \u201csit.\u201d\n\n[Illustration]\n\nI find that by the use of the initial \u201cs\u201d these names come to be\npractically entirely disconnected with the systematic names for the\nsquare from which they are derived. They are easy to learn, and when\nlearned can be used readily with the axes running in any direction.\n\nTo derive the limit names for a four-dimensional rectangular figure,\nlike the tesseract, is a simple extension of this process. These point,\nline, etc., names include those which apply to a cube, as will be\nevident on inspection of the first cube of the diagrams which follow.\n\nAll that is necessary is to place an \u201cs\u201d before each of the names given\nfor a tesseract block. We then obtain apellatives which, like the\ncolour names on page 174, fig. 103, apply to all the points, lines,\nfaces, solids, and to the hyper-solid of the tesseract. These names\nhave the advantage over the colour marks that each point, line, etc.,\nhas its own individual name.\n\nIn the diagrams I give the names corresponding to the positions shown\nin the coloured plate or described on p. 174. By comparing cubes 1, 2,\n3 with the first row of cubes in the coloured plate, the systematic\nnames of each of the points, lines, faces, etc., can be determined. The\nasterisk shows the origin from which the names run.\n\nThese point, line, face, etc., names should be used in connection with\nthe corresponding colours. The names should call up coloured images of\nthe parts named in their right connection.\n\n[Illustration]\n\nIt is found that a certain abbreviation adds vividness of distinction\nto these names. If the final \u201cen\u201d be dropped wherever it occurs the\nsystem is improved. Thus instead of \u201csenen,\u201d \u201cseten,\u201d \u201cselen,\u201d it is\npreferable to abbreviate to \u201csen,\u201d \u201cset,\u201d \u201csel,\u201d and also use \u201csan,\u201d\n\u201csin\u201d for \u201csanen,\u201d \u201csinen.\u201d\n\n[Illustration]\n\n[Illustration]\n\nWe can now name any section. Take _e.g._ the line in the first cube\nfrom senin to senel, we should call the line running from senin to\nsenel, senin senat senel, a line light yellow in colour with null\npoints.\n\n[Illustration]\n\nHere senat is the name for all of the line except its ends. Using\n\u201csenat\u201d in this way does not mean that the line is the whole of senat,\nbut what there is of it is senat. It is a part of the senat region.\nThus also the triangle, which has its three vertices in senin, senel,\nselen, is named thus:\n\n Area: setat.\n Sides: setan, senat, setet.\n Vertices: senin, senel, sel.\n\nThe tetrahedron section of the tesseract can be thought of as a series\nof plane sections in the successive sections of the tesseract shown in\nfig. 114, p. 191. In b_{0} the section is the one written above. In\nb_{1} the section is made by a plane which cuts the three edges from\nsanen intermediate of their lengths and thus will be:\n\n Area: satat.\n Sides: satan, sanat, satet.\n Vertices: sanan, sanet, sat.\n\nThe sections in b_{2}, b_{3} will be like the section in b_{1} but\nsmaller.\n\nFinally in b_{4} the section plane simply passes through the corner\nnamed sin.\n\nHence, putting these sections together in their right relation, from\nthe face setat, surrounded by the lines and points mentioned above,\nthere run:\n\n 3 faces: satan, sanat, satet\n 3 lines: sanan, sanet, sat\n\nand these faces and lines run to the point sin. Thus the tetrahedron is\ncompletely named.\n\nThe octahedron section of the tesseract, which can be traced from fig.\n72, p. 129 by extending the lines there drawn, is named:\n\nFront triangle selin, selat, selel, setal, senil, setit, selin with\narea setat.\n\nThe sections between the front and rear triangle, of which one is shown\nin 1b, another in 2b, are thus named, points and lines, salan, salat,\nsalet, satet, satel, satal, sanal, sanat, sanit, satit, satin, satan,\nsalan.\n\nThe rear triangle found in 3b by producing lines is sil, sitet, sinel,\nsinat, sinin, sitan, sil.\n\nThe assemblage of sections constitute the solid body of the octahedron\nsatat with triangular faces. The one from the line selat to the point\nsil, for instance, is named selin, selat, selel, salet, salat, salan,\nsil. The whole interior is salat.\n\nShapes can easily be cut out of cardboard which, when folded together,\nform not only the tetrahedron and the octahedron, but also samples of\nall the sections of the tesseract taken as it passes cornerwise through\nour space. To name and visualise with appropriate colours a series of\nthese sections is an admirable exercise for obtaining familiarity with\nthe subject.\n\n\n EXTENSION AND CONNECTION WITH NUMBERS.\n\nBy extending the letter sequence it is of course possible to name a\nlarger field. By using the limit names the corners of each square can\nbe named.\n\nThus \u201cen sen,\u201d \u201can sen,\u201d etc., will be the names of the points nearest\nthe origin in \u201cen\u201d and in \u201can.\u201d\n\nA field of points of which each one is indefinitely small is given by\nthe names written below.\n\n[Illustration]\n\nThe squares are shown in dotted lines, the names denote the points.\nThese points are not mathematical points, but really minute areas.\n\nInstead of starting with a set of squares and naming them, we can start\nwith a set of points.\n\nBy an easily remembered convention we can give names to such a region\nof points.\n\nLet the space names with a final \u201ce\u201d added denote the mathematical\npoints at the corner of each square nearest the origin. We have then\nfor the set of mathematical points indicated. This system is really\ncompletely independent of the area system and is connected with it\nmerely for the purpose of facilitating the memory processes. The word\n\u201cene\u201d is pronounced like \u201ceny,\u201d with just sufficient attention to the\nfinal vowel to distinguish it from the word \u201cen.\u201d\n\n[Illustration]\n\nNow, connecting the numbers 0, 1, 2 with the sequence e, a, i, and\nalso with the sequence n, t, l, we have a set of points named as with\nnumbers in a co-ordinate system. Thus \u201cene\u201d is (0, 0) \u201cate\u201d is (1,\n1) \u201cite\u201d is (2, 1). To pass to the area system the rule is that the\nname of the square is formed from the name of its point nearest to the\norigin by dropping the final e.\n\nBy using a notation analogous to the decimal system a larger field of\npoints can be named. It remains to assign a letter sequence to the\nnumbers from positive 0 to positive 9, and from negative 0 to negative\n9, to obtain a system which can be used to denote both the usual\nco-ordinate system of mapping and a system of named squares. The names\ndenoting the points all end with e. Those that denote squares end with\na consonant.\n\nThere are many considerations which must be attended to in extending\nthe sequences to be used, such as uniqueness in the meaning of the\nwords formed, ease of pronunciation, avoidance of awkward combinations.\n\nI drop \u201cs\u201d altogether from the consonant series and short \u201cu\u201d from\nthe vowel series. It is convenient to have unsignificant letters at\ndisposal. A double consonant like \u201cst\u201d for instance can be referred to\nwithout giving it a local significance by calling it \u201cust.\u201d I increase\nthe number of vowels by considering a sound like \u201cra\u201d to be a vowel,\nusing, that is, the letter \u201cr\u201d as forming a compound vowel.\n\nThe series is as follows:\u2014\n\n CONSONANTS.\n\n 0 1 2 3 4 5 6 7 8 9\n positive n t l p f sh k ch nt st\n negative z d th b v m g j nd sp\n\n VOWELS.\n\n 0 1 2 3 4 5 6 7 8 9\n positive e a i ee ae ai ar ra ri ree\n negative er o oo io oe iu or ro roo rio\n\n_Pronunciation._\u2014e as in men; a as in man; i as in in; ee as in\nbetween; ae as ay in may; ai as i in mine; ar as in art; er as ear in\nearth; o as in on; oo as oo in soon; io as in clarion; oe as oa in oat;\niu pronounced like yew.\n\nTo name a point such as (23, 41) it is considered as (3, 1) on from\n(20, 40) and is called \u201cifeete.\u201d It is the initial point of the square\nifeet of the area system.\n\nThe preceding amplification of a space language has been introduced\nmerely for the sake of completeness. As has already been said nine\nwords and their combinations, applied to a few simple models suffice\nfor the purposes of our present enquiry.\n\n\n _Printed by Hazell, Watson & Viney, Ld., London and Aylesbury._","title":"The Fourth Dimension","tradition":"philosophy-ethics","translated_text":null,"word_count":76609}