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Anyone seeking to utilize +this eBook outside of the United States should confirm copyright +status under the laws that apply to them. diff --git a/README.md b/README.md new file mode 100644 index 0000000..8ca7b27 --- /dev/null +++ b/README.md @@ -0,0 +1,2 @@ +Project Gutenberg (https://www.gutenberg.org) public repository for +eBook #67153 (https://www.gutenberg.org/ebooks/67153) diff --git a/old/67153-0.txt b/old/67153-0.txt deleted file mode 100644 index 5bbd70e..0000000 --- a/old/67153-0.txt +++ /dev/null @@ -1,8858 +0,0 @@ -The Project Gutenberg eBook of The Fourth Dimension, by C. Howard -Hinton - -This eBook is for the use of anyone anywhere in the United States and -most other parts of the world at no cost and with almost no restrictions -whatsoever. You may copy it, give it away or re-use it under the terms -of the Project Gutenberg License included with this eBook or online at -www.gutenberg.org. If you are not located in the United States, you -will have to check the laws of the country where you are located before -using this eBook. - -Title: The Fourth Dimension - -Author: C. Howard Hinton - -Release Date: January 12, 2022 [eBook #67153] - -Language: English - -Produced by: Chris Curnow, Les Galloway and the Online Distributed - Proofreading Team at https://www.pgdp.net (This file was - produced from images generously made available by The - Internet Archive) - -*** START OF THE PROJECT GUTENBERG EBOOK THE FOURTH DIMENSION *** - - Transcriber’s Notes - -Obvious typographical errors have been silently corrected. All other -spelling and punctuation remains unchanged. - -Italics are represented thus _italic_, bold thus =bold= and -superscripts thus y^{en}. - -It should be noted that much of the text is a discussion centred on the -many illustrations which have not been included. - - - - - THE FOURTH DIMENSION - - - - - SOME OPINIONS OF THE PRESS - - -“_Mr. C. H. Hinton discusses the subject of the higher dimensionality -of space, his aim being to avoid mathematical subtleties and -technicalities, and thus enable his argument to be followed by readers -who are not sufficiently conversant with mathematics to follow these -processes of reasoning._”—NOTTS GUARDIAN. - -“_The fourth dimension is a subject which has had a great fascination -for many teachers, and though one cannot pretend to have quite grasped -Mr. Hinton’s conceptions and arguments, yet it must be admitted that -he reveals the elusive idea in quite a fascinating light. Quite -apart from the main thesis of the book many chapters are of great -independent interest. Altogether an interesting, clever and ingenious -book._”—DUNDEE COURIER. - -“_The book will well repay the study of men who like to exercise their -wits upon the problems of abstract thought._”—SCOTSMAN. - -“_Professor Hinton has done well to attempt a treatise of moderate -size, which shall at once be clear in method and free from -technicalities of the schools._”—PALL MALL GAZETTE. - -“_A very interesting book he has made of it._”—PUBLISHERS’ CIRCULAR. - -“_Mr. Hinton tries to explain the theory of the fourth dimension so -that the ordinary reasoning mind can get a grasp of what metaphysical -mathematicians mean by it. If he is not altogether successful it is not -from want of clearness on his part, but because the whole theory comes -as such an absolute shock to all one’s preconceived ideas._”—BRISTOL -TIMES. - -“_Mr. Hinton’s enthusiasm is only the result of an exhaustive study, -which has enabled him to set his subject before the reader with far -more than the amount of lucidity to which it is accustomed._”—PALL MALL -GAZETTE. - -“_The book throughout is a very solid piece of reasoning in the domain -of higher mathematics._”—GLASGOW HERALD. - -“_Those who wish to grasp the meaning of this somewhat difficult -subject would do well to read_ The Fourth Dimension. _No mathematical -knowledge is demanded of the reader, and any one, who is not afraid of -a little hard thinking, should be able to follow the argument._”—LIGHT. - -“_A splendidly clear re-statement of the old problem of the fourth -dimension. All who are interested in this subject will find the -work not only fascinating, but lucid, it being written in a style -easily understandable. The illustrations make still more clear -the letterpress, and the whole is most admirably adapted to the -requirements of the novice or the student._”—TWO WORLDS. - -“_Those in search of mental gymnastics will find abundance of exercise -in Mr. C. H. Hinton’s_ Fourth Dimension.”—WESTMINSTER REVIEW. - - - FIRST EDITION, _April 1904_; SECOND EDITION, _May 1906_. - - - Views of the Tessaract. - - No. 1. No. 2. No. 3. - - No. 4. No. 5. No. 6. - - No. 7. No. 8. No. 9. - - No. 10. No. 11. No. 12. - - - - - THE - - FOURTH DIMENSION - - BY - - C. HOWARD HINTON, M.A. - - AUTHOR OF “SCIENTIFIC ROMANCES” - “A NEW ERA OF THOUGHT,” ETC., ETC. - - [Illustration: Colophon] - - - LONDON - SWAN SONNENSCHEIN & CO., LIMITED - 25 HIGH STREET, BLOOMSBURY - - 1906 - - - - - PRINTED BY - HAZELL, WATSON AND VINEY, LD., - LONDON AND AYLESBURY. - - - - - PREFACE - - -I have endeavoured to present the subject of the higher dimensionality -of space in a clear manner, devoid of mathematical subtleties and -technicalities. In order to engage the interest of the reader, I have -in the earlier chapters dwelt on the perspective the hypothesis of a -fourth dimension opens, and have treated of the many connections there -are between this hypothesis and the ordinary topics of our thoughts. - -A lack of mathematical knowledge will prove of no disadvantage to the -reader, for I have used no mathematical processes of reasoning. I have -taken the view that the space which we ordinarily think of, the space -of real things (which I would call permeable matter), is different from -the space treated of by mathematics. Mathematics will tell us a great -deal about space, just as the atomic theory will tell us a great deal -about the chemical combinations of bodies. But after all, a theory is -not precisely equivalent to the subject with regard to which it is -held. There is an opening, therefore, from the side of our ordinary -space perceptions for a simple, altogether rational, mechanical, and -observational way of treating this subject of higher space, and of -this opportunity I have availed myself. - -The details introduced in the earlier chapters, especially in -Chapters VIII., IX., X., may perhaps be found wearisome. They are of -no essential importance in the main line of argument, and if left -till Chapters XI. and XII. have been read, will be found to afford -interesting and obvious illustrations of the properties discussed in -the later chapters. - -My thanks are due to the friends who have assisted me in designing and -preparing the modifications of my previous models, and in no small -degree to the publisher of this volume, Mr. Sonnenschein, to whose -unique appreciation of the line of thought of this, as of my former -essays, their publication is owing. By the provision of a coloured -plate, in addition to the other illustrations, he has added greatly to -the convenience of the reader. - - C. HOWARD HINTON. - - - - - CONTENTS - - - CHAP. PAGE - - I. FOUR-DIMENSIONAL SPACE 1 - - II. THE ANALOGY OF A PLANE WORLD 6 - - III. THE SIGNIFICANCE OF A FOUR-DIMENSIONAL - EXISTENCE 15 - - IV. THE FIRST CHAPTER IN THE HISTORY OF FOUR - SPACE 23 - - V. THE SECOND CHAPTER IN THE HISTORY OF - FOUR SPACE 41 - - Lobatchewsky, Bolyai, and Gauss - Metageometry - - VI. THE HIGHER WORLD 61 - - VII. THE EVIDENCE FOR A FOURTH DIMENSION 76 - - VIII. THE USE OF FOUR DIMENSIONS IN THOUGHT 85 - - IX. APPLICATION TO KANT’S THEORY OF EXPERIENCE 107 - - X. A FOUR-DIMENSIONAL FIGURE 122 - - XI. NOMENCLATURE AND ANALOGIES 136 - - XII. THE SIMPLEST FOUR-DIMENSIONAL SOLID 157 - - XIII. REMARKS ON THE FIGURES 178 - - XIV. A RECAPITULATION AND EXTENSION OF THE - PHYSICAL ARGUMENT 203 - - APPENDIX I.—THE MODELS 231 - - " II.—A LANGUAGE OF SPACE 248 - - - - - THE FOURTH DIMENSION - - - - - CHAPTER I - - FOUR-DIMENSIONAL SPACE - - -There is nothing more indefinite, and at the same time more real, than -that which we indicate when we speak of the “higher.” In our social -life we see it evidenced in a greater complexity of relations. But this -complexity is not all. There is, at the same time, a contact with, an -apprehension of, something more fundamental, more real. - -With the greater development of man there comes a consciousness of -something more than all the forms in which it shows itself. There is -a readiness to give up all the visible and tangible for the sake of -those principles and values of which the visible and tangible are the -representation. The physical life of civilised man and of a mere savage -are practically the same, but the civilised man has discovered a depth -in his existence, which makes him feel that that which appears all to -the savage is a mere externality and appurtenage to his true being. - -Now, this higher—how shall we apprehend it? It is generally embraced -by our religious faculties, by our idealising tendency. But the higher -existence has two sides. It has a being as well as qualities. And in -trying to realise it through our emotions we are always taking the -subjective view. Our attention is always fixed on what we feel, what -we think. Is there any way of apprehending the higher after the purely -objective method of a natural science? I think that there is. - -Plato, in a wonderful allegory, speaks of some men living in such a -condition that they were practically reduced to be the denizens of -a shadow world. They were chained, and perceived but the shadows of -themselves and all real objects projected on a wall, towards which -their faces were turned. All movements to them were but movements -on the surface, all shapes but the shapes of outlines with no -substantiality. - -Plato uses this illustration to portray the relation between true -being and the illusions of the sense world. He says that just as a man -liberated from his chains could learn and discover that the world was -solid and real, and could go back and tell his bound companions of this -greater higher reality, so the philosopher who has been liberated, who -has gone into the thought of the ideal world, into the world of ideas -greater and more real than the things of sense, can come and tell his -fellow men of that which is more true than the visible sun—more noble -than Athens, the visible state. - -Now, I take Plato’s suggestion; but literally, not metaphorically. -He imagines a world which is lower than this world, in that shadow -figures and shadow motions are its constituents; and to it he contrasts -the real world. As the real world is to this shadow world, so is the -higher world to our world. I accept his analogy. As our world in three -dimensions is to a shadow or plane world, so is the higher world to our -three-dimensional world. That is, the higher world is four-dimensional; -the higher being is, so far as its existence is concerned apart from -its qualities, to be sought through the conception of an actual -existence spatially higher than that which we realise with our senses. - -Here you will observe I necessarily leave out all that gives its -charm and interest to Plato’s writings. All those conceptions of the -beautiful and good which live immortally in his pages. - -All that I keep from his great storehouse of wealth is this one thing -simply—a world spatially higher than this world, a world which can only -be approached through the stocks and stones of it, a world which must -be apprehended laboriously, patiently, through the material things of -it, the shapes, the movements, the figures of it. - -We must learn to realise the shapes of objects in this world of the -higher man; we must become familiar with the movements that objects -make in his world, so that we can learn something about his daily -experience, his thoughts of material objects, his machinery. - -The means for the prosecution of this enquiry are given in the -conception of space itself. - -It often happens that that which we consider to be unique and unrelated -gives us, within itself, those relations by means of which we are able -to see it as related to others, determining and determined by them. - -Thus, on the earth is given that phenomenon of weight by means of which -Newton brought the earth into its true relation to the sun and other -planets. Our terrestrial globe was determined in regard to other bodies -of the solar system by means of a relation which subsisted on the earth -itself. - -And so space itself bears within it relations of which we can -determine it as related to other space. For within space are given the -conceptions of point and line, line and plane, which really involve the -relation of space to a higher space. - -Where one segment of a straight line leaves off and another begins is -a point, and the straight line itself can be generated by the motion of -the point. - -One portion of a plane is bounded from another by a straight line, and -the plane itself can be generated by the straight line moving in a -direction not contained in itself. - -Again, two portions of solid space are limited with regard to each -other by a plane; and the plane, moving in a direction not contained in -itself, can generate solid space. - -Thus, going on, we may say that space is that which limits two portions -of higher space from each other, and that our space will generate the -higher space by moving in a direction not contained in itself. - -Another indication of the nature of four-dimensional space can be -gained by considering the problem of the arrangement of objects. - -If I have a number of swords of varying degrees of brightness, I can -represent them in respect of this quality by points arranged along a -straight line. - -If I place a sword at A, fig. 1, and regard it as having a certain -brightness, then the other swords can be arranged in a series along the -line, as at A, B, C, etc., according to their degrees of brightness. - -[Illustration: Fig. 1.] - -If now I take account of another quality, say length, they can be -arranged in a plane. Starting from A, B, C, I can find points to -represent different degrees of length along such lines as AF, BD, CE, -drawn from A and B and C. Points on these lines represent different -degrees of length with the same degree of brightness. Thus the whole -plane is occupied by points representing all conceivable varieties of -brightness and length. - -[Illustration: Fig. 2.] - -Bringing in a third quality, say sharpness, I can draw, as in fig. 3, -any number of upright lines. Let distances along these upright lines -represent degrees of sharpness, thus the points F and G will represent -swords of certain definite degrees of the three qualities mentioned, -and the whole of space will serve to represent all conceivable degrees -of these three qualities. - -[Illustration: Fig. 3.] - -If now I bring in a fourth quality, such as weight, and try to find a -means of representing it as I did the other three qualities, I find -a difficulty. Every point in space is taken up by some conceivable -combination of the three qualities already taken. - -To represent four qualities in the same way as that in which I have -represented three, I should need another dimension of space. - -Thus we may indicate the nature of four-dimensional space by saying -that it is a kind of space which would give positions representative -of four qualities, as three-dimensional space gives positions -representative of three qualities. - - - - - CHAPTER II - - THE ANALOGY OF A PLANE WORLD - - -At the risk of some prolixity I will go fully into the experience of -a hypothetical creature confined to motion on a plane surface. By so -doing I shall obtain an analogy which will serve in our subsequent -enquiries, because the change in our conception, which we make in -passing from the shapes and motions in two dimensions to those in -three, affords a pattern by which we can pass on still further to the -conception of an existence in four-dimensional space. - -A piece of paper on a smooth table affords a ready image of a -two-dimensional existence. If we suppose the being represented by -the piece of paper to have no knowledge of the thickness by which -he projects above the surface of the table, it is obvious that he -can have no knowledge of objects of a similar description, except by -the contact with their edges. His body and the objects in his world -have a thickness of which however, he has no consciousness. Since -the direction stretching up from the table is unknown to him he will -think of the objects of his world as extending in two dimensions only. -Figures are to him completely bounded by their lines, just as solid -objects are to us by their surfaces. He cannot conceive of approaching -the centre of a circle, except by breaking through the circumference, -for the circumference encloses the centre in the directions in which -motion is possible to him. The plane surface over which he slips and -with which he is always in contact will be unknown to him; there are no -differences by which he can recognise its existence. - -But for the purposes of our analogy this representation is deficient. - -A being as thus described has nothing about him to push off from, the -surface over which he slips affords no means by which he can move in -one direction rather than another. Placed on a surface over which he -slips freely, he is in a condition analogous to that in which we should -be if we were suspended free in space. There is nothing which he can -push off from in any direction known to him. - -Let us therefore modify our representation. Let us suppose a vertical -plane against which particles of thin matter slip, never leaving the -surface. Let these particles possess an attractive force and cohere -together into a disk; this disk will represent the globe of a plane -being. He must be conceived as existing on the rim. - -[Illustration: Fig. 4.] - -Let 1 represent this vertical disk of flat matter and 2 the plane being -on it, standing upon its rim as we stand on the surface of our earth. -The direction of the attractive force of his matter will give the -creature a knowledge of up and down, determining for him one direction -in his plane space. Also, since he can move along the surface of his -earth, he will have the sense of a direction parallel to its surface, -which we may call forwards and backwards. - -He will have no sense of right and left—that is, of the direction which -we recognise as extending out from the plane to our right and left. - -The distinction of right and left is the one that we must suppose to -be absent, in order to project ourselves into the condition of a plane -being. - -Let the reader imagine himself, as he looks along the plane, fig. 4, -to become more and more identified with the thin body on it, till he -finally looks along parallel to the surface of the plane earth, and up -and down, losing the sense of the direction which stretches right and -left. This direction will be an unknown dimension to him. - -Our space conceptions are so intimately connected with those which -we derive from the existence of gravitation that it is difficult to -realise the condition of a plane being, without picturing him as in -material surroundings with a definite direction of up and down. Hence -the necessity of our somewhat elaborate scheme of representation, -which, when its import has been grasped, can be dispensed with for the -simpler one of a thin object slipping over a smooth surface, which lies -in front of us. - -It is obvious that we must suppose some means by which the plane being -is kept in contact with the surface on which he slips. The simplest -supposition to make is that there is a transverse gravity, which keeps -him to the plane. This gravity must be thought of as different to the -attraction exercised by his matter, and as unperceived by him. - -At this stage of our enquiry I do not wish to enter into the question -of how a plane being could arrive at a knowledge of the third -dimension, but simply to investigate his plane consciousness. - -It is obvious that the existence of a plane being must be very limited. -A straight line standing up from the surface of his earth affords a bar -to his progress. An object like a wheel which rotates round an axis -would be unknown to him, for there is no conceivable way in which he -can get to the centre without going through the circumference. He would -have spinning disks, but could not get to the centre of them. The plane -being can represent the motion from any one point of his space to any -other, by means of two straight lines drawn at right angles to each -other. - -Let AX and AY be two such axes. He can accomplish the translation from -A to B by going along AX to C, and then from C along CB parallel to AY. - -The same result can of course be obtained by moving to D along AY and -then parallel to AX from D to B, or of course by any diagonal movement -compounded by these axial movements. - -[Illustration: Fig. 5.] - -By means of movements parallel to these two axes he can proceed (except -for material obstacles) from any one point of his space to any other. - -If now we suppose a third line drawn out from A at right angles to the -plane it is evident that no motion in either of the two dimensions he -knows will carry him in the least degree in the direction represented -by AZ. - -[Illustration: Fig. 6.] - -The lines AZ and AX determine a plane. If he could be taken off his -plane, and transferred to the plane AXZ, he would be in a world exactly -like his own. From every line in his world there goes off a space world -exactly like his own. - -[Illustration: Fig. 7.] - -From every point in his world a line can be drawn parallel to AZ in -the direction unknown to him. If we suppose the square in fig. 7 to be -a geometrical square from every point of it, inside as well as on the -contour, a straight line can be drawn parallel to AZ. The assemblage -of these lines constitute a solid figure, of which the square in the -plane is the base. If we consider the square to represent an object -in the plane being’s world then we must attribute to it a very small -thickness, for every real thing must possess all three dimensions. -This thickness he does not perceive, but thinks of this real object as -a geometrical square. He thinks of it as possessing area only, and no -degree of solidity. The edges which project from the plane to a very -small extent he thinks of as having merely length and no breadth—as -being, in fact, geometrical lines. - -With the first step in the apprehension of a third dimension there -would come to a plane being the conviction that he had previously -formed a wrong conception of the nature of his material objects. He -had conceived them as geometrical figures of two dimensions only. If a -third dimension exists, such figures are incapable of real existence. -Thus he would admit that all his real objects had a certain, though -very small thickness in the unknown dimension, and that the conditions -of his existence demanded the supposition of an extended sheet of -matter, from contact with which in their motion his objects never -diverge. - -Analogous conceptions must be formed by us on the supposition of a -four-dimensional existence. We must suppose a direction in which we can -never point extending from every point of our space. We must draw a -distinction between a geometrical cube and a cube of real matter. The -cube of real matter we must suppose to have an extension in an unknown -direction, real, but so small as to be imperceptible by us. From every -point of a cube, interior as well as exterior, we must imagine that it -is possible to draw a line in the unknown direction. The assemblage of -these lines would constitute a higher solid. The lines going off in -the unknown direction from the face of a cube would constitute a cube -starting from that face. Of this cube all that we should see in our -space would be the face. - -Again, just as the plane being can represent any motion in his space by -two axes, so we can represent any motion in our three-dimensional space -by means of three axes. There is no point in our space to which we -cannot move by some combination of movements on the directions marked -out by these axes. - -On the assumption of a fourth dimension we have to suppose a fourth -axis, which we will call AW. It must be supposed to be at right angles -to each and every one of the three axes AX, AY, AZ. Just as the two -axes, AX, AZ, determine a plane which is similar to the original plane -on which we supposed the plane being to exist, but which runs off from -it, and only meets it in a line; so in our space if we take any three -axes such as AX, AY, and AW, they determine a space like our space -world. This space runs off from our space, and if we were transferred -to it we should find ourselves in a space exactly similar to our own. - -We must give up any attempt to picture this space in its relation -to ours, just as a plane being would have to give up any attempt to -picture a plane at right angles to his plane. - -Such a space and ours run in different directions from the plane of AX -and AY. They meet in this plane but have nothing else in common, just -as the plane space of AX and AY and that of AX and AZ run in different -directions and have but the line AX in common. - -Omitting all discussion of the manner on which a plane being might be -conceived to form a theory of a three-dimensional existence, let us -examine how, with the means at his disposal, he could represent the -properties of three-dimensional objects. - -There are two ways in which the plane being can think of one of our -solid bodies. He can think of the cube, fig. 8, as composed of a number -of sections parallel to his plane, each lying in the third dimension -a little further off from his plane than the preceding one. These -sections he can represent as a series of plane figures lying in his -plane, but in so representing them he destroys the coherence of them -in the higher figure. The set of squares, A, B, C, D, represents the -section parallel to the plane of the cube shown in figure, but they are -not in their proper relative positions. - -[Illustration: Fig. 8.] - -The plane being can trace out a movement in the third dimension by -assuming discontinuous leaps from one section to another. Thus, -a motion along the edge of the cube from left to right would be -represented in the set of sections in the plane as the succession of -the corners of the sections A, B, C, D. A point moving from A through -BCD in our space must be represented in the plane as appearing in A, -then in B, and so on, without passing through the intervening plane -space. - -In these sections the plane being leaves out, of course, the extension -in the third dimension; the distance between any two sections is not -represented. In order to realise this distance the conception of motion -can be employed. - -[Illustration: Fig. 9.] - -Let fig. 9 represent a cube passing transverse to the plane. It will -appear to the plane being as a square object, but the matter of which -this object is composed will be continually altering. One material -particle takes the place of another, but it does not come from anywhere -or go anywhere in the space which the plane being knows. - -The analogous manner of representing a higher solid in our case, is to -conceive it as composed of a number of sections, each lying a little -further off in the unknown direction than the preceding. - -[Illustration: Fig. 10.] - -We can represent these sections as a number of solids. Thus the cubes -A, B, C, D, may be considered as the sections at different intervals in -the unknown dimension of a higher cube. Arranged thus their coherence -in the higher figure is destroyed, they are mere representations. - -A motion in the fourth dimension from A through B, C, etc., would be -continuous, but we can only represent it as the occupation of the -positions A, B, C, etc., in succession. We can exhibit the results of -the motion at different stages, but no more. - -In this representation we have left out the distance between one -section and another; we have considered the higher body merely as a -series of sections, and so left out its contents. The only way to -exhibit its contents is to call in the aid of the conception of motion. - -[Illustration: Fig. 11.] - -If a higher cube passes transverse to our space, it will appear as -a cube isolated in space, the part that has not come into our space -and the part that has passed through will not be visible. The gradual -passing through our space would appear as the change of the matter -of the cube before us. One material particle in it is succeeded by -another, neither coming nor going in any direction we can point to. In -this manner, by the duration of the figure, we can exhibit the higher -dimensionality of it; a cube of our matter, under the circumstances -supposed, namely, that it has a motion transverse to our space, would -instantly disappear. A higher cube would last till it had passed -transverse to our space by its whole distance of extension in the -fourth dimension. - -As the plane being can think of the cube as consisting of sections, -each like a figure he knows, extending away from his plane, so we can -think of a higher solid as composed of sections, each like a solid -which we know, but extending away from our space. - -Thus, taking a higher cube, we can look on it as starting from a cube -in our space and extending in the unknown dimension. - -[Illustration: Fig. 12.] - -Take the face A and conceive it to exist as simply a face, a square -with no thickness. From this face the cube in our space extends by the -occupation of space which we can see. - -But from this face there extends equally a cube in the unknown -dimension. We can think of the higher cube, then, by taking the set -of sections A, B, C, D, etc., and considering that from each of them -there runs a cube. These cubes have nothing in common with each other, -and of each of them in its actual position all that we can have in our -space is an isolated square. It is obvious that we can take our series -of sections in any manner we please. We can take them parallel, for -instance, to any one of the three isolated faces shown in the figure. -Corresponding to the three series of sections at right angles to each -other, which we can make of the cube in space, we must conceive of the -higher cube, as composed of cubes starting from squares parallel to the -faces of the cube, and of these cubes all that exist in our space are -the isolated squares from which they start. - - - - - CHAPTER III - - THE SIGNIFICANCE OF A FOUR-DIMENSIONAL EXISTENCE - - -Having now obtained the conception of a four-dimensional space, and -having formed the analogy which, without any further geometrical -difficulties, enables us to enquire into its properties, I will refer -the reader, whose interest is principally in the mechanical aspect, -to Chapters VI. and VII. In the present chapter I will deal with -the general significance of the enquiry, and in the next with the -historical origin of the idea. - -First, with regard to the question of whether there is any evidence -that we are really in four-dimensional space, I will go back to the -analogy of the plane world. - -A being in a plane world could not have any experience of -three-dimensional shapes, but he could have an experience of -three-dimensional movements. - -We have seen that his matter must be supposed to have an extension, -though a very small one, in the third dimension. And thus, in the -small particles of his matter, three-dimensional movements may well -be conceived to take place. Of these movements he would only perceive -the resultants. Since all movements of an observable size in the plane -world are two-dimensional, he would only perceive the resultants in -two dimensions of the small three-dimensional movements. Thus, there -would be phenomena which he could not explain by his theory of -mechanics—motions would take place which he could not explain by his -theory of motion. Hence, to determine if we are in a four-dimensional -world, we must examine the phenomena of motion in our space. If -movements occur which are not explicable on the suppositions of our -three-dimensional mechanics, we should have an indication of a possible -four-dimensional motion, and if, moreover, it could be shown that such -movements would be a consequence of a four-dimensional motion in the -minute particles of bodies or of the ether, we should have a strong -presumption in favour of the reality of the fourth dimension. - -By proceeding in the direction of finer and finer subdivision, we come -to forms of matter possessing properties different to those of the -larger masses. It is probable that at some stage in this process we -should come to a form of matter of such minute subdivision that its -particles possess a freedom of movement in four dimensions. This form -of matter I speak of as four-dimensional ether, and attribute to it -properties approximating to those of a perfect liquid. - -Deferring the detailed discussion of this form of matter to Chapter -VI., we will now examine the means by which a plane being would come to -the conclusion that three-dimensional movements existed in his world, -and point out the analogy by which we can conclude the existence of -four-dimensional movements in our world. Since the dimensions of the -matter in his world are small in the third direction, the phenomena in -which he would detect the motion would be those of the small particles -of matter. - -Suppose that there is a ring in his plane. We can imagine currents -flowing round the ring in either of two opposite directions. These -would produce unlike effects, and give rise to two different fields -of influence. If the ring with a current in it in one direction be -taken up and turned over, and put down again on the plane, it would be -identical with the ring having a current in the opposite direction. An -operation of this kind would be impossible to the plane being. Hence -he would have in his space two irreconcilable objects, namely, the -two fields of influence due to the two rings with currents in them in -opposite directions. By irreconcilable objects in the plane I mean -objects which cannot be thought of as transformed one into the other by -any movement in the plane. - -Instead of currents flowing in the rings we can imagine a different -kind of current. Imagine a number of small rings strung on the original -ring. A current round these secondary rings would give two varieties -of effect, or two different fields of influence, according to its -direction. These two varieties of current could be turned one into -the other by taking one of the rings up, turning it over, and putting -it down again in the plane. This operation is impossible to the plane -being, hence in this case also there would be two irreconcilable fields -in the plane. Now, if the plane being found two such irreconcilable -fields and could prove that they could not be accounted for by currents -in the rings, he would have to admit the existence of currents round -the rings—that is, in rings strung on the primary ring. Thus he would -come to admit the existence of a three-dimensional motion, for such a -disposition of currents is in three dimensions. - -Now in our space there are two fields of different properties, which -can be produced by an electric current flowing in a closed circuit or -ring. These two fields can be changed one into the other by reversing -the currents, but they cannot be changed one into the other by any -turning about of the rings in our space; for the disposition of the -field with regard to the ring itself is different when we turn the -ring, over and when we reverse the direction of the current in the ring. - -As hypotheses to explain the differences of these two fields and their -effects we can suppose the following kinds of space motions:—First, a -current along the conductor; second, a current round the conductor—that -is, of rings of currents strung on the conductor as an axis. Neither of -these suppositions accounts for facts of observation. - -Hence we have to make the supposition of a four-dimensional motion. -We find that a four-dimensional rotation of the nature explained in a -subsequent chapter, has the following characteristics:—First, it would -give us two fields of influence, the one of which could be turned into -the other by taking the circuit up into the fourth dimension, turning -it over, and putting it down in our space again, precisely as the two -kinds of fields in the plane could be turned one into the other by a -reversal of the current in our space. Second, it involves a phenomenon -precisely identical with that most remarkable and mysterious feature of -an electric current, namely that it is a field of action, the rim of -which necessarily abuts on a continuous boundary formed by a conductor. -Hence, on the assumption of a four-dimensional movement in the region -of the minute particles of matter, we should expect to find a motion -analogous to electricity. - -Now, a phenomenon of such universal occurrence as electricity cannot be -due to matter and motion in any very complex relation, but ought to be -seen as a simple and natural consequence of their properties. I infer -that the difficulty in its theory is due to the attempt to explain a -four-dimensional phenomenon by a three-dimensional geometry. - -In view of this piece of evidence we cannot disregard that afforded -by the existence of symmetry. In this connection I will allude to the -simple way of producing the images of insects, sometimes practised by -children. They put a few blots of ink in a straight line on a piece of -paper, fold the paper along the blots, and on opening it the lifelike -presentment of an insect is obtained. If we were to find a multitude -of these figures, we should conclude that they had originated from a -process of folding over; the chances against this kind of reduplication -of parts is too great to admit of the assumption that they had been -formed in any other way. - -The production of the symmetrical forms of organised beings, though not -of course due to a turning over of bodies of any appreciable size in -four-dimensional space, can well be imagined as due to a disposition in -that manner of the smallest living particles from which they are built -up. Thus, not only electricity, but life, and the processes by which we -think and feel, must be attributed to that region of magnitude in which -four-dimensional movements take place. - -I do not mean, however, that life can be explained as a -four-dimensional movement. It seems to me that the whole bias of -thought, which tends to explain the phenomena of life and volition, as -due to matter and motion in some peculiar relation, is adopted rather -in the interests of the explicability of things than with any regard to -probability. - -Of course, if we could show that life were a phenomenon of motion, we -should be able to explain a great deal that is at present obscure. But -there are two great difficulties in the way. It would be necessary to -show that in a germ capable of developing into a living being, there -were modifications of structure capable of determining in the developed -germ all the characteristics of its form, and not only this, but of -determining those of all the descendants of such a form in an infinite -series. Such a complexity of mechanical relations, undeniable though -it be, cannot surely be the best way of grouping the phenomena and -giving a practical account of them. And another difficulty is this, -that no amount of mechanical adaptation would give that element of -consciousness which we possess, and which is shared in to a modified -degree by the animal world. - -In those complex structures which men build up and direct, such as a -ship or a railway train (and which, if seen by an observer of such a -size that the men guiding them were invisible, would seem to present -some of the phenomena of life) the appearance of animation is not due -to any diffusion of life in the material parts of the structure, but to -the presence of a living being. - -The old hypothesis of a soul, a living organism within the visible one, -appears to me much more rational than the attempt to explain life as a -form of motion. And when we consider the region of extreme minuteness -characterised by four-dimensional motion the difficulty of conceiving -such an organism alongside the bodily one disappears. Lord Kelvin -supposes that matter is formed from the ether. We may very well suppose -that the living organisms directing the material ones are co-ordinate -with them, not composed of matter, but consisting of etherial bodies, -and as such capable of motion through the ether, and able to originate -material living bodies throughout the mineral. - -Hypotheses such as these find no immediate ground for proof or disproof -in the physical world. Let us, therefore, turn to a different field, -and, assuming that the human soul is a four-dimensional being, capable -in itself of four dimensional movements, but in its experiences through -the senses limited to three dimensions, ask if the history of thought, -of these productivities which characterise man, correspond to our -assumption. Let us pass in review those steps by which man, presumably -a four-dimensional being, despite his bodily environment, has come to -recognise the fact of four-dimensional existence. - -Deferring this enquiry to another chapter, I will here recapitulate the -argument in order to show that our purpose is entirely practical and -independent of any philosophical or metaphysical considerations. - -If two shots are fired at a target, and the second bullet hits it -at a different place to the first, we suppose that there was some -difference in the conditions under which the second shot was fired -from those affecting the first shot. The force of the powder, the -direction of aim, the strength of the wind, or some condition must -have been different in the second case, if the course of the bullet -was not exactly the same as in the first case. Corresponding to every -difference in a result there must be some difference in the antecedent -material conditions. By tracing out this chain of relations we explain -nature. - -But there is also another mode of explanation which we apply. If we ask -what was the cause that a certain ship was built, or that a certain -structure was erected, we might proceed to investigate the changes in -the brain cells of the men who designed the works. Every variation in -one ship or building from another ship or building is accompanied by -a variation in the processes that go on in the brain matter of the -designers. But practically this would be a very long task. - -A more effective mode of explaining the production of the ship or -building would be to enquire into the motives, plans, and aims of the -men who constructed them. We obtain a cumulative and consistent body of -knowledge much more easily and effectively in the latter way. - -Sometimes we apply the one, sometimes the other mode of explanation. - -But it must be observed that the method of explanation founded on -aim, purpose, volition, always presupposes a mechanical system on -which the volition and aim works. The conception of man as willing and -acting from motives involves that of a number of uniform processes of -nature which he can modify, and of which he can make application. In -the mechanical conditions of the three-dimensional world, the only -volitional agency which we can demonstrate is the human agency. But -when we consider the four-dimensional world the conclusion remains -perfectly open. - -The method of explanation founded on purpose and aim does not, surely, -suddenly begin with man and end with him. There is as much behind the -exhibition of will and motive which we see in man as there is behind -the phenomena of movement; they are co-ordinate, neither to be resolved -into the other. And the commencement of the investigation of that will -and motive which lies behind the will and motive manifested in the -three-dimensional mechanical field is in the conception of a soul—a -four-dimensional organism, which expresses its higher physical being -in the symmetry of the body, and gives the aims and motives of human -existence. - -Our primary task is to form a systematic knowledge of the phenomena -of a four-dimensional world and find those points in which this -knowledge must be called in to complete our mechanical explanation of -the universe. But a subsidiary contribution towards the verification -of the hypothesis may be made by passing in review the history of -human thought, and enquiring if it presents such features as would be -naturally expected on this assumption. - - - - - CHAPTER IV - - THE FIRST CHAPTER IN THE HISTORY OF FOUR SPACE - - -Parmenides, and the Asiatic thinkers with whom he is in close -affinity, propound a theory of existence which is in close accord -with a conception of a possible relation between a higher and a lower -dimensional space. This theory, prior and in marked contrast to the -main stream of thought, which we shall afterwards describe, forms a -closed circle by itself. It is one which in all ages has had a strong -attraction for pure intellect, and is the natural mode of thought for -those who refrain from projecting their own volition into nature under -the guise of causality. - -According to Parmenides of the school of Elea the all is one, unmoving -and unchanging. The permanent amid the transient—that foothold for -thought, that solid ground for feeling on the discovery of which -depends all our life—is no phantom; it is the image amidst deception of -true being, the eternal, the unmoved, the one. Thus says Parmenides. - -But how explain the shifting scene, these mutations of things! - -“Illusion,” answers Parmenides. Distinguishing between truth and -error, he tells of the true doctrine of the one—the false opinion of a -changing world. He is no less memorable for the manner of his advocacy -than for the cause he advocates. It is as if from his firm foothold -of being he could play with the thoughts under the burden of which -others laboured, for from him springs that fluency of supposition and -hypothesis which forms the texture of Plato’s dialectic. - -Can the mind conceive a more delightful intellectual picture than that -of Parmenides, pointing to the one, the true, the unchanging, and yet -on the other hand ready to discuss all manner of false opinion, forming -a cosmogony too, false “but mine own” after the fashion of the time? - -In support of the true opinion he proceeded by the negative way of -showing the self-contradictions in the ideas of change and motion. -It is doubtful if his criticism, save in minor points, has ever been -successfully refuted. To express his doctrine in the ponderous modern -way we must make the statement that motion is phenomenal, not real. - -Let us represent his doctrine. - -[Illustration: Fig. 13.] - -Imagine a sheet of still water into which a slanting stick is being -lowered with a motion vertically downwards. Let 1, 2, 3 (Fig. 13), -be three consecutive positions of the stick. A, B, C, will be three -consecutive positions of the meeting of the stick, with the surface of -the water. As the stick passes down, the meeting will move from A on to -B and C. - -Suppose now all the water to be removed except a film. At the meeting -of the film and the stick there will be an interruption of the film. -If we suppose the film to have a property, like that of a soap bubble, -of closing up round any penetrating object, then as the stick goes -vertically downwards the interruption in the film will move on. - -[Illustration: Fig. 14.] - -If we pass a spiral through the film the intersection will give a point -moving in a circle shown by the dotted lines in the figure. Suppose -now the spiral to be still and the film to move vertically upwards, -the whole spiral will be represented in the film of the consecutive -positions of the point of intersection. In the film the permanent -existence of the spiral is experienced as a time series—the record -of traversing the spiral is a point moving in a circle. If now we -suppose a consciousness connected with the film in such a way that the -intersection of the spiral with the film gives rise to a conscious -experience, we see that we shall have in the film a point moving in a -circle, conscious of its motion, knowing nothing of that real spiral -the record of the successive intersections of which by the film is the -motion of the point. - -It is easy to imagine complicated structures of the nature of the -spiral, structures consisting of filaments, and to suppose also that -these structures are distinguishable from each other at every section. -If we consider the intersections of these filaments with the film as it -passes to be the atoms constituting a filmar universe, we shall have in -the film a world of apparent motion; we shall have bodies corresponding -to the filamentary structure, and the positions of these structures -with regard to one another will give rise to bodies in the film moving -amongst one another. This mutual motion is apparent merely. The reality -is of permanent structures stationary, and all the relative motions -accounted for by one steady movement of the film as a whole. - -Thus we can imagine a plane world, in which all the variety of motion -is the phenomenon of structures consisting of filamentary atoms -traversed by a plane of consciousness. Passing to four dimensions and -our space, we can conceive that all things and movements in our world -are the reading off of a permanent reality by a space of consciousness. -Each atom at every moment is not what it was, but a new part of that -endless line which is itself. And all this system successively revealed -in the time which is but the succession of consciousness, separate -as it is in parts, in its entirety is one vast unity. Representing -Parmenides’ doctrine thus, we gain a firmer hold on it than if we -merely let his words rest, grand and massive, in our minds. And we have -gained the means also of representing phases of that Eastern thought -to which Parmenides was no stranger. Modifying his uncompromising -doctrine, let us suppose, to go back to the plane of consciousness -and the structure of filamentary atoms, that these structures are -themselves moving—are acting, living. Then, in the transverse motion -of the film, there would be two phenomena of motion, one due to the -reading off in the film of the permanent existences as they are in -themselves, and another phenomenon of motion due to the modification of -the record of the things themselves, by their proper motion during the -process of traversing them. - -Thus a conscious being in the plane would have, as it were, a -two-fold experience. In the complete traversing of the structure, the -intersection of which with the film gives his conscious all, the main -and principal movements and actions which he went through would be the -record of his higher self as it existed unmoved and unacting. Slight -modifications and deviations from these movements and actions would -represent the activity and self-determination of the complete being, of -his higher self. - -It is admissible to suppose that the consciousness in the plane has -a share in that volition by which the complete existence determines -itself. Thus the motive and will, the initiative and life, of the -higher being, would be represented in the case of the being in the -film by an initiative and a will capable, not of determining any great -things or important movements in his existence, but only of small and -relatively insignificant activities. In all the main features of his -life his experience would be representative of one state of the higher -being whose existence determines his as the film passes on. But in his -minute and apparently unimportant actions he would share in that will -and determination by which the whole of the being he really is acts and -lives. - -An alteration of the higher being would correspond to a different life -history for him. Let us now make the supposition that film after film -traverses these higher structures, that the life of the real being is -read off again and again in successive waves of consciousness. There -would be a succession of lives in the different advancing planes of -consciousness, each differing from the preceding, and differing in -virtue of that will and activity which in the preceding had not been -devoted to the greater and apparently most significant things in life, -but the minute and apparently unimportant. In all great things the -being of the film shares in the existence of his higher self as it is -at any one time. In the small things he shares in that volition by -which the higher being alters and changes, acts and lives. - -Thus we gain the conception of a life changing and developing as a -whole, a life in which our separation and cessation and fugitiveness -are merely apparent, but which in its events and course alters, -changes, develops; and the power of altering and changing this whole -lies in the will and power the limited being has of directing, guiding, -altering himself in the minute things of his existence. - -Transferring our conceptions to those of an existence in a higher -dimensionality traversed by a space of consciousness, we have an -illustration of a thought which has found frequent and varied -expression. When, however, we ask ourselves what degree of truth -there lies in it, we must admit that, as far as we can see, it is -merely symbolical. The true path in the investigation of a higher -dimensionality lies in another direction. - -The significance of the Parmenidean doctrine lies in this that here, as -again and again, we find that those conceptions which man introduces of -himself, which he does not derive from the mere record of his outward -experience, have a striking and significant correspondence to the -conception of a physical existence in a world of a higher space. How -close we come to Parmenides’ thought by this manner of representation -it is impossible to say. What I want to point out is the adequateness -of the illustration, not only to give a static model of his doctrine, -but one capable as it were, of a plastic modification into a -correspondence into kindred forms of thought. Either one of two things -must be true—that four-dimensional conceptions give a wonderful power -of representing the thought of the East, or that the thinkers of the -East must have been looking at and regarding four-dimensional existence. - -Coming now to the main stream of thought we must dwell in some detail -on Pythagoras, not because of his direct relation to the subject, but -because of his relation to investigators who came later. - -Pythagoras invented the two-way counting. Let us represent the -single-way counting by the posits _aa_, _ab_, _ac_, _ad_, using these -pairs of letters instead of the numbers 1, 2, 3, 4. I put an _a_ in -each case first for a reason which will immediately appear. - -We have a sequence and order. There is no conception of distance -necessarily involved. The difference between the posits is one of -order not of distance—only when identified with a number of equal -material things in juxtaposition does the notion of distance arise. - -Now, besides the simple series I can have, starting from _aa_, _ba_, -_ca_, _da_, from _ab_, _bb_, _cb_, _db_, and so on, and forming a -scheme: - - _da_ _db_ _dc_ _dd_ - _ca_ _cb_ _cc_ _cd_ - _ba_ _bb_ _bc_ _bd_ - _aa_ _ab_ _ac_ _ad_ - -This complex or manifold gives a two-way order. I can represent it by -a set of points, if I am on my guard against assuming any relation of -distance. - -[Illustration: Fig. 15.] - -Pythagoras studied this two-fold way of counting in reference to -material bodies, and discovered that most remarkable property of the -combination of number and matter that bears his name. - -The Pythagorean property of an extended material system can be -exhibited in a manner which will be of use to us afterwards, and which -therefore I will employ now instead of using the kind of figure which -he himself employed. - -Consider a two-fold field of points arranged in regular rows. Such a -field will be presupposed in the following argument. - -[Illustration: Fig. 16. 1 and 2] - -It is evident that in fig. 16 four of the points determine a square, -which square we may take as the unit of measurement for areas. But we -can also measure areas in another way. - -Fig. 16 (1) shows four points determining a square. - -But four squares also meet in a point, fig. 16 (2). - -Hence a point at the corner of a square belongs equally to four -squares. - -Thus we may say that the point value of the square shown is one point, -for if we take the square in fig. 16 (1) it has four points, but each -of these belong equally to four other squares. Hence one fourth of each -of them belongs to the square (1) in fig. 16. Thus the point value of -the square is one point. - -The result of counting the points is the same as that arrived at by -reckoning the square units enclosed. - -Hence, if we wish to measure the area of any square we can take the -number of points it encloses, count these as one each, and take -one-fourth of the number of points at its corners. - -[Illustration: Fig. 17.] - -Now draw a diagonal square as shown in fig. 17. It contains one point -and the four corners count for one point more; hence its point value is -2. The value is the measure of its area—the size of this square is two -of the unit squares. - -Looking now at the sides of this figure we see that there is a unit -square on each of them—the two squares contain no points, but have four -corner points each, which gives the point value of each as one point. - -Hence we see that the square on the diagonal is equal to the squares -on the two sides; or as it is generally expressed, the square on the -hypothenuse is equal to the sum of the squares on the sides. - -[Illustration: Fig. 18.] - -Noticing this fact we can proceed to ask if it is always true. Drawing -the square shown in fig. 18, we can count the number of its points. -There are five altogether. There are four points inside the square on -the diagonal, and hence, with the four points at its corners the point -value is 5—that is, the area is 5. Now the squares on the sides are -respectively of the area 4 and 1. Hence in this case also the square -on the diagonal is equal to the sum of the square on the sides. This -property of matter is one of the first great discoveries of applied -mathematics. We shall prove afterwards that it is not a property of -space. For the present it is enough to remark that the positions in -which the points are arranged is entirely experimental. It is by means -of equal pieces of some material, or the same piece of material moved -from one place to another, that the points are arranged. - -Pythagoras next enquired what the relation must be so that a square -drawn slanting-wise should be equal to one straight-wise. He found that -a square whose side is five can be placed either rectangularly along -the lines of points, or in a slanting position. And this square is -equivalent to two squares of sides 4 and 3. - -Here he came upon a numerical relation embodied in a property of -matter. Numbers immanent in the objects produced the equality so -satisfactory for intellectual apprehension. And he found that numbers -when immanent in sound—when the strings of a musical instrument were -given certain definite proportions of length—were no less captivating -to the ear than the equality of squares was to the reason. What wonder -then that he ascribed an active power to number! - -We must remember that, sharing like ourselves the search for the -permanent in changing phenomena, the Greeks had not that conception of -the permanent in matter that we have. To them material things were not -permanent. In fire solid things would vanish; absolutely disappear. -Rock and earth had a more stable existence, but they too grew and -decayed. The permanence of matter, the conservation of energy, were -unknown to them. And that distinction which we draw so readily between -the fleeting and permanent causes of sensation, between a sound and -a material object, for instance, had not the same meaning to them -which it has for us. Let us but imagine for a moment that material -things are fleeting, disappearing, and we shall enter with a far better -appreciation into that search for the permanent which, with the Greeks, -as with us, is the primary intellectual demand. - -What is that which amid a thousand forms is ever the same, which we can -recognise under all its vicissitudes, of which the diverse phenomena -are the appearances? - -To think that this is number is not so very wide of the mark. With -an intellectual apprehension which far outran the evidences for its -application, the atomists asserted that there were everlasting material -particles, which, by their union, produced all the varying forms and -states of bodies. But in view of the observed facts of nature as -then known, Aristotle, with perfect reason, refused to accept this -hypothesis. - -He expressly states that there is a change of quality, and that the -change due to motion is only one of the possible modes of change. - -With no permanent material world about us, with the fleeting, the -unpermanent, all around we should, I think, be ready to follow -Pythagoras in his identification of number with that principle which -subsists amidst all changes, which in multitudinous forms we apprehend -immanent in the changing and disappearing substance of things. - -And from the numerical idealism of Pythagoras there is but a step to -the more rich and full idealism of Plato. That which is apprehended by -the sense of touch we put as primary and real, and the other senses we -say are merely concerned with appearances. But Plato took them all as -valid, as giving qualities of existence. That the qualities were not -permanent in the world as given to the senses forced him to attribute -to them a different kind of permanence. He formed the conception of a -world of ideas, in which all that really is, all that affects us and -gives the rich and wonderful wealth of our experience, is not fleeting -and transitory, but eternal. And of this real and eternal we see in the -things about us the fleeting and transient images. - -And this world of ideas was no exclusive one, wherein was no place -for the innermost convictions of the soul and its most authoritative -assertions. Therein existed justice, beauty—the one, the good, all -that the soul demanded to be. The world of ideas, Plato’s wonderful -creation preserved for man, for his deliberate investigation and their -sure development, all that the rude incomprehensible changes of a harsh -experience scatters and destroys. - -Plato believed in the reality of ideas. He meets us fairly and -squarely. Divide a line into two parts, he says; one to represent -the real objects in the world, the other to represent the transitory -appearances, such as the image in still water, the glitter of the sun -on a bright surface, the shadows on the clouds. - - A B - ——————————————————————————————|————————————————————————————————- - Real things: Appearances: - _e.g._, the sun. _e.g._, the reflection of the sun. - -Take another line and divide it into two parts, one representing -our ideas, the ordinary occupants of our minds, such as whiteness, -equality, and the other representing our true knowledge, which is of -eternal principles, such as beauty, goodness. - - A^1 B^1 - ——————————————————————————————|————————————————————————————————- - Eternal principles, Appearances in the mind, - as beauty as whiteness, equality - -Then as A is to B, so is A^1 to B^1 - -That is, the soul can proceed, going away from real things to a region -of perfect certainty, where it beholds what is, not the scattered -reflections; beholds the sun, not the glitter on the sands; true being, -not chance opinion. - -Now, this is to us, as it was to Aristotle, absolutely inconceivable -from a scientific point of view. We can understand that a being is -known in the fulness of his relations; it is in his relations to his -circumstances that a man’s character is known; it is in his acts under -his conditions that his character exists. We cannot grasp or conceive -any principle of individuation apart from the fulness of the relations -to the surroundings. - -But suppose now that Plato is talking about the higher man—the -four-dimensional being that is limited in our external experience to a -three-dimensional world. Do not his words begin to have a meaning? Such -a being would have a consciousness of motion which is not as the motion -he can see with the eyes of the body. He, in his own being, knows a -reality to which the outward matter of this too solid earth is flimsy -superficiality. He too knows a mode of being, the fulness of relations, -in which can only be represented in the limited world of sense, as the -painter unsubstantially portrays the depths of woodland, plains, and -air. Thinking of such a being in man, was not Plato’s line well divided? - -It is noteworthy that, if Plato omitted his doctrine of the independent -origin of ideas, he would present exactly the four-dimensional -argument; a real thing as we think it is an idea. A plane being’s idea -of a square object is the idea of an abstraction, namely, a geometrical -square. Similarly our idea of a solid thing is an abstraction, for -in our idea there is not the four-dimensional thickness which is -necessary, however slight, to give reality. The argument would then -run, as a shadow is to a solid object, so is the solid object to the -reality. Thus A and B´ would be identified. - -In the allegory which I have already alluded to, Plato in almost as -many words shows forth the relation between existence in a superficies -and in solid space. And he uses this relation to point to the -conditions of a higher being. - -He imagines a number of men prisoners, chained so that they look at -the wall of a cavern in which they are confined, with their backs to -the road and the light. Over the road pass men and women, figures and -processions, but of all this pageant all that the prisoners behold -is the shadow of it on the wall whereon they gaze. Their own shadows -and the shadows of the things in the world are all that they see, and -identifying themselves with their shadows related as shadows to a world -of shadows, they live in a kind of dream. - -Plato imagines one of their number to pass out from amongst them -into the real space world, and then returning to tell them of their -condition. - -Here he presents most plainly the relation between existence in a plane -world and existence in a three-dimensional world. And he uses this -illustration as a type of the manner in which we are to proceed to a -higher state from the three-dimensional life we know. - -It must have hung upon the weight of a shadow which path he -took!—whether the one we shall follow toward the higher solid and the -four-dimensional existence, or the one which makes ideas the higher -realities, and the direct perception of them the contact with the truer -world. - -Passing on to Aristotle, we will touch on the points which most -immediately concern our enquiry. - -Just as a scientific man of the present day in reviewing the -speculations of the ancient world would treat them with a curiosity -half amused but wholly respectful, asking of each and all wherein lay -their relation to fact, so Aristotle, in discussing the philosophy -of Greece as he found it, asks, above all other things: “Does this -represent the world? In this system is there an adequate presentation -of what is?” - -He finds them all defective, some for the very reasons which we esteem -them most highly, as when he criticises the Atomic theory for its -reduction of all change to motion. But in the lofty march of his reason -he never loses sight of the whole; and that wherein our views differ -from his lies not so much in a superiority of our point of view, as -in the fact which he himself enunciates—that it is impossible for one -principle to be valid in all branches of enquiry. The conceptions -of one method of investigation are not those of another; and our -divergence lies in our exclusive attention to the conceptions useful -in one way of apprehending nature rather than in any possibility we -find in our theories of giving a view of the whole transcending that of -Aristotle. - -He takes account of everything; he does not separate matter and the -manifestation of matter; he fires all together in a conception of a -vast world process in which everything takes part—the motion of a grain -of dust, the unfolding of a leaf, the ordered motion of the spheres in -heaven—all are parts of one whole which he will not separate into dead -matter and adventitious modifications. - -And just as our theories, as representative of actuality, fall before -his unequalled grasp of fact, so the doctrine of ideas fell. It is -not an adequate account of existence, as Plato himself shows in his -“Parmenides”; it only explains things by putting their doubles beside -them. - -For his own part Aristotle invented a great marching definition which, -with a kind of power of its own, cleaves its way through phenomena -to limiting conceptions on either hand, towards whose existence all -experience points. - -In Aristotle’s definition of matter and form as the constituent of -reality, as in Plato’s mystical vision of the kingdom of ideas, the -existence of the higher dimensionality is implicitly involved. - -Substance according to Aristotle is relative, not absolute. In -everything that is there is the matter of which it is composed, the -form which it exhibits; but these are indissolubly connected, and -neither can be thought without the other. - -The blocks of stone out of which a house is built are the material for -the builder; but, as regards the quarrymen, they are the matter of the -rocks with the form he has imposed on them. Words are the final product -of the grammarian, but the mere matter of the orator or poet. The atom -is, with us, that out of which chemical substances are built up, but -looked at from another point of view is the result of complex processes. - -Nowhere do we find finality. The matter in one sphere is the matter, -plus form, of another sphere of thought. Making an obvious application -to geometry, plane figures exist as the limitation of different -portions of the plane by one another. In the bounding lines the -separated matter of the plane shows its determination into form. And -as the plane is the matter relatively to determinations in the plane, -so the plane itself exists in virtue of the determination of space. A -plane is that wherein formless space has form superimposed on it, and -gives an actuality of real relations. We cannot refuse to carry this -process of reasoning a step farther back, and say that space itself is -that which gives form to higher space. As a line is the determination -of a plane, and a plane of a solid, so solid space itself is the -determination of a higher space. - -As a line by itself is inconceivable without that plane which it -separates, so the plane is inconceivable without the solids which -it limits on either hand. And so space itself cannot be positively -defined. It is the negation of the possibility of movement in more than -three dimensions. The conception of space demands that of a higher -space. As a surface is thin and unsubstantial without the substance of -which it is the surface, so matter itself is thin without the higher -matter. - -Just as Aristotle invented that algebraical method of representing -unknown quantities by mere symbols, not by lines necessarily -determinate in length as was the habit of the Greek geometers, and so -struck out the path towards those objectifications of thought which, -like independent machines for reasoning, supply the mathematician -with his analytical weapons, so in the formulation of the doctrine -of matter and form, of potentiality and actuality, of the relativity -of substance, he produced another kind of objectification of mind—a -definition which had a vital force and an activity of its own. - -In none of his writings, as far as we know, did he carry it to its -legitimate conclusion on the side of matter, but in the direction of -the formal qualities he was led to his limiting conception of that -existence of pure form which lies beyond all known determination -of matter. The unmoved mover of all things is Aristotle’s highest -principle. Towards it, to partake of its perfection all things move. -The universe, according to Aristotle, is an active process—he does -not adopt the illogical conception that it was once set in motion -and has kept on ever since. There is room for activity, will, -self-determination, in Aristotle’s system, and for the contingent and -accidental as well. We do not follow him, because we are accustomed to -find in nature infinite series, and do not feel obliged to pass on to a -belief in the ultimate limits to which they seem to point. - -But apart from the pushing to the limit, as a relative principle -this doctrine of Aristotle’s as to the relativity of substance is -irrefragible in its logic. He was the first to show the necessity -of that path of thought which when followed leads to a belief in a -four-dimensional space. - -Antagonistic as he was to Plato in his conception of the practical -relation of reason to the world of phenomena, yet in one point he -coincided with him. And in this he showed the candour of his intellect. -He was more anxious to lose nothing than to explain everything. And -that wherein so many have detected an inconsistency, an inability to -free himself from the school of Plato, appears to us in connection with -our enquiry as an instance of the acuteness of his observation. For -beyond all knowledge given by the senses Aristotle held that there is -an active intelligence, a mind not the passive recipient of impressions -from without, but an active and originative being, capable of grasping -knowledge at first hand. In the active soul Aristotle recognised -something in man not produced by his physical surroundings, something -which creates, whose activity is a knowledge underived from sense. -This, he says, is the immortal and undying being in man. - -Thus we see that Aristotle was not far from the recognition of the -four-dimensional existence, both without and within man, and the -process of adequately realising the higher dimensional figures to which -we shall come subsequently is a simple reduction to practice of his -hypothesis of a soul. - -The next step in the unfolding of the drama of the recognition of -the soul as connected with our scientific conception of the world, -and, at the same time, the recognition of that higher of which a -three-dimensional world presents the superficial appearance, took place -many centuries later. If we pass over the intervening time without a -word it is because the soul was occupied with the assertion of itself -in other ways than that of knowledge. When it took up the task in -earnest of knowing this material world in which it found itself, and of -directing the course of inanimate nature, from that most objective aim -came, reflected back as from a mirror, its knowledge of itself. - - - - - CHAPTER V - - THE SECOND CHAPTER IN THE HISTORY OF FOUR SPACE - - - LOBATCHEWSKY, BOLYAI, AND GAUSS - -Before entering on a description of the work of Lobatchewsky and Bolyai -it will not be out of place to give a brief account of them, the -materials for which are to be found in an article by Franz Schmidt in -the forty-second volume of the _Mathematische Annalen_, and in Engel’s -edition of Lobatchewsky. - -Lobatchewsky was a man of the most complete and wonderful talents. -As a youth he was full of vivacity, carrying his exuberance so far -as to fall into serious trouble for hazing a professor, and other -freaks. Saved by the good offices of the mathematician Bartels, who -appreciated his ability, he managed to restrain himself within the -bounds of prudence. Appointed professor at his own University, Kasan, -he entered on his duties under the regime of a pietistic reactionary, -who surrounded himself with sycophants and hypocrites. Esteeming -probably the interests of his pupils as higher than any attempt at a -vain resistance, he made himself the tyrant’s right-hand man, doing an -incredible amount of teaching and performing the most varied official -duties. Amidst all his activities he found time to make important -contributions to science. His theory of parallels is most closely -connected with his name, but a study of his writings shows that he was -a man capable of carrying on mathematics in its main lines of advance, -and of a judgment equal to discerning what these lines were. Appointed -rector of his University, he died at an advanced age, surrounded by -friends, honoured, with the results of his beneficent activity all -around him. To him no subject came amiss, from the foundations of -geometry to the improvement of the stoves by which the peasants warmed -their houses. - -He was born in 1793. His scientific work was unnoticed till, in 1867, -Houel, the French mathematician, drew attention to its importance. - -Johann Bolyai de Bolyai was born in Klausenburg, a town in -Transylvania, December 15th, 1802. - -His father, Wolfgang Bolyai, a professor in the Reformed College of -Maros Vasarhely, retained the ardour in mathematical studies which had -made him a chosen companion of Gauss in their early student days at -Göttingen. - -He found an eager pupil in Johann. He relates that the boy sprang -before him like a devil. As soon as he had enunciated a problem the -child would give the solution and command him to go on further. As a -thirteen-year-old boy his father sometimes sent him to fill his place -when incapacitated from taking his classes. The pupils listened to him -with more attention than to his father for they found him clearer to -understand. - -In a letter to Gauss Wolfgang Bolyai writes:— - - “My boy is strongly built. He has learned to recognise many - constellations, and the ordinary figures of geometry. He makes apt - applications of his notions, drawing for instance the positions of the - stars with their constellations. Last winter in the country, seeing - Jupiter he asked: ‘How is it that we can see him from here as well as - from the town? He must be far off.’ And as to three different places - to which he had been he asked me to tell him about them in one word. I - did not know what he meant, and then he asked me if one was in a line - with the other and all in a row, or if they were in a triangle. - - “He enjoys cutting paper figures with a pair of scissors, and without - my ever having told him about triangles remarked that a right-angled - triangle which he had cut out was half of an oblong. I exercise his - body with care, he can dig well in the earth with his little hands. - The blossom can fall and no fruit left. When he is fifteen I want to - send him to you to be your pupil.” - -In Johann’s autobiography he says:— - - “My father called my attention to the imperfections and gaps in the - theory of parallels. He told me he had gained more satisfactory - results than his predecessors, but had obtained no perfect and - satisfying conclusion. None of his assumptions had the necessary - degree of geometrical certainty, although they sufficed to prove the - eleventh axiom and appeared acceptable on first sight. - - “He begged of me, anxious not without a reason, to hold myself aloof - and to shun all investigation on this subject, if I did not wish to - live all my life in vain.” - -Johann, in the failure of his father to obtain any response from Gauss, -in answer to a letter in which he asked the great mathematician to make -of his son “an apostle of truth in a far land,” entered the Engineering -School at Vienna. He writes from Temesvar, where he was appointed -sub-lieutenant September, 1823:— - - - “Temesvar, November 3rd, 1823. - - “DEAR GOOD FATHER, - - “I have so overwhelmingly much to write about my discovery that I know - no other way of checking myself than taking a quarter of a sheet only - to write on. I want an answer to my four-sheet letter. - - “I am unbroken in my determination to publish a work on Parallels, as - soon as I have put my material in order and have the means. - - “At present I have not made any discovery, but the way I have followed - almost certainly promises me the attainment of my object if any - possibility of it exists. - - “I have not got my object yet, but I have produced such stupendous - things that I was overwhelmed myself, and it would be an eternal shame - if they were lost. When you see them you will find that it is so. Now - I can only say that I have made a new world out of nothing. Everything - that I have sent you before is a house of cards in comparison with a - tower. I am convinced that it will be no less to my honour than if I - had already discovered it.” - -The discovery of which Johann here speaks was published as an appendix -to Wolfgang Bolyai’s _Tentamen_. - -Sending the book to Gauss, Wolfgang writes, after an interruption of -eighteen years in his correspondence:— - - “My son is first lieutenant of Engineers and will soon be captain. - He is a fine youth, a good violin player, a skilful fencer, and - brave, but has had many duels, and is wild even for a soldier. Yet he - is distinguished—light in darkness and darkness in light. He is an - impassioned mathematician with extraordinary capacities.... He will - think more of your judgment on his work than that of all Europe.” - -Wolfgang received no answer from Gauss to this letter, but sending a -second copy of the book received the following reply:— - - “You have rejoiced me, my unforgotten friend, by your letters. I - delayed answering the first because I wanted to wait for the arrival - of the promised little book. - - “Now something about your son’s work. - - “If I begin with saying that ‘I ought not to praise it,’ you will be - staggered for a moment. But I cannot say anything else. To praise it - is to praise myself, for the path your son has broken in upon and the - results to which he has been led are almost exactly the same as my own - reflections, some of which date from thirty to thirty-five years ago. - - “In fact I am astonished to the uttermost. My intention was to let - nothing be known in my lifetime about my own work, of which, for the - rest, but little is committed to writing. Most people have but little - perception of the problem, and I have found very few who took any - interest in the views I expressed to them. To be able to do that one - must first of all have had a real live feeling of what is wanting, and - as to that most men are completely in the dark. - - “Still it was my intention to commit everything to writing in the - course of time, so that at least it should not perish with me. - - “I am deeply surprised that this task can be spared me, and I am most - of all pleased in this that it is the son of my old friend who has in - so remarkable a manner preceded me.” - -The impression which we receive from Gauss’s inexplicable silence -towards his old friend is swept away by this letter. Hence we breathe -the clear air of the mountain tops. Gauss would not have failed to -perceive the vast significance of his thoughts, sure to be all the -greater in their effect on future ages from the want of comprehension -of the present. Yet there is not a word or a sign in his writing to -claim the thought for himself. He published no single line on the -subject. By the measure of what he thus silently relinquishes, by -such a measure of a world-transforming thought, we can appreciate his -greatness. - -It is a long step from Gauss’s serenity to the disturbed and passionate -life of Johann Bolyai—he and Galois, the two most interesting figures -in the history of mathematics. For Bolyai, the wild soldier, the -duellist, fell at odds with the world. It is related of him that he was -challenged by thirteen officers of his garrison, a thing not unlikely -to happen considering how differently he thought from every one else. -He fought them all in succession—making it his only condition that he -should be allowed to play on his violin for an interval between meeting -each opponent. He disarmed or wounded all his antagonists. It can be -easily imagined that a temperament such as his was one not congenial to -his military superiors. He was retired in 1833. - -His epoch-making discovery awoke no attention. He seems to have -conceived the idea that his father had betrayed him in some -inexplicable way by his communications with Gauss, and he challenged -the excellent Wolfgang to a duel. He passed his life in poverty, many a -time, says his biographer, seeking to snatch himself from dissipation -and apply himself again to mathematics. But his efforts had no result. -He died January 27th, 1860, fallen out with the world and with himself. - - - METAGEOMETRY - -The theories which are generally connected with the names of -Lobatchewsky and Bolyai bear a singular and curious relation to the -subject of higher space. - -In order to show what this relation is, I must ask the reader to be -at the pains to count carefully the sets of points by which I shall -estimate the volumes of certain figures. - -No mathematical processes beyond this simple one of counting will be -necessary. - -[Illustration: Fig. 19.] - -Let us suppose we have before us in fig. 19 a plane covered with points -at regular intervals, so placed that every four determine a square. - -Now it is evident that as four points determine a square, so four -squares meet in a point. - -[Illustration: Fig. 20.] - -Thus, considering a point inside a square as belonging to it, we may -say that a point on the corner of a square belongs to it and to three -others equally: belongs a quarter of it to each square. - -[Illustration: Fig. 21.] - -[Illustration: Fig. 22.] - -Thus the square ACDE (fig. 21) contains one point, and has four points -at the four corners. Since one-fourth of each of these four belongs to -the square, the four together count as one point, and the point value -of the square is two points—the one inside and the four at the corner -make two points belonging to it exclusively. - -Now the area of this square is two unit squares, as can be seen by -drawing two diagonals in fig. 22. - -We also notice that the square in question is equal to the sum of the -squares on the sides AB, BC, of the right-angled triangle ABC. Thus we -recognise the proposition that the square on the hypothenuse is equal -to the sum of the squares on the two sides of a right-angled triangle. - -Now suppose we set ourselves the question of determining the -whereabouts in the ordered system of points, the end of a line would -come when it turned about a point keeping one extremity fixed at the -point. - -We can solve this problem in a particular case. If we can find a square -lying slantwise amongst the dots which is equal to one which goes -regularly, we shall know that the two sides are equal, and that the -slanting side is equal to the straight-way side. Thus the volume and -shape of a figure remaining unchanged will be the test of its having -rotated about the point, so that we can say that its side in its first -position would turn into its side in the second position. - -Now, such a square can be found in the one whose side is five units in -length. - -[Illustration: Fig. 23.] - -In fig. 23, in the square on AB, there are— - - 9 points interior 9 - 4 at the corners 1 - 4 sides with 3 on each side, considered as - 1½ on each side, because belonging - equally to two squares 6 - -The total is 16. There are 9 points in the square on BC. - -In the square on AC there are— - - 24 points inside 24 - 4 at the corners 1 - -or 25 altogether. - -Hence we see again that the square on the hypothenuse is equal to the -squares on the sides. - -Now take the square AFHG, which is larger than the square on AB. It -contains 25 points. - - 16 inside 16 - 16 on the sides, counting as 8 - 4 on the corners 1 - -making 25 altogether. - -If two squares are equal we conclude the sides are equal. Hence, the -line AF turning round A would move so that it would after a certain -turning coincide with AC. - -This is preliminary, but it involves all the mathematical difficulties -that will present themselves. - -There are two alterations of a body by which its volume is not changed. - -One is the one we have just considered, rotation, the other is what is -called shear. - -Consider a book, or heap of loose pages. They can be slid so that each -one slips over the preceding one, and the whole assumes the shape _b_ -in fig. 24. - -[Illustration: Fig. 24.] - -This deformation is not shear alone, but shear accompanied by rotation. - -Shear can be considered as produced in another way. - -Take the square ABCD (fig. 25), and suppose that it is pulled out from -along one of its diagonals both ways, and proportionately compressed -along the other diagonal. It will assume the shape in fig. 26. - -This compression and expansion along two lines at right angles is what -is called shear; it is equivalent to the sliding illustrated above, -combined with a turning round. - -[Illustration: Fig. 25.] [Illustration: Fig. 26.] - -In pure shear a body is compressed and extended in two directions at -right angles to each other, so that its volume remains unchanged. - -Now we know that our material bodies resist shear—shear does violence -to the internal arrangement of their particles, but they turn as wholes -without such internal resistance. - -But there is an exception. In a liquid shear and rotation take place -equally easily, there is no more resistance against a shear than there -is against a rotation. - -Now, suppose all bodies were to be reduced to the liquid state, in -which they yield to shear and to rotation equally easily, and then -were to be reconstructed as solids, but in such a way that shear and -rotation had interchanged places. - -That is to say, let us suppose that when they had become solids again -they would shear without offering any internal resistance, but a -rotation would do violence to their internal arrangement. - -That is, we should have a world in which shear would have taken the -place of rotation. - -A shear does not alter the volume of a body: thus an inhabitant living -in such a world would look on a body sheared as we look on a body -rotated. He would say that it was of the same shape, but had turned a -bit round. - -Let us imagine a Pythagoras in this world going to work to investigate, -as is his wont. - -[Illustration: Fig. 27.] [Illustration: Fig. 28.] - -Fig. 27 represents a square unsheared. Fig. 28 represents a square -sheared. It is not the figure into which the square in fig. 27 would -turn, but the result of shear on some square not drawn. It is a simple -slanting placed figure, taken now as we took a simple slanting placed -square before. Now, since bodies in this world of shear offer no -internal resistance to shearing, and keep their volume when sheared, -an inhabitant accustomed to them would not consider that they altered -their shape under shear. He would call ACDE as much a square as the -square in fig. 27. We will call such figures shear squares. Counting -the dots in ACDE, we find— - - 2 inside = 2 - 4 at corners = 1 - -or a total of 3. - -Now, the square on the side AB has 4 points, that on BC has 1 point. -Here the shear square on the hypothenuse has not 5 points but 3; it is -not the sum of the squares on the sides, but the difference. - -This relation always holds. Look at fig. 29. - -[Illustration: Fig. 29.] - -Shear square on hypothenuse— - - 7 internal 7 - 4 at corners 1 - — - 8 - - -[Illustration: Fig. 29 _bis_.] - -Square on one side—which the reader can draw for himself— - - 4 internal 4 - 8 on sides 4 - 4 at corners 1 - — - 9 - -and the square on the other side is 1. Hence in this case again the -difference is equal to the shear square on the hypothenuse, 9 - 1 = 8. - -Thus in a world of shear the square on the hypothenuse would be equal -to the difference of the squares on the sides of a right-angled -triangle. - -In fig. 29 _bis_ another shear square is drawn on which the above -relation can be tested. - -What now would be the position a line on turning by shear would take up? - -We must settle this in the same way as previously with our turning. - -Since a body sheared remains the same, we must find two equal bodies, -one in the straight way, one in the slanting way, which have the same -volume. Then the side of one will by turning become the side of the -other, for the two figures are each what the other becomes by a shear -turning. - -We can solve the problem in a particular case— - -[Illustration: Fig. 30.] - -In the figure ACDE (fig. 30) there are— - - 15 inside 15 - 4 at corners 1 - -a total of 16. - -Now in the square ABGF, there are 16— - - 9 inside 9 - 12 on sides 6 - 4 at corners 1 - — - 16 - -Hence the square on AB would, by the shear turning, become the shear -square ACDE. - -And hence the inhabitant of this world would say that the line AB -turned into the line AC. These two lines would be to him two lines of -equal length, one turned a little way round from the other. - -That is, putting shear in place of rotation, we get a different kind -of figure, as the result of the shear rotation, from what we got with -our ordinary rotation. And as a consequence we get a position for the -end of a line of invariable length when it turns by the shear rotation, -different from the position which it would assume on turning by our -rotation. - -A real material rod in the shear world would, on turning about A, pass -from the position AB to the position AC. We say that its length alters -when it becomes AC, but this transformation of AB would seem to an -inhabitant of the shear world like a turning of AB without altering in -length. - -If now we suppose a communication of ideas that takes place between -one of ourselves and an inhabitant of the shear world, there would -evidently be a difference between his views of distance and ours. - -We should say that his line AB increased in length in turning to AC. He -would say that our line AF (fig. 23) decreased in length in turning to -AC. He would think that what we called an equal line was in reality a -shorter one. - -We should say that a rod turning round would have its extremities in -the positions we call at equal distances. So would he—but the positions -would be different. He could, like us, appeal to the properties of -matter. His rod to him alters as little as ours does to us. - -Now, is there any standard to which we could appeal, to say which of -the two is right in this argument? There is no standard. - -We should say that, with a change of position, the configuration and -shape of his objects altered. He would say that the configuration and -shape of our objects altered in what we called merely a change of -position. Hence distance independent of position is inconceivable, or -practically distance is solely a property of matter. - -There is no principle to which either party in this controversy could -appeal. There is nothing to connect the definition of distance with our -ideas rather than with his, except the behaviour of an actual piece of -matter. - -For the study of the processes which go on in our world the definition -of distance given by taking the sum of the squares is of paramount -importance to us. But as a question of pure space without making any -unnecessary assumptions the shear world is just as possible and just as -interesting as our world. - -It was the geometry of such conceivable worlds that Lobatchewsky and -Bolyai studied. - -This kind of geometry has evidently nothing to do directly with -four-dimensional space. - -But a connection arises in this way. It is evident that, instead of -taking a simple shear as I have done, and defining it as that change -of the arrangement of the particles of a solid which they will undergo -without offering any resistance due to their mutual action, I might -take a complex motion, composed of a shear and a rotation together, or -some other kind of deformation. - -Let us suppose such an alteration picked out and defined as the one -which means simple rotation, then the type, according to which all -bodies will alter by this rotation, is fixed. - -Looking at the movements of this kind, we should say that the objects -were altering their shape as well as rotating. But to the inhabitants -of that world they would seem to be unaltered, and our figures in their -motions would seem to them to alter. - -In such a world the features of geometry are different. We have seen -one such difference in the case of our illustration of the world of -shear, where the square on the hypothenuse was equal to the difference, -not the sum, of the squares on the sides. - -In our illustration we have the same laws of parallel lines as in our -ordinary rotation world, but in general the laws of parallel lines are -different. - -In one of these worlds of a different constitution of matter through -one point there can be two parallels to a given line, in another of -them there can be none, that is, although a line be drawn parallel to -another it will meet it after a time. - -Now it was precisely in this respect of parallels that Lobatchewsky and -Bolyai discovered these different worlds. They did not think of them as -worlds of matter, but they discovered that space did not necessarily -mean that our law of parallels is true. They made the distinction -between laws of space and laws of matter, although that is not the -form in which they stated their results. - -The way in which they were led to these results was the -following. Euclid had stated the existence of parallel lines as a -postulate—putting frankly this unproved proposition—that one line and -only one parallel to a given straight line can be drawn, as a demand, -as something that must be assumed. The words of his ninth postulate are -these: “If a straight line meeting two other straight lines makes the -interior angles on the same side of it equal to two right angles, the -two straight lines will never meet.” - -The mathematicians of later ages did not like this bald assumption, and -not being able to prove the proposition they called it an axiom—the -eleventh axiom. - -Many attempts were made to prove the axiom; no one doubted of its -truth, but no means could be found to demonstrate it. At last an -Italian, Sacchieri, unable to find a proof, said: “Let us suppose it -not true.” He deduced the results of there being possibly two parallels -to one given line through a given point, but feeling the waters too -deep for the human reason, he devoted the latter half of his book to -disproving what he had assumed in the first part. - -Then Bolyai and Lobatchewsky with firm step entered on the forbidden -path. There can be no greater evidence of the indomitable nature of -the human spirit, or of its manifest destiny to conquer all those -limitations which bind it down within the sphere of sense than this -grand assertion of Bolyai and Lobatchewsky. - - ─────────────────────────── - C D - ─────────────────────────────────── - A B -Take a line AB and a point C. We say and see and know that through C -can only be drawn one line parallel to AB. - -But Bolyai said: “I will draw two.” Let CD be parallel to AB, that -is, not meet AB however far produced, and let lines beyond CD also not -meet AB; let there be a certain region between CD and CE, in which no -line drawn meets AB. CE and CD produced backwards through C will give a -similar region on the other side of C. - -[Illustration: Fig. 32.] - -Nothing so triumphantly, one may almost say so insolently, ignoring -of sense had ever been written before. Men had struggled against the -limitations of the body, fought them, despised them, conquered them. -But no one had ever thought simply as if the body, the bodily eyes, -the organs of vision, all this vast experience of space, had never -existed. The age-long contest of the soul with the body, the struggle -for mastery, had come to a culmination. Bolyai and Lobatchewsky simply -thought as if the body was not. The struggle for dominion, the strife -and combat of the soul were over; they had mastered, and the Hungarian -drew his line. - -Can we point out any connection, as in the case of Parmenides, between -these speculations and higher space? Can we suppose it was any inner -perception by the soul of a motion not known to the senses, which -resulted in this theory so free from the bonds of sense? No such -supposition appears to be possible. - -Practically, however, metageometry had a great influence in bringing -the higher space to the front as a working hypothesis. This can -be traced to the tendency the mind has to move in the direction -of least resistance. The results of the new geometry could not be -neglected, the problem of parallels had occupied a place too prominent -in the development of mathematical thought for its final solution -to be neglected. But this utter independence of all mechanical -considerations, this perfect cutting loose from the familiar -intuitions, was so difficult that almost any other hypothesis was -more easy of acceptance, and when Beltrami showed that the geometry -of Lobatchewsky and Bolyai was the geometry of shortest lines drawn -on certain curved surfaces, the ordinary definitions of measurement -being retained, attention was drawn to the theory of a higher space. -An illustration of Beltrami’s theory is furnished by the simple -consideration of hypothetical beings living on a spherical surface. - -[Illustration: Fig. 33.] - -Let ABCD be the equator of a globe, and AP, BP, meridian lines drawn to -the pole, P. The lines AB, AP, BP would seem to be perfectly straight -to a person moving on the surface of the sphere, and unconscious of its -curvature. Now AP and BP both make right angles with AB. Hence they -satisfy the definition of parallels. Yet they meet in P. Hence a being -living on a spherical surface, and unconscious of its curvature, would -find that parallel lines would meet. He would also find that the angles -in a triangle were greater than two right angles. In the triangle PAB, -for instance, the angles at A and B are right angles, so the three -angles of the triangle PAB are greater than two right angles. - -Now in one of the systems of metageometry (for after Lobatchewsky had -shown the way it was found that other systems were possible besides -his) the angles of a triangle are greater than two right angles. - -Thus a being on a sphere would form conclusions about his space which -are the same as he would form if he lived on a plane, the matter in -which had such properties as are presupposed by one of these systems -of geometry. Beltrami also discovered a certain surface on which -there could be drawn more than one “straight” line through a point -which would not meet another given line. I use the word straight as -equivalent to the line having the property of giving the shortest path -between any two points on it. Hence, without giving up the ordinary -methods of measurement, it was possible to find conditions in which -a plane being would necessarily have an experience corresponding to -Lobatchewsky’s geometry. And by the consideration of a higher space, -and a solid curved in such a higher space, it was possible to account -for a similar experience in a space of three dimensions. - -Now, it is far more easy to conceive of a higher dimensionality to -space than to imagine that a rod in rotating does not move so that -its end describes a circle. Hence, a logical conception having been -found harder than that of a four dimensional space, thought turned -to the latter as a simple explanation of the possibilities to which -Lobatchewsky had awakened it. Thinkers became accustomed to deal with -the geometry of higher space—it was Kant, says Veronese, who first -used the expression of “different spaces”—and with familiarity the -inevitableness of the conception made itself felt. - -From this point it is but a small step to adapt the ordinary mechanical -conceptions to a higher spatial existence, and then the recognition of -its objective existence could be delayed no longer. Here, too, as in so -many cases, it turns out that the order and connection of our ideas is -the order and connection of things. - -What is the significance of Lobatchewsky’s and Bolyai’s work? - -It must be recognised as something totally different from the -conception of a higher space; it is applicable to spaces of any number -of dimensions. By immersing the conception of distance in matter to -which it properly belongs, it promises to be of the greatest aid in -analysis for the effective distance of any two particles is the -product of complex material conditions and cannot be measured by hard -and fast rules. Its ultimate significance is altogether unknown. It -is a cutting loose from the bonds of sense, not coincident with the -recognition of a higher dimensionality, but indirectly contributory -thereto. - -Thus, finally, we have come to accept what Plato held in the hollow -of his hand; what Aristotle’s doctrine of the relativity of substance -implies. The vast universe, too, has its higher, and in recognising it -we find that the directing being within us no longer stands inevitably -outside our systematic knowledge. - - - - - CHAPTER VI - - THE HIGHER WORLD - - -It is indeed strange, the manner in which we must begin to think about -the higher world. - -Those simplest objects analogous to those which are about us on every -side in our daily experience such as a door, a table, a wheel are -remote and uncognisable in the world of four dimensions, while the -abstract ideas of rotation, stress and strain, elasticity into which -analysis resolves the familiar elements of our daily experience are -transferable and applicable with no difficulty whatever. Thus we are -in the unwonted position of being obliged to construct the daily and -habitual experience of a four-dimensional being, from a knowledge of -the abstract theories of the space, the matter, the motion of it; -instead of, as in our case, passing to the abstract theories from the -richness of sensible things. - -What would a wheel be in four dimensions? What the shafting for the -transmission of power which a four-dimensional being would use. - -The four-dimensional wheel, and the four-dimensional shafting are -what will occupy us for these few pages. And it is no futile or -insignificant enquiry. For in the attempt to penetrate into the nature -of the higher, to grasp within our ken that which transcends all -analogies, because what we know are merely partial views of it, the -purely material and physical path affords a means of approach pursuing -which we are in less likelihood of error than if we use the more -frequently trodden path of framing conceptions which in their elevation -and beauty seem to us ideally perfect. - -For where we are concerned with our own thoughts, the development of -our own ideals, we are as it were on a curve, moving at any moment -in a direction of tangency. Whither we go, what we set up and exalt -as perfect, represents not the true trend of the curve, but our own -direction at the present—a tendency conditioned by the past, and by -a vital energy of motion essential but only true when perpetually -modified. That eternal corrector of our aspirations and ideals, the -material universe draws sublimely away from the simplest things we can -touch or handle to the infinite depths of starry space, in one and -all uninfluenced by what we think or feel, presenting unmoved fact -to which, think it good or think it evil, we can but conform, yet -out of all that impassivity with a reference to something beyond our -individual hopes and fears supporting us and giving us our being. - -And to this great being we come with the question: “You, too, what is -your higher?” - -Or to put it in a form which will leave our conclusions in the shape -of no barren formula, and attacking the problem on its most assailable -side: “What is the wheel and the shafting of the four-dimensional -mechanic?” - -In entering on this enquiry we must make a plan of procedure. The -method which I shall adopt is to trace out the steps of reasoning by -which a being confined to movement in a two-dimensional world could -arrive at a conception of our turning and rotation, and then to apply -an analogous process to the consideration of the higher movements. The -plane being must be imagined as no abstract figure, but as a real body -possessing all three dimensions. His limitation to a plane must be the -result of physical conditions. - -We will therefore think of him as of a figure cut out of paper placed -on a smooth plane. Sliding over this plane, and coming into contact -with other figures equally thin as he in the third dimension, he will -apprehend them only by their edges. To him they will be completely -bounded by lines. A “solid” body will be to him a two-dimensional -extent, the interior of which can only be reached by penetrating -through the bounding lines. - -Now such a plane being can think of our three-dimensional existence in -two ways. - -First, he can think of it as a series of sections, each like the solid -he knows of extending in a direction unknown to him, which stretches -transverse to his tangible universe, which lies in a direction at right -angles to every motion which he made. - -Secondly, relinquishing the attempt to think of the three-dimensional -solid body in its entirety he can regard it as consisting of a -number of plane sections, each of them in itself exactly like -the two-dimensional bodies he knows, but extending away from his -two-dimensional space. - -A square lying in his space he regards as a solid bounded by four -lines, each of which lies in his space. - -A square standing at right angles to his plane appears to him as simply -a line in his plane, for all of it except the line stretches in the -third dimension. - -He can think of a three-dimensional body as consisting of a number of -such sections, each of which starts from a line in his space. - -Now, since in his world he can make any drawing or model which involves -only two dimensions, he can represent each such upright section as it -actually is, and can represent a turning from a known into the unknown -dimension as a turning from one to another of his known dimensions. - -To see the whole he must relinquish part of that which he has, and take -the whole portion by portion. - -Consider now a plane being in front of a square, fig. 34. The square -can turn about any point in the plane—say the point A. But it cannot -turn about a line, as AB. For, in order to turn about the line AB, -the square must leave the plane and move in the third dimension. This -motion is out of his range of observation, and is therefore, except for -a process of reasoning, inconceivable to him. - -[Illustration: Fig. 34.] - -Rotation will therefore be to him rotation about a point. Rotation -about a line will be inconceivable to him. - -The result of rotation about a line he can apprehend. He can see the -first and last positions occupied in a half-revolution about the line -AC. The result of such a half revolution is to place the square ABCD -on the left hand instead of on the right hand of the line AC. It would -correspond to a pulling of the whole body ABCD through the line AC, -or to the production of a solid body which was the exact reflection -of it in the line AC. It would be as if the square ABCD turned into -its image, the line AB acting as a mirror. Such a reversal of the -positions of the parts of the square would be impossible in his space. -The occurrence of it would be a proof of the existence of a higher -dimensionality. - -Let him now, adopting the conception of a three-dimensional body as -a series of sections lying, each removed a little farther than the -preceding one, in direction at right angles to his plane, regard a -cube, fig. 36, as a series of sections, each like the square which -forms its base, all rigidly connected together. - -[Illustration: Fig. 35.] - -If now he turns the square about the point A in the plane of _xy_, -each parallel section turns with the square he moves. In each of the -sections there is a point at rest, that vertically over A. Hence he -would conclude that in the turning of a three-dimensional body there -is one line which is at rest. That is a three-dimensional turning in a -turning about a line. - - * * * * * - -In a similar way let us regard ourselves as limited to a -three-dimensional world by a physical condition. Let us imagine that -there is a direction at right angles to every direction in which we can -move, and that we are prevented from passing in this direction by a -vast solid, that against which in every movement we make we slip as the -plane being slips against his plane sheet. - -We can then consider a four-dimensional body as consisting of a series -of sections, each parallel to our space, and each a little farther off -than the preceding on the unknown dimension. - -Take the simplest four-dimensional body—one which begins as a cube, -fig. 36, in our space, and consists of sections, each a cube like fig. -36, lying away from our space. If we turn the cube which is its base in -our space about a line, if, _e.g._, in fig. 36 we turn the cube about -the line AB, not only it but each of the parallel cubes moves about a -line. The cube we see moves about the line AB, the cube beyond it about -a line parallel to AB and so on. Hence the whole four-dimensional body -moves about a plane, for the assemblage of these lines is our way of -thinking about the plane which, starting from the line AB in our space, -runs off in the unknown direction. - -[Illustration: Fig. 36.] - -In this case all that we see of the plane about which the turning takes -place is the line AB. - -But it is obvious that the axis plane may lie in our space. A point -near the plane determines with it a three-dimensional space. When it -begins to rotate round the plane it does not move anywhere in this -three-dimensional space, but moves out of it. A point can no more -rotate round a plane in three-dimensional space than a point can move -round a line in two-dimensional space. - -We will now apply the second of the modes of representation to this -case of turning about a plane, building up our analogy step by step -from the turning in a plane about a point and that in space about a -line, and so on. - -In order to reduce our considerations to those of the greatest -simplicity possible, let us realise how the plane being would think of -the motion by which a square is turned round a line. - -Let, fig. 34, ABCD be a square on his plane, and represent the two -dimensions of his space by the axes A_x_ A_y_. - -Now the motion by which the square is turned over about the line AC -involves the third dimension. - -He cannot represent the motion of the whole square in its turning, -but he can represent the motions of parts of it. Let the third axis -perpendicular to the plane of the paper be called the axis of _z_. Of -the three axes _x_, _y_, _z_, the plane being can represent any two in -his space. Let him then draw, in fig. 35, two axes, _x_ and _z_. Here -he has in his plane a representation of what exists in the plane which -goes off perpendicularly to his space. - -In this representation the square would not be shown, for in the plane -of _xz_ simply the line AB of the square is contained. - -The plane being then would have before him, in fig. 35, the -representation of one line AB of his square and two axes, _x_ and _z_, -at right angles. Now it would be obvious to him that, by a turning -such as he knows, by a rotation about a point, the line AB can turn -round A, and occupying all the intermediate positions, such as AB_{1}, -come after half a revolution to lie as A_x_ produced through A. - -Again, just as he can represent the vertical plane through AB, so he -can represent the vertical plane through A´B´, fig. 34, and in a like -manner can see that the line A´B´ can turn about the point A´ till it -lies in the opposite direction from that which it ran in at first. - -Now these two turnings are not inconsistent. In his plane, if AB -turned about A, and A´B´ about A´, the consistency of the square would -be destroyed, it would be an impossible motion for a rigid body to -perform. But in the turning which he studies portion by portion there -is nothing inconsistent. Each line in the square can turn in this way, -hence he would realise the turning of the whole square as the sum of -a number of turnings of isolated parts. Such turnings, if they took -place in his plane, would be inconsistent, but by virtue of a third -dimension they are consistent, and the result of them all is that the -square turns about the line AC and lies in a position in which it is -the mirror image of what it was in its first position. Thus he can -realise a turning about a line by relinquishing one of his axes, and -representing his body part by part. - -Let us apply this method to the turning of a cube so as to become the -mirror image of itself. In our space we can construct three independent -axes, _x_, _y_, _z_, shown in fig. 36. Suppose that there is a fourth -axis, _w_, at right angles to each and every one of them. We cannot, -keeping all three axes, _x_, _y_, _z_, represent _w_ in our space; but -if we relinquish one of our three axes we can let the fourth axis take -its place, and we can represent what lies in the space, determined by -the two axes we retain and the fourth axis. - -[Illustration: Fig. 37.] - -Let us suppose that we let the _y_ axis drop, and that we represent -the _w_ axis as occupying its direction. We have in fig. 37 a drawing -of what we should then see of the cube. The square ABCD, remains -unchanged, for that is in the plane of _xz_, and we still have that -plane. But from this plane the cube stretches out in the direction of -the _y_ axis. Now the _y_ axis is gone, and so we have no more of the -cube than the face ABCD. Considering now this face ABCD, we see that -it is free to turn about the line AB. It can rotate in the _x_ to _w_ -direction about this line. In fig. 38 it is shown on its way, and it -can evidently continue this rotation till it lies on the other side of -the _z_ axis in the plane of _xz_. - -We can also take a section parallel to the face ABCD, and then letting -drop all of our space except the plane of that section, introduce -the _w_ axis, running in the old _y_ direction. This section can be -represented by the same drawing, fig. 38, and we see that it can rotate -about the line on its left until it swings half way round and runs in -the opposite direction to that which it ran in before. These turnings -of the different sections are not inconsistent, and taken all together -they will bring the cube from the position shown in fig. 36 to that -shown in fig. 41. - -[Illustration: Fig. 38.] - -Since we have three axes at our disposal in our space, we are not -obliged to represent the _w_ axis by any particular one. We may let any -axis we like disappear, and let the fourth axis take its place. - -[Illustration: Fig. 39.] - -[Illustration: Fig. 40.] - -[Illustration: Fig. 41.] - -In fig. 36 suppose the _z_ axis to go. We have then simply the plane of -_xy_ and the square base of the cube ACEG, fig. 39, is all that could -be seen of it. Let now the _w_ axis take the place of the _z_ axis and -we have, in fig. 39 again, a representation of the space of _xyw_, in -which all that exists of the cube is its square base. Now, by a turning -of _x_ to _w_, this base can rotate around the line AE, it is shown -on its way in fig. 40, and finally it will, after half a revolution, -lie on the other side of the _y_ axis. In a similar way we may rotate -sections parallel to the base of the _xw_ rotation, and each of them -comes to run in the opposite direction from that which they occupied at -first. - -Thus again the cube comes from the position of fig. 36. to that of -fig. 41. In this _x_ to _w_ turning, we see that it takes place by -the rotations of sections parallel to the front face about lines -parallel to AB, or else we may consider it as consisting of the -rotation of sections parallel to the base about lines parallel to AE. -It is a rotation of the whole cube about the plane ABEF. Two separate -sections could not rotate about two separate lines in our space without -conflicting, but their motion is consistent when we consider another -dimension. Just, then, as a plane being can think of rotation about -a line as a rotation about a number of points, these rotations not -interfering as they would if they took place in his two-dimensional -space, so we can think of a rotation about a plane as the rotation -of a number of sections of a body about a number of lines in a plane, -these rotations not being inconsistent in a four-dimensional space as -they are in three-dimensional space. - -We are not limited to any particular direction for the lines in the -plane about which we suppose the rotation of the particular sections to -take place. Let us draw the section of the cube, fig. 36, through A, -F, C, H, forming a sloping plane. Now since the fourth dimension is at -right angles to every line in our space it is at right angles to this -section also. We can represent our space by drawing an axis at right -angles to the plane ACEG, our space is then determined by the plane -ACEG, and the perpendicular axis. If we let this axis drop and suppose -the fourth axis, _w_, to take its place, we have a representation of -the space which runs off in the fourth dimension from the plane ACEG. -In this space we shall see simply the section ACEG of the cube, and -nothing else, for one cube does not extend to any distance in the -fourth dimension. - -If, keeping this plane, we bring in the fourth dimension, we shall have -a space in which simply this section of the cube exists and nothing -else. The section can turn about the line AF, and parallel sections can -turn about parallel lines. Thus in considering the rotation about a -plane we can draw any lines we like and consider the rotation as taking -place in sections about them. - -[Illustration: Fig. 42.] - -To bring out this point more clearly let us take two parallel lines, -A and B, in the space of _xyz_, and let CD and EF be two rods running -above and below the plane of _xy_, from these lines. If we turn these -rods in our space about the lines A and B, as the upper end of one, -F, is going down, the lower end of the other, C, will be coming up. -They will meet and conflict. But it is quite possible for these two -rods each of them to turn about the two lines without altering their -relative distances. - -To see this suppose the _y_ axis to go, and let the _w_ axis take its -place. We shall see the lines A and B no longer, for they run in the -_y_ direction from the points G and H. - -[Illustration: Fig. 43.] - -Fig. 43 is a picture of the two rods seen in the space of _xzw_. If -they rotate in the direction shown by the arrows—in the _z_ to _w_ -direction—they move parallel to one another, keeping their relative -distances. Each will rotate about its own line, but their rotation will -not be inconsistent with their forming part of a rigid body. - -Now we have but to suppose a central plane with rods crossing it -at every point, like CD and EF cross the plane of _xy_, to have an -image of a mass of matter extending equal distances on each side of a -diametral plane. As two of these rods can rotate round, so can all, and -the whole mass of matter can rotate round its diametral plane. - -This rotation round a plane corresponds, in four dimensions, to the -rotation round an axis in three dimensions. Rotation of a body round a -plane is the analogue of rotation of a rod round an axis. - -In a plane we have rotation round a point, in three-space rotation -round an axis line, in four-space rotation round an axis plane. - -The four-dimensional being’s shaft by which he transmits power is a -disk rotating round its central plane—the whole contour corresponds -to the ends of an axis of rotation in our space. He can impart the -rotation at any point and take it off at any other point on the -contour, just as rotation round a line can in three-space be imparted -at one end of a rod and taken off at the other end. - -A four-dimensional wheel can easily be described from the analogy of -the representation which a plane being would form for himself of one of -our wheels. - -Suppose a wheel to move transverse to a plane, so that the whole disk, -which I will consider to be solid and without spokes, came at the same -time into contact with the plane. It would appear as a circular portion -of plane matter completely enclosing another and smaller portion—the -axle. - -This appearance would last, supposing the motion of the wheel to -continue until it had traversed the plane by the extent of its -thickness, when there would remain in the plane only the small disk -which is the section of the axle. There would be no means obvious in -the plane at first by which the axle could be reached, except by going -through the substance of the wheel. But the possibility of reaching it -without destroying the substance of the wheel would be shown by the -continued existence of the axle section after that of the wheel had -disappeared. - -In a similar way a four-dimensional wheel moving transverse to our -space would appear first as a solid sphere, completely surrounding -a smaller solid sphere. The outer sphere would represent the wheel, -and would last until the wheel has traversed our space by a distance -equal to its thickness. Then the small sphere alone would remain, -representing the section of the axle. The large sphere could move -round the small one quite freely. Any line in space could be taken as -an axis, and round this line the outer sphere could rotate, while the -inner sphere remained still. But in all these directions of revolution -there would be in reality one line which remained unaltered, that is -the line which stretches away in the fourth direction, forming the -axis of the axle. The four-dimensional wheel can rotate in any number -of planes, but all these planes are such that there is a line at right -angles to them all unaffected by rotation in them. - -An objection is sometimes experienced as to this mode of reasoning from -a plane world to a higher dimensionality. How artificial, it is argued, -this conception of a plane world is. If any real existence confined to -a superficies could be shown to exist, there would be an argument for -one relative to which our three-dimensional existence is superficial. -But, both on the one side and the other of the space we are familiar -with, spaces either with less or more than three dimensions are merely -arbitrary conceptions. - -In reply to this I would remark that a plane being having one less -dimension than our three would have one-third of our possibilities of -motion, while we have only one-fourth less than those of the higher -space. It may very well be that there may be a certain amount of -freedom of motion which is demanded as a condition of an organised -existence, and that no material existence is possible with a more -limited dimensionality than ours. This is well seen if we try to -construct the mechanics of a two-dimensional world. No tube could -exist, for unless joined together completely at one end two parallel -lines would be completely separate. The possibility of an organic -structure, subject to conditions such as this, is highly problematical; -yet, possibly in the convolutions of the brain there may be a mode of -existence to be described as two-dimensional. - -We have but to suppose the increase in surface and the diminution in -mass carried on to a certain extent to find a region which, though -without mobility of the constituents, would have to be described as -two-dimensional. - -But, however artificial the conception of a plane being may be, it is -none the less to be used in passing to the conception of a greater -dimensionality than ours, and hence the validity of the first part of -this objection altogether disappears directly we find evidence for such -a state of being. - -The second part of the objection has more weight. How is it possible -to conceive that in a four-dimensional space any creatures should be -confined to a three-dimensional existence? - -In reply I would say that we know as a matter of fact that life is -essentially a phenomenon of surface. The amplitude of the movements -which we can make is much greater along the surface of the earth than -it is up or down. - -Now we have but to conceive the extent of a solid surface increased, -while the motions possible tranverse to it are diminished in the -same proportion, to obtain the image of a three-dimensional world in -four-dimensional space. - -And as our habitat is the meeting of air and earth on the world, so -we must think of the meeting place of two as affording the condition -for our universe. The meeting of what two? What can that vastness be -in the higher space which stretches in such a perfect level that our -astronomical observations fail to detect the slightest curvature? - -The perfection of the level suggests a liquid—a lake amidst what vast -scenery!—whereon the matter of the universe floats speck-like. - -But this aspect of the problem is like what are called in mathematics -boundary conditions. - -We can trace out all the consequences of four-dimensional movements -down to their last detail. Then, knowing the mode of action which -would be characteristic of the minutest particles, if they were -free, we can draw conclusions from what they actually do of what the -constraint on them is. Of the two things, the material conditions and -the motion, one is known, and the other can be inferred. If the place -of this universe is a meeting of two, there would be a one-sideness -to space. If it lies so that what stretches away in one direction in -the unknown is unlike what stretches away in the other, then, as far -as the movements which participate in that dimension are concerned, -there would be a difference as to which way the motion took place. This -would be shown in the dissimilarity of phenomena, which, so far as -all three-space movements are concerned, were perfectly symmetrical. -To take an instance, merely, for the sake of precising our ideas, -not for any inherent probability in it; if it could be shown that -the electric current in the positive direction were exactly like the -electric current in the negative direction, except for a reversal of -the components of the motion in three-dimensional space, then the -dissimilarity of the discharge from the positive and negative poles -would be an indication of a one-sideness to our space. The only cause -of difference in the two discharges would be due to a component in -the fourth dimension, which directed in one direction transverse to -our space, met with a different resistance to that which it met when -directed in the opposite direction. - - - - - CHAPTER VII - - THE EVIDENCES FOR A FOURTH DIMENSION - - -The method necessarily to be employed in the search for the evidences -of a fourth dimension, consists primarily in the formation of the -conceptions of four-dimensional shapes and motions. When we are in -possession of these it is possible to call in the aid of observation, -without them we may have been all our lives in the familiar presence of -a four-dimensional phenomenon without ever recognising its nature. - -To take one of the conceptions we have already formed, the turning of a -real thing into its mirror image would be an occurrence which it would -be hard to explain, except on the assumption of a fourth dimension. - -We know of no such turning. But there exist a multitude of forms which -show a certain relation to a plane, a relation of symmetry, which -indicates more than an accidental juxtaposition of parts. In organic -life the universal type is of right- and left-handed symmetry, there -is a plane on each side of which the parts correspond. Now we have -seen that in four dimensions a plane takes the place of a line in -three dimensions. In our space, rotation about an axis is the type of -rotation, and the origin of bodies symmetrical about a line as the -earth is symmetrical about an axis can easily be explained. But where -there is symmetry about a plane no simple physical motion, such as we -are accustomed to, suffices to explain it. In our space a symmetrical -object must be built up by equal additions on each side of a central -plane. Such additions about such a plane are as little likely as any -other increments. The probability against the existence of symmetrical -form in inorganic nature is overwhelming in our space, and in organic -forms they would be as difficult of production as any other variety -of configuration. To illustrate this point we may take the child’s -amusement of making from dots of ink on a piece of paper a lifelike -representation of an insect by simply folding the paper over. The -dots spread out on a symmetrical line, and give the impression of a -segmented form with antennæ and legs. - -Now seeing a number of such figures we should naturally infer a folding -over. Can, then, a folding over in four-dimensional space account for -the symmetry of organic forms? The folding cannot of course be of the -bodies we see, but it may be of those minute constituents, the ultimate -elements of living matter which, turned in one way or the other, become -right- or left-handed, and so produce a corresponding structure. - -There is something in life not included in our conceptions of -mechanical movement. Is this something a four-dimensional movement? - -If we look at it from the broadest point of view, there is something -striking in the fact that where life comes in there arises an entirely -different set of phenomena to those of the inorganic world. - -The interest and values of life as we know it in ourselves, as we -know it existing around us in subordinate forms, is entirely and -completely different to anything which inorganic nature shows. And in -living beings we have a kind of form, a disposition of matter which -is entirely different from that shown in inorganic matter. Right- -and left-handed symmetry does not occur in the configurations of dead -matter. We have instances of symmetry about an axis, but not about -a plane. It can be argued that the occurrence of symmetry in two -dimensions involves the existence of a three-dimensional process, as -when a stone falls into water and makes rings of ripples, or as when -a mass of soft material rotates about an axis. It can be argued that -symmetry in any number of dimensions is the evidence of an action in -a higher dimensionality. Thus considering living beings, there is an -evidence both in their structure, and their different mode of activity, -of a something coming in from without into the inorganic world. - -And the objections which will readily occur, such as those derived from -the forms of twin crystals and the theoretical structure of chemical -molecules, do not invalidate the argument; for in these forms too the -presumable seat of the activity producing them lies in that very minute -region in which we necessarily place the seat of a four-dimensional -mobility. - -In another respect also the existence of symmetrical forms is -noteworthy. It is puzzling to conceive how two shapes exactly equal can -exist which are not superposible. Such a pair of symmetrical figures -as the two hands, right and left, show either a limitation in our -power of movement, by which we cannot superpose the one on the other, -or a definite influence and compulsion of space on matter, inflicting -limitations which are additional to those of the proportions of the -parts. - -We will, however, put aside the arguments to be drawn from the -consideration of symmetry as inconclusive, retaining one valuable -indication which they afford. If it is in virtue of a four-dimensional -motion that symmetry exists, it is only in the very minute particles -of bodies that that motion is to be found, for there is no such thing -as a bending over in four dimensions of any object of a size which we -can observe. The region of the extremely minute is the one, then, which -we shall have to investigate. We must look for some phenomenon which, -occasioning movements of the kind we know, still is itself inexplicable -as any form of motion which we know. - -Now in the theories of the actions of the minute particles of bodies -on one another, and in the motions of the ether, mathematicians -have tacitly assumed that the mechanical principles are the same as -those which prevail in the case of bodies which can be observed, it -has been assumed without proof that the conception of motion being -three-dimensional, holds beyond the region from observations in which -it was formed. - -Hence it is not from any phenomenon explained by mathematics that we -can derive a proof of four dimensions. Every phenomenon that has been -explained is explained as three-dimensional. And, moreover, since in -the region of the very minute we do not find rigid bodies acting on -each other at a distance, but elastic substances and continuous fluids -such as ether, we shall have a double task. - -We must form the conceptions of the possible movements of elastic and -liquid four-dimensional matter, before we can begin to observe. Let -us, therefore, take the four-dimensional rotation about a plane, and -enquire what it becomes in the case of extensible fluid substances. If -four-dimensional movements exist, this kind of rotation must exist, and -the finer portions of matter must exhibit it. - -Consider for a moment a rod of flexible and extensible material. It can -turn about an axis, even if not straight; a ring of india rubber can -turn inside out. - -What would this be in the case of four dimensions? - -Let us consider a sphere of our three-dimensional matter having a -definite thickness. To represent this thickness let us suppose that -from every point of the sphere in fig. 44 rods project both ways, in -and out, like D and F. We can only see the external portion, because -the internal parts are hidden by the sphere. - -[Illustration: Fig. 44. - -_Axis of x running towards the observer._] - -In this sphere the axis of _x_ is supposed to come towards the -observer, the axis of _z_ to run up, the axis of _y_ to go to the right. - -[Illustration: Fig. 45.] - -Now take the section determined by the _zy_ plane. This will be a -circle as shown in fig. 45. If we let drop the _x_ axis, this circle -is all we have of the sphere. Letting the _w_ axis now run in the -place of the old _x_ axis we have the space _yzw_, and in this space -all that we have of the sphere is the circle. Fig. 45 then represents -all that there is of the sphere in the space of _yzw_. In this space -it is evident that the rods CD and EF can turn round the circumference -as an axis. If the matter of the spherical shell is sufficiently -extensible to allow the particles C and E to become as widely separated -as they would be in the positions D and F, then the strip of matter -represented by CD and EF and a multitude of rods like them can turn -round the circular circumference. - -Thus this particular section of the sphere can turn inside out, and -what holds for any one section holds for all. Hence in four dimensions -the whole sphere can, if extensible turn inside out. Moreover, any part -of it—a bowl-shaped portion, for instance—can turn inside out, and so -on round and round. - -This is really no more than we had before in the rotation about a -plane, except that we see that the plane can, in the case of extensible -matter, be curved, and still play the part of an axis. - -If we suppose the spherical shell to be of four-dimensional matter, our -representation will be a little different. Let us suppose there to be a -small thickness to the matter in the fourth dimension. This would make -no difference in fig. 44, for that merely shows the view in the _xyz_ -space. But when the _x_ axis is let drop, and the _w_ axis comes in, -then the rods CD and EF which represent the matter of the shell, will -have a certain thickness perpendicular to the plane of the paper on -which they are drawn. If they have a thickness in the fourth dimension -they will show this thickness when looked at from the direction of the -_w_ axis. - -Supposing these rods, then, to be small slabs strung on the -circumference of the circle in fig. 45, we see that there will not -be in this case either any obstacle to their turning round the -circumference. We can have a shell of extensible material or of fluid -material turning inside out in four dimensions. - -And we must remember that in four dimensions there is no such thing as -rotation round an axis. If we want to investigate the motion of fluids -in four dimensions we must take a movement about an axis in our space, -and find the corresponding movement about a plane in four space. - -Now, of all the movements which take place in fluids, the most -important from a physical point of view is vortex motion. - -A vortex is a whirl or eddy—it is shown in the gyrating wreaths of -dust seen on a summer day; it is exhibited on a larger scale in the -destructive march of a cyclone. - -A wheel whirling round will throw off the water on it. But when -this circling motion takes place in a liquid itself it is strangely -persistent. There is, of course, a certain cohesion between the -particles of water by which they mutually impede their motions. But -in a liquid devoid of friction, such that every particle is free from -lateral cohesion on its path of motion, it can be shown that a vortex -or eddy separates from the mass of the fluid a certain portion, which -always remain in that vortex. - -The shape of the vortex may alter, but it always consists of the same -particles of the fluid. - -Now, a very remarkable fact about such a vortex is that the ends of the -vortex cannot remain suspended and isolated in the fluid. They must -always run to the boundary of the fluid. An eddy in water that remains -half way down without coming to the top is impossible. - -The ends of a vortex must reach the boundary of a fluid—the boundary -may be external or internal—a vortex may exist between two objects -in the fluid, terminating one end on each object, the objects being -internal boundaries of the fluid. Again, a vortex may have its ends -linked together, so that it forms a ring. Circular vortex rings of -this description are often seen in puffs of smoke, and that the smoke -travels on in the ring is a proof that the vortex always consists of -the same particles of air. - -Let us now enquire what a vortex would be in a four-dimensional fluid. - -We must replace the line axis by a plane axis. We should have therefore -a portion of fluid rotating round a plane. - -We have seen that the contour of this plane corresponds with the ends -of the axis line. Hence such a four-dimensional vortex must have its -rim on a boundary of the fluid. There would be a region of vorticity -with a contour. If such a rotation were started at one part of a -circular boundary, its edges would run round the boundary in both -directions till the whole interior region was filled with the vortex -sheet. - -A vortex in a three-dimensional liquid may consist of a number of -vortex filaments lying together producing a tube, or rod of vorticity. - -In the same way we can have in four dimensions a number of vortex -sheets alongside each other, each of which can be thought of as a -bowl-shaped portion of a spherical shell turning inside out. The -rotation takes place at any point not in the space occupied by the -shell, but from that space to the fourth dimension and round back again. - -Is there anything analogous to this within the range of our observation? - -An electric current answers this description in every respect. -Electricity does not flow through a wire. Its effect travels both ways -from the starting point along the wire. The spark which shows its -passing midway in its circuit is later than that which occurs at points -near its starting point on either side of it. - -Moreover, it is known that the action of the current is not in the -wire. It is in the region enclosed by the wire, this is the field of -force, the locus of the exhibition of the effects of the current. - -And the necessity of a conducting circuit for a current is exactly -that which we should expect if it were a four-dimensional vortex. -According to Maxwell every current forms a closed circuit, and this, -from the four-dimensional point of view, is the same as saying a vortex -must have its ends on a boundary of the fluid. - -Thus, on the hypothesis of a fourth dimension, the rotation of the -fluid ether would give the phenomenon of an electric current. We must -suppose the ether to be full of movement, for the more we examine into -the conditions which prevail in the obscurity of the minute, the more -we find that an unceasing and perpetual motion reigns. Thus we may say -that the conception of the fourth dimension means that there must be a -phenomenon which presents the characteristics of electricity. - -We know now that light is an electro-magnetic action, and that so far -from being a special and isolated phenomenon this electric action is -universal in the realm of the minute. Hence, may we not conclude that, -so far from the fourth dimension being remote and far away, being a -thing of symbolic import, a term for the explanation of dubious facts -by a more obscure theory, it is really the most important fact within -our knowledge. Our three-dimensional world is superficial. These -processes, which really lie at the basis of all phenomena of matter, -escape our observation by their minuteness, but reveal to our intellect -an amplitude of motion surpassing any that we can see. In such shapes -and motions there is a realm of the utmost intellectual beauty, and one -to which our symbolic methods apply with a better grace than they do to -those of three dimensions. - - - - - CHAPTER VIII - - THE USE OF FOUR DIMENSIONS IN THOUGHT - - -Having held before ourselves this outline of a conjecture of the world -as four-dimensional, having roughly thrown together those facts of -movement which we can see apply to our actual experience, let us pass -to another branch of our subject. - -The engineer uses drawings, graphical constructions, in a variety of -manners. He has, for instance, diagrams which represent the expansion -of steam, the efficiency of his valves. These exist alongside the -actual plans of his machines. They are not the pictures of anything -really existing, but enable him to think about the relations which -exist in his mechanisms. - -And so, besides showing us the actual existence of that world which -lies beneath the one of visible movements, four-dimensional space -enables us to make ideal constructions which serve to represent the -relations of things, and throw what would otherwise be obscure into a -definite and suggestive form. - -From amidst the great variety of instances which lies before me I will -select two, one dealing with a subject of slight intrinsic interest, -which however gives within a limited field a striking example of the -method of drawing conclusions and the use of higher space figures.[1] - - [1] It is suggestive also in another respect, because it shows very - clearly that in our processes of thought there are in play faculties - other than logical; in it the origin of the idea which proves to be - justified is drawn from the consideration of symmetry, a branch of the - beautiful. - -The other instance is chosen on account of the bearing it has on our -fundamental conceptions. In it I try to discover the real meaning of -Kant’s theory of experience. - -The investigation of the properties of numbers is much facilitated -by the fact that relations between numbers are themselves able to be -represented as numbers—_e.g._, 12, and 3 are both numbers, and the -relation between them is 4, another number. The way is thus opened for -a process of constructive theory, without there being any necessity for -a recourse to another class of concepts besides that which is given in -the phenomena to be studied. - -The discipline of number thus created is of great and varied -applicability, but it is not solely as quantitative that we learn to -understand the phenomena of nature. It is not possible to explain the -properties of matter by number simply, but all the activities of matter -are energies in space. They are numerically definite and also, we may -say, directedly definite, _i.e._ definite in direction. - -Is there, then, a body of doctrine about space which, like that of -number, is available in science? It is needless to answer: Yes; -geometry. But there is a method lying alongside the ordinary methods of -geometry, which tacitly used and presenting an analogy to the method of -numerical thought deserves to be brought into greater prominence than -it usually occupies. - -The relation of numbers is a number. - -Can we say in the same way that the relation of shapes is a shape? - -We can. - -To take an instance chosen on account of its ready availability. Let -us take two right-angled triangles of a given hypothenuse, but having -sides of different lengths (fig. 46). These triangles are shapes which -have a certain relation to each other. Let us exhibit their relation as -a figure. - -[Illustration: Fig. 46.] - -Draw two straight lines at right angles to each other, the one HL a -horizontal level, the other VL a vertical level (fig. 47). By means -of these two co-ordinating lines we can represent a double set of -magnitudes; one set as distances to the right of the vertical level, -the other as distances above the horizontal level, a suitable unit -being chosen. - -[Illustration: Fig. 47.] - -Thus the line marked 7 will pick out the assemblage of points whose -distance from the vertical level is 7, and the line marked 1 will pick -out the points whose distance above the horizontal level is 1. The -meeting point of these two lines, 7 and 1, will define a point which -with regard to the one set of magnitudes is 7, with regard to the -other is 1. Let us take the sides of our triangles as the two sets of -magnitudes in question. - -Then the point 7, 1, will represent the triangle whose sides are 7 and -1. Similarly the point 5, 5—5, that is, to the right of the vertical -level and 5 above the horizontal level—will represent the triangle -whose sides are 5 and 5 (fig. 48). - -[Illustration: Fig. 48.] - -Thus we have obtained a figure consisting of the two points 7, 1, and -5, 5, representative of our two triangles. But we can go further, -and, drawing an arc of a circle about O, the meeting point of the -horizontal and vertical levels, which passes through 7, 1, and 5, -5, assert that all the triangles which are right-angled and have a -hypothenuse whose square is 50 are represented by the points on this -arc. - -Thus, each individual of a class being represented by a point, the -whole class is represented by an assemblage of points forming a -figure. Accepting this representation we can attach a definite and -calculable significance to the expression, resemblance, or similarity -between two individuals of the class represented, the difference being -measured by the length of the line between two representative points. -It is needless to multiply examples, or to show how, corresponding to -different classes of triangles, we obtain different curves. - -A representation of this kind in which an object, a thing in space, -is represented as a point, and all its properties are left out, their -effect remaining only in the relative position which the representative -point bears to the representative points of the other objects, may be -called, after the analogy of Sir William R. Hamilton’s hodograph, a -“Poiograph.” - -Representations thus made have the character of natural objects; -they have a determinate and definite character of their own. Any -lack of completeness in them is probably due to a failure in point -of completeness of those observations which form the ground of their -construction. - -Every system of classification is a poiograph. In Mendeléeff’s scheme -of the elements, for instance, each element is represented by a point, -and the relations between the elements are represented by the relations -between the points. - -So far I have simply brought into prominence processes and -considerations with which we are all familiar. But it is worth while -to bring into the full light of our attention our habitual assumptions -and processes. It often happens that we find there are two of them -which have a bearing on each other, which, without this dragging into -the light, we should have allowed to remain without mutual influence. - -There is a fact which it concerns us to take into account in discussing -the theory of the poiograph. - -With respect to our knowledge of the world we are far from that -condition which Laplace imagined when he asserted that an all-knowing -mind could determine the future condition of every object, if he knew -the co-ordinates of its particles in space, and their velocity at any -particular moment. - -On the contrary, in the presence of any natural object, we have a great -complexity of conditions before us, which we cannot reduce to position -in space and date in time. - -There is mass, attraction apparently spontaneous, electrical and -magnetic properties which must be superadded to spatial configuration. -To cut the list short we must say that practically the phenomena of the -world present us problems involving many variables, which we must take -as independent. - -From this it follows that in making poiographs we must be prepared -to use space of more than three dimensions. If the symmetry and -completeness of our representation is to be of use to us we must be -prepared to appreciate and criticise figures of a complexity greater -than of those in three dimensions. It is impossible to give an example -of such a poiograph which will not be merely trivial, without going -into details of some kind irrelevant to our subject. I prefer to -introduce the irrelevant details rather than treat this part of the -subject perfunctorily. - -To take an instance of a poiograph which does not lead us into the -complexities incident on its application in classificatory science, -let us follow Mrs. Alicia Boole Stott in her representation of the -syllogism by its means. She will be interested to find that the curious -gap she detected has a significance. - -A syllogism consists of two statements, the major and the minor -premiss, with the conclusion that can be drawn from them. Thus, to take -an instance, fig. 49. It is evident, from looking at the successive -figures that, if we know that the region M lies altogether within the -region P, and also know that the region S lies altogether within the -region M, we can conclude that the region S lies altogether within -the region P. M is P, major premiss; S is M, minor premiss; S is P, -conclusion. Given the first two data we must conclude that S lies -in P. The conclusion S is P involves two terms, S and P, which are -respectively called the subject and the predicate, the letters S and -P being chosen with reference to the parts the notions they designate -play in the conclusion. S is the subject of the conclusion, P is the -predicate of the conclusion. The major premiss we take to be, that -which does not involve S, and here we always write it first. - -[Illustration: Fig. 49.] - -There are several varieties of statement possessing different degrees -of universality and manners of assertiveness. These different forms of -statement are called the moods. - -We will take the major premiss as one variable, as a thing capable of -different modifications of the same kind, the minor premiss as another, -and the different moods we will consider as defining the variations -which these variables undergo. - -There are four moods:— - - 1. The universal affirmative; all M is P, called mood A. - - 2. The universal negative; no M is P, mood E. - - 3. The particular affirmative; some M is P, mood I. - - 4. The particular negative; some M is not P, mood O. - -[Illustration: 1. 2. 3. 4. Mood A. Mood E. Mood I. Mood O. -Fig. 50.] - -The dotted lines in 3 and 4, fig. 50, denote that it is not known -whether or no any objects exist, corresponding to the space of which -the dotted line forms one delimiting boundary; thus, in mood I we do -not know if there are any M’S which are not P, we only know some M’S -are P. - -[Illustration: Fig. 51.] - -Representing the first premiss in its various moods by regions marked -by vertical lines to the right of PQ, we have in fig. 51, running up -from the four letters AEIO, four columns, each of which indicates that -the major premiss is in the mood denoted by the respective letter. In -the first column to the right of PQ is the mood A. Now above the line -RS let there be marked off four regions corresponding to the four moods -of the minor premiss. Thus, in the first row above RS all the region -between RS and the first horizontal line above it denotes that the -minor premiss is in the mood A. The letters E, I, O, in the same way -show the mood characterising the minor premiss in the rows opposite -these letters. - -We have still to exhibit the conclusion. To do this we must consider -the conclusion as a third variable, characterised in its different -varieties by four moods—this being the syllogistic classification. The -introduction of a third variable involves a change in our system of -representation. - -Before we started with the regions to the right of a certain line as -representing successively the major premiss in its moods; now we must -start with the regions to the right of a certain plane. Let LMNR be -the plane face of a cube, fig. 52, and let the cube be divided into -four parts by vertical sections parallel to LMNR. The variable, the -major premiss, is represented by the successive regions which occur to -the right of the plane LMNR—that region to which A stands opposite, -that slice of the cube, is significative of the mood A. This whole -quarter-part of the cube represents that for every part of it the major -premiss is in the mood A. - -[Illustration: Fig. 52.] - -In a similar manner the next section, the second with the letter E -opposite it, represents that for every one of the sixteen small cubic -spaces in it, the major premiss is in the mood E. The third and fourth -compartments made by the vertical sections denote the major premiss in -the moods I and O. But the cube can be divided in other ways by other -planes. Let the divisions, of which four stretch from the front face, -correspond to the minor premiss. The first wall of sixteen cubes, -facing the observer, has as its characteristic that in each of the -small cubes, whatever else may be the case, the minor premiss is in the -mood A. The variable—the minor premiss—varies through the phases A, E, -I, O, away from the front face of the cube, or the front plane of which -the front face is a part. - -And now we can represent the third variable in a precisely similar way. -We can take the conclusion as the third variable, going through its -four phases from the ground plane upwards. Each of the small cubes at -the base of the whole cube has this true about it, whatever else may -be the case, that the conclusion is, in it, in the mood A. Thus, to -recapitulate, the first wall of sixteen small cubes, the first of the -four walls which, proceeding from left to right, build up the whole -cube, is characterised in each part of it by this, that the major -premiss is in the mood A. - -The next wall denotes that the major premiss is in the mood E, and -so on. Proceeding from the front to the back the first wall presents -a region in every part of which the minor premiss is in the mood A. -The second wall is a region throughout which the minor premiss is in -the mood E, and so on. In the layers, from the bottom upwards, the -conclusion goes through its various moods beginning with A in the -lowest, E in the second, I in the third, O in the fourth. - -In the general case, in which the variables represented in the -poiograph pass through a wide range of values, the planes from which we -measure their degrees of variation in our representation are taken to -be indefinitely extended. In this case, however, all we are concerned -with is the finite region. - -We have now to represent, by some limitation of the complex we have -obtained, the fact that not every combination of premisses justifies -any kind of conclusion. This can be simply effected by marking the -regions in which the premisses, being such as are defined by the -positions, a conclusion which is valid is found. - -Taking the conjunction of the major premiss, all M is P, and the minor, -all S is M, we conclude that all S is P. Hence, that region must be -marked in which we have the conjunction of major premiss in mood A; -minor premiss, mood A; conclusion, mood A. This is the cube occupying -the lowest left-hand corner of the large cube. - -[Illustration: Fig. 53.] - -Proceeding in this way, we find that the regions which must be marked -are those shown in fig. 53. To discuss the case shown in the marked -cube which appears at the top of fig. 53. Here the major premiss is -in the second wall to the right—it is in the mood E and is of the -type no M is P. The minor premiss is in the mood characterised by the -third wall from the front. It is of the type some S is M. From these -premisses we draw the conclusion that some S is not P, a conclusion in -the mood O. Now the mood O of the conclusion is represented in the top -layer. Hence we see that the marking is correct in this respect. - -[Illustration: Fig. 54.] - -It would, of course, be possible to represent the cube on a plane by -means of four squares, as in fig. 54, if we consider each square to -represent merely the beginning of the region it stands for. Thus the -whole cube can be represented by four vertical squares, each standing -for a kind of vertical tray, and the markings would be as shown. In No. -1 the major premiss is in mood A for the whole of the region indicated -by the vertical square of sixteen divisions; in No. 2 it is in the mood -E, and so on. - -A creature confined to a plane would have to adopt some such -disjunctive way of representing the whole cube. He would be obliged to -represent that which we see as a whole in separate parts, and each part -would merely represent, would not be, that solid content which we see. - -The view of these four squares which the plane creature would have -would not be such as ours. He would not see the interior of the four -squares represented above, but each would be entirely contained within -its outline, the internal boundaries of the separate small squares he -could not see except by removing the outer squares. - -We are now ready to introduce the fourth variable involved in the -syllogism. - -In assigning letters to denote the terms of the syllogism we have taken -S and P to represent the subject and predicate in the conclusion, and -thus in the conclusion their order is invariable. But in the premisses -we have taken arbitrarily the order all M is P, and all S is M. There -is no reason why M instead of P should not be the predicate of the -major premiss, and so on. - -Accordingly we take the order of the terms in the premisses as the -fourth variable. Of this order there are four varieties, and these -varieties are called figures. - -Using the order in which the letters are written to denote that the -letter first written is subject, the one written second is predicate, -we have the following possibilities:— - - 1st Figure. 2nd Figure. 3rd Figure. 4th Figure. - Major M P P M M P P M - Minor S M S M M S M S - -There are therefore four possibilities with regard to this fourth -variable as with regard to the premisses. - -We have used up our dimensions of space in representing the phases of -the premisses and the conclusion in respect of mood, and to represent -in an analogous manner the variations in figure we require a fourth -dimension. - -Now in bringing in this fourth dimension we must make a change in our -origins of measurement analogous to that which we made in passing from -the plane to the solid. - -This fourth dimension is supposed to run at right angles to any of the -three space dimensions, as the third space dimension runs at right -angles to the two dimensions of a plane, and thus it gives us the -opportunity of generating a new kind of volume. If the whole cube moves -in this dimension, the solid itself traces out a path, each section of -which, made at right angles to the direction in which it moves, is a -solid, an exact repetition of the cube itself. - -The cube as we see it is the beginning of a solid of such a kind. It -represents a kind of tray, as the square face of the cube is a kind of -tray against which the cube rests. - -Suppose the cube to move in this fourth dimension in four stages, -and let the hyper-solid region traced out in the first stage of its -progress be characterised by this, that the terms of the syllogism -are in the first figure, then we can represent in each of the three -subsequent stages the remaining three figures. Thus the whole cube -forms the basis from which we measure the variation in figure. The -first figure holds good for the cube as we see it, and for that -hyper-solid which lies within the first stage; the second figure holds -good in the second stage, and so on. - -Thus we measure from the whole cube as far as figures are concerned. - -But we saw that when we measured in the cube itself having three -variables, namely, the two premisses and the conclusion, we measured -from three planes. The base from which we measured was in every case -the same. - -Hence, in measuring in this higher space we should have bases of the -same kind to measure from, we should have solid bases. - -The first solid base is easily seen, it is the cube itself. The other -can be found from this consideration. - -That solid from which we measure figure is that in which the remaining -variables run through their full range of varieties. - -Now, if we want to measure in respect of the moods of the major -premiss, we must let the minor premiss, the conclusion, run through -their range, and also the order of the terms. That is we must take as -basis of measurement in respect to the moods of the major that which -represents the variation of the moods of the minor, the conclusion and -the variation of the figures. - -Now the variation of the moods of the minor and of the conclusion are -represented in the square face on the left of the cube. Here are all -varieties of the minor premiss and the conclusion. The varieties of -the figures are represented by stages in a motion proceeding at right -angles to all space directions, at right angles consequently to the -face in question, the left-hand face of the cube. - -Consequently letting the left-hand face move in this direction we get -a cube, and in this cube all the varieties of the minor premiss, the -conclusion, and the figure are represented. - -Thus another cubic base of measurement is given to the cube, generated -by movement of the left-hand square in the fourth dimension. - -We find the other bases in a similar manner, one is the cube generated -by the front square moved in the fourth dimension so as to generate a -cube. From this cube variations in the mood of the minor are measured. -The fourth base is that found by moving the bottom square of the cube -in the fourth dimension. In this cube the variations of the major, -the minor, and the figure are given. Considering this as a basis in -the four stages proceeding from it, the variation in the moods of the -conclusion are given. - -Any one of these cubic bases can be represented in space, and then the -higher solid generated from them lies out of our space. It can only -be represented by a device analogous to that by which the plane being -represents a cube. - -He represents the cube shown above, by taking four square sections and -placing them arbitrarily at convenient distances the one from the other. - -So we must represent this higher solid by four cubes: each cube -represents only the beginning of the corresponding higher volume. - -It is sufficient for us, then, if we draw four cubes, the first -representing that region in which the figure is of the first kind, -the second that region in which the figure is of the second kind, -and so on. These cubes are the beginnings merely of the respective -regions—they are the trays, as it were, against which the real solids -must be conceived as resting, from which they start. The first one, as -it is the beginning of the region of the first figure, is characterised -by the order of the terms in the premisses being that of the first -figure. The second similarly has the terms of the premisses in the -order of the second figure, and so on. - -These cubes are shown below. - -For the sake of showing the properties of the method of representation, -not for the logical problem, I will make a digression. I will represent -in space the moods of the minor and of the conclusion and the different -figures, keeping the major always in mood A. Here we have three -variables in different stages, the minor, the conclusion, and the -figure. Let the square of the left-hand side of the original cube be -imagined to be standing by itself, without the solid part of the cube, -represented by (2) fig. 55. The A, E, I, O, which run away represent -the moods of the minor, the A, E, I, O, which run up represent the -moods of the conclusion. The whole square, since it is the beginning -of the region in the major premiss, mood A, is to be considered as in -major premiss, mood A. - -From this square, let it be supposed that that direction in which the -figures are represented runs to the left hand. Thus we have a cube (1) -running from the square above, in which the square itself is hidden, -but the letters A, E, I, O, of the conclusion are seen. In this cube -we have the minor premiss and the conclusion in all their moods, and -all the figures represented. With regard to the major premiss, since -the face (2) belongs to the first wall from the left in the original -arrangement, and in this arrangement was characterised by the major -premiss in the mood A, we may say that the whole of the cube we now -have put up represents the mood A of the major premiss. - -[Illustration: Fig. 55.] - -Hence the small cube at the bottom to the right in 1, nearest to the -spectator, is major premiss, mood A; minor premiss, mood A; conclusion, -mood A; and figure the first. The cube next to it, running to the left, -is major premiss, mood A; minor premiss, mood A; conclusion, mood A; -figure 2. - -So in this cube we have the representations of all the combinations -which can occur when the major premiss, remaining in the mood A, the -minor premiss, the conclusion, and the figures pass through their -varieties. - -In this case there is no room in space for a natural representation of -the moods of the major premiss. To represent them we must suppose as -before that there is a fourth dimension, and starting from this cube as -base in the fourth direction in four equal stages, all the first volume -corresponds to major premiss A, the second to major premiss, mood E, -the next to the mood I, and the last to mood O. - -The cube we see is as it were merely a tray against which the -four-dimensional figure rests. Its section at any stage is a cube. But -a transition in this direction being transverse to the whole of our -space is represented by no space motion. We can exhibit successive -stages of the result of transference of the cube in that direction, but -cannot exhibit the product of a transference, however small, in that -direction. - -[Illustration: Fig. 56.] - -To return to the original method of representing our variables, -consider fig. 56. These four cubes represent four sections of the -figure derived from the first of them by moving it in the fourth -dimension. The first portion of the motion, which begins with 1, traces -out a more than solid body, which is all in the first figure. The -beginning of this body is shown in 1. The next portion of the motion -traces out a more than solid body, all of which is in the second -figure; the beginning of this body is shown in 2; 3 and 4 follow on in -like manner. Here, then, in one four-dimensional figure we have all -the combinations of the four variables, major premiss, minor premiss, -figure, conclusion, represented, each variable going through its four -varieties. The disconnected cubes drawn are our representation in space -by means of disconnected sections of this higher body. - -Now it is only a limited number of conclusions which are true—their -truth depends on the particular combinations of the premisses and -figures which they accompany. The total figure thus represented may be -called the universe of thought in respect to these four constituents, -and out of the universe of possibly existing combinations it is the -province of logic to select those which correspond to the results of -our reasoning faculties. - -We can go over each of the premisses in each of the moods, and find out -what conclusion logically follows. But this is done in the works on -logic; most simply and clearly I believe in “Jevon’s Logic.” As we are -only concerned with a formal presentation of the results we will make -use of the mnemonic lines printed below, in which the words enclosed in -brackets refer to the figures, and are not significative:— - - Barbara celarent Darii ferio_que_ [prioris]. - Caesare Camestris Festino Baroko [secundae]. - [Tertia] darapti disamis datisi felapton. - Bokardo ferisson _habet_ [Quarta insuper addit]. - Bramantip camenes dimaris ferapton fresison. - -In these lines each significative word has three vowels, the first -vowel refers to the major premiss, and gives the mood of that premiss, -“a” signifying, for instance, that the major mood is in mood _a_. The -second vowel refers to the minor premiss, and gives its mood. The third -vowel refers to the conclusion, and gives its mood. Thus (prioris)—of -the first figure—the first mnemonic word is “barbara,” and this gives -major premiss, mood A; minor premiss, mood A; conclusion, mood A. -Accordingly in the first of our four cubes we mark the lowest left-hand -front cube. To take another instance in the third figure “Tertia,” -the word “ferisson” gives us major premiss mood E—_e.g._, no M is P, -minor premiss mood I; some M is S, conclusion, mood O; some S is not P. -The region to be marked then in the third representative cube is the -one in the second wall to the right for the major premiss, the third -wall from the front for the minor premiss, and the top layer for the -conclusion. - -It is easily seen that in the diagram this cube is marked, and so with -all the valid conclusions. The regions marked in the total region show -which combinations of the four variables, major premiss, minor premiss, -figure, and conclusion exist. - -That is to say, we objectify all possible conclusions, and build up an -ideal manifold, containing all possible combinations of them with the -premisses, and then out of this we eliminate all that do not satisfy -the laws of logic. The residue is the syllogism, considered as a canon -of reasoning. - -Looking at the shape which represents the totality of the valid -conclusions, it does not present any obvious symmetry, or easily -characterisable nature. A striking configuration, however, is -obtained, if we project the four-dimensional figure obtained into a -three-dimensional one; that is, if we take in the base cube all those -cubes which have a marked space anywhere in the series of four regions -which start from that cube. - -This corresponds to making abstraction of the figures, giving all the -conclusions which are valid whatever the figure may be. - -[Illustration: Fig. 57.] - -Proceeding in this way we obtain the arrangement of marked cubes shown -in fig. 57. We see that the valid conclusions are arranged almost -symmetrically round one cube—the one on the top of the column starting -from AAA. There is one breach of continuity however in this scheme. -One cube is unmarked, which if marked would give symmetry. It is the -one which would be denoted by the letters I, E, O, in the third -wall to the right, the second wall away, the topmost layer. Now this -combination of premisses in the mood IE, with a conclusion in the mood -O, is not noticed in any book on logic with which I am familiar. Let -us look at it for ourselves, as it seems that there must be something -curious in connection with this break of continuity in the poiograph. - -[Illustration: Fig. 58.] - -The propositions I, E, in the various figures are the following, as -shown in the accompanying scheme, fig. 58:—First figure: some M is P; -no S is M. Second figure: some P is M; no S is M. Third figure: some M -is P; no M is S. Fourth figure: some P is M; no M is S. - -Examining these figures, we see, taking the first, that if some M is P -and no S is M, we have no conclusion of the form S is P in the various -moods. It is quite indeterminate how the circle representing S lies -with regard to the circle representing P. It may lie inside, outside, -or partly inside P. The same is true in the other figures 2 and 3. -But when we come to the fourth figure, since M and S lie completely -outside each other, there cannot lie inside S that part of P which lies -inside M. Now we know by the major premiss that some of P does lie in -M. Hence S cannot contain the whole of P. In words, some P is M, no -M is S, therefore S does not contain the whole of P. If we take P as -the subject, this gives us a conclusion in the mood O about P. Some -P is not S. But it does not give us conclusion about S in any one of -the four forms recognised in the syllogism and called its moods. Hence -the breach of the continuity in the poiograph has enabled us to detect -a lack of completeness in the relations which are considered in the -syllogism. - -To take an instance:—Some Americans (P) are of African stock (M); No -Aryans (S) are of African stock (M); Aryans (S) do not include all of -Americans (P). - -In order to draw a conclusion about S we have to admit the statement, -“S does not contain the whole of P,” as a valid logical form—it is a -statement about S which can be made. The logic which gives us the form, -“some P is not S,” and which does not allow us to give the exactly -equivalent and equally primary form, “S does not contain the whole of -P,” is artificial. - -And I wish to point out that this artificiality leads to an error. - -If one trusted to the mnemonic lines given above, one would conclude -that no logical conclusion about S can be drawn from the statement, -“some P are M, no M are S.” - -But a conclusion can be drawn: S does not contain the whole of P. - -It is not that the result is given expressed in another form. The -mnemonic lines deny that any conclusion can be drawn from premisses in -the moods I, E, respectively. - -Thus a simple four-dimensional poiograph has enabled us to detect a -mistake in the mnemonic lines which have been handed down unchallenged -from mediæval times. To discuss the subject of these lines more fully a -logician defending them would probably say that a particular statement -cannot be a major premiss; and so deny the existence of the fourth -figure in the combination of moods. - -To take our instance: some Americans are of African stock; no Aryans -are of African stock. He would say that the conclusion is some -Americans are not Aryans; and that the second statement is the major. -He would refuse to say anything about Aryans, condemning us to an -eternal silence about them, as far as these premisses are concerned! -But, if there is a statement involving the relation of two classes, it -must be expressible as a statement about either of them. - -To bar the conclusion, “Aryans do not include the whole of Americans,” -is purely a makeshift in favour of a false classification. - -And the argument drawn from the universality of the major premiss -cannot be consistently maintained. It would preclude such combinations -as major O, minor A, conclusion O—_i.e._, such as some mountains (M) -are not permanent (P); all mountains (M) are scenery (S); some scenery -(S) is not permanent (P). - -This is allowed in “Jevon’s Logic,” and his omission to discuss I, E, -O, in the fourth figure, is inexplicable. A satisfactory poiograph -of the logical scheme can be made by admitting the use of the words -some, none, or all, about the predicate as well as about the subject. -Then we can express the statement, “Aryans do not include the whole of -Americans,” clumsily, but, when its obscurity is fathomed, correctly, -as “Some Aryans are not all Americans.” And this method is what is -called the “quantification of the predicate.” - -The laws of formal logic are coincident with the conclusions which -can be drawn about regions of space, which overlap one another in the -various possible ways. It is not difficult so to state the relations -or to obtain a symmetrical poiograph. But to enter into this branch -of geometry is beside our present purpose, which is to show the -application of the poiograph in a finite and limited region, without -any of those complexities which attend its use in regard to natural -objects. - -If we take the latter—plants, for instance—and, without assuming -fixed directions in space as representative of definite variations, -arrange the representative points in such a manner as to correspond to -the similarities of the objects, we obtain configuration of singular -interest; and perhaps in this way, in the making of shapes of shapes, -bodies with bodies omitted, some insight into the structure of the -species and genera might be obtained. - - - - - CHAPTER IX - - APPLICATION TO KANT’S THEORY OF EXPERIENCE - - -When we observe the heavenly bodies we become aware that they all -participate in one universal motion—a diurnal revolution round the -polar axis. - -In the case of fixed stars this is most unqualifiedly true, but in the -case of the sun, and the planets also, the single motion of revolution -can be discerned, modified, and slightly altered by other and secondary -motions. - -Hence the universal characteristic of the celestial bodies is that they -move in a diurnal circle. - -But we know that this one great fact which is true of them all has in -reality nothing to do with them. The diurnal revolution which they -visibly perform is the result of the condition of the observer. It is -because the observer is on a rotating earth that a universal statement -can be made about all the celestial bodies. - -The universal statement which is valid about every one of the celestial -bodies is that which does not concern them at all, and is but a -statement of the condition of the observer. - -Now there are universal statements of other kinds which we can make. We -can say that all objects of experience are in space and subject to the -laws of geometry. - -Does this mean that space and all that it means is due to a condition -of the observer? - -If a universal law in one case means nothing affecting the objects -themselves, but only a condition of observation, is this true in every -case? There is shown us in astronomy a _vera causa_ for the assertion -of a universal. Is the same cause to be traced everywhere? - -Such is a first approximation to the doctrine of Kant’s critique. - -It is the apprehension of a relation into which, on the one side and -the other, perfectly definite constituents enter—the human observer and -the stars—and a transference of this relation to a region in which the -constituents on either side are perfectly unknown. - -If spatiality is due to a condition of the observer, the observer -cannot be this bodily self of ours—the body, like the objects around -it, are equally in space. - -This conception Kant applied, not only to the intuitions of sense, but -to the concepts of reason—wherever a universal statement is made there -is afforded him an opportunity for the application of his principle. -He constructed a system in which one hardly knows which the most to -admire, the architectonic skill, or the reticence with regard to things -in themselves, and the observer in himself. - -His system can be compared to a garden, somewhat formal perhaps, but -with the charm of a quality more than intellectual, a _besonnenheit_, -an exquisite moderation over all. And from the ground he so carefully -prepared with that buried in obscurity, which it is fitting should be -obscure, science blossoms and the tree of real knowledge grows. - -The critique is a storehouse of ideas of profound interest. The one -of which I have given a partial statement leads, as we shall see -on studying it in detail, to a theory of mathematics suggestive of -enquiries in many directions. - -The justification for my treatment will be found amongst other passages -in that part of the transcendental analytic, in which Kant speaks of -objects of experience subject to the forms of sensibility, not subject -to the concepts of reason. - -Kant asserts that whenever we think we think of objects in space and -time, but he denies that the space and time exist as independent -entities. He goes about to explain them, and their universality, not by -assuming them, as most other philosophers do, but by postulating their -absence. How then does it come to pass that the world is in space and -time to us? - -Kant takes the same position with regard to what we call nature—a great -system subject to law and order. “How do you explain the law and order -in nature?” we ask the philosophers. All except Kant reply by assuming -law and order somewhere, and then showing how we can recognise it. - -In explaining our notions, philosophers from other than the Kantian -standpoint, assume the notions as existing outside us, and then it is -no difficult task to show how they come to us, either by inspiration or -by observation. - -We ask “Why do we have an idea of law in nature?” “Because natural -processes go according to law,” we are answered, “and experience -inherited or acquired, gives us this notion.” - -But when we speak about the law in nature we are speaking about a -notion of our own. So all that these expositors do is to explain our -notion by an assumption of it. - -Kant is very different. He supposes nothing. An experience such as ours -is very different from experience in the abstract. Imagine just simply -experience, succession of states, of consciousness! Why, there would be -no connecting any two together, there would be no personal identity, -no memory. It is out of a general experience such as this, which, in -respect to anything we call real, is less than a dream, that Kant shows -the genesis of an experience such as ours. - -Kant takes up the problem of the explanation of space, time, order, and -so quite logically does not presuppose them. - -But how, when every act of thought is of things in space, and time, -and ordered, shall we represent to ourselves that perfectly indefinite -somewhat which is Kant’s necessary hypothesis—that which is not in -space or time and is not ordered. That is our problem, to represent -that which Kant assumes not subject to any of our forms of thought, and -then show some function which working on that makes it into a “nature” -subject to law and order, in space and time. Such a function Kant -calls the “Unity of Apperception”; _i.e._, that which makes our state -of consciousness capable of being woven into a system with a self, an -outer world, memory, law, cause, and order. - -The difficulty that meets us in discussing Kant’s hypothesis is that -everything we think of is in space and time—how then shall we represent -in space an existence not in space, and in time an existence not in -time? This difficulty is still more evident when we come to construct -a poiograph, for a poiograph is essentially a space structure. But -because more evident the difficulty is nearer a solution. If we always -think in space, _i.e._ using space concepts, the first condition -requisite for adapting them to the representation of non-spatial -existence, is to be aware of the limitation of our thought, and so be -able to take the proper steps to overcome it. The problem before us, -then, is to represent in space an existence not in space. - -The solution is an easy one. It is provided by the conception of -alternativity. - -To get our ideas clear let us go right back behind the distinctions of -an inner and an outer world. Both of these, Kant says, are products. -Let us take merely states of consciousness, and not ask the question -whether they are produced or superinduced—to ask such a question is to -have got too far on, to have assumed something of which we have not -traced the origin. Of these states let us simply say that they occur. -Let us now use the word a “posit” for a phase of consciousness reduced -to its last possible stage of evanescence; let a posit be that phase of -consciousness of which all that can be said is that it occurs. - -Let _a_, _b_, _c_, be three such posits. We cannot represent them in -space without placing them in a certain order, as _a_, _b_, _c_. But -Kant distinguishes between the forms of sensibility and the concepts -of reason. A dream in which everything happens at haphazard would be -an experience subject to the form of sensibility and only partially -subject to the concepts of reason. It is partially subject to the -concepts of reason because, although there is no order of sequence, -still at any given time there is order. Perception of a thing as in -space is a form of sensibility, the perception of an order is a concept -of reason. - -We must, therefore, in order to get at that process which Kant supposes -to be constitutive of an ordered experience imagine the posits as in -space without order. - -As we know them they must be in some order, _abc_, _bca_, _cab_, _acb_, -_cba_, _bac_, one or another. - -To represent them as having no order conceive all these different -orders as equally existing. Introduce the conception of -alternativity—let us suppose that the order _abc_, and _bac_, for -example, exist equally, so that we cannot say about _a_ that it comes -before or after _b_. This would correspond to a sudden and arbitrary -change of _a_ into _b_ and _b_ into _a_, so that, to use Kant’s words, -it would be possible to call one thing by one name at one time and at -another time by another name. - -In an experience of this kind we have a kind of chaos, in which no -order exists; it is a manifold not subject to the concepts of reason. - -Now is there any process by which order can be introduced into such a -manifold—is there any function of consciousness in virtue of which an -ordered experience could arise? - -In the precise condition in which the posits are, as described above, -it does not seem to be possible. But if we imagine a duality to exist -in the manifold, a function of consciousness can be easily discovered -which will produce order out of no order. - -Let us imagine each posit, then, as having, a dual aspect. Let _a_ be -1_a_ in which the dual aspect is represented by the combination of -symbols. And similarly let _b_ be 2_b_, _c_ be 3_c_, in which 2 and _b_ -represent the dual aspects of _b_, 3 and _c_ those of _c_. - -Since _a_ can arbitrarily change into _b_, or into _c_, and so on, the -particular combinations written above cannot be kept. We have to assume -the equally possible occurrence of form such as 2_a_, 2_b_, and so on; -and in order to get a representation of all those combinations out of -which any set is alternatively possible, we must take every aspect with -every aspect. We must, that is, have every letter with every number. - -Let us now apply the method of space representation. - - _Note._—At the beginning of the next chapter the same structures as - those which follow are exhibited in more detail and a reference to - them will remove any obscurity which may be found in the immediately - following passages. They are there carried on to a greater - multiplicity of dimensions, and the significance of the process here - briefly explained becomes more apparent. - -[Illustration: Fig. 59.] - -Take three mutually rectangular axes in space 1, 2, 3 (fig. 59), and -on each mark three points, the common meeting point being the first on -each axis. Then by means of these three points on each axis we define -27 positions, 27 points in a cubical cluster, shown in fig. 60, the -same method of co-ordination being used as has been described before. -Each of these positions can be named by means of the axes and the -points combined. - -[Illustration: Fig. 60.] - -Thus, for instance, the one marked by an asterisk can be called 1_c_, -2_b_, 3_c_, because it is opposite to _c_ on 1, to _b_ on 2, to _c_ on -3. - -Let us now treat of the states of consciousness corresponding to -these positions. Each point represents a composite of posits, and -the manifold of consciousness corresponding to them is of a certain -complexity. - -Suppose now the constituents, the points on the axes, to interchange -arbitrarily, any one to become any other, and also the axes 1, 2, and -3, to interchange amongst themselves, any one to become any other, and -to be subject to no system or law, that is to say, that order does not -exist, and that the points which run _abc_ on each axis may run _bac_, -and so on. - -Then any one of the states of consciousness represented by the points -in the cluster can become any other. We have a representation of a -random consciousness of a certain degree of complexity. - -Now let us examine carefully one particular case of arbitrary -interchange of the points, _a_, _b_, _c_; as one such case, carefully -considered, makes the whole clear. - -[Illustration: Fig. 61.] - -Consider the points named in the figure 1_c_, 2_a_, 3_c_; 1_c_, 2_c_, -3_a_; 1_a_, 2_c_, 3_c_, and examine the effect on them when a change of -order takes place. Let us suppose, for instance, that _a_ changes into -_b_, and let us call the two sets of points we get, the one before and -the one after, their change conjugates. - - Before the change 1_c_ 2_a_ 3_c_ 1_c_ 2_c_ 3_a_ 1_a_ 2_c_ 3_c_}Conjug- - After the change 1_c_ 2_b_ 3_c_ 1_c_ 2_c_ 3_b_ 1_b_ 2_c_ 3_c_} ates. - -The points surrounded by rings represent the conjugate points. - -It is evident that as consciousness, represented first by the first -set of points and afterwards by the second set of points, would have -nothing in common in its two phases. It would not be capable of giving -an account of itself. There would be no identity. - -If, however, we can find any set of points in the cubical cluster, -which, when any arbitrary change takes place in the points on the -axes, or in the axes themselves, repeats itself, is reproduced, then a -consciousness represented by those points would have a permanence. It -would have a principle of identity. Despite the no law, the no order, -of the ultimate constituents, it would have an order, it would form a -system, the condition of a personal identity would be fulfilled. - -The question comes to this, then. Can we find a system of points -which is self-conjugate which is such that when any posit on the axes -becomes any other, or when any axis becomes any other, such a set -is transformed into itself, its identity is not submerged, but rises -superior to the chaos of its constituents? - -[Illustration: Fig. 62.] - -Such a set can be found. Consider the set represented in fig. 62, and -written down in the first of the two lines— - - Self- {1_a_ 2_b_ 3_c_ 1_b_ 2_a_ 3_c_ 1_c_ 2_a_ 3_b_ - conjugate. {1_c_ 2_b_ 3_a_ 1_b_ 2_c_ 3_a_ 1_a_ 2_c_ 3_b_ - - Self- {1_c_ 2_b_ 3_a_ 1_b_ 2_c_ 3_a_ 1_a_ 2_c_ 3_b_ - conjugate. {1_a_ 2_b_ 3_c_ 1_b_ 2_a_ 3_c_ 1_c_ 2_a_ 3_b_ - -If now _a_ change into _c_ and _c_ into _a_, we get the set in the -second line, which has the same members as are in the upper line. -Looking at the diagram we see that it would correspond simply to the -turning of the figures as a whole.[2] Any arbitrary change of the -points on the axes, or of the axes themselves, reproduces the same set. - - [2] These figures are described more fully, and extended, in the next - chapter. - -Thus, a function, by which a random, an unordered, consciousness -could give an ordered and systematic one, can be represented. It -is noteworthy that it is a system of selection. If out of all the -alternative forms that only is attended to which is self-conjugate, -an ordered consciousness is formed. A selection gives a feature of -permanence. - -Can we say that the permanent consciousness is this selection? - -An analogy between Kant and Darwin comes into light. That which is -swings clear of the fleeting, in virtue of its presenting a feature of -permanence. There is no need to suppose any function of “attending to.” -A consciousness capable of giving an account of itself is one which is -characterised by this combination. All combinations exist—of this kind -is the consciousness which can give an account of itself. And the very -duality which we have presupposed may be regarded as originated by a -process of selection. - -Darwin set himself to explain the origin of the fauna and flora of -the world. He denied specific tendencies. He assumed an indefinite -variability—that is, chance—but a chance confined within narrow limits -as regards the magnitude of any consecutive variations. He showed that -organisms possessing features of permanence, if they occurred would be -preserved. So his account of any structure or organised being was that -it possessed features of permanence. - -Kant, undertaking not the explanation of any particular phenomena but -of that which we call nature as a whole, had an origin of species -of his own, an account of the flora and fauna of consciousness. He -denied any specific tendency of the elements of consciousness, but -taking our own consciousness, pointed out that in which it resembled -any consciousness which could survive, which could give an account of -itself. - -He assumes a chance or random world, and as great and small were not -to him any given notions of which he could make use, he did not limit -the chance, the randomness, in any way. But any consciousness which -is permanent must possess certain features—those attributes namely -which give it permanence. Any consciousness like our own is simply a -consciousness which possesses those attributes. The main thing is that -which he calls the unity of apperception, which we have seen above is -simply the statement that a particular set of phases of consciousness -on the basis of complete randomness will be self-conjugate, and so -permanent. - -As with Darwin so with Kant, the reason for existence of any feature -comes to this—show that it tends to the permanence of that which -possesses it. - -We can thus regard Kant as the creator of the first of the modern -evolution theories. And, as is so often the case, the first effort was -the most stupendous in its scope. Kant does not investigate the origin -of any special part of the world, such as its organisms, its chemical -elements, its social communities of men. He simply investigates the -origin of the whole—of all that is included in consciousness, the -origin of that “thought thing” whose progressive realisation is the -knowable universe. - -This point of view is very different from the ordinary one, in which a -man is supposed to be placed in a world like that which he has come to -think of it, and then to learn what he has found out from this model -which he himself has placed on the scene. - -We all know that there are a number of questions in attempting an -answer to which such an assumption is not allowable. - -Mill, for instance, explains our notion of “law” by an invariable -sequence in nature. But what we call nature is something given in -thought. So he explains a thought of law and order by a thought of an -invariable sequence. He leaves the problem where he found it. - -Kant’s theory is not unique and alone. It is one of a number of -evolution theories. A notion of its import and significance can be -obtained by a comparison of it with other theories. - -Thus in Darwin’s theoretical world of natural selection a certain -assumption is made, the assumption of indefinite variability—slight -variability it is true, over any appreciable lapse of time, but -indefinite in the postulated epochs of transformation—and a whole chain -of results is shown to follow. - -This element of chance variation is not, however, an ultimate resting -place. It is a preliminary stage. This supposing the all is a -preliminary step towards finding out what is. If every kind of organism -can come into being, those that do survive will present such and such -characteristics. This is the necessary beginning for ascertaining what -kinds of organisms do come into existence. And so Kant’s hypothesis -of a random consciousness is the necessary beginning for the rational -investigation of consciousness as it is. His assumption supplies, as -it were, the space in which we can observe the phenomena. It gives the -general laws constitutive of any experience. If, on the assumption -of absolute randomness in the constituents, such and such would be -characteristic of the experience, then, whatever the constituents, -these characteristics must be universally valid. - -We will now proceed to examine more carefully the poiograph, -constructed for the purpose of exhibiting an illustration of Kant’s -unity of apperception. - -In order to show the derivation order out of non-order it has been -necessary to assume a principle of duality—we have had the axes and the -posits on the axes—there are two sets of elements, each non-ordered, -and it is in the reciprocal relation of them that the order, the -definite system, originates. - -Is there anything in our experience of the nature of a duality? - -There certainly are objects in our experience which have order and -those which are incapable of order. The two roots of a quadratic -equation have no order. No one can tell which comes first. If a body -rises vertically and then goes at right angles to its former course, -no one can assign any priority to the direction of the north or to -the east. There is no priority in directions of turning. We associate -turnings with no order progressions in a line with order. But in the -axes and points we have assumed above there is no such distinction. -It is the same, whether we assume an order among the turnings, and no -order among the points on the axes, or, _vice versa_, an order in the -points and no order in the turnings. A being with an infinite number of -axes mutually at right angles, with a definite sequence between them -and no sequence between the points on the axes, would be in a condition -formally indistinguishable from that of a creature who, according to an -assumption more natural to us, had on each axis an infinite number of -ordered points and no order of priority amongst the axes. A being in -such a constituted world would not be able to tell which was turning -and which was length along an axis, in order to distinguish between -them. Thus to take a pertinent illustration, we may be in a world -of an infinite number of dimensions, with three arbitrary points on -each—three points whose order is indifferent, or in a world of three -axes of arbitrary sequence with an infinite number of ordered points on -each. We can’t tell which is which, to distinguish it from the other. - -Thus it appears the mode of illustration which we have used is not an -artificial one. There really exists in nature a duality of the kind -which is necessary to explain the origin of order out of no order—the -duality, namely, of dimension and position. Let us use the term group -for that system of points which remains unchanged, whatever arbitrary -change of its constituents takes place. We notice that a group involves -a duality, is inconceivable without a duality. - -Thus, according to Kant, the primary element of experience is the -group, and the theory of groups would be the most fundamental branch -of science. Owing to an expression in the critique the authority of -Kant is sometimes adduced against the assumption of more than three -dimensions to space. It seems to me, however, that the whole tendency -of his theory lies in the opposite direction, and points to a perfect -duality between dimension and position in a dimension. - -If the order and the law we see is due to the conditions of conscious -experience, we must conceive nature as spontaneous, free, subject to no -predication that we can devise, but, however apprehended, subject to -our logic. - -And our logic is simply spatiality in the general sense—that resultant -of a selection of the permanent from the unpermanent, the ordered from -the unordered, by the means of the group and its underlying duality. - -We can predicate nothing about nature, only about the way in which -we can apprehend nature. All that we can say is that all that which -experience gives us will be conditioned as spatial, subject to our -logic. Thus, in exploring the facts of geometry from the simplest -logical relations to the properties of space of any number of -dimensions, we are merely observing ourselves, becoming aware of the -conditions under which we must perceive. Do any phenomena present -themselves incapable of explanation under the assumption of the space -we are dealing with, then we must habituate ourselves to the conception -of a higher space, in order that our logic may be equal to the task -before us. - -We gain a repetition of the thought that came before, experimentally -suggested. If the laws of the intellectual comprehension of nature are -those derived from considering her as absolute chance, subject to no -law save that derived from a process of selection, then, perhaps, the -order of nature requires different faculties from the intellectual to -apprehend it. The source and origin of ideas may have to be sought -elsewhere than in reasoning. - -The total outcome of the critique is to leave the ordinary man just -where he is, justified in his practical attitude towards nature, -liberated from the fetters of his own mental representations. - -The truth of a picture lies in its total effect. It is vain to seek -information about the landscape from an examination of the pigments. -And in any method of thought it is the complexity of the whole that -brings us to a knowledge of nature. Dimensions are artificial enough, -but in the multiplicity of them we catch some breath of nature. - -We must therefore, and this seems to me the practical conclusion of the -whole matter, proceed to form means of intellectual apprehension of a -greater and greater degree of complexity, both dimensionally and in -extent in any dimension. Such means of representation must always be -artificial, but in the multiplicity of the elements with which we deal, -however incipiently arbitrary, lies our chance of apprehending nature. - -And as a concluding chapter to this part of the book, I will extend -the figures, which have been used to represent Kant’s theory, two -steps, so that the reader may have the opportunity of looking at a -four-dimensional figure which can be delineated without any of the -special apparatus, to the consideration of which I shall subsequently -pass on. - - - - - CHAPTER X - - A FOUR-DIMENSIONAL FIGURE - - -The method used in the preceding chapter to illustrate the problem -of Kant’s critique, gives a singularly easy and direct mode of -constructing a series of important figures in any number of dimensions. - -We have seen that to represent our space a plane being must give up one -of his axes, and similarly to represent the higher shapes we must give -up one amongst our three axes. - -But there is another kind of giving up which reduces the construction -of higher shapes to a matter of the utmost simplicity. - -Ordinarily we have on a straight line any number of positions. The -wealth of space in position is illimitable, while there are only three -dimensions. - -I propose to give up this wealth of positions, and to consider the -figures obtained by taking just as many positions as dimensions. - -In this way I consider dimensions and positions as two “kinds,” and -applying the simple rule of selecting every one of one kind with every -other of every other kind, get a series of figures which are noteworthy -because they exactly fill space of any number of dimensions (as the -hexagon fills a plane) by equal repetitions of themselves. - -The rule will be made more evident by a simple application. - -Let us consider one dimension and one position. I will call the axis -_i_, and the position _o_. - - ———————————————-_i_ - _o_ - -Here the figure is the position _o_ on the line _i_. Take now two -dimensions and two positions on each. - -[Illustration: Fig. 63.] - -We have the two positions _o_; 1 on _i_, and the two positions _o_, 1 -on _j_, fig. 63. These give rise to a certain complexity. I will let -the two lines _i_ and _j_ meet in the position I call _o_ on each, and -I will consider _i_ as a direction starting equally from every position -on _j_, and _j_ as starting equally from every position on _i_. We thus -obtain the following figure:—A is both _oi_ and _oj_, B is 1_i_ and -_oj_, and so on as shown in fig. 63_b_. The positions on AC are all -_oi_ positions. They are, if we like to consider it in that way, points -at no distance in the _i_ direction from the line AC. We can call the -line AC the _oi_ line. Similarly the points on AB are those no distance -from AB in the _j_ direction, and we can call them _oj_ points and the -line AB the _oj_ line. Again, the line CD can be called the 1_j_ line -because the points on it are at a distance, 1 in the _j_ direction. - -[Illustration: Fig. 63_b_.] - -We have then four positions or points named as shown, and, considering -directions and positions as “kinds,” we have the combination of two -kinds with two kinds. Now, selecting every one of one kind with every -other of every other kind will mean that we take 1 of the kind _i_ and -with it _o_ of the kind _j_; and then, that we take _o_ of the kind _i_ -and with it 1 of the kind _j_. - -Thus we get a pair of positions lying in the straight line BC, fig. -64. We can call this pair 10 and 01 if we adopt the plan of mentally, -adding an _i_ to the first and a _j_ to the second of the symbols -written thus—01 is a short expression for O_i_, 1_j_. - -[Illustration: Fig. 64.] - -Coming now to our space, we have three dimensions, so we take three -positions on each. These positions I will suppose to be at equal -distances along each axis. The three axes and the three positions on -each are shown in the accompanying diagrams, fig. 65, of which the -first represents a cube with the front faces visible, the second the -rear faces of the same cube; the positions I will call 0, 1, 2; the -axes, _i_, _j_, _k_. I take the base ABC as the starting place, from -which to determine distances in the _k_ direction, and hence every -point in the base ABC will be an _ok_ position, and the base ABC can be -called an _ok_ plane. - -[Illustration: Fig. 65.] - -In the same way, measuring the distances from the face ADC, we see -that every position in the face ADC is an _oi_ position, and the whole -plane of the face may be called an _oi_ plane. Thus we see that with -the introduction of a new dimension the signification of a compound -symbol, such as “_oi_,” alters. In the plane it meant the line AC. In -space it means the whole plane ACD. - -Now, it is evident that we have twenty-seven positions, each of them -named. If the reader will follow this nomenclature in respect of the -positions marked in the figures he will have no difficulty in assigning -names to each one of the twenty-seven positions. A is _oi_, _oj_, _ok_. -It is at the distance 0 along _i_, 0 along _j_, 0 along _k_, and _io_ -can be written in short 000, where the _ijk_ symbols are omitted. - -The point immediately above is 001, for it is no distance in the _i_ -direction, and a distance of 1 in the _k_ direction. Again, looking at -B, it is at a distance of 2 from A, or from the plane ADC, in the _i_ -direction, 0 in the _j_ direction from the plane ABD, and 0 in the _k_ -direction, measured from the plane ABC. Hence it is 200 written for -2_i_, 0_j_, 0_k_. - -Now, out of these twenty-seven “things” or compounds of position and -dimension, select those which are given by the rule, every one of one -kind with every other of every other kind. - -Take 2 of the _i_ kind. With this we must have a 1 of the _j_ kind, and -then by the rule we can only have a 0 of the _k_ kind, for if we had -any other of the _k_ kind we should repeat one of the kinds we already -had. In 2_i_, 1_j_, 1_k_, for instance, 1 is repeated. The point we -obtain is that marked 210, fig. 66. - -[Illustration: Fig. 66.] - -Proceeding in this way, we pick out the following cluster of points, -fig. 67. They are joined by lines, dotted where they are hidden by the -body of the cube, and we see that they form a figure—a hexagon which -could be taken out of the cube and placed on a plane. It is a figure -which will fill a plane by equal repetitions of itself. The plane being -representing this construction in his plane would take three squares to -represent the cube. Let us suppose that he takes the _ij_ axes in his -space and _k_ represents the axis running out of his space, fig. 68. -In each of the three squares shown here as drawn separately he could -select the points given by the rule, and he would then have to try to -discover the figure determined by the three lines drawn. The line from -210 to 120 is given in the figure, but the line from 201 to 102 or GK -is not given. He can determine GK by making another set of drawings and -discovering in them what the relation between these two extremities is. - -[Illustration: Fig. 67.] - -[Illustration: Fig. 68.] - -[Illustration: Fig. 69.] - -Let him draw the _i_ and _k_ axes in his plane, fig. 69. The _j_ axis -then runs out and he has the accompanying figure. In the first of these -three squares, fig. 69, he can pick out by the rule the two points -201, 102—G, and K. Here they occur in one plane and he can measure the -distance between them. In his first representation they occur at G and -K in separate figures. - -Thus the plane being would find that the ends of each of the lines was -distant by the diagonal of a unit square from the corresponding end -of the last and he could then place the three lines in their right -relative position. Joining them he would have the figure of a hexagon. - -[Illustration: Fig. 70.] - -We may also notice that the plane being could make a representation of -the whole cube simultaneously. The three squares, shown in perspective -in fig. 70, all lie in one plane, and on these the plane being could -pick out any selection of points just as well as on three separate -squares. He would obtain a hexagon by joining the points marked. This -hexagon, as drawn, is of the right shape, but it would not be so if -actual squares were used instead of perspective, because the relation -between the separate squares as they lie in the plane figure is not -their real relation. The figure, however, as thus constructed, would -give him an idea of the correct figure, and he could determine it -accurately by remembering that distances in each square were correct, -but in passing from one square to another their distance in the third -dimension had to be taken into account. - -Coming now to the figure made by selecting according to our rule from -the whole mass of points given by four axes and four positions in each, -we must first draw a catalogue figure in which the whole assemblage is -shown. - -We can represent this assemblage of points by four solid figures. The -first giving all those positions which are at a distance O from our -space in the fourth dimension, the second showing all those that are at -a distance 1, and so on. - -These figures will each be cubes. The first two are drawn showing the -front faces, the second two the rear faces. We will mark the points 0, -1, 2, 3, putting points at those distances along each of these axes, -and suppose all the points thus determined to be contained in solid -models of which our drawings in fig. 71 are representatives. Here we -notice that as on the plane 0_i_ meant the whole line from which the -distances in the _i_ direction was measured, and as in space 0_i_ -means the whole plane from which distances in the _i_ direction are -measured, so now 0_h_ means the whole space in which the first cube -stands—measuring away from that space by a distance of one we come to -the second cube represented. - -[Illustration: Fig. 71.] - -Now selecting according to the rule every one of one kind with every -other of every other kind, we must take, for instance, 3_i_, 2_j_, -1_k_, 0_h_. This point is marked 3210 at the lower star in the figure. -It is 3 in the _i_ direction, 2 in the _j_ direction, 1 in the _k_ -direction, 0 in the _h_ direction. - -With 3_i_ we must also take 1_j_, 2_k_, 0_h_. This point is shown by -the second star in the cube 0_h_. - -[Illustration: Fig. 72.] - -In the first cube, since all the points are 0_h_ points, we can only -have varieties in which _i_, _j_, _k_, are accompanied by 3, 2, 1. - -The points determined are marked off in the diagram fig. 72, and lines -are drawn joining the adjacent pairs in each figure, the lines being -dotted when they pass within the substance of the cube in the first two -diagrams. - -Opposite each point, on one side or the other of each cube, is written -its name. It will be noticed that the figures are symmetrical right and -left; and right and left the first two numbers are simply interchanged. - -Now this being our selection of points, what figure do they make when -all are put together in their proper relative positions? - -To determine this we must find the distance between corresponding -corners of the separate hexagons. - -[Illustration: Fig. 73.] - -To do this let us keep the axes _i_, _j_, in our space, and draw _h_ -instead of _k_, letting _k_ run out in the fourth dimension, fig. 73. - -Here we have four cubes again, in the first of which all the points are -0_k_ points; that is, points at a distance zero in the _k_ direction -from the space of the three dimensions _ijh_. We have all the points -selected before, and some of the distances, which in the last diagram -led from figure to figure are shown here in the same figure, and so -capable of measurement. Take for instance the points 3120 to 3021, -which in the first diagram (fig. 72) lie in the first and second -figures. Their actual relation is shown in fig. 73 in the cube marked -2K, where the points in question are marked with a *. We see that the -distance in question is the diagonal of a unit square. In like manner -we find that the distance between corresponding points of any two -hexagonal figures is the diagonal of a unit square. The total figure -is now easily constructed. An idea of it may be gained by drawing all -the four cubes in the catalogue figure in one (fig. 74). These cubes -are exact repetitions of one another, so one drawing will serve as a -representation of the whole series, if we take care to remember where -we are, whether in a 0_h_, a 1_h_, a 2_h_, or a 3_h_ figure, when we -pick out the points required. Fig. 74 is a representation of all the -catalogue cubes put in one. For the sake of clearness the front faces -and the back faces of this cube are represented separately. - -[Illustration: Fig. 74.] - -The figure determined by the selected points is shown below. - -In putting the sections together some of the outlines in them -disappear. The line TW for instance is not wanted. - -We notice that PQTW and TWRS are each the half of a hexagon. Now QV and -VR lie in one straight line. Hence these two hexagons fit together, -forming one hexagon, and the line TW is only wanted when we consider a -section of the whole figure, we thus obtain the solid represented in -the lower part of fig. 74. Equal repetitions of this figure, called a -tetrakaidecagon, will fill up three-dimensional space. - -To make the corresponding four-dimensional figure we have to take five -axes mutually at right angles with five points on each. A catalogue of -the positions determined in five-dimensional space can be found thus. - -Take a cube with five points on each of its axes, the fifth point is -at a distance of four units of length from the first on any one of -the axes. And since the fourth dimension also stretches to a distance -of four we shall need to represent the successive sets of points at -distances 0, 1, 2, 3, 4, in the fourth dimensions, five cubes. Now -all of these extend to no distance at all in the fifth dimension. To -represent what lies in the fifth dimension we shall have to draw, -starting from each of our cubes, five similar cubes to represent the -four steps on in the fifth dimension. By this assemblage we get a -catalogue of all the points shown in fig. 75, in which _L_ represents -the fifth dimension. - -[Illustration: Fig. 75.] - -Now, as we saw before, there is nothing to prevent us from putting all -the cubes representing the different stages in the fourth dimension in -one figure, if we take note when we look at it, whether we consider -it as a 0_h_, a 1_h_, a 2_h_, etc., cube. Putting then the 0_h_, 1_h_, -2_h_, 3_h_, 4_h_ cubes of each row in one, we have five cubes with the -sides of each containing five positions, the first of these five cubes -represents the 0_l_ points, and has in it the _i_ points from 0 to 4, -the _j_ points from 0 to 4, the _k_ points from 0 to 4, while we have -to specify with regard to any selection we make from it, whether we -regard it as a 0_h_, a 1_h_, a 2_h_, a 3_h_, or a 4_h_ figure. In fig. -76 each cube is represented by two drawings, one of the front part, the -other of the rear part. - -Let then our five cubes be arranged before us and our selection be made -according to the rule. Take the first figure in which all points are -0_l_ points. We cannot have 0 with any other letter. Then, keeping in -the first figure, which is that of the 0_l_ positions, take first of -all that selection which always contains 1_h_. We suppose, therefore, -that the cube is a 1_h_ cube, and in it we take _i_, _j_, _k_ in -combination with 4, 3, 2 according to the rule. - -The figure we obtain is a hexagon, as shown, the one in front. The -points on the right hand have the same figures as those on the left, -with the first two numerals interchanged. Next keeping still to the -0_l_ figure let us suppose that the cube before us represents a section -at a distance of 2 in the _h_ direction. Let all the points in it be -considered as 2_h_ points. We then have a 0_l_, 2_h_ region, and have -the sets _ijk_ and 431 left over. We must then pick out in accordance -with our rule all such points as 4_i_, 3_j_, 1_k_. - -These are shown in the figure and we find that we can draw them without -confusion, forming the second hexagon from the front. Going on in this -way it will be seen that in each of the five figures a set of hexagons -is picked out, which put together form a three-space figure something -like the tetrakaidecagon. - -[Illustration: Fig. 76.] - -These separate figures are the successive stages in which the whole -four-dimensional figure in which they cohere can be apprehended. - -The first figure and the last are tetrakaidecagons. These are two -of the solid boundaries of the figure. The other solid boundaries -can be traced easily. Some of them are complete from one face in the -figure to the corresponding face in the next, as for instance the -solid which extends from the hexagonal base of the first figure to the -equal hexagonal base of the second figure. This kind of boundary is a -hexagonal prism. The hexagonal prism also occurs in another sectional -series, as for instance, in the square at the bottom of the first -figure, the oblong at the base of the second and the square at the base -of the third figure. - -Other solid boundaries can be traced through four of the five sectional -figures. Thus taking the hexagon at the top of the first figure we -find in the next a hexagon also, of which some alternate sides are -elongated. The top of the third figure is also a hexagon with the other -set of alternate rules elongated, and finally we come in the fourth -figure to a regular hexagon. - -These four sections are the sections of a tetrakaidecagon as can -be recognised from the sections of this figure which we have had -previously. Hence the boundaries are of two kinds, hexagonal prisms and -tetrakaidecagons. - -These four-dimensional figures exactly fill four-dimensional space by -equal repetitions of themselves. - - - - - CHAPTER XI - -NOMENCLATURE AND ANALOGIES PRELIMINARY TO THE STUDY OF FOUR-DIMENSIONAL - FIGURES - - -In the following pages a method of designating different regions of -space by a systematic colour scheme has been adopted. The explanations -have been given in such a manner as to involve no reference to models, -the diagrams will be found sufficient. But to facilitate the study a -description of a set of models is given in an appendix which the reader -can either make for himself or obtain. If models are used the diagrams -in Chapters XI. and XII. will form a guide sufficient to indicate their -use. Cubes of the colours designated by the diagrams should be picked -out and used to reinforce the diagrams. The reader, in the following -description, should suppose that a board or wall stretches away from -him, against which the figures are placed. - -[Illustration: Fig. 77.] - -Take a square, one of those shown in Fig. 77 and give it a neutral -colour, let this colour be called “null,” and be such that it makes no -appreciable difference to any colour with which it mixed. If there is -no such real colour let us imagine such a colour, and assign to it the -properties of the number zero, which makes no difference in any number -to which it is added. - -Above this square place a red square. Thus we symbolise the going up by -adding red to null. - -Away from this null square place a yellow square, and represent going -away by adding yellow to null. - -To complete the figure we need a fourth square. Colour this orange, -which is a mixture of red and yellow, and so appropriately represents a -going in a direction compounded of up and away. We have thus a colour -scheme which will serve to name the set of squares drawn. We have two -axes of colours—red and yellow—and they may occupy as in the figure -the direction up and away, or they may be turned about; in any case -they enable us to name the four squares drawn in their relation to one -another. - -Now take, in Fig. 78, nine squares, and suppose that at the end of the -going in any direction the colour started with repeats itself. - -[Illustration: Fig. 78.] - -We obtain a square named as shown. - -Let us now, in fig. 79, suppose the number of squares to be increased, -keeping still to the principle of colouring already used. - -Here the nulls remain four in number. There are three reds between the -first null and the null above it, three yellows between the first null -and the null beyond it, while the oranges increase in a double way. - -[Illustration: Fig. 79.] - -Suppose this process of enlarging the number of the squares to be -indefinitely pursued and the total figure obtained to be reduced in -size, we should obtain a square of which the interior was all orange, -while the lines round it were red and yellow, and merely the points -null colour, as in fig. 80. Thus all the points, lines, and the area -would have a colour. - -[Illustration: Fig. 80.] - -We can consider this scheme to originate thus:—Let a null point move -in a yellow direction and trace out a yellow line and end in a null -point. Then let the whole line thus traced move in a red direction. The -null points at the ends of the line will produce red lines, and end in -null points. The yellow line will trace out a yellow and red, or orange -square. - -Now, turning back to fig. 78, we see that these two ways of naming, the -one we started with and the one we arrived at, can be combined. - -By its position in the group of four squares, in fig. 77, the null -square has a relation to the yellow and to the red directions. We can -speak therefore of the red line of the null square without confusion, -meaning thereby the line AB, fig. 81, which runs up from the initial -null point A in the figure as drawn. The yellow line of the null square -is its lower horizontal line AC as it is situated in the figure. - -[Illustration: Fig. 81.] - -If we wish to denote the upper yellow line BD, fig. 81, we can speak -of it as the yellow γ line, meaning the yellow line which is separated -from the primary yellow line by the red movement. - -In a similar way each of the other squares has null points, red and -yellow lines. Although the yellow square is all yellow, its line CD, -for instance, can be referred to as its red line. - -This nomenclature can be extended. - -If the eight cubes drawn, in fig. 82, are put close together, as on -the right hand of the diagram, they form a cube, and in them, as thus -arranged, a going up is represented by adding red to the zero, or -null colour, a going away by adding yellow, a going to the right by -adding white. White is used as a colour, as a pigment, which produces -a colour change in the pigments with which it is mixed. From whatever -cube of the lower set we start, a motion up brings us to a cube showing -a change to red, thus light yellow becomes light yellow red, or light -orange, which is called ochre. And going to the right from the null on -the left we have a change involving the introduction of white, while -the yellow change runs from front to back. There are three colour -axes—the red, the white, the yellow—and these run in the position the -cubes occupy in the drawing—up, to the right, away—but they could be -turned about to occupy any positions in space. - -[Illustration: Fig. 82.] - -[Illustration: Fig. 83. The three layers.] - -We can conveniently represent a block of cubes by three sets of -squares, representing each the base of a cube. - -Thus the block, fig. 83, can be represented by the layers on the -right. Here, as in the case of the plane, the initial colours repeat -themselves at the end of the series. - -Proceeding now to increase the number of the cubes we obtain fig. 84, -in which the initial letters of the colours are given instead of their -full names. - -Here we see that there are four null cubes as before, but the series -which spring from the initial corner will tend to become lines of -cubes, as also the sets of cubes parallel to them, starting from other -corners. Thus, from the initial null springs a line of red cubes, a -line of white cubes, and a line of yellow cubes. - -If the number of the cubes is largely increased, and the size of the -whole cube is diminished, we get a cube with null points, and the edges -coloured with these three colours. - -[Illustration: Fig. 84.] - -The light yellow cubes increase in two ways, forming ultimately a sheet -of cubes, and the same is true of the orange and pink sets. Hence, -ultimately the cube thus formed would have red, white, and yellow -lines surrounding pink, orange, and light yellow faces. The ochre cubes -increase in three ways, and hence ultimately the whole interior of the -cube would be coloured ochre. - -We have thus a nomenclature for the points, lines, faces, and solid -content of a cube, and it can be named as exhibited in fig. 85. - -[Illustration: Fig. 85.] - -We can consider the cube to be produced in the following way. A null -point moves in a direction to which we attach the colour indication -yellow; it generates a yellow line and ends in a null point. The yellow -line thus generated moves in a direction to which we give the colour -indication red. This lies up in the figure. The yellow line traces out -a yellow, red, or orange square, and each of its null points trace out -a red line, and ends in a null point. - -This orange square moves in a direction to which we attribute the -colour indication white, in this case the direction is the right. The -square traces out a cube coloured orange, red, or ochre, the red lines -trace out red to white or pink squares, and the yellow lines trace out -light yellow squares, each line ending in a line of its own colour. -While the points each trace out a null + white, or white line to end in -a null point. - -Now returning to the first block of eight cubes we can name each point, -line, and square in them by reference to the colour scheme, which they -determine by their relation to each other. - -Thus, in fig. 86, the null cube touches the red cube by a light yellow -square; it touches the yellow cube by a pink square, and touches the -white cube by an orange square. - -There are three axes to which the colours red, yellow, and white are -assigned, the faces of each cube are designated by taking these colours -in pairs. Taking all the colours together we get a colour name for the -solidity of a cube. - -[Illustration: Fig. 86.] - -Let us now ask ourselves how the cube could be presented to the plane -being. Without going into the question of how he could have a real -experience of it, let us see how, if we could turn it about and show it -to him, he, under his limitations, could get information about it. If -the cube were placed with its red and yellow axes against a plane, that -is resting against it by its orange face, the plane being would observe -a square surrounded by red and yellow lines, and having null points. -See the dotted square, fig. 87. - -[Illustration: Fig. 87.] - -We could turn the cube about the red line so that a different face -comes into juxtaposition with the plane. - -Suppose the cube turned about the red line. As it is turning from its -first position all of it except the red line leaves the plane—goes -absolutely out of the range of the plane being’s apprehension. But when -the yellow line points straight out from the plane then the pink face -comes into contact with it. Thus the same red line remaining as he saw -it at first, now towards him comes a face surrounded by white and red -lines. - -If we call the direction to the right the unknown direction, then -the line he saw before, the yellow line, goes out into this unknown -direction, and the line which before went into the unknown direction, -comes in. It comes in in the opposite direction to that in which the -yellow line ran before; the interior of the face now against the plane -is pink. It is a property of two lines at right angles that, if one -turns out of a given direction and stands at right angles to it, then -the other of the two lines comes in, but runs the opposite way in that -given direction, as in fig. 88. - -[Illustration: Fig. 88.] - -Now these two presentations of the cube would seem, to the plane -creature like perfectly different material bodies, with only that line -in common in which they both meet. - -Again our cube can be turned about the yellow line. In this case the -yellow square would disappear as before, but a new square would come -into the plane after the cube had rotated by an angle of 90° about this -line. The bottom square of the cube would come in thus in figure 89. -The cube supposed in contact with the plane is rotated about the lower -yellow line and then the bottom face is in contact with the plane. - -Here, as before, the red line going out into the unknown dimension, -the white line which before ran in the unknown dimension would come -in downwards in the opposite sense to that in which the red line ran -before. - -[Illustration: Fig. 89.] - -Now if we use _i_, _j_, _k_, for the three space directions, _i_ left -to right, _j_ from near away, _k_ from below up; then, using the colour -names for the axes, we have that first of all white runs _i_, yellow -runs _j_, red runs _k_; then after the first turning round the _k_ -axis, white runs negative _j_, yellow runs _i_, red runs _k_; thus we -have the table:— - - _i_ _j_ _k_ - 1st position white yellow red - 2nd position yellow white— red - 3rd position red yellow white— - -Here white with a negative sign after it in the column under _j_ means -that white runs in the negative sense of the _j_ direction. - -We may express the fact in the following way:— In the plane there is -room for two axes while the body has three. Therefore in the plane we -can represent any two. If we want to keep the axis that goes in the -unknown dimension always running in the positive sense, then the axis -which originally ran in the unknown dimension (the white axis) must -come in in the negative sense of that axis which goes out of the plane -into the unknown dimension. - -It is obvious that the unknown direction, the direction in which the -white line runs at first, is quite distinct from any direction which -the plane creature knows. The white line may come in towards him, or -running down. If he is looking at a square, which is the face of a cube -(looking at it by a line), then any one of the bounding lines remaining -unmoved, another face of the cube may come in, any one of the faces, -namely, which have the white line in them. And the white line comes -sometimes in one of the space directions he knows, sometimes in another. - -Now this turning which leaves a line unchanged is something quite -unlike any turning he knows in the plane. In the plane a figure turns -round a point. The square can turn round the null point in his plane, -and the red and yellow lines change places, only of course, as with -every rotation of lines at right angles, if red goes where yellow went, -yellow comes in negative of red’s old direction. - -This turning, as the plane creature conceives it, we should call -turning about an axis perpendicular to the plane. What he calls turning -about the null point we call turning about the white line as it stands -out from his plane. There is no such thing as turning about a point, -there is always an axis, and really much more turns than the plane -being is aware of. - -Taking now a different point of view, let us suppose the cubes to be -presented to the plane being by being passed transverse to his plane. -Let us suppose the sheet of matter over which the plane being and all -objects in his world slide, to be of such a nature that objects can -pass through it without breaking it. Let us suppose it to be of the -same nature as the film of a soap bubble, so that it closes around -objects pushed through it, and, however the object alters its shape as -it passes through it, let us suppose this film to run up to the contour -of the object in every part, maintaining its plane surface unbroken. - -Then we can push a cube or any object through the film and the plane -being who slips about in the film will know the contour of the cube -just and exactly where the film meets it. - -[Illustration: Fig. 90.] - -Fig. 90 represents a cube passing through a plane film. The plane being -now comes into contact with a very thin slice of the cube somewhere -between the left and right hand faces. This very thin slice he thinks -of as having no thickness, and consequently his idea of it is what we -call a section. It is bounded by him by pink lines front and back, -coming from the part of the pink face he is in contact with, and above -and below, by light yellow lines. Its corners are not null-coloured -points, but white points, and its interior is ochre, the colour of the -interior of the cube. - -If now we suppose the cube to be an inch in each dimension, and to pass -across, from right to left, through the plane, then we should explain -the appearances presented to the plane being by saying: First of all -you have the face of a cube, this lasts only a moment; then you have a -figure of the same shape but differently coloured. This, which appears -not to move to you in any direction which you know of, is really moving -transverse to your plane world. Its appearance is unaltered, but each -moment it is something different—a section further on, in the white, -the unknown dimension. Finally, at the end of the minute, a face comes -in exactly like the face you first saw. This finishes up the cube—it is -the further face in the unknown dimension. - -The white line, which extends in length just like the red or the -yellow, you do not see as extensive; you apprehend it simply as an -enduring white point. The null point, under the condition of movement -of the cube, vanishes in a moment, the lasting white point is really -your apprehension of a white line, running in the unknown dimension. -In the same way the red line of the face by which the cube is first in -contact with the plane lasts only a moment, it is succeeded by the pink -line, and this pink line lasts for the inside of a minute. This lasting -pink line in your apprehension of a surface, which extends in two -dimensions just like the orange surface extends, as you know it, when -the cube is at rest. - -But the plane creature might answer, “This orange object is substance, -solid substance, bounded completely and on every side.” - -Here, of course, the difficulty comes in. His solid is our surface—his -notion of a solid is our notion of an abstract surface with no -thickness at all. - -We should have to explain to him that, from every point of what he -called a solid, a new dimension runs away. From every point a line -can be drawn in a direction unknown to him, and there is a solidity -of a kind greater than that which he knows. This solidity can only -be realised by him by his supposing an unknown direction, by motion -in which what he conceives to be solid matter instantly disappears. -The higher solid, however, which extends in this dimension as well -as in those which he knows, lasts when a motion of that kind takes -place, different sections of it come consecutively in the plane -of his apprehension, and take the place of the solid which he at -first conceives to be all. Thus, the higher solid—our solid in -contradistinction to his area solid, his two-dimensional solid, must -be conceived by him as something which has duration in it, under -circumstances in which his matter disappears out of his world. - -We may put the matter thus, using the conception of motion. - -A null point moving in a direction away generates a yellow line, and -the yellow line ends in a null point. We suppose, that is, a point -to move and mark out the products of this motion in such a manner. -Now suppose this whole line as thus produced to move in an upward -direction; it traces out the two-dimensional solid, and the plane being -gets an orange square. The null point moves in a red line and ends in -a null point, the yellow line moves and generates an orange square and -ends in a yellow line, the farther null point generates a red line and -ends in a null point. Thus, by movement in two successive directions -known to him, he can imagine his two-dimensional solid produced with -all its boundaries. - -Now we tell him: “This whole two-dimensional solid can move in a third -or unknown dimension to you. The null point moving in this dimension -out of your world generates a white line and ends in a null point. The -yellow line moving generates a light yellow two-dimensional solid and -ends in a yellow line, and this two-dimensional solid, lying end on to -your plane world, is bounded on the far side by the other yellow line. -In the same way each of the lines surrounding your square traces out an -area, just like the orange area you know. But there is something new -produced, something which you had no idea of before; it is that which -is produced by the movement of the orange square. That, than which you -can imagine nothing more solid, itself moves in a direction open to it -and produces a three-dimensional solid. Using the addition of white -to symbolise the products of this motion this new kind of solid will -be light orange or ochre, and it will be bounded on the far side by -the final position of the orange square which traced it out, and this -final position we suppose to be coloured like the square in its first -position, orange with yellow and red boundaries and null corners.” - -This product of movement, which it is so easy for us to describe, would -be difficult for him to conceive. But this difficulty is connected -rather with its totality than with any particular part of it. - -Any line, or plane of this, to him higher, solid we could show to him, -and put in his sensible world. - -We have already seen how the pink square could be put in his world by -a turning of the cube about the red line. And any section which we can -conceive made of the cube could be exhibited to him. You have simply to -turn the cube and push it through, so that the plane of his existence -is the plane which cuts out the given section of the cube, then the -section would appear to him as a solid. In his world he would see the -contour, get to any part of it by digging down into it. - - - THE PROCESS BY WHICH A PLANE BEING WOULD GAIN A NOTION OF A SOLID. - -If we suppose the plane being to have a general idea of the existence -of a higher solid—our solid—we must next trace out in detail the -method, the discipline, by which he would acquire a working familiarity -with our space existence. The process begins with an adequate -realisation of a simple solid figure. For this purpose we will suppose -eight cubes forming a larger cube, and first we will suppose each cube -to be coloured throughout uniformly. Let the cubes in fig. 91 be the -eight making a larger cube. - -[Illustration: Fig. 91.] - -Now, although each cube is supposed to be coloured entirely through -with the colour, the name of which is written on it, still we can -speak of the faces, edges, and corners of each cube as if the colour -scheme we have investigated held for it. Thus, on the null cube we can -speak of a null point, a red line, a white line, a pink face, and so -on. These colour designations are shown on No. 1 of the views of the -tesseract in the plate. Here these colour names are used simply in -their geometrical significance. They denote what the particular line, -etc., referred to would have as its colour, if in reference to the -particular cube the colour scheme described previously were carried out. - -If such a block of cubes were put against the plane and then passed -through it from right to left, at the rate of an inch a minute, each -cube being an inch each way, the plane being would have the following -appearances:— - -First of all, four squares null, yellow, red, orange, lasting each a -minute; and secondly, taking the exact places of these four squares, -four others, coloured white, light yellow, pink, ochre. Thus, to make -a catalogue of the solid body, he would have to put side by side in -his world two sets of four squares each, as in fig. 92. The first are -supposed to last a minute, and then the others to come in in place of -them, and also last a minute. - -[Illustration: Fig. 92.] - -In speaking of them he would have to denote what part of the respective -cube each square represents. Thus, at the beginning he would have null -cube orange face, and after the motion had begun he would have null -cube ochre section. As he could get the same coloured section whichever -way the cube passed through, it would be best for him to call this -section white section, meaning that it is transverse to the white axis. -These colour-names, of course, are merely used as names, and do not -imply in this case that the object is really coloured. Finally, after -a minute, as the first cube was passing beyond his plane he would have -null cube orange face again. - -The same names will hold for each of the other cubes, describing what -face or section of them the plane being has before him; and the second -wall of cubes will come on, continue, and go out in the same manner. In -the area he thus has he can represent any movement which we carry out -in the cubes, as long as it does not involve a motion in the direction -of the white axis. The relation of parts that succeed one another in -the direction of the white axis is realised by him as a consecution of -states. - -Now, his means of developing his space apprehension lies in this, that -that which is represented as a time sequence in one position of the -cubes, can become a real co-existence, _if something that has a real -co-existence becomes a time sequence_. - -We must suppose the cubes turned round each of the axes, the red line, -and the yellow line, then something, which was given as time before, -will now be given as the plane creature’s space; something, which was -given as space before, will now be given as a time series as the cube -is passed through the plane. - -The three positions in which the cubes must be studied are the one -given above and the two following ones. In each case the original null -point which was nearest to us at first is marked by an asterisk. In -figs. 93 and 94 the point marked with a star is the same in the cubes -and in the plane view. - -[Illustration: Fig. 93. The cube swung round the red line, so that the -white line points towards us.] - -In fig. 93 the cube is swung round the red line so as to point towards -us, and consequently the pink face comes next to the plane. As it -passes through there are two varieties of appearance designated by -the figures 1 and 2 in the plane. These appearances are named in the -figure, and are determined by the order in which the cubes come in the -motion of the whole block through the plane. - -With regard to these squares severally, however, different names must -be used, determined by their relations in the block. - -Thus, in fig. 93, when the cube first rests against the plane the null -cube is in contact by its pink face; as the block passes through we get -an ochre section of the null cube, but this is better called a yellow -section, as it is made by a plane perpendicular to the yellow line. -When the null cube has passed through the plane, as it is leaving it, -we get again a pink face. - -[Illustration: Fig. 94. The cube swung round yellow line, with red line -running from left to right, and white line running down.] - -The same series of changes take place with the cube appearances which -follow on those of the null cube. In this motion the yellow cube -follows on the null cube, and the square marked yellow in 2 in the -plane will be first “yellow pink face,” then “yellow yellow section,” -then “yellow pink face.” - -In fig. 94, in which the cube is turned about the yellow line, we have -a certain difficulty, for the plane being will find that the position -his squares are to be placed in will lie below that which they first -occupied. They will come where the support was on which he stood his -first set of squares. He will get over this difficulty by moving his -support. - -Then, since the cubes come upon his plane by the light yellow face, he -will have, taking the null cube as before for an example, null, light -yellow face; null, red section, because the section is perpendicular -to the red line; and finally, as the null cube leaves the plane, null, -light yellow face. Then, in this case red following on null, he will -have the same series of views of the red as he had of the null cube. - -[Illustration: Fig. 95.] - -There is another set of considerations which we will briefly allude to. - -Suppose there is a hollow cube, and a string is stretched across it -from null to null, _r_, _y_, _wh_, as we may call the far diagonal -point, how will this string appear to the plane being as the cube moves -transverse to his plane? - -Let us represent the cube as a number of sections, say 5, corresponding -to 4 equal divisions made along the white line perpendicular to it. - -We number these sections 0, 1, 2, 3, 4, corresponding to the distances -along the white line at which they are taken, and imagine each section -to come in successively, taking the place of the preceding one. - -These sections appear to the plane being, counting from the first, to -exactly coincide each with the preceding one. But the section of the -string occupies a different place in each to that which it does in the -preceding section. The section of the string appears in the position -marked by the dots. Hence the slant of the string appears as a motion -in the frame work marked out by the cube sides. If we suppose the -motion of the cube not to be recognised, then the string appears to the -plane being as a moving point. Hence extension on the unknown dimension -appears as duration. Extension sloping in the unknown direction appears -as continuous movement. - - - - - CHAPTER XII - - THE SIMPLEST FOUR-DIMENSIONAL SOLID - - -A plane being, in learning to apprehend solid existence, must first -of all realise that there is a sense of direction altogether wanting -to him. That which we call right and left does not exist in his -perception. He must assume a movement in a direction, and a distinction -of positive and negative in that direction, which has no reality -corresponding to it in the movements he can make. This direction, this -new dimension, he can only make sensible to himself by bringing in -time, and supposing that changes, which take place in time, are due -to objects of a definite configuration in three dimensions passing -transverse to his plane, and the different sections of it being -apprehended as changes of one and the same plane figure. - -He must also acquire a distinct notion about his plane world, he must -no longer believe that it is the all of space, but that space extends -on both sides of it. In order, then, to prevent his moving off in this -unknown direction, he must assume a sheet, an extended solid sheet, in -two dimensions, against which, in contact with which, all his movements -take place. - -When we come to think of a four-dimensional solid, what are the -corresponding assumptions which we must make? - -We must suppose a sense which we have not, a sense of direction -wanting in us, something which a being in a four-dimensional world -has, and which we have not. It is a sense corresponding to a new space -direction, a direction which extends positively and negatively from -every point of our space, and which goes right away from any space -direction we know of. The perpendicular to a plane is perpendicular, -not only to two lines in it, but to every line, and so we must conceive -this fourth dimension as running perpendicularly to each and every line -we can draw in our space. - -And as the plane being had to suppose something which prevented his -moving off in the third, the unknown dimension to him, so we have to -suppose something which prevents us moving off in the direction unknown -to us. This something, since we must be in contact with it in every one -of our movements, must not be a plane surface, but a solid; it must be -a solid, which in every one of our movements we are against, not in. -It must be supposed as stretching out in every space dimension that we -know; but we are not in it, we are against it, we are next to it, in -the fourth dimension. - -That is, as the plane being conceives himself as having a very small -thickness in the third dimension, of which he is not aware in his -sense experience, so we must suppose ourselves as having a very small -thickness in the fourth dimension, and, being thus four-dimensional -beings, to be prevented from realising that we are such beings by a -constraint which keeps us always in contact with a vast solid sheet, -which stretches on in every direction. We are against that sheet, so -that, if we had the power of four-dimensional movement, we should -either go away from it or through it; all our space movements as we -know them being such that, performing them, we keep in contact with -this solid sheet. - -Now consider the exposition a plane being would make for himself as to -the question of the enclosure of a square, and of a cube. - -He would say the square A, in Fig. 96, is completely enclosed by the -four squares, A far, A near, A above, A below, or as they are written -A_n_, A_f_, A_a_, A_b_. - -[Illustration: Fig. 96.] - -If now he conceives the square A to move in the, to him, unknown -dimension it will trace out a cube, and the bounding squares will -form cubes. Will these completely surround the cube generated by A? -No; there will be two faces of the cube made by A left uncovered; -the first, that face which coincides with the square A in its first -position; the next, that which coincides with the square A in its -final position. Against these two faces cubes must be placed in order -to completely enclose the cube A. These may be called the cubes left -and right or A_l_ and A_r_. Thus each of the enclosing squares of the -square A becomes a cube and two more cubes are wanted to enclose the -cube formed by the movement of A in the third dimension. - -[Illustration: Fig. 97.] - -The plane being could not see the square A with the squares A_n_, A_f_, -etc., placed about it, because they completely hide it from view; and -so we, in the analogous case in our three-dimensional world, cannot -see a cube A surrounded by six other cubes. These cubes we will call A -near A_n_, A far A_f_, A above A_a_, A below A_b_, A left A_l_, A right -A_r_, shown in fig. 97. If now the cube A moves in the fourth dimension -right out of space, it traces out a higher cube—a tesseract, as it may -be called. Each of the six surrounding cubes carried on in the same -motion will make a tesseract also, and these will be grouped around the -tesseract formed by A. But will they enclose it completely? - -All the cubes A_n_, A_f_, etc., lie in our space. But there is nothing -between the cube A and that solid sheet in contact with which every -particle of matter is. When the cube A moves in the fourth direction -it starts from its position, say A_k_, and ends in a final position -A_n_ (using the words “ana” and “kata” for up and down in the fourth -dimension). Now the movement in this fourth dimension is not bounded by -any of the cubes A_n_, A_f_, nor by what they form when thus moved. The -tesseract which A becomes is bounded in the positive and negative ways -in this new direction by the first position of A and the last position -of A. Or, if we ask how many tesseracts lie around the tesseract which -A forms, there are eight, of which one meets it by the cube A, and -another meets it by a cube like A at the end of its motion. - -We come here to a very curious thing. The whole solid cube A is to be -looked on merely as a boundary of the tesseract. - -Yet this is exactly analogous to what the plane being would come to in -his study of the solid world. The square A (fig. 96), which the plane -being looks on as a solid existence in his plane world, is merely the -boundary of the cube which he supposes generated by its motion. - -The fact is that we have to recognise that, if there is another -dimension of space, our present idea of a solid body, as one which -has three dimensions only, does not correspond to anything real, -but is the abstract idea of a three-dimensional boundary limiting a -four-dimensional solid, which a four-dimensional being would form. The -plane being’s thought of a square is not the thought of what we should -call a possibly existing real square, but the thought of an abstract -boundary, the face of a cube. - -Let us now take our eight coloured cubes, which form a cube in -space, and ask what additions we must make to them to represent -the simplest collection of four-dimensional bodies—namely, a group -of them of the same extent in every direction. In plane space we -have four squares. In solid space we have eight cubes. So we should -expect in four-dimensional space to have sixteen four-dimensional -bodies-bodies which in four-dimensional space correspond to cubes in -three-dimensional space, and these bodies we call tesseracts. - -Given then the null, white, red, yellow cubes, and those which make up -the block, we notice that we represent perfectly well the extension -in three directions (fig. 98). From the null point of the null cube, -travelling one inch, we come to the white cube; travelling one inch -away we come to the yellow cube; travelling one inch up we come to the -red cube. Now, if there is a fourth dimension, then travelling from the -same null point for one inch in that direction, we must come to the -body lying beyond the null region. - -[Illustration: Fig. 98.] - -I say null region, not cube; for with the introduction of the fourth -dimension each of our cubes must become something different from cubes. -If they are to have existence in the fourth dimension, they must be -“filled up from” in this fourth dimension. - -Now we will assume that as we get a transference from null to white -going in one way, from null to yellow going in another, so going -from null in the fourth direction we have a transference from null -to blue, using thus the colours white, yellow, red, blue, to denote -transferences in each of the four directions—right, away, up, unknown -or fourth dimension. - -[Illustration: Fig. 99. - -A plane being’s representation of a block of eight cubes by two sets of -four squares.] - -Hence, as the plane being must represent the solid regions, he would -come to by going right, as four squares lying in some position in his -plane, arbitrarily chosen, side by side with his original four squares, -so we must represent those eight four-dimensional regions, which we -should come to by going in the fourth dimension from each of our eight -cubes, by eight cubes placed in some arbitrary position relative to our -first eight cubes. - -[Illustration: Fig. 100.] - -Our representation of a block of sixteen tesseracts by two blocks of -eight cubes.[3] - - [3] The eight cubes used here in 2 can be found in the second of the - model blocks. They can be taken out and used. - -Hence, of the two sets of eight cubes, each one will serve us as a -representation of one of the sixteen tesseracts which form one single -block in four-dimensional space. Each cube, as we have it, is a tray, -as it were, against which the real four-dimensional figure rests—just -as each of the squares which the plane being has is a tray, so to -speak, against which the cube it represents could rest. - -If we suppose the cubes to be one inch each way, then the original -eight cubes will give eight tesseracts of the same colours, or the -cubes, extending each one inch in the fourth dimension. - -But after these there come, going on in the fourth dimension, eight -other bodies, eight other tesseracts. These must be there, if we -suppose the four-dimensional body we make up to have two divisions, one -inch each in each of four directions. - -The colour we choose to designate the transference to this second -region in the fourth dimension is blue. Thus, starting from the null -cube and going in the fourth dimension, we first go through one inch of -the null tesseract, then we come to a blue cube, which is the beginning -of a blue tesseract. This blue tesseract stretches one inch farther on -in the fourth dimension. - -Thus, beyond each of the eight tesseracts, which are of the same colour -as the cubes which are their bases, lie eight tesseracts whose colours -are derived from the colours of the first eight by adding blue. Thus— - - Null gives blue - Yellow ” green - Red ” purple - Orange ” brown - White ” light blue - Pink ” light purple - Light yellow ” light green - Ochre ” light brown - -The addition of blue to yellow gives green—this is a natural -supposition to make. It is also natural to suppose that blue added to -red makes purple. Orange and blue can be made to give a brown, by using -certain shades and proportions. And ochre and blue can be made to give -a light brown. - -But the scheme of colours is merely used for getting a definite and -realisable set of names and distinctions visible to the eye. Their -naturalness is apparent to any one in the habit of using colours, and -may be assumed to be justifiable, as the sole purpose is to devise a -set of names which are easy to remember, and which will give us a set -of colours by which diagrams may be made easy of comprehension. No -scientific classification of colours has been attempted. - -Starting, then, with these sixteen colour names, we have a catalogue of -the sixteen tesseracts, which form a four-dimensional block analogous -to the cubic block. But the cube which we can put in space and look at -is not one of the constituent tesseracts; it is merely the beginning, -the solid face, the side, the aspect, of a tesseract. - -We will now proceed to derive a name for each region, point, edge, -plane face, solid and a face of the tesseract. - -The system will be clear, if we look at a representation in the plane -of a tesseract with three, and one with four divisions in its side. - -The tesseract made up of three tesseracts each way corresponds to the -cube made up of three cubes each way, and will give us a complete -nomenclature. - -In this diagram, fig. 101, 1 represents a cube of 27 cubes, each of -which is the beginning of a tesseract. These cubes are represented -simply by their lowest squares, the solid content must be understood. 2 -represents the 27 cubes which are the beginnings of the 27 tesseracts -one inch on in the fourth dimension. These tesseracts are represented -as a block of cubes put side by side with the first block, but in -their proper positions they could not be in space with the first set. 3 -represents 27 cubes (forming a larger cube) which are the beginnings of -the tesseracts, which begin two inches in the fourth direction from our -space and continue another inch. - -[Illustration: Fig. 101.] - - -[Illustration: Fig. 102[4]] - - [4] The coloured plate, figs. 1, 2, 3, shows these relations more - conspicuously. - -In fig. 102, we have the representation of a block of 4 × 4 × 4 × 4 -or 256 tesseracts. They are given in four consecutive sections, each -supposed to be taken one inch apart in the fourth dimension, and so -giving four blocks of cubes, 64 in each block. Here we see, comparing -it with the figure of 81 tesseracts, that the number of the different -regions show a different tendency of increase. By taking five blocks of -five divisions each way this would become even more clear. - -We see, fig. 102, that starting from the point at any corner, the white -coloured regions only extend out in a line. The same is true for the -yellow, red, and blue. With regard to the latter it should be noticed -that the line of blues does not consist in regions next to each other -in the drawing, but in portions which come in in different cubes. -The portions which lie next to one another in the fourth dimension -must always be represented so, when we have a three-dimensional -representation. Again, those regions such as the pink one, go on -increasing in two dimensions. About the pink region this is seen -without going out of the cube itself, the pink regions increase in -length and height, but in no other dimension. In examining these -regions it is sufficient to take one as a sample. - -The purple increases in the same manner, for it comes in in a -succession from below to above in block 2, and in a succession from -block to block in 2 and 3. Now, a succession from below to above -represents a continuous extension upwards, and a succession from block -to block represents a continuous extension in the fourth dimension. -Thus the purple regions increase in two dimensions, the upward and -the fourth, so when we take a very great many divisions, and let each -become very small, the purple region forms a two-dimensional extension. - -In the same way, looking at the regions marked l. b. or light blue, -which starts nearest a corner, we see that the tesseracts occupying -it increase in length from left to right, forming a line, and that -there are as many lines of light blue tesseracts as there are sections -between the first and last section. Hence the light blue tesseracts -increase in number in two ways—in the right and left, and in the fourth -dimension. They ultimately form what we may call a plane surface. - -Now all those regions which contain a mixture of two simple colours, -white, yellow, red, blue, increase in two ways. On the other hand, -those which contain a mixture of three colours increase in three ways. -Take, for instance, the ochre region; this has three colours, white, -yellow, red; and in the cube itself it increases in three ways. - -Now regard the orange region; if we add blue to this we get a brown. -The region of the brown tesseracts extends in two ways on the left of -the second block, No. 2 in the figure. It extends also from left to -right in succession from one section to another, from section 2 to -section 3 in our figure. - -Hence the brown tesseracts increase in number in three dimensions -upwards, to and fro, fourth dimension. Hence they form a cubic, a -three-dimensional region; this region extends up and down, near -and far, and in the fourth direction, but is thin in the direction -from left to right. It is a cube which, when the complete tesseract -is represented in our space, appears as a series of faces on the -successive cubic sections of the tesseract. Compare fig. 103 in which -the middle block, 2, stands as representing a great number of sections -intermediate between 1 and 3. - -In a similar way from the pink region by addition of blue we have -the light purple region, which can be seen to increase in three ways -as the number of divisions becomes greater. The three ways in which -this region of tesseracts extends is up and down, right and left, -fourth dimension. Finally, therefore, it forms a cubic mass of very -small tesseracts, and when the tesseract is given in space sections -it appears on the faces containing the upward and the right and left -dimensions. - -We get then altogether, as three-dimensional regions, ochre, brown, -light purple, light green. - -Finally, there is the region which corresponds to a mixture of all the -colours; there is only one region such as this. It is the one that -springs from ochre by the addition of blue—this colour we call light -brown. - -Looking at the light brown region we see that it increases in four -ways. Hence, the tesseracts of which it is composed increase in -number in each of four dimensions, and the shape they form does not -remain thin in any of the four dimensions. Consequently this region -becomes the solid content of the block of tesseracts, itself; it -is the real four-dimensional solid. All the other regions are then -boundaries of this light brown region. If we suppose the process -of increasing the number of tesseracts and diminishing their size -carried on indefinitely, then the light brown coloured tesseracts -become the whole interior mass, the three-coloured tesseracts become -three-dimensional boundaries, thin in one dimension, and form the -ochre, the brown, the light purple, the light green. The two-coloured -tesseracts become two-dimensional boundaries, thin in two dimensions, -_e.g._, the pink, the green, the purple, the orange, the light blue, -the light yellow. The one-coloured tesseracts become bounding lines, -thin in three dimensions, and the null points become bounding corners, -thin in four dimensions. From these thin real boundaries we can pass in -thought to the abstractions—points, lines, faces, solids—bounding the -four-dimensional solid, which in this case is light brown coloured, and -under this supposition the light brown coloured region is the only real -one, is the only one which is not an abstraction. - -It should be observed that, in taking a square as the representation -of a cube on a plane, we only represent one face, or the section -between two faces. The squares, as drawn by a plane being, are not the -cubes themselves, but represent the faces or the sections of a cube. -Thus in the plane being’s diagram a cube of twenty-seven cubes “null” -represents a cube, but is really, in the normal position, the orange -square of a null cube, and may be called null, orange square. - -A plane being would save himself confusion if he named his -representative squares, not by using the names of the cubes simply, but -by adding to the names of the cubes a word to show what part of a cube -his representative square was. - -Thus a cube null standing against his plane touches it by null orange -face, passing through his plane it has in the plane a square as trace, -which is null white section, if we use the phrase white section to -mean a section drawn perpendicular to the white line. In the same way -the cubes which we take as representative of the tesseract are not -the tesseract itself, but definite faces or sections of it. In the -preceding figures we should say then, not null, but “null tesseract -ochre cube,” because the cube we actually have is the one determined by -the three axes, white, red, yellow. - -There is another way in which we can regard the colour nomenclature of -the boundaries of a tesseract. - -Consider a null point to move tracing out a white line one inch in -length, and terminating in a null point, see fig. 103 or in the -coloured plate. - -Then consider this white line with its terminal points itself to move -in a second dimension, each of the points traces out a line, the line -itself traces out an area, and gives two lines as well, its initial and -its final position. - -Thus, if we call “a region” any element of the figure, such as a point, -or a line, etc., every “region” in moving traces out a new kind of -region, “a higher region,” and gives two regions of its own kind, an -initial and a final position. The “higher region” means a region with -another dimension in it. - -Now the square can move and generate a cube. The square light yellow -moves and traces out the mass of the cube. Letting the addition of -red denote the region made by the motion in the upward direction we -get an ochre solid. The light yellow face in its initial and terminal -positions give the two square boundaries of the cube above and below. -Then each of the four lines of the light yellow square—white, yellow, -and the white, yellow opposite them—trace out a bounding square. So -there are in all six bounding squares, four of these squares being -designated in colour by adding red to the colour of the generating -lines. Finally, each point moving in the up direction gives rise to -a line coloured null + red, or red, and then there are the initial -and terminal positions of the points giving eight points. The number -of the lines is evidently twelve, for the four lines of this light -yellow square give four lines in their initial, four lines in their -final position, while the four points trace out four lines, that is -altogether twelve lines. - -Now the squares are each of them separate boundaries of the cube, while -the lines belong, each of them, to two squares, thus the red line is -that which is common to the orange and pink squares. - -Now suppose that there is a direction, the fourth dimension, which is -perpendicular alike to every one of the space dimensions already used—a -dimension perpendicular, for instance, to up and to right hand, so that -the pink square moving in this direction traces out a cube. - -A dimension, moreover, perpendicular to the up and away directions, -so that the orange square moving in this direction also traces out -a cube, and the light yellow square, too, moving in this direction -traces out a cube. Under this supposition, the whole cube moving in -the unknown dimension, traces out something new—a new kind of volume, -a higher volume. This higher volume is a four-dimensional volume, and -we designate it in colour by adding blue to the colour of that which by -moving generates it. - -It is generated by the motion of the ochre solid, and hence it is -of the colour we call light brown (white, yellow, red, blue, mixed -together). It is represented by a number of sections like 2 in fig. 103. - -Now this light brown higher solid has for boundaries: first, the ochre -cube in its initial position, second, the same cube in its final -position, 1 and 3, fig. 103. Each of the squares which bound the cube, -moreover, by movement in this new direction traces out a cube, so we -have from the front pink faces of the cube, third, a pink blue or -light purple cube, shown as a light purple face on cube 2 in fig. 103, -this cube standing for any number of intermediate sections; fourth, -a similar cube from the opposite pink face; fifth, a cube traced out -by the orange face—this is coloured brown and is represented by the -brown face of the section cube in fig. 103; sixth, a corresponding -brown cube on the right hand; seventh, a cube starting from the light -yellow square below; the unknown dimension is at right angles to this -also. This cube is coloured light yellow and blue or light green; and, -finally, eighth, a corresponding cube from the upper light yellow face, -shown as the light green square at the top of the section cube. - -The tesseract has thus eight cubic boundaries. These completely enclose -it, so that it would be invisible to a four-dimensional being. Now, as -to the other boundaries, just as the cube has squares, lines, points, -as boundaries, so the tesseract has cubes, squares, lines, points, as -boundaries. - -The number of squares is found thus—round the cube are six squares, -these will give six squares in their initial and six in their final -positions. Then each of the twelve lines of the cube trace out a square -in the motion in the fourth dimension. Hence there will be altogether -12 + 12 = 24 squares. - -If we look at any one of these squares we see that it is the meeting -surface of two of the cubic sides. Thus, the red line by its movement -in the fourth dimension, traces out a purple square—this is common -to two cubes, one of which is traced out by the pink square moving -in the fourth dimension, and the other is traced out by the orange -square moving in the same way. To take another square, the light yellow -one, this is common to the ochre cube and the light green cube. The -ochre cube comes from the light yellow square by moving it in the up -direction, the light green cube is made from the light yellow square by -moving it in the fourth dimension. The number of lines is thirty-two, -for the twelve lines of the cube give twelve lines of the tesseract -in their initial position, and twelve in their final position, making -twenty-four, while each of the eight points traces out a line, thus -forming thirty-two lines altogether. - -The lines are each of them common to three cubes, or to three square -faces; take, for instance, the red line. This is common to the orange -face, the pink face, and that face which is formed by moving the red -line in the sixth dimension, namely, the purple face. It is also common -to the ochre cube, the pale purple cube, and the brown cube. - -The points are common to six square faces and to four cubes; thus, -the null point from which we start is common to the three square -faces—pink, light yellow, orange, and to the three square faces made by -moving the three lines white, yellow, red, in the fourth dimension, -namely, the light blue, the light green, the purple faces—that is, to -six faces in all. The four cubes which meet in it are the ochre cube, -the light purple cube, the brown cube, and the light green cube. - -[Illustration: Fig. 103. - -The tesseract, red, white, yellow axes in space. In the lower line the -three rear faces are shown, the interior being removed.] - -[Illustration: Fig. 104. - -The tesseract, red, yellow, blue axes in space, the blue axis running -to the left, opposite faces are coloured identically.] - -A complete view of the tesseract in its various space presentations -is given in the following figures or catalogue cubes, figs. 103-106. -The first cube in each figure represents the view of a tesseract -coloured as described as it begins to pass transverse to our space. -The intermediate figure represents a sectional view when it is partly -through, and the final figure represents the far end as it is just -passing out. These figures will be explained in detail in the next -chapter. - -[Illustration: Fig. 105. - -The tesseract, with red, white, blue axes in space. Opposite faces are -coloured identically.] - -[Illustration: Fig. 106. - -The tesseract, with blue, white, yellow axes in space. The blue axis -runs downward from the base of the ochre cube as it stands originally. -Opposite faces are coloured identically.] - -We have thus obtained a nomenclature for each of the regions of a -tesseract; we can speak of any one of the eight bounding cubes, the -twenty square faces, the thirty-two lines, the sixteen points. - - - - - CHAPTER XIII - - REMARKS ON THE FIGURES - - -An inspection of above figures will give an answer to many questions -about the tesseract. If we have a tesseract one inch each way, then it -can be represented as a cube—a cube having white, yellow, red axes, -and from this cube as a beginning, a volume extending into the fourth -dimension. Now suppose the tesseract to pass transverse to our space, -the cube of the red, yellow, white axes disappears at once, it is -indefinitely thin in the fourth dimension. Its place is occupied by -those parts of the tesseract which lie further away from our space in -the fourth dimension. Each one of these sections will last only for -one moment, but the whole of them will take up some appreciable time -in passing. If we take the rate of one inch a minute the sections will -take the whole of the minute in their passage across our space, they -will take the whole of the minute except the moment which the beginning -cube and the end cube occupy in their crossing our space. In each one -of the cubes, the section cubes, we can draw lines in all directions -except in the direction occupied by the blue line, the fourth -dimension; lines in that direction are represented by the transition -from one section cube to another. Thus to give ourselves an adequate -representation of the tesseract we ought to have a limitless number of -section cubes intermediate between the first bounding cube, the ochre -cube, and the last bounding cube, the other ochre cube. Practically -three intermediate sectional cubes will be found sufficient for most -purposes. We will take then a series of five figures—two terminal -cubes, and three intermediate sections—and show how the different -regions appear in our space when we take each set of three out of the -four axes of the tesseract as lying in our space. - -In fig. 107 initial letters are used for the colours. A reference to -fig. 103 will show the complete nomenclature, which is merely indicated -here. - -[Illustration: Fig. 107.] - -In this figure the tesseract is shown in five stages distant from our -space: first, zero; second, 1/4 in.; third, 2/4 in.; fourth, 3/4 in.; -fifth, 1 in.; which are called _b_0, _b_1, _b_2, _b_3, _b_4, because -they are sections taken at distances 0, 1, 2, 3, 4 quarter inches along -the blue line. All the regions can be named from the first cube, the -_b_0 cube, as before, simply by remembering that transference along -the b axis gives the addition of blue to the colour of the region in -the ochre, the _b_0 cube. In the final cube _b_4, the colouring of the -original _b_0 cube is repeated. Thus the red line moved along the blue -axis gives a red and blue or purple square. This purple square appears -as the three purple lines in the sections _b_1, _b_2, _b_3, taken at -1/4, 2/4, 3/4 of an inch in the fourth dimension. If the tesseract -moves transverse to our space we have then in this particular region, -first of all a red line which lasts for a moment, secondly a purple -line which takes its place. This purple line lasts for a minute—that -is, all of a minute, except the moment taken by the crossing our space -of the initial and final red line. The purple line having lasted for -this period is succeeded by a red line, which lasts for a moment; then -this goes and the tesseract has passed across our space. The final red -line we call red bl., because it is separated from the initial red -line by a distance along the axis for which we use the colour blue. -Thus a line that lasts represents an area duration; is in this mode -of presentation equivalent to a dimension of space. In the same way -the white line, during the crossing our space by the tesseract, is -succeeded by a light blue line which lasts for the inside of a minute, -and as the tesseract leaves our space, having crossed it, the white bl. -line appears as the final termination. - -Take now the pink face. Moved in the blue direction it traces out a -light purple cube. This light purple cube is shown in sections in -_b__{1}, _b__{2}, _b__{3}, and the farther face of this cube in the -blue direction is shown in _b__{4}—a pink face, called pink _b_ because -it is distant from the pink face we began with in the blue direction. -Thus the cube which we colour light purple appears as a lasting square. -The square face itself, the pink face, vanishes instantly the tesseract -begins to move, but the light purple cube appears as a lasting square. -Here also duration is the equivalent of a dimension of space—a lasting -square is a cube. It is useful to connect these diagrams with the views -given in the coloured plate. - -Take again the orange face, that determined by the red and yellow axes; -from it goes a brown cube in the blue direction, for red and yellow -and blue are supposed to make brown. This brown cube is shown in three -sections in the faces _b__{1}, _b__{2}, _b__{3}. In _b__{4} is the -opposite orange face of the brown cube, the face called orange _b_, -for it is distant in the blue direction from the orange face. As the -tesseract passes transverse to our space, we have then in this region -an instantly vanishing orange square, followed by a lasting brown -square, and finally an orange face which vanishes instantly. - -Now, as any three axes will be in our space, let us send the white -axis out into the unknown, the fourth dimension, and take the blue -axis into our known space dimension. Since the white and blue axes are -perpendicular to each other, if the white axis goes out into the fourth -dimension in the positive sense, the blue axis will come into the -direction the white axis occupied, in the negative sense. - -[Illustration: Fig. 108.] - -Hence, not to complicate matters by having to think of two senses in -the unknown direction, let us send the white line into the positive -sense of the fourth dimension, and take the blue one as running in the -negative sense of that direction which the white line has left; let the -blue line, that is, run to the left. We have now the row of figures -in fig. 108. The dotted cube shows where we had a cube when the white -line ran in our space—now it has turned out of our space, and another -solid boundary, another cubic face of the tesseract comes into our -space. This cube has red and yellow axes as before; but now, instead -of a white axis running to the right, there is a blue axis running to -the left. Here we can distinguish the regions by colours in a perfectly -systematic way. The red line traces out a purple square in the -transference along the blue axis by which this cube is generated from -the orange face. This purple square made by the motion of the red line -is the same purple face that we saw before as a series of lines in the -sections _b__{1}, _b__{2}, _b__{3}. Here, since both red and blue axes -are in our space, we have no need of duration to represent the area -they determine. In the motion of the tesseract across space this purple -face would instantly disappear. - -From the orange face, which is common to the initial cubes in fig. 107 -and fig. 108, there goes in the blue direction a cube coloured brown. -This brown cube is now all in our space, because each of its three axes -run in space directions, up, away, to the left. It is the same brown -cube which appeared as the successive faces on the sections _b__{1}, -_b__{2}, _b__{3}. Having all its three axes in our space, it is given -in extension; no part of it needs to be represented as a succession. -The tesseract is now in a new position with regard to our space, and -when it moves across our space the brown cube instantly disappears. - -In order to exhibit the other regions of the tesseract we must remember -that now the white line runs in the unknown dimension. Where shall we -put the sections at distances along the line? Any arbitrary position in -our space will do: there is no way by which we can represent their real -position. - -However, as the brown cube comes off from the orange face to the left, -let us put these successive sections to the left. We can call them -_wh__{0}, _wh__{1}, _wh__{2}, _wh__{3}, _wh__{4}, because they are -sections along the white axis, which now runs in the unknown dimension. - -Running from the purple square in the white direction we find the light -purple cube. This is represented in the sections _wh__{1}, _wh__{2}, -_wh__{3}, _wh__{4}, fig. 108. It is the same cube that is represented -in the sections _b__{1}, _b__{2}, _b__{3}: in fig. 107 the red and -white axes are in our space, the blue out of it; in the other case, the -red and blue are in our space, the white out of it. It is evident that -the face pink _y_, opposite the pink face in fig. 107, makes a cube -shown in squares in _b__{1}, _b__{2}, _b__{3}, _b__{4}, on the opposite -side to the _l_ purple squares. Also the light yellow face at the base -of the cube _b__{0}, makes a light green cube, shown as a series of -base squares. - -The same light green cube can be found in fig. 107. The base square in -_wh__{0} is a green square, for it is enclosed by blue and yellow axes. -From it goes a cube in the white direction, this is then a light green -cube and the same as the one just mentioned as existing in the sections -_b__{0}, _b__{1}, _b__{2}, _b__{3}, _b__{4}. - -The case is, however, a little different with the brown cube. This cube -we have altogether in space in the section _wh__{0}, fig. 108, while -it exists as a series of squares, the left-hand ones, in the sections -_b__{0}, _b__{1}, _b__{2}, _b__{3}, _b__{4}. The brown cube exists as a -solid in our space, as shown in fig. 108. In the mode of representation -of the tesseract exhibited in fig. 107, the same brown cube appears as -a succession of squares. That is, as the tesseract moves across space, -the brown cube would actually be to us a square—it would be merely -the lasting boundary of another solid. It would have no thickness at -all, only extension in two dimensions, and its duration would show its -solidity in three dimensions. - -It is obvious that, if there is a four-dimensional space, matter in -three dimensions only is a mere abstraction; all material objects -must then have a slight four-dimensional thickness. In this case the -above statement will undergo modification. The material cube which is -used as the model of the boundary of a tesseract will have a slight -thickness in the fourth dimension, and when the cube is presented to -us in another aspect, it would not be a mere surface. But it is most -convenient to regard the cubes we use as having no extension at all in -the fourth dimension. This consideration serves to bring out a point -alluded to before, that, if there is a fourth dimension, our conception -of a solid is the conception of a mere abstraction, and our talking -about real three-dimensional objects would seem to a four-dimensional -being as incorrect as a two-dimensional being’s telling about real -squares, real triangles, etc., would seem to us. - -The consideration of the two views of the brown cube shows that any -section of a cube can be looked at by a presentation of the cube in -a different position in four-dimensional space. The brown faces in -_b__{1}, _b__{2}, _b__{3}, are the very same brown sections that would -be obtained by cutting the brown cube, _wh__{0}, across at the right -distances along the blue line, as shown in fig. 108. But as these -sections are placed in the brown cube, _wh__{0}, they come behind one -another in the blue direction. Now, in the sections _wh__{1}, _wh__{2}, -_wh__{3}, we are looking at these sections from the white direction—the -blue direction does not exist in these figures. So we see them in -a direction at right angles to that in which they occur behind one -another in _wh__{0}. There are intermediate views, which would come in -the rotation of a tesseract. These brown squares can be looked at from -directions intermediate between the white and blue axes. It must be -remembered that the fourth dimension is perpendicular equally to all -three space axes. Hence we must take the combinations of the blue axis, -with each two of our three axes, white, red, yellow, in turn. - -In fig. 109 we take red, white, and blue axes in space, sending yellow -into the fourth dimension. If it goes into the positive sense of the -fourth dimension the blue line will come in the opposite direction to -that in which the yellow line ran before. Hence, the cube determined -by the white, red, blue axes, will start from the pink plane and run -towards us. The dotted cube shows where the ochre cube was. When it is -turned out of space, the cube coming towards from its front face is -the one which comes into our space in this turning. Since the yellow -line now runs in the unknown dimension we call the sections _y__{0}, -_y__{1}, _y__{2}, _y__{3}, _y__{4}, as they are made at distances 0, 1, -2, 3, 4, quarter inches along the yellow line. We suppose these cubes -arranged in a line coming towards us—not that that is any more natural -than any other arbitrary series of positions, but it agrees with the -plan previously adopted. - -[Illustration: Fig. 109.] - -The interior of the first cube, _y__{0}, is that derived from pink by -adding blue, or, as we call it, light purple. The faces of the cube are -light blue, purple, pink. As drawn, we can only see the face nearest to -us, which is not the one from which the cube starts—but the face on the -opposite side has the same colour name as the face towards us. - -The successive sections of the series, _y__{0}, _y__{1}, _y__{2}, etc., -can be considered as derived from sections of the _b__{0} cube made at -distances along the yellow axis. What is distant a quarter inch from -the pink face in the yellow direction? This question is answered by -taking a section from a point a quarter inch along the yellow axis in -the cube _b__{0}, fig. 107. It is an ochre section with lines orange -and light yellow. This section will therefore take the place of the -pink face in _y__{1} when we go on in the yellow direction. Thus, the -first section, _y__{1}, will begin from an ochre face with light yellow -and orange lines. The colour of the axis which lies in space towards -us is blue, hence the regions of this section-cube are determined in -nomenclature, they will be found in full in fig. 105. - -There remains only one figure to be drawn, and that is the one in which -the red axis is replaced by the blue. Here, as before, if the red axis -goes out into the positive sense of the fourth dimension, the blue line -must come into our space in the negative sense of the direction which -the red line has left. Accordingly, the first cube will come in beneath -the position of our ochre cube, the one we have been in the habit of -starting with. - -[Illustration: Fig. 110.] - -To show these figures we must suppose the ochre cube to be on a movable -stand. When the red line swings out into the unknown dimension, and the -blue line comes in downwards, a cube appears below the place occupied -by the ochre cube. The dotted cube shows where the ochre cube was. -That cube has gone and a different cube runs downwards from its base. -This cube has white, yellow, and blue axes. Its top is a light yellow -square, and hence its interior is light yellow + blue or light green. -Its front face is formed by the white line moving along the blue axis, -and is therefore light blue, the left-hand side is formed by the yellow -line moving along the blue axis, and therefore green. - -As the red line now runs in the fourth dimension, the successive -sections can he called _r__{0}, _r__{1}, _r__{2}, _r__{3}, _r__{4}, -these letters indicating that at distances 0, 1/4, 2/4, 3/4, 1 inch -along the red axis we take all of the tesseract that can be found in a -three-dimensional space, this three-dimensional space extending not at -all in the fourth dimension, but up and down, right and left, far and -near. - -We can see what should replace the light yellow face of _r__{0}, when -the section _r__{1} comes in, by looking at the cube _b__{0}, fig. 107. -What is distant in it one-quarter of an inch from the light yellow face -in the red direction? It is an ochre section with orange and pink lines -and red points; see also fig. 103. - -This square then forms the top square of _r__{1}. Now we can determine -the nomenclature of all the regions of _r__{1} by considering what -would be formed by the motion of this square along a blue axis. - -But we can adopt another plan. Let us take a horizontal section of -_r__{0}, and finding that section in the figures, of fig. 107 or fig. -103, from them determine what will replace it, going on in the red -direction. - -A section of the _r__{0} cube has green, light blue, green, light blue -sides and blue points. - -Now this square occurs on the base of each of the section figures, -_b__{1}, _b__{2}, etc. In them we see that 1/4 inch in the red -direction from it lies a section with brown and light purple lines and -purple corners, the interior being of light brown. Hence this is the -nomenclature of the section which in _r__{1} replaces the section of -_r__{0} made from a point along the blue axis. - -Hence the colouring as given can be derived. - -We have thus obtained a perfectly named group of tesseracts. We can -take a group of eighty-one of them 3 × 3 × 3 × 3, in four dimensions, -and each tesseract will have its name null, red, white, yellow, blue, -etc., and whatever cubic view we take of them we can say exactly -what sides of the tesseracts we are handling, and how they touch each -other.[5] - - [5] At this point the reader will find it advantageous, if he has the - models, to go through the manipulations described in the appendix. - -Thus, for instance, if we have the sixteen tesseracts shown below, we -can ask how does null touch blue. - -[Illustration: Fig. 111.] - -In the arrangement given in fig. 111 we have the axes white, red, -yellow, in space, blue running in the fourth dimension. Hence we have -the ochre cubes as bases. Imagine now the tesseractic group to pass -transverse to our space—we have first of all null ochre cube, white -ochre cube, etc.; these instantly vanish, and we get the section shown -in the middle cube in fig. 103, and finally, just when the tesseract -block has moved one inch transverse to our space, we have null ochre -cube, and then immediately afterwards the ochre cube of blue comes in. -Hence the tesseract null touches the tesseract blue by its ochre cube, -which is in contact, each and every point of it, with the ochre cube of -blue. - -How does null touch white, we may ask? Looking at the beginning A, fig. -111, where we have the ochre cubes, we see that null ochre touches -white ochre by an orange face. Now let us generate the null and white -tesseracts by a motion in the blue direction of each of these cubes. -Each of them generates the corresponding tesseract, and the plane of -contact of the cubes generates the cube by which the tesseracts are -in contact. Now an orange plane carried along a blue axis generates a -brown cube. Hence null touches white by a brown cube. - -[Illustration: Fig. 112.] - -If we ask again how red touches light blue tesseract, let us rearrange -our group, fig. 112, or rather turn it about so that we have a -different space view of it; let the red axis and the white axis run -up and right, and let the blue axis come in space towards us, then -the yellow axis runs in the fourth dimension. We have then two blocks -in which the bounding cubes of the tesseracts are given, differently -arranged with regard to us—the arrangement is really the same, but it -appears different to us. Starting from the plane of the red and white -axes we have the four squares of the null, white, red, pink tesseracts -as shown in A, on the red, white plane, unaltered, only from them now -comes out towards us the blue axis. Hence we have null, white, red, -pink tesseracts in contact with our space by their cubes which have -the red, white, blue axis in them, that is by the light purple cubes. -Following on these four tesseracts we have that which comes next to -them in the blue direction, that is the four blue, light blue, purple, -light purple. These are likewise in contact with our space by their -light purple cubes, so we see a block as named in the figure, of which -each cube is the one determined by the red, white, blue, axes. - -The yellow line now runs out of space; accordingly one inch on in the -fourth dimension we come to the tesseracts which follow on the eight -named in C, fig. 112, in the yellow direction. - -These are shown in C.y_{1}, fig. 112. Between figure C and C.y_{1} is -that four-dimensional mass which is formed by moving each of the cubes -in C one inch in the fourth dimension—that is, along a yellow axis; for -the yellow axis now runs in the fourth dimension. - -In the block C we observe that red (light purple cube) touches light -blue (light purple cube) by a point. Now these two cubes moving -together remain in contact during the period in which they trace out -the tesseracts red and light blue. This motion is along the yellow -axis, consequently red and light blue touch by a yellow line. - -We have seen that the pink face moved in a yellow direction traces out -a cube; moved in the blue direction it also traces out a cube. Let us -ask what the pink face will trace out if it is moved in a direction -within the tesseract lying equally between the yellow and blue -directions. What section of the tesseract will it make? - -We will first consider the red line alone. Let us take a cube with the -red line in it and the yellow and blue axes. - -The cube with the yellow, red, blue axes is shown in fig. 113. If the -red line is moved equally in the yellow and in the blue direction by -four equal motions of ¼ inch each, it takes the positions 11, 22, 33, -and ends as a red line. - -[Illustration: Fig. 113.] - -Now, the whole of this red, yellow, blue, or brown cube appears as a -series of faces on the successive sections of the tesseract starting -from the ochre cube and letting the blue axis run in the fourth -dimension. Hence the plane traced out by the red line appears as a -series of lines in the successive sections, in our ordinary way of -representing the tesseract; these lines are in different places in each -successive section. - -[Illustration: Fig. 114.] - -Thus drawing our initial cube and the successive sections, calling them -_b__{0}, _b__{1}, _b__{2}, _b__{3}, _b__{4}, fig. 115, we have the red -line subject to this movement appearing in the positions indicated. - -We will now investigate what positions in the tesseract another line in -the pink face assumes when it is moved in a similar manner. - -Take a section of the original cube containing a vertical line, 4, -in the pink plane, fig. 115. We have, in the section, the yellow -direction, but not the blue. - -From this section a cube goes off in the fourth dimension, which is -formed by moving each point of the section in the blue direction. - -[Illustration: Fig. 115.] - -[Illustration: Fig. 116.] - -Drawing this cube we have fig. 116. - -Now this cube occurs as a series of sections in our original -representation of the tesseract. Taking four steps as before this cube -appears as the sections drawn in _b__{0}, _b__{1}, _b__{2}, _b__{3}, -_b__{4}, fig. 117, and if the line 4 is subjected to a movement equal -in the blue and yellow directions, it will occupy the positions -designated by 4, 4_{1}, 4_{2}, 4_{3}, 4_{4}. - -[Illustration: Fig. 117.] - -Hence, reasoning in a similar manner about every line, it is evident -that, moved equally in the blue and yellow directions, the pink plane -will trace out a space which is shown by the series of section planes -represented in the diagram. - -Thus the space traced out by the pink face, if it is moved equally in -the yellow and blue directions, is represented by the set of planes -delineated in Fig. 118, pink face or 0, then 1, 2, 3, and finally pink -face or 4. This solid is a diagonal solid of the tesseract, running -from a pink face to a pink face. Its length is the length of the -diagonal of a square, its side is a square. - -Let us now consider the unlimited space which springs from the pink -face extended. - -This space, if it goes off in the yellow direction, gives us in it the -ochre cube of the tesseract. Thus, if we have the pink face given and a -point in the ochre cube, we have determined this particular space. - -Similarly going off from the pink face in the blue direction is another -space, which gives us the light purple cube of the tesseract in it. And -any point being taken in the light purple cube, this space going off -from the pink face is fixed. - -[Illustration: Fig. 118.] - -The space we are speaking of can be conceived as swinging round the -pink face, and in each of its positions it cuts out a solid figure from -the tesseract, one of which we have seen represented in fig. 118. - -Each of these solid figures is given by one position of the swinging -space, and by one only. Hence in each of them, if one point is taken, -the particular one of the slanting spaces is fixed. Thus we see that -given a plane and a point out of it a space is determined. - -Now, two points determine a line. - -Again, think of a line and a point outside it. Imagine a plane rotating -round the line. At some time in its rotation it passes through the -point. Thus a line and a point, or three points, determine a plane. -And finally four points determine a space. We have seen that a plane -and a point determine a space, and that three points determine a plane; -so four points will determine a space. - -These four points may be any points, and we can take, for instance, the -four points at the extremities of the red, white, yellow, blue axes, in -the tesseract. These will determine a space slanting with regard to the -section spaces we have been previously considering. This space will cut -the tesseract in a certain figure. - -One of the simplest sections of a cube by a plane is that in which the -plane passes through the extremities of the three edges which meet in a -point. We see at once that this plane would cut the cube in a triangle, -but we will go through the process by which a plane being would most -conveniently treat the problem of the determination of this shape, in -order that we may apply the method to the determination of the figure -in which a space cuts a tesseract when it passes through the 4 points -at unit distance from a corner. - -We know that two points determine a line, three points determine a -plane, and given any two points in a plane the line between them lies -wholly in the plane. - -[Illustration: Fig. 119.] - -Let now the plane being study the section made by a plane passing -through the null _r_, null _wh_, and null _y_ points, fig. 119. Looking -at the orange square, which, as usual, we suppose to be initially in -his plane, he sees that the line from null _r_ to null _y_, which is -a line in the section plane, the plane, namely, through the three -extremities of the edges meeting in null, cuts the orange face in an -orange line with null points. This then is one of the boundaries of the -section figure. - -Let now the cube be so turned that the pink face comes in his plane. -The points null _r_ and null _wh_ are now visible. The line between -them is pink with null points, and since this line is common to the -surface of the cube and the cutting plane, it is a boundary of the -figure in which the plane cuts the cube. - -Again, suppose the cube turned so that the light yellow face is in -contact with the plane being’s plane. He sees two points, the null _wh_ -and the null _y_. The line between these lies in the cutting plane. -Hence, since the three cutting lines meet and enclose a portion of -the cube between them, he has determined the figure he sought. It is -a triangle with orange, pink, and light yellow sides, all equal, and -enclosing an ochre area. - -Let us now determine in what figure the space, determined by the four -points, null _r_, null _y_, null _wh_, null _b_, cuts the tesseract. We -can see three of these points in the primary position of the tesseract -resting against our solid sheet by the ochre cube. These three points -determine a plane which lies in the space we are considering, and this -plane cuts the ochre cube in a triangle, the interior of which is -ochre (fig. 119 will serve for this view), with pink, light yellow and -orange sides, and null points. Going in the fourth direction, in one -sense, from this plane we pass into the tesseract, in the other sense -we pass away from it. The whole area inside the triangle is common to -the cutting plane we see, and a boundary of the tesseract. Hence we -conclude that the triangle drawn is common to the tesseract and the -cutting space. - -Now let the ochre cube turn out and the brown cube come in. The dotted -lines show the position the ochre cube has left (fig. 120). - -[Illustration: Fig. 120.] - -Here we see three out of the four points through which the cutting -plane passes, null _r_, null _y_, and null _b_. The plane they -determine lies in the cutting space, and this plane cuts out of the -brown cube a triangle with orange, purple and green sides, and null -points. The orange line of this figure is the same as the orange line -in the last figure. - -Now let the light purple cube swing into our space, towards us, fig. -121. - -[Illustration: Fig. 121.] - -The cutting space which passes through the four points, null _r_, _y_, -_wh_, _b_, passes through the null _r_, _wh_, _b_, and therefore the -plane these determine lies in the cutting space. - -This triangle lies before us. It has a light purple interior and pink, -light blue, and purple edges with null points. - -This, since it is all of the plane that is common to it, and this -bounding of the tesseract, gives us one of the bounding faces of our -sectional figure. The pink line in it is the same as the pink line we -found in the first figure—that of the ochre cube. - -Finally, let the tesseract swing about the light yellow plane, so that -the light green cube comes into our space. It will point downwards. - -The three points, _n.y_, _n.wh_, _n.b_, are in the cutting space, and -the triangle they determine is common to the tesseract and the cutting -space. Hence this boundary is a triangle having a light yellow line, -which is the same as the light yellow line of the first figure, a light -blue line and a green line. - -[Illustration: Fig. 122.] - -We have now traced the cutting space between every set of three that -can be made out of the four points in which it cuts the tesseract, and -have got four faces which all join on to each other by lines. - -[Illustration: Fig. 123.] - -The triangles are shown in fig. 123 as they join on to the triangle -in the ochre cube. But they join on each to the other in an exactly -similar manner; their edges are all identical two and two. They form a -closed figure, a tetrahedron, enclosing a light brown portion which is -the portion of the cutting space which lies inside the tesseract. - -We cannot expect to see this light brown portion, any more than a plane -being could expect to see the inside of a cube if an angle of it were -pushed through his plane. All he can do is to come upon the boundaries -of it in a different way to that in which he would if it passed -straight through his plane. - -Thus in this solid section; the whole interior lies perfectly open in -the fourth dimension. Go round it as we may we are simply looking at -the boundaries of the tesseract which penetrates through our solid -sheet. If the tesseract were not to pass across so far, the triangle -would be smaller; if it were to pass farther, we should have a -different figure, the outlines of which can be determined in a similar -manner. - -The preceding method is open to the objection that it depends rather on -our inferring what must be, than our seeing what is. Let us therefore -consider our sectional space as consisting of a number of planes, each -very close to the last, and observe what is to be found in each plane. - -The corresponding method in the case of two dimensions is as -follows:—The plane being can see that line of the sectional plane -through null _y_, null _wh_, null _r_, which lies in the orange plane. -Let him now suppose the cube and the section plane to pass half way -through his plane. Replacing the red and yellow axes are lines parallel -to them, sections of the pink and light yellow faces. - -[Illustration: Fig. 124.] - -Where will the section plane cut these parallels to the red and yellow -axes? - -Let him suppose the cube, in the position of the drawing, fig. 124, -turned so that the pink face lies against his plane. He can see the -line from the null _r_ point to the null _wh_ point, and can see -(compare fig. 119) that it cuts AB a parallel to his red axis, drawn -at a point half way along the white line, in a point B, half way up. I -shall speak of the axis as having the length of an edge of the cube. -Similarly, by letting the cube turn so that the light yellow square -swings against his plane, he can see (compare fig. 119) that a parallel -to his yellow axis drawn from a point half-way along the white axis, is -cut at half its length by the trace of the section plane in the light -yellow face. - -Hence when the cube had passed half-way through he would have—instead -of the orange line with null points, which he had at first—an ochre -line of half its length, with pink and light yellow points. Thus, as -the cube passed slowly through his plane, he would have a succession -of lines gradually diminishing in length and forming an equilateral -triangle. The whole interior would be ochre, the line from which it -started would be orange. The succession of points at the ends of -the succeeding lines would form pink and light yellow lines and the -final point would be null. Thus looking at the successive lines in -the section plane as it and the cube passed across his plane he would -determine the figure cut out bit by bit. - -Coming now to the section of the tesseract, let us imagine that the -tesseract and its cutting _space_ pass slowly across our space; we can -examine portions of it, and their relation to portions of the cutting -space. Take the section space which passes through the four points, -null _r_, _wh_, _y_, _b_; we can see in the ochre cube (fig. 119) the -plane belonging to this section space, which passes through the three -extremities of the red, white, yellow axes. - -Now let the tesseract pass half way through our space. Instead of our -original axes we have parallels to them, purple, light blue, and green, -each of the same length as the first axes, for the section of the -tesseract is of exactly the same shape as its ochre cube. - -But the sectional space seen at this stage of the transference would -not cut the section of the tesseract in a plane disposed as at first. - -To see where the sectional space would cut these parallels to the -original axes let the tesseract swing so that, the orange face -remaining stationary, the blue line comes in to the left. - -Here (fig. 125) we have the null _r_, _y_, _b_ points, and of the -sectional space all we see is the plane through these three points in -it. - -[Illustration: Fig. 125.] - -In this figure we can draw the parallels to the red and yellow axes and -see that, if they started at a point half way along the blue axis, they -would each be cut at a point so as to be half of their previous length. - -Swinging the tesseract into our space about the pink face of the ochre -cube we likewise find that the parallel to the white axis is cut at -half its length by the sectional space. - -Hence in a section made when the tesseract had passed half across our -space the parallels to the red, white, yellow axes, which are now in -our space, are cut by the section space, each of them half way along, -and for this stage of the traversing motion we should have fig. 126. -The section made of this cube by the plane in which the sectional space -cuts it, is an equilateral triangle with purple, l. blue, green points, -and l. purple, brown, l. green lines. - -[Illustration: Fig. 126.] - -Thus the original ochre triangle, with null points and pink, orange, -light yellow lines, would be succeeded by a triangle coloured in manner -just described. - -This triangle would initially be only a very little smaller than the -original triangle, it would gradually diminish, until it ended in a -point, a null point. Each of its edges would be of the same length. -Thus the successive sections of the successive planes into which we -analyse the cutting space would be a tetrahedron of the description -shown (fig. 123), and the whole interior of the tetrahedron would be -light brown. - -[Illustration: Fig. 127. Front view. The rear faces.] - -In fig. 127 the tetrahedron is represented by means of its faces as -two triangles which meet in the p. line, and two rear triangles which -join on to them, the diagonal of the pink face being supposed to run -vertically upward. - -We have now reached a natural termination. The reader may pursue -the subject in further detail, but will find no essential novelty. -I conclude with an indication as to the manner in which figures -previously given may be used in determining sections by the method -developed above. - -Applying this method to the tesseract, as represented in Chapter IX., -sections made by a space cutting the axes equidistantly at any distance -can be drawn, and also the sections of tesseracts arranged in a block. - -If we draw a plane, cutting all four axes at a point six units distance -from null, we have a slanting space. This space cuts the red, white, -yellow axes in the points LMN (fig. 128), and so in the region of our -space before we go off into the fourth dimension, we have the plane -represented by LMN extended. This is what is common to the slanting -space and our space. - -[Illustration: Fig. 128.] - -This plane cuts the ochre cube in the triangle EFG. - -Comparing this with (fig. 72) _oh_, we see that the hexagon there drawn -is part of the triangle EFG. - -Let us now imagine the tesseract and the slanting space both together -to pass transverse to our space, a distance of one unit, we have in -1_h_ a section of the tesseract, whose axes are parallels to the -previous axes. The slanting space cuts them at a distance of five units -along each. Drawing the plane through these points in 1_h_ it will be -found to cut the cubical section of the tesseract in the hexagonal -figure drawn. In 2_h_ (fig. 72) the slanting space cuts the parallels -to the axes at a distance of four along each, and the hexagonal figure -is the section of this section of the tesseract by it. Finally when -3_h_ comes in the slanting space cuts the axes at a distance of three -along each, and the section is a triangle, of which the hexagon drawn -is a truncated portion. After this the tesseract, which extends only -three units in each of the four dimensions, has completely passed -transverse of our space, and there is no more of it to be cut. Hence, -putting the plane sections together in the right relations, we have -the section determined by the particular slanting space: namely an -octahedron. - - - - -CHAPTER XIV.[6] - -A RECAPITULATION AND EXTENSION OF THE PHYSICAL ARGUMENT - - -There are two directions of inquiry in which the research for the -physical reality of a fourth dimension can be prosecuted. One is the -investigation of the infinitely great, the other is the investigation -of the infinitely small. - - [6] The contents of this chapter are taken from a paper read before - the Philosophical Society of Washington. The mathematical portion - of the paper has appeared in part in the Proceedings of the Royal - Irish Academy under the title, “Cayley’s formulæ of orthogonal - transformation,” Nov. 29th, 1903. - -By the measurement of the angles of vast triangles, whose sides are the -distances between the stars, astronomers have sought to determine if -there is any deviation from the values given by geometrical deduction. -If the angles of a celestial triangle do not together equal two right -angles, there would be an evidence for the physical reality of a fourth -dimension. - -This conclusion deserves a word of explanation. If space is really -four-dimensional, certain conclusions follow which must be brought -clearly into evidence if we are to frame the questions definitely which -we put to Nature. To account for our limitation let us assume a solid -material sheet against which we move. This sheet must stretch alongside -every object in every direction in which it visibly moves. Every -material body must slip or slide along this sheet, not deviating from -contact with it in any motion which we can observe. - -The necessity for this assumption is clearly apparent, if we consider -the analogous case of a suppositionary plane world. If there were -any creatures whose experiences were confined to a plane, we must -account for their limitation. If they were free to move in every space -direction, they would have a three-dimensional motion; hence they must -be physically limited, and the only way in which we can conceive such -a limitation to exist is by means of a material surface against which -they slide. The existence of this surface could only be known to them -indirectly. It does not lie in any direction from them in which the -kinds of motion they know of leads them. If it were perfectly smooth -and always in contact with every material object, there would be no -difference in their relations to it which would direct their attention -to it. - -But if this surface were curved—if it were, say, in the form of a vast -sphere—the triangles they drew would really be triangles of a sphere, -and when these triangles are large enough the angles diverge from -the magnitudes they would have for the same lengths of sides if the -surface were plane. Hence by the measurement of triangles of very great -magnitude a plane being might detect a difference from the laws of a -plane world in his physical world, and so be led to the conclusion that -there was in reality another dimension to space—a third dimension—as -well as the two which his ordinary experience made him familiar with. - -Now, astronomers have thought it worth while to examine the -measurements of vast triangles drawn from one celestial body to another -with a view to determine if there is anything like a curvature in our -space—that is to say, they have tried astronomical measurements to -find out if the vast solid sheet against which, on the supposition of -a fourth dimension, everything slides is curved or not. These results -have been negative. The solid sheet, if it exists, is not curved or, -being curved, has not a sufficient curvature to cause any observable -deviation from the theoretical value of the angles calculated. - -Hence the examination of the infinitely great leads to no decisive -criterion. If it did we should have to decide between the present -theory and that of metageometry. - -Coming now to the prosecution of the inquiry in the direction of -the infinitely small, we have to state the question thus: Our laws -of movement are derived from the examination of bodies which move -in three-dimensional space. All our conceptions are founded on the -supposition of a space which is represented analytically by three -independent axes and variations along them—that is, it is a space in -which there are three independent movements. Any motion possible in it -can be compounded out of these three movements, which we may call: up, -right, away. - -To examine the actions of the very small portions of matter with the -view of ascertaining if there is any evidence in the phenomena for -the supposition of a fourth dimension of space, we must commence by -clearly defining what the laws of mechanics would be on the supposition -of a fourth dimension. It is of no use asking if the phenomena of the -smallest particles of matter are like—we do not know what. We must -have a definite conception of what the laws of motion would be on the -supposition of the fourth dimension, and then inquire if the phenomena -of the activity of the smaller particles of matter resemble the -conceptions which we have elaborated. - -Now, the task of forming these conceptions is by no means one to be -lightly dismissed. Movement in space has many features which differ -entirely from movement on a plane; and when we set about to form the -conception of motion in four dimensions, we find that there is at least -as great a step as from the plane to three-dimensional space. - -I do not say that the step is difficult, but I want to point out -that it must be taken. When we have formed the conception of -four-dimensional motion, we can ask a rational question of Nature. -Before we have elaborated our conceptions we are asking if an unknown -is like an unknown—a futile inquiry. - -As a matter of fact, four-dimensional movements are in every way simple -and more easy to calculate than three-dimensional movements, for -four-dimensional movements are simply two sets of plane movements put -together. - -Without the formation of an experience of four-dimensional bodies, -their shapes and motions, the subject can be but formal—logically -conclusive, not intuitively evident. It is to this logical apprehension -that I must appeal. - -It is perfectly simple to form an experiential familiarity with the -facts of four-dimensional movement. The method is analogous to that -which a plane being would have to adopt to form an experiential -familiarity with three-dimensional movements, and may be briefly summed -up as the formation of a compound sense by means of which duration is -regarded as equivalent to extension. - -Consider a being confined to a plane. A square enclosed by four lines -will be to him a solid, the interior of which can only be examined by -breaking through the lines. If such a square were to pass transverse to -his plane, it would immediately disappear. It would vanish, going in no -direction to which he could point. - -If, now, a cube be placed in contact with his plane, its surface of -contact would appear like the square which we have just mentioned. -But if it were to pass transverse to his plane, breaking through it, -it would appear as a lasting square. The three-dimensional matter will -give a lasting appearance in circumstances under which two-dimensional -matter will at once disappear. - -Similarly, a four-dimensional cube, or, as we may call it, a tesseract, -which is generated from a cube by a movement of every part of the cube -in a fourth direction at right angles to each of the three visible -directions in the cube, if it moved transverse to our space, would -appear as a lasting cube. - -A cube of three-dimensional matter, since it extends to no distance at -all in the fourth dimension, would instantly disappear, if subjected -to a motion transverse to our space. It would disappear and be gone, -without it being possible to point to any direction in which it had -moved. - -All attempts to visualise a fourth dimension are futile. It must be -connected with a time experience in three space. - -The most difficult notion for a plane being to acquire would be that of -rotation about a line. Consider a plane being facing a square. If he -were told that rotation about a line were possible, he would move his -square this way and that. A square in a plane can rotate about a point, -but to rotate about a line would seem to the plane being perfectly -impossible. How could those parts of his square which were on one side -of an edge come to the other side without the edge moving? He could -understand their reflection in the edge. He could form an idea of the -looking-glass image of his square lying on the opposite side of the -line of an edge, but by no motion that he knows of can he make the -actual square assume that position. The result of the rotation would be -like reflection in the edge, but it would be a physical impossibility -to produce it in the plane. - -The demonstration of rotation about a line must be to him purely -formal. If he conceived the notion of a cube stretching out in an -unknown direction away from his plane, then he can see the base of -it, his square in the plane, rotating round a point. He can likewise -apprehend that every parallel section taken at successive intervals in -the unknown direction rotates in like manner round a point. Thus he -would come to conclude that the whole body rotates round a line—the -line consisting of the succession of points round which the plane -sections rotate. Thus, given three axes, _x_, _y_, _z_, if _x_ rotates -to take the place of _y_, and _y_ turns so as to point to negative -_x_, then the third axis remaining unaffected by this turning is the -axis about which the rotation takes place. This, then, would have to be -his criterion of the axis of a rotation—that which remains unchanged -when a rotation of every plane section of a body takes place. - -There is another way in which a plane being can think about -three-dimensional movements; and, as it affords the type by which we -can most conveniently think about four-dimensional movements, it will -be no loss of time to consider it in detail. - -[Illustration: Fig. 1 (129).] - -We can represent the plane being and his object by figures cut out of -paper, which slip on a smooth surface. The thickness of these bodies -must be taken as so minute that their extension in the third dimension -escapes the observation of the plane being, and he thinks about them -as if they were mathematical plane figures in a plane instead of being -material bodies capable of moving on a plane surface. Let A_x_, A_y_ -be two axes and ABCD a square. As far as movements in the plane are -concerned, the square can rotate about a point A, for example. It -cannot rotate about a side, such as AC. - -But if the plane being is aware of the existence of a third dimension -he can study the movements possible in the ample space, taking his -figure portion by portion. - -His plane can only hold two axes. But, since it can hold two, he is -able to represent a turning into the third dimension if he neglects one -of his axes and represents the third axis as lying in his plane. He can -make a drawing in his plane of what stands up perpendicularly from his -plane. Let A_z_ be the axis, which stands perpendicular to his plane at -A. He can draw in his plane two lines to represent the two axes, A_x_ -and A_z_. Let Fig. 2 be this drawing. Here the _z_ axis has taken the -place of the _y_ axis, and the plane of A_x_ A_z_ is represented in his -plane. In this figure all that exists of the square ABCD will be the -line AB. - -[Illustration: Fig. 2 (130).] - -The square extends from this line in the _y_ direction, but more of -that direction is represented in Fig. 2. The plane being can study the -turning of the line AB in this diagram. It is simply a case of plane -turning around the point A. The line AB occupies intermediate portions -like AB_{1} and after half a revolution will lie on A_x_ produced -through A. - -Now, in the same way, the plane being can take another point, A´, and -another line, A´B´, in his square. He can make the drawing of the two -directions at A´, one along A´B´, the other perpendicular to his plane. -He will obtain a figure precisely similar to Fig. 2, and will see that, -as AB can turn around A, so A´C´ around A. - -In this turning AB and A´B´ would not interfere with each other, as -they would if they moved in the plane around the separate points A and -A´. - -Hence the plane being would conclude that a rotation round a line was -possible. He could see his square as it began to make this turning. He -could see it half way round when it came to lie on the opposite side of -the line AC. But in intermediate portions he could not see it, for it -runs out of the plane. - -Coming now to the question of a four-dimensional body, let us conceive -of it as a series of cubic sections, the first in our space, the rest -at intervals, stretching away from our space in the unknown direction. - -We must not think of a four-dimensional body as formed by moving a -three-dimensional body in any direction which we can see. - -Refer for a moment to Fig. 3. The point A, moving to the right, traces -out the line AC. The line AC, moving away in a new direction, traces -out the square ACEG at the base of the cube. The square AEGC, moving -in a new direction, will trace out the cube ACEGBDHF. The vertical -direction of this last motion is not identical with any motion possible -in the plane of the base of the cube. It is an entirely new direction, -at right angles to every line that can be drawn in the base. To trace -out a tesseract the cube must move in a new direction—a direction at -right angles to any and every line that can be drawn in the space of -the cube. - -The cubic sections of the tesseract are related to the cube we see, as -the square sections of the cube are related to the square of its base -which a plane being sees. - -Let us imagine the cube in our space, which is the base of a tesseract, -to turn about one of its edges. The rotation will carry the whole body -with it, and each of the cubic sections will rotate. The axis we see -in our space will remain unchanged, and likewise the series of axes -parallel to it about which each of the parallel cubic sections rotates. -The assemblage of all of these is a plane. - -Hence in four dimensions a body rotates about a plane. There is no such -thing as rotation round an axis. - -We may regard the rotation from a different point of view. Consider -four independent axes each at right angles to all the others, drawn in -a four-dimensional body. Of these four axes we can see any three. The -fourth extends normal to our space. - -Rotation is the turning of one axis into a second, and the second -turning to take the place of the negative of the first. It involves -two axes. Thus, in this rotation of a four-dimensional body, two axes -change and two remain at rest. Four-dimensional rotation is therefore a -turning about a plane. - -As in the case of a plane being, the result of rotation about a -line would appear as the production of a looking-glass image of the -original object on the other side of the line, so to us the result -of a four-dimensional rotation would appear like the production of a -looking-glass image of a body on the other side of a plane. The plane -would be the axis of the rotation, and the path of the body between its -two appearances would be unimaginable in three-dimensional space. - -[Illustration: Fig. 3 (131).] - -Let us now apply the method by which a plane being could examine -the nature of rotation about a line in our examination of rotation -about a plane. Fig. 3 represents a cube in our space, the three axes -_x_, _y_, _z_ denoting its three dimensions. Let _w_ represent the -fourth dimension. Now, since in our space we can represent any three -dimensions, we can, if we choose, make a representation of what is -in the space determined by the three axes _x_, _z_, _w_. This is a -three-dimensional space determined by two of the axes we have drawn, -_x_ and _z_, and in place of _y_ the fourth axis, _w_. We cannot, -keeping _x_ and _z_, have both _y_ and _w_ in our space; so we will -let _y_ go and draw _w_ in its place. What will be our view of the cube? - -Evidently we shall have simply the square that is in the plane of _xz_, -the square ACDB. The rest of the cube stretches in the _y_ direction, -and, as we have none of the space so determined, we have only the face -of the cube. This is represented in fig. 4. - -[Illustration: Fig. 4 (132).] - -Now, suppose the whole cube to be turned from the _x_ to the _w_ -direction. Conformably with our method, we will not take the whole of -the cube into consideration at once, but will begin with the face ABCD. - -Let this face begin to turn. Fig. 5 represents one of the positions it -will occupy; the line AB remains on the _z_ axis. The rest of the face -extends between the _x_ and the _w_ direction. - -[Illustration: Fig. 5 (133).] - -Now, since we can take any three axes, let us look at what lies in the -space of _zyw_, and examine the turning there. We must now let the _z_ -axis disappear and let the _w_ axis run in the direction in which the -_z_ ran. - -Making this representation, what do we see of the cube? Obviously we -see only the lower face. The rest of the cube lies in the space of -_xyz_. In the space of _xyz_ we have merely the base of the cube lying -in the plane of _xy_, as shown in fig. 6. - -[Illustration: Fig. 6 (134).] - -Now let the _x_ to _w_ turning take place. The square ACEG will turn -about the line AE. This edge will remain along the _y_ axis and will be -stationary, however far the square turns. - -Thus, if the cube be turned by an _x_ to _w_ turning, both the edge AB -and the edge AC remain stationary; hence the whole face ABEF in the -_yz_ plane remains fixed. The turning has taken place about the face -ABEF. - -[Illustration: Fig. 7 (135).] - -Suppose this turning to continue till AC runs to the left from -A. The cube will occupy the position shown in fig. 8. This is -the looking-glass image of the cube in fig. 3. By no rotation in -three-dimensional space can the cube be brought from the position in -fig. 3 to that shown in fig. 8. - -[Illustration: Fig. 8 (136).] - -We can think of this turning as a turning of the face ABCD about AB, -and a turning of each section parallel to ABCD round the vertical line -in which it intersects the face ABEF, the space in which the turning -takes place being a different one from that in which the cube lies. - -One of the conditions, then, of our inquiry in the direction of the -infinitely small is that we form the conception of a rotation about -a plane. The production of a body in a state in which it presents -the appearance of a looking-glass image of its former state is the -criterion for a four-dimensional rotation. - -There is some evidence for the occurrence of such transformations -of bodies in the change of bodies from those which produce a -right-handed polarisation of light to those which produce a left-handed -polarisation; but this is not a point to which any very great -importance can be attached. - -Still, in this connection, let me quote a remark from Prof. John G. -McKendrick’s address on Physiology before the British Association -at Glasgow. Discussing the possibility of the hereditary production -of characteristics through the material structure of the ovum, he -estimates that in it there exist 12,000,000,000 biophors, or ultimate -particles of living matter, a sufficient number to account for -hereditary transmission, and observes: “Thus it is conceivable that -vital activities may also be determined by the kind of motion that -takes place in the molecules of that which we speak of as living -matter. It may be different in kind from some of the motions known to -physicists, and it is conceivable that life may be the transmission -to dead matter, the molecules of which have already a special kind of -motion, of a form of motion _sui generis_.” - -Now, in the realm of organic beings symmetrical structures—those with a -right and left symmetry—are everywhere in evidence. Granted that four -dimensions exist, the simplest turning produces the image form, and by -a folding-over structures could be produced, duplicated right and left, -just as is the case of symmetry in a plane. - -Thus one very general characteristic of the forms of organisms could -be accounted for by the supposition that a four-dimensional motion was -involved in the process of life. - -But whether four-dimensional motions correspond in other respects to -the physiologist’s demand for a special kind of motion, or not, I -do not know. Our business is with the evidence for their existence -in physics. For this purpose it is necessary to examine into the -significance of rotation round a plane in the case of extensible and of -fluid matter. - -Let us dwell a moment longer on the rotation of a rigid body. Looking -at the cube in fig. 3, which turns about the face of ABFE, we see that -any line in the face can take the place of the vertical and horizontal -lines we have examined. Take the diagonal line AF and the section -through it to GH. The portions of matter which were on one side of AF -in this section in fig. 3 are on the opposite side of it in fig. 8. -They have gone round the line AF. Thus the rotation round a face can be -considered as a number of rotations of sections round parallel lines in -it. - -The turning about two different lines is impossible in -three-dimensional space. To take another illustration, suppose A and -B are two parallel lines in the _xy_ plane, and let CD and EF be two -rods crossing them. Now, in the space of _xyz_ if the rods turn round -the lines A and B in the same direction they will make two independent -circles. - -When the end F is going down the end C will be coming up. They will -meet and conflict. - -[Illustration: Fig. 9 (137).] - -But if we rotate the rods about the plane of AB by the _z_ to _w_ -rotation these movements will not conflict. Suppose all the figure -removed with the exception of the plane _xz_, and from this plane draw -the axis of _w_, so that we are looking at the space of _xzw_. - -Here, fig. 10, we cannot see the lines A and B. We see the points G and -H, in which A and B intercept the _x_ axis, but we cannot see the lines -themselves, for they run in the _y_ direction, and that is not in our -drawing. - -Now, if the rods move with the _z_ to _w_ rotation they will turn in -parallel planes, keeping their relative positions. The point D, for -instance, will describe a circle. At one time it will be above the line -A, at another time below it. Hence it rotates round A. - -[Illustration: Fig. 10 (138).] - -Not only two rods but any number of rods crossing the plane will move -round it harmoniously. We can think of this rotation by supposing the -rods standing up from one line to move round that line and remembering -that it is not inconsistent with this rotation for the rods standing up -along another line also to move round it, the relative positions of all -the rods being preserved. Now, if the rods are thick together, they may -represent a disk of matter, and we see that a disk of matter can rotate -round a central plane. - -Rotation round a plane is exactly analogous to rotation round an axis -in three dimensions. If we want a rod to turn round, the ends must be -free; so if we want a disk of matter to turn round its central plane -by a four-dimensional turning, all the contour must be free. The whole -contour corresponds to the ends of the rod. Each point of the contour -can be looked on as the extremity of an axis in the body, round each -point of which there is a rotation of the matter in the disk. - -If the one end of a rod be clamped, we can twist the rod, but not turn -it round; so if any part of the contour of a disk is clamped we can -impart a twist to the disk, but not turn it round its central plane. In -the case of extensible materials a long, thin rod will twist round its -axis, even when the axis is curved, as, for instance, in the case of a -ring of India rubber. - -In an analogous manner, in four dimensions we can have rotation round -a curved plane, if I may use the expression. A sphere can be turned -inside out in four dimensions. - -[Illustration: Fig. 11 (139).] - -Let fig. 11 represent a spherical surface, on each side of which a -layer of matter exists. The thickness of the matter is represented by -the rods CD and EF, extending equally without and within. - -[Illustration: Fig. 12 (140).] - -Now, take the section of the sphere by the _yz_ plane we have a -circle—fig. 12. Now, let the _w_ axis be drawn in place of the _x_ axis -so that we have the space of _yzw_ represented. In this space all that -there will be seen of the sphere is the circle drawn. - -Here we see that there is no obstacle to prevent the rods turning -round. If the matter is so elastic that it will give enough for the -particles at E and C to be separated as they are at F and D, they -can rotate round to the position D and F, and a similar motion is -possible for all other particles. There is no matter or obstacle to -prevent them from moving out in the _w_ direction, and then on round -the circumference as an axis. Now, what will hold for one section will -hold for all, as the fourth dimension is at right angles to all the -sections which can be made of the sphere. - -We have supposed the matter of which the sphere is composed to be -three-dimensional. If the matter had a small thickness in the fourth -dimension, there would be a slight thickness in fig. 12 above the -plane of the paper—a thickness equal to the thickness of the matter -in the fourth dimension. The rods would have to be replaced by thin -slabs. But this would make no difference as to the possibility of the -rotation. This motion is discussed by Newcomb in the first volume of -the _American Journal of Mathematics_. - -Let us now consider, not a merely extensible body, but a liquid one. A -mass of rotating liquid, a whirl, eddy, or vortex, has many remarkable -properties. On first consideration we should expect the rotating mass -of liquid immediately to spread off and lose itself in the surrounding -liquid. The water flies off a wheel whirled round, and we should expect -the rotating liquid to be dispersed. But see the eddies in a river -strangely persistent. The rings that occur in puffs of smoke and last -so long are whirls or vortices curved round so that their opposite ends -join together. A cyclone will travel over great distances. - -Helmholtz was the first to investigate the properties of vortices. -He studied them as they would occur in a perfect fluid—that is, one -without friction of one moving portion or another. In such a medium -vortices would be indestructible. They would go on for ever, altering -their shape, but consisting always of the same portion of the fluid. -But a straight vortex could not exist surrounded entirely by the fluid. -The ends of a vortex must reach to some boundary inside or outside the -fluid. - -A vortex which is bent round so that its opposite ends join is capable -of existing, but no vortex has a free end in the fluid. The fluid -round the vortex is always in motion, and one produces a definite -movement in another. - -Lord Kelvin has proposed the hypothesis that portions of a fluid -segregated in vortices account for the origin of matter. The properties -of the ether in respect of its capacity of propagating disturbances -can be explained by the assumption of vortices in it instead of by a -property of rigidity. It is difficult to conceive, however, of any -arrangement of the vortex rings and endless vortex filaments in the -ether. - -Now, the further consideration of four-dimensional rotations shows the -existence of a kind of vortex which would make an ether filled with a -homogeneous vortex motion easily thinkable. - -To understand the nature of this vortex, we must go on and take a -step by which we accept the full significance of the four-dimensional -hypothesis. Granted four-dimensional axes, we have seen that a rotation -of one into another leaves two unaltered, and these two form the axial -plane about which the rotation takes place. But what about these two? -Do they necessarily remain motionless? There is nothing to prevent a -rotation of these two, one into the other, taking place concurrently -with the first rotation. This possibility of a double rotation deserves -the most careful attention, for it is the kind of movement which is -distinctly typical of four dimensions. - -Rotation round a plane is analogous to rotation round an axis. But in -three-dimensional space there is no motion analogous to the double -rotation, in which, while axis 1 changes into axis 2, axis 3 changes -into axis 4. - -Consider a four-dimensional body, with four independent axes, _x_, -_y_, _z_, _w_. A point in it can move in only one direction at a given -moment. If the body has a velocity of rotation by which the _x_ axis -changes into the _y_ axis and all parallel sections move in a similar -manner, then the point will describe a circle. If, now, in addition -to the rotation by which the _x_ axis changes into the _y_ axis the -body has a rotation by which the _z_ axis turns into the _w_ axis, the -point in question will have a double motion in consequence of the two -turnings. The motions will compound, and the point will describe a -circle, but not the same circle which it would describe in virtue of -either rotation separately. - -We know that if a body in three-dimensional space is given two -movements of rotation they will combine into a single movement of -rotation round a definite axis. It is in no different condition -from that in which it is subjected to one movement of rotation. The -direction of the axis changes; that is all. The same is not true about -a four-dimensional body. The two rotations, _x_ to _y_ and _z_ to _w_, -are independent. A body subject to the two is in a totally different -condition to that which it is in when subject to one only. When subject -to a rotation such as that of _x_ to _y_, a whole plane in the body, -as we have seen, is stationary. When subject to the double rotation -no part of the body is stationary except the point common to the two -planes of rotation. - -If the two rotations are equal in velocity, every point in the body -describes a circle. All points equally distant from the stationary -point describe circles of equal size. - -We can represent a four-dimensional sphere by means of two diagrams, -in one of which we take the three axes, _x_, _y_, _z_; in the -other the axes _x_, _w_, and _z_. In fig. 13 we have the view of a -four-dimensional sphere in the space of _xyz_. Fig. 13 shows all that -we can see of the four sphere in the space of _xyz_, for it represents -all the points in that space, which are at an equal distance from the -centre. - -Let us now take the _xz_ section, and let the axis of _w_ take the -place of the _y_ axis. Here, in fig. 14, we have the space of _xzw_. -In this space we have to take all the points which are at the same -distance from the centre, consequently we have another sphere. If we -had a three-dimensional sphere, as has been shown before, we should -have merely a circle in the _xzw_ space, the _xz_ circle seen in the -space of _xzw_. But now, taking the view in the space of _xzw_, we have -a sphere in that space also. In a similar manner, whichever set of -three axes we take, we obtain a sphere. - -[Illustration: _Showing axes xyz_ -Fig. 13 (141).] - -[Illustration: _Showing axes xwz_ -Fig. 14 (142).] - -In fig. 13, let us imagine the rotation in the direction _xy_ to be -taking place. The point _x_ will turn to _y_, and _p_ to _p´_. The axis -_zz´_ remains stationary, and this axis is all of the plane _zw_ which -we can see in the space section exhibited in the figure. - -In fig. 14, imagine the rotation from _z_ to _w_ to be taking place. -The _w_ axis now occupies the position previously occupied by the _y_ -axis. This does not mean that the _w_ axis can coincide with the _y_ -axis. It indicates that we are looking at the four-dimensional sphere -from a different point of view. Any three-space view will show us three -axes, and in fig. 14 we are looking at _xzw_. - -The only part that is identical in the two diagrams is the circle of -the _x_ and _z_ axes, which axes are contained in both diagrams. Thus -the plane _zxz´_ is the same in both, and the point _p_ represents the -same point in both diagrams. Now, in fig. 14 let the _zw_ rotation -take place, the _z_ axis will turn toward the point _w_ of the _w_ -axis, and the point _p_ will move in a circle about the point _x_. - -Thus in fig. 13 the point _p_ moves in a circle parallel to the _xy_ -plane; in fig. 14 it moves in a circle parallel to the _zw_ plane, -indicated by the arrow. - -Now, suppose both of these independent rotations compounded, the point -_p_ will move in a circle, but this circle will coincide with neither -of the circles in which either one of the rotations will take it. The -circle the point _p_ will move in will depend on its position on the -surface of the four sphere. - -In this double rotation, possible in four-dimensional space, there -is a kind of movement totally unlike any with which we are familiar -in three-dimensional space. It is a requisite preliminary to the -discussion of the behaviour of the small particles of matter, -with a view to determining whether they show the characteristics -of four-dimensional movements, to become familiar with the main -characteristics of this double rotation. And here I must rely on a -formal and logical assent rather than on the intuitive apprehension, -which can only be obtained by a more detailed study. - -In the first place this double rotation consists in two varieties or -kinds, which we will call the A and B kinds. Consider four axes, _x_, -_y_, _z_, _w_. The rotation of _x_ to _y_ can be accompanied with the -rotation of _z_ to _w_. Call this the A kind. - -But also the rotation of _x_ to _y_ can be accompanied by the rotation, -of not _z_ to _w_, but _w_ to _z_. Call this the B kind. - -They differ in only one of the component rotations. One is not the -negative of the other. It is the semi-negative. The opposite of an -_x_ to _y_, _z_ to _w_ rotation would be _y_ to _x_, _w_ to _z_. The -semi-negative is _x_ to _y_ and _w_ to _z_. - -If four dimensions exist and we cannot perceive them, because the -extension of matter is so small in the fourth dimension that all -movements are withheld from direct observation except those which are -three-dimensional, we should not observe these double rotations, but -only the effects of them in three-dimensional movements of the type -with which we are familiar. - -If matter in its small particles is four-dimensional, we should expect -this double rotation to be a universal characteristic of the atoms -and molecules, for no portion of matter is at rest. The consequences -of this corpuscular motion can be perceived, but only under the form -of ordinary rotation or displacement. Thus, if the theory of four -dimensions is true, we have in the corpuscles of matter a whole world -of movement, which we can never study directly, but only by means of -inference. - -The rotation A, as I have defined it, consists of two equal -rotations—one about the plane of _zw_, the other about the plane -of _xy_. It is evident that these rotations are not necessarily -equal. A body may be moving with a double rotation, in which these -two independent components are not equal; but in such a case we can -consider the body to be moving with a composite rotation—a rotation of -the A or B kind and, in addition, a rotation about a plane. - -If we combine an A and a B movement, we obtain a rotation about a -plane; for, the first being _x_ to _y_ and _z_ to _w_, and the second -being _x_ to _y_ and _w_ to _z_, when they are put together the _z_ -to _w_ and _w_ to _z_ rotations neutralise each other, and we obtain -an _x_ to _y_ rotation only, which is a rotation about the plane of -_zw_. Similarly, if we take a B rotation, _y_ to _x_ and _z_ to _w_, -we get, on combining this with the A rotation, a rotation of _z_ to -_w_ about the _xy_ plane. In this case the plane of rotation is in the -three-dimensional space of _xyz_, and we have—what has been described -before—a twisting about a plane in our space. - -Consider now a portion of a perfect liquid having an A motion. It -can be proved that it possesses the properties of a vortex. It -forms a permanent individuality—a separated-out portion of the -liquid—accompanied by a motion of the surrounding liquid. It has -properties analogous to those of a vortex filament. But it is not -necessary for its existence that its ends should reach the boundary of -the liquid. It is self-contained and, unless disturbed, is circular in -every section. - -[Illustration: Fig. 15 (143).] - -If we suppose the ether to have its properties of transmitting -vibration given it by such vortices, we must inquire how they lie -together in four-dimensional space. Placing a circular disk on a plane -and surrounding it by six others, we find that if the central one is -given a motion of rotation, it imparts to the others a rotation which -is antagonistic in every two adjacent ones. If A goes round, as shown -by the arrow, B and C will be moving in opposite ways, and each tends -to destroy the motion of the other. - -Now, if we suppose spheres to be arranged in a corresponding manner -in three-dimensional space, they will be grouped in figures which are -for three-dimensional space what hexagons are for plane space. If a -number of spheres of soft clay be pressed together, so as to fill up -the interstices, each will assume the form of a fourteen-sided figure -called a tetrakaidecagon. - -Now, assuming space to be filled with such tetrakaidecagons, and -placing a sphere in each, it will be found that one sphere is touched -by eight others. The remaining six spheres of the fourteen which -surround the central one will not touch it, but will touch three of -those in contact with it. Hence, if the central sphere rotates, it -will not necessarily drive those around it so that their motions will -be antagonistic to each other, but the velocities will not arrange -themselves in a systematic manner. - -In four-dimensional space the figure which forms the next term of the -series hexagon, tetrakaidecagon, is a thirty-sided figure. It has for -its faces ten solid tetrakaidecagons and twenty hexagonal prisms. Such -figures will exactly fill four-dimensional space, five of them meeting -at every point. If, now, in each of these figures we suppose a solid -four-dimensional sphere to be placed, any one sphere is surrounded by -thirty others. Of these it touches ten, and, if it rotates, it drives -the rest by means of these. Now, if we imagine the central sphere to be -given an A or a B rotation, it will turn the whole mass of sphere round -in a systematic manner. Suppose four-dimensional space to be filled -with such spheres, each rotating with a double rotation, the whole mass -would form one consistent system of motion, in which each one drove -every other one, with no friction or lagging behind. - -Every sphere would have the same kind of rotation. In three-dimensional -space, if one body drives another round the second body rotates -with the opposite kind of rotation; but in four-dimensional space -these four-dimensional spheres would each have the double negative -of the rotation of the one next it, and we have seen that the -double negative of an A or B rotation is still an A or B rotation. -Thus four-dimensional space could be filled with a system of -self-preservative living energy. If we imagine the four-dimensional -spheres to be of liquid and not of solid matter, then, even if the -liquid were not quite perfect and there were a slight retarding effect -of one vortex on another, the system would still maintain itself. - -In this hypothesis we must look on the ether as possessing energy, -and its transmission of vibrations, not as the conveying of a motion -imparted from without, but as a modification of its own motion. - -We are now in possession of some of the conceptions of four-dimensional -mechanics, and will turn aside from the line of their development -to inquire if there is any evidence of their applicability to the -processes of nature. - -Is there any mode of motion in the region of the minute which, giving -three-dimensional movements for its effect, still in itself escapes the -grasp of our mechanical theories? I would point to electricity. Through -the labours of Faraday and Maxwell we are convinced that the phenomena -of electricity are of the nature of the stress and strain of a medium; -but there is still a gap to be bridged over in their explanation—the -laws of elasticity, which Maxwell assumes, are not those of ordinary -matter. And, to take another instance: a magnetic pole in the -neighbourhood of a current tends to move. Maxwell has shown that the -pressures on it are analogous to the velocities in a liquid which would -exist if a vortex took the place of the electric current: but we cannot -point out the definite mechanical explanation of these pressures. There -must be some mode of motion of a body or of the medium in virtue of -which a body is said to be electrified. - -Take the ions which convey charges of electricity 500 times greater in -proportion to their mass than are carried by the molecules of hydrogen -in electrolysis. In respect of what motion can these ions be said to -be electrified? It can be shown that the energy they possess is not -energy of rotation. Think of a short rod rotating. If it is turned -over it is found to be rotating in the opposite direction. Now, if -rotation in one direction corresponds to positive electricity, rotation -in the opposite direction corresponds to negative electricity, and the -smallest electrified particles would have their charges reversed by -being turned over—an absurd supposition. - -If we fix on a mode of motion as a definition of electricity, we must -have two varieties of it, one for positive and one for negative; and a -body possessing the one kind must not become possessed of the other by -any change in its position. - -All three-dimensional motions are compounded of rotations and -translations, and none of them satisfy this first condition for serving -as a definition of electricity. - -But consider the double rotation of the A and B kinds. A body rotating -with the A motion cannot have its motion transformed into the B kind -by being turned over in any way. Suppose a body has the rotation _x_ -to _y_ and _z_ to _w_. Turning it about the _xy_ plane, we reverse the -direction of the motion _x_ to _y_. But we also reverse the _z_ to _w_ -motion, for the point at the extremity of the positive _z_ axis is -now at the extremity of the negative _z_ axis, and since we have not -interfered with its motion it goes in the direction of position _w_. -Hence we have _y_ to _x_ and _w_ to _z_, which is the same as _x_ to -_y_ and _z_ to _w_. Thus both components are reversed, and there is the -A motion over again. The B kind is the semi-negative, with only one -component reversed. - -Hence a system of molecules with the A motion would not destroy it in -one another, and would impart it to a body in contact with them. Thus A -and B motions possess the first requisite which must be demanded in any -mode of motion representative of electricity. - -Let us trace out the consequences of defining positive electricity as -an A motion and negative electricity as a B motion. The combination of -positive and negative electricity produces a current. Imagine a vortex -in the ether of the A kind and unite with this one of the B kind. An -A motion and B motion produce rotation round a plane, which is in the -ether a vortex round an axial surface. It is a vortex of the kind we -represent as a part of a sphere turning inside out. Now such a vortex -must have its rim on a boundary of the ether—on a body in the ether. - -Let us suppose that a conductor is a body which has the property of -serving as the terminal abutment of such a vortex. Then the conception -we must form of a closed current is of a vortex sheet having its edge -along the circuit of the conducting wire. The whole wire will then be -like the centres on which a spindle turns in three-dimensional space, -and any interruption of the continuity of the wire will produce a -tension in place of a continuous revolution. - -As the direction of the rotation of the vortex is from a three-space -direction into the fourth dimension and back again, there will be no -direction of flow to the current; but it will have two sides, according -to whether _z_ goes to _w_ or _z_ goes to negative _w_. - -We can draw any line from one part of the circuit to another; then the -ether along that line is rotating round its points. - -This geometric image corresponds to the definition of an electric -circuit. It is known that the action does not lie in the wire, but in -the medium, and it is known that there is no direction of flow in the -wire. - -No explanation has been offered in three-dimensional mechanics of how -an action can be impressed throughout a region and yet necessarily -run itself out along a closed boundary, as is the case in an electric -current. But this phenomenon corresponds exactly to the definition of a -four-dimensional vortex. - -If we take a very long magnet, so long that one of its poles is -practically isolated, and put this pole in the vicinity of an electric -circuit, we find that it moves. - -Now, assuming for the sake of simplicity that the wire which determines -the current is in the form of a circle, if we take a number of small -magnets and place them all pointing in the same direction normal to -the plane of the circle, so that they fill it and the wire binds them -round, we find that this sheet of magnets has the same effect on -the magnetic pole that the current has. The sheet of magnets may be -curved, but the edge of it must coincide with the wire. The collection -of magnets is then equivalent to the vortex sheet, and an elementary -magnet to a part of it. Thus, we must think of a magnet as conditioning -a rotation in the ether round the plane which bisects at right angles -the line joining its poles. - -If a current is started in a circuit, we must imagine vortices like -bowls turning themselves inside out, starting from the contour. In -reaching a parallel circuit, if the vortex sheet were interrupted and -joined momentarily to the second circuit by a free rim, the axis plane -would lie between the two circuits, and a point on the second circuit -opposite a point on the first would correspond to a point opposite -to it on the first; hence we should expect a current in the opposite -direction in the second circuit. Thus the phenomena of induction are -not inconsistent with the hypothesis of a vortex about an axial plane. - -In four-dimensional space, in which all four dimensions were -commensurable, the intensity of the action transmitted by the medium -would vary inversely as the cube of the distance. Now, the action of -a current on a magnetic pole varies inversely as the square of the -distance; hence, over measurable distances the extension of the ether -in the fourth dimension cannot be assumed as other than small in -comparison with those distances. - -If we suppose the ether to be filled with vortices in the shape of -four-dimensional spheres rotating with the A motion, the B motion would -correspond to electricity in the one-fluid theory. There would thus -be a possibility of electricity existing in two forms, statically, -by itself, and, combined with the universal motion, in the form of a -current. - -To arrive at a definite conclusion it will be necessary to investigate -the resultant pressures which accompany the collocation of solid -vortices with surface ones. - -To recapitulate: - -The movements and mechanics of four-dimensional space are definite and -intelligible. A vortex with a surface as its axis affords a geometric -image of a closed circuit, and there are rotations which by their -polarity afford a possible definition of statical electricity.[7] - - [7] These double rotations of the A and B kinds I should like to call - Hamiltons and co-Hamiltons, for it is a singular fact that in his - “Quaternions” Sir Wm. Rowan Hamilton has given the theory of either - the A or the B kind. They follow the laws of his symbols, I, J, K. - -Hamiltons and co-Hamiltons seem to be natural units of geometrical -expression. In the paper in the “Proceedings of the Royal Irish -Academy,” Nov. 1903, already alluded to, I have shown something of the -remarkable facility which is gained in dealing with the composition of -three- and four-dimensional rotations by an alteration in Hamilton’s -notation, which enables his system to be applied to both the A and B -kinds of rotations. - -The objection which has been often made to Hamilton’s system, namely, -that it is only under special conditions of application that his -processes give geometrically interpretable results, can be removed, if -we assume that he was really dealing with a four-dimensional motion, -and alter his notation to bring this circumstance into explicit -recognition. - - - - - APPENDIX I - - THE MODELS - - -In Chapter XI. a description has been given which will enable any -one to make a set of models illustrative of the tesseract and its -properties. The set here supposed to be employed consists of:— - - 1. Three sets of twenty-seven cubes each. - 2. Twenty-seven slabs. - 3. Twelve cubes with points, lines, faces, distinguished by colours, - which will be called the catalogue cubes. - -The preparation of the twelve catalogue cubes involves the expenditure -of a considerable amount of time. It is advantageous to use them, but -they can be replaced by the drawing of the views of the tesseract or by -a reference to figs. 103, 104, 105, 106 of the text. - -The slabs are coloured like the twenty-seven cubes of the first cubic -block in fig. 101, the one with red, white, yellow axes. - -The colours of the three sets of twenty-seven cubes are those of the -cubes shown in fig. 101. - -The slabs are used to form the representation of a cube in a plane, and -can well be dispensed with by any one who is accustomed to deal with -solid figures. But the whole theory depends on a careful observation of -how the cube would be represented by these slabs. - -In the first step, that of forming a clear idea how a plane being -would represent three-dimensional space, only one of the catalogue -cubes and one of the three blocks is needed. - - - APPLICATION TO THE STEP FROM PLANE TO SOLID. - -Look at fig. 1 of the views of the tesseract, or, what comes to the -same thing, take catalogue cube No. 1 and place it before you with the -red line running up, the white line running to the right, the yellow -line running away. The three dimensions of space are then marked out -by these lines or axes. Now take a piece of cardboard, or a book, and -place it so that it forms a wall extending up and down not opposite to -you, but running away parallel to the wall of the room on your left -hand. - -Placing the catalogue cube against this wall we see that it comes into -contact with it by the red and yellow lines, and by the included orange -face. - -In the plane being’s world the aspect he has of the cube would be a -square surrounded by red and yellow lines with grey points. - -Now, keeping the red line fixed, turn the cube about it so that the -yellow line goes out to the right, and the white line comes into -contact with the plane. - -In this case a different aspect is presented to the plane being, a -square, namely, surrounded by red and white lines and grey points. You -should particularly notice that when the yellow line goes out, at right -angles to the plane, and the white comes in, the latter does not run in -the same sense that the yellow did. - -From the fixed grey point at the base of the red line the yellow line -ran away from you. The white line now runs towards you. This turning -at right angles makes the line which was out of the plane before, come -into it in an opposite sense to that in which the line ran which has -just left the plane. If the cube does not break through the plane this -is always the rule. - -Again turn the cube back to the normal position with red running up, -white to the right, and yellow away, and try another turning. - -You can keep the yellow line fixed, and turn the cube about it. In this -case the red line going out to the right the white line will come in -pointing downwards. - -You will be obliged to elevate the cube from the table in order to -carry out this turning. It is always necessary when a vertical axis -goes out of a space to imagine a movable support which will allow the -line which ran out before to come in below. - -Having looked at the three ways of turning the cube so as to present -different faces to the plane, examine what would be the appearance if -a square hole were cut in the piece of cardboard, and the cube were to -pass through it. A hole can be actually cut, and it will be seen that -in the normal position, with red axis running up, yellow away, and -white to the right, the square first perceived by the plane being—the -one contained by red and yellow lines—would be replaced by another -square of which the line towards you is pink—the section line of the -pink face. The line above is light yellow, below is light yellow and on -the opposite side away from you is pink. - -In the same way the cube can be pushed through a square opening in the -plane from any of the positions which you have already turned it into. -In each case the plane being will perceive a different set of contour -lines. - -Having observed these facts about the catalogue cube, turn now to the -first block of twenty-seven cubes. - -You notice that the colour scheme on the catalogue cube and that of -this set of blocks is the same. - -Place them before you, a grey or null cube on the table, above it a -red cube, and on the top a null cube again. Then away from you place a -yellow cube, and beyond it a null cube. Then to the right place a white -cube and beyond it another null. Then complete the block, according to -the scheme of the catalogue cube, putting in the centre of all an ochre -cube. - -You have now a cube like that which is described in the text. For the -sake of simplicity, in some cases, this cubic block can be reduced to -one of eight cubes, by leaving out the terminations in each direction. -Thus, instead of null, red, null, three cubes, you can take null, red, -two cubes, and so on. - -It is useful, however, to practise the representation in a plane of a -block of twenty-seven cubes. For this purpose take the slabs, and build -them up against the piece of cardboard, or the book in such a way as to -represent the different aspects of the cube. - -Proceed as follows:— - -First, cube in normal position. - -Place nine slabs against the cardboard to represent the nine cubes -in the wall of the red and yellow axes, facing the cardboard; these -represent the aspect of the cube as it touches the plane. - -Now push these along the cardboard and make a different set of nine -slabs to represent the appearance which the cube would present to a -plane being, if it were to pass half way through the plane. - -There would be a white slab, above it a pink one, above that another -white one, and six others, representing what would be the nature of a -section across the middle of the block of cubes. The section can be -thought of as a thin slice cut out by two parallel cuts across the -cube. Having arranged these nine slabs, push them along the plane, and -make another set of nine to represent what would be the appearance of -the cube when it had almost completely gone through. This set of nine -will be the same as the first set of nine. - -Now we have in the plane three sets of nine slabs each, which represent -three sections of the twenty-seven block. - -They are put alongside one another. We see that it does not matter in -what order the sets of nine are put. As the cube passes through the -plane they represent appearances which follow the one after the other. -If they were what they represented, they could not exist in the same -plane together. - -This is a rather important point, namely, to notice that they should -not co-exist on the plane, and that the order in which they are placed -is indifferent. When we represent a four-dimensional body our solid -cubes are to us in the same position that the slabs are to the plane -being. You should also notice that each of these slabs represents only -the very thinnest slice of a cube. The set of nine slabs first set up -represents the side surface of the block. It is, as it were, a kind -of tray—a beginning from which the solid cube goes off. The slabs -as we use them have thickness, but this thickness is a necessity of -construction. They are to be thought of as merely of the thickness of a -line. - -If now the block of cubes passed through the plane at the rate of an -inch a minute the appearance to a plane being would be represented by:— - -1. The first set of nine slabs lasting for one minute. - -2. The second set of nine slabs lasting for one minute. - -3. The third set of nine slabs lasting for one minute. - -Now the appearances which the cube would present to the plane being -in other positions can be shown by means of these slabs. The use of -such slabs would be the means by which a plane being could acquire a -familiarity with our cube. Turn the catalogue cube (or imagine the -coloured figure turned) so that the red line runs up, the yellow line -out to the right, and the white line towards you. Then turn the block -of cubes to occupy a similar position. - -The block has now a different wall in contact with the plane. Its -appearance to a plane being will not be the same as before. He has, -however, enough slabs to represent this new set of appearances. But he -must remodel his former arrangement of them. - -He must take a null, a red, and a null slab from the first of his sets -of slabs, then a white, a pink, and a white from the second, and then a -null, a red, and a null from the third set of slabs. - -He takes the first column from the first set, the first column from the -second set, and the first column from the third set. - -To represent the half-way-through appearance, which is as if a very -thin slice were cut out half way through the block, he must take the -second column of each of his sets of slabs, and to represent the final -appearance, the third column of each set. - -Now turn the catalogue cube back to the normal position, and also the -block of cubes. - -There is another turning—a turning about the yellow line, in which the -white axis comes below the support. - -You cannot break through the surface of the table, so you must imagine -the old support to be raised. Then the top of the block of cubes in its -new position is at the level at which the base of it was before. - -Now representing the appearance on the plane, we must draw a horizontal -line to represent the old base. The line should be drawn three inches -high on the cardboard. - -Below this the representative slabs can be arranged. - -It is easy to see what they are. The old arrangements have to be -broken up, and the layers taken in order, the first layer of each for -the representation of the aspect of the block as it touches the plane. - -Then the second layers will represent the appearance half way through, -and the third layers will represent the final appearance. - -It is evident that the slabs individually do not represent the same -portion of the cube in these different presentations. - -In the first case each slab represents a section or a face -perpendicular to the white axis, in the second case a face or a section -which runs perpendicularly to the yellow axis, and in the third case a -section or a face perpendicular to the red axis. - -But by means of these nine slabs the plane being can represent the -whole of the cubic block. He can touch and handle each portion of the -cubic block, there is no part of it which he cannot observe. Taking it -bit by bit, two axes at a time, he can examine the whole of it. - - - OUR REPRESENTATION OF A BLOCK OF TESSERACTS. - -Look at the views of the tesseract 1, 2, 3, or take the catalogue cubes -1, 2, 3, and place them in front of you, in any order, say running from -left to right, placing 1 in the normal position, the red axis running -up, the white to the right, and yellow away. - -Now notice that in catalogue cube 2 the colours of each region are -derived from those of the corresponding region of cube 1 by the -addition of blue. Thus null + blue = blue, and the corners of number 2 -are blue. Again, red + blue = purple, and the vertical lines of 2 are -purple. Blue + yellow = green, and the line which runs away is coloured -green. - -By means of these observations you may be sure that catalogue cube 2 -is rightly placed. Catalogue cube 3 is just like number 1. - -Having these cubes in what we may call their normal position, proceed -to build up the three sets of blocks. - -This is easily done in accordance with the colour scheme on the -catalogue cubes. - -The first block we already know. Build up the second block, beginning -with a blue corner cube, placing a purple on it, and so on. - -Having these three blocks we have the means of representing the -appearances of a group of eighty-one tesseracts. - -Let us consider a moment what the analogy in the case of the plane -being is. - -He has his three sets of nine slabs each. We have our three sets of -twenty-seven cubes each. - -Our cubes are like his slabs. As his slabs are not the things which -they represent to him, so our cubes are not the things they represent -to us. - -The plane being’s slabs are to him the faces of cubes. - -Our cubes then are the faces of tesseracts, the cubes by which they are -in contact with our space. - -As each set of slabs in the case of the plane being might be considered -as a sort of tray from which the solid contents of the cubes came out, -so our three blocks of cubes may be considered as three-space trays, -each of which is the beginning of an inch of the solid contents of the -four-dimensional solids starting from them. - -We want now to use the names null, red, white, etc., for tesseracts. -The cubes we use are only tesseract faces. Let us denote that fact -by calling the cube of null colour, null face; or, shortly, null f., -meaning that it is the face of a tesseract. - -To determine which face it is let us look at the catalogue cube 1 or -the first of the views of the tesseract, which can be used instead of -the models. It has three axes, red, white, yellow, in our space. Hence -the cube determined by these axes is the face of the tesseract which we -now have before us. It is the ochre face. It is enough, however, simply -to say null f., red f. for the cubes which we use. - -To impress this in your mind, imagine that tesseracts do actually run -from each cube. Then, when you move the cubes about, you move the -tesseracts about with them. You move the face but the tesseract follows -with it, as the cube follows when its face is shifted in a plane. - -The cube null in the normal position is the cube which has in it the -red, yellow, white axes. It is the face having these, but wanting the -blue. In this way you can define which face it is you are handling. I -will write an “f.” after the name of each tesseract just as the plane -being might call each of his slabs null slab, yellow slab, etc., to -denote that they were representations. - -We have then in the first block of twenty-seven cubes, the -following—null f., red f., null f., going up; white f., null f., lying -to the right, and so on. Starting from the null point and travelling -up one inch we are in the null region, the same for the away and the -right-hand directions. And if we were to travel in the fourth dimension -for an inch we should still be in a null region. The tesseract -stretches equally all four ways. Hence the appearance we have in this -first block would do equally well if the tesseract block were to move -across our space for a certain distance. For anything less than an inch -of their transverse motion we should still have the same appearance. -You must notice, however, that we should not have null face after the -motion had begun. - -When the tesseract, null for instance, had moved ever so little we -should not have a face of null but a section of null in our space. -Hence, when we think of the motion across our space we must call our -cubes tesseract sections. Thus on null passing across we should see -first null f., then null s., and then, finally, null f. again. - -Imagine now the whole first block of twenty-seven tesseracts to have -moved tranverse to our space a distance of one inch. Then the second -set of tesseracts, which originally were an inch distant from our -space, would be ready to come in. - -Their colours are shown in the second block of twenty-seven cubes which -you have before you. These represent the tesseract faces of the set of -tesseracts that lay before an inch away from our space. They are ready -now to come in, and we can observe their colours. In the place which -null f. occupied before we have blue f., in place of red f. we have -purple f., and so on. Each tesseract is coloured like the one whose -place it takes in this motion with the addition of blue. - -Now if the tesseract block goes on moving at the rate of an inch a -minute, this next set of tesseracts will occupy a minute in passing -across. We shall see, to take the null one for instance, first of all -null face, then null section, then null face again. - -At the end of the second minute the second set of tesseracts has gone -through, and the third set comes in. This, as you see, is coloured just -like the first. Altogether, these three sets extend three inches in the -fourth dimension, making the tesseract block of equal magnitude in all -dimensions. - -We have now before us a complete catalogue of all the tesseracts in our -group. We have seen them all, and we shall refer to this arrangement -of the blocks as the “normal position.” We have seen as much of each -tesseract at a time as could be done in a three-dimensional space. Each -part of each tesseract has been in our space, and we could have touched -it. - -The fourth dimension appeared to us as the duration of the block. - -If a bit of our matter were to be subjected to the same motion it -would be instantly removed out of our space. Being thin in the fourth -dimension it is at once taken out of our space by a motion in the -fourth dimension. - -But the tesseract block we represent having length in the fourth -dimension remains steadily before our eyes for three minutes, when it -is subjected to this transverse motion. - -We have now to form representations of the other views of the same -tesseract group which are possible in our space. - -Let us then turn the block of tesseracts so that another face of it -comes into contact with our space, and then by observing what we have, -and what changes come when the block traverses our space, we shall have -another view of it. The dimension which appeared as duration before -will become extension in one of our known dimensions, and a dimension -which coincided with one of our space dimensions will appear as -duration. - -Leaving catalogue cube 1 in the normal position, remove the other two, -or suppose them removed. We have in space the red, the yellow, and the -white axes. Let the white axis go out into the unknown, and occupy the -position the blue axis holds. Then the blue axis, which runs in that -direction now will come into space. But it will not come in pointing -in the same way that the white axis does now. It will point in the -opposite sense. It will come in running to the left instead of running -to the right as the white axis does now. - -When this turning takes place every part of the cube 1 will disappear -except the left-hand face—the orange face. - -And the new cube that appears in our space will run to the left from -this orange face, having axes, red, yellow, blue. - -Take models 4, 5, 6. Place 4, or suppose No. 4 of the tesseract views -placed, with its orange face coincident with the orange face of 1, red -line to red line, and yellow line to yellow line, with the blue line -pointing to the left. Then remove cube 1 and we have the tesseract face -which comes in when the white axis runs in the positive unknown, and -the blue axis comes into our space. - -Now place catalogue cube 5 in some position, it does not matter which, -say to the left; and place it so that there is a correspondence of -colour corresponding to the colour of the line that runs out of space. -The line that runs out of space is white, hence, every part of this -cube 5 should differ from the corresponding part of 4 by an alteration -in the direction of white. - -Thus we have white points in 5 corresponding to the null points in -4. We have a pink line corresponding to a red line, a light yellow -line corresponding to a yellow line, an ochre face corresponding to -an orange face. This cube section is completely named in Chapter XI. -Finally cube 6 is a replica of 1. - -These catalogue cubes will enable us to set up our models of the block -of tesseracts. - -First of all for the set of tesseracts, which beginning in our space -reach out one inch in the unknown, we have the pattern of catalogue -cube 4. - -We see that we can build up a block of twenty-seven tesseract faces -after the colour scheme of cube 4, by taking the left-hand wall of -block 1, then the left-hand wall of block 2, and finally that of block -3. We take, that is, the three first walls of our previous arrangement -to form the first cubic block of this new one. - -This will represent the cubic faces by which the group of tesseracts in -its new position touches our space. We have running up, null f., red -f., null f. In the next vertical line, on the side remote from us, we -have yellow f., orange f., yellow f., and then the first colours over -again. Then the three following columns are, blue f., purple f., blue -f.; green f., brown f., green f.; blue f., purple f., blue f. The last -three columns are like the first. - -These tesseracts touch our space, and none of them are by any part of -them distant more than an inch from it. What lies beyond them in the -unknown? - -This can be told by looking at catalogue cube 5. According to its -scheme of colour we see that the second wall of each of our old -arrangements must be taken. Putting them together we have, as the -corner, white f. above it, pink f. above it, white f. The column next -to this remote from us is as follows:—light yellow f., ochre f., light -yellow f., and beyond this a column like the first. Then for the middle -of the block, light blue f., above it light purple, then light blue. -The centre column has, at the bottom, light green f., light brown f. -in the centre and at the top light green f. The last wall is like the -first. - -The third block is made by taking the third walls of our previous -arrangement, which we called the normal one. - -You may ask what faces and what sections our cubes represent. To answer -this question look at what axes you have in our space. You have red, -yellow, blue. Now these determine brown. The colours red, yellow, blue -are supposed by us when mixed to produce a brown colour. And that cube -which is determined by the red, yellow, blue axes we call the brown -cube. - -When the tesseract block in its new position begins to move across our -space each tesseract in it gives a section in our space. This section -is transverse to the white axis, which now runs in the unknown. - -As the tesseract in its present position passes across our space, we -should see first of all the first of the blocks of cubic faces we have -put up—these would last for a minute, then would come the second block -and then the third. At first we should have a cube of tesseract faces, -each of which would be brown. Directly the movement began, we should -have tesseract sections transverse to the white line. - -There are two more analogous positions in which the block of tesseracts -can be placed. To find the third position, restore the blocks to the -normal arrangement. - -Let us make the yellow axis go out into the positive unknown, and let -the blue axis, consequently, come in running towards us. The yellow ran -away, so the blue will come in running towards us. - -Put catalogue cube 1 in its normal position. Take catalogue cube 7 -and place it so that its pink face coincides with the pink face of -cube 1, making also its red axis coincide with the red axis of 1 and -its white with the white. Moreover, make cube 7 come towards us from -cube 1. Looking at it we see in our space, red, white, and blue axes. -The yellow runs out. Place catalogue cube 8 in the neighbourhood -of 7—observe that every region in 8 has a change in the direction -of yellow from the corresponding region in 7. This is because it -represents what you come to now in going in the unknown, when the -yellow axis runs out of our space. Finally catalogue cube 9, which is -like number 7, shows the colours of the third set of tesseracts. Now -evidently, starting from the normal position, to make up our three -blocks of tesseract faces we have to take the near wall from the first -block, the near wall from the second, and then the near wall from the -third block. This gives us the cubic block formed by the faces of the -twenty-seven tesseracts which are now immediately touching our space. - -Following the colour scheme of catalogue cube 8, we make the next set -of twenty-seven tesseract faces, representing the tesseracts, each of -which begins one inch off from our space, by putting the second walls -of our previous arrangement together, and the representation of the -third set of tesseracts is the cubic block formed of the remaining -three walls. - -Since we have red, white, blue axes in our space to begin with, the -cubes we see at first are light purple tesseract faces, and after the -transverse motion begins we have cubic sections transverse to the -yellow line. - -Restore the blocks to the normal position, there remains the case in -which the red axis turns out of space. In this case the blue axis will -come in downwards, opposite to the sense in which the red axis ran. - -In this case take catalogue cubes 10, 11, 12. Lift up catalogue cube 1 -and put 10 underneath it, imagining that it goes down from the previous -position of 1. - -We have to keep in space the white and the yellow axes, and let the red -go out, the blue come in. - -Now, you will find on cube 10 a light yellow face; this should coincide -with the base of 1, and the white and yellow lines on the two cubes -should coincide. Then the blue axis running down you have the catalogue -cube correctly placed, and it forms a guide for putting up the first -representative block. - -Catalogue cube 11 will represent what lies in the fourth dimension—now -the red line runs in the fourth dimension. Thus the change from 10 to -11 should be towards red, corresponding to a null point is a red point, -to a white line is a pink line, to a yellow line an orange line, and so -on. - -Catalogue cube 12 is like 10. Hence we see that to build up our blocks -of tesseract faces we must take the bottom layer of the first block, -hold that up in the air, underneath it place the bottom layer of the -second block, and finally underneath this last the bottom layer of the -last of our normal blocks. - -Similarly we make the second representative group by taking the middle -courses of our three blocks. The last is made by taking the three -topmost layers. The three axes in our space before the transverse -motion begins are blue, white, yellow, so we have light green tesseract -faces, and after the motion begins sections transverse to the red light. - -These three blocks represent the appearances as the tesseract group in -its new position passes across our space. The cubes of contact in this -case are those determinal by the three axes in our space, namely, the -white, the yellow, the blue. Hence they are light green. - -It follows from this that light green is the interior cube of the first -block of representative cubic faces. - -Practice in the manipulations described, with a realization in each -case of the face or section which is in our space, is one of the best -means of a thorough comprehension of the subject. - -We have to learn how to get any part of these four-dimensional figures -into space, so that we can look at them. We must first learn to swing a -tesseract, and a group of tesseracts about in any way. - -When these operations have been repeated and the method of arrangement -of the set of blocks has become familiar, it is a good plan to rotate -the axes of the normal cube 1 about a diagonal, and then repeat the -whole series of turnings. - -Thus, in the normal position, red goes up, white to the right, yellow -away. Make white go up, yellow to the right, and red away. Learn the -cube in this position by putting up the set of blocks of the normal -cube, over and over again till it becomes as familiar to you as in the -normal position. Then when this is learned, and the corresponding -changes in the arrangements of the tesseract groups are made, another -change should be made: let, in the normal cube, yellow go up, red to -the right, and white away. - -Learn the normal block of cubes in this new position by arranging them -and re-arranging them till you know without thought where each one -goes. Then carry out all the tesseract arrangements and turnings. - -If you want to understand the subject, but do not see your way clearly, -if it does not seem natural and easy to you, practise these turnings. -Practise, first of all, the turning of a block of cubes round, so that -you know it in every position as well as in the normal one. Practise by -gradually putting up the set of cubes in their new arrangements. Then -put up the tesseract blocks in their arrangements. This will give you -a working conception of higher space, you will gain the feeling of it, -whether you take up the mathematical treatment of it or not. - - - - - APPENDIX II - - A LANGUAGE OF SPACE - - -The mere naming the parts of the figures we consider involves a certain -amount of time and attention. This time and attention leads to no -result, for with each new figure the nomenclature applied is completely -changed, every letter or symbol is used in a different significance. - -Surely it must be possible in some way to utilise the labour thus at -present wasted! - -Why should we not make a language for space itself, so that every -position we want to refer to would have its own name? Then every time -we named a figure in order to demonstrate its properties we should be -exercising ourselves in the vocabulary of place. - -If we use a definite system of names, and always refer to the same -space position by the same name, we create as it were a multitude of -little hands, each prepared to grasp a special point, position, or -element, and hold it for us in its proper relations. - -We make, to use another analogy, a kind of mental paper, which has -somewhat of the properties of a sensitive plate, in that it will -register, without effort, complex, visual, or tactual impressions. - -But of far more importance than the applications of a space language to -the plane and to solid space is the facilitation it brings with it to -the study of four-dimensional shapes. - -I have delayed introducing a space language because all the systems I -made turned out, after giving them a fair trial, to be intolerable. I -have now come upon one which seems to present features of permanence, -and I will here give an outline of it, so that it can be applied to the -subject of the text, and in order that it may be subjected to criticism. - -The principle on which the language is constructed is to sacrifice -every other consideration for brevity. - -It is indeed curious that we are able to talk and converse on every -subject of thought except the fundamental one of space. The only way of -speaking about the spatial configurations that underlie every subject -of discursive thought is a co-ordinate system of numbers. This is so -awkward and incommodious that it is never used. In thinking also, in -realising shapes, we do not use it; we confine ourselves to a direct -visualisation. - -Now, the use of words corresponds to the storing up of our experience -in a definite brain structure. A child, in the endless tactual, visual, -mental manipulations it makes for itself, is best left to itself, but -in the course of instruction the introduction of space names would -make the teachers work more cumulative, and the child’s knowledge more -social. - -Their full use can only be appreciated, if they are introduced early -in the course of education; but in a minor degree any one can convince -himself of their utility, especially in our immediate subject of -handling four-dimensional shapes. The sum total of the results obtained -in the preceding pages can be compendiously and accurately expressed in -nine words of the Space Language. - -In one of Plato’s dialogues Socrates makes an experiment on a slave boy -standing by. He makes certain perceptions of space awake in the mind -of Meno’s slave by directing his close attention on some simple facts -of geometry. - -By means of a few words and some simple forms we can repeat Plato’s -experiment on new ground. - -Do we by directing our close attention on the facts of four dimensions -awaken a latent faculty in ourselves? The old experiment of Plato’s, it -seems to me, has come down to us as novel as on the day he incepted it, -and its significance not better understood through all the discussion -of which it has been the subject. - -Imagine a voiceless people living in a region where everything had -a velvety surface, and who were thus deprived of all opportunity of -experiencing what sound is. They could observe the slow pulsations -of the air caused by their movements, and arguing from analogy, they -would no doubt infer that more rapid vibrations were possible. From -the theoretical side they could determine all about these more rapid -vibrations. They merely differ, they would say, from slower ones, -by the number that occur in a given time; there is a merely formal -difference. - -But suppose they were to take the trouble, go to the pains of producing -these more rapid vibrations, then a totally new sensation would fall -on their rudimentary ears. Probably at first they would only be dimly -conscious of Sound, but even from the first they would become aware -that a merely formal difference, a mere difference in point of number -in this particular respect, made a great difference practically, as -related to them. And to us the difference between three and four -dimensions is merely formal, numerical. We can tell formally all about -four dimensions, calculate the relations that would exist. But that -the difference is merely formal does not prove that it is a futile and -empty task, to present to ourselves as closely as we can the phenomena -of four dimensions. In our formal knowledge of it, the whole question -of its actual relation to us, as we are, is left in abeyance. - -Possibly a new apprehension of nature may come to us through the -practical, as distinguished from the mathematical and formal, study -of four dimensions. As a child handles and examines the objects with -which he comes in contact, so we can mentally handle and examine -four-dimensional objects. The point to be determined is this. Do we -find something cognate and natural to our faculties, or are we merely -building up an artificial presentation of a scheme only formally -possible, conceivable, but which has no real connection with any -existing or possible experience? - -This, it seems to me, is a question which can only be settled by -actually trying. This practical attempt is the logical and direct -continuation of the experiment Plato devised in the “Meno.” - -Why do we think true? Why, by our processes of thought, can we predict -what will happen, and correctly conjecture the constitution of the -things around us? This is a problem which every modern philosopher has -considered, and of which Descartes, Leibnitz, Kant, to name a few, -have given memorable solutions. Plato was the first to suggest it. -And as he had the unique position of being the first devisor of the -problem, so his solution is the most unique. Later philosophers have -talked about consciousness and its laws, sensations, categories. But -Plato never used such words. Consciousness apart from a conscious being -meant nothing to him. His was always an objective search. He made man’s -intuitions the basis of a new kind of natural history. - -In a few simple words Plato puts us in an attitude with regard to -psychic phenomena—the mind—the ego—“what we are,” which is analogous -to the attitude scientific men of the present day have with regard -to the phenomena of outward nature. Behind this first apprehension -of ours of nature, there is an infinite depth to be learned and -known. Plato said that behind the phenomena of mind that Meno’s slave -boy exhibited, there was a vast, an infinite perspective. And his -singularity, his originality, comes out most strongly marked in this, -that the perspective, the complex phenomena beyond were, according to -him, phenomena of personal experience. A footprint in the sand means a -man to a being that has the conception of a man. But to a creature that -has no such conception, it means a curious mark, somehow resulting from -the concatenation of ordinary occurrences. Such a being would attempt -merely to explain how causes known to him could so coincide as to -produce such a result; he would not recognise its significance. - -Plato introduced the conception which made a new kind of natural -history possible. He said that Meno’s slave boy thought true about -things he had never learned, because his “soul” had experience. I -know this will sound absurd to some people, and it flies straight in -the face of the maxim, that explanation consists in showing how an -effect depends on simple causes. But what a mistaken maxim that is! -Can any single instance be shown of a simple cause? Take the behaviour -of spheres for instance; say those ivory spheres, billiard balls, -for example. We can explain their behaviour by supposing they are -homogeneous elastic solids. We can give formulæ which will account for -their movements in every variety. But are they homogeneous elastic -solids? No, certainly not. They are complex in physical and molecular -structure, and atoms and ions beyond open an endless vista. Our simple -explanation is false, false as it can be. The balls act as if they -were homogeneous elastic spheres. There is a statistical simplicity in -the resultant of very complex conditions, which makes that artificial -conception useful. But its usefulness must not blind us to the fact -that it is artificial. If we really look deep into nature, we find a -much greater complexity than we at first suspect. And so behind this -simple “I,” this myself, is there not a parallel complexity? Plato’s -“soul” would be quite acceptable to a large class of thinkers, if by -“soul” and the complexity he attributes to it, he meant the product of -a long course of evolutionary changes, whereby simple forms of living -matter endowed with rudimentary sensation had gradually developed into -fully conscious beings. - -But Plato does not mean by “soul” a being of such a kind. His soul is -a being whose faculties are clogged by its bodily environment, or at -least hampered by the difficulty of directing its bodily frame—a being -which is essentially higher than the account it gives of itself through -its organs. At the same time Plato’s soul is not incorporeal. It is a -real being with a real experience. The question of whether Plato had -the conception of non-spatial existence has been much discussed. The -verdict is, I believe, that even his “ideas” were conceived by him as -beings in space, or, as we should say, real. Plato’s attitude is that -of Science, inasmuch as he thinks of a world in Space. But, granting -this, it cannot be denied that there is a fundamental divergence -between Plato’s conception and the evolutionary theory, and also an -absolute divergence between his conception and the genetic account of -the origin of the human faculties. The functions and capacities of -Plato’s “soul” are not derived by the interaction of the body and its -environment. - -Plato was engaged on a variety of problems, and his religious and -ethical thoughts were so keen and fertile that the experimental -investigation of his soul appears involved with many other motives. -In one passage Plato will combine matter of thought of all kinds and -from all sources, overlapping, interrunning. And in no case is he more -involved and rich than in this question of the soul. In fact, I wish -there were two words, one denoting that being, corporeal and real, but -with higher faculties than we manifest in our bodily actions, which is -to be taken as the subject of experimental investigation; and the other -word denoting “soul” in the sense in which it is made the recipient and -the promise of so much that men desire. It is the soul in the former -sense that I wish to investigate, and in a limited sphere only. I wish -to find out, in continuation of the experiment in the Meno, what the -“soul” in us thinks about extension, experimenting on the grounds laid -down by Plato. He made, to state the matter briefly, the hypothesis -with regard to the thinking power of a being in us, a “soul.” This -soul is not accessible to observation by sight or touch, but it can be -observed by its functions; it is the object of a new kind of natural -history, the materials for constructing which lie in what it is natural -to us to think. With Plato “thought” was a very wide-reaching term, but -still I would claim in his general plan of procedure a place for the -particular question of extension. - -The problem comes to be, “What is it natural to us to think about -matter _qua_ extended?” - -First of all, I find that the ordinary intuition of any simple object -is extremely imperfect. Take a block of differently marked cubes, for -instance, and become acquainted with them in their positions. You may -think you know them quite well, but when you turn them round—rotate -the block round a diagonal, for instance—you will find that you have -lost track of the individuals in their new positions. You can mentally -construct the block in its new position, by a rule, by taking the -remembered sequences, but you don’t know it intuitively. By observation -of a block of cubes in various positions, and very expeditiously -by a use of Space names applied to the cubes in their different -presentations, it is possible to get an intuitive knowledge of the -block of cubes, which is not disturbed by any displacement. Now, with -regard to this intuition, we moderns would say that I had formed it by -my tactual visual experiences (aided by hereditary pre-disposition). -Plato would say that the soul had been stimulated to recognise an -instance of shape which it knew. Plato would consider the operation -of learning merely as a stimulus; we as completely accounting for -the result. The latter is the more common-sense view. But, on the -other hand, it presupposes the generation of experience from physical -changes. The world of sentient experience, according to the modern -view, is closed and limited; only the physical world is ample and large -and of ever-to-be-discovered complexity. Plato’s world of soul, on the -other hand, is at least as large and ample as the world of things. - -Let us now try a crucial experiment. Can I form an intuition of a -four-dimensional object? Such an object is not given in the physical -range of my sense contacts. All I can do is to present to myself the -sequences of solids, which would mean the presentation to me under my -conditions of a four-dimensional object. All I can do is to visualise -and tactualise different series of solids which are alternative sets of -sectional views of a four-dimensional shape. - -If now, on presenting these sequences, I find a power in me of -intuitively passing from one of these sets of sequences to another, of, -being given one, intuitively constructing another, not using a rule, -but directly apprehending it, then I have found a new fact about my -soul, that it has a four-dimensional experience; I have observed it by -a function it has. - -I do not like to speak positively, for I might occasion a loss of time -on the part of others, if, as may very well be, I am mistaken. But for -my own part, I think there are indications of such an intuition; from -the results of my experiments, I adopt the hypothesis that that which -thinks in us has an ample experience, of which the intuitions we use in -dealing with the world of real objects are a part; of which experience, -the intuition of four-dimensional forms and motions is also a part. The -process we are engaged in intellectually is the reading the obscure -signals of our nerves into a world of reality, by means of intuitions -derived from the inner experience. - -The image I form is as follows. Imagine the captain of a modern -battle-ship directing its course. He has his charts before him; he -is in communication with his associates and subordinates; can convey -his messages and commands to every part of the ship, and receive -information from the conning-tower and the engine-room. Now suppose the -captain immersed in the problem of the navigation of his ship over the -ocean, to have so absorbed himself in the problem of the direction of -his craft over the plane surface of the sea that he forgets himself. -All that occupies his attention is the kind of movement that his ship -makes. The operations by which that movement is produced have sunk -below the threshold of his consciousness, his own actions, by which -he pushes the buttons, gives the orders, are so familiar as to be -automatic, his mind is on the motion of the ship as a whole. In such a -case we can imagine that he identifies himself with his ship; all that -enters his conscious thought is the direction of its movement over the -plane surface of the ocean. - -Such is the relation, as I imagine it, of the soul to the body. A -relation which we can imagine as existing momentarily in the case -of the captain is the normal one in the case of the soul with its -craft. As the captain is capable of a kind of movement, an amplitude -of motion, which does not enter into his thoughts with regard to the -directing the ship over the plane surface of the ocean, so the soul is -capable of a kind of movement, has an amplitude of motion, which is -not used in its task of directing the body in the three-dimensional -region in which the body’s activity lies. If for any reason it became -necessary for the captain to consider three-dimensional motions with -regard to his ship, it would not be difficult for him to gain the -materials for thinking about such motions; all he has to do is to -call his own intimate experience into play. As far as the navigation -of the ship, however, is concerned, he is not obliged to call on -such experience. The ship as a whole simply moves on a surface. The -problem of three-dimensional movement does not ordinarily concern its -steering. And thus with regard to ourselves all those movements and -activities which characterise our bodily organs are three-dimensional; -we never need to consider the ampler movements. But we do more than -use the movements of our body to effect our aims by direct means; we -have now come to the pass when we act indirectly on nature, when we -call processes into play which lie beyond the reach of any explanation -we can give by the kind of thought which has been sufficient for the -steering of our craft as a whole. When we come to the problem of what -goes on in the minute, and apply ourselves to the mechanism of the -minute, we find our habitual conceptions inadequate. - -The captain in us must wake up to his own intimate nature, realise -those functions of movement which are his own, and in virtue of his -knowledge of them apprehend how to deal with the problems he has come -to. - -Think of the history of man. When has there been a time, in which his -thoughts of form and movement were not exclusively of such varieties as -were adapted for his bodily performance? We have never had a demand to -conceive what our own most intimate powers are. But, just as little as -by immersing himself in the steering of his ship over the plane surface -of the ocean, a captain can lose the faculty of thinking about what he -actually does, so little can the soul lose its own nature. It can be -roused to an intuition that is not derived from the experience which -the senses give. All that is necessary is to present some few of those -appearances which, while inconsistent with three-dimensional matter, -are yet consistent with our formal knowledge of four-dimensional -matter, in order for the soul to wake up and not begin to learn, but of -its own intimate feeling fill up the gaps in the presentiment, grasp -the full orb of possibilities from the isolated points presented to -it. In relation to this question of our perceptions, let me suggest -another illustration, not taking it too seriously, only propounding it -to exhibit the possibilities in a broad and general way. - -In the heavens, amongst the multitude of stars, there are some which, -when the telescope is directed on them, seem not to be single stars, -but to be split up into two. Regarding these twin stars through a -spectroscope, an astronomer sees in each a spectrum of bands of colour -and black lines. Comparing these spectrums with one another, he finds -that there is a slight relative shifting of the dark lines, and from -that shifting he knows that the stars are rotating round one another, -and can tell their relative velocity with regard to the earth. By -means of his terrestrial physics he reads this signal of the skies. -This shifting of lines, the mere slight variation of a black line in a -spectrum, is very unlike that which the astronomer knows it means. But -it is probably much more like what it means than the signals which the -nerves deliver are like the phenomena of the outer world. - -No picture of an object is conveyed through the nerves. No picture of -motion, in the sense in which we postulate its existence, is conveyed -through the nerves. The actual deliverances of which our consciousness -takes account are probably identical for eye and ear, sight and touch. - -If for a moment I take the whole earth together and regard it as a -sentient being, I find that the problem of its apprehension is a very -complex one, and involves a long series of personal and physical -events. Similarly the problem of our apprehension is a very complex -one. I only use this illustration to exhibit my meaning. It has this -especial merit, that, as the process of conscious apprehension takes -place in our case in the minute, so, with regard to this earth being, -the corresponding process takes place in what is relatively to it very -minute. - -Now, Plato’s view of a soul leads us to the hypothesis that that -which we designate as an act of apprehension may be a very complex -event, both physically and personally. He does not seek to explain -what an intuition is; he makes it a basis from whence he sets out on -a voyage of discovery. Knowledge means knowledge; he puts conscious -being to account for conscious being. He makes an hypothesis of the -kind that is so fertile in physical science—an hypothesis making no -claim to finality, which marks out a vista of possible determination -behind determination, like the hypothesis of space itself, the type of -serviceable hypotheses. - -And, above all, Plato’s hypothesis is conducive to experiment. He -gives the perspective in which real objects can be determined; and, -in our present enquiry, we are making the simplest of all possible -experiments—we are enquiring what it is natural to the soul to think of -matter as extended. - -Aristotle says we always use a “phantasm” in thinking, a phantasm of -our corporeal senses a visualisation or a tactualisation. But we can -so modify that visualisation or tactualisation that it represents -something not known by the senses. Do we by that representation wake -up an intuition of the soul? Can we by the presentation of these -hypothetical forms, that are the subject of our present discussion, -wake ourselves up to higher intuitions? And can we explain the world -around by a motion that we only know by our souls? - -Apart from all speculation, however, it seems to me that the interest -of these four-dimensional shapes and motions is sufficient reason for -studying them, and that they are the way by which we can grow into a -fuller apprehension of the world as a concrete whole. - - - SPACE NAMES. - -If the words written in the squares drawn in fig. 1 are used as the -names of the squares in the positions in which they are placed, it is -evident that a combination of these names will denote a figure composed -of the designated squares. It is found to be most convenient to take as -the initial square that marked with an asterisk, so that the directions -of progression are towards the observer and to his right. The -directions of progression, however, are arbitrary, and can be chosen at -will. - -[Illustration: Fig. 1.] - -Thus _et_, _at_, _it_, _an_, _al_ will denote a figure in the form of a -cross composed of five squares. - -Here, by means of the double sequence, _e_, _a_, _i_ and _n_, _t_, _l_, -it is possible to name a limited collection of space elements. - -The system can obviously be extended by using letter sequences of more -members. - -But, without introducing such a complexity, the principles of a space -language can be exhibited, and a nomenclature obtained adequate to all -the considerations of the preceding pages. - - -1. _Extension._ - -Call the large squares in fig. 2 by the name written in them. It is -evident that each can be divided as shown in fig. 1. Then the small -square marked 1 will be “en” in “En,” or “Enen.” The square marked 2 -will be “et” in “En” or “Enet,” while the square marked 4 will be “en” -in “Et” or “Eten.” Thus the square 5 will be called “Ilil.” - -[Illustration: Fig. 2.] - -This principle of extension can be applied in any number of dimensions. - - -2. _Application to Three-Dimensional Space._ - -To name a three-dimensional collocation of cubes take the upward -direction first, secondly the direction towards the observer, thirdly -the direction to his right hand. - -[Illustration] - -These form a word in which the first letter gives the place of the cube -upwards, the second letter its place towards the observer, the third -letter its place to the right. - -We have thus the following scheme, which represents the set of cubes of -column 1, fig. 101, page 165. - -We begin with the remote lowest cube at the left hand, where the -asterisk is placed (this proves to be by far the most convenient origin -to take for the normal system). - -Thus “nen” is a “null” cube, “ten” a red cube on it, and “len” a “null” -cube above “ten.” - -By using a more extended sequence of consonants and vowels a larger set -of cubes can be named. - -To name a four-dimensional block of tesseracts it is simply necessary -to prefix an “e,” an “a,” or an “i” to the cube names. - -Thus the tesseract blocks schematically represented on page 165, fig. -101 are named as follows:— - -[Illustration: 1 2 3] - - -2. DERIVATION OF POINT, LINE, FACE, ETC., NAMES. - -[Illustration] - -The principle of derivation can be shown as follows: Taking the square -of squares the number of squares in it can be enlarged and the whole -kept the same size. - -[Illustration] - -Compare fig. 79, p. 138, for instance, or the bottom layer of fig. 84. - -Now use an initial “s” to denote the result of carrying this process on -to a great extent, and we obtain the limit names, that is the point, -line, area names for a square. “Sat” is the whole interior. The corners -are “sen,” “sel,” “sin,” “sil,” while the lines are “san,” “sal,” -“set,” “sit.” - -[Illustration] - -I find that by the use of the initial “s” these names come to be -practically entirely disconnected with the systematic names for the -square from which they are derived. They are easy to learn, and when -learned can be used readily with the axes running in any direction. - -To derive the limit names for a four-dimensional rectangular figure, -like the tesseract, is a simple extension of this process. These point, -line, etc., names include those which apply to a cube, as will be -evident on inspection of the first cube of the diagrams which follow. - -All that is necessary is to place an “s” before each of the names given -for a tesseract block. We then obtain apellatives which, like the -colour names on page 174, fig. 103, apply to all the points, lines, -faces, solids, and to the hyper-solid of the tesseract. These names -have the advantage over the colour marks that each point, line, etc., -has its own individual name. - -In the diagrams I give the names corresponding to the positions shown -in the coloured plate or described on p. 174. By comparing cubes 1, 2, -3 with the first row of cubes in the coloured plate, the systematic -names of each of the points, lines, faces, etc., can be determined. The -asterisk shows the origin from which the names run. - -These point, line, face, etc., names should be used in connection with -the corresponding colours. The names should call up coloured images of -the parts named in their right connection. - -[Illustration] - -It is found that a certain abbreviation adds vividness of distinction -to these names. If the final “en” be dropped wherever it occurs the -system is improved. Thus instead of “senen,” “seten,” “selen,” it is -preferable to abbreviate to “sen,” “set,” “sel,” and also use “san,” -“sin” for “sanen,” “sinen.” - -[Illustration] - -[Illustration] - -We can now name any section. Take _e.g._ the line in the first cube -from senin to senel, we should call the line running from senin to -senel, senin senat senel, a line light yellow in colour with null -points. - -[Illustration] - -Here senat is the name for all of the line except its ends. Using -“senat” in this way does not mean that the line is the whole of senat, -but what there is of it is senat. It is a part of the senat region. -Thus also the triangle, which has its three vertices in senin, senel, -selen, is named thus: - - Area: setat. - Sides: setan, senat, setet. - Vertices: senin, senel, sel. - -The tetrahedron section of the tesseract can be thought of as a series -of plane sections in the successive sections of the tesseract shown in -fig. 114, p. 191. In b_{0} the section is the one written above. In -b_{1} the section is made by a plane which cuts the three edges from -sanen intermediate of their lengths and thus will be: - - Area: satat. - Sides: satan, sanat, satet. - Vertices: sanan, sanet, sat. - -The sections in b_{2}, b_{3} will be like the section in b_{1} but -smaller. - -Finally in b_{4} the section plane simply passes through the corner -named sin. - -Hence, putting these sections together in their right relation, from -the face setat, surrounded by the lines and points mentioned above, -there run: - - 3 faces: satan, sanat, satet - 3 lines: sanan, sanet, sat - -and these faces and lines run to the point sin. Thus the tetrahedron is -completely named. - -The octahedron section of the tesseract, which can be traced from fig. -72, p. 129 by extending the lines there drawn, is named: - -Front triangle selin, selat, selel, setal, senil, setit, selin with -area setat. - -The sections between the front and rear triangle, of which one is shown -in 1b, another in 2b, are thus named, points and lines, salan, salat, -salet, satet, satel, satal, sanal, sanat, sanit, satit, satin, satan, -salan. - -The rear triangle found in 3b by producing lines is sil, sitet, sinel, -sinat, sinin, sitan, sil. - -The assemblage of sections constitute the solid body of the octahedron -satat with triangular faces. The one from the line selat to the point -sil, for instance, is named selin, selat, selel, salet, salat, salan, -sil. The whole interior is salat. - -Shapes can easily be cut out of cardboard which, when folded together, -form not only the tetrahedron and the octahedron, but also samples of -all the sections of the tesseract taken as it passes cornerwise through -our space. To name and visualise with appropriate colours a series of -these sections is an admirable exercise for obtaining familiarity with -the subject. - - - EXTENSION AND CONNECTION WITH NUMBERS. - -By extending the letter sequence it is of course possible to name a -larger field. By using the limit names the corners of each square can -be named. - -Thus “en sen,” “an sen,” etc., will be the names of the points nearest -the origin in “en” and in “an.” - -A field of points of which each one is indefinitely small is given by -the names written below. - -[Illustration] - -The squares are shown in dotted lines, the names denote the points. -These points are not mathematical points, but really minute areas. - -Instead of starting with a set of squares and naming them, we can start -with a set of points. - -By an easily remembered convention we can give names to such a region -of points. - -Let the space names with a final “e” added denote the mathematical -points at the corner of each square nearest the origin. We have then -for the set of mathematical points indicated. This system is really -completely independent of the area system and is connected with it -merely for the purpose of facilitating the memory processes. The word -“ene” is pronounced like “eny,” with just sufficient attention to the -final vowel to distinguish it from the word “en.” - -[Illustration] - -Now, connecting the numbers 0, 1, 2 with the sequence e, a, i, and -also with the sequence n, t, l, we have a set of points named as with -numbers in a co-ordinate system. Thus “ene” is (0, 0) “ate” is (1, -1) “ite” is (2, 1). To pass to the area system the rule is that the -name of the square is formed from the name of its point nearest to the -origin by dropping the final e. - -By using a notation analogous to the decimal system a larger field of -points can be named. It remains to assign a letter sequence to the -numbers from positive 0 to positive 9, and from negative 0 to negative -9, to obtain a system which can be used to denote both the usual -co-ordinate system of mapping and a system of named squares. The names -denoting the points all end with e. Those that denote squares end with -a consonant. - -There are many considerations which must be attended to in extending -the sequences to be used, such as uniqueness in the meaning of the -words formed, ease of pronunciation, avoidance of awkward combinations. - -I drop “s” altogether from the consonant series and short “u” from -the vowel series. It is convenient to have unsignificant letters at -disposal. A double consonant like “st” for instance can be referred to -without giving it a local significance by calling it “ust.” I increase -the number of vowels by considering a sound like “ra” to be a vowel, -using, that is, the letter “r” as forming a compound vowel. - -The series is as follows:— - - CONSONANTS. - - 0 1 2 3 4 5 6 7 8 9 - positive n t l p f sh k ch nt st - negative z d th b v m g j nd sp - - VOWELS. - - 0 1 2 3 4 5 6 7 8 9 - positive e a i ee ae ai ar ra ri ree - negative er o oo io oe iu or ro roo rio - -_Pronunciation._—e as in men; a as in man; i as in in; ee as in -between; ae as ay in may; ai as i in mine; ar as in art; er as ear in -earth; o as in on; oo as oo in soon; io as in clarion; oe as oa in oat; -iu pronounced like yew. - -To name a point such as (23, 41) it is considered as (3, 1) on from -(20, 40) and is called “ifeete.” It is the initial point of the square -ifeet of the area system. - -The preceding amplification of a space language has been introduced -merely for the sake of completeness. As has already been said nine -words and their combinations, applied to a few simple models suffice -for the purposes of our present enquiry. - - - _Printed by Hazell, Watson & Viney, Ld., London and Aylesbury._ - -*** END OF THE PROJECT GUTENBERG EBOOK THE FOURTH DIMENSION *** - -Updated editions will replace the previous one--the old editions will -be renamed. - -Creating the works from print editions not protected by U.S. copyright -law means that no one owns a United States copyright in these works, -so the Foundation (and you!) can copy and distribute it in the -United States without permission and without paying copyright -royalties. Special rules, set forth in the General Terms of Use part -of this license, apply to copying and distributing Project -Gutenberg-tm electronic works to protect the PROJECT GUTENBERG-tm -concept and trademark. 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Howard Hinton</p> -<div style='display:block; margin:1em 0'> -This eBook is for the use of anyone anywhere in the United States and -most other parts of the world at no cost and with almost no restrictions -whatsoever. You may copy it, give it away or re-use it under the terms -of the Project Gutenberg License included with this eBook or online -at <a href="https://www.gutenberg.org">www.gutenberg.org</a>. If you -are not located in the United States, you will have to check the laws of the -country where you are located before using this eBook. -</div> - -<p style='display:block; margin-top:1em; margin-bottom:1em; margin-left:2em; text-indent:-2em'>Title: The Fourth Dimension</p> -<p style='display:block; margin-top:1em; margin-bottom:0; margin-left:2em; text-indent:-2em'>Author: C. Howard Hinton</p> -<p style='display:block; text-indent:0; margin:1em 0'>Release Date: January 12, 2022 [eBook #67153]</p> -<p style='display:block; text-indent:0; margin:1em 0'>Language: English</p> - <p style='display:block; margin-top:1em; margin-bottom:0; margin-left:2em; text-indent:-2em; text-align:left'>Produced by: Chris Curnow, Les Galloway and the Online Distributed Proofreading Team at https://www.pgdp.net (This file was produced from images generously made available by The Internet Archive)</p> -<div style='margin-top:2em; margin-bottom:4em'>*** START OF THE PROJECT GUTENBERG EBOOK THE FOURTH DIMENSION ***</div> - -<div class="transnote"> -<h3> Transcriber’s Notes</h3> - -<p>Obvious typographical errors have been silently corrected. All other -spelling and punctuation remains unchanged.</p> - -<p>The cover was prepared by the transcriber and is placed in the public -domain.</p> -</div> -<hr class="chap" /> - - -<div class="half-title">THE FOURTH DIMENSION</div> - - -<hr class="chap x-ebookmaker-drop" /> - -<div class="chapter"> -<h2 class="nobreak" id="SOME_OPINIONS_OF_THE_PRESS">SOME OPINIONS OF THE PRESS</h2> -</div> - - -<p>“<i>Mr. C. H. Hinton discusses the subject of the higher dimensionality of -space, his aim being to avoid mathematical subtleties and technicalities, and -thus enable his argument to be followed by readers who are not sufficiently -conversant with mathematics to follow these processes of reasoning.</i>”—<span class="smcap">Notts -Guardian.</span></p> - -<p>“<i>The fourth dimension is a subject which has had a great fascination for -many teachers, and though one cannot pretend to have quite grasped -Mr. Hinton’s conceptions and arguments, yet it must be admitted that he -reveals the elusive idea in quite a fascinating light. Quite apart from the -main thesis of the book many chapters are of great independent interest. -Altogether an interesting, clever and ingenious book.</i>”—<span class="smcap">Dundee Courier.</span></p> - -<p>“<i>The book will well repay the study of men who like to exercise their wits -upon the problems of abstract thought.</i>”—<span class="smcap">Scotsman.</span></p> - -<p>“<i>Professor Hinton has done well to attempt a treatise of moderate size, -which shall at once be clear in method and free from technicalities of the -schools.</i>”—<span class="smcap">Pall Mall Gazette.</span></p> - -<p>“<i>A very interesting book he has made of it.</i>”—<span class="smcap">Publishers’ Circular.</span></p> - -<p>“<i>Mr. Hinton tries to explain the theory of the fourth dimension so that -the ordinary reasoning mind can get a grasp of what metaphysical -mathematicians mean by it. If he is not altogether successful it is not from -want of clearness on his part, but because the whole theory comes as such an -absolute shock to all one’s preconceived ideas.</i>”—<span class="smcap">Bristol Times.</span></p> - -<p>“<i>Mr. Hinton’s enthusiasm is only the result of an exhaustive study, which -has enabled him to set his subject before the reader with far more than the -amount of lucidity to which it is accustomed.</i>”—<span class="smcap">Pall Mall Gazette.</span></p> - -<p>“<i>The book throughout is a very solid piece of reasoning in the domain of -higher mathematics.</i>”—<span class="smcap">Glasgow Herald.</span></p> - -<p>“<i>Those who wish to grasp the meaning of this somewhat difficult subject -would do well to read</i> The Fourth Dimension. <i>No mathematical knowledge -is demanded of the reader, and any one, who is not afraid of a little hard -thinking, should be able to follow the argument.</i>”—<span class="smcap">Light.</span></p> - -<p>“<i>A splendidly clear re-statement of the old problem of the fourth dimension. -All who are interested in this subject will find the work not only fascinating, -but lucid, it being written in a style easily understandable. The illustrations -make still more clear the letterpress, and the whole is most admirably adapted -to the requirements of the novice or the student.</i>”—<span class="smcap">Two Worlds.</span></p> - -<p>“<i>Those in search of mental gymnastics will find abundance of exercise in -Mr. C. H. Hinton’s</i> Fourth Dimension.”—<span class="smcap">Westminster Review.</span></p> - - -<p><span class="smcap">First Edition</span>, <i>April 1904</i>; <span class="smcap">Second Edition</span>, <i>May 1906</i>.</p> - - -<hr class="chap x-ebookmaker-drop" /> - -<div class="figcenter illowp100" id="i_frontis" style="max-width: 50em;"> - <img src="images/i_frontis.jpg" alt="" /> - <div class="caption">Views of the Tessaract.</div> -</div> - -<div class="chapter"></div> - - -<h1> -<small>THE</small><br /> - -FOURTH DIMENSION</h1> - -<p class="center small">BY</p> - -<p class="center">C. HOWARD HINTON, M.A.<br /> - -<small>AUTHOR OF “SCIENTIFIC ROMANCES”<br /> -“A NEW ERA OF THOUGHT,” ETC., ETC.</small></p> - -<div class="figcenter illowp20" id="colop" style="max-width: 9.375em;"> - <img src="images/colop.png" alt="Colophon" /> -</div> - -<p class="center"><small>LONDON</small><br /> -SWAN SONNENSCHEIN & CO., LIMITED<br /> -25 HIGH STREET, BLOOMSBURY<br /> -<br /> -<small>1906</small><br /> -</p> - - -<p class="center small spaced"> -PRINTED BY<br /> -HAZELL, WATSON AND VINEY, LD.,<br /> -LONDON AND AYLESBURY.</p> -<hr class="chap x-ebookmaker-drop" /> - -<div class="chapter"> -<p><span class="pagenum" id="Page_v">[Pg v]</span></p> - -<h2 class="nobreak" id="PREFACE">PREFACE</h2> -</div> - - -<p>I have endeavoured to present the subject of the higher -dimensionality of space in a clear manner, devoid of -mathematical subtleties and technicalities. In order to -engage the interest of the reader, I have in the earlier -chapters dwelt on the perspective the hypothesis of a -fourth dimension opens, and have treated of the many -connections there are between this hypothesis and the -ordinary topics of our thoughts.</p> - -<p>A lack of mathematical knowledge will prove of no -disadvantage to the reader, for I have used no mathematical -processes of reasoning. I have taken the view -that the space which we ordinarily think of, the space -of real things (which I would call permeable matter), -is different from the space treated of by mathematics. -Mathematics will tell us a great deal about space, just -as the atomic theory will tell us a great deal about the -chemical combinations of bodies. But after all, a theory -is not precisely equivalent to the subject with regard -to which it is held. There is an opening, therefore, from -the side of our ordinary space perceptions for a simple, -altogether rational, mechanical, and observational way<span class="pagenum" id="Page_vi">[Pg vi]</span> -of treating this subject of higher space, and of this -opportunity I have availed myself.</p> - -<p>The details introduced in the earlier chapters, especially -in Chapters VIII., IX., X., may perhaps be found -wearisome. They are of no essential importance in the -main line of argument, and if left till Chapters XI. -and XII. have been read, will be found to afford -interesting and obvious illustrations of the properties -discussed in the later chapters.</p> - -<p>My thanks are due to the friends who have assisted -me in designing and preparing the modifications of -my previous models, and in no small degree to the -publisher of this volume, Mr. Sonnenschein, to whose -unique appreciation of the line of thought of this, as -of my former essays, their publication is owing. By -the provision of a coloured plate, in addition to the other -illustrations, he has added greatly to the convenience -of the reader.</p> - -<p class="psig"> -<span class="smcap">C. Howard Hinton.</span></p> - -<hr class="chap x-ebookmaker-drop" /> - -<div class="chapter"> -<p><span class="pagenum" id="Page_vii">[Pg vii]</span></p> - -<h2 class="nobreak" id="CONTENTS">CONTENTS</h2> -</div> - - -<table class="standard" summary=""> -<tr> -<td class="tdr"><small>CHAP</small>.</td> -<td></td> -<td class="tdr"><small>PAGE</small></td> -</tr> -<tr> -<td class="tdr"><a href="#CHAPTER_I">I.</a></td> -<td class="tdh"><span class="smcap">Four-Dimensional Space</span></td> -<td class="tdr">1</td> -</tr> -<tr> -<td class="tdr"><a href="#CHAPTER_II">II.</a></td> -<td class="tdh"><span class="smcap">The Analogy of a Plane World</span></td> -<td class="tdr">6</td> -</tr> -<tr> -<td class="tdr"><a href="#CHAPTER_III">III.</a></td> -<td class="tdh"><span class="smcap">The Significance of a Four-Dimensional -Existence</span></td> -<td class="tdr">15</td> -</tr> -<tr> -<td class="tdr"><a href="#CHAPTER_IV">IV.</a></td> -<td class="tdh"><span class="smcap">The First Chapter in the History of Four -Space</span></td> -<td class="tdr">23</td> -</tr> -<tr> -<td class="tdr"><a href="#CHAPTER_V">V.</a></td> -<td class="tdh"><span class="smcap">The Second Chapter in the History Of Four Space</span></td> -<td class="tdr">41</td> -</tr> -<tr> -<td></td> -<td class="tdh"><small>Lobatchewsky, Bolyai, and Gauss<br />Metageometry</small></td> -</tr> -<tr> -<td class="tdr"><a href="#CHAPTER_VI">VI.</a></td> -<td class="tdh"><span class="smcap">The Higher World</span></td> -<td class="tdr">61</td> -</tr> -<tr> -<td class="tdr"><a href="#CHAPTER_VII">VII.</a></td> -<td class="tdh"><span class="smcap">The Evidence for a Fourth Dimension</span></td> -<td class="tdr">76</td> -</tr> -<tr> -<td class="tdr"><a href="#CHAPTER_VIII">VIII.</a></td> -<td class="tdh"><span class="smcap">The Use of Four Dimensions in Thought</span></td> -<td class="tdr">85</td> -</tr> -<tr> -<td class="tdr"><a href="#CHAPTER_IX">IX.</a></td> -<td class="tdh"><span class="smcap">Application to Kant’s Theory of Experience</span></td> -<td class="tdr">107</td> -</tr> -<tr> -<td class="tdr"><a href="#CHAPTER_X">X.</a></td> -<td class="tdh"><span class="smcap">A Four-Dimensional Figure</span></td> -<td class="tdr">122</td> -</tr> -<tr> -<td class="tdr"><a href="#CHAPTER_XI">XI.</a></td> -<td class="tdh"><span class="smcap">Nomenclature and Analogies</span></td> -<td class="tdr">136<span class="pagenum" id="Page_viii">[Pg viii]</span></td> -</tr> -<tr> -<td class="tdr"><a href="#CHAPTER_XII">XII.</a></td> -<td class="tdh"><span class="smcap">The Simplest Four-Dimensional Solid</span></td> -<td class="tdr">157</td> -</tr> -<tr> -<td class="tdr"><a href="#CHAPTER_XIII">XIII.</a></td> -<td class="tdh"><span class="smcap">Remarks on the Figures</span></td> -<td class="tdr">178</td> -</tr> -<tr> -<td class="tdr"><a href="#CHAPTER_XIV">XIV.</a></td> -<td class="tdh"><span class="smcap">A Recapitulation and Extension of the -Physical Argument</span></td> -<td class="tdr">203</td> -</tr> -<tr> -<td class="tdl" colspan="2"><a href="#APPENDIX_I">APPENDIX I.</a>—<span class="smcap">The Models</span></td> -<td class="tdr">231</td> -</tr> -<tr> -<td class="tdl" colspan="2"><a href="#APPENDIX_II">APPENDIX II.</a>—<span class="smcap">A Language of Space</span></td> -<td class="tdr">248</td> -</tr> -</table> - -<hr class="chap x-ebookmaker-drop" /> - -<div class="chapter"> -<p><span class="pagenum" id="Page_1">[Pg 1]</span></p> - -<p class="half-title">THE FOURTH DIMENSION</p> - - -<hr class="small" /> - - -<h2 class="nobreak" id="CHAPTER_I">CHAPTER I<br /> - - -<small>FOUR-DIMENSIONAL SPACE</small></h2> -</div> - -<p>There is nothing more indefinite, and at the same time -more real, than that which we indicate when we speak -of the “higher.” In our social life we see it evidenced -in a greater complexity of relations. But this complexity -is not all. There is, at the same time, a contact -with, an apprehension of, something more fundamental, -more real.</p> - -<p>With the greater development of man there comes -a consciousness of something more than all the forms -in which it shows itself. There is a readiness to give -up all the visible and tangible for the sake of those -principles and values of which the visible and tangible -are the representation. The physical life of civilised -man and of a mere savage are practically the same, but -the civilised man has discovered a depth in his existence, -which makes him feel that that which appears all to -the savage is a mere externality and appurtenage to his -true being.</p> - -<p>Now, this higher—how shall we apprehend it? It is -generally embraced by our religious faculties, by our -idealising tendency. But the higher existence has two -sides. It has a being as well as qualities. And in trying<span class="pagenum" id="Page_2">[Pg 2]</span> -to realise it through our emotions we are always taking the -subjective view. Our attention is always fixed on what we -feel, what we think. Is there any way of apprehending -the higher after the purely objective method of a natural -science? I think that there is.</p> - -<p>Plato, in a wonderful allegory, speaks of some men -living in such a condition that they were practically -reduced to be the denizens of a shadow world. They -were chained, and perceived but the shadows of themselves -and all real objects projected on a wall, towards -which their faces were turned. All movements to them -were but movements on the surface, all shapes but the -shapes of outlines with no substantiality.</p> - -<p>Plato uses this illustration to portray the relation -between true being and the illusions of the sense world. -He says that just as a man liberated from his chains -could learn and discover that the world was solid and -real, and could go back and tell his bound companions of -this greater higher reality, so the philosopher who has -been liberated, who has gone into the thought of the -ideal world, into the world of ideas greater and more -real than the things of sense, can come and tell his fellow -men of that which is more true than the visible sun—more -noble than Athens, the visible state.</p> - -<p>Now, I take Plato’s suggestion; but literally, not -metaphorically. He imagines a world which is lower -than this world, in that shadow figures and shadow -motions are its constituents; and to it he contrasts the real -world. As the real world is to this shadow world, so is the -higher world to our world. I accept his analogy. As our -world in three dimensions is to a shadow or plane world, -so is the higher world to our three-dimensional world. -That is, the higher world is four-dimensional; the higher -being is, so far as its existence is concerned apart from its -qualities, to be sought through the conception of an actual<span class="pagenum" id="Page_3">[Pg 3]</span> -existence spatially higher than that which we realise with -our senses.</p> - -<p>Here you will observe I necessarily leave out all that -gives its charm and interest to Plato’s writings. All -those conceptions of the beautiful and good which live -immortally in his pages.</p> - -<p>All that I keep from his great storehouse of wealth is -this one thing simply—a world spatially higher than this -world, a world which can only be approached through the -stocks and stones of it, a world which must be apprehended -laboriously, patiently, through the material things -of it, the shapes, the movements, the figures of it.</p> - -<p>We must learn to realise the shapes of objects in -this world of the higher man; we must become familiar -with the movements that objects make in his world, so -that we can learn something about his daily experience, -his thoughts of material objects, his machinery.</p> - -<p>The means for the prosecution of this enquiry are given -in the conception of space itself.</p> - -<p>It often happens that that which we consider to be -unique and unrelated gives us, within itself, those relations -by means of which we are able to see it as related to -others, determining and determined by them.</p> - -<p>Thus, on the earth is given that phenomenon of weight -by means of which Newton brought the earth into its -true relation to the sun and other planets. Our terrestrial -globe was determined in regard to other bodies of the -solar system by means of a relation which subsisted on -the earth itself.</p> - -<p>And so space itself bears within it relations of which -we can determine it as related to other space. For within -space are given the conceptions of point and line, line and -plane, which really involve the relation of space to a -higher space.</p> - -<p>Where one segment of a straight line leaves off and<span class="pagenum" id="Page_4">[Pg 4]</span> -another begins is a point, and the straight line itself can -be generated by the motion of the point.</p> - -<p>One portion of a plane is bounded from another by a -straight line, and the plane itself can be generated by -the straight line moving in a direction not contained -in itself.</p> - -<p>Again, two portions of solid space are limited with -regard to each other by a plane; and the plane, moving -in a direction not contained in itself, can generate solid -space.</p> - -<p>Thus, going on, we may say that space is that which -limits two portions of higher space from each other, and -that our space will generate the higher space by moving -in a direction not contained in itself.</p> - -<p>Another indication of the nature of four-dimensional -space can be gained by considering the problem of the -arrangement of objects.</p> - -<p>If I have a number of swords of varying degrees of -brightness, I can represent them in respect of this quality -by points arranged along a straight line.</p> - -<div class="figleft illowp25" id="fig_1" style="max-width: 10em;"> - <img src="images/fig_1.png" alt="" /> - <div class="caption">Fig. 1.</div> -</div> - -<p>If I place a sword at <span class="allsmcap">A</span>, <a href="#fig_1">fig. 1</a>, and regard it as having -a certain brightness, then the other swords -can be arranged in a series along the -line, as at <span class="allsmcap">A</span>, <span class="allsmcap">B</span>, <span class="allsmcap">C</span>, etc., according to -their degrees of brightness.</p> - -<div class="figleft illowp25" id="fig_2" style="max-width: 10em;"> - <img src="images/fig_2.png" alt="" /> - <div class="caption">Fig. 2.</div> -</div> - -<p>If now I take account of another quality, say length, -they can be arranged in a plane. Starting from <span class="allsmcap">A</span>, <span class="allsmcap">B</span>, <span class="allsmcap">C</span>, I -can find points to represent different -degrees of length along such lines as -<span class="allsmcap">AF</span>, <span class="allsmcap">BD</span>, <span class="allsmcap">CE</span>, drawn from <span class="allsmcap">A</span> and <span class="allsmcap">B</span> and <span class="allsmcap">C</span>. -Points on these lines represent different -degrees of length with the same degree of -brightness. Thus the whole plane is occupied by points -representing all conceivable varieties of brightness and -length.</p> - -<p><span class="pagenum" id="Page_5">[Pg 5]</span></p> - -<div class="figleft illowp30" id="fig_3" style="max-width: 10em;"> - <img src="images/fig_3.png" alt="" /> - <div class="caption">Fig. 3.</div> -</div> - -<p>Bringing in a third quality, say sharpness, I can draw, -as in <a href="#fig_3">fig. 3</a>, any number of upright -lines. Let distances along these -upright lines represent degrees of -sharpness, thus the points <span class="allsmcap">F</span> and <span class="allsmcap">G</span> -will represent swords of certain -definite degrees of the three qualities -mentioned, and the whole of space will serve to represent -all conceivable degrees of these three qualities.</p> - -<p>If now I bring in a fourth quality, such as weight, and -try to find a means of representing it as I did the other -three qualities, I find a difficulty. Every point in space is -taken up by some conceivable combination of the three -qualities already taken.</p> - -<p>To represent four qualities in the same way as that in -which I have represented three, I should need another -dimension of space.</p> - -<p>Thus we may indicate the nature of four-dimensional -space by saying that it is a kind of space which would -give positions representative of four qualities, as three-dimensional -space gives positions representative of three -qualities.</p> -<hr class="chap x-ebookmaker-drop" /> - -<div class="chapter"> -<p><span class="pagenum" id="Page_6">[Pg 6]</span></p> - -<h2 class="nobreak" id="CHAPTER_II">CHAPTER II<br /> - -<small><i>THE ANALOGY OF A PLANE WORLD</i></small></h2></div> - - -<p>At the risk of some prolixity I will go fully into the -experience of a hypothetical creature confined to motion -on a plane surface. By so doing I shall obtain an analogy -which will serve in our subsequent enquiries, because the -change in our conception, which we make in passing from -the shapes and motions in two dimensions to those in -three, affords a pattern by which we can pass on still -further to the conception of an existence in four-dimensional -space.</p> - -<p>A piece of paper on a smooth table affords a ready -image of a two-dimensional existence. If we suppose the -being represented by the piece of paper to have no -knowledge of the thickness by which he projects above the -surface of the table, it is obvious that he can have no -knowledge of objects of a similar description, except by -the contact with their edges. His body and the objects -in his world have a thickness of which however, he has no -consciousness. Since the direction stretching up from -the table is unknown to him he will think of the objects -of his world as extending in two dimensions only. Figures -are to him completely bounded by their lines, just as solid -objects are to us by their surfaces. He cannot conceive -of approaching the centre of a circle, except by breaking -through the circumference, for the circumference encloses -the centre in the directions in which motion is possible to<span class="pagenum" id="Page_7">[Pg 7]</span> -him. The plane surface over which he slips and with -which he is always in contact will be unknown to him; -there are no differences by which he can recognise its -existence.</p> - -<p>But for the purposes of our analogy this representation -is deficient.</p> - -<p>A being as thus described has nothing about him to -push off from, the surface over which he slips affords no -means by which he can move in one direction rather than -another. Placed on a surface over which he slips freely, -he is in a condition analogous to that in which we should -be if we were suspended free in space. There is nothing -which he can push off from in any direction known to him.</p> - -<p>Let us therefore modify our representation. Let us -suppose a vertical plane against which particles of thin -matter slip, never leaving the surface. Let these particles -possess an attractive force and cohere together into a disk; -this disk will represent the globe of a plane being. He -must be conceived as existing on the rim.</p> - -<div class="figleft illowp25" id="fig_4" style="max-width: 10.9375em;"> - <img src="images/fig_4.png" alt="" /> - <div class="caption">Fig. 4.</div> -</div> - -<p>Let 1 represent this vertical disk of flat matter and 2 -the plane being on it, standing upon its -rim as we stand on the surface of our earth. -The direction of the attractive force of his -matter will give the creature a knowledge -of up and down, determining for him one -direction in his plane space. Also, since -he can move along the surface of his earth, -he will have the sense of a direction parallel to its surface, -which we may call forwards and backwards.</p> - -<p>He will have no sense of right and left—that is, of the -direction which we recognise as extending out from the -plane to our right and left.</p> - -<p>The distinction of right and left is the one that we -must suppose to be absent, in order to project ourselves -into the condition of a plane being.</p> - -<p><span class="pagenum" id="Page_8">[Pg 8]</span></p> - -<p>Let the reader imagine himself, as he looks along the -plane, <a href="#fig_4">fig. 4</a>, to become more and more identified with -the thin body on it, till he finally looks along parallel to -the surface of the plane earth, and up and down, losing -the sense of the direction which stretches right and left. -This direction will be an unknown dimension to him.</p> - -<p>Our space conceptions are so intimately connected with -those which we derive from the existence of gravitation -that it is difficult to realise the condition of a plane being, -without picturing him as in material surroundings with -a definite direction of up and down. Hence the necessity -of our somewhat elaborate scheme of representation, which, -when its import has been grasped, can be dispensed with -for the simpler one of a thin object slipping over a -smooth surface, which lies in front of us.</p> - -<p>It is obvious that we must suppose some means by -which the plane being is kept in contact with the surface -on which he slips. The simplest supposition to make is -that there is a transverse gravity, which keeps him to the -plane. This gravity must be thought of as different to -the attraction exercised by his matter, and as unperceived -by him.</p> - -<p>At this stage of our enquiry I do not wish to enter -into the question of how a plane being could arrive at -a knowledge of the third dimension, but simply to investigate -his plane consciousness.</p> - -<p>It is obvious that the existence of a plane being must -be very limited. A straight line standing up from the -surface of his earth affords a bar to his progress. An -object like a wheel which rotates round an axis would -be unknown to him, for there is no conceivable way in -which he can get to the centre without going through -the circumference. He would have spinning disks, but -could not get to the centre of them. The plane being -can represent the motion from any one point of his space<span class="pagenum" id="Page_9">[Pg 9]</span> -to any other, by means of two straight lines drawn at -right angles to each other.</p> - -<div class="figleft illowp35" id="fig_5" style="max-width: 26.6875em;"> - <img src="images/fig_5.png" alt="" /> - <div class="caption">Fig. 5.</div> -</div> - -<p>Let <span class="allsmcap">AX</span> and <span class="allsmcap">AY</span> be two such axes. He can accomplish -the translation from <span class="allsmcap">A</span> to <span class="allsmcap">B</span> by going along <span class="allsmcap">AX</span> to <span class="allsmcap">C</span>, and -then from <span class="allsmcap">C</span> along <span class="allsmcap">CB</span> parallel to <span class="allsmcap">AY</span>.</p> - -<p>The same result can of course be obtained -by moving to <span class="allsmcap">D</span> along <span class="allsmcap">AY</span> and then parallel -to <span class="allsmcap">AX</span> from <span class="allsmcap">D</span> to <span class="allsmcap">B</span>, or of course by any -diagonal movement compounded by these -axial movements.</p> - -<p>By means of movements parallel to -these two axes he can proceed (except for -material obstacles) from any one point of his space to -any other.</p> - -<div class="figleft illowp35" id="fig_6" style="max-width: 16.875em;"> - <img src="images/fig_6.png" alt="" /> - <div class="caption">Fig. 6.</div> -</div> - -<p>If now we suppose a third line drawn -out from <span class="allsmcap">A</span> at right angles to the plane -it is evident that no motion in either -of the two dimensions he knows will -carry him in the least degree in the -direction represented by <span class="allsmcap">AZ</span>.</p> - -<p>The lines <span class="allsmcap">AZ</span> and <span class="allsmcap">AX</span> determine a -plane. If he could be taken off his plane, and transferred -to the plane <span class="allsmcap">AXZ</span>, he would be in a world exactly -like his own. From every line in his -world there goes off a space world exactly -like his own.</p> - -<div class="figleft illowp25" id="fig_7" style="max-width: 12.5em;"> - <img src="images/fig_7.png" alt="" /> - <div class="caption">Fig. 7.</div> -</div> - -<p>From every point in his world a line can -be drawn parallel to <span class="allsmcap">AZ</span> in the direction -unknown to him. If we suppose the square -in <a href="#fig_7">fig. 7</a> to be a geometrical square from -every point of it, inside as well as on the -contour, a straight line can be drawn parallel -to <span class="allsmcap">AZ</span>. The assemblage of these lines constitute a solid -figure, of which the square in the plane is the base. If -we consider the square to represent an object in the plane<span class="pagenum" id="Page_10">[Pg 10]</span> -being’s world then we must attribute to it a very small -thickness, for every real thing must possess all three -dimensions. This thickness he does not perceive, but -thinks of this real object as a geometrical square. He -thinks of it as possessing area only, and no degree of -solidity. The edges which project from the plane to a -very small extent he thinks of as having merely length -and no breadth—as being, in fact, geometrical lines.</p> - -<p>With the first step in the apprehension of a third -dimension there would come to a plane being the conviction -that he had previously formed a wrong conception -of the nature of his material objects. He had conceived -them as geometrical figures of two dimensions only. -If a third dimension exists, such figures are incapable -of real existence. Thus he would admit that all his real -objects had a certain, though very small thickness in the -unknown dimension, and that the conditions of his -existence demanded the supposition of an extended sheet -of matter, from contact with which in their motion his -objects never diverge.</p> - -<p>Analogous conceptions must be formed by us on the -supposition of a four-dimensional existence. We must -suppose a direction in which we can never point extending -from every point of our space. We must draw a distinction -between a geometrical cube and a cube of real -matter. The cube of real matter we must suppose to -have an extension in an unknown direction, real, but so -small as to be imperceptible by us. From every point -of a cube, interior as well as exterior, we must imagine -that it is possible to draw a line in the unknown direction. -The assemblage of these lines would constitute a higher -solid. The lines going off in the unknown direction from -the face of a cube would constitute a cube starting from -that face. Of this cube all that we should see in our -space would be the face.</p> - -<p><span class="pagenum" id="Page_11">[Pg 11]</span></p> - -<p>Again, just as the plane being can represent any -motion in his space by two axes, so we can represent any -motion in our three-dimensional space by means of three -axes. There is no point in our space to which we cannot -move by some combination of movements on the directions -marked out by these axes.</p> - -<p>On the assumption of a fourth dimension we have -to suppose a fourth axis, which we will call <span class="allsmcap">AW</span>. It must -be supposed to be at right angles to each and every -one of the three axes <span class="allsmcap">AX</span>, <span class="allsmcap">AY</span>, <span class="allsmcap">AZ</span>. Just as the two axes, -<span class="allsmcap">AX</span>, <span class="allsmcap">AZ</span>, determine a plane which is similar to the original -plane on which we supposed the plane being to exist, but -which runs off from it, and only meets it in a line; so in -our space if we take any three axes such as <span class="allsmcap">AX</span>, <span class="allsmcap">AY</span>, and -<span class="allsmcap">AW</span>, they determine a space like our space world. This -space runs off from our space, and if we were transferred -to it we should find ourselves in a space exactly similar to -our own.</p> - -<p>We must give up any attempt to picture this space in -its relation to ours, just as a plane being would have to -give up any attempt to picture a plane at right angles -to his plane.</p> - -<p>Such a space and ours run in different directions from -the plane of <span class="allsmcap">AX</span> and <span class="allsmcap">AY</span>. They meet in this plane but -have nothing else in common, just as the plane space -of <span class="allsmcap">AX</span> and <span class="allsmcap">AY</span> and that of <span class="allsmcap">AX</span> and <span class="allsmcap">AZ</span> run in different -directions and have but the line <span class="allsmcap">AX</span> in common.</p> - -<p>Omitting all discussion of the manner on which a plane -being might be conceived to form a theory of a three-dimensional -existence, let us examine how, with the means -at his disposal, he could represent the properties of three-dimensional -objects.</p> - -<div class="figleft illowp40" id="fig_8" style="max-width: 25em;"> - <img src="images/fig_8.png" alt="" /> - <div class="caption">Fig. 8.</div> -</div> - -<p>There are two ways in which the plane being can think -of one of our solid bodies. He can think of the cube, -<a href="#fig_8">fig. 8</a>, as composed of a number of sections parallel to<span class="pagenum" id="Page_12">[Pg 12]</span> -his plane, each lying in the third dimension a little -further off from his plane than -the preceding one. These sections -he can represent as a -series of plane figures lying in -his plane, but in so representing -them he destroys the coherence -of them in the higher figure. -The set of squares, <span class="allsmcap">A</span>, <span class="allsmcap">B</span>, <span class="allsmcap">C</span>, <span class="allsmcap">D</span>, -represents the section parallel -to the plane of the cube shown in figure, but they are -not in their proper relative positions.</p> - -<p>The plane being can trace out a movement in the third -dimension by assuming discontinuous leaps from one -section to another. Thus, a motion along the edge of -the cube from left to right would be represented in the -set of sections in the plane as the succession of the -corners of the sections <span class="allsmcap">A</span>, <span class="allsmcap">B</span>, <span class="allsmcap">C</span>, <span class="allsmcap">D</span>. A point moving from -<span class="allsmcap">A</span> through <span class="allsmcap">BCD</span> in our space must be represented in the -plane as appearing in <span class="allsmcap">A</span>, then in <span class="allsmcap">B</span>, and so on, without -passing through the intervening plane space.</p> - -<p>In these sections the plane being leaves out, of course, -the extension in the third dimension; the distance between -any two sections is not represented. In order to realise -this distance the conception of motion can be employed.</p> - -<div class="figleft illowp25" id="fig_9" style="max-width: 12.5em;"> - <img src="images/fig_9.png" alt="" /> - <div class="caption">Fig. 9.</div> -</div> - -<p>Let <a href="#fig_9">fig. 9</a> represent a cube passing transverse to the -plane. It will appear to the plane being as a -square object, but the matter of which this -object is composed will be continually altering. -One material particle takes the place of another, -but it does not come from anywhere or go -anywhere in the space which the plane being -knows.</p> - -<p>The analogous manner of representing a higher solid in -our case, is to conceive it as composed of a number of<span class="pagenum" id="Page_13">[Pg 13]</span> -sections, each lying a little further off in the unknown -direction than the preceding.</p> - -<div class="figleft illowp75" id="fig_10" style="max-width: 31.25em;"> - <img src="images/fig_10.png" alt="" /> - <div class="caption">Fig. 10.</div> -</div> - -<p>We can represent these sections as a number of solids. -Thus the cubes <span class="allsmcap">A</span>, <span class="allsmcap">B</span>, <span class="allsmcap">C</span>, <span class="allsmcap">D</span>, -may be considered as -the sections at different -intervals in the unknown -dimension of a higher -cube. Arranged thus their coherence in the higher figure -is destroyed, they are mere representations.</p> - -<p>A motion in the fourth dimension from <span class="allsmcap">A</span> through <span class="allsmcap">B</span>, <span class="allsmcap">C</span>, -etc., would be continuous, but we can only represent it as -the occupation of the positions <span class="allsmcap">A</span>, <span class="allsmcap">B</span>, <span class="allsmcap">C</span>, etc., in succession. -We can exhibit the results of the motion at different -stages, but no more.</p> - -<p>In this representation we have left out the distance -between one section and another; we have considered the -higher body merely as a series of sections, and so left out -its contents. The only way to exhibit its contents is to -call in the aid of the conception of motion.</p> - -<div class="figleft illowp25" id="fig_11" style="max-width: 9.375em;"> - <img src="images/fig_11.png" alt="" /> - <div class="caption">Fig. 11.</div> -</div> - -<p>If a higher cube passes transverse to our space, it will -appear as a cube isolated in space, the part -that has not come into our space and the part -that has passed through will not be visible. -The gradual passing through our space would -appear as the change of the matter of the cube -before us. One material particle in it is succeeded by -another, neither coming nor going in any direction we can -point to. In this manner, by the duration of the figure, -we can exhibit the higher dimensionality of it; a cube of -our matter, under the circumstances supposed, namely, -that it has a motion transverse to our space, would instantly -disappear. A higher cube would last till it had passed -transverse to our space by its whole distance of extension -in the fourth dimension.</p> - -<p><span class="pagenum" id="Page_14">[Pg 14]</span></p> - -<p>As the plane being can think of the cube as consisting -of sections, each like a figure he knows, extending away -from his plane, so we can think of a higher solid as composed -of sections, each like a solid which we know, but -extending away from our space.</p> - -<p>Thus, taking a higher cube, we can look on it as -starting from a cube in our space and extending in the -unknown dimension.</p> - -<div class="figcenter illowp100" id="fig_12" style="max-width: 25em;"> - <img src="images/fig_12.png" alt="" /> - <div class="caption">Fig. 12.</div> -</div> - -<p>Take the face <span class="allsmcap">A</span> and conceive it to exist as simply a -face, a square with no thickness. From this face the -cube in our space extends by the occupation of space -which we can see.</p> - -<p>But from this face there extends equally a cube in the -unknown dimension. We can think of the higher cube, -then, by taking the set of sections <span class="allsmcap">A</span>, <span class="allsmcap">B</span>, <span class="allsmcap">C</span>, <span class="allsmcap">D</span>, etc., and -considering that from each of them there runs a cube. -These cubes have nothing in common with each other, -and of each of them in its actual position all that we can -have in our space is an isolated square. It is obvious that -we can take our series of sections in any manner we -please. We can take them parallel, for instance, to any -one of the three isolated faces shown in the figure. -Corresponding to the three series of sections at right -angles to each other, which we can make of the cube -in space, we must conceive of the higher cube, as composed -of cubes starting from squares parallel to the faces -of the cube, and of these cubes all that exist in our space -are the isolated squares from which they start.</p> -<hr class="chap x-ebookmaker-drop" /> - -<div class="chapter"> -<p><span class="pagenum" id="Page_15">[Pg 15]</span></p> - -<h2 class="nobreak" id="CHAPTER_III">CHAPTER III<br /> - -<small><i>THE SIGNIFICANCE OF A FOUR-DIMENSIONAL -EXISTENCE</i></small></h2></div> - - -<p>Having now obtained the conception of a four-dimensional -space, and having formed the analogy which, without -any further geometrical difficulties, enables us to enquire -into its properties, I will refer the reader, whose interest -is principally in the mechanical aspect, to Chapters VI. -and VII. In the present chapter I will deal with the -general significance of the enquiry, and in the next -with the historical origin of the idea.</p> - -<p>First, with regard to the question of whether there -is any evidence that we are really in four-dimensional -space, I will go back to the analogy of the plane world.</p> - -<p>A being in a plane world could not have any experience -of three-dimensional shapes, but he could have -an experience of three-dimensional movements.</p> - -<p>We have seen that his matter must be supposed to -have an extension, though a very small one, in the third -dimension. And thus, in the small particles of his -matter, three-dimensional movements may well be conceived -to take place. Of these movements he would only -perceive the resultants. Since all movements of an -observable size in the plane world are two-dimensional, -he would only perceive the resultants in two dimensions -of the small three-dimensional movements. Thus, there -would be phenomena which he could not explain by his<span class="pagenum" id="Page_16">[Pg 16]</span> -theory of mechanics—motions would take place which -he could not explain by his theory of motion. Hence, -to determine if we are in a four-dimensional world, we -must examine the phenomena of motion in our space. -If movements occur which are not explicable on the suppositions -of our three-dimensional mechanics, we should -have an indication of a possible four-dimensional motion, -and if, moreover, it could be shown that such movements -would be a consequence of a four-dimensional motion in -the minute particles of bodies or of the ether, we should -have a strong presumption in favour of the reality of -the fourth dimension.</p> - -<p>By proceeding in the direction of finer and finer subdivision, -we come to forms of matter possessing properties -different to those of the larger masses. It is probable that -at some stage in this process we should come to a form -of matter of such minute subdivision that its particles -possess a freedom of movement in four dimensions. This -form of matter I speak of as four-dimensional ether, and -attribute to it properties approximating to those of a -perfect liquid.</p> - -<p>Deferring the detailed discussion of this form of matter -to Chapter VI., we will now examine the means by which -a plane being would come to the conclusion that three-dimensional -movements existed in his world, and point -out the analogy by which we can conclude the existence -of four-dimensional movements in our world. Since the -dimensions of the matter in his world are small in the -third direction, the phenomena in which he would detect -the motion would be those of the small particles of -matter.</p> - -<p>Suppose that there is a ring in his plane. We can -imagine currents flowing round the ring in either of two -opposite directions. These would produce unlike effects, -and give rise to two different fields of influence. If the<span class="pagenum" id="Page_17">[Pg 17]</span> -ring with a current in it in one direction be taken up -and turned over, and put down again on the plane, it -would be identical with the ring having a current in the -opposite direction. An operation of this kind would be -impossible to the plane being. Hence he would have -in his space two irreconcilable objects, namely, the two -fields of influence due to the two rings with currents in -them in opposite directions. By irreconcilable objects -in the plane I mean objects which cannot be thought -of as transformed one into the other by any movement -in the plane.</p> - -<p>Instead of currents flowing in the rings we can imagine -a different kind of current. Imagine a number of small -rings strung on the original ring. A current round these -secondary rings would give two varieties of effect, or two -different fields of influence, according to its direction. -These two varieties of current could be turned one into -the other by taking one of the rings up, turning it over, -and putting it down again in the plane. This operation -is impossible to the plane being, hence in this case also -there would be two irreconcilable fields in the plane. -Now, if the plane being found two such irreconcilable -fields and could prove that they could not be accounted -for by currents in the rings, he would have to admit the -existence of currents round the rings—that is, in rings -strung on the primary ring. Thus he would come to -admit the existence of a three-dimensional motion, for -such a disposition of currents is in three dimensions.</p> - -<p>Now in our space there are two fields of different -properties, which can be produced by an electric current -flowing in a closed circuit or ring. These two fields can -be changed one into the other by reversing the currents, but -they cannot be changed one into the other by any turning -about of the rings in our space; for the disposition of the -field with regard to the ring itself is different when we<span class="pagenum" id="Page_18">[Pg 18]</span> -turn the ring, over and when we reverse the direction of -the current in the ring.</p> - -<p>As hypotheses to explain the differences of these two -fields and their effects we can suppose the following kinds -of space motions:—First, a current along the conductor; -second, a current round the conductor—that is, of rings of -currents strung on the conductor as an axis. Neither of -these suppositions accounts for facts of observation.</p> - -<p>Hence we have to make the supposition of a four-dimensional -motion. We find that a four-dimensional -rotation of the nature explained in a subsequent chapter, -has the following characteristics:—First, it would give us -two fields of influence, the one of which could be turned -into the other by taking the circuit up into the fourth -dimension, turning it over, and putting it down in our -space again, precisely as the two kinds of fields in the -plane could be turned one into the other by a reversal of -the current in our space. Second, it involves a phenomenon -precisely identical with that most remarkable and -mysterious feature of an electric current, namely that it -is a field of action, the rim of which necessarily abuts on a -continuous boundary formed by a conductor. Hence, on -the assumption of a four-dimensional movement in the -region of the minute particles of matter, we should expect -to find a motion analogous to electricity.</p> - -<p>Now, a phenomenon of such universal occurrence as -electricity cannot be due to matter and motion in any -very complex relation, but ought to be seen as a simple -and natural consequence of their properties. I infer that -the difficulty in its theory is due to the attempt to explain -a four-dimensional phenomenon by a three-dimensional -geometry.</p> - -<p>In view of this piece of evidence we cannot disregard -that afforded by the existence of symmetry. In this -connection I will allude to the simple way of producing<span class="pagenum" id="Page_19">[Pg 19]</span> -the images of insects, sometimes practised by children. -They put a few blots of ink in a straight line on a piece of -paper, fold the paper along the blots, and on opening it the -lifelike presentment of an insect is obtained. If we were -to find a multitude of these figures, we should conclude -that they had originated from a process of folding over; -the chances against this kind of reduplication of parts -is too great to admit of the assumption that they had -been formed in any other way.</p> - -<p>The production of the symmetrical forms of organised -beings, though not of course due to a turning over of -bodies of any appreciable size in four-dimensional space, -can well be imagined as due to a disposition in that -manner of the smallest living particles from which they -are built up. Thus, not only electricity, but life, and the -processes by which we think and feel, must be attributed -to that region of magnitude in which four-dimensional -movements take place.</p> - -<p>I do not mean, however, that life can be explained as a -four-dimensional movement. It seems to me that the -whole bias of thought, which tends to explain the -phenomena of life and volition, as due to matter and -motion in some peculiar relation, is adopted rather in the -interests of the explicability of things than with any -regard to probability.</p> - -<p>Of course, if we could show that life were a phenomenon -of motion, we should be able to explain a great deal that is -at present obscure. But there are two great difficulties in -the way. It would be necessary to show that in a germ -capable of developing into a living being, there were -modifications of structure capable of determining in the -developed germ all the characteristics of its form, and not -only this, but of determining those of all the descendants -of such a form in an infinite series. Such a complexity of -mechanical relations, undeniable though it be, cannot<span class="pagenum" id="Page_20">[Pg 20]</span> -surely be the best way of grouping the phenomena and -giving a practical account of them. And another difficulty -is this, that no amount of mechanical adaptation would -give that element of consciousness which we possess, and -which is shared in to a modified degree by the animal -world.</p> - -<p>In those complex structures which men build up and -direct, such as a ship or a railway train (and which, if seen -by an observer of such a size that the men guiding them -were invisible, would seem to present some of the -phenomena of life) the appearance of animation is not -due to any diffusion of life in the material parts of the -structure, but to the presence of a living being.</p> - -<p>The old hypothesis of a soul, a living organism within -the visible one, appears to me much more rational than the -attempt to explain life as a form of motion. And when we -consider the region of extreme minuteness characterised -by four-dimensional motion the difficulty of conceiving -such an organism alongside the bodily one disappears. -Lord Kelvin supposes that matter is formed from the -ether. We may very well suppose that the living -organisms directing the material ones are co-ordinate -with them, not composed of matter, but consisting of -etherial bodies, and as such capable of motion through -the ether, and able to originate material living bodies -throughout the mineral.</p> - -<p>Hypotheses such as these find no immediate ground for -proof or disproof in the physical world. Let us, therefore, -turn to a different field, and, assuming that the human -soul is a four-dimensional being, capable in itself of four -dimensional movements, but in its experiences through -the senses limited to three dimensions, ask if the history -of thought, of these productivities which characterise man, -correspond to our assumption. Let us pass in review -those steps by which man, presumably a four-dimensional<span class="pagenum" id="Page_21">[Pg 21]</span> -being, despite his bodily environment, has come to recognise -the fact of four-dimensional existence.</p> - -<p>Deferring this enquiry to another chapter, I will here -recapitulate the argument in order to show that our -purpose is entirely practical and independent of any -philosophical or metaphysical considerations.</p> - -<p>If two shots are fired at a target, and the second bullet -hits it at a different place to the first, we suppose that -there was some difference in the conditions under which -the second shot was fired from those affecting the first -shot. The force of the powder, the direction of aim, the -strength of the wind, or some condition must have been -different in the second case, if the course of the bullet was -not exactly the same as in the first case. Corresponding -to every difference in a result there must be some difference -in the antecedent material conditions. By tracing -out this chain of relations we explain nature.</p> - -<p>But there is also another mode of explanation which we -apply. If we ask what was the cause that a certain ship -was built, or that a certain structure was erected, we might -proceed to investigate the changes in the brain cells of -the men who designed the works. Every variation in one -ship or building from another ship or building is accompanied -by a variation in the processes that go on in the -brain matter of the designers. But practically this would -be a very long task.</p> - -<p>A more effective mode of explaining the production of -the ship or building would be to enquire into the motives, -plans, and aims of the men who constructed them. We -obtain a cumulative and consistent body of knowledge -much more easily and effectively in the latter way.</p> - -<p>Sometimes we apply the one, sometimes the other -mode of explanation.</p> - -<p>But it must be observed that the method of explanation -founded on aim, purpose, volition, always presupposes<span class="pagenum" id="Page_22">[Pg 22]</span> -a mechanical system on which the volition and aim -works. The conception of man as willing and acting -from motives involves that of a number of uniform processes -of nature which he can modify, and of which he -can make application. In the mechanical conditions of -the three-dimensional world, the only volitional agency -which we can demonstrate is the human agency. But -when we consider the four-dimensional world the -conclusion remains perfectly open.</p> - -<p>The method of explanation founded on purpose and aim -does not, surely, suddenly begin with man and end with -him. There is as much behind the exhibition of will and -motive which we see in man as there is behind the -phenomena of movement; they are co-ordinate, neither -to be resolved into the other. And the commencement -of the investigation of that will and motive which lies -behind the will and motive manifested in the three-dimensional -mechanical field is in the conception of a -soul—a four-dimensional organism, which expresses its -higher physical being in the symmetry of the body, and -gives the aims and motives of human existence.</p> - -<p>Our primary task is to form a systematic knowledge of -the phenomena of a four-dimensional world and find those -points in which this knowledge must be called in to -complete our mechanical explanation of the universe. -But a subsidiary contribution towards the verification of -the hypothesis may be made by passing in review the -history of human thought, and enquiring if it presents -such features as would be naturally expected on this -assumption.</p> -<hr class="chap x-ebookmaker-drop" /> - -<div class="chapter"> -<p><span class="pagenum" id="Page_23">[Pg 23]</span></p> - -<h2 class="nobreak" id="CHAPTER_IV">CHAPTER IV<br /> - -<small><i>THE FIRST CHAPTER IN THE HISTORY -OF FOUR SPACE</i></small></h2></div> - - -<p>Parmenides, and the Asiatic thinkers with whom he is -in close affinity, propound a theory of existence which -is in close accord with a conception of a possible relation -between a higher and a lower dimensional space. This -theory, prior and in marked contrast to the main stream -of thought, which we shall afterwards describe, forms a -closed circle by itself. It is one which in all ages has -had a strong attraction for pure intellect, and is the -natural mode of thought for those who refrain from -projecting their own volition into nature under the guise -of causality.</p> - -<p>According to Parmenides of the school of Elea the all -is one, unmoving and unchanging. The permanent amid -the transient—that foothold for thought, that solid ground -for feeling on the discovery of which depends all our life—is -no phantom; it is the image amidst deception of true -being, the eternal, the unmoved, the one. Thus says -Parmenides.</p> - -<p>But how explain the shifting scene, these mutations -of things!</p> - -<p>“Illusion,” answers Parmenides. Distinguishing between -truth and error, he tells of the true doctrine of the -one—the false opinion of a changing world. He is no -less memorable for the manner of his advocacy than for<span class="pagenum" id="Page_24">[Pg 24]</span> -the cause he advocates. It is as if from his firm foothold -of being he could play with the thoughts under the -burden of which others laboured, for from him springs -that fluency of supposition and hypothesis which forms -the texture of Plato’s dialectic.</p> - -<p>Can the mind conceive a more delightful intellectual -picture than that of Parmenides, pointing to the one, the -true, the unchanging, and yet on the other hand ready to -discuss all manner of false opinion, forming a cosmogony -too, false “but mine own” after the fashion of the time?</p> - -<p>In support of the true opinion he proceeded by the -negative way of showing the self-contradictions in the -ideas of change and motion. It is doubtful if his criticism, -save in minor points, has ever been successfully refuted. -To express his doctrine in the ponderous modern way we -must make the statement that motion is phenomenal, -not real.</p> - -<p>Let us represent his doctrine.</p> - -<div class="figleft illowp35" id="fig_13" style="max-width: 9.375em;"> - <img src="images/fig_13.png" alt="" /> - <div class="caption">Fig. 13.</div> -</div> - -<p>Imagine a sheet of still water into which a slanting stick -is being lowered with a motion vertically -downwards. Let 1, 2, 3 (Fig. 13), -be three consecutive positions of the -stick. <span class="allsmcap">A</span>, <span class="allsmcap">B</span>, <span class="allsmcap">C</span>, will be three consecutive -positions of the meeting of the stick, -with the surface of the water. As -the stick passes down, the meeting will -move from <span class="allsmcap">A</span> on to <span class="allsmcap">B</span> and <span class="allsmcap">C</span>.</p> - -<p>Suppose now all the water to be -removed except a film. At the meeting -of the film and the stick there -will be an interruption of the film. -If we suppose the film to have a property, -like that of a soap bubble, of closing up round any -penetrating object, then as the stick goes vertically -downwards the interruption in the film will move on.</p> - -<p><span class="pagenum" id="Page_25">[Pg 25]</span></p> - -<div class="figleft illowp35" id="fig_14" style="max-width: 10em;"> - <img src="images/fig_14.png" alt="" /> - <div class="caption">Fig. 14.</div> -</div> - -<p>If we pass a spiral through the film the intersection -will give a point moving in a circle shown by the dotted -lines in the figure. Suppose -now the spiral to be still and -the film to move vertically -upwards, the whole spiral will -be represented in the film of -the consecutive positions of the -point of intersection. In the -film the permanent existence -of the spiral is experienced as -a time series—the record of -traversing the spiral is a point -moving in a circle. If now -we suppose a consciousness connected -with the film in such a way that the intersection of -the spiral with the film gives rise to a conscious experience, -we see that we shall have in the film a point moving in a -circle, conscious of its motion, knowing nothing of that -real spiral the record of the successive intersections of -which by the film is the motion of the point.</p> - -<p>It is easy to imagine complicated structures of the -nature of the spiral, structures consisting of filaments, -and to suppose also that these structures are distinguishable -from each other at every section. If we consider -the intersections of these filaments with the film as it -passes to be the atoms constituting a filmar universe, -we shall have in the film a world of apparent motion; -we shall have bodies corresponding to the filamentary -structure, and the positions of these structures with -regard to one another will give rise to bodies in the -film moving amongst one another. This mutual motion -is apparent merely. The reality is of permanent structures -stationary, and all the relative motions accounted for by -one steady movement of the film as a whole.</p> - -<p><span class="pagenum" id="Page_26">[Pg 26]</span></p> - -<p>Thus we can imagine a plane world, in which all the -variety of motion is the phenomenon of structures consisting -of filamentary atoms traversed by a plane of -consciousness. Passing to four dimensions and our -space, we can conceive that all things and movements -in our world are the reading off of a permanent reality -by a space of consciousness. Each atom at every moment -is not what it was, but a new part of that endless line -which is itself. And all this system successively revealed -in the time which is but the succession of consciousness, -separate as it is in parts, in its entirety is one vast unity. -Representing Parmenides’ doctrine thus, we gain a firmer -hold on it than if we merely let his words rest, grand and -massive, in our minds. And we have gained the means also -of representing phases of that Eastern thought to which -Parmenides was no stranger. Modifying his uncompromising -doctrine, let us suppose, to go back to the plane -of consciousness and the structure of filamentary atoms, -that these structures are themselves moving—are acting, -living. Then, in the transverse motion of the film, there -would be two phenomena of motion, one due to the reading -off in the film of the permanent existences as they are in -themselves, and another phenomenon of motion due to -the modification of the record of the things themselves, by -their proper motion during the process of traversing them.</p> - -<p>Thus a conscious being in the plane would have, as it -were, a two-fold experience. In the complete traversing -of the structure, the intersection of which with the film -gives his conscious all, the main and principal movements -and actions which he went through would be the record -of his higher self as it existed unmoved and unacting. -Slight modifications and deviations from these movements -and actions would represent the activity and self-determination -of the complete being, of his higher self.</p> - -<p>It is admissible to suppose that the consciousness in<span class="pagenum" id="Page_27">[Pg 27]</span> -the plane has a share in that volition by which the -complete existence determines itself. Thus the motive -and will, the initiative and life, of the higher being, would -be represented in the case of the being in the film by an -initiative and a will capable, not of determining any great -things or important movements in his existence, but only -of small and relatively insignificant activities. In all the -main features of his life his experience would be representative -of one state of the higher being whose existence -determines his as the film passes on. But in his minute -and apparently unimportant actions he would share in -that will and determination by which the whole of the -being he really is acts and lives.</p> - -<p>An alteration of the higher being would correspond to -a different life history for him. Let us now make the -supposition that film after film traverses these higher -structures, that the life of the real being is read off again -and again in successive waves of consciousness. There -would be a succession of lives in the different advancing -planes of consciousness, each differing from the preceding, -and differing in virtue of that will and activity which in -the preceding had not been devoted to the greater and -apparently most significant things in life, but the minute -and apparently unimportant. In all great things the -being of the film shares in the existence of his higher -self as it is at any one time. In the small things he -shares in that volition by which the higher being alters -and changes, acts and lives.</p> - -<p>Thus we gain the conception of a life changing and -developing as a whole, a life in which our separation and -cessation and fugitiveness are merely apparent, but which -in its events and course alters, changes, develops; and -the power of altering and changing this whole lies in the -will and power the limited being has of directing, guiding, -altering himself in the minute things of his existence.</p> - -<p><span class="pagenum" id="Page_28">[Pg 28]</span></p> - -<p>Transferring our conceptions to those of an existence in -a higher dimensionality traversed by a space of consciousness, -we have an illustration of a thought which has -found frequent and varied expression. When, however, -we ask ourselves what degree of truth there lies in it, we -must admit that, as far as we can see, it is merely symbolical. -The true path in the investigation of a higher -dimensionality lies in another direction.</p> - -<p>The significance of the Parmenidean doctrine lies in -this that here, as again and again, we find that those conceptions -which man introduces of himself, which he does -not derive from the mere record of his outward experience, -have a striking and significant correspondence to the -conception of a physical existence in a world of a higher -space. How close we come to Parmenides’ thought by -this manner of representation it is impossible to say. -What I want to point out is the adequateness of the -illustration, not only to give a static model of his doctrine, -but one capable as it were, of a plastic modification into a -correspondence into kindred forms of thought. Either one -of two things must be true—that four-dimensional conceptions -give a wonderful power of representing the thought -of the East, or that the thinkers of the East must have been -looking at and regarding four-dimensional existence.</p> - -<p>Coming now to the main stream of thought we must -dwell in some detail on Pythagoras, not because of his -direct relation to the subject, but because of his relation -to investigators who came later.</p> - -<p>Pythagoras invented the two-way counting. Let us -represent the single-way counting by the posits <i>aa</i>, -<i>ab</i>, <i>ac</i>, <i>ad</i>, using these pairs of letters instead of the -numbers 1, 2, 3, 4. I put an <i>a</i> in each case first for a -reason which will immediately appear.</p> - -<p>We have a sequence and order. There is no conception -of distance necessarily involved. The difference<span class="pagenum" id="Page_29">[Pg 29]</span> -between the posits is one of order not of distance—only -when identified with a number of equal material -things in juxtaposition does the notion of distance arise.</p> - -<p>Now, besides the simple series I can have, starting from -<i>aa</i>, <i>ba</i>, <i>ca</i>, <i>da</i>, from <i>ab</i>, <i>bb</i>, <i>cb</i>, <i>db</i>, and so on, and forming -a scheme:</p> - - -<table class="standard" summary=""> -<tr> -<td class="tdlp"><i>da</i></td> -<td class="tdlp"><i>db</i></td> -<td class="tdlp"><i>dc</i></td> -<td class="tdlp"><i>dd</i></td> -</tr> -<tr> -<td class="tdlp"><i>ca</i></td> -<td class="tdlp"><i>cb</i></td> -<td class="tdlp"><i>cc</i></td> -<td class="tdlp"><i>cd</i></td> -</tr> -<tr> -<td class="tdlp"><i>ba</i></td> -<td class="tdlp"><i>bb</i></td> -<td class="tdlp"><i>bc</i></td> -<td class="tdlp"><i>bd</i></td> -</tr> -<tr> -<td class="tdlp"><i>aa</i></td> -<td class="tdlp"><i>ab</i></td> -<td class="tdlp"><i>ac</i></td> -<td class="tdlp"><i>ad</i></td> -</tr> -</table> - - -<p>This complex or manifold gives a two-way order. I can -represent it by a set of points, if I am on my guard -against assuming any relation of distance.</p> - -<div class="figleft illowp25" id="fig_15" style="max-width: 10em;"> - <img src="images/fig_15.png" alt="" /> - <div class="caption">Fig. 15.</div> -</div> - -<p>Pythagoras studied this two-fold way of -counting in reference to material bodies, and -discovered that most remarkable property of -the combination of number and matter that -bears his name.</p> - -<p>The Pythagorean property of an extended material -system can be exhibited in a manner which will be of -use to us afterwards, and which therefore I will employ -now instead of using the kind of figure which he himself -employed.</p> - -<p>Consider a two-fold field of points arranged in regular -rows. Such a field will be presupposed in the following -argument.</p> - -<div class="figleft illowp40" id="fig_16" style="max-width: 21.25em;"> - <img src="images/fig_16.png" alt="" /> - <div class="caption">Fig. 16.</div> -</div> - -<p>It is evident that in <a href="#fig_16">fig. 16</a> four -of the points determine a square, -which square we may take as the -unit of measurement for areas. -But we can also measure areas -in another way.</p> - -<p>Fig. 16 (1) shows four points determining a square.</p> - -<p>But four squares also meet in a point, <a href="#fig_16">fig. 16</a> (2).</p> - -<p>Hence a point at the corner of a square belongs equally -to four squares.</p> - -<p><span class="pagenum" id="Page_30">[Pg 30]</span></p> - -<p>Thus we may say that the point value of the square -shown is one point, for if we take the square in <a href="#fig_16">fig. 16</a> (1) -it has four points, but each of these belong equally to -four other squares. Hence one fourth of each of them -belongs to the square (1) in <a href="#fig_16">fig. 16</a>. Thus the point -value of the square is one point.</p> - -<p>The result of counting the points is the same as that -arrived at by reckoning the square units enclosed.</p> - -<p>Hence, if we wish to measure the area of any square -we can take the number of points it encloses, count these -as one each, and take one-fourth of the number of points -at its corners.</p> - -<div class="figleft illowp25" id="fig_17" style="max-width: 12.5em;"> - <img src="images/fig_17.png" alt="" /> - <div class="caption">Fig. 17.</div> -</div> - -<p>Now draw a diagonal square as shown in <a href="#fig_17">fig. 17</a>. It -contains one point and the four corners count for one -point more; hence its point value is 2. -The value is the measure of its area—the -size of this square is two of the unit squares.</p> - -<p>Looking now at the sides of this figure -we see that there is a unit square on each -of them—the two squares contain no points, -but have four corner points each, which gives the point -value of each as one point.</p> - -<p>Hence we see that the square on the diagonal is equal -to the squares on the two sides; or as it is generally -expressed, the square on the hypothenuse is equal to -the sum of the squares on the sides.</p> - -<div class="figleft illowp25" id="fig_18" style="max-width: 12.5em;"> - <img src="images/fig_18.png" alt="" /> - <div class="caption">Fig. 18.</div> -</div> - -<p>Noticing this fact we can proceed to ask if it is always -true. Drawing the square shown in <a href="#fig_18">fig. 18</a>, we can count -the number of its points. There are five -altogether. There are four points inside -the square on the diagonal, and hence, with -the four points at its corners the point -value is 5—that is, the area is 5. Now -the squares on the sides are respectively -of the area 4 and 1. Hence in this case also the square<span class="pagenum" id="Page_31">[Pg 31]</span> -on the diagonal is equal to the sum of the square on -the sides. This property of matter is one of the first -great discoveries of applied mathematics. We shall prove -afterwards that it is not a property of space. For the -present it is enough to remark that the positions in -which the points are arranged is entirely experimental. -It is by means of equal pieces of some material, or the -same piece of material moved from one place to another, -that the points are arranged.</p> - -<p>Pythagoras next enquired what the relation must be -so that a square drawn slanting-wise should be equal to -one straight-wise. He found that a square whose side is -five can be placed either rectangularly along the lines -of points, or in a slanting position. And this square is -equivalent to two squares of sides 4 and 3.</p> - -<p>Here he came upon a numerical relation embodied in -a property of matter. Numbers immanent in the objects -produced the equality so satisfactory for intellectual apprehension. -And he found that numbers when immanent -in sound—when the strings of a musical instrument -were given certain definite proportions of length—were -no less captivating to the ear than the equality of squares -was to the reason. What wonder then that he ascribed -an active power to number!</p> - -<p>We must remember that, sharing like ourselves the -search for the permanent in changing phenomena, the -Greeks had not that conception of the permanent in -matter that we have. To them material things were not -permanent. In fire solid things would vanish; absolutely -disappear. Rock and earth had a more stable existence, -but they too grew and decayed. The permanence of -matter, the conservation of energy, were unknown to -them. And that distinction which we draw so readily -between the fleeting and permanent causes of sensation, -between a sound and a material object, for instance, had<span class="pagenum" id="Page_32">[Pg 32]</span> -not the same meaning to them which it has for us. -Let us but imagine for a moment that material things -are fleeting, disappearing, and we shall enter with a far -better appreciation into that search for the permanent -which, with the Greeks, as with us, is the primary -intellectual demand.</p> - -<p>What is that which amid a thousand forms is ever the -same, which we can recognise under all its vicissitudes, -of which the diverse phenomena are the appearances?</p> - -<p>To think that this is number is not so very wide of -the mark. With an intellectual apprehension which far -outran the evidences for its application, the atomists -asserted that there were everlasting material particles, -which, by their union, produced all the varying forms and -states of bodies. But in view of the observed facts of -nature as then known, Aristotle, with perfect reason, -refused to accept this hypothesis.</p> - -<p>He expressly states that there is a change of quality, -and that the change due to motion is only one of the -possible modes of change.</p> - -<p>With no permanent material world about us, with -the fleeting, the unpermanent, all around we should, I -think, be ready to follow Pythagoras in his identification -of number with that principle which subsists amidst -all changes, which in multitudinous forms we apprehend -immanent in the changing and disappearing substance -of things.</p> - -<p>And from the numerical idealism of Pythagoras there -is but a step to the more rich and full idealism of Plato. -That which is apprehended by the sense of touch we -put as primary and real, and the other senses we say -are merely concerned with appearances. But Plato took -them all as valid, as giving qualities of existence. That -the qualities were not permanent in the world as given -to the senses forced him to attribute to them a different<span class="pagenum" id="Page_33">[Pg 33]</span> -kind of permanence. He formed the conception of a -world of ideas, in which all that really is, all that affects -us and gives the rich and wonderful wealth of our -experience, is not fleeting and transitory, but eternal. -And of this real and eternal we see in the things about -us the fleeting and transient images.</p> - -<p>And this world of ideas was no exclusive one, wherein -was no place for the innermost convictions of the soul and -its most authoritative assertions. Therein existed justice, -beauty—the one, the good, all that the soul demanded -to be. The world of ideas, Plato’s wonderful creation -preserved for man, for his deliberate investigation and -their sure development, all that the rude incomprehensible -changes of a harsh experience scatters and -destroys.</p> - -<p>Plato believed in the reality of ideas. He meets us -fairly and squarely. Divide a line into two parts, he -says; one to represent the real objects in the world, the -other to represent the transitory appearances, such as the -image in still water, the glitter of the sun on a bright -surface, the shadows on the clouds.</p> - -<div class="figcenter illowp100" id="i_033a" style="max-width: 50em;"> - <img src="images/i_033a.png" alt="" /> - <div class="caption"><table class="standard" summary=""> -<col width="30%" /><col width="20%" /><col width="30%" /> -<tr> -<td class="tdc">Real things:<br /> <i>e.g.</i>, the sun.</td> -<td></td> -<td class="tdc">Appearances:<br /> <i>e.g.</i>, the reflection of the sun.</td> -</tr> -</table> -</div> -</div> - -<p>Take another line and divide it into two parts, one -representing our ideas, the ordinary occupants of our -minds, such as whiteness, equality, and the other representing -our true knowledge, which is of eternal principles, -such as beauty, goodness.</p> - -<div class="figcenter illowp100" id="i_033b" style="max-width: 50em;"> - <img src="images/i_033b.png" alt="" /> - <div class="caption"><table class="standard" summary=""> -<col width="30%" /><col width="20%" /><col width="30%" /> -<tr> -<td class="tdc">Eternal principles,<br />as beauty.</td> -<td></td> -<td class="tdc"> Appearances in the mind,<br />as whiteness, equality</td> -</tr> -</table> -</div> -</div> - -<p>Then as A is to B, so is A<sup>1</sup> to B<sup>1</sup></p> - -<p>That is, the soul can proceed, going away from real<span class="pagenum" id="Page_34">[Pg 34]</span> -things to a region of perfect certainty, where it beholds -what is, not the scattered reflections; beholds the sun, not -the glitter on the sands; true being, not chance opinion.</p> - -<p>Now, this is to us, as it was to Aristotle, absolutely -inconceivable from a scientific point of view. We can -understand that a being is known in the fulness of his -relations; it is in his relations to his circumstances that -a man’s character is known; it is in his acts under his -conditions that his character exists. We cannot grasp or -conceive any principle of individuation apart from the -fulness of the relations to the surroundings.</p> - -<p>But suppose now that Plato is talking about the higher -man—the four-dimensional being that is limited in our -external experience to a three-dimensional world. Do not -his words begin to have a meaning? Such a being -would have a consciousness of motion which is not as -the motion he can see with the eyes of the body. He, -in his own being, knows a reality to which the outward -matter of this too solid earth is flimsy superficiality. He -too knows a mode of being, the fulness of relations, in -which can only be represented in the limited world of -sense, as the painter unsubstantially portrays the depths -of woodland, plains, and air. Thinking of such a being -in man, was not Plato’s line well divided?</p> - -<p>It is noteworthy that, if Plato omitted his doctrine of -the independent origin of ideas, he would present exactly -the four-dimensional argument; a real thing as we think -it is an idea. A plane being’s idea of a square object is -the idea of an abstraction, namely, a geometrical square. -Similarly our idea of a solid thing is an abstraction, for in -our idea there is not the four-dimensional thickness which -is necessary, however slight, to give reality. The argument -would then run, as a shadow is to a solid object, so -is the solid object to the reality. Thus A and B´ would -be identified.</p> - -<p><span class="pagenum" id="Page_35">[Pg 35]</span></p> - -<p>In the allegory which I have already alluded to, Plato -in almost as many words shows forth the relation between -existence in a superficies and in solid space. And he -uses this relation to point to the conditions of a higher -being.</p> - -<p>He imagines a number of men prisoners, chained so -that they look at the wall of a cavern in which they are -confined, with their backs to the road and the light. -Over the road pass men and women, figures and processions, -but of all this pageant all that the prisoners -behold is the shadow of it on the wall whereon they gaze. -Their own shadows and the shadows of the things in the -world are all that they see, and identifying themselves -with their shadows related as shadows to a world of -shadows, they live in a kind of dream.</p> - -<p>Plato imagines one of their number to pass out from -amongst them into the real space world, and then returning -to tell them of their condition.</p> - -<p>Here he presents most plainly the relation between -existence in a plane world and existence in a three-dimensional -world. And he uses this illustration as a -type of the manner in which we are to proceed to a -higher state from the three-dimensional life we know.</p> - -<p>It must have hung upon the weight of a shadow which -path he took!—whether the one we shall follow toward -the higher solid and the four-dimensional existence, or -the one which makes ideas the higher realities, and the -direct perception of them the contact with the truer -world.</p> - -<p>Passing on to Aristotle, we will touch on the points -which most immediately concern our enquiry.</p> - -<p>Just as a scientific man of the present day in -reviewing the speculations of the ancient world would -treat them with a curiosity half amused but wholly -respectful, asking of each and all wherein lay their<span class="pagenum" id="Page_36">[Pg 36]</span> -relation to fact, so Aristotle, in discussing the philosophy -of Greece as he found it, asks, above all other things: -“Does this represent the world? In this system is there -an adequate presentation of what is?”</p> - -<p>He finds them all defective, some for the very reasons -which we esteem them most highly, as when he criticises -the Atomic theory for its reduction of all change to motion. -But in the lofty march of his reason he never loses sight -of the whole; and that wherein our views differ from his -lies not so much in a superiority of our point of view, as -in the fact which he himself enunciates—that it is impossible -for one principle to be valid in all branches of -enquiry. The conceptions of one method of investigation -are not those of another; and our divergence lies in our -exclusive attention to the conceptions useful in one way -of apprehending nature rather than in any possibility we -find in our theories of giving a view of the whole transcending -that of Aristotle.</p> - -<p>He takes account of everything; he does not separate -matter and the manifestation of matter; he fires all -together in a conception of a vast world process in -which everything takes part—the motion of a grain of -dust, the unfolding of a leaf, the ordered motion of the -spheres in heaven—all are parts of one whole which -he will not separate into dead matter and adventitious -modifications.</p> - -<p>And just as our theories, as representative of actuality, -fall before his unequalled grasp of fact, so the doctrine -of ideas fell. It is not an adequate account of existence, -as Plato himself shows in his “Parmenides”; -it only explains things by putting their doubles beside -them.</p> - -<p>For his own part Aristotle invented a great marching -definition which, with a kind of power of its own, cleaves -its way through phenomena to limiting conceptions on<span class="pagenum" id="Page_37">[Pg 37]</span> -either hand, towards whose existence all experience -points.</p> - -<p>In Aristotle’s definition of matter and form as the -constituent of reality, as in Plato’s mystical vision of the -kingdom of ideas, the existence of the higher dimensionality -is implicitly involved.</p> - -<p>Substance according to Aristotle is relative, not absolute. -In everything that is there is the matter of which it -is composed, the form which it exhibits; but these are -indissolubly connected, and neither can be thought -without the other.</p> - -<p>The blocks of stone out of which a house is built are the -material for the builder; but, as regards the quarrymen, -they are the matter of the rocks with the form he has -imposed on them. Words are the final product of the -grammarian, but the mere matter of the orator or poet. -The atom is, with us, that out of which chemical substances -are built up, but looked at from another point of view is -the result of complex processes.</p> - -<p>Nowhere do we find finality. The matter in one sphere -is the matter, plus form, of another sphere of thought. -Making an obvious application to geometry, plane figures -exist as the limitation of different portions of the plane -by one another. In the bounding lines the separated -matter of the plane shows its determination into form. -And as the plane is the matter relatively to determinations -in the plane, so the plane itself exists in virtue of the -determination of space. A plane is that wherein formless -space has form superimposed on it, and gives an actuality -of real relations. We cannot refuse to carry this process -of reasoning a step farther back, and say that space itself -is that which gives form to higher space. As a line is -the determination of a plane, and a plane of a solid, so -solid space itself is the determination of a higher space.</p> - -<p>As a line by itself is inconceivable without that plane<span class="pagenum" id="Page_38">[Pg 38]</span> -which it separates, so the plane is inconceivable without -the solids which it limits on either hand. And so space -itself cannot be positively defined. It is the negation -of the possibility of movement in more than three -dimensions. The conception of space demands that of -a higher space. As a surface is thin and unsubstantial -without the substance of which it is the surface, so matter -itself is thin without the higher matter.</p> - -<p>Just as Aristotle invented that algebraical method of -representing unknown quantities by mere symbols, not by -lines necessarily determinate in length as was the habit -of the Greek geometers, and so struck out the path -towards those objectifications of thought which, like -independent machines for reasoning, supply the mathematician -with his analytical weapons, so in the formulation -of the doctrine of matter and form, of potentiality and -actuality, of the relativity of substance, he produced -another kind of objectification of mind—a definition -which had a vital force and an activity of its own.</p> - -<p>In none of his writings, as far as we know, did he carry it -to its legitimate conclusion on the side of matter, but in -the direction of the formal qualities he was led to his -limiting conception of that existence of pure form which -lies beyond all known determination of matter. The -unmoved mover of all things is Aristotle’s highest -principle. Towards it, to partake of its perfection all -things move. The universe, according to Aristotle, is an -active process—he does not adopt the illogical conception -that it was once set in motion and has kept on ever since. -There is room for activity, will, self-determination, in -Aristotle’s system, and for the contingent and accidental -as well. We do not follow him, because we are accustomed -to find in nature infinite series, and do not feel -obliged to pass on to a belief in the ultimate limits to -which they seem to point.</p> - -<p><span class="pagenum" id="Page_39">[Pg 39]</span></p> - -<p>But apart from the pushing to the limit, as a relative -principle this doctrine of Aristotle’s as to the relativity of -substance is irrefragible in its logic. He was the first to -show the necessity of that path of thought which when -followed leads to a belief in a four-dimensional space.</p> - -<p>Antagonistic as he was to Plato in his conception -of the practical relation of reason to the world of -phenomena, yet in one point he coincided with him. -And in this he showed the candour of his intellect. He -was more anxious to lose nothing than to explain everything. -And that wherein so many have detected an -inconsistency, an inability to free himself from the school -of Plato, appears to us in connection with our enquiry -as an instance of the acuteness of his observation. For -beyond all knowledge given by the senses Aristotle held -that there is an active intelligence, a mind not the passive -recipient of impressions from without, but an active and -originative being, capable of grasping knowledge at first -hand. In the active soul Aristotle recognised something -in man not produced by his physical surroundings, something -which creates, whose activity is a knowledge -underived from sense. This, he says, is the immortal and -undying being in man.</p> - -<p>Thus we see that Aristotle was not far from the -recognition of the four-dimensional existence, both -without and within man, and the process of adequately -realising the higher dimensional figures to which we -shall come subsequently is a simple reduction to practice -of his hypothesis of a soul.</p> - -<p>The next step in the unfolding of the drama of the -recognition of the soul as connected with our scientific -conception of the world, and, at the same time, the -recognition of that higher of which a three-dimensional -world presents the superficial appearance, took place many -centuries later. If we pass over the intervening time<span class="pagenum" id="Page_40">[Pg 40]</span> -without a word it is because the soul was occupied with -the assertion of itself in other ways than that of knowledge. -When it took up the task in earnest of knowing this -material world in which it found itself, and of directing -the course of inanimate nature, from that most objective -aim came, reflected back as from a mirror, its knowledge -of itself.</p> -<hr class="chap x-ebookmaker-drop" /> - -<div class="chapter"> -<p><span class="pagenum" id="Page_41">[Pg 41]</span></p> - -<h2 class="nobreak" id="CHAPTER_V">CHAPTER V<br /> - -<small><i>THE SECOND CHAPTER IN THE HISTORY -OF FOUR SPACE</i></small></h2></div> - - -<p><span class="smcap">Lobatchewsky, Bolyai, and Gauss</span> -Before entering on a description of the work of -Lobatchewsky and Bolyai it will not be out of place -to give a brief account of them, the materials for which -are to be found in an article by Franz Schmidt in the -forty-second volume of the <i>Mathematische Annalen</i>, -and in Engel’s edition of Lobatchewsky.</p> - -<p>Lobatchewsky was a man of the most complete and -wonderful talents. As a youth he was full of vivacity, -carrying his exuberance so far as to fall into serious -trouble for hazing a professor, and other freaks. Saved -by the good offices of the mathematician Bartels, who -appreciated his ability, he managed to restrain himself -within the bounds of prudence. Appointed professor at -his own University, Kasan, he entered on his duties under -the regime of a pietistic reactionary, who surrounded -himself with sycophants and hypocrites. Esteeming -probably the interests of his pupils as higher than any -attempt at a vain resistance, he made himself the tyrant’s -right-hand man, doing an incredible amount of teaching -and performing the most varied official duties. Amidst -all his activities he found time to make important contributions -to science. His theory of parallels is most<span class="pagenum" id="Page_42">[Pg 42]</span> -closely connected with his name, but a study of his -writings shows that he was a man capable of carrying -on mathematics in its main lines of advance, and of a -judgment equal to discerning what these lines were. -Appointed rector of his University, he died at an -advanced age, surrounded by friends, honoured, with the -results of his beneficent activity all around him. To him -no subject came amiss, from the foundations of geometry -to the improvement of the stoves by which the peasants -warmed their houses.</p> - -<p>He was born in 1793. His scientific work was -unnoticed till, in 1867, Houel, the French mathematician, -drew attention to its importance.</p> - -<p>Johann Bolyai de Bolyai was born in Klausenburg, -a town in Transylvania, December 15th, 1802.</p> - -<p>His father, Wolfgang Bolyai, a professor in the -Reformed College of Maros Vasarhely, retained the ardour -in mathematical studies which had made him a chosen -companion of Gauss in their early student days at -Göttingen.</p> - -<p>He found an eager pupil in Johann. He relates that -the boy sprang before him like a devil. As soon as he -had enunciated a problem the child would give the -solution and command him to go on further. As a -thirteen-year-old boy his father sometimes sent him to fill -his place when incapacitated from taking his classes. -The pupils listened to him with more attention than to -his father for they found him clearer to understand.</p> - -<p>In a letter to Gauss Wolfgang Bolyai writes:—</p> - -<p>“My boy is strongly built. He has learned to recognise -many constellations, and the ordinary figures of geometry. -He makes apt applications of his notions, drawing for -instance the positions of the stars with their constellations. -Last winter in the country, seeing Jupiter he asked: -‘How is it that we can see him from here as well as from<span class="pagenum" id="Page_43">[Pg 43]</span> -the town? He must be far off.’ And as to three -different places to which he had been he asked me to tell -him about them in one word. I did not know what he -meant, and then he asked me if one was in a line with -the other and all in a row, or if they were in a triangle.</p> - -<p>“He enjoys cutting paper figures with a pair of scissors, -and without my ever having told him about triangles -remarked that a right-angled triangle which he had cut -out was half of an oblong. I exercise his body with care, -he can dig well in the earth with his little hands. The -blossom can fall and no fruit left. When he is fifteen -I want to send him to you to be your pupil.”</p> - -<p>In Johann’s autobiography he says:—</p> - -<p>“My father called my attention to the imperfections -and gaps in the theory of parallels. He told me he had -gained more satisfactory results than his predecessors, -but had obtained no perfect and satisfying conclusion. -None of his assumptions had the necessary degree of -geometrical certainty, although they sufficed to prove the -eleventh axiom and appeared acceptable on first sight.</p> - -<p>“He begged of me, anxious not without a reason, to -hold myself aloof and to shun all investigation on this -subject, if I did not wish to live all my life in vain.”</p> - -<p>Johann, in the failure of his father to obtain any -response from Gauss, in answer to a letter in which he -asked the great mathematician to make of his son “an -apostle of truth in a far land,” entered the Engineering -School at Vienna. He writes from Temesvar, where he -was appointed sub-lieutenant September, 1823:—</p> - -<div class="blockquote"> -<p class="psig"> -“Temesvar, November 3rd, 1823.</p> - -<p>“<span class="smcap">Dear Good Father</span>, -</p> - -<p>“I have so overwhelmingly much to write -about my discovery that I know no other way of checking -myself than taking a quarter of a sheet only to write on. -I want an answer to my four-sheet letter.</p> - -<p><span class="pagenum" id="Page_44">[Pg 44]</span></p> - -<p>“I am unbroken in my determination to publish a -work on Parallels, as soon as I have put my material in -order and have the means.</p> - -<p>“At present I have not made any discovery, but -the way I have followed almost certainly promises me -the attainment of my object if any possibility of it -exists.</p> - -<p>“I have not got my object yet, but I have produced -such stupendous things that I was overwhelmed myself, -and it would be an eternal shame if they were lost. -When you see them you will find that it is so. Now -I can only say that I have made a new world out of -nothing. Everything that I have sent you before is a -house of cards in comparison with a tower. I am convinced -that it will be no less to my honour than if I had -already discovered it.”</p> -</div> - -<p>The discovery of which Johann here speaks was -published as an appendix to Wolfgang Bolyai’s <i>Tentamen</i>.</p> - -<p>Sending the book to Gauss, Wolfgang writes, after an -interruption of eighteen years in his correspondence:—</p> - -<div class="blockquote"> - -<p>“My son is first lieutenant of Engineers and will soon -be captain. He is a fine youth, a good violin player, -a skilful fencer, and brave, but has had many duels, and -is wild even for a soldier. Yet he is distinguished—light -in darkness and darkness in light. He is an impassioned -mathematician with extraordinary capacities.... He -will think more of your judgment on his work than that -of all Europe.”</p> -</div> - -<p>Wolfgang received no answer from Gauss to this letter, -but sending a second copy of the book received the -following reply:—</p> - -<div class="blockquote"> -<p>“You have rejoiced me, my unforgotten friend, by your -letters. I delayed answering the first because I wanted -to wait for the arrival of the promised little book.</p> - -<p>“Now something about your son’s work.</p> - -<p><span class="pagenum" id="Page_45">[Pg 45]</span></p> - -<p>“If I begin with saying that ‘I ought not to praise it,’ -you will be staggered for a moment. But I cannot say -anything else. To praise it is to praise myself, for the -path your son has broken in upon and the results to which -he has been led are almost exactly the same as my own -reflections, some of which date from thirty to thirty-five -years ago.</p> - -<p>“In fact I am astonished to the uttermost. My intention -was to let nothing be known in my lifetime about -my own work, of which, for the rest, but little is committed -to writing. Most people have but little perception -of the problem, and I have found very few who took any -interest in the views I expressed to them. To be able to -do that one must first of all have had a real live feeling -of what is wanting, and as to that most men are completely -in the dark.</p> - -<p>“Still it was my intention to commit everything to -writing in the course of time, so that at least it should -not perish with me.</p> - -<p>“I am deeply surprised that this task can be spared -me, and I am most of all pleased in this that it is the son -of my old friend who has in so remarkable a manner -preceded me.”</p> -</div> - -<p>The impression which we receive from Gauss’s inexplicable -silence towards his old friend is swept away -by this letter. Hence we breathe the clear air of the -mountain tops. Gauss would not have failed to perceive -the vast significance of his thoughts, sure to be all the -greater in their effect on future ages from the want of -comprehension of the present. Yet there is not a word -or a sign in his writing to claim the thought for himself. -He published no single line on the subject. By the -measure of what he thus silently relinquishes, by such a -measure of a world-transforming thought, we can appreciate -his greatness.</p> - -<p><span class="pagenum" id="Page_46">[Pg 46]</span></p> - -<p>It is a long step from Gauss’s serenity to the disturbed -and passionate life of Johann Bolyai—he and Galois, -the two most interesting figures in the history of mathematics. -For Bolyai, the wild soldier, the duellist, fell -at odds with the world. It is related of him that he was -challenged by thirteen officers of his garrison, a thing not -unlikely to happen considering how differently he thought -from every one else. He fought them all in succession—making -it his only condition that he should be allowed -to play on his violin for an interval between meeting each -opponent. He disarmed or wounded all his antagonists. -It can be easily imagined that a temperament such as -his was one not congenial to his military superiors. He -was retired in 1833.</p> - -<p>His epoch-making discovery awoke no attention. He -seems to have conceived the idea that his father had -betrayed him in some inexplicable way by his communications -with Gauss, and he challenged the excellent -Wolfgang to a duel. He passed his life in poverty, -many a time, says his biographer, seeking to snatch -himself from dissipation and apply himself again to -mathematics. But his efforts had no result. He died -January 27th, 1860, fallen out with the world and with -himself.</p> - - -<h3><span class="smcap">Metageometry</span></h3> - -<p>The theories which are generally connected with the -names of Lobatchewsky and Bolyai bear a singular and -curious relation to the subject of higher space.</p> - -<p>In order to show what this relation is, I must ask the -reader to be at the pains to count carefully the sets of -points by which I shall estimate the volumes of certain -figures.</p> - -<p><span class="pagenum" id="Page_47">[Pg 47]</span></p> - -<p>No mathematical processes beyond this simple one of -counting will be necessary.</p> - -<div class="figleft illowp25" id="fig_19" style="max-width: 12.5em;"> - <img src="images/fig_19.png" alt="" /> - <div class="caption">Fig. 19.</div> -</div> - -<p>Let us suppose we have before us in -<a href="#fig_19">fig. 19</a> a plane covered with points at regular -intervals, so placed that every four determine -a square.</p> - -<p>Now it is evident that as four points -determine a square, so four squares meet in a point.</p> - -<div class="figleft illowp25" id="fig_20" style="max-width: 12.5em;"> - <img src="images/fig_20.png" alt="" /> - <div class="caption">Fig. 20.</div> -</div> - -<p>Thus, considering a point inside a square as -belonging to it, we may say that a point on -the corner of a square belongs to it and to -three others equally: belongs a quarter of it -to each square.</p> - -<p>Thus the square <span class="allsmcap">ACDE</span> (<a href="#fig_21">fig. 21</a>) contains one point, and -has four points at the four corners. Since one-fourth of -each of these four belongs to the square, the four together -count as one point, and the point value of the square is -two points—the one inside and the four at the corner -make two points belonging to it exclusively.</p> - -<div class="figleft illowp25" id="fig_21" style="max-width: 12.5em;"> - <img src="images/fig_21.png" alt="" /> - <div class="caption">Fig. 21.</div> -</div> - -<div class="figright illowp25" id="fig_22" style="max-width: 12.8125em;"> - <img src="images/fig_22.png" alt="" /> - <div class="caption">Fig. 22.</div> -</div> - -<p>Now the area of this square is two unit squares, as can -be seen by drawing two diagonals in <a href="#fig_22">fig. 22</a>.</p> - -<p>We also notice that the square in question is equal to -the sum of the squares on the sides <span class="allsmcap">AB</span>, <span class="allsmcap">BC</span>, of the right-angled -triangle <span class="allsmcap">ABC</span>. Thus we recognise the proposition -that the square on the hypothenuse is equal to the sum -of the squares on the two sides of a right-angled triangle.</p> - -<p>Now suppose we set ourselves the question of determining -the whereabouts in the ordered system of points,<span class="pagenum" id="Page_48">[Pg 48]</span> -the end of a line would come when it turned about a -point keeping one extremity fixed at the point.</p> - -<p>We can solve this problem in a particular case. If we -can find a square lying slantwise amongst the dots which is -equal to one which goes regularly, we shall know that the -two sides are equal, and that the slanting side is equal to the -straight-way side. Thus the volume and shape of a figure -remaining unchanged will be the test of its having rotated -about the point, so that we can say that its side in its first -position would turn into its side in the second position.</p> - -<p>Now, such a square can be found in the one whose side -is five units in length.</p> - -<div class="figcenter illowp66" id="fig_23" style="max-width: 25em;"> - <img src="images/fig_23.png" alt="" /> - <div class="caption">Fig. 23.</div> -</div> - -<p>In <a href="#fig_23">fig. 23</a>, in the square on <span class="allsmcap">AB</span>, there are—</p> - -<table class="standard" summary=""> -<tr> -<td class="tdl">9 points interior</td> -<td class="tdr">9</td> -</tr> -<tr> -<td class="tdl">4 at the corners</td> -<td class="tdr">1</td> -</tr> -<tr> -<td class="tdh"> 4 sides with 3 on each side, considered as -1½ on each side, because belonging -equally to two squares</td> -<td class="tdrb">6</td> -</tr> -</table> - -<p>The total is 16. There are 9 points in the square -on <span class="allsmcap">BC</span>.</p> - -<p><span class="pagenum" id="Page_49">[Pg 49]</span></p> - -<p>In the square on <span class="allsmcap">AC</span> there are—</p> - - -<table class="standard" summary=""> -<tr> -<td class="tdl">24 points inside</td> -<td class="tdr">24</td> -</tr> -<tr> -<td class="tdl"> 4 at the corners</td> -<td class="tdr">1</td> -</tr> -</table> - -<p>or 25 altogether.</p> - -<p>Hence we see again that the square on the hypothenuse -is equal to the squares on the sides.</p> - -<p>Now take the square <span class="allsmcap">AFHG</span>, which is larger than the -square on <span class="allsmcap">AB</span>. It contains 25 points.</p> - - -<table class="standard" summary=""> -<tr> -<td class="tdl">16 inside</td> -<td class="tdr">16</td> -</tr> -<tr> -<td class="tdl">16 on the sides, counting as</td> -<td class="tdr">8</td> -</tr> -<tr> -<td class="tdl"> 4 on the corners</td> -<td class="tdr">1</td> -</tr> -</table> - -<p>making 25 altogether.</p> - -<p>If two squares are equal we conclude the sides are -equal. Hence, the line <span class="allsmcap">AF</span> turning round <span class="allsmcap">A</span> would -move so that it would after a certain turning coincide -with <span class="allsmcap">AC</span>.</p> - -<p>This is preliminary, but it involves all the mathematical -difficulties that will present themselves.</p> - -<p>There are two alterations of a body by which its volume -is not changed.</p> - -<p>One is the one we have just considered, rotation, the -other is what is called shear.</p> - -<p>Consider a book, or heap of loose pages. They can be -slid so that each one slips -over the preceding one, -and the whole assumes -the shape <i>b</i> in <a href="#fig_24">fig. 24</a>.</p> - -<div class="figleft illowp50" id="fig_24" style="max-width: 25em;"> - <img src="images/fig_24.png" alt="" /> - <div class="caption">Fig. 24.</div> -</div> - -<p>This deformation is not shear alone, but shear accompanied -by rotation.</p> - -<p>Shear can be considered as produced in another way.</p> - -<p>Take the square <span class="allsmcap">ABCD</span> (<a href="#fig_25">fig. 25</a>), and suppose that it -is pulled out from along one of its diagonals both ways, -and proportionately compressed along the other diagonal. -It will assume the shape in <a href="#fig_26">fig. 26</a>.</p> - -<p><span class="pagenum" id="Page_50">[Pg 50]</span></p> - -<p>This compression and expansion along two lines at right -angles is what is called shear; it is equivalent to the -sliding illustrated above, combined with a turning round.</p> - -<div class="figleft illowp45" id="fig_25" style="max-width: 12.5em;"> - <img src="images/fig_25.png" alt="" /> - <div class="caption">Fig. 25.</div> -</div> - -<div class="figright illowp50" id="fig_26" style="max-width: 18.75em;"> - <img src="images/fig_26.png" alt="" /> - <div class="caption">Fig. 26.</div> -</div> - -<p>In pure shear a body is compressed and extended in -two directions at right angles to each other, so that its -volume remains unchanged.</p> - -<p>Now we know that our material bodies resist shear—shear -does violence to the internal arrangement of their -particles, but they turn as wholes without such internal -resistance.</p> - -<p>But there is an exception. In a liquid shear and -rotation take place equally easily, there is no more -resistance against a shear than there is against a -rotation.</p> - -<p>Now, suppose all bodies were to be reduced to the liquid -state, in which they yield to shear and to rotation equally -easily, and then were to be reconstructed as solids, but in -such a way that shear and rotation had interchanged -places.</p> - -<p>That is to say, let us suppose that when they had -become solids again they would shear without offering -any internal resistance, but a rotation would do violence -to their internal arrangement.</p> - -<p>That is, we should have a world in which shear would -have taken the place of rotation.</p> - -<p><span class="pagenum" id="Page_51">[Pg 51]</span></p> - -<p>A shear does not alter the volume of a body: thus an -inhabitant living in such a world would look on a body -sheared as we look on a body rotated. He would say -that it was of the same shape, but had turned a bit -round.</p> - -<p>Let us imagine a Pythagoras in this world going to -work to investigate, as is his wont.</p> - -<div class="figleft illowp40" id="fig_27" style="max-width: 12.5em;"> - <img src="images/fig_27.png" alt="" /> - <div class="caption">Fig. 27.</div> -</div> -<div class="figright illowp40" id="fig_28" style="max-width: 13.125em;"> - <img src="images/fig_28.png" alt="" /> - <div class="caption">Fig. 28.</div> -</div> - -<p>Fig. 27 represents a square unsheared. Fig. 28 -represents a square sheared. It is not the figure into -which the square in <a href="#fig_27">fig. 27</a> would turn, but the result of -shear on some square not drawn. It is a simple slanting -placed figure, taken now as we took a simple slanting -placed square before. Now, since bodies in this world of -shear offer no internal resistance to shearing, and keep -their volume when sheared, an inhabitant accustomed to -them would not consider that they altered their shape -under shear. He would call <span class="allsmcap">ACDE</span> as much a square as -the square in <a href="#fig_27">fig. 27</a>. We will call such figures shear -squares. Counting the dots in <span class="allsmcap">ACDE</span>, we find—</p> - - -<table class="standard" summary=""> -<tr> -<td class="tdl">2 inside</td> -<td class="tdc">=</td> -<td class="tdc">2</td> -</tr> -<tr> -<td class="tdl">4 at corners</td> -<td class="tdc">=</td> -<td class="tdc">1</td> -</tr> -</table> - -<p>or a total of 3.</p> - -<p>Now, the square on the side <span class="allsmcap">AB</span> has 4 points, that on <span class="allsmcap">BC</span> -has 1 point. Here the shear square on the hypothenuse -has not 5 points but 3; it is not the sum of the squares on -the sides, but the difference.</p> - -<p><span class="pagenum" id="Page_52">[Pg 52]</span></p> - -<div class="figleft illowp25" id="fig_29" style="max-width: 13.75em;"> - <img src="images/fig_29.png" alt="" /> - <div class="caption">Fig. 29.</div> -</div> - -<p>This relation always holds. Look at -<a href="#fig_29">fig. 29</a>.</p> - -<p>Shear square on hypothenuse—</p> - -<table class="standard" summary=""> -<tr> -<td class="tdl">7 internal</td> -<td class="tdr"> 7</td> -</tr> -<tr> -<td class="tdl">4 at corners</td> -<td class="tdr">1</td> -</tr> -<tr> -<td class="tdl"></td> -<td class="tdr_bt">8</td> -</tr> -</table> - - -<div class="figleft illowp50" id="fig_29bis" style="max-width: 25em;"> - <img src="images/fig_29bis.png" alt="" /> - <div class="caption">Fig. 29 <i>bis</i>.</div> -</div> - -<p>Square on one side—which the reader can draw for -himself—</p> - -<table class="standard" summary=""> -<tr> -<td class="tdl">4 internal</td> -<td class="tdr"> 4</td> -</tr> -<tr> -<td class="tdl">8 on sides</td> -<td class="tdr">4</td> -</tr> -<tr> -<td class="tdl">4 at corners</td> -<td class="tdr">1</td> -</tr> -<tr> -<td class="tdl"></td> -<td class="tdr_bt">9</td> -</tr> -</table> - - -<p>and the square on the other -side is 1. Hence in this -case again the difference is -equal to the shear square on -the hypothenuse, 9 - 1 = 8.</p> - -<p>Thus in a world of shear -the square on the hypothenuse -would be equal to the -difference of the squares on -the sides of a right-angled -triangle.</p> - -<p>In <a href="#fig_29">fig. 29</a> <i>bis</i> another shear square is drawn on which -the above relation can be tested.</p> - -<p>What now would be the position a line on turning by -shear would take up?</p> - -<p>We must settle this in the same way as previously with -our turning.</p> - -<p>Since a body sheared remains the same, we must find two -equal bodies, one in the straight way, one in the slanting -way, which have the same volume. Then the side of one -will by turning become the side of the other, for the two -figures are each what the other becomes by a shear turning.</p> - -<p><span class="pagenum" id="Page_53">[Pg 53]</span></p> - -<p>We can solve the problem in a particular case—</p> - -<div class="figleft illowp50" id="fig_30" style="max-width: 25em;"> - <img src="images/fig_30.png" alt="" /> - <div class="caption">Fig. 30.</div> -</div> - -<p>In the figure <span class="allsmcap">ACDE</span> -(<a href="#fig_30">fig. 30</a>) there are—</p> - -<table class="standard" summary=""> -<tr> -<td class="tdr">15 inside</td> -<td class="tdl">15</td> -</tr> -<tr> -<td class="tdl">4 at corners</td> -<td class="tdr"> 1</td> -</tr> -</table> - -<p>a total of 16.</p> - -<p>Now in the square <span class="allsmcap">ABGF</span>, -there are 16—</p> - -<table class="standard" summary=""> -<tr> -<td class="tdl">9 inside</td> -<td class="tdr"> 9</td> -</tr> -<tr> -<td class="tdl">12 on sides</td> -<td class="tdr">6</td> -</tr> -<tr> -<td class="tdl">4 at corners</td> -<td class="tdr">1</td> -</tr> -<tr> -<td></td> -<td class="tdr_bt">16</td> -</tr> -</table> - -<p>Hence the square on <span class="allsmcap">AB</span> -would, by the shear turning, -become the shear square -<span class="allsmcap">ACDE</span>.</p> - -<p>And hence the inhabitant of this world would say that -the line <span class="allsmcap">AB</span> turned into the line <span class="allsmcap">AC</span>. These two lines -would be to him two lines of equal length, one turned -a little way round from the other.</p> - -<p>That is, putting shear in place of rotation, we get a -different kind of figure, as the result of the shear rotation, -from what we got with our ordinary rotation. And as a -consequence we get a position for the end of a line of -invariable length when it turns by the shear rotation, -different from the position which it would assume on -turning by our rotation.</p> - -<p>A real material rod in the shear world would, on turning -about <span class="allsmcap">A</span>, pass from the position <span class="allsmcap">AB</span> to the position <span class="allsmcap">AC</span>. -We say that its length alters when it becomes <span class="allsmcap">AC</span>, but this -transformation of <span class="allsmcap">AB</span> would seem to an inhabitant of the -shear world like a turning of <span class="allsmcap">AB</span> without altering in -length.</p> - -<p>If now we suppose a communication of ideas that takes -place between one of ourselves and an inhabitant of the<span class="pagenum" id="Page_54">[Pg 54]</span> -shear world, there would evidently be a difference between -his views of distance and ours.</p> - -<p>We should say that his line <span class="allsmcap">AB</span> increased in length in -turning to <span class="allsmcap">AC</span>. He would say that our line <span class="allsmcap">AF</span> (<a href="#fig_23">fig. 23</a>) -decreased in length in turning to <span class="allsmcap">AC</span>. He would think -that what we called an equal line was in reality a shorter -one.</p> - -<p>We should say that a rod turning round would have its -extremities in the positions we call at equal distances. -So would he—but the positions would be different. He -could, like us, appeal to the properties of matter. His -rod to him alters as little as ours does to us.</p> - -<p>Now, is there any standard to which we could appeal, to -say which of the two is right in this argument? There -is no standard.</p> - -<p>We should say that, with a change of position, the -configuration and shape of his objects altered. He would -say that the configuration and shape of our objects altered -in what we called merely a change of position. Hence -distance independent of position is inconceivable, or -practically distance is solely a property of matter.</p> - -<p>There is no principle to which either party in this -controversy could appeal. There is nothing to connect -the definition of distance with our ideas rather than with -his, except the behaviour of an actual piece of matter.</p> - -<p>For the study of the processes which go on in our world -the definition of distance given by taking the sum of the -squares is of paramount importance to us. But as a question -of pure space without making any unnecessary -assumptions the shear world is just as possible and just as -interesting as our world.</p> - -<p>It was the geometry of such conceivable worlds that -Lobatchewsky and Bolyai studied.</p> - -<p>This kind of geometry has evidently nothing to do -directly with four-dimensional space.</p> - -<p><span class="pagenum" id="Page_55">[Pg 55]</span></p> - -<p>But a connection arises in this way. It is evident that, -instead of taking a simple shear as I have done, and -defining it as that change of the arrangement of the -particles of a solid which they will undergo without -offering any resistance due to their mutual action, I -might take a complex motion, composed of a shear and -a rotation together, or some other kind of deformation.</p> - -<p>Let us suppose such an alteration picked out and -defined as the one which means simple rotation, then the -type, according to which all bodies will alter by this -rotation, is fixed.</p> - -<p>Looking at the movements of this kind, we should say -that the objects were altering their shape as well as -rotating. But to the inhabitants of that world they -would seem to be unaltered, and our figures in their -motions would seem to them to alter.</p> - -<p>In such a world the features of geometry are different. -We have seen one such difference in the case of our illustration -of the world of shear, where the square on the -hypothenuse was equal to the difference, not the sum, of -the squares on the sides.</p> - -<p>In our illustration we have the same laws of parallel -lines as in our ordinary rotation world, but in general the -laws of parallel lines are different.</p> - -<p>In one of these worlds of a different constitution of -matter through one point there can be two parallels to -a given line, in another of them there can be none, that -is, although a line be drawn parallel to another it will -meet it after a time.</p> - -<p>Now it was precisely in this respect of parallels that -Lobatchewsky and Bolyai discovered these different -worlds. They did not think of them as worlds of matter, -but they discovered that space did not necessarily mean -that our law of parallels is true. They made the -distinction between laws of space and laws of matter,<span class="pagenum" id="Page_56">[Pg 56]</span> -although that is not the form in which they stated their -results.</p> - -<p>The way in which they were led to these results was the -following. Euclid had stated the existence of parallel lines -as a postulate—putting frankly this unproved proposition—that -one line and only one parallel to a given straight -line can be drawn, as a demand, as something that must -be assumed. The words of his ninth postulate are these: -“If a straight line meeting two other straight lines -makes the interior angles on the same side of it equal -to two right angles, the two straight lines will never -meet.”</p> - -<p>The mathematicians of later ages did not like this bald -assumption, and not being able to prove the proposition -they called it an axiom—the eleventh axiom.</p> - -<p>Many attempts were made to prove the axiom; no one -doubted of its truth, but no means could be found to -demonstrate it. At last an Italian, Sacchieri, unable to -find a proof, said: “Let us suppose it not true.” He deduced -the results of there being possibly two parallels to one -given line through a given point, but feeling the waters -too deep for the human reason, he devoted the latter half -of his book to disproving what he had assumed in the first -part.</p> - -<p>Then Bolyai and Lobatchewsky with firm step entered -on the forbidden path. There can be no greater evidence -of the indomitable nature of the human spirit, or of its -manifest destiny to conquer all those limitations which -bind it down within the sphere of sense than this grand -assertion of Bolyai and Lobatchewsky.</p> - -<div class="figleft illowp25" id="fig_31" style="max-width: 12.5em;"> - <img src="images/fig_31.png" alt="" /> - <div class="caption">Fig. 31.</div> -</div> - -<p>Take a line <span class="allsmcap">AB</span> and a point <span class="allsmcap">C</span>. We -say and see and know that through <span class="allsmcap">C</span> -can only be drawn one line parallel -to <span class="allsmcap">AB</span>.</p> - -<p>But Bolyai said: “I will draw two.” Let <span class="allsmcap">CD</span> be parallel<span class="pagenum" id="Page_57">[Pg 57]</span> -to <span class="allsmcap">AB</span>, that is, not meet <span class="allsmcap">AB</span> however far produced, and let -lines beyond <span class="allsmcap">CD</span> also not meet -<span class="allsmcap">AB</span>; let there be a certain -region between <span class="allsmcap">CD</span> and <span class="allsmcap">CE</span>, -in which no line drawn meets -<span class="allsmcap">AB</span>. <span class="allsmcap">CE</span> and <span class="allsmcap">CD</span> produced -backwards through <span class="allsmcap">C</span> will give a similar region on the -other side of <span class="allsmcap">C</span>.</p> - -<div class="figleft illowp40" id="fig_32" style="max-width: 21.875em;"> - <img src="images/fig_32.png" alt="" /> - <div class="caption">Fig. 32.</div> -</div> - -<p>Nothing so triumphantly, one may almost say so -insolently, ignoring of sense had ever been written before. -Men had struggled against the limitations of the body, -fought them, despised them, conquered them. But no -one had ever thought simply as if the body, the bodily -eyes, the organs of vision, all this vast experience of space, -had never existed. The age-long contest of the soul with -the body, the struggle for mastery, had come to a culmination. -Bolyai and Lobatchewsky simply thought as -if the body was not. The struggle for dominion, the strife -and combat of the soul were over; they had mastered, -and the Hungarian drew his line.</p> - -<p>Can we point out any connection, as in the case of -Parmenides, between these speculations and higher -space? Can we suppose it was any inner perception by -the soul of a motion not known to the senses, which resulted -in this theory so free from the bonds of sense? No -such supposition appears to be possible.</p> - -<p>Practically, however, metageometry had a great influence -in bringing the higher space to the front as a -working hypothesis. This can be traced to the tendency -the mind has to move in the direction of least resistance. -The results of the new geometry could not be neglected, -the problem of parallels had occupied a place too prominent -in the development of mathematical thought for its final -solution to be neglected. But this utter independence of -all mechanical considerations, this perfect cutting loose<span class="pagenum" id="Page_58">[Pg 58]</span> -from the familiar intuitions, was so difficult that almost -any other hypothesis was more easy of acceptance, and -when Beltrami showed that the geometry of Lobatchewsky -and Bolyai was the geometry of shortest lines drawn on -certain curved surfaces, the ordinary definitions of measurement -being retained, attention was drawn to the theory of -a higher space. An illustration of Beltrami’s theory is -furnished by the simple consideration of hypothetical -beings living on a spherical surface.</p> - -<div class="figleft illowp35" id="fig_33" style="max-width: 15.625em;"> - <img src="images/fig_33.png" alt="" /> - <div class="caption">Fig. 33.</div> -</div> - -<p>Let <span class="allsmcap">ABCD</span> be the equator of a globe, and <span class="allsmcap">AP</span>, <span class="allsmcap">BP</span>, -meridian lines drawn to the pole, <span class="allsmcap">P</span>. -The lines <span class="allsmcap">AB</span>, <span class="allsmcap">AP</span>, <span class="allsmcap">BP</span> would seem to be -perfectly straight to a person moving -on the surface of the sphere, and -unconscious of its curvature. Now -<span class="allsmcap">AP</span> and <span class="allsmcap">BP</span> both make right angles -with <span class="allsmcap">AB</span>. Hence they satisfy the -definition of parallels. Yet they -meet in <span class="allsmcap">P</span>. Hence a being living on a spherical surface, -and unconscious of its curvature, would find that parallel -lines would meet. He would also find that the angles -in a triangle were greater than two right angles. In -the triangle <span class="allsmcap">PAB</span>, for instance, the angles at <span class="allsmcap">A</span> and <span class="allsmcap">B</span> -are right angles, so the three angles of the triangle -<span class="allsmcap">PAB</span> are greater than two right angles.</p> - -<p>Now in one of the systems of metageometry (for after -Lobatchewsky had shown the way it was found that other -systems were possible besides his) the angles of a triangle -are greater than two right angles.</p> - -<p>Thus a being on a sphere would form conclusions about -his space which are the same as he would form if he lived -on a plane, the matter in which had such properties as -are presupposed by one of these systems of geometry. -Beltrami also discovered a certain surface on which there -could be drawn more than one “straight” line through a<span class="pagenum" id="Page_59">[Pg 59]</span> -point which would not meet another given line. I use -the word straight as equivalent to the line having the -property of giving the shortest path between any two -points on it. Hence, without giving up the ordinary -methods of measurement, it was possible to find conditions -in which a plane being would necessarily have an experience -corresponding to Lobatchewsky’s geometry. -And by the consideration of a higher space, and a solid -curved in such a higher space, it was possible to account -for a similar experience in a space of three dimensions.</p> - -<p>Now, it is far more easy to conceive of a higher dimensionality -to space than to imagine that a rod in rotating -does not move so that its end describes a circle. Hence, -a logical conception having been found harder than that -of a four dimensional space, thought turned to the latter -as a simple explanation of the possibilities to which -Lobatchewsky had awakened it. Thinkers became accustomed -to deal with the geometry of higher space—it was -Kant, says Veronese, who first used the expression of -“different spaces”—and with familiarity the inevitableness -of the conception made itself felt.</p> - -<p>From this point it is but a small step to adapt the -ordinary mechanical conceptions to a higher spatial -existence, and then the recognition of its objective -existence could be delayed no longer. Here, too, as in so -many cases, it turns out that the order and connection of -our ideas is the order and connection of things.</p> - -<p>What is the significance of Lobatchewsky’s and Bolyai’s -work?</p> - -<p>It must be recognised as something totally different -from the conception of a higher space; it is applicable to -spaces of any number of dimensions. By immersing the -conception of distance in matter to which it properly -belongs, it promises to be of the greatest aid in analysis -for the effective distance of any two particles is the<span class="pagenum" id="Page_60">[Pg 60]</span> -product of complex material conditions and cannot be -measured by hard and fast rules. Its ultimate significance -is altogether unknown. It is a cutting loose -from the bonds of sense, not coincident with the recognition -of a higher dimensionality, but indirectly contributory -thereto.</p> - -<p>Thus, finally, we have come to accept what Plato held -in the hollow of his hand; what Aristotle’s doctrine of -the relativity of substance implies. The vast universe, too, -has its higher, and in recognising it we find that the -directing being within us no longer stands inevitably -outside our systematic knowledge.</p> -<hr class="chap x-ebookmaker-drop" /> - -<div class="chapter"> -<p><span class="pagenum" id="Page_61">[Pg 61]</span></p> - -<h2 class="nobreak" id="CHAPTER_VI">CHAPTER VI<br /> - -<small><i>THE HIGHER WORLD</i></small></h2></div> - - -<p>It is indeed strange, the manner in which we must begin -to think about the higher world.</p> - -<p>Those simplest objects analogous to those which are -about us on every side in our daily experience such as a -door, a table, a wheel are remote and uncognisable in the -world of four dimensions, while the abstract ideas of -rotation, stress and strain, elasticity into which analysis -resolves the familiar elements of our daily experience are -transferable and applicable with no difficulty whatever. -Thus we are in the unwonted position of being obliged -to construct the daily and habitual experience of a four-dimensional -being, from a knowledge of the abstract -theories of the space, the matter, the motion of it; -instead of, as in our case, passing to the abstract theories -from the richness of sensible things.</p> - -<p>What would a wheel be in four dimensions? What -the shafting for the transmission of power which a -four-dimensional being would use.</p> - -<p>The four-dimensional wheel, and the four-dimensional -shafting are what will occupy us for these few pages. And -it is no futile or insignificant enquiry. For in the attempt -to penetrate into the nature of the higher, to grasp within -our ken that which transcends all analogies, because what -we know are merely partial views of it, the purely -material and physical path affords a means of approach<span class="pagenum" id="Page_62">[Pg 62]</span> -pursuing which we are in less likelihood of error than if -we use the more frequently trodden path of framing -conceptions which in their elevation and beauty seem to -us ideally perfect.</p> - -<p>For where we are concerned with our own thoughts, the -development of our own ideals, we are as it were on a -curve, moving at any moment in a direction of tangency. -Whither we go, what we set up and exalt as perfect, -represents not the true trend of the curve, but our own -direction at the present—a tendency conditioned by the -past, and by a vital energy of motion essential but -only true when perpetually modified. That eternal corrector -of our aspirations and ideals, the material universe -draws sublimely away from the simplest things we can -touch or handle to the infinite depths of starry space, -in one and all uninfluenced by what we think or feel, -presenting unmoved fact to which, think it good or -think it evil, we can but conform, yet out of all that -impassivity with a reference to something beyond our -individual hopes and fears supporting us and giving us -our being.</p> - -<p>And to this great being we come with the question: -“You, too, what is your higher?”</p> - -<p>Or to put it in a form which will leave our conclusions in -the shape of no barren formula, and attacking the problem -on its most assailable side: “What is the wheel and the -shafting of the four-dimensional mechanic?”</p> - -<p>In entering on this enquiry we must make a plan of -procedure. The method which I shall adopt is to trace -out the steps of reasoning by which a being confined -to movement in a two-dimensional world could arrive at a -conception of our turning and rotation, and then to apply -an analogous process to the consideration of the higher -movements. The plane being must be imagined as no -abstract figure, but as a real body possessing all three<span class="pagenum" id="Page_63">[Pg 63]</span> -dimensions. His limitation to a plane must be the result -of physical conditions.</p> - -<p>We will therefore think of him as of a figure cut out of -paper placed on a smooth plane. Sliding over this plane, -and coming into contact with other figures equally thin -as he in the third dimension, he will apprehend them only -by their edges. To him they will be completely bounded -by lines. A “solid” body will be to him a two-dimensional -extent, the interior of which can only be reached by -penetrating through the bounding lines.</p> - -<p>Now such a plane being can think of our three-dimensional -existence in two ways.</p> - -<p>First, he can think of it as a series of sections, each like -the solid he knows of extending in a direction unknown -to him, which stretches transverse to his tangible -universe, which lies in a direction at right angles to every -motion which he made.</p> - -<p>Secondly, relinquishing the attempt to think of the -three-dimensional solid body in its entirety he can regard -it as consisting of a number of plane sections, each of them -in itself exactly like the two-dimensional bodies he knows, -but extending away from his two-dimensional space.</p> - -<p>A square lying in his space he regards as a solid -bounded by four lines, each of which lies in his space.</p> - -<p>A square standing at right angles to his plane appears -to him as simply a line in his plane, for all of it except -the line stretches in the third dimension.</p> - -<p>He can think of a three-dimensional body as consisting -of a number of such sections, each of which starts from a -line in his space.</p> - -<p>Now, since in his world he can make any drawing or -model which involves only two dimensions, he can represent -each such upright section as it actually is, and can represent -a turning from a known into the unknown dimension -as a turning from one to another of his known dimensions.</p> - -<p><span class="pagenum" id="Page_64">[Pg 64]</span></p> - -<p>To see the whole he must relinquish part of that which -he has, and take the whole portion by portion.</p> - -<div class= "figleft illowp30" id="fig_34" style="max-width: 15.625em;"> - <img src="images/fig_34.png" alt="" /> - <div class="caption">Fig. 34.</div> -</div> - -<p>Consider now a plane being in front of a square, <a href="#fig_34">fig. 34</a>. -The square can turn about any point -in the plane—say the point <span class="allsmcap">A</span>. But it -cannot turn about a line, as <span class="allsmcap">AB</span>. For, -in order to turn about the line <span class="allsmcap">AB</span>, -the square must leave the plane and -move in the third dimension. This -motion is out of his range of observation, -and is therefore, except for a -process of reasoning, inconceivable to him.</p> - -<p>Rotation will therefore be to him rotation about a point. -Rotation about a line will be inconceivable to him.</p> - -<p>The result of rotation about a line he can apprehend. -He can see the first and last positions occupied in a half-revolution -about the line <span class="allsmcap">AC</span>. The result of such a half revolution -is to place the square <span class="allsmcap">ABCD</span> on the left hand instead -of on the right hand of the line <span class="allsmcap">AC</span>. It would correspond -to a pulling of the whole body <span class="allsmcap">ABCD</span> through the line <span class="allsmcap">AC</span>, -or to the production of a solid body which was the exact -reflection of it in the line <span class="allsmcap">AC</span>. It would be as if the square -<span class="allsmcap">ABCD</span> turned into its image, the line <span class="allsmcap">AB</span> acting as a mirror. -Such a reversal of the positions of the parts of the square -would be impossible in his space. The occurrence of it -would be a proof of the existence of a higher dimensionality.</p> - -<div class="figleft illowp30" id="fig_35" style="max-width: 18.75em;"> - <img src="images/fig_35.png" alt="" /> - <div class="caption">Fig. 35.</div> -</div> - -<p>Let him now, adopting the conception of a three-dimensional -body as a series of -sections lying, each removed a little -farther than the preceding one, in -direction at right angles to his -plane, regard a cube, <a href="#fig_36">fig. 36</a>, as a -series of sections, each like the -square which forms its base, all -rigidly connected together.</p> - -<p><span class="pagenum" id="Page_65">[Pg 65]</span></p> - -<p>If now he turns the square about the point <span class="allsmcap">A</span> in the -plane of <i>xy</i>, each parallel section turns with the square -he moves. In each of the sections there is a point at -rest, that vertically over <span class="allsmcap">A</span>. Hence he would conclude -that in the turning of a three-dimensional body there is -one line which is at rest. That is a three-dimensional -turning in a turning about a line.</p> - -<hr class="tb" /> - -<p>In a similar way let us regard ourselves as limited to a -three-dimensional world by a physical condition. Let us -imagine that there is a direction at right angles to every -direction in which we can move, and that we are prevented -from passing in this direction by a vast solid, that -against which in every movement we make we slip as -the plane being slips against his plane sheet.</p> - -<p>We can then consider a four-dimensional body as consisting -of a series of sections, each parallel to our space, -and each a little farther off than the preceding on the -unknown dimension.</p> - -<div class="figleft illowp35" id="fig_36" style="max-width: 18.75em;"> - <img src="images/fig_36.png" alt="" /> - <div class="caption">Fig. 36.</div> -</div> - -<p>Take the simplest four-dimensional body—one which -begins as a cube, <a href="#fig_36">fig. 36</a>, in our -space, and consists of sections, each -a cube like <a href="#fig_36">fig. 36</a>, lying away from -our space. If we turn the cube -which is its base in our space -about a line, if, <i>e.g.</i>, in <a href="#fig_36">fig. 36</a> we -turn the cube about the line <span class="allsmcap">AB</span>, -not only it but each of the parallel -cubes moves about a line. The -cube we see moves about the line <span class="allsmcap">AB</span>, the cube beyond it -about a line parallel to <span class="allsmcap">AB</span> and so on. Hence the whole -four-dimensional body moves about a plane, for the -assemblage of these lines is our way of thinking about the -plane which, starting from the line <span class="allsmcap">AB</span> in our space, runs -off in the unknown direction.</p> - -<p><span class="pagenum" id="Page_66">[Pg 66]</span></p> - -<p>In this case all that we see of the plane about which -the turning takes place is the line <span class="allsmcap">AB</span>.</p> - -<p>But it is obvious that the axis plane may lie in our -space. A point near the plane determines with it a three-dimensional -space. When it begins to rotate round the -plane it does not move anywhere in this three-dimensional -space, but moves out of it. A point can no more rotate -round a plane in three-dimensional space than a point -can move round a line in two-dimensional space.</p> - -<p>We will now apply the second of the modes of representation -to this case of turning about a plane, building -up our analogy step by step from the turning in a plane -about a point and that in space about a line, and so on.</p> - -<p>In order to reduce our considerations to those of the -greatest simplicity possible, let us realise how the plane -being would think of the motion by which a square is -turned round a line.</p> - -<p>Let, <a href="#fig_34">fig. 34</a>, <span class="allsmcap">ABCD</span> be a square on his plane, and represent -the two dimensions of his space by the axes <span class="allsmcap">A</span><i>x</i> <span class="allsmcap">A</span><i>y</i>.</p> - -<p>Now the motion by which the square is turned over -about the line <span class="allsmcap">AC</span> involves the third dimension.</p> - -<p>He cannot represent the motion of the whole square in -its turning, but he can represent the motions of parts of -it. Let the third axis perpendicular to the plane of the -paper be called the axis of <i>z</i>. Of the three axes <i>x</i>, <i>y</i>, <i>z</i>, -the plane being can represent any two in his space. Let -him then draw, in <a href="#fig_35">fig. 35</a>, two axes, <i>x</i> and <i>z</i>. Here he has -in his plane a representation of what exists in the plane -which goes off perpendicularly to his space.</p> - -<p>In this representation the square would not be shown, -for in the plane of <i>xz</i> simply the line <span class="allsmcap">AB</span> of the square is -contained.</p> - -<p>The plane being then would have before him, in <a href="#fig_35">fig. 35</a>, -the representation of one line <span class="allsmcap">AB</span> of his square and two -axes, <i>x</i> and <i>z</i>, at right angles. Now it would be obvious<span class="pagenum" id="Page_67">[Pg 67]</span> -to him that, by a turning such as he knows, by a rotation -about a point, the line <span class="allsmcap">AB</span> can turn round <span class="allsmcap">A</span>, and occupying -all the intermediate positions, such as <span class="allsmcap">AB</span><sub>1</sub>, come -after half a revolution to lie as <span class="allsmcap">A</span><i>x</i> produced through <span class="allsmcap">A</span>.</p> - -<p>Again, just as he can represent the vertical plane -through <span class="allsmcap">AB</span>, so he can represent the vertical plane -through <span class="allsmcap">A´B´</span>, <a href="#fig_34">fig. 34</a>, and in a like manner can see that -the line <span class="allsmcap">A´B´</span> can turn about the point <span class="allsmcap">A´</span> till it lies in the -opposite direction from that which it ran in at first.</p> - -<p>Now these two turnings are not inconsistent. In his -plane, if <span class="allsmcap">AB</span> turned about <span class="allsmcap">A</span>, and <span class="allsmcap">A´B´</span> about <span class="allsmcap">A´</span>, the consistency -of the square would be destroyed, it would be an -impossible motion for a rigid body to perform. But in -the turning which he studies portion by portion there is -nothing inconsistent. Each line in the square can turn -in this way, hence he would realise the turning of the -whole square as the sum of a number of turnings of -isolated parts. Such turnings, if they took place in his -plane, would be inconsistent, but by virtue of a third -dimension they are consistent, and the result of them all -is that the square turns about the line <span class="allsmcap">AC</span> and lies in a -position in which it is the mirror image of what it was in -its first position. Thus he can realise a turning about a -line by relinquishing one of his axes, and representing his -body part by part.</p> - -<p>Let us apply this method to the turning of a cube so as -to become the mirror image of itself. In our space we can -construct three independent axes, <i>x</i>, <i>y</i>, <i>z</i>, shown in <a href="#fig_36">fig. 36</a>. -Suppose that there is a fourth axis, <i>w</i>, at right angles to -each and every one of them. We cannot, keeping all -three axes, <i>x</i>, <i>y</i>, <i>z</i>, represent <i>w</i> in our space; but if we -relinquish one of our three axes we can let the fourth axis -take its place, and we can represent what lies in the -space, determined by the two axes we retain and the -fourth axis.</p> - -<p><span class="pagenum" id="Page_68">[Pg 68]</span></p> - -<div class="figleft illowp35" id="fig_37" style="max-width: 18.75em;"> - <img src="images/fig_37.png" alt="" /> - <div class="caption">Fig. 37.</div> -</div> - -<p>Let us suppose that we let the <i>y</i> axis drop, and that -we represent the <i>w</i> axis as occupying -its direction. We have in fig. -37 a drawing of what we should -then see of the cube. The square -<span class="allsmcap">ABCD</span>, remains unchanged, for that -is in the plane of <i>xz</i>, and we -still have that plane. But from -this plane the cube stretches out -in the direction of the <i>y</i> axis. Now the <i>y</i> axis is gone, -and so we have no more of the cube than the face <span class="allsmcap">ABCD</span>. -Considering now this face <span class="allsmcap">ABCD</span>, we -see that it is free to turn about the -line <span class="allsmcap">AB</span>. It can rotate in the <i>x</i> to <i>w</i> -direction about this line. In <a href="#fig_38">fig. 38</a> -it is shown on its way, and it can -evidently continue this rotation till -it lies on the other side of the <i>z</i> -axis in the plane of <i>xz</i>.</p> - -<div class="figleft illowp35" id="fig_38" style="max-width: 18.75em;"> - <img src="images/fig_38.png" alt="" /> - <div class="caption">Fig. 38.</div> -</div> - -<p>We can also take a section parallel to the face <span class="allsmcap">ABCD</span>, -and then letting drop all of our space except the plane of -that section, introduce the <i>w</i> axis, running in the old <i>y</i> -direction. This section can be represented by the same -drawing, <a href="#fig_38">fig. 38</a>, and we see that it can rotate about the -line on its left until it swings half way round and runs in -the opposite direction to that which it ran in before. -These turnings of the different sections are not inconsistent, -and taken all together they will bring the cube -from the position shown in <a href="#fig_36">fig. 36</a> to that shown in -<a href="#fig_41">fig. 41</a>.</p> - -<p>Since we have three axes at our disposal in our space, -we are not obliged to represent the <i>w</i> axis by any particular -one. We may let any axis we like disappear, and let the -fourth axis take its place.</p> - -<div class="figleft illowp40" id="fig_39" style="max-width: 18.75em;"> - <img src="images/fig_39.png" alt="" /> - <div class="caption">Fig. 39.</div> -</div> -<div class="figleft illowp40" id="fig_40" style="max-width: 18.75em;"> - <img src="images/fig_40.png" alt="" /> - <div class="caption">Fig. 40.</div> -</div> - -<div class="figleft illowp40" id="fig_41" style="max-width: 21.875em;"> - <img src="images/fig_41.png" alt="" /> - <div class="caption">Fig. 41.</div> -</div> - -<p>In <a href="#fig_36">fig. 36</a> suppose the <i>z</i> axis to go. We have then<span class="pagenum" id="Page_69">[Pg 69]</span> -simply the plane of <i>xy</i> and the square base of the -cube <span class="allsmcap">ACEG</span>, <a href="#fig_39">fig. 39</a>, is all that could -be seen of it. Let now the <i>w</i> axis -take the place of the <i>z</i> axis and -we have, in <a href="#fig_39">fig. 39</a> again, a representation -of the space of <i>xyw</i>, in -which all that exists of the cube is -its square base. Now, by a turning -of <i>x</i> to <i>w</i>, this base can rotate around the line <span class="allsmcap">AE</span>, it is -shown on its way in <a href="#fig_40">fig. 40</a>, and -finally it will, after half a revolution, -lie on the other side of the <i>y</i> axis. -In a similar way we may rotate -sections parallel to the base of the -<i>xw</i> rotation, and each of them comes -to run in the opposite direction from -that which they occupied at first.</p> - -<p>Thus again the cube comes from the position of <a href="#fig_36">fig. 36</a>. -to that of <a href="#fig_41">fig. 41</a>. In this <i>x</i> -to <i>w</i> turning, we see that it -takes place by the rotations of -sections parallel to the front -face about lines parallel to <span class="allsmcap">AB</span>, -or else we may consider it as -consisting of the rotation of -sections parallel to the base -about lines parallel to <span class="allsmcap">AE</span>. It -is a rotation of the whole cube about the plane <span class="allsmcap">ABEF</span>. -Two separate sections could not rotate about two separate -lines in our space without conflicting, but their motion is -consistent when we consider another dimension. Just, -then, as a plane being can think of rotation about a line as -a rotation about a number of points, these rotations not -interfering as they would if they took place in his two-dimensional -space, so we can think of a rotation about a<span class="pagenum" id="Page_70">[Pg 70]</span> -plane as the rotation of a number of sections of a body -about a number of lines in a plane, these rotations not -being inconsistent in a four-dimensional space as they are -in three-dimensional space.</p> - -<p>We are not limited to any particular direction for the -lines in the plane about which we suppose the rotation -of the particular sections to take place. Let us draw -the section of the cube, <a href="#fig_36">fig. 36</a>, through <span class="allsmcap">A</span>, <span class="allsmcap">F</span>, <span class="allsmcap">C</span>, <span class="allsmcap">H</span>, forming a -sloping plane. Now since the fourth dimension is at -right angles to every line in our space it is at right -angles to this section also. We can represent our space -by drawing an axis at right angles to the plane <span class="allsmcap">ACEG</span>, our -space is then determined by the plane <span class="allsmcap">ACEG</span>, and the perpendicular -axis. If we let this axis drop and suppose the -fourth axis, <i>w</i>, to take its place, we have a representation of -the space which runs off in the fourth dimension from the -plane <span class="allsmcap">ACEG</span>. In this space we shall see simply the section -<span class="allsmcap">ACEG</span> of the cube, and nothing else, for one cube does not -extend to any distance in the fourth dimension.</p> - -<div class="figleft illowp40" id="fig_42" style="max-width: 25em;"> - <img src="images/fig_42.png" alt="" /> - <div class="caption">Fig. 42.</div> -</div> - -<p>If, keeping this plane, we bring in the fourth dimension, -we shall have a space in which simply this section of -the cube exists and nothing else. The section can turn -about the line <span class="allsmcap">AF</span>, and parallel sections can turn about -parallel lines. Thus in considering -the rotation about -a plane we can draw any -lines we like and consider -the rotation as taking place -in sections about them.</p> - -<p>To bring out this point -more clearly let us take two -parallel lines, <span class="allsmcap">A</span> and <span class="allsmcap">B</span>, in -the space of <i>xyz</i>, and let <span class="allsmcap">CD</span> -and <span class="allsmcap">EF</span> be two rods running -above and below the plane of <i>xy</i>, from these lines. If we<span class="pagenum" id="Page_71">[Pg 71]</span> -turn these rods in our space about the lines <span class="allsmcap">A</span> and <span class="allsmcap">B</span>, as -the upper end of one, <span class="allsmcap">F</span>, is going down, the lower end of -the other, <span class="allsmcap">C</span>, will be coming up. They will meet and -conflict. But it is quite possible for these two rods -each of them to turn about the two lines without altering -their relative distances.</p> - -<p>To see this suppose the <i>y</i> axis to go, and let the <i>w</i> axis -take its place. We shall see the lines <span class="allsmcap">A</span> and <span class="allsmcap">B</span> no longer, -for they run in the <i>y</i> direction from the points <span class="allsmcap">G</span> and <span class="allsmcap">H</span>.</p> - -<div class="figleft illowp40" id="fig_43" style="max-width: 21.875em;"> - <img src="images/fig_43.png" alt="" /> - <div class="caption">Fig. 43.</div> -</div> - -<p>Fig. 43 is a picture of the two rods seen in the space -of <i>xzw</i>. If they rotate in the -direction shown by the arrows—in -the <i>z</i> to <i>w</i> direction—they -move parallel to one another, -keeping their relative distances. -Each will rotate about its own -line, but their rotation will not -be inconsistent with their forming -part of a rigid body.</p> - -<p>Now we have but to suppose -a central plane with rods crossing -it at every point, like <span class="allsmcap">CD</span> and <span class="allsmcap">EF</span> cross the plane of <i>xy</i>, -to have an image of a mass of matter extending equal -distances on each side of a diametral plane. As two of -these rods can rotate round, so can all, and the whole -mass of matter can rotate round its diametral plane.</p> - -<p>This rotation round a plane corresponds, in four -dimensions, to the rotation round an axis in three -dimensions. Rotation of a body round a plane is the -analogue of rotation of a rod round an axis.</p> - -<p>In a plane we have rotation round a point, in three-space -rotation round an axis line, in four-space rotation -round an axis plane.</p> - -<p>The four-dimensional being’s shaft by which he transmits -power is a disk rotating round its central<span class="pagenum" id="Page_72">[Pg 72]</span> -plane—the whole contour corresponds to the ends of an axis -of rotation in our space. He can impart the rotation at -any point and take it off at any other point on the contour, -just as rotation round a line can in three-space be imparted -at one end of a rod and taken off at the other end.</p> - -<p>A four-dimensional wheel can easily be described from -the analogy of the representation which a plane being -would form for himself of one of our wheels.</p> - -<p>Suppose a wheel to move transverse to a plane, so that -the whole disk, which I will consider to be solid and -without spokes, came at the same time into contact with -the plane. It would appear as a circular portion of plane -matter completely enclosing another and smaller portion—the -axle.</p> - -<p>This appearance would last, supposing the motion of -the wheel to continue until it had traversed the plane by -the extent of its thickness, when there would remain in -the plane only the small disk which is the section of the -axle. There would be no means obvious in the plane -at first by which the axle could be reached, except by -going through the substance of the wheel. But the -possibility of reaching it without destroying the substance -of the wheel would be shown by the continued existence -of the axle section after that of the wheel had disappeared.</p> - -<p>In a similar way a four-dimensional wheel moving -transverse to our space would appear first as a solid sphere, -completely surrounding a smaller solid sphere. The -outer sphere would represent the wheel, and would last -until the wheel has traversed our space by a distance -equal to its thickness. Then the small sphere alone -would remain, representing the section of the axle. The -large sphere could move round the small one quite freely. -Any line in space could be taken as an axis, and round -this line the outer sphere could rotate, while the inner -sphere remained still. But in all these directions of<span class="pagenum" id="Page_73">[Pg 73]</span> -revolution there would be in reality one line which -remained unaltered, that is the line which stretches away -in the fourth direction, forming the axis of the axle. The -four-dimensional wheel can rotate in any number of planes, -but all these planes are such that there is a line at right -angles to them all unaffected by rotation in them.</p> - -<p>An objection is sometimes experienced as to this mode -of reasoning from a plane world to a higher dimensionality. -How artificial, it is argued, this conception of a plane -world is. If any real existence confined to a superficies -could be shown to exist, there would be an argument for -one relative to which our three-dimensional existence is -superficial. But, both on the one side and the other of -the space we are familiar with, spaces either with less -or more than three dimensions are merely arbitrary -conceptions.</p> - -<p>In reply to this I would remark that a plane being -having one less dimension than our three would have one-third -of our possibilities of motion, while we have only -one-fourth less than those of the higher space. It may -very well be that there may be a certain amount of -freedom of motion which is demanded as a condition of an -organised existence, and that no material existence is -possible with a more limited dimensionality than ours. -This is well seen if we try to construct the mechanics of a -two-dimensional world. No tube could exist, for unless -joined together completely at one end two parallel lines -would be completely separate. The possibility of an -organic structure, subject to conditions such as this, is -highly problematical; yet, possibly in the convolutions -of the brain there may be a mode of existence to be -described as two-dimensional.</p> - -<p>We have but to suppose the increase in surface and -the diminution in mass carried on to a certain extent -to find a region which, though without mobility of the<span class="pagenum" id="Page_74">[Pg 74]</span> -constituents, would have to be described as two-dimensional.</p> - -<p>But, however artificial the conception of a plane being -may be, it is none the less to be used in passing to the -conception of a greater dimensionality than ours, and -hence the validity of the first part of this objection -altogether disappears directly we find evidence for such a -state of being.</p> - -<p>The second part of the objection has more weight. -How is it possible to conceive that in a four-dimensional -space any creatures should be confined to a three-dimensional -existence?</p> - -<p>In reply I would say that we know as a matter of fact -that life is essentially a phenomenon of surface. The -amplitude of the movements which we can make is much -greater along the surface of the earth than it is up -or down.</p> - -<p>Now we have but to conceive the extent of a solid -surface increased, while the motions possible tranverse to -it are diminished in the same proportion, to obtain the -image of a three-dimensional world in four-dimensional -space.</p> - -<p>And as our habitat is the meeting of air and earth on -the world, so we must think of the meeting place of two -as affording the condition for our universe. The meeting -of what two? What can that vastness be in the higher -space which stretches in such a perfect level that our -astronomical observations fail to detect the slightest -curvature?</p> - -<p>The perfection of the level suggests a liquid—a lake -amidst what vast scenery!—whereon the matter of the -universe floats speck-like.</p> - -<p>But this aspect of the problem is like what are called -in mathematics boundary conditions.</p> - -<p>We can trace out all the consequences of four-dimensional -movements down to their last detail. Then, knowing<span class="pagenum" id="Page_75">[Pg 75]</span> -the mode of action which would be characteristic of the -minutest particles, if they were free, we can draw conclusions -from what they actually do of what the constraint -on them is. Of the two things, the material conditions and -the motion, one is known, and the other can be inferred. -If the place of this universe is a meeting of two, there -would be a one-sideness to space. If it lies so that what -stretches away in one direction in the unknown is unlike -what stretches away in the other, then, as far as the -movements which participate in that dimension are concerned, -there would be a difference as to which way the -motion took place. This would be shown in the dissimilarity -of phenomena, which, so far as all three-space -movements are concerned, were perfectly symmetrical. -To take an instance, merely, for the sake of precising -our ideas, not for any inherent probability in it; if it could -be shown that the electric current in the positive direction -were exactly like the electric current in the negative -direction, except for a reversal of the components of the -motion in three-dimensional space, then the dissimilarity -of the discharge from the positive and negative poles -would be an indication of a one-sideness to our space. -The only cause of difference in the two discharges would -be due to a component in the fourth dimension, which -directed in one direction transverse to our space, met with -a different resistance to that which it met when directed -in the opposite direction.</p> -<hr class="chap x-ebookmaker-drop" /> - -<div class="chapter"> -<p><span class="pagenum" id="Page_76">[Pg 76]</span></p> - -<h2 class="nobreak" id="CHAPTER_VII">CHAPTER VII<br /> - -<small><i>THE EVIDENCES FOR A FOURTH DIMENSION</i></small></h2></div> - - -<p>The method necessarily to be employed in the search for -the evidences of a fourth dimension, consists primarily in -the formation of the conceptions of four-dimensional -shapes and motions. When we are in possession of these -it is possible to call in the aid of observation, without -them we may have been all our lives in the familiar -presence of a four-dimensional phenomenon without ever -recognising its nature.</p> - -<p>To take one of the conceptions we have already formed, -the turning of a real thing into its mirror image would be -an occurrence which it would be hard to explain, except on -the assumption of a fourth dimension.</p> - -<p>We know of no such turning. But there exist a multitude -of forms which show a certain relation to a plane, -a relation of symmetry, which indicates more than an accidental -juxtaposition of parts. In organic life the universal -type is of right- and left-handed symmetry, there is a plane -on each side of which the parts correspond. Now we have -seen that in four dimensions a plane takes the place of a -line in three dimensions. In our space, rotation about an -axis is the type of rotation, and the origin of bodies symmetrical -about a line as the earth is symmetrical about an -axis can easily be explained. But where there is symmetry -about a plane no simple physical motion, such as we<span class="pagenum" id="Page_77">[Pg 77]</span> -are accustomed to, suffices to explain it. In our space a -symmetrical object must be built up by equal additions -on each side of a central plane. Such additions about -such a plane are as little likely as any other increments. -The probability against the existence of symmetrical -form in inorganic nature is overwhelming in our space, -and in organic forms they would be as difficult of production -as any other variety of configuration. To illustrate -this point we may take the child’s amusement of making -from dots of ink on a piece of paper a lifelike representation -of an insect by simply folding the paper -over. The dots spread out on a symmetrical line, and -give the impression of a segmented form with antennæ -and legs.</p> - -<p>Now seeing a number of such figures we should -naturally infer a folding over. Can, then, a folding over -in four-dimensional space account for the symmetry of -organic forms? The folding cannot of course be of the -bodies we see, but it may be of those minute constituents, -the ultimate elements of living matter which, turned in one -way or the other, become right- or left-handed, and so -produce a corresponding structure.</p> - -<p>There is something in life not included in our conceptions -of mechanical movement. Is this something a four-dimensional -movement?</p> - -<p>If we look at it from the broadest point of view, there is -something striking in the fact that where life comes in -there arises an entirely different set of phenomena to -those of the inorganic world.</p> - -<p>The interest and values of life as we know it in ourselves, -as we know it existing around us in subordinate -forms, is entirely and completely different to anything -which inorganic nature shows. And in living beings we -have a kind of form, a disposition of matter which is -entirely different from that shown in inorganic matter.<span class="pagenum" id="Page_78">[Pg 78]</span> -Right- and left-handed symmetry does not occur in the -configurations of dead matter. We have instances of -symmetry about an axis, but not about a plane. It can -be argued that the occurrence of symmetry in two dimensions -involves the existence of a three-dimensional process, -as when a stone falls into water and makes rings of ripples, -or as when a mass of soft material rotates about an axis. -It can be argued that symmetry in any number of dimensions -is the evidence of an action in a higher dimensionality. -Thus considering living beings, there is an evidence both -in their structure, and their different mode of activity, of a -something coming in from without into the inorganic -world.</p> - -<p>And the objections which will readily occur, such as -those derived from the forms of twin crystals and the -theoretical structure of chemical molecules, do not invalidate -the argument; for in these forms too the -presumable seat of the activity producing them lies in that -very minute region in which we necessarily place the seat -of a four-dimensional mobility.</p> - -<p>In another respect also the existence of symmetrical forms -is noteworthy. It is puzzling to conceive how two shapes -exactly equal can exist which are not superposible. Such -a pair of symmetrical figures as the two hands, right and -left, show either a limitation in our power of movement, -by which we cannot superpose the one on the other, or a -definite influence and compulsion of space on matter, -inflicting limitations which are additional to those of the -proportions of the parts.</p> - -<p>We will, however, put aside the arguments to be drawn -from the consideration of symmetry as inconclusive, -retaining one valuable indication which they afford. If -it is in virtue of a four-dimensional motion that symmetry -exists, it is only in the very minute particles -of bodies that that motion is to be found, for there is<span class="pagenum" id="Page_79">[Pg 79]</span> -no such thing as a bending over in four dimensions of -any object of a size which we can observe. The region -of the extremely minute is the one, then, which we -shall have to investigate. We must look for some -phenomenon which, occasioning movements of the kind -we know, still is itself inexplicable as any form of motion -which we know.</p> - -<p>Now in the theories of the actions of the minute -particles of bodies on one another, and in the motions of -the ether, mathematicians have tacitly assumed that the -mechanical principles are the same as those which prevail -in the case of bodies which can be observed, it has been -assumed without proof that the conception of motion being -three-dimensional, holds beyond the region from observations -in which it was formed.</p> - -<p>Hence it is not from any phenomenon explained by -mathematics that we can derive a proof of four dimensions. -Every phenomenon that has been explained is explained -as three-dimensional. And, moreover, since in the region -of the very minute we do not find rigid bodies acting -on each other at a distance, but elastic substances and -continuous fluids such as ether, we shall have a double -task.</p> - -<p>We must form the conceptions of the possible movements -of elastic and liquid four-dimensional matter, before -we can begin to observe. Let us, therefore, take the four-dimensional -rotation about a plane, and enquire what it -becomes in the case of extensible fluid substances. If -four-dimensional movements exist, this kind of rotation -must exist, and the finer portions of matter must exhibit -it.</p> - -<p>Consider for a moment a rod of flexible and extensible -material. It can turn about an axis, even if not straight; -a ring of india rubber can turn inside out.</p> - -<p>What would this be in the case of four dimensions?</p> - -<p><span class="pagenum" id="Page_80">[Pg 80]</span></p> -<div class="figleft illowp50" id="fig_44" style="max-width: 25em;"> - <img src="images/fig_44.png" alt="" /> - <div class="caption">Fig. 44.<br /> -<i>Axis of x running towards -the observer.</i></div> -</div> - -<p>Let us consider a sphere of our three-dimensional -matter having a definite -thickness. To represent -this thickness let us suppose -that from every point -of the sphere in <a href="#fig_44">fig. 44</a> rods -project both ways, in and -out, like <span class="allsmcap">D</span> and <span class="allsmcap">F</span>. We can -only see the external portion, -because the internal -parts are hidden by the -sphere.</p> - -<p>In this sphere the axis -of <i>x</i> is supposed to come -towards the observer, the -axis of <i>z</i> to run up, the axis of <i>y</i> to go to the right.</p> - -<div class="figleft illowp50" id="fig_45" style="max-width: 25em;"> - <img src="images/fig_45.png" alt="" /> - <div class="caption">Fig. 45.</div> -</div> - -<p>Now take the section determined by the <i>zy</i> plane. -This will be a circle as -shown in <a href="#fig_45">fig. 45</a>. If we -let drop the <i>x</i> axis, this -circle is all we have of -the sphere. Letting the -<i>w</i> axis now run in the -place of the old <i>x</i> axis -we have the space <i>yzw</i>, -and in this space all that -we have of the sphere is -the circle. Fig. 45 then -represents all that there -is of the sphere in the -space of <i>yzw</i>. In this space it is evident that the rods -<span class="allsmcap">CD</span> and <span class="allsmcap">EF</span> can turn round the circumference as an axis. -If the matter of the spherical shell is sufficiently extensible -to allow the particles <span class="allsmcap">C</span> and <span class="allsmcap">E</span> to become as widely -separated as they would be in the positions <span class="allsmcap">D</span> and <span class="allsmcap">F</span>, then<span class="pagenum" id="Page_81">[Pg 81]</span> -the strip of matter represented by <span class="allsmcap">CD</span> and <span class="allsmcap">EF</span> and a -multitude of rods like them can turn round the circular -circumference.</p> - -<p>Thus this particular section of the sphere can turn -inside out, and what holds for any one section holds for -all. Hence in four dimensions the whole sphere can, if -extensible turn inside out. Moreover, any part of it—a -bowl-shaped portion, for instance—can turn inside out, -and so on round and round.</p> - -<p>This is really no more than we had before in the -rotation about a plane, except that we see that the plane -can, in the case of extensible matter, be curved, and still -play the part of an axis.</p> - -<p>If we suppose the spherical shell to be of four-dimensional -matter, our representation will be a little different. -Let us suppose there to be a small thickness to the matter -in the fourth dimension. This would make no difference -in <a href="#fig_44">fig. 44</a>, for that merely shows the view in the <i>xyz</i> -space. But when the <i>x</i> axis is let drop, and the <i>w</i> axis -comes in, then the rods <span class="allsmcap">CD</span> and <span class="allsmcap">EF</span> which represent the -matter of the shell, will have a certain thickness perpendicular -to the plane of the paper on which they are drawn. -If they have a thickness in the fourth dimension they will -show this thickness when looked at from the direction of -the <i>w</i> axis.</p> - -<p>Supposing these rods, then, to be small slabs strung on -the circumference of the circle in <a href="#fig_45">fig. 45</a>, we see that -there will not be in this case either any obstacle to their -turning round the circumference. We can have a shell -of extensible material or of fluid material turning inside -out in four dimensions.</p> - -<p>And we must remember that in four dimensions there -is no such thing as rotation round an axis. If we want to -investigate the motion of fluids in four dimensions we -must take a movement about an axis in our space, and<span class="pagenum" id="Page_82">[Pg 82]</span> -find the corresponding movement about a plane in -four space.</p> - -<p>Now, of all the movements which take place in fluids, -the most important from a physical point of view is -vortex motion.</p> - -<p>A vortex is a whirl or eddy—it is shown in the gyrating -wreaths of dust seen on a summer day; it is exhibited on -a larger scale in the destructive march of a cyclone.</p> - -<p>A wheel whirling round will throw off the water on it. -But when this circling motion takes place in a liquid -itself it is strangely persistent. There is, of course, a -certain cohesion between the particles of water by which -they mutually impede their motions. But in a liquid -devoid of friction, such that every particle is free from -lateral cohesion on its path of motion, it can be shown -that a vortex or eddy separates from the mass of the -fluid a certain portion, which always remain in that -vortex.</p> - -<p>The shape of the vortex may alter, but it always consists -of the same particles of the fluid.</p> - -<p>Now, a very remarkable fact about such a vortex is that -the ends of the vortex cannot remain suspended and -isolated in the fluid. They must always run to the -boundary of the fluid. An eddy in water that remains -half way down without coming to the top is impossible.</p> - -<p>The ends of a vortex must reach the boundary of a -fluid—the boundary may be external or internal—a vortex -may exist between two objects in the fluid, terminating -one end on each object, the objects being internal -boundaries of the fluid. Again, a vortex may have its -ends linked together, so that it forms a ring. Circular -vortex rings of this description are often seen in puffs of -smoke, and that the smoke travels on in the ring is a -proof that the vortex always consists of the same particles -of air.</p> - -<p><span class="pagenum" id="Page_83">[Pg 83]</span></p> - -<p>Let us now enquire what a vortex would be in a four-dimensional -fluid.</p> - -<p>We must replace the line axis by a plane axis. We -should have therefore a portion of fluid rotating round -a plane.</p> - -<p>We have seen that the contour of this plane corresponds -with the ends of the axis line. Hence such a four-dimensional -vortex must have its rim on a boundary of -the fluid. There would be a region of vorticity with a -contour. If such a rotation were started at one part of a -circular boundary, its edges would run round the boundary -in both directions till the whole interior region was filled -with the vortex sheet.</p> - -<p>A vortex in a three-dimensional liquid may consist of a -number of vortex filaments lying together producing a -tube, or rod of vorticity.</p> - -<p>In the same way we can have in four dimensions a -number of vortex sheets alongside each other, each of which -can be thought of as a bowl-shaped portion of a spherical -shell turning inside out. The rotation takes place at any -point not in the space occupied by the shell, but from -that space to the fourth dimension and round back again.</p> - -<p>Is there anything analogous to this within the range -of our observation?</p> - -<p>An electric current answers this description in every -respect. Electricity does not flow through a wire. Its effect -travels both ways from the starting point along the wire. -The spark which shows its passing midway in its circuit -is later than that which occurs at points near its starting -point on either side of it.</p> - -<p>Moreover, it is known that the action of the current -is not in the wire. It is in the region enclosed by the -wire, this is the field of force, the locus of the exhibition -of the effects of the current.</p> - -<p>And the necessity of a conducting circuit for a current is<span class="pagenum" id="Page_84">[Pg 84]</span> -exactly that which we should expect if it were a four-dimensional -vortex. According to Maxwell every current forms -a closed circuit, and this, from the four-dimensional point -of view, is the same as saying a vortex must have its ends -on a boundary of the fluid.</p> - -<p>Thus, on the hypothesis of a fourth dimension, the rotation -of the fluid ether would give the phenomenon of an -electric current. We must suppose the ether to be full of -movement, for the more we examine into the conditions -which prevail in the obscurity of the minute, the more we -find that an unceasing and perpetual motion reigns. Thus -we may say that the conception of the fourth dimension -means that there must be a phenomenon which presents -the characteristics of electricity.</p> - -<p>We know now that light is an electro-magnetic action, -and that so far from being a special and isolated phenomenon -this electric action is universal in the realm of the -minute. Hence, may we not conclude that, so far from -the fourth dimension being remote and far away, being a -thing of symbolic import, a term for the explanation of -dubious facts by a more obscure theory, it is really the -most important fact within our knowledge. Our three-dimensional -world is superficial. These processes, which -really lie at the basis of all phenomena of matter, -escape our observation by their minuteness, but reveal -to our intellect an amplitude of motion surpassing any -that we can see. In such shapes and motions there is a -realm of the utmost intellectual beauty, and one to -which our symbolic methods apply with a better grace -than they do to those of three dimensions.</p> -<hr class="chap x-ebookmaker-drop" /> - -<div class="chapter"> -<p><span class="pagenum" id="Page_85">[Pg 85]</span></p> - -<h2 class="nobreak" id="CHAPTER_VIII">CHAPTER VIII<br /> - -<small><i>THE USE OF FOUR DIMENSIONS IN -THOUGHT</i></small></h2></div> - - -<p>Having held before ourselves this outline of a conjecture -of the world as four-dimensional, having roughly thrown -together those facts of movement which we can see apply -to our actual experience, let us pass to another branch -of our subject.</p> - -<p>The engineer uses drawings, graphical constructions, -in a variety of manners. He has, for instance, diagrams -which represent the expansion of steam, the efficiency -of his valves. These exist alongside the actual plans of -his machines. They are not the pictures of anything -really existing, but enable him to think about the relations -which exist in his mechanisms.</p> - -<p>And so, besides showing us the actual existence of that -world which lies beneath the one of visible movements, -four-dimensional space enables us to make ideal constructions -which serve to represent the relations of things, -and throw what would otherwise be obscure into a definite -and suggestive form.</p> - -<p>From amidst the great variety of instances which lies -before me I will select two, one dealing with a subject -of slight intrinsic interest, which however gives within -a limited field a striking example of the method<span class="pagenum" id="Page_86">[Pg 86]</span> -of drawing conclusions and the use of higher space -figures.<a id="FNanchor_1" href="#Footnote_1" class="fnanchor">[1]</a></p> - -<div class="footnotes"> -<div class="footnote"> - -<p><a id="Footnote_1" href="#FNanchor_1" class="label">[1]</a> It is suggestive also in another respect, because it shows very -clearly that in our processes of thought there are in play faculties other -than logical; in it the origin of the idea which proves to be justified is -drawn from the consideration of symmetry, a branch of the beautiful.</p> - -</div></div> - -<p>The other instance is chosen on account of the bearing -it has on our fundamental conceptions. In it I try to -discover the real meaning of Kant’s theory of experience.</p> - -<p>The investigation of the properties of numbers is much -facilitated by the fact that relations between numbers are -themselves able to be represented as numbers—<i>e.g.</i>, 12, -and 3 are both numbers, and the relation between them -is 4, another number. The way is thus opened for a -process of constructive theory, without there being any -necessity for a recourse to another class of concepts -besides that which is given in the phenomena to be -studied.</p> - -<p>The discipline of number thus created is of great and -varied applicability, but it is not solely as quantitative -that we learn to understand the phenomena of nature. -It is not possible to explain the properties of matter -by number simply, but all the activities of matter are -energies in space. They are numerically definite and also, -we may say, directedly definite, <i>i.e.</i> definite in direction.</p> - -<p>Is there, then, a body of doctrine about space which, like -that of number, is available in science? It is needless -to answer: Yes; geometry. But there is a method -lying alongside the ordinary methods of geometry, which -tacitly used and presenting an analogy to the method -of numerical thought deserves to be brought into greater -prominence than it usually occupies.</p> - -<p>The relation of numbers is a number.</p> - -<p>Can we say in the same way that the relation of -shapes is a shape?</p> - -<p>We can.</p> - -<p><span class="pagenum" id="Page_87">[Pg 87]</span></p> -<div class="figleft illowp50" id="fig_46" style="max-width: 25em;"> - <img src="images/fig_46.png" alt="" /> - <div class="caption">Fig. 46.</div> -</div> - -<p>To take an instance chosen on account of its ready -availability. Let us take -two right-angled triangles of -a given hypothenuse, but -having sides of different -lengths (<a href="#fig_46">fig. 46</a>). These -triangles are shapes which have a certain relation to each -other. Let us exhibit their relation as a figure.</p> - -<div class="figleft illowp40" id="fig_47" style="max-width: 18.75em;"> - <img src="images/fig_47.png" alt="" /> - <div class="caption">Fig. 47.</div> -</div> - -<p>Draw two straight lines at right angles to each other, -the one <span class="allsmcap">HL</span> a horizontal level, the -other <span class="allsmcap">VL</span> a vertical level (<a href="#fig_47">fig. 47</a>). -By means of these two co-ordinating -lines we can represent a -double set of magnitudes; one set -as distances to the right of the vertical -level, the other as distances -above the horizontal level, a suitable unit being chosen.</p> - -<p>Thus the line marked 7 will pick out the assemblage -of points whose distance from the vertical level is 7, -and the line marked 1 will pick out the points whose -distance above the horizontal level is 1. The meeting -point of these two lines, 7 and 1, will define a point -which with regard to the one set of magnitudes is 7, -with regard to the other is 1. Let us take the sides of -our triangles as the two sets of magnitudes in question.</p> - -<div class="figleft illowp40" id="fig_48" style="max-width: 18.75em;"> - <img src="images/fig_48.png" alt="" /> - <div class="caption">Fig. 48.</div> -</div> - -<p>Then the point 7, 1, will represent the triangle whose -sides are 7 and 1. Similarly the point 5, 5—5, that -is, to the right of the vertical level and 5 above the -horizontal level—will represent the -triangle whose sides are 5 and 5 -(<a href="#fig_48">fig. 48</a>).</p> - -<p>Thus we have obtained a figure -consisting of the two points 7, 1, -and 5, 5, representative of our two -triangles. But we can go further, and, drawing an arc<span class="pagenum" id="Page_88">[Pg 88]</span> -of a circle about <span class="allsmcap">O</span>, the meeting point of the horizontal -and vertical levels, which passes through 7, 1, and 5, 5, -assert that all the triangles which are right-angled and -have a hypothenuse whose square is 50 are represented -by the points on this arc.</p> - -<p>Thus, each individual of a class being represented by a -point, the whole class is represented by an assemblage of -points forming a figure. Accepting this representation -we can attach a definite and calculable significance to the -expression, resemblance, or similarity between two individuals -of the class represented, the difference being -measured by the length of the line between two representative -points. It is needless to multiply examples, or -to show how, corresponding to different classes of triangles, -we obtain different curves.</p> - -<p>A representation of this kind in which an object, a -thing in space, is represented as a point, and all its properties -are left out, their effect remaining only in the -relative position which the representative point bears -to the representative points of the other objects, may be -called, after the analogy of Sir William R. Hamilton’s -hodograph, a “Poiograph.”</p> - -<p>Representations thus made have the character of -natural objects; they have a determinate and definite -character of their own. Any lack of completeness in them -is probably due to a failure in point of completeness -of those observations which form the ground of their -construction.</p> - -<p>Every system of classification is a poiograph. In -Mendeléeff’s scheme of the elements, for instance, each -element is represented by a point, and the relations -between the elements are represented by the relations -between the points.</p> - -<p>So far I have simply brought into prominence processes -and considerations with which we are all familiar. But<span class="pagenum" id="Page_89">[Pg 89]</span> -it is worth while to bring into the full light of our attention -our habitual assumptions and processes. It often -happens that we find there are two of them which have -a bearing on each other, which, without this dragging into -the light, we should have allowed to remain without -mutual influence.</p> - -<p>There is a fact which it concerns us to take into account -in discussing the theory of the poiograph.</p> - -<p>With respect to our knowledge of the world we are -far from that condition which Laplace imagined when he -asserted that an all-knowing mind could determine the -future condition of every object, if he knew the co-ordinates -of its particles in space, and their velocity at any -particular moment.</p> - -<p>On the contrary, in the presence of any natural object, -we have a great complexity of conditions before us, -which we cannot reduce to position in space and date -in time.</p> - -<p>There is mass, attraction apparently spontaneous, electrical -and magnetic properties which must be superadded -to spatial configuration. To cut the list short we must -say that practically the phenomena of the world present -us problems involving many variables, which we must -take as independent.</p> - -<p>From this it follows that in making poiographs we -must be prepared to use space of more than three dimensions. -If the symmetry and completeness of our representation -is to be of use to us we must be prepared to -appreciate and criticise figures of a complexity greater -than of those in three dimensions. It is impossible to give -an example of such a poiograph which will not be merely -trivial, without going into details of some kind irrelevant -to our subject. I prefer to introduce the irrelevant details -rather than treat this part of the subject perfunctorily.</p> - -<p>To take an instance of a poiograph which does not lead<span class="pagenum" id="Page_90">[Pg 90]</span> -us into the complexities incident on its application in -classificatory science, let us follow Mrs. Alicia Boole Stott -in her representation of the syllogism by its means. She -will be interested to find that the curious gap she detected -has a significance.</p> - -<div class= "figleft illowp40" id="fig_49" style="max-width: 13.75em;"> - <img src="images/fig_49.png" alt="" /> - <div class="caption">Fig. 49.</div> -</div> - -<p>A syllogism consists of two statements, the major and -the minor premiss, with the conclusion that can be drawn -from them. Thus, to take an instance, <a href="#fig_49">fig. 49</a>. It is -evident, from looking at the successive figures that, if we -know that the region <span class="allsmcap">M</span> lies altogether within the region -<span class="allsmcap">P</span>, and also know that the region <span class="allsmcap">S</span> lies altogether within -the region <span class="allsmcap">M</span>, we can conclude that the region <span class="allsmcap">S</span> lies -altogether within the region <span class="allsmcap">P</span>. <span class="allsmcap">M</span> is <span class="allsmcap">P</span>, -major premiss; <span class="allsmcap">S</span> is <span class="allsmcap">M</span>, minor premiss; <span class="allsmcap">S</span> -is <span class="allsmcap">P</span>, conclusion. Given the first two data -we must conclude that <span class="allsmcap">S</span> lies in <span class="allsmcap">P</span>. The -conclusion <span class="allsmcap">S</span> is <span class="allsmcap">P</span> involves two terms, <span class="allsmcap">S</span> and -<span class="allsmcap">P</span>, which are respectively called the subject -and the predicate, the letters <span class="allsmcap">S</span> and <span class="allsmcap">P</span> -being chosen with reference to the parts -the notions they designate play in the -conclusion. <span class="allsmcap">S</span> is the subject of the conclusion, -<span class="allsmcap">P</span> is the predicate of the conclusion. -The major premiss we take to be, that -which does not involve <span class="allsmcap">S</span>, and here we -always write it first.</p> - -<p>There are several varieties of statement -possessing different degrees of universality and manners of -assertiveness. These different forms of statement are -called the moods.</p> - -<p>We will take the major premiss as one variable, as a -thing capable of different modifications of the same kind, -the minor premiss as another, and the different moods we -will consider as defining the variations which these -variables undergo.</p> - -<p><span class="pagenum" id="Page_91">[Pg 91]</span></p> - -<p>There are four moods:—</p> - -<p>1. The universal affirmative; all <span class="allsmcap">M</span> is <span class="allsmcap">P</span>, called mood <span class="allsmcap">A</span>.</p> - -<p>2. The universal negative; no <span class="allsmcap">M</span> is <span class="allsmcap">P</span>, mood <span class="allsmcap">E</span>.</p> - -<p>3. The particular affirmative; some <span class="allsmcap">M</span> is <span class="allsmcap">P</span>, mood <span class="allsmcap">I</span>.</p> - -<p>4. The particular negative; some <span class="allsmcap">M</span> is not <span class="allsmcap">P</span>, mood <span class="allsmcap">O</span>.</p> - -<div class="figcenter illowp100" id="fig_50" style="max-width: 62.5em;"> - <img src="images/fig_50.png" alt="" /> - <div class="caption">Figure 50. -</div></div> - - -<p>The dotted lines in 3 and 4, <a href="#fig_50">fig. 50</a>, denote that it is -not known whether or no any objects exist, corresponding -to the space of which the dotted line forms one delimiting -boundary; thus, in mood <span class="allsmcap">I</span> we do not know if there are -any <span class="smcap">M’s</span> which are not <span class="allsmcap">P</span>, we only know some <span class="smcap">M’s</span> are <span class="allsmcap">P</span>.</p> - -<div class="figleft illowp30" id="fig_51" style="max-width: 15.625em;"> - <img src="images/fig_51.png" alt="" /> - <div class="caption">Fig. 51.</div> -</div> - -<p>Representing the first premiss in its various moods by -regions marked by vertical lines to -the right of <span class="allsmcap">PQ</span>, we have in <a href="#fig_51">fig. 51</a>, -running up from the four letters <span class="allsmcap">AEIO</span>, -four columns, each of which indicates -that the major premiss is in the mood -denoted by the respective letter. In -the first column to the right of <span class="allsmcap">PQ</span> is -the mood <span class="allsmcap">A</span>. Now above the line <span class="allsmcap">RS</span> let there be marked -off four regions corresponding to the four moods of the -minor premiss. Thus, in the first row above <span class="allsmcap">RS</span> all the -region between <span class="allsmcap">RS</span> and the first horizontal line above it -denotes that the minor premiss is in the mood <span class="allsmcap">A</span>. The<span class="pagenum" id="Page_92">[Pg 92]</span> -letters <span class="allsmcap">E</span>, <span class="allsmcap">I</span>, <span class="allsmcap">O</span>, in the same way show the mood characterising -the minor premiss in the rows opposite these letters.</p> - -<p>We have still to exhibit the conclusion. To do this we -must consider the conclusion as a third variable, characterised -in its different varieties by four moods—this being -the syllogistic classification. The introduction of a third -variable involves a change in our system of representation.</p> - -<div class="figleft illowp25" id="fig_52" style="max-width: 12.5em;"> - <img src="images/fig_52.png" alt="" /> - <div class="caption">Fig. 52.</div> -</div> - -<p>Before we started with the regions to the right of a -certain line as representing successively the major premiss -in its moods; now we must start with the regions to the -right of a certain plane. Let <span class="allsmcap">LMNR</span> -be the plane face of a cube, <a href="#fig_52">fig. 52</a>, and -let the cube be divided into four parts -by vertical sections parallel to <span class="allsmcap">LMNR</span>. -The variable, the major premiss, is represented -by the successive regions -which occur to the right of the plane -<span class="allsmcap">LMNR</span>—that region to which <span class="allsmcap">A</span> stands opposite, that -slice of the cube, is significative of the mood <span class="allsmcap">A</span>. This -whole quarter-part of the cube represents that for every -part of it the major premiss is in the mood <span class="allsmcap">A</span>.</p> - -<p>In a similar manner the next section, the second with -the letter <span class="allsmcap">E</span> opposite it, represents that for every one of -the sixteen small cubic spaces in it, the major premiss is -in the mood <span class="allsmcap">E</span>. The third and fourth compartments made -by the vertical sections denote the major premiss in the -moods <span class="allsmcap">I</span> and <span class="allsmcap">O</span>. But the cube can be divided in other -ways by other planes. Let the divisions, of which four -stretch from the front face, correspond to the minor -premiss. The first wall of sixteen cubes, facing the -observer, has as its characteristic that in each of the small -cubes, whatever else may be the case, the minor premiss is -in the mood <span class="allsmcap">A</span>. The variable—the minor premiss—varies -through the phases <span class="allsmcap">A</span>, <span class="allsmcap">E</span>, <span class="allsmcap">I</span>, <span class="allsmcap">O</span>, away from the front face of the -cube, or the front plane of which the front face is a part.</p> - -<p><span class="pagenum" id="Page_93">[Pg 93]</span></p> - -<p>And now we can represent the third variable in a precisely -similar way. We can take the conclusion as the third -variable, going through its four phases from the ground -plane upwards. Each of the small cubes at the base of -the whole cube has this true about it, whatever else may -be the case, that the conclusion is, in it, in the mood <span class="allsmcap">A</span>. -Thus, to recapitulate, the first wall of sixteen small cubes, -the first of the four walls which, proceeding from left to -right, build up the whole cube, is characterised in each -part of it by this, that the major premiss is in the mood <span class="allsmcap">A</span>.</p> - -<p>The next wall denotes that the major premiss is in the -mood <span class="allsmcap">E</span>, and so on. Proceeding from the front to the -back the first wall presents a region in every part of -which the minor premiss is in the mood <span class="allsmcap">A</span>. The second -wall is a region throughout which the minor premiss is in -the mood <span class="allsmcap">E</span>, and so on. In the layers, from the bottom -upwards, the conclusion goes through its various moods -beginning with <span class="allsmcap">A</span> in the lowest, <span class="allsmcap">E</span> in the second, <span class="allsmcap">I</span> in the -third, <span class="allsmcap">O</span> in the fourth.</p> - -<p>In the general case, in which the variables represented -in the poiograph pass through a wide range of values, the -planes from which we measure their degrees of variation -in our representation are taken to be indefinitely extended. -In this case, however, all we are concerned with is the -finite region.</p> - -<p>We have now to represent, by some limitation of the -complex we have obtained, the fact that not every combination -of premisses justifies any kind of conclusion. -This can be simply effected by marking the regions in -which the premisses, being such as are defined by the -positions, a conclusion which is valid is found.</p> - -<p>Taking the conjunction of the major premiss, all <span class="allsmcap">M</span> is -<span class="allsmcap">P</span>, and the minor, all <span class="allsmcap">S</span> is <span class="allsmcap">M</span>, we conclude that all <span class="allsmcap">S</span> is <span class="allsmcap">P</span>. -Hence, that region must be marked in which we have the -conjunction of major premiss in mood <span class="allsmcap">A</span>; minor premiss,<span class="pagenum" id="Page_94">[Pg 94]</span> -mood <span class="allsmcap">A</span>; conclusion, mood <span class="allsmcap">A</span>. This is the cube occupying -the lowest left-hand corner of the large cube.</p> - -<div class="figleft illowp25" id="fig_53" style="max-width: 12.5em;"> - <img src="images/fig_53.png" alt="" /> - <div class="caption">Fig. 53.</div> -</div> - - -<p>Proceeding in this way, we find that the regions which -must be marked are those shown in <a href="#fig_53">fig. 53</a>. -To discuss the case shown in the marked -cube which appears at the top of <a href="#fig_53">fig. 53</a>. -Here the major premiss is in the second -wall to the right—it is in the mood <span class="allsmcap">E</span> and -is of the type no <span class="allsmcap">M</span> is <span class="allsmcap">P</span>. The minor -premiss is in the mood characterised by -the third wall from the front. It is of -the type some <span class="allsmcap">S</span> is <span class="allsmcap">M</span>. From these premisses we draw -the conclusion that some <span class="allsmcap">S</span> is not <span class="allsmcap">P</span>, a conclusion in the -mood <span class="allsmcap">O</span>. Now the mood <span class="allsmcap">O</span> of the conclusion is represented -in the top layer. Hence we see that the marking is -correct in this respect.</p> - -<div class="figleft illowp50" id="fig_54" style="max-width: 25em;"> - <img src="images/fig_54.png" alt="" /> - <div class="caption">Fig. 54.</div> -</div> - -<p>It would, of course, be possible to represent the cube on -a plane by means of four -squares, as in <a href="#fig_54">fig. 54</a>, if we -consider each square to represent -merely the beginning -of the region it stands for. -Thus the whole cube can be -represented by four vertical -squares, each standing for a -kind of vertical tray, and the -markings would be as shown. In No. 1 the major premiss -is in mood <span class="allsmcap">A</span> for the whole of the region indicated by the -vertical square of sixteen divisions; in No. 2 it is in the -mood <span class="allsmcap">E</span>, and so on.</p> - -<p>A creature confined to a plane would have to adopt some -such disjunctive way of representing the whole cube. He -would be obliged to represent that which we see as a -whole in separate parts, and each part would merely -represent, would not be, that solid content which we see.</p> - -<p><span class="pagenum" id="Page_95">[Pg 95]</span></p> - -<p>The view of these four squares which the plane creature -would have would not be such as ours. He would not -see the interior of the four squares represented above, but -each would be entirely contained within its outline, the -internal boundaries of the separate small squares he could -not see except by removing the outer squares.</p> - -<p>We are now ready to introduce the fourth variable -involved in the syllogism.</p> - -<p>In assigning letters to denote the terms of the syllogism -we have taken <span class="allsmcap">S</span> and <span class="allsmcap">P</span> to represent the subject and -predicate in the conclusion, and thus in the conclusion -their order is invariable. But in the premisses we have -taken arbitrarily the order all <span class="allsmcap">M</span> is <span class="allsmcap">P</span>, and all <span class="allsmcap">S</span> is <span class="allsmcap">M</span>. -There is no reason why <span class="allsmcap">M</span> instead of <span class="allsmcap">P</span> should not be the -predicate of the major premiss, and so on.</p> - -<p>Accordingly we take the order of the terms in the premisses -as the fourth variable. Of this order there are four -varieties, and these varieties are called figures.</p> - -<p>Using the order in which the letters are written to -denote that the letter first written is subject, the one -written second is predicate, we have the following possibilities:—</p> - - -<table class="standard" summary=""> -<tr> -<td class="tdc"></td> -<td class="tdc">1st Figure.</td> -<td class="tdc">2nd Figure.</td> -<td class="tdc">3rd Figure.</td> -<td class="tdc">4th Figure.</td> -</tr> -<tr> -<td class="tdc">Major</td> -<td class="tdc"><span class="allsmcap">M P</span></td> -<td class="tdc"><span class="allsmcap">P M</span></td> -<td class="tdc"><span class="allsmcap">M P</span></td> -<td class="tdc"><span class="allsmcap">P M</span></td> -</tr> -<tr> -<td class="tdc">Minor</td> -<td class="tdc"><span class="allsmcap">S M</span></td> -<td class="tdc"><span class="allsmcap">S M</span></td> -<td class="tdc"><span class="allsmcap">M S</span></td> -<td class="tdc"><span class="allsmcap">M S</span></td> -</tr> -</table> - -<p>There are therefore four possibilities with regard to -this fourth variable as with regard to the premisses.</p> - -<p>We have used up our dimensions of space in representing -the phases of the premisses and the conclusion in -respect of mood, and to represent in an analogous manner -the variations in figure we require a fourth dimension.</p> - -<p>Now in bringing in this fourth dimension we must -make a change in our origins of measurement analogous -to that which we made in passing from the plane to the -solid.</p> - -<p><span class="pagenum" id="Page_96">[Pg 96]</span></p> - -<p>This fourth dimension is supposed to run at right -angles to any of the three space dimensions, as the third -space dimension runs at right angles to the two dimensions -of a plane, and thus it gives us the opportunity of -generating a new kind of volume. If the whole cube -moves in this dimension, the solid itself traces out a path, -each section of which, made at right angles to the -direction in which it moves, is a solid, an exact repetition -of the cube itself.</p> - -<p>The cube as we see it is the beginning of a solid of such -a kind. It represents a kind of tray, as the square face of -the cube is a kind of tray against which the cube rests.</p> - -<p>Suppose the cube to move in this fourth dimension in -four stages, and let the hyper-solid region traced out in -the first stage of its progress be characterised by this, that -the terms of the syllogism are in the first figure, then we -can represent in each of the three subsequent stages the -remaining three figures. Thus the whole cube forms -the basis from which we measure the variation in figure. -The first figure holds good for the cube as we see it, and -for that hyper-solid which lies within the first stage; -the second figure holds good in the second stage, and -so on.</p> - -<p>Thus we measure from the whole cube as far as figures -are concerned.</p> - -<p>But we saw that when we measured in the cube itself -having three variables, namely, the two premisses and -the conclusion, we measured from three planes. The base -from which we measured was in every case the same.</p> - -<p>Hence, in measuring in this higher space we should -have bases of the same kind to measure from, we should -have solid bases.</p> - -<p>The first solid base is easily seen, it is the cube itself. -The other can be found from this consideration.</p> - -<p>That solid from which we measure figure is that in<span class="pagenum" id="Page_97">[Pg 97]</span> -which the remaining variables run through their full -range of varieties.</p> - -<p>Now, if we want to measure in respect of the moods of -the major premiss, we must let the minor premiss, the -conclusion, run through their range, and also the order -of the terms. That is we must take as basis of measurement -in respect to the moods of the major that which -represents the variation of the moods of the minor, the -conclusion and the variation of the figures.</p> - -<p>Now the variation of the moods of the minor and of the -conclusion are represented in the square face on the left -of the cube. Here are all varieties of the minor premiss -and the conclusion. The varieties of the figures are -represented by stages in a motion proceeding at right -angles to all space directions, at right angles consequently -to the face in question, the left-hand face of the cube.</p> - -<p>Consequently letting the left-hand face move in this -direction we get a cube, and in this cube all the varieties -of the minor premiss, the conclusion, and the figure are -represented.</p> - -<p>Thus another cubic base of measurement is given to -the cube, generated by movement of the left-hand square -in the fourth dimension.</p> - -<p>We find the other bases in a similar manner, one is the -cube generated by the front square moved in the fourth -dimension so as to generate a cube. From this cube -variations in the mood of the minor are measured. The -fourth base is that found by moving the bottom square of -the cube in the fourth dimension. In this cube the -variations of the major, the minor, and the figure are given. -Considering this as a basis in the four stages proceeding -from it, the variation in the moods of the conclusion are -given.</p> - -<p>Any one of these cubic bases can be represented in space, -and then the higher solid generated from them lies out of<span class="pagenum" id="Page_98">[Pg 98]</span> -our space. It can only be represented by a device analogous -to that by which the plane being represents a cube.</p> - -<p>He represents the cube shown above, by taking four -square sections and placing them arbitrarily at convenient -distances the one from the other.</p> - -<p>So we must represent this higher solid by four cubes: -each cube represents only the beginning of the corresponding -higher volume.</p> - -<p>It is sufficient for us, then, if we draw four cubes, the -first representing that region in which the figure is of the -first kind, the second that region in which the figure is -of the second kind, and so on. These cubes are the -beginnings merely of the respective regions—they are -the trays, as it were, against which the real solids must -be conceived as resting, from which they start. The first -one, as it is the beginning of the region of the first figure, -is characterised by the order of the terms in the premisses -being that of the first figure. The second similarly has -the terms of the premisses in the order of the second -figure, and so on.</p> - -<p>These cubes are shown below.</p> - -<p>For the sake of showing the properties of the method -of representation, not for the logical problem, I will make -a digression. I will represent in space the moods of the -minor and of the conclusion and the different figures, -keeping the major always in mood <span class="allsmcap">A</span>. Here we have -three variables in different stages, the minor, the conclusion, -and the figure. Let the square of the left-hand -side of the original cube be imagined to be standing by -itself, without the solid part of the cube, represented by -(2) <a href="#fig_55">fig. 55</a>. The <span class="allsmcap">A</span>, <span class="allsmcap">E</span>, <span class="allsmcap">I</span>, <span class="allsmcap">O</span>, which run away represent the -moods of the minor, the <span class="allsmcap">A</span>, <span class="allsmcap">E</span>, <span class="allsmcap">I</span>, <span class="allsmcap">O</span>, which run up represent -the moods of the conclusion. The whole square, since it -is the beginning of the region in the major premiss, mood -<span class="allsmcap">A</span>, is to be considered as in major premiss, mood <span class="allsmcap">A</span>.</p> - -<p><span class="pagenum" id="Page_99">[Pg 99]</span></p> - -<p>From this square, let it be supposed that that direction -in which the figures are represented runs to the -left hand. Thus we have a cube (1) running from the -square above, in which the square itself is hidden, but -the letters <span class="allsmcap">A</span>, <span class="allsmcap">E</span>, <span class="allsmcap">I</span>, <span class="allsmcap">O</span>, of the conclusion are seen. In this -cube we have the minor premiss and the conclusion in all -their moods, and all the figures represented. With regard -to the major premiss, since the face (2) belongs to the first -wall from the left in the original arrangement, and in this -arrangement was characterised by the major premiss in the -mood <span class="allsmcap">A</span>, we may say that the whole of the cube we now -have put up represents the mood <span class="allsmcap">A</span> of the major premiss.</p> - -<div class="figcenter illowp100" id="fig_55" style="max-width: 50em;"> - <img src="images/fig_55.png" alt="" /> - <div class="caption">Fig. 55.</div> -</div> - -<p>Hence the small cube at the bottom to the right in 1, -nearest to the spectator, is major premiss, mood <span class="allsmcap">A</span>; minor -premiss, mood <span class="allsmcap">A</span>; conclusion, mood <span class="allsmcap">A</span>; and figure the first. -The cube next to it, running to the left, is major premiss, -mood <span class="allsmcap">A</span>; minor premiss, mood <span class="allsmcap">A</span>; conclusion, mood <span class="allsmcap">A</span>; -figure 2.</p> - -<p>So in this cube we have the representations of all the -combinations which can occur when the major premiss, -remaining in the mood <span class="allsmcap">A</span>, the minor premiss, the conclusion, -and the figures pass through their varieties.</p> - -<p>In this case there is no room in space for a natural -representation of the moods of the major premiss. To -represent them we must suppose as before that there is a -fourth dimension, and starting from this cube as base in -the fourth direction in four equal stages, all the first volume -corresponds to major premiss <span class="allsmcap">A</span>, the second to major<span class="pagenum" id="Page_100">[Pg 100]</span> -premiss, mood <span class="allsmcap">E</span>, the next to the mood <span class="allsmcap">I</span>, and the last -to mood <span class="allsmcap">O</span>.</p> - -<p>The cube we see is as it were merely a tray against -which the four-dimensional figure rests. Its section at -any stage is a cube. But a transition in this direction -being transverse to the whole of our space is represented -by no space motion. We can exhibit successive stages of -the result of transference of the cube in that direction, -but cannot exhibit the product of a transference, however -small, in that direction.</p> - -<div class="figcenter illowp100" id="fig_56" style="max-width: 62.5em;"> - <img src="images/fig_56.png" alt="" /> - <div class="caption">Fig. 56.</div> -</div> - -<p>To return to the original method of representing our -variables, consider <a href="#fig_56">fig. 56</a>. These four cubes represent -four sections of the figure derived from the first of them -by moving it in the fourth dimension. The first portion -of the motion, which begins with 1, traces out a -more than solid body, which is all in the first figure. -The beginning of this body is shown in 1. The next -portion of the motion traces out a more than solid body, -all of which is in the second figure; the beginning of -this body is shown in 2; 3 and 4 follow on in like -manner. Here, then, in one four-dimensional figure we -have all the combinations of the four variables, major -premiss, minor premiss, figure, conclusion, represented, -each variable going through its four varieties. The disconnected -cubes drawn are our representation in space by -means of disconnected sections of this higher body.</p> - -<p><span class="pagenum" id="Page_101">[Pg 101]</span></p> - -<p>Now it is only a limited number of conclusions which -are true—their truth depends on the particular combinations -of the premisses and figures which they accompany. -The total figure thus represented may be called the -universe of thought in respect to these four constituents, -and out of the universe of possibly existing combinations -it is the province of logic to select those which correspond -to the results of our reasoning faculties.</p> - -<p>We can go over each of the premisses in each of the -moods, and find out what conclusion logically follows. -But this is done in the works on logic; most simply and -clearly I believe in “Jevon’s Logic.” As we are only concerned -with a formal presentation of the results we will -make use of the mnemonic lines printed below, in which -the words enclosed in brackets refer to the figures, and -are not significative:—</p> - -<ul> -<li>Barbara celarent Darii ferio<i>que</i> [prioris].</li> -<li>Caesare Camestris Festino Baroko [secundae].</li> -<li>[Tertia] darapti disamis datisi felapton.</li> -<li>Bokardo ferisson <i>habet</i> [Quarta insuper addit].</li> -<li>Bramantip camenes dimaris ferapton fresison.</li> -</ul> - -<p>In these lines each significative word has three vowels, -the first vowel refers to the major premiss, and gives the -mood of that premiss, “a” signifying, for instance, that -the major mood is in mood <i>a</i>. The second vowel refers -to the minor premiss, and gives its mood. The third -vowel refers to the conclusion, and gives its mood. Thus -(prioris)—of the first figure—the first mnemonic word is -“barbara,” and this gives major premiss, mood <span class="allsmcap">A</span>; minor -premiss, mood <span class="allsmcap">A</span>; conclusion, mood <span class="allsmcap">A</span>. Accordingly in the -first of our four cubes we mark the lowest left-hand front -cube. To take another instance in the third figure “Tertia,” -the word “ferisson” gives us major premiss mood <span class="allsmcap">E</span>—<i>e.g.</i>, -no <span class="allsmcap">M</span> is <span class="allsmcap">P</span>, minor premiss mood <span class="allsmcap">I</span>; some <span class="allsmcap">M</span> is <span class="allsmcap">S</span>, conclusion, -mood <span class="allsmcap">O</span>; some <span class="allsmcap">S</span> is not <span class="allsmcap">P</span>. The region to be marked then<span class="pagenum" id="Page_102">[Pg 102]</span> -in the third representative cube is the one in the second -wall to the right for the major premiss, the third wall -from the front for the minor premiss, and the top layer -for the conclusion.</p> - -<p>It is easily seen that in the diagram this cube is -marked, and so with all the valid conclusions. The -regions marked in the total region show which combinations -of the four variables, major premiss, minor -premiss, figure, and conclusion exist.</p> - -<p>That is to say, we objectify all possible conclusions, and -build up an ideal manifold, containing all possible combinations -of them with the premisses, and then out of -this we eliminate all that do not satisfy the laws of logic. -The residue is the syllogism, considered as a canon of -reasoning.</p> - -<p>Looking at the shape which represents the totality -of the valid conclusions, it does not present any obvious -symmetry, or easily characterisable nature. A striking -configuration, however, is obtained, if we project the four-dimensional -figure obtained into a three-dimensional one; -that is, if we take in the base cube all those cubes which -have a marked space anywhere in the series of four -regions which start from that cube.</p> - -<p>This corresponds to making abstraction of the figures, -giving all the conclusions which are valid whatever the -figure may be.</p> - -<div class="figcenter illowp25" id="fig_57" style="max-width: 12.5em;"> - <img src="images/fig_57.png" alt="" /> - <div class="caption">Fig. 57.</div> -</div> - -<p>Proceeding in this way we obtain the arrangement of -marked cubes shown in <a href="#fig_57">fig. 57</a>. We see -that the valid conclusions are arranged -almost symmetrically round one cube—the -one on the top of the column starting from -<span class="allsmcap">AAA</span>. There is one breach of continuity -however in this scheme. One cube is -unmarked, which if marked would give -symmetry. It is the one which would be denoted by the<span class="pagenum" id="Page_103">[Pg 103]</span> -letters <span class="allsmcap">I</span>, <span class="allsmcap">E</span>, <span class="allsmcap">O</span>, in the third wall to the right, the second -wall away, the topmost layer. Now this combination of -premisses in the mood <span class="allsmcap">IE</span>, with a conclusion in the mood -<span class="allsmcap">O</span>, is not noticed in any book on logic with which I am -familiar. Let us look at it for ourselves, as it seems -that there must be something curious in connection with -this break of continuity in the poiograph.</p> - -<div class="figcenter illowp100" id="fig_58" style="max-width: 62.5em;"> - <img src="images/fig_58.png" alt="" /> - <div class="caption">Fig. 58.</div> -</div> - -<p>The propositions <span class="allsmcap">I</span>, <span class="allsmcap">E</span>, in the various figures are the -following, as shown in the accompanying scheme, <a href="#fig_58">fig. 58</a>:—First -figure: some <span class="allsmcap">M</span> is <span class="allsmcap">P</span>; no <span class="allsmcap">S</span> is <span class="allsmcap">M</span>. Second figure: -some <span class="allsmcap">P</span> is <span class="allsmcap">M</span>; no <span class="allsmcap">S</span> is <span class="allsmcap">M</span>. Third figure: some <span class="allsmcap">M</span> is <span class="allsmcap">P</span>; no -<span class="allsmcap">M</span> is <span class="allsmcap">S</span>. Fourth figure: some <span class="allsmcap">P</span> is <span class="allsmcap">M</span>; no <span class="allsmcap">M</span> is <span class="allsmcap">S</span>.</p> - -<p>Examining these figures, we see, taking the first, that -if some <span class="allsmcap">M</span> is <span class="allsmcap">P</span> and no <span class="allsmcap">S</span> is <span class="allsmcap">M</span>, we have no conclusion of<span class="pagenum" id="Page_104">[Pg 104]</span> -the form <span class="allsmcap">S</span> is <span class="allsmcap">P</span> in the various moods. It is quite indeterminate -how the circle representing <span class="allsmcap">S</span> lies with regard -to the circle representing <span class="allsmcap">P</span>. It may lie inside, outside, -or partly inside <span class="allsmcap">P</span>. The same is true in the other figures -2 and 3. But when we come to the fourth figure, since -<span class="allsmcap">M</span> and <span class="allsmcap">S</span> lie completely outside each other, there cannot -lie inside <span class="allsmcap">S</span> that part of <span class="allsmcap">P</span> which lies inside <span class="allsmcap">M</span>. Now -we know by the major premiss that some of <span class="allsmcap">P</span> does lie -in <span class="allsmcap">M</span>. Hence <span class="allsmcap">S</span> cannot contain the whole of <span class="allsmcap">P</span>. In -words, some <span class="allsmcap">P</span> is <span class="allsmcap">M</span>, no <span class="allsmcap">M</span> is <span class="allsmcap">S</span>, therefore <span class="allsmcap">S</span> does not contain -the whole of <span class="allsmcap">P</span>. If we take <span class="allsmcap">P</span> as the subject, this gives -us a conclusion in the mood <span class="allsmcap">O</span> about <span class="allsmcap">P</span>. Some <span class="allsmcap">P</span> is not <span class="allsmcap">S</span>. -But it does not give us conclusion about <span class="allsmcap">S</span> in any one -of the four forms recognised in the syllogism and called -its moods. Hence the breach of the continuity in the -poiograph has enabled us to detect a lack of completeness -in the relations which are considered in the syllogism.</p> - -<p>To take an instance:—Some Americans (<span class="allsmcap">P</span>) are of -African stock (<span class="allsmcap">M</span>); No Aryans (<span class="allsmcap">S</span>) are of African stock -(<span class="allsmcap">M</span>); Aryans (<span class="allsmcap">S</span>) do not include all of Americans (<span class="allsmcap">P</span>).</p> - -<p>In order to draw a conclusion about <span class="allsmcap">S</span> we have to admit -the statement, “<span class="allsmcap">S</span> does not contain the whole of <span class="allsmcap">P</span>,” as -a valid logical form—it is a statement about <span class="allsmcap">S</span> which can -be made. The logic which gives us the form, “some <span class="allsmcap">P</span> -is not <span class="allsmcap">S</span>,” and which does not allow us to give the exactly -equivalent and equally primary form, “<span class="allsmcap">S</span> does not contain -the whole of <span class="allsmcap">P</span>,” is artificial.</p> - -<p>And I wish to point out that this artificiality leads -to an error.</p> - -<p>If one trusted to the mnemonic lines given above, one -would conclude that no logical conclusion about <span class="allsmcap">S</span> can -be drawn from the statement, “some <span class="allsmcap">P</span> are <span class="allsmcap">M</span>, no <span class="allsmcap">M</span> are <span class="allsmcap">S</span>.”</p> - -<p>But a conclusion can be drawn: <span class="allsmcap">S</span> does not contain -the whole of <span class="allsmcap">P</span>.</p> - -<p>It is not that the result is given expressed in another<span class="pagenum" id="Page_105">[Pg 105]</span> -form. The mnemonic lines deny that any conclusion -can be drawn from premisses in the moods <span class="allsmcap">I</span>, <span class="allsmcap">E</span>, respectively.</p> - -<p>Thus a simple four-dimensional poiograph has enabled -us to detect a mistake in the mnemonic lines which have -been handed down unchallenged from mediæval times. -To discuss the subject of these lines more fully a logician -defending them would probably say that a particular -statement cannot be a major premiss; and so deny the -existence of the fourth figure in the combination of moods.</p> - -<p>To take our instance: some Americans are of African -stock; no Aryans are of African stock. He would say -that the conclusion is some Americans are not Aryans; -and that the second statement is the major. He would -refuse to say anything about Aryans, condemning us to -an eternal silence about them, as far as these premisses -are concerned! But, if there is a statement involving -the relation of two classes, it must be expressible as a -statement about either of them.</p> - -<p>To bar the conclusion, “Aryans do not include the -whole of Americans,” is purely a makeshift in favour of -a false classification.</p> - -<p>And the argument drawn from the universality of the -major premiss cannot be consistently maintained. It -would preclude such combinations as major <span class="allsmcap">O</span>, minor <span class="allsmcap">A</span>, -conclusion <span class="allsmcap">O</span>—<i>i.e.</i>, such as some mountains (<span class="allsmcap">M</span>) are not -permanent (<span class="allsmcap">P</span>); all mountains (<span class="allsmcap">M</span>) are scenery (<span class="allsmcap">S</span>); some -scenery (<span class="allsmcap">S</span>) is not permanent (<span class="allsmcap">P</span>).</p> - -<p>This is allowed in “Jevon’s Logic,” and his omission to -discuss <span class="allsmcap">I</span>, <span class="allsmcap">E</span>, <span class="allsmcap">O</span>, in the fourth figure, is inexplicable. A -satisfactory poiograph of the logical scheme can be made -by admitting the use of the words some, none, or all, -about the predicate as well as about the subject. Then -we can express the statement, “Aryans do not include the -whole of Americans,” clumsily, but, when its obscurity -is fathomed, correctly, as “Some Aryans are not all<span class="pagenum" id="Page_106">[Pg 106]</span> -Americans.” And this method is what is called the -“quantification of the predicate.”</p> - -<p>The laws of formal logic are coincident with the conclusions -which can be drawn about regions of space, which -overlap one another in the various possible ways. It is -not difficult so to state the relations or to obtain a -symmetrical poiograph. But to enter into this branch of -geometry is beside our present purpose, which is to show -the application of the poiograph in a finite and limited -region, without any of those complexities which attend its -use in regard to natural objects.</p> - -<p>If we take the latter—plants, for instance—and, without -assuming fixed directions in space as representative of -definite variations, arrange the representative points in -such a manner as to correspond to the similarities of the -objects, we obtain configuration of singular interest; and -perhaps in this way, in the making of shapes of shapes, -bodies with bodies omitted, some insight into the structure -of the species and genera might be obtained.</p> -<hr class="chap x-ebookmaker-drop" /> - -<div class="chapter"> -<p><span class="pagenum" id="Page_107">[Pg 107]</span></p> - -<h2 class="nobreak" id="CHAPTER_IX">CHAPTER IX<br /> - -<small><i>APPLICATION TO KANT’S THEORY OF -EXPERIENCE</i></small></h2></div> - - -<p>When we observe the heavenly bodies we become aware -that they all participate in one universal motion—a -diurnal revolution round the polar axis.</p> - -<p>In the case of fixed stars this is most unqualifiedly true, -but in the case of the sun, and the planets also, the single -motion of revolution can be discerned, modified, and -slightly altered by other and secondary motions.</p> - -<p>Hence the universal characteristic of the celestial bodies -is that they move in a diurnal circle.</p> - -<p>But we know that this one great fact which is true of -them all has in reality nothing to do with them. The -diurnal revolution which they visibly perform is the result -of the condition of the observer. It is because the -observer is on a rotating earth that a universal statement -can be made about all the celestial bodies.</p> - -<p>The universal statement which is valid about every one -of the celestial bodies is that which does not concern -them at all, and is but a statement of the condition of -the observer.</p> - -<p>Now there are universal statements of other kinds -which we can make. We can say that all objects of -experience are in space and subject to the laws of -geometry.</p> - -<p><span class="pagenum" id="Page_108">[Pg 108]</span></p> - -<p>Does this mean that space and all that it means is due -to a condition of the observer?</p> - -<p>If a universal law in one case means nothing affecting -the objects themselves, but only a condition of observation, -is this true in every case? There is shown us in -astronomy a <i>vera causa</i> for the assertion of a universal. -Is the same cause to be traced everywhere?</p> - -<p>Such is a first approximation to the doctrine of Kant’s -critique.</p> - -<p>It is the apprehension of a relation into which, on the -one side and the other, perfectly definite constituents -enter—the human observer and the stars—and a transference -of this relation to a region in which the constituents -on either side are perfectly unknown.</p> - -<p>If spatiality is due to a condition of the observer, the -observer cannot be this bodily self of ours—the body, like -the objects around it, are equally in space.</p> - -<p>This conception Kant applied, not only to the intuitions -of sense, but to the concepts of reason—wherever a universal -statement is made there is afforded him an opportunity -for the application of his principle. He constructed a -system in which one hardly knows which the most to -admire, the architectonic skill, or the reticence with regard -to things in themselves, and the observer in himself.</p> - -<p>His system can be compared to a garden, somewhat -formal perhaps, but with the charm of a quality more -than intellectual, a <i>besonnenheit</i>, an exquisite moderation -over all. And from the ground he so carefully prepared -with that buried in obscurity, which it is fitting should -be obscure, science blossoms and the tree of real knowledge -grows.</p> - -<p>The critique is a storehouse of ideas of profound interest. -The one of which I have given a partial statement leads, -as we shall see on studying it in detail, to a theory of -mathematics suggestive of enquiries in many directions.</p> - -<p><span class="pagenum" id="Page_109">[Pg 109]</span></p> - -<p>The justification for my treatment will be found -amongst other passages in that part of the transcendental -analytic, in which Kant speaks of objects of experience -subject to the forms of sensibility, not subject to the -concepts of reason.</p> - -<p>Kant asserts that whenever we think we think of -objects in space and time, but he denies that the space -and time exist as independent entities. He goes about -to explain them, and their universality, not by assuming -them, as most other philosophers do, but by postulating -their absence. How then does it come to pass that the -world is in space and time to us?</p> - -<p>Kant takes the same position with regard to what we -call nature—a great system subject to law and order. -“How do you explain the law and order in nature?” we -ask the philosophers. All except Kant reply by assuming -law and order somewhere, and then showing how we can -recognise it.</p> - -<p>In explaining our notions, philosophers from other than -the Kantian standpoint, assume the notions as existing -outside us, and then it is no difficult task to show how -they come to us, either by inspiration or by observation.</p> - -<p>We ask “Why do we have an idea of law in nature?” -“Because natural processes go according to law,” we are -answered, “and experience inherited or acquired, gives us -this notion.”</p> - -<p>But when we speak about the law in nature we are -speaking about a notion of our own. So all that these -expositors do is to explain our notion by an assumption -of it.</p> - -<p>Kant is very different. He supposes nothing. An experience -such as ours is very different from experience -in the abstract. Imagine just simply experience, succession -of states, of consciousness! Why, there would -be no connecting any two together, there would be no<span class="pagenum" id="Page_110">[Pg 110]</span> -personal identity, no memory. It is out of a general -experience such as this, which, in respect to anything we -call real, is less than a dream, that Kant shows the -genesis of an experience such as ours.</p> - -<p>Kant takes up the problem of the explanation of space, -time, order, and so quite logically does not presuppose -them.</p> - -<p>But how, when every act of thought is of things in -space, and time, and ordered, shall we represent to ourselves -that perfectly indefinite somewhat which is Kant’s -necessary hypothesis—that which is not in space or time -and is not ordered. That is our problem, to represent -that which Kant assumes not subject to any of our forms -of thought, and then show some function which working -on that makes it into a “nature” subject to law and -order, in space and time. Such a function Kant calls the -“Unity of Apperception”; <i>i.e.</i>, that which makes our state -of consciousness capable of being woven into a system -with a self, an outer world, memory, law, cause, and order.</p> - -<p>The difficulty that meets us in discussing Kant’s -hypothesis is that everything we think of is in space -and time—how then shall we represent in space an existence -not in space, and in time an existence not in time? -This difficulty is still more evident when we come to -construct a poiograph, for a poiograph is essentially a -space structure. But because more evident the difficulty -is nearer a solution. If we always think in space, <i>i.e.</i> -using space concepts, the first condition requisite for -adapting them to the representation of non-spatial existence, -is to be aware of the limitation of our thought, -and so be able to take the proper steps to overcome it. -The problem before us, then, is to represent in space an -existence not in space.</p> - -<p>The solution is an easy one. It is provided by the -conception of alternativity.</p> - -<p><span class="pagenum" id="Page_111">[Pg 111]</span></p> - -<p>To get our ideas clear let us go right back behind the -distinctions of an inner and an outer world. Both of -these, Kant says, are products. Let us take merely states -of consciousness, and not ask the question whether they are -produced or superinduced—to ask such a question is to -have got too far on, to have assumed something of which -we have not traced the origin. Of these states let us -simply say that they occur. Let us now use the word -a “posit” for a phase of consciousness reduced to its -last possible stage of evanescence; let a posit be that -phase of consciousness of which all that can be said is -that it occurs.</p> - -<p>Let <i>a</i>, <i>b</i>, <i>c</i>, be three such posits. We cannot represent -them in space without placing them in a certain order, -as <i>a</i>, <i>b</i>, <i>c</i>. But Kant distinguishes between the forms -of sensibility and the concepts of reason. A dream in -which everything happens at haphazard would be an -experience subject to the form of sensibility and only -partially subject to the concepts of reason. It is partially -subject to the concepts of reason because, although -there is no order of sequence, still at any given time -there is order. Perception of a thing as in space is a -form of sensibility, the perception of an order is a concept -of reason.</p> - -<p>We must, therefore, in order to get at that process -which Kant supposes to be constitutive of an ordered -experience imagine the posits as in space without -order.</p> - -<p>As we know them they must be in some order, <i>abc</i>, -<i>bca</i>, <i>cab</i>, <i>acb</i>, <i>cba</i>, <i>bac</i>, one or another.</p> - -<p>To represent them as having no order conceive all -these different orders as equally existing. Introduce the -conception of alternativity—let us suppose that the order -<i>abc</i>, and <i>bac</i>, for example, exist equally, so that we -cannot say about <i>a</i> that it comes before or after <i>b</i>. This<span class="pagenum" id="Page_112">[Pg 112]</span> -would correspond to a sudden and arbitrary change of <i>a</i> -into <i>b</i> and <i>b</i> into <i>a</i>, so that, to use Kant’s words, it would -be possible to call one thing by one name at one time -and at another time by another name.</p> - -<p>In an experience of this kind we have a kind of chaos, -in which no order exists; it is a manifold not subject to -the concepts of reason.</p> - -<p>Now is there any process by which order can be introduced -into such a manifold—is there any function of -consciousness in virtue of which an ordered experience -could arise?</p> - -<p>In the precise condition in which the posits are, as -described above, it does not seem to be possible. But -if we imagine a duality to exist in the manifold, a -function of consciousness can be easily discovered which -will produce order out of no order.</p> - -<p>Let us imagine each posit, then, as having, a dual aspect. -Let <i>a</i> be 1<i>a</i> in which the dual aspect is represented by the -combination of symbols. And similarly let <i>b</i> be 2<i>b</i>, -<i>c</i> be 3<i>c</i>, in which 2 and <i>b</i> represent the dual aspects -of <i>b</i>, 3 and <i>c</i> those of <i>c</i>.</p> - -<p>Since <i>a</i> can arbitrarily change into <i>b</i>, or into <i>c</i>, and -so on, the particular combinations written above cannot -be kept. We have to assume the equally possible occurrence -of form such as 2<i>a</i>, 2<i>b</i>, and so on; and in order -to get a representation of all those combinations out of -which any set is alternatively possible, we must take -every aspect with every aspect. We must, that is, have -every letter with every number.</p> - -<p>Let us now apply the method of space representation.</p> - -<div class="blockquote"> - -<p><i>Note.</i>—At the beginning of the next chapter the same -structures as those which follow are exhibited in -more detail and a reference to them will remove -any obscurity which may be found in the immediately -following passages. They are there carried</p> - -<p><span class="pagenum" id="Page_113">[Pg 113]</span></p> - -<p>on to a greater multiplicity of dimensions, and the -significance of the process here briefly explained -becomes more apparent.</p> -</div> -<div class="figleft illowp25" id="fig_59" style="max-width: 12.5em;"> - <img src="images/fig_59.png" alt="" /> - <div class="caption">Fig. 59.</div> -</div> - -<p>Take three mutually rectangular axes in space 1, 2, 3 -(<a href="#fig_59">fig. 59</a>), and on each mark three points, -the common meeting point being the -first on each axis. Then by means of -these three points on each axis we -define 27 positions, 27 points in a -cubical cluster, shown in <a href="#fig_60">fig. 60</a>, the -same method of co-ordination being -used as has been described before. -Each of these positions can be named by means of the -axes and the points combined.</p> - -<div class="figleft illowp30" id="fig_60" style="max-width: 18.75em;"> - <img src="images/fig_60.png" alt="" /> - <div class="caption">Fig. 60.</div> -</div> - - -<p>Thus, for instance, the one marked by an asterisk can -be called 1<i>c</i>, 2<i>b</i>, 3<i>c</i>, because it is -opposite to <i>c</i> on 1, to <i>b</i> on 2, to -<i>c</i> on 3.</p> - -<p>Let us now treat of the states of -consciousness corresponding to these -positions. Each point represents a -composite of posits, and the manifold -of consciousness corresponding -to them is of a certain complexity.</p> - -<p>Suppose now the constituents, the points on the axes, -to interchange arbitrarily, any one to become any other, -and also the axes 1, 2, and 3, to interchange amongst -themselves, any one to become any other, and to be subject -to no system or law, that is to say, that order does -not exist, and that the points which run <i>abc</i> on each axis -may run <i>bac</i>, and so on.</p> - -<p>Then any one of the states of consciousness represented -by the points in the cluster can become any other. We -have a representation of a random consciousness of a -certain degree of complexity.</p> - -<p><span class="pagenum" id="Page_114">[Pg 114]</span></p> - -<p>Now let us examine carefully one particular case of -arbitrary interchange of the points, <i>a</i>, <i>b</i>, <i>c</i>; as one such -case, carefully considered, makes the whole clear.</p> - -<div class="figleft illowp40" id="fig_61" style="max-width: 15.625em;"> - <img src="images/fig_61.png" alt="" /> - <div class="caption">Fig. 61.</div> -</div> - -<p>Consider the points named in the figure 1<i>c</i>, 2<i>a</i>, 3<i>c</i>; -1<i>c</i>, 2<i>c</i>, 3<i>a</i>; 1<i>a</i>, 2<i>c</i>, 3<i>c</i>, and -examine the effect on them -when a change of order takes -place. Let us suppose, for -instance, that <i>a</i> changes into <i>b</i>, -and let us call the two sets of -points we get, the one before -and the one after, their change -conjugates.</p> - - -<table class="standard" summary=""> -<tr> -<td class="tdl">Before the change</td> - -<td class="tdl">1<i>c</i> 2<i>a</i> 3<i>c</i></td> -<td class="tdlp">1<i>c</i> 2<i>c</i> 3<i>a</i></td> -<td class="tdlp">1<i>a</i> 2<i>c</i> 3<i>c</i></td> -<td class="tdl" rowspan="2">} Conjugates.</td> -</tr> -<tr> -<td class="tdl">After the change</td> -<td class="tdl">1<i>c</i> 2<i>b</i> 3<i>c</i></td> -<td class="tdlp">1<i>c</i> 2<i>c</i> 3<i>b</i></td> -<td class="tdlp">1<i>b</i> 2<i>c</i> 3<i>c</i></td> -</tr> -</table> - -<p>The points surrounded by rings represent the conjugate -points.</p> - -<p>It is evident that as consciousness, represented first by -the first set of points and afterwards by the second set of -points, would have nothing in common in its two phases. -It would not be capable of giving an account of itself. -There would be no identity.</p> - -<div class="figleft illowp35" id="fig_62" style="max-width: 18.75em;"> - <img src="images/fig_62.png" alt="" /> - <div class="caption">Fig. 62.</div> -</div> - -<p>If, however, we can find any set of points in the -cubical cluster, which, when any arbitrary change takes -place in the points on the axes, or in the axes themselves, -repeats itself, is reproduced, then a consciousness represented -by those points would have a permanence. It -would have a principle of identity. Despite the no law, -the no order, of the ultimate constituents, it would have -an order, it would form a system, the condition of a -personal identity would be fulfilled.</p> - -<p>The question comes to this, then. Can we find a -system of points which is self-conjugate which is such -that when any posit on the axes becomes any other, or<span class="pagenum" id="Page_115">[Pg 115]</span> -when any axis becomes any other, such a set is transformed -into itself, its identity -is not submerged, but rises -superior to the chaos of its -constituents?</p> - -<p>Such a set can be found. -Consider the set represented -in <a href="#fig_62">fig. 62</a>, and written down in -the first of the two lines—</p> - - -<table class="standard" summary=""> -<tr> -<td class="tdl" rowspan="2">Self-<br />conjugate</td> -<td class="tdl" rowspan="2">{</td> -<td class="tdl">1<i>a</i> 2<i>b</i> 3<i>c</i></td> -<td class="tdlp">1<i>b</i> 2<i>a</i> 3<i>c</i></td> -<td class="tdlp">1<i>c</i> 2<i>a</i> 3<i>b</i></td> -<td class="tdlp">1<i>c</i> 2<i>b</i> 3<i>a</i></td> -<td class="tdlp">1<i>b</i> 2<i>c</i> 3<i>a</i></td> -<td class="tdlp">1<i>a</i> 2<i>c</i> 3<i>b</i></td> -</tr> -<tr> -<td class="tdl">1<i>c</i> 2<i>b</i> 3<i>a</i></td> -<td class="tdlp">1<i>b</i> 2<i>c</i> 3<i>a</i></td> -<td class="tdlp">1<i>a</i> 2<i>c</i> 3<i>b</i></td> -<td class="tdlp">1<i>a</i> 2<i>b</i> 3<i>c</i></td> -<td class="tdlp">1<i>b</i> 2<i>a</i> 3<i>c</i></td> -<td class="tdlp">1<i>c</i> 2<i>a</i> 3<i>b</i></td> -</tr> -</table> - -<p>If now <i>a</i> change into <i>c</i> and <i>c</i> into <i>a</i>, we get the set in -the second line, which has the same members as are in the -upper line. Looking at the diagram we see that it would -correspond simply to the turning of the figures as a -whole.<a id="FNanchor_2" href="#Footnote_2" class="fnanchor">[2]</a> Any arbitrary change of the points on the axes, -or of the axes themselves, reproduces the same set.</p> - -<div class="footnotes"> -<div class="footnote"> - -<p><a id="Footnote_2" href="#FNanchor_2" class="label">[2]</a> These figures are described more fully, and extended, in the next -chapter.</p> - -</div></div> - -<p>Thus, a function, by which a random, an unordered, consciousness -could give an ordered and systematic one, can -be represented. It is noteworthy that it is a system of -selection. If out of all the alternative forms that only is -attended to which is self-conjugate, an ordered consciousness -is formed. A selection gives a feature of permanence.</p> - -<p>Can we say that the permanent consciousness is this -selection?</p> - -<p>An analogy between Kant and Darwin comes into light. -That which is swings clear of the fleeting, in virtue of its -presenting a feature of permanence. There is no need -to suppose any function of “attending to.” A consciousness -capable of giving an account of itself is one -which is characterised by this combination. All combinations -exist—of this kind is the consciousness which -can give an account of itself. And the very duality which<span class="pagenum" id="Page_116">[Pg 116]</span> -we have presupposed may be regarded as originated by -a process of selection.</p> - -<p>Darwin set himself to explain the origin of the fauna -and flora of the world. He denied specific tendencies. -He assumed an indefinite variability—that is, chance—but -a chance confined within narrow limits as regards the -magnitude of any consecutive variations. He showed that -organisms possessing features of permanence, if they -occurred would be preserved. So his account of any -structure or organised being was that it possessed features -of permanence.</p> - -<p>Kant, undertaking not the explanation of any particular -phenomena but of that which we call nature as a whole, -had an origin of species of his own, an account of the -flora and fauna of consciousness. He denied any specific -tendency of the elements of consciousness, but taking our -own consciousness, pointed out that in which it resembled -any consciousness which could survive, which could give -an account of itself.</p> - -<p>He assumes a chance or random world, and as great -and small were not to him any given notions of which he -could make use, he did not limit the chance, the randomness, -in any way. But any consciousness which is permanent -must possess certain features—those attributes -namely which give it permanence. Any consciousness -like our own is simply a consciousness which possesses -those attributes. The main thing is that which he calls -the unity of apperception, which we have seen above is -simply the statement that a particular set of phases of -consciousness on the basis of complete randomness will be -self-conjugate, and so permanent.</p> - -<p>As with Darwin so with Kant, the reason for existence -of any feature comes to this—show that it tends to the -permanence of that which possesses it.</p> - -<p>We can thus regard Kant as the creator of the first of<span class="pagenum" id="Page_117">[Pg 117]</span> -the modern evolution theories. And, as is so often the -case, the first effort was the most stupendous in its scope. -Kant does not investigate the origin of any special part -of the world, such as its organisms, its chemical elements, -its social communities of men. He simply investigates -the origin of the whole—of all that is included in consciousness, -the origin of that “thought thing” whose -progressive realisation is the knowable universe.</p> - -<p>This point of view is very different from the ordinary -one, in which a man is supposed to be placed in a world -like that which he has come to think of it, and then to -learn what he has found out from this model which he -himself has placed on the scene.</p> - -<p>We all know that there are a number of questions in -attempting an answer to which such an assumption is not -allowable.</p> - -<p>Mill, for instance, explains our notion of “law” by an -invariable sequence in nature. But what we call nature -is something given in thought. So he explains a thought -of law and order by a thought of an invariable sequence. -He leaves the problem where he found it.</p> - -<p>Kant’s theory is not unique and alone. It is one of -a number of evolution theories. A notion of its import -and significance can be obtained by a comparison of it -with other theories.</p> - -<p>Thus in Darwin’s theoretical world of natural selection -a certain assumption is made, the assumption of indefinite -variability—slight variability it is true, over any appreciable -lapse of time, but indefinite in the postulated -epochs of transformation—and a whole chain of results -is shown to follow.</p> - -<p>This element of chance variation is not, however, an -ultimate resting place. It is a preliminary stage. This -supposing the all is a preliminary step towards finding -out what is. If every kind of organism can come into<span class="pagenum" id="Page_118">[Pg 118]</span> -being, those that do survive will present such and such -characteristics. This is the necessary beginning for ascertaining -what kinds of organisms do come into existence. -And so Kant’s hypothesis of a random consciousness is -the necessary beginning for the rational investigation -of consciousness as it is. His assumption supplies, as -it were, the space in which we can observe the phenomena. -It gives the general laws constitutive of any -experience. If, on the assumption of absolute randomness -in the constituents, such and such would be -characteristic of the experience, then, whatever the constituents, -these characteristics must be universally valid.</p> - -<p>We will now proceed to examine more carefully the -poiograph, constructed for the purpose of exhibiting an -illustration of Kant’s unity of apperception.</p> - -<p>In order to show the derivation order out of non-order -it has been necessary to assume a principle of duality—we -have had the axes and the posits on the axes—there -are two sets of elements, each non-ordered, and it is in -the reciprocal relation of them that the order, the definite -system, originates.</p> - -<p>Is there anything in our experience of the nature of a -duality?</p> - -<p>There certainly are objects in our experience which -have order and those which are incapable of order. The -two roots of a quadratic equation have no order. No one -can tell which comes first. If a body rises vertically and -then goes at right angles to its former course, no one can -assign any priority to the direction of the north or to the -east. There is no priority in directions of turning. We -associate turnings with no order progressions in a line -with order. But in the axes and points we have assumed -above there is no such distinction. It is the same, whether -we assume an order among the turnings, and no order -among the points on the axes, or, <i>vice versa</i>, an order in<span class="pagenum" id="Page_119">[Pg 119]</span> -the points and no order in the turnings. A being with -an infinite number of axes mutually at right angles, -with a definite sequence between them and no sequence -between the points on the axes, would be in a condition -formally indistinguishable from that of a creature who, -according to an assumption more natural to us, had on -each axis an infinite number of ordered points and no -order of priority amongst the axes. A being in such -a constituted world would not be able to tell which -was turning and which was length along an axis, in -order to distinguish between them. Thus to take a pertinent -illustration, we may be in a world of an infinite -number of dimensions, with three arbitrary points on -each—three points whose order is indifferent, or in a -world of three axes of arbitrary sequence with an infinite -number of ordered points on each. We can’t tell which -is which, to distinguish it from the other.</p> - -<p>Thus it appears the mode of illustration which we -have used is not an artificial one. There really exists -in nature a duality of the kind which is necessary to -explain the origin of order out of no order—the duality, -namely, of dimension and position. Let us use the term -group for that system of points which remains unchanged, -whatever arbitrary change of its constituents takes place. -We notice that a group involves a duality, is inconceivable -without a duality.</p> - -<p>Thus, according to Kant, the primary element of experience -is the group, and the theory of groups would be -the most fundamental branch of science. Owing to an -expression in the critique the authority of Kant is sometimes -adduced against the assumption of more than three -dimensions to space. It seems to me, however, that the -whole tendency of his theory lies in the opposite direction, -and points to a perfect duality between dimension and -position in a dimension.</p> - -<p><span class="pagenum" id="Page_120">[Pg 120]</span></p> - -<p>If the order and the law we see is due to the conditions -of conscious experience, we must conceive nature as -spontaneous, free, subject to no predication that we can -devise, but, however apprehended, subject to our logic.</p> - -<p>And our logic is simply spatiality in the general sense—that -resultant of a selection of the permanent from the -unpermanent, the ordered from the unordered, by the -means of the group and its underlying duality.</p> - -<p>We can predicate nothing about nature, only about the -way in which we can apprehend nature. All that we can -say is that all that which experience gives us will be conditioned -as spatial, subject to our logic. Thus, in exploring -the facts of geometry from the simplest logical relations -to the properties of space of any number of dimensions, -we are merely observing ourselves, becoming aware of -the conditions under which we must perceive. Do any -phenomena present themselves incapable of explanation -under the assumption of the space we are dealing with, -then we must habituate ourselves to the conception of a -higher space, in order that our logic may be equal to the -task before us.</p> - -<p>We gain a repetition of the thought that came before, -experimentally suggested. If the laws of the intellectual -comprehension of nature are those derived from considering -her as absolute chance, subject to no law save -that derived from a process of selection, then, perhaps, the -order of nature requires different faculties from the intellectual -to apprehend it. The source and origin of -ideas may have to be sought elsewhere than in reasoning.</p> - -<p>The total outcome of the critique is to leave the -ordinary man just where he is, justified in his practical -attitude towards nature, liberated from the fetters of his -own mental representations.</p> - -<p>The truth of a picture lies in its total effect. It is vain -to seek information about the landscape from an examina<span class="pagenum" id="Page_121">[Pg 121]</span>tion -of the pigments. And in any method of thought it -is the complexity of the whole that brings us to a knowledge -of nature. Dimensions are artificial enough, but in -the multiplicity of them we catch some breath of nature.</p> - -<p>We must therefore, and this seems to me the practical -conclusion of the whole matter, proceed to form means of -intellectual apprehension of a greater and greater degree -of complexity, both dimensionally and in extent in any -dimension. Such means of representation must always -be artificial, but in the multiplicity of the elements with -which we deal, however incipiently arbitrary, lies our -chance of apprehending nature.</p> - -<p>And as a concluding chapter to this part of the book, -I will extend the figures, which have been used to represent -Kant’s theory, two steps, so that the reader may -have the opportunity of looking at a four-dimensional -figure which can be delineated without any of the special -apparatus, to the consideration of which I shall subsequently -pass on.</p> -<hr class="chap x-ebookmaker-drop" /> - -<div class="chapter"> -<p><span class="pagenum" id="Page_122">[Pg 122]</span></p> - -<h2 class="nobreak" id="CHAPTER_X">CHAPTER X<br /> - -<small><i>A FOUR-DIMENSIONAL FIGURE</i></small></h2></div> - - -<p>The method used in the preceding chapter to illustrate -the problem of Kant’s critique, gives a singularly easy -and direct mode of constructing a series of important -figures in any number of dimensions.</p> - -<p>We have seen that to represent our space a plane being -must give up one of his axes, and similarly to represent -the higher shapes we must give up one amongst our -three axes.</p> - -<p>But there is another kind of giving up which reduces -the construction of higher shapes to a matter of the -utmost simplicity.</p> - -<p>Ordinarily we have on a straight line any number of -positions. The wealth of space in position is illimitable, -while there are only three dimensions.</p> - -<p>I propose to give up this wealth of positions, and to -consider the figures obtained by taking just as many -positions as dimensions.</p> - -<p>In this way I consider dimensions and positions as two -“kinds,” and applying the simple rule of selecting every -one of one kind with every other of every other kind, -get a series of figures which are noteworthy because -they exactly fill space of any number of dimensions -(as the hexagon fills a plane) by equal repetitions of -themselves.</p> - -<p><span class="pagenum" id="Page_123">[Pg 123]</span></p> - -<p>The rule will be made more evident by a simple -application.</p> - -<p>Let us consider one dimension and one position. I will -call the axis <i>i</i>, and the position <i>o</i>.</p> - -<p class="center"> -———————————————-<i>i</i><br /> -<span style="margin-left: 3.5em;"><i>o</i></span> -</p> - -<p>Here the figure is the position <i>o</i> on the line <i>i</i>. Take -now two dimensions and two positions on each.</p> - -<div class="figleft illowp30" id="fig_63" style="max-width: 12.125em;"> - <img src="images/fig_63.png" alt="" /> - <div class="caption">Fig. 63.</div> -</div> - -<p>We have the two positions <i>o</i>; 1 on <i>i</i>, and the two -positions <i>o</i>, 1 on <i>j</i>, <a href="#fig_63">fig. 63</a>. These give -rise to a certain complexity. I will -let the two lines <i>i</i> and <i>j</i> meet in the -position I call <i>o</i> on each, and I will -consider <i>i</i> as a direction starting equally -from every position on <i>j</i>, and <i>j</i> as -starting equally from every position on <i>i</i>. We thus -obtain the following figure:—<span class="allsmcap">A</span> is both <i>oi</i> and <i>oj</i>, <span class="allsmcap">B</span> is 1<i>i</i> -and <i>oj</i>, and so on as shown in <a href="#fig_63">fig. 63</a><i>b</i>. -The positions on <span class="allsmcap">AC</span> are all <i>oi</i> positions. -They are, if we like to consider it in -that way, points at no distance in the <i>i</i> -direction from the line <span class="allsmcap">AC</span>. We can -call the line <span class="allsmcap">AC</span> the <i>oi</i> line. Similarly -the points on <span class="allsmcap">AB</span> are those no distance -from <span class="allsmcap">AB</span> in the <i>j</i> direction, and we can -call them <i>oj</i> points and the line <span class="allsmcap">AB</span> the <i>oj</i> line. Again, -the line <span class="allsmcap">CD</span> can be called the 1<i>j</i> line because the points -on it are at a distance, 1 in the <i>j</i> direction.</p> - -<div class="figleft illowp30" id="fig_63b" style="max-width: 12.5em;"> - <img src="images/fig_63b.png" alt="" /> - <div class="caption">Fig. 63<i>b</i>.</div> -</div> - -<p>We have then four positions or points named as shown, -and, considering directions and positions as “kinds,” we -have the combination of two kinds with two kinds. Now, -selecting every one of one kind with every other of every -other kind will mean that we take 1 of the kind <i>i</i> and<span class="pagenum" id="Page_124">[Pg 124]</span> -with it <i>o</i> of the kind <i>j</i>; and then, that we take <i>o</i> of the -kind <i>i</i> and with it 1 of the kind <i>j</i>.</p> - -<div class="figleft illowp25" id="fig_64" style="max-width: 12.5em;"> - <img src="images/fig_64.png" alt="" /> - <div class="caption">Fig. 64.</div> -</div> - -<p>Thus we get a pair of positions lying in the straight -line <span class="allsmcap">BC</span>, <a href="#fig_64">fig. 64</a>. We can call this pair 10 -and 01 if we adopt the plan of mentally, -adding an <i>i</i> to the first and a <i>j</i> to the -second of the symbols written thus—01 -is a short expression for O<i>i</i>, 1<i>j</i>.</p> - -<div class="figcenter illowp80" id="fig_65" style="max-width: 62.5em;"> - <img src="images/fig_65.png" alt="" /> - <div class="caption">Fig. 65.</div> -</div> - -<p>Coming now to our space, we have three -dimensions, so we take three positions on each. These -positions I will suppose to be at equal distances along each -axis. The three axes and the three positions on each are -shown in the accompanying diagrams, <a href="#fig_65">fig. 65</a>, of which -the first represents a cube with the front faces visible, the -second the rear faces of the same cube; the positions I -will call 0, 1, 2; the axes, <i>i</i>, <i>j</i>, <i>k</i>. I take the base <span class="allsmcap">ABC</span> as -the starting place, from which to determine distances in -the <i>k</i> direction, and hence every point in the base <span class="allsmcap">ABC</span> -will be an <i>ok</i> position, and the base <span class="allsmcap">ABC</span> can be called an -<i>ok</i> plane.</p> - -<p>In the same way, measuring the distances from the face -<span class="allsmcap">ADC</span>, we see that every position in the face <span class="allsmcap">ADC</span> is an <i>oi</i> -position, and the whole plane of the face may be called an -<i>oi</i> plane. Thus we see that with the introduction of a<span class="pagenum" id="Page_125">[Pg 125]</span> -new dimension the signification of a compound symbol, -such as “<i>oi</i>,” alters. In the plane it meant the line <span class="allsmcap">AC</span>. -In space it means the whole plane <span class="allsmcap">ACD</span>.</p> - -<p>Now, it is evident that we have twenty-seven positions, -each of them named. If the reader will follow this -nomenclature in respect of the positions marked in the -figures he will have no difficulty in assigning names to -each one of the twenty-seven positions. <span class="allsmcap">A</span> is <i>oi</i>, <i>oj</i>, <i>ok</i>. -It is at the distance 0 along <i>i</i>, 0 along <i>j</i>, 0 along <i>k</i>, and -<i>io</i> can be written in short 000, where the <i>ijk</i> symbols -are omitted.</p> - -<p>The point immediately above is 001, for it is no distance -in the <i>i</i> direction, and a distance of 1 in the <i>k</i> -direction. Again, looking at <span class="allsmcap">B</span>, it is at a distance of 2 -from <span class="allsmcap">A</span>, or from the plane <span class="allsmcap">ADC</span>, in the <i>i</i> direction, 0 in the -<i>j</i> direction from the plane <span class="allsmcap">ABD</span>, and 0 in the <i>k</i> direction, -measured from the plane <span class="allsmcap">ABC</span>. Hence it is 200 written -for 2<i>i</i>, 0<i>j</i>, 0<i>k</i>.</p> - -<p>Now, out of these twenty-seven “things” or compounds -of position and dimension, select those which are given by -the rule, every one of one kind with every other of every -other kind.</p> - -<div class="figleft illowp30" id="fig_66" style="max-width: 15.625em;"> - <img src="images/fig_66.png" alt="" /> - <div class="caption">Fig. 66.</div> -</div> - -<p>Take 2 of the <i>i</i> kind. With this -we must have a 1 of the <i>j</i> kind, -and then by the rule we can only -have a 0 of the <i>k</i> kind, for if we -had any other of the <i>k</i> kind we -should repeat one of the kinds we -already had. In 2<i>i</i>, 1<i>j</i>, 1<i>k</i>, for -instance, 1 is repeated. The point -we obtain is that marked 210, <a href="#fig_66">fig. 66</a>.</p> - -<div class="figleft illowp30" id="fig_67" style="max-width: 15.625em;"> - <img src="images/fig_67.png" alt="" /> - <div class="caption">Fig. 67.</div> -</div> - -<p>Proceeding in this way, we pick out the following -cluster of points, <a href="#fig_67">fig. 67</a>. They are joined by lines, -dotted where they are hidden by the body of the cube, -and we see that they form a figure—a hexagon which<span class="pagenum" id="Page_126">[Pg 126]</span> -could be taken out of the cube and placed on a plane. -It is a figure which will fill a -plane by equal repetitions of itself. -The plane being representing this -construction in his plane would -take three squares to represent the -cube. Let us suppose that he -takes the <i>ij</i> axes in his space and -<i>k</i> represents the axis running out -of his space, <a href="#fig_68">fig. 68</a>. In each of -the three squares shown here as drawn separately he -could select the points given by the rule, and he would -then have to try to discover the figure determined by -the three lines drawn. The line from 210 to 120 is -given in the figure, but the line from 201 to 102 or <span class="allsmcap">GK</span> -is not given. He can determine <span class="allsmcap">GK</span> by making another -set of drawings and discovering in them what the relation -between these two extremities is.</p> - -<div class="figcenter illowp100" id="fig_68" style="max-width: 62.5em;"> - <img src="images/fig_68.png" alt="" /> - <div class="caption">Fig. 68.</div> -</div> - -<div class="figcenter illowp80" id="fig_69" style="max-width: 50em;"> - <img src="images/fig_69.png" alt="" /> - <div class="caption">Fig. 69.</div> -</div> - -<p>Let him draw the <i>i</i> and <i>k</i> axes in his plane, <a href="#fig_69">fig. 69</a>. -The <i>j</i> axis then runs out and he has the accompanying -figure. In the first of these three squares, <a href="#fig_69">fig. 69</a>, he can<span class="pagenum" id="Page_127">[Pg 127]</span> -pick out by the rule the two points 201, 102—<span class="allsmcap">G</span>, and <span class="allsmcap">K</span>. -Here they occur in one plane and he can measure the -distance between them. In his first representation they -occur at <span class="allsmcap">G</span> and <span class="allsmcap">K</span> in separate figures.</p> - -<p>Thus the plane being would find that the ends of each -of the lines was distant by the diagonal of a unit square -from the corresponding end of the last and he could then -place the three lines in their right relative position. -Joining them he would have the figure of a hexagon.</p> - -<div class="figleft illowp30" id="fig_70" style="max-width: 15.625em;"> - <img src="images/fig_70.png" alt="" /> - <div class="caption">Fig. 70.</div> -</div> - -<p>We may also notice that the plane being could make -a representation of the whole cube -simultaneously. The three squares, -shown in perspective in <a href="#fig_70">fig. 70</a>, all -lie in one plane, and on these the -plane being could pick out any -selection of points just as well as on -three separate squares. He would -obtain a hexagon by joining the -points marked. This hexagon, as -drawn, is of the right shape, but it would not be so if -actual squares were used instead of perspective, because -the relation between the separate squares as they lie in -the plane figure is not their real relation. The figure, -however, as thus constructed, would give him an idea of -the correct figure, and he could determine it accurately -by remembering that distances in each square were -correct, but in passing from one square to another their -distance in the third dimension had to be taken into -account.</p> - -<p>Coming now to the figure made by selecting according -to our rule from the whole mass of points given by four -axes and four positions in each, we must first draw a -catalogue figure in which the whole assemblage is shown.</p> - -<p>We can represent this assemblage of points by four -solid figures. The first giving all those positions which<span class="pagenum" id="Page_128">[Pg 128]</span> -are at a distance <span class="allsmcap">O</span> from our space in the fourth dimension, -the second showing all those that are at a distance 1, -and so on.</p> - -<p>These figures will each be cubes. The first two are -drawn showing the front faces, the second two the rear -faces. We will mark the points 0, 1, 2, 3, putting points -at those distances along each of these axes, and suppose -all the points thus determined to be contained in solid -models of which our drawings in <a href="#fig_71">fig. 71</a> are representatives. -Here we notice that as on the plane 0<i>i</i> meant -the whole line from which the distances in the <i>i</i> direction -was measured, and as in space 0<i>i</i> means the whole plane -from which distances in the <i>i</i> direction are measured, so -now 0<i>h</i> means the whole space in which the first cube -stands—measuring away from that space by a distance -of one we come to the second cube represented.</p> - -<div class="figcenter illowp80" id="fig_71" style="max-width: 62.5em;"> - <img src="images/fig_71.png" alt="" /> - <div class="caption">Fig. 71.</div> -</div> - -<p><span class="pagenum" id="Page_129">[Pg 129]</span></p> - -<p>Now selecting according to the rule every one of one -kind with every other of every other kind, we must take, -for instance, 3<i>i</i>, 2<i>j</i>, 1<i>k</i>, 0<i>h</i>. This point is marked -3210 at the lower star in the figure. It is 3 in the -<i>i</i> direction, 2 in the <i>j</i> direction, 1 in the <i>k</i> direction, -0 in the <i>h</i> direction.</p> - -<p>With 3<i>i</i> we must also take 1<i>j</i>, 2<i>k</i>, 0<i>h</i>. This point -is shown by the second star in the cube 0<i>h</i>.</p> - -<div class="figcenter illowp80" id="fig_72" style="max-width: 62.5em;"> - <img src="images/fig_72.png" alt="" /> - <div class="caption">Fig. 72.</div> -</div> - -<p>In the first cube, since all the points are 0<i>h</i> points, -we can only have varieties in which <i>i</i>, <i>j</i>, <i>k</i>, are accompanied -by 3, 2, 1.</p> - -<p>The points determined are marked off in the diagram -fig. 72, and lines are drawn joining the adjacent pairs -in each figure, the lines being dotted when they pass -within the substance of the cube in the first two diagrams.</p> - -<p>Opposite each point, on one side or the other of each<span class="pagenum" id="Page_130">[Pg 130]</span> -cube, is written its name. It will be noticed that the -figures are symmetrical right and left; and right and -left the first two numbers are simply interchanged.</p> - -<p>Now this being our selection of points, what figure do -they make when all are put together in their proper -relative positions?</p> - -<p>To determine this we must find the distance between -corresponding corners of the separate hexagons.</p> -<div class="figcenter illowp80" id="fig_73" style="max-width: 62.5em;"> - <img src="images/fig_73.png" alt="" /> - <div class="caption">Fig. 73.</div> -</div> - - -<p>To do this let us keep the axes <i>i</i>, <i>j</i>, in our space, and -draw <i>h</i> instead of <i>k</i>, letting <i>k</i> run out in the fourth -dimension, <a href="#fig_73">fig. 73</a>.</p> - -<div class="figright illowp50" id="fig_74" style="max-width: 37.5em;"> - <img src="images/fig_74.png" alt="" /> - <div class="caption">Fig. 74.</div> -</div> - -<p>Here we have four cubes again, in the first of which all -the points are 0<i>k</i> points; that is, points at a distance zero -in the <i>k</i> direction from the space of the three dimensions -<i>ijh</i>. We have all the points selected before, and some -of the distances, which in the last diagram led from figure -to figure are shown here in the same figure, and so capable<span class="pagenum" id="Page_131">[Pg 131]</span> -of measurement. Take for instance the points 3120 to -3021, which in the first diagram (<a href="#fig_72">fig. 72</a>) lie in the first -and second figures. Their actual relation is shown in -fig. 73 in the cube marked 2<span class="allsmcap">K</span>, where the points in question -are marked with a *. We see that the -distance in question is the diagonal of a unit square. In -like manner we find that the distance between corresponding -points of any two hexagonal figures is the -diagonal of a unit square. The total figure is now easily -constructed. An idea -of it may be gained by -drawing all the four -cubes in the catalogue -figure in one (fig. 74). -These cubes are exact -repetitions of one -another, so one drawing -will serve as a -representation of the -whole series, if we -take care to remember -where we are, whether -in a 0<i>h</i>, a 1<i>h</i>, a 2<i>h</i>, -or a 3<i>h</i> figure, when -we pick out the points required. Fig. 74 is a representation -of all the catalogue cubes put in one. For the -sake of clearness the front faces and the back faces of -this cube are represented separately.</p> - -<p>The figure determined by the selected points is shown -below.</p> - -<p>In putting the sections together some of the outlines -in them disappear. The line <span class="allsmcap">TW</span> for instance is not -wanted.</p> - -<p>We notice that <span class="allsmcap">PQTW</span> and <span class="allsmcap">TWRS</span> are each the half -of a hexagon. Now <span class="allsmcap">QV</span> and <span class="allsmcap">VR</span> lie in one straight line.<span class="pagenum" id="Page_132">[Pg 132]</span> -Hence these two hexagons fit together, forming one -hexagon, and the line <span class="allsmcap">TW</span> is only wanted when we consider -a section of the whole figure, we thus obtain the -solid represented in the lower part of <a href="#fig_74">fig. 74</a>. Equal -repetitions of this figure, called a tetrakaidecagon, will -fill up three-dimensional space.</p> - -<p>To make the corresponding four-dimensional figure we -have to take five axes mutually at right angles with five -points on each. A catalogue of the positions determined -in five-dimensional space can be found thus.</p> -<div class="figleft illowp60" id="fig_75" style="max-width: 37.5em;"> - <img src="images/fig_75.png" alt="" /> - <div class="caption">Fig. 75.</div> -</div> - -<p>Take a cube with five points on each of its axes, the -fifth point is at a distance of four units of length from the -first on any one of the axes. And since the fourth dimension -also stretches to a distance of four we shall need to -represent the successive -sets of points at -distances 0, 1, 2, 3, 4, -in the fourth dimensions, -five cubes. Now -all of these extend to -no distance at all in -the fifth dimension. -To represent what -lies in the fifth dimension -we shall have to -draw, starting from -each of our cubes, five -similar cubes to represent -the four steps -on in the fifth dimension. By this assemblage we get a -catalogue of all the points shown in <a href="#fig_75">fig. 75</a>, in which -<i>L</i> represents the fifth dimension.</p> - -<p>Now, as we saw before, there is nothing to prevent us -from putting all the cubes representing the different -stages in the fourth dimension in one figure, if we take<span class="pagenum" id="Page_133">[Pg 133]</span> -note when we look at it, whether we consider it as a 0<i>h</i>, a -1<i>h</i>, a 2<i>h</i>, etc., cube. Putting then the 0<i>h</i>, 1<i>h</i>, 2<i>h</i>, 3<i>h</i>, 4<i>h</i> -cubes of each row in one, we have five cubes with the sides -of each containing five positions, the first of these five -cubes represents the 0<i>l</i> points, and has in it the <i>i</i> points -from 0 to 4, the <i>j</i> points from 0 to 4, the <i>k</i> points from -0 to 4, while we have to specify with regard to any -selection we make from it, whether we regard it as a 0<i>h</i>, -a 1<i>h</i>, a 2<i>h</i>, a 3<i>h</i>, or a 4<i>h</i> figure. In <a href="#fig_76">fig. 76</a> each cube -is represented by two drawings, one of the front part, the -other of the rear part.</p> - -<p>Let then our five cubes be arranged before us and our -selection be made according to the rule. Take the first -figure in which all points are 0<i>l</i> points. We cannot -have 0 with any other letter. Then, keeping in the first -figure, which is that of the 0<i>l</i> positions, take first of all -that selection which always contains 1<i>h</i>. We suppose, -therefore, that the cube is a 1<i>h</i> cube, and in it we take -<i>i</i>, <i>j</i>, <i>k</i> in combination with 4, 3, 2 according to the rule.</p> - -<p>The figure we obtain is a hexagon, as shown, the one -in front. The points on the right hand have the same -figures as those on the left, with the first two numerals -interchanged. Next keeping still to the 0<i>l</i> figure let -us suppose that the cube before us represents a section -at a distance of 2 in the <i>h</i> direction. Let all the points -in it be considered as 2<i>h</i> points. We then have a 0<i>l</i>, 2<i>h</i> -region, and have the sets <i>ijk</i> and 431 left over. We -must then pick out in accordance with our rule all such -points as 4<i>i</i>, 3<i>j</i>, 1<i>k</i>.</p> - -<p>These are shown in the figure and we find that we can -draw them without confusion, forming the second hexagon -from the front. Going on in this way it will be seen -that in each of the five figures a set of hexagons is picked -out, which put together form a three-space figure something -like the tetrakaidecagon.</p> - -<p><span class="pagenum" id="Page_134">[Pg 134]</span></p> - -<div class="figcenter illowp100" id="fig_76" style="max-width: 93.75em;"> - <img src="images/fig_76.png" alt="" /> - <div class="caption">Fig. 76.</div> -</div> - -<p><span class="pagenum" id="Page_135">[Pg 135]</span></p> - -<p>These separate figures are the successive stages in -which the whole four-dimensional figure in which they -cohere can be apprehended.</p> - -<p>The first figure and the last are tetrakaidecagons. -These are two of the solid boundaries of the figure. The -other solid boundaries can be traced easily. Some of -them are complete from one face in the figure to the -corresponding face in the next, as for instance the solid -which extends from the hexagonal base of the first figure -to the equal hexagonal base of the second figure. This -kind of boundary is a hexagonal prism. The hexagonal -prism also occurs in another sectional series, as for -instance, in the square at the bottom of the first figure, -the oblong at the base of the second and the square at -the base of the third figure.</p> - -<p>Other solid boundaries can be traced through four of -the five sectional figures. Thus taking the hexagon at -the top of the first figure we find in the next a hexagon -also, of which some alternate sides are elongated. The -top of the third figure is also a hexagon with the other -set of alternate rules elongated, and finally we come in -the fourth figure to a regular hexagon.</p> - -<p>These four sections are the sections of a tetrakaidecagon -as can be recognised from the sections of this figure -which we have had previously. Hence the boundaries -are of two kinds, hexagonal prisms and tetrakaidecagons.</p> - -<p>These four-dimensional figures exactly fill four-dimensional -space by equal repetitions of themselves.</p> -<hr class="chap x-ebookmaker-drop" /> - -<div class="chapter"> -<p><span class="pagenum" id="Page_136">[Pg 136]</span></p> - -<h2 class="nobreak" id="CHAPTER_XI">CHAPTER XI<br /> - -<small><i>NOMENCLATURE AND ANALOGIES PRELIMINARY -TO THE STUDY OF FOUR-DIMENSIONAL -FIGURES</i></small></h2></div> - - -<p>In the following pages a method of designating different -regions of space by a systematic colour scheme has been -adopted. The explanations have been given in such a -manner as to involve no reference to models, the diagrams -will be found sufficient. But to facilitate the study a -description of a set of models is given in an appendix -which the reader can either make for himself or obtain. -If models are used the diagrams in Chapters XI. and XII. -will form a guide sufficient to indicate their use. Cubes -of the colours designated by the diagrams should be picked -out and used to reinforce the diagrams. The reader, -in the following description, should -suppose that a board or wall -stretches away from him, against -which the figures are placed.</p> - -<div class="figleft illowp30" id="fig_77" style="max-width: 15.625em;"> - <img src="images/fig_77.png" alt="" /> - <div class="caption">Fig. 77.</div> -</div> - -<p>Take a square, one of those -shown in Fig. 77 and give it a -neutral colour, let this colour be -called “null,” and be such that it -makes no appreciable difference<span class="pagenum" id="Page_137">[Pg 137]</span> -to any colour with which it mixed. If there is no -such real colour let us imagine such a colour, and -assign to it the properties of the number zero, which -makes no difference in any number to which it is -added.</p> - -<p>Above this square place a red square. Thus we symbolise -the going up by adding red to null.</p> - -<p>Away from this null square place a yellow square, and -represent going away by adding yellow to null.</p> - -<div class="figleft illowp40" id="fig_78" style="max-width: 15.625em;"> - <img src="images/fig_78.png" alt="" /> - <div class="caption">Fig. 78.</div> -</div> - -<p>To complete the figure we need a fourth square. -Colour this orange, which is a mixture of red and -yellow, and so appropriately represents a going in a -direction compounded of up and away. We have thus -a colour scheme which will serve to name the set of -squares drawn. We have two axes of colours—red and -yellow—and they may occupy -as in the figure the -direction up and away, or -they may be turned about; -in any case they enable us -to name the four squares -drawn in their relation to -one another.</p> - -<p>Now take, in Fig. 78, -nine squares, and suppose -that at the end of the -going in any direction the -colour started with repeats itself.</p> - -<p>We obtain a square named as shown.</p> - -<p>Let us now, in <a href="#fig_79">fig. 79</a>, suppose the number of squares to -be increased, keeping still to the principle of colouring -already used.</p> - -<p>Here the nulls remain four in number. There -are three reds between the first null and the null -above it, three yellows between the first null and the<span class="pagenum" id="Page_138">[Pg 138]</span> -null beyond it, while the oranges increase in a double -way.</p> - -<div class="figcenter illowp80" id="fig_79" style="max-width: 62.5em;"> - <img src="images/fig_79.png" alt="" /> - <div class="caption">Fig. 79.</div> -</div> - -<p>Suppose this process of enlarging the number of the -squares to be indefinitely pursued and -the total figure obtained to be reduced -in size, we should obtain a square of -which the interior was all orange, -while the lines round it were red and -yellow, and merely the points null -colour, as in <a href="#fig_80">fig. 80</a>. Thus all the points, lines, and the -area would have a colour.</p> - -<div class="figleft illowp25" id="fig_80" style="max-width: 15.625em;"> - <img src="images/fig_80.png" alt="" /> - <div class="caption">Fig. 80.</div> -</div> - - -<p>We can consider this scheme to originate thus:—Let -a null point move in a yellow direction and trace out a -yellow line and end in a null point. Then let the whole -line thus traced move in a red direction. The null points -at the ends of the line will produce red lines, and end in<span class="pagenum" id="Page_139">[Pg 139]</span> -null points. The yellow line will trace out a yellow and -red, or orange square.</p> - -<p>Now, turning back to <a href="#fig_78">fig. 78</a>, we see that these two -ways of naming, the one we started with and the one we -arrived at, can be combined.</p> - -<p>By its position in the group of four squares, in <a href="#fig_77">fig. 77</a>, -the null square has a relation to the yellow and to the red -directions. We can speak therefore of the red line of the -null square without confusion, meaning thereby the line -<span class="allsmcap">AB</span>, <a href="#fig_81">fig. 81</a>, which runs up from the -initial null point <span class="allsmcap">A</span> in the figure as -drawn. The yellow line of the null -square is its lower horizontal line <span class="allsmcap">AC</span> -as it is situated in the figure.</p> - -<div class="figleft illowp30" id="fig_81" style="max-width: 15.625em;"> - <img src="images/fig_81.png" alt="" /> - <div class="caption">Fig. 81.</div> -</div> - -<p>If we wish to denote the upper -yellow line <span class="allsmcap">BD</span>, <a href="#fig_81">fig. 81</a>, we can speak -of it as the yellow γ line, meaning -the yellow line which is separated -from the primary yellow line by the red movement.</p> - -<p>In a similar way each of the other squares has null -points, red and yellow lines. Although the yellow square -is all yellow, its line <span class="allsmcap">CD</span>, for instance, can be referred to as -its red line.</p> - -<p>This nomenclature can be extended.</p> - -<p>If the eight cubes drawn, in <a href="#fig_82">fig. 82</a>, are put close -together, as on the right hand of the diagram, they form -a cube, and in them, as thus arranged, a going up is -represented by adding red to the zero, or null colour, a -going away by adding yellow, a going to the right by -adding white. White is used as a colour, as a pigment, -which produces a colour change in the pigments with which -it is mixed. From whatever cube of the lower set we -start, a motion up brings us to a cube showing a change -to red, thus light yellow becomes light yellow red, or -light orange, which is called ochre. And going to the<span class="pagenum" id="Page_140">[Pg 140]</span> -right from the null on the left we have a change involving -the introduction of white, while the yellow change runs -from front to back. There are three colour axes—the red, -the white, the yellow—and these run in the position the -cubes occupy in the drawing—up, to the right, away—but -they could be turned about to occupy any positions in space.</p> - -<div class="figcenter illowp100" id="fig_82" style="max-width: 62.5em;"> - <img src="images/fig_82.png" alt="" /> - <div class="caption">Fig. 82.</div> -</div> - - -<div class="figcenter illowp100" id="fig_83" style="max-width: 62.5em;"> - <img src="images/fig_83.png" alt="" /> - <div class="caption">Fig. 83.</div> -</div> - -<p>We can conveniently represent a block of cubes by -three sets of squares, representing each the base of a cube.</p> - -<p>Thus the block, <a href="#fig_83">fig. 83</a>, can be represented by the<span class="pagenum" id="Page_141">[Pg 141]</span> -layers on the right. Here, as in the case of the plane, -the initial colours repeat themselves at the end of the -series.</p> - -<div class="figleft illowp50" id="fig_84" style="max-width: 31.25em;"> - <img src="images/fig_84.png" alt="" /> - <div class="caption">Fig. 84.</div> -</div> - -<p>Proceeding now to increase the number of the cubes -we obtain <a href="#fig_84">fig. 84</a>, -in which the initial -letters of the colours -are given instead of -their full names.</p> - -<p>Here we see that -there are four null -cubes as before, but -the series which spring -from the initial corner -will tend to become -lines of cubes, as also -the sets of cubes -parallel to them, starting -from other corners. -Thus, from the initial -null springs a line of -red cubes, a line of -white cubes, and a line -of yellow cubes.</p> - -<p>If the number of the -cubes is largely increased, -and the size -of the whole cube is -diminished, we get -a cube with null -points, and the edges -coloured with these three colours.</p> - -<p>The light yellow cubes increase in two ways, forming -ultimately a sheet of cubes, and the same is true of -the orange and pink sets. Hence, ultimately the cube<span class="pagenum" id="Page_142">[Pg 142]</span> -thus formed would have red, white, and yellow lines -surrounding pink, orange, and light yellow faces. The -ochre cubes increase in three ways, and hence ultimately -the whole interior of the cube would be coloured -ochre.</p> - -<p>We have thus a nomenclature for the points, lines, -faces, and solid content of a cube, and it can be named -as exhibited in <a href="#fig_85">fig. 85</a>.</p> - -<div class="figleft illowp30" id="fig_85" style="max-width: 15.625em;"> - <img src="images/fig_85.png" alt="" /> - <div class="caption">Fig. 85.</div> -</div> - -<p>We can consider the cube to be produced in the -following way. A null point -moves in a direction to which -we attach the colour indication -yellow; it generates a yellow line -and ends in a null point. The -yellow line thus generated moves -in a direction to which we give -the colour indication red. This -lies up in the figure. The yellow -line traces out a yellow, red, or -orange square, and each of its null points trace out a -red line, and ends in a null point.</p> - -<p>This orange square moves in a direction to which -we attribute the colour indication white, in this case -the direction is the right. The square traces out a -cube coloured orange, red, or ochre, the red lines trace -out red to white or pink squares, and the yellow -lines trace out light yellow squares, each line ending -in a line of its own colour. While the points each -trace out a null + white, or white line to end in a null -point.</p> - -<p>Now returning to the first block of eight cubes we can -name each point, line, and square in them by reference to -the colour scheme, which they determine by their relation -to each other.</p> - -<p>Thus, in <a href="#fig_86">fig. 86</a>, the null cube touches the red cube by<span class="pagenum" id="Page_143">[Pg 143]</span> -a light yellow square; it touches the yellow cube by a -pink square, and touches -the white cube by an -orange square.</p> - -<div class="figleft illowp50" id="fig_86" style="max-width: 25em;"> - <img src="images/fig_86.png" alt="" /> - <div class="caption">Fig. 86.</div> -</div> - -<p>There are three axes -to which the colours red, -yellow, and white are -assigned, the faces of -each cube are designated -by taking these colours in pairs. Taking all the colours -together we get a colour name for the solidity of a cube.</p> - - -<p>Let us now ask ourselves how the cube could be presented -to the plane being. Without going into the question -of how he could have a real experience of it, let us see -how, if we could turn it about and show it to him, he, -under his limitations, could get information about it. -If the cube were placed with its red and yellow axes -against a plane, that is resting against it by its orange -face, the plane being would observe a square surrounded -by red and yellow lines, and having null points. See the -dotted square, <a href="#fig_87">fig. 87</a>.</p> - -<div class="figcenter illowp80" id="fig_87" style="max-width: 37.5em;"> - <img src="images/fig_87.png" alt="" /> - <div class="caption">Fig. 87.</div> -</div> - -<p>We could turn the cube about the red line so that -a different face comes into juxtaposition with the plane.</p> - -<p>Suppose the cube turned about the red line. As it<span class="pagenum" id="Page_144">[Pg 144]</span> -is turning from its first position all of it except the red -line leaves the plane—goes absolutely out of the range -of the plane being’s apprehension. But when the yellow -line points straight out from the plane then the pink -face comes into contact with it. Thus the same red line -remaining as he saw it at first, now towards him comes -a face surrounded by white and red lines.</p> - -<div class="figleft illowp35" id="fig_88" style="max-width: 18.75em;"> - <img src="images/fig_88.png" alt="" /> - <div class="caption">Fig. 88.</div> -</div> - -<p>If we call the direction to the right the unknown -direction, then the line he saw before, the yellow line, -goes out into this unknown direction, and the line which -before went into the unknown direction, comes in. It -comes in in the opposite direction to that in which the -yellow line ran before; the interior of the face now -against the plane is pink. It is -a property of two lines at right -angles that, if one turns out of -a given direction and stands at -right angles to it, then the other -of the two lines comes in, but -runs the opposite way in that -given direction, as in <a href="#fig_88">fig. 88</a>.</p> - -<p>Now these two presentations of the cube would seem, -to the plane creature like perfectly different material -bodies, with only that line in common in which they -both meet.</p> - -<p>Again our cube can be turned about the yellow line. -In this case the yellow square would disappear as before, -but a new square would come into the plane after the -cube had rotated by an angle of 90° about this line. -The bottom square of the cube would come in thus -in figure 89. The cube supposed in contact with the -plane is rotated about the lower yellow line and then -the bottom face is in contact with the plane.</p> - -<p>Here, as before, the red line going out into the unknown -dimension, the white line which before ran in the<span class="pagenum" id="Page_145">[Pg 145]</span> -unknown dimension would come in downwards in the -opposite sense to that in which the red line ran before.</p> - -<div class="figcenter illowp80" id="fig_89" style="max-width: 62.5em;"> - <img src="images/fig_89.png" alt="" /> - <div class="caption">Fig. 89.</div> -</div> - -<p>Now if we use <i>i</i>, <i>j</i>, <i>k</i>, for the three space directions, -<i>i</i> left to right, <i>j</i> from near away, <i>k</i> from below up; then, -using the colour names for the axes, we have that first -of all white runs <i>i</i>, yellow runs <i>j</i>, red runs <i>k</i>; then after -the first turning round the <i>k</i> axis, white runs negative <i>j</i>, -yellow runs <i>i</i>, red runs <i>k</i>; thus we have the table:—</p> - -<table class="standard" summary=""> -<tr> -<td class="tdc"></td> -<td class="tdc"><i>i</i></td> -<td class="tdc"><i>j</i></td> -<td class="tdc"><i>k</i></td> -</tr> -<tr> -<td class="tdl">1st position</td> -<td class="tdc">white</td> -<td class="tdc">yellow</td> -<td class="tdc">red</td> -</tr> -<tr> -<td class="tdl">2nd position</td> -<td class="tdc">yellow</td> -<td class="tdc">white—</td> -<td class="tdc">red</td> -</tr> -<tr> -<td class="tdl">3rd position</td> -<td class="tdc">red</td> -<td class="tdc">yellow</td> -<td class="tdc">white—</td> -</tr> -</table> - - -<p>Here white with a negative sign after it in the column -under <i>j</i> means that white runs in the negative sense of -the <i>j</i> direction.</p> - -<p>We may express the fact in the following way:— -In the plane there is room for two axes while the body -has three. Therefore in the plane we can represent any -two. If we want to keep the axis that goes in the -unknown dimension always running in the positive sense, -then the axis which originally ran in the unknown<span class="pagenum" id="Page_146">[Pg 146]</span> -dimension (the white axis) must come in in the negative -sense of that axis which goes out of the plane into the -unknown dimension.</p> - -<p>It is obvious that the unknown direction, the direction -in which the white line runs at first, is quite distinct from -any direction which the plane creature knows. The white -line may come in towards him, or running down. If he -is looking at a square, which is the face of a cube -(looking at it by a line), then any one of the bounding lines -remaining unmoved, another face of the cube may come -in, any one of the faces, namely, which have the white line -in them. And the white line comes sometimes in one -of the space directions he knows, sometimes in another.</p> - -<p>Now this turning which leaves a line unchanged is -something quite unlike any turning he knows in the -plane. In the plane a figure turns round a point. The -square can turn round the null point in his plane, and -the red and yellow lines change places, only of course, as -with every rotation of lines at right angles, if red goes -where yellow went, yellow comes in negative of red’s old -direction.</p> - -<p>This turning, as the plane creature conceives it, we -should call turning about an axis perpendicular to the -plane. What he calls turning about the null point we -call turning about the white line as it stands out from -his plane. There is no such thing as turning about a -point, there is always an axis, and really much more turns -than the plane being is aware of.</p> - -<p>Taking now a different point of view, let us suppose the -cubes to be presented to the plane being by being passed -transverse to his plane. Let us suppose the sheet of -matter over which the plane being and all objects in his -world slide, to be of such a nature that objects can pass -through it without breaking it. Let us suppose it to be -of the same nature as the film of a soap bubble, so that<span class="pagenum" id="Page_147">[Pg 147]</span> -it closes around objects pushed through it, and, however -the object alters its shape as it passes through it, let us -suppose this film to run up to the contour of the object -in every part, maintaining its plane surface unbroken.</p> - -<p>Then we can push a cube or any object through the -film and the plane being who slips about in the film -will know the contour of the cube just and exactly where -the film meets it.</p> - -<div class="figleft illowp40" id="fig_90" style="max-width: 18.75em;"> - <img src="images/fig_90.png" alt="" /> - <div class="caption">Fig. 90.</div> -</div> - -<p>Fig. 90 represents a cube passing through a plane film. -The plane being now comes into -contact with a very thin slice -of the cube somewhere between -the left and right hand faces. -This very thin slice he thinks -of as having no thickness, and -consequently his idea of it is -what we call a section. It is -bounded by him by pink lines -front and back, coming from -the part of the pink face he is -in contact with, and above and below, by light yellow -lines. Its corners are not null-coloured points, but white -points, and its interior is ochre, the colour of the interior -of the cube.</p> - -<p>If now we suppose the cube to be an inch in each -dimension, and to pass across, from right to left, through -the plane, then we should explain the appearances presented -to the plane being by saying: First of all you -have the face of a cube, this lasts only a moment; then -you have a figure of the same shape but differently -coloured. This, which appears not to move to you in any -direction which you know of, is really moving transverse -to your plane world. Its appearance is unaltered, but -each moment it is something different—a section further -on, in the white, the unknown dimension. Finally, at the<span class="pagenum" id="Page_148">[Pg 148]</span> -end of the minute, a face comes in exactly like the face -you first saw. This finishes up the cube—it is the further -face in the unknown dimension.</p> - -<p>The white line, which extends in length just like the -red or the yellow, you do not see as extensive; you apprehend -it simply as an enduring white point. The null -point, under the condition of movement of the cube, -vanishes in a moment, the lasting white point is really -your apprehension of a white line, running in the unknown -dimension. In the same way the red line of the face by -which the cube is first in contact with the plane lasts only -a moment, it is succeeded by the pink line, and this pink -line lasts for the inside of a minute. This lasting pink -line in your apprehension of a surface, which extends in -two dimensions just like the orange surface extends, as you -know it, when the cube is at rest.</p> - -<p>But the plane creature might answer, “This orange -object is substance, solid substance, bounded completely -and on every side.”</p> - -<p>Here, of course, the difficulty comes in. His solid is our -surface—his notion of a solid is our notion of an abstract -surface with no thickness at all.</p> - -<p>We should have to explain to him that, from every point -of what he called a solid, a new dimension runs away. -From every point a line can be drawn in a direction -unknown to him, and there is a solidity of a kind greater -than that which he knows. This solidity can only be -realised by him by his supposing an unknown direction, -by motion in which what he conceives to be solid matter -instantly disappears. The higher solid, however, which -extends in this dimension as well as in those which he -knows, lasts when a motion of that kind takes place, -different sections of it come consecutively in the plane of -his apprehension, and take the place of the solid which he -at first conceives to be all. Thus, the higher solid—our<span class="pagenum" id="Page_149">[Pg 149]</span> -solid in contradistinction to his area solid, his two-dimensional -solid, must be conceived by him as something -which has duration in it, under circumstances in which his -matter disappears out of his world.</p> - -<p>We may put the matter thus, using the conception of -motion.</p> - -<p>A null point moving in a direction away generates a -yellow line, and the yellow line ends in a null point. We -suppose, that is, a point to move and mark out the -products of this motion in such a manner. Now -suppose this whole line as thus produced to move in -an upward direction; it traces out the two-dimensional -solid, and the plane being gets an orange square. The -null point moves in a red line and ends in a null point, -the yellow line moves and generates an orange square and -ends in a yellow line, the farther null point generates -a red line and ends in a null point. Thus, by movement -in two successive directions known to him, he -can imagine his two-dimensional solid produced with all -its boundaries.</p> - -<p>Now we tell him: “This whole two-dimensional solid -can move in a third or unknown dimension to you. The -null point moving in this dimension out of your world -generates a white line and ends in a null point. The -yellow line moving generates a light yellow two-dimensional -solid and ends in a yellow line, and this -two-dimensional solid, lying end on to your plane world, is -bounded on the far side by the other yellow line. In -the same way each of the lines surrounding your square -traces out an area, just like the orange area you know. -But there is something new produced, something which -you had no idea of before; it is that which is produced by -the movement of the orange square. That, than which -you can imagine nothing more solid, itself moves in a -direction open to it and produces a three-dimensional<span class="pagenum" id="Page_150">[Pg 150]</span> -solid. Using the addition of white to symbolise the -products of this motion this new kind of solid will be light -orange or ochre, and it will be bounded on the far side by -the final position of the orange square which traced it -out, and this final position we suppose to be coloured like -the square in its first position, orange with yellow and -red boundaries and null corners.”</p> - -<p>This product of movement, which it is so easy for us to -describe, would be difficult for him to conceive. But this -difficulty is connected rather with its totality than with -any particular part of it.</p> - -<p>Any line, or plane of this, to him higher, solid we could -show to him, and put in his sensible world.</p> - -<p>We have already seen how the pink square could be put -in his world by a turning of the cube about the red line. -And any section which we can conceive made of the cube -could be exhibited to him. You have simply to turn the -cube and push it through, so that the plane of his existence -is the plane which cuts out the given section of the cube, -then the section would appear to him as a solid. In his -world he would see the contour, get to any part of it by -digging down into it.</p> - - -<p><span class="smcap">The Process by which a Plane Being would gain -a Notion of a Solid.</span></p> - -<p>If we suppose the plane being to have a general idea of -the existence of a higher solid—our solid—we must next -trace out in detail the method, the discipline, by which -he would acquire a working familiarity with our space -existence. The process begins with an adequate realisation -of a simple solid figure. For this purpose we will -suppose eight cubes forming a larger cube, and first we -will suppose each cube to be coloured throughout uniformly.<span class="pagenum" id="Page_151">[Pg 151]</span> -Let the cubes in <a href="#fig_91">fig. 91</a> be the eight making a larger -cube.</p> - -<div class="figcenter illowp80" id="fig_91" style="max-width: 62.5em;"> - <img src="images/fig_91.png" alt="" /> - <div class="caption">Fig. 91.</div> -</div> - - -<p>Now, although each cube is supposed to be coloured -entirely through with the colour, the name of which is -written on it, still we can speak of the faces, edges, and -corners of each cube as if the colour scheme we have -investigated held for it. Thus, on the null cube we can -speak of a null point, a red line, a white line, a pink face, and -so on. These colour designations are shown on No. 1 of -the views of the tesseract in the plate. Here these colour -names are used simply in their geometrical significance. -They denote what the particular line, etc., referred to would -have as its colour, if in reference to the particular cube -the colour scheme described previously were carried out.</p> - -<p>If such a block of cubes were put against the plane and -then passed through it from right to left, at the rate of an -inch a minute, each cube being an inch each way, the -plane being would have the following appearances:—</p> - -<p>First of all, four squares null, yellow, red, orange, lasting -each a minute; and secondly, taking the exact places -of these four squares, four others, coloured white, light -yellow, pink, ochre. Thus, to make a catalogue of the -solid body, he would have to put side by side in his world -two sets of four squares each, as in <a href="#fig_92">fig. 92</a>. The first<span class="pagenum" id="Page_152">[Pg 152]</span> -are supposed to last a minute, and then the others to -come in in place of them, -and also last a minute.</p> - -<div class="figleft illowp50" id="fig_92" style="max-width: 25em;"> - <img src="images/fig_92.png" alt="" /> - <div class="caption">Fig. 92.</div> -</div> - -<p>In speaking of them -he would have to denote -what part of the respective -cube each square represents. -Thus, at the beginning -he would have null -cube orange face, and after -the motion had begun he -would have null cube ochre -section. As he could get -the same coloured section whichever way the cube passed -through, it would be best for him to call this section white -section, meaning that it is transverse to the white axis. -These colour-names, of course, are merely used as names, -and do not imply in this case that the object is really -coloured. Finally, after a minute, as the first cube was -passing beyond his plane he would have null cube orange -face again.</p> - -<p>The same names will hold for each of the other cubes, -describing what face or section of them the plane being -has before him; and the second wall of cubes will come -on, continue, and go out in the same manner. In the -area he thus has he can represent any movement which -we carry out in the cubes, as long as it does not involve -a motion in the direction of the white axis. The relation -of parts that succeed one another in the direction of the -white axis is realised by him as a consecution of states.</p> - -<p>Now, his means of developing his space apprehension -lies in this, that that which is represented as a time -sequence in one position of the cubes, can become a real -co-existence, <i>if something that has a real co-existence -becomes a time sequence</i>.</p> - -<p><span class="pagenum" id="Page_153">[Pg 153]</span></p> - -<p>We must suppose the cubes turned round each of the -axes, the red line, and the yellow line, then something, -which was given as time before, will now be given as the -plane creature’s space; something, which was given as space -before, will now be given as a time series as the cube is -passed through the plane.</p> - -<p>The three positions in which the cubes must be studied -are the one given above and the two following ones. In -each case the original null point which was nearest to us -at first is marked by an asterisk. In figs. 93 and 94 the -point marked with a star is the same in the cubes and in -the plane view.</p> - -<div class="figcenter illowp100" id="fig_93" style="max-width: 62.5em;"> - <img src="images/fig_93.png" alt="" /> - <div class="caption">Fig. 93.<br /> -The cube swung round the red line, so that the white line points -towards us.</div> -</div> - -<p>In <a href="#fig_93">fig. 93</a> the cube is swung round the red line so as to -point towards us, and consequently the pink face comes -next to the plane. As it passes through there are two -varieties of appearance designated by the figures 1 and 2 -in the plane. These appearances are named in the figure, -and are determined by the order in which the cubes<span class="pagenum" id="Page_154">[Pg 154]</span> -come in the motion of the whole block through the -plane.</p> - -<p>With regard to these squares severally, however, -different names must be used, determined by their -relations in the block.</p> - -<p>Thus, in <a href="#fig_93">fig. 93</a>, when the cube first rests against the -plane the null cube is in contact by its pink face; as the -block passes through we get an ochre section of the null -cube, but this is better called a yellow section, as it is -made by a plane perpendicular to the yellow line. When -the null cube has passed through the plane, as it is -leaving it, we get again a pink face.</p> - -<div class="figcenter illowp100" id="fig_94" style="max-width: 62.5em;"> - <img src="images/fig_94.png" alt="" /> - <div class="caption">Fig. 94.<br /> -The cube swung round yellow line, with red line running from left -to right, and white line running down.</div> -</div> - -<p>The same series of changes take place with the cube -appearances which follow on those of the null cube. In -this motion the yellow cube follows on the null cube, and -the square marked yellow in 2 in the plane will be first -“yellow pink face,” then “yellow yellow section,” then -“yellow pink face.”</p> - -<p>In <a href="#fig_94">fig. 94</a>, in which the cube is turned about the yellow -line, we have a certain difficulty, for the plane being will<span class="pagenum" id="Page_155">[Pg 155]</span> -find that the position his squares are to be placed in will -lie below that which they first occupied. They will come -where the support was on which he stood his first set of -squares. He will get over this difficulty by moving his -support.</p> - -<p>Then, since the cubes come upon his plane by the light -yellow face, he will have, taking the null cube as before for -an example, null, light yellow face; null, red section, -because the section is perpendicular to the red line; and -finally, as the null cube leaves the plane, null, light yellow -face. Then, in this case red following on null, he will -have the same series of views of the red as he had of the -null cube.</p> - -<div class="figcenter illowp100" id="fig_95" style="max-width: 62.5em;"> - <img src="images/fig_95.png" alt="" /> - <div class="caption">Fig. 95.</div> -</div> - -<p>There is another set of considerations which we will -briefly allude to.</p> - -<p>Suppose there is a hollow cube, and a string is stretched -across it from null to null, <i>r</i>, <i>y</i>, <i>wh</i>, as we may call the -far diagonal point, how will this string appear to the -plane being as the cube moves transverse to his plane?</p> - -<p>Let us represent the cube as a number of sections, say -5, corresponding to 4 equal divisions made along the white -line perpendicular to it.</p> - -<p>We number these sections 0, 1, 2, 3, 4, corresponding -to the distances along the white line at which they are<span class="pagenum" id="Page_156">[Pg 156]</span> -taken, and imagine each section to come in successively, -taking the place of the preceding one.</p> - -<p>These sections appear to the plane being, counting from -the first, to exactly coincide each with the preceding one. -But the section of the string occupies a different place in -each to that which it does in the preceding section. The -section of the string appears in the position marked by -the dots. Hence the slant of the string appears as a -motion in the frame work marked out by the cube sides. -If we suppose the motion of the cube not to be recognised, -then the string appears to the plane being as a moving -point. Hence extension on the unknown dimension -appears as duration. Extension sloping in the unknown -direction appears as continuous movement.</p> -<hr class="chap x-ebookmaker-drop" /> - -<div class="chapter"> -<p><span class="pagenum" id="Page_157">[Pg 157]</span></p> - -<h2 class="nobreak" id="CHAPTER_XII">CHAPTER XII<br /> - -<small><i>THE SIMPLEST FOUR-DIMENSIONAL SOLID</i></small></h2></div> - - -<p>A plane being, in learning to apprehend solid existence, -must first of all realise that there is a sense of direction -altogether wanting to him. That which we call right -and left does not exist in his perception. He must -assume a movement in a direction, and a distinction of -positive and negative in that direction, which has no -reality corresponding to it in the movements he can -make. This direction, this new dimension, he can only -make sensible to himself by bringing in time, and supposing -that changes, which take place in time, are due to -objects of a definite configuration in three dimensions -passing transverse to his plane, and the different sections -of it being apprehended as changes of one and the same -plane figure.</p> - -<p>He must also acquire a distinct notion about his plane -world, he must no longer believe that it is the all of -space, but that space extends on both sides of it. In -order, then, to prevent his moving off in this unknown -direction, he must assume a sheet, an extended solid sheet, -in two dimensions, against which, in contact with which, -all his movements take place.</p> - -<p>When we come to think of a four-dimensional solid, -what are the corresponding assumptions which we must -make?</p> - -<p>We must suppose a sense which we have not, a sense<span class="pagenum" id="Page_158">[Pg 158]</span> -of direction wanting in us, something which a being in -a four-dimensional world has, and which we have not. It -is a sense corresponding to a new space direction, a -direction which extends positively and negatively from -every point of our space, and which goes right away from -any space direction we know of. The perpendicular to a -plane is perpendicular, not only to two lines in it, but to -every line, and so we must conceive this fourth dimension -as running perpendicularly to each and every line we can -draw in our space.</p> - -<p>And as the plane being had to suppose something -which prevented his moving off in the third, the -unknown dimension to him, so we have to suppose -something which prevents us moving off in the direction -unknown to us. This something, since we must be in -contact with it in every one of our movements, must not -be a plane surface, but a solid; it must be a solid, which -in every one of our movements we are against, not in. It -must be supposed as stretching out in every space dimension -that we know; but we are not in it, we are against it, we -are next to it, in the fourth dimension.</p> - -<p>That is, as the plane being conceives himself as having -a very small thickness in the third dimension, of which -he is not aware in his sense experience, so we must -suppose ourselves as having a very small thickness in -the fourth dimension, and, being thus four-dimensional -beings, to be prevented from realising that we are -such beings by a constraint which keeps us always in -contact with a vast solid sheet, which stretches on in -every direction. We are against that sheet, so that, if we -had the power of four-dimensional movement, we should -either go away from it or through it; all our space -movements as we know them being such that, performing -them, we keep in contact with this solid sheet.</p> - -<p>Now consider the exposition a plane being would make<span class="pagenum" id="Page_159">[Pg 159]</span> -for himself as to the question of the enclosure of a square, -and of a cube.</p> - -<p>He would say the square <span class="allsmcap">A</span>, in Fig. 96, is completely -enclosed by the four squares, <span class="allsmcap">A</span> far, -<span class="allsmcap">A</span> near, <span class="allsmcap">A</span> above, <span class="allsmcap">A</span> below, or as they -are written <span class="allsmcap">A</span><i>n</i>, <span class="allsmcap">A</span><i>f</i>, <span class="allsmcap">A</span><i>a</i>, <span class="allsmcap">A</span><i>b</i>.</p> - -<div class="figleft illowp30" id="fig_96" style="max-width: 15.625em;"> - <img src="images/fig_96.png" alt="" /> - <div class="caption">Fig. 96.</div> -</div> - -<p>If now he conceives the square <span class="allsmcap">A</span> -to move in the, to him, unknown -dimension it will trace out a cube, -and the bounding squares will form -cubes. Will these completely surround -the cube generated by <span class="allsmcap">A</span>? No; -there will be two faces of the cube -made by <span class="allsmcap">A</span> left uncovered; the first, -that face which coincides with the -square <span class="allsmcap">A</span> in its first position; the next, that which coincides -with the square <span class="allsmcap">A</span> in its final position. Against these -two faces cubes must be placed in order to completely -enclose the cube <span class="allsmcap">A</span>. These may be called the cubes left -and right or <span class="allsmcap">A</span><i>l</i> and <span class="allsmcap">A</span><i>r</i>. Thus each of the enclosing -squares of the square <span class="allsmcap">A</span> becomes a cube and two more -cubes are wanted to enclose the cube formed by the -movement of <span class="allsmcap">A</span> in the third dimension.</p> - -<div class="figleft illowp30" id="fig_97" style="max-width: 34.6875em;"> - <img src="images/fig_97.png" alt="" /> - <div class="caption">Fig. 97.</div> -</div> - -<p>The plane being could not see the square <span class="allsmcap">A</span> with the -squares <span class="allsmcap">A</span><i>n</i>, <span class="allsmcap">A</span><i>f</i>, etc., placed about it, -because they completely hide it from -view; and so we, in the analogous -case in our three-dimensional world, -cannot see a cube <span class="allsmcap">A</span> surrounded by -six other cubes. These cubes we -will call <span class="allsmcap">A</span> near <span class="allsmcap">A</span><i>n</i>, <span class="allsmcap">A</span> far <span class="allsmcap">A</span><i>f</i>, <span class="allsmcap">A</span> above -<span class="allsmcap">A</span><i>a</i>, <span class="allsmcap">A</span> below <span class="allsmcap">A</span><i>b</i>, <span class="allsmcap">A</span> left <span class="allsmcap">A</span><i>l</i>, <span class="allsmcap">A</span> right <span class="allsmcap">A</span><i>r</i>, -shown in <a href="#fig_97">fig. 97</a>. If now the cube <span class="allsmcap">A</span> -moves in the fourth dimension right out of space, it traces -out a higher cube—a tesseract, as it may be called.<span class="pagenum" id="Page_160">[Pg 160]</span> -Each of the six surrounding cubes carried on in the same -motion will make a tesseract also, and these will be -grouped around the tesseract formed by <span class="allsmcap">A</span>. But will they -enclose it completely?</p> - -<p>All the cubes <span class="allsmcap">A</span><i>n</i>, <span class="allsmcap">A</span><i>f</i>, etc., lie in our space. But there is -nothing between the cube <span class="allsmcap">A</span> and that solid sheet in contact -with which every particle of matter is. When the cube <span class="allsmcap">A</span> -moves in the fourth direction it starts from its position, -say <span class="allsmcap">A</span><i>k</i>, and ends in a final position <span class="allsmcap">A</span><i>n</i> (using the words -“ana” and “kata” for up and down in the fourth dimension). -Now the movement in this fourth dimension is -not bounded by any of the cubes <span class="allsmcap">A</span><i>n</i>, <span class="allsmcap">A</span><i>f</i>, nor by what -they form when thus moved. The tesseract which <span class="allsmcap">A</span> -becomes is bounded in the positive and negative ways in -this new direction by the first position of <span class="allsmcap">A</span> and the last -position of <span class="allsmcap">A</span>. Or, if we ask how many tesseracts lie -around the tesseract which <span class="allsmcap">A</span> forms, there are eight, of -which one meets it by the cube <span class="allsmcap">A</span>, and another meets it -by a cube like <span class="allsmcap">A</span> at the end of its motion.</p> - -<p>We come here to a very curious thing. The whole -solid cube <span class="allsmcap">A</span> is to be looked on merely as a boundary of -the tesseract.</p> - -<p>Yet this is exactly analogous to what the plane being -would come to in his study of the solid world. The -square <span class="allsmcap">A</span> (<a href="#fig_96">fig. 96</a>), which the plane being looks on as a -solid existence in his plane world, is merely the boundary -of the cube which he supposes generated by its motion.</p> - -<p>The fact is that we have to recognise that, if there is -another dimension of space, our present idea of a solid -body, as one which has three dimensions only, does not -correspond to anything real, but is the abstract idea of a -three-dimensional boundary limiting a four-dimensional -solid, which a four-dimensional being would form. The -plane being’s thought of a square is not the thought -of what we should call a possibly existing real square,<span class="pagenum" id="Page_161">[Pg 161]</span> -but the thought of an abstract boundary, the face of -a cube.</p> - -<p>Let us now take our eight coloured cubes, which form -a cube in space, and ask what additions we must make -to them to represent the simplest collection of four-dimensional -bodies—namely, a group of them of the same extent -in every direction. In plane space we have four squares. -In solid space we have eight cubes. So we should expect -in four-dimensional space to have sixteen four-dimensional -bodies-bodies which in four-dimensional space -correspond to cubes in three-dimensional space, and these -bodies we call tesseracts.</p> - -<div class="figleft illowp30" id="fig_98" style="max-width: 15.625em;"> - <img src="images/fig_98.png" alt="" /> - <div class="caption">Fig. 98.</div> -</div> - -<p>Given then the null, white, red, yellow cubes, and -those which make up the block, we -notice that we represent perfectly -well the extension in three directions -(fig. 98). From the null point of -the null cube, travelling one inch, we -come to the white cube; travelling -one inch away we come to the yellow -cube; travelling one inch up we come -to the red cube. Now, if there is -a fourth dimension, then travelling -from the same null point for one -inch in that direction, we must come to the body lying -beyond the null region.</p> - -<p>I say null region, not cube; for with the introduction -of the fourth dimension each of our cubes must become -something different from cubes. If they are to have -existence in the fourth dimension, they must be “filled -up from” in this fourth dimension.</p> - -<p>Now we will assume that as we get a transference from -null to white going in one way, from null to yellow going -in another, so going from null in the fourth direction we -have a transference from null to blue, using thus the<span class="pagenum" id="Page_162">[Pg 162]</span> -colours white, yellow, red, blue, to denote transferences in -each of the four directions—right, away, up, unknown or -fourth dimension.</p> - -<div class="figleft illowp60" id="fig_99" style="max-width: 25em;"> - <img src="images/fig_99.png" alt="" /> - <div class="caption">Fig. 99.<br /> -A plane being’s representation of a block -of eight cubes by two sets of four squares.</div> -</div> - -<p>Hence, as the plane being must represent the solid regions, -he would come to by going right, as four squares lying -in some position in -his plane, arbitrarily -chosen, side by side -with his original four -squares, so we must -represent those eight -four-dimensional regions, -which we -should come to by -going in the fourth -dimension from each -of our eight cubes, by eight cubes placed in some arbitrary -position relative to our first eight cubes.</p> - -<div class="figcenter illowp80" id="fig_100" style="max-width: 50em;"> - <img src="images/fig_100.png" alt="" /> - <div class="caption">Fig. 100.</div> -</div> - -<p>Our representation of a block of sixteen tesseracts by -two blocks of eight cubes.<a id="FNanchor_3" href="#Footnote_3" class="fnanchor">[3]</a></p> - - -<div class="footnotes"><div class="footnote"> - -<p><a id="Footnote_3" href="#FNanchor_3" class="label">[3]</a> The eight cubes used here in 2 can be found in the second of the -model blocks. They can be taken out and used.</p> - -</div></div> - -<p>Hence, of the two sets of eight cubes, each one will serve<span class="pagenum" id="Page_163">[Pg 163]</span> -us as a representation of one of the sixteen tesseracts -which form one single block in four-dimensional space. -Each cube, as we have it, is a tray, as it were, against -which the real four-dimensional figure rests—just as each -of the squares which the plane being has is a tray, so to -speak, against which the cube it represents could rest.</p> - -<p>If we suppose the cubes to be one inch each way, then -the original eight cubes will give eight tesseracts of the -same colours, or the cubes, extending each one inch in the -fourth dimension.</p> - -<p>But after these there come, going on in the fourth dimension, -eight other bodies, eight other tesseracts. These -must be there, if we suppose the four-dimensional body -we make up to have two divisions, one inch each in each -of four directions.</p> - -<p>The colour we choose to designate the transference to -this second region in the fourth dimension is blue. Thus, -starting from the null cube and going in the fourth -dimension, we first go through one inch of the null -tesseract, then we come to a blue cube, which is the -beginning of a blue tesseract. This blue tesseract stretches -one inch farther on in the fourth dimension.</p> - -<p>Thus, beyond each of the eight tesseracts, which are of -the same colour as the cubes which are their bases, lie -eight tesseracts whose colours are derived from the colours -of the first eight by adding blue. Thus—</p> - -<table class="standard" summary=""> -<tr> -<td class="tdl">Null</td> -<td class="tdc">gives</td> -<td class="tdl">blue</td> -</tr> -<tr> -<td class="tdl">Yellow</td> -<td class="tdc">”</td> -<td class="tdl">green</td> -</tr> -<tr> -<td class="tdl">Red</td> -<td class="tdc">”</td> -<td class="tdl">purple</td> -</tr> -<tr> -<td class="tdl">Orange</td> -<td class="tdc">”</td> -<td class="tdl">brown</td> -</tr> -<tr> -<td class="tdl">White</td> -<td class="tdc">”</td> -<td class="tdl">light blue</td> -</tr> -<tr> -<td class="tdl">Pink</td> -<td class="tdc">”</td> -<td class="tdl">light purple</td> -</tr> -<tr> -<td class="tdl">Light yellow</td> -<td class="tdc">”</td> -<td class="tdl">light green</td> -</tr> -<tr> -<td class="tdl">Ochre</td> -<td class="tdc">”</td> -<td class="tdl">light brown</td> -</tr> -</table> - -<p>The addition of blue to yellow gives green—this is a<span class="pagenum" id="Page_164">[Pg 164]</span> -natural supposition to make. It is also natural to suppose -that blue added to red makes purple. Orange and blue -can be made to give a brown, by using certain shades and -proportions. And ochre and blue can be made to give a -light brown.</p> - -<p>But the scheme of colours is merely used for getting -a definite and realisable set of names and distinctions -visible to the eye. Their naturalness is apparent to any -one in the habit of using colours, and may be assumed to -be justifiable, as the sole purpose is to devise a set of -names which are easy to remember, and which will give -us a set of colours by which diagrams may be made easy -of comprehension. No scientific classification of colours -has been attempted.</p> - -<p>Starting, then, with these sixteen colour names, we have -a catalogue of the sixteen tesseracts, which form a four-dimensional -block analogous to the cubic block. But -the cube which we can put in space and look at is not one -of the constituent tesseracts; it is merely the beginning, -the solid face, the side, the aspect, of a tesseract.</p> - -<p>We will now proceed to derive a name for each region, -point, edge, plane face, solid and a face of the tesseract.</p> - -<p>The system will be clear, if we look at a representation -in the plane of a tesseract with three, and one with four -divisions in its side.</p> - -<p>The tesseract made up of three tesseracts each way -corresponds to the cube made up of three cubes each way, -and will give us a complete nomenclature.</p> - -<p>In this diagram, <a href="#fig_101">fig. 101</a>, 1 represents a cube of 27 -cubes, each of which is the beginning of a tesseract. -These cubes are represented simply by their lowest squares, -the solid content must be understood. 2 represents the -27 cubes which are the beginnings of the 27 tesseracts -one inch on in the fourth dimension. These tesseracts -are represented as a block of cubes put side by side with<span class="pagenum" id="Page_165">[Pg 165]</span> -the first block, but in their proper positions they could -not be in space with the first set. 3 represents 27 cubes -(forming a larger cube) which are the beginnings of the -tesseracts, which begin two inches in the fourth direction -from our space and continue another inch.</p> - -<div class="figcenter illowp100" id="fig_101" style="max-width: 62.5em;"> - <img src="images/fig_101.png" alt="" /> - <div class="caption">Fig. 101.<br /> - - -<table class="standard" summary=""> -<col width="30%" /> <col width="30%" /> <col width="30%" /> -<tr> -<td class="tdc">1</td> -<td class="tdc">2</td> -<td class="tdc">3</td> -</tr> -<tr> -<td class="tdc">Each cube is the -beginning of the first -tesseract going in the -fourth dimension. -</td> -<td class="tdc">Each cube is the -beginning of the -second tesseract. -</td> -<td class="tdc">Each cube is the -beginning of the -third tesseract. -</td> -</tr> -</table></div> -</div> - - -<p><span class="pagenum" id="Page_166">[Pg 166]</span></p> - - -<div class="figcenter illowp100" id="fig_102" style="max-width: 62.5em;"> - <img src="images/fig_102.png" alt="" /> - <div class="caption">Fig. 102.<a id="FNanchor_4" href="#Footnote_4" class="fnanchor">[4]</a></div> -<table class="standard" summary=""> -<col width="25%" /> <col width="25%" /> <col width="25%" /> <col width="25%" /> -<tr> -<td class="tdc">1</td> -<td class="tdc">2</td> -<td class="tdc">3</td> -<td class="tdc">4</td> -</tr> -<tr> -<td class="tdl">A cube of 64 cubes -each 1. in × 1 in., the beginning of a tesseract. -</td> -<td class="tdl">A cube of 64 cubes, -each 1 in. × 1 in. × 1 in. the beginning -of tesseracts 1 in. from our space -in the 4th dimension. -</td> -<td class="tdl">A cube of 64 cubes, -each 1 in. × 1 in. × 1 in. the beginning -of tesseracts 2 in. from our space -in the 4th dimension. -</td> -<td class="tdl">A cube of 64 cubes, -each 1 in. × 1 in. × 1 in. the beginning -of tesseracts 3 in. from our space -in the 4th dimension. -</td> -</tr> -</table></div> - - -<div class="footnotes"> -<div class="footnote"> - -<p><a id="Footnote_4" href="#FNanchor_4" class="label">[4]</a> The coloured plate, figs. 1, 2, 3, shows these relations more -conspicuously.</p> - -</div></div> - - -<p>In <a href="#fig_102">fig. 102</a>, we have the representation of a block of -4 × 4 × 4 × 4 or 256 tesseracts. They are given in -four consecutive sections, each supposed to be taken one -inch apart in the fourth dimension, and so giving four<span class="pagenum" id="Page_167">[Pg 167]</span> -blocks of cubes, 64 in each block. Here we see, comparing -it with the figure of 81 tesseracts, that the number -of the different regions show a different tendency of -increase. By taking five blocks of five divisions each way -this would become even more clear.</p> - -<p>We see, <a href="#fig_102">fig. 102</a>, that starting from the point at any -corner, the white coloured regions only extend out in -a line. The same is true for the yellow, red, and blue. -With regard to the latter it should be noticed that the -line of blues does not consist in regions next to each -other in the drawing, but in portions which come in in -different cubes. The portions which lie next to one -another in the fourth dimension must always be represented -so, when we have a three-dimensional representation. -Again, those regions such as the pink one, go on increasing -in two dimensions. About the pink region this is seen -without going out of the cube itself, the pink regions -increase in length and height, but in no other dimension. -In examining these regions it is sufficient to take one as -a sample.</p> - -<p>The purple increases in the same manner, for it comes -in in a succession from below to above in block 2, and in -a succession from block to block in 2 and 3. Now, a -succession from below to above represents a continuous -extension upwards, and a succession from block to block -represents a continuous extension in the fourth dimension. -Thus the purple regions increase in two dimensions, the -upward and the fourth, so when we take a very great -many divisions, and let each become very small, the -purple region forms a two-dimensional extension.</p> - -<p>In the same way, looking at the regions marked l. b. or -light blue, which starts nearest a corner, we see that the -tesseracts occupying it increase in length from left to -right, forming a line, and that there are as many lines of -light blue tesseracts as there are sections between the<span class="pagenum" id="Page_168">[Pg 168]</span> -first and last section. Hence the light blue tesseracts -increase in number in two ways—in the right and left, -and in the fourth dimension. They ultimately form -what we may call a plane surface.</p> - -<p>Now all those regions which contain a mixture of two -simple colours, white, yellow, red, blue, increase in two -ways. On the other hand, those which contain a mixture -of three colours increase in three ways. Take, for instance, -the ochre region; this has three colours, white, yellow, -red; and in the cube itself it increases in three ways.</p> - -<p>Now regard the orange region; if we add blue to this -we get a brown. The region of the brown tesseracts -extends in two ways on the left of the second block, -No. 2 in the figure. It extends also from left to right in -succession from one section to another, from section 2 -to section 3 in our figure.</p> - -<p>Hence the brown tesseracts increase in number in three -dimensions upwards, to and fro, fourth dimension. Hence -they form a cubic, a three-dimensional region; this region -extends up and down, near and far, and in the fourth -direction, but is thin in the direction from left to right. -It is a cube which, when the complete tesseract is represented -in our space, appears as a series of faces on the -successive cubic sections of the tesseract. Compare fig. -103 in which the middle block, 2, stands as representing a -great number of sections intermediate between 1 and 3.</p> - -<p>In a similar way from the pink region by addition of -blue we have the light purple region, which can be seen -to increase in three ways as the number of divisions -becomes greater. The three ways in which this region of -tesseracts extends is up and down, right and left, fourth -dimension. Finally, therefore, it forms a cubic mass of -very small tesseracts, and when the tesseract is given in -space sections it appears on the faces containing the -upward and the right and left dimensions.</p> - -<p><span class="pagenum" id="Page_169">[Pg 169]</span></p> - -<p>We get then altogether, as three-dimensional regions, -ochre, brown, light purple, light green.</p> - -<p>Finally, there is the region which corresponds to a -mixture of all the colours; there is only one region such -as this. It is the one that springs from ochre by the -addition of blue—this colour we call light brown.</p> - -<p>Looking at the light brown region we see that it -increases in four ways. Hence, the tesseracts of which it -is composed increase in number in each of four dimensions, -and the shape they form does not remain thin in -any of the four dimensions. Consequently this region -becomes the solid content of the block of tesseracts, itself; -it is the real four-dimensional solid. All the other regions -are then boundaries of this light brown region. If we -suppose the process of increasing the number of tesseracts -and diminishing their size carried on indefinitely, then -the light brown coloured tesseracts become the whole -interior mass, the three-coloured tesseracts become three-dimensional -boundaries, thin in one dimension, and form -the ochre, the brown, the light purple, the light green. -The two-coloured tesseracts become two-dimensional -boundaries, thin in two dimensions, <i>e.g.</i>, the pink, the -green, the purple, the orange, the light blue, the light -yellow. The one-coloured tesseracts become bounding -lines, thin in three dimensions, and the null points become -bounding corners, thin in four dimensions. From these -thin real boundaries we can pass in thought to the -abstractions—points, lines, faces, solids—bounding the -four-dimensional solid, which in this case is light brown -coloured, and under this supposition the light brown -coloured region is the only real one, is the only one which -is not an abstraction.</p> - -<p>It should be observed that, in taking a square as the -representation of a cube on a plane, we only represent -one face, or the section between two faces. The squares,<span class="pagenum" id="Page_170">[Pg 170]</span> -as drawn by a plane being, are not the cubes themselves, -but represent the faces or the sections of a cube. Thus -in the plane being’s diagram a cube of twenty-seven cubes -“null” represents a cube, but is really, in the normal -position, the orange square of a null cube, and may be -called null, orange square.</p> - -<p>A plane being would save himself confusion if he named -his representative squares, not by using the names of the -cubes simply, but by adding to the names of the cubes a -word to show what part of a cube his representative square -was.</p> - -<p>Thus a cube null standing against his plane touches it -by null orange face, passing through his plane it has in -the plane a square as trace, which is null white section, if -we use the phrase white section to mean a section drawn -perpendicular to the white line. In the same way the -cubes which we take as representative of the tesseract are -not the tesseract itself, but definite faces or sections of it. -In the preceding figures we should say then, not null, but -“null tesseract ochre cube,” because the cube we actually -have is the one determined by the three axes, white, red, -yellow.</p> - -<p>There is another way in which we can regard the colour -nomenclature of the boundaries of a tesseract.</p> - -<p>Consider a null point to move tracing out a white line -one inch in length, and terminating in a null point, -see <a href="#fig_103">fig. 103</a> or in the coloured plate.</p> - -<p>Then consider this white line with its terminal points -itself to move in a second dimension, each of the points -traces out a line, the line itself traces out an area, and -gives two lines as well, its initial and its final position.</p> - -<p>Thus, if we call “a region” any element of the figure, -such as a point, or a line, etc., every “region” in moving -traces out a new kind of region, “a higher region,” and -gives two regions of its own kind, an initial and a final<span class="pagenum" id="Page_171">[Pg 171]</span> -position. The “higher region” means a region with -another dimension in it.</p> - -<p>Now the square can move and generate a cube. The -square light yellow moves and traces out the mass of the -cube. Letting the addition of red denote the region -made by the motion in the upward direction we get an -ochre solid. The light yellow face in its initial and -terminal positions give the two square boundaries of the -cube above and below. Then each of the four lines of the -light yellow square—white, yellow, and the white, yellow -opposite them—trace out a bounding square. So there -are in all six bounding squares, four of these squares being -designated in colour by adding red to the colour of the -generating lines. Finally, each point moving in the up -direction gives rise to a line coloured null + red, or red, -and then there are the initial and terminal positions of the -points giving eight points. The number of the lines is -evidently twelve, for the four lines of this light yellow -square give four lines in their initial, four lines in their -final position, while the four points trace out four lines, -that is altogether twelve lines.</p> - -<p>Now the squares are each of them separate boundaries -of the cube, while the lines belong, each of them, to two -squares, thus the red line is that which is common to the -orange and pink squares.</p> - -<p>Now suppose that there is a direction, the fourth -dimension, which is perpendicular alike to every one -of the space dimensions already used—a dimension -perpendicular, for instance, to up and to right hand, -so that the pink square moving in this direction traces -out a cube.</p> - -<p>A dimension, moreover, perpendicular to the up and -away directions, so that the orange square moving in this -direction also traces out a cube, and the light yellow -square, too, moving in this direction traces out a cube.<span class="pagenum" id="Page_172">[Pg 172]</span> -Under this supposition, the whole cube moving in the -unknown dimension, traces out something new—a new -kind of volume, a higher volume. This higher volume -is a four-dimensional volume, and we designate it in colour -by adding blue to the colour of that which by moving -generates it.</p> - -<p>It is generated by the motion of the ochre solid, and -hence it is of the colour we call light brown (white, yellow, -red, blue, mixed together). It is represented by a number -of sections like 2 in <a href="#fig_103">fig. 103</a>.</p> - -<p>Now this light brown higher solid has for boundaries: -first, the ochre cube in its initial position, second, the -same cube in its final position, 1 and 3, <a href="#fig_103">fig. 103</a>. Each -of the squares which bound the cube, moreover, by movement -in this new direction traces out a cube, so we have -from the front pink faces of the cube, third, a pink blue or -light purple cube, shown as a light purple face on cube 2 -in <a href="#fig_103">fig. 103</a>, this cube standing for any number of intermediate -sections; fourth, a similar cube from the opposite -pink face; fifth, a cube traced out by the orange face—this -is coloured brown and is represented by the brown -face of the section cube in <a href="#fig_103">fig. 103</a>; sixth, a corresponding -brown cube on the right hand; seventh, a cube -starting from the light yellow square below; the unknown -dimension is at right angles to this also. This cube is -coloured light yellow and blue or light green; and, -finally, eighth, a corresponding cube from the upper -light yellow face, shown as the light green square at the -top of the section cube.</p> - -<p>The tesseract has thus eight cubic boundaries. These -completely enclose it, so that it would be invisible to a -four-dimensional being. Now, as to the other boundaries, -just as the cube has squares, lines, points, as boundaries, -so the tesseract has cubes, squares, lines, points, as -boundaries.</p> - -<p><span class="pagenum" id="Page_173">[Pg 173]</span></p> - -<p>The number of squares is found thus—round the cube -are six squares, these will give six squares in their initial -and six in their final positions. Then each of the twelve -lines of the cube trace out a square in the motion in -the fourth dimension. Hence there will be altogether -12 + 12 = 24 squares.</p> - -<p>If we look at any one of these squares we see that it -is the meeting surface of two of the cubic sides. Thus, -the red line by its movement in the fourth dimension, -traces out a purple square—this is common to two -cubes, one of which is traced out by the pink square -moving in the fourth dimension, and the other is -traced out by the orange square moving in the same -way. To take another square, the light yellow one, this -is common to the ochre cube and the light green cube. -The ochre cube comes from the light yellow square -by moving it in the up direction, the light green cube -is made from the light yellow square by moving it in -the fourth dimension. The number of lines is thirty-two, -for the twelve lines of the cube give twelve lines -of the tesseract in their initial position, and twelve in -their final position, making twenty-four, while each of -the eight points traces out a line, thus forming thirty-two -lines altogether.</p> - -<p>The lines are each of them common to three cubes, or -to three square faces; take, for instance, the red line. -This is common to the orange face, the pink face, and -that face which is formed by moving the red line in the -sixth dimension, namely, the purple face. It is also -common to the ochre cube, the pale purple cube, and the -brown cube.</p> - -<p>The points are common to six square faces and to four -cubes; thus, the null point from which we start is common -to the three square faces—pink, light yellow, orange, and -to the three square faces made by moving the three lines<span class="pagenum" id="Page_174">[Pg 174]</span> -white, yellow, red, in the fourth dimension, namely, the -light blue, the light green, the purple faces—that is, to -six faces in all. The four cubes which meet in it are the -ochre cube, the light purple cube, the brown cube, and -the light green cube.</p> - -<div class="figcenter illowp100" id="fig_103" style="max-width: 62.5em;"> - <img src="images/fig_103.png" alt="" /> - <div class="caption">Fig. 103.</div> -</div> - - -<p>The tesseract, red, white, yellow axes in space. In the lower line the three rear faces -are shown, the interior being removed.]</p> - -<p><span class="pagenum" id="Page_175">[Pg 175]</span></p> - -<div class="figcenter illowp100" id="fig_104" style="max-width: 62.5em;"> - <img src="images/fig_104.png" alt="" /> - <div class="caption">Fig. 104.<br /> -The tesseract, red, yellow, blue axes in space, -the blue axis running to the left, -opposite faces are coloured identically.</div> -</div> - -<p>A complete view of the tesseract in its various space -presentations is given in the following figures or catalogue -cubes, figs. 103-106. The first cube in each figure<span class="pagenum" id="Page_176">[Pg 176]</span> -represents the view of a tesseract coloured as described as -it begins to pass transverse to our space. The intermediate -figure represents a sectional view when it is partly through, -and the final figure represents the far end as it is just -passing out. These figures will be explained in detail in -the next chapter.</p> - -<div class="figcenter illowp100" id="fig_105" style="max-width: 62.5em;"> - <img src="images/fig_105.png" alt="" /> - <div class="caption">Fig. 105.<br /> -The tesseract, with red, white, blue axes in space. Opposite faces are coloured identically.</div> -</div> - -<p><span class="pagenum" id="Page_177">[Pg 177]</span></p> - -<div class="figcenter illowp100" id="fig_106" style="max-width: 62.5em;"> - <img src="images/fig_106.png" alt="" /> - <div class="caption">Fig. 106.<br /> -The tesseract, with blue, white, yellow axes in space. The blue axis runs downward -from the base of the ochre cube as it stands originally. Opposite faces are coloured -identically.</div> -</div> - -<p>We have thus obtained a nomenclature for each of the -regions of a tesseract; we can speak of any one of the -eight bounding cubes, the twenty square faces, the thirty-two -lines, the sixteen points.</p> -<hr class="chap x-ebookmaker-drop" /> - -<div class="chapter"> -<p><span class="pagenum" id="Page_178">[Pg 178]</span></p> - -<h2 class="nobreak" id="CHAPTER_XIII">CHAPTER XIII<br /> - -<small><i>REMARKS ON THE FIGURES</i></small></h2></div> - - -<p>An inspection of above figures will give an answer to -many questions about the tesseract. If we have a -tesseract one inch each way, then it can be represented -as a cube—a cube having white, yellow, red axes, and -from this cube as a beginning, a volume extending into -the fourth dimension. Now suppose the tesseract to pass -transverse to our space, the cube of the red, yellow, white -axes disappears at once, it is indefinitely thin in the -fourth dimension. Its place is occupied by those parts -of the tesseract which lie further away from our space -in the fourth dimension. Each one of these sections -will last only for one moment, but the whole of them -will take up some appreciable time in passing. If we -take the rate of one inch a minute the sections will take -the whole of the minute in their passage across our -space, they will take the whole of the minute except the -moment which the beginning cube and the end cube -occupy in their crossing our space. In each one of the -cubes, the section cubes, we can draw lines in all directions -except in the direction occupied by the blue line, the -fourth dimension; lines in that direction are represented -by the transition from one section cube to another. Thus -to give ourselves an adequate representation of the -tesseract we ought to have a limitless number of section -cubes intermediate between the first bounding cube, the<span class="pagenum" id="Page_179">[Pg 179]</span> -ochre cube, and the last bounding cube, the other ochre -cube. Practically three intermediate sectional cubes will -be found sufficient for most purposes. We will take then -a series of five figures—two terminal cubes, and three -intermediate sections—and show how the different regions -appear in our space when we take each set of three out -of the four axes of the tesseract as lying in our space.</p> - -<p>In <a href="#fig_107">fig. 107</a> initial letters are used for the colours. -A reference to <a href="#fig_103">fig. 103</a> will show the complete nomenclature, -which is merely indicated here.</p> - -<div class="figcenter illowp100" id="fig_107" style="max-width: 62.5em;"> - <img src="images/fig_107.png" alt="" /> - <div class="caption">Fig. 107.</div> -</div> - -<p>In this figure the tesseract is shown in five stages -distant from our space: first, zero; second, 1/4 in.; third, -2/4 in.; fourth, 3/4 in.; fifth, 1 in.; which are called <i>b</i>0, <i>b</i>1, -<i>b</i>2, <i>b</i>3, <i>b</i>4, because they are sections taken at distances -0, 1, 2, 3, 4 quarter inches along the blue line. All the -regions can be named from the first cube, the <i>b</i>0 cube, -as before, simply by remembering that transference along -the b axis gives the addition of blue to the colour of -the region in the ochre, the <i>b</i>0 cube. In the final cube -<i>b</i>4, the colouring of the original <i>b</i>0 cube is repeated. -Thus the red line moved along the blue axis gives a red -and blue or purple square. This purple square appears -as the three purple lines in the sections <i>b</i>1, <i>b</i>2, <i>b</i>3, taken -at 1/4, 2/4, 3/4 of an inch in the fourth dimension. If the -tesseract moves transverse to our space we have then in -this particular region, first of all a red line which lasts -for a moment, secondly a purple line which takes its<span class="pagenum" id="Page_180">[Pg 180]</span> -place. This purple line lasts for a minute—that is, all -of a minute, except the moment taken by the crossing -our space of the initial and final red line. The purple -line having lasted for this period is succeeded by a red -line, which lasts for a moment; then this goes and the -tesseract has passed across our space. The final red line -we call red bl., because it is separated from the initial -red line by a distance along the axis for which we use -the colour blue. Thus a line that lasts represents an -area duration; is in this mode of presentation equivalent -to a dimension of space. In the same way the white -line, during the crossing our space by the tesseract, is -succeeded by a light blue line which lasts for the inside -of a minute, and as the tesseract leaves our space, having -crossed it, the white bl. line appears as the final -termination.</p> - -<p>Take now the pink face. Moved in the blue direction -it traces out a light purple cube. This light purple -cube is shown in sections in <i>b</i><sub>1</sub>, <i>b</i><sub>2</sub>, <i>b</i><sub>3</sub>, and the farther -face of this cube in the blue direction is shown in <i>b</i><sub>4</sub>—a -pink face, called pink <i>b</i> because it is distant from the -pink face we began with in the blue direction. Thus -the cube which we colour light purple appears as a lasting -square. The square face itself, the pink face, vanishes -instantly the tesseract begins to move, but the light -purple cube appears as a lasting square. Here also -duration is the equivalent of a dimension of space—a -lasting square is a cube. It is useful to connect these -diagrams with the views given in the coloured plate.</p> - -<p>Take again the orange face, that determined by the -red and yellow axes; from it goes a brown cube in the -blue direction, for red and yellow and blue are supposed -to make brown. This brown cube is shown in three -sections in the faces <i>b</i><sub>1</sub>, <i>b</i><sub>2</sub>, <i>b</i><sub>3</sub>. In <i>b</i><sub>4</sub> is the opposite -orange face of the brown cube, the face called orange <i>b</i>,<span class="pagenum" id="Page_181">[Pg 181]</span> -for it is distant in the blue direction from the orange -face. As the tesseract passes transverse to our space, -we have then in this region an instantly vanishing orange -square, followed by a lasting brown square, and finally -an orange face which vanishes instantly.</p> - -<p>Now, as any three axes will be in our space, let us send -the white axis out into the unknown, the fourth dimension, -and take the blue axis into our known space -dimension. Since the white and blue axes are perpendicular -to each other, if the white axis goes out into -the fourth dimension in the positive sense, the blue axis -will come into the direction the white axis occupied, -in the negative sense.</p> - -<div class="figcenter illowp100" id="fig_108" style="max-width: 62.5em;"> - <img src="images/fig_108.png" alt="" /> - <div class="caption">Fig. 108.</div> -</div> - -<p>Hence, not to complicate matters by having to think -of two senses in the unknown direction, let us send the -white line into the positive sense of the fourth dimension, -and take the blue one as running in the negative -sense of that direction which the white line has left; -let the blue line, that is, run to the left. We have -now the row of figures in <a href="#fig_108">fig. 108</a>. The dotted cube -shows where we had a cube when the white line ran -in our space—now it has turned out of our space, and -another solid boundary, another cubic face of the tesseract -comes into our space. This cube has red and yellow -axes as before; but now, instead of a white axis running -to the right, there is a blue axis running to the left. -Here we can distinguish the regions by colours in a perfectly -systematic way. The red line traces out a purple<span class="pagenum" id="Page_182">[Pg 182]</span> -square in the transference along the blue axis by which -this cube is generated from the orange face. This -purple square made by the motion of the red line is -the same purple face that we saw before as a series of -lines in the sections <i>b</i><sub>1</sub>, <i>b</i><sub>2</sub>, <i>b</i><sub>3</sub>. Here, since both red and -blue axes are in our space, we have no need of duration -to represent the area they determine. In the motion -of the tesseract across space this purple face would -instantly disappear.</p> - -<p>From the orange face, which is common to the initial -cubes in <a href="#fig_107">fig. 107</a> and <a href="#fig_108">fig. 108</a>, there goes in the blue -direction a cube coloured brown. This brown cube is -now all in our space, because each of its three axes run -in space directions, up, away, to the left. It is the same -brown cube which appeared as the successive faces on the -sections <i>b</i><sub>1</sub>, <i>b</i><sub>2</sub>, <i>b</i><sub>3</sub>. Having all its three axes in our -space, it is given in extension; no part of it needs to -be represented as a succession. The tesseract is now -in a new position with regard to our space, and when -it moves across our space the brown cube instantly -disappears.</p> - -<p>In order to exhibit the other regions of the tesseract -we must remember that now the white line runs in the -unknown dimension. Where shall we put the sections -at distances along the line? Any arbitrary position in -our space will do: there is no way by which we can -represent their real position.</p> - -<p>However, as the brown cube comes off from the orange -face to the left, let us put these successive sections to -the left. We can call them <i>wh</i><sub>0</sub>, <i>wh</i><sub>1</sub>, <i>wh</i><sub>2</sub>, <i>wh</i><sub>3</sub>, <i>wh</i><sub>4</sub>, -because they are sections along the white axis, which -now runs in the unknown dimension.</p> - -<p>Running from the purple square in the white direction -we find the light purple cube. This is represented in the -<span class="pagenum" id="Page_183">[Pg 183]</span>sections <i>wh</i><sub>1</sub>, <i>wh</i><sub>2</sub>, <i>wh</i><sub>3</sub>, <i>wh</i><sub>4</sub>, <a href="#fig_108">fig. 108</a>. It is the same cube -that is represented in the sections <i>b</i><sub>1</sub>, <i>b</i><sub>2</sub>, <i>b</i><sub>3</sub>: in <a href="#fig_107">fig. 107</a> -the red and white axes are in our space, the blue out of -it; in the other case, the red and blue are in our space, -the white out of it. It is evident that the face pink <i>y</i>, -opposite the pink face in <a href="#fig_107">fig. 107</a>, makes a cube shown -in squares in <i>b</i><sub>1</sub>, <i>b</i><sub>2</sub>, <i>b</i><sub>3</sub>, <i>b</i><sub>4</sub>, on the opposite side to the <i>l</i> -purple squares. Also the light yellow face at the base -of the cube <i>b</i><sub>0</sub>, makes a light green cube, shown as a series -of base squares.</p> - -<p>The same light green cube can be found in <a href="#fig_107">fig. 107</a>. -The base square in <i>wh</i><sub>0</sub> is a green square, for it is enclosed -by blue and yellow axes. From it goes a cube in the -white direction, this is then a light green cube and the -same as the one just mentioned as existing in the sections -<i>b</i><sub>0</sub>, <i>b</i><sub>1</sub>, <i>b</i><sub>2</sub>, <i>b</i><sub>3</sub>, <i>b</i><sub>4</sub>.</p> - -<p>The case is, however, a little different with the brown -cube. This cube we have altogether in space in the -section <i>wh</i><sub>0</sub>, <a href="#fig_108">fig. 108</a>, while it exists as a series of squares, -the left-hand ones, in the sections <i>b</i><sub>0</sub>, <i>b</i><sub>1</sub>, <i>b</i><sub>2</sub>, <i>b</i><sub>3</sub>, <i>b</i><sub>4</sub>. The -brown cube exists as a solid in our space, as shown in -<a href="#fig_108">fig. 108</a>. In the mode of representation of the tesseract -exhibited in <a href="#fig_107">fig. 107</a>, the same brown cube appears as a -succession of squares. That is, as the tesseract moves -across space, the brown cube would actually be to us a -square—it would be merely the lasting boundary of another -solid. It would have no thickness at all, only extension -in two dimensions, and its duration would show its solidity -in three dimensions.</p> - -<p>It is obvious that, if there is a four-dimensional space, -matter in three dimensions only is a mere abstraction; all -material objects must then have a slight four-dimensional -thickness. In this case the above statement will undergo -modification. The material cube which is used as the -model of the boundary of a tesseract will have a slight -thickness in the fourth dimension, and when the cube is<span class="pagenum" id="Page_184">[Pg 184]</span> -presented to us in another aspect, it would not be a mere -surface. But it is most convenient to regard the cubes -we use as having no extension at all in the fourth -dimension. This consideration serves to bring out a point -alluded to before, that, if there is a fourth dimension, our -conception of a solid is the conception of a mere abstraction, -and our talking about real three-dimensional objects would -seem to a four-dimensional being as incorrect as a two-dimensional -being’s telling about real squares, real -triangles, etc., would seem to us.</p> - -<p>The consideration of the two views of the brown cube -shows that any section of a cube can be looked at by a -presentation of the cube in a different position in four-dimensional -space. The brown faces in <i>b</i><sub>1</sub>, <i>b</i><sub>2</sub>, <i>b</i><sub>3</sub>, are the -very same brown sections that would be obtained by -cutting the brown cube, <i>wh</i><sub>0</sub>, across at the right distances -along the blue line, as shown in <a href="#fig_108">fig. 108</a>. But as these -sections are placed in the brown cube, <i>wh</i><sub>0</sub>, they come -behind one another in the blue direction. Now, in the -sections <i>wh</i><sub>1</sub>, <i>wh</i><sub>2</sub>, <i>wh</i><sub>3</sub>, we are looking at these sections -from the white direction—the blue direction does not -exist in these figures. So we see them in a direction at -right angles to that in which they occur behind one -another in <i>wh</i><sub>0</sub>. There are intermediate views, which -would come in the rotation of a tesseract. These brown -squares can be looked at from directions intermediate -between the white and blue axes. It must be remembered -that the fourth dimension is perpendicular equally to all -three space axes. Hence we must take the combinations -of the blue axis, with each two of our three axes, white, -red, yellow, in turn.</p> - -<p>In <a href="#fig_109">fig. 109</a> we take red, white, and blue axes in space, -sending yellow into the fourth dimension. If it goes into -the positive sense of the fourth dimension the blue line -will come in the opposite direction to that in which the<span class="pagenum" id="Page_185">[Pg 185]</span> -yellow line ran before. Hence, the cube determined by -the white, red, blue axes, will start from the pink plane -and run towards us. The dotted cube shows where the -ochre cube was. When it is turned out of space, the cube -coming towards from its front face is the one which comes -into our space in this turning. Since the yellow line now -runs in the unknown dimension we call the sections -<i>y</i><sub>0</sub>, <i>y</i><sub>1</sub>, <i>y</i><sub>2</sub>, <i>y</i><sub>3</sub>, <i>y</i><sub>4</sub>, as they are made at distances 0, 1, 2, 3, 4, -quarter inches along the yellow line. We suppose these -cubes arranged in a line coming towards us—not that -that is any more natural than any other arbitrary series -of positions, but it agrees with the plan previously adopted.</p> - -<div class="figcenter illowp100" id="fig_109" style="max-width: 62.5em;"> - <img src="images/fig_109.png" alt="" /> - <div class="caption">Fig. 109.</div> -</div> - -<p>The interior of the first cube, <i>y</i><sub>0</sub>, is that derived from -pink by adding blue, or, as we call it, light purple. The -faces of the cube are light blue, purple, pink. As drawn, -we can only see the face nearest to us, which is not the -one from which the cube starts—but the face on the -opposite side has the same colour name as the face -towards us.</p> - -<p>The successive sections of the series, <i>y</i><sub>0</sub>, <i>y</i><sub>1</sub>, <i>y</i><sub>2</sub>, etc., can -be considered as derived from sections of the <i>b</i><sub>0</sub> cube -made at distances along the yellow axis. What is distant -a quarter inch from the pink face in the yellow direction? -This question is answered by taking a section from a point -a quarter inch along the yellow axis in the cube <i>b</i><sub>0</sub>, <a href="#fig_107">fig. 107</a>. -It is an ochre section with lines orange and light yellow. -This section will therefore take the place of the pink face<span class="pagenum" id="Page_186">[Pg 186]</span> -in <i>y</i><sub>1</sub> when we go on in the yellow direction. Thus, the -first section, <i>y</i><sub>1</sub>, will begin from an ochre face with light -yellow and orange lines. The colour of the axis which -lies in space towards us is blue, hence the regions of this -section-cube are determined in nomenclature, they will be -found in full in <a href="#fig_105">fig. 105</a>.</p> - -<p>There remains only one figure to be drawn, and that is -the one in which the red axis is replaced by the blue. -Here, as before, if the red axis goes out into the positive -sense of the fourth dimension, the blue line must come -into our space in the negative sense of the direction which -the red line has left. Accordingly, the first cube will -come in beneath the position of our ochre cube, the one -we have been in the habit of starting with.</p> - -<div class="figcenter illowp100" id="fig_110" style="max-width: 62.5em;"> - <img src="images/fig_110.png" alt="" /> - <div class="caption">Fig. 110.</div> -</div> - -<p>To show these figures we must suppose the ochre cube -to be on a movable stand. When the red line swings out -into the unknown dimension, and the blue line comes in -downwards, a cube appears below the place occupied by -the ochre cube. The dotted cube shows where the ochre -cube was. That cube has gone and a different cube runs -downwards from its base. This cube has white, yellow, -and blue axes. Its top is a light yellow square, and hence -its interior is light yellow + blue or light green. Its front -face is formed by the white line moving along the blue -axis, and is therefore light blue, the left-hand side is -formed by the yellow line moving along the blue axis, and -therefore green.</p> - -<p><span class="pagenum" id="Page_187">[Pg 187]</span></p> - -<p>As the red line now runs in the fourth dimension, the -successive sections can he called <i>r</i><sub>0</sub>, <i>r</i><sub>1</sub>, <i>r</i><sub>2</sub>, <i>r</i><sub>3</sub>, <i>r</i><sub>4</sub>, these -letters indicating that at distances 0, 1/4, 2/4, 3/4, 1 inch along -the red axis we take all of the tesseract that can be found -in a three-dimensional space, this three-dimensional space -extending not at all in the fourth dimension, but up and -down, right and left, far and near.</p> - -<p>We can see what should replace the light yellow face of -<i>r</i><sub>0</sub>, when the section <i>r</i><sub>1</sub> comes in, by looking at the cube -<i>b</i><sub>0</sub>, <a href="#fig_107">fig. 107</a>. What is distant in it one-quarter of an inch -from the light yellow face in the red direction? It is an -ochre section with orange and pink lines and red points; -see also <a href="#fig_103">fig. 103</a>.</p> - -<p>This square then forms the top square of <i>r</i><sub>1</sub>. Now we -can determine the nomenclature of all the regions of <i>r</i><sub>1</sub> by -considering what would be formed by the motion of this -square along a blue axis.</p> - -<p>But we can adopt another plan. Let us take a horizontal -section of <i>r</i><sub>0</sub>, and finding that section in the figures, -of <a href="#fig_107">fig. 107</a> or <a href="#fig_103">fig. 103</a>, from them determine what will -replace it, going on in the red direction.</p> - -<p>A section of the <i>r</i><sub>0</sub> cube has green, light blue, green, -light blue sides and blue points.</p> - -<p>Now this square occurs on the base of each of the -section figures, <i>b</i><sub>1</sub>, <i>b</i><sub>2</sub>, etc. In them we see that 1/4 inch in -the red direction from it lies a section with brown and -light purple lines and purple corners, the interior being -of light brown. Hence this is the nomenclature of the -section which in <i>r</i><sub>1</sub> replaces the section of <i>r</i><sub>0</sub> made from a -point along the blue axis.</p> - -<p>Hence the colouring as given can be derived.</p> - -<p>We have thus obtained a perfectly named group of -tesseracts. We can take a group of eighty-one of them -3 × 3 × 3 × 3, in four dimensions, and each tesseract will -have its name null, red, white, yellow, blue, etc., and<span class="pagenum" id="Page_188">[Pg 188]</span> -whatever cubic view we take of them we can say exactly -what sides of the tesseracts we are handling, and how -they touch each other.<a id="FNanchor_5" href="#Footnote_5" class="fnanchor">[5]</a></p> - -<div class="footnotes"> -<div class="footnote"> - -<p><a id="Footnote_5" href="#FNanchor_5" class="label">[5]</a> At this point the reader will find it advantageous, if he has the -models, to go through the manipulations described in the appendix.</p> - -</div></div> - -<p>Thus, for instance, if we have the sixteen tesseracts -shown below, we can ask how does null touch blue.</p> - -<div class="figcenter illowp100" id="fig_111" style="max-width: 62.5em;"> - <img src="images/fig_111.png" alt="" /> - <div class="caption">Fig. 111.</div> -</div> - -<p>In the arrangement given in <a href="#fig_111">fig. 111</a> we have the axes -white, red, yellow, in space, blue running in the fourth -dimension. Hence we have the ochre cubes as bases. -Imagine now the tesseractic group to pass transverse to -our space—we have first of all null ochre cube, white -ochre cube, etc.; these instantly vanish, and we get the -section shown in the middle cube in <a href="#fig_103">fig. 103</a>, and finally, -just when the tesseract block has moved one inch transverse -to our space, we have null ochre cube, and then -immediately afterwards the ochre cube of blue comes in. -Hence the tesseract null touches the tesseract blue by its -ochre cube, which is in contact, each and every point -of it, with the ochre cube of blue.</p> - -<p>How does null touch white, we may ask? Looking at -the beginning A, <a href="#fig_111">fig. 111</a>, where we have the ochre<span class="pagenum" id="Page_189">[Pg 189]</span> -cubes, we see that null ochre touches white ochre by an -orange face. Now let us generate the null and white -tesseracts by a motion in the blue direction of each of -these cubes. Each of them generates the corresponding -tesseract, and the plane of contact of the cubes generates -the cube by which the tesseracts are in contact. Now an -orange plane carried along a blue axis generates a brown -cube. Hence null touches white by a brown cube.</p> - -<div class="figcenter illowp100" id="fig_112" style="max-width: 62.5em;"> - <img src="images/fig_112.png" alt="" /> - <div class="caption">Fig. 112.</div> -</div> - -<p>If we ask again how red touches light blue tesseract, -let us rearrange our group, <a href="#fig_112">fig. 112</a>, or rather turn it -about so that we have a different space view of it; let -the red axis and the white axis run up and right, and let -the blue axis come in space towards us, then the yellow -axis runs in the fourth dimension. We have then two -blocks in which the bounding cubes of the tesseracts are -given, differently arranged with regard to us—the arrangement -is really the same, but it appears different to us. -Starting from the plane of the red and white axes we -have the four squares of the null, white, red, pink tesseracts -as shown in A, on the red, white plane, unaltered, only -from them now comes out towards us the blue axis.<span class="pagenum" id="Page_190">[Pg 190]</span> -Hence we have null, white, red, pink tesseracts in contact -with our space by their cubes which have the red, white, -blue axis in them, that is by the light purple cubes. -Following on these four tesseracts we have that which -comes next to them in the blue direction, that is the -four blue, light blue, purple, light purple. These are -likewise in contact with our space by their light purple -cubes, so we see a block as named in the figure, of which -each cube is the one determined by the red, white, blue, -axes.</p> - -<p>The yellow line now runs out of space; accordingly one -inch on in the fourth dimension we come to the tesseracts -which follow on the eight named in C, <a href="#fig_112">fig. 112</a>, in the -yellow direction.</p> - -<p>These are shown in C.y<sub>1</sub>, <a href="#fig_112">fig. 112</a>. Between figure C -and C.y<sub>1</sub> is that four-dimensional mass which is formed -by moving each of the cubes in C one inch in the fourth -dimension—that is, along a yellow axis; for the yellow -axis now runs in the fourth dimension.</p> - -<p>In the block C we observe that red (light purple -cube) touches light blue (light purple cube) by a point. -Now these two cubes moving together remain in contact -during the period in which they trace out the tesseracts -red and light blue. This motion is along the yellow -axis, consequently red and light blue touch by a yellow -line.</p> - -<p>We have seen that the pink face moved in a yellow -direction traces out a cube; moved in the blue direction it -also traces out a cube. Let us ask what the pink face -will trace out if it is moved in a direction within the -tesseract lying equally between the yellow and blue -directions. What section of the tesseract will it make?</p> - -<p>We will first consider the red line alone. Let us take -a cube with the red line in it and the yellow and blue -axes.</p> - -<p><span class="pagenum" id="Page_191">[Pg 191]</span></p> - -<div class="figleft illowp35" id="fig_113" style="max-width: 15.625em;"> - <img src="images/fig_113.png" alt="" /> - <div class="caption">Fig. 113.</div> -</div> - -<p>The cube with the yellow, red, blue axes is shown in -<a href="#fig_113">fig. 113</a>. If the red line is -moved equally in the yellow and -in the blue direction by four -equal motions of ¼ inch each, it -takes the positions 11, 22, 33, -and ends as a red line.</p> - -<p>Now, the whole of this red, -yellow, blue, or brown cube appears -as a series of faces on the -successive sections of the tesseract -starting from the ochre cube and letting the blue -axis run in the fourth dimension. Hence the plane -traced out by the red line appears as a series of lines in -the successive sections, in our ordinary way of representing -the tesseract; these lines are in different places in each -successive section.</p> - -<div class="figcenter illowp100" id="fig_114" style="max-width: 62.5em;"> - <img src="images/fig_114.png" alt="" /> - <div class="caption">Fig. 114.</div> -</div> - -<p>Thus drawing our initial cube and the successive -sections, calling them <i>b</i><sub>0</sub>, <i>b</i><sub>1</sub>, <i>b</i><sub>2</sub>, <i>b</i><sub>3</sub>, <i>b</i><sub>4</sub>, <a href="#fig_115">fig. 115</a>, we have -the red line subject to this movement appearing in the -positions indicated.</p> - -<p>We will now investigate what positions in the tesseract -another line in the pink face assumes when it is moved in -a similar manner.</p> - -<p>Take a section of the original cube containing a vertical -line, 4, in the pink plane, <a href="#fig_115">fig. 115</a>. We have, in the -section, the yellow direction, but not the blue.</p> - -<p><span class="pagenum" id="Page_192">[Pg 192]</span></p> - -<p>From this section a cube goes off in the fourth dimension, -which is formed by moving each point of the section -in the blue direction.</p> - -<div class="figleft illowp40" id="fig_115" style="max-width: 15.625em;"> - <img src="images/fig_115.png" alt="" /> - <div class="caption">Fig. 115.</div> -</div> - -<div class="figright illowp40" id="fig_116" style="max-width: 18.75em;"> - <img src="images/fig_116.png" alt="" /> - <div class="caption">Fig. 116.</div> -</div> - -<p>Drawing this cube we have <a href="#fig_116">fig. 116</a>.</p> - -<p>Now this cube occurs as a series of sections in our -original representation of the tesseract. Taking four steps -as before this cube appears as the sections drawn in <i>b</i><sub>0</sub>, <i>b</i><sub>1</sub>, -<i>b</i><sub>2</sub>, <i>b</i><sub>3</sub>, <i>b</i><sub>4</sub>, <a href="#fig_117">fig. 117</a>, and if the line 4 is subjected to a -movement equal in the blue and yellow directions, it will -occupy the positions designated by 4, 4<sub>1</sub>, 4<sub>2</sub>, 4<sub>3</sub>, 4<sub>4</sub>.</p> - -<div class="figcenter illowp100" id="fig_117" style="max-width: 62.5em;"> - <img src="images/fig_117.png" alt="" /> - <div class="caption">Fig. 117.</div> -</div> - -<p>Hence, reasoning in a similar manner about every line, -it is evident that, moved equally in the blue and yellow -directions, the pink plane will trace out a space which is -shown by the series of section planes represented in the -diagram.</p> - -<p>Thus the space traced out by the pink face, if it is -moved equally in the yellow and blue directions, is represented -by the set of planes delineated in Fig. 118, pink<span class="pagenum" id="Page_193">[Pg 193]</span> -face or 0, then 1, 2, 3, and finally pink face or 4. This -solid is a diagonal solid of the tesseract, running from a -pink face to a pink face. Its length is the length of the -diagonal of a square, its side is a square.</p> - -<p>Let us now consider the unlimited space which springs -from the pink face extended.</p> - -<p>This space, if it goes off in the yellow direction, gives -us in it the ochre cube of the tesseract. Thus, if we have -the pink face given and a point in the ochre cube, we -have determined this particular space.</p> - -<p>Similarly going off from the pink face in the blue -direction is another space, which gives us the light purple -cube of the tesseract in it. And any point being taken in -the light purple cube, this space going off from the pink -face is fixed.</p> - -<div class="figcenter illowp100" id="fig_118" style="max-width: 62.5em;"> - <img src="images/fig_118.png" alt="" /> - <div class="caption">Fig. 118.</div> -</div> - -<p>The space we are speaking of can be conceived as -swinging round the pink face, and in each of its positions -it cuts out a solid figure from the tesseract, one of which -we have seen represented in <a href="#fig_118">fig. 118</a>.</p> - -<p>Each of these solid figures is given by one position of -the swinging space, and by one only. Hence in each of -them, if one point is taken, the particular one of the -slanting spaces is fixed. Thus we see that given a plane -and a point out of it a space is determined.</p> - -<p>Now, two points determine a line.</p> - -<p>Again, think of a line and a point outside it. Imagine -a plane rotating round the line. At some time in its -rotation it passes through the point. Thus a line and a<span class="pagenum" id="Page_194">[Pg 194]</span> -point, or three points, determine a plane. And finally -four points determine a space. We have seen that a -plane and a point determine a space, and that three -points determine a plane; so four points will determine -a space.</p> - -<p>These four points may be any points, and we can take, -for instance, the four points at the extremities of the red, -white, yellow, blue axes, in the tesseract. These will -determine a space slanting with regard to the section -spaces we have been previously considering. This space -will cut the tesseract in a certain figure.</p> - -<p>One of the simplest sections of a cube by a plane is -that in which the plane passes through the extremities -of the three edges which meet in a point. We see at -once that this plane would cut the cube in a triangle, but -we will go through the process by which a plane being -would most conveniently treat the problem of the determination -of this shape, in order that we may apply the -method to the determination of the figure in which a -space cuts a tesseract when it passes through the 4 -points at unit distance from a corner.</p> - -<p>We know that two points determine a line, three points -determine a plane, and given any two points in a plane -the line between them lies wholly in the plane.</p> -<div class="figleft illowp40" id="fig_119" style="max-width: 18.75em;"> - <img src="images/fig_119.png" alt="" /> - <div class="caption">Fig. 119.</div> -</div> - -<p>Let now the plane being study the section made by -a plane passing through the -null <i>r</i>, null <i>wh</i>, and null <i>y</i> -points, <a href="#fig_119">fig. 119</a>. Looking at -the orange square, which, as -usual, we suppose to be initially -in his plane, he sees -that the line from null <i>r</i> to -null <i>y</i>, which is a line in the -section plane, the plane, namely, through the three -extremities of the edges meeting in null, cuts the orange<span class="pagenum" id="Page_195">[Pg 195]</span> -face in an orange line with null points. This then is one -of the boundaries of the section figure.</p> - -<p>Let now the cube be so turned that the pink face -comes in his plane. The points null <i>r</i> and null <i>wh</i> -are now visible. The line between them is pink -with null points, and since this line is common to -the surface of the cube and the cutting plane, it is -a boundary of the figure in which the plane cuts the -cube.</p> - -<p>Again, suppose the cube turned so that the light -yellow face is in contact with the plane being’s plane. -He sees two points, the null <i>wh</i> and the null <i>y</i>. The -line between these lies in the cutting plane. Hence, -since the three cutting lines meet and enclose a portion -of the cube between them, he has determined the -figure he sought. It is a triangle with orange, pink, -and light yellow sides, all equal, and enclosing an -ochre area.</p> - -<p>Let us now determine in what figure the space, -determined by the four points, null <i>r</i>, null <i>y</i>, null -<i>wh</i>, null <i>b</i>, cuts the tesseract. We can see three -of these points in the primary position of the tesseract -resting against our solid sheet by the ochre cube. -These three points determine a plane which lies in -the space we are considering, and this plane cuts -the ochre cube in a triangle, the interior of which -is ochre (<a href="#fig_119">fig. 119</a> will serve for this view), with pink, -light yellow and orange sides, and null points. Going -in the fourth direction, in one sense, from this plane -we pass into the tesseract, in the other sense we pass -away from it. The whole area inside the triangle is -common to the cutting plane we see, and a boundary -of the tesseract. Hence we conclude that the triangle -drawn is common to the tesseract and the cutting -space.</p> - -<p><span class="pagenum" id="Page_196">[Pg 196]</span></p> - -<div class="figleft illowp50" id="fig_120" style="max-width: 21.875em;"> - <img src="images/fig_120.png" alt="" /> - <div class="caption">Fig. 120.</div> -</div> - -<p>Now let the ochre cube turn out and the brown cube -come in. The dotted lines -show the position the ochre -cube has left (<a href="#fig_120">fig. 120</a>).</p> - -<p>Here we see three out -of the four points through -which the cutting plane -passes, null <i>r</i>, null <i>y</i>, and -null <i>b</i>. The plane they -determine lies in the cutting space, and this plane -cuts out of the brown cube a triangle with orange, -purple and green sides, and null points. The orange -line of this figure is the same as the orange line in -the last figure.</p> - -<p>Now let the light purple cube swing into our space, -towards us, <a href="#fig_121">fig. 121</a>.</p> - -<div class="figleft illowp40" id="fig_121" style="max-width: 21.875em;"> - <img src="images/fig_121.png" alt="" /> - <div class="caption">Fig. 121.</div> -</div> - -<p>The cutting space which passes through the four points, -null <i>r</i>, <i>y</i>, <i>wh</i>, <i>b</i>, passes through -the null <i>r</i>, <i>wh</i>, <i>b</i>, and therefore -the plane these determine -lies in the cutting space.</p> - -<p>This triangle lies before us. -It has a light purple interior -and pink, light blue, and -purple edges with null points.</p> - -<p>This, since it is all of the -plane that is common to it, and this bounding of the -tesseract, gives us one of the bounding faces of our sectional -figure. The pink line in it is the same as the -pink line we found in the first figure—that of the ochre -cube.</p> - -<p>Finally, let the tesseract swing about the light yellow -plane, so that the light green cube comes into our space. -It will point downwards.</p> - -<div class="figleft illowp40" id="fig_122" style="max-width: 21.875em;"> - <img src="images/fig_122.png" alt="" /> - <div class="caption">Fig. 122.</div> -</div> - -<p>The three points, <i>n.y</i>, <i>n.wh</i>, <i>n.b</i>, are in the cutting<span class="pagenum" id="Page_197">[Pg 197]</span> -space, and the triangle they determine is common to -the tesseract and the cutting -space. Hence this -boundary is a triangle having -a light yellow line, -which is the same as the -light yellow line of the first -figure, a light blue line and -a green line.</p> - -<p>We have now traced the -cutting space between every -set of three that can be -made out of the four points -in which it cuts the tesseract, and have got four faces -which all join on to each other by lines.</p> - -<div class="figleft illowp35" id="fig_123" style="max-width: 18.75em;"> - <img src="images/fig_123.png" alt="" /> - <div class="caption">Fig. 123.</div> -</div> - -<p>The triangles are shown in <a href="#fig_123">fig. 123</a> as they join on to -the triangle in the ochre cube. But -they join on each to the other in an -exactly similar manner; their edges -are all identical two and two. They -form a closed figure, a tetrahedron, -enclosing a light brown portion which -is the portion of the cutting space -which lies inside the tesseract.</p> - -<p>We cannot expect to see this light brown portion, any -more than a plane being could expect to see the inside -of a cube if an angle of it were pushed through his -plane. All he can do is to come upon the boundaries -of it in a different way to that in which he would if it -passed straight through his plane.</p> - -<p>Thus in this solid section; the whole interior lies perfectly -open in the fourth dimension. Go round it as -we may we are simply looking at the boundaries of the -tesseract which penetrates through our solid sheet. If -the tesseract were not to pass across so far, the triangle<span class="pagenum" id="Page_198">[Pg 198]</span> -would be smaller; if it were to pass farther, we should -have a different figure, the outlines of which can be -determined in a similar manner.</p> - -<p>The preceding method is open to the objection that -it depends rather on our inferring what must be, than -our seeing what is. Let us therefore consider our -sectional space as consisting of a number of planes, each -very close to the last, and observe what is to be found -in each plane.</p> - -<div class="figleft illowp40" id="fig_124" style="max-width: 21.875em;"> - <img src="images/fig_124.png" alt="" /> - <div class="caption">Fig. 124.</div> -</div> - -<p>The corresponding method in the case of two dimensions -is as follows:—The plane -being can see that line of the -sectional plane through null <i>y</i>, -null <i>wh</i>, null <i>r</i>, which lies in -the orange plane. Let him -now suppose the cube and the -section plane to pass half way -through his plane. Replacing -the red and yellow axes are lines parallel to them, sections -of the pink and light yellow faces.</p> - -<p>Where will the section plane cut these parallels to -the red and yellow axes?</p> - -<p>Let him suppose the cube, in the position of the -drawing, <a href="#fig_124">fig. 124</a>, turned so that the pink face lies -against his plane. He can see the line from the null <i>r</i> -point to the null <i>wh</i> point, and can see (compare <a href="#fig_119">fig. 119</a>) -that it cuts <span class="allsmcap">AB</span> a parallel to his red axis, drawn at a point -half way along the white line, in a point <span class="allsmcap">B</span>, half way up. -I shall speak of the axis as having the length of an edge -of the cube. Similarly, by letting the cube turn so that -the light yellow square swings against his plane, he can -see (compare <a href="#fig_119">fig. 119</a>) that a parallel to his yellow axis -drawn from a point half-way along the white axis, is cut -at half its length by the trace of the section plane in the -light yellow face.</p> - -<p><span class="pagenum" id="Page_199">[Pg 199]</span></p> - -<p>Hence when the cube had passed half-way through he -would have—instead of the orange line with null points, -which he had at first—an ochre line of half its length, -with pink and light yellow points. Thus, as the cube -passed slowly through his plane, he would have a succession -of lines gradually diminishing in length and -forming an equilateral triangle. The whole interior would -be ochre, the line from which it started would be orange. -The succession of points at the ends of the succeeding -lines would form pink and light yellow lines and the -final point would be null. Thus looking at the successive -lines in the section plane as it and the cube passed across -his plane he would determine the figure cut out bit -by bit.</p> - -<p>Coming now to the section of the tesseract, let us -imagine that the tesseract and its cutting <i>space</i> pass -slowly across our space; we can examine portions of it, -and their relation to portions of the cutting space. Take -the section space which passes through the four points, -null <i>r</i>, <i>wh</i>, <i>y</i>, <i>b</i>; we can see in the ochre cube (<a href="#fig_119">fig. 119</a>) -the plane belonging to this section space, which passes -through the three extremities of the red, white, yellow -axes.</p> - -<p>Now let the tesseract pass half way through our space. -Instead of our original axes we have parallels to them, -purple, light blue, and green, each of the same length as -the first axes, for the section of the tesseract is of exactly -the same shape as its ochre cube.</p> - -<p>But the sectional space seen at this stage of the transference -would not cut the section of the tesseract in a -plane disposed as at first.</p> - -<p>To see where the sectional space would cut these -parallels to the original axes let the tesseract swing so -that, the orange face remaining stationary, the blue line -comes in to the left.</p> - -<p><span class="pagenum" id="Page_200">[Pg 200]</span></p> - -<div class="figleft illowp45" id="fig_125" style="max-width: 25em;"> - <img src="images/fig_125.png" alt="" /> - <div class="caption">Fig. 125.</div> -</div> - -<p>Here (<a href="#fig_125">fig. 125</a>) we have the null <i>r</i>, <i>y</i>, <i>b</i> points, and of -the sectional space all we -see is the plane through these -three points in it.</p> - -<p>In this figure we can draw -the parallels to the red and -yellow axes and see that, if -they started at a point half -way along the blue axis, they -would each be cut at a point so as to be half of their -previous length.</p> - -<p>Swinging the tesseract into our space about the pink -face of the ochre cube we likewise find that the parallel -to the white axis is cut at half its length by the sectional -space.</p> - -<div class="figleft illowp40" id="fig_126" style="max-width: 25em;"> - <img src="images/fig_126.png" alt="" /> - <div class="caption">Fig. 126.</div> -</div> - -<p>Hence in a section made when the tesseract had passed -half across our space the parallels to the red, white, yellow -axes, which are now in our -space, are cut by the section -space, each of them half way -along, and for this stage of -the traversing motion we -should have <a href="#fig_126">fig. 126</a>. The -section made of this cube by -the plane in which the sectional -space cuts it, is an -equilateral triangle with purple, l. blue, green points, and -l. purple, brown, l. green lines.</p> - -<p>Thus the original ochre triangle, with null points and -pink, orange, light yellow lines, would be succeeded by a -triangle coloured in manner just described.</p> - -<p>This triangle would initially be only a very little smaller -than the original triangle, it would gradually diminish, -until it ended in a point, a null point. Each of its -edges would be of the same length. Thus the successive<span class="pagenum" id="Page_201">[Pg 201]</span> -sections of the successive planes into which we analyse the -cutting space would be a tetrahedron of the description -shown (<a href="#fig_123">fig. 123</a>), and the whole interior of the tetrahedron -would be light brown.</p> - -<div class="figcenter illowp100" id="fig_127" style="max-width: 50em;"> - <img src="images/fig_127.png" alt="" /> - <div class="caption">Front view. <span class="gap8l"> The rear faces.</span><br /> -Fig. 127.</div> -</div> - - -<p>In <a href="#fig_127">fig. 127</a> the tetrahedron is represented by means of -its faces as two triangles which meet in the p. line, and -two rear triangles which join on to them, the diagonal -of the pink face being supposed to run vertically -upward.</p> - -<p>We have now reached a natural termination. The -reader may pursue the subject in further detail, but will -find no essential novelty. I conclude with an indication -as to the manner in which figures previously given may -be used in determining sections by the method developed -above.</p> - -<p>Applying this method to the tesseract, as represented -in Chapter IX., sections made by a space cutting the axes -equidistantly at any distance can be drawn, and also the -sections of tesseracts arranged in a block.</p> - -<p>If we draw a plane, cutting all four axes at a point -six units distance from null, we have a slanting space. -This space cuts the red, white, yellow axes in the<span class="pagenum" id="Page_202">[Pg 202]</span> -points <span class="allsmcap">LMN</span> (<a href="#fig_128">fig. 128</a>), and so in the region of our space -before we go off into -the fourth dimension, -we have the plane -represented by <span class="allsmcap">LMN</span> -extended. This is what -is common to the -slanting space and our -space.</p> - -<div class="figleft illowp50" id="fig_128" style="max-width: 31.25em;"> - <img src="images/fig_128.png" alt="" /> - <div class="caption">Fig. 128.</div> -</div> - -<p>This plane cuts the -ochre cube in the triangle <span class="allsmcap">EFG</span>.</p> - -<p>Comparing this with (<a href="#fig_72">fig. 72</a>) <i>oh</i>, we see that the -hexagon there drawn is part of the triangle <span class="allsmcap">EFG</span>.</p> - -<p>Let us now imagine the tesseract and the slanting -space both together to pass transverse to our space, a -distance of one unit, we have in 1<i>h</i> a section of the -tesseract, whose axes are parallels to the previous axes. -The slanting space cuts them at a distance of five units -along each. Drawing the plane through these points in -1<i>h</i> it will be found to cut the cubical section of the -tesseract in the hexagonal figure drawn. In 2<i>h</i> (<a href="#fig_72">fig. 72</a>) the -slanting space cuts the parallels to the axes at a distance -of four along each, and the hexagonal figure is the section -of this section of the tesseract by it. Finally when 3<i>h</i> -comes in the slanting space cuts the axes at a distance -of three along each, and the section is a triangle, of which -the hexagon drawn is a truncated portion. After this -the tesseract, which extends only three units in each of -the four dimensions, has completely passed transverse -of our space, and there is no more of it to be cut. Hence, -putting the plane sections together in the right relations, -we have the section determined by the particular slanting -space: namely an octahedron.</p> -<hr class="chap x-ebookmaker-drop" /> - -<div class="chapter"> -<p><span class="pagenum" id="Page_203">[Pg 203]</span></p> - -<h2 class="nobreak" id="CHAPTER_XIV">CHAPTER XIV.<a id="FNanchor_6" href="#Footnote_6" class="fnanchor">[6]</a><br /> - -<small><i>A RECAPITULATION AND EXTENSION OF -THE PHYSICAL ARGUMENT</i></small></h2></div> - -<div class="footnotes"> -<div class="footnote"> - -<p><a id="Footnote_6" href="#FNanchor_6" class="label">[6]</a> The contents of this chapter are taken from a paper read before -the Philosophical Society of Washington. The mathematical portion -of the paper has appeared in part in the Proceedings of the Royal -Irish Academy under the title, “Cayley’s formulæ of orthogonal -transformation,” Nov. 29th, 1903.</p> - -</div></div> - -<p>There are two directions of inquiry in which the -research for the physical reality of a fourth dimension -can be prosecuted. One is the investigation of the -infinitely great, the other is the investigation of the -infinitely small.</p> - -<p>By the measurement of the angles of vast triangles, -whose sides are the distances between the stars, astronomers -have sought to determine if there is any deviation from -the values given by geometrical deduction. If the angles -of a celestial triangle do not together equal two right -angles, there would be an evidence for the physical reality -of a fourth dimension.</p> - -<p>This conclusion deserves a word of explanation. If -space is really four-dimensional, certain conclusions follow -which must be brought clearly into evidence if we are to -frame the questions definitely which we put to Nature. -To account for our limitation let us assume a solid material -sheet against which we move. This sheet must stretch -alongside every object in every direction in which it -visibly moves. Every material body must slip or slide -along this sheet, not deviating from contact with it in -any motion which we can observe.</p> - -<p><span class="pagenum" id="Page_204">[Pg 204]</span></p> - -<p>The necessity for this assumption is clearly apparent, if -we consider the analogous case of a suppositionary plane -world. If there were any creatures whose experiences -were confined to a plane, we must account for their -limitation. If they were free to move in every space -direction, they would have a three-dimensional motion; -hence they must be physically limited, and the only way -in which we can conceive such a limitation to exist is by -means of a material surface against which they slide. -The existence of this surface could only be known to -them indirectly. It does not lie in any direction from -them in which the kinds of motion they know of leads -them. If it were perfectly smooth and always in contact -with every material object, there would be no difference in -their relations to it which would direct their attention to it.</p> - -<p>But if this surface were curved—if it were, say, in the -form of a vast sphere—the triangles they drew would -really be triangles of a sphere, and when these triangles -are large enough the angles diverge from the magnitudes -they would have for the same lengths of sides if the -surface were plane. Hence by the measurement of -triangles of very great magnitude a plane being might -detect a difference from the laws of a plane world in his -physical world, and so be led to the conclusion that there -was in reality another dimension to space—a third -dimension—as well as the two which his ordinary experience -made him familiar with.</p> - -<p>Now, astronomers have thought it worth while to -examine the measurements of vast triangles drawn from -one celestial body to another with a view to determine if -there is anything like a curvature in our space—that is to -say, they have tried astronomical measurements to find<span class="pagenum" id="Page_205">[Pg 205]</span> -out if the vast solid sheet against which, on the supposition -of a fourth dimension, everything slides is -curved or not. These results have been negative. The -solid sheet, if it exists, is not curved or, being curved, has -not a sufficient curvature to cause any observable deviation -from the theoretical value of the angles calculated.</p> - -<p>Hence the examination of the infinitely great leads to -no decisive criterion. If it did we should have to decide -between the present theory and that of metageometry.</p> - -<p>Coming now to the prosecution of the inquiry in the -direction of the infinitely small, we have to state the -question thus: Our laws of movement are derived from -the examination of bodies which move in three-dimensional -space. All our conceptions are founded on the supposition -of a space which is represented analytically by -three independent axes and variations along them—that -is, it is a space in which there are three independent -movements. Any motion possible in it can be compounded -out of these three movements, which we may call: up, -right, away.</p> - -<p>To examine the actions of the very small portions of -matter with the view of ascertaining if there is any -evidence in the phenomena for the supposition of a fourth -dimension of space, we must commence by clearly defining -what the laws of mechanics would be on the supposition -of a fourth dimension. It is of no use asking if the -phenomena of the smallest particles of matter are like—we -do not know what. We must have a definite conception -of what the laws of motion would be on the -supposition of the fourth dimension, and then inquire if -the phenomena of the activity of the smaller particles of -matter resemble the conceptions which we have elaborated.</p> - -<p>Now, the task of forming these conceptions is by no -means one to be lightly dismissed. Movement in space -has many features which differ entirely from movement<span class="pagenum" id="Page_206">[Pg 206]</span> -on a plane; and when we set about to form the conception -of motion in four dimensions, we find that there -is at least as great a step as from the plane to three-dimensional -space.</p> - -<p>I do not say that the step is difficult, but I want to -point out that it must be taken. When we have formed -the conception of four-dimensional motion, we can ask a -rational question of Nature. Before we have elaborated -our conceptions we are asking if an unknown is like an -unknown—a futile inquiry.</p> - -<p>As a matter of fact, four-dimensional movements are in -every way simple and more easy to calculate than three-dimensional -movements, for four-dimensional movements -are simply two sets of plane movements put together.</p> - -<p>Without the formation of an experience of four-dimensional -bodies, their shapes and motions, the subject -can be but formal—logically conclusive, not intuitively -evident. It is to this logical apprehension that I must -appeal.</p> - -<p>It is perfectly simple to form an experiential familiarity -with the facts of four-dimensional movement. The -method is analogous to that which a plane being would -have to adopt to form an experiential familiarity with -three-dimensional movements, and may be briefly -summed up as the formation of a compound sense by -means of which duration is regarded as equivalent to -extension.</p> - -<p>Consider a being confined to a plane. A square enclosed -by four lines will be to him a solid, the interior of which -can only be examined by breaking through the lines. -If such a square were to pass transverse to his plane, it -would immediately disappear. It would vanish, going in -no direction to which he could point.</p> - -<p>If, now, a cube be placed in contact with his plane, its -surface of contact would appear like the square which we<span class="pagenum" id="Page_207">[Pg 207]</span> -have just mentioned. But if it were to pass transverse to -his plane, breaking through it, it would appear as a lasting -square. The three-dimensional matter will give a lasting -appearance in circumstances under which two-dimensional -matter will at once disappear.</p> - -<p>Similarly, a four-dimensional cube, or, as we may call -it, a tesseract, which is generated from a cube by a -movement of every part of the cube in a fourth direction -at right angles to each of the three visible directions in -the cube, if it moved transverse to our space, would -appear as a lasting cube.</p> - -<p>A cube of three-dimensional matter, since it extends to -no distance at all in the fourth dimension, would instantly -disappear, if subjected to a motion transverse to our space. -It would disappear and be gone, without it being possible -to point to any direction in which it had moved.</p> - -<p>All attempts to visualise a fourth dimension are futile. It -must be connected with a time experience in three space.</p> - -<p>The most difficult notion for a plane being to acquire -would be that of rotation about a line. Consider a plane -being facing a square. If he were told that rotation -about a line were possible, he would move his square this -way and that. A square in a plane can rotate about a -point, but to rotate about a line would seem to the plane -being perfectly impossible. How could those parts of his -square which were on one side of an edge come to the -other side without the edge moving? He could understand -their reflection in the edge. He could form an -idea of the looking-glass image of his square lying on the -opposite side of the line of an edge, but by no motion -that he knows of can he make the actual square assume -that position. The result of the rotation would be like -reflection in the edge, but it would be a physical impossibility -to produce it in the plane.</p> - -<p>The demonstration of rotation about a line must be to<span class="pagenum" id="Page_208">[Pg 208]</span> -him purely formal. If he conceived the notion of a cube -stretching out in an unknown direction away from his -plane, then he can see the base of it, his square in the -plane, rotating round a point. He can likewise apprehend -that every parallel section taken at successive intervals in -the unknown direction rotates in like manner round a -point. Thus he would come to conclude that the whole -body rotates round a line—the line consisting of the -succession of points round which the plane sections rotate. -Thus, given three axes, <i>x</i>, <i>y</i>, <i>z</i>, if <i>x</i> rotates to take -the place of <i>y</i>, and <i>y</i> turns so as to point to negative <i>x</i>, -then the third axis remaining unaffected by this turning -is the axis about which the rotation takes place. This, -then, would have to be his criterion of the axis of a -rotation—that which remains unchanged when a rotation -of every plane section of a body takes place.</p> - -<p>There is another way in which a plane being can think -about three-dimensional movements; and, as it affords -the type by which we can most conveniently think about -four-dimensional movements, it will be no loss of time to -consider it in detail.</p> -<div class="figleft illowp30" id="fig_129" style="max-width: 18.75em;"> - <img src="images/fig_129.png" alt="" /> - <div class="caption">Fig. 1 (129).</div> -</div> - -<p>We can represent the plane being and his object by -figures cut out of paper, which slip on a smooth surface. -The thickness of these bodies must be taken as so minute -that their extension in the third dimension -escapes the observation of the -plane being, and he thinks about them -as if they were mathematical plane -figures in a plane instead of being -material bodies capable of moving on -a plane surface. Let <span class="allsmcap">A</span><i>x</i>, <span class="allsmcap">A</span><i>y</i> be two -axes and <span class="allsmcap">ABCD</span> a square. As far as -movements in the plane are concerned, the square can -rotate about a point <span class="allsmcap">A</span>, for example. It cannot rotate -about a side, such as <span class="allsmcap">AC</span>.</p> - -<p><span class="pagenum" id="Page_209">[Pg 209]</span></p> - -<p>But if the plane being is aware of the existence of a -third dimension he can study the movements possible in -the ample space, taking his figure portion by portion.</p> - -<p>His plane can only hold two axes. But, since it can -hold two, he is able to represent a turning into the third -dimension if he neglects one of his axes and represents the -third axis as lying in his plane. He can make a drawing -in his plane of what stands up perpendicularly from his -plane. Let <span class="allsmcap">A</span><i>z</i> be the axis, which -stands perpendicular to his plane at -<span class="allsmcap">A</span>. He can draw in his plane two -lines to represent the two axes, <span class="allsmcap">A</span><i>x</i> -and <span class="allsmcap">A</span><i>z</i>. Let Fig. 2 be this drawing. -Here the <i>z</i> axis has taken -the place of the <i>y</i> axis, and the -plane of <span class="allsmcap">A</span><i>x</i> <span class="allsmcap">A</span><i>z</i> is represented in his -plane. In this figure all that exists of the square <span class="allsmcap">ABCD</span> -will be the line <span class="allsmcap">AB</span>.</p> - -<div class="figleft illowp30" id="fig_130" style="max-width: 18.75em;"> - <img src="images/fig_130.png" alt="" /> - <div class="caption">Fig. 2 (130).</div> -</div> - -<p>The square extends from this line in the <i>y</i> direction, -but more of that direction is represented in Fig. 2. The -plane being can study the turning of the line <span class="allsmcap">AB</span> in this -diagram. It is simply a case of plane turning around the -point <span class="allsmcap">A</span>. The line <span class="allsmcap">AB</span> occupies intermediate portions like <span class="allsmcap">AB</span><sub>1</sub> -and after half a revolution will lie on <span class="allsmcap">A</span><i>x</i> produced through <span class="allsmcap">A</span>.</p> - -<p>Now, in the same way, the plane being can take -another point, <span class="allsmcap">A´</span>, and another line, <span class="allsmcap">A´B´</span>, in his square. -He can make the drawing of the two directions at <span class="allsmcap">A´</span>, one -along <span class="allsmcap">A´B´</span>, the other perpendicular to his plane. He -will obtain a figure precisely similar to Fig. 2, and will -see that, as <span class="allsmcap">AB</span> can turn around <span class="allsmcap">A</span>, so <span class="allsmcap">A´C´</span> around <span class="allsmcap">A</span>.</p> - -<p>In this turning <span class="allsmcap">AB</span> and <span class="allsmcap">A´B´</span> would not interfere with -each other, as they would if they moved in the plane -around the separate points <span class="allsmcap">A</span> and <span class="allsmcap">A´</span>.</p> - -<p>Hence the plane being would conclude that a rotation -round a line was possible. He could see his square as it<span class="pagenum" id="Page_210">[Pg 210]</span> -began to make this turning. He could see it half way -round when it came to lie on the opposite side of the line -<span class="allsmcap">AC</span>. But in intermediate portions he could not see it, -for it runs out of the plane.</p> - -<p>Coming now to the question of a four-dimensional body, -let us conceive of it as a series of cubic sections, the first -in our space, the rest at intervals, stretching away from -our space in the unknown direction.</p> - -<p>We must not think of a four-dimensional body as -formed by moving a three-dimensional body in any -direction which we can see.</p> - -<p>Refer for a moment to Fig. 3. The point <span class="allsmcap">A</span>, moving to -the right, traces out the line <span class="allsmcap">AC</span>. The line <span class="allsmcap">AC</span>, moving -away in a new direction, traces out the square <span class="allsmcap">ACEG</span> at -the base of the cube. The square <span class="allsmcap">AEGC</span>, moving in a -new direction, will trace out the cube <span class="allsmcap">ACEGBDHF</span>. The -vertical direction of this last motion is not identical with -any motion possible in the plane of the base of the cube. -It is an entirely new direction, at right angles to every -line that can be drawn in the base. To trace out a -tesseract the cube must move in a new direction—a -direction at right angles to any and every line that can -be drawn in the space of the cube.</p> - -<p>The cubic sections of the tesseract are related to the -cube we see, as the square sections of the cube are related -to the square of its base which a plane being sees.</p> - -<p>Let us imagine the cube in our space, which is the base -of a tesseract, to turn about one of its edges. The rotation -will carry the whole body with it, and each of the cubic -sections will rotate. The axis we see in our space will -remain unchanged, and likewise the series of axes parallel -to it about which each of the parallel cubic sections -rotates. The assemblage of all of these is a plane.</p> - -<p>Hence in four dimensions a body rotates about a plane. -There is no such thing as rotation round an axis.</p> - -<p><span class="pagenum" id="Page_211">[Pg 211]</span></p> - -<p>We may regard the rotation from a different point of -view. Consider four independent axes each at right -angles to all the others, drawn in a four-dimensional body. -Of these four axes we can see any three. The fourth -extends normal to our space.</p> - -<p>Rotation is the turning of one axis into a second, and -the second turning to take the place of the negative of -the first. It involves two axes. Thus, in this rotation of -a four-dimensional body, two axes change and two remain -at rest. Four-dimensional rotation is therefore a turning -about a plane.</p> - -<p>As in the case of a plane being, the result of rotation -about a line would appear as the production of a looking-glass -image of the original object on the other side of the -line, so to us the result of a four-dimensional rotation -would appear like the production of a looking-glass image -of a body on the other side of a plane. The plane would -be the axis of the rotation, and the path of the body -between its two appearances would be unimaginable in -three-dimensional space.</p> - -<div class="figleft illowp30" id="fig_131" style="max-width: 18.75em;"> - <img src="images/fig_131.png" alt="" /> - <div class="caption">Fig. 3 (131).</div> -</div> - -<p>Let us now apply the method by which a plane being -could examine the nature of rotation -about a line in our examination -of rotation about a plane. Fig. 3 -represents a cube in our space, the -three axes <i>x</i>, <i>y</i>, <i>z</i> denoting its -three dimensions. Let <i>w</i> represent -the fourth dimension. Now, since -in our space we can represent any -three dimensions, we can, if we -choose, make a representation of what is in the space -determined by the three axes <i>x</i>, <i>z</i>, <i>w</i>. This is a three-dimensional -space determined by two of the axes we have -drawn, <i>x</i> and <i>z</i>, and in place of <i>y</i> the fourth axis, <i>w</i>. We -cannot, keeping <i>x</i> and <i>z</i>, have both <i>y</i> and <i>w</i> in our space;<span class="pagenum" id="Page_212">[Pg 212]</span> -so we will let <i>y</i> go and draw <i>w</i> in its place. What will be -our view of the cube?</p> - -<div class="figleft illowp30" id="fig_132" style="max-width: 18.75em;"> - <img src="images/fig_132.png" alt="" /> - <div class="caption">Fig. 4 (132).</div> -</div> - -<p>Evidently we shall have simply the square that is in -the plane of <i>xz</i>, the square <span class="allsmcap">ACDB</span>. -The rest of the cube stretches in -the <i>y</i> direction, and, as we have -none of the space so determined, -we have only the face of the cube. -This is represented in <a href="#fig_132">fig. 4</a>.</p> - -<p>Now, suppose the whole cube to -be turned from the <i>x</i> to the <i>w</i> -direction. Conformably with our method, we will not -take the whole of the cube into consideration at once, but -will begin with the face <span class="allsmcap">ABCD</span>.</p> - -<div class="figleft illowp30" id="fig_133" style="max-width: 18.75em;"> - <img src="images/fig_133.png" alt="" /> - <div class="caption">Fig. 5 (133).</div> -</div> - -<p>Let this face begin to turn. Fig. 5 -represents one of the positions it will -occupy; the line <span class="allsmcap">AB</span> remains on the -<i>z</i> axis. The rest of the face extends -between the <i>x</i> and the <i>w</i> direction.</p> - -<p>Now, since we can take any three -axes, let us look at what lies in -the space of <i>zyw</i>, and examine the -turning there. We must now let the <i>z</i> axis disappear -and let the <i>w</i> axis run in the direction in which the <i>z</i> ran.</p> - -<div class="figleft illowp30" id="fig_134" style="max-width: 18.75em;"> - <img src="images/fig_134.png" alt="" /> - <div class="caption">Fig. 6 (134).</div> -</div> - -<p>Making this representation, what -do we see of the cube? Obviously -we see only the lower face. The rest -of the cube lies in the space of <i>xyz</i>. -In the space of <i>xyz</i> we have merely -the base of the cube lying in the -plane of <i>xy</i>, as shown in <a href="#fig_134">fig. 6</a>.</p> - -<p>Now let the <i>x</i> to <i>w</i> turning take place. The square -<span class="allsmcap">ACEG</span> will turn about the line <span class="allsmcap">AE</span>. This edge will -remain along the <i>y</i> axis and will be stationary, however -far the square turns.</p> - -<p><span class="pagenum" id="Page_213">[Pg 213]</span></p> - -<div class="figleft illowp30" id="fig_135" style="max-width: 18.75em;"> - <img src="images/fig_135.png" alt="" /> - <div class="caption">Fig. 7 (135).</div> -</div> - -<p>Thus, if the cube be turned by an <i>x</i> to <i>w</i> turning, both -the edge <span class="allsmcap">AB</span> and the edge <span class="allsmcap">AC</span> remain -stationary; hence the whole face -<span class="allsmcap">ABEF</span> in the <i>yz</i> plane remains fixed. -The turning has taken place about -the face <span class="allsmcap">ABEF</span>.</p> - -<p>Suppose this turning to continue -till <span class="allsmcap">AC</span> runs to the left from <span class="allsmcap">A</span>. -The cube will occupy the position -shown in <a href="#fig_136">fig. 8</a>. This is the looking-glass image of the -cube in <a href="#fig_131">fig. 3</a>. By no rotation in three-dimensional space -can the cube be brought from -the position in <a href="#fig_131">fig. 3</a> to that -shown in <a href="#fig_136">fig. 8</a>.</p> - -<div class="figleft illowp40" id="fig_136" style="max-width: 21.875em;"> - <img src="images/fig_136.png" alt="" /> - <div class="caption">Fig. 8 (136).</div> -</div> - -<p>We can think of this turning -as a turning of the face <span class="allsmcap">ABCD</span> -about <span class="allsmcap">AB</span>, and a turning of each -section parallel to <span class="allsmcap">ABCD</span> round -the vertical line in which it -intersects the face <span class="allsmcap">ABEF</span>, the -space in which the turning takes place being a different -one from that in which the cube lies.</p> - -<p>One of the conditions, then, of our inquiry in the -direction of the infinitely small is that we form the conception -of a rotation about a plane. The production of a -body in a state in which it presents the appearance of a -looking-glass image of its former state is the criterion -for a four-dimensional rotation.</p> - -<p>There is some evidence for the occurrence of such transformations -of bodies in the change of bodies from those -which produce a right-handed polarisation of light to -those which produce a left-handed polarisation; but this -is not a point to which any very great importance can -be attached.</p> - -<p>Still, in this connection, let me quote a remark from<span class="pagenum" id="Page_214">[Pg 214]</span> -Prof. John G. McKendrick’s address on Physiology before -the British Association at Glasgow. Discussing the -possibility of the hereditary production of characteristics -through the material structure of the ovum, he estimates -that in it there exist 12,000,000,000 biophors, or ultimate -particles of living matter, a sufficient number to account -for hereditary transmission, and observes: “Thus it is -conceivable that vital activities may also be determined -by the kind of motion that takes place in the molecules -of that which we speak of as living matter. It may be -different in kind from some of the motions known to -physicists, and it is conceivable that life may be the -transmission to dead matter, the molecules of which have -already a special kind of motion, of a form of motion -<i>sui generis</i>.”</p> - -<p>Now, in the realm of organic beings symmetrical structures—those -with a right and left symmetry—are everywhere -in evidence. Granted that four dimensions exist, -the simplest turning produces the image form, and by a -folding-over structures could be produced, duplicated -right and left, just as is the case of symmetry in a -plane.</p> - -<p>Thus one very general characteristic of the forms of -organisms could be accounted for by the supposition that -a four-dimensional motion was involved in the process of -life.</p> - -<p>But whether four-dimensional motions correspond in -other respects to the physiologist’s demand for a special -kind of motion, or not, I do not know. Our business is -with the evidence for their existence in physics. For -this purpose it is necessary to examine into the significance -of rotation round a plane in the case of extensible -and of fluid matter.</p> - -<p>Let us dwell a moment longer on the rotation of a rigid -body. Looking at the cube in <a href="#fig_131">fig. 3</a>, which turns about<span class="pagenum" id="Page_215">[Pg 215]</span> -the face of <span class="allsmcap">ABFE</span>, we see that any line in the face can -take the place of the vertical and horizontal lines we have -examined. Take the diagonal line <span class="allsmcap">AF</span> and the section -through it to <span class="allsmcap">GH</span>. The portions of matter which were on -one side of <span class="allsmcap">AF</span> in this section in <a href="#fig_131">fig. 3</a> are on the -opposite side of it in <a href="#fig_136">fig. 8</a>. They have gone round the -line <span class="allsmcap">AF</span>. Thus the rotation round a face can be considered -as a number of rotations of sections round parallel lines -in it.</p> - -<p>The turning about two different lines is impossible in -three-dimensional space. To take another illustration, -suppose <span class="allsmcap">A</span> and <span class="allsmcap">B</span> are two parallel lines in the <i>xy</i> plane, -and let <span class="allsmcap">CD</span> and <span class="allsmcap">EF</span> be two rods crossing them. Now, in -the space of <i>xyz</i> if the rods turn round the lines <span class="allsmcap">A</span> and <span class="allsmcap">B</span> -in the same direction they -will make two independent -circles.</p> - -<div class="figleft illowp40" id="fig_137" style="max-width: 21.875em;"> - <img src="images/fig_137.png" alt="" /> - <div class="caption">Fig. 9 (137).</div> -</div> - -<p>When the end <span class="allsmcap">F</span> is going -down the end <span class="allsmcap">C</span> will be coming -up. They will meet and conflict.</p> - -<p>But if we rotate the rods -about the plane of <span class="allsmcap">AB</span> by the -<i>z</i> to <i>w</i> rotation these movements -will not conflict. Suppose -all the figure removed -with the exception of the plane <i>xz</i>, and from this plane -draw the axis of <i>w</i>, so that we are looking at the space -of <i>xzw</i>.</p> - -<p>Here, <a href="#fig_138">fig. 10</a>, we cannot see the lines <span class="allsmcap">A</span> and <span class="allsmcap">B</span>. We -see the points <span class="allsmcap">G</span> and <span class="allsmcap">H</span>, in which <span class="allsmcap">A</span> and <span class="allsmcap">B</span> intercept -the <i>x</i> axis, but we cannot see the lines themselves, for -they run in the <i>y</i> direction, and that is not in our -drawing.</p> - -<p>Now, if the rods move with the <i>z</i> to <i>w</i> rotation they will<span class="pagenum" id="Page_216">[Pg 216]</span> -turn in parallel planes, keeping their relative positions. -The point <span class="allsmcap">D</span>, for instance, will -describe a circle. At one time -it will be above the line <span class="allsmcap">A</span>, at -another time below it. Hence -it rotates round <span class="allsmcap">A</span>.</p> - -<div class="figleft illowp40" id="fig_138" style="max-width: 21.875em;"> - <img src="images/fig_138.png" alt="" /> - <div class="caption">Fig. 10 (138).</div> -</div> - -<p>Not only two rods but any -number of rods crossing the -plane will move round it harmoniously. -We can think of -this rotation by supposing the -rods standing up from one line -to move round that line and remembering that it is -not inconsistent with this rotation for the rods standing -up along another line also to move round it, the relative -positions of all the rods being preserved. Now, if the -rods are thick together, they may represent a disk of -matter, and we see that a disk of matter can rotate -round a central plane.</p> - -<p>Rotation round a plane is exactly analogous to rotation -round an axis in three dimensions. If we want a rod to -turn round, the ends must be free; so if we want a disk -of matter to turn round its central plane by a four-dimensional -turning, all the contour must be free. The whole -contour corresponds to the ends of the rod. Each point -of the contour can be looked on as the extremity of an -axis in the body, round each point of which there is a -rotation of the matter in the disk.</p> - -<p>If the one end of a rod be clamped, we can twist the -rod, but not turn it round; so if any part of the contour -of a disk is clamped we can impart a twist to the disk, -but not turn it round its central plane. In the case of -extensible materials a long, thin rod will twist round its -axis, even when the axis is curved, as, for instance, in the -case of a ring of India rubber.</p> - -<p><span class="pagenum" id="Page_217">[Pg 217]</span></p> - -<p>In an analogous manner, in four dimensions we can have -rotation round a curved plane, if I may use the expression. -A sphere can be turned inside out in four dimensions.</p> - -<div class="figleft illowp45" id="fig_139" style="max-width: 25em;"> - <img src="images/fig_139.png" alt="" /> - <div class="caption">Fig. 11 (139).</div> -</div> - -<p>Let <a href="#fig_139">fig. 11</a> represent a -spherical surface, on each -side of which a layer of -matter exists. The thickness -of the matter is represented -by the rods <span class="allsmcap">CD</span> and -<span class="allsmcap">EF</span>, extending equally without -and within.</p> - -<p>Now, take the section of -the sphere by the <i>yz</i> plane -we have a circle—<a href="#fig_140">fig. 12</a>. -Now, let the <i>w</i> axis be drawn -in place of the <i>x</i> axis so that -we have the space of <i>yzw</i> -represented. In this space all that there will be seen of -the sphere is the circle drawn.</p> - -<div class="figleft illowp45" id="fig_140" style="max-width: 25em;"> - <img src="images/fig_140.png" alt="" /> - <div class="caption">Fig. 12 (140).</div> -</div> - -<p>Here we see that there is no obstacle to prevent the -rods turning round. If -the matter is so elastic -that it will give enough -for the particles at <span class="allsmcap">E</span> and -<span class="allsmcap">C</span> to be separated as they -are at <span class="allsmcap">F</span> and <span class="allsmcap">D</span>, they -can rotate round to the -position <span class="allsmcap">D</span> and <span class="allsmcap">F</span>, and a -similar motion is possible -for all other particles. -There is no matter or -obstacle to prevent them -from moving out in the -<i>w</i> direction, and then on round the circumference as an -axis. Now, what will hold for one section will hold for<span class="pagenum" id="Page_218">[Pg 218]</span> -all, as the fourth dimension is at right angles to all the -sections which can be made of the sphere.</p> - -<p>We have supposed the matter of which the sphere is -composed to be three-dimensional. If the matter had a -small thickness in the fourth dimension, there would be -a slight thickness in <a href="#fig_140">fig. 12</a> above the plane of the paper—a -thickness equal to the thickness of the matter in the -fourth dimension. The rods would have to be replaced -by thin slabs. But this would make no difference as to -the possibility of the rotation. This motion is discussed -by Newcomb in the first volume of the <i>American Journal -of Mathematics</i>.</p> - -<p>Let us now consider, not a merely extensible body, but -a liquid one. A mass of rotating liquid, a whirl, eddy, -or vortex, has many remarkable properties. On first -consideration we should expect the rotating mass of -liquid immediately to spread off and lose itself in the -surrounding liquid. The water flies off a wheel whirled -round, and we should expect the rotating liquid to be -dispersed. But see the eddies in a river strangely persistent. -The rings that occur in puffs of smoke and last -so long are whirls or vortices curved round so that their -opposite ends join together. A cyclone will travel over -great distances.</p> - -<p>Helmholtz was the first to investigate the properties of -vortices. He studied them as they would occur in a perfect -fluid—that is, one without friction of one moving portion -or another. In such a medium vortices would be indestructible. -They would go on for ever, altering their -shape, but consisting always of the same portion of the -fluid. But a straight vortex could not exist surrounded -entirely by the fluid. The ends of a vortex must reach to -some boundary inside or outside the fluid.</p> - -<p>A vortex which is bent round so that its opposite ends -join is capable of existing, but no vortex has a free end in<span class="pagenum" id="Page_219">[Pg 219]</span> -the fluid. The fluid round the vortex is always in motion, -and one produces a definite movement in another.</p> - -<p>Lord Kelvin has proposed the hypothesis that portions -of a fluid segregated in vortices account for the origin of -matter. The properties of the ether in respect of its -capacity of propagating disturbances can be explained -by the assumption of vortices in it instead of by a property -of rigidity. It is difficult to conceive, however, -of any arrangement of the vortex rings and endless vortex -filaments in the ether.</p> - -<p>Now, the further consideration of four-dimensional -rotations shows the existence of a kind of vortex which -would make an ether filled with a homogeneous vortex -motion easily thinkable.</p> - -<p>To understand the nature of this vortex, we must go -on and take a step by which we accept the full significance -of the four-dimensional hypothesis. Granted four-dimensional -axes, we have seen that a rotation of one into -another leaves two unaltered, and these two form the -axial plane about which the rotation takes place. But -what about these two? Do they necessarily remain -motionless? There is nothing to prevent a rotation of -these two, one into the other, taking place concurrently -with the first rotation. This possibility of a double -rotation deserves the most careful attention, for it is the -kind of movement which is distinctly typical of four -dimensions.</p> - -<p>Rotation round a plane is analogous to rotation round -an axis. But in three-dimensional space there is no -motion analogous to the double rotation, in which, while -axis 1 changes into axis 2, axis 3 changes into axis 4.</p> - -<p>Consider a four-dimensional body, with four independent -axes, <i>x</i>, <i>y</i>, <i>z</i>, <i>w</i>. A point in it can move in only one -direction at a given moment. If the body has a velocity -of rotation by which the <i>x</i> axis changes into the <i>y</i> axis<span class="pagenum" id="Page_220">[Pg 220]</span> -and all parallel sections move in a similar manner, then -the point will describe a circle. If, now, in addition to -the rotation by which the <i>x</i> axis changes into the <i>y</i> axis the -body has a rotation by which the <i>z</i> axis turns into the -<i>w</i> axis, the point in question will have a double motion -in consequence of the two turnings. The motions will -compound, and the point will describe a circle, but not -the same circle which it would describe in virtue of either -rotation separately.</p> - -<p>We know that if a body in three-dimensional space is -given two movements of rotation they will combine into a -single movement of rotation round a definite axis. It is -in no different condition from that in which it is subjected -to one movement of rotation. The direction of -the axis changes; that is all. The same is not true about -a four-dimensional body. The two rotations, <i>x</i> to <i>y</i> and -<i>z</i> to <i>w</i>, are independent. A body subject to the two is in -a totally different condition to that which it is in when -subject to one only. When subject to a rotation such as -that of <i>x</i> to <i>y</i>, a whole plane in the body, as we have -seen, is stationary. When subject to the double rotation -no part of the body is stationary except the point common -to the two planes of rotation.</p> - -<p>If the two rotations are equal in velocity, every point -in the body describes a circle. All points equally distant -from the stationary point describe circles of equal size.</p> - -<p>We can represent a four-dimensional sphere by means -of two diagrams, in one of which we take the three axes, -<i>x</i>, <i>y</i>, <i>z</i>; in the other the axes <i>x</i>, <i>w</i>, and <i>z</i>. In <a href="#fig_141">fig. 13</a> we -have the view of a four-dimensional sphere in the space of -<i>xyz</i>. Fig. 13 shows all that we can see of the four -sphere in the space of <i>xyz</i>, for it represents all the -points in that space, which are at an equal distance from -the centre.</p> - -<p>Let us now take the <i>xz</i> section, and let the axis of <i>w</i><span class="pagenum" id="Page_221">[Pg 221]</span> -take the place of the <i>y</i> axis. Here, in <a href="#fig_142">fig. 14</a>, we have -the space of <i>xzw</i>. In this space we have to take all the -points which are at the same distance from the centre, -consequently we have another sphere. If we had a three-dimensional -sphere, as has been shown before, we should -have merely a circle in the <i>xzw</i> space, the <i>xz</i> circle seen -in the space of <i>xzw</i>. But now, taking the view in the -space of <i>xzw</i>, we have a sphere in that space also. In a -similar manner, whichever set of three axes we take, we -obtain a sphere.</p> - -<div class="figleft illowp40" id="fig_141" style="max-width: 28.125em;"> - <img src="images/fig_141.png" alt="" /> - <div class="caption"><i>Showing axes xyz</i><br /> -Fig. 13 (141).</div> -</div> - -<div class="figright illowp40" id="fig_142" style="max-width: 28.125em;"> - <img src="images/fig_142.png" alt="" /> - <div class="caption"><i>Showing axes xwz</i><br /> -Fig. 14 (142).</div> -</div> - -<p>In <a href="#fig_141">fig. 13</a>, let us imagine the rotation in the direction -<i>xy</i> to be taking place. The point <i>x</i> will turn to <i>y</i>, and <i>p</i> -to <i>p´</i>. The axis <i>zz´</i> remains stationary, and this axis is all -of the plane <i>zw</i> which we can see in the space section -exhibited in the figure.</p> - -<p>In <a href="#fig_142">fig. 14</a>, imagine the rotation from <i>z</i> to <i>w</i> to be taking -place. The <i>w</i> axis now occupies the position previously -occupied by the <i>y</i> axis. This does not mean that the -<i>w</i> axis can coincide with the <i>y</i> axis. It indicates that we -are looking at the four-dimensional sphere from a different -point of view. Any three-space view will show us three -axes, and in <a href="#fig_142">fig. 14</a> we are looking at <i>xzw</i>.</p> - -<p>The only part that is identical in the two diagrams is -the circle of the <i>x</i> and <i>z</i> axes, which axes are contained -in both diagrams. Thus the plane <i>zxz´</i> is the same in -both, and the point <i>p</i> represents the same point in both<span class="pagenum" id="Page_222">[Pg 222]</span> -diagrams. Now, in <a href="#fig_142">fig. 14</a> let the <i>zw</i> rotation take place, -the <i>z</i> axis will turn toward the point <i>w</i> of the <i>w</i> axis, and -the point <i>p</i> will move in a circle about the point <i>x</i>.</p> - -<p>Thus in <a href="#fig_141">fig. 13</a> the point <i>p</i> moves in a circle parallel to -the <i>xy</i> plane; in <a href="#fig_142">fig. 14</a> it moves in a circle parallel to the -<i>zw</i> plane, indicated by the arrow.</p> - -<p>Now, suppose both of these independent rotations compounded, -the point <i>p</i> will move in a circle, but this circle -will coincide with neither of the circles in which either -one of the rotations will take it. The circle the point <i>p</i> -will move in will depend on its position on the surface of -the four sphere.</p> - -<p>In this double rotation, possible in four-dimensional -space, there is a kind of movement totally unlike any -with which we are familiar in three-dimensional space. -It is a requisite preliminary to the discussion of the -behaviour of the small particles of matter, with a view to -determining whether they show the characteristics of four-dimensional -movements, to become familiar with the main -characteristics of this double rotation. And here I must -rely on a formal and logical assent rather than on the -intuitive apprehension, which can only be obtained by a -more detailed study.</p> - -<p>In the first place this double rotation consists in two -varieties or kinds, which we will call the A and B kinds. -Consider four axes, <i>x</i>, <i>y</i>, <i>z</i>, <i>w</i>. The rotation of <i>x</i> to <i>y</i> can -be accompanied with the rotation of <i>z</i> to <i>w</i>. Call this -the A kind.</p> - -<p>But also the rotation of <i>x</i> to <i>y</i> can be accompanied by -the rotation, of not <i>z</i> to <i>w</i>, but <i>w</i> to <i>z</i>. Call this the -B kind.</p> - -<p>They differ in only one of the component rotations. One -is not the negative of the other. It is the semi-negative. -The opposite of an <i>x</i> to <i>y</i>, <i>z</i> to <i>w</i> rotation would be <i>y</i> to <i>x</i>, -<i>w</i> to <i>z</i>. The semi-negative is <i>x</i> to <i>y</i> and <i>w</i> to <i>z</i>.</p> - -<p><span class="pagenum" id="Page_223">[Pg 223]</span></p> - -<p>If four dimensions exist and we cannot perceive them, -because the extension of matter is so small in the fourth -dimension that all movements are withheld from direct -observation except those which are three-dimensional, we -should not observe these double rotations, but only the -effects of them in three-dimensional movements of the -type with which we are familiar.</p> - -<p>If matter in its small particles is four-dimensional, -we should expect this double rotation to be a universal -characteristic of the atoms and molecules, for no portion -of matter is at rest. The consequences of this corpuscular -motion can be perceived, but only under the form -of ordinary rotation or displacement. Thus, if the theory -of four dimensions is true, we have in the corpuscles of -matter a whole world of movement, which we can never -study directly, but only by means of inference.</p> - -<p>The rotation A, as I have defined it, consists of two -equal rotations—one about the plane of <i>zw</i>, the other -about the plane of <i>xy</i>. It is evident that these rotations -are not necessarily equal. A body may be moving with a -double rotation, in which these two independent components -are not equal; but in such a case we can consider -the body to be moving with a composite rotation—a -rotation of the A or B kind and, in addition, a rotation -about a plane.</p> - -<p>If we combine an A and a B movement, we obtain a -rotation about a plane; for, the first being <i>x</i> to <i>y</i> and -<i>z</i> to <i>w</i>, and the second being <i>x</i> to <i>y</i> and <i>w</i> to <i>z</i>, when they -are put together the <i>z</i> to <i>w</i> and <i>w</i> to <i>z</i> rotations neutralise -each other, and we obtain an <i>x</i> to <i>y</i> rotation only, which -is a rotation about the plane of <i>zw</i>. Similarly, if we -take a B rotation, <i>y</i> to <i>x</i> and <i>z</i> to <i>w</i>, we get, on combining -this with the A rotation, a rotation of <i>z</i> to <i>w</i> about the -<i>xy</i> plane. In this case the plane of rotation is in the -three-dimensional space of <i>xyz</i>, and we have—what has<span class="pagenum" id="Page_224">[Pg 224]</span> -been described before—a twisting about a plane in our -space.</p> - -<p>Consider now a portion of a perfect liquid having an A -motion. It can be proved that it possesses the properties -of a vortex. It forms a permanent individuality—a -separated-out portion of the liquid—accompanied by a -motion of the surrounding liquid. It has properties -analogous to those of a vortex filament. But it is not -necessary for its existence that its ends should reach the -boundary of the liquid. It is self-contained and, unless -disturbed, is circular in every section.</p> - -<div class="figleft illowp50" id="fig_143" style="max-width: 28.125em;"> - <img src="images/fig_143.png" alt="" /> - <div class="caption">Fig. 15 (143).</div> -</div> - -<p>If we suppose the ether to have its properties of transmitting -vibration given it by such vortices, we must -inquire how they lie together in four-dimensional space. -Placing a circular disk on a plane and surrounding it by -six others, we find that if the central one is given a motion -of rotation, it imparts to the others a rotation which is -antagonistic in every two adjacent -ones. If <span class="allsmcap">A</span> goes round, -as shown by the arrow, <span class="allsmcap">B</span> and -<span class="allsmcap">C</span> will be moving in opposite -ways, and each tends to destroy -the motion of the other.</p> - -<p>Now, if we suppose spheres -to be arranged in a corresponding -manner in three-dimensional -space, they will -be grouped in figures which -are for three-dimensional space what hexagons are for -plane space. If a number of spheres of soft clay be -pressed together, so as to fill up the interstices, each will -assume the form of a fourteen-sided figure called a -tetrakaidecagon.</p> - -<p>Now, assuming space to be filled with such tetrakaidecagons, -and placing a sphere in each, it will be found<span class="pagenum" id="Page_225">[Pg 225]</span> -that one sphere is touched by eight others. The remaining -six spheres of the fourteen which surround the -central one will not touch it, but will touch three of -those in contact with it. Hence, if the central sphere -rotates, it will not necessarily drive those around it so -that their motions will be antagonistic to each other, -but the velocities will not arrange themselves in a -systematic manner.</p> - -<p>In four-dimensional space the figure which forms the -next term of the series hexagon, tetrakaidecagon, is a -thirty-sided figure. It has for its faces ten solid tetrakaidecagons -and twenty hexagonal prisms. Such figures -will exactly fill four-dimensional space, five of them meeting -at every point. If, now, in each of these figures we -suppose a solid four-dimensional sphere to be placed, any -one sphere is surrounded by thirty others. Of these it -touches ten, and, if it rotates, it drives the rest by means -of these. Now, if we imagine the central sphere to be -given an A or a B rotation, it will turn the whole mass of -sphere round in a systematic manner. Suppose four-dimensional -space to be filled with such spheres, each -rotating with a double rotation, the whole mass would -form one consistent system of motion, in which each one -drove every other one, with no friction or lagging behind.</p> - -<p>Every sphere would have the same kind of rotation. In -three-dimensional space, if one body drives another round -the second body rotates with the opposite kind of rotation; -but in four-dimensional space these four-dimensional -spheres would each have the double negative of the rotation -of the one next it, and we have seen that the double -negative of an A or B rotation is still an A or B rotation. -Thus four-dimensional space could be filled with a system -of self-preservative living energy. If we imagine the -four-dimensional spheres to be of liquid and not of solid -matter, then, even if the liquid were not quite perfect and<span class="pagenum" id="Page_226">[Pg 226]</span> -there were a slight retarding effect of one vortex on -another, the system would still maintain itself.</p> - -<p>In this hypothesis we must look on the ether as -possessing energy, and its transmission of vibrations, not -as the conveying of a motion imparted from without, but -as a modification of its own motion.</p> - -<p>We are now in possession of some of the conceptions of -four-dimensional mechanics, and will turn aside from the -line of their development to inquire if there is any -evidence of their applicability to the processes of nature.</p> - -<p>Is there any mode of motion in the region of the -minute which, giving three-dimensional movements for -its effect, still in itself escapes the grasp of our mechanical -theories? I would point to electricity. Through the -labours of Faraday and Maxwell we are convinced that the -phenomena of electricity are of the nature of the stress -and strain of a medium; but there is still a gap to be -bridged over in their explanation—the laws of elasticity, -which Maxwell assumes, are not those of ordinary matter. -And, to take another instance: a magnetic pole in the -neighbourhood of a current tends to move. Maxwell has -shown that the pressures on it are analogous to the -velocities in a liquid which would exist if a vortex took -the place of the electric current: but we cannot point out -the definite mechanical explanation of these pressures. -There must be some mode of motion of a body or of the -medium in virtue of which a body is said to be -electrified.</p> - -<p>Take the ions which convey charges of electricity 500 -times greater in proportion to their mass than are carried -by the molecules of hydrogen in electrolysis. In respect -of what motion can these ions be said to be electrified? -It can be shown that the energy they possess is not -energy of rotation. Think of a short rod rotating. If it -is turned over it is found to be rotating in the opposite<span class="pagenum" id="Page_227">[Pg 227]</span> -direction. Now, if rotation in one direction corresponds to -positive electricity, rotation in the opposite direction corresponds -to negative electricity, and the smallest electrified -particles would have their charges reversed by being -turned over—an absurd supposition.</p> - -<p>If we fix on a mode of motion as a definition of -electricity, we must have two varieties of it, one for -positive and one for negative; and a body possessing the -one kind must not become possessed of the other by any -change in its position.</p> - -<p>All three-dimensional motions are compounded of rotations -and translations, and none of them satisfy this first -condition for serving as a definition of electricity.</p> - -<p>But consider the double rotation of the A and B kinds. -A body rotating with the A motion cannot have its -motion transformed into the B kind by being turned over -in any way. Suppose a body has the rotation <i>x</i> to <i>y</i> and -<i>z</i> to <i>w</i>. Turning it about the <i>xy</i> plane, we reverse the -direction of the motion <i>x</i> to <i>y</i>. But we also reverse the -<i>z</i> to <i>w</i> motion, for the point at the extremity of the -positive <i>z</i> axis is now at the extremity of the negative <i>z</i> -axis, and since we have not interfered with its motion it -goes in the direction of position <i>w</i>. Hence we have <i>y</i> to -<i>x</i> and <i>w</i> to <i>z</i>, which is the same as <i>x</i> to <i>y</i> and <i>z</i> to <i>w</i>. -Thus both components are reversed, and there is the A -motion over again. The B kind is the semi-negative, -with only one component reversed.</p> - -<p>Hence a system of molecules with the A motion would -not destroy it in one another, and would impart it to a -body in contact with them. Thus A and B motions -possess the first requisite which must be demanded in -any mode of motion representative of electricity.</p> - -<p>Let us trace out the consequences of defining positive -electricity as an A motion and negative electricity as a B -motion. The combination of positive and negative<span class="pagenum" id="Page_228">[Pg 228]</span> -electricity produces a current. Imagine a vortex in the -ether of the A kind and unite with this one of the B kind. -An A motion and B motion produce rotation round a plane, -which is in the ether a vortex round an axial surface. -It is a vortex of the kind we represent as a part of a -sphere turning inside out. Now such a vortex must have -its rim on a boundary of the ether—on a body in the -ether.</p> - -<p>Let us suppose that a conductor is a body which has -the property of serving as the terminal abutment of such -a vortex. Then the conception we must form of a closed -current is of a vortex sheet having its edge along the -circuit of the conducting wire. The whole wire will then -be like the centres on which a spindle turns in three-dimensional -space, and any interruption of the continuity -of the wire will produce a tension in place of a continuous -revolution.</p> - -<p>As the direction of the rotation of the vortex is from a -three-space direction into the fourth dimension and back -again, there will be no direction of flow to the current; -but it will have two sides, according to whether <i>z</i> goes -to <i>w</i> or <i>z</i> goes to negative <i>w</i>.</p> - -<p>We can draw any line from one part of the circuit to -another; then the ether along that line is rotating round -its points.</p> - -<p>This geometric image corresponds to the definition of -an electric circuit. It is known that the action does not -lie in the wire, but in the medium, and it is known that -there is no direction of flow in the wire.</p> - -<p>No explanation has been offered in three-dimensional -mechanics of how an action can be impressed throughout -a region and yet necessarily run itself out along a closed -boundary, as is the case in an electric current. But this -phenomenon corresponds exactly to the definition of a -four-dimensional vortex.</p> - -<p><span class="pagenum" id="Page_229">[Pg 229]</span></p> - -<p>If we take a very long magnet, so long that one of its -poles is practically isolated, and put this pole in the -vicinity of an electric circuit, we find that it moves.</p> - -<p>Now, assuming for the sake of simplicity that the wire -which determines the current is in the form of a circle, -if we take a number of small magnets and place them all -pointing in the same direction normal to the plane of the -circle, so that they fill it and the wire binds them round, -we find that this sheet of magnets has the same effect on -the magnetic pole that the current has. The sheet of -magnets may be curved, but the edge of it must coincide -with the wire. The collection of magnets is then -equivalent to the vortex sheet, and an elementary magnet -to a part of it. Thus, we must think of a magnet as -conditioning a rotation in the ether round the plane -which bisects at right angles the line joining its poles.</p> - -<p>If a current is started in a circuit, we must imagine -vortices like bowls turning themselves inside out, starting -from the contour. In reaching a parallel circuit, if the -vortex sheet were interrupted and joined momentarily to -the second circuit by a free rim, the axis plane would lie -between the two circuits, and a point on the second circuit -opposite a point on the first would correspond to a point -opposite to it on the first; hence we should expect a -current in the opposite direction in the second circuit. -Thus the phenomena of induction are not inconsistent -with the hypothesis of a vortex about an axial plane.</p> - -<p>In four-dimensional space, in which all four dimensions -were commensurable, the intensity of the action transmitted -by the medium would vary inversely as the cube of the -distance. Now, the action of a current on a magnetic -pole varies inversely as the square of the distance; hence, -over measurable distances the extension of the ether in -the fourth dimension cannot be assumed as other than -small in comparison with those distances.</p> - -<p><span class="pagenum" id="Page_230">[Pg 230]</span></p> - -<p>If we suppose the ether to be filled with vortices in the -shape of four-dimensional spheres rotating with the A -motion, the B motion would correspond to electricity in -the one-fluid theory. There would thus be a possibility -of electricity existing in two forms, statically, by itself, -and, combined with the universal motion, in the form of -a current.</p> - -<p>To arrive at a definite conclusion it will be necessary to -investigate the resultant pressures which accompany the -collocation of solid vortices with surface ones.</p> - -<p>To recapitulate:</p> - -<p>The movements and mechanics of four-dimensional -space are definite and intelligible. A vortex with a -surface as its axis affords a geometric image of a closed -circuit, and there are rotations which by their polarity -afford a possible definition of statical electricity.<a id="FNanchor_7" href="#Footnote_7" class="fnanchor">[7]</a></p> - -<div class="footnotes"> -<div class="footnote"> - -<p><a id="Footnote_7" href="#FNanchor_7" class="label">[7]</a> These double rotations of the A and B kinds I should like to call -Hamiltons and co-Hamiltons, for it is a singular fact that in his -“Quaternions” Sir Wm. Rowan Hamilton has given the theory of -either the A or the B kind. They follow the laws of his symbols, -I, J, K.</p> - -<p>Hamiltons and co-Hamiltons seem to be natural units of geometrical -expression. In the paper in the “Proceedings of the Royal Irish -Academy,” Nov. 1903, already alluded to, I have shown something of -the remarkable facility which is gained in dealing with the composition -of three- and four-dimensional rotations by an alteration in Hamilton’s -notation, which enables his system to be applied to both the A and B -kinds of rotations.</p> - -<p>The objection which has been often made to Hamilton’s system, -namely, that it is only under special conditions of application that his -processes give geometrically interpretable results, can be removed, if -we assume that he was really dealing with a four-dimensional motion, -and alter his notation to bring this circumstance into explicit -recognition.</p> - -</div></div> - -<hr class="chap x-ebookmaker-drop" /> - -<div class="chapter"> -<p><span class="pagenum" id="Page_231">[Pg 231]</span></p> - -<h2 class="nobreak" id="APPENDIX_I">APPENDIX I<br /> - -<small><i>THE MODELS</i></small></h2></div> - - -<p>In Chapter XI. a description has been given which will -enable any one to make a set of models illustrative of the -tesseract and its properties. The set here supposed to be -employed consists of:—</p> - -<div class="blockquote"> - -<p>1. Three sets of twenty-seven cubes each.</p> - -<p>2. Twenty-seven slabs.</p> - -<p>3. Twelve cubes with points, lines, faces, distinguished -by colours, which will be called the catalogue cubes.</p> -</div> - -<p>The preparation of the twelve catalogue cubes involves -the expenditure of a considerable amount of time. It is -advantageous to use them, but they can be replaced by -the drawing of the views of the tesseract or by a reference -to figs. <a href="#fig_103">103</a>, <a href="#fig_104">104</a>, <a href="#fig_105">105</a>, <a href="#fig_106">106</a> of the text.</p> - -<p>The slabs are coloured like the twenty-seven cubes of -the first cubic block in <a href="#fig_101">fig. 101</a>, the one with red, -white, yellow axes.</p> - -<p>The colours of the three sets of twenty-seven cubes are -those of the cubes shown in <a href="#fig_101">fig. 101</a>.</p> - -<p>The slabs are used to form the representation of a cube -in a plane, and can well be dispensed with by any one -who is accustomed to deal with solid figures. But the -whole theory depends on a careful observation of how the -cube would be represented by these slabs.</p> - -<p>In the first step, that of forming a clear idea how a<span class="pagenum" id="Page_232">[Pg 232]</span> -plane being would represent three-dimensional space, only -one of the catalogue cubes and one of the three blocks is -needed.</p> - - -<h3><span class="smcap">Application to the Step from Plane to Solid.</span></h3> - -<p>Look at <a href="#fig_1">fig. 1</a> of the views of the tesseract, or, what -comes to the same thing, take catalogue cube No. 1 and -place it before you with the red line running up, the -white line running to the right, the yellow line running -away. The three dimensions of space are then marked -out by these lines or axes. Now take a piece of cardboard, -or a book, and place it so that it forms a wall -extending up and down not opposite to you, but running -away parallel to the wall of the room on your -left hand.</p> - -<p>Placing the catalogue cube against this wall we see -that it comes into contact with it by the red and yellow -lines, and by the included orange face.</p> - -<p>In the plane being’s world the aspect he has of the -cube would be a square surrounded by red and yellow -lines with grey points.</p> - -<p>Now, keeping the red line fixed, turn the cube about it -so that the yellow line goes out to the right, and the -white line comes into contact with the plane.</p> - -<p>In this case a different aspect is presented to the plane -being, a square, namely, surrounded by red and white -lines and grey points. You should particularly notice -that when the yellow line goes out, at right angles to the -plane, and the white comes in, the latter does not run in -the same sense that the yellow did.</p> - -<p>From the fixed grey point at the base of the red line -the yellow line ran away from you. The white line now -runs towards you. This turning at right angles makes -the line which was out of the plane before, come into it<span class="pagenum" id="Page_233">[Pg 233]</span> -in an opposite sense to that in which the line ran which -has just left the plane. If the cube does not break -through the plane this is always the rule.</p> - -<p>Again turn the cube back to the normal position with -red running up, white to the right, and yellow away, and -try another turning.</p> - -<p>You can keep the yellow line fixed, and turn the cube -about it. In this case the red line going out to the -right the white line will come in pointing downwards.</p> - -<p>You will be obliged to elevate the cube from the table -in order to carry out this turning. It is always necessary -when a vertical axis goes out of a space to imagine a -movable support which will allow the line which ran out -before to come in below.</p> - -<p>Having looked at the three ways of turning the cube -so as to present different faces to the plane, examine what -would be the appearance if a square hole were cut in the -piece of cardboard, and the cube were to pass through it. -A hole can be actually cut, and it will be seen that in the -normal position, with red axis running up, yellow away, -and white to the right, the square first perceived by the -plane being—the one contained by red and yellow lines—would -be replaced by another square of which the line -towards you is pink—the section line of the pink face. -The line above is light yellow, below is light yellow and -on the opposite side away from you is pink.</p> - -<p>In the same way the cube can be pushed through a -square opening in the plane from any of the positions -which you have already turned it into. In each case -the plane being will perceive a different set of contour -lines.</p> - -<p>Having observed these facts about the catalogue cube, -turn now to the first block of twenty-seven cubes.</p> - -<p>You notice that the colour scheme on the catalogue cube -and that of this set of blocks is the same.</p> - -<p><span class="pagenum" id="Page_234">[Pg 234]</span></p> - -<p>Place them before you, a grey or null cube on the -table, above it a red cube, and on the top a null cube -again. Then away from you place a yellow cube, and -beyond it a null cube. Then to the right place a white -cube and beyond it another null. Then complete the -block, according to the scheme of the catalogue cube, -putting in the centre of all an ochre cube.</p> - -<p>You have now a cube like that which is described in -the text. For the sake of simplicity, in some cases, this -cubic block can be reduced to one of eight cubes, by -leaving out the terminations in each direction. Thus, -instead of null, red, null, three cubes, you can take null, -red, two cubes, and so on.</p> - -<p>It is useful, however, to practise the representation in -a plane of a block of twenty-seven cubes. For this -purpose take the slabs, and build them up against the -piece of cardboard, or the book in such a way as to -represent the different aspects of the cube.</p> - -<p>Proceed as follows:—</p> - -<p>First, cube in normal position.</p> - -<p>Place nine slabs against the cardboard to represent the -nine cubes in the wall of the red and yellow axes, facing -the cardboard; these represent the aspect of the cube as it -touches the plane.</p> - -<p>Now push these along the cardboard and make a -different set of nine slabs to represent the appearance -which the cube would present to a plane being, if it were -to pass half way through the plane.</p> - -<p>There would be a white slab, above it a pink one, above -that another white one, and six others, representing what -would be the nature of a section across the middle of the -block of cubes. The section can be thought of as a thin -slice cut out by two parallel cuts across the cube. -Having arranged these nine slabs, push them along the -plane, and make another set of nine to represent what<span class="pagenum" id="Page_235">[Pg 235]</span> -would be the appearance of the cube when it had almost -completely gone through. This set of nine will be the -same as the first set of nine.</p> - -<p>Now we have in the plane three sets of nine slabs -each, which represent three sections of the twenty-seven -block.</p> - -<p>They are put alongside one another. We see that it -does not matter in what order the sets of nine are put. -As the cube passes through the plane they represent appearances -which follow the one after the other. If they -were what they represented, they could not exist in the -same plane together.</p> - -<p>This is a rather important point, namely, to notice that -they should not co-exist on the plane, and that the order -in which they are placed is indifferent. When we -represent a four-dimensional body our solid cubes are to -us in the same position that the slabs are to the plane -being. You should also notice that each of these slabs -represents only the very thinnest slice of a cube. The -set of nine slabs first set up represents the side surface of -the block. It is, as it were, a kind of tray—a beginning -from which the solid cube goes off. The slabs as we use -them have thickness, but this thickness is a necessity of -construction. They are to be thought of as merely of the -thickness of a line.</p> - -<p>If now the block of cubes passed through the plane at -the rate of an inch a minute the appearance to a plane -being would be represented by:—</p> - -<p>1. The first set of nine slabs lasting for one minute.</p> - -<p>2. The second set of nine slabs lasting for one minute.</p> - -<p>3. The third set of nine slabs lasting for one minute.</p> - -<p>Now the appearances which the cube would present -to the plane being in other positions can be shown by -means of these slabs. The use of such slabs would be -the means by which a plane being could acquire a<span class="pagenum" id="Page_236">[Pg 236]</span> -familiarity with our cube. Turn the catalogue cube (or -imagine the coloured figure turned) so that the red line -runs up, the yellow line out to the right, and the white -line towards you. Then turn the block of cubes to -occupy a similar position.</p> - -<p>The block has now a different wall in contact with -the plane. Its appearance to a plane being will not be -the same as before. He has, however, enough slabs to -represent this new set of appearances. But he must -remodel his former arrangement of them.</p> - -<p>He must take a null, a red, and a null slab from the first -of his sets of slabs, then a white, a pink, and a white from -the second, and then a null, a red, and a null from the -third set of slabs.</p> - -<p>He takes the first column from the first set, the first -column from the second set, and the first column from -the third set.</p> - -<p>To represent the half-way-through appearance, which -is as if a very thin slice were cut out half way through the -block, he must take the second column of each of his -sets of slabs, and to represent the final appearance, the -third column of each set.</p> - -<p>Now turn the catalogue cube back to the normal -position, and also the block of cubes.</p> - -<p>There is another turning—a turning about the yellow -line, in which the white axis comes below the support.</p> - -<p>You cannot break through the surface of the table, so -you must imagine the old support to be raised. Then -the top of the block of cubes in its new position is at the -level at which the base of it was before.</p> - -<p>Now representing the appearance on the plane, we must -draw a horizontal line to represent the old base. The -line should be drawn three inches high on the cardboard.</p> - -<p>Below this the representative slabs can be arranged.</p> - -<p>It is easy to see what they are. The old arrangements<span class="pagenum" id="Page_237">[Pg 237]</span> -have to be broken up, and the layers taken in order, the -first layer of each for the representation of the aspect of -the block as it touches the plane.</p> - -<p>Then the second layers will represent the appearance -half way through, and the third layers will represent the -final appearance.</p> - -<p>It is evident that the slabs individually do not represent -the same portion of the cube in these different presentations.</p> - -<p>In the first case each slab represents a section or a face -perpendicular to the white axis, in the second case a -face or a section which runs perpendicularly to the yellow -axis, and in the third case a section or a face perpendicular -to the red axis.</p> - -<p>But by means of these nine slabs the plane being can -represent the whole of the cubic block. He can touch -and handle each portion of the cubic block, there is no -part of it which he cannot observe. Taking it bit by bit, -two axes at a time, he can examine the whole of it.</p> - - -<h3><span class="smcap">Our Representation of a Block of Tesseracts.</span></h3> - -<p>Look at the views of the tesseract 1, 2, 3, or take the -catalogue cubes 1, 2, 3, and place them in front of you, -in any order, say running from left to right, placing 1 in -the normal position, the red axis running up, the white -to the right, and yellow away.</p> - -<p>Now notice that in catalogue cube 2 the colours of each -region are derived from those of the corresponding region -of cube 1 by the addition of blue. Thus null + blue = -blue, and the corners of number 2 are blue. Again, -red + blue = purple, and the vertical lines of 2 are purple. -Blue + yellow = green, and the line which runs away is -coloured green.</p> - -<p>By means of these observations you may be sure that<span class="pagenum" id="Page_238">[Pg 238]</span> -catalogue cube 2 is rightly placed. Catalogue cube 3 is -just like number 1.</p> - -<p>Having these cubes in what we may call their normal -position, proceed to build up the three sets of blocks.</p> - -<p>This is easily done in accordance with the colour scheme -on the catalogue cubes.</p> - -<p>The first block we already know. Build up the second -block, beginning with a blue corner cube, placing a purple -on it, and so on.</p> - -<p>Having these three blocks we have the means of -representing the appearances of a group of eighty-one -tesseracts.</p> - -<p>Let us consider a moment what the analogy in the case -of the plane being is.</p> - -<p>He has his three sets of nine slabs each. We have our -three sets of twenty-seven cubes each.</p> - -<p>Our cubes are like his slabs. As his slabs are not the -things which they represent to him, so our cubes are not -the things they represent to us.</p> - -<p>The plane being’s slabs are to him the faces of cubes.</p> - -<p>Our cubes then are the faces of tesseracts, the cubes by -which they are in contact with our space.</p> - -<p>As each set of slabs in the case of the plane being -might be considered as a sort of tray from which the solid -contents of the cubes came out, so our three blocks of -cubes may be considered as three-space trays, each of -which is the beginning of an inch of the solid contents -of the four-dimensional solids starting from them.</p> - -<p>We want now to use the names null, red, white, etc., -for tesseracts. The cubes we use are only tesseract faces. -Let us denote that fact by calling the cube of null colour, -null face; or, shortly, null f., meaning that it is the face -of a tesseract.</p> - -<p>To determine which face it is let us look at the catalogue -cube 1 or the first of the views of the tesseract, which<span class="pagenum" id="Page_239">[Pg 239]</span> -can be used instead of the models. It has three axes, -red, white, yellow, in our space. Hence the cube determined -by these axes is the face of the tesseract which we -now have before us. It is the ochre face. It is enough, -however, simply to say null f., red f. for the cubes which -we use.</p> - -<p>To impress this in your mind, imagine that tesseracts -do actually run from each cube. Then, when you move the -cubes about, you move the tesseracts about with them. -You move the face but the tesseract follows with it, as the -cube follows when its face is shifted in a plane.</p> - -<p>The cube null in the normal position is the cube which -has in it the red, yellow, white axes. It is the face -having these, but wanting the blue. In this way you can -define which face it is you are handling. I will write an -“f.” after the name of each tesseract just as the plane -being might call each of his slabs null slab, yellow slab, -etc., to denote that they were representations.</p> - -<p>We have then in the first block of twenty-seven cubes, -the following—null f., red f., null f., going up; white f., null -f., lying to the right, and so on. Starting from the null -point and travelling up one inch we are in the null region, -the same for the away and the right-hand directions. -And if we were to travel in the fourth dimension for an -inch we should still be in a null region. The tesseract -stretches equally all four ways. Hence the appearance we -have in this first block would do equally well if the -tesseract block were to move across our space for a certain -distance. For anything less than an inch of their transverse -motion we should still have the same appearance. -You must notice, however, that we should not have null -face after the motion had begun.</p> - -<p>When the tesseract, null for instance, had moved ever -so little we should not have a face of null but a section of -null in our space. Hence, when we think of the motion<span class="pagenum" id="Page_240">[Pg 240]</span> -across our space we must call our cubes tesseract sections. -Thus on null passing across we should see first null f., then -null s., and then, finally, null f. again.</p> - -<p>Imagine now the whole first block of twenty-seven -tesseracts to have moved tranverse to our space a distance -of one inch. Then the second set of tesseracts, which -originally were an inch distant from our space, would be -ready to come in.</p> - -<p>Their colours are shown in the second block of twenty-seven -cubes which you have before you. These represent -the tesseract faces of the set of tesseracts that lay before -an inch away from our space. They are ready now to -come in, and we can observe their colours. In the place -which null f. occupied before we have blue f., in place of -red f. we have purple f., and so on. Each tesseract is -coloured like the one whose place it takes in this motion -with the addition of blue.</p> - -<p>Now if the tesseract block goes on moving at the rate -of an inch a minute, this next set of tesseracts will occupy -a minute in passing across. We shall see, to take the null -one for instance, first of all null face, then null section, -then null face again.</p> - -<p>At the end of the second minute the second set of -tesseracts has gone through, and the third set comes in. -This, as you see, is coloured just like the first. Altogether, -these three sets extend three inches in the fourth dimension, -making the tesseract block of equal magnitude in all -dimensions.</p> - -<p>We have now before us a complete catalogue of all the -tesseracts in our group. We have seen them all, and we -shall refer to this arrangement of the blocks as the -“normal position.” We have seen as much of each -tesseract at a time as could be done in a three-dimensional -space. Each part of each tesseract has been in -our space, and we could have touched it.</p> - -<p><span class="pagenum" id="Page_241">[Pg 241]</span></p> - -<p>The fourth dimension appeared to us as the duration -of the block.</p> - -<p>If a bit of our matter were to be subjected to the same -motion it would be instantly removed out of our space. -Being thin in the fourth dimension it is at once taken -out of our space by a motion in the fourth dimension.</p> - -<p>But the tesseract block we represent having length in -the fourth dimension remains steadily before our eyes for -three minutes, when it is subjected to this transverse -motion.</p> - -<p>We have now to form representations of the other -views of the same tesseract group which are possible in -our space.</p> - -<p>Let us then turn the block of tesseracts so that another -face of it comes into contact with our space, and then -by observing what we have, and what changes come when -the block traverses our space, we shall have another view -of it. The dimension which appeared as duration before -will become extension in one of our known dimensions, -and a dimension which coincided with one of our space -dimensions will appear as duration.</p> - -<p>Leaving catalogue cube 1 in the normal position, -remove the other two, or suppose them removed. We -have in space the red, the yellow, and the white axes. -Let the white axis go out into the unknown, and occupy -the position the blue axis holds. Then the blue axis, -which runs in that direction now will come into space. -But it will not come in pointing in the same way that -the white axis does now. It will point in the opposite -sense. It will come in running to the left instead of -running to the right as the white axis does now.</p> - -<p>When this turning takes place every part of the cube 1 -will disappear except the left-hand face—the orange face.</p> - -<p>And the new cube that appears in our space will run to -the left from this orange face, having axes, red, yellow, blue.</p> - -<p><span class="pagenum" id="Page_242">[Pg 242]</span></p> - -<p>Take models 4, 5, 6. Place 4, or suppose No. 4 of the -tesseract views placed, with its orange face coincident with -the orange face of 1, red line to red line, and yellow line -to yellow line, with the blue line pointing to the left. -Then remove cube 1 and we have the tesseract face -which comes in when the white axis runs in the positive -unknown, and the blue axis comes into our space.</p> - -<p>Now place catalogue cube 5 in some position, it does -not matter which, say to the left; and place it so that -there is a correspondence of colour corresponding to the -colour of the line that runs out of space. The line that -runs out of space is white, hence, every part of this -cube 5 should differ from the corresponding part of 4 by -an alteration in the direction of white.</p> - -<p>Thus we have white points in 5 corresponding to the -null points in 4. We have a pink line corresponding to -a red line, a light yellow line corresponding to a yellow -line, an ochre face corresponding to an orange face. This -cube section is completely named in Chapter XI. Finally -cube 6 is a replica of 1.</p> - -<p>These catalogue cubes will enable us to set up our -models of the block of tesseracts.</p> - -<p>First of all for the set of tesseracts, which beginning -in our space reach out one inch in the unknown, we have -the pattern of catalogue cube 4.</p> - -<p>We see that we can build up a block of twenty-seven -tesseract faces after the colour scheme of cube 4, by -taking the left-hand wall of block 1, then the left-hand -wall of block 2, and finally that of block 3. We take, -that is, the three first walls of our previous arrangement -to form the first cubic block of this new one.</p> - -<p>This will represent the cubic faces by which the group -of tesseracts in its new position touches our space. -We have running up, null f., red f., null f. In the next -vertical line, on the side remote from us, we have yellow f.,<span class="pagenum" id="Page_243">[Pg 243]</span> -orange f., yellow f., and then the first colours over again. -Then the three following columns are, blue f., purple f., -blue f.; green f., brown f., green f.; blue f., purple f., blue f. -The last three columns are like the first.</p> - -<p>These tesseracts touch our space, and none of them are -by any part of them distant more than an inch from it. -What lies beyond them in the unknown?</p> - -<p>This can be told by looking at catalogue cube 5. -According to its scheme of colour we see that the second -wall of each of our old arrangements must be taken. -Putting them together we have, as the corner, white f. -above it, pink f. above it, white f. The column next to -this remote from us is as follows:—light yellow f., ochre f., -light yellow f., and beyond this a column like the first. -Then for the middle of the block, light blue f., above -it light purple, then light blue. The centre column has, -at the bottom, light green f., light brown f. in the centre -and at the top light green f. The last wall is like the -first.</p> - -<p>The third block is made by taking the third walls of -our previous arrangement, which we called the normal -one.</p> - -<p>You may ask what faces and what sections our cubes -represent. To answer this question look at what axes -you have in our space. You have red, yellow, blue. -Now these determine brown. The colours red, -yellow, blue are supposed by us when mixed to produce -a brown colour. And that cube which is determined -by the red, yellow, blue axes we call the brown cube.</p> - -<p>When the tesseract block in its new position begins to -move across our space each tesseract in it gives a section -in our space. This section is transverse to the white -axis, which now runs in the unknown.</p> - -<p>As the tesseract in its present position passes across -our space, we should see first of all the first of the blocks<span class="pagenum" id="Page_244">[Pg 244]</span> -of cubic faces we have put up—these would last for a -minute, then would come the second block and then the -third. At first we should have a cube of tesseract faces, -each of which would be brown. Directly the movement -began, we should have tesseract sections transverse to the -white line.</p> - -<p>There are two more analogous positions in which the -block of tesseracts can be placed. To find the third -position, restore the blocks to the normal arrangement.</p> - -<p>Let us make the yellow axis go out into the positive -unknown, and let the blue axis, consequently, come in -running towards us. The yellow ran away, so the blue -will come in running towards us.</p> - -<p>Put catalogue cube 1 in its normal position. Take -catalogue cube 7 and place it so that its pink face -coincides with the pink face of cube 1, making also its -red axis coincide with the red axis of 1 and its white -with the white. Moreover, make cube 7 come -towards us from cube 1. Looking at it we see in our -space, red, white, and blue axes. The yellow runs out. -Place catalogue cube 8 in the neighbourhood of -7—observe that every region in 8 has a change in -the direction of yellow from the corresponding region -in 7. This is because it represents what you come -to now in going in the unknown, when the yellow axis -runs out of our space. Finally catalogue cube 9, -which is like number 7, shows the colours of the third -set of tesseracts. Now evidently, starting from the -normal position, to make up our three blocks of tesseract -faces we have to take the near wall from the first block, -the near wall from the second, and then the near wall -from the third block. This gives us the cubic block -formed by the faces of the twenty-seven tesseracts which -are now immediately touching our space.</p> - -<p>Following the colour scheme of catalogue cube 8,<span class="pagenum" id="Page_245">[Pg 245]</span> -we make the next set of twenty-seven tesseract faces, -representing the tesseracts, each of which begins one inch -off from our space, by putting the second walls of our -previous arrangement together, and the representation -of the third set of tesseracts is the cubic block formed of -the remaining three walls.</p> - -<p>Since we have red, white, blue axes in our space to -begin with, the cubes we see at first are light purple -tesseract faces, and after the transverse motion begins -we have cubic sections transverse to the yellow line.</p> - -<p>Restore the blocks to the normal position, there -remains the case in which the red axis turns out of -space. In this case the blue axis will come in downwards, -opposite to the sense in which the red axis ran.</p> - -<p>In this case take catalogue cubes 10, 11, 12. Lift up -catalogue cube 1 and put 10 underneath it, imagining -that it goes down from the previous position of 1.</p> - -<p>We have to keep in space the white and the yellow -axes, and let the red go out, the blue come in.</p> - -<p>Now, you will find on cube 10 a light yellow face; this -should coincide with the base of 1, and the white and -yellow lines on the two cubes should coincide. Then the -blue axis running down you have the catalogue cube -correctly placed, and it forms a guide for putting up the -first representative block.</p> - -<p>Catalogue cube 11 will represent what lies in the fourth -dimension—now the red line runs in the fourth dimension. -Thus the change from 10 to 11 should be towards -red, corresponding to a null point is a red point, to a -white line is a pink line, to a yellow line an orange -line, and so on.</p> - -<p>Catalogue cube 12 is like 10. Hence we see that to -build up our blocks of tesseract faces we must take the -bottom layer of the first block, hold that up in the air, -underneath it place the bottom layer of the second block,<span class="pagenum" id="Page_246">[Pg 246]</span> -and finally underneath this last the bottom layer of the -last of our normal blocks.</p> - -<p>Similarly we make the second representative group by -taking the middle courses of our three blocks. The last -is made by taking the three topmost layers. The three -axes in our space before the transverse motion begins are -blue, white, yellow, so we have light green tesseract -faces, and after the motion begins sections transverse to -the red light.</p> - -<p>These three blocks represent the appearances as the -tesseract group in its new position passes across our space. -The cubes of contact in this case are those determinal by -the three axes in our space, namely, the white, the -yellow, the blue. Hence they are light green.</p> - -<p>It follows from this that light green is the interior -cube of the first block of representative cubic faces.</p> - -<p>Practice in the manipulations described, with a -realization in each case of the face or section which -is in our space, is one of the best means of a thorough -comprehension of the subject.</p> - -<p>We have to learn how to get any part of these four-dimensional -figures into space, so that we can look at -them. We must first learn to swing a tesseract, and a -group of tesseracts about in any way.</p> - -<p>When these operations have been repeated and the -method of arrangement of the set of blocks has become -familiar, it is a good plan to rotate the axes of the normal -cube 1 about a diagonal, and then repeat the whole series -of turnings.</p> - -<p>Thus, in the normal position, red goes up, white to the -right, yellow away. Make white go up, yellow to the right, -and red away. Learn the cube in this position by putting -up the set of blocks of the normal cube, over and over -again till it becomes as familiar to you as in the normal -position. Then when this is learned, and the corre<span class="pagenum" id="Page_247">[Pg 247]</span>sponding -changes in the arrangements of the tesseract -groups are made, another change should be made: let, -in the normal cube, yellow go up, red to the right, and -white away.</p> - -<p>Learn the normal block of cubes in this new position -by arranging them and re-arranging them till you know -without thought where each one goes. Then carry out -all the tesseract arrangements and turnings.</p> - -<p>If you want to understand the subject, but do not see -your way clearly, if it does not seem natural and easy to -you, practise these turnings. Practise, first of all, the -turning of a block of cubes round, so that you know it -in every position as well as in the normal one. Practise -by gradually putting up the set of cubes in their new -arrangements. Then put up the tesseract blocks in their -arrangements. This will give you a working conception -of higher space, you will gain the feeling of it, whether -you take up the mathematical treatment of it or not.</p> -<hr class="chap x-ebookmaker-drop" /> - -<div class="chapter"> -<p><span class="pagenum" id="Page_248">[Pg 248]</span></p> - -<h2 class="nobreak" id="APPENDIX_II">APPENDIX II<br /> - -<small><i>A LANGUAGE OF SPACE</i></small></h2></div> - - -<p>The mere naming the parts of the figures we consider -involves a certain amount of time and attention. This -time and attention leads to no result, for with each -new figure the nomenclature applied is completely -changed, every letter or symbol is used in a different -significance.</p> - -<p>Surely it must be possible in some way to utilise the -labour thus at present wasted!</p> - -<p>Why should we not make a language for space itself, so -that every position we want to refer to would have its own -name? Then every time we named a figure in order to -demonstrate its properties we should be exercising -ourselves in the vocabulary of place.</p> - -<p>If we use a definite system of names, and always refer -to the same space position by the same name, we create -as it were a multitude of little hands, each prepared to -grasp a special point, position, or element, and hold it -for us in its proper relations.</p> - -<p>We make, to use another analogy, a kind of mental -paper, which has somewhat of the properties of a sensitive -plate, in that it will register, without effort, complex, -visual, or tactual impressions.</p> - -<p>But of far more importance than the applications of a -space language to the plane and to solid space is the<span class="pagenum" id="Page_249">[Pg 249]</span> -facilitation it brings with it to the study of four-dimensional -shapes.</p> - -<p>I have delayed introducing a space language because -all the systems I made turned out, after giving them a -fair trial, to be intolerable. I have now come upon one -which seems to present features of permanence, and I will -here give an outline of it, so that it can be applied to -the subject of the text, and in order that it may be -subjected to criticism.</p> - -<p>The principle on which the language is constructed is -to sacrifice every other consideration for brevity.</p> - -<p>It is indeed curious that we are able to talk and -converse on every subject of thought except the fundamental -one of space. The only way of speaking about -the spatial configurations that underlie every subject -of discursive thought is a co-ordinate system of numbers. -This is so awkward and incommodious that it is never -used. In thinking also, in realising shapes, we do not -use it; we confine ourselves to a direct visualisation.</p> - -<p>Now, the use of words corresponds to the storing up -of our experience in a definite brain structure. A child, -in the endless tactual, visual, mental manipulations it -makes for itself, is best left to itself, but in the course -of instruction the introduction of space names would -make the teachers work more cumulative, and the child’s -knowledge more social.</p> - -<p>Their full use can only be appreciated, if they are -introduced early in the course of education; but in a -minor degree any one can convince himself of their -utility, especially in our immediate subject of handling -four-dimensional shapes. The sum total of the results -obtained in the preceding pages can be compendiously and -accurately expressed in nine words of the Space Language.</p> - -<p>In one of Plato’s dialogues Socrates makes an experiment -on a slave boy standing by. He makes certain<span class="pagenum" id="Page_250">[Pg 250]</span> -perceptions of space awake in the mind of Meno’s slave -by directing his close attention on some simple facts of -geometry.</p> - -<p>By means of a few words and some simple forms we can -repeat Plato’s experiment on new ground.</p> - -<p>Do we by directing our close attention on the facts of -four dimensions awaken a latent faculty in ourselves? -The old experiment of Plato’s, it seems to me, has come -down to us as novel as on the day he incepted it, and its -significance not better understood through all the discussion -of which it has been the subject.</p> - -<p>Imagine a voiceless people living in a region where -everything had a velvety surface, and who were thus -deprived of all opportunity of experiencing what sound is. -They could observe the slow pulsations of the air caused -by their movements, and arguing from analogy, they -would no doubt infer that more rapid vibrations were -possible. From the theoretical side they could determine -all about these more rapid vibrations. They merely differ, -they would say, from slower ones, by the number that -occur in a given time; there is a merely formal difference.</p> - -<p>But suppose they were to take the trouble, go to the -pains of producing these more rapid vibrations, then a -totally new sensation would fall on their rudimentary ears. -Probably at first they would only be dimly conscious of -Sound, but even from the first they would become aware -that a merely formal difference, a mere difference in point -of number in this particular respect, made a great difference -practically, as related to them. And to us the difference -between three and four dimensions is merely formal, -numerical. We can tell formally all about four dimensions, -calculate the relations that would exist. But that the -difference is merely formal does not prove that it is a -futile and empty task, to present to ourselves as closely as -we can the phenomena of four dimensions. In our formal<span class="pagenum" id="Page_251">[Pg 251]</span> -knowledge of it, the whole question of its actual relation -to us, as we are, is left in abeyance.</p> - -<p>Possibly a new apprehension of nature may come to us -through the practical, as distinguished from the mathematical -and formal, study of four dimensions. As a child -handles and examines the objects with which he comes in -contact, so we can mentally handle and examine four-dimensional -objects. The point to be determined is this. -Do we find something cognate and natural to our faculties, -or are we merely building up an artificial presentation of -a scheme only formally possible, conceivable, but which -has no real connection with any existing or possible -experience?</p> - -<p>This, it seems to me, is a question which can only be -settled by actually trying. This practical attempt is the -logical and direct continuation of the experiment Plato -devised in the “Meno.”</p> - -<p>Why do we think true? Why, by our processes of -thought, can we predict what will happen, and correctly -conjecture the constitution of the things around us? -This is a problem which every modern philosopher has -considered, and of which Descartes, Leibnitz, Kant, to -name a few, have given memorable solutions. Plato was -the first to suggest it. And as he had the unique position -of being the first devisor of the problem, so his solution -is the most unique. Later philosophers have talked about -consciousness and its laws, sensations, categories. But -Plato never used such words. Consciousness apart from a -conscious being meant nothing to him. His was always -an objective search. He made man’s intuitions the basis -of a new kind of natural history.</p> - -<p>In a few simple words Plato puts us in an attitude -with regard to psychic phenomena—the mind—the ego—“what -we are,” which is analogous to the attitude scientific -men of the present day have with regard to the phenomena<span class="pagenum" id="Page_252">[Pg 252]</span> -of outward nature. Behind this first apprehension of ours -of nature, there is an infinite depth to be learned and -known. Plato said that behind the phenomena of mind -that Meno’s slave boy exhibited, there was a vast, an -infinite perspective. And his singularity, his originality, -comes out most strongly marked in this, that the perspective, -the complex phenomena beyond were, according -to him, phenomena of personal experience. A footprint -in the sand means a man to a being that has the conception -of a man. But to a creature that has no such -conception, it means a curious mark, somehow resulting -from the concatenation of ordinary occurrences. Such a -being would attempt merely to explain how causes known -to him could so coincide as to produce such a result; -he would not recognise its significance.</p> - -<p>Plato introduced the conception which made a new -kind of natural history possible. He said that Meno’s -slave boy thought true about things he had never -learned, because his “soul” had experience. I know this -will sound absurd to some people, and it flies straight -in the face of the maxim, that explanation consists in -showing how an effect depends on simple causes. But -what a mistaken maxim that is! Can any single instance -be shown of a simple cause? Take the behaviour of -spheres for instance; say those ivory spheres, billiard balls, -for example. We can explain their behaviour by supposing -they are homogeneous elastic solids. We can give formulæ -which will account for their movements in every variety. -But are they homogeneous elastic solids? No, certainly -not. They are complex in physical and molecular structure, -and atoms and ions beyond open an endless vista. Our -simple explanation is false, false as it can be. The balls -act as if they were homogeneous elastic spheres. There is -a statistical simplicity in the resultant of very complex -conditions, which makes that artificial conception useful.<span class="pagenum" id="Page_253">[Pg 253]</span> -But its usefulness must not blind us to the fact that it is -artificial. If we really look deep into nature, we find a -much greater complexity than we at first suspect. And -so behind this simple “I,” this myself, is there not a -parallel complexity? Plato’s “soul” would be quite -acceptable to a large class of thinkers, if by “soul” and -the complexity he attributes to it, he meant the product -of a long course of evolutionary changes, whereby simple -forms of living matter endowed with rudimentary sensation -had gradually developed into fully conscious beings.</p> - -<p>But Plato does not mean by “soul” a being of such a -kind. His soul is a being whose faculties are clogged by -its bodily environment, or at least hampered by the -difficulty of directing its bodily frame—a being which -is essentially higher than the account it gives of itself -through its organs. At the same time Plato’s soul is -not incorporeal. It is a real being with a real experience. -The question of whether Plato had the conception of non-spatial -existence has been much discussed. The verdict -is, I believe, that even his “ideas” were conceived by him -as beings in space, or, as we should say, real. Plato’s -attitude is that of Science, inasmuch as he thinks of a -world in Space. But, granting this, it cannot be denied -that there is a fundamental divergence between Plato’s -conception and the evolutionary theory, and also an -absolute divergence between his conception and the -genetic account of the origin of the human faculties. -The functions and capacities of Plato’s “soul” are not -derived by the interaction of the body and its environment.</p> - -<p>Plato was engaged on a variety of problems, and his -religious and ethical thoughts were so keen and fertile -that the experimental investigation of his soul appears -involved with many other motives. In one passage Plato -will combine matter of thought of all kinds and from all -sources, overlapping, interrunning. And in no case is he<span class="pagenum" id="Page_254">[Pg 254]</span> -more involved and rich than in this question of the soul. -In fact, I wish there were two words, one denoting that -being, corporeal and real, but with higher faculties than -we manifest in our bodily actions, which is to be taken as -the subject of experimental investigation; and the other -word denoting “soul” in the sense in which it is made -the recipient and the promise of so much that men desire. -It is the soul in the former sense that I wish to investigate, -and in a limited sphere only. I wish to find out, in continuation -of the experiment in the Meno, what the “soul” -in us thinks about extension, experimenting on the -grounds laid down by Plato. He made, to state the -matter briefly, the hypothesis with regard to the thinking -power of a being in us, a “soul.” This soul is not accessible -to observation by sight or touch, but it can be -observed by its functions; it is the object of a new kind -of natural history, the materials for constructing which -lie in what it is natural to us to think. With Plato -“thought” was a very wide-reaching term, but still I -would claim in his general plan of procedure a place for -the particular question of extension.</p> - -<p>The problem comes to be, “What is it natural to us to -think about matter <i>qua</i> extended?”</p> - -<p>First of all, I find that the ordinary intuition of any -simple object is extremely imperfect. Take a block of -differently marked cubes, for instance, and become acquainted -with them in their positions. You may think -you know them quite well, but when you turn them round—rotate -the block round a diagonal, for instance—you -will find that you have lost track of the individuals in -their new positions. You can mentally construct the -block in its new position, by a rule, by taking the remembered -sequences, but you don’t know it intuitively. By -observation of a block of cubes in various positions, and -very expeditiously by a use of Space names applied to the<span class="pagenum" id="Page_255">[Pg 255]</span> -cubes in their different presentations, it is possible to get -an intuitive knowledge of the block of cubes, which is not -disturbed by any displacement. Now, with regard to this -intuition, we moderns would say that I had formed it by -my tactual visual experiences (aided by hereditary pre-disposition). -Plato would say that the soul had been -stimulated to recognise an instance of shape which it -knew. Plato would consider the operation of learning -merely as a stimulus; we as completely accounting for -the result. The latter is the more common-sense view. -But, on the other hand, it presupposes the generation of -experience from physical changes. The world of sentient -experience, according to the modern view, is closed and -limited; only the physical world is ample and large and -of ever-to-be-discovered complexity. Plato’s world of soul, -on the other hand, is at least as large and ample as the -world of things.</p> - -<p>Let us now try a crucial experiment. Can I form an -intuition of a four-dimensional object? Such an object -is not given in the physical range of my sense contacts. -All I can do is to present to myself the sequences of solids, -which would mean the presentation to me under my conditions -of a four-dimensional object. All I can do is to -visualise and tactualise different series of solids which are -alternative sets of sectional views of a four-dimensional -shape.</p> - -<p>If now, on presenting these sequences, I find a power -in me of intuitively passing from one of these sets of -sequences to another, of, being given one, intuitively -constructing another, not using a rule, but directly apprehending -it, then I have found a new fact about my soul, -that it has a four-dimensional experience; I have observed -it by a function it has.</p> - -<p>I do not like to speak positively, for I might occasion -a loss of time on the part of others, if, as may very well<span class="pagenum" id="Page_256">[Pg 256]</span> -be, I am mistaken. But for my own part, I think there -are indications of such an intuition; from the results of -my experiments, I adopt the hypothesis that that which -thinks in us has an ample experience, of which the intuitions -we use in dealing with the world of real objects -are a part; of which experience, the intuition of four-dimensional -forms and motions is also a part. The process -we are engaged in intellectually is the reading the obscure -signals of our nerves into a world of reality, by means of -intuitions derived from the inner experience.</p> - -<p>The image I form is as follows. Imagine the captain -of a modern battle-ship directing its course. He has -his charts before him; he is in communication with his -associates and subordinates; can convey his messages and -commands to every part of the ship, and receive information -from the conning-tower and the engine-room. Now -suppose the captain immersed in the problem of the -navigation of his ship over the ocean, to have so absorbed -himself in the problem of the direction of his craft over -the plane surface of the sea that he forgets himself. All -that occupies his attention is the kind of movement that -his ship makes. The operations by which that movement -is produced have sunk below the threshold of his consciousness, -his own actions, by which he pushes the buttons, -gives the orders, are so familiar as to be automatic, his -mind is on the motion of the ship as a whole. In such -a case we can imagine that he identifies himself with his -ship; all that enters his conscious thought is the direction -of its movement over the plane surface of the ocean.</p> - -<p>Such is the relation, as I imagine it, of the soul to the -body. A relation which we can imagine as existing -momentarily in the case of the captain is the normal -one in the case of the soul with its craft. As the captain -is capable of a kind of movement, an amplitude of motion, -which does not enter into his thoughts with regard to the<span class="pagenum" id="Page_257">[Pg 257]</span> -directing the ship over the plane surface of the ocean, so -the soul is capable of a kind of movement, has an amplitude -of motion, which is not used in its task of directing -the body in the three-dimensional region in which the -body’s activity lies. If for any reason it became necessary -for the captain to consider three-dimensional motions with -regard to his ship, it would not be difficult for him to -gain the materials for thinking about such motions; all -he has to do is to call his own intimate experience into -play. As far as the navigation of the ship, however, is -concerned, he is not obliged to call on such experience. -The ship as a whole simply moves on a surface. The -problem of three-dimensional movement does not ordinarily -concern its steering. And thus with regard to ourselves -all those movements and activities which characterise our -bodily organs are three-dimensional; we never need to -consider the ampler movements. But we do more than -use the movements of our body to effect our aims by -direct means; we have now come to the pass when we act -indirectly on nature, when we call processes into play -which lie beyond the reach of any explanation we can -give by the kind of thought which has been sufficient for -the steering of our craft as a whole. When we come to -the problem of what goes on in the minute, and apply -ourselves to the mechanism of the minute, we find our -habitual conceptions inadequate.</p> - -<p>The captain in us must wake up to his own intimate -nature, realise those functions of movement which are his -own, and in virtue of his knowledge of them apprehend -how to deal with the problems he has come to.</p> - -<p>Think of the history of man. When has there been a -time, in which his thoughts of form and movement were -not exclusively of such varieties as were adapted for his -bodily performance? We have never had a demand to -conceive what our own most intimate powers are. But,<span class="pagenum" id="Page_258">[Pg 258]</span> -just as little as by immersing himself in the steering of -his ship over the plane surface of the ocean, a captain -can lose the faculty of thinking about what he actually -does, so little can the soul lose its own nature. It -can be roused to an intuition that is not derived from -the experience which the senses give. All that is -necessary is to present some few of those appearances -which, while inconsistent with three-dimensional matter, -are yet consistent with our formal knowledge of four-dimensional -matter, in order for the soul to wake up and -not begin to learn, but of its own intimate feeling fill up -the gaps in the presentiment, grasp the full orb of possibilities -from the isolated points presented to it. In relation -to this question of our perceptions, let me suggest another -illustration, not taking it too seriously, only propounding -it to exhibit the possibilities in a broad and general way.</p> - -<p>In the heavens, amongst the multitude of stars, there -are some which, when the telescope is directed on them, -seem not to be single stars, but to be split up into two. -Regarding these twin stars through a spectroscope, an -astronomer sees in each a spectrum of bands of colour and -black lines. Comparing these spectrums with one another, -he finds that there is a slight relative shifting of the dark -lines, and from that shifting he knows that the stars are -rotating round one another, and can tell their relative -velocity with regard to the earth. By means of his -terrestrial physics he reads this signal of the skies. This -shifting of lines, the mere slight variation of a black line -in a spectrum, is very unlike that which the astronomer -knows it means. But it is probably much more like what -it means than the signals which the nerves deliver are -like the phenomena of the outer world.</p> - -<p>No picture of an object is conveyed through the nerves. -No picture of motion, in the sense in which we postulate -its existence, is conveyed through the nerves. The actual<span class="pagenum" id="Page_259">[Pg 259]</span> -deliverances of which our consciousness takes account are -probably identical for eye and ear, sight and touch.</p> - -<p>If for a moment I take the whole earth together and -regard it as a sentient being, I find that the problem of -its apprehension is a very complex one, and involves a -long series of personal and physical events. Similarly the -problem of our apprehension is a very complex one. I -only use this illustration to exhibit my meaning. It has -this especial merit, that, as the process of conscious -apprehension takes place in our case in the minute, so, -with regard to this earth being, the corresponding process -takes place in what is relatively to it very minute.</p> - -<p>Now, Plato’s view of a soul leads us to the hypothesis -that that which we designate as an act of apprehension -may be a very complex event, both physically and personally. -He does not seek to explain what an intuition -is; he makes it a basis from whence he sets out on a -voyage of discovery. Knowledge means knowledge; he -puts conscious being to account for conscious being. He -makes an hypothesis of the kind that is so fertile in -physical science—an hypothesis making no claim to -finality, which marks out a vista of possible determination -behind determination, like the hypothesis of space itself, -the type of serviceable hypotheses.</p> - -<p>And, above all, Plato’s hypothesis is conducive to experiment. -He gives the perspective in which real objects -can be determined; and, in our present enquiry, we are -making the simplest of all possible experiments—we are -enquiring what it is natural to the soul to think of matter -as extended.</p> - -<p>Aristotle says we always use a “phantasm” in thinking, -a phantasm of our corporeal senses a visualisation or a -tactualisation. But we can so modify that visualisation -or tactualisation that it represents something not known -by the senses. Do we by that representation wake up an<span class="pagenum" id="Page_260">[Pg 260]</span> -intuition of the soul? Can we by the presentation of -these hypothetical forms, that are the subject of our -present discussion, wake ourselves up to higher intuitions? -And can we explain the world around by a motion that we -only know by our souls?</p> - -<p>Apart from all speculation, however, it seems to me -that the interest of these four-dimensional shapes and -motions is sufficient reason for studying them, and that -they are the way by which we can grow into a fuller -apprehension of the world as a concrete whole.</p> - - -<h3><span class="smcap">Space Names.</span></h3> - -<p>If the words written in the squares drawn in <a href="#fig_144">fig. 1</a> are -used as the names of the squares in the positions in -which they are placed, it is evident that -a combination of these names will denote -a figure composed of the designated -squares. It is found to be most convenient -to take as the initial square that -marked with an asterisk, so that the -directions of progression are towards the -observer and to his right. The directions -of progression, however, are arbitrary, and can be chosen -at will.</p> - -<div class="figleft illowp25" id="fig_144" style="max-width: 12.5em;"> - <img src="images/fig_144.png" alt="" /> - <div class="caption">Fig. 1.</div> -</div> - -<p>Thus <i>et</i>, <i>at</i>, <i>it</i>, <i>an</i>, <i>al</i> will denote a figure in the form -of a cross composed of five squares.</p> - -<p>Here, by means of the double sequence, <i>e</i>, <i>a</i>, <i>i</i> and <i>n</i>, <i>t</i>, <i>l</i>, it -is possible to name a limited collection of space elements.</p> - -<p>The system can obviously be extended by using letter -sequences of more members.</p> - -<p>But, without introducing such a complexity, the -principles of a space language can be exhibited, and a -nomenclature obtained adequate to all the considerations -of the preceding pages.</p> - -<p><span class="pagenum" id="Page_261">[Pg 261]</span></p> - - -<p>1. <i>Extension.</i></p> - -<div class="figleft illowp35" id="fig_145" style="max-width: 15.625em;"> - <img src="images/fig_145.png" alt="" /> - <div class="caption">Fig. 2.</div> -</div> - -<p>Call the large squares in <a href="#fig_145">2</a> by the name written -in them. It is evident that each -can be divided as shown in <a href="#fig_144">fig. 1</a>. -Then the small square marked 1 -will be “en” in “En,” or “Enen.” -The square marked 2 will be “et” -in “En” or “Enet,” while the -square marked 4 will be “en” in -“Et” or “Eten.” Thus the square -5 will be called “Ilil.”</p> - -<p>This principle of extension can -be applied in any number of dimensions.</p> - - -<p>2. <i>Application to Three-Dimensional Space.</i></p> - -<div class="figleft illowp25" id="fig_146" style="max-width: 12.5em;"> - <img src="images/fig_146.png" alt="Three cube faces" /> -</div> - -<p>To name a three-dimensional collocation of cubes take -the upward direction first, secondly the -direction towards the observer, thirdly the -direction to his right hand.</p> - -<p>These form a word in which the first -letter gives the place of the cube upwards, -the second letter its place towards the -observer, the third letter its place to the -right.</p> - -<p>We have thus the following scheme, -which represents the set of cubes of -column 1, <a href="#fig_101">fig. 101</a>, page 165.</p> - -<p>We begin with the remote lowest cube -at the left hand, where the asterisk is -placed (this proves to be by far the most -convenient origin to take for the normal -system).</p> - -<p>Thus “nen” is a “null” cube, “ten” -a red cube on it, and “len” a “null” -cube above “ten.”</p> - -<p><span class="pagenum" id="Page_262">[Pg 262]</span></p> - -<p>By using a more extended sequence of consonants and -vowels a larger set of cubes can be named.</p> - -<p>To name a four-dimensional block of tesseracts it is -simply necessary to prefix an “e,” an “a,” or an “i” to -the cube names.</p> - -<p>Thus the tesseract blocks schematically represented on -page 165, <a href="#fig_101">fig. 101</a> are named as follows:—</p> - -<div class="figcenter illowp80" id="fig_147" style="max-width: 62.5em;"> - <img src="images/fig_147.png" alt="Nine cube faces" /> -</div> - -<p>2. <span class="smcap">Derivation of Point, Line, Face, etc., Names.</span></p> - -<p>The principle of derivation can be shown as follows: -Taking the square of squares<span class="pagenum" id="Page_263">[Pg 263]</span></p> - -<div class="figcenter illowp35" id="fig_148" style="max-width: 15.625em;"> - <img src="images/fig_148.png" alt="Cube face" /> -</div> -<p class="pnind">the number of squares in it can be enlarged and the -whole kept the same size.</p> - -<div class="figcenter illowp35" id="fig_149" style="max-width: 15.625em;"> - <img src="images/fig_149.png" alt="Cube face" /> -</div> - -<p>Compare <a href="#fig_79">fig. 79</a>, p. 138, for instance, or the bottom layer -of <a href="#fig_84">fig. 84</a>.</p> - -<p>Now use an initial “s” to denote the result of carrying -this process on to a great extent, and we obtain the limit -names, that is the point, line, area names for a square. -“Sat” is the whole interior. The corners are “sen,” -“sel,” “sin,” “sil,” while the lines -are “san,” “sal,” “set,” “sit.”</p> - -<div class="figleft illowp30" id="fig_150" style="max-width: 15.625em;"> - <img src="images/fig_150.png" alt="see para above" /> -</div> - -<p>I find that by the use of the -initial “s” these names come to be -practically entirely disconnected with -the systematic names for the square -from which they are derived. They -are easy to learn, and when learned -can be used readily with the axes running in any -direction.</p> - -<p>To derive the limit names for a four-dimensional rectangular -figure, like the tesseract, is a simple extension of -this process. These point, line, etc., names include those -which apply to a cube, as will be evident on inspection -of the first cube of the diagrams which follow.</p> - -<p>All that is necessary is to place an “s” before each of the -names given for a tesseract block. We then obtain -apellatives which, like the colour names on page 174, -<a href="#fig_103">fig. 103</a>, apply to all the points, lines, faces, solids, and to<span class="pagenum" id="Page_264">[Pg 264]</span> -the hyper-solid of the tesseract. These names have the -advantage over the colour marks that each point, line, etc., -has its own individual name.</p> - -<p>In the diagrams I give the names corresponding to -the positions shown in the coloured plate or described on -p. 174. By comparing cubes 1, 2, 3 with the first row of -cubes in the coloured plate, the systematic names of each -of the points, lines, faces, etc., can be determined. The -asterisk shows the origin from which the names run.</p> - -<p>These point, line, face, etc., names should be used in -connection with the corresponding colours. The names -should call up coloured images of the parts named in their -right connection.</p> - -<p>It is found that a certain abbreviation adds vividness of -distinction to these names. If the final “en” be dropped -wherever it occurs the system is improved. Thus instead -of “senen,” “seten,” “selen,” it is preferable to abbreviate -to “sen,” “set,” “sel,” and also use “san,” “sin” for -“sanen,” “sinen.”</p> -<div class="figcenter illowp100" id="fig_151" style="max-width: 62.5em;"> - <img src="images/fig_151.png" alt="See above" /> -</div> -<p><span class="pagenum" id="Page_265">[Pg 265]</span></p> - -<div class="figcenter illowp100" id="fig_152" style="max-width: 62.5em;"> - <img src="images/fig_152.png" alt="see above" /> -</div> - -<div class="figcenter illowp100" id="fig_153" style="max-width: 62.5em;"> - <img src="images/fig_153.png" alt="see above" /> -</div> - -<p><span class="pagenum" id="Page_266">[Pg 266]</span></p> - -<div class="figcenter illowp100" id="fig_154" style="max-width: 62.5em;"> - <img src="images/fig_154.png" alt="see above" /> -</div> - -<p>We can now name any section. Take <i>e.g.</i> the line in -the first cube from senin to senel, we should call the line -running from senin to senel, senin senat senel, a line -light yellow in colour with null points.</p> - -<p>Here senat is the name for all of the line except its ends. -Using “senat” in this way does not mean that the line is -the whole of senat, but what there is of it is senat. It is -a part of the senat region. Thus also the triangle, which -has its three vertices in senin, senel, selen, is named thus:</p> - - -<ul> -<li>Area: setat.</li> -<li>Sides: setan, senat, setet.</li> -<li>Vertices: senin, senel, sel.</li> -</ul> - -<p>The tetrahedron section of the tesseract can be thought -of as a series of plane sections in the successive sections of -the tesseract shown in <a href="#fig_114">fig. 114</a>, p. 191. In b<sub>0</sub> the section -<span class="pagenum" id="Page_267">[Pg 267]</span>is the one written above. In b<sub>1</sub> the section is made by a -plane which cuts the three edges from sanen intermediate -of their lengths and thus will be:</p> - - -<ul> -<li>Area: satat.</li> -<li>Sides: satan, sanat, satet.</li> -<li>Vertices: sanan, sanet, sat.</li> -</ul> - - -<p>The sections in b<sub>2</sub>, b<sub>3</sub> will be like the section in b<sub>1</sub> but -smaller.</p> - -<p>Finally in b<sub>4</sub> the section plane simply passes through the -corner named sin.</p> - -<p>Hence, putting these sections together in their right -relation, from the face setat, surrounded by the lines and -points mentioned above, there run:</p> - - -<ul> -<li>3 faces: satan, sanat, satet</li> -<li>3 lines: sanan, sanet, sat</li> -</ul> - - -<p>and these faces and lines run to the point sin. Thus -the tetrahedron is completely named.</p> - -<p>The octahedron section of the tesseract, which can be -traced from <a href="#fig_72">fig. 72</a>, p. 129 by extending the lines there -drawn, is named:</p> - -<p>Front triangle selin, selat, selel, setal, senil, setit, selin -with area setat.</p> - -<p>The sections between the front and rear triangle, of -which one is shown in 1b, another in 2b, are thus named, -points and lines, salan, salat, salet, satet, satel, satal, sanal, -sanat, sanit, satit, satin, satan, salan.</p> - -<p>The rear triangle found in 3b by producing lines is sil, -sitet, sinel, sinat, sinin, sitan, sil.</p> - -<p>The assemblage of sections constitute the solid body of -the octahedron satat with triangular faces. The one from -the line selat to the point sil, for instance, is named<span class="pagenum" id="Page_268">[Pg 268]</span> -selin, selat, selel, salet, salat, salan, sil. The whole -interior is salat.</p> - -<p>Shapes can easily be cut out of cardboard which, when -folded together, form not only the tetrahedron and the -octahedron, but also samples of all the sections of the -tesseract taken as it passes cornerwise through our space. -To name and visualise with appropriate colours a series of -these sections is an admirable exercise for obtaining -familiarity with the subject.</p> - - -<h3><span class="smcap">Extension and Connection with Numbers.</span></h3> - -<p>By extending the letter sequence it is of course possible -to name a larger field. By using the limit names the -corners of each square can be named.</p> - -<p>Thus “en sen,” “an sen,” etc., will be the names of the -points nearest the origin in “en” and in “an.”</p> - -<p>A field of points of which each one is indefinitely small -is given by the names written below.</p> - -<div class="figcenter illowp30" id="fig_155" style="max-width: 12.5em;"> - <img src="images/fig_155.png" alt="Field of points" /> -</div> - -<p>The squares are shown in dotted lines, the names -denote the points. These points are not mathematical -points, but really minute areas.</p> - -<p>Instead of starting with a set of squares and naming -them, we can start with a set of points.</p> - -<p>By an easily remembered convention we can give -names to such a region of points.</p> - -<p><span class="pagenum" id="Page_269">[Pg 269]</span></p> - -<p>Let the space names with a final “e” added denote the -mathematical points at the corner of each square nearest -the origin. We have then</p> - -<div class="figcenter illowp25" id="i_269" style="max-width: 15.625em;"> - <img src="images/i_269.png" alt="illustrating immediate text" /> -</div> -<p class="pnind">for the set of mathematical points indicated. This -system is really completely independent of the area -system and is connected with it merely for the purpose -of facilitating the memory processes. The word “ene” is -pronounced like “eny,” with just sufficient attention to -the final vowel to distinguish it from the word “en.”</p> - -<p>Now, connecting the numbers 0, 1, 2 with the sequence -e, a, i, and also with the sequence n, t, l, we have a set of -points named as with numbers in a co-ordinate system. -Thus “ene” is (0, 0) “ate” is (1, 1) “ite” is (2, 1). -To pass to the area system the rule is that the name of -the square is formed from the name of its point nearest -to the origin by dropping the final e.</p> - -<p>By using a notation analogous to the decimal system -a larger field of points can be named. It remains to -assign a letter sequence to the numbers from positive 0 -to positive 9, and from negative 0 to negative 9, to obtain -a system which can be used to denote both the usual -co-ordinate system of mapping and a system of named -squares. The names denoting the points all end with e. -Those that denote squares end with a consonant.</p> - -<p>There are many considerations which must be attended -to in extending the sequences to be used, such as -uniqueness in the meaning of the words formed, ease -of pronunciation, avoidance of awkward combinations.</p> - -<p><span class="pagenum" id="Page_270">[Pg 270]</span></p> - -<p>I drop “s” altogether from the consonant series and -short “u” from the vowel series. It is convenient to -have unsignificant letters at disposal. A double consonant -like “st” for instance can be referred to without giving it -a local significance by calling it “ust.” I increase the -number of vowels by considering a sound like “ra” to -be a vowel, using, that is, the letter “r” as forming a -compound vowel.</p> - -<p>The series is as follows:—</p> - -<table class="standard" summary=""> -<tr> -<td class="tdc" colspan="11"><span class="smcap">Consonants.</span></td> -</tr> -<tr> -<td class="tdc"></td> -<td class="tdc">0</td> -<td class="tdc">1</td> -<td class="tdc">2</td> -<td class="tdc">3</td> -<td class="tdc">4</td> -<td class="tdc">5</td> -<td class="tdc">6</td> -<td class="tdc">7</td> -<td class="tdc">8</td> -<td class="tdc">9</td> -</tr> -<tr> -<td class="tdl">positive</td> -<td class="tdc">n</td> -<td class="tdc">t</td> -<td class="tdc">l</td> -<td class="tdc">p</td> -<td class="tdc">f</td> -<td class="tdc">sh</td> -<td class="tdc">k</td> -<td class="tdc">ch</td> -<td class="tdc">nt</td> -<td class="tdc">st</td> -</tr> -<tr> -<td class="tdl">negative</td> -<td class="tdc">z</td> -<td class="tdc">d</td> -<td class="tdc">th</td> -<td class="tdc">b</td> -<td class="tdc">v</td> -<td class="tdc">m</td> -<td class="tdc">g</td> -<td class="tdc">j</td> -<td class="tdc">nd</td> -<td class="tdc">sp</td> -</tr> -<tr> -<td class="tdc" colspan="11"><span class="smcap">Vowels.</span></td> -</tr> -<tr> -<td class="tdc"></td> -<td class="tdc">0</td> -<td class="tdc">1</td> -<td class="tdc">2</td> -<td class="tdc">3</td> -<td class="tdc">4</td> -<td class="tdc">5</td> -<td class="tdc">6</td> -<td class="tdc">7</td> -<td class="tdc">8</td> -<td class="tdc">9</td> -</tr> -<tr> -<td class="tdc">positive</td> -<td class="tdc">e</td> -<td class="tdc">a</td> -<td class="tdc">i</td> -<td class="tdc">ee</td> -<td class="tdc">ae</td> -<td class="tdc">ai</td> -<td class="tdc">ar</td> -<td class="tdc">ra</td> -<td class="tdc">ri</td> -<td class="tdc">ree</td> -</tr> -<tr> -<td class="tdc">negative</td> -<td class="tdc">er</td> -<td class="tdc">o</td> -<td class="tdc">oo</td> -<td class="tdc">io</td> -<td class="tdc">oe</td> -<td class="tdc">iu</td> -<td class="tdc">or</td> -<td class="tdc">ro</td> -<td class="tdc">roo rio</td> -</tr> -</table> - - -<p><i>Pronunciation.</i>—e as in men; a as in man; i as in in; -ee as in between; ae as ay in may; ai as i in mine; ar as -in art; er as ear in earth; o as in on; oo as oo in soon; -io as in clarion; oe as oa in oat; iu pronounced like yew.</p> - -<p>To name a point such as (23, 41) it is considered as -(3, 1) on from (20, 40) and is called “ifeete.” It is the -initial point of the square ifeet of the area system.</p> - -<p>The preceding amplification of a space language has -been introduced merely for the sake of completeness. As -has already been said nine words and their combinations, -applied to a few simple models suffice for the purposes of -our present enquiry.</p> - - -<p class="center small"><i>Printed by Hazell, Watson & Viney, Ld., London and Aylesbury.</i></p> - -<div style='display:block; margin-top:4em'>*** END OF THE PROJECT GUTENBERG EBOOK THE FOURTH DIMENSION ***</div> -<div style='text-align:left'> - -<div style='display:block; margin:1em 0'> -Updated editions will replace the previous one—the old editions will -be renamed. -</div> - -<div style='display:block; margin:1em 0'> -Creating the works from print editions not protected by U.S. copyright -law means that no one owns a United States copyright in these works, -so the Foundation (and you!) can copy and distribute it in the United -States without permission and without paying copyright -royalties. 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All other -spelling and punctuation remains unchanged. - -Italics are represented thus _italic_, bold thus =bold= and -superscripts thus y^{en}. - -It should be noted that much of the text is a discussion centred on the -many illustrations which have not been included. - - - - - THE FOURTH DIMENSION - - - - - SOME OPINIONS OF THE PRESS - - -“_Mr. C. H. Hinton discusses the subject of the higher dimensionality -of space, his aim being to avoid mathematical subtleties and -technicalities, and thus enable his argument to be followed by readers -who are not sufficiently conversant with mathematics to follow these -processes of reasoning._”—NOTTS GUARDIAN. - -“_The fourth dimension is a subject which has had a great fascination -for many teachers, and though one cannot pretend to have quite grasped -Mr. Hinton’s conceptions and arguments, yet it must be admitted that -he reveals the elusive idea in quite a fascinating light. Quite -apart from the main thesis of the book many chapters are of great -independent interest. Altogether an interesting, clever and ingenious -book._”—DUNDEE COURIER. - -“_The book will well repay the study of men who like to exercise their -wits upon the problems of abstract thought._”—SCOTSMAN. - -“_Professor Hinton has done well to attempt a treatise of moderate -size, which shall at once be clear in method and free from -technicalities of the schools._”—PALL MALL GAZETTE. - -“_A very interesting book he has made of it._”—PUBLISHERS’ CIRCULAR. - -“_Mr. Hinton tries to explain the theory of the fourth dimension so -that the ordinary reasoning mind can get a grasp of what metaphysical -mathematicians mean by it. If he is not altogether successful it is not -from want of clearness on his part, but because the whole theory comes -as such an absolute shock to all one’s preconceived ideas._”—BRISTOL -TIMES. - -“_Mr. Hinton’s enthusiasm is only the result of an exhaustive study, -which has enabled him to set his subject before the reader with far -more than the amount of lucidity to which it is accustomed._”—PALL MALL -GAZETTE. - -“_The book throughout is a very solid piece of reasoning in the domain -of higher mathematics._”—GLASGOW HERALD. - -“_Those who wish to grasp the meaning of this somewhat difficult -subject would do well to read_ The Fourth Dimension. _No mathematical -knowledge is demanded of the reader, and any one, who is not afraid of -a little hard thinking, should be able to follow the argument._”—LIGHT. - -“_A splendidly clear re-statement of the old problem of the fourth -dimension. All who are interested in this subject will find the -work not only fascinating, but lucid, it being written in a style -easily understandable. The illustrations make still more clear -the letterpress, and the whole is most admirably adapted to the -requirements of the novice or the student._”—TWO WORLDS. - -“_Those in search of mental gymnastics will find abundance of exercise -in Mr. C. H. Hinton’s_ Fourth Dimension.”—WESTMINSTER REVIEW. - - - FIRST EDITION, _April 1904_; SECOND EDITION, _May 1906_. - - - Views of the Tessaract. - - No. 1. No. 2. No. 3. - - No. 4. No. 5. No. 6. - - No. 7. No. 8. No. 9. - - No. 10. No. 11. No. 12. - - - - - THE - - FOURTH DIMENSION - - BY - - C. HOWARD HINTON, M.A. - - AUTHOR OF “SCIENTIFIC ROMANCES” - “A NEW ERA OF THOUGHT,” ETC., ETC. - - [Illustration: Colophon] - - - LONDON - SWAN SONNENSCHEIN & CO., LIMITED - 25 HIGH STREET, BLOOMSBURY - - 1906 - - - - - PRINTED BY - HAZELL, WATSON AND VINEY, LD., - LONDON AND AYLESBURY. - - - - - PREFACE - - -I have endeavoured to present the subject of the higher dimensionality -of space in a clear manner, devoid of mathematical subtleties and -technicalities. In order to engage the interest of the reader, I have -in the earlier chapters dwelt on the perspective the hypothesis of a -fourth dimension opens, and have treated of the many connections there -are between this hypothesis and the ordinary topics of our thoughts. - -A lack of mathematical knowledge will prove of no disadvantage to the -reader, for I have used no mathematical processes of reasoning. I have -taken the view that the space which we ordinarily think of, the space -of real things (which I would call permeable matter), is different from -the space treated of by mathematics. Mathematics will tell us a great -deal about space, just as the atomic theory will tell us a great deal -about the chemical combinations of bodies. But after all, a theory is -not precisely equivalent to the subject with regard to which it is -held. There is an opening, therefore, from the side of our ordinary -space perceptions for a simple, altogether rational, mechanical, and -observational way of treating this subject of higher space, and of -this opportunity I have availed myself. - -The details introduced in the earlier chapters, especially in -Chapters VIII., IX., X., may perhaps be found wearisome. They are of -no essential importance in the main line of argument, and if left -till Chapters XI. and XII. have been read, will be found to afford -interesting and obvious illustrations of the properties discussed in -the later chapters. - -My thanks are due to the friends who have assisted me in designing and -preparing the modifications of my previous models, and in no small -degree to the publisher of this volume, Mr. Sonnenschein, to whose -unique appreciation of the line of thought of this, as of my former -essays, their publication is owing. By the provision of a coloured -plate, in addition to the other illustrations, he has added greatly to -the convenience of the reader. - - C. HOWARD HINTON. - - - - - CONTENTS - - - CHAP. PAGE - - I. FOUR-DIMENSIONAL SPACE 1 - - II. THE ANALOGY OF A PLANE WORLD 6 - - III. THE SIGNIFICANCE OF A FOUR-DIMENSIONAL - EXISTENCE 15 - - IV. THE FIRST CHAPTER IN THE HISTORY OF FOUR - SPACE 23 - - V. THE SECOND CHAPTER IN THE HISTORY OF - FOUR SPACE 41 - - Lobatchewsky, Bolyai, and Gauss - Metageometry - - VI. THE HIGHER WORLD 61 - - VII. THE EVIDENCE FOR A FOURTH DIMENSION 76 - - VIII. THE USE OF FOUR DIMENSIONS IN THOUGHT 85 - - IX. APPLICATION TO KANT’S THEORY OF EXPERIENCE 107 - - X. A FOUR-DIMENSIONAL FIGURE 122 - - XI. NOMENCLATURE AND ANALOGIES 136 - - XII. THE SIMPLEST FOUR-DIMENSIONAL SOLID 157 - - XIII. REMARKS ON THE FIGURES 178 - - XIV. A RECAPITULATION AND EXTENSION OF THE - PHYSICAL ARGUMENT 203 - - APPENDIX I.—THE MODELS 231 - - " II.—A LANGUAGE OF SPACE 248 - - - - - THE FOURTH DIMENSION - - - - - CHAPTER I - - FOUR-DIMENSIONAL SPACE - - -There is nothing more indefinite, and at the same time more real, than -that which we indicate when we speak of the “higher.” In our social -life we see it evidenced in a greater complexity of relations. But this -complexity is not all. There is, at the same time, a contact with, an -apprehension of, something more fundamental, more real. - -With the greater development of man there comes a consciousness of -something more than all the forms in which it shows itself. There is -a readiness to give up all the visible and tangible for the sake of -those principles and values of which the visible and tangible are the -representation. The physical life of civilised man and of a mere savage -are practically the same, but the civilised man has discovered a depth -in his existence, which makes him feel that that which appears all to -the savage is a mere externality and appurtenage to his true being. - -Now, this higher—how shall we apprehend it? It is generally embraced -by our religious faculties, by our idealising tendency. But the higher -existence has two sides. It has a being as well as qualities. And in -trying to realise it through our emotions we are always taking the -subjective view. Our attention is always fixed on what we feel, what -we think. Is there any way of apprehending the higher after the purely -objective method of a natural science? I think that there is. - -Plato, in a wonderful allegory, speaks of some men living in such a -condition that they were practically reduced to be the denizens of -a shadow world. They were chained, and perceived but the shadows of -themselves and all real objects projected on a wall, towards which -their faces were turned. All movements to them were but movements -on the surface, all shapes but the shapes of outlines with no -substantiality. - -Plato uses this illustration to portray the relation between true -being and the illusions of the sense world. He says that just as a man -liberated from his chains could learn and discover that the world was -solid and real, and could go back and tell his bound companions of this -greater higher reality, so the philosopher who has been liberated, who -has gone into the thought of the ideal world, into the world of ideas -greater and more real than the things of sense, can come and tell his -fellow men of that which is more true than the visible sun—more noble -than Athens, the visible state. - -Now, I take Plato’s suggestion; but literally, not metaphorically. -He imagines a world which is lower than this world, in that shadow -figures and shadow motions are its constituents; and to it he contrasts -the real world. As the real world is to this shadow world, so is the -higher world to our world. I accept his analogy. As our world in three -dimensions is to a shadow or plane world, so is the higher world to our -three-dimensional world. That is, the higher world is four-dimensional; -the higher being is, so far as its existence is concerned apart from -its qualities, to be sought through the conception of an actual -existence spatially higher than that which we realise with our senses. - -Here you will observe I necessarily leave out all that gives its -charm and interest to Plato’s writings. All those conceptions of the -beautiful and good which live immortally in his pages. - -All that I keep from his great storehouse of wealth is this one thing -simply—a world spatially higher than this world, a world which can only -be approached through the stocks and stones of it, a world which must -be apprehended laboriously, patiently, through the material things of -it, the shapes, the movements, the figures of it. - -We must learn to realise the shapes of objects in this world of the -higher man; we must become familiar with the movements that objects -make in his world, so that we can learn something about his daily -experience, his thoughts of material objects, his machinery. - -The means for the prosecution of this enquiry are given in the -conception of space itself. - -It often happens that that which we consider to be unique and unrelated -gives us, within itself, those relations by means of which we are able -to see it as related to others, determining and determined by them. - -Thus, on the earth is given that phenomenon of weight by means of which -Newton brought the earth into its true relation to the sun and other -planets. Our terrestrial globe was determined in regard to other bodies -of the solar system by means of a relation which subsisted on the earth -itself. - -And so space itself bears within it relations of which we can -determine it as related to other space. For within space are given the -conceptions of point and line, line and plane, which really involve the -relation of space to a higher space. - -Where one segment of a straight line leaves off and another begins is -a point, and the straight line itself can be generated by the motion of -the point. - -One portion of a plane is bounded from another by a straight line, and -the plane itself can be generated by the straight line moving in a -direction not contained in itself. - -Again, two portions of solid space are limited with regard to each -other by a plane; and the plane, moving in a direction not contained in -itself, can generate solid space. - -Thus, going on, we may say that space is that which limits two portions -of higher space from each other, and that our space will generate the -higher space by moving in a direction not contained in itself. - -Another indication of the nature of four-dimensional space can be -gained by considering the problem of the arrangement of objects. - -If I have a number of swords of varying degrees of brightness, I can -represent them in respect of this quality by points arranged along a -straight line. - -If I place a sword at A, fig. 1, and regard it as having a certain -brightness, then the other swords can be arranged in a series along the -line, as at A, B, C, etc., according to their degrees of brightness. - -[Illustration: Fig. 1.] - -If now I take account of another quality, say length, they can be -arranged in a plane. Starting from A, B, C, I can find points to -represent different degrees of length along such lines as AF, BD, CE, -drawn from A and B and C. Points on these lines represent different -degrees of length with the same degree of brightness. Thus the whole -plane is occupied by points representing all conceivable varieties of -brightness and length. - -[Illustration: Fig. 2.] - -Bringing in a third quality, say sharpness, I can draw, as in fig. 3, -any number of upright lines. Let distances along these upright lines -represent degrees of sharpness, thus the points F and G will represent -swords of certain definite degrees of the three qualities mentioned, -and the whole of space will serve to represent all conceivable degrees -of these three qualities. - -[Illustration: Fig. 3.] - -If now I bring in a fourth quality, such as weight, and try to find a -means of representing it as I did the other three qualities, I find -a difficulty. Every point in space is taken up by some conceivable -combination of the three qualities already taken. - -To represent four qualities in the same way as that in which I have -represented three, I should need another dimension of space. - -Thus we may indicate the nature of four-dimensional space by saying -that it is a kind of space which would give positions representative -of four qualities, as three-dimensional space gives positions -representative of three qualities. - - - - - CHAPTER II - - THE ANALOGY OF A PLANE WORLD - - -At the risk of some prolixity I will go fully into the experience of -a hypothetical creature confined to motion on a plane surface. By so -doing I shall obtain an analogy which will serve in our subsequent -enquiries, because the change in our conception, which we make in -passing from the shapes and motions in two dimensions to those in -three, affords a pattern by which we can pass on still further to the -conception of an existence in four-dimensional space. - -A piece of paper on a smooth table affords a ready image of a -two-dimensional existence. If we suppose the being represented by -the piece of paper to have no knowledge of the thickness by which -he projects above the surface of the table, it is obvious that he -can have no knowledge of objects of a similar description, except by -the contact with their edges. His body and the objects in his world -have a thickness of which however, he has no consciousness. Since -the direction stretching up from the table is unknown to him he will -think of the objects of his world as extending in two dimensions only. -Figures are to him completely bounded by their lines, just as solid -objects are to us by their surfaces. He cannot conceive of approaching -the centre of a circle, except by breaking through the circumference, -for the circumference encloses the centre in the directions in which -motion is possible to him. The plane surface over which he slips and -with which he is always in contact will be unknown to him; there are no -differences by which he can recognise its existence. - -But for the purposes of our analogy this representation is deficient. - -A being as thus described has nothing about him to push off from, the -surface over which he slips affords no means by which he can move in -one direction rather than another. Placed on a surface over which he -slips freely, he is in a condition analogous to that in which we should -be if we were suspended free in space. There is nothing which he can -push off from in any direction known to him. - -Let us therefore modify our representation. Let us suppose a vertical -plane against which particles of thin matter slip, never leaving the -surface. Let these particles possess an attractive force and cohere -together into a disk; this disk will represent the globe of a plane -being. He must be conceived as existing on the rim. - -[Illustration: Fig. 4.] - -Let 1 represent this vertical disk of flat matter and 2 the plane being -on it, standing upon its rim as we stand on the surface of our earth. -The direction of the attractive force of his matter will give the -creature a knowledge of up and down, determining for him one direction -in his plane space. Also, since he can move along the surface of his -earth, he will have the sense of a direction parallel to its surface, -which we may call forwards and backwards. - -He will have no sense of right and left—that is, of the direction which -we recognise as extending out from the plane to our right and left. - -The distinction of right and left is the one that we must suppose to -be absent, in order to project ourselves into the condition of a plane -being. - -Let the reader imagine himself, as he looks along the plane, fig. 4, -to become more and more identified with the thin body on it, till he -finally looks along parallel to the surface of the plane earth, and up -and down, losing the sense of the direction which stretches right and -left. This direction will be an unknown dimension to him. - -Our space conceptions are so intimately connected with those which -we derive from the existence of gravitation that it is difficult to -realise the condition of a plane being, without picturing him as in -material surroundings with a definite direction of up and down. Hence -the necessity of our somewhat elaborate scheme of representation, -which, when its import has been grasped, can be dispensed with for the -simpler one of a thin object slipping over a smooth surface, which lies -in front of us. - -It is obvious that we must suppose some means by which the plane being -is kept in contact with the surface on which he slips. The simplest -supposition to make is that there is a transverse gravity, which keeps -him to the plane. This gravity must be thought of as different to the -attraction exercised by his matter, and as unperceived by him. - -At this stage of our enquiry I do not wish to enter into the question -of how a plane being could arrive at a knowledge of the third -dimension, but simply to investigate his plane consciousness. - -It is obvious that the existence of a plane being must be very limited. -A straight line standing up from the surface of his earth affords a bar -to his progress. An object like a wheel which rotates round an axis -would be unknown to him, for there is no conceivable way in which he -can get to the centre without going through the circumference. He would -have spinning disks, but could not get to the centre of them. The plane -being can represent the motion from any one point of his space to any -other, by means of two straight lines drawn at right angles to each -other. - -Let AX and AY be two such axes. He can accomplish the translation from -A to B by going along AX to C, and then from C along CB parallel to AY. - -The same result can of course be obtained by moving to D along AY and -then parallel to AX from D to B, or of course by any diagonal movement -compounded by these axial movements. - -[Illustration: Fig. 5.] - -By means of movements parallel to these two axes he can proceed (except -for material obstacles) from any one point of his space to any other. - -If now we suppose a third line drawn out from A at right angles to the -plane it is evident that no motion in either of the two dimensions he -knows will carry him in the least degree in the direction represented -by AZ. - -[Illustration: Fig. 6.] - -The lines AZ and AX determine a plane. If he could be taken off his -plane, and transferred to the plane AXZ, he would be in a world exactly -like his own. From every line in his world there goes off a space world -exactly like his own. - -[Illustration: Fig. 7.] - -From every point in his world a line can be drawn parallel to AZ in -the direction unknown to him. If we suppose the square in fig. 7 to be -a geometrical square from every point of it, inside as well as on the -contour, a straight line can be drawn parallel to AZ. The assemblage -of these lines constitute a solid figure, of which the square in the -plane is the base. If we consider the square to represent an object -in the plane being’s world then we must attribute to it a very small -thickness, for every real thing must possess all three dimensions. -This thickness he does not perceive, but thinks of this real object as -a geometrical square. He thinks of it as possessing area only, and no -degree of solidity. The edges which project from the plane to a very -small extent he thinks of as having merely length and no breadth—as -being, in fact, geometrical lines. - -With the first step in the apprehension of a third dimension there -would come to a plane being the conviction that he had previously -formed a wrong conception of the nature of his material objects. He -had conceived them as geometrical figures of two dimensions only. If a -third dimension exists, such figures are incapable of real existence. -Thus he would admit that all his real objects had a certain, though -very small thickness in the unknown dimension, and that the conditions -of his existence demanded the supposition of an extended sheet of -matter, from contact with which in their motion his objects never -diverge. - -Analogous conceptions must be formed by us on the supposition of a -four-dimensional existence. We must suppose a direction in which we can -never point extending from every point of our space. We must draw a -distinction between a geometrical cube and a cube of real matter. The -cube of real matter we must suppose to have an extension in an unknown -direction, real, but so small as to be imperceptible by us. From every -point of a cube, interior as well as exterior, we must imagine that it -is possible to draw a line in the unknown direction. The assemblage of -these lines would constitute a higher solid. The lines going off in -the unknown direction from the face of a cube would constitute a cube -starting from that face. Of this cube all that we should see in our -space would be the face. - -Again, just as the plane being can represent any motion in his space by -two axes, so we can represent any motion in our three-dimensional space -by means of three axes. There is no point in our space to which we -cannot move by some combination of movements on the directions marked -out by these axes. - -On the assumption of a fourth dimension we have to suppose a fourth -axis, which we will call AW. It must be supposed to be at right angles -to each and every one of the three axes AX, AY, AZ. Just as the two -axes, AX, AZ, determine a plane which is similar to the original plane -on which we supposed the plane being to exist, but which runs off from -it, and only meets it in a line; so in our space if we take any three -axes such as AX, AY, and AW, they determine a space like our space -world. This space runs off from our space, and if we were transferred -to it we should find ourselves in a space exactly similar to our own. - -We must give up any attempt to picture this space in its relation -to ours, just as a plane being would have to give up any attempt to -picture a plane at right angles to his plane. - -Such a space and ours run in different directions from the plane of AX -and AY. They meet in this plane but have nothing else in common, just -as the plane space of AX and AY and that of AX and AZ run in different -directions and have but the line AX in common. - -Omitting all discussion of the manner on which a plane being might be -conceived to form a theory of a three-dimensional existence, let us -examine how, with the means at his disposal, he could represent the -properties of three-dimensional objects. - -There are two ways in which the plane being can think of one of our -solid bodies. He can think of the cube, fig. 8, as composed of a number -of sections parallel to his plane, each lying in the third dimension -a little further off from his plane than the preceding one. These -sections he can represent as a series of plane figures lying in his -plane, but in so representing them he destroys the coherence of them -in the higher figure. The set of squares, A, B, C, D, represents the -section parallel to the plane of the cube shown in figure, but they are -not in their proper relative positions. - -[Illustration: Fig. 8.] - -The plane being can trace out a movement in the third dimension by -assuming discontinuous leaps from one section to another. Thus, -a motion along the edge of the cube from left to right would be -represented in the set of sections in the plane as the succession of -the corners of the sections A, B, C, D. A point moving from A through -BCD in our space must be represented in the plane as appearing in A, -then in B, and so on, without passing through the intervening plane -space. - -In these sections the plane being leaves out, of course, the extension -in the third dimension; the distance between any two sections is not -represented. In order to realise this distance the conception of motion -can be employed. - -[Illustration: Fig. 9.] - -Let fig. 9 represent a cube passing transverse to the plane. It will -appear to the plane being as a square object, but the matter of which -this object is composed will be continually altering. One material -particle takes the place of another, but it does not come from anywhere -or go anywhere in the space which the plane being knows. - -The analogous manner of representing a higher solid in our case, is to -conceive it as composed of a number of sections, each lying a little -further off in the unknown direction than the preceding. - -[Illustration: Fig. 10.] - -We can represent these sections as a number of solids. Thus the cubes -A, B, C, D, may be considered as the sections at different intervals in -the unknown dimension of a higher cube. Arranged thus their coherence -in the higher figure is destroyed, they are mere representations. - -A motion in the fourth dimension from A through B, C, etc., would be -continuous, but we can only represent it as the occupation of the -positions A, B, C, etc., in succession. We can exhibit the results of -the motion at different stages, but no more. - -In this representation we have left out the distance between one -section and another; we have considered the higher body merely as a -series of sections, and so left out its contents. The only way to -exhibit its contents is to call in the aid of the conception of motion. - -[Illustration: Fig. 11.] - -If a higher cube passes transverse to our space, it will appear as -a cube isolated in space, the part that has not come into our space -and the part that has passed through will not be visible. The gradual -passing through our space would appear as the change of the matter -of the cube before us. One material particle in it is succeeded by -another, neither coming nor going in any direction we can point to. In -this manner, by the duration of the figure, we can exhibit the higher -dimensionality of it; a cube of our matter, under the circumstances -supposed, namely, that it has a motion transverse to our space, would -instantly disappear. A higher cube would last till it had passed -transverse to our space by its whole distance of extension in the -fourth dimension. - -As the plane being can think of the cube as consisting of sections, -each like a figure he knows, extending away from his plane, so we can -think of a higher solid as composed of sections, each like a solid -which we know, but extending away from our space. - -Thus, taking a higher cube, we can look on it as starting from a cube -in our space and extending in the unknown dimension. - -[Illustration: Fig. 12.] - -Take the face A and conceive it to exist as simply a face, a square -with no thickness. From this face the cube in our space extends by the -occupation of space which we can see. - -But from this face there extends equally a cube in the unknown -dimension. We can think of the higher cube, then, by taking the set -of sections A, B, C, D, etc., and considering that from each of them -there runs a cube. These cubes have nothing in common with each other, -and of each of them in its actual position all that we can have in our -space is an isolated square. It is obvious that we can take our series -of sections in any manner we please. We can take them parallel, for -instance, to any one of the three isolated faces shown in the figure. -Corresponding to the three series of sections at right angles to each -other, which we can make of the cube in space, we must conceive of the -higher cube, as composed of cubes starting from squares parallel to the -faces of the cube, and of these cubes all that exist in our space are -the isolated squares from which they start. - - - - - CHAPTER III - - THE SIGNIFICANCE OF A FOUR-DIMENSIONAL EXISTENCE - - -Having now obtained the conception of a four-dimensional space, and -having formed the analogy which, without any further geometrical -difficulties, enables us to enquire into its properties, I will refer -the reader, whose interest is principally in the mechanical aspect, -to Chapters VI. and VII. In the present chapter I will deal with -the general significance of the enquiry, and in the next with the -historical origin of the idea. - -First, with regard to the question of whether there is any evidence -that we are really in four-dimensional space, I will go back to the -analogy of the plane world. - -A being in a plane world could not have any experience of -three-dimensional shapes, but he could have an experience of -three-dimensional movements. - -We have seen that his matter must be supposed to have an extension, -though a very small one, in the third dimension. And thus, in the -small particles of his matter, three-dimensional movements may well -be conceived to take place. Of these movements he would only perceive -the resultants. Since all movements of an observable size in the plane -world are two-dimensional, he would only perceive the resultants in -two dimensions of the small three-dimensional movements. Thus, there -would be phenomena which he could not explain by his theory of -mechanics—motions would take place which he could not explain by his -theory of motion. Hence, to determine if we are in a four-dimensional -world, we must examine the phenomena of motion in our space. If -movements occur which are not explicable on the suppositions of our -three-dimensional mechanics, we should have an indication of a possible -four-dimensional motion, and if, moreover, it could be shown that such -movements would be a consequence of a four-dimensional motion in the -minute particles of bodies or of the ether, we should have a strong -presumption in favour of the reality of the fourth dimension. - -By proceeding in the direction of finer and finer subdivision, we come -to forms of matter possessing properties different to those of the -larger masses. It is probable that at some stage in this process we -should come to a form of matter of such minute subdivision that its -particles possess a freedom of movement in four dimensions. This form -of matter I speak of as four-dimensional ether, and attribute to it -properties approximating to those of a perfect liquid. - -Deferring the detailed discussion of this form of matter to Chapter -VI., we will now examine the means by which a plane being would come to -the conclusion that three-dimensional movements existed in his world, -and point out the analogy by which we can conclude the existence of -four-dimensional movements in our world. Since the dimensions of the -matter in his world are small in the third direction, the phenomena in -which he would detect the motion would be those of the small particles -of matter. - -Suppose that there is a ring in his plane. We can imagine currents -flowing round the ring in either of two opposite directions. These -would produce unlike effects, and give rise to two different fields -of influence. If the ring with a current in it in one direction be -taken up and turned over, and put down again on the plane, it would be -identical with the ring having a current in the opposite direction. An -operation of this kind would be impossible to the plane being. Hence -he would have in his space two irreconcilable objects, namely, the -two fields of influence due to the two rings with currents in them in -opposite directions. By irreconcilable objects in the plane I mean -objects which cannot be thought of as transformed one into the other by -any movement in the plane. - -Instead of currents flowing in the rings we can imagine a different -kind of current. Imagine a number of small rings strung on the original -ring. A current round these secondary rings would give two varieties -of effect, or two different fields of influence, according to its -direction. These two varieties of current could be turned one into -the other by taking one of the rings up, turning it over, and putting -it down again in the plane. This operation is impossible to the plane -being, hence in this case also there would be two irreconcilable fields -in the plane. Now, if the plane being found two such irreconcilable -fields and could prove that they could not be accounted for by currents -in the rings, he would have to admit the existence of currents round -the rings—that is, in rings strung on the primary ring. Thus he would -come to admit the existence of a three-dimensional motion, for such a -disposition of currents is in three dimensions. - -Now in our space there are two fields of different properties, which -can be produced by an electric current flowing in a closed circuit or -ring. These two fields can be changed one into the other by reversing -the currents, but they cannot be changed one into the other by any -turning about of the rings in our space; for the disposition of the -field with regard to the ring itself is different when we turn the -ring, over and when we reverse the direction of the current in the ring. - -As hypotheses to explain the differences of these two fields and their -effects we can suppose the following kinds of space motions:—First, a -current along the conductor; second, a current round the conductor—that -is, of rings of currents strung on the conductor as an axis. Neither of -these suppositions accounts for facts of observation. - -Hence we have to make the supposition of a four-dimensional motion. -We find that a four-dimensional rotation of the nature explained in a -subsequent chapter, has the following characteristics:—First, it would -give us two fields of influence, the one of which could be turned into -the other by taking the circuit up into the fourth dimension, turning -it over, and putting it down in our space again, precisely as the two -kinds of fields in the plane could be turned one into the other by a -reversal of the current in our space. Second, it involves a phenomenon -precisely identical with that most remarkable and mysterious feature of -an electric current, namely that it is a field of action, the rim of -which necessarily abuts on a continuous boundary formed by a conductor. -Hence, on the assumption of a four-dimensional movement in the region -of the minute particles of matter, we should expect to find a motion -analogous to electricity. - -Now, a phenomenon of such universal occurrence as electricity cannot be -due to matter and motion in any very complex relation, but ought to be -seen as a simple and natural consequence of their properties. I infer -that the difficulty in its theory is due to the attempt to explain a -four-dimensional phenomenon by a three-dimensional geometry. - -In view of this piece of evidence we cannot disregard that afforded -by the existence of symmetry. In this connection I will allude to the -simple way of producing the images of insects, sometimes practised by -children. They put a few blots of ink in a straight line on a piece of -paper, fold the paper along the blots, and on opening it the lifelike -presentment of an insect is obtained. If we were to find a multitude -of these figures, we should conclude that they had originated from a -process of folding over; the chances against this kind of reduplication -of parts is too great to admit of the assumption that they had been -formed in any other way. - -The production of the symmetrical forms of organised beings, though not -of course due to a turning over of bodies of any appreciable size in -four-dimensional space, can well be imagined as due to a disposition in -that manner of the smallest living particles from which they are built -up. Thus, not only electricity, but life, and the processes by which we -think and feel, must be attributed to that region of magnitude in which -four-dimensional movements take place. - -I do not mean, however, that life can be explained as a -four-dimensional movement. It seems to me that the whole bias of -thought, which tends to explain the phenomena of life and volition, as -due to matter and motion in some peculiar relation, is adopted rather -in the interests of the explicability of things than with any regard to -probability. - -Of course, if we could show that life were a phenomenon of motion, we -should be able to explain a great deal that is at present obscure. But -there are two great difficulties in the way. It would be necessary to -show that in a germ capable of developing into a living being, there -were modifications of structure capable of determining in the developed -germ all the characteristics of its form, and not only this, but of -determining those of all the descendants of such a form in an infinite -series. Such a complexity of mechanical relations, undeniable though -it be, cannot surely be the best way of grouping the phenomena and -giving a practical account of them. And another difficulty is this, -that no amount of mechanical adaptation would give that element of -consciousness which we possess, and which is shared in to a modified -degree by the animal world. - -In those complex structures which men build up and direct, such as a -ship or a railway train (and which, if seen by an observer of such a -size that the men guiding them were invisible, would seem to present -some of the phenomena of life) the appearance of animation is not due -to any diffusion of life in the material parts of the structure, but to -the presence of a living being. - -The old hypothesis of a soul, a living organism within the visible one, -appears to me much more rational than the attempt to explain life as a -form of motion. And when we consider the region of extreme minuteness -characterised by four-dimensional motion the difficulty of conceiving -such an organism alongside the bodily one disappears. Lord Kelvin -supposes that matter is formed from the ether. We may very well suppose -that the living organisms directing the material ones are co-ordinate -with them, not composed of matter, but consisting of etherial bodies, -and as such capable of motion through the ether, and able to originate -material living bodies throughout the mineral. - -Hypotheses such as these find no immediate ground for proof or disproof -in the physical world. Let us, therefore, turn to a different field, -and, assuming that the human soul is a four-dimensional being, capable -in itself of four dimensional movements, but in its experiences through -the senses limited to three dimensions, ask if the history of thought, -of these productivities which characterise man, correspond to our -assumption. Let us pass in review those steps by which man, presumably -a four-dimensional being, despite his bodily environment, has come to -recognise the fact of four-dimensional existence. - -Deferring this enquiry to another chapter, I will here recapitulate the -argument in order to show that our purpose is entirely practical and -independent of any philosophical or metaphysical considerations. - -If two shots are fired at a target, and the second bullet hits it -at a different place to the first, we suppose that there was some -difference in the conditions under which the second shot was fired -from those affecting the first shot. The force of the powder, the -direction of aim, the strength of the wind, or some condition must -have been different in the second case, if the course of the bullet -was not exactly the same as in the first case. Corresponding to every -difference in a result there must be some difference in the antecedent -material conditions. By tracing out this chain of relations we explain -nature. - -But there is also another mode of explanation which we apply. If we ask -what was the cause that a certain ship was built, or that a certain -structure was erected, we might proceed to investigate the changes in -the brain cells of the men who designed the works. Every variation in -one ship or building from another ship or building is accompanied by -a variation in the processes that go on in the brain matter of the -designers. But practically this would be a very long task. - -A more effective mode of explaining the production of the ship or -building would be to enquire into the motives, plans, and aims of the -men who constructed them. We obtain a cumulative and consistent body of -knowledge much more easily and effectively in the latter way. - -Sometimes we apply the one, sometimes the other mode of explanation. - -But it must be observed that the method of explanation founded on -aim, purpose, volition, always presupposes a mechanical system on -which the volition and aim works. The conception of man as willing and -acting from motives involves that of a number of uniform processes of -nature which he can modify, and of which he can make application. In -the mechanical conditions of the three-dimensional world, the only -volitional agency which we can demonstrate is the human agency. But -when we consider the four-dimensional world the conclusion remains -perfectly open. - -The method of explanation founded on purpose and aim does not, surely, -suddenly begin with man and end with him. There is as much behind the -exhibition of will and motive which we see in man as there is behind -the phenomena of movement; they are co-ordinate, neither to be resolved -into the other. And the commencement of the investigation of that will -and motive which lies behind the will and motive manifested in the -three-dimensional mechanical field is in the conception of a soul—a -four-dimensional organism, which expresses its higher physical being -in the symmetry of the body, and gives the aims and motives of human -existence. - -Our primary task is to form a systematic knowledge of the phenomena -of a four-dimensional world and find those points in which this -knowledge must be called in to complete our mechanical explanation of -the universe. But a subsidiary contribution towards the verification -of the hypothesis may be made by passing in review the history of -human thought, and enquiring if it presents such features as would be -naturally expected on this assumption. - - - - - CHAPTER IV - - THE FIRST CHAPTER IN THE HISTORY OF FOUR SPACE - - -Parmenides, and the Asiatic thinkers with whom he is in close -affinity, propound a theory of existence which is in close accord -with a conception of a possible relation between a higher and a lower -dimensional space. This theory, prior and in marked contrast to the -main stream of thought, which we shall afterwards describe, forms a -closed circle by itself. It is one which in all ages has had a strong -attraction for pure intellect, and is the natural mode of thought for -those who refrain from projecting their own volition into nature under -the guise of causality. - -According to Parmenides of the school of Elea the all is one, unmoving -and unchanging. The permanent amid the transient—that foothold for -thought, that solid ground for feeling on the discovery of which -depends all our life—is no phantom; it is the image amidst deception of -true being, the eternal, the unmoved, the one. Thus says Parmenides. - -But how explain the shifting scene, these mutations of things! - -“Illusion,” answers Parmenides. Distinguishing between truth and -error, he tells of the true doctrine of the one—the false opinion of a -changing world. He is no less memorable for the manner of his advocacy -than for the cause he advocates. It is as if from his firm foothold -of being he could play with the thoughts under the burden of which -others laboured, for from him springs that fluency of supposition and -hypothesis which forms the texture of Plato’s dialectic. - -Can the mind conceive a more delightful intellectual picture than that -of Parmenides, pointing to the one, the true, the unchanging, and yet -on the other hand ready to discuss all manner of false opinion, forming -a cosmogony too, false “but mine own” after the fashion of the time? - -In support of the true opinion he proceeded by the negative way of -showing the self-contradictions in the ideas of change and motion. -It is doubtful if his criticism, save in minor points, has ever been -successfully refuted. To express his doctrine in the ponderous modern -way we must make the statement that motion is phenomenal, not real. - -Let us represent his doctrine. - -[Illustration: Fig. 13.] - -Imagine a sheet of still water into which a slanting stick is being -lowered with a motion vertically downwards. Let 1, 2, 3 (Fig. 13), -be three consecutive positions of the stick. A, B, C, will be three -consecutive positions of the meeting of the stick, with the surface of -the water. As the stick passes down, the meeting will move from A on to -B and C. - -Suppose now all the water to be removed except a film. At the meeting -of the film and the stick there will be an interruption of the film. -If we suppose the film to have a property, like that of a soap bubble, -of closing up round any penetrating object, then as the stick goes -vertically downwards the interruption in the film will move on. - -[Illustration: Fig. 14.] - -If we pass a spiral through the film the intersection will give a point -moving in a circle shown by the dotted lines in the figure. Suppose -now the spiral to be still and the film to move vertically upwards, -the whole spiral will be represented in the film of the consecutive -positions of the point of intersection. In the film the permanent -existence of the spiral is experienced as a time series—the record -of traversing the spiral is a point moving in a circle. If now we -suppose a consciousness connected with the film in such a way that the -intersection of the spiral with the film gives rise to a conscious -experience, we see that we shall have in the film a point moving in a -circle, conscious of its motion, knowing nothing of that real spiral -the record of the successive intersections of which by the film is the -motion of the point. - -It is easy to imagine complicated structures of the nature of the -spiral, structures consisting of filaments, and to suppose also that -these structures are distinguishable from each other at every section. -If we consider the intersections of these filaments with the film as it -passes to be the atoms constituting a filmar universe, we shall have in -the film a world of apparent motion; we shall have bodies corresponding -to the filamentary structure, and the positions of these structures -with regard to one another will give rise to bodies in the film moving -amongst one another. This mutual motion is apparent merely. The reality -is of permanent structures stationary, and all the relative motions -accounted for by one steady movement of the film as a whole. - -Thus we can imagine a plane world, in which all the variety of motion -is the phenomenon of structures consisting of filamentary atoms -traversed by a plane of consciousness. Passing to four dimensions and -our space, we can conceive that all things and movements in our world -are the reading off of a permanent reality by a space of consciousness. -Each atom at every moment is not what it was, but a new part of that -endless line which is itself. And all this system successively revealed -in the time which is but the succession of consciousness, separate -as it is in parts, in its entirety is one vast unity. Representing -Parmenides’ doctrine thus, we gain a firmer hold on it than if we -merely let his words rest, grand and massive, in our minds. And we have -gained the means also of representing phases of that Eastern thought -to which Parmenides was no stranger. Modifying his uncompromising -doctrine, let us suppose, to go back to the plane of consciousness -and the structure of filamentary atoms, that these structures are -themselves moving—are acting, living. Then, in the transverse motion -of the film, there would be two phenomena of motion, one due to the -reading off in the film of the permanent existences as they are in -themselves, and another phenomenon of motion due to the modification of -the record of the things themselves, by their proper motion during the -process of traversing them. - -Thus a conscious being in the plane would have, as it were, a -two-fold experience. In the complete traversing of the structure, the -intersection of which with the film gives his conscious all, the main -and principal movements and actions which he went through would be the -record of his higher self as it existed unmoved and unacting. Slight -modifications and deviations from these movements and actions would -represent the activity and self-determination of the complete being, of -his higher self. - -It is admissible to suppose that the consciousness in the plane has -a share in that volition by which the complete existence determines -itself. Thus the motive and will, the initiative and life, of the -higher being, would be represented in the case of the being in the -film by an initiative and a will capable, not of determining any great -things or important movements in his existence, but only of small and -relatively insignificant activities. In all the main features of his -life his experience would be representative of one state of the higher -being whose existence determines his as the film passes on. But in his -minute and apparently unimportant actions he would share in that will -and determination by which the whole of the being he really is acts and -lives. - -An alteration of the higher being would correspond to a different life -history for him. Let us now make the supposition that film after film -traverses these higher structures, that the life of the real being is -read off again and again in successive waves of consciousness. There -would be a succession of lives in the different advancing planes of -consciousness, each differing from the preceding, and differing in -virtue of that will and activity which in the preceding had not been -devoted to the greater and apparently most significant things in life, -but the minute and apparently unimportant. In all great things the -being of the film shares in the existence of his higher self as it is -at any one time. In the small things he shares in that volition by -which the higher being alters and changes, acts and lives. - -Thus we gain the conception of a life changing and developing as a -whole, a life in which our separation and cessation and fugitiveness -are merely apparent, but which in its events and course alters, -changes, develops; and the power of altering and changing this whole -lies in the will and power the limited being has of directing, guiding, -altering himself in the minute things of his existence. - -Transferring our conceptions to those of an existence in a higher -dimensionality traversed by a space of consciousness, we have an -illustration of a thought which has found frequent and varied -expression. When, however, we ask ourselves what degree of truth -there lies in it, we must admit that, as far as we can see, it is -merely symbolical. The true path in the investigation of a higher -dimensionality lies in another direction. - -The significance of the Parmenidean doctrine lies in this that here, as -again and again, we find that those conceptions which man introduces of -himself, which he does not derive from the mere record of his outward -experience, have a striking and significant correspondence to the -conception of a physical existence in a world of a higher space. How -close we come to Parmenides’ thought by this manner of representation -it is impossible to say. What I want to point out is the adequateness -of the illustration, not only to give a static model of his doctrine, -but one capable as it were, of a plastic modification into a -correspondence into kindred forms of thought. Either one of two things -must be true—that four-dimensional conceptions give a wonderful power -of representing the thought of the East, or that the thinkers of the -East must have been looking at and regarding four-dimensional existence. - -Coming now to the main stream of thought we must dwell in some detail -on Pythagoras, not because of his direct relation to the subject, but -because of his relation to investigators who came later. - -Pythagoras invented the two-way counting. Let us represent the -single-way counting by the posits _aa_, _ab_, _ac_, _ad_, using these -pairs of letters instead of the numbers 1, 2, 3, 4. I put an _a_ in -each case first for a reason which will immediately appear. - -We have a sequence and order. There is no conception of distance -necessarily involved. The difference between the posits is one of -order not of distance—only when identified with a number of equal -material things in juxtaposition does the notion of distance arise. - -Now, besides the simple series I can have, starting from _aa_, _ba_, -_ca_, _da_, from _ab_, _bb_, _cb_, _db_, and so on, and forming a -scheme: - - _da_ _db_ _dc_ _dd_ - _ca_ _cb_ _cc_ _cd_ - _ba_ _bb_ _bc_ _bd_ - _aa_ _ab_ _ac_ _ad_ - -This complex or manifold gives a two-way order. I can represent it by -a set of points, if I am on my guard against assuming any relation of -distance. - -[Illustration: Fig. 15.] - -Pythagoras studied this two-fold way of counting in reference to -material bodies, and discovered that most remarkable property of the -combination of number and matter that bears his name. - -The Pythagorean property of an extended material system can be -exhibited in a manner which will be of use to us afterwards, and which -therefore I will employ now instead of using the kind of figure which -he himself employed. - -Consider a two-fold field of points arranged in regular rows. Such a -field will be presupposed in the following argument. - -[Illustration: Fig. 16. 1 and 2] - -It is evident that in fig. 16 four of the points determine a square, -which square we may take as the unit of measurement for areas. But we -can also measure areas in another way. - -Fig. 16 (1) shows four points determining a square. - -But four squares also meet in a point, fig. 16 (2). - -Hence a point at the corner of a square belongs equally to four -squares. - -Thus we may say that the point value of the square shown is one point, -for if we take the square in fig. 16 (1) it has four points, but each -of these belong equally to four other squares. Hence one fourth of each -of them belongs to the square (1) in fig. 16. Thus the point value of -the square is one point. - -The result of counting the points is the same as that arrived at by -reckoning the square units enclosed. - -Hence, if we wish to measure the area of any square we can take the -number of points it encloses, count these as one each, and take -one-fourth of the number of points at its corners. - -[Illustration: Fig. 17.] - -Now draw a diagonal square as shown in fig. 17. It contains one point -and the four corners count for one point more; hence its point value is -2. The value is the measure of its area—the size of this square is two -of the unit squares. - -Looking now at the sides of this figure we see that there is a unit -square on each of them—the two squares contain no points, but have four -corner points each, which gives the point value of each as one point. - -Hence we see that the square on the diagonal is equal to the squares -on the two sides; or as it is generally expressed, the square on the -hypothenuse is equal to the sum of the squares on the sides. - -[Illustration: Fig. 18.] - -Noticing this fact we can proceed to ask if it is always true. Drawing -the square shown in fig. 18, we can count the number of its points. -There are five altogether. There are four points inside the square on -the diagonal, and hence, with the four points at its corners the point -value is 5—that is, the area is 5. Now the squares on the sides are -respectively of the area 4 and 1. Hence in this case also the square -on the diagonal is equal to the sum of the square on the sides. This -property of matter is one of the first great discoveries of applied -mathematics. We shall prove afterwards that it is not a property of -space. For the present it is enough to remark that the positions in -which the points are arranged is entirely experimental. It is by means -of equal pieces of some material, or the same piece of material moved -from one place to another, that the points are arranged. - -Pythagoras next enquired what the relation must be so that a square -drawn slanting-wise should be equal to one straight-wise. He found that -a square whose side is five can be placed either rectangularly along -the lines of points, or in a slanting position. And this square is -equivalent to two squares of sides 4 and 3. - -Here he came upon a numerical relation embodied in a property of -matter. Numbers immanent in the objects produced the equality so -satisfactory for intellectual apprehension. And he found that numbers -when immanent in sound—when the strings of a musical instrument were -given certain definite proportions of length—were no less captivating -to the ear than the equality of squares was to the reason. What wonder -then that he ascribed an active power to number! - -We must remember that, sharing like ourselves the search for the -permanent in changing phenomena, the Greeks had not that conception of -the permanent in matter that we have. To them material things were not -permanent. In fire solid things would vanish; absolutely disappear. -Rock and earth had a more stable existence, but they too grew and -decayed. The permanence of matter, the conservation of energy, were -unknown to them. And that distinction which we draw so readily between -the fleeting and permanent causes of sensation, between a sound and -a material object, for instance, had not the same meaning to them -which it has for us. Let us but imagine for a moment that material -things are fleeting, disappearing, and we shall enter with a far better -appreciation into that search for the permanent which, with the Greeks, -as with us, is the primary intellectual demand. - -What is that which amid a thousand forms is ever the same, which we can -recognise under all its vicissitudes, of which the diverse phenomena -are the appearances? - -To think that this is number is not so very wide of the mark. With -an intellectual apprehension which far outran the evidences for its -application, the atomists asserted that there were everlasting material -particles, which, by their union, produced all the varying forms and -states of bodies. But in view of the observed facts of nature as -then known, Aristotle, with perfect reason, refused to accept this -hypothesis. - -He expressly states that there is a change of quality, and that the -change due to motion is only one of the possible modes of change. - -With no permanent material world about us, with the fleeting, the -unpermanent, all around we should, I think, be ready to follow -Pythagoras in his identification of number with that principle which -subsists amidst all changes, which in multitudinous forms we apprehend -immanent in the changing and disappearing substance of things. - -And from the numerical idealism of Pythagoras there is but a step to -the more rich and full idealism of Plato. That which is apprehended by -the sense of touch we put as primary and real, and the other senses we -say are merely concerned with appearances. But Plato took them all as -valid, as giving qualities of existence. That the qualities were not -permanent in the world as given to the senses forced him to attribute -to them a different kind of permanence. He formed the conception of a -world of ideas, in which all that really is, all that affects us and -gives the rich and wonderful wealth of our experience, is not fleeting -and transitory, but eternal. And of this real and eternal we see in the -things about us the fleeting and transient images. - -And this world of ideas was no exclusive one, wherein was no place -for the innermost convictions of the soul and its most authoritative -assertions. Therein existed justice, beauty—the one, the good, all -that the soul demanded to be. The world of ideas, Plato’s wonderful -creation preserved for man, for his deliberate investigation and their -sure development, all that the rude incomprehensible changes of a harsh -experience scatters and destroys. - -Plato believed in the reality of ideas. He meets us fairly and -squarely. Divide a line into two parts, he says; one to represent -the real objects in the world, the other to represent the transitory -appearances, such as the image in still water, the glitter of the sun -on a bright surface, the shadows on the clouds. - - A B - ——————————————————————————————|————————————————————————————————- - Real things: Appearances: - _e.g._, the sun. _e.g._, the reflection of the sun. - -Take another line and divide it into two parts, one representing -our ideas, the ordinary occupants of our minds, such as whiteness, -equality, and the other representing our true knowledge, which is of -eternal principles, such as beauty, goodness. - - A^1 B^1 - ——————————————————————————————|————————————————————————————————- - Eternal principles, Appearances in the mind, - as beauty as whiteness, equality - -Then as A is to B, so is A^1 to B^1 - -That is, the soul can proceed, going away from real things to a region -of perfect certainty, where it beholds what is, not the scattered -reflections; beholds the sun, not the glitter on the sands; true being, -not chance opinion. - -Now, this is to us, as it was to Aristotle, absolutely inconceivable -from a scientific point of view. We can understand that a being is -known in the fulness of his relations; it is in his relations to his -circumstances that a man’s character is known; it is in his acts under -his conditions that his character exists. We cannot grasp or conceive -any principle of individuation apart from the fulness of the relations -to the surroundings. - -But suppose now that Plato is talking about the higher man—the -four-dimensional being that is limited in our external experience to a -three-dimensional world. Do not his words begin to have a meaning? Such -a being would have a consciousness of motion which is not as the motion -he can see with the eyes of the body. He, in his own being, knows a -reality to which the outward matter of this too solid earth is flimsy -superficiality. He too knows a mode of being, the fulness of relations, -in which can only be represented in the limited world of sense, as the -painter unsubstantially portrays the depths of woodland, plains, and -air. Thinking of such a being in man, was not Plato’s line well divided? - -It is noteworthy that, if Plato omitted his doctrine of the independent -origin of ideas, he would present exactly the four-dimensional -argument; a real thing as we think it is an idea. A plane being’s idea -of a square object is the idea of an abstraction, namely, a geometrical -square. Similarly our idea of a solid thing is an abstraction, for -in our idea there is not the four-dimensional thickness which is -necessary, however slight, to give reality. The argument would then -run, as a shadow is to a solid object, so is the solid object to the -reality. Thus A and B´ would be identified. - -In the allegory which I have already alluded to, Plato in almost as -many words shows forth the relation between existence in a superficies -and in solid space. And he uses this relation to point to the -conditions of a higher being. - -He imagines a number of men prisoners, chained so that they look at -the wall of a cavern in which they are confined, with their backs to -the road and the light. Over the road pass men and women, figures and -processions, but of all this pageant all that the prisoners behold -is the shadow of it on the wall whereon they gaze. Their own shadows -and the shadows of the things in the world are all that they see, and -identifying themselves with their shadows related as shadows to a world -of shadows, they live in a kind of dream. - -Plato imagines one of their number to pass out from amongst them -into the real space world, and then returning to tell them of their -condition. - -Here he presents most plainly the relation between existence in a plane -world and existence in a three-dimensional world. And he uses this -illustration as a type of the manner in which we are to proceed to a -higher state from the three-dimensional life we know. - -It must have hung upon the weight of a shadow which path he -took!—whether the one we shall follow toward the higher solid and the -four-dimensional existence, or the one which makes ideas the higher -realities, and the direct perception of them the contact with the truer -world. - -Passing on to Aristotle, we will touch on the points which most -immediately concern our enquiry. - -Just as a scientific man of the present day in reviewing the -speculations of the ancient world would treat them with a curiosity -half amused but wholly respectful, asking of each and all wherein lay -their relation to fact, so Aristotle, in discussing the philosophy -of Greece as he found it, asks, above all other things: “Does this -represent the world? In this system is there an adequate presentation -of what is?” - -He finds them all defective, some for the very reasons which we esteem -them most highly, as when he criticises the Atomic theory for its -reduction of all change to motion. But in the lofty march of his reason -he never loses sight of the whole; and that wherein our views differ -from his lies not so much in a superiority of our point of view, as -in the fact which he himself enunciates—that it is impossible for one -principle to be valid in all branches of enquiry. The conceptions -of one method of investigation are not those of another; and our -divergence lies in our exclusive attention to the conceptions useful -in one way of apprehending nature rather than in any possibility we -find in our theories of giving a view of the whole transcending that of -Aristotle. - -He takes account of everything; he does not separate matter and the -manifestation of matter; he fires all together in a conception of a -vast world process in which everything takes part—the motion of a grain -of dust, the unfolding of a leaf, the ordered motion of the spheres in -heaven—all are parts of one whole which he will not separate into dead -matter and adventitious modifications. - -And just as our theories, as representative of actuality, fall before -his unequalled grasp of fact, so the doctrine of ideas fell. It is -not an adequate account of existence, as Plato himself shows in his -“Parmenides”; it only explains things by putting their doubles beside -them. - -For his own part Aristotle invented a great marching definition which, -with a kind of power of its own, cleaves its way through phenomena -to limiting conceptions on either hand, towards whose existence all -experience points. - -In Aristotle’s definition of matter and form as the constituent of -reality, as in Plato’s mystical vision of the kingdom of ideas, the -existence of the higher dimensionality is implicitly involved. - -Substance according to Aristotle is relative, not absolute. In -everything that is there is the matter of which it is composed, the -form which it exhibits; but these are indissolubly connected, and -neither can be thought without the other. - -The blocks of stone out of which a house is built are the material for -the builder; but, as regards the quarrymen, they are the matter of the -rocks with the form he has imposed on them. Words are the final product -of the grammarian, but the mere matter of the orator or poet. The atom -is, with us, that out of which chemical substances are built up, but -looked at from another point of view is the result of complex processes. - -Nowhere do we find finality. The matter in one sphere is the matter, -plus form, of another sphere of thought. Making an obvious application -to geometry, plane figures exist as the limitation of different -portions of the plane by one another. In the bounding lines the -separated matter of the plane shows its determination into form. And -as the plane is the matter relatively to determinations in the plane, -so the plane itself exists in virtue of the determination of space. A -plane is that wherein formless space has form superimposed on it, and -gives an actuality of real relations. We cannot refuse to carry this -process of reasoning a step farther back, and say that space itself is -that which gives form to higher space. As a line is the determination -of a plane, and a plane of a solid, so solid space itself is the -determination of a higher space. - -As a line by itself is inconceivable without that plane which it -separates, so the plane is inconceivable without the solids which -it limits on either hand. And so space itself cannot be positively -defined. It is the negation of the possibility of movement in more than -three dimensions. The conception of space demands that of a higher -space. As a surface is thin and unsubstantial without the substance of -which it is the surface, so matter itself is thin without the higher -matter. - -Just as Aristotle invented that algebraical method of representing -unknown quantities by mere symbols, not by lines necessarily -determinate in length as was the habit of the Greek geometers, and so -struck out the path towards those objectifications of thought which, -like independent machines for reasoning, supply the mathematician -with his analytical weapons, so in the formulation of the doctrine -of matter and form, of potentiality and actuality, of the relativity -of substance, he produced another kind of objectification of mind—a -definition which had a vital force and an activity of its own. - -In none of his writings, as far as we know, did he carry it to its -legitimate conclusion on the side of matter, but in the direction of -the formal qualities he was led to his limiting conception of that -existence of pure form which lies beyond all known determination -of matter. The unmoved mover of all things is Aristotle’s highest -principle. Towards it, to partake of its perfection all things move. -The universe, according to Aristotle, is an active process—he does -not adopt the illogical conception that it was once set in motion -and has kept on ever since. There is room for activity, will, -self-determination, in Aristotle’s system, and for the contingent and -accidental as well. We do not follow him, because we are accustomed to -find in nature infinite series, and do not feel obliged to pass on to a -belief in the ultimate limits to which they seem to point. - -But apart from the pushing to the limit, as a relative principle -this doctrine of Aristotle’s as to the relativity of substance is -irrefragible in its logic. He was the first to show the necessity -of that path of thought which when followed leads to a belief in a -four-dimensional space. - -Antagonistic as he was to Plato in his conception of the practical -relation of reason to the world of phenomena, yet in one point he -coincided with him. And in this he showed the candour of his intellect. -He was more anxious to lose nothing than to explain everything. And -that wherein so many have detected an inconsistency, an inability to -free himself from the school of Plato, appears to us in connection with -our enquiry as an instance of the acuteness of his observation. For -beyond all knowledge given by the senses Aristotle held that there is -an active intelligence, a mind not the passive recipient of impressions -from without, but an active and originative being, capable of grasping -knowledge at first hand. In the active soul Aristotle recognised -something in man not produced by his physical surroundings, something -which creates, whose activity is a knowledge underived from sense. -This, he says, is the immortal and undying being in man. - -Thus we see that Aristotle was not far from the recognition of the -four-dimensional existence, both without and within man, and the -process of adequately realising the higher dimensional figures to which -we shall come subsequently is a simple reduction to practice of his -hypothesis of a soul. - -The next step in the unfolding of the drama of the recognition of -the soul as connected with our scientific conception of the world, -and, at the same time, the recognition of that higher of which a -three-dimensional world presents the superficial appearance, took place -many centuries later. If we pass over the intervening time without a -word it is because the soul was occupied with the assertion of itself -in other ways than that of knowledge. When it took up the task in -earnest of knowing this material world in which it found itself, and of -directing the course of inanimate nature, from that most objective aim -came, reflected back as from a mirror, its knowledge of itself. - - - - - CHAPTER V - - THE SECOND CHAPTER IN THE HISTORY OF FOUR SPACE - - - LOBATCHEWSKY, BOLYAI, AND GAUSS - -Before entering on a description of the work of Lobatchewsky and Bolyai -it will not be out of place to give a brief account of them, the -materials for which are to be found in an article by Franz Schmidt in -the forty-second volume of the _Mathematische Annalen_, and in Engel’s -edition of Lobatchewsky. - -Lobatchewsky was a man of the most complete and wonderful talents. -As a youth he was full of vivacity, carrying his exuberance so far -as to fall into serious trouble for hazing a professor, and other -freaks. Saved by the good offices of the mathematician Bartels, who -appreciated his ability, he managed to restrain himself within the -bounds of prudence. Appointed professor at his own University, Kasan, -he entered on his duties under the regime of a pietistic reactionary, -who surrounded himself with sycophants and hypocrites. Esteeming -probably the interests of his pupils as higher than any attempt at a -vain resistance, he made himself the tyrant’s right-hand man, doing an -incredible amount of teaching and performing the most varied official -duties. Amidst all his activities he found time to make important -contributions to science. His theory of parallels is most closely -connected with his name, but a study of his writings shows that he was -a man capable of carrying on mathematics in its main lines of advance, -and of a judgment equal to discerning what these lines were. Appointed -rector of his University, he died at an advanced age, surrounded by -friends, honoured, with the results of his beneficent activity all -around him. To him no subject came amiss, from the foundations of -geometry to the improvement of the stoves by which the peasants warmed -their houses. - -He was born in 1793. His scientific work was unnoticed till, in 1867, -Houel, the French mathematician, drew attention to its importance. - -Johann Bolyai de Bolyai was born in Klausenburg, a town in -Transylvania, December 15th, 1802. - -His father, Wolfgang Bolyai, a professor in the Reformed College of -Maros Vasarhely, retained the ardour in mathematical studies which had -made him a chosen companion of Gauss in their early student days at -Göttingen. - -He found an eager pupil in Johann. He relates that the boy sprang -before him like a devil. As soon as he had enunciated a problem the -child would give the solution and command him to go on further. As a -thirteen-year-old boy his father sometimes sent him to fill his place -when incapacitated from taking his classes. The pupils listened to him -with more attention than to his father for they found him clearer to -understand. - -In a letter to Gauss Wolfgang Bolyai writes:— - - “My boy is strongly built. He has learned to recognise many - constellations, and the ordinary figures of geometry. He makes apt - applications of his notions, drawing for instance the positions of the - stars with their constellations. Last winter in the country, seeing - Jupiter he asked: ‘How is it that we can see him from here as well as - from the town? He must be far off.’ And as to three different places - to which he had been he asked me to tell him about them in one word. I - did not know what he meant, and then he asked me if one was in a line - with the other and all in a row, or if they were in a triangle. - - “He enjoys cutting paper figures with a pair of scissors, and without - my ever having told him about triangles remarked that a right-angled - triangle which he had cut out was half of an oblong. I exercise his - body with care, he can dig well in the earth with his little hands. - The blossom can fall and no fruit left. When he is fifteen I want to - send him to you to be your pupil.” - -In Johann’s autobiography he says:— - - “My father called my attention to the imperfections and gaps in the - theory of parallels. He told me he had gained more satisfactory - results than his predecessors, but had obtained no perfect and - satisfying conclusion. None of his assumptions had the necessary - degree of geometrical certainty, although they sufficed to prove the - eleventh axiom and appeared acceptable on first sight. - - “He begged of me, anxious not without a reason, to hold myself aloof - and to shun all investigation on this subject, if I did not wish to - live all my life in vain.” - -Johann, in the failure of his father to obtain any response from Gauss, -in answer to a letter in which he asked the great mathematician to make -of his son “an apostle of truth in a far land,” entered the Engineering -School at Vienna. He writes from Temesvar, where he was appointed -sub-lieutenant September, 1823:— - - - “Temesvar, November 3rd, 1823. - - “DEAR GOOD FATHER, - - “I have so overwhelmingly much to write about my discovery that I know - no other way of checking myself than taking a quarter of a sheet only - to write on. I want an answer to my four-sheet letter. - - “I am unbroken in my determination to publish a work on Parallels, as - soon as I have put my material in order and have the means. - - “At present I have not made any discovery, but the way I have followed - almost certainly promises me the attainment of my object if any - possibility of it exists. - - “I have not got my object yet, but I have produced such stupendous - things that I was overwhelmed myself, and it would be an eternal shame - if they were lost. When you see them you will find that it is so. Now - I can only say that I have made a new world out of nothing. Everything - that I have sent you before is a house of cards in comparison with a - tower. I am convinced that it will be no less to my honour than if I - had already discovered it.” - -The discovery of which Johann here speaks was published as an appendix -to Wolfgang Bolyai’s _Tentamen_. - -Sending the book to Gauss, Wolfgang writes, after an interruption of -eighteen years in his correspondence:— - - “My son is first lieutenant of Engineers and will soon be captain. - He is a fine youth, a good violin player, a skilful fencer, and - brave, but has had many duels, and is wild even for a soldier. Yet he - is distinguished—light in darkness and darkness in light. He is an - impassioned mathematician with extraordinary capacities.... He will - think more of your judgment on his work than that of all Europe.” - -Wolfgang received no answer from Gauss to this letter, but sending a -second copy of the book received the following reply:— - - “You have rejoiced me, my unforgotten friend, by your letters. I - delayed answering the first because I wanted to wait for the arrival - of the promised little book. - - “Now something about your son’s work. - - “If I begin with saying that ‘I ought not to praise it,’ you will be - staggered for a moment. But I cannot say anything else. To praise it - is to praise myself, for the path your son has broken in upon and the - results to which he has been led are almost exactly the same as my own - reflections, some of which date from thirty to thirty-five years ago. - - “In fact I am astonished to the uttermost. My intention was to let - nothing be known in my lifetime about my own work, of which, for the - rest, but little is committed to writing. Most people have but little - perception of the problem, and I have found very few who took any - interest in the views I expressed to them. To be able to do that one - must first of all have had a real live feeling of what is wanting, and - as to that most men are completely in the dark. - - “Still it was my intention to commit everything to writing in the - course of time, so that at least it should not perish with me. - - “I am deeply surprised that this task can be spared me, and I am most - of all pleased in this that it is the son of my old friend who has in - so remarkable a manner preceded me.” - -The impression which we receive from Gauss’s inexplicable silence -towards his old friend is swept away by this letter. Hence we breathe -the clear air of the mountain tops. Gauss would not have failed to -perceive the vast significance of his thoughts, sure to be all the -greater in their effect on future ages from the want of comprehension -of the present. Yet there is not a word or a sign in his writing to -claim the thought for himself. He published no single line on the -subject. By the measure of what he thus silently relinquishes, by -such a measure of a world-transforming thought, we can appreciate his -greatness. - -It is a long step from Gauss’s serenity to the disturbed and passionate -life of Johann Bolyai—he and Galois, the two most interesting figures -in the history of mathematics. For Bolyai, the wild soldier, the -duellist, fell at odds with the world. It is related of him that he was -challenged by thirteen officers of his garrison, a thing not unlikely -to happen considering how differently he thought from every one else. -He fought them all in succession—making it his only condition that he -should be allowed to play on his violin for an interval between meeting -each opponent. He disarmed or wounded all his antagonists. It can be -easily imagined that a temperament such as his was one not congenial to -his military superiors. He was retired in 1833. - -His epoch-making discovery awoke no attention. He seems to have -conceived the idea that his father had betrayed him in some -inexplicable way by his communications with Gauss, and he challenged -the excellent Wolfgang to a duel. He passed his life in poverty, many a -time, says his biographer, seeking to snatch himself from dissipation -and apply himself again to mathematics. But his efforts had no result. -He died January 27th, 1860, fallen out with the world and with himself. - - - METAGEOMETRY - -The theories which are generally connected with the names of -Lobatchewsky and Bolyai bear a singular and curious relation to the -subject of higher space. - -In order to show what this relation is, I must ask the reader to be -at the pains to count carefully the sets of points by which I shall -estimate the volumes of certain figures. - -No mathematical processes beyond this simple one of counting will be -necessary. - -[Illustration: Fig. 19.] - -Let us suppose we have before us in fig. 19 a plane covered with points -at regular intervals, so placed that every four determine a square. - -Now it is evident that as four points determine a square, so four -squares meet in a point. - -[Illustration: Fig. 20.] - -Thus, considering a point inside a square as belonging to it, we may -say that a point on the corner of a square belongs to it and to three -others equally: belongs a quarter of it to each square. - -[Illustration: Fig. 21.] - -[Illustration: Fig. 22.] - -Thus the square ACDE (fig. 21) contains one point, and has four points -at the four corners. Since one-fourth of each of these four belongs to -the square, the four together count as one point, and the point value -of the square is two points—the one inside and the four at the corner -make two points belonging to it exclusively. - -Now the area of this square is two unit squares, as can be seen by -drawing two diagonals in fig. 22. - -We also notice that the square in question is equal to the sum of the -squares on the sides AB, BC, of the right-angled triangle ABC. Thus we -recognise the proposition that the square on the hypothenuse is equal -to the sum of the squares on the two sides of a right-angled triangle. - -Now suppose we set ourselves the question of determining the -whereabouts in the ordered system of points, the end of a line would -come when it turned about a point keeping one extremity fixed at the -point. - -We can solve this problem in a particular case. If we can find a square -lying slantwise amongst the dots which is equal to one which goes -regularly, we shall know that the two sides are equal, and that the -slanting side is equal to the straight-way side. Thus the volume and -shape of a figure remaining unchanged will be the test of its having -rotated about the point, so that we can say that its side in its first -position would turn into its side in the second position. - -Now, such a square can be found in the one whose side is five units in -length. - -[Illustration: Fig. 23.] - -In fig. 23, in the square on AB, there are— - - 9 points interior 9 - 4 at the corners 1 - 4 sides with 3 on each side, considered as - 1½ on each side, because belonging - equally to two squares 6 - -The total is 16. There are 9 points in the square on BC. - -In the square on AC there are— - - 24 points inside 24 - 4 at the corners 1 - -or 25 altogether. - -Hence we see again that the square on the hypothenuse is equal to the -squares on the sides. - -Now take the square AFHG, which is larger than the square on AB. It -contains 25 points. - - 16 inside 16 - 16 on the sides, counting as 8 - 4 on the corners 1 - -making 25 altogether. - -If two squares are equal we conclude the sides are equal. Hence, the -line AF turning round A would move so that it would after a certain -turning coincide with AC. - -This is preliminary, but it involves all the mathematical difficulties -that will present themselves. - -There are two alterations of a body by which its volume is not changed. - -One is the one we have just considered, rotation, the other is what is -called shear. - -Consider a book, or heap of loose pages. They can be slid so that each -one slips over the preceding one, and the whole assumes the shape _b_ -in fig. 24. - -[Illustration: Fig. 24.] - -This deformation is not shear alone, but shear accompanied by rotation. - -Shear can be considered as produced in another way. - -Take the square ABCD (fig. 25), and suppose that it is pulled out from -along one of its diagonals both ways, and proportionately compressed -along the other diagonal. It will assume the shape in fig. 26. - -This compression and expansion along two lines at right angles is what -is called shear; it is equivalent to the sliding illustrated above, -combined with a turning round. - -[Illustration: Fig. 25.] [Illustration: Fig. 26.] - -In pure shear a body is compressed and extended in two directions at -right angles to each other, so that its volume remains unchanged. - -Now we know that our material bodies resist shear—shear does violence -to the internal arrangement of their particles, but they turn as wholes -without such internal resistance. - -But there is an exception. In a liquid shear and rotation take place -equally easily, there is no more resistance against a shear than there -is against a rotation. - -Now, suppose all bodies were to be reduced to the liquid state, in -which they yield to shear and to rotation equally easily, and then -were to be reconstructed as solids, but in such a way that shear and -rotation had interchanged places. - -That is to say, let us suppose that when they had become solids again -they would shear without offering any internal resistance, but a -rotation would do violence to their internal arrangement. - -That is, we should have a world in which shear would have taken the -place of rotation. - -A shear does not alter the volume of a body: thus an inhabitant living -in such a world would look on a body sheared as we look on a body -rotated. He would say that it was of the same shape, but had turned a -bit round. - -Let us imagine a Pythagoras in this world going to work to investigate, -as is his wont. - -[Illustration: Fig. 27.] [Illustration: Fig. 28.] - -Fig. 27 represents a square unsheared. Fig. 28 represents a square -sheared. It is not the figure into which the square in fig. 27 would -turn, but the result of shear on some square not drawn. It is a simple -slanting placed figure, taken now as we took a simple slanting placed -square before. Now, since bodies in this world of shear offer no -internal resistance to shearing, and keep their volume when sheared, -an inhabitant accustomed to them would not consider that they altered -their shape under shear. He would call ACDE as much a square as the -square in fig. 27. We will call such figures shear squares. Counting -the dots in ACDE, we find— - - 2 inside = 2 - 4 at corners = 1 - -or a total of 3. - -Now, the square on the side AB has 4 points, that on BC has 1 point. -Here the shear square on the hypothenuse has not 5 points but 3; it is -not the sum of the squares on the sides, but the difference. - -This relation always holds. Look at fig. 29. - -[Illustration: Fig. 29.] - -Shear square on hypothenuse— - - 7 internal 7 - 4 at corners 1 - — - 8 - - -[Illustration: Fig. 29 _bis_.] - -Square on one side—which the reader can draw for himself— - - 4 internal 4 - 8 on sides 4 - 4 at corners 1 - — - 9 - -and the square on the other side is 1. Hence in this case again the -difference is equal to the shear square on the hypothenuse, 9 - 1 = 8. - -Thus in a world of shear the square on the hypothenuse would be equal -to the difference of the squares on the sides of a right-angled -triangle. - -In fig. 29 _bis_ another shear square is drawn on which the above -relation can be tested. - -What now would be the position a line on turning by shear would take up? - -We must settle this in the same way as previously with our turning. - -Since a body sheared remains the same, we must find two equal bodies, -one in the straight way, one in the slanting way, which have the same -volume. Then the side of one will by turning become the side of the -other, for the two figures are each what the other becomes by a shear -turning. - -We can solve the problem in a particular case— - -[Illustration: Fig. 30.] - -In the figure ACDE (fig. 30) there are— - - 15 inside 15 - 4 at corners 1 - -a total of 16. - -Now in the square ABGF, there are 16— - - 9 inside 9 - 12 on sides 6 - 4 at corners 1 - — - 16 - -Hence the square on AB would, by the shear turning, become the shear -square ACDE. - -And hence the inhabitant of this world would say that the line AB -turned into the line AC. These two lines would be to him two lines of -equal length, one turned a little way round from the other. - -That is, putting shear in place of rotation, we get a different kind -of figure, as the result of the shear rotation, from what we got with -our ordinary rotation. And as a consequence we get a position for the -end of a line of invariable length when it turns by the shear rotation, -different from the position which it would assume on turning by our -rotation. - -A real material rod in the shear world would, on turning about A, pass -from the position AB to the position AC. We say that its length alters -when it becomes AC, but this transformation of AB would seem to an -inhabitant of the shear world like a turning of AB without altering in -length. - -If now we suppose a communication of ideas that takes place between -one of ourselves and an inhabitant of the shear world, there would -evidently be a difference between his views of distance and ours. - -We should say that his line AB increased in length in turning to AC. He -would say that our line AF (fig. 23) decreased in length in turning to -AC. He would think that what we called an equal line was in reality a -shorter one. - -We should say that a rod turning round would have its extremities in -the positions we call at equal distances. So would he—but the positions -would be different. He could, like us, appeal to the properties of -matter. His rod to him alters as little as ours does to us. - -Now, is there any standard to which we could appeal, to say which of -the two is right in this argument? There is no standard. - -We should say that, with a change of position, the configuration and -shape of his objects altered. He would say that the configuration and -shape of our objects altered in what we called merely a change of -position. Hence distance independent of position is inconceivable, or -practically distance is solely a property of matter. - -There is no principle to which either party in this controversy could -appeal. There is nothing to connect the definition of distance with our -ideas rather than with his, except the behaviour of an actual piece of -matter. - -For the study of the processes which go on in our world the definition -of distance given by taking the sum of the squares is of paramount -importance to us. But as a question of pure space without making any -unnecessary assumptions the shear world is just as possible and just as -interesting as our world. - -It was the geometry of such conceivable worlds that Lobatchewsky and -Bolyai studied. - -This kind of geometry has evidently nothing to do directly with -four-dimensional space. - -But a connection arises in this way. It is evident that, instead of -taking a simple shear as I have done, and defining it as that change -of the arrangement of the particles of a solid which they will undergo -without offering any resistance due to their mutual action, I might -take a complex motion, composed of a shear and a rotation together, or -some other kind of deformation. - -Let us suppose such an alteration picked out and defined as the one -which means simple rotation, then the type, according to which all -bodies will alter by this rotation, is fixed. - -Looking at the movements of this kind, we should say that the objects -were altering their shape as well as rotating. But to the inhabitants -of that world they would seem to be unaltered, and our figures in their -motions would seem to them to alter. - -In such a world the features of geometry are different. We have seen -one such difference in the case of our illustration of the world of -shear, where the square on the hypothenuse was equal to the difference, -not the sum, of the squares on the sides. - -In our illustration we have the same laws of parallel lines as in our -ordinary rotation world, but in general the laws of parallel lines are -different. - -In one of these worlds of a different constitution of matter through -one point there can be two parallels to a given line, in another of -them there can be none, that is, although a line be drawn parallel to -another it will meet it after a time. - -Now it was precisely in this respect of parallels that Lobatchewsky and -Bolyai discovered these different worlds. They did not think of them as -worlds of matter, but they discovered that space did not necessarily -mean that our law of parallels is true. They made the distinction -between laws of space and laws of matter, although that is not the -form in which they stated their results. - -The way in which they were led to these results was the -following. Euclid had stated the existence of parallel lines as a -postulate—putting frankly this unproved proposition—that one line and -only one parallel to a given straight line can be drawn, as a demand, -as something that must be assumed. The words of his ninth postulate are -these: “If a straight line meeting two other straight lines makes the -interior angles on the same side of it equal to two right angles, the -two straight lines will never meet.” - -The mathematicians of later ages did not like this bald assumption, and -not being able to prove the proposition they called it an axiom—the -eleventh axiom. - -Many attempts were made to prove the axiom; no one doubted of its -truth, but no means could be found to demonstrate it. At last an -Italian, Sacchieri, unable to find a proof, said: “Let us suppose it -not true.” He deduced the results of there being possibly two parallels -to one given line through a given point, but feeling the waters too -deep for the human reason, he devoted the latter half of his book to -disproving what he had assumed in the first part. - -Then Bolyai and Lobatchewsky with firm step entered on the forbidden -path. There can be no greater evidence of the indomitable nature of -the human spirit, or of its manifest destiny to conquer all those -limitations which bind it down within the sphere of sense than this -grand assertion of Bolyai and Lobatchewsky. - - ─────────────────────────── - C D - ─────────────────────────────────── - A B -Take a line AB and a point C. We say and see and know that through C -can only be drawn one line parallel to AB. - -But Bolyai said: “I will draw two.” Let CD be parallel to AB, that -is, not meet AB however far produced, and let lines beyond CD also not -meet AB; let there be a certain region between CD and CE, in which no -line drawn meets AB. CE and CD produced backwards through C will give a -similar region on the other side of C. - -[Illustration: Fig. 32.] - -Nothing so triumphantly, one may almost say so insolently, ignoring -of sense had ever been written before. Men had struggled against the -limitations of the body, fought them, despised them, conquered them. -But no one had ever thought simply as if the body, the bodily eyes, -the organs of vision, all this vast experience of space, had never -existed. The age-long contest of the soul with the body, the struggle -for mastery, had come to a culmination. Bolyai and Lobatchewsky simply -thought as if the body was not. The struggle for dominion, the strife -and combat of the soul were over; they had mastered, and the Hungarian -drew his line. - -Can we point out any connection, as in the case of Parmenides, between -these speculations and higher space? Can we suppose it was any inner -perception by the soul of a motion not known to the senses, which -resulted in this theory so free from the bonds of sense? No such -supposition appears to be possible. - -Practically, however, metageometry had a great influence in bringing -the higher space to the front as a working hypothesis. This can -be traced to the tendency the mind has to move in the direction -of least resistance. The results of the new geometry could not be -neglected, the problem of parallels had occupied a place too prominent -in the development of mathematical thought for its final solution -to be neglected. But this utter independence of all mechanical -considerations, this perfect cutting loose from the familiar -intuitions, was so difficult that almost any other hypothesis was -more easy of acceptance, and when Beltrami showed that the geometry -of Lobatchewsky and Bolyai was the geometry of shortest lines drawn -on certain curved surfaces, the ordinary definitions of measurement -being retained, attention was drawn to the theory of a higher space. -An illustration of Beltrami’s theory is furnished by the simple -consideration of hypothetical beings living on a spherical surface. - -[Illustration: Fig. 33.] - -Let ABCD be the equator of a globe, and AP, BP, meridian lines drawn to -the pole, P. The lines AB, AP, BP would seem to be perfectly straight -to a person moving on the surface of the sphere, and unconscious of its -curvature. Now AP and BP both make right angles with AB. Hence they -satisfy the definition of parallels. Yet they meet in P. Hence a being -living on a spherical surface, and unconscious of its curvature, would -find that parallel lines would meet. He would also find that the angles -in a triangle were greater than two right angles. In the triangle PAB, -for instance, the angles at A and B are right angles, so the three -angles of the triangle PAB are greater than two right angles. - -Now in one of the systems of metageometry (for after Lobatchewsky had -shown the way it was found that other systems were possible besides -his) the angles of a triangle are greater than two right angles. - -Thus a being on a sphere would form conclusions about his space which -are the same as he would form if he lived on a plane, the matter in -which had such properties as are presupposed by one of these systems -of geometry. Beltrami also discovered a certain surface on which -there could be drawn more than one “straight” line through a point -which would not meet another given line. I use the word straight as -equivalent to the line having the property of giving the shortest path -between any two points on it. Hence, without giving up the ordinary -methods of measurement, it was possible to find conditions in which -a plane being would necessarily have an experience corresponding to -Lobatchewsky’s geometry. And by the consideration of a higher space, -and a solid curved in such a higher space, it was possible to account -for a similar experience in a space of three dimensions. - -Now, it is far more easy to conceive of a higher dimensionality to -space than to imagine that a rod in rotating does not move so that -its end describes a circle. Hence, a logical conception having been -found harder than that of a four dimensional space, thought turned -to the latter as a simple explanation of the possibilities to which -Lobatchewsky had awakened it. Thinkers became accustomed to deal with -the geometry of higher space—it was Kant, says Veronese, who first -used the expression of “different spaces”—and with familiarity the -inevitableness of the conception made itself felt. - -From this point it is but a small step to adapt the ordinary mechanical -conceptions to a higher spatial existence, and then the recognition of -its objective existence could be delayed no longer. Here, too, as in so -many cases, it turns out that the order and connection of our ideas is -the order and connection of things. - -What is the significance of Lobatchewsky’s and Bolyai’s work? - -It must be recognised as something totally different from the -conception of a higher space; it is applicable to spaces of any number -of dimensions. By immersing the conception of distance in matter to -which it properly belongs, it promises to be of the greatest aid in -analysis for the effective distance of any two particles is the -product of complex material conditions and cannot be measured by hard -and fast rules. Its ultimate significance is altogether unknown. It -is a cutting loose from the bonds of sense, not coincident with the -recognition of a higher dimensionality, but indirectly contributory -thereto. - -Thus, finally, we have come to accept what Plato held in the hollow -of his hand; what Aristotle’s doctrine of the relativity of substance -implies. The vast universe, too, has its higher, and in recognising it -we find that the directing being within us no longer stands inevitably -outside our systematic knowledge. - - - - - CHAPTER VI - - THE HIGHER WORLD - - -It is indeed strange, the manner in which we must begin to think about -the higher world. - -Those simplest objects analogous to those which are about us on every -side in our daily experience such as a door, a table, a wheel are -remote and uncognisable in the world of four dimensions, while the -abstract ideas of rotation, stress and strain, elasticity into which -analysis resolves the familiar elements of our daily experience are -transferable and applicable with no difficulty whatever. Thus we are -in the unwonted position of being obliged to construct the daily and -habitual experience of a four-dimensional being, from a knowledge of -the abstract theories of the space, the matter, the motion of it; -instead of, as in our case, passing to the abstract theories from the -richness of sensible things. - -What would a wheel be in four dimensions? What the shafting for the -transmission of power which a four-dimensional being would use. - -The four-dimensional wheel, and the four-dimensional shafting are -what will occupy us for these few pages. And it is no futile or -insignificant enquiry. For in the attempt to penetrate into the nature -of the higher, to grasp within our ken that which transcends all -analogies, because what we know are merely partial views of it, the -purely material and physical path affords a means of approach pursuing -which we are in less likelihood of error than if we use the more -frequently trodden path of framing conceptions which in their elevation -and beauty seem to us ideally perfect. - -For where we are concerned with our own thoughts, the development of -our own ideals, we are as it were on a curve, moving at any moment -in a direction of tangency. Whither we go, what we set up and exalt -as perfect, represents not the true trend of the curve, but our own -direction at the present—a tendency conditioned by the past, and by -a vital energy of motion essential but only true when perpetually -modified. That eternal corrector of our aspirations and ideals, the -material universe draws sublimely away from the simplest things we can -touch or handle to the infinite depths of starry space, in one and -all uninfluenced by what we think or feel, presenting unmoved fact -to which, think it good or think it evil, we can but conform, yet -out of all that impassivity with a reference to something beyond our -individual hopes and fears supporting us and giving us our being. - -And to this great being we come with the question: “You, too, what is -your higher?” - -Or to put it in a form which will leave our conclusions in the shape -of no barren formula, and attacking the problem on its most assailable -side: “What is the wheel and the shafting of the four-dimensional -mechanic?” - -In entering on this enquiry we must make a plan of procedure. The -method which I shall adopt is to trace out the steps of reasoning by -which a being confined to movement in a two-dimensional world could -arrive at a conception of our turning and rotation, and then to apply -an analogous process to the consideration of the higher movements. The -plane being must be imagined as no abstract figure, but as a real body -possessing all three dimensions. His limitation to a plane must be the -result of physical conditions. - -We will therefore think of him as of a figure cut out of paper placed -on a smooth plane. Sliding over this plane, and coming into contact -with other figures equally thin as he in the third dimension, he will -apprehend them only by their edges. To him they will be completely -bounded by lines. A “solid” body will be to him a two-dimensional -extent, the interior of which can only be reached by penetrating -through the bounding lines. - -Now such a plane being can think of our three-dimensional existence in -two ways. - -First, he can think of it as a series of sections, each like the solid -he knows of extending in a direction unknown to him, which stretches -transverse to his tangible universe, which lies in a direction at right -angles to every motion which he made. - -Secondly, relinquishing the attempt to think of the three-dimensional -solid body in its entirety he can regard it as consisting of a -number of plane sections, each of them in itself exactly like -the two-dimensional bodies he knows, but extending away from his -two-dimensional space. - -A square lying in his space he regards as a solid bounded by four -lines, each of which lies in his space. - -A square standing at right angles to his plane appears to him as simply -a line in his plane, for all of it except the line stretches in the -third dimension. - -He can think of a three-dimensional body as consisting of a number of -such sections, each of which starts from a line in his space. - -Now, since in his world he can make any drawing or model which involves -only two dimensions, he can represent each such upright section as it -actually is, and can represent a turning from a known into the unknown -dimension as a turning from one to another of his known dimensions. - -To see the whole he must relinquish part of that which he has, and take -the whole portion by portion. - -Consider now a plane being in front of a square, fig. 34. The square -can turn about any point in the plane—say the point A. But it cannot -turn about a line, as AB. For, in order to turn about the line AB, -the square must leave the plane and move in the third dimension. This -motion is out of his range of observation, and is therefore, except for -a process of reasoning, inconceivable to him. - -[Illustration: Fig. 34.] - -Rotation will therefore be to him rotation about a point. Rotation -about a line will be inconceivable to him. - -The result of rotation about a line he can apprehend. He can see the -first and last positions occupied in a half-revolution about the line -AC. The result of such a half revolution is to place the square ABCD -on the left hand instead of on the right hand of the line AC. It would -correspond to a pulling of the whole body ABCD through the line AC, -or to the production of a solid body which was the exact reflection -of it in the line AC. It would be as if the square ABCD turned into -its image, the line AB acting as a mirror. Such a reversal of the -positions of the parts of the square would be impossible in his space. -The occurrence of it would be a proof of the existence of a higher -dimensionality. - -Let him now, adopting the conception of a three-dimensional body as -a series of sections lying, each removed a little farther than the -preceding one, in direction at right angles to his plane, regard a -cube, fig. 36, as a series of sections, each like the square which -forms its base, all rigidly connected together. - -[Illustration: Fig. 35.] - -If now he turns the square about the point A in the plane of _xy_, -each parallel section turns with the square he moves. In each of the -sections there is a point at rest, that vertically over A. Hence he -would conclude that in the turning of a three-dimensional body there -is one line which is at rest. That is a three-dimensional turning in a -turning about a line. - - * * * * * - -In a similar way let us regard ourselves as limited to a -three-dimensional world by a physical condition. Let us imagine that -there is a direction at right angles to every direction in which we can -move, and that we are prevented from passing in this direction by a -vast solid, that against which in every movement we make we slip as the -plane being slips against his plane sheet. - -We can then consider a four-dimensional body as consisting of a series -of sections, each parallel to our space, and each a little farther off -than the preceding on the unknown dimension. - -Take the simplest four-dimensional body—one which begins as a cube, -fig. 36, in our space, and consists of sections, each a cube like fig. -36, lying away from our space. If we turn the cube which is its base in -our space about a line, if, _e.g._, in fig. 36 we turn the cube about -the line AB, not only it but each of the parallel cubes moves about a -line. The cube we see moves about the line AB, the cube beyond it about -a line parallel to AB and so on. Hence the whole four-dimensional body -moves about a plane, for the assemblage of these lines is our way of -thinking about the plane which, starting from the line AB in our space, -runs off in the unknown direction. - -[Illustration: Fig. 36.] - -In this case all that we see of the plane about which the turning takes -place is the line AB. - -But it is obvious that the axis plane may lie in our space. A point -near the plane determines with it a three-dimensional space. When it -begins to rotate round the plane it does not move anywhere in this -three-dimensional space, but moves out of it. A point can no more -rotate round a plane in three-dimensional space than a point can move -round a line in two-dimensional space. - -We will now apply the second of the modes of representation to this -case of turning about a plane, building up our analogy step by step -from the turning in a plane about a point and that in space about a -line, and so on. - -In order to reduce our considerations to those of the greatest -simplicity possible, let us realise how the plane being would think of -the motion by which a square is turned round a line. - -Let, fig. 34, ABCD be a square on his plane, and represent the two -dimensions of his space by the axes A_x_ A_y_. - -Now the motion by which the square is turned over about the line AC -involves the third dimension. - -He cannot represent the motion of the whole square in its turning, -but he can represent the motions of parts of it. Let the third axis -perpendicular to the plane of the paper be called the axis of _z_. Of -the three axes _x_, _y_, _z_, the plane being can represent any two in -his space. Let him then draw, in fig. 35, two axes, _x_ and _z_. Here -he has in his plane a representation of what exists in the plane which -goes off perpendicularly to his space. - -In this representation the square would not be shown, for in the plane -of _xz_ simply the line AB of the square is contained. - -The plane being then would have before him, in fig. 35, the -representation of one line AB of his square and two axes, _x_ and _z_, -at right angles. Now it would be obvious to him that, by a turning -such as he knows, by a rotation about a point, the line AB can turn -round A, and occupying all the intermediate positions, such as AB_{1}, -come after half a revolution to lie as A_x_ produced through A. - -Again, just as he can represent the vertical plane through AB, so he -can represent the vertical plane through A´B´, fig. 34, and in a like -manner can see that the line A´B´ can turn about the point A´ till it -lies in the opposite direction from that which it ran in at first. - -Now these two turnings are not inconsistent. In his plane, if AB -turned about A, and A´B´ about A´, the consistency of the square would -be destroyed, it would be an impossible motion for a rigid body to -perform. But in the turning which he studies portion by portion there -is nothing inconsistent. Each line in the square can turn in this way, -hence he would realise the turning of the whole square as the sum of -a number of turnings of isolated parts. Such turnings, if they took -place in his plane, would be inconsistent, but by virtue of a third -dimension they are consistent, and the result of them all is that the -square turns about the line AC and lies in a position in which it is -the mirror image of what it was in its first position. Thus he can -realise a turning about a line by relinquishing one of his axes, and -representing his body part by part. - -Let us apply this method to the turning of a cube so as to become the -mirror image of itself. In our space we can construct three independent -axes, _x_, _y_, _z_, shown in fig. 36. Suppose that there is a fourth -axis, _w_, at right angles to each and every one of them. We cannot, -keeping all three axes, _x_, _y_, _z_, represent _w_ in our space; but -if we relinquish one of our three axes we can let the fourth axis take -its place, and we can represent what lies in the space, determined by -the two axes we retain and the fourth axis. - -[Illustration: Fig. 37.] - -Let us suppose that we let the _y_ axis drop, and that we represent -the _w_ axis as occupying its direction. We have in fig. 37 a drawing -of what we should then see of the cube. The square ABCD, remains -unchanged, for that is in the plane of _xz_, and we still have that -plane. But from this plane the cube stretches out in the direction of -the _y_ axis. Now the _y_ axis is gone, and so we have no more of the -cube than the face ABCD. Considering now this face ABCD, we see that -it is free to turn about the line AB. It can rotate in the _x_ to _w_ -direction about this line. In fig. 38 it is shown on its way, and it -can evidently continue this rotation till it lies on the other side of -the _z_ axis in the plane of _xz_. - -We can also take a section parallel to the face ABCD, and then letting -drop all of our space except the plane of that section, introduce -the _w_ axis, running in the old _y_ direction. This section can be -represented by the same drawing, fig. 38, and we see that it can rotate -about the line on its left until it swings half way round and runs in -the opposite direction to that which it ran in before. These turnings -of the different sections are not inconsistent, and taken all together -they will bring the cube from the position shown in fig. 36 to that -shown in fig. 41. - -[Illustration: Fig. 38.] - -Since we have three axes at our disposal in our space, we are not -obliged to represent the _w_ axis by any particular one. We may let any -axis we like disappear, and let the fourth axis take its place. - -[Illustration: Fig. 39.] - -[Illustration: Fig. 40.] - -[Illustration: Fig. 41.] - -In fig. 36 suppose the _z_ axis to go. We have then simply the plane of -_xy_ and the square base of the cube ACEG, fig. 39, is all that could -be seen of it. Let now the _w_ axis take the place of the _z_ axis and -we have, in fig. 39 again, a representation of the space of _xyw_, in -which all that exists of the cube is its square base. Now, by a turning -of _x_ to _w_, this base can rotate around the line AE, it is shown -on its way in fig. 40, and finally it will, after half a revolution, -lie on the other side of the _y_ axis. In a similar way we may rotate -sections parallel to the base of the _xw_ rotation, and each of them -comes to run in the opposite direction from that which they occupied at -first. - -Thus again the cube comes from the position of fig. 36. to that of -fig. 41. In this _x_ to _w_ turning, we see that it takes place by -the rotations of sections parallel to the front face about lines -parallel to AB, or else we may consider it as consisting of the -rotation of sections parallel to the base about lines parallel to AE. -It is a rotation of the whole cube about the plane ABEF. Two separate -sections could not rotate about two separate lines in our space without -conflicting, but their motion is consistent when we consider another -dimension. Just, then, as a plane being can think of rotation about -a line as a rotation about a number of points, these rotations not -interfering as they would if they took place in his two-dimensional -space, so we can think of a rotation about a plane as the rotation -of a number of sections of a body about a number of lines in a plane, -these rotations not being inconsistent in a four-dimensional space as -they are in three-dimensional space. - -We are not limited to any particular direction for the lines in the -plane about which we suppose the rotation of the particular sections to -take place. Let us draw the section of the cube, fig. 36, through A, -F, C, H, forming a sloping plane. Now since the fourth dimension is at -right angles to every line in our space it is at right angles to this -section also. We can represent our space by drawing an axis at right -angles to the plane ACEG, our space is then determined by the plane -ACEG, and the perpendicular axis. If we let this axis drop and suppose -the fourth axis, _w_, to take its place, we have a representation of -the space which runs off in the fourth dimension from the plane ACEG. -In this space we shall see simply the section ACEG of the cube, and -nothing else, for one cube does not extend to any distance in the -fourth dimension. - -If, keeping this plane, we bring in the fourth dimension, we shall have -a space in which simply this section of the cube exists and nothing -else. The section can turn about the line AF, and parallel sections can -turn about parallel lines. Thus in considering the rotation about a -plane we can draw any lines we like and consider the rotation as taking -place in sections about them. - -[Illustration: Fig. 42.] - -To bring out this point more clearly let us take two parallel lines, -A and B, in the space of _xyz_, and let CD and EF be two rods running -above and below the plane of _xy_, from these lines. If we turn these -rods in our space about the lines A and B, as the upper end of one, -F, is going down, the lower end of the other, C, will be coming up. -They will meet and conflict. But it is quite possible for these two -rods each of them to turn about the two lines without altering their -relative distances. - -To see this suppose the _y_ axis to go, and let the _w_ axis take its -place. We shall see the lines A and B no longer, for they run in the -_y_ direction from the points G and H. - -[Illustration: Fig. 43.] - -Fig. 43 is a picture of the two rods seen in the space of _xzw_. If -they rotate in the direction shown by the arrows—in the _z_ to _w_ -direction—they move parallel to one another, keeping their relative -distances. Each will rotate about its own line, but their rotation will -not be inconsistent with their forming part of a rigid body. - -Now we have but to suppose a central plane with rods crossing it -at every point, like CD and EF cross the plane of _xy_, to have an -image of a mass of matter extending equal distances on each side of a -diametral plane. As two of these rods can rotate round, so can all, and -the whole mass of matter can rotate round its diametral plane. - -This rotation round a plane corresponds, in four dimensions, to the -rotation round an axis in three dimensions. Rotation of a body round a -plane is the analogue of rotation of a rod round an axis. - -In a plane we have rotation round a point, in three-space rotation -round an axis line, in four-space rotation round an axis plane. - -The four-dimensional being’s shaft by which he transmits power is a -disk rotating round its central plane—the whole contour corresponds -to the ends of an axis of rotation in our space. He can impart the -rotation at any point and take it off at any other point on the -contour, just as rotation round a line can in three-space be imparted -at one end of a rod and taken off at the other end. - -A four-dimensional wheel can easily be described from the analogy of -the representation which a plane being would form for himself of one of -our wheels. - -Suppose a wheel to move transverse to a plane, so that the whole disk, -which I will consider to be solid and without spokes, came at the same -time into contact with the plane. It would appear as a circular portion -of plane matter completely enclosing another and smaller portion—the -axle. - -This appearance would last, supposing the motion of the wheel to -continue until it had traversed the plane by the extent of its -thickness, when there would remain in the plane only the small disk -which is the section of the axle. There would be no means obvious in -the plane at first by which the axle could be reached, except by going -through the substance of the wheel. But the possibility of reaching it -without destroying the substance of the wheel would be shown by the -continued existence of the axle section after that of the wheel had -disappeared. - -In a similar way a four-dimensional wheel moving transverse to our -space would appear first as a solid sphere, completely surrounding -a smaller solid sphere. The outer sphere would represent the wheel, -and would last until the wheel has traversed our space by a distance -equal to its thickness. Then the small sphere alone would remain, -representing the section of the axle. The large sphere could move -round the small one quite freely. Any line in space could be taken as -an axis, and round this line the outer sphere could rotate, while the -inner sphere remained still. But in all these directions of revolution -there would be in reality one line which remained unaltered, that is -the line which stretches away in the fourth direction, forming the -axis of the axle. The four-dimensional wheel can rotate in any number -of planes, but all these planes are such that there is a line at right -angles to them all unaffected by rotation in them. - -An objection is sometimes experienced as to this mode of reasoning from -a plane world to a higher dimensionality. How artificial, it is argued, -this conception of a plane world is. If any real existence confined to -a superficies could be shown to exist, there would be an argument for -one relative to which our three-dimensional existence is superficial. -But, both on the one side and the other of the space we are familiar -with, spaces either with less or more than three dimensions are merely -arbitrary conceptions. - -In reply to this I would remark that a plane being having one less -dimension than our three would have one-third of our possibilities of -motion, while we have only one-fourth less than those of the higher -space. It may very well be that there may be a certain amount of -freedom of motion which is demanded as a condition of an organised -existence, and that no material existence is possible with a more -limited dimensionality than ours. This is well seen if we try to -construct the mechanics of a two-dimensional world. No tube could -exist, for unless joined together completely at one end two parallel -lines would be completely separate. The possibility of an organic -structure, subject to conditions such as this, is highly problematical; -yet, possibly in the convolutions of the brain there may be a mode of -existence to be described as two-dimensional. - -We have but to suppose the increase in surface and the diminution in -mass carried on to a certain extent to find a region which, though -without mobility of the constituents, would have to be described as -two-dimensional. - -But, however artificial the conception of a plane being may be, it is -none the less to be used in passing to the conception of a greater -dimensionality than ours, and hence the validity of the first part of -this objection altogether disappears directly we find evidence for such -a state of being. - -The second part of the objection has more weight. How is it possible -to conceive that in a four-dimensional space any creatures should be -confined to a three-dimensional existence? - -In reply I would say that we know as a matter of fact that life is -essentially a phenomenon of surface. The amplitude of the movements -which we can make is much greater along the surface of the earth than -it is up or down. - -Now we have but to conceive the extent of a solid surface increased, -while the motions possible tranverse to it are diminished in the -same proportion, to obtain the image of a three-dimensional world in -four-dimensional space. - -And as our habitat is the meeting of air and earth on the world, so -we must think of the meeting place of two as affording the condition -for our universe. The meeting of what two? What can that vastness be -in the higher space which stretches in such a perfect level that our -astronomical observations fail to detect the slightest curvature? - -The perfection of the level suggests a liquid—a lake amidst what vast -scenery!—whereon the matter of the universe floats speck-like. - -But this aspect of the problem is like what are called in mathematics -boundary conditions. - -We can trace out all the consequences of four-dimensional movements -down to their last detail. Then, knowing the mode of action which -would be characteristic of the minutest particles, if they were -free, we can draw conclusions from what they actually do of what the -constraint on them is. Of the two things, the material conditions and -the motion, one is known, and the other can be inferred. If the place -of this universe is a meeting of two, there would be a one-sideness -to space. If it lies so that what stretches away in one direction in -the unknown is unlike what stretches away in the other, then, as far -as the movements which participate in that dimension are concerned, -there would be a difference as to which way the motion took place. This -would be shown in the dissimilarity of phenomena, which, so far as -all three-space movements are concerned, were perfectly symmetrical. -To take an instance, merely, for the sake of precising our ideas, -not for any inherent probability in it; if it could be shown that -the electric current in the positive direction were exactly like the -electric current in the negative direction, except for a reversal of -the components of the motion in three-dimensional space, then the -dissimilarity of the discharge from the positive and negative poles -would be an indication of a one-sideness to our space. The only cause -of difference in the two discharges would be due to a component in -the fourth dimension, which directed in one direction transverse to -our space, met with a different resistance to that which it met when -directed in the opposite direction. - - - - - CHAPTER VII - - THE EVIDENCES FOR A FOURTH DIMENSION - - -The method necessarily to be employed in the search for the evidences -of a fourth dimension, consists primarily in the formation of the -conceptions of four-dimensional shapes and motions. When we are in -possession of these it is possible to call in the aid of observation, -without them we may have been all our lives in the familiar presence of -a four-dimensional phenomenon without ever recognising its nature. - -To take one of the conceptions we have already formed, the turning of a -real thing into its mirror image would be an occurrence which it would -be hard to explain, except on the assumption of a fourth dimension. - -We know of no such turning. But there exist a multitude of forms which -show a certain relation to a plane, a relation of symmetry, which -indicates more than an accidental juxtaposition of parts. In organic -life the universal type is of right- and left-handed symmetry, there -is a plane on each side of which the parts correspond. Now we have -seen that in four dimensions a plane takes the place of a line in -three dimensions. In our space, rotation about an axis is the type of -rotation, and the origin of bodies symmetrical about a line as the -earth is symmetrical about an axis can easily be explained. But where -there is symmetry about a plane no simple physical motion, such as we -are accustomed to, suffices to explain it. In our space a symmetrical -object must be built up by equal additions on each side of a central -plane. Such additions about such a plane are as little likely as any -other increments. The probability against the existence of symmetrical -form in inorganic nature is overwhelming in our space, and in organic -forms they would be as difficult of production as any other variety -of configuration. To illustrate this point we may take the child’s -amusement of making from dots of ink on a piece of paper a lifelike -representation of an insect by simply folding the paper over. The -dots spread out on a symmetrical line, and give the impression of a -segmented form with antennæ and legs. - -Now seeing a number of such figures we should naturally infer a folding -over. Can, then, a folding over in four-dimensional space account for -the symmetry of organic forms? The folding cannot of course be of the -bodies we see, but it may be of those minute constituents, the ultimate -elements of living matter which, turned in one way or the other, become -right- or left-handed, and so produce a corresponding structure. - -There is something in life not included in our conceptions of -mechanical movement. Is this something a four-dimensional movement? - -If we look at it from the broadest point of view, there is something -striking in the fact that where life comes in there arises an entirely -different set of phenomena to those of the inorganic world. - -The interest and values of life as we know it in ourselves, as we -know it existing around us in subordinate forms, is entirely and -completely different to anything which inorganic nature shows. And in -living beings we have a kind of form, a disposition of matter which -is entirely different from that shown in inorganic matter. Right- -and left-handed symmetry does not occur in the configurations of dead -matter. We have instances of symmetry about an axis, but not about -a plane. It can be argued that the occurrence of symmetry in two -dimensions involves the existence of a three-dimensional process, as -when a stone falls into water and makes rings of ripples, or as when -a mass of soft material rotates about an axis. It can be argued that -symmetry in any number of dimensions is the evidence of an action in -a higher dimensionality. Thus considering living beings, there is an -evidence both in their structure, and their different mode of activity, -of a something coming in from without into the inorganic world. - -And the objections which will readily occur, such as those derived from -the forms of twin crystals and the theoretical structure of chemical -molecules, do not invalidate the argument; for in these forms too the -presumable seat of the activity producing them lies in that very minute -region in which we necessarily place the seat of a four-dimensional -mobility. - -In another respect also the existence of symmetrical forms is -noteworthy. It is puzzling to conceive how two shapes exactly equal can -exist which are not superposible. Such a pair of symmetrical figures -as the two hands, right and left, show either a limitation in our -power of movement, by which we cannot superpose the one on the other, -or a definite influence and compulsion of space on matter, inflicting -limitations which are additional to those of the proportions of the -parts. - -We will, however, put aside the arguments to be drawn from the -consideration of symmetry as inconclusive, retaining one valuable -indication which they afford. If it is in virtue of a four-dimensional -motion that symmetry exists, it is only in the very minute particles -of bodies that that motion is to be found, for there is no such thing -as a bending over in four dimensions of any object of a size which we -can observe. The region of the extremely minute is the one, then, which -we shall have to investigate. We must look for some phenomenon which, -occasioning movements of the kind we know, still is itself inexplicable -as any form of motion which we know. - -Now in the theories of the actions of the minute particles of bodies -on one another, and in the motions of the ether, mathematicians -have tacitly assumed that the mechanical principles are the same as -those which prevail in the case of bodies which can be observed, it -has been assumed without proof that the conception of motion being -three-dimensional, holds beyond the region from observations in which -it was formed. - -Hence it is not from any phenomenon explained by mathematics that we -can derive a proof of four dimensions. Every phenomenon that has been -explained is explained as three-dimensional. And, moreover, since in -the region of the very minute we do not find rigid bodies acting on -each other at a distance, but elastic substances and continuous fluids -such as ether, we shall have a double task. - -We must form the conceptions of the possible movements of elastic and -liquid four-dimensional matter, before we can begin to observe. Let -us, therefore, take the four-dimensional rotation about a plane, and -enquire what it becomes in the case of extensible fluid substances. If -four-dimensional movements exist, this kind of rotation must exist, and -the finer portions of matter must exhibit it. - -Consider for a moment a rod of flexible and extensible material. It can -turn about an axis, even if not straight; a ring of india rubber can -turn inside out. - -What would this be in the case of four dimensions? - -Let us consider a sphere of our three-dimensional matter having a -definite thickness. To represent this thickness let us suppose that -from every point of the sphere in fig. 44 rods project both ways, in -and out, like D and F. We can only see the external portion, because -the internal parts are hidden by the sphere. - -[Illustration: Fig. 44. - -_Axis of x running towards the observer._] - -In this sphere the axis of _x_ is supposed to come towards the -observer, the axis of _z_ to run up, the axis of _y_ to go to the right. - -[Illustration: Fig. 45.] - -Now take the section determined by the _zy_ plane. This will be a -circle as shown in fig. 45. If we let drop the _x_ axis, this circle -is all we have of the sphere. Letting the _w_ axis now run in the -place of the old _x_ axis we have the space _yzw_, and in this space -all that we have of the sphere is the circle. Fig. 45 then represents -all that there is of the sphere in the space of _yzw_. In this space -it is evident that the rods CD and EF can turn round the circumference -as an axis. If the matter of the spherical shell is sufficiently -extensible to allow the particles C and E to become as widely separated -as they would be in the positions D and F, then the strip of matter -represented by CD and EF and a multitude of rods like them can turn -round the circular circumference. - -Thus this particular section of the sphere can turn inside out, and -what holds for any one section holds for all. Hence in four dimensions -the whole sphere can, if extensible turn inside out. Moreover, any part -of it—a bowl-shaped portion, for instance—can turn inside out, and so -on round and round. - -This is really no more than we had before in the rotation about a -plane, except that we see that the plane can, in the case of extensible -matter, be curved, and still play the part of an axis. - -If we suppose the spherical shell to be of four-dimensional matter, our -representation will be a little different. Let us suppose there to be a -small thickness to the matter in the fourth dimension. This would make -no difference in fig. 44, for that merely shows the view in the _xyz_ -space. But when the _x_ axis is let drop, and the _w_ axis comes in, -then the rods CD and EF which represent the matter of the shell, will -have a certain thickness perpendicular to the plane of the paper on -which they are drawn. If they have a thickness in the fourth dimension -they will show this thickness when looked at from the direction of the -_w_ axis. - -Supposing these rods, then, to be small slabs strung on the -circumference of the circle in fig. 45, we see that there will not -be in this case either any obstacle to their turning round the -circumference. We can have a shell of extensible material or of fluid -material turning inside out in four dimensions. - -And we must remember that in four dimensions there is no such thing as -rotation round an axis. If we want to investigate the motion of fluids -in four dimensions we must take a movement about an axis in our space, -and find the corresponding movement about a plane in four space. - -Now, of all the movements which take place in fluids, the most -important from a physical point of view is vortex motion. - -A vortex is a whirl or eddy—it is shown in the gyrating wreaths of -dust seen on a summer day; it is exhibited on a larger scale in the -destructive march of a cyclone. - -A wheel whirling round will throw off the water on it. But when -this circling motion takes place in a liquid itself it is strangely -persistent. There is, of course, a certain cohesion between the -particles of water by which they mutually impede their motions. But -in a liquid devoid of friction, such that every particle is free from -lateral cohesion on its path of motion, it can be shown that a vortex -or eddy separates from the mass of the fluid a certain portion, which -always remain in that vortex. - -The shape of the vortex may alter, but it always consists of the same -particles of the fluid. - -Now, a very remarkable fact about such a vortex is that the ends of the -vortex cannot remain suspended and isolated in the fluid. They must -always run to the boundary of the fluid. An eddy in water that remains -half way down without coming to the top is impossible. - -The ends of a vortex must reach the boundary of a fluid—the boundary -may be external or internal—a vortex may exist between two objects -in the fluid, terminating one end on each object, the objects being -internal boundaries of the fluid. Again, a vortex may have its ends -linked together, so that it forms a ring. Circular vortex rings of -this description are often seen in puffs of smoke, and that the smoke -travels on in the ring is a proof that the vortex always consists of -the same particles of air. - -Let us now enquire what a vortex would be in a four-dimensional fluid. - -We must replace the line axis by a plane axis. We should have therefore -a portion of fluid rotating round a plane. - -We have seen that the contour of this plane corresponds with the ends -of the axis line. Hence such a four-dimensional vortex must have its -rim on a boundary of the fluid. There would be a region of vorticity -with a contour. If such a rotation were started at one part of a -circular boundary, its edges would run round the boundary in both -directions till the whole interior region was filled with the vortex -sheet. - -A vortex in a three-dimensional liquid may consist of a number of -vortex filaments lying together producing a tube, or rod of vorticity. - -In the same way we can have in four dimensions a number of vortex -sheets alongside each other, each of which can be thought of as a -bowl-shaped portion of a spherical shell turning inside out. The -rotation takes place at any point not in the space occupied by the -shell, but from that space to the fourth dimension and round back again. - -Is there anything analogous to this within the range of our observation? - -An electric current answers this description in every respect. -Electricity does not flow through a wire. Its effect travels both ways -from the starting point along the wire. The spark which shows its -passing midway in its circuit is later than that which occurs at points -near its starting point on either side of it. - -Moreover, it is known that the action of the current is not in the -wire. It is in the region enclosed by the wire, this is the field of -force, the locus of the exhibition of the effects of the current. - -And the necessity of a conducting circuit for a current is exactly -that which we should expect if it were a four-dimensional vortex. -According to Maxwell every current forms a closed circuit, and this, -from the four-dimensional point of view, is the same as saying a vortex -must have its ends on a boundary of the fluid. - -Thus, on the hypothesis of a fourth dimension, the rotation of the -fluid ether would give the phenomenon of an electric current. We must -suppose the ether to be full of movement, for the more we examine into -the conditions which prevail in the obscurity of the minute, the more -we find that an unceasing and perpetual motion reigns. Thus we may say -that the conception of the fourth dimension means that there must be a -phenomenon which presents the characteristics of electricity. - -We know now that light is an electro-magnetic action, and that so far -from being a special and isolated phenomenon this electric action is -universal in the realm of the minute. Hence, may we not conclude that, -so far from the fourth dimension being remote and far away, being a -thing of symbolic import, a term for the explanation of dubious facts -by a more obscure theory, it is really the most important fact within -our knowledge. Our three-dimensional world is superficial. These -processes, which really lie at the basis of all phenomena of matter, -escape our observation by their minuteness, but reveal to our intellect -an amplitude of motion surpassing any that we can see. In such shapes -and motions there is a realm of the utmost intellectual beauty, and one -to which our symbolic methods apply with a better grace than they do to -those of three dimensions. - - - - - CHAPTER VIII - - THE USE OF FOUR DIMENSIONS IN THOUGHT - - -Having held before ourselves this outline of a conjecture of the world -as four-dimensional, having roughly thrown together those facts of -movement which we can see apply to our actual experience, let us pass -to another branch of our subject. - -The engineer uses drawings, graphical constructions, in a variety of -manners. He has, for instance, diagrams which represent the expansion -of steam, the efficiency of his valves. These exist alongside the -actual plans of his machines. They are not the pictures of anything -really existing, but enable him to think about the relations which -exist in his mechanisms. - -And so, besides showing us the actual existence of that world which -lies beneath the one of visible movements, four-dimensional space -enables us to make ideal constructions which serve to represent the -relations of things, and throw what would otherwise be obscure into a -definite and suggestive form. - -From amidst the great variety of instances which lies before me I will -select two, one dealing with a subject of slight intrinsic interest, -which however gives within a limited field a striking example of the -method of drawing conclusions and the use of higher space figures.[1] - - [1] It is suggestive also in another respect, because it shows very - clearly that in our processes of thought there are in play faculties - other than logical; in it the origin of the idea which proves to be - justified is drawn from the consideration of symmetry, a branch of the - beautiful. - -The other instance is chosen on account of the bearing it has on our -fundamental conceptions. In it I try to discover the real meaning of -Kant’s theory of experience. - -The investigation of the properties of numbers is much facilitated -by the fact that relations between numbers are themselves able to be -represented as numbers—_e.g._, 12, and 3 are both numbers, and the -relation between them is 4, another number. The way is thus opened for -a process of constructive theory, without there being any necessity for -a recourse to another class of concepts besides that which is given in -the phenomena to be studied. - -The discipline of number thus created is of great and varied -applicability, but it is not solely as quantitative that we learn to -understand the phenomena of nature. It is not possible to explain the -properties of matter by number simply, but all the activities of matter -are energies in space. They are numerically definite and also, we may -say, directedly definite, _i.e._ definite in direction. - -Is there, then, a body of doctrine about space which, like that of -number, is available in science? It is needless to answer: Yes; -geometry. But there is a method lying alongside the ordinary methods of -geometry, which tacitly used and presenting an analogy to the method of -numerical thought deserves to be brought into greater prominence than -it usually occupies. - -The relation of numbers is a number. - -Can we say in the same way that the relation of shapes is a shape? - -We can. - -To take an instance chosen on account of its ready availability. Let -us take two right-angled triangles of a given hypothenuse, but having -sides of different lengths (fig. 46). These triangles are shapes which -have a certain relation to each other. Let us exhibit their relation as -a figure. - -[Illustration: Fig. 46.] - -Draw two straight lines at right angles to each other, the one HL a -horizontal level, the other VL a vertical level (fig. 47). By means -of these two co-ordinating lines we can represent a double set of -magnitudes; one set as distances to the right of the vertical level, -the other as distances above the horizontal level, a suitable unit -being chosen. - -[Illustration: Fig. 47.] - -Thus the line marked 7 will pick out the assemblage of points whose -distance from the vertical level is 7, and the line marked 1 will pick -out the points whose distance above the horizontal level is 1. The -meeting point of these two lines, 7 and 1, will define a point which -with regard to the one set of magnitudes is 7, with regard to the -other is 1. Let us take the sides of our triangles as the two sets of -magnitudes in question. - -Then the point 7, 1, will represent the triangle whose sides are 7 and -1. Similarly the point 5, 5—5, that is, to the right of the vertical -level and 5 above the horizontal level—will represent the triangle -whose sides are 5 and 5 (fig. 48). - -[Illustration: Fig. 48.] - -Thus we have obtained a figure consisting of the two points 7, 1, and -5, 5, representative of our two triangles. But we can go further, -and, drawing an arc of a circle about O, the meeting point of the -horizontal and vertical levels, which passes through 7, 1, and 5, -5, assert that all the triangles which are right-angled and have a -hypothenuse whose square is 50 are represented by the points on this -arc. - -Thus, each individual of a class being represented by a point, the -whole class is represented by an assemblage of points forming a -figure. Accepting this representation we can attach a definite and -calculable significance to the expression, resemblance, or similarity -between two individuals of the class represented, the difference being -measured by the length of the line between two representative points. -It is needless to multiply examples, or to show how, corresponding to -different classes of triangles, we obtain different curves. - -A representation of this kind in which an object, a thing in space, -is represented as a point, and all its properties are left out, their -effect remaining only in the relative position which the representative -point bears to the representative points of the other objects, may be -called, after the analogy of Sir William R. Hamilton’s hodograph, a -“Poiograph.” - -Representations thus made have the character of natural objects; -they have a determinate and definite character of their own. Any -lack of completeness in them is probably due to a failure in point -of completeness of those observations which form the ground of their -construction. - -Every system of classification is a poiograph. In Mendeléeff’s scheme -of the elements, for instance, each element is represented by a point, -and the relations between the elements are represented by the relations -between the points. - -So far I have simply brought into prominence processes and -considerations with which we are all familiar. But it is worth while -to bring into the full light of our attention our habitual assumptions -and processes. It often happens that we find there are two of them -which have a bearing on each other, which, without this dragging into -the light, we should have allowed to remain without mutual influence. - -There is a fact which it concerns us to take into account in discussing -the theory of the poiograph. - -With respect to our knowledge of the world we are far from that -condition which Laplace imagined when he asserted that an all-knowing -mind could determine the future condition of every object, if he knew -the co-ordinates of its particles in space, and their velocity at any -particular moment. - -On the contrary, in the presence of any natural object, we have a great -complexity of conditions before us, which we cannot reduce to position -in space and date in time. - -There is mass, attraction apparently spontaneous, electrical and -magnetic properties which must be superadded to spatial configuration. -To cut the list short we must say that practically the phenomena of the -world present us problems involving many variables, which we must take -as independent. - -From this it follows that in making poiographs we must be prepared -to use space of more than three dimensions. If the symmetry and -completeness of our representation is to be of use to us we must be -prepared to appreciate and criticise figures of a complexity greater -than of those in three dimensions. It is impossible to give an example -of such a poiograph which will not be merely trivial, without going -into details of some kind irrelevant to our subject. I prefer to -introduce the irrelevant details rather than treat this part of the -subject perfunctorily. - -To take an instance of a poiograph which does not lead us into the -complexities incident on its application in classificatory science, -let us follow Mrs. Alicia Boole Stott in her representation of the -syllogism by its means. She will be interested to find that the curious -gap she detected has a significance. - -A syllogism consists of two statements, the major and the minor -premiss, with the conclusion that can be drawn from them. Thus, to take -an instance, fig. 49. It is evident, from looking at the successive -figures that, if we know that the region M lies altogether within the -region P, and also know that the region S lies altogether within the -region M, we can conclude that the region S lies altogether within -the region P. M is P, major premiss; S is M, minor premiss; S is P, -conclusion. Given the first two data we must conclude that S lies -in P. The conclusion S is P involves two terms, S and P, which are -respectively called the subject and the predicate, the letters S and -P being chosen with reference to the parts the notions they designate -play in the conclusion. S is the subject of the conclusion, P is the -predicate of the conclusion. The major premiss we take to be, that -which does not involve S, and here we always write it first. - -[Illustration: Fig. 49.] - -There are several varieties of statement possessing different degrees -of universality and manners of assertiveness. These different forms of -statement are called the moods. - -We will take the major premiss as one variable, as a thing capable of -different modifications of the same kind, the minor premiss as another, -and the different moods we will consider as defining the variations -which these variables undergo. - -There are four moods:— - - 1. The universal affirmative; all M is P, called mood A. - - 2. The universal negative; no M is P, mood E. - - 3. The particular affirmative; some M is P, mood I. - - 4. The particular negative; some M is not P, mood O. - -[Illustration: 1. 2. 3. 4. Mood A. Mood E. Mood I. Mood O. -Fig. 50.] - -The dotted lines in 3 and 4, fig. 50, denote that it is not known -whether or no any objects exist, corresponding to the space of which -the dotted line forms one delimiting boundary; thus, in mood I we do -not know if there are any M’S which are not P, we only know some M’S -are P. - -[Illustration: Fig. 51.] - -Representing the first premiss in its various moods by regions marked -by vertical lines to the right of PQ, we have in fig. 51, running up -from the four letters AEIO, four columns, each of which indicates that -the major premiss is in the mood denoted by the respective letter. In -the first column to the right of PQ is the mood A. Now above the line -RS let there be marked off four regions corresponding to the four moods -of the minor premiss. Thus, in the first row above RS all the region -between RS and the first horizontal line above it denotes that the -minor premiss is in the mood A. The letters E, I, O, in the same way -show the mood characterising the minor premiss in the rows opposite -these letters. - -We have still to exhibit the conclusion. To do this we must consider -the conclusion as a third variable, characterised in its different -varieties by four moods—this being the syllogistic classification. The -introduction of a third variable involves a change in our system of -representation. - -Before we started with the regions to the right of a certain line as -representing successively the major premiss in its moods; now we must -start with the regions to the right of a certain plane. Let LMNR be -the plane face of a cube, fig. 52, and let the cube be divided into -four parts by vertical sections parallel to LMNR. The variable, the -major premiss, is represented by the successive regions which occur to -the right of the plane LMNR—that region to which A stands opposite, -that slice of the cube, is significative of the mood A. This whole -quarter-part of the cube represents that for every part of it the major -premiss is in the mood A. - -[Illustration: Fig. 52.] - -In a similar manner the next section, the second with the letter E -opposite it, represents that for every one of the sixteen small cubic -spaces in it, the major premiss is in the mood E. The third and fourth -compartments made by the vertical sections denote the major premiss in -the moods I and O. But the cube can be divided in other ways by other -planes. Let the divisions, of which four stretch from the front face, -correspond to the minor premiss. The first wall of sixteen cubes, -facing the observer, has as its characteristic that in each of the -small cubes, whatever else may be the case, the minor premiss is in the -mood A. The variable—the minor premiss—varies through the phases A, E, -I, O, away from the front face of the cube, or the front plane of which -the front face is a part. - -And now we can represent the third variable in a precisely similar way. -We can take the conclusion as the third variable, going through its -four phases from the ground plane upwards. Each of the small cubes at -the base of the whole cube has this true about it, whatever else may -be the case, that the conclusion is, in it, in the mood A. Thus, to -recapitulate, the first wall of sixteen small cubes, the first of the -four walls which, proceeding from left to right, build up the whole -cube, is characterised in each part of it by this, that the major -premiss is in the mood A. - -The next wall denotes that the major premiss is in the mood E, and -so on. Proceeding from the front to the back the first wall presents -a region in every part of which the minor premiss is in the mood A. -The second wall is a region throughout which the minor premiss is in -the mood E, and so on. In the layers, from the bottom upwards, the -conclusion goes through its various moods beginning with A in the -lowest, E in the second, I in the third, O in the fourth. - -In the general case, in which the variables represented in the -poiograph pass through a wide range of values, the planes from which we -measure their degrees of variation in our representation are taken to -be indefinitely extended. In this case, however, all we are concerned -with is the finite region. - -We have now to represent, by some limitation of the complex we have -obtained, the fact that not every combination of premisses justifies -any kind of conclusion. This can be simply effected by marking the -regions in which the premisses, being such as are defined by the -positions, a conclusion which is valid is found. - -Taking the conjunction of the major premiss, all M is P, and the minor, -all S is M, we conclude that all S is P. Hence, that region must be -marked in which we have the conjunction of major premiss in mood A; -minor premiss, mood A; conclusion, mood A. This is the cube occupying -the lowest left-hand corner of the large cube. - -[Illustration: Fig. 53.] - -Proceeding in this way, we find that the regions which must be marked -are those shown in fig. 53. To discuss the case shown in the marked -cube which appears at the top of fig. 53. Here the major premiss is -in the second wall to the right—it is in the mood E and is of the -type no M is P. The minor premiss is in the mood characterised by the -third wall from the front. It is of the type some S is M. From these -premisses we draw the conclusion that some S is not P, a conclusion in -the mood O. Now the mood O of the conclusion is represented in the top -layer. Hence we see that the marking is correct in this respect. - -[Illustration: Fig. 54.] - -It would, of course, be possible to represent the cube on a plane by -means of four squares, as in fig. 54, if we consider each square to -represent merely the beginning of the region it stands for. Thus the -whole cube can be represented by four vertical squares, each standing -for a kind of vertical tray, and the markings would be as shown. In No. -1 the major premiss is in mood A for the whole of the region indicated -by the vertical square of sixteen divisions; in No. 2 it is in the mood -E, and so on. - -A creature confined to a plane would have to adopt some such -disjunctive way of representing the whole cube. He would be obliged to -represent that which we see as a whole in separate parts, and each part -would merely represent, would not be, that solid content which we see. - -The view of these four squares which the plane creature would have -would not be such as ours. He would not see the interior of the four -squares represented above, but each would be entirely contained within -its outline, the internal boundaries of the separate small squares he -could not see except by removing the outer squares. - -We are now ready to introduce the fourth variable involved in the -syllogism. - -In assigning letters to denote the terms of the syllogism we have taken -S and P to represent the subject and predicate in the conclusion, and -thus in the conclusion their order is invariable. But in the premisses -we have taken arbitrarily the order all M is P, and all S is M. There -is no reason why M instead of P should not be the predicate of the -major premiss, and so on. - -Accordingly we take the order of the terms in the premisses as the -fourth variable. Of this order there are four varieties, and these -varieties are called figures. - -Using the order in which the letters are written to denote that the -letter first written is subject, the one written second is predicate, -we have the following possibilities:— - - 1st Figure. 2nd Figure. 3rd Figure. 4th Figure. - Major M P P M M P P M - Minor S M S M M S M S - -There are therefore four possibilities with regard to this fourth -variable as with regard to the premisses. - -We have used up our dimensions of space in representing the phases of -the premisses and the conclusion in respect of mood, and to represent -in an analogous manner the variations in figure we require a fourth -dimension. - -Now in bringing in this fourth dimension we must make a change in our -origins of measurement analogous to that which we made in passing from -the plane to the solid. - -This fourth dimension is supposed to run at right angles to any of the -three space dimensions, as the third space dimension runs at right -angles to the two dimensions of a plane, and thus it gives us the -opportunity of generating a new kind of volume. If the whole cube moves -in this dimension, the solid itself traces out a path, each section of -which, made at right angles to the direction in which it moves, is a -solid, an exact repetition of the cube itself. - -The cube as we see it is the beginning of a solid of such a kind. It -represents a kind of tray, as the square face of the cube is a kind of -tray against which the cube rests. - -Suppose the cube to move in this fourth dimension in four stages, -and let the hyper-solid region traced out in the first stage of its -progress be characterised by this, that the terms of the syllogism -are in the first figure, then we can represent in each of the three -subsequent stages the remaining three figures. Thus the whole cube -forms the basis from which we measure the variation in figure. The -first figure holds good for the cube as we see it, and for that -hyper-solid which lies within the first stage; the second figure holds -good in the second stage, and so on. - -Thus we measure from the whole cube as far as figures are concerned. - -But we saw that when we measured in the cube itself having three -variables, namely, the two premisses and the conclusion, we measured -from three planes. The base from which we measured was in every case -the same. - -Hence, in measuring in this higher space we should have bases of the -same kind to measure from, we should have solid bases. - -The first solid base is easily seen, it is the cube itself. The other -can be found from this consideration. - -That solid from which we measure figure is that in which the remaining -variables run through their full range of varieties. - -Now, if we want to measure in respect of the moods of the major -premiss, we must let the minor premiss, the conclusion, run through -their range, and also the order of the terms. That is we must take as -basis of measurement in respect to the moods of the major that which -represents the variation of the moods of the minor, the conclusion and -the variation of the figures. - -Now the variation of the moods of the minor and of the conclusion are -represented in the square face on the left of the cube. Here are all -varieties of the minor premiss and the conclusion. The varieties of -the figures are represented by stages in a motion proceeding at right -angles to all space directions, at right angles consequently to the -face in question, the left-hand face of the cube. - -Consequently letting the left-hand face move in this direction we get -a cube, and in this cube all the varieties of the minor premiss, the -conclusion, and the figure are represented. - -Thus another cubic base of measurement is given to the cube, generated -by movement of the left-hand square in the fourth dimension. - -We find the other bases in a similar manner, one is the cube generated -by the front square moved in the fourth dimension so as to generate a -cube. From this cube variations in the mood of the minor are measured. -The fourth base is that found by moving the bottom square of the cube -in the fourth dimension. In this cube the variations of the major, -the minor, and the figure are given. Considering this as a basis in -the four stages proceeding from it, the variation in the moods of the -conclusion are given. - -Any one of these cubic bases can be represented in space, and then the -higher solid generated from them lies out of our space. It can only -be represented by a device analogous to that by which the plane being -represents a cube. - -He represents the cube shown above, by taking four square sections and -placing them arbitrarily at convenient distances the one from the other. - -So we must represent this higher solid by four cubes: each cube -represents only the beginning of the corresponding higher volume. - -It is sufficient for us, then, if we draw four cubes, the first -representing that region in which the figure is of the first kind, -the second that region in which the figure is of the second kind, -and so on. These cubes are the beginnings merely of the respective -regions—they are the trays, as it were, against which the real solids -must be conceived as resting, from which they start. The first one, as -it is the beginning of the region of the first figure, is characterised -by the order of the terms in the premisses being that of the first -figure. The second similarly has the terms of the premisses in the -order of the second figure, and so on. - -These cubes are shown below. - -For the sake of showing the properties of the method of representation, -not for the logical problem, I will make a digression. I will represent -in space the moods of the minor and of the conclusion and the different -figures, keeping the major always in mood A. Here we have three -variables in different stages, the minor, the conclusion, and the -figure. Let the square of the left-hand side of the original cube be -imagined to be standing by itself, without the solid part of the cube, -represented by (2) fig. 55. The A, E, I, O, which run away represent -the moods of the minor, the A, E, I, O, which run up represent the -moods of the conclusion. The whole square, since it is the beginning -of the region in the major premiss, mood A, is to be considered as in -major premiss, mood A. - -From this square, let it be supposed that that direction in which the -figures are represented runs to the left hand. Thus we have a cube (1) -running from the square above, in which the square itself is hidden, -but the letters A, E, I, O, of the conclusion are seen. In this cube -we have the minor premiss and the conclusion in all their moods, and -all the figures represented. With regard to the major premiss, since -the face (2) belongs to the first wall from the left in the original -arrangement, and in this arrangement was characterised by the major -premiss in the mood A, we may say that the whole of the cube we now -have put up represents the mood A of the major premiss. - -[Illustration: Fig. 55.] - -Hence the small cube at the bottom to the right in 1, nearest to the -spectator, is major premiss, mood A; minor premiss, mood A; conclusion, -mood A; and figure the first. The cube next to it, running to the left, -is major premiss, mood A; minor premiss, mood A; conclusion, mood A; -figure 2. - -So in this cube we have the representations of all the combinations -which can occur when the major premiss, remaining in the mood A, the -minor premiss, the conclusion, and the figures pass through their -varieties. - -In this case there is no room in space for a natural representation of -the moods of the major premiss. To represent them we must suppose as -before that there is a fourth dimension, and starting from this cube as -base in the fourth direction in four equal stages, all the first volume -corresponds to major premiss A, the second to major premiss, mood E, -the next to the mood I, and the last to mood O. - -The cube we see is as it were merely a tray against which the -four-dimensional figure rests. Its section at any stage is a cube. But -a transition in this direction being transverse to the whole of our -space is represented by no space motion. We can exhibit successive -stages of the result of transference of the cube in that direction, but -cannot exhibit the product of a transference, however small, in that -direction. - -[Illustration: Fig. 56.] - -To return to the original method of representing our variables, -consider fig. 56. These four cubes represent four sections of the -figure derived from the first of them by moving it in the fourth -dimension. The first portion of the motion, which begins with 1, traces -out a more than solid body, which is all in the first figure. The -beginning of this body is shown in 1. The next portion of the motion -traces out a more than solid body, all of which is in the second -figure; the beginning of this body is shown in 2; 3 and 4 follow on in -like manner. Here, then, in one four-dimensional figure we have all -the combinations of the four variables, major premiss, minor premiss, -figure, conclusion, represented, each variable going through its four -varieties. The disconnected cubes drawn are our representation in space -by means of disconnected sections of this higher body. - -Now it is only a limited number of conclusions which are true—their -truth depends on the particular combinations of the premisses and -figures which they accompany. The total figure thus represented may be -called the universe of thought in respect to these four constituents, -and out of the universe of possibly existing combinations it is the -province of logic to select those which correspond to the results of -our reasoning faculties. - -We can go over each of the premisses in each of the moods, and find out -what conclusion logically follows. But this is done in the works on -logic; most simply and clearly I believe in “Jevon’s Logic.” As we are -only concerned with a formal presentation of the results we will make -use of the mnemonic lines printed below, in which the words enclosed in -brackets refer to the figures, and are not significative:— - - Barbara celarent Darii ferio_que_ [prioris]. - Caesare Camestris Festino Baroko [secundae]. - [Tertia] darapti disamis datisi felapton. - Bokardo ferisson _habet_ [Quarta insuper addit]. - Bramantip camenes dimaris ferapton fresison. - -In these lines each significative word has three vowels, the first -vowel refers to the major premiss, and gives the mood of that premiss, -“a” signifying, for instance, that the major mood is in mood _a_. The -second vowel refers to the minor premiss, and gives its mood. The third -vowel refers to the conclusion, and gives its mood. Thus (prioris)—of -the first figure—the first mnemonic word is “barbara,” and this gives -major premiss, mood A; minor premiss, mood A; conclusion, mood A. -Accordingly in the first of our four cubes we mark the lowest left-hand -front cube. To take another instance in the third figure “Tertia,” -the word “ferisson” gives us major premiss mood E—_e.g._, no M is P, -minor premiss mood I; some M is S, conclusion, mood O; some S is not P. -The region to be marked then in the third representative cube is the -one in the second wall to the right for the major premiss, the third -wall from the front for the minor premiss, and the top layer for the -conclusion. - -It is easily seen that in the diagram this cube is marked, and so with -all the valid conclusions. The regions marked in the total region show -which combinations of the four variables, major premiss, minor premiss, -figure, and conclusion exist. - -That is to say, we objectify all possible conclusions, and build up an -ideal manifold, containing all possible combinations of them with the -premisses, and then out of this we eliminate all that do not satisfy -the laws of logic. The residue is the syllogism, considered as a canon -of reasoning. - -Looking at the shape which represents the totality of the valid -conclusions, it does not present any obvious symmetry, or easily -characterisable nature. A striking configuration, however, is -obtained, if we project the four-dimensional figure obtained into a -three-dimensional one; that is, if we take in the base cube all those -cubes which have a marked space anywhere in the series of four regions -which start from that cube. - -This corresponds to making abstraction of the figures, giving all the -conclusions which are valid whatever the figure may be. - -[Illustration: Fig. 57.] - -Proceeding in this way we obtain the arrangement of marked cubes shown -in fig. 57. We see that the valid conclusions are arranged almost -symmetrically round one cube—the one on the top of the column starting -from AAA. There is one breach of continuity however in this scheme. -One cube is unmarked, which if marked would give symmetry. It is the -one which would be denoted by the letters I, E, O, in the third -wall to the right, the second wall away, the topmost layer. Now this -combination of premisses in the mood IE, with a conclusion in the mood -O, is not noticed in any book on logic with which I am familiar. Let -us look at it for ourselves, as it seems that there must be something -curious in connection with this break of continuity in the poiograph. - -[Illustration: Fig. 58.] - -The propositions I, E, in the various figures are the following, as -shown in the accompanying scheme, fig. 58:—First figure: some M is P; -no S is M. Second figure: some P is M; no S is M. Third figure: some M -is P; no M is S. Fourth figure: some P is M; no M is S. - -Examining these figures, we see, taking the first, that if some M is P -and no S is M, we have no conclusion of the form S is P in the various -moods. It is quite indeterminate how the circle representing S lies -with regard to the circle representing P. It may lie inside, outside, -or partly inside P. The same is true in the other figures 2 and 3. -But when we come to the fourth figure, since M and S lie completely -outside each other, there cannot lie inside S that part of P which lies -inside M. Now we know by the major premiss that some of P does lie in -M. Hence S cannot contain the whole of P. In words, some P is M, no -M is S, therefore S does not contain the whole of P. If we take P as -the subject, this gives us a conclusion in the mood O about P. Some -P is not S. But it does not give us conclusion about S in any one of -the four forms recognised in the syllogism and called its moods. Hence -the breach of the continuity in the poiograph has enabled us to detect -a lack of completeness in the relations which are considered in the -syllogism. - -To take an instance:—Some Americans (P) are of African stock (M); No -Aryans (S) are of African stock (M); Aryans (S) do not include all of -Americans (P). - -In order to draw a conclusion about S we have to admit the statement, -“S does not contain the whole of P,” as a valid logical form—it is a -statement about S which can be made. The logic which gives us the form, -“some P is not S,” and which does not allow us to give the exactly -equivalent and equally primary form, “S does not contain the whole of -P,” is artificial. - -And I wish to point out that this artificiality leads to an error. - -If one trusted to the mnemonic lines given above, one would conclude -that no logical conclusion about S can be drawn from the statement, -“some P are M, no M are S.” - -But a conclusion can be drawn: S does not contain the whole of P. - -It is not that the result is given expressed in another form. The -mnemonic lines deny that any conclusion can be drawn from premisses in -the moods I, E, respectively. - -Thus a simple four-dimensional poiograph has enabled us to detect a -mistake in the mnemonic lines which have been handed down unchallenged -from mediæval times. To discuss the subject of these lines more fully a -logician defending them would probably say that a particular statement -cannot be a major premiss; and so deny the existence of the fourth -figure in the combination of moods. - -To take our instance: some Americans are of African stock; no Aryans -are of African stock. He would say that the conclusion is some -Americans are not Aryans; and that the second statement is the major. -He would refuse to say anything about Aryans, condemning us to an -eternal silence about them, as far as these premisses are concerned! -But, if there is a statement involving the relation of two classes, it -must be expressible as a statement about either of them. - -To bar the conclusion, “Aryans do not include the whole of Americans,” -is purely a makeshift in favour of a false classification. - -And the argument drawn from the universality of the major premiss -cannot be consistently maintained. It would preclude such combinations -as major O, minor A, conclusion O—_i.e._, such as some mountains (M) -are not permanent (P); all mountains (M) are scenery (S); some scenery -(S) is not permanent (P). - -This is allowed in “Jevon’s Logic,” and his omission to discuss I, E, -O, in the fourth figure, is inexplicable. A satisfactory poiograph -of the logical scheme can be made by admitting the use of the words -some, none, or all, about the predicate as well as about the subject. -Then we can express the statement, “Aryans do not include the whole of -Americans,” clumsily, but, when its obscurity is fathomed, correctly, -as “Some Aryans are not all Americans.” And this method is what is -called the “quantification of the predicate.” - -The laws of formal logic are coincident with the conclusions which -can be drawn about regions of space, which overlap one another in the -various possible ways. It is not difficult so to state the relations -or to obtain a symmetrical poiograph. But to enter into this branch -of geometry is beside our present purpose, which is to show the -application of the poiograph in a finite and limited region, without -any of those complexities which attend its use in regard to natural -objects. - -If we take the latter—plants, for instance—and, without assuming -fixed directions in space as representative of definite variations, -arrange the representative points in such a manner as to correspond to -the similarities of the objects, we obtain configuration of singular -interest; and perhaps in this way, in the making of shapes of shapes, -bodies with bodies omitted, some insight into the structure of the -species and genera might be obtained. - - - - - CHAPTER IX - - APPLICATION TO KANT’S THEORY OF EXPERIENCE - - -When we observe the heavenly bodies we become aware that they all -participate in one universal motion—a diurnal revolution round the -polar axis. - -In the case of fixed stars this is most unqualifiedly true, but in the -case of the sun, and the planets also, the single motion of revolution -can be discerned, modified, and slightly altered by other and secondary -motions. - -Hence the universal characteristic of the celestial bodies is that they -move in a diurnal circle. - -But we know that this one great fact which is true of them all has in -reality nothing to do with them. The diurnal revolution which they -visibly perform is the result of the condition of the observer. It is -because the observer is on a rotating earth that a universal statement -can be made about all the celestial bodies. - -The universal statement which is valid about every one of the celestial -bodies is that which does not concern them at all, and is but a -statement of the condition of the observer. - -Now there are universal statements of other kinds which we can make. We -can say that all objects of experience are in space and subject to the -laws of geometry. - -Does this mean that space and all that it means is due to a condition -of the observer? - -If a universal law in one case means nothing affecting the objects -themselves, but only a condition of observation, is this true in every -case? There is shown us in astronomy a _vera causa_ for the assertion -of a universal. Is the same cause to be traced everywhere? - -Such is a first approximation to the doctrine of Kant’s critique. - -It is the apprehension of a relation into which, on the one side and -the other, perfectly definite constituents enter—the human observer and -the stars—and a transference of this relation to a region in which the -constituents on either side are perfectly unknown. - -If spatiality is due to a condition of the observer, the observer -cannot be this bodily self of ours—the body, like the objects around -it, are equally in space. - -This conception Kant applied, not only to the intuitions of sense, but -to the concepts of reason—wherever a universal statement is made there -is afforded him an opportunity for the application of his principle. -He constructed a system in which one hardly knows which the most to -admire, the architectonic skill, or the reticence with regard to things -in themselves, and the observer in himself. - -His system can be compared to a garden, somewhat formal perhaps, but -with the charm of a quality more than intellectual, a _besonnenheit_, -an exquisite moderation over all. And from the ground he so carefully -prepared with that buried in obscurity, which it is fitting should be -obscure, science blossoms and the tree of real knowledge grows. - -The critique is a storehouse of ideas of profound interest. The one -of which I have given a partial statement leads, as we shall see -on studying it in detail, to a theory of mathematics suggestive of -enquiries in many directions. - -The justification for my treatment will be found amongst other passages -in that part of the transcendental analytic, in which Kant speaks of -objects of experience subject to the forms of sensibility, not subject -to the concepts of reason. - -Kant asserts that whenever we think we think of objects in space and -time, but he denies that the space and time exist as independent -entities. He goes about to explain them, and their universality, not by -assuming them, as most other philosophers do, but by postulating their -absence. How then does it come to pass that the world is in space and -time to us? - -Kant takes the same position with regard to what we call nature—a great -system subject to law and order. “How do you explain the law and order -in nature?” we ask the philosophers. All except Kant reply by assuming -law and order somewhere, and then showing how we can recognise it. - -In explaining our notions, philosophers from other than the Kantian -standpoint, assume the notions as existing outside us, and then it is -no difficult task to show how they come to us, either by inspiration or -by observation. - -We ask “Why do we have an idea of law in nature?” “Because natural -processes go according to law,” we are answered, “and experience -inherited or acquired, gives us this notion.” - -But when we speak about the law in nature we are speaking about a -notion of our own. So all that these expositors do is to explain our -notion by an assumption of it. - -Kant is very different. He supposes nothing. An experience such as ours -is very different from experience in the abstract. Imagine just simply -experience, succession of states, of consciousness! Why, there would be -no connecting any two together, there would be no personal identity, -no memory. It is out of a general experience such as this, which, in -respect to anything we call real, is less than a dream, that Kant shows -the genesis of an experience such as ours. - -Kant takes up the problem of the explanation of space, time, order, and -so quite logically does not presuppose them. - -But how, when every act of thought is of things in space, and time, -and ordered, shall we represent to ourselves that perfectly indefinite -somewhat which is Kant’s necessary hypothesis—that which is not in -space or time and is not ordered. That is our problem, to represent -that which Kant assumes not subject to any of our forms of thought, and -then show some function which working on that makes it into a “nature” -subject to law and order, in space and time. Such a function Kant -calls the “Unity of Apperception”; _i.e._, that which makes our state -of consciousness capable of being woven into a system with a self, an -outer world, memory, law, cause, and order. - -The difficulty that meets us in discussing Kant’s hypothesis is that -everything we think of is in space and time—how then shall we represent -in space an existence not in space, and in time an existence not in -time? This difficulty is still more evident when we come to construct -a poiograph, for a poiograph is essentially a space structure. But -because more evident the difficulty is nearer a solution. If we always -think in space, _i.e._ using space concepts, the first condition -requisite for adapting them to the representation of non-spatial -existence, is to be aware of the limitation of our thought, and so be -able to take the proper steps to overcome it. The problem before us, -then, is to represent in space an existence not in space. - -The solution is an easy one. It is provided by the conception of -alternativity. - -To get our ideas clear let us go right back behind the distinctions of -an inner and an outer world. Both of these, Kant says, are products. -Let us take merely states of consciousness, and not ask the question -whether they are produced or superinduced—to ask such a question is to -have got too far on, to have assumed something of which we have not -traced the origin. Of these states let us simply say that they occur. -Let us now use the word a “posit” for a phase of consciousness reduced -to its last possible stage of evanescence; let a posit be that phase of -consciousness of which all that can be said is that it occurs. - -Let _a_, _b_, _c_, be three such posits. We cannot represent them in -space without placing them in a certain order, as _a_, _b_, _c_. But -Kant distinguishes between the forms of sensibility and the concepts -of reason. A dream in which everything happens at haphazard would be -an experience subject to the form of sensibility and only partially -subject to the concepts of reason. It is partially subject to the -concepts of reason because, although there is no order of sequence, -still at any given time there is order. Perception of a thing as in -space is a form of sensibility, the perception of an order is a concept -of reason. - -We must, therefore, in order to get at that process which Kant supposes -to be constitutive of an ordered experience imagine the posits as in -space without order. - -As we know them they must be in some order, _abc_, _bca_, _cab_, _acb_, -_cba_, _bac_, one or another. - -To represent them as having no order conceive all these different -orders as equally existing. Introduce the conception of -alternativity—let us suppose that the order _abc_, and _bac_, for -example, exist equally, so that we cannot say about _a_ that it comes -before or after _b_. This would correspond to a sudden and arbitrary -change of _a_ into _b_ and _b_ into _a_, so that, to use Kant’s words, -it would be possible to call one thing by one name at one time and at -another time by another name. - -In an experience of this kind we have a kind of chaos, in which no -order exists; it is a manifold not subject to the concepts of reason. - -Now is there any process by which order can be introduced into such a -manifold—is there any function of consciousness in virtue of which an -ordered experience could arise? - -In the precise condition in which the posits are, as described above, -it does not seem to be possible. But if we imagine a duality to exist -in the manifold, a function of consciousness can be easily discovered -which will produce order out of no order. - -Let us imagine each posit, then, as having, a dual aspect. Let _a_ be -1_a_ in which the dual aspect is represented by the combination of -symbols. And similarly let _b_ be 2_b_, _c_ be 3_c_, in which 2 and _b_ -represent the dual aspects of _b_, 3 and _c_ those of _c_. - -Since _a_ can arbitrarily change into _b_, or into _c_, and so on, the -particular combinations written above cannot be kept. We have to assume -the equally possible occurrence of form such as 2_a_, 2_b_, and so on; -and in order to get a representation of all those combinations out of -which any set is alternatively possible, we must take every aspect with -every aspect. We must, that is, have every letter with every number. - -Let us now apply the method of space representation. - - _Note._—At the beginning of the next chapter the same structures as - those which follow are exhibited in more detail and a reference to - them will remove any obscurity which may be found in the immediately - following passages. They are there carried on to a greater - multiplicity of dimensions, and the significance of the process here - briefly explained becomes more apparent. - -[Illustration: Fig. 59.] - -Take three mutually rectangular axes in space 1, 2, 3 (fig. 59), and -on each mark three points, the common meeting point being the first on -each axis. Then by means of these three points on each axis we define -27 positions, 27 points in a cubical cluster, shown in fig. 60, the -same method of co-ordination being used as has been described before. -Each of these positions can be named by means of the axes and the -points combined. - -[Illustration: Fig. 60.] - -Thus, for instance, the one marked by an asterisk can be called 1_c_, -2_b_, 3_c_, because it is opposite to _c_ on 1, to _b_ on 2, to _c_ on -3. - -Let us now treat of the states of consciousness corresponding to -these positions. Each point represents a composite of posits, and -the manifold of consciousness corresponding to them is of a certain -complexity. - -Suppose now the constituents, the points on the axes, to interchange -arbitrarily, any one to become any other, and also the axes 1, 2, and -3, to interchange amongst themselves, any one to become any other, and -to be subject to no system or law, that is to say, that order does not -exist, and that the points which run _abc_ on each axis may run _bac_, -and so on. - -Then any one of the states of consciousness represented by the points -in the cluster can become any other. We have a representation of a -random consciousness of a certain degree of complexity. - -Now let us examine carefully one particular case of arbitrary -interchange of the points, _a_, _b_, _c_; as one such case, carefully -considered, makes the whole clear. - -[Illustration: Fig. 61.] - -Consider the points named in the figure 1_c_, 2_a_, 3_c_; 1_c_, 2_c_, -3_a_; 1_a_, 2_c_, 3_c_, and examine the effect on them when a change of -order takes place. Let us suppose, for instance, that _a_ changes into -_b_, and let us call the two sets of points we get, the one before and -the one after, their change conjugates. - - Before the change 1_c_ 2_a_ 3_c_ 1_c_ 2_c_ 3_a_ 1_a_ 2_c_ 3_c_}Conjug- - After the change 1_c_ 2_b_ 3_c_ 1_c_ 2_c_ 3_b_ 1_b_ 2_c_ 3_c_} ates. - -The points surrounded by rings represent the conjugate points. - -It is evident that as consciousness, represented first by the first -set of points and afterwards by the second set of points, would have -nothing in common in its two phases. It would not be capable of giving -an account of itself. There would be no identity. - -If, however, we can find any set of points in the cubical cluster, -which, when any arbitrary change takes place in the points on the -axes, or in the axes themselves, repeats itself, is reproduced, then a -consciousness represented by those points would have a permanence. It -would have a principle of identity. Despite the no law, the no order, -of the ultimate constituents, it would have an order, it would form a -system, the condition of a personal identity would be fulfilled. - -The question comes to this, then. Can we find a system of points -which is self-conjugate which is such that when any posit on the axes -becomes any other, or when any axis becomes any other, such a set -is transformed into itself, its identity is not submerged, but rises -superior to the chaos of its constituents? - -[Illustration: Fig. 62.] - -Such a set can be found. Consider the set represented in fig. 62, and -written down in the first of the two lines— - - Self- {1_a_ 2_b_ 3_c_ 1_b_ 2_a_ 3_c_ 1_c_ 2_a_ 3_b_ - conjugate. {1_c_ 2_b_ 3_a_ 1_b_ 2_c_ 3_a_ 1_a_ 2_c_ 3_b_ - - Self- {1_c_ 2_b_ 3_a_ 1_b_ 2_c_ 3_a_ 1_a_ 2_c_ 3_b_ - conjugate. {1_a_ 2_b_ 3_c_ 1_b_ 2_a_ 3_c_ 1_c_ 2_a_ 3_b_ - -If now _a_ change into _c_ and _c_ into _a_, we get the set in the -second line, which has the same members as are in the upper line. -Looking at the diagram we see that it would correspond simply to the -turning of the figures as a whole.[2] Any arbitrary change of the -points on the axes, or of the axes themselves, reproduces the same set. - - [2] These figures are described more fully, and extended, in the next - chapter. - -Thus, a function, by which a random, an unordered, consciousness -could give an ordered and systematic one, can be represented. It -is noteworthy that it is a system of selection. If out of all the -alternative forms that only is attended to which is self-conjugate, -an ordered consciousness is formed. A selection gives a feature of -permanence. - -Can we say that the permanent consciousness is this selection? - -An analogy between Kant and Darwin comes into light. That which is -swings clear of the fleeting, in virtue of its presenting a feature of -permanence. There is no need to suppose any function of “attending to.” -A consciousness capable of giving an account of itself is one which is -characterised by this combination. All combinations exist—of this kind -is the consciousness which can give an account of itself. And the very -duality which we have presupposed may be regarded as originated by a -process of selection. - -Darwin set himself to explain the origin of the fauna and flora of -the world. He denied specific tendencies. He assumed an indefinite -variability—that is, chance—but a chance confined within narrow limits -as regards the magnitude of any consecutive variations. He showed that -organisms possessing features of permanence, if they occurred would be -preserved. So his account of any structure or organised being was that -it possessed features of permanence. - -Kant, undertaking not the explanation of any particular phenomena but -of that which we call nature as a whole, had an origin of species -of his own, an account of the flora and fauna of consciousness. He -denied any specific tendency of the elements of consciousness, but -taking our own consciousness, pointed out that in which it resembled -any consciousness which could survive, which could give an account of -itself. - -He assumes a chance or random world, and as great and small were not -to him any given notions of which he could make use, he did not limit -the chance, the randomness, in any way. But any consciousness which -is permanent must possess certain features—those attributes namely -which give it permanence. Any consciousness like our own is simply a -consciousness which possesses those attributes. The main thing is that -which he calls the unity of apperception, which we have seen above is -simply the statement that a particular set of phases of consciousness -on the basis of complete randomness will be self-conjugate, and so -permanent. - -As with Darwin so with Kant, the reason for existence of any feature -comes to this—show that it tends to the permanence of that which -possesses it. - -We can thus regard Kant as the creator of the first of the modern -evolution theories. And, as is so often the case, the first effort was -the most stupendous in its scope. Kant does not investigate the origin -of any special part of the world, such as its organisms, its chemical -elements, its social communities of men. He simply investigates the -origin of the whole—of all that is included in consciousness, the -origin of that “thought thing” whose progressive realisation is the -knowable universe. - -This point of view is very different from the ordinary one, in which a -man is supposed to be placed in a world like that which he has come to -think of it, and then to learn what he has found out from this model -which he himself has placed on the scene. - -We all know that there are a number of questions in attempting an -answer to which such an assumption is not allowable. - -Mill, for instance, explains our notion of “law” by an invariable -sequence in nature. But what we call nature is something given in -thought. So he explains a thought of law and order by a thought of an -invariable sequence. He leaves the problem where he found it. - -Kant’s theory is not unique and alone. It is one of a number of -evolution theories. A notion of its import and significance can be -obtained by a comparison of it with other theories. - -Thus in Darwin’s theoretical world of natural selection a certain -assumption is made, the assumption of indefinite variability—slight -variability it is true, over any appreciable lapse of time, but -indefinite in the postulated epochs of transformation—and a whole chain -of results is shown to follow. - -This element of chance variation is not, however, an ultimate resting -place. It is a preliminary stage. This supposing the all is a -preliminary step towards finding out what is. If every kind of organism -can come into being, those that do survive will present such and such -characteristics. This is the necessary beginning for ascertaining what -kinds of organisms do come into existence. And so Kant’s hypothesis -of a random consciousness is the necessary beginning for the rational -investigation of consciousness as it is. His assumption supplies, as -it were, the space in which we can observe the phenomena. It gives the -general laws constitutive of any experience. If, on the assumption -of absolute randomness in the constituents, such and such would be -characteristic of the experience, then, whatever the constituents, -these characteristics must be universally valid. - -We will now proceed to examine more carefully the poiograph, -constructed for the purpose of exhibiting an illustration of Kant’s -unity of apperception. - -In order to show the derivation order out of non-order it has been -necessary to assume a principle of duality—we have had the axes and the -posits on the axes—there are two sets of elements, each non-ordered, -and it is in the reciprocal relation of them that the order, the -definite system, originates. - -Is there anything in our experience of the nature of a duality? - -There certainly are objects in our experience which have order and -those which are incapable of order. The two roots of a quadratic -equation have no order. No one can tell which comes first. If a body -rises vertically and then goes at right angles to its former course, -no one can assign any priority to the direction of the north or to -the east. There is no priority in directions of turning. We associate -turnings with no order progressions in a line with order. But in the -axes and points we have assumed above there is no such distinction. -It is the same, whether we assume an order among the turnings, and no -order among the points on the axes, or, _vice versa_, an order in the -points and no order in the turnings. A being with an infinite number of -axes mutually at right angles, with a definite sequence between them -and no sequence between the points on the axes, would be in a condition -formally indistinguishable from that of a creature who, according to an -assumption more natural to us, had on each axis an infinite number of -ordered points and no order of priority amongst the axes. A being in -such a constituted world would not be able to tell which was turning -and which was length along an axis, in order to distinguish between -them. Thus to take a pertinent illustration, we may be in a world -of an infinite number of dimensions, with three arbitrary points on -each—three points whose order is indifferent, or in a world of three -axes of arbitrary sequence with an infinite number of ordered points on -each. We can’t tell which is which, to distinguish it from the other. - -Thus it appears the mode of illustration which we have used is not an -artificial one. There really exists in nature a duality of the kind -which is necessary to explain the origin of order out of no order—the -duality, namely, of dimension and position. Let us use the term group -for that system of points which remains unchanged, whatever arbitrary -change of its constituents takes place. We notice that a group involves -a duality, is inconceivable without a duality. - -Thus, according to Kant, the primary element of experience is the -group, and the theory of groups would be the most fundamental branch -of science. Owing to an expression in the critique the authority of -Kant is sometimes adduced against the assumption of more than three -dimensions to space. It seems to me, however, that the whole tendency -of his theory lies in the opposite direction, and points to a perfect -duality between dimension and position in a dimension. - -If the order and the law we see is due to the conditions of conscious -experience, we must conceive nature as spontaneous, free, subject to no -predication that we can devise, but, however apprehended, subject to -our logic. - -And our logic is simply spatiality in the general sense—that resultant -of a selection of the permanent from the unpermanent, the ordered from -the unordered, by the means of the group and its underlying duality. - -We can predicate nothing about nature, only about the way in which -we can apprehend nature. All that we can say is that all that which -experience gives us will be conditioned as spatial, subject to our -logic. Thus, in exploring the facts of geometry from the simplest -logical relations to the properties of space of any number of -dimensions, we are merely observing ourselves, becoming aware of the -conditions under which we must perceive. Do any phenomena present -themselves incapable of explanation under the assumption of the space -we are dealing with, then we must habituate ourselves to the conception -of a higher space, in order that our logic may be equal to the task -before us. - -We gain a repetition of the thought that came before, experimentally -suggested. If the laws of the intellectual comprehension of nature are -those derived from considering her as absolute chance, subject to no -law save that derived from a process of selection, then, perhaps, the -order of nature requires different faculties from the intellectual to -apprehend it. The source and origin of ideas may have to be sought -elsewhere than in reasoning. - -The total outcome of the critique is to leave the ordinary man just -where he is, justified in his practical attitude towards nature, -liberated from the fetters of his own mental representations. - -The truth of a picture lies in its total effect. It is vain to seek -information about the landscape from an examination of the pigments. -And in any method of thought it is the complexity of the whole that -brings us to a knowledge of nature. Dimensions are artificial enough, -but in the multiplicity of them we catch some breath of nature. - -We must therefore, and this seems to me the practical conclusion of the -whole matter, proceed to form means of intellectual apprehension of a -greater and greater degree of complexity, both dimensionally and in -extent in any dimension. Such means of representation must always be -artificial, but in the multiplicity of the elements with which we deal, -however incipiently arbitrary, lies our chance of apprehending nature. - -And as a concluding chapter to this part of the book, I will extend -the figures, which have been used to represent Kant’s theory, two -steps, so that the reader may have the opportunity of looking at a -four-dimensional figure which can be delineated without any of the -special apparatus, to the consideration of which I shall subsequently -pass on. - - - - - CHAPTER X - - A FOUR-DIMENSIONAL FIGURE - - -The method used in the preceding chapter to illustrate the problem -of Kant’s critique, gives a singularly easy and direct mode of -constructing a series of important figures in any number of dimensions. - -We have seen that to represent our space a plane being must give up one -of his axes, and similarly to represent the higher shapes we must give -up one amongst our three axes. - -But there is another kind of giving up which reduces the construction -of higher shapes to a matter of the utmost simplicity. - -Ordinarily we have on a straight line any number of positions. The -wealth of space in position is illimitable, while there are only three -dimensions. - -I propose to give up this wealth of positions, and to consider the -figures obtained by taking just as many positions as dimensions. - -In this way I consider dimensions and positions as two “kinds,” and -applying the simple rule of selecting every one of one kind with every -other of every other kind, get a series of figures which are noteworthy -because they exactly fill space of any number of dimensions (as the -hexagon fills a plane) by equal repetitions of themselves. - -The rule will be made more evident by a simple application. - -Let us consider one dimension and one position. I will call the axis -_i_, and the position _o_. - - ———————————————-_i_ - _o_ - -Here the figure is the position _o_ on the line _i_. Take now two -dimensions and two positions on each. - -[Illustration: Fig. 63.] - -We have the two positions _o_; 1 on _i_, and the two positions _o_, 1 -on _j_, fig. 63. These give rise to a certain complexity. I will let -the two lines _i_ and _j_ meet in the position I call _o_ on each, and -I will consider _i_ as a direction starting equally from every position -on _j_, and _j_ as starting equally from every position on _i_. We thus -obtain the following figure:—A is both _oi_ and _oj_, B is 1_i_ and -_oj_, and so on as shown in fig. 63_b_. The positions on AC are all -_oi_ positions. They are, if we like to consider it in that way, points -at no distance in the _i_ direction from the line AC. We can call the -line AC the _oi_ line. Similarly the points on AB are those no distance -from AB in the _j_ direction, and we can call them _oj_ points and the -line AB the _oj_ line. Again, the line CD can be called the 1_j_ line -because the points on it are at a distance, 1 in the _j_ direction. - -[Illustration: Fig. 63_b_.] - -We have then four positions or points named as shown, and, considering -directions and positions as “kinds,” we have the combination of two -kinds with two kinds. Now, selecting every one of one kind with every -other of every other kind will mean that we take 1 of the kind _i_ and -with it _o_ of the kind _j_; and then, that we take _o_ of the kind _i_ -and with it 1 of the kind _j_. - -Thus we get a pair of positions lying in the straight line BC, fig. -64. We can call this pair 10 and 01 if we adopt the plan of mentally, -adding an _i_ to the first and a _j_ to the second of the symbols -written thus—01 is a short expression for O_i_, 1_j_. - -[Illustration: Fig. 64.] - -Coming now to our space, we have three dimensions, so we take three -positions on each. These positions I will suppose to be at equal -distances along each axis. The three axes and the three positions on -each are shown in the accompanying diagrams, fig. 65, of which the -first represents a cube with the front faces visible, the second the -rear faces of the same cube; the positions I will call 0, 1, 2; the -axes, _i_, _j_, _k_. I take the base ABC as the starting place, from -which to determine distances in the _k_ direction, and hence every -point in the base ABC will be an _ok_ position, and the base ABC can be -called an _ok_ plane. - -[Illustration: Fig. 65.] - -In the same way, measuring the distances from the face ADC, we see -that every position in the face ADC is an _oi_ position, and the whole -plane of the face may be called an _oi_ plane. Thus we see that with -the introduction of a new dimension the signification of a compound -symbol, such as “_oi_,” alters. In the plane it meant the line AC. In -space it means the whole plane ACD. - -Now, it is evident that we have twenty-seven positions, each of them -named. If the reader will follow this nomenclature in respect of the -positions marked in the figures he will have no difficulty in assigning -names to each one of the twenty-seven positions. A is _oi_, _oj_, _ok_. -It is at the distance 0 along _i_, 0 along _j_, 0 along _k_, and _io_ -can be written in short 000, where the _ijk_ symbols are omitted. - -The point immediately above is 001, for it is no distance in the _i_ -direction, and a distance of 1 in the _k_ direction. Again, looking at -B, it is at a distance of 2 from A, or from the plane ADC, in the _i_ -direction, 0 in the _j_ direction from the plane ABD, and 0 in the _k_ -direction, measured from the plane ABC. Hence it is 200 written for -2_i_, 0_j_, 0_k_. - -Now, out of these twenty-seven “things” or compounds of position and -dimension, select those which are given by the rule, every one of one -kind with every other of every other kind. - -Take 2 of the _i_ kind. With this we must have a 1 of the _j_ kind, and -then by the rule we can only have a 0 of the _k_ kind, for if we had -any other of the _k_ kind we should repeat one of the kinds we already -had. In 2_i_, 1_j_, 1_k_, for instance, 1 is repeated. The point we -obtain is that marked 210, fig. 66. - -[Illustration: Fig. 66.] - -Proceeding in this way, we pick out the following cluster of points, -fig. 67. They are joined by lines, dotted where they are hidden by the -body of the cube, and we see that they form a figure—a hexagon which -could be taken out of the cube and placed on a plane. It is a figure -which will fill a plane by equal repetitions of itself. The plane being -representing this construction in his plane would take three squares to -represent the cube. Let us suppose that he takes the _ij_ axes in his -space and _k_ represents the axis running out of his space, fig. 68. -In each of the three squares shown here as drawn separately he could -select the points given by the rule, and he would then have to try to -discover the figure determined by the three lines drawn. The line from -210 to 120 is given in the figure, but the line from 201 to 102 or GK -is not given. He can determine GK by making another set of drawings and -discovering in them what the relation between these two extremities is. - -[Illustration: Fig. 67.] - -[Illustration: Fig. 68.] - -[Illustration: Fig. 69.] - -Let him draw the _i_ and _k_ axes in his plane, fig. 69. The _j_ axis -then runs out and he has the accompanying figure. In the first of these -three squares, fig. 69, he can pick out by the rule the two points -201, 102—G, and K. Here they occur in one plane and he can measure the -distance between them. In his first representation they occur at G and -K in separate figures. - -Thus the plane being would find that the ends of each of the lines was -distant by the diagonal of a unit square from the corresponding end -of the last and he could then place the three lines in their right -relative position. Joining them he would have the figure of a hexagon. - -[Illustration: Fig. 70.] - -We may also notice that the plane being could make a representation of -the whole cube simultaneously. The three squares, shown in perspective -in fig. 70, all lie in one plane, and on these the plane being could -pick out any selection of points just as well as on three separate -squares. He would obtain a hexagon by joining the points marked. This -hexagon, as drawn, is of the right shape, but it would not be so if -actual squares were used instead of perspective, because the relation -between the separate squares as they lie in the plane figure is not -their real relation. The figure, however, as thus constructed, would -give him an idea of the correct figure, and he could determine it -accurately by remembering that distances in each square were correct, -but in passing from one square to another their distance in the third -dimension had to be taken into account. - -Coming now to the figure made by selecting according to our rule from -the whole mass of points given by four axes and four positions in each, -we must first draw a catalogue figure in which the whole assemblage is -shown. - -We can represent this assemblage of points by four solid figures. The -first giving all those positions which are at a distance O from our -space in the fourth dimension, the second showing all those that are at -a distance 1, and so on. - -These figures will each be cubes. The first two are drawn showing the -front faces, the second two the rear faces. We will mark the points 0, -1, 2, 3, putting points at those distances along each of these axes, -and suppose all the points thus determined to be contained in solid -models of which our drawings in fig. 71 are representatives. Here we -notice that as on the plane 0_i_ meant the whole line from which the -distances in the _i_ direction was measured, and as in space 0_i_ -means the whole plane from which distances in the _i_ direction are -measured, so now 0_h_ means the whole space in which the first cube -stands—measuring away from that space by a distance of one we come to -the second cube represented. - -[Illustration: Fig. 71.] - -Now selecting according to the rule every one of one kind with every -other of every other kind, we must take, for instance, 3_i_, 2_j_, -1_k_, 0_h_. This point is marked 3210 at the lower star in the figure. -It is 3 in the _i_ direction, 2 in the _j_ direction, 1 in the _k_ -direction, 0 in the _h_ direction. - -With 3_i_ we must also take 1_j_, 2_k_, 0_h_. This point is shown by -the second star in the cube 0_h_. - -[Illustration: Fig. 72.] - -In the first cube, since all the points are 0_h_ points, we can only -have varieties in which _i_, _j_, _k_, are accompanied by 3, 2, 1. - -The points determined are marked off in the diagram fig. 72, and lines -are drawn joining the adjacent pairs in each figure, the lines being -dotted when they pass within the substance of the cube in the first two -diagrams. - -Opposite each point, on one side or the other of each cube, is written -its name. It will be noticed that the figures are symmetrical right and -left; and right and left the first two numbers are simply interchanged. - -Now this being our selection of points, what figure do they make when -all are put together in their proper relative positions? - -To determine this we must find the distance between corresponding -corners of the separate hexagons. - -[Illustration: Fig. 73.] - -To do this let us keep the axes _i_, _j_, in our space, and draw _h_ -instead of _k_, letting _k_ run out in the fourth dimension, fig. 73. - -Here we have four cubes again, in the first of which all the points are -0_k_ points; that is, points at a distance zero in the _k_ direction -from the space of the three dimensions _ijh_. We have all the points -selected before, and some of the distances, which in the last diagram -led from figure to figure are shown here in the same figure, and so -capable of measurement. Take for instance the points 3120 to 3021, -which in the first diagram (fig. 72) lie in the first and second -figures. Their actual relation is shown in fig. 73 in the cube marked -2K, where the points in question are marked with a *. We see that the -distance in question is the diagonal of a unit square. In like manner -we find that the distance between corresponding points of any two -hexagonal figures is the diagonal of a unit square. The total figure -is now easily constructed. An idea of it may be gained by drawing all -the four cubes in the catalogue figure in one (fig. 74). These cubes -are exact repetitions of one another, so one drawing will serve as a -representation of the whole series, if we take care to remember where -we are, whether in a 0_h_, a 1_h_, a 2_h_, or a 3_h_ figure, when we -pick out the points required. Fig. 74 is a representation of all the -catalogue cubes put in one. For the sake of clearness the front faces -and the back faces of this cube are represented separately. - -[Illustration: Fig. 74.] - -The figure determined by the selected points is shown below. - -In putting the sections together some of the outlines in them -disappear. The line TW for instance is not wanted. - -We notice that PQTW and TWRS are each the half of a hexagon. Now QV and -VR lie in one straight line. Hence these two hexagons fit together, -forming one hexagon, and the line TW is only wanted when we consider a -section of the whole figure, we thus obtain the solid represented in -the lower part of fig. 74. Equal repetitions of this figure, called a -tetrakaidecagon, will fill up three-dimensional space. - -To make the corresponding four-dimensional figure we have to take five -axes mutually at right angles with five points on each. A catalogue of -the positions determined in five-dimensional space can be found thus. - -Take a cube with five points on each of its axes, the fifth point is -at a distance of four units of length from the first on any one of -the axes. And since the fourth dimension also stretches to a distance -of four we shall need to represent the successive sets of points at -distances 0, 1, 2, 3, 4, in the fourth dimensions, five cubes. Now -all of these extend to no distance at all in the fifth dimension. To -represent what lies in the fifth dimension we shall have to draw, -starting from each of our cubes, five similar cubes to represent the -four steps on in the fifth dimension. By this assemblage we get a -catalogue of all the points shown in fig. 75, in which _L_ represents -the fifth dimension. - -[Illustration: Fig. 75.] - -Now, as we saw before, there is nothing to prevent us from putting all -the cubes representing the different stages in the fourth dimension in -one figure, if we take note when we look at it, whether we consider -it as a 0_h_, a 1_h_, a 2_h_, etc., cube. Putting then the 0_h_, 1_h_, -2_h_, 3_h_, 4_h_ cubes of each row in one, we have five cubes with the -sides of each containing five positions, the first of these five cubes -represents the 0_l_ points, and has in it the _i_ points from 0 to 4, -the _j_ points from 0 to 4, the _k_ points from 0 to 4, while we have -to specify with regard to any selection we make from it, whether we -regard it as a 0_h_, a 1_h_, a 2_h_, a 3_h_, or a 4_h_ figure. In fig. -76 each cube is represented by two drawings, one of the front part, the -other of the rear part. - -Let then our five cubes be arranged before us and our selection be made -according to the rule. Take the first figure in which all points are -0_l_ points. We cannot have 0 with any other letter. Then, keeping in -the first figure, which is that of the 0_l_ positions, take first of -all that selection which always contains 1_h_. We suppose, therefore, -that the cube is a 1_h_ cube, and in it we take _i_, _j_, _k_ in -combination with 4, 3, 2 according to the rule. - -The figure we obtain is a hexagon, as shown, the one in front. The -points on the right hand have the same figures as those on the left, -with the first two numerals interchanged. Next keeping still to the -0_l_ figure let us suppose that the cube before us represents a section -at a distance of 2 in the _h_ direction. Let all the points in it be -considered as 2_h_ points. We then have a 0_l_, 2_h_ region, and have -the sets _ijk_ and 431 left over. We must then pick out in accordance -with our rule all such points as 4_i_, 3_j_, 1_k_. - -These are shown in the figure and we find that we can draw them without -confusion, forming the second hexagon from the front. Going on in this -way it will be seen that in each of the five figures a set of hexagons -is picked out, which put together form a three-space figure something -like the tetrakaidecagon. - -[Illustration: Fig. 76.] - -These separate figures are the successive stages in which the whole -four-dimensional figure in which they cohere can be apprehended. - -The first figure and the last are tetrakaidecagons. These are two -of the solid boundaries of the figure. The other solid boundaries -can be traced easily. Some of them are complete from one face in the -figure to the corresponding face in the next, as for instance the -solid which extends from the hexagonal base of the first figure to the -equal hexagonal base of the second figure. This kind of boundary is a -hexagonal prism. The hexagonal prism also occurs in another sectional -series, as for instance, in the square at the bottom of the first -figure, the oblong at the base of the second and the square at the base -of the third figure. - -Other solid boundaries can be traced through four of the five sectional -figures. Thus taking the hexagon at the top of the first figure we -find in the next a hexagon also, of which some alternate sides are -elongated. The top of the third figure is also a hexagon with the other -set of alternate rules elongated, and finally we come in the fourth -figure to a regular hexagon. - -These four sections are the sections of a tetrakaidecagon as can -be recognised from the sections of this figure which we have had -previously. Hence the boundaries are of two kinds, hexagonal prisms and -tetrakaidecagons. - -These four-dimensional figures exactly fill four-dimensional space by -equal repetitions of themselves. - - - - - CHAPTER XI - -NOMENCLATURE AND ANALOGIES PRELIMINARY TO THE STUDY OF FOUR-DIMENSIONAL - FIGURES - - -In the following pages a method of designating different regions of -space by a systematic colour scheme has been adopted. The explanations -have been given in such a manner as to involve no reference to models, -the diagrams will be found sufficient. But to facilitate the study a -description of a set of models is given in an appendix which the reader -can either make for himself or obtain. If models are used the diagrams -in Chapters XI. and XII. will form a guide sufficient to indicate their -use. Cubes of the colours designated by the diagrams should be picked -out and used to reinforce the diagrams. The reader, in the following -description, should suppose that a board or wall stretches away from -him, against which the figures are placed. - -[Illustration: Fig. 77.] - -Take a square, one of those shown in Fig. 77 and give it a neutral -colour, let this colour be called “null,” and be such that it makes no -appreciable difference to any colour with which it mixed. If there is -no such real colour let us imagine such a colour, and assign to it the -properties of the number zero, which makes no difference in any number -to which it is added. - -Above this square place a red square. Thus we symbolise the going up by -adding red to null. - -Away from this null square place a yellow square, and represent going -away by adding yellow to null. - -To complete the figure we need a fourth square. Colour this orange, -which is a mixture of red and yellow, and so appropriately represents a -going in a direction compounded of up and away. We have thus a colour -scheme which will serve to name the set of squares drawn. We have two -axes of colours—red and yellow—and they may occupy as in the figure -the direction up and away, or they may be turned about; in any case -they enable us to name the four squares drawn in their relation to one -another. - -Now take, in Fig. 78, nine squares, and suppose that at the end of the -going in any direction the colour started with repeats itself. - -[Illustration: Fig. 78.] - -We obtain a square named as shown. - -Let us now, in fig. 79, suppose the number of squares to be increased, -keeping still to the principle of colouring already used. - -Here the nulls remain four in number. There are three reds between the -first null and the null above it, three yellows between the first null -and the null beyond it, while the oranges increase in a double way. - -[Illustration: Fig. 79.] - -Suppose this process of enlarging the number of the squares to be -indefinitely pursued and the total figure obtained to be reduced in -size, we should obtain a square of which the interior was all orange, -while the lines round it were red and yellow, and merely the points -null colour, as in fig. 80. Thus all the points, lines, and the area -would have a colour. - -[Illustration: Fig. 80.] - -We can consider this scheme to originate thus:—Let a null point move -in a yellow direction and trace out a yellow line and end in a null -point. Then let the whole line thus traced move in a red direction. The -null points at the ends of the line will produce red lines, and end in -null points. The yellow line will trace out a yellow and red, or orange -square. - -Now, turning back to fig. 78, we see that these two ways of naming, the -one we started with and the one we arrived at, can be combined. - -By its position in the group of four squares, in fig. 77, the null -square has a relation to the yellow and to the red directions. We can -speak therefore of the red line of the null square without confusion, -meaning thereby the line AB, fig. 81, which runs up from the initial -null point A in the figure as drawn. The yellow line of the null square -is its lower horizontal line AC as it is situated in the figure. - -[Illustration: Fig. 81.] - -If we wish to denote the upper yellow line BD, fig. 81, we can speak -of it as the yellow γ line, meaning the yellow line which is separated -from the primary yellow line by the red movement. - -In a similar way each of the other squares has null points, red and -yellow lines. Although the yellow square is all yellow, its line CD, -for instance, can be referred to as its red line. - -This nomenclature can be extended. - -If the eight cubes drawn, in fig. 82, are put close together, as on -the right hand of the diagram, they form a cube, and in them, as thus -arranged, a going up is represented by adding red to the zero, or -null colour, a going away by adding yellow, a going to the right by -adding white. White is used as a colour, as a pigment, which produces -a colour change in the pigments with which it is mixed. From whatever -cube of the lower set we start, a motion up brings us to a cube showing -a change to red, thus light yellow becomes light yellow red, or light -orange, which is called ochre. And going to the right from the null on -the left we have a change involving the introduction of white, while -the yellow change runs from front to back. There are three colour -axes—the red, the white, the yellow—and these run in the position the -cubes occupy in the drawing—up, to the right, away—but they could be -turned about to occupy any positions in space. - -[Illustration: Fig. 82.] - -[Illustration: Fig. 83. The three layers.] - -We can conveniently represent a block of cubes by three sets of -squares, representing each the base of a cube. - -Thus the block, fig. 83, can be represented by the layers on the -right. Here, as in the case of the plane, the initial colours repeat -themselves at the end of the series. - -Proceeding now to increase the number of the cubes we obtain fig. 84, -in which the initial letters of the colours are given instead of their -full names. - -Here we see that there are four null cubes as before, but the series -which spring from the initial corner will tend to become lines of -cubes, as also the sets of cubes parallel to them, starting from other -corners. Thus, from the initial null springs a line of red cubes, a -line of white cubes, and a line of yellow cubes. - -If the number of the cubes is largely increased, and the size of the -whole cube is diminished, we get a cube with null points, and the edges -coloured with these three colours. - -[Illustration: Fig. 84.] - -The light yellow cubes increase in two ways, forming ultimately a sheet -of cubes, and the same is true of the orange and pink sets. Hence, -ultimately the cube thus formed would have red, white, and yellow -lines surrounding pink, orange, and light yellow faces. The ochre cubes -increase in three ways, and hence ultimately the whole interior of the -cube would be coloured ochre. - -We have thus a nomenclature for the points, lines, faces, and solid -content of a cube, and it can be named as exhibited in fig. 85. - -[Illustration: Fig. 85.] - -We can consider the cube to be produced in the following way. A null -point moves in a direction to which we attach the colour indication -yellow; it generates a yellow line and ends in a null point. The yellow -line thus generated moves in a direction to which we give the colour -indication red. This lies up in the figure. The yellow line traces out -a yellow, red, or orange square, and each of its null points trace out -a red line, and ends in a null point. - -This orange square moves in a direction to which we attribute the -colour indication white, in this case the direction is the right. The -square traces out a cube coloured orange, red, or ochre, the red lines -trace out red to white or pink squares, and the yellow lines trace out -light yellow squares, each line ending in a line of its own colour. -While the points each trace out a null + white, or white line to end in -a null point. - -Now returning to the first block of eight cubes we can name each point, -line, and square in them by reference to the colour scheme, which they -determine by their relation to each other. - -Thus, in fig. 86, the null cube touches the red cube by a light yellow -square; it touches the yellow cube by a pink square, and touches the -white cube by an orange square. - -There are three axes to which the colours red, yellow, and white are -assigned, the faces of each cube are designated by taking these colours -in pairs. Taking all the colours together we get a colour name for the -solidity of a cube. - -[Illustration: Fig. 86.] - -Let us now ask ourselves how the cube could be presented to the plane -being. Without going into the question of how he could have a real -experience of it, let us see how, if we could turn it about and show it -to him, he, under his limitations, could get information about it. If -the cube were placed with its red and yellow axes against a plane, that -is resting against it by its orange face, the plane being would observe -a square surrounded by red and yellow lines, and having null points. -See the dotted square, fig. 87. - -[Illustration: Fig. 87.] - -We could turn the cube about the red line so that a different face -comes into juxtaposition with the plane. - -Suppose the cube turned about the red line. As it is turning from its -first position all of it except the red line leaves the plane—goes -absolutely out of the range of the plane being’s apprehension. But when -the yellow line points straight out from the plane then the pink face -comes into contact with it. Thus the same red line remaining as he saw -it at first, now towards him comes a face surrounded by white and red -lines. - -If we call the direction to the right the unknown direction, then -the line he saw before, the yellow line, goes out into this unknown -direction, and the line which before went into the unknown direction, -comes in. It comes in in the opposite direction to that in which the -yellow line ran before; the interior of the face now against the plane -is pink. It is a property of two lines at right angles that, if one -turns out of a given direction and stands at right angles to it, then -the other of the two lines comes in, but runs the opposite way in that -given direction, as in fig. 88. - -[Illustration: Fig. 88.] - -Now these two presentations of the cube would seem, to the plane -creature like perfectly different material bodies, with only that line -in common in which they both meet. - -Again our cube can be turned about the yellow line. In this case the -yellow square would disappear as before, but a new square would come -into the plane after the cube had rotated by an angle of 90° about this -line. The bottom square of the cube would come in thus in figure 89. -The cube supposed in contact with the plane is rotated about the lower -yellow line and then the bottom face is in contact with the plane. - -Here, as before, the red line going out into the unknown dimension, -the white line which before ran in the unknown dimension would come -in downwards in the opposite sense to that in which the red line ran -before. - -[Illustration: Fig. 89.] - -Now if we use _i_, _j_, _k_, for the three space directions, _i_ left -to right, _j_ from near away, _k_ from below up; then, using the colour -names for the axes, we have that first of all white runs _i_, yellow -runs _j_, red runs _k_; then after the first turning round the _k_ -axis, white runs negative _j_, yellow runs _i_, red runs _k_; thus we -have the table:— - - _i_ _j_ _k_ - 1st position white yellow red - 2nd position yellow white— red - 3rd position red yellow white— - -Here white with a negative sign after it in the column under _j_ means -that white runs in the negative sense of the _j_ direction. - -We may express the fact in the following way:— In the plane there is -room for two axes while the body has three. Therefore in the plane we -can represent any two. If we want to keep the axis that goes in the -unknown dimension always running in the positive sense, then the axis -which originally ran in the unknown dimension (the white axis) must -come in in the negative sense of that axis which goes out of the plane -into the unknown dimension. - -It is obvious that the unknown direction, the direction in which the -white line runs at first, is quite distinct from any direction which -the plane creature knows. The white line may come in towards him, or -running down. If he is looking at a square, which is the face of a cube -(looking at it by a line), then any one of the bounding lines remaining -unmoved, another face of the cube may come in, any one of the faces, -namely, which have the white line in them. And the white line comes -sometimes in one of the space directions he knows, sometimes in another. - -Now this turning which leaves a line unchanged is something quite -unlike any turning he knows in the plane. In the plane a figure turns -round a point. The square can turn round the null point in his plane, -and the red and yellow lines change places, only of course, as with -every rotation of lines at right angles, if red goes where yellow went, -yellow comes in negative of red’s old direction. - -This turning, as the plane creature conceives it, we should call -turning about an axis perpendicular to the plane. What he calls turning -about the null point we call turning about the white line as it stands -out from his plane. There is no such thing as turning about a point, -there is always an axis, and really much more turns than the plane -being is aware of. - -Taking now a different point of view, let us suppose the cubes to be -presented to the plane being by being passed transverse to his plane. -Let us suppose the sheet of matter over which the plane being and all -objects in his world slide, to be of such a nature that objects can -pass through it without breaking it. Let us suppose it to be of the -same nature as the film of a soap bubble, so that it closes around -objects pushed through it, and, however the object alters its shape as -it passes through it, let us suppose this film to run up to the contour -of the object in every part, maintaining its plane surface unbroken. - -Then we can push a cube or any object through the film and the plane -being who slips about in the film will know the contour of the cube -just and exactly where the film meets it. - -[Illustration: Fig. 90.] - -Fig. 90 represents a cube passing through a plane film. The plane being -now comes into contact with a very thin slice of the cube somewhere -between the left and right hand faces. This very thin slice he thinks -of as having no thickness, and consequently his idea of it is what we -call a section. It is bounded by him by pink lines front and back, -coming from the part of the pink face he is in contact with, and above -and below, by light yellow lines. Its corners are not null-coloured -points, but white points, and its interior is ochre, the colour of the -interior of the cube. - -If now we suppose the cube to be an inch in each dimension, and to pass -across, from right to left, through the plane, then we should explain -the appearances presented to the plane being by saying: First of all -you have the face of a cube, this lasts only a moment; then you have a -figure of the same shape but differently coloured. This, which appears -not to move to you in any direction which you know of, is really moving -transverse to your plane world. Its appearance is unaltered, but each -moment it is something different—a section further on, in the white, -the unknown dimension. Finally, at the end of the minute, a face comes -in exactly like the face you first saw. This finishes up the cube—it is -the further face in the unknown dimension. - -The white line, which extends in length just like the red or the -yellow, you do not see as extensive; you apprehend it simply as an -enduring white point. The null point, under the condition of movement -of the cube, vanishes in a moment, the lasting white point is really -your apprehension of a white line, running in the unknown dimension. -In the same way the red line of the face by which the cube is first in -contact with the plane lasts only a moment, it is succeeded by the pink -line, and this pink line lasts for the inside of a minute. This lasting -pink line in your apprehension of a surface, which extends in two -dimensions just like the orange surface extends, as you know it, when -the cube is at rest. - -But the plane creature might answer, “This orange object is substance, -solid substance, bounded completely and on every side.” - -Here, of course, the difficulty comes in. His solid is our surface—his -notion of a solid is our notion of an abstract surface with no -thickness at all. - -We should have to explain to him that, from every point of what he -called a solid, a new dimension runs away. From every point a line -can be drawn in a direction unknown to him, and there is a solidity -of a kind greater than that which he knows. This solidity can only -be realised by him by his supposing an unknown direction, by motion -in which what he conceives to be solid matter instantly disappears. -The higher solid, however, which extends in this dimension as well -as in those which he knows, lasts when a motion of that kind takes -place, different sections of it come consecutively in the plane -of his apprehension, and take the place of the solid which he at -first conceives to be all. Thus, the higher solid—our solid in -contradistinction to his area solid, his two-dimensional solid, must -be conceived by him as something which has duration in it, under -circumstances in which his matter disappears out of his world. - -We may put the matter thus, using the conception of motion. - -A null point moving in a direction away generates a yellow line, and -the yellow line ends in a null point. We suppose, that is, a point -to move and mark out the products of this motion in such a manner. -Now suppose this whole line as thus produced to move in an upward -direction; it traces out the two-dimensional solid, and the plane being -gets an orange square. The null point moves in a red line and ends in -a null point, the yellow line moves and generates an orange square and -ends in a yellow line, the farther null point generates a red line and -ends in a null point. Thus, by movement in two successive directions -known to him, he can imagine his two-dimensional solid produced with -all its boundaries. - -Now we tell him: “This whole two-dimensional solid can move in a third -or unknown dimension to you. The null point moving in this dimension -out of your world generates a white line and ends in a null point. The -yellow line moving generates a light yellow two-dimensional solid and -ends in a yellow line, and this two-dimensional solid, lying end on to -your plane world, is bounded on the far side by the other yellow line. -In the same way each of the lines surrounding your square traces out an -area, just like the orange area you know. But there is something new -produced, something which you had no idea of before; it is that which -is produced by the movement of the orange square. That, than which you -can imagine nothing more solid, itself moves in a direction open to it -and produces a three-dimensional solid. Using the addition of white -to symbolise the products of this motion this new kind of solid will -be light orange or ochre, and it will be bounded on the far side by -the final position of the orange square which traced it out, and this -final position we suppose to be coloured like the square in its first -position, orange with yellow and red boundaries and null corners.” - -This product of movement, which it is so easy for us to describe, would -be difficult for him to conceive. But this difficulty is connected -rather with its totality than with any particular part of it. - -Any line, or plane of this, to him higher, solid we could show to him, -and put in his sensible world. - -We have already seen how the pink square could be put in his world by -a turning of the cube about the red line. And any section which we can -conceive made of the cube could be exhibited to him. You have simply to -turn the cube and push it through, so that the plane of his existence -is the plane which cuts out the given section of the cube, then the -section would appear to him as a solid. In his world he would see the -contour, get to any part of it by digging down into it. - - - THE PROCESS BY WHICH A PLANE BEING WOULD GAIN A NOTION OF A SOLID. - -If we suppose the plane being to have a general idea of the existence -of a higher solid—our solid—we must next trace out in detail the -method, the discipline, by which he would acquire a working familiarity -with our space existence. The process begins with an adequate -realisation of a simple solid figure. For this purpose we will suppose -eight cubes forming a larger cube, and first we will suppose each cube -to be coloured throughout uniformly. Let the cubes in fig. 91 be the -eight making a larger cube. - -[Illustration: Fig. 91.] - -Now, although each cube is supposed to be coloured entirely through -with the colour, the name of which is written on it, still we can -speak of the faces, edges, and corners of each cube as if the colour -scheme we have investigated held for it. Thus, on the null cube we can -speak of a null point, a red line, a white line, a pink face, and so -on. These colour designations are shown on No. 1 of the views of the -tesseract in the plate. Here these colour names are used simply in -their geometrical significance. They denote what the particular line, -etc., referred to would have as its colour, if in reference to the -particular cube the colour scheme described previously were carried out. - -If such a block of cubes were put against the plane and then passed -through it from right to left, at the rate of an inch a minute, each -cube being an inch each way, the plane being would have the following -appearances:— - -First of all, four squares null, yellow, red, orange, lasting each a -minute; and secondly, taking the exact places of these four squares, -four others, coloured white, light yellow, pink, ochre. Thus, to make -a catalogue of the solid body, he would have to put side by side in -his world two sets of four squares each, as in fig. 92. The first are -supposed to last a minute, and then the others to come in in place of -them, and also last a minute. - -[Illustration: Fig. 92.] - -In speaking of them he would have to denote what part of the respective -cube each square represents. Thus, at the beginning he would have null -cube orange face, and after the motion had begun he would have null -cube ochre section. As he could get the same coloured section whichever -way the cube passed through, it would be best for him to call this -section white section, meaning that it is transverse to the white axis. -These colour-names, of course, are merely used as names, and do not -imply in this case that the object is really coloured. Finally, after -a minute, as the first cube was passing beyond his plane he would have -null cube orange face again. - -The same names will hold for each of the other cubes, describing what -face or section of them the plane being has before him; and the second -wall of cubes will come on, continue, and go out in the same manner. In -the area he thus has he can represent any movement which we carry out -in the cubes, as long as it does not involve a motion in the direction -of the white axis. The relation of parts that succeed one another in -the direction of the white axis is realised by him as a consecution of -states. - -Now, his means of developing his space apprehension lies in this, that -that which is represented as a time sequence in one position of the -cubes, can become a real co-existence, _if something that has a real -co-existence becomes a time sequence_. - -We must suppose the cubes turned round each of the axes, the red line, -and the yellow line, then something, which was given as time before, -will now be given as the plane creature’s space; something, which was -given as space before, will now be given as a time series as the cube -is passed through the plane. - -The three positions in which the cubes must be studied are the one -given above and the two following ones. In each case the original null -point which was nearest to us at first is marked by an asterisk. In -figs. 93 and 94 the point marked with a star is the same in the cubes -and in the plane view. - -[Illustration: Fig. 93. The cube swung round the red line, so that the -white line points towards us.] - -In fig. 93 the cube is swung round the red line so as to point towards -us, and consequently the pink face comes next to the plane. As it -passes through there are two varieties of appearance designated by -the figures 1 and 2 in the plane. These appearances are named in the -figure, and are determined by the order in which the cubes come in the -motion of the whole block through the plane. - -With regard to these squares severally, however, different names must -be used, determined by their relations in the block. - -Thus, in fig. 93, when the cube first rests against the plane the null -cube is in contact by its pink face; as the block passes through we get -an ochre section of the null cube, but this is better called a yellow -section, as it is made by a plane perpendicular to the yellow line. -When the null cube has passed through the plane, as it is leaving it, -we get again a pink face. - -[Illustration: Fig. 94. The cube swung round yellow line, with red line -running from left to right, and white line running down.] - -The same series of changes take place with the cube appearances which -follow on those of the null cube. In this motion the yellow cube -follows on the null cube, and the square marked yellow in 2 in the -plane will be first “yellow pink face,” then “yellow yellow section,” -then “yellow pink face.” - -In fig. 94, in which the cube is turned about the yellow line, we have -a certain difficulty, for the plane being will find that the position -his squares are to be placed in will lie below that which they first -occupied. They will come where the support was on which he stood his -first set of squares. He will get over this difficulty by moving his -support. - -Then, since the cubes come upon his plane by the light yellow face, he -will have, taking the null cube as before for an example, null, light -yellow face; null, red section, because the section is perpendicular -to the red line; and finally, as the null cube leaves the plane, null, -light yellow face. Then, in this case red following on null, he will -have the same series of views of the red as he had of the null cube. - -[Illustration: Fig. 95.] - -There is another set of considerations which we will briefly allude to. - -Suppose there is a hollow cube, and a string is stretched across it -from null to null, _r_, _y_, _wh_, as we may call the far diagonal -point, how will this string appear to the plane being as the cube moves -transverse to his plane? - -Let us represent the cube as a number of sections, say 5, corresponding -to 4 equal divisions made along the white line perpendicular to it. - -We number these sections 0, 1, 2, 3, 4, corresponding to the distances -along the white line at which they are taken, and imagine each section -to come in successively, taking the place of the preceding one. - -These sections appear to the plane being, counting from the first, to -exactly coincide each with the preceding one. But the section of the -string occupies a different place in each to that which it does in the -preceding section. The section of the string appears in the position -marked by the dots. Hence the slant of the string appears as a motion -in the frame work marked out by the cube sides. If we suppose the -motion of the cube not to be recognised, then the string appears to the -plane being as a moving point. Hence extension on the unknown dimension -appears as duration. Extension sloping in the unknown direction appears -as continuous movement. - - - - - CHAPTER XII - - THE SIMPLEST FOUR-DIMENSIONAL SOLID - - -A plane being, in learning to apprehend solid existence, must first -of all realise that there is a sense of direction altogether wanting -to him. That which we call right and left does not exist in his -perception. He must assume a movement in a direction, and a distinction -of positive and negative in that direction, which has no reality -corresponding to it in the movements he can make. This direction, this -new dimension, he can only make sensible to himself by bringing in -time, and supposing that changes, which take place in time, are due -to objects of a definite configuration in three dimensions passing -transverse to his plane, and the different sections of it being -apprehended as changes of one and the same plane figure. - -He must also acquire a distinct notion about his plane world, he must -no longer believe that it is the all of space, but that space extends -on both sides of it. In order, then, to prevent his moving off in this -unknown direction, he must assume a sheet, an extended solid sheet, in -two dimensions, against which, in contact with which, all his movements -take place. - -When we come to think of a four-dimensional solid, what are the -corresponding assumptions which we must make? - -We must suppose a sense which we have not, a sense of direction -wanting in us, something which a being in a four-dimensional world -has, and which we have not. It is a sense corresponding to a new space -direction, a direction which extends positively and negatively from -every point of our space, and which goes right away from any space -direction we know of. The perpendicular to a plane is perpendicular, -not only to two lines in it, but to every line, and so we must conceive -this fourth dimension as running perpendicularly to each and every line -we can draw in our space. - -And as the plane being had to suppose something which prevented his -moving off in the third, the unknown dimension to him, so we have to -suppose something which prevents us moving off in the direction unknown -to us. This something, since we must be in contact with it in every one -of our movements, must not be a plane surface, but a solid; it must be -a solid, which in every one of our movements we are against, not in. -It must be supposed as stretching out in every space dimension that we -know; but we are not in it, we are against it, we are next to it, in -the fourth dimension. - -That is, as the plane being conceives himself as having a very small -thickness in the third dimension, of which he is not aware in his -sense experience, so we must suppose ourselves as having a very small -thickness in the fourth dimension, and, being thus four-dimensional -beings, to be prevented from realising that we are such beings by a -constraint which keeps us always in contact with a vast solid sheet, -which stretches on in every direction. We are against that sheet, so -that, if we had the power of four-dimensional movement, we should -either go away from it or through it; all our space movements as we -know them being such that, performing them, we keep in contact with -this solid sheet. - -Now consider the exposition a plane being would make for himself as to -the question of the enclosure of a square, and of a cube. - -He would say the square A, in Fig. 96, is completely enclosed by the -four squares, A far, A near, A above, A below, or as they are written -A_n_, A_f_, A_a_, A_b_. - -[Illustration: Fig. 96.] - -If now he conceives the square A to move in the, to him, unknown -dimension it will trace out a cube, and the bounding squares will -form cubes. Will these completely surround the cube generated by A? -No; there will be two faces of the cube made by A left uncovered; -the first, that face which coincides with the square A in its first -position; the next, that which coincides with the square A in its -final position. Against these two faces cubes must be placed in order -to completely enclose the cube A. These may be called the cubes left -and right or A_l_ and A_r_. Thus each of the enclosing squares of the -square A becomes a cube and two more cubes are wanted to enclose the -cube formed by the movement of A in the third dimension. - -[Illustration: Fig. 97.] - -The plane being could not see the square A with the squares A_n_, A_f_, -etc., placed about it, because they completely hide it from view; and -so we, in the analogous case in our three-dimensional world, cannot -see a cube A surrounded by six other cubes. These cubes we will call A -near A_n_, A far A_f_, A above A_a_, A below A_b_, A left A_l_, A right -A_r_, shown in fig. 97. If now the cube A moves in the fourth dimension -right out of space, it traces out a higher cube—a tesseract, as it may -be called. Each of the six surrounding cubes carried on in the same -motion will make a tesseract also, and these will be grouped around the -tesseract formed by A. But will they enclose it completely? - -All the cubes A_n_, A_f_, etc., lie in our space. But there is nothing -between the cube A and that solid sheet in contact with which every -particle of matter is. When the cube A moves in the fourth direction -it starts from its position, say A_k_, and ends in a final position -A_n_ (using the words “ana” and “kata” for up and down in the fourth -dimension). Now the movement in this fourth dimension is not bounded by -any of the cubes A_n_, A_f_, nor by what they form when thus moved. The -tesseract which A becomes is bounded in the positive and negative ways -in this new direction by the first position of A and the last position -of A. Or, if we ask how many tesseracts lie around the tesseract which -A forms, there are eight, of which one meets it by the cube A, and -another meets it by a cube like A at the end of its motion. - -We come here to a very curious thing. The whole solid cube A is to be -looked on merely as a boundary of the tesseract. - -Yet this is exactly analogous to what the plane being would come to in -his study of the solid world. The square A (fig. 96), which the plane -being looks on as a solid existence in his plane world, is merely the -boundary of the cube which he supposes generated by its motion. - -The fact is that we have to recognise that, if there is another -dimension of space, our present idea of a solid body, as one which -has three dimensions only, does not correspond to anything real, -but is the abstract idea of a three-dimensional boundary limiting a -four-dimensional solid, which a four-dimensional being would form. The -plane being’s thought of a square is not the thought of what we should -call a possibly existing real square, but the thought of an abstract -boundary, the face of a cube. - -Let us now take our eight coloured cubes, which form a cube in -space, and ask what additions we must make to them to represent -the simplest collection of four-dimensional bodies—namely, a group -of them of the same extent in every direction. In plane space we -have four squares. In solid space we have eight cubes. So we should -expect in four-dimensional space to have sixteen four-dimensional -bodies-bodies which in four-dimensional space correspond to cubes in -three-dimensional space, and these bodies we call tesseracts. - -Given then the null, white, red, yellow cubes, and those which make up -the block, we notice that we represent perfectly well the extension -in three directions (fig. 98). From the null point of the null cube, -travelling one inch, we come to the white cube; travelling one inch -away we come to the yellow cube; travelling one inch up we come to the -red cube. Now, if there is a fourth dimension, then travelling from the -same null point for one inch in that direction, we must come to the -body lying beyond the null region. - -[Illustration: Fig. 98.] - -I say null region, not cube; for with the introduction of the fourth -dimension each of our cubes must become something different from cubes. -If they are to have existence in the fourth dimension, they must be -“filled up from” in this fourth dimension. - -Now we will assume that as we get a transference from null to white -going in one way, from null to yellow going in another, so going -from null in the fourth direction we have a transference from null -to blue, using thus the colours white, yellow, red, blue, to denote -transferences in each of the four directions—right, away, up, unknown -or fourth dimension. - -[Illustration: Fig. 99. - -A plane being’s representation of a block of eight cubes by two sets of -four squares.] - -Hence, as the plane being must represent the solid regions, he would -come to by going right, as four squares lying in some position in his -plane, arbitrarily chosen, side by side with his original four squares, -so we must represent those eight four-dimensional regions, which we -should come to by going in the fourth dimension from each of our eight -cubes, by eight cubes placed in some arbitrary position relative to our -first eight cubes. - -[Illustration: Fig. 100.] - -Our representation of a block of sixteen tesseracts by two blocks of -eight cubes.[3] - - [3] The eight cubes used here in 2 can be found in the second of the - model blocks. They can be taken out and used. - -Hence, of the two sets of eight cubes, each one will serve us as a -representation of one of the sixteen tesseracts which form one single -block in four-dimensional space. Each cube, as we have it, is a tray, -as it were, against which the real four-dimensional figure rests—just -as each of the squares which the plane being has is a tray, so to -speak, against which the cube it represents could rest. - -If we suppose the cubes to be one inch each way, then the original -eight cubes will give eight tesseracts of the same colours, or the -cubes, extending each one inch in the fourth dimension. - -But after these there come, going on in the fourth dimension, eight -other bodies, eight other tesseracts. These must be there, if we -suppose the four-dimensional body we make up to have two divisions, one -inch each in each of four directions. - -The colour we choose to designate the transference to this second -region in the fourth dimension is blue. Thus, starting from the null -cube and going in the fourth dimension, we first go through one inch of -the null tesseract, then we come to a blue cube, which is the beginning -of a blue tesseract. This blue tesseract stretches one inch farther on -in the fourth dimension. - -Thus, beyond each of the eight tesseracts, which are of the same colour -as the cubes which are their bases, lie eight tesseracts whose colours -are derived from the colours of the first eight by adding blue. Thus— - - Null gives blue - Yellow ” green - Red ” purple - Orange ” brown - White ” light blue - Pink ” light purple - Light yellow ” light green - Ochre ” light brown - -The addition of blue to yellow gives green—this is a natural -supposition to make. It is also natural to suppose that blue added to -red makes purple. Orange and blue can be made to give a brown, by using -certain shades and proportions. And ochre and blue can be made to give -a light brown. - -But the scheme of colours is merely used for getting a definite and -realisable set of names and distinctions visible to the eye. Their -naturalness is apparent to any one in the habit of using colours, and -may be assumed to be justifiable, as the sole purpose is to devise a -set of names which are easy to remember, and which will give us a set -of colours by which diagrams may be made easy of comprehension. No -scientific classification of colours has been attempted. - -Starting, then, with these sixteen colour names, we have a catalogue of -the sixteen tesseracts, which form a four-dimensional block analogous -to the cubic block. But the cube which we can put in space and look at -is not one of the constituent tesseracts; it is merely the beginning, -the solid face, the side, the aspect, of a tesseract. - -We will now proceed to derive a name for each region, point, edge, -plane face, solid and a face of the tesseract. - -The system will be clear, if we look at a representation in the plane -of a tesseract with three, and one with four divisions in its side. - -The tesseract made up of three tesseracts each way corresponds to the -cube made up of three cubes each way, and will give us a complete -nomenclature. - -In this diagram, fig. 101, 1 represents a cube of 27 cubes, each of -which is the beginning of a tesseract. These cubes are represented -simply by their lowest squares, the solid content must be understood. 2 -represents the 27 cubes which are the beginnings of the 27 tesseracts -one inch on in the fourth dimension. These tesseracts are represented -as a block of cubes put side by side with the first block, but in -their proper positions they could not be in space with the first set. 3 -represents 27 cubes (forming a larger cube) which are the beginnings of -the tesseracts, which begin two inches in the fourth direction from our -space and continue another inch. - -[Illustration: Fig. 101.] - - -[Illustration: Fig. 102[4]] - - [4] The coloured plate, figs. 1, 2, 3, shows these relations more - conspicuously. - -In fig. 102, we have the representation of a block of 4 × 4 × 4 × 4 -or 256 tesseracts. They are given in four consecutive sections, each -supposed to be taken one inch apart in the fourth dimension, and so -giving four blocks of cubes, 64 in each block. Here we see, comparing -it with the figure of 81 tesseracts, that the number of the different -regions show a different tendency of increase. By taking five blocks of -five divisions each way this would become even more clear. - -We see, fig. 102, that starting from the point at any corner, the white -coloured regions only extend out in a line. The same is true for the -yellow, red, and blue. With regard to the latter it should be noticed -that the line of blues does not consist in regions next to each other -in the drawing, but in portions which come in in different cubes. -The portions which lie next to one another in the fourth dimension -must always be represented so, when we have a three-dimensional -representation. Again, those regions such as the pink one, go on -increasing in two dimensions. About the pink region this is seen -without going out of the cube itself, the pink regions increase in -length and height, but in no other dimension. In examining these -regions it is sufficient to take one as a sample. - -The purple increases in the same manner, for it comes in in a -succession from below to above in block 2, and in a succession from -block to block in 2 and 3. Now, a succession from below to above -represents a continuous extension upwards, and a succession from block -to block represents a continuous extension in the fourth dimension. -Thus the purple regions increase in two dimensions, the upward and -the fourth, so when we take a very great many divisions, and let each -become very small, the purple region forms a two-dimensional extension. - -In the same way, looking at the regions marked l. b. or light blue, -which starts nearest a corner, we see that the tesseracts occupying -it increase in length from left to right, forming a line, and that -there are as many lines of light blue tesseracts as there are sections -between the first and last section. Hence the light blue tesseracts -increase in number in two ways—in the right and left, and in the fourth -dimension. They ultimately form what we may call a plane surface. - -Now all those regions which contain a mixture of two simple colours, -white, yellow, red, blue, increase in two ways. On the other hand, -those which contain a mixture of three colours increase in three ways. -Take, for instance, the ochre region; this has three colours, white, -yellow, red; and in the cube itself it increases in three ways. - -Now regard the orange region; if we add blue to this we get a brown. -The region of the brown tesseracts extends in two ways on the left of -the second block, No. 2 in the figure. It extends also from left to -right in succession from one section to another, from section 2 to -section 3 in our figure. - -Hence the brown tesseracts increase in number in three dimensions -upwards, to and fro, fourth dimension. Hence they form a cubic, a -three-dimensional region; this region extends up and down, near -and far, and in the fourth direction, but is thin in the direction -from left to right. It is a cube which, when the complete tesseract -is represented in our space, appears as a series of faces on the -successive cubic sections of the tesseract. Compare fig. 103 in which -the middle block, 2, stands as representing a great number of sections -intermediate between 1 and 3. - -In a similar way from the pink region by addition of blue we have -the light purple region, which can be seen to increase in three ways -as the number of divisions becomes greater. The three ways in which -this region of tesseracts extends is up and down, right and left, -fourth dimension. Finally, therefore, it forms a cubic mass of very -small tesseracts, and when the tesseract is given in space sections -it appears on the faces containing the upward and the right and left -dimensions. - -We get then altogether, as three-dimensional regions, ochre, brown, -light purple, light green. - -Finally, there is the region which corresponds to a mixture of all the -colours; there is only one region such as this. It is the one that -springs from ochre by the addition of blue—this colour we call light -brown. - -Looking at the light brown region we see that it increases in four -ways. Hence, the tesseracts of which it is composed increase in -number in each of four dimensions, and the shape they form does not -remain thin in any of the four dimensions. Consequently this region -becomes the solid content of the block of tesseracts, itself; it -is the real four-dimensional solid. All the other regions are then -boundaries of this light brown region. If we suppose the process -of increasing the number of tesseracts and diminishing their size -carried on indefinitely, then the light brown coloured tesseracts -become the whole interior mass, the three-coloured tesseracts become -three-dimensional boundaries, thin in one dimension, and form the -ochre, the brown, the light purple, the light green. The two-coloured -tesseracts become two-dimensional boundaries, thin in two dimensions, -_e.g._, the pink, the green, the purple, the orange, the light blue, -the light yellow. The one-coloured tesseracts become bounding lines, -thin in three dimensions, and the null points become bounding corners, -thin in four dimensions. From these thin real boundaries we can pass in -thought to the abstractions—points, lines, faces, solids—bounding the -four-dimensional solid, which in this case is light brown coloured, and -under this supposition the light brown coloured region is the only real -one, is the only one which is not an abstraction. - -It should be observed that, in taking a square as the representation -of a cube on a plane, we only represent one face, or the section -between two faces. The squares, as drawn by a plane being, are not the -cubes themselves, but represent the faces or the sections of a cube. -Thus in the plane being’s diagram a cube of twenty-seven cubes “null” -represents a cube, but is really, in the normal position, the orange -square of a null cube, and may be called null, orange square. - -A plane being would save himself confusion if he named his -representative squares, not by using the names of the cubes simply, but -by adding to the names of the cubes a word to show what part of a cube -his representative square was. - -Thus a cube null standing against his plane touches it by null orange -face, passing through his plane it has in the plane a square as trace, -which is null white section, if we use the phrase white section to -mean a section drawn perpendicular to the white line. In the same way -the cubes which we take as representative of the tesseract are not -the tesseract itself, but definite faces or sections of it. In the -preceding figures we should say then, not null, but “null tesseract -ochre cube,” because the cube we actually have is the one determined by -the three axes, white, red, yellow. - -There is another way in which we can regard the colour nomenclature of -the boundaries of a tesseract. - -Consider a null point to move tracing out a white line one inch in -length, and terminating in a null point, see fig. 103 or in the -coloured plate. - -Then consider this white line with its terminal points itself to move -in a second dimension, each of the points traces out a line, the line -itself traces out an area, and gives two lines as well, its initial and -its final position. - -Thus, if we call “a region” any element of the figure, such as a point, -or a line, etc., every “region” in moving traces out a new kind of -region, “a higher region,” and gives two regions of its own kind, an -initial and a final position. The “higher region” means a region with -another dimension in it. - -Now the square can move and generate a cube. The square light yellow -moves and traces out the mass of the cube. Letting the addition of -red denote the region made by the motion in the upward direction we -get an ochre solid. The light yellow face in its initial and terminal -positions give the two square boundaries of the cube above and below. -Then each of the four lines of the light yellow square—white, yellow, -and the white, yellow opposite them—trace out a bounding square. So -there are in all six bounding squares, four of these squares being -designated in colour by adding red to the colour of the generating -lines. Finally, each point moving in the up direction gives rise to -a line coloured null + red, or red, and then there are the initial -and terminal positions of the points giving eight points. The number -of the lines is evidently twelve, for the four lines of this light -yellow square give four lines in their initial, four lines in their -final position, while the four points trace out four lines, that is -altogether twelve lines. - -Now the squares are each of them separate boundaries of the cube, while -the lines belong, each of them, to two squares, thus the red line is -that which is common to the orange and pink squares. - -Now suppose that there is a direction, the fourth dimension, which is -perpendicular alike to every one of the space dimensions already used—a -dimension perpendicular, for instance, to up and to right hand, so that -the pink square moving in this direction traces out a cube. - -A dimension, moreover, perpendicular to the up and away directions, -so that the orange square moving in this direction also traces out -a cube, and the light yellow square, too, moving in this direction -traces out a cube. Under this supposition, the whole cube moving in -the unknown dimension, traces out something new—a new kind of volume, -a higher volume. This higher volume is a four-dimensional volume, and -we designate it in colour by adding blue to the colour of that which by -moving generates it. - -It is generated by the motion of the ochre solid, and hence it is -of the colour we call light brown (white, yellow, red, blue, mixed -together). It is represented by a number of sections like 2 in fig. 103. - -Now this light brown higher solid has for boundaries: first, the ochre -cube in its initial position, second, the same cube in its final -position, 1 and 3, fig. 103. Each of the squares which bound the cube, -moreover, by movement in this new direction traces out a cube, so we -have from the front pink faces of the cube, third, a pink blue or -light purple cube, shown as a light purple face on cube 2 in fig. 103, -this cube standing for any number of intermediate sections; fourth, -a similar cube from the opposite pink face; fifth, a cube traced out -by the orange face—this is coloured brown and is represented by the -brown face of the section cube in fig. 103; sixth, a corresponding -brown cube on the right hand; seventh, a cube starting from the light -yellow square below; the unknown dimension is at right angles to this -also. This cube is coloured light yellow and blue or light green; and, -finally, eighth, a corresponding cube from the upper light yellow face, -shown as the light green square at the top of the section cube. - -The tesseract has thus eight cubic boundaries. These completely enclose -it, so that it would be invisible to a four-dimensional being. Now, as -to the other boundaries, just as the cube has squares, lines, points, -as boundaries, so the tesseract has cubes, squares, lines, points, as -boundaries. - -The number of squares is found thus—round the cube are six squares, -these will give six squares in their initial and six in their final -positions. Then each of the twelve lines of the cube trace out a square -in the motion in the fourth dimension. Hence there will be altogether -12 + 12 = 24 squares. - -If we look at any one of these squares we see that it is the meeting -surface of two of the cubic sides. Thus, the red line by its movement -in the fourth dimension, traces out a purple square—this is common -to two cubes, one of which is traced out by the pink square moving -in the fourth dimension, and the other is traced out by the orange -square moving in the same way. To take another square, the light yellow -one, this is common to the ochre cube and the light green cube. The -ochre cube comes from the light yellow square by moving it in the up -direction, the light green cube is made from the light yellow square by -moving it in the fourth dimension. The number of lines is thirty-two, -for the twelve lines of the cube give twelve lines of the tesseract -in their initial position, and twelve in their final position, making -twenty-four, while each of the eight points traces out a line, thus -forming thirty-two lines altogether. - -The lines are each of them common to three cubes, or to three square -faces; take, for instance, the red line. This is common to the orange -face, the pink face, and that face which is formed by moving the red -line in the sixth dimension, namely, the purple face. It is also common -to the ochre cube, the pale purple cube, and the brown cube. - -The points are common to six square faces and to four cubes; thus, -the null point from which we start is common to the three square -faces—pink, light yellow, orange, and to the three square faces made by -moving the three lines white, yellow, red, in the fourth dimension, -namely, the light blue, the light green, the purple faces—that is, to -six faces in all. The four cubes which meet in it are the ochre cube, -the light purple cube, the brown cube, and the light green cube. - -[Illustration: Fig. 103. - -The tesseract, red, white, yellow axes in space. In the lower line the -three rear faces are shown, the interior being removed.] - -[Illustration: Fig. 104. - -The tesseract, red, yellow, blue axes in space, the blue axis running -to the left, opposite faces are coloured identically.] - -A complete view of the tesseract in its various space presentations -is given in the following figures or catalogue cubes, figs. 103-106. -The first cube in each figure represents the view of a tesseract -coloured as described as it begins to pass transverse to our space. -The intermediate figure represents a sectional view when it is partly -through, and the final figure represents the far end as it is just -passing out. These figures will be explained in detail in the next -chapter. - -[Illustration: Fig. 105. - -The tesseract, with red, white, blue axes in space. Opposite faces are -coloured identically.] - -[Illustration: Fig. 106. - -The tesseract, with blue, white, yellow axes in space. The blue axis -runs downward from the base of the ochre cube as it stands originally. -Opposite faces are coloured identically.] - -We have thus obtained a nomenclature for each of the regions of a -tesseract; we can speak of any one of the eight bounding cubes, the -twenty square faces, the thirty-two lines, the sixteen points. - - - - - CHAPTER XIII - - REMARKS ON THE FIGURES - - -An inspection of above figures will give an answer to many questions -about the tesseract. If we have a tesseract one inch each way, then it -can be represented as a cube—a cube having white, yellow, red axes, -and from this cube as a beginning, a volume extending into the fourth -dimension. Now suppose the tesseract to pass transverse to our space, -the cube of the red, yellow, white axes disappears at once, it is -indefinitely thin in the fourth dimension. Its place is occupied by -those parts of the tesseract which lie further away from our space in -the fourth dimension. Each one of these sections will last only for -one moment, but the whole of them will take up some appreciable time -in passing. If we take the rate of one inch a minute the sections will -take the whole of the minute in their passage across our space, they -will take the whole of the minute except the moment which the beginning -cube and the end cube occupy in their crossing our space. In each one -of the cubes, the section cubes, we can draw lines in all directions -except in the direction occupied by the blue line, the fourth -dimension; lines in that direction are represented by the transition -from one section cube to another. Thus to give ourselves an adequate -representation of the tesseract we ought to have a limitless number of -section cubes intermediate between the first bounding cube, the ochre -cube, and the last bounding cube, the other ochre cube. Practically -three intermediate sectional cubes will be found sufficient for most -purposes. We will take then a series of five figures—two terminal -cubes, and three intermediate sections—and show how the different -regions appear in our space when we take each set of three out of the -four axes of the tesseract as lying in our space. - -In fig. 107 initial letters are used for the colours. A reference to -fig. 103 will show the complete nomenclature, which is merely indicated -here. - -[Illustration: Fig. 107.] - -In this figure the tesseract is shown in five stages distant from our -space: first, zero; second, 1/4 in.; third, 2/4 in.; fourth, 3/4 in.; -fifth, 1 in.; which are called _b_0, _b_1, _b_2, _b_3, _b_4, because -they are sections taken at distances 0, 1, 2, 3, 4 quarter inches along -the blue line. All the regions can be named from the first cube, the -_b_0 cube, as before, simply by remembering that transference along -the b axis gives the addition of blue to the colour of the region in -the ochre, the _b_0 cube. In the final cube _b_4, the colouring of the -original _b_0 cube is repeated. Thus the red line moved along the blue -axis gives a red and blue or purple square. This purple square appears -as the three purple lines in the sections _b_1, _b_2, _b_3, taken at -1/4, 2/4, 3/4 of an inch in the fourth dimension. If the tesseract -moves transverse to our space we have then in this particular region, -first of all a red line which lasts for a moment, secondly a purple -line which takes its place. This purple line lasts for a minute—that -is, all of a minute, except the moment taken by the crossing our space -of the initial and final red line. The purple line having lasted for -this period is succeeded by a red line, which lasts for a moment; then -this goes and the tesseract has passed across our space. The final red -line we call red bl., because it is separated from the initial red -line by a distance along the axis for which we use the colour blue. -Thus a line that lasts represents an area duration; is in this mode -of presentation equivalent to a dimension of space. In the same way -the white line, during the crossing our space by the tesseract, is -succeeded by a light blue line which lasts for the inside of a minute, -and as the tesseract leaves our space, having crossed it, the white bl. -line appears as the final termination. - -Take now the pink face. Moved in the blue direction it traces out a -light purple cube. This light purple cube is shown in sections in -_b__{1}, _b__{2}, _b__{3}, and the farther face of this cube in the -blue direction is shown in _b__{4}—a pink face, called pink _b_ because -it is distant from the pink face we began with in the blue direction. -Thus the cube which we colour light purple appears as a lasting square. -The square face itself, the pink face, vanishes instantly the tesseract -begins to move, but the light purple cube appears as a lasting square. -Here also duration is the equivalent of a dimension of space—a lasting -square is a cube. It is useful to connect these diagrams with the views -given in the coloured plate. - -Take again the orange face, that determined by the red and yellow axes; -from it goes a brown cube in the blue direction, for red and yellow -and blue are supposed to make brown. This brown cube is shown in three -sections in the faces _b__{1}, _b__{2}, _b__{3}. In _b__{4} is the -opposite orange face of the brown cube, the face called orange _b_, -for it is distant in the blue direction from the orange face. As the -tesseract passes transverse to our space, we have then in this region -an instantly vanishing orange square, followed by a lasting brown -square, and finally an orange face which vanishes instantly. - -Now, as any three axes will be in our space, let us send the white -axis out into the unknown, the fourth dimension, and take the blue -axis into our known space dimension. Since the white and blue axes are -perpendicular to each other, if the white axis goes out into the fourth -dimension in the positive sense, the blue axis will come into the -direction the white axis occupied, in the negative sense. - -[Illustration: Fig. 108.] - -Hence, not to complicate matters by having to think of two senses in -the unknown direction, let us send the white line into the positive -sense of the fourth dimension, and take the blue one as running in the -negative sense of that direction which the white line has left; let the -blue line, that is, run to the left. We have now the row of figures -in fig. 108. The dotted cube shows where we had a cube when the white -line ran in our space—now it has turned out of our space, and another -solid boundary, another cubic face of the tesseract comes into our -space. This cube has red and yellow axes as before; but now, instead -of a white axis running to the right, there is a blue axis running to -the left. Here we can distinguish the regions by colours in a perfectly -systematic way. The red line traces out a purple square in the -transference along the blue axis by which this cube is generated from -the orange face. This purple square made by the motion of the red line -is the same purple face that we saw before as a series of lines in the -sections _b__{1}, _b__{2}, _b__{3}. Here, since both red and blue axes -are in our space, we have no need of duration to represent the area -they determine. In the motion of the tesseract across space this purple -face would instantly disappear. - -From the orange face, which is common to the initial cubes in fig. 107 -and fig. 108, there goes in the blue direction a cube coloured brown. -This brown cube is now all in our space, because each of its three axes -run in space directions, up, away, to the left. It is the same brown -cube which appeared as the successive faces on the sections _b__{1}, -_b__{2}, _b__{3}. Having all its three axes in our space, it is given -in extension; no part of it needs to be represented as a succession. -The tesseract is now in a new position with regard to our space, and -when it moves across our space the brown cube instantly disappears. - -In order to exhibit the other regions of the tesseract we must remember -that now the white line runs in the unknown dimension. Where shall we -put the sections at distances along the line? Any arbitrary position in -our space will do: there is no way by which we can represent their real -position. - -However, as the brown cube comes off from the orange face to the left, -let us put these successive sections to the left. We can call them -_wh__{0}, _wh__{1}, _wh__{2}, _wh__{3}, _wh__{4}, because they are -sections along the white axis, which now runs in the unknown dimension. - -Running from the purple square in the white direction we find the light -purple cube. This is represented in the sections _wh__{1}, _wh__{2}, -_wh__{3}, _wh__{4}, fig. 108. It is the same cube that is represented -in the sections _b__{1}, _b__{2}, _b__{3}: in fig. 107 the red and -white axes are in our space, the blue out of it; in the other case, the -red and blue are in our space, the white out of it. It is evident that -the face pink _y_, opposite the pink face in fig. 107, makes a cube -shown in squares in _b__{1}, _b__{2}, _b__{3}, _b__{4}, on the opposite -side to the _l_ purple squares. Also the light yellow face at the base -of the cube _b__{0}, makes a light green cube, shown as a series of -base squares. - -The same light green cube can be found in fig. 107. The base square in -_wh__{0} is a green square, for it is enclosed by blue and yellow axes. -From it goes a cube in the white direction, this is then a light green -cube and the same as the one just mentioned as existing in the sections -_b__{0}, _b__{1}, _b__{2}, _b__{3}, _b__{4}. - -The case is, however, a little different with the brown cube. This cube -we have altogether in space in the section _wh__{0}, fig. 108, while -it exists as a series of squares, the left-hand ones, in the sections -_b__{0}, _b__{1}, _b__{2}, _b__{3}, _b__{4}. The brown cube exists as a -solid in our space, as shown in fig. 108. In the mode of representation -of the tesseract exhibited in fig. 107, the same brown cube appears as -a succession of squares. That is, as the tesseract moves across space, -the brown cube would actually be to us a square—it would be merely -the lasting boundary of another solid. It would have no thickness at -all, only extension in two dimensions, and its duration would show its -solidity in three dimensions. - -It is obvious that, if there is a four-dimensional space, matter in -three dimensions only is a mere abstraction; all material objects -must then have a slight four-dimensional thickness. In this case the -above statement will undergo modification. The material cube which is -used as the model of the boundary of a tesseract will have a slight -thickness in the fourth dimension, and when the cube is presented to -us in another aspect, it would not be a mere surface. But it is most -convenient to regard the cubes we use as having no extension at all in -the fourth dimension. This consideration serves to bring out a point -alluded to before, that, if there is a fourth dimension, our conception -of a solid is the conception of a mere abstraction, and our talking -about real three-dimensional objects would seem to a four-dimensional -being as incorrect as a two-dimensional being’s telling about real -squares, real triangles, etc., would seem to us. - -The consideration of the two views of the brown cube shows that any -section of a cube can be looked at by a presentation of the cube in -a different position in four-dimensional space. The brown faces in -_b__{1}, _b__{2}, _b__{3}, are the very same brown sections that would -be obtained by cutting the brown cube, _wh__{0}, across at the right -distances along the blue line, as shown in fig. 108. But as these -sections are placed in the brown cube, _wh__{0}, they come behind one -another in the blue direction. Now, in the sections _wh__{1}, _wh__{2}, -_wh__{3}, we are looking at these sections from the white direction—the -blue direction does not exist in these figures. So we see them in -a direction at right angles to that in which they occur behind one -another in _wh__{0}. There are intermediate views, which would come in -the rotation of a tesseract. These brown squares can be looked at from -directions intermediate between the white and blue axes. It must be -remembered that the fourth dimension is perpendicular equally to all -three space axes. Hence we must take the combinations of the blue axis, -with each two of our three axes, white, red, yellow, in turn. - -In fig. 109 we take red, white, and blue axes in space, sending yellow -into the fourth dimension. If it goes into the positive sense of the -fourth dimension the blue line will come in the opposite direction to -that in which the yellow line ran before. Hence, the cube determined -by the white, red, blue axes, will start from the pink plane and run -towards us. The dotted cube shows where the ochre cube was. When it is -turned out of space, the cube coming towards from its front face is -the one which comes into our space in this turning. Since the yellow -line now runs in the unknown dimension we call the sections _y__{0}, -_y__{1}, _y__{2}, _y__{3}, _y__{4}, as they are made at distances 0, 1, -2, 3, 4, quarter inches along the yellow line. We suppose these cubes -arranged in a line coming towards us—not that that is any more natural -than any other arbitrary series of positions, but it agrees with the -plan previously adopted. - -[Illustration: Fig. 109.] - -The interior of the first cube, _y__{0}, is that derived from pink by -adding blue, or, as we call it, light purple. The faces of the cube are -light blue, purple, pink. As drawn, we can only see the face nearest to -us, which is not the one from which the cube starts—but the face on the -opposite side has the same colour name as the face towards us. - -The successive sections of the series, _y__{0}, _y__{1}, _y__{2}, etc., -can be considered as derived from sections of the _b__{0} cube made at -distances along the yellow axis. What is distant a quarter inch from -the pink face in the yellow direction? This question is answered by -taking a section from a point a quarter inch along the yellow axis in -the cube _b__{0}, fig. 107. It is an ochre section with lines orange -and light yellow. This section will therefore take the place of the -pink face in _y__{1} when we go on in the yellow direction. Thus, the -first section, _y__{1}, will begin from an ochre face with light yellow -and orange lines. The colour of the axis which lies in space towards -us is blue, hence the regions of this section-cube are determined in -nomenclature, they will be found in full in fig. 105. - -There remains only one figure to be drawn, and that is the one in which -the red axis is replaced by the blue. Here, as before, if the red axis -goes out into the positive sense of the fourth dimension, the blue line -must come into our space in the negative sense of the direction which -the red line has left. Accordingly, the first cube will come in beneath -the position of our ochre cube, the one we have been in the habit of -starting with. - -[Illustration: Fig. 110.] - -To show these figures we must suppose the ochre cube to be on a movable -stand. When the red line swings out into the unknown dimension, and the -blue line comes in downwards, a cube appears below the place occupied -by the ochre cube. The dotted cube shows where the ochre cube was. -That cube has gone and a different cube runs downwards from its base. -This cube has white, yellow, and blue axes. Its top is a light yellow -square, and hence its interior is light yellow + blue or light green. -Its front face is formed by the white line moving along the blue axis, -and is therefore light blue, the left-hand side is formed by the yellow -line moving along the blue axis, and therefore green. - -As the red line now runs in the fourth dimension, the successive -sections can he called _r__{0}, _r__{1}, _r__{2}, _r__{3}, _r__{4}, -these letters indicating that at distances 0, 1/4, 2/4, 3/4, 1 inch -along the red axis we take all of the tesseract that can be found in a -three-dimensional space, this three-dimensional space extending not at -all in the fourth dimension, but up and down, right and left, far and -near. - -We can see what should replace the light yellow face of _r__{0}, when -the section _r__{1} comes in, by looking at the cube _b__{0}, fig. 107. -What is distant in it one-quarter of an inch from the light yellow face -in the red direction? It is an ochre section with orange and pink lines -and red points; see also fig. 103. - -This square then forms the top square of _r__{1}. Now we can determine -the nomenclature of all the regions of _r__{1} by considering what -would be formed by the motion of this square along a blue axis. - -But we can adopt another plan. Let us take a horizontal section of -_r__{0}, and finding that section in the figures, of fig. 107 or fig. -103, from them determine what will replace it, going on in the red -direction. - -A section of the _r__{0} cube has green, light blue, green, light blue -sides and blue points. - -Now this square occurs on the base of each of the section figures, -_b__{1}, _b__{2}, etc. In them we see that 1/4 inch in the red -direction from it lies a section with brown and light purple lines and -purple corners, the interior being of light brown. Hence this is the -nomenclature of the section which in _r__{1} replaces the section of -_r__{0} made from a point along the blue axis. - -Hence the colouring as given can be derived. - -We have thus obtained a perfectly named group of tesseracts. We can -take a group of eighty-one of them 3 × 3 × 3 × 3, in four dimensions, -and each tesseract will have its name null, red, white, yellow, blue, -etc., and whatever cubic view we take of them we can say exactly -what sides of the tesseracts we are handling, and how they touch each -other.[5] - - [5] At this point the reader will find it advantageous, if he has the - models, to go through the manipulations described in the appendix. - -Thus, for instance, if we have the sixteen tesseracts shown below, we -can ask how does null touch blue. - -[Illustration: Fig. 111.] - -In the arrangement given in fig. 111 we have the axes white, red, -yellow, in space, blue running in the fourth dimension. Hence we have -the ochre cubes as bases. Imagine now the tesseractic group to pass -transverse to our space—we have first of all null ochre cube, white -ochre cube, etc.; these instantly vanish, and we get the section shown -in the middle cube in fig. 103, and finally, just when the tesseract -block has moved one inch transverse to our space, we have null ochre -cube, and then immediately afterwards the ochre cube of blue comes in. -Hence the tesseract null touches the tesseract blue by its ochre cube, -which is in contact, each and every point of it, with the ochre cube of -blue. - -How does null touch white, we may ask? Looking at the beginning A, fig. -111, where we have the ochre cubes, we see that null ochre touches -white ochre by an orange face. Now let us generate the null and white -tesseracts by a motion in the blue direction of each of these cubes. -Each of them generates the corresponding tesseract, and the plane of -contact of the cubes generates the cube by which the tesseracts are -in contact. Now an orange plane carried along a blue axis generates a -brown cube. Hence null touches white by a brown cube. - -[Illustration: Fig. 112.] - -If we ask again how red touches light blue tesseract, let us rearrange -our group, fig. 112, or rather turn it about so that we have a -different space view of it; let the red axis and the white axis run -up and right, and let the blue axis come in space towards us, then -the yellow axis runs in the fourth dimension. We have then two blocks -in which the bounding cubes of the tesseracts are given, differently -arranged with regard to us—the arrangement is really the same, but it -appears different to us. Starting from the plane of the red and white -axes we have the four squares of the null, white, red, pink tesseracts -as shown in A, on the red, white plane, unaltered, only from them now -comes out towards us the blue axis. Hence we have null, white, red, -pink tesseracts in contact with our space by their cubes which have -the red, white, blue axis in them, that is by the light purple cubes. -Following on these four tesseracts we have that which comes next to -them in the blue direction, that is the four blue, light blue, purple, -light purple. These are likewise in contact with our space by their -light purple cubes, so we see a block as named in the figure, of which -each cube is the one determined by the red, white, blue, axes. - -The yellow line now runs out of space; accordingly one inch on in the -fourth dimension we come to the tesseracts which follow on the eight -named in C, fig. 112, in the yellow direction. - -These are shown in C.y_{1}, fig. 112. Between figure C and C.y_{1} is -that four-dimensional mass which is formed by moving each of the cubes -in C one inch in the fourth dimension—that is, along a yellow axis; for -the yellow axis now runs in the fourth dimension. - -In the block C we observe that red (light purple cube) touches light -blue (light purple cube) by a point. Now these two cubes moving -together remain in contact during the period in which they trace out -the tesseracts red and light blue. This motion is along the yellow -axis, consequently red and light blue touch by a yellow line. - -We have seen that the pink face moved in a yellow direction traces out -a cube; moved in the blue direction it also traces out a cube. Let us -ask what the pink face will trace out if it is moved in a direction -within the tesseract lying equally between the yellow and blue -directions. What section of the tesseract will it make? - -We will first consider the red line alone. Let us take a cube with the -red line in it and the yellow and blue axes. - -The cube with the yellow, red, blue axes is shown in fig. 113. If the -red line is moved equally in the yellow and in the blue direction by -four equal motions of ¼ inch each, it takes the positions 11, 22, 33, -and ends as a red line. - -[Illustration: Fig. 113.] - -Now, the whole of this red, yellow, blue, or brown cube appears as a -series of faces on the successive sections of the tesseract starting -from the ochre cube and letting the blue axis run in the fourth -dimension. Hence the plane traced out by the red line appears as a -series of lines in the successive sections, in our ordinary way of -representing the tesseract; these lines are in different places in each -successive section. - -[Illustration: Fig. 114.] - -Thus drawing our initial cube and the successive sections, calling them -_b__{0}, _b__{1}, _b__{2}, _b__{3}, _b__{4}, fig. 115, we have the red -line subject to this movement appearing in the positions indicated. - -We will now investigate what positions in the tesseract another line in -the pink face assumes when it is moved in a similar manner. - -Take a section of the original cube containing a vertical line, 4, -in the pink plane, fig. 115. We have, in the section, the yellow -direction, but not the blue. - -From this section a cube goes off in the fourth dimension, which is -formed by moving each point of the section in the blue direction. - -[Illustration: Fig. 115.] - -[Illustration: Fig. 116.] - -Drawing this cube we have fig. 116. - -Now this cube occurs as a series of sections in our original -representation of the tesseract. Taking four steps as before this cube -appears as the sections drawn in _b__{0}, _b__{1}, _b__{2}, _b__{3}, -_b__{4}, fig. 117, and if the line 4 is subjected to a movement equal -in the blue and yellow directions, it will occupy the positions -designated by 4, 4_{1}, 4_{2}, 4_{3}, 4_{4}. - -[Illustration: Fig. 117.] - -Hence, reasoning in a similar manner about every line, it is evident -that, moved equally in the blue and yellow directions, the pink plane -will trace out a space which is shown by the series of section planes -represented in the diagram. - -Thus the space traced out by the pink face, if it is moved equally in -the yellow and blue directions, is represented by the set of planes -delineated in Fig. 118, pink face or 0, then 1, 2, 3, and finally pink -face or 4. This solid is a diagonal solid of the tesseract, running -from a pink face to a pink face. Its length is the length of the -diagonal of a square, its side is a square. - -Let us now consider the unlimited space which springs from the pink -face extended. - -This space, if it goes off in the yellow direction, gives us in it the -ochre cube of the tesseract. Thus, if we have the pink face given and a -point in the ochre cube, we have determined this particular space. - -Similarly going off from the pink face in the blue direction is another -space, which gives us the light purple cube of the tesseract in it. And -any point being taken in the light purple cube, this space going off -from the pink face is fixed. - -[Illustration: Fig. 118.] - -The space we are speaking of can be conceived as swinging round the -pink face, and in each of its positions it cuts out a solid figure from -the tesseract, one of which we have seen represented in fig. 118. - -Each of these solid figures is given by one position of the swinging -space, and by one only. Hence in each of them, if one point is taken, -the particular one of the slanting spaces is fixed. Thus we see that -given a plane and a point out of it a space is determined. - -Now, two points determine a line. - -Again, think of a line and a point outside it. Imagine a plane rotating -round the line. At some time in its rotation it passes through the -point. Thus a line and a point, or three points, determine a plane. -And finally four points determine a space. We have seen that a plane -and a point determine a space, and that three points determine a plane; -so four points will determine a space. - -These four points may be any points, and we can take, for instance, the -four points at the extremities of the red, white, yellow, blue axes, in -the tesseract. These will determine a space slanting with regard to the -section spaces we have been previously considering. This space will cut -the tesseract in a certain figure. - -One of the simplest sections of a cube by a plane is that in which the -plane passes through the extremities of the three edges which meet in a -point. We see at once that this plane would cut the cube in a triangle, -but we will go through the process by which a plane being would most -conveniently treat the problem of the determination of this shape, in -order that we may apply the method to the determination of the figure -in which a space cuts a tesseract when it passes through the 4 points -at unit distance from a corner. - -We know that two points determine a line, three points determine a -plane, and given any two points in a plane the line between them lies -wholly in the plane. - -[Illustration: Fig. 119.] - -Let now the plane being study the section made by a plane passing -through the null _r_, null _wh_, and null _y_ points, fig. 119. Looking -at the orange square, which, as usual, we suppose to be initially in -his plane, he sees that the line from null _r_ to null _y_, which is -a line in the section plane, the plane, namely, through the three -extremities of the edges meeting in null, cuts the orange face in an -orange line with null points. This then is one of the boundaries of the -section figure. - -Let now the cube be so turned that the pink face comes in his plane. -The points null _r_ and null _wh_ are now visible. The line between -them is pink with null points, and since this line is common to the -surface of the cube and the cutting plane, it is a boundary of the -figure in which the plane cuts the cube. - -Again, suppose the cube turned so that the light yellow face is in -contact with the plane being’s plane. He sees two points, the null _wh_ -and the null _y_. The line between these lies in the cutting plane. -Hence, since the three cutting lines meet and enclose a portion of -the cube between them, he has determined the figure he sought. It is -a triangle with orange, pink, and light yellow sides, all equal, and -enclosing an ochre area. - -Let us now determine in what figure the space, determined by the four -points, null _r_, null _y_, null _wh_, null _b_, cuts the tesseract. We -can see three of these points in the primary position of the tesseract -resting against our solid sheet by the ochre cube. These three points -determine a plane which lies in the space we are considering, and this -plane cuts the ochre cube in a triangle, the interior of which is -ochre (fig. 119 will serve for this view), with pink, light yellow and -orange sides, and null points. Going in the fourth direction, in one -sense, from this plane we pass into the tesseract, in the other sense -we pass away from it. The whole area inside the triangle is common to -the cutting plane we see, and a boundary of the tesseract. Hence we -conclude that the triangle drawn is common to the tesseract and the -cutting space. - -Now let the ochre cube turn out and the brown cube come in. The dotted -lines show the position the ochre cube has left (fig. 120). - -[Illustration: Fig. 120.] - -Here we see three out of the four points through which the cutting -plane passes, null _r_, null _y_, and null _b_. The plane they -determine lies in the cutting space, and this plane cuts out of the -brown cube a triangle with orange, purple and green sides, and null -points. The orange line of this figure is the same as the orange line -in the last figure. - -Now let the light purple cube swing into our space, towards us, fig. -121. - -[Illustration: Fig. 121.] - -The cutting space which passes through the four points, null _r_, _y_, -_wh_, _b_, passes through the null _r_, _wh_, _b_, and therefore the -plane these determine lies in the cutting space. - -This triangle lies before us. It has a light purple interior and pink, -light blue, and purple edges with null points. - -This, since it is all of the plane that is common to it, and this -bounding of the tesseract, gives us one of the bounding faces of our -sectional figure. The pink line in it is the same as the pink line we -found in the first figure—that of the ochre cube. - -Finally, let the tesseract swing about the light yellow plane, so that -the light green cube comes into our space. It will point downwards. - -The three points, _n.y_, _n.wh_, _n.b_, are in the cutting space, and -the triangle they determine is common to the tesseract and the cutting -space. Hence this boundary is a triangle having a light yellow line, -which is the same as the light yellow line of the first figure, a light -blue line and a green line. - -[Illustration: Fig. 122.] - -We have now traced the cutting space between every set of three that -can be made out of the four points in which it cuts the tesseract, and -have got four faces which all join on to each other by lines. - -[Illustration: Fig. 123.] - -The triangles are shown in fig. 123 as they join on to the triangle -in the ochre cube. But they join on each to the other in an exactly -similar manner; their edges are all identical two and two. They form a -closed figure, a tetrahedron, enclosing a light brown portion which is -the portion of the cutting space which lies inside the tesseract. - -We cannot expect to see this light brown portion, any more than a plane -being could expect to see the inside of a cube if an angle of it were -pushed through his plane. All he can do is to come upon the boundaries -of it in a different way to that in which he would if it passed -straight through his plane. - -Thus in this solid section; the whole interior lies perfectly open in -the fourth dimension. Go round it as we may we are simply looking at -the boundaries of the tesseract which penetrates through our solid -sheet. If the tesseract were not to pass across so far, the triangle -would be smaller; if it were to pass farther, we should have a -different figure, the outlines of which can be determined in a similar -manner. - -The preceding method is open to the objection that it depends rather on -our inferring what must be, than our seeing what is. Let us therefore -consider our sectional space as consisting of a number of planes, each -very close to the last, and observe what is to be found in each plane. - -The corresponding method in the case of two dimensions is as -follows:—The plane being can see that line of the sectional plane -through null _y_, null _wh_, null _r_, which lies in the orange plane. -Let him now suppose the cube and the section plane to pass half way -through his plane. Replacing the red and yellow axes are lines parallel -to them, sections of the pink and light yellow faces. - -[Illustration: Fig. 124.] - -Where will the section plane cut these parallels to the red and yellow -axes? - -Let him suppose the cube, in the position of the drawing, fig. 124, -turned so that the pink face lies against his plane. He can see the -line from the null _r_ point to the null _wh_ point, and can see -(compare fig. 119) that it cuts AB a parallel to his red axis, drawn -at a point half way along the white line, in a point B, half way up. I -shall speak of the axis as having the length of an edge of the cube. -Similarly, by letting the cube turn so that the light yellow square -swings against his plane, he can see (compare fig. 119) that a parallel -to his yellow axis drawn from a point half-way along the white axis, is -cut at half its length by the trace of the section plane in the light -yellow face. - -Hence when the cube had passed half-way through he would have—instead -of the orange line with null points, which he had at first—an ochre -line of half its length, with pink and light yellow points. Thus, as -the cube passed slowly through his plane, he would have a succession -of lines gradually diminishing in length and forming an equilateral -triangle. The whole interior would be ochre, the line from which it -started would be orange. The succession of points at the ends of -the succeeding lines would form pink and light yellow lines and the -final point would be null. Thus looking at the successive lines in -the section plane as it and the cube passed across his plane he would -determine the figure cut out bit by bit. - -Coming now to the section of the tesseract, let us imagine that the -tesseract and its cutting _space_ pass slowly across our space; we can -examine portions of it, and their relation to portions of the cutting -space. Take the section space which passes through the four points, -null _r_, _wh_, _y_, _b_; we can see in the ochre cube (fig. 119) the -plane belonging to this section space, which passes through the three -extremities of the red, white, yellow axes. - -Now let the tesseract pass half way through our space. Instead of our -original axes we have parallels to them, purple, light blue, and green, -each of the same length as the first axes, for the section of the -tesseract is of exactly the same shape as its ochre cube. - -But the sectional space seen at this stage of the transference would -not cut the section of the tesseract in a plane disposed as at first. - -To see where the sectional space would cut these parallels to the -original axes let the tesseract swing so that, the orange face -remaining stationary, the blue line comes in to the left. - -Here (fig. 125) we have the null _r_, _y_, _b_ points, and of the -sectional space all we see is the plane through these three points in -it. - -[Illustration: Fig. 125.] - -In this figure we can draw the parallels to the red and yellow axes and -see that, if they started at a point half way along the blue axis, they -would each be cut at a point so as to be half of their previous length. - -Swinging the tesseract into our space about the pink face of the ochre -cube we likewise find that the parallel to the white axis is cut at -half its length by the sectional space. - -Hence in a section made when the tesseract had passed half across our -space the parallels to the red, white, yellow axes, which are now in -our space, are cut by the section space, each of them half way along, -and for this stage of the traversing motion we should have fig. 126. -The section made of this cube by the plane in which the sectional space -cuts it, is an equilateral triangle with purple, l. blue, green points, -and l. purple, brown, l. green lines. - -[Illustration: Fig. 126.] - -Thus the original ochre triangle, with null points and pink, orange, -light yellow lines, would be succeeded by a triangle coloured in manner -just described. - -This triangle would initially be only a very little smaller than the -original triangle, it would gradually diminish, until it ended in a -point, a null point. Each of its edges would be of the same length. -Thus the successive sections of the successive planes into which we -analyse the cutting space would be a tetrahedron of the description -shown (fig. 123), and the whole interior of the tetrahedron would be -light brown. - -[Illustration: Fig. 127. Front view. The rear faces.] - -In fig. 127 the tetrahedron is represented by means of its faces as -two triangles which meet in the p. line, and two rear triangles which -join on to them, the diagonal of the pink face being supposed to run -vertically upward. - -We have now reached a natural termination. The reader may pursue -the subject in further detail, but will find no essential novelty. -I conclude with an indication as to the manner in which figures -previously given may be used in determining sections by the method -developed above. - -Applying this method to the tesseract, as represented in Chapter IX., -sections made by a space cutting the axes equidistantly at any distance -can be drawn, and also the sections of tesseracts arranged in a block. - -If we draw a plane, cutting all four axes at a point six units distance -from null, we have a slanting space. This space cuts the red, white, -yellow axes in the points LMN (fig. 128), and so in the region of our -space before we go off into the fourth dimension, we have the plane -represented by LMN extended. This is what is common to the slanting -space and our space. - -[Illustration: Fig. 128.] - -This plane cuts the ochre cube in the triangle EFG. - -Comparing this with (fig. 72) _oh_, we see that the hexagon there drawn -is part of the triangle EFG. - -Let us now imagine the tesseract and the slanting space both together -to pass transverse to our space, a distance of one unit, we have in -1_h_ a section of the tesseract, whose axes are parallels to the -previous axes. The slanting space cuts them at a distance of five units -along each. Drawing the plane through these points in 1_h_ it will be -found to cut the cubical section of the tesseract in the hexagonal -figure drawn. In 2_h_ (fig. 72) the slanting space cuts the parallels -to the axes at a distance of four along each, and the hexagonal figure -is the section of this section of the tesseract by it. Finally when -3_h_ comes in the slanting space cuts the axes at a distance of three -along each, and the section is a triangle, of which the hexagon drawn -is a truncated portion. After this the tesseract, which extends only -three units in each of the four dimensions, has completely passed -transverse of our space, and there is no more of it to be cut. Hence, -putting the plane sections together in the right relations, we have -the section determined by the particular slanting space: namely an -octahedron. - - - - -CHAPTER XIV.[6] - -A RECAPITULATION AND EXTENSION OF THE PHYSICAL ARGUMENT - - -There are two directions of inquiry in which the research for the -physical reality of a fourth dimension can be prosecuted. One is the -investigation of the infinitely great, the other is the investigation -of the infinitely small. - - [6] The contents of this chapter are taken from a paper read before - the Philosophical Society of Washington. The mathematical portion - of the paper has appeared in part in the Proceedings of the Royal - Irish Academy under the title, “Cayley’s formulæ of orthogonal - transformation,” Nov. 29th, 1903. - -By the measurement of the angles of vast triangles, whose sides are the -distances between the stars, astronomers have sought to determine if -there is any deviation from the values given by geometrical deduction. -If the angles of a celestial triangle do not together equal two right -angles, there would be an evidence for the physical reality of a fourth -dimension. - -This conclusion deserves a word of explanation. If space is really -four-dimensional, certain conclusions follow which must be brought -clearly into evidence if we are to frame the questions definitely which -we put to Nature. To account for our limitation let us assume a solid -material sheet against which we move. This sheet must stretch alongside -every object in every direction in which it visibly moves. Every -material body must slip or slide along this sheet, not deviating from -contact with it in any motion which we can observe. - -The necessity for this assumption is clearly apparent, if we consider -the analogous case of a suppositionary plane world. If there were -any creatures whose experiences were confined to a plane, we must -account for their limitation. If they were free to move in every space -direction, they would have a three-dimensional motion; hence they must -be physically limited, and the only way in which we can conceive such -a limitation to exist is by means of a material surface against which -they slide. The existence of this surface could only be known to them -indirectly. It does not lie in any direction from them in which the -kinds of motion they know of leads them. If it were perfectly smooth -and always in contact with every material object, there would be no -difference in their relations to it which would direct their attention -to it. - -But if this surface were curved—if it were, say, in the form of a vast -sphere—the triangles they drew would really be triangles of a sphere, -and when these triangles are large enough the angles diverge from -the magnitudes they would have for the same lengths of sides if the -surface were plane. Hence by the measurement of triangles of very great -magnitude a plane being might detect a difference from the laws of a -plane world in his physical world, and so be led to the conclusion that -there was in reality another dimension to space—a third dimension—as -well as the two which his ordinary experience made him familiar with. - -Now, astronomers have thought it worth while to examine the -measurements of vast triangles drawn from one celestial body to another -with a view to determine if there is anything like a curvature in our -space—that is to say, they have tried astronomical measurements to -find out if the vast solid sheet against which, on the supposition of -a fourth dimension, everything slides is curved or not. These results -have been negative. The solid sheet, if it exists, is not curved or, -being curved, has not a sufficient curvature to cause any observable -deviation from the theoretical value of the angles calculated. - -Hence the examination of the infinitely great leads to no decisive -criterion. If it did we should have to decide between the present -theory and that of metageometry. - -Coming now to the prosecution of the inquiry in the direction of -the infinitely small, we have to state the question thus: Our laws -of movement are derived from the examination of bodies which move -in three-dimensional space. All our conceptions are founded on the -supposition of a space which is represented analytically by three -independent axes and variations along them—that is, it is a space in -which there are three independent movements. Any motion possible in it -can be compounded out of these three movements, which we may call: up, -right, away. - -To examine the actions of the very small portions of matter with the -view of ascertaining if there is any evidence in the phenomena for -the supposition of a fourth dimension of space, we must commence by -clearly defining what the laws of mechanics would be on the supposition -of a fourth dimension. It is of no use asking if the phenomena of the -smallest particles of matter are like—we do not know what. We must -have a definite conception of what the laws of motion would be on the -supposition of the fourth dimension, and then inquire if the phenomena -of the activity of the smaller particles of matter resemble the -conceptions which we have elaborated. - -Now, the task of forming these conceptions is by no means one to be -lightly dismissed. Movement in space has many features which differ -entirely from movement on a plane; and when we set about to form the -conception of motion in four dimensions, we find that there is at least -as great a step as from the plane to three-dimensional space. - -I do not say that the step is difficult, but I want to point out -that it must be taken. When we have formed the conception of -four-dimensional motion, we can ask a rational question of Nature. -Before we have elaborated our conceptions we are asking if an unknown -is like an unknown—a futile inquiry. - -As a matter of fact, four-dimensional movements are in every way simple -and more easy to calculate than three-dimensional movements, for -four-dimensional movements are simply two sets of plane movements put -together. - -Without the formation of an experience of four-dimensional bodies, -their shapes and motions, the subject can be but formal—logically -conclusive, not intuitively evident. It is to this logical apprehension -that I must appeal. - -It is perfectly simple to form an experiential familiarity with the -facts of four-dimensional movement. The method is analogous to that -which a plane being would have to adopt to form an experiential -familiarity with three-dimensional movements, and may be briefly summed -up as the formation of a compound sense by means of which duration is -regarded as equivalent to extension. - -Consider a being confined to a plane. A square enclosed by four lines -will be to him a solid, the interior of which can only be examined by -breaking through the lines. If such a square were to pass transverse to -his plane, it would immediately disappear. It would vanish, going in no -direction to which he could point. - -If, now, a cube be placed in contact with his plane, its surface of -contact would appear like the square which we have just mentioned. -But if it were to pass transverse to his plane, breaking through it, -it would appear as a lasting square. The three-dimensional matter will -give a lasting appearance in circumstances under which two-dimensional -matter will at once disappear. - -Similarly, a four-dimensional cube, or, as we may call it, a tesseract, -which is generated from a cube by a movement of every part of the cube -in a fourth direction at right angles to each of the three visible -directions in the cube, if it moved transverse to our space, would -appear as a lasting cube. - -A cube of three-dimensional matter, since it extends to no distance at -all in the fourth dimension, would instantly disappear, if subjected -to a motion transverse to our space. It would disappear and be gone, -without it being possible to point to any direction in which it had -moved. - -All attempts to visualise a fourth dimension are futile. It must be -connected with a time experience in three space. - -The most difficult notion for a plane being to acquire would be that of -rotation about a line. Consider a plane being facing a square. If he -were told that rotation about a line were possible, he would move his -square this way and that. A square in a plane can rotate about a point, -but to rotate about a line would seem to the plane being perfectly -impossible. How could those parts of his square which were on one side -of an edge come to the other side without the edge moving? He could -understand their reflection in the edge. He could form an idea of the -looking-glass image of his square lying on the opposite side of the -line of an edge, but by no motion that he knows of can he make the -actual square assume that position. The result of the rotation would be -like reflection in the edge, but it would be a physical impossibility -to produce it in the plane. - -The demonstration of rotation about a line must be to him purely -formal. If he conceived the notion of a cube stretching out in an -unknown direction away from his plane, then he can see the base of -it, his square in the plane, rotating round a point. He can likewise -apprehend that every parallel section taken at successive intervals in -the unknown direction rotates in like manner round a point. Thus he -would come to conclude that the whole body rotates round a line—the -line consisting of the succession of points round which the plane -sections rotate. Thus, given three axes, _x_, _y_, _z_, if _x_ rotates -to take the place of _y_, and _y_ turns so as to point to negative -_x_, then the third axis remaining unaffected by this turning is the -axis about which the rotation takes place. This, then, would have to be -his criterion of the axis of a rotation—that which remains unchanged -when a rotation of every plane section of a body takes place. - -There is another way in which a plane being can think about -three-dimensional movements; and, as it affords the type by which we -can most conveniently think about four-dimensional movements, it will -be no loss of time to consider it in detail. - -[Illustration: Fig. 1 (129).] - -We can represent the plane being and his object by figures cut out of -paper, which slip on a smooth surface. The thickness of these bodies -must be taken as so minute that their extension in the third dimension -escapes the observation of the plane being, and he thinks about them -as if they were mathematical plane figures in a plane instead of being -material bodies capable of moving on a plane surface. Let A_x_, A_y_ -be two axes and ABCD a square. As far as movements in the plane are -concerned, the square can rotate about a point A, for example. It -cannot rotate about a side, such as AC. - -But if the plane being is aware of the existence of a third dimension -he can study the movements possible in the ample space, taking his -figure portion by portion. - -His plane can only hold two axes. But, since it can hold two, he is -able to represent a turning into the third dimension if he neglects one -of his axes and represents the third axis as lying in his plane. He can -make a drawing in his plane of what stands up perpendicularly from his -plane. Let A_z_ be the axis, which stands perpendicular to his plane at -A. He can draw in his plane two lines to represent the two axes, A_x_ -and A_z_. Let Fig. 2 be this drawing. Here the _z_ axis has taken the -place of the _y_ axis, and the plane of A_x_ A_z_ is represented in his -plane. In this figure all that exists of the square ABCD will be the -line AB. - -[Illustration: Fig. 2 (130).] - -The square extends from this line in the _y_ direction, but more of -that direction is represented in Fig. 2. The plane being can study the -turning of the line AB in this diagram. It is simply a case of plane -turning around the point A. The line AB occupies intermediate portions -like AB_{1} and after half a revolution will lie on A_x_ produced -through A. - -Now, in the same way, the plane being can take another point, A´, and -another line, A´B´, in his square. He can make the drawing of the two -directions at A´, one along A´B´, the other perpendicular to his plane. -He will obtain a figure precisely similar to Fig. 2, and will see that, -as AB can turn around A, so A´C´ around A. - -In this turning AB and A´B´ would not interfere with each other, as -they would if they moved in the plane around the separate points A and -A´. - -Hence the plane being would conclude that a rotation round a line was -possible. He could see his square as it began to make this turning. He -could see it half way round when it came to lie on the opposite side of -the line AC. But in intermediate portions he could not see it, for it -runs out of the plane. - -Coming now to the question of a four-dimensional body, let us conceive -of it as a series of cubic sections, the first in our space, the rest -at intervals, stretching away from our space in the unknown direction. - -We must not think of a four-dimensional body as formed by moving a -three-dimensional body in any direction which we can see. - -Refer for a moment to Fig. 3. The point A, moving to the right, traces -out the line AC. The line AC, moving away in a new direction, traces -out the square ACEG at the base of the cube. The square AEGC, moving -in a new direction, will trace out the cube ACEGBDHF. The vertical -direction of this last motion is not identical with any motion possible -in the plane of the base of the cube. It is an entirely new direction, -at right angles to every line that can be drawn in the base. To trace -out a tesseract the cube must move in a new direction—a direction at -right angles to any and every line that can be drawn in the space of -the cube. - -The cubic sections of the tesseract are related to the cube we see, as -the square sections of the cube are related to the square of its base -which a plane being sees. - -Let us imagine the cube in our space, which is the base of a tesseract, -to turn about one of its edges. The rotation will carry the whole body -with it, and each of the cubic sections will rotate. The axis we see -in our space will remain unchanged, and likewise the series of axes -parallel to it about which each of the parallel cubic sections rotates. -The assemblage of all of these is a plane. - -Hence in four dimensions a body rotates about a plane. There is no such -thing as rotation round an axis. - -We may regard the rotation from a different point of view. Consider -four independent axes each at right angles to all the others, drawn in -a four-dimensional body. Of these four axes we can see any three. The -fourth extends normal to our space. - -Rotation is the turning of one axis into a second, and the second -turning to take the place of the negative of the first. It involves -two axes. Thus, in this rotation of a four-dimensional body, two axes -change and two remain at rest. Four-dimensional rotation is therefore a -turning about a plane. - -As in the case of a plane being, the result of rotation about a -line would appear as the production of a looking-glass image of the -original object on the other side of the line, so to us the result -of a four-dimensional rotation would appear like the production of a -looking-glass image of a body on the other side of a plane. The plane -would be the axis of the rotation, and the path of the body between its -two appearances would be unimaginable in three-dimensional space. - -[Illustration: Fig. 3 (131).] - -Let us now apply the method by which a plane being could examine -the nature of rotation about a line in our examination of rotation -about a plane. Fig. 3 represents a cube in our space, the three axes -_x_, _y_, _z_ denoting its three dimensions. Let _w_ represent the -fourth dimension. Now, since in our space we can represent any three -dimensions, we can, if we choose, make a representation of what is -in the space determined by the three axes _x_, _z_, _w_. This is a -three-dimensional space determined by two of the axes we have drawn, -_x_ and _z_, and in place of _y_ the fourth axis, _w_. We cannot, -keeping _x_ and _z_, have both _y_ and _w_ in our space; so we will -let _y_ go and draw _w_ in its place. What will be our view of the cube? - -Evidently we shall have simply the square that is in the plane of _xz_, -the square ACDB. The rest of the cube stretches in the _y_ direction, -and, as we have none of the space so determined, we have only the face -of the cube. This is represented in fig. 4. - -[Illustration: Fig. 4 (132).] - -Now, suppose the whole cube to be turned from the _x_ to the _w_ -direction. Conformably with our method, we will not take the whole of -the cube into consideration at once, but will begin with the face ABCD. - -Let this face begin to turn. Fig. 5 represents one of the positions it -will occupy; the line AB remains on the _z_ axis. The rest of the face -extends between the _x_ and the _w_ direction. - -[Illustration: Fig. 5 (133).] - -Now, since we can take any three axes, let us look at what lies in the -space of _zyw_, and examine the turning there. We must now let the _z_ -axis disappear and let the _w_ axis run in the direction in which the -_z_ ran. - -Making this representation, what do we see of the cube? Obviously we -see only the lower face. The rest of the cube lies in the space of -_xyz_. In the space of _xyz_ we have merely the base of the cube lying -in the plane of _xy_, as shown in fig. 6. - -[Illustration: Fig. 6 (134).] - -Now let the _x_ to _w_ turning take place. The square ACEG will turn -about the line AE. This edge will remain along the _y_ axis and will be -stationary, however far the square turns. - -Thus, if the cube be turned by an _x_ to _w_ turning, both the edge AB -and the edge AC remain stationary; hence the whole face ABEF in the -_yz_ plane remains fixed. The turning has taken place about the face -ABEF. - -[Illustration: Fig. 7 (135).] - -Suppose this turning to continue till AC runs to the left from -A. The cube will occupy the position shown in fig. 8. This is -the looking-glass image of the cube in fig. 3. By no rotation in -three-dimensional space can the cube be brought from the position in -fig. 3 to that shown in fig. 8. - -[Illustration: Fig. 8 (136).] - -We can think of this turning as a turning of the face ABCD about AB, -and a turning of each section parallel to ABCD round the vertical line -in which it intersects the face ABEF, the space in which the turning -takes place being a different one from that in which the cube lies. - -One of the conditions, then, of our inquiry in the direction of the -infinitely small is that we form the conception of a rotation about -a plane. The production of a body in a state in which it presents -the appearance of a looking-glass image of its former state is the -criterion for a four-dimensional rotation. - -There is some evidence for the occurrence of such transformations -of bodies in the change of bodies from those which produce a -right-handed polarisation of light to those which produce a left-handed -polarisation; but this is not a point to which any very great -importance can be attached. - -Still, in this connection, let me quote a remark from Prof. John G. -McKendrick’s address on Physiology before the British Association -at Glasgow. Discussing the possibility of the hereditary production -of characteristics through the material structure of the ovum, he -estimates that in it there exist 12,000,000,000 biophors, or ultimate -particles of living matter, a sufficient number to account for -hereditary transmission, and observes: “Thus it is conceivable that -vital activities may also be determined by the kind of motion that -takes place in the molecules of that which we speak of as living -matter. It may be different in kind from some of the motions known to -physicists, and it is conceivable that life may be the transmission -to dead matter, the molecules of which have already a special kind of -motion, of a form of motion _sui generis_.” - -Now, in the realm of organic beings symmetrical structures—those with a -right and left symmetry—are everywhere in evidence. Granted that four -dimensions exist, the simplest turning produces the image form, and by -a folding-over structures could be produced, duplicated right and left, -just as is the case of symmetry in a plane. - -Thus one very general characteristic of the forms of organisms could -be accounted for by the supposition that a four-dimensional motion was -involved in the process of life. - -But whether four-dimensional motions correspond in other respects to -the physiologist’s demand for a special kind of motion, or not, I -do not know. Our business is with the evidence for their existence -in physics. For this purpose it is necessary to examine into the -significance of rotation round a plane in the case of extensible and of -fluid matter. - -Let us dwell a moment longer on the rotation of a rigid body. Looking -at the cube in fig. 3, which turns about the face of ABFE, we see that -any line in the face can take the place of the vertical and horizontal -lines we have examined. Take the diagonal line AF and the section -through it to GH. The portions of matter which were on one side of AF -in this section in fig. 3 are on the opposite side of it in fig. 8. -They have gone round the line AF. Thus the rotation round a face can be -considered as a number of rotations of sections round parallel lines in -it. - -The turning about two different lines is impossible in -three-dimensional space. To take another illustration, suppose A and -B are two parallel lines in the _xy_ plane, and let CD and EF be two -rods crossing them. Now, in the space of _xyz_ if the rods turn round -the lines A and B in the same direction they will make two independent -circles. - -When the end F is going down the end C will be coming up. They will -meet and conflict. - -[Illustration: Fig. 9 (137).] - -But if we rotate the rods about the plane of AB by the _z_ to _w_ -rotation these movements will not conflict. Suppose all the figure -removed with the exception of the plane _xz_, and from this plane draw -the axis of _w_, so that we are looking at the space of _xzw_. - -Here, fig. 10, we cannot see the lines A and B. We see the points G and -H, in which A and B intercept the _x_ axis, but we cannot see the lines -themselves, for they run in the _y_ direction, and that is not in our -drawing. - -Now, if the rods move with the _z_ to _w_ rotation they will turn in -parallel planes, keeping their relative positions. The point D, for -instance, will describe a circle. At one time it will be above the line -A, at another time below it. Hence it rotates round A. - -[Illustration: Fig. 10 (138).] - -Not only two rods but any number of rods crossing the plane will move -round it harmoniously. We can think of this rotation by supposing the -rods standing up from one line to move round that line and remembering -that it is not inconsistent with this rotation for the rods standing up -along another line also to move round it, the relative positions of all -the rods being preserved. Now, if the rods are thick together, they may -represent a disk of matter, and we see that a disk of matter can rotate -round a central plane. - -Rotation round a plane is exactly analogous to rotation round an axis -in three dimensions. If we want a rod to turn round, the ends must be -free; so if we want a disk of matter to turn round its central plane -by a four-dimensional turning, all the contour must be free. The whole -contour corresponds to the ends of the rod. Each point of the contour -can be looked on as the extremity of an axis in the body, round each -point of which there is a rotation of the matter in the disk. - -If the one end of a rod be clamped, we can twist the rod, but not turn -it round; so if any part of the contour of a disk is clamped we can -impart a twist to the disk, but not turn it round its central plane. In -the case of extensible materials a long, thin rod will twist round its -axis, even when the axis is curved, as, for instance, in the case of a -ring of India rubber. - -In an analogous manner, in four dimensions we can have rotation round -a curved plane, if I may use the expression. A sphere can be turned -inside out in four dimensions. - -[Illustration: Fig. 11 (139).] - -Let fig. 11 represent a spherical surface, on each side of which a -layer of matter exists. The thickness of the matter is represented by -the rods CD and EF, extending equally without and within. - -[Illustration: Fig. 12 (140).] - -Now, take the section of the sphere by the _yz_ plane we have a -circle—fig. 12. Now, let the _w_ axis be drawn in place of the _x_ axis -so that we have the space of _yzw_ represented. In this space all that -there will be seen of the sphere is the circle drawn. - -Here we see that there is no obstacle to prevent the rods turning -round. If the matter is so elastic that it will give enough for the -particles at E and C to be separated as they are at F and D, they -can rotate round to the position D and F, and a similar motion is -possible for all other particles. There is no matter or obstacle to -prevent them from moving out in the _w_ direction, and then on round -the circumference as an axis. Now, what will hold for one section will -hold for all, as the fourth dimension is at right angles to all the -sections which can be made of the sphere. - -We have supposed the matter of which the sphere is composed to be -three-dimensional. If the matter had a small thickness in the fourth -dimension, there would be a slight thickness in fig. 12 above the -plane of the paper—a thickness equal to the thickness of the matter -in the fourth dimension. The rods would have to be replaced by thin -slabs. But this would make no difference as to the possibility of the -rotation. This motion is discussed by Newcomb in the first volume of -the _American Journal of Mathematics_. - -Let us now consider, not a merely extensible body, but a liquid one. A -mass of rotating liquid, a whirl, eddy, or vortex, has many remarkable -properties. On first consideration we should expect the rotating mass -of liquid immediately to spread off and lose itself in the surrounding -liquid. The water flies off a wheel whirled round, and we should expect -the rotating liquid to be dispersed. But see the eddies in a river -strangely persistent. The rings that occur in puffs of smoke and last -so long are whirls or vortices curved round so that their opposite ends -join together. A cyclone will travel over great distances. - -Helmholtz was the first to investigate the properties of vortices. -He studied them as they would occur in a perfect fluid—that is, one -without friction of one moving portion or another. In such a medium -vortices would be indestructible. They would go on for ever, altering -their shape, but consisting always of the same portion of the fluid. -But a straight vortex could not exist surrounded entirely by the fluid. -The ends of a vortex must reach to some boundary inside or outside the -fluid. - -A vortex which is bent round so that its opposite ends join is capable -of existing, but no vortex has a free end in the fluid. The fluid -round the vortex is always in motion, and one produces a definite -movement in another. - -Lord Kelvin has proposed the hypothesis that portions of a fluid -segregated in vortices account for the origin of matter. The properties -of the ether in respect of its capacity of propagating disturbances -can be explained by the assumption of vortices in it instead of by a -property of rigidity. It is difficult to conceive, however, of any -arrangement of the vortex rings and endless vortex filaments in the -ether. - -Now, the further consideration of four-dimensional rotations shows the -existence of a kind of vortex which would make an ether filled with a -homogeneous vortex motion easily thinkable. - -To understand the nature of this vortex, we must go on and take a -step by which we accept the full significance of the four-dimensional -hypothesis. Granted four-dimensional axes, we have seen that a rotation -of one into another leaves two unaltered, and these two form the axial -plane about which the rotation takes place. But what about these two? -Do they necessarily remain motionless? There is nothing to prevent a -rotation of these two, one into the other, taking place concurrently -with the first rotation. This possibility of a double rotation deserves -the most careful attention, for it is the kind of movement which is -distinctly typical of four dimensions. - -Rotation round a plane is analogous to rotation round an axis. But in -three-dimensional space there is no motion analogous to the double -rotation, in which, while axis 1 changes into axis 2, axis 3 changes -into axis 4. - -Consider a four-dimensional body, with four independent axes, _x_, -_y_, _z_, _w_. A point in it can move in only one direction at a given -moment. If the body has a velocity of rotation by which the _x_ axis -changes into the _y_ axis and all parallel sections move in a similar -manner, then the point will describe a circle. If, now, in addition -to the rotation by which the _x_ axis changes into the _y_ axis the -body has a rotation by which the _z_ axis turns into the _w_ axis, the -point in question will have a double motion in consequence of the two -turnings. The motions will compound, and the point will describe a -circle, but not the same circle which it would describe in virtue of -either rotation separately. - -We know that if a body in three-dimensional space is given two -movements of rotation they will combine into a single movement of -rotation round a definite axis. It is in no different condition -from that in which it is subjected to one movement of rotation. The -direction of the axis changes; that is all. The same is not true about -a four-dimensional body. The two rotations, _x_ to _y_ and _z_ to _w_, -are independent. A body subject to the two is in a totally different -condition to that which it is in when subject to one only. When subject -to a rotation such as that of _x_ to _y_, a whole plane in the body, -as we have seen, is stationary. When subject to the double rotation -no part of the body is stationary except the point common to the two -planes of rotation. - -If the two rotations are equal in velocity, every point in the body -describes a circle. All points equally distant from the stationary -point describe circles of equal size. - -We can represent a four-dimensional sphere by means of two diagrams, -in one of which we take the three axes, _x_, _y_, _z_; in the -other the axes _x_, _w_, and _z_. In fig. 13 we have the view of a -four-dimensional sphere in the space of _xyz_. Fig. 13 shows all that -we can see of the four sphere in the space of _xyz_, for it represents -all the points in that space, which are at an equal distance from the -centre. - -Let us now take the _xz_ section, and let the axis of _w_ take the -place of the _y_ axis. Here, in fig. 14, we have the space of _xzw_. -In this space we have to take all the points which are at the same -distance from the centre, consequently we have another sphere. If we -had a three-dimensional sphere, as has been shown before, we should -have merely a circle in the _xzw_ space, the _xz_ circle seen in the -space of _xzw_. But now, taking the view in the space of _xzw_, we have -a sphere in that space also. In a similar manner, whichever set of -three axes we take, we obtain a sphere. - -[Illustration: _Showing axes xyz_ -Fig. 13 (141).] - -[Illustration: _Showing axes xwz_ -Fig. 14 (142).] - -In fig. 13, let us imagine the rotation in the direction _xy_ to be -taking place. The point _x_ will turn to _y_, and _p_ to _p´_. The axis -_zz´_ remains stationary, and this axis is all of the plane _zw_ which -we can see in the space section exhibited in the figure. - -In fig. 14, imagine the rotation from _z_ to _w_ to be taking place. -The _w_ axis now occupies the position previously occupied by the _y_ -axis. This does not mean that the _w_ axis can coincide with the _y_ -axis. It indicates that we are looking at the four-dimensional sphere -from a different point of view. Any three-space view will show us three -axes, and in fig. 14 we are looking at _xzw_. - -The only part that is identical in the two diagrams is the circle of -the _x_ and _z_ axes, which axes are contained in both diagrams. Thus -the plane _zxz´_ is the same in both, and the point _p_ represents the -same point in both diagrams. Now, in fig. 14 let the _zw_ rotation -take place, the _z_ axis will turn toward the point _w_ of the _w_ -axis, and the point _p_ will move in a circle about the point _x_. - -Thus in fig. 13 the point _p_ moves in a circle parallel to the _xy_ -plane; in fig. 14 it moves in a circle parallel to the _zw_ plane, -indicated by the arrow. - -Now, suppose both of these independent rotations compounded, the point -_p_ will move in a circle, but this circle will coincide with neither -of the circles in which either one of the rotations will take it. The -circle the point _p_ will move in will depend on its position on the -surface of the four sphere. - -In this double rotation, possible in four-dimensional space, there -is a kind of movement totally unlike any with which we are familiar -in three-dimensional space. It is a requisite preliminary to the -discussion of the behaviour of the small particles of matter, -with a view to determining whether they show the characteristics -of four-dimensional movements, to become familiar with the main -characteristics of this double rotation. And here I must rely on a -formal and logical assent rather than on the intuitive apprehension, -which can only be obtained by a more detailed study. - -In the first place this double rotation consists in two varieties or -kinds, which we will call the A and B kinds. Consider four axes, _x_, -_y_, _z_, _w_. The rotation of _x_ to _y_ can be accompanied with the -rotation of _z_ to _w_. Call this the A kind. - -But also the rotation of _x_ to _y_ can be accompanied by the rotation, -of not _z_ to _w_, but _w_ to _z_. Call this the B kind. - -They differ in only one of the component rotations. One is not the -negative of the other. It is the semi-negative. The opposite of an -_x_ to _y_, _z_ to _w_ rotation would be _y_ to _x_, _w_ to _z_. The -semi-negative is _x_ to _y_ and _w_ to _z_. - -If four dimensions exist and we cannot perceive them, because the -extension of matter is so small in the fourth dimension that all -movements are withheld from direct observation except those which are -three-dimensional, we should not observe these double rotations, but -only the effects of them in three-dimensional movements of the type -with which we are familiar. - -If matter in its small particles is four-dimensional, we should expect -this double rotation to be a universal characteristic of the atoms -and molecules, for no portion of matter is at rest. The consequences -of this corpuscular motion can be perceived, but only under the form -of ordinary rotation or displacement. Thus, if the theory of four -dimensions is true, we have in the corpuscles of matter a whole world -of movement, which we can never study directly, but only by means of -inference. - -The rotation A, as I have defined it, consists of two equal -rotations—one about the plane of _zw_, the other about the plane -of _xy_. It is evident that these rotations are not necessarily -equal. A body may be moving with a double rotation, in which these -two independent components are not equal; but in such a case we can -consider the body to be moving with a composite rotation—a rotation of -the A or B kind and, in addition, a rotation about a plane. - -If we combine an A and a B movement, we obtain a rotation about a -plane; for, the first being _x_ to _y_ and _z_ to _w_, and the second -being _x_ to _y_ and _w_ to _z_, when they are put together the _z_ -to _w_ and _w_ to _z_ rotations neutralise each other, and we obtain -an _x_ to _y_ rotation only, which is a rotation about the plane of -_zw_. Similarly, if we take a B rotation, _y_ to _x_ and _z_ to _w_, -we get, on combining this with the A rotation, a rotation of _z_ to -_w_ about the _xy_ plane. In this case the plane of rotation is in the -three-dimensional space of _xyz_, and we have—what has been described -before—a twisting about a plane in our space. - -Consider now a portion of a perfect liquid having an A motion. It -can be proved that it possesses the properties of a vortex. It -forms a permanent individuality—a separated-out portion of the -liquid—accompanied by a motion of the surrounding liquid. It has -properties analogous to those of a vortex filament. But it is not -necessary for its existence that its ends should reach the boundary of -the liquid. It is self-contained and, unless disturbed, is circular in -every section. - -[Illustration: Fig. 15 (143).] - -If we suppose the ether to have its properties of transmitting -vibration given it by such vortices, we must inquire how they lie -together in four-dimensional space. Placing a circular disk on a plane -and surrounding it by six others, we find that if the central one is -given a motion of rotation, it imparts to the others a rotation which -is antagonistic in every two adjacent ones. If A goes round, as shown -by the arrow, B and C will be moving in opposite ways, and each tends -to destroy the motion of the other. - -Now, if we suppose spheres to be arranged in a corresponding manner -in three-dimensional space, they will be grouped in figures which are -for three-dimensional space what hexagons are for plane space. If a -number of spheres of soft clay be pressed together, so as to fill up -the interstices, each will assume the form of a fourteen-sided figure -called a tetrakaidecagon. - -Now, assuming space to be filled with such tetrakaidecagons, and -placing a sphere in each, it will be found that one sphere is touched -by eight others. The remaining six spheres of the fourteen which -surround the central one will not touch it, but will touch three of -those in contact with it. Hence, if the central sphere rotates, it -will not necessarily drive those around it so that their motions will -be antagonistic to each other, but the velocities will not arrange -themselves in a systematic manner. - -In four-dimensional space the figure which forms the next term of the -series hexagon, tetrakaidecagon, is a thirty-sided figure. It has for -its faces ten solid tetrakaidecagons and twenty hexagonal prisms. Such -figures will exactly fill four-dimensional space, five of them meeting -at every point. If, now, in each of these figures we suppose a solid -four-dimensional sphere to be placed, any one sphere is surrounded by -thirty others. Of these it touches ten, and, if it rotates, it drives -the rest by means of these. Now, if we imagine the central sphere to be -given an A or a B rotation, it will turn the whole mass of sphere round -in a systematic manner. Suppose four-dimensional space to be filled -with such spheres, each rotating with a double rotation, the whole mass -would form one consistent system of motion, in which each one drove -every other one, with no friction or lagging behind. - -Every sphere would have the same kind of rotation. In three-dimensional -space, if one body drives another round the second body rotates -with the opposite kind of rotation; but in four-dimensional space -these four-dimensional spheres would each have the double negative -of the rotation of the one next it, and we have seen that the -double negative of an A or B rotation is still an A or B rotation. -Thus four-dimensional space could be filled with a system of -self-preservative living energy. If we imagine the four-dimensional -spheres to be of liquid and not of solid matter, then, even if the -liquid were not quite perfect and there were a slight retarding effect -of one vortex on another, the system would still maintain itself. - -In this hypothesis we must look on the ether as possessing energy, -and its transmission of vibrations, not as the conveying of a motion -imparted from without, but as a modification of its own motion. - -We are now in possession of some of the conceptions of four-dimensional -mechanics, and will turn aside from the line of their development -to inquire if there is any evidence of their applicability to the -processes of nature. - -Is there any mode of motion in the region of the minute which, giving -three-dimensional movements for its effect, still in itself escapes the -grasp of our mechanical theories? I would point to electricity. Through -the labours of Faraday and Maxwell we are convinced that the phenomena -of electricity are of the nature of the stress and strain of a medium; -but there is still a gap to be bridged over in their explanation—the -laws of elasticity, which Maxwell assumes, are not those of ordinary -matter. And, to take another instance: a magnetic pole in the -neighbourhood of a current tends to move. Maxwell has shown that the -pressures on it are analogous to the velocities in a liquid which would -exist if a vortex took the place of the electric current: but we cannot -point out the definite mechanical explanation of these pressures. There -must be some mode of motion of a body or of the medium in virtue of -which a body is said to be electrified. - -Take the ions which convey charges of electricity 500 times greater in -proportion to their mass than are carried by the molecules of hydrogen -in electrolysis. In respect of what motion can these ions be said to -be electrified? It can be shown that the energy they possess is not -energy of rotation. Think of a short rod rotating. If it is turned -over it is found to be rotating in the opposite direction. Now, if -rotation in one direction corresponds to positive electricity, rotation -in the opposite direction corresponds to negative electricity, and the -smallest electrified particles would have their charges reversed by -being turned over—an absurd supposition. - -If we fix on a mode of motion as a definition of electricity, we must -have two varieties of it, one for positive and one for negative; and a -body possessing the one kind must not become possessed of the other by -any change in its position. - -All three-dimensional motions are compounded of rotations and -translations, and none of them satisfy this first condition for serving -as a definition of electricity. - -But consider the double rotation of the A and B kinds. A body rotating -with the A motion cannot have its motion transformed into the B kind -by being turned over in any way. Suppose a body has the rotation _x_ -to _y_ and _z_ to _w_. Turning it about the _xy_ plane, we reverse the -direction of the motion _x_ to _y_. But we also reverse the _z_ to _w_ -motion, for the point at the extremity of the positive _z_ axis is -now at the extremity of the negative _z_ axis, and since we have not -interfered with its motion it goes in the direction of position _w_. -Hence we have _y_ to _x_ and _w_ to _z_, which is the same as _x_ to -_y_ and _z_ to _w_. Thus both components are reversed, and there is the -A motion over again. The B kind is the semi-negative, with only one -component reversed. - -Hence a system of molecules with the A motion would not destroy it in -one another, and would impart it to a body in contact with them. Thus A -and B motions possess the first requisite which must be demanded in any -mode of motion representative of electricity. - -Let us trace out the consequences of defining positive electricity as -an A motion and negative electricity as a B motion. The combination of -positive and negative electricity produces a current. Imagine a vortex -in the ether of the A kind and unite with this one of the B kind. An -A motion and B motion produce rotation round a plane, which is in the -ether a vortex round an axial surface. It is a vortex of the kind we -represent as a part of a sphere turning inside out. Now such a vortex -must have its rim on a boundary of the ether—on a body in the ether. - -Let us suppose that a conductor is a body which has the property of -serving as the terminal abutment of such a vortex. Then the conception -we must form of a closed current is of a vortex sheet having its edge -along the circuit of the conducting wire. The whole wire will then be -like the centres on which a spindle turns in three-dimensional space, -and any interruption of the continuity of the wire will produce a -tension in place of a continuous revolution. - -As the direction of the rotation of the vortex is from a three-space -direction into the fourth dimension and back again, there will be no -direction of flow to the current; but it will have two sides, according -to whether _z_ goes to _w_ or _z_ goes to negative _w_. - -We can draw any line from one part of the circuit to another; then the -ether along that line is rotating round its points. - -This geometric image corresponds to the definition of an electric -circuit. It is known that the action does not lie in the wire, but in -the medium, and it is known that there is no direction of flow in the -wire. - -No explanation has been offered in three-dimensional mechanics of how -an action can be impressed throughout a region and yet necessarily -run itself out along a closed boundary, as is the case in an electric -current. But this phenomenon corresponds exactly to the definition of a -four-dimensional vortex. - -If we take a very long magnet, so long that one of its poles is -practically isolated, and put this pole in the vicinity of an electric -circuit, we find that it moves. - -Now, assuming for the sake of simplicity that the wire which determines -the current is in the form of a circle, if we take a number of small -magnets and place them all pointing in the same direction normal to -the plane of the circle, so that they fill it and the wire binds them -round, we find that this sheet of magnets has the same effect on -the magnetic pole that the current has. The sheet of magnets may be -curved, but the edge of it must coincide with the wire. The collection -of magnets is then equivalent to the vortex sheet, and an elementary -magnet to a part of it. Thus, we must think of a magnet as conditioning -a rotation in the ether round the plane which bisects at right angles -the line joining its poles. - -If a current is started in a circuit, we must imagine vortices like -bowls turning themselves inside out, starting from the contour. In -reaching a parallel circuit, if the vortex sheet were interrupted and -joined momentarily to the second circuit by a free rim, the axis plane -would lie between the two circuits, and a point on the second circuit -opposite a point on the first would correspond to a point opposite -to it on the first; hence we should expect a current in the opposite -direction in the second circuit. Thus the phenomena of induction are -not inconsistent with the hypothesis of a vortex about an axial plane. - -In four-dimensional space, in which all four dimensions were -commensurable, the intensity of the action transmitted by the medium -would vary inversely as the cube of the distance. Now, the action of -a current on a magnetic pole varies inversely as the square of the -distance; hence, over measurable distances the extension of the ether -in the fourth dimension cannot be assumed as other than small in -comparison with those distances. - -If we suppose the ether to be filled with vortices in the shape of -four-dimensional spheres rotating with the A motion, the B motion would -correspond to electricity in the one-fluid theory. There would thus -be a possibility of electricity existing in two forms, statically, -by itself, and, combined with the universal motion, in the form of a -current. - -To arrive at a definite conclusion it will be necessary to investigate -the resultant pressures which accompany the collocation of solid -vortices with surface ones. - -To recapitulate: - -The movements and mechanics of four-dimensional space are definite and -intelligible. A vortex with a surface as its axis affords a geometric -image of a closed circuit, and there are rotations which by their -polarity afford a possible definition of statical electricity.[7] - - [7] These double rotations of the A and B kinds I should like to call - Hamiltons and co-Hamiltons, for it is a singular fact that in his - “Quaternions” Sir Wm. Rowan Hamilton has given the theory of either - the A or the B kind. They follow the laws of his symbols, I, J, K. - -Hamiltons and co-Hamiltons seem to be natural units of geometrical -expression. In the paper in the “Proceedings of the Royal Irish -Academy,” Nov. 1903, already alluded to, I have shown something of the -remarkable facility which is gained in dealing with the composition of -three- and four-dimensional rotations by an alteration in Hamilton’s -notation, which enables his system to be applied to both the A and B -kinds of rotations. - -The objection which has been often made to Hamilton’s system, namely, -that it is only under special conditions of application that his -processes give geometrically interpretable results, can be removed, if -we assume that he was really dealing with a four-dimensional motion, -and alter his notation to bring this circumstance into explicit -recognition. - - - - - APPENDIX I - - THE MODELS - - -In Chapter XI. a description has been given which will enable any -one to make a set of models illustrative of the tesseract and its -properties. The set here supposed to be employed consists of:— - - 1. Three sets of twenty-seven cubes each. - 2. Twenty-seven slabs. - 3. Twelve cubes with points, lines, faces, distinguished by colours, - which will be called the catalogue cubes. - -The preparation of the twelve catalogue cubes involves the expenditure -of a considerable amount of time. It is advantageous to use them, but -they can be replaced by the drawing of the views of the tesseract or by -a reference to figs. 103, 104, 105, 106 of the text. - -The slabs are coloured like the twenty-seven cubes of the first cubic -block in fig. 101, the one with red, white, yellow axes. - -The colours of the three sets of twenty-seven cubes are those of the -cubes shown in fig. 101. - -The slabs are used to form the representation of a cube in a plane, and -can well be dispensed with by any one who is accustomed to deal with -solid figures. But the whole theory depends on a careful observation of -how the cube would be represented by these slabs. - -In the first step, that of forming a clear idea how a plane being -would represent three-dimensional space, only one of the catalogue -cubes and one of the three blocks is needed. - - - APPLICATION TO THE STEP FROM PLANE TO SOLID. - -Look at fig. 1 of the views of the tesseract, or, what comes to the -same thing, take catalogue cube No. 1 and place it before you with the -red line running up, the white line running to the right, the yellow -line running away. The three dimensions of space are then marked out -by these lines or axes. Now take a piece of cardboard, or a book, and -place it so that it forms a wall extending up and down not opposite to -you, but running away parallel to the wall of the room on your left -hand. - -Placing the catalogue cube against this wall we see that it comes into -contact with it by the red and yellow lines, and by the included orange -face. - -In the plane being’s world the aspect he has of the cube would be a -square surrounded by red and yellow lines with grey points. - -Now, keeping the red line fixed, turn the cube about it so that the -yellow line goes out to the right, and the white line comes into -contact with the plane. - -In this case a different aspect is presented to the plane being, a -square, namely, surrounded by red and white lines and grey points. You -should particularly notice that when the yellow line goes out, at right -angles to the plane, and the white comes in, the latter does not run in -the same sense that the yellow did. - -From the fixed grey point at the base of the red line the yellow line -ran away from you. The white line now runs towards you. This turning -at right angles makes the line which was out of the plane before, come -into it in an opposite sense to that in which the line ran which has -just left the plane. If the cube does not break through the plane this -is always the rule. - -Again turn the cube back to the normal position with red running up, -white to the right, and yellow away, and try another turning. - -You can keep the yellow line fixed, and turn the cube about it. In this -case the red line going out to the right the white line will come in -pointing downwards. - -You will be obliged to elevate the cube from the table in order to -carry out this turning. It is always necessary when a vertical axis -goes out of a space to imagine a movable support which will allow the -line which ran out before to come in below. - -Having looked at the three ways of turning the cube so as to present -different faces to the plane, examine what would be the appearance if -a square hole were cut in the piece of cardboard, and the cube were to -pass through it. A hole can be actually cut, and it will be seen that -in the normal position, with red axis running up, yellow away, and -white to the right, the square first perceived by the plane being—the -one contained by red and yellow lines—would be replaced by another -square of which the line towards you is pink—the section line of the -pink face. The line above is light yellow, below is light yellow and on -the opposite side away from you is pink. - -In the same way the cube can be pushed through a square opening in the -plane from any of the positions which you have already turned it into. -In each case the plane being will perceive a different set of contour -lines. - -Having observed these facts about the catalogue cube, turn now to the -first block of twenty-seven cubes. - -You notice that the colour scheme on the catalogue cube and that of -this set of blocks is the same. - -Place them before you, a grey or null cube on the table, above it a -red cube, and on the top a null cube again. Then away from you place a -yellow cube, and beyond it a null cube. Then to the right place a white -cube and beyond it another null. Then complete the block, according to -the scheme of the catalogue cube, putting in the centre of all an ochre -cube. - -You have now a cube like that which is described in the text. For the -sake of simplicity, in some cases, this cubic block can be reduced to -one of eight cubes, by leaving out the terminations in each direction. -Thus, instead of null, red, null, three cubes, you can take null, red, -two cubes, and so on. - -It is useful, however, to practise the representation in a plane of a -block of twenty-seven cubes. For this purpose take the slabs, and build -them up against the piece of cardboard, or the book in such a way as to -represent the different aspects of the cube. - -Proceed as follows:— - -First, cube in normal position. - -Place nine slabs against the cardboard to represent the nine cubes -in the wall of the red and yellow axes, facing the cardboard; these -represent the aspect of the cube as it touches the plane. - -Now push these along the cardboard and make a different set of nine -slabs to represent the appearance which the cube would present to a -plane being, if it were to pass half way through the plane. - -There would be a white slab, above it a pink one, above that another -white one, and six others, representing what would be the nature of a -section across the middle of the block of cubes. The section can be -thought of as a thin slice cut out by two parallel cuts across the -cube. Having arranged these nine slabs, push them along the plane, and -make another set of nine to represent what would be the appearance of -the cube when it had almost completely gone through. This set of nine -will be the same as the first set of nine. - -Now we have in the plane three sets of nine slabs each, which represent -three sections of the twenty-seven block. - -They are put alongside one another. We see that it does not matter in -what order the sets of nine are put. As the cube passes through the -plane they represent appearances which follow the one after the other. -If they were what they represented, they could not exist in the same -plane together. - -This is a rather important point, namely, to notice that they should -not co-exist on the plane, and that the order in which they are placed -is indifferent. When we represent a four-dimensional body our solid -cubes are to us in the same position that the slabs are to the plane -being. You should also notice that each of these slabs represents only -the very thinnest slice of a cube. The set of nine slabs first set up -represents the side surface of the block. It is, as it were, a kind -of tray—a beginning from which the solid cube goes off. The slabs -as we use them have thickness, but this thickness is a necessity of -construction. They are to be thought of as merely of the thickness of a -line. - -If now the block of cubes passed through the plane at the rate of an -inch a minute the appearance to a plane being would be represented by:— - -1. The first set of nine slabs lasting for one minute. - -2. The second set of nine slabs lasting for one minute. - -3. The third set of nine slabs lasting for one minute. - -Now the appearances which the cube would present to the plane being -in other positions can be shown by means of these slabs. The use of -such slabs would be the means by which a plane being could acquire a -familiarity with our cube. Turn the catalogue cube (or imagine the -coloured figure turned) so that the red line runs up, the yellow line -out to the right, and the white line towards you. Then turn the block -of cubes to occupy a similar position. - -The block has now a different wall in contact with the plane. Its -appearance to a plane being will not be the same as before. He has, -however, enough slabs to represent this new set of appearances. But he -must remodel his former arrangement of them. - -He must take a null, a red, and a null slab from the first of his sets -of slabs, then a white, a pink, and a white from the second, and then a -null, a red, and a null from the third set of slabs. - -He takes the first column from the first set, the first column from the -second set, and the first column from the third set. - -To represent the half-way-through appearance, which is as if a very -thin slice were cut out half way through the block, he must take the -second column of each of his sets of slabs, and to represent the final -appearance, the third column of each set. - -Now turn the catalogue cube back to the normal position, and also the -block of cubes. - -There is another turning—a turning about the yellow line, in which the -white axis comes below the support. - -You cannot break through the surface of the table, so you must imagine -the old support to be raised. Then the top of the block of cubes in its -new position is at the level at which the base of it was before. - -Now representing the appearance on the plane, we must draw a horizontal -line to represent the old base. The line should be drawn three inches -high on the cardboard. - -Below this the representative slabs can be arranged. - -It is easy to see what they are. The old arrangements have to be -broken up, and the layers taken in order, the first layer of each for -the representation of the aspect of the block as it touches the plane. - -Then the second layers will represent the appearance half way through, -and the third layers will represent the final appearance. - -It is evident that the slabs individually do not represent the same -portion of the cube in these different presentations. - -In the first case each slab represents a section or a face -perpendicular to the white axis, in the second case a face or a section -which runs perpendicularly to the yellow axis, and in the third case a -section or a face perpendicular to the red axis. - -But by means of these nine slabs the plane being can represent the -whole of the cubic block. He can touch and handle each portion of the -cubic block, there is no part of it which he cannot observe. Taking it -bit by bit, two axes at a time, he can examine the whole of it. - - - OUR REPRESENTATION OF A BLOCK OF TESSERACTS. - -Look at the views of the tesseract 1, 2, 3, or take the catalogue cubes -1, 2, 3, and place them in front of you, in any order, say running from -left to right, placing 1 in the normal position, the red axis running -up, the white to the right, and yellow away. - -Now notice that in catalogue cube 2 the colours of each region are -derived from those of the corresponding region of cube 1 by the -addition of blue. Thus null + blue = blue, and the corners of number 2 -are blue. Again, red + blue = purple, and the vertical lines of 2 are -purple. Blue + yellow = green, and the line which runs away is coloured -green. - -By means of these observations you may be sure that catalogue cube 2 -is rightly placed. Catalogue cube 3 is just like number 1. - -Having these cubes in what we may call their normal position, proceed -to build up the three sets of blocks. - -This is easily done in accordance with the colour scheme on the -catalogue cubes. - -The first block we already know. Build up the second block, beginning -with a blue corner cube, placing a purple on it, and so on. - -Having these three blocks we have the means of representing the -appearances of a group of eighty-one tesseracts. - -Let us consider a moment what the analogy in the case of the plane -being is. - -He has his three sets of nine slabs each. We have our three sets of -twenty-seven cubes each. - -Our cubes are like his slabs. As his slabs are not the things which -they represent to him, so our cubes are not the things they represent -to us. - -The plane being’s slabs are to him the faces of cubes. - -Our cubes then are the faces of tesseracts, the cubes by which they are -in contact with our space. - -As each set of slabs in the case of the plane being might be considered -as a sort of tray from which the solid contents of the cubes came out, -so our three blocks of cubes may be considered as three-space trays, -each of which is the beginning of an inch of the solid contents of the -four-dimensional solids starting from them. - -We want now to use the names null, red, white, etc., for tesseracts. -The cubes we use are only tesseract faces. Let us denote that fact -by calling the cube of null colour, null face; or, shortly, null f., -meaning that it is the face of a tesseract. - -To determine which face it is let us look at the catalogue cube 1 or -the first of the views of the tesseract, which can be used instead of -the models. It has three axes, red, white, yellow, in our space. Hence -the cube determined by these axes is the face of the tesseract which we -now have before us. It is the ochre face. It is enough, however, simply -to say null f., red f. for the cubes which we use. - -To impress this in your mind, imagine that tesseracts do actually run -from each cube. Then, when you move the cubes about, you move the -tesseracts about with them. You move the face but the tesseract follows -with it, as the cube follows when its face is shifted in a plane. - -The cube null in the normal position is the cube which has in it the -red, yellow, white axes. It is the face having these, but wanting the -blue. In this way you can define which face it is you are handling. I -will write an “f.” after the name of each tesseract just as the plane -being might call each of his slabs null slab, yellow slab, etc., to -denote that they were representations. - -We have then in the first block of twenty-seven cubes, the -following—null f., red f., null f., going up; white f., null f., lying -to the right, and so on. Starting from the null point and travelling -up one inch we are in the null region, the same for the away and the -right-hand directions. And if we were to travel in the fourth dimension -for an inch we should still be in a null region. The tesseract -stretches equally all four ways. Hence the appearance we have in this -first block would do equally well if the tesseract block were to move -across our space for a certain distance. For anything less than an inch -of their transverse motion we should still have the same appearance. -You must notice, however, that we should not have null face after the -motion had begun. - -When the tesseract, null for instance, had moved ever so little we -should not have a face of null but a section of null in our space. -Hence, when we think of the motion across our space we must call our -cubes tesseract sections. Thus on null passing across we should see -first null f., then null s., and then, finally, null f. again. - -Imagine now the whole first block of twenty-seven tesseracts to have -moved tranverse to our space a distance of one inch. Then the second -set of tesseracts, which originally were an inch distant from our -space, would be ready to come in. - -Their colours are shown in the second block of twenty-seven cubes which -you have before you. These represent the tesseract faces of the set of -tesseracts that lay before an inch away from our space. They are ready -now to come in, and we can observe their colours. In the place which -null f. occupied before we have blue f., in place of red f. we have -purple f., and so on. Each tesseract is coloured like the one whose -place it takes in this motion with the addition of blue. - -Now if the tesseract block goes on moving at the rate of an inch a -minute, this next set of tesseracts will occupy a minute in passing -across. We shall see, to take the null one for instance, first of all -null face, then null section, then null face again. - -At the end of the second minute the second set of tesseracts has gone -through, and the third set comes in. This, as you see, is coloured just -like the first. Altogether, these three sets extend three inches in the -fourth dimension, making the tesseract block of equal magnitude in all -dimensions. - -We have now before us a complete catalogue of all the tesseracts in our -group. We have seen them all, and we shall refer to this arrangement -of the blocks as the “normal position.” We have seen as much of each -tesseract at a time as could be done in a three-dimensional space. Each -part of each tesseract has been in our space, and we could have touched -it. - -The fourth dimension appeared to us as the duration of the block. - -If a bit of our matter were to be subjected to the same motion it -would be instantly removed out of our space. Being thin in the fourth -dimension it is at once taken out of our space by a motion in the -fourth dimension. - -But the tesseract block we represent having length in the fourth -dimension remains steadily before our eyes for three minutes, when it -is subjected to this transverse motion. - -We have now to form representations of the other views of the same -tesseract group which are possible in our space. - -Let us then turn the block of tesseracts so that another face of it -comes into contact with our space, and then by observing what we have, -and what changes come when the block traverses our space, we shall have -another view of it. The dimension which appeared as duration before -will become extension in one of our known dimensions, and a dimension -which coincided with one of our space dimensions will appear as -duration. - -Leaving catalogue cube 1 in the normal position, remove the other two, -or suppose them removed. We have in space the red, the yellow, and the -white axes. Let the white axis go out into the unknown, and occupy the -position the blue axis holds. Then the blue axis, which runs in that -direction now will come into space. But it will not come in pointing -in the same way that the white axis does now. It will point in the -opposite sense. It will come in running to the left instead of running -to the right as the white axis does now. - -When this turning takes place every part of the cube 1 will disappear -except the left-hand face—the orange face. - -And the new cube that appears in our space will run to the left from -this orange face, having axes, red, yellow, blue. - -Take models 4, 5, 6. Place 4, or suppose No. 4 of the tesseract views -placed, with its orange face coincident with the orange face of 1, red -line to red line, and yellow line to yellow line, with the blue line -pointing to the left. Then remove cube 1 and we have the tesseract face -which comes in when the white axis runs in the positive unknown, and -the blue axis comes into our space. - -Now place catalogue cube 5 in some position, it does not matter which, -say to the left; and place it so that there is a correspondence of -colour corresponding to the colour of the line that runs out of space. -The line that runs out of space is white, hence, every part of this -cube 5 should differ from the corresponding part of 4 by an alteration -in the direction of white. - -Thus we have white points in 5 corresponding to the null points in -4. We have a pink line corresponding to a red line, a light yellow -line corresponding to a yellow line, an ochre face corresponding to -an orange face. This cube section is completely named in Chapter XI. -Finally cube 6 is a replica of 1. - -These catalogue cubes will enable us to set up our models of the block -of tesseracts. - -First of all for the set of tesseracts, which beginning in our space -reach out one inch in the unknown, we have the pattern of catalogue -cube 4. - -We see that we can build up a block of twenty-seven tesseract faces -after the colour scheme of cube 4, by taking the left-hand wall of -block 1, then the left-hand wall of block 2, and finally that of block -3. We take, that is, the three first walls of our previous arrangement -to form the first cubic block of this new one. - -This will represent the cubic faces by which the group of tesseracts in -its new position touches our space. We have running up, null f., red -f., null f. In the next vertical line, on the side remote from us, we -have yellow f., orange f., yellow f., and then the first colours over -again. Then the three following columns are, blue f., purple f., blue -f.; green f., brown f., green f.; blue f., purple f., blue f. The last -three columns are like the first. - -These tesseracts touch our space, and none of them are by any part of -them distant more than an inch from it. What lies beyond them in the -unknown? - -This can be told by looking at catalogue cube 5. According to its -scheme of colour we see that the second wall of each of our old -arrangements must be taken. Putting them together we have, as the -corner, white f. above it, pink f. above it, white f. The column next -to this remote from us is as follows:—light yellow f., ochre f., light -yellow f., and beyond this a column like the first. Then for the middle -of the block, light blue f., above it light purple, then light blue. -The centre column has, at the bottom, light green f., light brown f. -in the centre and at the top light green f. The last wall is like the -first. - -The third block is made by taking the third walls of our previous -arrangement, which we called the normal one. - -You may ask what faces and what sections our cubes represent. To answer -this question look at what axes you have in our space. You have red, -yellow, blue. Now these determine brown. The colours red, yellow, blue -are supposed by us when mixed to produce a brown colour. And that cube -which is determined by the red, yellow, blue axes we call the brown -cube. - -When the tesseract block in its new position begins to move across our -space each tesseract in it gives a section in our space. This section -is transverse to the white axis, which now runs in the unknown. - -As the tesseract in its present position passes across our space, we -should see first of all the first of the blocks of cubic faces we have -put up—these would last for a minute, then would come the second block -and then the third. At first we should have a cube of tesseract faces, -each of which would be brown. Directly the movement began, we should -have tesseract sections transverse to the white line. - -There are two more analogous positions in which the block of tesseracts -can be placed. To find the third position, restore the blocks to the -normal arrangement. - -Let us make the yellow axis go out into the positive unknown, and let -the blue axis, consequently, come in running towards us. The yellow ran -away, so the blue will come in running towards us. - -Put catalogue cube 1 in its normal position. Take catalogue cube 7 -and place it so that its pink face coincides with the pink face of -cube 1, making also its red axis coincide with the red axis of 1 and -its white with the white. Moreover, make cube 7 come towards us from -cube 1. Looking at it we see in our space, red, white, and blue axes. -The yellow runs out. Place catalogue cube 8 in the neighbourhood -of 7—observe that every region in 8 has a change in the direction -of yellow from the corresponding region in 7. This is because it -represents what you come to now in going in the unknown, when the -yellow axis runs out of our space. Finally catalogue cube 9, which is -like number 7, shows the colours of the third set of tesseracts. Now -evidently, starting from the normal position, to make up our three -blocks of tesseract faces we have to take the near wall from the first -block, the near wall from the second, and then the near wall from the -third block. This gives us the cubic block formed by the faces of the -twenty-seven tesseracts which are now immediately touching our space. - -Following the colour scheme of catalogue cube 8, we make the next set -of twenty-seven tesseract faces, representing the tesseracts, each of -which begins one inch off from our space, by putting the second walls -of our previous arrangement together, and the representation of the -third set of tesseracts is the cubic block formed of the remaining -three walls. - -Since we have red, white, blue axes in our space to begin with, the -cubes we see at first are light purple tesseract faces, and after the -transverse motion begins we have cubic sections transverse to the -yellow line. - -Restore the blocks to the normal position, there remains the case in -which the red axis turns out of space. In this case the blue axis will -come in downwards, opposite to the sense in which the red axis ran. - -In this case take catalogue cubes 10, 11, 12. Lift up catalogue cube 1 -and put 10 underneath it, imagining that it goes down from the previous -position of 1. - -We have to keep in space the white and the yellow axes, and let the red -go out, the blue come in. - -Now, you will find on cube 10 a light yellow face; this should coincide -with the base of 1, and the white and yellow lines on the two cubes -should coincide. Then the blue axis running down you have the catalogue -cube correctly placed, and it forms a guide for putting up the first -representative block. - -Catalogue cube 11 will represent what lies in the fourth dimension—now -the red line runs in the fourth dimension. Thus the change from 10 to -11 should be towards red, corresponding to a null point is a red point, -to a white line is a pink line, to a yellow line an orange line, and so -on. - -Catalogue cube 12 is like 10. Hence we see that to build up our blocks -of tesseract faces we must take the bottom layer of the first block, -hold that up in the air, underneath it place the bottom layer of the -second block, and finally underneath this last the bottom layer of the -last of our normal blocks. - -Similarly we make the second representative group by taking the middle -courses of our three blocks. The last is made by taking the three -topmost layers. The three axes in our space before the transverse -motion begins are blue, white, yellow, so we have light green tesseract -faces, and after the motion begins sections transverse to the red light. - -These three blocks represent the appearances as the tesseract group in -its new position passes across our space. The cubes of contact in this -case are those determinal by the three axes in our space, namely, the -white, the yellow, the blue. Hence they are light green. - -It follows from this that light green is the interior cube of the first -block of representative cubic faces. - -Practice in the manipulations described, with a realization in each -case of the face or section which is in our space, is one of the best -means of a thorough comprehension of the subject. - -We have to learn how to get any part of these four-dimensional figures -into space, so that we can look at them. We must first learn to swing a -tesseract, and a group of tesseracts about in any way. - -When these operations have been repeated and the method of arrangement -of the set of blocks has become familiar, it is a good plan to rotate -the axes of the normal cube 1 about a diagonal, and then repeat the -whole series of turnings. - -Thus, in the normal position, red goes up, white to the right, yellow -away. Make white go up, yellow to the right, and red away. Learn the -cube in this position by putting up the set of blocks of the normal -cube, over and over again till it becomes as familiar to you as in the -normal position. Then when this is learned, and the corresponding -changes in the arrangements of the tesseract groups are made, another -change should be made: let, in the normal cube, yellow go up, red to -the right, and white away. - -Learn the normal block of cubes in this new position by arranging them -and re-arranging them till you know without thought where each one -goes. Then carry out all the tesseract arrangements and turnings. - -If you want to understand the subject, but do not see your way clearly, -if it does not seem natural and easy to you, practise these turnings. -Practise, first of all, the turning of a block of cubes round, so that -you know it in every position as well as in the normal one. Practise by -gradually putting up the set of cubes in their new arrangements. Then -put up the tesseract blocks in their arrangements. This will give you -a working conception of higher space, you will gain the feeling of it, -whether you take up the mathematical treatment of it or not. - - - - - APPENDIX II - - A LANGUAGE OF SPACE - - -The mere naming the parts of the figures we consider involves a certain -amount of time and attention. This time and attention leads to no -result, for with each new figure the nomenclature applied is completely -changed, every letter or symbol is used in a different significance. - -Surely it must be possible in some way to utilise the labour thus at -present wasted! - -Why should we not make a language for space itself, so that every -position we want to refer to would have its own name? Then every time -we named a figure in order to demonstrate its properties we should be -exercising ourselves in the vocabulary of place. - -If we use a definite system of names, and always refer to the same -space position by the same name, we create as it were a multitude of -little hands, each prepared to grasp a special point, position, or -element, and hold it for us in its proper relations. - -We make, to use another analogy, a kind of mental paper, which has -somewhat of the properties of a sensitive plate, in that it will -register, without effort, complex, visual, or tactual impressions. - -But of far more importance than the applications of a space language to -the plane and to solid space is the facilitation it brings with it to -the study of four-dimensional shapes. - -I have delayed introducing a space language because all the systems I -made turned out, after giving them a fair trial, to be intolerable. I -have now come upon one which seems to present features of permanence, -and I will here give an outline of it, so that it can be applied to the -subject of the text, and in order that it may be subjected to criticism. - -The principle on which the language is constructed is to sacrifice -every other consideration for brevity. - -It is indeed curious that we are able to talk and converse on every -subject of thought except the fundamental one of space. The only way of -speaking about the spatial configurations that underlie every subject -of discursive thought is a co-ordinate system of numbers. This is so -awkward and incommodious that it is never used. In thinking also, in -realising shapes, we do not use it; we confine ourselves to a direct -visualisation. - -Now, the use of words corresponds to the storing up of our experience -in a definite brain structure. A child, in the endless tactual, visual, -mental manipulations it makes for itself, is best left to itself, but -in the course of instruction the introduction of space names would -make the teachers work more cumulative, and the child’s knowledge more -social. - -Their full use can only be appreciated, if they are introduced early -in the course of education; but in a minor degree any one can convince -himself of their utility, especially in our immediate subject of -handling four-dimensional shapes. The sum total of the results obtained -in the preceding pages can be compendiously and accurately expressed in -nine words of the Space Language. - -In one of Plato’s dialogues Socrates makes an experiment on a slave boy -standing by. He makes certain perceptions of space awake in the mind -of Meno’s slave by directing his close attention on some simple facts -of geometry. - -By means of a few words and some simple forms we can repeat Plato’s -experiment on new ground. - -Do we by directing our close attention on the facts of four dimensions -awaken a latent faculty in ourselves? The old experiment of Plato’s, it -seems to me, has come down to us as novel as on the day he incepted it, -and its significance not better understood through all the discussion -of which it has been the subject. - -Imagine a voiceless people living in a region where everything had -a velvety surface, and who were thus deprived of all opportunity of -experiencing what sound is. They could observe the slow pulsations -of the air caused by their movements, and arguing from analogy, they -would no doubt infer that more rapid vibrations were possible. From -the theoretical side they could determine all about these more rapid -vibrations. They merely differ, they would say, from slower ones, -by the number that occur in a given time; there is a merely formal -difference. - -But suppose they were to take the trouble, go to the pains of producing -these more rapid vibrations, then a totally new sensation would fall -on their rudimentary ears. Probably at first they would only be dimly -conscious of Sound, but even from the first they would become aware -that a merely formal difference, a mere difference in point of number -in this particular respect, made a great difference practically, as -related to them. And to us the difference between three and four -dimensions is merely formal, numerical. We can tell formally all about -four dimensions, calculate the relations that would exist. But that -the difference is merely formal does not prove that it is a futile and -empty task, to present to ourselves as closely as we can the phenomena -of four dimensions. In our formal knowledge of it, the whole question -of its actual relation to us, as we are, is left in abeyance. - -Possibly a new apprehension of nature may come to us through the -practical, as distinguished from the mathematical and formal, study -of four dimensions. As a child handles and examines the objects with -which he comes in contact, so we can mentally handle and examine -four-dimensional objects. The point to be determined is this. Do we -find something cognate and natural to our faculties, or are we merely -building up an artificial presentation of a scheme only formally -possible, conceivable, but which has no real connection with any -existing or possible experience? - -This, it seems to me, is a question which can only be settled by -actually trying. This practical attempt is the logical and direct -continuation of the experiment Plato devised in the “Meno.” - -Why do we think true? Why, by our processes of thought, can we predict -what will happen, and correctly conjecture the constitution of the -things around us? This is a problem which every modern philosopher has -considered, and of which Descartes, Leibnitz, Kant, to name a few, -have given memorable solutions. Plato was the first to suggest it. -And as he had the unique position of being the first devisor of the -problem, so his solution is the most unique. Later philosophers have -talked about consciousness and its laws, sensations, categories. But -Plato never used such words. Consciousness apart from a conscious being -meant nothing to him. His was always an objective search. He made man’s -intuitions the basis of a new kind of natural history. - -In a few simple words Plato puts us in an attitude with regard to -psychic phenomena—the mind—the ego—“what we are,” which is analogous -to the attitude scientific men of the present day have with regard -to the phenomena of outward nature. Behind this first apprehension -of ours of nature, there is an infinite depth to be learned and -known. Plato said that behind the phenomena of mind that Meno’s slave -boy exhibited, there was a vast, an infinite perspective. And his -singularity, his originality, comes out most strongly marked in this, -that the perspective, the complex phenomena beyond were, according to -him, phenomena of personal experience. A footprint in the sand means a -man to a being that has the conception of a man. But to a creature that -has no such conception, it means a curious mark, somehow resulting from -the concatenation of ordinary occurrences. Such a being would attempt -merely to explain how causes known to him could so coincide as to -produce such a result; he would not recognise its significance. - -Plato introduced the conception which made a new kind of natural -history possible. He said that Meno’s slave boy thought true about -things he had never learned, because his “soul” had experience. I -know this will sound absurd to some people, and it flies straight in -the face of the maxim, that explanation consists in showing how an -effect depends on simple causes. But what a mistaken maxim that is! -Can any single instance be shown of a simple cause? Take the behaviour -of spheres for instance; say those ivory spheres, billiard balls, -for example. We can explain their behaviour by supposing they are -homogeneous elastic solids. We can give formulæ which will account for -their movements in every variety. But are they homogeneous elastic -solids? No, certainly not. They are complex in physical and molecular -structure, and atoms and ions beyond open an endless vista. Our simple -explanation is false, false as it can be. The balls act as if they -were homogeneous elastic spheres. There is a statistical simplicity in -the resultant of very complex conditions, which makes that artificial -conception useful. But its usefulness must not blind us to the fact -that it is artificial. If we really look deep into nature, we find a -much greater complexity than we at first suspect. And so behind this -simple “I,” this myself, is there not a parallel complexity? Plato’s -“soul” would be quite acceptable to a large class of thinkers, if by -“soul” and the complexity he attributes to it, he meant the product of -a long course of evolutionary changes, whereby simple forms of living -matter endowed with rudimentary sensation had gradually developed into -fully conscious beings. - -But Plato does not mean by “soul” a being of such a kind. His soul is -a being whose faculties are clogged by its bodily environment, or at -least hampered by the difficulty of directing its bodily frame—a being -which is essentially higher than the account it gives of itself through -its organs. At the same time Plato’s soul is not incorporeal. It is a -real being with a real experience. The question of whether Plato had -the conception of non-spatial existence has been much discussed. The -verdict is, I believe, that even his “ideas” were conceived by him as -beings in space, or, as we should say, real. Plato’s attitude is that -of Science, inasmuch as he thinks of a world in Space. But, granting -this, it cannot be denied that there is a fundamental divergence -between Plato’s conception and the evolutionary theory, and also an -absolute divergence between his conception and the genetic account of -the origin of the human faculties. The functions and capacities of -Plato’s “soul” are not derived by the interaction of the body and its -environment. - -Plato was engaged on a variety of problems, and his religious and -ethical thoughts were so keen and fertile that the experimental -investigation of his soul appears involved with many other motives. -In one passage Plato will combine matter of thought of all kinds and -from all sources, overlapping, interrunning. And in no case is he more -involved and rich than in this question of the soul. In fact, I wish -there were two words, one denoting that being, corporeal and real, but -with higher faculties than we manifest in our bodily actions, which is -to be taken as the subject of experimental investigation; and the other -word denoting “soul” in the sense in which it is made the recipient and -the promise of so much that men desire. It is the soul in the former -sense that I wish to investigate, and in a limited sphere only. I wish -to find out, in continuation of the experiment in the Meno, what the -“soul” in us thinks about extension, experimenting on the grounds laid -down by Plato. He made, to state the matter briefly, the hypothesis -with regard to the thinking power of a being in us, a “soul.” This -soul is not accessible to observation by sight or touch, but it can be -observed by its functions; it is the object of a new kind of natural -history, the materials for constructing which lie in what it is natural -to us to think. With Plato “thought” was a very wide-reaching term, but -still I would claim in his general plan of procedure a place for the -particular question of extension. - -The problem comes to be, “What is it natural to us to think about -matter _qua_ extended?” - -First of all, I find that the ordinary intuition of any simple object -is extremely imperfect. Take a block of differently marked cubes, for -instance, and become acquainted with them in their positions. You may -think you know them quite well, but when you turn them round—rotate -the block round a diagonal, for instance—you will find that you have -lost track of the individuals in their new positions. You can mentally -construct the block in its new position, by a rule, by taking the -remembered sequences, but you don’t know it intuitively. By observation -of a block of cubes in various positions, and very expeditiously -by a use of Space names applied to the cubes in their different -presentations, it is possible to get an intuitive knowledge of the -block of cubes, which is not disturbed by any displacement. Now, with -regard to this intuition, we moderns would say that I had formed it by -my tactual visual experiences (aided by hereditary pre-disposition). -Plato would say that the soul had been stimulated to recognise an -instance of shape which it knew. Plato would consider the operation -of learning merely as a stimulus; we as completely accounting for -the result. The latter is the more common-sense view. But, on the -other hand, it presupposes the generation of experience from physical -changes. The world of sentient experience, according to the modern -view, is closed and limited; only the physical world is ample and large -and of ever-to-be-discovered complexity. Plato’s world of soul, on the -other hand, is at least as large and ample as the world of things. - -Let us now try a crucial experiment. Can I form an intuition of a -four-dimensional object? Such an object is not given in the physical -range of my sense contacts. All I can do is to present to myself the -sequences of solids, which would mean the presentation to me under my -conditions of a four-dimensional object. All I can do is to visualise -and tactualise different series of solids which are alternative sets of -sectional views of a four-dimensional shape. - -If now, on presenting these sequences, I find a power in me of -intuitively passing from one of these sets of sequences to another, of, -being given one, intuitively constructing another, not using a rule, -but directly apprehending it, then I have found a new fact about my -soul, that it has a four-dimensional experience; I have observed it by -a function it has. - -I do not like to speak positively, for I might occasion a loss of time -on the part of others, if, as may very well be, I am mistaken. But for -my own part, I think there are indications of such an intuition; from -the results of my experiments, I adopt the hypothesis that that which -thinks in us has an ample experience, of which the intuitions we use in -dealing with the world of real objects are a part; of which experience, -the intuition of four-dimensional forms and motions is also a part. The -process we are engaged in intellectually is the reading the obscure -signals of our nerves into a world of reality, by means of intuitions -derived from the inner experience. - -The image I form is as follows. Imagine the captain of a modern -battle-ship directing its course. He has his charts before him; he -is in communication with his associates and subordinates; can convey -his messages and commands to every part of the ship, and receive -information from the conning-tower and the engine-room. Now suppose the -captain immersed in the problem of the navigation of his ship over the -ocean, to have so absorbed himself in the problem of the direction of -his craft over the plane surface of the sea that he forgets himself. -All that occupies his attention is the kind of movement that his ship -makes. The operations by which that movement is produced have sunk -below the threshold of his consciousness, his own actions, by which -he pushes the buttons, gives the orders, are so familiar as to be -automatic, his mind is on the motion of the ship as a whole. In such a -case we can imagine that he identifies himself with his ship; all that -enters his conscious thought is the direction of its movement over the -plane surface of the ocean. - -Such is the relation, as I imagine it, of the soul to the body. A -relation which we can imagine as existing momentarily in the case -of the captain is the normal one in the case of the soul with its -craft. As the captain is capable of a kind of movement, an amplitude -of motion, which does not enter into his thoughts with regard to the -directing the ship over the plane surface of the ocean, so the soul is -capable of a kind of movement, has an amplitude of motion, which is -not used in its task of directing the body in the three-dimensional -region in which the body’s activity lies. If for any reason it became -necessary for the captain to consider three-dimensional motions with -regard to his ship, it would not be difficult for him to gain the -materials for thinking about such motions; all he has to do is to -call his own intimate experience into play. As far as the navigation -of the ship, however, is concerned, he is not obliged to call on -such experience. The ship as a whole simply moves on a surface. The -problem of three-dimensional movement does not ordinarily concern its -steering. And thus with regard to ourselves all those movements and -activities which characterise our bodily organs are three-dimensional; -we never need to consider the ampler movements. But we do more than -use the movements of our body to effect our aims by direct means; we -have now come to the pass when we act indirectly on nature, when we -call processes into play which lie beyond the reach of any explanation -we can give by the kind of thought which has been sufficient for the -steering of our craft as a whole. When we come to the problem of what -goes on in the minute, and apply ourselves to the mechanism of the -minute, we find our habitual conceptions inadequate. - -The captain in us must wake up to his own intimate nature, realise -those functions of movement which are his own, and in virtue of his -knowledge of them apprehend how to deal with the problems he has come -to. - -Think of the history of man. When has there been a time, in which his -thoughts of form and movement were not exclusively of such varieties as -were adapted for his bodily performance? We have never had a demand to -conceive what our own most intimate powers are. But, just as little as -by immersing himself in the steering of his ship over the plane surface -of the ocean, a captain can lose the faculty of thinking about what he -actually does, so little can the soul lose its own nature. It can be -roused to an intuition that is not derived from the experience which -the senses give. All that is necessary is to present some few of those -appearances which, while inconsistent with three-dimensional matter, -are yet consistent with our formal knowledge of four-dimensional -matter, in order for the soul to wake up and not begin to learn, but of -its own intimate feeling fill up the gaps in the presentiment, grasp -the full orb of possibilities from the isolated points presented to -it. In relation to this question of our perceptions, let me suggest -another illustration, not taking it too seriously, only propounding it -to exhibit the possibilities in a broad and general way. - -In the heavens, amongst the multitude of stars, there are some which, -when the telescope is directed on them, seem not to be single stars, -but to be split up into two. Regarding these twin stars through a -spectroscope, an astronomer sees in each a spectrum of bands of colour -and black lines. Comparing these spectrums with one another, he finds -that there is a slight relative shifting of the dark lines, and from -that shifting he knows that the stars are rotating round one another, -and can tell their relative velocity with regard to the earth. By -means of his terrestrial physics he reads this signal of the skies. -This shifting of lines, the mere slight variation of a black line in a -spectrum, is very unlike that which the astronomer knows it means. But -it is probably much more like what it means than the signals which the -nerves deliver are like the phenomena of the outer world. - -No picture of an object is conveyed through the nerves. No picture of -motion, in the sense in which we postulate its existence, is conveyed -through the nerves. The actual deliverances of which our consciousness -takes account are probably identical for eye and ear, sight and touch. - -If for a moment I take the whole earth together and regard it as a -sentient being, I find that the problem of its apprehension is a very -complex one, and involves a long series of personal and physical -events. Similarly the problem of our apprehension is a very complex -one. I only use this illustration to exhibit my meaning. It has this -especial merit, that, as the process of conscious apprehension takes -place in our case in the minute, so, with regard to this earth being, -the corresponding process takes place in what is relatively to it very -minute. - -Now, Plato’s view of a soul leads us to the hypothesis that that -which we designate as an act of apprehension may be a very complex -event, both physically and personally. He does not seek to explain -what an intuition is; he makes it a basis from whence he sets out on -a voyage of discovery. Knowledge means knowledge; he puts conscious -being to account for conscious being. He makes an hypothesis of the -kind that is so fertile in physical science—an hypothesis making no -claim to finality, which marks out a vista of possible determination -behind determination, like the hypothesis of space itself, the type of -serviceable hypotheses. - -And, above all, Plato’s hypothesis is conducive to experiment. He -gives the perspective in which real objects can be determined; and, -in our present enquiry, we are making the simplest of all possible -experiments—we are enquiring what it is natural to the soul to think of -matter as extended. - -Aristotle says we always use a “phantasm” in thinking, a phantasm of -our corporeal senses a visualisation or a tactualisation. But we can -so modify that visualisation or tactualisation that it represents -something not known by the senses. Do we by that representation wake -up an intuition of the soul? Can we by the presentation of these -hypothetical forms, that are the subject of our present discussion, -wake ourselves up to higher intuitions? And can we explain the world -around by a motion that we only know by our souls? - -Apart from all speculation, however, it seems to me that the interest -of these four-dimensional shapes and motions is sufficient reason for -studying them, and that they are the way by which we can grow into a -fuller apprehension of the world as a concrete whole. - - - SPACE NAMES. - -If the words written in the squares drawn in fig. 1 are used as the -names of the squares in the positions in which they are placed, it is -evident that a combination of these names will denote a figure composed -of the designated squares. It is found to be most convenient to take as -the initial square that marked with an asterisk, so that the directions -of progression are towards the observer and to his right. The -directions of progression, however, are arbitrary, and can be chosen at -will. - -[Illustration: Fig. 1.] - -Thus _et_, _at_, _it_, _an_, _al_ will denote a figure in the form of a -cross composed of five squares. - -Here, by means of the double sequence, _e_, _a_, _i_ and _n_, _t_, _l_, -it is possible to name a limited collection of space elements. - -The system can obviously be extended by using letter sequences of more -members. - -But, without introducing such a complexity, the principles of a space -language can be exhibited, and a nomenclature obtained adequate to all -the considerations of the preceding pages. - - -1. _Extension._ - -Call the large squares in fig. 2 by the name written in them. It is -evident that each can be divided as shown in fig. 1. Then the small -square marked 1 will be “en” in “En,” or “Enen.” The square marked 2 -will be “et” in “En” or “Enet,” while the square marked 4 will be “en” -in “Et” or “Eten.” Thus the square 5 will be called “Ilil.” - -[Illustration: Fig. 2.] - -This principle of extension can be applied in any number of dimensions. - - -2. _Application to Three-Dimensional Space._ - -To name a three-dimensional collocation of cubes take the upward -direction first, secondly the direction towards the observer, thirdly -the direction to his right hand. - -[Illustration] - -These form a word in which the first letter gives the place of the cube -upwards, the second letter its place towards the observer, the third -letter its place to the right. - -We have thus the following scheme, which represents the set of cubes of -column 1, fig. 101, page 165. - -We begin with the remote lowest cube at the left hand, where the -asterisk is placed (this proves to be by far the most convenient origin -to take for the normal system). - -Thus “nen” is a “null” cube, “ten” a red cube on it, and “len” a “null” -cube above “ten.” - -By using a more extended sequence of consonants and vowels a larger set -of cubes can be named. - -To name a four-dimensional block of tesseracts it is simply necessary -to prefix an “e,” an “a,” or an “i” to the cube names. - -Thus the tesseract blocks schematically represented on page 165, fig. -101 are named as follows:— - -[Illustration: 1 2 3] - - -2. DERIVATION OF POINT, LINE, FACE, ETC., NAMES. - -[Illustration] - -The principle of derivation can be shown as follows: Taking the square -of squares the number of squares in it can be enlarged and the whole -kept the same size. - -[Illustration] - -Compare fig. 79, p. 138, for instance, or the bottom layer of fig. 84. - -Now use an initial “s” to denote the result of carrying this process on -to a great extent, and we obtain the limit names, that is the point, -line, area names for a square. “Sat” is the whole interior. The corners -are “sen,” “sel,” “sin,” “sil,” while the lines are “san,” “sal,” -“set,” “sit.” - -[Illustration] - -I find that by the use of the initial “s” these names come to be -practically entirely disconnected with the systematic names for the -square from which they are derived. They are easy to learn, and when -learned can be used readily with the axes running in any direction. - -To derive the limit names for a four-dimensional rectangular figure, -like the tesseract, is a simple extension of this process. These point, -line, etc., names include those which apply to a cube, as will be -evident on inspection of the first cube of the diagrams which follow. - -All that is necessary is to place an “s” before each of the names given -for a tesseract block. We then obtain apellatives which, like the -colour names on page 174, fig. 103, apply to all the points, lines, -faces, solids, and to the hyper-solid of the tesseract. These names -have the advantage over the colour marks that each point, line, etc., -has its own individual name. - -In the diagrams I give the names corresponding to the positions shown -in the coloured plate or described on p. 174. By comparing cubes 1, 2, -3 with the first row of cubes in the coloured plate, the systematic -names of each of the points, lines, faces, etc., can be determined. The -asterisk shows the origin from which the names run. - -These point, line, face, etc., names should be used in connection with -the corresponding colours. The names should call up coloured images of -the parts named in their right connection. - -[Illustration] - -It is found that a certain abbreviation adds vividness of distinction -to these names. If the final “en” be dropped wherever it occurs the -system is improved. Thus instead of “senen,” “seten,” “selen,” it is -preferable to abbreviate to “sen,” “set,” “sel,” and also use “san,” -“sin” for “sanen,” “sinen.” - -[Illustration] - -[Illustration] - -We can now name any section. Take _e.g._ the line in the first cube -from senin to senel, we should call the line running from senin to -senel, senin senat senel, a line light yellow in colour with null -points. - -[Illustration] - -Here senat is the name for all of the line except its ends. Using -“senat” in this way does not mean that the line is the whole of senat, -but what there is of it is senat. It is a part of the senat region. -Thus also the triangle, which has its three vertices in senin, senel, -selen, is named thus: - - Area: setat. - Sides: setan, senat, setet. - Vertices: senin, senel, sel. - -The tetrahedron section of the tesseract can be thought of as a series -of plane sections in the successive sections of the tesseract shown in -fig. 114, p. 191. In b_{0} the section is the one written above. In -b_{1} the section is made by a plane which cuts the three edges from -sanen intermediate of their lengths and thus will be: - - Area: satat. - Sides: satan, sanat, satet. - Vertices: sanan, sanet, sat. - -The sections in b_{2}, b_{3} will be like the section in b_{1} but -smaller. - -Finally in b_{4} the section plane simply passes through the corner -named sin. - -Hence, putting these sections together in their right relation, from -the face setat, surrounded by the lines and points mentioned above, -there run: - - 3 faces: satan, sanat, satet - 3 lines: sanan, sanet, sat - -and these faces and lines run to the point sin. Thus the tetrahedron is -completely named. - -The octahedron section of the tesseract, which can be traced from fig. -72, p. 129 by extending the lines there drawn, is named: - -Front triangle selin, selat, selel, setal, senil, setit, selin with -area setat. - -The sections between the front and rear triangle, of which one is shown -in 1b, another in 2b, are thus named, points and lines, salan, salat, -salet, satet, satel, satal, sanal, sanat, sanit, satit, satin, satan, -salan. - -The rear triangle found in 3b by producing lines is sil, sitet, sinel, -sinat, sinin, sitan, sil. - -The assemblage of sections constitute the solid body of the octahedron -satat with triangular faces. The one from the line selat to the point -sil, for instance, is named selin, selat, selel, salet, salat, salan, -sil. The whole interior is salat. - -Shapes can easily be cut out of cardboard which, when folded together, -form not only the tetrahedron and the octahedron, but also samples of -all the sections of the tesseract taken as it passes cornerwise through -our space. To name and visualise with appropriate colours a series of -these sections is an admirable exercise for obtaining familiarity with -the subject. - - - EXTENSION AND CONNECTION WITH NUMBERS. - -By extending the letter sequence it is of course possible to name a -larger field. By using the limit names the corners of each square can -be named. - -Thus “en sen,” “an sen,” etc., will be the names of the points nearest -the origin in “en” and in “an.” - -A field of points of which each one is indefinitely small is given by -the names written below. - -[Illustration] - -The squares are shown in dotted lines, the names denote the points. -These points are not mathematical points, but really minute areas. - -Instead of starting with a set of squares and naming them, we can start -with a set of points. - -By an easily remembered convention we can give names to such a region -of points. - -Let the space names with a final “e” added denote the mathematical -points at the corner of each square nearest the origin. We have then -for the set of mathematical points indicated. This system is really -completely independent of the area system and is connected with it -merely for the purpose of facilitating the memory processes. The word -“ene” is pronounced like “eny,” with just sufficient attention to the -final vowel to distinguish it from the word “en.” - -[Illustration] - -Now, connecting the numbers 0, 1, 2 with the sequence e, a, i, and -also with the sequence n, t, l, we have a set of points named as with -numbers in a co-ordinate system. Thus “ene” is (0, 0) “ate” is (1, -1) “ite” is (2, 1). To pass to the area system the rule is that the -name of the square is formed from the name of its point nearest to the -origin by dropping the final e. - -By using a notation analogous to the decimal system a larger field of -points can be named. It remains to assign a letter sequence to the -numbers from positive 0 to positive 9, and from negative 0 to negative -9, to obtain a system which can be used to denote both the usual -co-ordinate system of mapping and a system of named squares. The names -denoting the points all end with e. Those that denote squares end with -a consonant. - -There are many considerations which must be attended to in extending -the sequences to be used, such as uniqueness in the meaning of the -words formed, ease of pronunciation, avoidance of awkward combinations. - -I drop “s” altogether from the consonant series and short “u” from -the vowel series. It is convenient to have unsignificant letters at -disposal. A double consonant like “st” for instance can be referred to -without giving it a local significance by calling it “ust.” I increase -the number of vowels by considering a sound like “ra” to be a vowel, -using, that is, the letter “r” as forming a compound vowel. - -The series is as follows:— - - CONSONANTS. - - 0 1 2 3 4 5 6 7 8 9 - positive n t l p f sh k ch nt st - negative z d th b v m g j nd sp - - VOWELS. - - 0 1 2 3 4 5 6 7 8 9 - positive e a i ee ae ai ar ra ri ree - negative er o oo io oe iu or ro roo rio - -_Pronunciation._—e as in men; a as in man; i as in in; ee as in -between; ae as ay in may; ai as i in mine; ar as in art; er as ear in -earth; o as in on; oo as oo in soon; io as in clarion; oe as oa in oat; -iu pronounced like yew. - -To name a point such as (23, 41) it is considered as (3, 1) on from -(20, 40) and is called “ifeete.” It is the initial point of the square -ifeet of the area system. - -The preceding amplification of a space language has been introduced -merely for the sake of completeness. As has already been said nine -words and their combinations, applied to a few simple models suffice -for the purposes of our present enquiry. - - - _Printed by Hazell, Watson & Viney, Ld., London and Aylesbury._ - -*** END OF THE PROJECT GUTENBERG EBOOK 67153 *** diff --git a/old/old-2024-12-23/67153-h/67153-h.htm b/old/old-2024-12-23/67153-h/67153-h.htm deleted file mode 100644 index b68b881..0000000 --- a/old/old-2024-12-23/67153-h/67153-h.htm +++ /dev/null @@ -1,11586 +0,0 @@ -<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN" - "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> -<html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en" lang="en"> - <head> - <meta http-equiv="Content-Type" content="text/html;charset=UTF-8" /> - <meta http-equiv="Content-Style-Type" content="text/css" /> - <title> - The Project Gutenberg eBook of The Fourth Dimension, by C. Howard Hinton. - </title> - <link rel="coverpage" href="images/i_cover.jpg" /> - <style type="text/css"> - -body { - margin-left: 10%; - margin-right: 10%; -} - -h1 -{ - margin-top: 2em; margin-bottom: 2em; - text-align: center; - font-size: x-large; - font-weight: normal; - line-height: 1.6; -} - - h2, h3 { - text-align: center; - clear: both; - } - -.half-title { - margin-top: 2em; margin-bottom: 2em; - text-align: center; - font-size: x-large; - font-weight: normal; - line-height: 1.6; - } - -div.chapter {page-break-before: always;} -h2.nobreak {page-break-before: avoid;} - -/* Paragraphs */ - -p {text-indent: 1em; - margin-top: .75em; - text-align: justify; - margin-bottom: .75em; - } - -.pnind {text-indent: 0em;} -.psig {text-align: right; margin-right: 2em;} -.spaced {margin-top: 3em; margin-bottom: 3em;} - -hr { - width: 33%; - margin-top: 2em; - margin-bottom: 2em; - margin-left: 33.5%; - margin-right: 33.5%; - clear: both; -} - - -hr.tb {width: 45%; margin-left: 27.5%; margin-right: 27.5%;} -hr.chap {width: 65%; margin-left: 17.5%; margin-right: 17.5%;} -hr.small {width: 25%; margin-left: 37.5%; margin-right: 37.5%;} -@media print { hr.chap {display: none; visibility: hidden;} } - -ul {list-style-type: none; } -li {text-indent: 5em;} - -table { - margin-left: auto; - margin-right: auto; - } -.standard { font-size: .9em; border-collapse: collapse; } -td {padding-left: 5px;} - -.tdl {text-align: left;} -.tdr {text-align: right;} -.tdc {text-align: center;} -.tdrb {text-align: right; vertical-align: bottom;} -.tdh {text-align: justify; padding-left: 1.75em; - text-indent: -1.75em;} -.tdr_bt {text-align: right; border-top: 1px solid black;} -.tdlp {text-align: left; padding-left: 15px;} - -.pagenum { /* uncomment the next line for invisible page numbers */ - /* visibility: hidden; */ - position: absolute; - left: 92%; - font-size: smaller; - text-align: right; -} /* page numbers */ - - -.blockquote { - margin-left: 5%; - margin-right: 10%; -} -.gap8l {padding-left: 8em;} - -.center {text-align: center;} - -.smcap {font-variant: small-caps;} -.allsmcap {font-variant: small-caps; text-transform: lowercase;} - -.small {font-size: small;} - - -/* Images */ - -img {border: none; max-width: 100%} -.caption {font-size: smaller; font-weight: bold;} - -.figcenter { - margin: auto; - text-align: center; - page-break-inside: avoid; - max-width: 100%; -} -.figleft { - float: left; - clear: left; - margin-left: 0; - margin-bottom: 1em; - margin-top: 1em; - margin-right: 1em; - padding: 0; - text-align: center; - page-break-inside: avoid; - max-width: 100%; -} -/* comment out next line and uncomment the following one for floating figleft on ebookmaker output */ -/*.x-ebookmaker .figleft {float: none; text-align: center; margin-right: 0;}*/ - .x-ebookmaker .figleft {float: left;} - -.figright { - float: right; - clear: right; - margin-left: 1em; - margin-bottom: 1em; - margin-top: 1em; - margin-right: 0; - padding: 0; - text-align: center; - page-break-inside: avoid; - max-width: 100%; -} -/* comment out next line and uncomment the following one for floating figright on ebookmaker output */ -/*.x-ebookmaker .figright {float: none; text-align: center; margin-left: 0;}*/ - .x-ebookmaker .figright {float: right;} - - -/* Footnotes */ - - .footnotes {border: dashed 1px;} - - .footnote { - margin-left: 10%; - margin-right: 10%; - font-size: 0.9em; - } - -.footnote .label { - position: absolute; - right: 84%; - text-align: right; - } - -.fnanchor { - vertical-align: super; - font-size: .8em; - text-decoration: none; - white-space: nowrap - } - - -/* Transcriber's notes */ - -.transnote { - background-color: #E6E6FA; - color: black; - font-size:smaller; - padding:0.5em; - margin-bottom:5em; - font-family:sans-serif, serif; - } - -/* Illustration classes */ -.illowp100 {width: 100%;} -.illowp20 {width: 20%;} -.illowp25 {width: 25%;} -.illowp30 {width: 30%;} -.illowp35 {width: 35%;} -.illowp40 {width: 40%;} -.illowp45 {width: 45%;} -.illowp50 {width: 50%;} -.illowp60 {width: 60%;} -.x-ebookmaker .illowp60 {width: 100%;} -.illowp66 {width: 66%;} -.x-ebookmaker .illowp66 {width: 100%;} -.illowp75 {width: 75%;} -.x-ebookmaker .illowp75 {width: 100%;} -.illowp80 {width: 80%;} -.x-ebookmaker .illowp80 {width: 100%;} - - </style> - </head> -<body> -<div>*** START OF THE PROJECT GUTENBERG EBOOK 67153 ***</div> - -<div class="transnote"> -<h3> Transcriber’s Notes</h3> - -<p>Obvious typographical errors have been silently corrected. All other -spelling and punctuation remains unchanged.</p> - -<p>The cover was prepared by the transcriber and is placed in the public -domain.</p> -</div> -<hr class="chap" /> - - -<div class="half-title">THE FOURTH DIMENSION</div> - - -<hr class="chap x-ebookmaker-drop" /> - -<div class="chapter"> -<h2 class="nobreak" id="SOME_OPINIONS_OF_THE_PRESS">SOME OPINIONS OF THE PRESS</h2> -</div> - - -<p>“<i>Mr. C. H. Hinton discusses the subject of the higher dimensionality of -space, his aim being to avoid mathematical subtleties and technicalities, and -thus enable his argument to be followed by readers who are not sufficiently -conversant with mathematics to follow these processes of reasoning.</i>”—<span class="smcap">Notts -Guardian.</span></p> - -<p>“<i>The fourth dimension is a subject which has had a great fascination for -many teachers, and though one cannot pretend to have quite grasped -Mr. Hinton’s conceptions and arguments, yet it must be admitted that he -reveals the elusive idea in quite a fascinating light. Quite apart from the -main thesis of the book many chapters are of great independent interest. -Altogether an interesting, clever and ingenious book.</i>”—<span class="smcap">Dundee Courier.</span></p> - -<p>“<i>The book will well repay the study of men who like to exercise their wits -upon the problems of abstract thought.</i>”—<span class="smcap">Scotsman.</span></p> - -<p>“<i>Professor Hinton has done well to attempt a treatise of moderate size, -which shall at once be clear in method and free from technicalities of the -schools.</i>”—<span class="smcap">Pall Mall Gazette.</span></p> - -<p>“<i>A very interesting book he has made of it.</i>”—<span class="smcap">Publishers’ Circular.</span></p> - -<p>“<i>Mr. Hinton tries to explain the theory of the fourth dimension so that -the ordinary reasoning mind can get a grasp of what metaphysical -mathematicians mean by it. If he is not altogether successful it is not from -want of clearness on his part, but because the whole theory comes as such an -absolute shock to all one’s preconceived ideas.</i>”—<span class="smcap">Bristol Times.</span></p> - -<p>“<i>Mr. Hinton’s enthusiasm is only the result of an exhaustive study, which -has enabled him to set his subject before the reader with far more than the -amount of lucidity to which it is accustomed.</i>”—<span class="smcap">Pall Mall Gazette.</span></p> - -<p>“<i>The book throughout is a very solid piece of reasoning in the domain of -higher mathematics.</i>”—<span class="smcap">Glasgow Herald.</span></p> - -<p>“<i>Those who wish to grasp the meaning of this somewhat difficult subject -would do well to read</i> The Fourth Dimension. <i>No mathematical knowledge -is demanded of the reader, and any one, who is not afraid of a little hard -thinking, should be able to follow the argument.</i>”—<span class="smcap">Light.</span></p> - -<p>“<i>A splendidly clear re-statement of the old problem of the fourth dimension. -All who are interested in this subject will find the work not only fascinating, -but lucid, it being written in a style easily understandable. The illustrations -make still more clear the letterpress, and the whole is most admirably adapted -to the requirements of the novice or the student.</i>”—<span class="smcap">Two Worlds.</span></p> - -<p>“<i>Those in search of mental gymnastics will find abundance of exercise in -Mr. C. H. Hinton’s</i> Fourth Dimension.”—<span class="smcap">Westminster Review.</span></p> - - -<p><span class="smcap">First Edition</span>, <i>April 1904</i>; <span class="smcap">Second Edition</span>, <i>May 1906</i>.</p> - - -<hr class="chap x-ebookmaker-drop" /> - -<div class="figcenter illowp100" id="i_frontis" style="max-width: 50em;"> - <img src="images/i_frontis.jpg" alt="" /> - <div class="caption">Views of the Tessaract.</div> -</div> - -<div class="chapter"></div> - - -<h1> -<small>THE</small><br /> - -FOURTH DIMENSION</h1> - -<p class="center small">BY</p> - -<p class="center">C. HOWARD HINTON, M.A.<br /> - -<small>AUTHOR OF “SCIENTIFIC ROMANCES”<br /> -“A NEW ERA OF THOUGHT,” ETC., ETC.</small></p> - -<div class="figcenter illowp20" id="colop" style="max-width: 9.375em;"> - <img src="images/colop.png" alt="Colophon" /> -</div> - -<p class="center"><small>LONDON</small><br /> -SWAN SONNENSCHEIN & CO., LIMITED<br /> -25 HIGH STREET, BLOOMSBURY<br /> -<br /> -<small>1906</small><br /> -</p> - - -<p class="center small spaced"> -PRINTED BY<br /> -HAZELL, WATSON AND VINEY, LD.,<br /> -LONDON AND AYLESBURY.</p> -<hr class="chap x-ebookmaker-drop" /> - -<div class="chapter"> -<p><span class="pagenum" id="Page_v">[Pg v]</span></p> - -<h2 class="nobreak" id="PREFACE">PREFACE</h2> -</div> - - -<p>I have endeavoured to present the subject of the higher -dimensionality of space in a clear manner, devoid of -mathematical subtleties and technicalities. In order to -engage the interest of the reader, I have in the earlier -chapters dwelt on the perspective the hypothesis of a -fourth dimension opens, and have treated of the many -connections there are between this hypothesis and the -ordinary topics of our thoughts.</p> - -<p>A lack of mathematical knowledge will prove of no -disadvantage to the reader, for I have used no mathematical -processes of reasoning. I have taken the view -that the space which we ordinarily think of, the space -of real things (which I would call permeable matter), -is different from the space treated of by mathematics. -Mathematics will tell us a great deal about space, just -as the atomic theory will tell us a great deal about the -chemical combinations of bodies. But after all, a theory -is not precisely equivalent to the subject with regard -to which it is held. There is an opening, therefore, from -the side of our ordinary space perceptions for a simple, -altogether rational, mechanical, and observational way<span class="pagenum" id="Page_vi">[Pg vi]</span> -of treating this subject of higher space, and of this -opportunity I have availed myself.</p> - -<p>The details introduced in the earlier chapters, especially -in Chapters VIII., IX., X., may perhaps be found -wearisome. They are of no essential importance in the -main line of argument, and if left till Chapters XI. -and XII. have been read, will be found to afford -interesting and obvious illustrations of the properties -discussed in the later chapters.</p> - -<p>My thanks are due to the friends who have assisted -me in designing and preparing the modifications of -my previous models, and in no small degree to the -publisher of this volume, Mr. Sonnenschein, to whose -unique appreciation of the line of thought of this, as -of my former essays, their publication is owing. By -the provision of a coloured plate, in addition to the other -illustrations, he has added greatly to the convenience -of the reader.</p> - -<p class="psig"> -<span class="smcap">C. Howard Hinton.</span></p> - -<hr class="chap x-ebookmaker-drop" /> - -<div class="chapter"> -<p><span class="pagenum" id="Page_vii">[Pg vii]</span></p> - -<h2 class="nobreak" id="CONTENTS">CONTENTS</h2> -</div> - - -<table class="standard" summary=""> -<tr> -<td class="tdr"><small>CHAP</small>.</td> -<td></td> -<td class="tdr"><small>PAGE</small></td> -</tr> -<tr> -<td class="tdr"><a href="#CHAPTER_I">I.</a></td> -<td class="tdh"><span class="smcap">Four-Dimensional Space</span></td> -<td class="tdr">1</td> -</tr> -<tr> -<td class="tdr"><a href="#CHAPTER_II">II.</a></td> -<td class="tdh"><span class="smcap">The Analogy of a Plane World</span></td> -<td class="tdr">6</td> -</tr> -<tr> -<td class="tdr"><a href="#CHAPTER_III">III.</a></td> -<td class="tdh"><span class="smcap">The Significance of a Four-Dimensional -Existence</span></td> -<td class="tdr">15</td> -</tr> -<tr> -<td class="tdr"><a href="#CHAPTER_IV">IV.</a></td> -<td class="tdh"><span class="smcap">The First Chapter in the History of Four -Space</span></td> -<td class="tdr">23</td> -</tr> -<tr> -<td class="tdr"><a href="#CHAPTER_V">V.</a></td> -<td class="tdh"><span class="smcap">The Second Chapter in the History Of Four Space</span></td> -<td class="tdr">41</td> -</tr> -<tr> -<td></td> -<td class="tdh"><small>Lobatchewsky, Bolyai, and Gauss<br />Metageometry</small></td> -</tr> -<tr> -<td class="tdr"><a href="#CHAPTER_VI">VI.</a></td> -<td class="tdh"><span class="smcap">The Higher World</span></td> -<td class="tdr">61</td> -</tr> -<tr> -<td class="tdr"><a href="#CHAPTER_VII">VII.</a></td> -<td class="tdh"><span class="smcap">The Evidence for a Fourth Dimension</span></td> -<td class="tdr">76</td> -</tr> -<tr> -<td class="tdr"><a href="#CHAPTER_VIII">VIII.</a></td> -<td class="tdh"><span class="smcap">The Use of Four Dimensions in Thought</span></td> -<td class="tdr">85</td> -</tr> -<tr> -<td class="tdr"><a href="#CHAPTER_IX">IX.</a></td> -<td class="tdh"><span class="smcap">Application to Kant’s Theory of Experience</span></td> -<td class="tdr">107</td> -</tr> -<tr> -<td class="tdr"><a href="#CHAPTER_X">X.</a></td> -<td class="tdh"><span class="smcap">A Four-Dimensional Figure</span></td> -<td class="tdr">122</td> -</tr> -<tr> -<td class="tdr"><a href="#CHAPTER_XI">XI.</a></td> -<td class="tdh"><span class="smcap">Nomenclature and Analogies</span></td> -<td class="tdr">136<span class="pagenum" id="Page_viii">[Pg viii]</span></td> -</tr> -<tr> -<td class="tdr"><a href="#CHAPTER_XII">XII.</a></td> -<td class="tdh"><span class="smcap">The Simplest Four-Dimensional Solid</span></td> -<td class="tdr">157</td> -</tr> -<tr> -<td class="tdr"><a href="#CHAPTER_XIII">XIII.</a></td> -<td class="tdh"><span class="smcap">Remarks on the Figures</span></td> -<td class="tdr">178</td> -</tr> -<tr> -<td class="tdr"><a href="#CHAPTER_XIV">XIV.</a></td> -<td class="tdh"><span class="smcap">A Recapitulation and Extension of the -Physical Argument</span></td> -<td class="tdr">203</td> -</tr> -<tr> -<td class="tdl" colspan="2"><a href="#APPENDIX_I">APPENDIX I.</a>—<span class="smcap">The Models</span></td> -<td class="tdr">231</td> -</tr> -<tr> -<td class="tdl" colspan="2"><a href="#APPENDIX_II">APPENDIX II.</a>—<span class="smcap">A Language of Space</span></td> -<td class="tdr">248</td> -</tr> -</table> - -<hr class="chap x-ebookmaker-drop" /> - -<div class="chapter"> -<p><span class="pagenum" id="Page_1">[Pg 1]</span></p> - -<p class="half-title">THE FOURTH DIMENSION</p> - - -<hr class="small" /> - - -<h2 class="nobreak" id="CHAPTER_I">CHAPTER I<br /> - - -<small>FOUR-DIMENSIONAL SPACE</small></h2> -</div> - -<p>There is nothing more indefinite, and at the same time -more real, than that which we indicate when we speak -of the “higher.” In our social life we see it evidenced -in a greater complexity of relations. But this complexity -is not all. There is, at the same time, a contact -with, an apprehension of, something more fundamental, -more real.</p> - -<p>With the greater development of man there comes -a consciousness of something more than all the forms -in which it shows itself. There is a readiness to give -up all the visible and tangible for the sake of those -principles and values of which the visible and tangible -are the representation. The physical life of civilised -man and of a mere savage are practically the same, but -the civilised man has discovered a depth in his existence, -which makes him feel that that which appears all to -the savage is a mere externality and appurtenage to his -true being.</p> - -<p>Now, this higher—how shall we apprehend it? It is -generally embraced by our religious faculties, by our -idealising tendency. But the higher existence has two -sides. It has a being as well as qualities. And in trying<span class="pagenum" id="Page_2">[Pg 2]</span> -to realise it through our emotions we are always taking the -subjective view. Our attention is always fixed on what we -feel, what we think. Is there any way of apprehending -the higher after the purely objective method of a natural -science? I think that there is.</p> - -<p>Plato, in a wonderful allegory, speaks of some men -living in such a condition that they were practically -reduced to be the denizens of a shadow world. They -were chained, and perceived but the shadows of themselves -and all real objects projected on a wall, towards -which their faces were turned. All movements to them -were but movements on the surface, all shapes but the -shapes of outlines with no substantiality.</p> - -<p>Plato uses this illustration to portray the relation -between true being and the illusions of the sense world. -He says that just as a man liberated from his chains -could learn and discover that the world was solid and -real, and could go back and tell his bound companions of -this greater higher reality, so the philosopher who has -been liberated, who has gone into the thought of the -ideal world, into the world of ideas greater and more -real than the things of sense, can come and tell his fellow -men of that which is more true than the visible sun—more -noble than Athens, the visible state.</p> - -<p>Now, I take Plato’s suggestion; but literally, not -metaphorically. He imagines a world which is lower -than this world, in that shadow figures and shadow -motions are its constituents; and to it he contrasts the real -world. As the real world is to this shadow world, so is the -higher world to our world. I accept his analogy. As our -world in three dimensions is to a shadow or plane world, -so is the higher world to our three-dimensional world. -That is, the higher world is four-dimensional; the higher -being is, so far as its existence is concerned apart from its -qualities, to be sought through the conception of an actual<span class="pagenum" id="Page_3">[Pg 3]</span> -existence spatially higher than that which we realise with -our senses.</p> - -<p>Here you will observe I necessarily leave out all that -gives its charm and interest to Plato’s writings. All -those conceptions of the beautiful and good which live -immortally in his pages.</p> - -<p>All that I keep from his great storehouse of wealth is -this one thing simply—a world spatially higher than this -world, a world which can only be approached through the -stocks and stones of it, a world which must be apprehended -laboriously, patiently, through the material things -of it, the shapes, the movements, the figures of it.</p> - -<p>We must learn to realise the shapes of objects in -this world of the higher man; we must become familiar -with the movements that objects make in his world, so -that we can learn something about his daily experience, -his thoughts of material objects, his machinery.</p> - -<p>The means for the prosecution of this enquiry are given -in the conception of space itself.</p> - -<p>It often happens that that which we consider to be -unique and unrelated gives us, within itself, those relations -by means of which we are able to see it as related to -others, determining and determined by them.</p> - -<p>Thus, on the earth is given that phenomenon of weight -by means of which Newton brought the earth into its -true relation to the sun and other planets. Our terrestrial -globe was determined in regard to other bodies of the -solar system by means of a relation which subsisted on -the earth itself.</p> - -<p>And so space itself bears within it relations of which -we can determine it as related to other space. For within -space are given the conceptions of point and line, line and -plane, which really involve the relation of space to a -higher space.</p> - -<p>Where one segment of a straight line leaves off and<span class="pagenum" id="Page_4">[Pg 4]</span> -another begins is a point, and the straight line itself can -be generated by the motion of the point.</p> - -<p>One portion of a plane is bounded from another by a -straight line, and the plane itself can be generated by -the straight line moving in a direction not contained -in itself.</p> - -<p>Again, two portions of solid space are limited with -regard to each other by a plane; and the plane, moving -in a direction not contained in itself, can generate solid -space.</p> - -<p>Thus, going on, we may say that space is that which -limits two portions of higher space from each other, and -that our space will generate the higher space by moving -in a direction not contained in itself.</p> - -<p>Another indication of the nature of four-dimensional -space can be gained by considering the problem of the -arrangement of objects.</p> - -<p>If I have a number of swords of varying degrees of -brightness, I can represent them in respect of this quality -by points arranged along a straight line.</p> - -<div class="figleft illowp25" id="fig_1" style="max-width: 10em;"> - <img src="images/fig_1.png" alt="" /> - <div class="caption">Fig. 1.</div> -</div> - -<p>If I place a sword at <span class="allsmcap">A</span>, <a href="#fig_1">fig. 1</a>, and regard it as having -a certain brightness, then the other swords -can be arranged in a series along the -line, as at <span class="allsmcap">A</span>, <span class="allsmcap">B</span>, <span class="allsmcap">C</span>, etc., according to -their degrees of brightness.</p> - -<div class="figleft illowp25" id="fig_2" style="max-width: 10em;"> - <img src="images/fig_2.png" alt="" /> - <div class="caption">Fig. 2.</div> -</div> - -<p>If now I take account of another quality, say length, -they can be arranged in a plane. Starting from <span class="allsmcap">A</span>, <span class="allsmcap">B</span>, <span class="allsmcap">C</span>, I -can find points to represent different -degrees of length along such lines as -<span class="allsmcap">AF</span>, <span class="allsmcap">BD</span>, <span class="allsmcap">CE</span>, drawn from <span class="allsmcap">A</span> and <span class="allsmcap">B</span> and <span class="allsmcap">C</span>. -Points on these lines represent different -degrees of length with the same degree of -brightness. Thus the whole plane is occupied by points -representing all conceivable varieties of brightness and -length.</p> - -<p><span class="pagenum" id="Page_5">[Pg 5]</span></p> - -<div class="figleft illowp30" id="fig_3" style="max-width: 10em;"> - <img src="images/fig_3.png" alt="" /> - <div class="caption">Fig. 3.</div> -</div> - -<p>Bringing in a third quality, say sharpness, I can draw, -as in <a href="#fig_3">fig. 3</a>, any number of upright -lines. Let distances along these -upright lines represent degrees of -sharpness, thus the points <span class="allsmcap">F</span> and <span class="allsmcap">G</span> -will represent swords of certain -definite degrees of the three qualities -mentioned, and the whole of space will serve to represent -all conceivable degrees of these three qualities.</p> - -<p>If now I bring in a fourth quality, such as weight, and -try to find a means of representing it as I did the other -three qualities, I find a difficulty. Every point in space is -taken up by some conceivable combination of the three -qualities already taken.</p> - -<p>To represent four qualities in the same way as that in -which I have represented three, I should need another -dimension of space.</p> - -<p>Thus we may indicate the nature of four-dimensional -space by saying that it is a kind of space which would -give positions representative of four qualities, as three-dimensional -space gives positions representative of three -qualities.</p> -<hr class="chap x-ebookmaker-drop" /> - -<div class="chapter"> -<p><span class="pagenum" id="Page_6">[Pg 6]</span></p> - -<h2 class="nobreak" id="CHAPTER_II">CHAPTER II<br /> - -<small><i>THE ANALOGY OF A PLANE WORLD</i></small></h2></div> - - -<p>At the risk of some prolixity I will go fully into the -experience of a hypothetical creature confined to motion -on a plane surface. By so doing I shall obtain an analogy -which will serve in our subsequent enquiries, because the -change in our conception, which we make in passing from -the shapes and motions in two dimensions to those in -three, affords a pattern by which we can pass on still -further to the conception of an existence in four-dimensional -space.</p> - -<p>A piece of paper on a smooth table affords a ready -image of a two-dimensional existence. If we suppose the -being represented by the piece of paper to have no -knowledge of the thickness by which he projects above the -surface of the table, it is obvious that he can have no -knowledge of objects of a similar description, except by -the contact with their edges. His body and the objects -in his world have a thickness of which however, he has no -consciousness. Since the direction stretching up from -the table is unknown to him he will think of the objects -of his world as extending in two dimensions only. Figures -are to him completely bounded by their lines, just as solid -objects are to us by their surfaces. He cannot conceive -of approaching the centre of a circle, except by breaking -through the circumference, for the circumference encloses -the centre in the directions in which motion is possible to<span class="pagenum" id="Page_7">[Pg 7]</span> -him. The plane surface over which he slips and with -which he is always in contact will be unknown to him; -there are no differences by which he can recognise its -existence.</p> - -<p>But for the purposes of our analogy this representation -is deficient.</p> - -<p>A being as thus described has nothing about him to -push off from, the surface over which he slips affords no -means by which he can move in one direction rather than -another. Placed on a surface over which he slips freely, -he is in a condition analogous to that in which we should -be if we were suspended free in space. There is nothing -which he can push off from in any direction known to him.</p> - -<p>Let us therefore modify our representation. Let us -suppose a vertical plane against which particles of thin -matter slip, never leaving the surface. Let these particles -possess an attractive force and cohere together into a disk; -this disk will represent the globe of a plane being. He -must be conceived as existing on the rim.</p> - -<div class="figleft illowp25" id="fig_4" style="max-width: 10.9375em;"> - <img src="images/fig_4.png" alt="" /> - <div class="caption">Fig. 4.</div> -</div> - -<p>Let 1 represent this vertical disk of flat matter and 2 -the plane being on it, standing upon its -rim as we stand on the surface of our earth. -The direction of the attractive force of his -matter will give the creature a knowledge -of up and down, determining for him one -direction in his plane space. Also, since -he can move along the surface of his earth, -he will have the sense of a direction parallel to its surface, -which we may call forwards and backwards.</p> - -<p>He will have no sense of right and left—that is, of the -direction which we recognise as extending out from the -plane to our right and left.</p> - -<p>The distinction of right and left is the one that we -must suppose to be absent, in order to project ourselves -into the condition of a plane being.</p> - -<p><span class="pagenum" id="Page_8">[Pg 8]</span></p> - -<p>Let the reader imagine himself, as he looks along the -plane, <a href="#fig_4">fig. 4</a>, to become more and more identified with -the thin body on it, till he finally looks along parallel to -the surface of the plane earth, and up and down, losing -the sense of the direction which stretches right and left. -This direction will be an unknown dimension to him.</p> - -<p>Our space conceptions are so intimately connected with -those which we derive from the existence of gravitation -that it is difficult to realise the condition of a plane being, -without picturing him as in material surroundings with -a definite direction of up and down. Hence the necessity -of our somewhat elaborate scheme of representation, which, -when its import has been grasped, can be dispensed with -for the simpler one of a thin object slipping over a -smooth surface, which lies in front of us.</p> - -<p>It is obvious that we must suppose some means by -which the plane being is kept in contact with the surface -on which he slips. The simplest supposition to make is -that there is a transverse gravity, which keeps him to the -plane. This gravity must be thought of as different to -the attraction exercised by his matter, and as unperceived -by him.</p> - -<p>At this stage of our enquiry I do not wish to enter -into the question of how a plane being could arrive at -a knowledge of the third dimension, but simply to investigate -his plane consciousness.</p> - -<p>It is obvious that the existence of a plane being must -be very limited. A straight line standing up from the -surface of his earth affords a bar to his progress. An -object like a wheel which rotates round an axis would -be unknown to him, for there is no conceivable way in -which he can get to the centre without going through -the circumference. He would have spinning disks, but -could not get to the centre of them. The plane being -can represent the motion from any one point of his space<span class="pagenum" id="Page_9">[Pg 9]</span> -to any other, by means of two straight lines drawn at -right angles to each other.</p> - -<div class="figleft illowp35" id="fig_5" style="max-width: 26.6875em;"> - <img src="images/fig_5.png" alt="" /> - <div class="caption">Fig. 5.</div> -</div> - -<p>Let <span class="allsmcap">AX</span> and <span class="allsmcap">AY</span> be two such axes. He can accomplish -the translation from <span class="allsmcap">A</span> to <span class="allsmcap">B</span> by going along <span class="allsmcap">AX</span> to <span class="allsmcap">C</span>, and -then from <span class="allsmcap">C</span> along <span class="allsmcap">CB</span> parallel to <span class="allsmcap">AY</span>.</p> - -<p>The same result can of course be obtained -by moving to <span class="allsmcap">D</span> along <span class="allsmcap">AY</span> and then parallel -to <span class="allsmcap">AX</span> from <span class="allsmcap">D</span> to <span class="allsmcap">B</span>, or of course by any -diagonal movement compounded by these -axial movements.</p> - -<p>By means of movements parallel to -these two axes he can proceed (except for -material obstacles) from any one point of his space to -any other.</p> - -<div class="figleft illowp35" id="fig_6" style="max-width: 16.875em;"> - <img src="images/fig_6.png" alt="" /> - <div class="caption">Fig. 6.</div> -</div> - -<p>If now we suppose a third line drawn -out from <span class="allsmcap">A</span> at right angles to the plane -it is evident that no motion in either -of the two dimensions he knows will -carry him in the least degree in the -direction represented by <span class="allsmcap">AZ</span>.</p> - -<p>The lines <span class="allsmcap">AZ</span> and <span class="allsmcap">AX</span> determine a -plane. If he could be taken off his plane, and transferred -to the plane <span class="allsmcap">AXZ</span>, he would be in a world exactly -like his own. From every line in his -world there goes off a space world exactly -like his own.</p> - -<div class="figleft illowp25" id="fig_7" style="max-width: 12.5em;"> - <img src="images/fig_7.png" alt="" /> - <div class="caption">Fig. 7.</div> -</div> - -<p>From every point in his world a line can -be drawn parallel to <span class="allsmcap">AZ</span> in the direction -unknown to him. If we suppose the square -in <a href="#fig_7">fig. 7</a> to be a geometrical square from -every point of it, inside as well as on the -contour, a straight line can be drawn parallel -to <span class="allsmcap">AZ</span>. The assemblage of these lines constitute a solid -figure, of which the square in the plane is the base. If -we consider the square to represent an object in the plane<span class="pagenum" id="Page_10">[Pg 10]</span> -being’s world then we must attribute to it a very small -thickness, for every real thing must possess all three -dimensions. This thickness he does not perceive, but -thinks of this real object as a geometrical square. He -thinks of it as possessing area only, and no degree of -solidity. The edges which project from the plane to a -very small extent he thinks of as having merely length -and no breadth—as being, in fact, geometrical lines.</p> - -<p>With the first step in the apprehension of a third -dimension there would come to a plane being the conviction -that he had previously formed a wrong conception -of the nature of his material objects. He had conceived -them as geometrical figures of two dimensions only. -If a third dimension exists, such figures are incapable -of real existence. Thus he would admit that all his real -objects had a certain, though very small thickness in the -unknown dimension, and that the conditions of his -existence demanded the supposition of an extended sheet -of matter, from contact with which in their motion his -objects never diverge.</p> - -<p>Analogous conceptions must be formed by us on the -supposition of a four-dimensional existence. We must -suppose a direction in which we can never point extending -from every point of our space. We must draw a distinction -between a geometrical cube and a cube of real -matter. The cube of real matter we must suppose to -have an extension in an unknown direction, real, but so -small as to be imperceptible by us. From every point -of a cube, interior as well as exterior, we must imagine -that it is possible to draw a line in the unknown direction. -The assemblage of these lines would constitute a higher -solid. The lines going off in the unknown direction from -the face of a cube would constitute a cube starting from -that face. Of this cube all that we should see in our -space would be the face.</p> - -<p><span class="pagenum" id="Page_11">[Pg 11]</span></p> - -<p>Again, just as the plane being can represent any -motion in his space by two axes, so we can represent any -motion in our three-dimensional space by means of three -axes. There is no point in our space to which we cannot -move by some combination of movements on the directions -marked out by these axes.</p> - -<p>On the assumption of a fourth dimension we have -to suppose a fourth axis, which we will call <span class="allsmcap">AW</span>. It must -be supposed to be at right angles to each and every -one of the three axes <span class="allsmcap">AX</span>, <span class="allsmcap">AY</span>, <span class="allsmcap">AZ</span>. Just as the two axes, -<span class="allsmcap">AX</span>, <span class="allsmcap">AZ</span>, determine a plane which is similar to the original -plane on which we supposed the plane being to exist, but -which runs off from it, and only meets it in a line; so in -our space if we take any three axes such as <span class="allsmcap">AX</span>, <span class="allsmcap">AY</span>, and -<span class="allsmcap">AW</span>, they determine a space like our space world. This -space runs off from our space, and if we were transferred -to it we should find ourselves in a space exactly similar to -our own.</p> - -<p>We must give up any attempt to picture this space in -its relation to ours, just as a plane being would have to -give up any attempt to picture a plane at right angles -to his plane.</p> - -<p>Such a space and ours run in different directions from -the plane of <span class="allsmcap">AX</span> and <span class="allsmcap">AY</span>. They meet in this plane but -have nothing else in common, just as the plane space -of <span class="allsmcap">AX</span> and <span class="allsmcap">AY</span> and that of <span class="allsmcap">AX</span> and <span class="allsmcap">AZ</span> run in different -directions and have but the line <span class="allsmcap">AX</span> in common.</p> - -<p>Omitting all discussion of the manner on which a plane -being might be conceived to form a theory of a three-dimensional -existence, let us examine how, with the means -at his disposal, he could represent the properties of three-dimensional -objects.</p> - -<div class="figleft illowp40" id="fig_8" style="max-width: 25em;"> - <img src="images/fig_8.png" alt="" /> - <div class="caption">Fig. 8.</div> -</div> - -<p>There are two ways in which the plane being can think -of one of our solid bodies. He can think of the cube, -<a href="#fig_8">fig. 8</a>, as composed of a number of sections parallel to<span class="pagenum" id="Page_12">[Pg 12]</span> -his plane, each lying in the third dimension a little -further off from his plane than -the preceding one. These sections -he can represent as a -series of plane figures lying in -his plane, but in so representing -them he destroys the coherence -of them in the higher figure. -The set of squares, <span class="allsmcap">A</span>, <span class="allsmcap">B</span>, <span class="allsmcap">C</span>, <span class="allsmcap">D</span>, -represents the section parallel -to the plane of the cube shown in figure, but they are -not in their proper relative positions.</p> - -<p>The plane being can trace out a movement in the third -dimension by assuming discontinuous leaps from one -section to another. Thus, a motion along the edge of -the cube from left to right would be represented in the -set of sections in the plane as the succession of the -corners of the sections <span class="allsmcap">A</span>, <span class="allsmcap">B</span>, <span class="allsmcap">C</span>, <span class="allsmcap">D</span>. A point moving from -<span class="allsmcap">A</span> through <span class="allsmcap">BCD</span> in our space must be represented in the -plane as appearing in <span class="allsmcap">A</span>, then in <span class="allsmcap">B</span>, and so on, without -passing through the intervening plane space.</p> - -<p>In these sections the plane being leaves out, of course, -the extension in the third dimension; the distance between -any two sections is not represented. In order to realise -this distance the conception of motion can be employed.</p> - -<div class="figleft illowp25" id="fig_9" style="max-width: 12.5em;"> - <img src="images/fig_9.png" alt="" /> - <div class="caption">Fig. 9.</div> -</div> - -<p>Let <a href="#fig_9">fig. 9</a> represent a cube passing transverse to the -plane. It will appear to the plane being as a -square object, but the matter of which this -object is composed will be continually altering. -One material particle takes the place of another, -but it does not come from anywhere or go -anywhere in the space which the plane being -knows.</p> - -<p>The analogous manner of representing a higher solid in -our case, is to conceive it as composed of a number of<span class="pagenum" id="Page_13">[Pg 13]</span> -sections, each lying a little further off in the unknown -direction than the preceding.</p> - -<div class="figleft illowp75" id="fig_10" style="max-width: 31.25em;"> - <img src="images/fig_10.png" alt="" /> - <div class="caption">Fig. 10.</div> -</div> - -<p>We can represent these sections as a number of solids. -Thus the cubes <span class="allsmcap">A</span>, <span class="allsmcap">B</span>, <span class="allsmcap">C</span>, <span class="allsmcap">D</span>, -may be considered as -the sections at different -intervals in the unknown -dimension of a higher -cube. Arranged thus their coherence in the higher figure -is destroyed, they are mere representations.</p> - -<p>A motion in the fourth dimension from <span class="allsmcap">A</span> through <span class="allsmcap">B</span>, <span class="allsmcap">C</span>, -etc., would be continuous, but we can only represent it as -the occupation of the positions <span class="allsmcap">A</span>, <span class="allsmcap">B</span>, <span class="allsmcap">C</span>, etc., in succession. -We can exhibit the results of the motion at different -stages, but no more.</p> - -<p>In this representation we have left out the distance -between one section and another; we have considered the -higher body merely as a series of sections, and so left out -its contents. The only way to exhibit its contents is to -call in the aid of the conception of motion.</p> - -<div class="figleft illowp25" id="fig_11" style="max-width: 9.375em;"> - <img src="images/fig_11.png" alt="" /> - <div class="caption">Fig. 11.</div> -</div> - -<p>If a higher cube passes transverse to our space, it will -appear as a cube isolated in space, the part -that has not come into our space and the part -that has passed through will not be visible. -The gradual passing through our space would -appear as the change of the matter of the cube -before us. One material particle in it is succeeded by -another, neither coming nor going in any direction we can -point to. In this manner, by the duration of the figure, -we can exhibit the higher dimensionality of it; a cube of -our matter, under the circumstances supposed, namely, -that it has a motion transverse to our space, would instantly -disappear. A higher cube would last till it had passed -transverse to our space by its whole distance of extension -in the fourth dimension.</p> - -<p><span class="pagenum" id="Page_14">[Pg 14]</span></p> - -<p>As the plane being can think of the cube as consisting -of sections, each like a figure he knows, extending away -from his plane, so we can think of a higher solid as composed -of sections, each like a solid which we know, but -extending away from our space.</p> - -<p>Thus, taking a higher cube, we can look on it as -starting from a cube in our space and extending in the -unknown dimension.</p> - -<div class="figcenter illowp100" id="fig_12" style="max-width: 25em;"> - <img src="images/fig_12.png" alt="" /> - <div class="caption">Fig. 12.</div> -</div> - -<p>Take the face <span class="allsmcap">A</span> and conceive it to exist as simply a -face, a square with no thickness. From this face the -cube in our space extends by the occupation of space -which we can see.</p> - -<p>But from this face there extends equally a cube in the -unknown dimension. We can think of the higher cube, -then, by taking the set of sections <span class="allsmcap">A</span>, <span class="allsmcap">B</span>, <span class="allsmcap">C</span>, <span class="allsmcap">D</span>, etc., and -considering that from each of them there runs a cube. -These cubes have nothing in common with each other, -and of each of them in its actual position all that we can -have in our space is an isolated square. It is obvious that -we can take our series of sections in any manner we -please. We can take them parallel, for instance, to any -one of the three isolated faces shown in the figure. -Corresponding to the three series of sections at right -angles to each other, which we can make of the cube -in space, we must conceive of the higher cube, as composed -of cubes starting from squares parallel to the faces -of the cube, and of these cubes all that exist in our space -are the isolated squares from which they start.</p> -<hr class="chap x-ebookmaker-drop" /> - -<div class="chapter"> -<p><span class="pagenum" id="Page_15">[Pg 15]</span></p> - -<h2 class="nobreak" id="CHAPTER_III">CHAPTER III<br /> - -<small><i>THE SIGNIFICANCE OF A FOUR-DIMENSIONAL -EXISTENCE</i></small></h2></div> - - -<p>Having now obtained the conception of a four-dimensional -space, and having formed the analogy which, without -any further geometrical difficulties, enables us to enquire -into its properties, I will refer the reader, whose interest -is principally in the mechanical aspect, to Chapters VI. -and VII. In the present chapter I will deal with the -general significance of the enquiry, and in the next -with the historical origin of the idea.</p> - -<p>First, with regard to the question of whether there -is any evidence that we are really in four-dimensional -space, I will go back to the analogy of the plane world.</p> - -<p>A being in a plane world could not have any experience -of three-dimensional shapes, but he could have -an experience of three-dimensional movements.</p> - -<p>We have seen that his matter must be supposed to -have an extension, though a very small one, in the third -dimension. And thus, in the small particles of his -matter, three-dimensional movements may well be conceived -to take place. Of these movements he would only -perceive the resultants. Since all movements of an -observable size in the plane world are two-dimensional, -he would only perceive the resultants in two dimensions -of the small three-dimensional movements. Thus, there -would be phenomena which he could not explain by his<span class="pagenum" id="Page_16">[Pg 16]</span> -theory of mechanics—motions would take place which -he could not explain by his theory of motion. Hence, -to determine if we are in a four-dimensional world, we -must examine the phenomena of motion in our space. -If movements occur which are not explicable on the suppositions -of our three-dimensional mechanics, we should -have an indication of a possible four-dimensional motion, -and if, moreover, it could be shown that such movements -would be a consequence of a four-dimensional motion in -the minute particles of bodies or of the ether, we should -have a strong presumption in favour of the reality of -the fourth dimension.</p> - -<p>By proceeding in the direction of finer and finer subdivision, -we come to forms of matter possessing properties -different to those of the larger masses. It is probable that -at some stage in this process we should come to a form -of matter of such minute subdivision that its particles -possess a freedom of movement in four dimensions. This -form of matter I speak of as four-dimensional ether, and -attribute to it properties approximating to those of a -perfect liquid.</p> - -<p>Deferring the detailed discussion of this form of matter -to Chapter VI., we will now examine the means by which -a plane being would come to the conclusion that three-dimensional -movements existed in his world, and point -out the analogy by which we can conclude the existence -of four-dimensional movements in our world. Since the -dimensions of the matter in his world are small in the -third direction, the phenomena in which he would detect -the motion would be those of the small particles of -matter.</p> - -<p>Suppose that there is a ring in his plane. We can -imagine currents flowing round the ring in either of two -opposite directions. These would produce unlike effects, -and give rise to two different fields of influence. If the<span class="pagenum" id="Page_17">[Pg 17]</span> -ring with a current in it in one direction be taken up -and turned over, and put down again on the plane, it -would be identical with the ring having a current in the -opposite direction. An operation of this kind would be -impossible to the plane being. Hence he would have -in his space two irreconcilable objects, namely, the two -fields of influence due to the two rings with currents in -them in opposite directions. By irreconcilable objects -in the plane I mean objects which cannot be thought -of as transformed one into the other by any movement -in the plane.</p> - -<p>Instead of currents flowing in the rings we can imagine -a different kind of current. Imagine a number of small -rings strung on the original ring. A current round these -secondary rings would give two varieties of effect, or two -different fields of influence, according to its direction. -These two varieties of current could be turned one into -the other by taking one of the rings up, turning it over, -and putting it down again in the plane. This operation -is impossible to the plane being, hence in this case also -there would be two irreconcilable fields in the plane. -Now, if the plane being found two such irreconcilable -fields and could prove that they could not be accounted -for by currents in the rings, he would have to admit the -existence of currents round the rings—that is, in rings -strung on the primary ring. Thus he would come to -admit the existence of a three-dimensional motion, for -such a disposition of currents is in three dimensions.</p> - -<p>Now in our space there are two fields of different -properties, which can be produced by an electric current -flowing in a closed circuit or ring. These two fields can -be changed one into the other by reversing the currents, but -they cannot be changed one into the other by any turning -about of the rings in our space; for the disposition of the -field with regard to the ring itself is different when we<span class="pagenum" id="Page_18">[Pg 18]</span> -turn the ring, over and when we reverse the direction of -the current in the ring.</p> - -<p>As hypotheses to explain the differences of these two -fields and their effects we can suppose the following kinds -of space motions:—First, a current along the conductor; -second, a current round the conductor—that is, of rings of -currents strung on the conductor as an axis. Neither of -these suppositions accounts for facts of observation.</p> - -<p>Hence we have to make the supposition of a four-dimensional -motion. We find that a four-dimensional -rotation of the nature explained in a subsequent chapter, -has the following characteristics:—First, it would give us -two fields of influence, the one of which could be turned -into the other by taking the circuit up into the fourth -dimension, turning it over, and putting it down in our -space again, precisely as the two kinds of fields in the -plane could be turned one into the other by a reversal of -the current in our space. Second, it involves a phenomenon -precisely identical with that most remarkable and -mysterious feature of an electric current, namely that it -is a field of action, the rim of which necessarily abuts on a -continuous boundary formed by a conductor. Hence, on -the assumption of a four-dimensional movement in the -region of the minute particles of matter, we should expect -to find a motion analogous to electricity.</p> - -<p>Now, a phenomenon of such universal occurrence as -electricity cannot be due to matter and motion in any -very complex relation, but ought to be seen as a simple -and natural consequence of their properties. I infer that -the difficulty in its theory is due to the attempt to explain -a four-dimensional phenomenon by a three-dimensional -geometry.</p> - -<p>In view of this piece of evidence we cannot disregard -that afforded by the existence of symmetry. In this -connection I will allude to the simple way of producing<span class="pagenum" id="Page_19">[Pg 19]</span> -the images of insects, sometimes practised by children. -They put a few blots of ink in a straight line on a piece of -paper, fold the paper along the blots, and on opening it the -lifelike presentment of an insect is obtained. If we were -to find a multitude of these figures, we should conclude -that they had originated from a process of folding over; -the chances against this kind of reduplication of parts -is too great to admit of the assumption that they had -been formed in any other way.</p> - -<p>The production of the symmetrical forms of organised -beings, though not of course due to a turning over of -bodies of any appreciable size in four-dimensional space, -can well be imagined as due to a disposition in that -manner of the smallest living particles from which they -are built up. Thus, not only electricity, but life, and the -processes by which we think and feel, must be attributed -to that region of magnitude in which four-dimensional -movements take place.</p> - -<p>I do not mean, however, that life can be explained as a -four-dimensional movement. It seems to me that the -whole bias of thought, which tends to explain the -phenomena of life and volition, as due to matter and -motion in some peculiar relation, is adopted rather in the -interests of the explicability of things than with any -regard to probability.</p> - -<p>Of course, if we could show that life were a phenomenon -of motion, we should be able to explain a great deal that is -at present obscure. But there are two great difficulties in -the way. It would be necessary to show that in a germ -capable of developing into a living being, there were -modifications of structure capable of determining in the -developed germ all the characteristics of its form, and not -only this, but of determining those of all the descendants -of such a form in an infinite series. Such a complexity of -mechanical relations, undeniable though it be, cannot<span class="pagenum" id="Page_20">[Pg 20]</span> -surely be the best way of grouping the phenomena and -giving a practical account of them. And another difficulty -is this, that no amount of mechanical adaptation would -give that element of consciousness which we possess, and -which is shared in to a modified degree by the animal -world.</p> - -<p>In those complex structures which men build up and -direct, such as a ship or a railway train (and which, if seen -by an observer of such a size that the men guiding them -were invisible, would seem to present some of the -phenomena of life) the appearance of animation is not -due to any diffusion of life in the material parts of the -structure, but to the presence of a living being.</p> - -<p>The old hypothesis of a soul, a living organism within -the visible one, appears to me much more rational than the -attempt to explain life as a form of motion. And when we -consider the region of extreme minuteness characterised -by four-dimensional motion the difficulty of conceiving -such an organism alongside the bodily one disappears. -Lord Kelvin supposes that matter is formed from the -ether. We may very well suppose that the living -organisms directing the material ones are co-ordinate -with them, not composed of matter, but consisting of -etherial bodies, and as such capable of motion through -the ether, and able to originate material living bodies -throughout the mineral.</p> - -<p>Hypotheses such as these find no immediate ground for -proof or disproof in the physical world. Let us, therefore, -turn to a different field, and, assuming that the human -soul is a four-dimensional being, capable in itself of four -dimensional movements, but in its experiences through -the senses limited to three dimensions, ask if the history -of thought, of these productivities which characterise man, -correspond to our assumption. Let us pass in review -those steps by which man, presumably a four-dimensional<span class="pagenum" id="Page_21">[Pg 21]</span> -being, despite his bodily environment, has come to recognise -the fact of four-dimensional existence.</p> - -<p>Deferring this enquiry to another chapter, I will here -recapitulate the argument in order to show that our -purpose is entirely practical and independent of any -philosophical or metaphysical considerations.</p> - -<p>If two shots are fired at a target, and the second bullet -hits it at a different place to the first, we suppose that -there was some difference in the conditions under which -the second shot was fired from those affecting the first -shot. The force of the powder, the direction of aim, the -strength of the wind, or some condition must have been -different in the second case, if the course of the bullet was -not exactly the same as in the first case. Corresponding -to every difference in a result there must be some difference -in the antecedent material conditions. By tracing -out this chain of relations we explain nature.</p> - -<p>But there is also another mode of explanation which we -apply. If we ask what was the cause that a certain ship -was built, or that a certain structure was erected, we might -proceed to investigate the changes in the brain cells of -the men who designed the works. Every variation in one -ship or building from another ship or building is accompanied -by a variation in the processes that go on in the -brain matter of the designers. But practically this would -be a very long task.</p> - -<p>A more effective mode of explaining the production of -the ship or building would be to enquire into the motives, -plans, and aims of the men who constructed them. We -obtain a cumulative and consistent body of knowledge -much more easily and effectively in the latter way.</p> - -<p>Sometimes we apply the one, sometimes the other -mode of explanation.</p> - -<p>But it must be observed that the method of explanation -founded on aim, purpose, volition, always presupposes<span class="pagenum" id="Page_22">[Pg 22]</span> -a mechanical system on which the volition and aim -works. The conception of man as willing and acting -from motives involves that of a number of uniform processes -of nature which he can modify, and of which he -can make application. In the mechanical conditions of -the three-dimensional world, the only volitional agency -which we can demonstrate is the human agency. But -when we consider the four-dimensional world the -conclusion remains perfectly open.</p> - -<p>The method of explanation founded on purpose and aim -does not, surely, suddenly begin with man and end with -him. There is as much behind the exhibition of will and -motive which we see in man as there is behind the -phenomena of movement; they are co-ordinate, neither -to be resolved into the other. And the commencement -of the investigation of that will and motive which lies -behind the will and motive manifested in the three-dimensional -mechanical field is in the conception of a -soul—a four-dimensional organism, which expresses its -higher physical being in the symmetry of the body, and -gives the aims and motives of human existence.</p> - -<p>Our primary task is to form a systematic knowledge of -the phenomena of a four-dimensional world and find those -points in which this knowledge must be called in to -complete our mechanical explanation of the universe. -But a subsidiary contribution towards the verification of -the hypothesis may be made by passing in review the -history of human thought, and enquiring if it presents -such features as would be naturally expected on this -assumption.</p> -<hr class="chap x-ebookmaker-drop" /> - -<div class="chapter"> -<p><span class="pagenum" id="Page_23">[Pg 23]</span></p> - -<h2 class="nobreak" id="CHAPTER_IV">CHAPTER IV<br /> - -<small><i>THE FIRST CHAPTER IN THE HISTORY -OF FOUR SPACE</i></small></h2></div> - - -<p>Parmenides, and the Asiatic thinkers with whom he is -in close affinity, propound a theory of existence which -is in close accord with a conception of a possible relation -between a higher and a lower dimensional space. This -theory, prior and in marked contrast to the main stream -of thought, which we shall afterwards describe, forms a -closed circle by itself. It is one which in all ages has -had a strong attraction for pure intellect, and is the -natural mode of thought for those who refrain from -projecting their own volition into nature under the guise -of causality.</p> - -<p>According to Parmenides of the school of Elea the all -is one, unmoving and unchanging. The permanent amid -the transient—that foothold for thought, that solid ground -for feeling on the discovery of which depends all our life—is -no phantom; it is the image amidst deception of true -being, the eternal, the unmoved, the one. Thus says -Parmenides.</p> - -<p>But how explain the shifting scene, these mutations -of things!</p> - -<p>“Illusion,” answers Parmenides. Distinguishing between -truth and error, he tells of the true doctrine of the -one—the false opinion of a changing world. He is no -less memorable for the manner of his advocacy than for<span class="pagenum" id="Page_24">[Pg 24]</span> -the cause he advocates. It is as if from his firm foothold -of being he could play with the thoughts under the -burden of which others laboured, for from him springs -that fluency of supposition and hypothesis which forms -the texture of Plato’s dialectic.</p> - -<p>Can the mind conceive a more delightful intellectual -picture than that of Parmenides, pointing to the one, the -true, the unchanging, and yet on the other hand ready to -discuss all manner of false opinion, forming a cosmogony -too, false “but mine own” after the fashion of the time?</p> - -<p>In support of the true opinion he proceeded by the -negative way of showing the self-contradictions in the -ideas of change and motion. It is doubtful if his criticism, -save in minor points, has ever been successfully refuted. -To express his doctrine in the ponderous modern way we -must make the statement that motion is phenomenal, -not real.</p> - -<p>Let us represent his doctrine.</p> - -<div class="figleft illowp35" id="fig_13" style="max-width: 9.375em;"> - <img src="images/fig_13.png" alt="" /> - <div class="caption">Fig. 13.</div> -</div> - -<p>Imagine a sheet of still water into which a slanting stick -is being lowered with a motion vertically -downwards. Let 1, 2, 3 (Fig. 13), -be three consecutive positions of the -stick. <span class="allsmcap">A</span>, <span class="allsmcap">B</span>, <span class="allsmcap">C</span>, will be three consecutive -positions of the meeting of the stick, -with the surface of the water. As -the stick passes down, the meeting will -move from <span class="allsmcap">A</span> on to <span class="allsmcap">B</span> and <span class="allsmcap">C</span>.</p> - -<p>Suppose now all the water to be -removed except a film. At the meeting -of the film and the stick there -will be an interruption of the film. -If we suppose the film to have a property, -like that of a soap bubble, of closing up round any -penetrating object, then as the stick goes vertically -downwards the interruption in the film will move on.</p> - -<p><span class="pagenum" id="Page_25">[Pg 25]</span></p> - -<div class="figleft illowp35" id="fig_14" style="max-width: 10em;"> - <img src="images/fig_14.png" alt="" /> - <div class="caption">Fig. 14.</div> -</div> - -<p>If we pass a spiral through the film the intersection -will give a point moving in a circle shown by the dotted -lines in the figure. Suppose -now the spiral to be still and -the film to move vertically -upwards, the whole spiral will -be represented in the film of -the consecutive positions of the -point of intersection. In the -film the permanent existence -of the spiral is experienced as -a time series—the record of -traversing the spiral is a point -moving in a circle. If now -we suppose a consciousness connected -with the film in such a way that the intersection of -the spiral with the film gives rise to a conscious experience, -we see that we shall have in the film a point moving in a -circle, conscious of its motion, knowing nothing of that -real spiral the record of the successive intersections of -which by the film is the motion of the point.</p> - -<p>It is easy to imagine complicated structures of the -nature of the spiral, structures consisting of filaments, -and to suppose also that these structures are distinguishable -from each other at every section. If we consider -the intersections of these filaments with the film as it -passes to be the atoms constituting a filmar universe, -we shall have in the film a world of apparent motion; -we shall have bodies corresponding to the filamentary -structure, and the positions of these structures with -regard to one another will give rise to bodies in the -film moving amongst one another. This mutual motion -is apparent merely. The reality is of permanent structures -stationary, and all the relative motions accounted for by -one steady movement of the film as a whole.</p> - -<p><span class="pagenum" id="Page_26">[Pg 26]</span></p> - -<p>Thus we can imagine a plane world, in which all the -variety of motion is the phenomenon of structures consisting -of filamentary atoms traversed by a plane of -consciousness. Passing to four dimensions and our -space, we can conceive that all things and movements -in our world are the reading off of a permanent reality -by a space of consciousness. Each atom at every moment -is not what it was, but a new part of that endless line -which is itself. And all this system successively revealed -in the time which is but the succession of consciousness, -separate as it is in parts, in its entirety is one vast unity. -Representing Parmenides’ doctrine thus, we gain a firmer -hold on it than if we merely let his words rest, grand and -massive, in our minds. And we have gained the means also -of representing phases of that Eastern thought to which -Parmenides was no stranger. Modifying his uncompromising -doctrine, let us suppose, to go back to the plane -of consciousness and the structure of filamentary atoms, -that these structures are themselves moving—are acting, -living. Then, in the transverse motion of the film, there -would be two phenomena of motion, one due to the reading -off in the film of the permanent existences as they are in -themselves, and another phenomenon of motion due to -the modification of the record of the things themselves, by -their proper motion during the process of traversing them.</p> - -<p>Thus a conscious being in the plane would have, as it -were, a two-fold experience. In the complete traversing -of the structure, the intersection of which with the film -gives his conscious all, the main and principal movements -and actions which he went through would be the record -of his higher self as it existed unmoved and unacting. -Slight modifications and deviations from these movements -and actions would represent the activity and self-determination -of the complete being, of his higher self.</p> - -<p>It is admissible to suppose that the consciousness in<span class="pagenum" id="Page_27">[Pg 27]</span> -the plane has a share in that volition by which the -complete existence determines itself. Thus the motive -and will, the initiative and life, of the higher being, would -be represented in the case of the being in the film by an -initiative and a will capable, not of determining any great -things or important movements in his existence, but only -of small and relatively insignificant activities. In all the -main features of his life his experience would be representative -of one state of the higher being whose existence -determines his as the film passes on. But in his minute -and apparently unimportant actions he would share in -that will and determination by which the whole of the -being he really is acts and lives.</p> - -<p>An alteration of the higher being would correspond to -a different life history for him. Let us now make the -supposition that film after film traverses these higher -structures, that the life of the real being is read off again -and again in successive waves of consciousness. There -would be a succession of lives in the different advancing -planes of consciousness, each differing from the preceding, -and differing in virtue of that will and activity which in -the preceding had not been devoted to the greater and -apparently most significant things in life, but the minute -and apparently unimportant. In all great things the -being of the film shares in the existence of his higher -self as it is at any one time. In the small things he -shares in that volition by which the higher being alters -and changes, acts and lives.</p> - -<p>Thus we gain the conception of a life changing and -developing as a whole, a life in which our separation and -cessation and fugitiveness are merely apparent, but which -in its events and course alters, changes, develops; and -the power of altering and changing this whole lies in the -will and power the limited being has of directing, guiding, -altering himself in the minute things of his existence.</p> - -<p><span class="pagenum" id="Page_28">[Pg 28]</span></p> - -<p>Transferring our conceptions to those of an existence in -a higher dimensionality traversed by a space of consciousness, -we have an illustration of a thought which has -found frequent and varied expression. When, however, -we ask ourselves what degree of truth there lies in it, we -must admit that, as far as we can see, it is merely symbolical. -The true path in the investigation of a higher -dimensionality lies in another direction.</p> - -<p>The significance of the Parmenidean doctrine lies in -this that here, as again and again, we find that those conceptions -which man introduces of himself, which he does -not derive from the mere record of his outward experience, -have a striking and significant correspondence to the -conception of a physical existence in a world of a higher -space. How close we come to Parmenides’ thought by -this manner of representation it is impossible to say. -What I want to point out is the adequateness of the -illustration, not only to give a static model of his doctrine, -but one capable as it were, of a plastic modification into a -correspondence into kindred forms of thought. Either one -of two things must be true—that four-dimensional conceptions -give a wonderful power of representing the thought -of the East, or that the thinkers of the East must have been -looking at and regarding four-dimensional existence.</p> - -<p>Coming now to the main stream of thought we must -dwell in some detail on Pythagoras, not because of his -direct relation to the subject, but because of his relation -to investigators who came later.</p> - -<p>Pythagoras invented the two-way counting. Let us -represent the single-way counting by the posits <i>aa</i>, -<i>ab</i>, <i>ac</i>, <i>ad</i>, using these pairs of letters instead of the -numbers 1, 2, 3, 4. I put an <i>a</i> in each case first for a -reason which will immediately appear.</p> - -<p>We have a sequence and order. There is no conception -of distance necessarily involved. The difference<span class="pagenum" id="Page_29">[Pg 29]</span> -between the posits is one of order not of distance—only -when identified with a number of equal material -things in juxtaposition does the notion of distance arise.</p> - -<p>Now, besides the simple series I can have, starting from -<i>aa</i>, <i>ba</i>, <i>ca</i>, <i>da</i>, from <i>ab</i>, <i>bb</i>, <i>cb</i>, <i>db</i>, and so on, and forming -a scheme:</p> - - -<table class="standard" summary=""> -<tr> -<td class="tdlp"><i>da</i></td> -<td class="tdlp"><i>db</i></td> -<td class="tdlp"><i>dc</i></td> -<td class="tdlp"><i>dd</i></td> -</tr> -<tr> -<td class="tdlp"><i>ca</i></td> -<td class="tdlp"><i>cb</i></td> -<td class="tdlp"><i>cc</i></td> -<td class="tdlp"><i>cd</i></td> -</tr> -<tr> -<td class="tdlp"><i>ba</i></td> -<td class="tdlp"><i>bb</i></td> -<td class="tdlp"><i>bc</i></td> -<td class="tdlp"><i>bd</i></td> -</tr> -<tr> -<td class="tdlp"><i>aa</i></td> -<td class="tdlp"><i>ab</i></td> -<td class="tdlp"><i>ac</i></td> -<td class="tdlp"><i>ad</i></td> -</tr> -</table> - - -<p>This complex or manifold gives a two-way order. I can -represent it by a set of points, if I am on my guard -against assuming any relation of distance.</p> - -<div class="figleft illowp25" id="fig_15" style="max-width: 10em;"> - <img src="images/fig_15.png" alt="" /> - <div class="caption">Fig. 15.</div> -</div> - -<p>Pythagoras studied this two-fold way of -counting in reference to material bodies, and -discovered that most remarkable property of -the combination of number and matter that -bears his name.</p> - -<p>The Pythagorean property of an extended material -system can be exhibited in a manner which will be of -use to us afterwards, and which therefore I will employ -now instead of using the kind of figure which he himself -employed.</p> - -<p>Consider a two-fold field of points arranged in regular -rows. Such a field will be presupposed in the following -argument.</p> - -<div class="figleft illowp40" id="fig_16" style="max-width: 21.25em;"> - <img src="images/fig_16.png" alt="" /> - <div class="caption">Fig. 16.</div> -</div> - -<p>It is evident that in <a href="#fig_16">fig. 16</a> four -of the points determine a square, -which square we may take as the -unit of measurement for areas. -But we can also measure areas -in another way.</p> - -<p>Fig. 16 (1) shows four points determining a square.</p> - -<p>But four squares also meet in a point, <a href="#fig_16">fig. 16</a> (2).</p> - -<p>Hence a point at the corner of a square belongs equally -to four squares.</p> - -<p><span class="pagenum" id="Page_30">[Pg 30]</span></p> - -<p>Thus we may say that the point value of the square -shown is one point, for if we take the square in <a href="#fig_16">fig. 16</a> (1) -it has four points, but each of these belong equally to -four other squares. Hence one fourth of each of them -belongs to the square (1) in <a href="#fig_16">fig. 16</a>. Thus the point -value of the square is one point.</p> - -<p>The result of counting the points is the same as that -arrived at by reckoning the square units enclosed.</p> - -<p>Hence, if we wish to measure the area of any square -we can take the number of points it encloses, count these -as one each, and take one-fourth of the number of points -at its corners.</p> - -<div class="figleft illowp25" id="fig_17" style="max-width: 12.5em;"> - <img src="images/fig_17.png" alt="" /> - <div class="caption">Fig. 17.</div> -</div> - -<p>Now draw a diagonal square as shown in <a href="#fig_17">fig. 17</a>. It -contains one point and the four corners count for one -point more; hence its point value is 2. -The value is the measure of its area—the -size of this square is two of the unit squares.</p> - -<p>Looking now at the sides of this figure -we see that there is a unit square on each -of them—the two squares contain no points, -but have four corner points each, which gives the point -value of each as one point.</p> - -<p>Hence we see that the square on the diagonal is equal -to the squares on the two sides; or as it is generally -expressed, the square on the hypothenuse is equal to -the sum of the squares on the sides.</p> - -<div class="figleft illowp25" id="fig_18" style="max-width: 12.5em;"> - <img src="images/fig_18.png" alt="" /> - <div class="caption">Fig. 18.</div> -</div> - -<p>Noticing this fact we can proceed to ask if it is always -true. Drawing the square shown in <a href="#fig_18">fig. 18</a>, we can count -the number of its points. There are five -altogether. There are four points inside -the square on the diagonal, and hence, with -the four points at its corners the point -value is 5—that is, the area is 5. Now -the squares on the sides are respectively -of the area 4 and 1. Hence in this case also the square<span class="pagenum" id="Page_31">[Pg 31]</span> -on the diagonal is equal to the sum of the square on -the sides. This property of matter is one of the first -great discoveries of applied mathematics. We shall prove -afterwards that it is not a property of space. For the -present it is enough to remark that the positions in -which the points are arranged is entirely experimental. -It is by means of equal pieces of some material, or the -same piece of material moved from one place to another, -that the points are arranged.</p> - -<p>Pythagoras next enquired what the relation must be -so that a square drawn slanting-wise should be equal to -one straight-wise. He found that a square whose side is -five can be placed either rectangularly along the lines -of points, or in a slanting position. And this square is -equivalent to two squares of sides 4 and 3.</p> - -<p>Here he came upon a numerical relation embodied in -a property of matter. Numbers immanent in the objects -produced the equality so satisfactory for intellectual apprehension. -And he found that numbers when immanent -in sound—when the strings of a musical instrument -were given certain definite proportions of length—were -no less captivating to the ear than the equality of squares -was to the reason. What wonder then that he ascribed -an active power to number!</p> - -<p>We must remember that, sharing like ourselves the -search for the permanent in changing phenomena, the -Greeks had not that conception of the permanent in -matter that we have. To them material things were not -permanent. In fire solid things would vanish; absolutely -disappear. Rock and earth had a more stable existence, -but they too grew and decayed. The permanence of -matter, the conservation of energy, were unknown to -them. And that distinction which we draw so readily -between the fleeting and permanent causes of sensation, -between a sound and a material object, for instance, had<span class="pagenum" id="Page_32">[Pg 32]</span> -not the same meaning to them which it has for us. -Let us but imagine for a moment that material things -are fleeting, disappearing, and we shall enter with a far -better appreciation into that search for the permanent -which, with the Greeks, as with us, is the primary -intellectual demand.</p> - -<p>What is that which amid a thousand forms is ever the -same, which we can recognise under all its vicissitudes, -of which the diverse phenomena are the appearances?</p> - -<p>To think that this is number is not so very wide of -the mark. With an intellectual apprehension which far -outran the evidences for its application, the atomists -asserted that there were everlasting material particles, -which, by their union, produced all the varying forms and -states of bodies. But in view of the observed facts of -nature as then known, Aristotle, with perfect reason, -refused to accept this hypothesis.</p> - -<p>He expressly states that there is a change of quality, -and that the change due to motion is only one of the -possible modes of change.</p> - -<p>With no permanent material world about us, with -the fleeting, the unpermanent, all around we should, I -think, be ready to follow Pythagoras in his identification -of number with that principle which subsists amidst -all changes, which in multitudinous forms we apprehend -immanent in the changing and disappearing substance -of things.</p> - -<p>And from the numerical idealism of Pythagoras there -is but a step to the more rich and full idealism of Plato. -That which is apprehended by the sense of touch we -put as primary and real, and the other senses we say -are merely concerned with appearances. But Plato took -them all as valid, as giving qualities of existence. That -the qualities were not permanent in the world as given -to the senses forced him to attribute to them a different<span class="pagenum" id="Page_33">[Pg 33]</span> -kind of permanence. He formed the conception of a -world of ideas, in which all that really is, all that affects -us and gives the rich and wonderful wealth of our -experience, is not fleeting and transitory, but eternal. -And of this real and eternal we see in the things about -us the fleeting and transient images.</p> - -<p>And this world of ideas was no exclusive one, wherein -was no place for the innermost convictions of the soul and -its most authoritative assertions. Therein existed justice, -beauty—the one, the good, all that the soul demanded -to be. The world of ideas, Plato’s wonderful creation -preserved for man, for his deliberate investigation and -their sure development, all that the rude incomprehensible -changes of a harsh experience scatters and -destroys.</p> - -<p>Plato believed in the reality of ideas. He meets us -fairly and squarely. Divide a line into two parts, he -says; one to represent the real objects in the world, the -other to represent the transitory appearances, such as the -image in still water, the glitter of the sun on a bright -surface, the shadows on the clouds.</p> - -<div class="figcenter illowp100" id="i_033a" style="max-width: 50em;"> - <img src="images/i_033a.png" alt="" /> - <div class="caption"><table class="standard" summary=""> -<col width="30%" /><col width="20%" /><col width="30%" /> -<tr> -<td class="tdc">Real things:<br /> <i>e.g.</i>, the sun.</td> -<td></td> -<td class="tdc">Appearances:<br /> <i>e.g.</i>, the reflection of the sun.</td> -</tr> -</table> -</div> -</div> - -<p>Take another line and divide it into two parts, one -representing our ideas, the ordinary occupants of our -minds, such as whiteness, equality, and the other representing -our true knowledge, which is of eternal principles, -such as beauty, goodness.</p> - -<div class="figcenter illowp100" id="i_033b" style="max-width: 50em;"> - <img src="images/i_033b.png" alt="" /> - <div class="caption"><table class="standard" summary=""> -<col width="30%" /><col width="20%" /><col width="30%" /> -<tr> -<td class="tdc">Eternal principles,<br />as beauty.</td> -<td></td> -<td class="tdc"> Appearances in the mind,<br />as whiteness, equality</td> -</tr> -</table> -</div> -</div> - -<p>Then as A is to B, so is A<sup>1</sup> to B<sup>1</sup></p> - -<p>That is, the soul can proceed, going away from real<span class="pagenum" id="Page_34">[Pg 34]</span> -things to a region of perfect certainty, where it beholds -what is, not the scattered reflections; beholds the sun, not -the glitter on the sands; true being, not chance opinion.</p> - -<p>Now, this is to us, as it was to Aristotle, absolutely -inconceivable from a scientific point of view. We can -understand that a being is known in the fulness of his -relations; it is in his relations to his circumstances that -a man’s character is known; it is in his acts under his -conditions that his character exists. We cannot grasp or -conceive any principle of individuation apart from the -fulness of the relations to the surroundings.</p> - -<p>But suppose now that Plato is talking about the higher -man—the four-dimensional being that is limited in our -external experience to a three-dimensional world. Do not -his words begin to have a meaning? Such a being -would have a consciousness of motion which is not as -the motion he can see with the eyes of the body. He, -in his own being, knows a reality to which the outward -matter of this too solid earth is flimsy superficiality. He -too knows a mode of being, the fulness of relations, in -which can only be represented in the limited world of -sense, as the painter unsubstantially portrays the depths -of woodland, plains, and air. Thinking of such a being -in man, was not Plato’s line well divided?</p> - -<p>It is noteworthy that, if Plato omitted his doctrine of -the independent origin of ideas, he would present exactly -the four-dimensional argument; a real thing as we think -it is an idea. A plane being’s idea of a square object is -the idea of an abstraction, namely, a geometrical square. -Similarly our idea of a solid thing is an abstraction, for in -our idea there is not the four-dimensional thickness which -is necessary, however slight, to give reality. The argument -would then run, as a shadow is to a solid object, so -is the solid object to the reality. Thus A and B´ would -be identified.</p> - -<p><span class="pagenum" id="Page_35">[Pg 35]</span></p> - -<p>In the allegory which I have already alluded to, Plato -in almost as many words shows forth the relation between -existence in a superficies and in solid space. And he -uses this relation to point to the conditions of a higher -being.</p> - -<p>He imagines a number of men prisoners, chained so -that they look at the wall of a cavern in which they are -confined, with their backs to the road and the light. -Over the road pass men and women, figures and processions, -but of all this pageant all that the prisoners -behold is the shadow of it on the wall whereon they gaze. -Their own shadows and the shadows of the things in the -world are all that they see, and identifying themselves -with their shadows related as shadows to a world of -shadows, they live in a kind of dream.</p> - -<p>Plato imagines one of their number to pass out from -amongst them into the real space world, and then returning -to tell them of their condition.</p> - -<p>Here he presents most plainly the relation between -existence in a plane world and existence in a three-dimensional -world. And he uses this illustration as a -type of the manner in which we are to proceed to a -higher state from the three-dimensional life we know.</p> - -<p>It must have hung upon the weight of a shadow which -path he took!—whether the one we shall follow toward -the higher solid and the four-dimensional existence, or -the one which makes ideas the higher realities, and the -direct perception of them the contact with the truer -world.</p> - -<p>Passing on to Aristotle, we will touch on the points -which most immediately concern our enquiry.</p> - -<p>Just as a scientific man of the present day in -reviewing the speculations of the ancient world would -treat them with a curiosity half amused but wholly -respectful, asking of each and all wherein lay their<span class="pagenum" id="Page_36">[Pg 36]</span> -relation to fact, so Aristotle, in discussing the philosophy -of Greece as he found it, asks, above all other things: -“Does this represent the world? In this system is there -an adequate presentation of what is?”</p> - -<p>He finds them all defective, some for the very reasons -which we esteem them most highly, as when he criticises -the Atomic theory for its reduction of all change to motion. -But in the lofty march of his reason he never loses sight -of the whole; and that wherein our views differ from his -lies not so much in a superiority of our point of view, as -in the fact which he himself enunciates—that it is impossible -for one principle to be valid in all branches of -enquiry. The conceptions of one method of investigation -are not those of another; and our divergence lies in our -exclusive attention to the conceptions useful in one way -of apprehending nature rather than in any possibility we -find in our theories of giving a view of the whole transcending -that of Aristotle.</p> - -<p>He takes account of everything; he does not separate -matter and the manifestation of matter; he fires all -together in a conception of a vast world process in -which everything takes part—the motion of a grain of -dust, the unfolding of a leaf, the ordered motion of the -spheres in heaven—all are parts of one whole which -he will not separate into dead matter and adventitious -modifications.</p> - -<p>And just as our theories, as representative of actuality, -fall before his unequalled grasp of fact, so the doctrine -of ideas fell. It is not an adequate account of existence, -as Plato himself shows in his “Parmenides”; -it only explains things by putting their doubles beside -them.</p> - -<p>For his own part Aristotle invented a great marching -definition which, with a kind of power of its own, cleaves -its way through phenomena to limiting conceptions on<span class="pagenum" id="Page_37">[Pg 37]</span> -either hand, towards whose existence all experience -points.</p> - -<p>In Aristotle’s definition of matter and form as the -constituent of reality, as in Plato’s mystical vision of the -kingdom of ideas, the existence of the higher dimensionality -is implicitly involved.</p> - -<p>Substance according to Aristotle is relative, not absolute. -In everything that is there is the matter of which it -is composed, the form which it exhibits; but these are -indissolubly connected, and neither can be thought -without the other.</p> - -<p>The blocks of stone out of which a house is built are the -material for the builder; but, as regards the quarrymen, -they are the matter of the rocks with the form he has -imposed on them. Words are the final product of the -grammarian, but the mere matter of the orator or poet. -The atom is, with us, that out of which chemical substances -are built up, but looked at from another point of view is -the result of complex processes.</p> - -<p>Nowhere do we find finality. The matter in one sphere -is the matter, plus form, of another sphere of thought. -Making an obvious application to geometry, plane figures -exist as the limitation of different portions of the plane -by one another. In the bounding lines the separated -matter of the plane shows its determination into form. -And as the plane is the matter relatively to determinations -in the plane, so the plane itself exists in virtue of the -determination of space. A plane is that wherein formless -space has form superimposed on it, and gives an actuality -of real relations. We cannot refuse to carry this process -of reasoning a step farther back, and say that space itself -is that which gives form to higher space. As a line is -the determination of a plane, and a plane of a solid, so -solid space itself is the determination of a higher space.</p> - -<p>As a line by itself is inconceivable without that plane<span class="pagenum" id="Page_38">[Pg 38]</span> -which it separates, so the plane is inconceivable without -the solids which it limits on either hand. And so space -itself cannot be positively defined. It is the negation -of the possibility of movement in more than three -dimensions. The conception of space demands that of -a higher space. As a surface is thin and unsubstantial -without the substance of which it is the surface, so matter -itself is thin without the higher matter.</p> - -<p>Just as Aristotle invented that algebraical method of -representing unknown quantities by mere symbols, not by -lines necessarily determinate in length as was the habit -of the Greek geometers, and so struck out the path -towards those objectifications of thought which, like -independent machines for reasoning, supply the mathematician -with his analytical weapons, so in the formulation -of the doctrine of matter and form, of potentiality and -actuality, of the relativity of substance, he produced -another kind of objectification of mind—a definition -which had a vital force and an activity of its own.</p> - -<p>In none of his writings, as far as we know, did he carry it -to its legitimate conclusion on the side of matter, but in -the direction of the formal qualities he was led to his -limiting conception of that existence of pure form which -lies beyond all known determination of matter. The -unmoved mover of all things is Aristotle’s highest -principle. Towards it, to partake of its perfection all -things move. The universe, according to Aristotle, is an -active process—he does not adopt the illogical conception -that it was once set in motion and has kept on ever since. -There is room for activity, will, self-determination, in -Aristotle’s system, and for the contingent and accidental -as well. We do not follow him, because we are accustomed -to find in nature infinite series, and do not feel -obliged to pass on to a belief in the ultimate limits to -which they seem to point.</p> - -<p><span class="pagenum" id="Page_39">[Pg 39]</span></p> - -<p>But apart from the pushing to the limit, as a relative -principle this doctrine of Aristotle’s as to the relativity of -substance is irrefragible in its logic. He was the first to -show the necessity of that path of thought which when -followed leads to a belief in a four-dimensional space.</p> - -<p>Antagonistic as he was to Plato in his conception -of the practical relation of reason to the world of -phenomena, yet in one point he coincided with him. -And in this he showed the candour of his intellect. He -was more anxious to lose nothing than to explain everything. -And that wherein so many have detected an -inconsistency, an inability to free himself from the school -of Plato, appears to us in connection with our enquiry -as an instance of the acuteness of his observation. For -beyond all knowledge given by the senses Aristotle held -that there is an active intelligence, a mind not the passive -recipient of impressions from without, but an active and -originative being, capable of grasping knowledge at first -hand. In the active soul Aristotle recognised something -in man not produced by his physical surroundings, something -which creates, whose activity is a knowledge -underived from sense. This, he says, is the immortal and -undying being in man.</p> - -<p>Thus we see that Aristotle was not far from the -recognition of the four-dimensional existence, both -without and within man, and the process of adequately -realising the higher dimensional figures to which we -shall come subsequently is a simple reduction to practice -of his hypothesis of a soul.</p> - -<p>The next step in the unfolding of the drama of the -recognition of the soul as connected with our scientific -conception of the world, and, at the same time, the -recognition of that higher of which a three-dimensional -world presents the superficial appearance, took place many -centuries later. If we pass over the intervening time<span class="pagenum" id="Page_40">[Pg 40]</span> -without a word it is because the soul was occupied with -the assertion of itself in other ways than that of knowledge. -When it took up the task in earnest of knowing this -material world in which it found itself, and of directing -the course of inanimate nature, from that most objective -aim came, reflected back as from a mirror, its knowledge -of itself.</p> -<hr class="chap x-ebookmaker-drop" /> - -<div class="chapter"> -<p><span class="pagenum" id="Page_41">[Pg 41]</span></p> - -<h2 class="nobreak" id="CHAPTER_V">CHAPTER V<br /> - -<small><i>THE SECOND CHAPTER IN THE HISTORY -OF FOUR SPACE</i></small></h2></div> - - -<p><span class="smcap">Lobatchewsky, Bolyai, and Gauss</span> -Before entering on a description of the work of -Lobatchewsky and Bolyai it will not be out of place -to give a brief account of them, the materials for which -are to be found in an article by Franz Schmidt in the -forty-second volume of the <i>Mathematische Annalen</i>, -and in Engel’s edition of Lobatchewsky.</p> - -<p>Lobatchewsky was a man of the most complete and -wonderful talents. As a youth he was full of vivacity, -carrying his exuberance so far as to fall into serious -trouble for hazing a professor, and other freaks. Saved -by the good offices of the mathematician Bartels, who -appreciated his ability, he managed to restrain himself -within the bounds of prudence. Appointed professor at -his own University, Kasan, he entered on his duties under -the regime of a pietistic reactionary, who surrounded -himself with sycophants and hypocrites. Esteeming -probably the interests of his pupils as higher than any -attempt at a vain resistance, he made himself the tyrant’s -right-hand man, doing an incredible amount of teaching -and performing the most varied official duties. Amidst -all his activities he found time to make important contributions -to science. His theory of parallels is most<span class="pagenum" id="Page_42">[Pg 42]</span> -closely connected with his name, but a study of his -writings shows that he was a man capable of carrying -on mathematics in its main lines of advance, and of a -judgment equal to discerning what these lines were. -Appointed rector of his University, he died at an -advanced age, surrounded by friends, honoured, with the -results of his beneficent activity all around him. To him -no subject came amiss, from the foundations of geometry -to the improvement of the stoves by which the peasants -warmed their houses.</p> - -<p>He was born in 1793. His scientific work was -unnoticed till, in 1867, Houel, the French mathematician, -drew attention to its importance.</p> - -<p>Johann Bolyai de Bolyai was born in Klausenburg, -a town in Transylvania, December 15th, 1802.</p> - -<p>His father, Wolfgang Bolyai, a professor in the -Reformed College of Maros Vasarhely, retained the ardour -in mathematical studies which had made him a chosen -companion of Gauss in their early student days at -Göttingen.</p> - -<p>He found an eager pupil in Johann. He relates that -the boy sprang before him like a devil. As soon as he -had enunciated a problem the child would give the -solution and command him to go on further. As a -thirteen-year-old boy his father sometimes sent him to fill -his place when incapacitated from taking his classes. -The pupils listened to him with more attention than to -his father for they found him clearer to understand.</p> - -<p>In a letter to Gauss Wolfgang Bolyai writes:—</p> - -<p>“My boy is strongly built. He has learned to recognise -many constellations, and the ordinary figures of geometry. -He makes apt applications of his notions, drawing for -instance the positions of the stars with their constellations. -Last winter in the country, seeing Jupiter he asked: -‘How is it that we can see him from here as well as from<span class="pagenum" id="Page_43">[Pg 43]</span> -the town? He must be far off.’ And as to three -different places to which he had been he asked me to tell -him about them in one word. I did not know what he -meant, and then he asked me if one was in a line with -the other and all in a row, or if they were in a triangle.</p> - -<p>“He enjoys cutting paper figures with a pair of scissors, -and without my ever having told him about triangles -remarked that a right-angled triangle which he had cut -out was half of an oblong. I exercise his body with care, -he can dig well in the earth with his little hands. The -blossom can fall and no fruit left. When he is fifteen -I want to send him to you to be your pupil.”</p> - -<p>In Johann’s autobiography he says:—</p> - -<p>“My father called my attention to the imperfections -and gaps in the theory of parallels. He told me he had -gained more satisfactory results than his predecessors, -but had obtained no perfect and satisfying conclusion. -None of his assumptions had the necessary degree of -geometrical certainty, although they sufficed to prove the -eleventh axiom and appeared acceptable on first sight.</p> - -<p>“He begged of me, anxious not without a reason, to -hold myself aloof and to shun all investigation on this -subject, if I did not wish to live all my life in vain.”</p> - -<p>Johann, in the failure of his father to obtain any -response from Gauss, in answer to a letter in which he -asked the great mathematician to make of his son “an -apostle of truth in a far land,” entered the Engineering -School at Vienna. He writes from Temesvar, where he -was appointed sub-lieutenant September, 1823:—</p> - -<div class="blockquote"> -<p class="psig"> -“Temesvar, November 3rd, 1823.</p> - -<p>“<span class="smcap">Dear Good Father</span>, -</p> - -<p>“I have so overwhelmingly much to write -about my discovery that I know no other way of checking -myself than taking a quarter of a sheet only to write on. -I want an answer to my four-sheet letter.</p> - -<p><span class="pagenum" id="Page_44">[Pg 44]</span></p> - -<p>“I am unbroken in my determination to publish a -work on Parallels, as soon as I have put my material in -order and have the means.</p> - -<p>“At present I have not made any discovery, but -the way I have followed almost certainly promises me -the attainment of my object if any possibility of it -exists.</p> - -<p>“I have not got my object yet, but I have produced -such stupendous things that I was overwhelmed myself, -and it would be an eternal shame if they were lost. -When you see them you will find that it is so. Now -I can only say that I have made a new world out of -nothing. Everything that I have sent you before is a -house of cards in comparison with a tower. I am convinced -that it will be no less to my honour than if I had -already discovered it.”</p> -</div> - -<p>The discovery of which Johann here speaks was -published as an appendix to Wolfgang Bolyai’s <i>Tentamen</i>.</p> - -<p>Sending the book to Gauss, Wolfgang writes, after an -interruption of eighteen years in his correspondence:—</p> - -<div class="blockquote"> - -<p>“My son is first lieutenant of Engineers and will soon -be captain. He is a fine youth, a good violin player, -a skilful fencer, and brave, but has had many duels, and -is wild even for a soldier. Yet he is distinguished—light -in darkness and darkness in light. He is an impassioned -mathematician with extraordinary capacities.... He -will think more of your judgment on his work than that -of all Europe.”</p> -</div> - -<p>Wolfgang received no answer from Gauss to this letter, -but sending a second copy of the book received the -following reply:—</p> - -<div class="blockquote"> -<p>“You have rejoiced me, my unforgotten friend, by your -letters. I delayed answering the first because I wanted -to wait for the arrival of the promised little book.</p> - -<p>“Now something about your son’s work.</p> - -<p><span class="pagenum" id="Page_45">[Pg 45]</span></p> - -<p>“If I begin with saying that ‘I ought not to praise it,’ -you will be staggered for a moment. But I cannot say -anything else. To praise it is to praise myself, for the -path your son has broken in upon and the results to which -he has been led are almost exactly the same as my own -reflections, some of which date from thirty to thirty-five -years ago.</p> - -<p>“In fact I am astonished to the uttermost. My intention -was to let nothing be known in my lifetime about -my own work, of which, for the rest, but little is committed -to writing. Most people have but little perception -of the problem, and I have found very few who took any -interest in the views I expressed to them. To be able to -do that one must first of all have had a real live feeling -of what is wanting, and as to that most men are completely -in the dark.</p> - -<p>“Still it was my intention to commit everything to -writing in the course of time, so that at least it should -not perish with me.</p> - -<p>“I am deeply surprised that this task can be spared -me, and I am most of all pleased in this that it is the son -of my old friend who has in so remarkable a manner -preceded me.”</p> -</div> - -<p>The impression which we receive from Gauss’s inexplicable -silence towards his old friend is swept away -by this letter. Hence we breathe the clear air of the -mountain tops. Gauss would not have failed to perceive -the vast significance of his thoughts, sure to be all the -greater in their effect on future ages from the want of -comprehension of the present. Yet there is not a word -or a sign in his writing to claim the thought for himself. -He published no single line on the subject. By the -measure of what he thus silently relinquishes, by such a -measure of a world-transforming thought, we can appreciate -his greatness.</p> - -<p><span class="pagenum" id="Page_46">[Pg 46]</span></p> - -<p>It is a long step from Gauss’s serenity to the disturbed -and passionate life of Johann Bolyai—he and Galois, -the two most interesting figures in the history of mathematics. -For Bolyai, the wild soldier, the duellist, fell -at odds with the world. It is related of him that he was -challenged by thirteen officers of his garrison, a thing not -unlikely to happen considering how differently he thought -from every one else. He fought them all in succession—making -it his only condition that he should be allowed -to play on his violin for an interval between meeting each -opponent. He disarmed or wounded all his antagonists. -It can be easily imagined that a temperament such as -his was one not congenial to his military superiors. He -was retired in 1833.</p> - -<p>His epoch-making discovery awoke no attention. He -seems to have conceived the idea that his father had -betrayed him in some inexplicable way by his communications -with Gauss, and he challenged the excellent -Wolfgang to a duel. He passed his life in poverty, -many a time, says his biographer, seeking to snatch -himself from dissipation and apply himself again to -mathematics. But his efforts had no result. He died -January 27th, 1860, fallen out with the world and with -himself.</p> - - -<h3><span class="smcap">Metageometry</span></h3> - -<p>The theories which are generally connected with the -names of Lobatchewsky and Bolyai bear a singular and -curious relation to the subject of higher space.</p> - -<p>In order to show what this relation is, I must ask the -reader to be at the pains to count carefully the sets of -points by which I shall estimate the volumes of certain -figures.</p> - -<p><span class="pagenum" id="Page_47">[Pg 47]</span></p> - -<p>No mathematical processes beyond this simple one of -counting will be necessary.</p> - -<div class="figleft illowp25" id="fig_19" style="max-width: 12.5em;"> - <img src="images/fig_19.png" alt="" /> - <div class="caption">Fig. 19.</div> -</div> - -<p>Let us suppose we have before us in -<a href="#fig_19">fig. 19</a> a plane covered with points at regular -intervals, so placed that every four determine -a square.</p> - -<p>Now it is evident that as four points -determine a square, so four squares meet in a point.</p> - -<div class="figleft illowp25" id="fig_20" style="max-width: 12.5em;"> - <img src="images/fig_20.png" alt="" /> - <div class="caption">Fig. 20.</div> -</div> - -<p>Thus, considering a point inside a square as -belonging to it, we may say that a point on -the corner of a square belongs to it and to -three others equally: belongs a quarter of it -to each square.</p> - -<p>Thus the square <span class="allsmcap">ACDE</span> (<a href="#fig_21">fig. 21</a>) contains one point, and -has four points at the four corners. Since one-fourth of -each of these four belongs to the square, the four together -count as one point, and the point value of the square is -two points—the one inside and the four at the corner -make two points belonging to it exclusively.</p> - -<div class="figleft illowp25" id="fig_21" style="max-width: 12.5em;"> - <img src="images/fig_21.png" alt="" /> - <div class="caption">Fig. 21.</div> -</div> - -<div class="figright illowp25" id="fig_22" style="max-width: 12.8125em;"> - <img src="images/fig_22.png" alt="" /> - <div class="caption">Fig. 22.</div> -</div> - -<p>Now the area of this square is two unit squares, as can -be seen by drawing two diagonals in <a href="#fig_22">fig. 22</a>.</p> - -<p>We also notice that the square in question is equal to -the sum of the squares on the sides <span class="allsmcap">AB</span>, <span class="allsmcap">BC</span>, of the right-angled -triangle <span class="allsmcap">ABC</span>. Thus we recognise the proposition -that the square on the hypothenuse is equal to the sum -of the squares on the two sides of a right-angled triangle.</p> - -<p>Now suppose we set ourselves the question of determining -the whereabouts in the ordered system of points,<span class="pagenum" id="Page_48">[Pg 48]</span> -the end of a line would come when it turned about a -point keeping one extremity fixed at the point.</p> - -<p>We can solve this problem in a particular case. If we -can find a square lying slantwise amongst the dots which is -equal to one which goes regularly, we shall know that the -two sides are equal, and that the slanting side is equal to the -straight-way side. Thus the volume and shape of a figure -remaining unchanged will be the test of its having rotated -about the point, so that we can say that its side in its first -position would turn into its side in the second position.</p> - -<p>Now, such a square can be found in the one whose side -is five units in length.</p> - -<div class="figcenter illowp66" id="fig_23" style="max-width: 25em;"> - <img src="images/fig_23.png" alt="" /> - <div class="caption">Fig. 23.</div> -</div> - -<p>In <a href="#fig_23">fig. 23</a>, in the square on <span class="allsmcap">AB</span>, there are—</p> - -<table class="standard" summary=""> -<tr> -<td class="tdl">9 points interior</td> -<td class="tdr">9</td> -</tr> -<tr> -<td class="tdl">4 at the corners</td> -<td class="tdr">1</td> -</tr> -<tr> -<td class="tdh"> 4 sides with 3 on each side, considered as -1½ on each side, because belonging -equally to two squares</td> -<td class="tdrb">6</td> -</tr> -</table> - -<p>The total is 16. There are 9 points in the square -on <span class="allsmcap">BC</span>.</p> - -<p><span class="pagenum" id="Page_49">[Pg 49]</span></p> - -<p>In the square on <span class="allsmcap">AC</span> there are—</p> - - -<table class="standard" summary=""> -<tr> -<td class="tdl">24 points inside</td> -<td class="tdr">24</td> -</tr> -<tr> -<td class="tdl"> 4 at the corners</td> -<td class="tdr">1</td> -</tr> -</table> - -<p>or 25 altogether.</p> - -<p>Hence we see again that the square on the hypothenuse -is equal to the squares on the sides.</p> - -<p>Now take the square <span class="allsmcap">AFHG</span>, which is larger than the -square on <span class="allsmcap">AB</span>. It contains 25 points.</p> - - -<table class="standard" summary=""> -<tr> -<td class="tdl">16 inside</td> -<td class="tdr">16</td> -</tr> -<tr> -<td class="tdl">16 on the sides, counting as</td> -<td class="tdr">8</td> -</tr> -<tr> -<td class="tdl"> 4 on the corners</td> -<td class="tdr">1</td> -</tr> -</table> - -<p>making 25 altogether.</p> - -<p>If two squares are equal we conclude the sides are -equal. Hence, the line <span class="allsmcap">AF</span> turning round <span class="allsmcap">A</span> would -move so that it would after a certain turning coincide -with <span class="allsmcap">AC</span>.</p> - -<p>This is preliminary, but it involves all the mathematical -difficulties that will present themselves.</p> - -<p>There are two alterations of a body by which its volume -is not changed.</p> - -<p>One is the one we have just considered, rotation, the -other is what is called shear.</p> - -<p>Consider a book, or heap of loose pages. They can be -slid so that each one slips -over the preceding one, -and the whole assumes -the shape <i>b</i> in <a href="#fig_24">fig. 24</a>.</p> - -<div class="figleft illowp50" id="fig_24" style="max-width: 25em;"> - <img src="images/fig_24.png" alt="" /> - <div class="caption">Fig. 24.</div> -</div> - -<p>This deformation is not shear alone, but shear accompanied -by rotation.</p> - -<p>Shear can be considered as produced in another way.</p> - -<p>Take the square <span class="allsmcap">ABCD</span> (<a href="#fig_25">fig. 25</a>), and suppose that it -is pulled out from along one of its diagonals both ways, -and proportionately compressed along the other diagonal. -It will assume the shape in <a href="#fig_26">fig. 26</a>.</p> - -<p><span class="pagenum" id="Page_50">[Pg 50]</span></p> - -<p>This compression and expansion along two lines at right -angles is what is called shear; it is equivalent to the -sliding illustrated above, combined with a turning round.</p> - -<div class="figleft illowp45" id="fig_25" style="max-width: 12.5em;"> - <img src="images/fig_25.png" alt="" /> - <div class="caption">Fig. 25.</div> -</div> - -<div class="figright illowp50" id="fig_26" style="max-width: 18.75em;"> - <img src="images/fig_26.png" alt="" /> - <div class="caption">Fig. 26.</div> -</div> - -<p>In pure shear a body is compressed and extended in -two directions at right angles to each other, so that its -volume remains unchanged.</p> - -<p>Now we know that our material bodies resist shear—shear -does violence to the internal arrangement of their -particles, but they turn as wholes without such internal -resistance.</p> - -<p>But there is an exception. In a liquid shear and -rotation take place equally easily, there is no more -resistance against a shear than there is against a -rotation.</p> - -<p>Now, suppose all bodies were to be reduced to the liquid -state, in which they yield to shear and to rotation equally -easily, and then were to be reconstructed as solids, but in -such a way that shear and rotation had interchanged -places.</p> - -<p>That is to say, let us suppose that when they had -become solids again they would shear without offering -any internal resistance, but a rotation would do violence -to their internal arrangement.</p> - -<p>That is, we should have a world in which shear would -have taken the place of rotation.</p> - -<p><span class="pagenum" id="Page_51">[Pg 51]</span></p> - -<p>A shear does not alter the volume of a body: thus an -inhabitant living in such a world would look on a body -sheared as we look on a body rotated. He would say -that it was of the same shape, but had turned a bit -round.</p> - -<p>Let us imagine a Pythagoras in this world going to -work to investigate, as is his wont.</p> - -<div class="figleft illowp40" id="fig_27" style="max-width: 12.5em;"> - <img src="images/fig_27.png" alt="" /> - <div class="caption">Fig. 27.</div> -</div> -<div class="figright illowp40" id="fig_28" style="max-width: 13.125em;"> - <img src="images/fig_28.png" alt="" /> - <div class="caption">Fig. 28.</div> -</div> - -<p>Fig. 27 represents a square unsheared. Fig. 28 -represents a square sheared. It is not the figure into -which the square in <a href="#fig_27">fig. 27</a> would turn, but the result of -shear on some square not drawn. It is a simple slanting -placed figure, taken now as we took a simple slanting -placed square before. Now, since bodies in this world of -shear offer no internal resistance to shearing, and keep -their volume when sheared, an inhabitant accustomed to -them would not consider that they altered their shape -under shear. He would call <span class="allsmcap">ACDE</span> as much a square as -the square in <a href="#fig_27">fig. 27</a>. We will call such figures shear -squares. Counting the dots in <span class="allsmcap">ACDE</span>, we find—</p> - - -<table class="standard" summary=""> -<tr> -<td class="tdl">2 inside</td> -<td class="tdc">=</td> -<td class="tdc">2</td> -</tr> -<tr> -<td class="tdl">4 at corners</td> -<td class="tdc">=</td> -<td class="tdc">1</td> -</tr> -</table> - -<p>or a total of 3.</p> - -<p>Now, the square on the side <span class="allsmcap">AB</span> has 4 points, that on <span class="allsmcap">BC</span> -has 1 point. Here the shear square on the hypothenuse -has not 5 points but 3; it is not the sum of the squares on -the sides, but the difference.</p> - -<p><span class="pagenum" id="Page_52">[Pg 52]</span></p> - -<div class="figleft illowp25" id="fig_29" style="max-width: 13.75em;"> - <img src="images/fig_29.png" alt="" /> - <div class="caption">Fig. 29.</div> -</div> - -<p>This relation always holds. Look at -<a href="#fig_29">fig. 29</a>.</p> - -<p>Shear square on hypothenuse—</p> - -<table class="standard" summary=""> -<tr> -<td class="tdl">7 internal</td> -<td class="tdr"> 7</td> -</tr> -<tr> -<td class="tdl">4 at corners</td> -<td class="tdr">1</td> -</tr> -<tr> -<td class="tdl"></td> -<td class="tdr_bt">8</td> -</tr> -</table> - - -<div class="figleft illowp50" id="fig_29bis" style="max-width: 25em;"> - <img src="images/fig_29bis.png" alt="" /> - <div class="caption">Fig. 29 <i>bis</i>.</div> -</div> - -<p>Square on one side—which the reader can draw for -himself—</p> - -<table class="standard" summary=""> -<tr> -<td class="tdl">4 internal</td> -<td class="tdr"> 4</td> -</tr> -<tr> -<td class="tdl">8 on sides</td> -<td class="tdr">4</td> -</tr> -<tr> -<td class="tdl">4 at corners</td> -<td class="tdr">1</td> -</tr> -<tr> -<td class="tdl"></td> -<td class="tdr_bt">9</td> -</tr> -</table> - - -<p>and the square on the other -side is 1. Hence in this -case again the difference is -equal to the shear square on -the hypothenuse, 9 - 1 = 8.</p> - -<p>Thus in a world of shear -the square on the hypothenuse -would be equal to the -difference of the squares on -the sides of a right-angled -triangle.</p> - -<p>In <a href="#fig_29">fig. 29</a> <i>bis</i> another shear square is drawn on which -the above relation can be tested.</p> - -<p>What now would be the position a line on turning by -shear would take up?</p> - -<p>We must settle this in the same way as previously with -our turning.</p> - -<p>Since a body sheared remains the same, we must find two -equal bodies, one in the straight way, one in the slanting -way, which have the same volume. Then the side of one -will by turning become the side of the other, for the two -figures are each what the other becomes by a shear turning.</p> - -<p><span class="pagenum" id="Page_53">[Pg 53]</span></p> - -<p>We can solve the problem in a particular case—</p> - -<div class="figleft illowp50" id="fig_30" style="max-width: 25em;"> - <img src="images/fig_30.png" alt="" /> - <div class="caption">Fig. 30.</div> -</div> - -<p>In the figure <span class="allsmcap">ACDE</span> -(<a href="#fig_30">fig. 30</a>) there are—</p> - -<table class="standard" summary=""> -<tr> -<td class="tdr">15 inside</td> -<td class="tdl">15</td> -</tr> -<tr> -<td class="tdl">4 at corners</td> -<td class="tdr"> 1</td> -</tr> -</table> - -<p>a total of 16.</p> - -<p>Now in the square <span class="allsmcap">ABGF</span>, -there are 16—</p> - -<table class="standard" summary=""> -<tr> -<td class="tdl">9 inside</td> -<td class="tdr"> 9</td> -</tr> -<tr> -<td class="tdl">12 on sides</td> -<td class="tdr">6</td> -</tr> -<tr> -<td class="tdl">4 at corners</td> -<td class="tdr">1</td> -</tr> -<tr> -<td></td> -<td class="tdr_bt">16</td> -</tr> -</table> - -<p>Hence the square on <span class="allsmcap">AB</span> -would, by the shear turning, -become the shear square -<span class="allsmcap">ACDE</span>.</p> - -<p>And hence the inhabitant of this world would say that -the line <span class="allsmcap">AB</span> turned into the line <span class="allsmcap">AC</span>. These two lines -would be to him two lines of equal length, one turned -a little way round from the other.</p> - -<p>That is, putting shear in place of rotation, we get a -different kind of figure, as the result of the shear rotation, -from what we got with our ordinary rotation. And as a -consequence we get a position for the end of a line of -invariable length when it turns by the shear rotation, -different from the position which it would assume on -turning by our rotation.</p> - -<p>A real material rod in the shear world would, on turning -about <span class="allsmcap">A</span>, pass from the position <span class="allsmcap">AB</span> to the position <span class="allsmcap">AC</span>. -We say that its length alters when it becomes <span class="allsmcap">AC</span>, but this -transformation of <span class="allsmcap">AB</span> would seem to an inhabitant of the -shear world like a turning of <span class="allsmcap">AB</span> without altering in -length.</p> - -<p>If now we suppose a communication of ideas that takes -place between one of ourselves and an inhabitant of the<span class="pagenum" id="Page_54">[Pg 54]</span> -shear world, there would evidently be a difference between -his views of distance and ours.</p> - -<p>We should say that his line <span class="allsmcap">AB</span> increased in length in -turning to <span class="allsmcap">AC</span>. He would say that our line <span class="allsmcap">AF</span> (<a href="#fig_23">fig. 23</a>) -decreased in length in turning to <span class="allsmcap">AC</span>. He would think -that what we called an equal line was in reality a shorter -one.</p> - -<p>We should say that a rod turning round would have its -extremities in the positions we call at equal distances. -So would he—but the positions would be different. He -could, like us, appeal to the properties of matter. His -rod to him alters as little as ours does to us.</p> - -<p>Now, is there any standard to which we could appeal, to -say which of the two is right in this argument? There -is no standard.</p> - -<p>We should say that, with a change of position, the -configuration and shape of his objects altered. He would -say that the configuration and shape of our objects altered -in what we called merely a change of position. Hence -distance independent of position is inconceivable, or -practically distance is solely a property of matter.</p> - -<p>There is no principle to which either party in this -controversy could appeal. There is nothing to connect -the definition of distance with our ideas rather than with -his, except the behaviour of an actual piece of matter.</p> - -<p>For the study of the processes which go on in our world -the definition of distance given by taking the sum of the -squares is of paramount importance to us. But as a question -of pure space without making any unnecessary -assumptions the shear world is just as possible and just as -interesting as our world.</p> - -<p>It was the geometry of such conceivable worlds that -Lobatchewsky and Bolyai studied.</p> - -<p>This kind of geometry has evidently nothing to do -directly with four-dimensional space.</p> - -<p><span class="pagenum" id="Page_55">[Pg 55]</span></p> - -<p>But a connection arises in this way. It is evident that, -instead of taking a simple shear as I have done, and -defining it as that change of the arrangement of the -particles of a solid which they will undergo without -offering any resistance due to their mutual action, I -might take a complex motion, composed of a shear and -a rotation together, or some other kind of deformation.</p> - -<p>Let us suppose such an alteration picked out and -defined as the one which means simple rotation, then the -type, according to which all bodies will alter by this -rotation, is fixed.</p> - -<p>Looking at the movements of this kind, we should say -that the objects were altering their shape as well as -rotating. But to the inhabitants of that world they -would seem to be unaltered, and our figures in their -motions would seem to them to alter.</p> - -<p>In such a world the features of geometry are different. -We have seen one such difference in the case of our illustration -of the world of shear, where the square on the -hypothenuse was equal to the difference, not the sum, of -the squares on the sides.</p> - -<p>In our illustration we have the same laws of parallel -lines as in our ordinary rotation world, but in general the -laws of parallel lines are different.</p> - -<p>In one of these worlds of a different constitution of -matter through one point there can be two parallels to -a given line, in another of them there can be none, that -is, although a line be drawn parallel to another it will -meet it after a time.</p> - -<p>Now it was precisely in this respect of parallels that -Lobatchewsky and Bolyai discovered these different -worlds. They did not think of them as worlds of matter, -but they discovered that space did not necessarily mean -that our law of parallels is true. They made the -distinction between laws of space and laws of matter,<span class="pagenum" id="Page_56">[Pg 56]</span> -although that is not the form in which they stated their -results.</p> - -<p>The way in which they were led to these results was the -following. Euclid had stated the existence of parallel lines -as a postulate—putting frankly this unproved proposition—that -one line and only one parallel to a given straight -line can be drawn, as a demand, as something that must -be assumed. The words of his ninth postulate are these: -“If a straight line meeting two other straight lines -makes the interior angles on the same side of it equal -to two right angles, the two straight lines will never -meet.”</p> - -<p>The mathematicians of later ages did not like this bald -assumption, and not being able to prove the proposition -they called it an axiom—the eleventh axiom.</p> - -<p>Many attempts were made to prove the axiom; no one -doubted of its truth, but no means could be found to -demonstrate it. At last an Italian, Sacchieri, unable to -find a proof, said: “Let us suppose it not true.” He deduced -the results of there being possibly two parallels to one -given line through a given point, but feeling the waters -too deep for the human reason, he devoted the latter half -of his book to disproving what he had assumed in the first -part.</p> - -<p>Then Bolyai and Lobatchewsky with firm step entered -on the forbidden path. There can be no greater evidence -of the indomitable nature of the human spirit, or of its -manifest destiny to conquer all those limitations which -bind it down within the sphere of sense than this grand -assertion of Bolyai and Lobatchewsky.</p> - -<div class="figleft illowp25" id="fig_31" style="max-width: 12.5em;"> - <img src="images/fig_31.png" alt="" /> - <div class="caption">Fig. 31.</div> -</div> - -<p>Take a line <span class="allsmcap">AB</span> and a point <span class="allsmcap">C</span>. We -say and see and know that through <span class="allsmcap">C</span> -can only be drawn one line parallel -to <span class="allsmcap">AB</span>.</p> - -<p>But Bolyai said: “I will draw two.” Let <span class="allsmcap">CD</span> be parallel<span class="pagenum" id="Page_57">[Pg 57]</span> -to <span class="allsmcap">AB</span>, that is, not meet <span class="allsmcap">AB</span> however far produced, and let -lines beyond <span class="allsmcap">CD</span> also not meet -<span class="allsmcap">AB</span>; let there be a certain -region between <span class="allsmcap">CD</span> and <span class="allsmcap">CE</span>, -in which no line drawn meets -<span class="allsmcap">AB</span>. <span class="allsmcap">CE</span> and <span class="allsmcap">CD</span> produced -backwards through <span class="allsmcap">C</span> will give a similar region on the -other side of <span class="allsmcap">C</span>.</p> - -<div class="figleft illowp40" id="fig_32" style="max-width: 21.875em;"> - <img src="images/fig_32.png" alt="" /> - <div class="caption">Fig. 32.</div> -</div> - -<p>Nothing so triumphantly, one may almost say so -insolently, ignoring of sense had ever been written before. -Men had struggled against the limitations of the body, -fought them, despised them, conquered them. But no -one had ever thought simply as if the body, the bodily -eyes, the organs of vision, all this vast experience of space, -had never existed. The age-long contest of the soul with -the body, the struggle for mastery, had come to a culmination. -Bolyai and Lobatchewsky simply thought as -if the body was not. The struggle for dominion, the strife -and combat of the soul were over; they had mastered, -and the Hungarian drew his line.</p> - -<p>Can we point out any connection, as in the case of -Parmenides, between these speculations and higher -space? Can we suppose it was any inner perception by -the soul of a motion not known to the senses, which resulted -in this theory so free from the bonds of sense? No -such supposition appears to be possible.</p> - -<p>Practically, however, metageometry had a great influence -in bringing the higher space to the front as a -working hypothesis. This can be traced to the tendency -the mind has to move in the direction of least resistance. -The results of the new geometry could not be neglected, -the problem of parallels had occupied a place too prominent -in the development of mathematical thought for its final -solution to be neglected. But this utter independence of -all mechanical considerations, this perfect cutting loose<span class="pagenum" id="Page_58">[Pg 58]</span> -from the familiar intuitions, was so difficult that almost -any other hypothesis was more easy of acceptance, and -when Beltrami showed that the geometry of Lobatchewsky -and Bolyai was the geometry of shortest lines drawn on -certain curved surfaces, the ordinary definitions of measurement -being retained, attention was drawn to the theory of -a higher space. An illustration of Beltrami’s theory is -furnished by the simple consideration of hypothetical -beings living on a spherical surface.</p> - -<div class="figleft illowp35" id="fig_33" style="max-width: 15.625em;"> - <img src="images/fig_33.png" alt="" /> - <div class="caption">Fig. 33.</div> -</div> - -<p>Let <span class="allsmcap">ABCD</span> be the equator of a globe, and <span class="allsmcap">AP</span>, <span class="allsmcap">BP</span>, -meridian lines drawn to the pole, <span class="allsmcap">P</span>. -The lines <span class="allsmcap">AB</span>, <span class="allsmcap">AP</span>, <span class="allsmcap">BP</span> would seem to be -perfectly straight to a person moving -on the surface of the sphere, and -unconscious of its curvature. Now -<span class="allsmcap">AP</span> and <span class="allsmcap">BP</span> both make right angles -with <span class="allsmcap">AB</span>. Hence they satisfy the -definition of parallels. Yet they -meet in <span class="allsmcap">P</span>. Hence a being living on a spherical surface, -and unconscious of its curvature, would find that parallel -lines would meet. He would also find that the angles -in a triangle were greater than two right angles. In -the triangle <span class="allsmcap">PAB</span>, for instance, the angles at <span class="allsmcap">A</span> and <span class="allsmcap">B</span> -are right angles, so the three angles of the triangle -<span class="allsmcap">PAB</span> are greater than two right angles.</p> - -<p>Now in one of the systems of metageometry (for after -Lobatchewsky had shown the way it was found that other -systems were possible besides his) the angles of a triangle -are greater than two right angles.</p> - -<p>Thus a being on a sphere would form conclusions about -his space which are the same as he would form if he lived -on a plane, the matter in which had such properties as -are presupposed by one of these systems of geometry. -Beltrami also discovered a certain surface on which there -could be drawn more than one “straight” line through a<span class="pagenum" id="Page_59">[Pg 59]</span> -point which would not meet another given line. I use -the word straight as equivalent to the line having the -property of giving the shortest path between any two -points on it. Hence, without giving up the ordinary -methods of measurement, it was possible to find conditions -in which a plane being would necessarily have an experience -corresponding to Lobatchewsky’s geometry. -And by the consideration of a higher space, and a solid -curved in such a higher space, it was possible to account -for a similar experience in a space of three dimensions.</p> - -<p>Now, it is far more easy to conceive of a higher dimensionality -to space than to imagine that a rod in rotating -does not move so that its end describes a circle. Hence, -a logical conception having been found harder than that -of a four dimensional space, thought turned to the latter -as a simple explanation of the possibilities to which -Lobatchewsky had awakened it. Thinkers became accustomed -to deal with the geometry of higher space—it was -Kant, says Veronese, who first used the expression of -“different spaces”—and with familiarity the inevitableness -of the conception made itself felt.</p> - -<p>From this point it is but a small step to adapt the -ordinary mechanical conceptions to a higher spatial -existence, and then the recognition of its objective -existence could be delayed no longer. Here, too, as in so -many cases, it turns out that the order and connection of -our ideas is the order and connection of things.</p> - -<p>What is the significance of Lobatchewsky’s and Bolyai’s -work?</p> - -<p>It must be recognised as something totally different -from the conception of a higher space; it is applicable to -spaces of any number of dimensions. By immersing the -conception of distance in matter to which it properly -belongs, it promises to be of the greatest aid in analysis -for the effective distance of any two particles is the<span class="pagenum" id="Page_60">[Pg 60]</span> -product of complex material conditions and cannot be -measured by hard and fast rules. Its ultimate significance -is altogether unknown. It is a cutting loose -from the bonds of sense, not coincident with the recognition -of a higher dimensionality, but indirectly contributory -thereto.</p> - -<p>Thus, finally, we have come to accept what Plato held -in the hollow of his hand; what Aristotle’s doctrine of -the relativity of substance implies. The vast universe, too, -has its higher, and in recognising it we find that the -directing being within us no longer stands inevitably -outside our systematic knowledge.</p> -<hr class="chap x-ebookmaker-drop" /> - -<div class="chapter"> -<p><span class="pagenum" id="Page_61">[Pg 61]</span></p> - -<h2 class="nobreak" id="CHAPTER_VI">CHAPTER VI<br /> - -<small><i>THE HIGHER WORLD</i></small></h2></div> - - -<p>It is indeed strange, the manner in which we must begin -to think about the higher world.</p> - -<p>Those simplest objects analogous to those which are -about us on every side in our daily experience such as a -door, a table, a wheel are remote and uncognisable in the -world of four dimensions, while the abstract ideas of -rotation, stress and strain, elasticity into which analysis -resolves the familiar elements of our daily experience are -transferable and applicable with no difficulty whatever. -Thus we are in the unwonted position of being obliged -to construct the daily and habitual experience of a four-dimensional -being, from a knowledge of the abstract -theories of the space, the matter, the motion of it; -instead of, as in our case, passing to the abstract theories -from the richness of sensible things.</p> - -<p>What would a wheel be in four dimensions? What -the shafting for the transmission of power which a -four-dimensional being would use.</p> - -<p>The four-dimensional wheel, and the four-dimensional -shafting are what will occupy us for these few pages. And -it is no futile or insignificant enquiry. For in the attempt -to penetrate into the nature of the higher, to grasp within -our ken that which transcends all analogies, because what -we know are merely partial views of it, the purely -material and physical path affords a means of approach<span class="pagenum" id="Page_62">[Pg 62]</span> -pursuing which we are in less likelihood of error than if -we use the more frequently trodden path of framing -conceptions which in their elevation and beauty seem to -us ideally perfect.</p> - -<p>For where we are concerned with our own thoughts, the -development of our own ideals, we are as it were on a -curve, moving at any moment in a direction of tangency. -Whither we go, what we set up and exalt as perfect, -represents not the true trend of the curve, but our own -direction at the present—a tendency conditioned by the -past, and by a vital energy of motion essential but -only true when perpetually modified. That eternal corrector -of our aspirations and ideals, the material universe -draws sublimely away from the simplest things we can -touch or handle to the infinite depths of starry space, -in one and all uninfluenced by what we think or feel, -presenting unmoved fact to which, think it good or -think it evil, we can but conform, yet out of all that -impassivity with a reference to something beyond our -individual hopes and fears supporting us and giving us -our being.</p> - -<p>And to this great being we come with the question: -“You, too, what is your higher?”</p> - -<p>Or to put it in a form which will leave our conclusions in -the shape of no barren formula, and attacking the problem -on its most assailable side: “What is the wheel and the -shafting of the four-dimensional mechanic?”</p> - -<p>In entering on this enquiry we must make a plan of -procedure. The method which I shall adopt is to trace -out the steps of reasoning by which a being confined -to movement in a two-dimensional world could arrive at a -conception of our turning and rotation, and then to apply -an analogous process to the consideration of the higher -movements. The plane being must be imagined as no -abstract figure, but as a real body possessing all three<span class="pagenum" id="Page_63">[Pg 63]</span> -dimensions. His limitation to a plane must be the result -of physical conditions.</p> - -<p>We will therefore think of him as of a figure cut out of -paper placed on a smooth plane. Sliding over this plane, -and coming into contact with other figures equally thin -as he in the third dimension, he will apprehend them only -by their edges. To him they will be completely bounded -by lines. A “solid” body will be to him a two-dimensional -extent, the interior of which can only be reached by -penetrating through the bounding lines.</p> - -<p>Now such a plane being can think of our three-dimensional -existence in two ways.</p> - -<p>First, he can think of it as a series of sections, each like -the solid he knows of extending in a direction unknown -to him, which stretches transverse to his tangible -universe, which lies in a direction at right angles to every -motion which he made.</p> - -<p>Secondly, relinquishing the attempt to think of the -three-dimensional solid body in its entirety he can regard -it as consisting of a number of plane sections, each of them -in itself exactly like the two-dimensional bodies he knows, -but extending away from his two-dimensional space.</p> - -<p>A square lying in his space he regards as a solid -bounded by four lines, each of which lies in his space.</p> - -<p>A square standing at right angles to his plane appears -to him as simply a line in his plane, for all of it except -the line stretches in the third dimension.</p> - -<p>He can think of a three-dimensional body as consisting -of a number of such sections, each of which starts from a -line in his space.</p> - -<p>Now, since in his world he can make any drawing or -model which involves only two dimensions, he can represent -each such upright section as it actually is, and can represent -a turning from a known into the unknown dimension -as a turning from one to another of his known dimensions.</p> - -<p><span class="pagenum" id="Page_64">[Pg 64]</span></p> - -<p>To see the whole he must relinquish part of that which -he has, and take the whole portion by portion.</p> - -<div class= "figleft illowp30" id="fig_34" style="max-width: 15.625em;"> - <img src="images/fig_34.png" alt="" /> - <div class="caption">Fig. 34.</div> -</div> - -<p>Consider now a plane being in front of a square, <a href="#fig_34">fig. 34</a>. -The square can turn about any point -in the plane—say the point <span class="allsmcap">A</span>. But it -cannot turn about a line, as <span class="allsmcap">AB</span>. For, -in order to turn about the line <span class="allsmcap">AB</span>, -the square must leave the plane and -move in the third dimension. This -motion is out of his range of observation, -and is therefore, except for a -process of reasoning, inconceivable to him.</p> - -<p>Rotation will therefore be to him rotation about a point. -Rotation about a line will be inconceivable to him.</p> - -<p>The result of rotation about a line he can apprehend. -He can see the first and last positions occupied in a half-revolution -about the line <span class="allsmcap">AC</span>. The result of such a half revolution -is to place the square <span class="allsmcap">ABCD</span> on the left hand instead -of on the right hand of the line <span class="allsmcap">AC</span>. It would correspond -to a pulling of the whole body <span class="allsmcap">ABCD</span> through the line <span class="allsmcap">AC</span>, -or to the production of a solid body which was the exact -reflection of it in the line <span class="allsmcap">AC</span>. It would be as if the square -<span class="allsmcap">ABCD</span> turned into its image, the line <span class="allsmcap">AB</span> acting as a mirror. -Such a reversal of the positions of the parts of the square -would be impossible in his space. The occurrence of it -would be a proof of the existence of a higher dimensionality.</p> - -<div class="figleft illowp30" id="fig_35" style="max-width: 18.75em;"> - <img src="images/fig_35.png" alt="" /> - <div class="caption">Fig. 35.</div> -</div> - -<p>Let him now, adopting the conception of a three-dimensional -body as a series of -sections lying, each removed a little -farther than the preceding one, in -direction at right angles to his -plane, regard a cube, <a href="#fig_36">fig. 36</a>, as a -series of sections, each like the -square which forms its base, all -rigidly connected together.</p> - -<p><span class="pagenum" id="Page_65">[Pg 65]</span></p> - -<p>If now he turns the square about the point <span class="allsmcap">A</span> in the -plane of <i>xy</i>, each parallel section turns with the square -he moves. In each of the sections there is a point at -rest, that vertically over <span class="allsmcap">A</span>. Hence he would conclude -that in the turning of a three-dimensional body there is -one line which is at rest. That is a three-dimensional -turning in a turning about a line.</p> - -<hr class="tb" /> - -<p>In a similar way let us regard ourselves as limited to a -three-dimensional world by a physical condition. Let us -imagine that there is a direction at right angles to every -direction in which we can move, and that we are prevented -from passing in this direction by a vast solid, that -against which in every movement we make we slip as -the plane being slips against his plane sheet.</p> - -<p>We can then consider a four-dimensional body as consisting -of a series of sections, each parallel to our space, -and each a little farther off than the preceding on the -unknown dimension.</p> - -<div class="figleft illowp35" id="fig_36" style="max-width: 18.75em;"> - <img src="images/fig_36.png" alt="" /> - <div class="caption">Fig. 36.</div> -</div> - -<p>Take the simplest four-dimensional body—one which -begins as a cube, <a href="#fig_36">fig. 36</a>, in our -space, and consists of sections, each -a cube like <a href="#fig_36">fig. 36</a>, lying away from -our space. If we turn the cube -which is its base in our space -about a line, if, <i>e.g.</i>, in <a href="#fig_36">fig. 36</a> we -turn the cube about the line <span class="allsmcap">AB</span>, -not only it but each of the parallel -cubes moves about a line. The -cube we see moves about the line <span class="allsmcap">AB</span>, the cube beyond it -about a line parallel to <span class="allsmcap">AB</span> and so on. Hence the whole -four-dimensional body moves about a plane, for the -assemblage of these lines is our way of thinking about the -plane which, starting from the line <span class="allsmcap">AB</span> in our space, runs -off in the unknown direction.</p> - -<p><span class="pagenum" id="Page_66">[Pg 66]</span></p> - -<p>In this case all that we see of the plane about which -the turning takes place is the line <span class="allsmcap">AB</span>.</p> - -<p>But it is obvious that the axis plane may lie in our -space. A point near the plane determines with it a three-dimensional -space. When it begins to rotate round the -plane it does not move anywhere in this three-dimensional -space, but moves out of it. A point can no more rotate -round a plane in three-dimensional space than a point -can move round a line in two-dimensional space.</p> - -<p>We will now apply the second of the modes of representation -to this case of turning about a plane, building -up our analogy step by step from the turning in a plane -about a point and that in space about a line, and so on.</p> - -<p>In order to reduce our considerations to those of the -greatest simplicity possible, let us realise how the plane -being would think of the motion by which a square is -turned round a line.</p> - -<p>Let, <a href="#fig_34">fig. 34</a>, <span class="allsmcap">ABCD</span> be a square on his plane, and represent -the two dimensions of his space by the axes <span class="allsmcap">A</span><i>x</i> <span class="allsmcap">A</span><i>y</i>.</p> - -<p>Now the motion by which the square is turned over -about the line <span class="allsmcap">AC</span> involves the third dimension.</p> - -<p>He cannot represent the motion of the whole square in -its turning, but he can represent the motions of parts of -it. Let the third axis perpendicular to the plane of the -paper be called the axis of <i>z</i>. Of the three axes <i>x</i>, <i>y</i>, <i>z</i>, -the plane being can represent any two in his space. Let -him then draw, in <a href="#fig_35">fig. 35</a>, two axes, <i>x</i> and <i>z</i>. Here he has -in his plane a representation of what exists in the plane -which goes off perpendicularly to his space.</p> - -<p>In this representation the square would not be shown, -for in the plane of <i>xz</i> simply the line <span class="allsmcap">AB</span> of the square is -contained.</p> - -<p>The plane being then would have before him, in <a href="#fig_35">fig. 35</a>, -the representation of one line <span class="allsmcap">AB</span> of his square and two -axes, <i>x</i> and <i>z</i>, at right angles. Now it would be obvious<span class="pagenum" id="Page_67">[Pg 67]</span> -to him that, by a turning such as he knows, by a rotation -about a point, the line <span class="allsmcap">AB</span> can turn round <span class="allsmcap">A</span>, and occupying -all the intermediate positions, such as <span class="allsmcap">AB</span><sub>1</sub>, come -after half a revolution to lie as <span class="allsmcap">A</span><i>x</i> produced through <span class="allsmcap">A</span>.</p> - -<p>Again, just as he can represent the vertical plane -through <span class="allsmcap">AB</span>, so he can represent the vertical plane -through <span class="allsmcap">A´B´</span>, <a href="#fig_34">fig. 34</a>, and in a like manner can see that -the line <span class="allsmcap">A´B´</span> can turn about the point <span class="allsmcap">A´</span> till it lies in the -opposite direction from that which it ran in at first.</p> - -<p>Now these two turnings are not inconsistent. In his -plane, if <span class="allsmcap">AB</span> turned about <span class="allsmcap">A</span>, and <span class="allsmcap">A´B´</span> about <span class="allsmcap">A´</span>, the consistency -of the square would be destroyed, it would be an -impossible motion for a rigid body to perform. But in -the turning which he studies portion by portion there is -nothing inconsistent. Each line in the square can turn -in this way, hence he would realise the turning of the -whole square as the sum of a number of turnings of -isolated parts. Such turnings, if they took place in his -plane, would be inconsistent, but by virtue of a third -dimension they are consistent, and the result of them all -is that the square turns about the line <span class="allsmcap">AC</span> and lies in a -position in which it is the mirror image of what it was in -its first position. Thus he can realise a turning about a -line by relinquishing one of his axes, and representing his -body part by part.</p> - -<p>Let us apply this method to the turning of a cube so as -to become the mirror image of itself. In our space we can -construct three independent axes, <i>x</i>, <i>y</i>, <i>z</i>, shown in <a href="#fig_36">fig. 36</a>. -Suppose that there is a fourth axis, <i>w</i>, at right angles to -each and every one of them. We cannot, keeping all -three axes, <i>x</i>, <i>y</i>, <i>z</i>, represent <i>w</i> in our space; but if we -relinquish one of our three axes we can let the fourth axis -take its place, and we can represent what lies in the -space, determined by the two axes we retain and the -fourth axis.</p> - -<p><span class="pagenum" id="Page_68">[Pg 68]</span></p> - -<div class="figleft illowp35" id="fig_37" style="max-width: 18.75em;"> - <img src="images/fig_37.png" alt="" /> - <div class="caption">Fig. 37.</div> -</div> - -<p>Let us suppose that we let the <i>y</i> axis drop, and that -we represent the <i>w</i> axis as occupying -its direction. We have in fig. -37 a drawing of what we should -then see of the cube. The square -<span class="allsmcap">ABCD</span>, remains unchanged, for that -is in the plane of <i>xz</i>, and we -still have that plane. But from -this plane the cube stretches out -in the direction of the <i>y</i> axis. Now the <i>y</i> axis is gone, -and so we have no more of the cube than the face <span class="allsmcap">ABCD</span>. -Considering now this face <span class="allsmcap">ABCD</span>, we -see that it is free to turn about the -line <span class="allsmcap">AB</span>. It can rotate in the <i>x</i> to <i>w</i> -direction about this line. In <a href="#fig_38">fig. 38</a> -it is shown on its way, and it can -evidently continue this rotation till -it lies on the other side of the <i>z</i> -axis in the plane of <i>xz</i>.</p> - -<div class="figleft illowp35" id="fig_38" style="max-width: 18.75em;"> - <img src="images/fig_38.png" alt="" /> - <div class="caption">Fig. 38.</div> -</div> - -<p>We can also take a section parallel to the face <span class="allsmcap">ABCD</span>, -and then letting drop all of our space except the plane of -that section, introduce the <i>w</i> axis, running in the old <i>y</i> -direction. This section can be represented by the same -drawing, <a href="#fig_38">fig. 38</a>, and we see that it can rotate about the -line on its left until it swings half way round and runs in -the opposite direction to that which it ran in before. -These turnings of the different sections are not inconsistent, -and taken all together they will bring the cube -from the position shown in <a href="#fig_36">fig. 36</a> to that shown in -<a href="#fig_41">fig. 41</a>.</p> - -<p>Since we have three axes at our disposal in our space, -we are not obliged to represent the <i>w</i> axis by any particular -one. We may let any axis we like disappear, and let the -fourth axis take its place.</p> - -<div class="figleft illowp40" id="fig_39" style="max-width: 18.75em;"> - <img src="images/fig_39.png" alt="" /> - <div class="caption">Fig. 39.</div> -</div> -<div class="figleft illowp40" id="fig_40" style="max-width: 18.75em;"> - <img src="images/fig_40.png" alt="" /> - <div class="caption">Fig. 40.</div> -</div> - -<div class="figleft illowp40" id="fig_41" style="max-width: 21.875em;"> - <img src="images/fig_41.png" alt="" /> - <div class="caption">Fig. 41.</div> -</div> - -<p>In <a href="#fig_36">fig. 36</a> suppose the <i>z</i> axis to go. We have then<span class="pagenum" id="Page_69">[Pg 69]</span> -simply the plane of <i>xy</i> and the square base of the -cube <span class="allsmcap">ACEG</span>, <a href="#fig_39">fig. 39</a>, is all that could -be seen of it. Let now the <i>w</i> axis -take the place of the <i>z</i> axis and -we have, in <a href="#fig_39">fig. 39</a> again, a representation -of the space of <i>xyw</i>, in -which all that exists of the cube is -its square base. Now, by a turning -of <i>x</i> to <i>w</i>, this base can rotate around the line <span class="allsmcap">AE</span>, it is -shown on its way in <a href="#fig_40">fig. 40</a>, and -finally it will, after half a revolution, -lie on the other side of the <i>y</i> axis. -In a similar way we may rotate -sections parallel to the base of the -<i>xw</i> rotation, and each of them comes -to run in the opposite direction from -that which they occupied at first.</p> - -<p>Thus again the cube comes from the position of <a href="#fig_36">fig. 36</a>. -to that of <a href="#fig_41">fig. 41</a>. In this <i>x</i> -to <i>w</i> turning, we see that it -takes place by the rotations of -sections parallel to the front -face about lines parallel to <span class="allsmcap">AB</span>, -or else we may consider it as -consisting of the rotation of -sections parallel to the base -about lines parallel to <span class="allsmcap">AE</span>. It -is a rotation of the whole cube about the plane <span class="allsmcap">ABEF</span>. -Two separate sections could not rotate about two separate -lines in our space without conflicting, but their motion is -consistent when we consider another dimension. Just, -then, as a plane being can think of rotation about a line as -a rotation about a number of points, these rotations not -interfering as they would if they took place in his two-dimensional -space, so we can think of a rotation about a<span class="pagenum" id="Page_70">[Pg 70]</span> -plane as the rotation of a number of sections of a body -about a number of lines in a plane, these rotations not -being inconsistent in a four-dimensional space as they are -in three-dimensional space.</p> - -<p>We are not limited to any particular direction for the -lines in the plane about which we suppose the rotation -of the particular sections to take place. Let us draw -the section of the cube, <a href="#fig_36">fig. 36</a>, through <span class="allsmcap">A</span>, <span class="allsmcap">F</span>, <span class="allsmcap">C</span>, <span class="allsmcap">H</span>, forming a -sloping plane. Now since the fourth dimension is at -right angles to every line in our space it is at right -angles to this section also. We can represent our space -by drawing an axis at right angles to the plane <span class="allsmcap">ACEG</span>, our -space is then determined by the plane <span class="allsmcap">ACEG</span>, and the perpendicular -axis. If we let this axis drop and suppose the -fourth axis, <i>w</i>, to take its place, we have a representation of -the space which runs off in the fourth dimension from the -plane <span class="allsmcap">ACEG</span>. In this space we shall see simply the section -<span class="allsmcap">ACEG</span> of the cube, and nothing else, for one cube does not -extend to any distance in the fourth dimension.</p> - -<div class="figleft illowp40" id="fig_42" style="max-width: 25em;"> - <img src="images/fig_42.png" alt="" /> - <div class="caption">Fig. 42.</div> -</div> - -<p>If, keeping this plane, we bring in the fourth dimension, -we shall have a space in which simply this section of -the cube exists and nothing else. The section can turn -about the line <span class="allsmcap">AF</span>, and parallel sections can turn about -parallel lines. Thus in considering -the rotation about -a plane we can draw any -lines we like and consider -the rotation as taking place -in sections about them.</p> - -<p>To bring out this point -more clearly let us take two -parallel lines, <span class="allsmcap">A</span> and <span class="allsmcap">B</span>, in -the space of <i>xyz</i>, and let <span class="allsmcap">CD</span> -and <span class="allsmcap">EF</span> be two rods running -above and below the plane of <i>xy</i>, from these lines. If we<span class="pagenum" id="Page_71">[Pg 71]</span> -turn these rods in our space about the lines <span class="allsmcap">A</span> and <span class="allsmcap">B</span>, as -the upper end of one, <span class="allsmcap">F</span>, is going down, the lower end of -the other, <span class="allsmcap">C</span>, will be coming up. They will meet and -conflict. But it is quite possible for these two rods -each of them to turn about the two lines without altering -their relative distances.</p> - -<p>To see this suppose the <i>y</i> axis to go, and let the <i>w</i> axis -take its place. We shall see the lines <span class="allsmcap">A</span> and <span class="allsmcap">B</span> no longer, -for they run in the <i>y</i> direction from the points <span class="allsmcap">G</span> and <span class="allsmcap">H</span>.</p> - -<div class="figleft illowp40" id="fig_43" style="max-width: 21.875em;"> - <img src="images/fig_43.png" alt="" /> - <div class="caption">Fig. 43.</div> -</div> - -<p>Fig. 43 is a picture of the two rods seen in the space -of <i>xzw</i>. If they rotate in the -direction shown by the arrows—in -the <i>z</i> to <i>w</i> direction—they -move parallel to one another, -keeping their relative distances. -Each will rotate about its own -line, but their rotation will not -be inconsistent with their forming -part of a rigid body.</p> - -<p>Now we have but to suppose -a central plane with rods crossing -it at every point, like <span class="allsmcap">CD</span> and <span class="allsmcap">EF</span> cross the plane of <i>xy</i>, -to have an image of a mass of matter extending equal -distances on each side of a diametral plane. As two of -these rods can rotate round, so can all, and the whole -mass of matter can rotate round its diametral plane.</p> - -<p>This rotation round a plane corresponds, in four -dimensions, to the rotation round an axis in three -dimensions. Rotation of a body round a plane is the -analogue of rotation of a rod round an axis.</p> - -<p>In a plane we have rotation round a point, in three-space -rotation round an axis line, in four-space rotation -round an axis plane.</p> - -<p>The four-dimensional being’s shaft by which he transmits -power is a disk rotating round its central<span class="pagenum" id="Page_72">[Pg 72]</span> -plane—the whole contour corresponds to the ends of an axis -of rotation in our space. He can impart the rotation at -any point and take it off at any other point on the contour, -just as rotation round a line can in three-space be imparted -at one end of a rod and taken off at the other end.</p> - -<p>A four-dimensional wheel can easily be described from -the analogy of the representation which a plane being -would form for himself of one of our wheels.</p> - -<p>Suppose a wheel to move transverse to a plane, so that -the whole disk, which I will consider to be solid and -without spokes, came at the same time into contact with -the plane. It would appear as a circular portion of plane -matter completely enclosing another and smaller portion—the -axle.</p> - -<p>This appearance would last, supposing the motion of -the wheel to continue until it had traversed the plane by -the extent of its thickness, when there would remain in -the plane only the small disk which is the section of the -axle. There would be no means obvious in the plane -at first by which the axle could be reached, except by -going through the substance of the wheel. But the -possibility of reaching it without destroying the substance -of the wheel would be shown by the continued existence -of the axle section after that of the wheel had disappeared.</p> - -<p>In a similar way a four-dimensional wheel moving -transverse to our space would appear first as a solid sphere, -completely surrounding a smaller solid sphere. The -outer sphere would represent the wheel, and would last -until the wheel has traversed our space by a distance -equal to its thickness. Then the small sphere alone -would remain, representing the section of the axle. The -large sphere could move round the small one quite freely. -Any line in space could be taken as an axis, and round -this line the outer sphere could rotate, while the inner -sphere remained still. But in all these directions of<span class="pagenum" id="Page_73">[Pg 73]</span> -revolution there would be in reality one line which -remained unaltered, that is the line which stretches away -in the fourth direction, forming the axis of the axle. The -four-dimensional wheel can rotate in any number of planes, -but all these planes are such that there is a line at right -angles to them all unaffected by rotation in them.</p> - -<p>An objection is sometimes experienced as to this mode -of reasoning from a plane world to a higher dimensionality. -How artificial, it is argued, this conception of a plane -world is. If any real existence confined to a superficies -could be shown to exist, there would be an argument for -one relative to which our three-dimensional existence is -superficial. But, both on the one side and the other of -the space we are familiar with, spaces either with less -or more than three dimensions are merely arbitrary -conceptions.</p> - -<p>In reply to this I would remark that a plane being -having one less dimension than our three would have one-third -of our possibilities of motion, while we have only -one-fourth less than those of the higher space. It may -very well be that there may be a certain amount of -freedom of motion which is demanded as a condition of an -organised existence, and that no material existence is -possible with a more limited dimensionality than ours. -This is well seen if we try to construct the mechanics of a -two-dimensional world. No tube could exist, for unless -joined together completely at one end two parallel lines -would be completely separate. The possibility of an -organic structure, subject to conditions such as this, is -highly problematical; yet, possibly in the convolutions -of the brain there may be a mode of existence to be -described as two-dimensional.</p> - -<p>We have but to suppose the increase in surface and -the diminution in mass carried on to a certain extent -to find a region which, though without mobility of the<span class="pagenum" id="Page_74">[Pg 74]</span> -constituents, would have to be described as two-dimensional.</p> - -<p>But, however artificial the conception of a plane being -may be, it is none the less to be used in passing to the -conception of a greater dimensionality than ours, and -hence the validity of the first part of this objection -altogether disappears directly we find evidence for such a -state of being.</p> - -<p>The second part of the objection has more weight. -How is it possible to conceive that in a four-dimensional -space any creatures should be confined to a three-dimensional -existence?</p> - -<p>In reply I would say that we know as a matter of fact -that life is essentially a phenomenon of surface. The -amplitude of the movements which we can make is much -greater along the surface of the earth than it is up -or down.</p> - -<p>Now we have but to conceive the extent of a solid -surface increased, while the motions possible tranverse to -it are diminished in the same proportion, to obtain the -image of a three-dimensional world in four-dimensional -space.</p> - -<p>And as our habitat is the meeting of air and earth on -the world, so we must think of the meeting place of two -as affording the condition for our universe. The meeting -of what two? What can that vastness be in the higher -space which stretches in such a perfect level that our -astronomical observations fail to detect the slightest -curvature?</p> - -<p>The perfection of the level suggests a liquid—a lake -amidst what vast scenery!—whereon the matter of the -universe floats speck-like.</p> - -<p>But this aspect of the problem is like what are called -in mathematics boundary conditions.</p> - -<p>We can trace out all the consequences of four-dimensional -movements down to their last detail. Then, knowing<span class="pagenum" id="Page_75">[Pg 75]</span> -the mode of action which would be characteristic of the -minutest particles, if they were free, we can draw conclusions -from what they actually do of what the constraint -on them is. Of the two things, the material conditions and -the motion, one is known, and the other can be inferred. -If the place of this universe is a meeting of two, there -would be a one-sideness to space. If it lies so that what -stretches away in one direction in the unknown is unlike -what stretches away in the other, then, as far as the -movements which participate in that dimension are concerned, -there would be a difference as to which way the -motion took place. This would be shown in the dissimilarity -of phenomena, which, so far as all three-space -movements are concerned, were perfectly symmetrical. -To take an instance, merely, for the sake of precising -our ideas, not for any inherent probability in it; if it could -be shown that the electric current in the positive direction -were exactly like the electric current in the negative -direction, except for a reversal of the components of the -motion in three-dimensional space, then the dissimilarity -of the discharge from the positive and negative poles -would be an indication of a one-sideness to our space. -The only cause of difference in the two discharges would -be due to a component in the fourth dimension, which -directed in one direction transverse to our space, met with -a different resistance to that which it met when directed -in the opposite direction.</p> -<hr class="chap x-ebookmaker-drop" /> - -<div class="chapter"> -<p><span class="pagenum" id="Page_76">[Pg 76]</span></p> - -<h2 class="nobreak" id="CHAPTER_VII">CHAPTER VII<br /> - -<small><i>THE EVIDENCES FOR A FOURTH DIMENSION</i></small></h2></div> - - -<p>The method necessarily to be employed in the search for -the evidences of a fourth dimension, consists primarily in -the formation of the conceptions of four-dimensional -shapes and motions. When we are in possession of these -it is possible to call in the aid of observation, without -them we may have been all our lives in the familiar -presence of a four-dimensional phenomenon without ever -recognising its nature.</p> - -<p>To take one of the conceptions we have already formed, -the turning of a real thing into its mirror image would be -an occurrence which it would be hard to explain, except on -the assumption of a fourth dimension.</p> - -<p>We know of no such turning. But there exist a multitude -of forms which show a certain relation to a plane, -a relation of symmetry, which indicates more than an accidental -juxtaposition of parts. In organic life the universal -type is of right- and left-handed symmetry, there is a plane -on each side of which the parts correspond. Now we have -seen that in four dimensions a plane takes the place of a -line in three dimensions. In our space, rotation about an -axis is the type of rotation, and the origin of bodies symmetrical -about a line as the earth is symmetrical about an -axis can easily be explained. But where there is symmetry -about a plane no simple physical motion, such as we<span class="pagenum" id="Page_77">[Pg 77]</span> -are accustomed to, suffices to explain it. In our space a -symmetrical object must be built up by equal additions -on each side of a central plane. Such additions about -such a plane are as little likely as any other increments. -The probability against the existence of symmetrical -form in inorganic nature is overwhelming in our space, -and in organic forms they would be as difficult of production -as any other variety of configuration. To illustrate -this point we may take the child’s amusement of making -from dots of ink on a piece of paper a lifelike representation -of an insect by simply folding the paper -over. The dots spread out on a symmetrical line, and -give the impression of a segmented form with antennæ -and legs.</p> - -<p>Now seeing a number of such figures we should -naturally infer a folding over. Can, then, a folding over -in four-dimensional space account for the symmetry of -organic forms? The folding cannot of course be of the -bodies we see, but it may be of those minute constituents, -the ultimate elements of living matter which, turned in one -way or the other, become right- or left-handed, and so -produce a corresponding structure.</p> - -<p>There is something in life not included in our conceptions -of mechanical movement. Is this something a four-dimensional -movement?</p> - -<p>If we look at it from the broadest point of view, there is -something striking in the fact that where life comes in -there arises an entirely different set of phenomena to -those of the inorganic world.</p> - -<p>The interest and values of life as we know it in ourselves, -as we know it existing around us in subordinate -forms, is entirely and completely different to anything -which inorganic nature shows. And in living beings we -have a kind of form, a disposition of matter which is -entirely different from that shown in inorganic matter.<span class="pagenum" id="Page_78">[Pg 78]</span> -Right- and left-handed symmetry does not occur in the -configurations of dead matter. We have instances of -symmetry about an axis, but not about a plane. It can -be argued that the occurrence of symmetry in two dimensions -involves the existence of a three-dimensional process, -as when a stone falls into water and makes rings of ripples, -or as when a mass of soft material rotates about an axis. -It can be argued that symmetry in any number of dimensions -is the evidence of an action in a higher dimensionality. -Thus considering living beings, there is an evidence both -in their structure, and their different mode of activity, of a -something coming in from without into the inorganic -world.</p> - -<p>And the objections which will readily occur, such as -those derived from the forms of twin crystals and the -theoretical structure of chemical molecules, do not invalidate -the argument; for in these forms too the -presumable seat of the activity producing them lies in that -very minute region in which we necessarily place the seat -of a four-dimensional mobility.</p> - -<p>In another respect also the existence of symmetrical forms -is noteworthy. It is puzzling to conceive how two shapes -exactly equal can exist which are not superposible. Such -a pair of symmetrical figures as the two hands, right and -left, show either a limitation in our power of movement, -by which we cannot superpose the one on the other, or a -definite influence and compulsion of space on matter, -inflicting limitations which are additional to those of the -proportions of the parts.</p> - -<p>We will, however, put aside the arguments to be drawn -from the consideration of symmetry as inconclusive, -retaining one valuable indication which they afford. If -it is in virtue of a four-dimensional motion that symmetry -exists, it is only in the very minute particles -of bodies that that motion is to be found, for there is<span class="pagenum" id="Page_79">[Pg 79]</span> -no such thing as a bending over in four dimensions of -any object of a size which we can observe. The region -of the extremely minute is the one, then, which we -shall have to investigate. We must look for some -phenomenon which, occasioning movements of the kind -we know, still is itself inexplicable as any form of motion -which we know.</p> - -<p>Now in the theories of the actions of the minute -particles of bodies on one another, and in the motions of -the ether, mathematicians have tacitly assumed that the -mechanical principles are the same as those which prevail -in the case of bodies which can be observed, it has been -assumed without proof that the conception of motion being -three-dimensional, holds beyond the region from observations -in which it was formed.</p> - -<p>Hence it is not from any phenomenon explained by -mathematics that we can derive a proof of four dimensions. -Every phenomenon that has been explained is explained -as three-dimensional. And, moreover, since in the region -of the very minute we do not find rigid bodies acting -on each other at a distance, but elastic substances and -continuous fluids such as ether, we shall have a double -task.</p> - -<p>We must form the conceptions of the possible movements -of elastic and liquid four-dimensional matter, before -we can begin to observe. Let us, therefore, take the four-dimensional -rotation about a plane, and enquire what it -becomes in the case of extensible fluid substances. If -four-dimensional movements exist, this kind of rotation -must exist, and the finer portions of matter must exhibit -it.</p> - -<p>Consider for a moment a rod of flexible and extensible -material. It can turn about an axis, even if not straight; -a ring of india rubber can turn inside out.</p> - -<p>What would this be in the case of four dimensions?</p> - -<p><span class="pagenum" id="Page_80">[Pg 80]</span></p> -<div class="figleft illowp50" id="fig_44" style="max-width: 25em;"> - <img src="images/fig_44.png" alt="" /> - <div class="caption">Fig. 44.<br /> -<i>Axis of x running towards -the observer.</i></div> -</div> - -<p>Let us consider a sphere of our three-dimensional -matter having a definite -thickness. To represent -this thickness let us suppose -that from every point -of the sphere in <a href="#fig_44">fig. 44</a> rods -project both ways, in and -out, like <span class="allsmcap">D</span> and <span class="allsmcap">F</span>. We can -only see the external portion, -because the internal -parts are hidden by the -sphere.</p> - -<p>In this sphere the axis -of <i>x</i> is supposed to come -towards the observer, the -axis of <i>z</i> to run up, the axis of <i>y</i> to go to the right.</p> - -<div class="figleft illowp50" id="fig_45" style="max-width: 25em;"> - <img src="images/fig_45.png" alt="" /> - <div class="caption">Fig. 45.</div> -</div> - -<p>Now take the section determined by the <i>zy</i> plane. -This will be a circle as -shown in <a href="#fig_45">fig. 45</a>. If we -let drop the <i>x</i> axis, this -circle is all we have of -the sphere. Letting the -<i>w</i> axis now run in the -place of the old <i>x</i> axis -we have the space <i>yzw</i>, -and in this space all that -we have of the sphere is -the circle. Fig. 45 then -represents all that there -is of the sphere in the -space of <i>yzw</i>. In this space it is evident that the rods -<span class="allsmcap">CD</span> and <span class="allsmcap">EF</span> can turn round the circumference as an axis. -If the matter of the spherical shell is sufficiently extensible -to allow the particles <span class="allsmcap">C</span> and <span class="allsmcap">E</span> to become as widely -separated as they would be in the positions <span class="allsmcap">D</span> and <span class="allsmcap">F</span>, then<span class="pagenum" id="Page_81">[Pg 81]</span> -the strip of matter represented by <span class="allsmcap">CD</span> and <span class="allsmcap">EF</span> and a -multitude of rods like them can turn round the circular -circumference.</p> - -<p>Thus this particular section of the sphere can turn -inside out, and what holds for any one section holds for -all. Hence in four dimensions the whole sphere can, if -extensible turn inside out. Moreover, any part of it—a -bowl-shaped portion, for instance—can turn inside out, -and so on round and round.</p> - -<p>This is really no more than we had before in the -rotation about a plane, except that we see that the plane -can, in the case of extensible matter, be curved, and still -play the part of an axis.</p> - -<p>If we suppose the spherical shell to be of four-dimensional -matter, our representation will be a little different. -Let us suppose there to be a small thickness to the matter -in the fourth dimension. This would make no difference -in <a href="#fig_44">fig. 44</a>, for that merely shows the view in the <i>xyz</i> -space. But when the <i>x</i> axis is let drop, and the <i>w</i> axis -comes in, then the rods <span class="allsmcap">CD</span> and <span class="allsmcap">EF</span> which represent the -matter of the shell, will have a certain thickness perpendicular -to the plane of the paper on which they are drawn. -If they have a thickness in the fourth dimension they will -show this thickness when looked at from the direction of -the <i>w</i> axis.</p> - -<p>Supposing these rods, then, to be small slabs strung on -the circumference of the circle in <a href="#fig_45">fig. 45</a>, we see that -there will not be in this case either any obstacle to their -turning round the circumference. We can have a shell -of extensible material or of fluid material turning inside -out in four dimensions.</p> - -<p>And we must remember that in four dimensions there -is no such thing as rotation round an axis. If we want to -investigate the motion of fluids in four dimensions we -must take a movement about an axis in our space, and<span class="pagenum" id="Page_82">[Pg 82]</span> -find the corresponding movement about a plane in -four space.</p> - -<p>Now, of all the movements which take place in fluids, -the most important from a physical point of view is -vortex motion.</p> - -<p>A vortex is a whirl or eddy—it is shown in the gyrating -wreaths of dust seen on a summer day; it is exhibited on -a larger scale in the destructive march of a cyclone.</p> - -<p>A wheel whirling round will throw off the water on it. -But when this circling motion takes place in a liquid -itself it is strangely persistent. There is, of course, a -certain cohesion between the particles of water by which -they mutually impede their motions. But in a liquid -devoid of friction, such that every particle is free from -lateral cohesion on its path of motion, it can be shown -that a vortex or eddy separates from the mass of the -fluid a certain portion, which always remain in that -vortex.</p> - -<p>The shape of the vortex may alter, but it always consists -of the same particles of the fluid.</p> - -<p>Now, a very remarkable fact about such a vortex is that -the ends of the vortex cannot remain suspended and -isolated in the fluid. They must always run to the -boundary of the fluid. An eddy in water that remains -half way down without coming to the top is impossible.</p> - -<p>The ends of a vortex must reach the boundary of a -fluid—the boundary may be external or internal—a vortex -may exist between two objects in the fluid, terminating -one end on each object, the objects being internal -boundaries of the fluid. Again, a vortex may have its -ends linked together, so that it forms a ring. Circular -vortex rings of this description are often seen in puffs of -smoke, and that the smoke travels on in the ring is a -proof that the vortex always consists of the same particles -of air.</p> - -<p><span class="pagenum" id="Page_83">[Pg 83]</span></p> - -<p>Let us now enquire what a vortex would be in a four-dimensional -fluid.</p> - -<p>We must replace the line axis by a plane axis. We -should have therefore a portion of fluid rotating round -a plane.</p> - -<p>We have seen that the contour of this plane corresponds -with the ends of the axis line. Hence such a four-dimensional -vortex must have its rim on a boundary of -the fluid. There would be a region of vorticity with a -contour. If such a rotation were started at one part of a -circular boundary, its edges would run round the boundary -in both directions till the whole interior region was filled -with the vortex sheet.</p> - -<p>A vortex in a three-dimensional liquid may consist of a -number of vortex filaments lying together producing a -tube, or rod of vorticity.</p> - -<p>In the same way we can have in four dimensions a -number of vortex sheets alongside each other, each of which -can be thought of as a bowl-shaped portion of a spherical -shell turning inside out. The rotation takes place at any -point not in the space occupied by the shell, but from -that space to the fourth dimension and round back again.</p> - -<p>Is there anything analogous to this within the range -of our observation?</p> - -<p>An electric current answers this description in every -respect. Electricity does not flow through a wire. Its effect -travels both ways from the starting point along the wire. -The spark which shows its passing midway in its circuit -is later than that which occurs at points near its starting -point on either side of it.</p> - -<p>Moreover, it is known that the action of the current -is not in the wire. It is in the region enclosed by the -wire, this is the field of force, the locus of the exhibition -of the effects of the current.</p> - -<p>And the necessity of a conducting circuit for a current is<span class="pagenum" id="Page_84">[Pg 84]</span> -exactly that which we should expect if it were a four-dimensional -vortex. According to Maxwell every current forms -a closed circuit, and this, from the four-dimensional point -of view, is the same as saying a vortex must have its ends -on a boundary of the fluid.</p> - -<p>Thus, on the hypothesis of a fourth dimension, the rotation -of the fluid ether would give the phenomenon of an -electric current. We must suppose the ether to be full of -movement, for the more we examine into the conditions -which prevail in the obscurity of the minute, the more we -find that an unceasing and perpetual motion reigns. Thus -we may say that the conception of the fourth dimension -means that there must be a phenomenon which presents -the characteristics of electricity.</p> - -<p>We know now that light is an electro-magnetic action, -and that so far from being a special and isolated phenomenon -this electric action is universal in the realm of the -minute. Hence, may we not conclude that, so far from -the fourth dimension being remote and far away, being a -thing of symbolic import, a term for the explanation of -dubious facts by a more obscure theory, it is really the -most important fact within our knowledge. Our three-dimensional -world is superficial. These processes, which -really lie at the basis of all phenomena of matter, -escape our observation by their minuteness, but reveal -to our intellect an amplitude of motion surpassing any -that we can see. In such shapes and motions there is a -realm of the utmost intellectual beauty, and one to -which our symbolic methods apply with a better grace -than they do to those of three dimensions.</p> -<hr class="chap x-ebookmaker-drop" /> - -<div class="chapter"> -<p><span class="pagenum" id="Page_85">[Pg 85]</span></p> - -<h2 class="nobreak" id="CHAPTER_VIII">CHAPTER VIII<br /> - -<small><i>THE USE OF FOUR DIMENSIONS IN -THOUGHT</i></small></h2></div> - - -<p>Having held before ourselves this outline of a conjecture -of the world as four-dimensional, having roughly thrown -together those facts of movement which we can see apply -to our actual experience, let us pass to another branch -of our subject.</p> - -<p>The engineer uses drawings, graphical constructions, -in a variety of manners. He has, for instance, diagrams -which represent the expansion of steam, the efficiency -of his valves. These exist alongside the actual plans of -his machines. They are not the pictures of anything -really existing, but enable him to think about the relations -which exist in his mechanisms.</p> - -<p>And so, besides showing us the actual existence of that -world which lies beneath the one of visible movements, -four-dimensional space enables us to make ideal constructions -which serve to represent the relations of things, -and throw what would otherwise be obscure into a definite -and suggestive form.</p> - -<p>From amidst the great variety of instances which lies -before me I will select two, one dealing with a subject -of slight intrinsic interest, which however gives within -a limited field a striking example of the method<span class="pagenum" id="Page_86">[Pg 86]</span> -of drawing conclusions and the use of higher space -figures.<a id="FNanchor_1" href="#Footnote_1" class="fnanchor">[1]</a></p> - -<div class="footnotes"> -<div class="footnote"> - -<p><a id="Footnote_1" href="#FNanchor_1" class="label">[1]</a> It is suggestive also in another respect, because it shows very -clearly that in our processes of thought there are in play faculties other -than logical; in it the origin of the idea which proves to be justified is -drawn from the consideration of symmetry, a branch of the beautiful.</p> - -</div></div> - -<p>The other instance is chosen on account of the bearing -it has on our fundamental conceptions. In it I try to -discover the real meaning of Kant’s theory of experience.</p> - -<p>The investigation of the properties of numbers is much -facilitated by the fact that relations between numbers are -themselves able to be represented as numbers—<i>e.g.</i>, 12, -and 3 are both numbers, and the relation between them -is 4, another number. The way is thus opened for a -process of constructive theory, without there being any -necessity for a recourse to another class of concepts -besides that which is given in the phenomena to be -studied.</p> - -<p>The discipline of number thus created is of great and -varied applicability, but it is not solely as quantitative -that we learn to understand the phenomena of nature. -It is not possible to explain the properties of matter -by number simply, but all the activities of matter are -energies in space. They are numerically definite and also, -we may say, directedly definite, <i>i.e.</i> definite in direction.</p> - -<p>Is there, then, a body of doctrine about space which, like -that of number, is available in science? It is needless -to answer: Yes; geometry. But there is a method -lying alongside the ordinary methods of geometry, which -tacitly used and presenting an analogy to the method -of numerical thought deserves to be brought into greater -prominence than it usually occupies.</p> - -<p>The relation of numbers is a number.</p> - -<p>Can we say in the same way that the relation of -shapes is a shape?</p> - -<p>We can.</p> - -<p><span class="pagenum" id="Page_87">[Pg 87]</span></p> -<div class="figleft illowp50" id="fig_46" style="max-width: 25em;"> - <img src="images/fig_46.png" alt="" /> - <div class="caption">Fig. 46.</div> -</div> - -<p>To take an instance chosen on account of its ready -availability. Let us take -two right-angled triangles of -a given hypothenuse, but -having sides of different -lengths (<a href="#fig_46">fig. 46</a>). These -triangles are shapes which have a certain relation to each -other. Let us exhibit their relation as a figure.</p> - -<div class="figleft illowp40" id="fig_47" style="max-width: 18.75em;"> - <img src="images/fig_47.png" alt="" /> - <div class="caption">Fig. 47.</div> -</div> - -<p>Draw two straight lines at right angles to each other, -the one <span class="allsmcap">HL</span> a horizontal level, the -other <span class="allsmcap">VL</span> a vertical level (<a href="#fig_47">fig. 47</a>). -By means of these two co-ordinating -lines we can represent a -double set of magnitudes; one set -as distances to the right of the vertical -level, the other as distances -above the horizontal level, a suitable unit being chosen.</p> - -<p>Thus the line marked 7 will pick out the assemblage -of points whose distance from the vertical level is 7, -and the line marked 1 will pick out the points whose -distance above the horizontal level is 1. The meeting -point of these two lines, 7 and 1, will define a point -which with regard to the one set of magnitudes is 7, -with regard to the other is 1. Let us take the sides of -our triangles as the two sets of magnitudes in question.</p> - -<div class="figleft illowp40" id="fig_48" style="max-width: 18.75em;"> - <img src="images/fig_48.png" alt="" /> - <div class="caption">Fig. 48.</div> -</div> - -<p>Then the point 7, 1, will represent the triangle whose -sides are 7 and 1. Similarly the point 5, 5—5, that -is, to the right of the vertical level and 5 above the -horizontal level—will represent the -triangle whose sides are 5 and 5 -(<a href="#fig_48">fig. 48</a>).</p> - -<p>Thus we have obtained a figure -consisting of the two points 7, 1, -and 5, 5, representative of our two -triangles. But we can go further, and, drawing an arc<span class="pagenum" id="Page_88">[Pg 88]</span> -of a circle about <span class="allsmcap">O</span>, the meeting point of the horizontal -and vertical levels, which passes through 7, 1, and 5, 5, -assert that all the triangles which are right-angled and -have a hypothenuse whose square is 50 are represented -by the points on this arc.</p> - -<p>Thus, each individual of a class being represented by a -point, the whole class is represented by an assemblage of -points forming a figure. Accepting this representation -we can attach a definite and calculable significance to the -expression, resemblance, or similarity between two individuals -of the class represented, the difference being -measured by the length of the line between two representative -points. It is needless to multiply examples, or -to show how, corresponding to different classes of triangles, -we obtain different curves.</p> - -<p>A representation of this kind in which an object, a -thing in space, is represented as a point, and all its properties -are left out, their effect remaining only in the -relative position which the representative point bears -to the representative points of the other objects, may be -called, after the analogy of Sir William R. Hamilton’s -hodograph, a “Poiograph.”</p> - -<p>Representations thus made have the character of -natural objects; they have a determinate and definite -character of their own. Any lack of completeness in them -is probably due to a failure in point of completeness -of those observations which form the ground of their -construction.</p> - -<p>Every system of classification is a poiograph. In -Mendeléeff’s scheme of the elements, for instance, each -element is represented by a point, and the relations -between the elements are represented by the relations -between the points.</p> - -<p>So far I have simply brought into prominence processes -and considerations with which we are all familiar. But<span class="pagenum" id="Page_89">[Pg 89]</span> -it is worth while to bring into the full light of our attention -our habitual assumptions and processes. It often -happens that we find there are two of them which have -a bearing on each other, which, without this dragging into -the light, we should have allowed to remain without -mutual influence.</p> - -<p>There is a fact which it concerns us to take into account -in discussing the theory of the poiograph.</p> - -<p>With respect to our knowledge of the world we are -far from that condition which Laplace imagined when he -asserted that an all-knowing mind could determine the -future condition of every object, if he knew the co-ordinates -of its particles in space, and their velocity at any -particular moment.</p> - -<p>On the contrary, in the presence of any natural object, -we have a great complexity of conditions before us, -which we cannot reduce to position in space and date -in time.</p> - -<p>There is mass, attraction apparently spontaneous, electrical -and magnetic properties which must be superadded -to spatial configuration. To cut the list short we must -say that practically the phenomena of the world present -us problems involving many variables, which we must -take as independent.</p> - -<p>From this it follows that in making poiographs we -must be prepared to use space of more than three dimensions. -If the symmetry and completeness of our representation -is to be of use to us we must be prepared to -appreciate and criticise figures of a complexity greater -than of those in three dimensions. It is impossible to give -an example of such a poiograph which will not be merely -trivial, without going into details of some kind irrelevant -to our subject. I prefer to introduce the irrelevant details -rather than treat this part of the subject perfunctorily.</p> - -<p>To take an instance of a poiograph which does not lead<span class="pagenum" id="Page_90">[Pg 90]</span> -us into the complexities incident on its application in -classificatory science, let us follow Mrs. Alicia Boole Stott -in her representation of the syllogism by its means. She -will be interested to find that the curious gap she detected -has a significance.</p> - -<div class= "figleft illowp40" id="fig_49" style="max-width: 13.75em;"> - <img src="images/fig_49.png" alt="" /> - <div class="caption">Fig. 49.</div> -</div> - -<p>A syllogism consists of two statements, the major and -the minor premiss, with the conclusion that can be drawn -from them. Thus, to take an instance, <a href="#fig_49">fig. 49</a>. It is -evident, from looking at the successive figures that, if we -know that the region <span class="allsmcap">M</span> lies altogether within the region -<span class="allsmcap">P</span>, and also know that the region <span class="allsmcap">S</span> lies altogether within -the region <span class="allsmcap">M</span>, we can conclude that the region <span class="allsmcap">S</span> lies -altogether within the region <span class="allsmcap">P</span>. <span class="allsmcap">M</span> is <span class="allsmcap">P</span>, -major premiss; <span class="allsmcap">S</span> is <span class="allsmcap">M</span>, minor premiss; <span class="allsmcap">S</span> -is <span class="allsmcap">P</span>, conclusion. Given the first two data -we must conclude that <span class="allsmcap">S</span> lies in <span class="allsmcap">P</span>. The -conclusion <span class="allsmcap">S</span> is <span class="allsmcap">P</span> involves two terms, <span class="allsmcap">S</span> and -<span class="allsmcap">P</span>, which are respectively called the subject -and the predicate, the letters <span class="allsmcap">S</span> and <span class="allsmcap">P</span> -being chosen with reference to the parts -the notions they designate play in the -conclusion. <span class="allsmcap">S</span> is the subject of the conclusion, -<span class="allsmcap">P</span> is the predicate of the conclusion. -The major premiss we take to be, that -which does not involve <span class="allsmcap">S</span>, and here we -always write it first.</p> - -<p>There are several varieties of statement -possessing different degrees of universality and manners of -assertiveness. These different forms of statement are -called the moods.</p> - -<p>We will take the major premiss as one variable, as a -thing capable of different modifications of the same kind, -the minor premiss as another, and the different moods we -will consider as defining the variations which these -variables undergo.</p> - -<p><span class="pagenum" id="Page_91">[Pg 91]</span></p> - -<p>There are four moods:—</p> - -<p>1. The universal affirmative; all <span class="allsmcap">M</span> is <span class="allsmcap">P</span>, called mood <span class="allsmcap">A</span>.</p> - -<p>2. The universal negative; no <span class="allsmcap">M</span> is <span class="allsmcap">P</span>, mood <span class="allsmcap">E</span>.</p> - -<p>3. The particular affirmative; some <span class="allsmcap">M</span> is <span class="allsmcap">P</span>, mood <span class="allsmcap">I</span>.</p> - -<p>4. The particular negative; some <span class="allsmcap">M</span> is not <span class="allsmcap">P</span>, mood <span class="allsmcap">O</span>.</p> - -<div class="figcenter illowp100" id="fig_50" style="max-width: 62.5em;"> - <img src="images/fig_50.png" alt="" /> - <div class="caption">Figure 50. -</div></div> - - -<p>The dotted lines in 3 and 4, <a href="#fig_50">fig. 50</a>, denote that it is -not known whether or no any objects exist, corresponding -to the space of which the dotted line forms one delimiting -boundary; thus, in mood <span class="allsmcap">I</span> we do not know if there are -any <span class="smcap">M’s</span> which are not <span class="allsmcap">P</span>, we only know some <span class="smcap">M’s</span> are <span class="allsmcap">P</span>.</p> - -<div class="figleft illowp30" id="fig_51" style="max-width: 15.625em;"> - <img src="images/fig_51.png" alt="" /> - <div class="caption">Fig. 51.</div> -</div> - -<p>Representing the first premiss in its various moods by -regions marked by vertical lines to -the right of <span class="allsmcap">PQ</span>, we have in <a href="#fig_51">fig. 51</a>, -running up from the four letters <span class="allsmcap">AEIO</span>, -four columns, each of which indicates -that the major premiss is in the mood -denoted by the respective letter. In -the first column to the right of <span class="allsmcap">PQ</span> is -the mood <span class="allsmcap">A</span>. Now above the line <span class="allsmcap">RS</span> let there be marked -off four regions corresponding to the four moods of the -minor premiss. Thus, in the first row above <span class="allsmcap">RS</span> all the -region between <span class="allsmcap">RS</span> and the first horizontal line above it -denotes that the minor premiss is in the mood <span class="allsmcap">A</span>. The<span class="pagenum" id="Page_92">[Pg 92]</span> -letters <span class="allsmcap">E</span>, <span class="allsmcap">I</span>, <span class="allsmcap">O</span>, in the same way show the mood characterising -the minor premiss in the rows opposite these letters.</p> - -<p>We have still to exhibit the conclusion. To do this we -must consider the conclusion as a third variable, characterised -in its different varieties by four moods—this being -the syllogistic classification. The introduction of a third -variable involves a change in our system of representation.</p> - -<div class="figleft illowp25" id="fig_52" style="max-width: 12.5em;"> - <img src="images/fig_52.png" alt="" /> - <div class="caption">Fig. 52.</div> -</div> - -<p>Before we started with the regions to the right of a -certain line as representing successively the major premiss -in its moods; now we must start with the regions to the -right of a certain plane. Let <span class="allsmcap">LMNR</span> -be the plane face of a cube, <a href="#fig_52">fig. 52</a>, and -let the cube be divided into four parts -by vertical sections parallel to <span class="allsmcap">LMNR</span>. -The variable, the major premiss, is represented -by the successive regions -which occur to the right of the plane -<span class="allsmcap">LMNR</span>—that region to which <span class="allsmcap">A</span> stands opposite, that -slice of the cube, is significative of the mood <span class="allsmcap">A</span>. This -whole quarter-part of the cube represents that for every -part of it the major premiss is in the mood <span class="allsmcap">A</span>.</p> - -<p>In a similar manner the next section, the second with -the letter <span class="allsmcap">E</span> opposite it, represents that for every one of -the sixteen small cubic spaces in it, the major premiss is -in the mood <span class="allsmcap">E</span>. The third and fourth compartments made -by the vertical sections denote the major premiss in the -moods <span class="allsmcap">I</span> and <span class="allsmcap">O</span>. But the cube can be divided in other -ways by other planes. Let the divisions, of which four -stretch from the front face, correspond to the minor -premiss. The first wall of sixteen cubes, facing the -observer, has as its characteristic that in each of the small -cubes, whatever else may be the case, the minor premiss is -in the mood <span class="allsmcap">A</span>. The variable—the minor premiss—varies -through the phases <span class="allsmcap">A</span>, <span class="allsmcap">E</span>, <span class="allsmcap">I</span>, <span class="allsmcap">O</span>, away from the front face of the -cube, or the front plane of which the front face is a part.</p> - -<p><span class="pagenum" id="Page_93">[Pg 93]</span></p> - -<p>And now we can represent the third variable in a precisely -similar way. We can take the conclusion as the third -variable, going through its four phases from the ground -plane upwards. Each of the small cubes at the base of -the whole cube has this true about it, whatever else may -be the case, that the conclusion is, in it, in the mood <span class="allsmcap">A</span>. -Thus, to recapitulate, the first wall of sixteen small cubes, -the first of the four walls which, proceeding from left to -right, build up the whole cube, is characterised in each -part of it by this, that the major premiss is in the mood <span class="allsmcap">A</span>.</p> - -<p>The next wall denotes that the major premiss is in the -mood <span class="allsmcap">E</span>, and so on. Proceeding from the front to the -back the first wall presents a region in every part of -which the minor premiss is in the mood <span class="allsmcap">A</span>. The second -wall is a region throughout which the minor premiss is in -the mood <span class="allsmcap">E</span>, and so on. In the layers, from the bottom -upwards, the conclusion goes through its various moods -beginning with <span class="allsmcap">A</span> in the lowest, <span class="allsmcap">E</span> in the second, <span class="allsmcap">I</span> in the -third, <span class="allsmcap">O</span> in the fourth.</p> - -<p>In the general case, in which the variables represented -in the poiograph pass through a wide range of values, the -planes from which we measure their degrees of variation -in our representation are taken to be indefinitely extended. -In this case, however, all we are concerned with is the -finite region.</p> - -<p>We have now to represent, by some limitation of the -complex we have obtained, the fact that not every combination -of premisses justifies any kind of conclusion. -This can be simply effected by marking the regions in -which the premisses, being such as are defined by the -positions, a conclusion which is valid is found.</p> - -<p>Taking the conjunction of the major premiss, all <span class="allsmcap">M</span> is -<span class="allsmcap">P</span>, and the minor, all <span class="allsmcap">S</span> is <span class="allsmcap">M</span>, we conclude that all <span class="allsmcap">S</span> is <span class="allsmcap">P</span>. -Hence, that region must be marked in which we have the -conjunction of major premiss in mood <span class="allsmcap">A</span>; minor premiss,<span class="pagenum" id="Page_94">[Pg 94]</span> -mood <span class="allsmcap">A</span>; conclusion, mood <span class="allsmcap">A</span>. This is the cube occupying -the lowest left-hand corner of the large cube.</p> - -<div class="figleft illowp25" id="fig_53" style="max-width: 12.5em;"> - <img src="images/fig_53.png" alt="" /> - <div class="caption">Fig. 53.</div> -</div> - - -<p>Proceeding in this way, we find that the regions which -must be marked are those shown in <a href="#fig_53">fig. 53</a>. -To discuss the case shown in the marked -cube which appears at the top of <a href="#fig_53">fig. 53</a>. -Here the major premiss is in the second -wall to the right—it is in the mood <span class="allsmcap">E</span> and -is of the type no <span class="allsmcap">M</span> is <span class="allsmcap">P</span>. The minor -premiss is in the mood characterised by -the third wall from the front. It is of -the type some <span class="allsmcap">S</span> is <span class="allsmcap">M</span>. From these premisses we draw -the conclusion that some <span class="allsmcap">S</span> is not <span class="allsmcap">P</span>, a conclusion in the -mood <span class="allsmcap">O</span>. Now the mood <span class="allsmcap">O</span> of the conclusion is represented -in the top layer. Hence we see that the marking is -correct in this respect.</p> - -<div class="figleft illowp50" id="fig_54" style="max-width: 25em;"> - <img src="images/fig_54.png" alt="" /> - <div class="caption">Fig. 54.</div> -</div> - -<p>It would, of course, be possible to represent the cube on -a plane by means of four -squares, as in <a href="#fig_54">fig. 54</a>, if we -consider each square to represent -merely the beginning -of the region it stands for. -Thus the whole cube can be -represented by four vertical -squares, each standing for a -kind of vertical tray, and the -markings would be as shown. In No. 1 the major premiss -is in mood <span class="allsmcap">A</span> for the whole of the region indicated by the -vertical square of sixteen divisions; in No. 2 it is in the -mood <span class="allsmcap">E</span>, and so on.</p> - -<p>A creature confined to a plane would have to adopt some -such disjunctive way of representing the whole cube. He -would be obliged to represent that which we see as a -whole in separate parts, and each part would merely -represent, would not be, that solid content which we see.</p> - -<p><span class="pagenum" id="Page_95">[Pg 95]</span></p> - -<p>The view of these four squares which the plane creature -would have would not be such as ours. He would not -see the interior of the four squares represented above, but -each would be entirely contained within its outline, the -internal boundaries of the separate small squares he could -not see except by removing the outer squares.</p> - -<p>We are now ready to introduce the fourth variable -involved in the syllogism.</p> - -<p>In assigning letters to denote the terms of the syllogism -we have taken <span class="allsmcap">S</span> and <span class="allsmcap">P</span> to represent the subject and -predicate in the conclusion, and thus in the conclusion -their order is invariable. But in the premisses we have -taken arbitrarily the order all <span class="allsmcap">M</span> is <span class="allsmcap">P</span>, and all <span class="allsmcap">S</span> is <span class="allsmcap">M</span>. -There is no reason why <span class="allsmcap">M</span> instead of <span class="allsmcap">P</span> should not be the -predicate of the major premiss, and so on.</p> - -<p>Accordingly we take the order of the terms in the premisses -as the fourth variable. Of this order there are four -varieties, and these varieties are called figures.</p> - -<p>Using the order in which the letters are written to -denote that the letter first written is subject, the one -written second is predicate, we have the following possibilities:—</p> - - -<table class="standard" summary=""> -<tr> -<td class="tdc"></td> -<td class="tdc">1st Figure.</td> -<td class="tdc">2nd Figure.</td> -<td class="tdc">3rd Figure.</td> -<td class="tdc">4th Figure.</td> -</tr> -<tr> -<td class="tdc">Major</td> -<td class="tdc"><span class="allsmcap">M P</span></td> -<td class="tdc"><span class="allsmcap">P M</span></td> -<td class="tdc"><span class="allsmcap">M P</span></td> -<td class="tdc"><span class="allsmcap">P M</span></td> -</tr> -<tr> -<td class="tdc">Minor</td> -<td class="tdc"><span class="allsmcap">S M</span></td> -<td class="tdc"><span class="allsmcap">S M</span></td> -<td class="tdc"><span class="allsmcap">M S</span></td> -<td class="tdc"><span class="allsmcap">M S</span></td> -</tr> -</table> - -<p>There are therefore four possibilities with regard to -this fourth variable as with regard to the premisses.</p> - -<p>We have used up our dimensions of space in representing -the phases of the premisses and the conclusion in -respect of mood, and to represent in an analogous manner -the variations in figure we require a fourth dimension.</p> - -<p>Now in bringing in this fourth dimension we must -make a change in our origins of measurement analogous -to that which we made in passing from the plane to the -solid.</p> - -<p><span class="pagenum" id="Page_96">[Pg 96]</span></p> - -<p>This fourth dimension is supposed to run at right -angles to any of the three space dimensions, as the third -space dimension runs at right angles to the two dimensions -of a plane, and thus it gives us the opportunity of -generating a new kind of volume. If the whole cube -moves in this dimension, the solid itself traces out a path, -each section of which, made at right angles to the -direction in which it moves, is a solid, an exact repetition -of the cube itself.</p> - -<p>The cube as we see it is the beginning of a solid of such -a kind. It represents a kind of tray, as the square face of -the cube is a kind of tray against which the cube rests.</p> - -<p>Suppose the cube to move in this fourth dimension in -four stages, and let the hyper-solid region traced out in -the first stage of its progress be characterised by this, that -the terms of the syllogism are in the first figure, then we -can represent in each of the three subsequent stages the -remaining three figures. Thus the whole cube forms -the basis from which we measure the variation in figure. -The first figure holds good for the cube as we see it, and -for that hyper-solid which lies within the first stage; -the second figure holds good in the second stage, and -so on.</p> - -<p>Thus we measure from the whole cube as far as figures -are concerned.</p> - -<p>But we saw that when we measured in the cube itself -having three variables, namely, the two premisses and -the conclusion, we measured from three planes. The base -from which we measured was in every case the same.</p> - -<p>Hence, in measuring in this higher space we should -have bases of the same kind to measure from, we should -have solid bases.</p> - -<p>The first solid base is easily seen, it is the cube itself. -The other can be found from this consideration.</p> - -<p>That solid from which we measure figure is that in<span class="pagenum" id="Page_97">[Pg 97]</span> -which the remaining variables run through their full -range of varieties.</p> - -<p>Now, if we want to measure in respect of the moods of -the major premiss, we must let the minor premiss, the -conclusion, run through their range, and also the order -of the terms. That is we must take as basis of measurement -in respect to the moods of the major that which -represents the variation of the moods of the minor, the -conclusion and the variation of the figures.</p> - -<p>Now the variation of the moods of the minor and of the -conclusion are represented in the square face on the left -of the cube. Here are all varieties of the minor premiss -and the conclusion. The varieties of the figures are -represented by stages in a motion proceeding at right -angles to all space directions, at right angles consequently -to the face in question, the left-hand face of the cube.</p> - -<p>Consequently letting the left-hand face move in this -direction we get a cube, and in this cube all the varieties -of the minor premiss, the conclusion, and the figure are -represented.</p> - -<p>Thus another cubic base of measurement is given to -the cube, generated by movement of the left-hand square -in the fourth dimension.</p> - -<p>We find the other bases in a similar manner, one is the -cube generated by the front square moved in the fourth -dimension so as to generate a cube. From this cube -variations in the mood of the minor are measured. The -fourth base is that found by moving the bottom square of -the cube in the fourth dimension. In this cube the -variations of the major, the minor, and the figure are given. -Considering this as a basis in the four stages proceeding -from it, the variation in the moods of the conclusion are -given.</p> - -<p>Any one of these cubic bases can be represented in space, -and then the higher solid generated from them lies out of<span class="pagenum" id="Page_98">[Pg 98]</span> -our space. It can only be represented by a device analogous -to that by which the plane being represents a cube.</p> - -<p>He represents the cube shown above, by taking four -square sections and placing them arbitrarily at convenient -distances the one from the other.</p> - -<p>So we must represent this higher solid by four cubes: -each cube represents only the beginning of the corresponding -higher volume.</p> - -<p>It is sufficient for us, then, if we draw four cubes, the -first representing that region in which the figure is of the -first kind, the second that region in which the figure is -of the second kind, and so on. These cubes are the -beginnings merely of the respective regions—they are -the trays, as it were, against which the real solids must -be conceived as resting, from which they start. The first -one, as it is the beginning of the region of the first figure, -is characterised by the order of the terms in the premisses -being that of the first figure. The second similarly has -the terms of the premisses in the order of the second -figure, and so on.</p> - -<p>These cubes are shown below.</p> - -<p>For the sake of showing the properties of the method -of representation, not for the logical problem, I will make -a digression. I will represent in space the moods of the -minor and of the conclusion and the different figures, -keeping the major always in mood <span class="allsmcap">A</span>. Here we have -three variables in different stages, the minor, the conclusion, -and the figure. Let the square of the left-hand -side of the original cube be imagined to be standing by -itself, without the solid part of the cube, represented by -(2) <a href="#fig_55">fig. 55</a>. The <span class="allsmcap">A</span>, <span class="allsmcap">E</span>, <span class="allsmcap">I</span>, <span class="allsmcap">O</span>, which run away represent the -moods of the minor, the <span class="allsmcap">A</span>, <span class="allsmcap">E</span>, <span class="allsmcap">I</span>, <span class="allsmcap">O</span>, which run up represent -the moods of the conclusion. The whole square, since it -is the beginning of the region in the major premiss, mood -<span class="allsmcap">A</span>, is to be considered as in major premiss, mood <span class="allsmcap">A</span>.</p> - -<p><span class="pagenum" id="Page_99">[Pg 99]</span></p> - -<p>From this square, let it be supposed that that direction -in which the figures are represented runs to the -left hand. Thus we have a cube (1) running from the -square above, in which the square itself is hidden, but -the letters <span class="allsmcap">A</span>, <span class="allsmcap">E</span>, <span class="allsmcap">I</span>, <span class="allsmcap">O</span>, of the conclusion are seen. In this -cube we have the minor premiss and the conclusion in all -their moods, and all the figures represented. With regard -to the major premiss, since the face (2) belongs to the first -wall from the left in the original arrangement, and in this -arrangement was characterised by the major premiss in the -mood <span class="allsmcap">A</span>, we may say that the whole of the cube we now -have put up represents the mood <span class="allsmcap">A</span> of the major premiss.</p> - -<div class="figcenter illowp100" id="fig_55" style="max-width: 50em;"> - <img src="images/fig_55.png" alt="" /> - <div class="caption">Fig. 55.</div> -</div> - -<p>Hence the small cube at the bottom to the right in 1, -nearest to the spectator, is major premiss, mood <span class="allsmcap">A</span>; minor -premiss, mood <span class="allsmcap">A</span>; conclusion, mood <span class="allsmcap">A</span>; and figure the first. -The cube next to it, running to the left, is major premiss, -mood <span class="allsmcap">A</span>; minor premiss, mood <span class="allsmcap">A</span>; conclusion, mood <span class="allsmcap">A</span>; -figure 2.</p> - -<p>So in this cube we have the representations of all the -combinations which can occur when the major premiss, -remaining in the mood <span class="allsmcap">A</span>, the minor premiss, the conclusion, -and the figures pass through their varieties.</p> - -<p>In this case there is no room in space for a natural -representation of the moods of the major premiss. To -represent them we must suppose as before that there is a -fourth dimension, and starting from this cube as base in -the fourth direction in four equal stages, all the first volume -corresponds to major premiss <span class="allsmcap">A</span>, the second to major<span class="pagenum" id="Page_100">[Pg 100]</span> -premiss, mood <span class="allsmcap">E</span>, the next to the mood <span class="allsmcap">I</span>, and the last -to mood <span class="allsmcap">O</span>.</p> - -<p>The cube we see is as it were merely a tray against -which the four-dimensional figure rests. Its section at -any stage is a cube. But a transition in this direction -being transverse to the whole of our space is represented -by no space motion. We can exhibit successive stages of -the result of transference of the cube in that direction, -but cannot exhibit the product of a transference, however -small, in that direction.</p> - -<div class="figcenter illowp100" id="fig_56" style="max-width: 62.5em;"> - <img src="images/fig_56.png" alt="" /> - <div class="caption">Fig. 56.</div> -</div> - -<p>To return to the original method of representing our -variables, consider <a href="#fig_56">fig. 56</a>. These four cubes represent -four sections of the figure derived from the first of them -by moving it in the fourth dimension. The first portion -of the motion, which begins with 1, traces out a -more than solid body, which is all in the first figure. -The beginning of this body is shown in 1. The next -portion of the motion traces out a more than solid body, -all of which is in the second figure; the beginning of -this body is shown in 2; 3 and 4 follow on in like -manner. Here, then, in one four-dimensional figure we -have all the combinations of the four variables, major -premiss, minor premiss, figure, conclusion, represented, -each variable going through its four varieties. The disconnected -cubes drawn are our representation in space by -means of disconnected sections of this higher body.</p> - -<p><span class="pagenum" id="Page_101">[Pg 101]</span></p> - -<p>Now it is only a limited number of conclusions which -are true—their truth depends on the particular combinations -of the premisses and figures which they accompany. -The total figure thus represented may be called the -universe of thought in respect to these four constituents, -and out of the universe of possibly existing combinations -it is the province of logic to select those which correspond -to the results of our reasoning faculties.</p> - -<p>We can go over each of the premisses in each of the -moods, and find out what conclusion logically follows. -But this is done in the works on logic; most simply and -clearly I believe in “Jevon’s Logic.” As we are only concerned -with a formal presentation of the results we will -make use of the mnemonic lines printed below, in which -the words enclosed in brackets refer to the figures, and -are not significative:—</p> - -<ul> -<li>Barbara celarent Darii ferio<i>que</i> [prioris].</li> -<li>Caesare Camestris Festino Baroko [secundae].</li> -<li>[Tertia] darapti disamis datisi felapton.</li> -<li>Bokardo ferisson <i>habet</i> [Quarta insuper addit].</li> -<li>Bramantip camenes dimaris ferapton fresison.</li> -</ul> - -<p>In these lines each significative word has three vowels, -the first vowel refers to the major premiss, and gives the -mood of that premiss, “a” signifying, for instance, that -the major mood is in mood <i>a</i>. The second vowel refers -to the minor premiss, and gives its mood. The third -vowel refers to the conclusion, and gives its mood. Thus -(prioris)—of the first figure—the first mnemonic word is -“barbara,” and this gives major premiss, mood <span class="allsmcap">A</span>; minor -premiss, mood <span class="allsmcap">A</span>; conclusion, mood <span class="allsmcap">A</span>. Accordingly in the -first of our four cubes we mark the lowest left-hand front -cube. To take another instance in the third figure “Tertia,” -the word “ferisson” gives us major premiss mood <span class="allsmcap">E</span>—<i>e.g.</i>, -no <span class="allsmcap">M</span> is <span class="allsmcap">P</span>, minor premiss mood <span class="allsmcap">I</span>; some <span class="allsmcap">M</span> is <span class="allsmcap">S</span>, conclusion, -mood <span class="allsmcap">O</span>; some <span class="allsmcap">S</span> is not <span class="allsmcap">P</span>. The region to be marked then<span class="pagenum" id="Page_102">[Pg 102]</span> -in the third representative cube is the one in the second -wall to the right for the major premiss, the third wall -from the front for the minor premiss, and the top layer -for the conclusion.</p> - -<p>It is easily seen that in the diagram this cube is -marked, and so with all the valid conclusions. The -regions marked in the total region show which combinations -of the four variables, major premiss, minor -premiss, figure, and conclusion exist.</p> - -<p>That is to say, we objectify all possible conclusions, and -build up an ideal manifold, containing all possible combinations -of them with the premisses, and then out of -this we eliminate all that do not satisfy the laws of logic. -The residue is the syllogism, considered as a canon of -reasoning.</p> - -<p>Looking at the shape which represents the totality -of the valid conclusions, it does not present any obvious -symmetry, or easily characterisable nature. A striking -configuration, however, is obtained, if we project the four-dimensional -figure obtained into a three-dimensional one; -that is, if we take in the base cube all those cubes which -have a marked space anywhere in the series of four -regions which start from that cube.</p> - -<p>This corresponds to making abstraction of the figures, -giving all the conclusions which are valid whatever the -figure may be.</p> - -<div class="figcenter illowp25" id="fig_57" style="max-width: 12.5em;"> - <img src="images/fig_57.png" alt="" /> - <div class="caption">Fig. 57.</div> -</div> - -<p>Proceeding in this way we obtain the arrangement of -marked cubes shown in <a href="#fig_57">fig. 57</a>. We see -that the valid conclusions are arranged -almost symmetrically round one cube—the -one on the top of the column starting from -<span class="allsmcap">AAA</span>. There is one breach of continuity -however in this scheme. One cube is -unmarked, which if marked would give -symmetry. It is the one which would be denoted by the<span class="pagenum" id="Page_103">[Pg 103]</span> -letters <span class="allsmcap">I</span>, <span class="allsmcap">E</span>, <span class="allsmcap">O</span>, in the third wall to the right, the second -wall away, the topmost layer. Now this combination of -premisses in the mood <span class="allsmcap">IE</span>, with a conclusion in the mood -<span class="allsmcap">O</span>, is not noticed in any book on logic with which I am -familiar. Let us look at it for ourselves, as it seems -that there must be something curious in connection with -this break of continuity in the poiograph.</p> - -<div class="figcenter illowp100" id="fig_58" style="max-width: 62.5em;"> - <img src="images/fig_58.png" alt="" /> - <div class="caption">Fig. 58.</div> -</div> - -<p>The propositions <span class="allsmcap">I</span>, <span class="allsmcap">E</span>, in the various figures are the -following, as shown in the accompanying scheme, <a href="#fig_58">fig. 58</a>:—First -figure: some <span class="allsmcap">M</span> is <span class="allsmcap">P</span>; no <span class="allsmcap">S</span> is <span class="allsmcap">M</span>. Second figure: -some <span class="allsmcap">P</span> is <span class="allsmcap">M</span>; no <span class="allsmcap">S</span> is <span class="allsmcap">M</span>. Third figure: some <span class="allsmcap">M</span> is <span class="allsmcap">P</span>; no -<span class="allsmcap">M</span> is <span class="allsmcap">S</span>. Fourth figure: some <span class="allsmcap">P</span> is <span class="allsmcap">M</span>; no <span class="allsmcap">M</span> is <span class="allsmcap">S</span>.</p> - -<p>Examining these figures, we see, taking the first, that -if some <span class="allsmcap">M</span> is <span class="allsmcap">P</span> and no <span class="allsmcap">S</span> is <span class="allsmcap">M</span>, we have no conclusion of<span class="pagenum" id="Page_104">[Pg 104]</span> -the form <span class="allsmcap">S</span> is <span class="allsmcap">P</span> in the various moods. It is quite indeterminate -how the circle representing <span class="allsmcap">S</span> lies with regard -to the circle representing <span class="allsmcap">P</span>. It may lie inside, outside, -or partly inside <span class="allsmcap">P</span>. The same is true in the other figures -2 and 3. But when we come to the fourth figure, since -<span class="allsmcap">M</span> and <span class="allsmcap">S</span> lie completely outside each other, there cannot -lie inside <span class="allsmcap">S</span> that part of <span class="allsmcap">P</span> which lies inside <span class="allsmcap">M</span>. Now -we know by the major premiss that some of <span class="allsmcap">P</span> does lie -in <span class="allsmcap">M</span>. Hence <span class="allsmcap">S</span> cannot contain the whole of <span class="allsmcap">P</span>. In -words, some <span class="allsmcap">P</span> is <span class="allsmcap">M</span>, no <span class="allsmcap">M</span> is <span class="allsmcap">S</span>, therefore <span class="allsmcap">S</span> does not contain -the whole of <span class="allsmcap">P</span>. If we take <span class="allsmcap">P</span> as the subject, this gives -us a conclusion in the mood <span class="allsmcap">O</span> about <span class="allsmcap">P</span>. Some <span class="allsmcap">P</span> is not <span class="allsmcap">S</span>. -But it does not give us conclusion about <span class="allsmcap">S</span> in any one -of the four forms recognised in the syllogism and called -its moods. Hence the breach of the continuity in the -poiograph has enabled us to detect a lack of completeness -in the relations which are considered in the syllogism.</p> - -<p>To take an instance:—Some Americans (<span class="allsmcap">P</span>) are of -African stock (<span class="allsmcap">M</span>); No Aryans (<span class="allsmcap">S</span>) are of African stock -(<span class="allsmcap">M</span>); Aryans (<span class="allsmcap">S</span>) do not include all of Americans (<span class="allsmcap">P</span>).</p> - -<p>In order to draw a conclusion about <span class="allsmcap">S</span> we have to admit -the statement, “<span class="allsmcap">S</span> does not contain the whole of <span class="allsmcap">P</span>,” as -a valid logical form—it is a statement about <span class="allsmcap">S</span> which can -be made. The logic which gives us the form, “some <span class="allsmcap">P</span> -is not <span class="allsmcap">S</span>,” and which does not allow us to give the exactly -equivalent and equally primary form, “<span class="allsmcap">S</span> does not contain -the whole of <span class="allsmcap">P</span>,” is artificial.</p> - -<p>And I wish to point out that this artificiality leads -to an error.</p> - -<p>If one trusted to the mnemonic lines given above, one -would conclude that no logical conclusion about <span class="allsmcap">S</span> can -be drawn from the statement, “some <span class="allsmcap">P</span> are <span class="allsmcap">M</span>, no <span class="allsmcap">M</span> are <span class="allsmcap">S</span>.”</p> - -<p>But a conclusion can be drawn: <span class="allsmcap">S</span> does not contain -the whole of <span class="allsmcap">P</span>.</p> - -<p>It is not that the result is given expressed in another<span class="pagenum" id="Page_105">[Pg 105]</span> -form. The mnemonic lines deny that any conclusion -can be drawn from premisses in the moods <span class="allsmcap">I</span>, <span class="allsmcap">E</span>, respectively.</p> - -<p>Thus a simple four-dimensional poiograph has enabled -us to detect a mistake in the mnemonic lines which have -been handed down unchallenged from mediæval times. -To discuss the subject of these lines more fully a logician -defending them would probably say that a particular -statement cannot be a major premiss; and so deny the -existence of the fourth figure in the combination of moods.</p> - -<p>To take our instance: some Americans are of African -stock; no Aryans are of African stock. He would say -that the conclusion is some Americans are not Aryans; -and that the second statement is the major. He would -refuse to say anything about Aryans, condemning us to -an eternal silence about them, as far as these premisses -are concerned! But, if there is a statement involving -the relation of two classes, it must be expressible as a -statement about either of them.</p> - -<p>To bar the conclusion, “Aryans do not include the -whole of Americans,” is purely a makeshift in favour of -a false classification.</p> - -<p>And the argument drawn from the universality of the -major premiss cannot be consistently maintained. It -would preclude such combinations as major <span class="allsmcap">O</span>, minor <span class="allsmcap">A</span>, -conclusion <span class="allsmcap">O</span>—<i>i.e.</i>, such as some mountains (<span class="allsmcap">M</span>) are not -permanent (<span class="allsmcap">P</span>); all mountains (<span class="allsmcap">M</span>) are scenery (<span class="allsmcap">S</span>); some -scenery (<span class="allsmcap">S</span>) is not permanent (<span class="allsmcap">P</span>).</p> - -<p>This is allowed in “Jevon’s Logic,” and his omission to -discuss <span class="allsmcap">I</span>, <span class="allsmcap">E</span>, <span class="allsmcap">O</span>, in the fourth figure, is inexplicable. A -satisfactory poiograph of the logical scheme can be made -by admitting the use of the words some, none, or all, -about the predicate as well as about the subject. Then -we can express the statement, “Aryans do not include the -whole of Americans,” clumsily, but, when its obscurity -is fathomed, correctly, as “Some Aryans are not all<span class="pagenum" id="Page_106">[Pg 106]</span> -Americans.” And this method is what is called the -“quantification of the predicate.”</p> - -<p>The laws of formal logic are coincident with the conclusions -which can be drawn about regions of space, which -overlap one another in the various possible ways. It is -not difficult so to state the relations or to obtain a -symmetrical poiograph. But to enter into this branch of -geometry is beside our present purpose, which is to show -the application of the poiograph in a finite and limited -region, without any of those complexities which attend its -use in regard to natural objects.</p> - -<p>If we take the latter—plants, for instance—and, without -assuming fixed directions in space as representative of -definite variations, arrange the representative points in -such a manner as to correspond to the similarities of the -objects, we obtain configuration of singular interest; and -perhaps in this way, in the making of shapes of shapes, -bodies with bodies omitted, some insight into the structure -of the species and genera might be obtained.</p> -<hr class="chap x-ebookmaker-drop" /> - -<div class="chapter"> -<p><span class="pagenum" id="Page_107">[Pg 107]</span></p> - -<h2 class="nobreak" id="CHAPTER_IX">CHAPTER IX<br /> - -<small><i>APPLICATION TO KANT’S THEORY OF -EXPERIENCE</i></small></h2></div> - - -<p>When we observe the heavenly bodies we become aware -that they all participate in one universal motion—a -diurnal revolution round the polar axis.</p> - -<p>In the case of fixed stars this is most unqualifiedly true, -but in the case of the sun, and the planets also, the single -motion of revolution can be discerned, modified, and -slightly altered by other and secondary motions.</p> - -<p>Hence the universal characteristic of the celestial bodies -is that they move in a diurnal circle.</p> - -<p>But we know that this one great fact which is true of -them all has in reality nothing to do with them. The -diurnal revolution which they visibly perform is the result -of the condition of the observer. It is because the -observer is on a rotating earth that a universal statement -can be made about all the celestial bodies.</p> - -<p>The universal statement which is valid about every one -of the celestial bodies is that which does not concern -them at all, and is but a statement of the condition of -the observer.</p> - -<p>Now there are universal statements of other kinds -which we can make. We can say that all objects of -experience are in space and subject to the laws of -geometry.</p> - -<p><span class="pagenum" id="Page_108">[Pg 108]</span></p> - -<p>Does this mean that space and all that it means is due -to a condition of the observer?</p> - -<p>If a universal law in one case means nothing affecting -the objects themselves, but only a condition of observation, -is this true in every case? There is shown us in -astronomy a <i>vera causa</i> for the assertion of a universal. -Is the same cause to be traced everywhere?</p> - -<p>Such is a first approximation to the doctrine of Kant’s -critique.</p> - -<p>It is the apprehension of a relation into which, on the -one side and the other, perfectly definite constituents -enter—the human observer and the stars—and a transference -of this relation to a region in which the constituents -on either side are perfectly unknown.</p> - -<p>If spatiality is due to a condition of the observer, the -observer cannot be this bodily self of ours—the body, like -the objects around it, are equally in space.</p> - -<p>This conception Kant applied, not only to the intuitions -of sense, but to the concepts of reason—wherever a universal -statement is made there is afforded him an opportunity -for the application of his principle. He constructed a -system in which one hardly knows which the most to -admire, the architectonic skill, or the reticence with regard -to things in themselves, and the observer in himself.</p> - -<p>His system can be compared to a garden, somewhat -formal perhaps, but with the charm of a quality more -than intellectual, a <i>besonnenheit</i>, an exquisite moderation -over all. And from the ground he so carefully prepared -with that buried in obscurity, which it is fitting should -be obscure, science blossoms and the tree of real knowledge -grows.</p> - -<p>The critique is a storehouse of ideas of profound interest. -The one of which I have given a partial statement leads, -as we shall see on studying it in detail, to a theory of -mathematics suggestive of enquiries in many directions.</p> - -<p><span class="pagenum" id="Page_109">[Pg 109]</span></p> - -<p>The justification for my treatment will be found -amongst other passages in that part of the transcendental -analytic, in which Kant speaks of objects of experience -subject to the forms of sensibility, not subject to the -concepts of reason.</p> - -<p>Kant asserts that whenever we think we think of -objects in space and time, but he denies that the space -and time exist as independent entities. He goes about -to explain them, and their universality, not by assuming -them, as most other philosophers do, but by postulating -their absence. How then does it come to pass that the -world is in space and time to us?</p> - -<p>Kant takes the same position with regard to what we -call nature—a great system subject to law and order. -“How do you explain the law and order in nature?” we -ask the philosophers. All except Kant reply by assuming -law and order somewhere, and then showing how we can -recognise it.</p> - -<p>In explaining our notions, philosophers from other than -the Kantian standpoint, assume the notions as existing -outside us, and then it is no difficult task to show how -they come to us, either by inspiration or by observation.</p> - -<p>We ask “Why do we have an idea of law in nature?” -“Because natural processes go according to law,” we are -answered, “and experience inherited or acquired, gives us -this notion.”</p> - -<p>But when we speak about the law in nature we are -speaking about a notion of our own. So all that these -expositors do is to explain our notion by an assumption -of it.</p> - -<p>Kant is very different. He supposes nothing. An experience -such as ours is very different from experience -in the abstract. Imagine just simply experience, succession -of states, of consciousness! Why, there would -be no connecting any two together, there would be no<span class="pagenum" id="Page_110">[Pg 110]</span> -personal identity, no memory. It is out of a general -experience such as this, which, in respect to anything we -call real, is less than a dream, that Kant shows the -genesis of an experience such as ours.</p> - -<p>Kant takes up the problem of the explanation of space, -time, order, and so quite logically does not presuppose -them.</p> - -<p>But how, when every act of thought is of things in -space, and time, and ordered, shall we represent to ourselves -that perfectly indefinite somewhat which is Kant’s -necessary hypothesis—that which is not in space or time -and is not ordered. That is our problem, to represent -that which Kant assumes not subject to any of our forms -of thought, and then show some function which working -on that makes it into a “nature” subject to law and -order, in space and time. Such a function Kant calls the -“Unity of Apperception”; <i>i.e.</i>, that which makes our state -of consciousness capable of being woven into a system -with a self, an outer world, memory, law, cause, and order.</p> - -<p>The difficulty that meets us in discussing Kant’s -hypothesis is that everything we think of is in space -and time—how then shall we represent in space an existence -not in space, and in time an existence not in time? -This difficulty is still more evident when we come to -construct a poiograph, for a poiograph is essentially a -space structure. But because more evident the difficulty -is nearer a solution. If we always think in space, <i>i.e.</i> -using space concepts, the first condition requisite for -adapting them to the representation of non-spatial existence, -is to be aware of the limitation of our thought, -and so be able to take the proper steps to overcome it. -The problem before us, then, is to represent in space an -existence not in space.</p> - -<p>The solution is an easy one. It is provided by the -conception of alternativity.</p> - -<p><span class="pagenum" id="Page_111">[Pg 111]</span></p> - -<p>To get our ideas clear let us go right back behind the -distinctions of an inner and an outer world. Both of -these, Kant says, are products. Let us take merely states -of consciousness, and not ask the question whether they are -produced or superinduced—to ask such a question is to -have got too far on, to have assumed something of which -we have not traced the origin. Of these states let us -simply say that they occur. Let us now use the word -a “posit” for a phase of consciousness reduced to its -last possible stage of evanescence; let a posit be that -phase of consciousness of which all that can be said is -that it occurs.</p> - -<p>Let <i>a</i>, <i>b</i>, <i>c</i>, be three such posits. We cannot represent -them in space without placing them in a certain order, -as <i>a</i>, <i>b</i>, <i>c</i>. But Kant distinguishes between the forms -of sensibility and the concepts of reason. A dream in -which everything happens at haphazard would be an -experience subject to the form of sensibility and only -partially subject to the concepts of reason. It is partially -subject to the concepts of reason because, although -there is no order of sequence, still at any given time -there is order. Perception of a thing as in space is a -form of sensibility, the perception of an order is a concept -of reason.</p> - -<p>We must, therefore, in order to get at that process -which Kant supposes to be constitutive of an ordered -experience imagine the posits as in space without -order.</p> - -<p>As we know them they must be in some order, <i>abc</i>, -<i>bca</i>, <i>cab</i>, <i>acb</i>, <i>cba</i>, <i>bac</i>, one or another.</p> - -<p>To represent them as having no order conceive all -these different orders as equally existing. Introduce the -conception of alternativity—let us suppose that the order -<i>abc</i>, and <i>bac</i>, for example, exist equally, so that we -cannot say about <i>a</i> that it comes before or after <i>b</i>. This<span class="pagenum" id="Page_112">[Pg 112]</span> -would correspond to a sudden and arbitrary change of <i>a</i> -into <i>b</i> and <i>b</i> into <i>a</i>, so that, to use Kant’s words, it would -be possible to call one thing by one name at one time -and at another time by another name.</p> - -<p>In an experience of this kind we have a kind of chaos, -in which no order exists; it is a manifold not subject to -the concepts of reason.</p> - -<p>Now is there any process by which order can be introduced -into such a manifold—is there any function of -consciousness in virtue of which an ordered experience -could arise?</p> - -<p>In the precise condition in which the posits are, as -described above, it does not seem to be possible. But -if we imagine a duality to exist in the manifold, a -function of consciousness can be easily discovered which -will produce order out of no order.</p> - -<p>Let us imagine each posit, then, as having, a dual aspect. -Let <i>a</i> be 1<i>a</i> in which the dual aspect is represented by the -combination of symbols. And similarly let <i>b</i> be 2<i>b</i>, -<i>c</i> be 3<i>c</i>, in which 2 and <i>b</i> represent the dual aspects -of <i>b</i>, 3 and <i>c</i> those of <i>c</i>.</p> - -<p>Since <i>a</i> can arbitrarily change into <i>b</i>, or into <i>c</i>, and -so on, the particular combinations written above cannot -be kept. We have to assume the equally possible occurrence -of form such as 2<i>a</i>, 2<i>b</i>, and so on; and in order -to get a representation of all those combinations out of -which any set is alternatively possible, we must take -every aspect with every aspect. We must, that is, have -every letter with every number.</p> - -<p>Let us now apply the method of space representation.</p> - -<div class="blockquote"> - -<p><i>Note.</i>—At the beginning of the next chapter the same -structures as those which follow are exhibited in -more detail and a reference to them will remove -any obscurity which may be found in the immediately -following passages. They are there carried</p> - -<p><span class="pagenum" id="Page_113">[Pg 113]</span></p> - -<p>on to a greater multiplicity of dimensions, and the -significance of the process here briefly explained -becomes more apparent.</p> -</div> -<div class="figleft illowp25" id="fig_59" style="max-width: 12.5em;"> - <img src="images/fig_59.png" alt="" /> - <div class="caption">Fig. 59.</div> -</div> - -<p>Take three mutually rectangular axes in space 1, 2, 3 -(<a href="#fig_59">fig. 59</a>), and on each mark three points, -the common meeting point being the -first on each axis. Then by means of -these three points on each axis we -define 27 positions, 27 points in a -cubical cluster, shown in <a href="#fig_60">fig. 60</a>, the -same method of co-ordination being -used as has been described before. -Each of these positions can be named by means of the -axes and the points combined.</p> - -<div class="figleft illowp30" id="fig_60" style="max-width: 18.75em;"> - <img src="images/fig_60.png" alt="" /> - <div class="caption">Fig. 60.</div> -</div> - - -<p>Thus, for instance, the one marked by an asterisk can -be called 1<i>c</i>, 2<i>b</i>, 3<i>c</i>, because it is -opposite to <i>c</i> on 1, to <i>b</i> on 2, to -<i>c</i> on 3.</p> - -<p>Let us now treat of the states of -consciousness corresponding to these -positions. Each point represents a -composite of posits, and the manifold -of consciousness corresponding -to them is of a certain complexity.</p> - -<p>Suppose now the constituents, the points on the axes, -to interchange arbitrarily, any one to become any other, -and also the axes 1, 2, and 3, to interchange amongst -themselves, any one to become any other, and to be subject -to no system or law, that is to say, that order does -not exist, and that the points which run <i>abc</i> on each axis -may run <i>bac</i>, and so on.</p> - -<p>Then any one of the states of consciousness represented -by the points in the cluster can become any other. We -have a representation of a random consciousness of a -certain degree of complexity.</p> - -<p><span class="pagenum" id="Page_114">[Pg 114]</span></p> - -<p>Now let us examine carefully one particular case of -arbitrary interchange of the points, <i>a</i>, <i>b</i>, <i>c</i>; as one such -case, carefully considered, makes the whole clear.</p> - -<div class="figleft illowp40" id="fig_61" style="max-width: 15.625em;"> - <img src="images/fig_61.png" alt="" /> - <div class="caption">Fig. 61.</div> -</div> - -<p>Consider the points named in the figure 1<i>c</i>, 2<i>a</i>, 3<i>c</i>; -1<i>c</i>, 2<i>c</i>, 3<i>a</i>; 1<i>a</i>, 2<i>c</i>, 3<i>c</i>, and -examine the effect on them -when a change of order takes -place. Let us suppose, for -instance, that <i>a</i> changes into <i>b</i>, -and let us call the two sets of -points we get, the one before -and the one after, their change -conjugates.</p> - - -<table class="standard" summary=""> -<tr> -<td class="tdl">Before the change</td> - -<td class="tdl">1<i>c</i> 2<i>a</i> 3<i>c</i></td> -<td class="tdlp">1<i>c</i> 2<i>c</i> 3<i>a</i></td> -<td class="tdlp">1<i>a</i> 2<i>c</i> 3<i>c</i></td> -<td class="tdl" rowspan="2">} Conjugates.</td> -</tr> -<tr> -<td class="tdl">After the change</td> -<td class="tdl">1<i>c</i> 2<i>b</i> 3<i>c</i></td> -<td class="tdlp">1<i>c</i> 2<i>c</i> 3<i>b</i></td> -<td class="tdlp">1<i>b</i> 2<i>c</i> 3<i>c</i></td> -</tr> -</table> - -<p>The points surrounded by rings represent the conjugate -points.</p> - -<p>It is evident that as consciousness, represented first by -the first set of points and afterwards by the second set of -points, would have nothing in common in its two phases. -It would not be capable of giving an account of itself. -There would be no identity.</p> - -<div class="figleft illowp35" id="fig_62" style="max-width: 18.75em;"> - <img src="images/fig_62.png" alt="" /> - <div class="caption">Fig. 62.</div> -</div> - -<p>If, however, we can find any set of points in the -cubical cluster, which, when any arbitrary change takes -place in the points on the axes, or in the axes themselves, -repeats itself, is reproduced, then a consciousness represented -by those points would have a permanence. It -would have a principle of identity. Despite the no law, -the no order, of the ultimate constituents, it would have -an order, it would form a system, the condition of a -personal identity would be fulfilled.</p> - -<p>The question comes to this, then. Can we find a -system of points which is self-conjugate which is such -that when any posit on the axes becomes any other, or<span class="pagenum" id="Page_115">[Pg 115]</span> -when any axis becomes any other, such a set is transformed -into itself, its identity -is not submerged, but rises -superior to the chaos of its -constituents?</p> - -<p>Such a set can be found. -Consider the set represented -in <a href="#fig_62">fig. 62</a>, and written down in -the first of the two lines—</p> - - -<table class="standard" summary=""> -<tr> -<td class="tdl" rowspan="2">Self-<br />conjugate</td> -<td class="tdl" rowspan="2">{</td> -<td class="tdl">1<i>a</i> 2<i>b</i> 3<i>c</i></td> -<td class="tdlp">1<i>b</i> 2<i>a</i> 3<i>c</i></td> -<td class="tdlp">1<i>c</i> 2<i>a</i> 3<i>b</i></td> -<td class="tdlp">1<i>c</i> 2<i>b</i> 3<i>a</i></td> -<td class="tdlp">1<i>b</i> 2<i>c</i> 3<i>a</i></td> -<td class="tdlp">1<i>a</i> 2<i>c</i> 3<i>b</i></td> -</tr> -<tr> -<td class="tdl">1<i>c</i> 2<i>b</i> 3<i>a</i></td> -<td class="tdlp">1<i>b</i> 2<i>c</i> 3<i>a</i></td> -<td class="tdlp">1<i>a</i> 2<i>c</i> 3<i>b</i></td> -<td class="tdlp">1<i>a</i> 2<i>b</i> 3<i>c</i></td> -<td class="tdlp">1<i>b</i> 2<i>a</i> 3<i>c</i></td> -<td class="tdlp">1<i>c</i> 2<i>a</i> 3<i>b</i></td> -</tr> -</table> - -<p>If now <i>a</i> change into <i>c</i> and <i>c</i> into <i>a</i>, we get the set in -the second line, which has the same members as are in the -upper line. Looking at the diagram we see that it would -correspond simply to the turning of the figures as a -whole.<a id="FNanchor_2" href="#Footnote_2" class="fnanchor">[2]</a> Any arbitrary change of the points on the axes, -or of the axes themselves, reproduces the same set.</p> - -<div class="footnotes"> -<div class="footnote"> - -<p><a id="Footnote_2" href="#FNanchor_2" class="label">[2]</a> These figures are described more fully, and extended, in the next -chapter.</p> - -</div></div> - -<p>Thus, a function, by which a random, an unordered, consciousness -could give an ordered and systematic one, can -be represented. It is noteworthy that it is a system of -selection. If out of all the alternative forms that only is -attended to which is self-conjugate, an ordered consciousness -is formed. A selection gives a feature of permanence.</p> - -<p>Can we say that the permanent consciousness is this -selection?</p> - -<p>An analogy between Kant and Darwin comes into light. -That which is swings clear of the fleeting, in virtue of its -presenting a feature of permanence. There is no need -to suppose any function of “attending to.” A consciousness -capable of giving an account of itself is one -which is characterised by this combination. All combinations -exist—of this kind is the consciousness which -can give an account of itself. And the very duality which<span class="pagenum" id="Page_116">[Pg 116]</span> -we have presupposed may be regarded as originated by -a process of selection.</p> - -<p>Darwin set himself to explain the origin of the fauna -and flora of the world. He denied specific tendencies. -He assumed an indefinite variability—that is, chance—but -a chance confined within narrow limits as regards the -magnitude of any consecutive variations. He showed that -organisms possessing features of permanence, if they -occurred would be preserved. So his account of any -structure or organised being was that it possessed features -of permanence.</p> - -<p>Kant, undertaking not the explanation of any particular -phenomena but of that which we call nature as a whole, -had an origin of species of his own, an account of the -flora and fauna of consciousness. He denied any specific -tendency of the elements of consciousness, but taking our -own consciousness, pointed out that in which it resembled -any consciousness which could survive, which could give -an account of itself.</p> - -<p>He assumes a chance or random world, and as great -and small were not to him any given notions of which he -could make use, he did not limit the chance, the randomness, -in any way. But any consciousness which is permanent -must possess certain features—those attributes -namely which give it permanence. Any consciousness -like our own is simply a consciousness which possesses -those attributes. The main thing is that which he calls -the unity of apperception, which we have seen above is -simply the statement that a particular set of phases of -consciousness on the basis of complete randomness will be -self-conjugate, and so permanent.</p> - -<p>As with Darwin so with Kant, the reason for existence -of any feature comes to this—show that it tends to the -permanence of that which possesses it.</p> - -<p>We can thus regard Kant as the creator of the first of<span class="pagenum" id="Page_117">[Pg 117]</span> -the modern evolution theories. And, as is so often the -case, the first effort was the most stupendous in its scope. -Kant does not investigate the origin of any special part -of the world, such as its organisms, its chemical elements, -its social communities of men. He simply investigates -the origin of the whole—of all that is included in consciousness, -the origin of that “thought thing” whose -progressive realisation is the knowable universe.</p> - -<p>This point of view is very different from the ordinary -one, in which a man is supposed to be placed in a world -like that which he has come to think of it, and then to -learn what he has found out from this model which he -himself has placed on the scene.</p> - -<p>We all know that there are a number of questions in -attempting an answer to which such an assumption is not -allowable.</p> - -<p>Mill, for instance, explains our notion of “law” by an -invariable sequence in nature. But what we call nature -is something given in thought. So he explains a thought -of law and order by a thought of an invariable sequence. -He leaves the problem where he found it.</p> - -<p>Kant’s theory is not unique and alone. It is one of -a number of evolution theories. A notion of its import -and significance can be obtained by a comparison of it -with other theories.</p> - -<p>Thus in Darwin’s theoretical world of natural selection -a certain assumption is made, the assumption of indefinite -variability—slight variability it is true, over any appreciable -lapse of time, but indefinite in the postulated -epochs of transformation—and a whole chain of results -is shown to follow.</p> - -<p>This element of chance variation is not, however, an -ultimate resting place. It is a preliminary stage. This -supposing the all is a preliminary step towards finding -out what is. If every kind of organism can come into<span class="pagenum" id="Page_118">[Pg 118]</span> -being, those that do survive will present such and such -characteristics. This is the necessary beginning for ascertaining -what kinds of organisms do come into existence. -And so Kant’s hypothesis of a random consciousness is -the necessary beginning for the rational investigation -of consciousness as it is. His assumption supplies, as -it were, the space in which we can observe the phenomena. -It gives the general laws constitutive of any -experience. If, on the assumption of absolute randomness -in the constituents, such and such would be -characteristic of the experience, then, whatever the constituents, -these characteristics must be universally valid.</p> - -<p>We will now proceed to examine more carefully the -poiograph, constructed for the purpose of exhibiting an -illustration of Kant’s unity of apperception.</p> - -<p>In order to show the derivation order out of non-order -it has been necessary to assume a principle of duality—we -have had the axes and the posits on the axes—there -are two sets of elements, each non-ordered, and it is in -the reciprocal relation of them that the order, the definite -system, originates.</p> - -<p>Is there anything in our experience of the nature of a -duality?</p> - -<p>There certainly are objects in our experience which -have order and those which are incapable of order. The -two roots of a quadratic equation have no order. No one -can tell which comes first. If a body rises vertically and -then goes at right angles to its former course, no one can -assign any priority to the direction of the north or to the -east. There is no priority in directions of turning. We -associate turnings with no order progressions in a line -with order. But in the axes and points we have assumed -above there is no such distinction. It is the same, whether -we assume an order among the turnings, and no order -among the points on the axes, or, <i>vice versa</i>, an order in<span class="pagenum" id="Page_119">[Pg 119]</span> -the points and no order in the turnings. A being with -an infinite number of axes mutually at right angles, -with a definite sequence between them and no sequence -between the points on the axes, would be in a condition -formally indistinguishable from that of a creature who, -according to an assumption more natural to us, had on -each axis an infinite number of ordered points and no -order of priority amongst the axes. A being in such -a constituted world would not be able to tell which -was turning and which was length along an axis, in -order to distinguish between them. Thus to take a pertinent -illustration, we may be in a world of an infinite -number of dimensions, with three arbitrary points on -each—three points whose order is indifferent, or in a -world of three axes of arbitrary sequence with an infinite -number of ordered points on each. We can’t tell which -is which, to distinguish it from the other.</p> - -<p>Thus it appears the mode of illustration which we -have used is not an artificial one. There really exists -in nature a duality of the kind which is necessary to -explain the origin of order out of no order—the duality, -namely, of dimension and position. Let us use the term -group for that system of points which remains unchanged, -whatever arbitrary change of its constituents takes place. -We notice that a group involves a duality, is inconceivable -without a duality.</p> - -<p>Thus, according to Kant, the primary element of experience -is the group, and the theory of groups would be -the most fundamental branch of science. Owing to an -expression in the critique the authority of Kant is sometimes -adduced against the assumption of more than three -dimensions to space. It seems to me, however, that the -whole tendency of his theory lies in the opposite direction, -and points to a perfect duality between dimension and -position in a dimension.</p> - -<p><span class="pagenum" id="Page_120">[Pg 120]</span></p> - -<p>If the order and the law we see is due to the conditions -of conscious experience, we must conceive nature as -spontaneous, free, subject to no predication that we can -devise, but, however apprehended, subject to our logic.</p> - -<p>And our logic is simply spatiality in the general sense—that -resultant of a selection of the permanent from the -unpermanent, the ordered from the unordered, by the -means of the group and its underlying duality.</p> - -<p>We can predicate nothing about nature, only about the -way in which we can apprehend nature. All that we can -say is that all that which experience gives us will be conditioned -as spatial, subject to our logic. Thus, in exploring -the facts of geometry from the simplest logical relations -to the properties of space of any number of dimensions, -we are merely observing ourselves, becoming aware of -the conditions under which we must perceive. Do any -phenomena present themselves incapable of explanation -under the assumption of the space we are dealing with, -then we must habituate ourselves to the conception of a -higher space, in order that our logic may be equal to the -task before us.</p> - -<p>We gain a repetition of the thought that came before, -experimentally suggested. If the laws of the intellectual -comprehension of nature are those derived from considering -her as absolute chance, subject to no law save -that derived from a process of selection, then, perhaps, the -order of nature requires different faculties from the intellectual -to apprehend it. The source and origin of -ideas may have to be sought elsewhere than in reasoning.</p> - -<p>The total outcome of the critique is to leave the -ordinary man just where he is, justified in his practical -attitude towards nature, liberated from the fetters of his -own mental representations.</p> - -<p>The truth of a picture lies in its total effect. It is vain -to seek information about the landscape from an examina<span class="pagenum" id="Page_121">[Pg 121]</span>tion -of the pigments. And in any method of thought it -is the complexity of the whole that brings us to a knowledge -of nature. Dimensions are artificial enough, but in -the multiplicity of them we catch some breath of nature.</p> - -<p>We must therefore, and this seems to me the practical -conclusion of the whole matter, proceed to form means of -intellectual apprehension of a greater and greater degree -of complexity, both dimensionally and in extent in any -dimension. Such means of representation must always -be artificial, but in the multiplicity of the elements with -which we deal, however incipiently arbitrary, lies our -chance of apprehending nature.</p> - -<p>And as a concluding chapter to this part of the book, -I will extend the figures, which have been used to represent -Kant’s theory, two steps, so that the reader may -have the opportunity of looking at a four-dimensional -figure which can be delineated without any of the special -apparatus, to the consideration of which I shall subsequently -pass on.</p> -<hr class="chap x-ebookmaker-drop" /> - -<div class="chapter"> -<p><span class="pagenum" id="Page_122">[Pg 122]</span></p> - -<h2 class="nobreak" id="CHAPTER_X">CHAPTER X<br /> - -<small><i>A FOUR-DIMENSIONAL FIGURE</i></small></h2></div> - - -<p>The method used in the preceding chapter to illustrate -the problem of Kant’s critique, gives a singularly easy -and direct mode of constructing a series of important -figures in any number of dimensions.</p> - -<p>We have seen that to represent our space a plane being -must give up one of his axes, and similarly to represent -the higher shapes we must give up one amongst our -three axes.</p> - -<p>But there is another kind of giving up which reduces -the construction of higher shapes to a matter of the -utmost simplicity.</p> - -<p>Ordinarily we have on a straight line any number of -positions. The wealth of space in position is illimitable, -while there are only three dimensions.</p> - -<p>I propose to give up this wealth of positions, and to -consider the figures obtained by taking just as many -positions as dimensions.</p> - -<p>In this way I consider dimensions and positions as two -“kinds,” and applying the simple rule of selecting every -one of one kind with every other of every other kind, -get a series of figures which are noteworthy because -they exactly fill space of any number of dimensions -(as the hexagon fills a plane) by equal repetitions of -themselves.</p> - -<p><span class="pagenum" id="Page_123">[Pg 123]</span></p> - -<p>The rule will be made more evident by a simple -application.</p> - -<p>Let us consider one dimension and one position. I will -call the axis <i>i</i>, and the position <i>o</i>.</p> - -<p class="center"> -———————————————-<i>i</i><br /> -<span style="margin-left: 3.5em;"><i>o</i></span> -</p> - -<p>Here the figure is the position <i>o</i> on the line <i>i</i>. Take -now two dimensions and two positions on each.</p> - -<div class="figleft illowp30" id="fig_63" style="max-width: 12.125em;"> - <img src="images/fig_63.png" alt="" /> - <div class="caption">Fig. 63.</div> -</div> - -<p>We have the two positions <i>o</i>; 1 on <i>i</i>, and the two -positions <i>o</i>, 1 on <i>j</i>, <a href="#fig_63">fig. 63</a>. These give -rise to a certain complexity. I will -let the two lines <i>i</i> and <i>j</i> meet in the -position I call <i>o</i> on each, and I will -consider <i>i</i> as a direction starting equally -from every position on <i>j</i>, and <i>j</i> as -starting equally from every position on <i>i</i>. We thus -obtain the following figure:—<span class="allsmcap">A</span> is both <i>oi</i> and <i>oj</i>, <span class="allsmcap">B</span> is 1<i>i</i> -and <i>oj</i>, and so on as shown in <a href="#fig_63">fig. 63</a><i>b</i>. -The positions on <span class="allsmcap">AC</span> are all <i>oi</i> positions. -They are, if we like to consider it in -that way, points at no distance in the <i>i</i> -direction from the line <span class="allsmcap">AC</span>. We can -call the line <span class="allsmcap">AC</span> the <i>oi</i> line. Similarly -the points on <span class="allsmcap">AB</span> are those no distance -from <span class="allsmcap">AB</span> in the <i>j</i> direction, and we can -call them <i>oj</i> points and the line <span class="allsmcap">AB</span> the <i>oj</i> line. Again, -the line <span class="allsmcap">CD</span> can be called the 1<i>j</i> line because the points -on it are at a distance, 1 in the <i>j</i> direction.</p> - -<div class="figleft illowp30" id="fig_63b" style="max-width: 12.5em;"> - <img src="images/fig_63b.png" alt="" /> - <div class="caption">Fig. 63<i>b</i>.</div> -</div> - -<p>We have then four positions or points named as shown, -and, considering directions and positions as “kinds,” we -have the combination of two kinds with two kinds. Now, -selecting every one of one kind with every other of every -other kind will mean that we take 1 of the kind <i>i</i> and<span class="pagenum" id="Page_124">[Pg 124]</span> -with it <i>o</i> of the kind <i>j</i>; and then, that we take <i>o</i> of the -kind <i>i</i> and with it 1 of the kind <i>j</i>.</p> - -<div class="figleft illowp25" id="fig_64" style="max-width: 12.5em;"> - <img src="images/fig_64.png" alt="" /> - <div class="caption">Fig. 64.</div> -</div> - -<p>Thus we get a pair of positions lying in the straight -line <span class="allsmcap">BC</span>, <a href="#fig_64">fig. 64</a>. We can call this pair 10 -and 01 if we adopt the plan of mentally, -adding an <i>i</i> to the first and a <i>j</i> to the -second of the symbols written thus—01 -is a short expression for O<i>i</i>, 1<i>j</i>.</p> - -<div class="figcenter illowp80" id="fig_65" style="max-width: 62.5em;"> - <img src="images/fig_65.png" alt="" /> - <div class="caption">Fig. 65.</div> -</div> - -<p>Coming now to our space, we have three -dimensions, so we take three positions on each. These -positions I will suppose to be at equal distances along each -axis. The three axes and the three positions on each are -shown in the accompanying diagrams, <a href="#fig_65">fig. 65</a>, of which -the first represents a cube with the front faces visible, the -second the rear faces of the same cube; the positions I -will call 0, 1, 2; the axes, <i>i</i>, <i>j</i>, <i>k</i>. I take the base <span class="allsmcap">ABC</span> as -the starting place, from which to determine distances in -the <i>k</i> direction, and hence every point in the base <span class="allsmcap">ABC</span> -will be an <i>ok</i> position, and the base <span class="allsmcap">ABC</span> can be called an -<i>ok</i> plane.</p> - -<p>In the same way, measuring the distances from the face -<span class="allsmcap">ADC</span>, we see that every position in the face <span class="allsmcap">ADC</span> is an <i>oi</i> -position, and the whole plane of the face may be called an -<i>oi</i> plane. Thus we see that with the introduction of a<span class="pagenum" id="Page_125">[Pg 125]</span> -new dimension the signification of a compound symbol, -such as “<i>oi</i>,” alters. In the plane it meant the line <span class="allsmcap">AC</span>. -In space it means the whole plane <span class="allsmcap">ACD</span>.</p> - -<p>Now, it is evident that we have twenty-seven positions, -each of them named. If the reader will follow this -nomenclature in respect of the positions marked in the -figures he will have no difficulty in assigning names to -each one of the twenty-seven positions. <span class="allsmcap">A</span> is <i>oi</i>, <i>oj</i>, <i>ok</i>. -It is at the distance 0 along <i>i</i>, 0 along <i>j</i>, 0 along <i>k</i>, and -<i>io</i> can be written in short 000, where the <i>ijk</i> symbols -are omitted.</p> - -<p>The point immediately above is 001, for it is no distance -in the <i>i</i> direction, and a distance of 1 in the <i>k</i> -direction. Again, looking at <span class="allsmcap">B</span>, it is at a distance of 2 -from <span class="allsmcap">A</span>, or from the plane <span class="allsmcap">ADC</span>, in the <i>i</i> direction, 0 in the -<i>j</i> direction from the plane <span class="allsmcap">ABD</span>, and 0 in the <i>k</i> direction, -measured from the plane <span class="allsmcap">ABC</span>. Hence it is 200 written -for 2<i>i</i>, 0<i>j</i>, 0<i>k</i>.</p> - -<p>Now, out of these twenty-seven “things” or compounds -of position and dimension, select those which are given by -the rule, every one of one kind with every other of every -other kind.</p> - -<div class="figleft illowp30" id="fig_66" style="max-width: 15.625em;"> - <img src="images/fig_66.png" alt="" /> - <div class="caption">Fig. 66.</div> -</div> - -<p>Take 2 of the <i>i</i> kind. With this -we must have a 1 of the <i>j</i> kind, -and then by the rule we can only -have a 0 of the <i>k</i> kind, for if we -had any other of the <i>k</i> kind we -should repeat one of the kinds we -already had. In 2<i>i</i>, 1<i>j</i>, 1<i>k</i>, for -instance, 1 is repeated. The point -we obtain is that marked 210, <a href="#fig_66">fig. 66</a>.</p> - -<div class="figleft illowp30" id="fig_67" style="max-width: 15.625em;"> - <img src="images/fig_67.png" alt="" /> - <div class="caption">Fig. 67.</div> -</div> - -<p>Proceeding in this way, we pick out the following -cluster of points, <a href="#fig_67">fig. 67</a>. They are joined by lines, -dotted where they are hidden by the body of the cube, -and we see that they form a figure—a hexagon which<span class="pagenum" id="Page_126">[Pg 126]</span> -could be taken out of the cube and placed on a plane. -It is a figure which will fill a -plane by equal repetitions of itself. -The plane being representing this -construction in his plane would -take three squares to represent the -cube. Let us suppose that he -takes the <i>ij</i> axes in his space and -<i>k</i> represents the axis running out -of his space, <a href="#fig_68">fig. 68</a>. In each of -the three squares shown here as drawn separately he -could select the points given by the rule, and he would -then have to try to discover the figure determined by -the three lines drawn. The line from 210 to 120 is -given in the figure, but the line from 201 to 102 or <span class="allsmcap">GK</span> -is not given. He can determine <span class="allsmcap">GK</span> by making another -set of drawings and discovering in them what the relation -between these two extremities is.</p> - -<div class="figcenter illowp100" id="fig_68" style="max-width: 62.5em;"> - <img src="images/fig_68.png" alt="" /> - <div class="caption">Fig. 68.</div> -</div> - -<div class="figcenter illowp80" id="fig_69" style="max-width: 50em;"> - <img src="images/fig_69.png" alt="" /> - <div class="caption">Fig. 69.</div> -</div> - -<p>Let him draw the <i>i</i> and <i>k</i> axes in his plane, <a href="#fig_69">fig. 69</a>. -The <i>j</i> axis then runs out and he has the accompanying -figure. In the first of these three squares, <a href="#fig_69">fig. 69</a>, he can<span class="pagenum" id="Page_127">[Pg 127]</span> -pick out by the rule the two points 201, 102—<span class="allsmcap">G</span>, and <span class="allsmcap">K</span>. -Here they occur in one plane and he can measure the -distance between them. In his first representation they -occur at <span class="allsmcap">G</span> and <span class="allsmcap">K</span> in separate figures.</p> - -<p>Thus the plane being would find that the ends of each -of the lines was distant by the diagonal of a unit square -from the corresponding end of the last and he could then -place the three lines in their right relative position. -Joining them he would have the figure of a hexagon.</p> - -<div class="figleft illowp30" id="fig_70" style="max-width: 15.625em;"> - <img src="images/fig_70.png" alt="" /> - <div class="caption">Fig. 70.</div> -</div> - -<p>We may also notice that the plane being could make -a representation of the whole cube -simultaneously. The three squares, -shown in perspective in <a href="#fig_70">fig. 70</a>, all -lie in one plane, and on these the -plane being could pick out any -selection of points just as well as on -three separate squares. He would -obtain a hexagon by joining the -points marked. This hexagon, as -drawn, is of the right shape, but it would not be so if -actual squares were used instead of perspective, because -the relation between the separate squares as they lie in -the plane figure is not their real relation. The figure, -however, as thus constructed, would give him an idea of -the correct figure, and he could determine it accurately -by remembering that distances in each square were -correct, but in passing from one square to another their -distance in the third dimension had to be taken into -account.</p> - -<p>Coming now to the figure made by selecting according -to our rule from the whole mass of points given by four -axes and four positions in each, we must first draw a -catalogue figure in which the whole assemblage is shown.</p> - -<p>We can represent this assemblage of points by four -solid figures. The first giving all those positions which<span class="pagenum" id="Page_128">[Pg 128]</span> -are at a distance <span class="allsmcap">O</span> from our space in the fourth dimension, -the second showing all those that are at a distance 1, -and so on.</p> - -<p>These figures will each be cubes. The first two are -drawn showing the front faces, the second two the rear -faces. We will mark the points 0, 1, 2, 3, putting points -at those distances along each of these axes, and suppose -all the points thus determined to be contained in solid -models of which our drawings in <a href="#fig_71">fig. 71</a> are representatives. -Here we notice that as on the plane 0<i>i</i> meant -the whole line from which the distances in the <i>i</i> direction -was measured, and as in space 0<i>i</i> means the whole plane -from which distances in the <i>i</i> direction are measured, so -now 0<i>h</i> means the whole space in which the first cube -stands—measuring away from that space by a distance -of one we come to the second cube represented.</p> - -<div class="figcenter illowp80" id="fig_71" style="max-width: 62.5em;"> - <img src="images/fig_71.png" alt="" /> - <div class="caption">Fig. 71.</div> -</div> - -<p><span class="pagenum" id="Page_129">[Pg 129]</span></p> - -<p>Now selecting according to the rule every one of one -kind with every other of every other kind, we must take, -for instance, 3<i>i</i>, 2<i>j</i>, 1<i>k</i>, 0<i>h</i>. This point is marked -3210 at the lower star in the figure. It is 3 in the -<i>i</i> direction, 2 in the <i>j</i> direction, 1 in the <i>k</i> direction, -0 in the <i>h</i> direction.</p> - -<p>With 3<i>i</i> we must also take 1<i>j</i>, 2<i>k</i>, 0<i>h</i>. This point -is shown by the second star in the cube 0<i>h</i>.</p> - -<div class="figcenter illowp80" id="fig_72" style="max-width: 62.5em;"> - <img src="images/fig_72.png" alt="" /> - <div class="caption">Fig. 72.</div> -</div> - -<p>In the first cube, since all the points are 0<i>h</i> points, -we can only have varieties in which <i>i</i>, <i>j</i>, <i>k</i>, are accompanied -by 3, 2, 1.</p> - -<p>The points determined are marked off in the diagram -fig. 72, and lines are drawn joining the adjacent pairs -in each figure, the lines being dotted when they pass -within the substance of the cube in the first two diagrams.</p> - -<p>Opposite each point, on one side or the other of each<span class="pagenum" id="Page_130">[Pg 130]</span> -cube, is written its name. It will be noticed that the -figures are symmetrical right and left; and right and -left the first two numbers are simply interchanged.</p> - -<p>Now this being our selection of points, what figure do -they make when all are put together in their proper -relative positions?</p> - -<p>To determine this we must find the distance between -corresponding corners of the separate hexagons.</p> -<div class="figcenter illowp80" id="fig_73" style="max-width: 62.5em;"> - <img src="images/fig_73.png" alt="" /> - <div class="caption">Fig. 73.</div> -</div> - - -<p>To do this let us keep the axes <i>i</i>, <i>j</i>, in our space, and -draw <i>h</i> instead of <i>k</i>, letting <i>k</i> run out in the fourth -dimension, <a href="#fig_73">fig. 73</a>.</p> - -<div class="figright illowp50" id="fig_74" style="max-width: 37.5em;"> - <img src="images/fig_74.png" alt="" /> - <div class="caption">Fig. 74.</div> -</div> - -<p>Here we have four cubes again, in the first of which all -the points are 0<i>k</i> points; that is, points at a distance zero -in the <i>k</i> direction from the space of the three dimensions -<i>ijh</i>. We have all the points selected before, and some -of the distances, which in the last diagram led from figure -to figure are shown here in the same figure, and so capable<span class="pagenum" id="Page_131">[Pg 131]</span> -of measurement. Take for instance the points 3120 to -3021, which in the first diagram (<a href="#fig_72">fig. 72</a>) lie in the first -and second figures. Their actual relation is shown in -fig. 73 in the cube marked 2<span class="allsmcap">K</span>, where the points in question -are marked with a *. We see that the -distance in question is the diagonal of a unit square. In -like manner we find that the distance between corresponding -points of any two hexagonal figures is the -diagonal of a unit square. The total figure is now easily -constructed. An idea -of it may be gained by -drawing all the four -cubes in the catalogue -figure in one (fig. 74). -These cubes are exact -repetitions of one -another, so one drawing -will serve as a -representation of the -whole series, if we -take care to remember -where we are, whether -in a 0<i>h</i>, a 1<i>h</i>, a 2<i>h</i>, -or a 3<i>h</i> figure, when -we pick out the points required. Fig. 74 is a representation -of all the catalogue cubes put in one. For the -sake of clearness the front faces and the back faces of -this cube are represented separately.</p> - -<p>The figure determined by the selected points is shown -below.</p> - -<p>In putting the sections together some of the outlines -in them disappear. The line <span class="allsmcap">TW</span> for instance is not -wanted.</p> - -<p>We notice that <span class="allsmcap">PQTW</span> and <span class="allsmcap">TWRS</span> are each the half -of a hexagon. Now <span class="allsmcap">QV</span> and <span class="allsmcap">VR</span> lie in one straight line.<span class="pagenum" id="Page_132">[Pg 132]</span> -Hence these two hexagons fit together, forming one -hexagon, and the line <span class="allsmcap">TW</span> is only wanted when we consider -a section of the whole figure, we thus obtain the -solid represented in the lower part of <a href="#fig_74">fig. 74</a>. Equal -repetitions of this figure, called a tetrakaidecagon, will -fill up three-dimensional space.</p> - -<p>To make the corresponding four-dimensional figure we -have to take five axes mutually at right angles with five -points on each. A catalogue of the positions determined -in five-dimensional space can be found thus.</p> -<div class="figleft illowp60" id="fig_75" style="max-width: 37.5em;"> - <img src="images/fig_75.png" alt="" /> - <div class="caption">Fig. 75.</div> -</div> - -<p>Take a cube with five points on each of its axes, the -fifth point is at a distance of four units of length from the -first on any one of the axes. And since the fourth dimension -also stretches to a distance of four we shall need to -represent the successive -sets of points at -distances 0, 1, 2, 3, 4, -in the fourth dimensions, -five cubes. Now -all of these extend to -no distance at all in -the fifth dimension. -To represent what -lies in the fifth dimension -we shall have to -draw, starting from -each of our cubes, five -similar cubes to represent -the four steps -on in the fifth dimension. By this assemblage we get a -catalogue of all the points shown in <a href="#fig_75">fig. 75</a>, in which -<i>L</i> represents the fifth dimension.</p> - -<p>Now, as we saw before, there is nothing to prevent us -from putting all the cubes representing the different -stages in the fourth dimension in one figure, if we take<span class="pagenum" id="Page_133">[Pg 133]</span> -note when we look at it, whether we consider it as a 0<i>h</i>, a -1<i>h</i>, a 2<i>h</i>, etc., cube. Putting then the 0<i>h</i>, 1<i>h</i>, 2<i>h</i>, 3<i>h</i>, 4<i>h</i> -cubes of each row in one, we have five cubes with the sides -of each containing five positions, the first of these five -cubes represents the 0<i>l</i> points, and has in it the <i>i</i> points -from 0 to 4, the <i>j</i> points from 0 to 4, the <i>k</i> points from -0 to 4, while we have to specify with regard to any -selection we make from it, whether we regard it as a 0<i>h</i>, -a 1<i>h</i>, a 2<i>h</i>, a 3<i>h</i>, or a 4<i>h</i> figure. In <a href="#fig_76">fig. 76</a> each cube -is represented by two drawings, one of the front part, the -other of the rear part.</p> - -<p>Let then our five cubes be arranged before us and our -selection be made according to the rule. Take the first -figure in which all points are 0<i>l</i> points. We cannot -have 0 with any other letter. Then, keeping in the first -figure, which is that of the 0<i>l</i> positions, take first of all -that selection which always contains 1<i>h</i>. We suppose, -therefore, that the cube is a 1<i>h</i> cube, and in it we take -<i>i</i>, <i>j</i>, <i>k</i> in combination with 4, 3, 2 according to the rule.</p> - -<p>The figure we obtain is a hexagon, as shown, the one -in front. The points on the right hand have the same -figures as those on the left, with the first two numerals -interchanged. Next keeping still to the 0<i>l</i> figure let -us suppose that the cube before us represents a section -at a distance of 2 in the <i>h</i> direction. Let all the points -in it be considered as 2<i>h</i> points. We then have a 0<i>l</i>, 2<i>h</i> -region, and have the sets <i>ijk</i> and 431 left over. We -must then pick out in accordance with our rule all such -points as 4<i>i</i>, 3<i>j</i>, 1<i>k</i>.</p> - -<p>These are shown in the figure and we find that we can -draw them without confusion, forming the second hexagon -from the front. Going on in this way it will be seen -that in each of the five figures a set of hexagons is picked -out, which put together form a three-space figure something -like the tetrakaidecagon.</p> - -<p><span class="pagenum" id="Page_134">[Pg 134]</span></p> - -<div class="figcenter illowp100" id="fig_76" style="max-width: 93.75em;"> - <img src="images/fig_76.png" alt="" /> - <div class="caption">Fig. 76.</div> -</div> - -<p><span class="pagenum" id="Page_135">[Pg 135]</span></p> - -<p>These separate figures are the successive stages in -which the whole four-dimensional figure in which they -cohere can be apprehended.</p> - -<p>The first figure and the last are tetrakaidecagons. -These are two of the solid boundaries of the figure. The -other solid boundaries can be traced easily. Some of -them are complete from one face in the figure to the -corresponding face in the next, as for instance the solid -which extends from the hexagonal base of the first figure -to the equal hexagonal base of the second figure. This -kind of boundary is a hexagonal prism. The hexagonal -prism also occurs in another sectional series, as for -instance, in the square at the bottom of the first figure, -the oblong at the base of the second and the square at -the base of the third figure.</p> - -<p>Other solid boundaries can be traced through four of -the five sectional figures. Thus taking the hexagon at -the top of the first figure we find in the next a hexagon -also, of which some alternate sides are elongated. The -top of the third figure is also a hexagon with the other -set of alternate rules elongated, and finally we come in -the fourth figure to a regular hexagon.</p> - -<p>These four sections are the sections of a tetrakaidecagon -as can be recognised from the sections of this figure -which we have had previously. Hence the boundaries -are of two kinds, hexagonal prisms and tetrakaidecagons.</p> - -<p>These four-dimensional figures exactly fill four-dimensional -space by equal repetitions of themselves.</p> -<hr class="chap x-ebookmaker-drop" /> - -<div class="chapter"> -<p><span class="pagenum" id="Page_136">[Pg 136]</span></p> - -<h2 class="nobreak" id="CHAPTER_XI">CHAPTER XI<br /> - -<small><i>NOMENCLATURE AND ANALOGIES PRELIMINARY -TO THE STUDY OF FOUR-DIMENSIONAL -FIGURES</i></small></h2></div> - - -<p>In the following pages a method of designating different -regions of space by a systematic colour scheme has been -adopted. The explanations have been given in such a -manner as to involve no reference to models, the diagrams -will be found sufficient. But to facilitate the study a -description of a set of models is given in an appendix -which the reader can either make for himself or obtain. -If models are used the diagrams in Chapters XI. and XII. -will form a guide sufficient to indicate their use. Cubes -of the colours designated by the diagrams should be picked -out and used to reinforce the diagrams. The reader, -in the following description, should -suppose that a board or wall -stretches away from him, against -which the figures are placed.</p> - -<div class="figleft illowp30" id="fig_77" style="max-width: 15.625em;"> - <img src="images/fig_77.png" alt="" /> - <div class="caption">Fig. 77.</div> -</div> - -<p>Take a square, one of those -shown in Fig. 77 and give it a -neutral colour, let this colour be -called “null,” and be such that it -makes no appreciable difference<span class="pagenum" id="Page_137">[Pg 137]</span> -to any colour with which it mixed. If there is no -such real colour let us imagine such a colour, and -assign to it the properties of the number zero, which -makes no difference in any number to which it is -added.</p> - -<p>Above this square place a red square. Thus we symbolise -the going up by adding red to null.</p> - -<p>Away from this null square place a yellow square, and -represent going away by adding yellow to null.</p> - -<div class="figleft illowp40" id="fig_78" style="max-width: 15.625em;"> - <img src="images/fig_78.png" alt="" /> - <div class="caption">Fig. 78.</div> -</div> - -<p>To complete the figure we need a fourth square. -Colour this orange, which is a mixture of red and -yellow, and so appropriately represents a going in a -direction compounded of up and away. We have thus -a colour scheme which will serve to name the set of -squares drawn. We have two axes of colours—red and -yellow—and they may occupy -as in the figure the -direction up and away, or -they may be turned about; -in any case they enable us -to name the four squares -drawn in their relation to -one another.</p> - -<p>Now take, in Fig. 78, -nine squares, and suppose -that at the end of the -going in any direction the -colour started with repeats itself.</p> - -<p>We obtain a square named as shown.</p> - -<p>Let us now, in <a href="#fig_79">fig. 79</a>, suppose the number of squares to -be increased, keeping still to the principle of colouring -already used.</p> - -<p>Here the nulls remain four in number. There -are three reds between the first null and the null -above it, three yellows between the first null and the<span class="pagenum" id="Page_138">[Pg 138]</span> -null beyond it, while the oranges increase in a double -way.</p> - -<div class="figcenter illowp80" id="fig_79" style="max-width: 62.5em;"> - <img src="images/fig_79.png" alt="" /> - <div class="caption">Fig. 79.</div> -</div> - -<p>Suppose this process of enlarging the number of the -squares to be indefinitely pursued and -the total figure obtained to be reduced -in size, we should obtain a square of -which the interior was all orange, -while the lines round it were red and -yellow, and merely the points null -colour, as in <a href="#fig_80">fig. 80</a>. Thus all the points, lines, and the -area would have a colour.</p> - -<div class="figleft illowp25" id="fig_80" style="max-width: 15.625em;"> - <img src="images/fig_80.png" alt="" /> - <div class="caption">Fig. 80.</div> -</div> - - -<p>We can consider this scheme to originate thus:—Let -a null point move in a yellow direction and trace out a -yellow line and end in a null point. Then let the whole -line thus traced move in a red direction. The null points -at the ends of the line will produce red lines, and end in<span class="pagenum" id="Page_139">[Pg 139]</span> -null points. The yellow line will trace out a yellow and -red, or orange square.</p> - -<p>Now, turning back to <a href="#fig_78">fig. 78</a>, we see that these two -ways of naming, the one we started with and the one we -arrived at, can be combined.</p> - -<p>By its position in the group of four squares, in <a href="#fig_77">fig. 77</a>, -the null square has a relation to the yellow and to the red -directions. We can speak therefore of the red line of the -null square without confusion, meaning thereby the line -<span class="allsmcap">AB</span>, <a href="#fig_81">fig. 81</a>, which runs up from the -initial null point <span class="allsmcap">A</span> in the figure as -drawn. The yellow line of the null -square is its lower horizontal line <span class="allsmcap">AC</span> -as it is situated in the figure.</p> - -<div class="figleft illowp30" id="fig_81" style="max-width: 15.625em;"> - <img src="images/fig_81.png" alt="" /> - <div class="caption">Fig. 81.</div> -</div> - -<p>If we wish to denote the upper -yellow line <span class="allsmcap">BD</span>, <a href="#fig_81">fig. 81</a>, we can speak -of it as the yellow γ line, meaning -the yellow line which is separated -from the primary yellow line by the red movement.</p> - -<p>In a similar way each of the other squares has null -points, red and yellow lines. Although the yellow square -is all yellow, its line <span class="allsmcap">CD</span>, for instance, can be referred to as -its red line.</p> - -<p>This nomenclature can be extended.</p> - -<p>If the eight cubes drawn, in <a href="#fig_82">fig. 82</a>, are put close -together, as on the right hand of the diagram, they form -a cube, and in them, as thus arranged, a going up is -represented by adding red to the zero, or null colour, a -going away by adding yellow, a going to the right by -adding white. White is used as a colour, as a pigment, -which produces a colour change in the pigments with which -it is mixed. From whatever cube of the lower set we -start, a motion up brings us to a cube showing a change -to red, thus light yellow becomes light yellow red, or -light orange, which is called ochre. And going to the<span class="pagenum" id="Page_140">[Pg 140]</span> -right from the null on the left we have a change involving -the introduction of white, while the yellow change runs -from front to back. There are three colour axes—the red, -the white, the yellow—and these run in the position the -cubes occupy in the drawing—up, to the right, away—but -they could be turned about to occupy any positions in space.</p> - -<div class="figcenter illowp100" id="fig_82" style="max-width: 62.5em;"> - <img src="images/fig_82.png" alt="" /> - <div class="caption">Fig. 82.</div> -</div> - - -<div class="figcenter illowp100" id="fig_83" style="max-width: 62.5em;"> - <img src="images/fig_83.png" alt="" /> - <div class="caption">Fig. 83.</div> -</div> - -<p>We can conveniently represent a block of cubes by -three sets of squares, representing each the base of a cube.</p> - -<p>Thus the block, <a href="#fig_83">fig. 83</a>, can be represented by the<span class="pagenum" id="Page_141">[Pg 141]</span> -layers on the right. Here, as in the case of the plane, -the initial colours repeat themselves at the end of the -series.</p> - -<div class="figleft illowp50" id="fig_84" style="max-width: 31.25em;"> - <img src="images/fig_84.png" alt="" /> - <div class="caption">Fig. 84.</div> -</div> - -<p>Proceeding now to increase the number of the cubes -we obtain <a href="#fig_84">fig. 84</a>, -in which the initial -letters of the colours -are given instead of -their full names.</p> - -<p>Here we see that -there are four null -cubes as before, but -the series which spring -from the initial corner -will tend to become -lines of cubes, as also -the sets of cubes -parallel to them, starting -from other corners. -Thus, from the initial -null springs a line of -red cubes, a line of -white cubes, and a line -of yellow cubes.</p> - -<p>If the number of the -cubes is largely increased, -and the size -of the whole cube is -diminished, we get -a cube with null -points, and the edges -coloured with these three colours.</p> - -<p>The light yellow cubes increase in two ways, forming -ultimately a sheet of cubes, and the same is true of -the orange and pink sets. Hence, ultimately the cube<span class="pagenum" id="Page_142">[Pg 142]</span> -thus formed would have red, white, and yellow lines -surrounding pink, orange, and light yellow faces. The -ochre cubes increase in three ways, and hence ultimately -the whole interior of the cube would be coloured -ochre.</p> - -<p>We have thus a nomenclature for the points, lines, -faces, and solid content of a cube, and it can be named -as exhibited in <a href="#fig_85">fig. 85</a>.</p> - -<div class="figleft illowp30" id="fig_85" style="max-width: 15.625em;"> - <img src="images/fig_85.png" alt="" /> - <div class="caption">Fig. 85.</div> -</div> - -<p>We can consider the cube to be produced in the -following way. A null point -moves in a direction to which -we attach the colour indication -yellow; it generates a yellow line -and ends in a null point. The -yellow line thus generated moves -in a direction to which we give -the colour indication red. This -lies up in the figure. The yellow -line traces out a yellow, red, or -orange square, and each of its null points trace out a -red line, and ends in a null point.</p> - -<p>This orange square moves in a direction to which -we attribute the colour indication white, in this case -the direction is the right. The square traces out a -cube coloured orange, red, or ochre, the red lines trace -out red to white or pink squares, and the yellow -lines trace out light yellow squares, each line ending -in a line of its own colour. While the points each -trace out a null + white, or white line to end in a null -point.</p> - -<p>Now returning to the first block of eight cubes we can -name each point, line, and square in them by reference to -the colour scheme, which they determine by their relation -to each other.</p> - -<p>Thus, in <a href="#fig_86">fig. 86</a>, the null cube touches the red cube by<span class="pagenum" id="Page_143">[Pg 143]</span> -a light yellow square; it touches the yellow cube by a -pink square, and touches -the white cube by an -orange square.</p> - -<div class="figleft illowp50" id="fig_86" style="max-width: 25em;"> - <img src="images/fig_86.png" alt="" /> - <div class="caption">Fig. 86.</div> -</div> - -<p>There are three axes -to which the colours red, -yellow, and white are -assigned, the faces of -each cube are designated -by taking these colours in pairs. Taking all the colours -together we get a colour name for the solidity of a cube.</p> - - -<p>Let us now ask ourselves how the cube could be presented -to the plane being. Without going into the question -of how he could have a real experience of it, let us see -how, if we could turn it about and show it to him, he, -under his limitations, could get information about it. -If the cube were placed with its red and yellow axes -against a plane, that is resting against it by its orange -face, the plane being would observe a square surrounded -by red and yellow lines, and having null points. See the -dotted square, <a href="#fig_87">fig. 87</a>.</p> - -<div class="figcenter illowp80" id="fig_87" style="max-width: 37.5em;"> - <img src="images/fig_87.png" alt="" /> - <div class="caption">Fig. 87.</div> -</div> - -<p>We could turn the cube about the red line so that -a different face comes into juxtaposition with the plane.</p> - -<p>Suppose the cube turned about the red line. As it<span class="pagenum" id="Page_144">[Pg 144]</span> -is turning from its first position all of it except the red -line leaves the plane—goes absolutely out of the range -of the plane being’s apprehension. But when the yellow -line points straight out from the plane then the pink -face comes into contact with it. Thus the same red line -remaining as he saw it at first, now towards him comes -a face surrounded by white and red lines.</p> - -<div class="figleft illowp35" id="fig_88" style="max-width: 18.75em;"> - <img src="images/fig_88.png" alt="" /> - <div class="caption">Fig. 88.</div> -</div> - -<p>If we call the direction to the right the unknown -direction, then the line he saw before, the yellow line, -goes out into this unknown direction, and the line which -before went into the unknown direction, comes in. It -comes in in the opposite direction to that in which the -yellow line ran before; the interior of the face now -against the plane is pink. It is -a property of two lines at right -angles that, if one turns out of -a given direction and stands at -right angles to it, then the other -of the two lines comes in, but -runs the opposite way in that -given direction, as in <a href="#fig_88">fig. 88</a>.</p> - -<p>Now these two presentations of the cube would seem, -to the plane creature like perfectly different material -bodies, with only that line in common in which they -both meet.</p> - -<p>Again our cube can be turned about the yellow line. -In this case the yellow square would disappear as before, -but a new square would come into the plane after the -cube had rotated by an angle of 90° about this line. -The bottom square of the cube would come in thus -in figure 89. The cube supposed in contact with the -plane is rotated about the lower yellow line and then -the bottom face is in contact with the plane.</p> - -<p>Here, as before, the red line going out into the unknown -dimension, the white line which before ran in the<span class="pagenum" id="Page_145">[Pg 145]</span> -unknown dimension would come in downwards in the -opposite sense to that in which the red line ran before.</p> - -<div class="figcenter illowp80" id="fig_89" style="max-width: 62.5em;"> - <img src="images/fig_89.png" alt="" /> - <div class="caption">Fig. 89.</div> -</div> - -<p>Now if we use <i>i</i>, <i>j</i>, <i>k</i>, for the three space directions, -<i>i</i> left to right, <i>j</i> from near away, <i>k</i> from below up; then, -using the colour names for the axes, we have that first -of all white runs <i>i</i>, yellow runs <i>j</i>, red runs <i>k</i>; then after -the first turning round the <i>k</i> axis, white runs negative <i>j</i>, -yellow runs <i>i</i>, red runs <i>k</i>; thus we have the table:—</p> - -<table class="standard" summary=""> -<tr> -<td class="tdc"></td> -<td class="tdc"><i>i</i></td> -<td class="tdc"><i>j</i></td> -<td class="tdc"><i>k</i></td> -</tr> -<tr> -<td class="tdl">1st position</td> -<td class="tdc">white</td> -<td class="tdc">yellow</td> -<td class="tdc">red</td> -</tr> -<tr> -<td class="tdl">2nd position</td> -<td class="tdc">yellow</td> -<td class="tdc">white—</td> -<td class="tdc">red</td> -</tr> -<tr> -<td class="tdl">3rd position</td> -<td class="tdc">red</td> -<td class="tdc">yellow</td> -<td class="tdc">white—</td> -</tr> -</table> - - -<p>Here white with a negative sign after it in the column -under <i>j</i> means that white runs in the negative sense of -the <i>j</i> direction.</p> - -<p>We may express the fact in the following way:— -In the plane there is room for two axes while the body -has three. Therefore in the plane we can represent any -two. If we want to keep the axis that goes in the -unknown dimension always running in the positive sense, -then the axis which originally ran in the unknown<span class="pagenum" id="Page_146">[Pg 146]</span> -dimension (the white axis) must come in in the negative -sense of that axis which goes out of the plane into the -unknown dimension.</p> - -<p>It is obvious that the unknown direction, the direction -in which the white line runs at first, is quite distinct from -any direction which the plane creature knows. The white -line may come in towards him, or running down. If he -is looking at a square, which is the face of a cube -(looking at it by a line), then any one of the bounding lines -remaining unmoved, another face of the cube may come -in, any one of the faces, namely, which have the white line -in them. And the white line comes sometimes in one -of the space directions he knows, sometimes in another.</p> - -<p>Now this turning which leaves a line unchanged is -something quite unlike any turning he knows in the -plane. In the plane a figure turns round a point. The -square can turn round the null point in his plane, and -the red and yellow lines change places, only of course, as -with every rotation of lines at right angles, if red goes -where yellow went, yellow comes in negative of red’s old -direction.</p> - -<p>This turning, as the plane creature conceives it, we -should call turning about an axis perpendicular to the -plane. What he calls turning about the null point we -call turning about the white line as it stands out from -his plane. There is no such thing as turning about a -point, there is always an axis, and really much more turns -than the plane being is aware of.</p> - -<p>Taking now a different point of view, let us suppose the -cubes to be presented to the plane being by being passed -transverse to his plane. Let us suppose the sheet of -matter over which the plane being and all objects in his -world slide, to be of such a nature that objects can pass -through it without breaking it. Let us suppose it to be -of the same nature as the film of a soap bubble, so that<span class="pagenum" id="Page_147">[Pg 147]</span> -it closes around objects pushed through it, and, however -the object alters its shape as it passes through it, let us -suppose this film to run up to the contour of the object -in every part, maintaining its plane surface unbroken.</p> - -<p>Then we can push a cube or any object through the -film and the plane being who slips about in the film -will know the contour of the cube just and exactly where -the film meets it.</p> - -<div class="figleft illowp40" id="fig_90" style="max-width: 18.75em;"> - <img src="images/fig_90.png" alt="" /> - <div class="caption">Fig. 90.</div> -</div> - -<p>Fig. 90 represents a cube passing through a plane film. -The plane being now comes into -contact with a very thin slice -of the cube somewhere between -the left and right hand faces. -This very thin slice he thinks -of as having no thickness, and -consequently his idea of it is -what we call a section. It is -bounded by him by pink lines -front and back, coming from -the part of the pink face he is -in contact with, and above and below, by light yellow -lines. Its corners are not null-coloured points, but white -points, and its interior is ochre, the colour of the interior -of the cube.</p> - -<p>If now we suppose the cube to be an inch in each -dimension, and to pass across, from right to left, through -the plane, then we should explain the appearances presented -to the plane being by saying: First of all you -have the face of a cube, this lasts only a moment; then -you have a figure of the same shape but differently -coloured. This, which appears not to move to you in any -direction which you know of, is really moving transverse -to your plane world. Its appearance is unaltered, but -each moment it is something different—a section further -on, in the white, the unknown dimension. Finally, at the<span class="pagenum" id="Page_148">[Pg 148]</span> -end of the minute, a face comes in exactly like the face -you first saw. This finishes up the cube—it is the further -face in the unknown dimension.</p> - -<p>The white line, which extends in length just like the -red or the yellow, you do not see as extensive; you apprehend -it simply as an enduring white point. The null -point, under the condition of movement of the cube, -vanishes in a moment, the lasting white point is really -your apprehension of a white line, running in the unknown -dimension. In the same way the red line of the face by -which the cube is first in contact with the plane lasts only -a moment, it is succeeded by the pink line, and this pink -line lasts for the inside of a minute. This lasting pink -line in your apprehension of a surface, which extends in -two dimensions just like the orange surface extends, as you -know it, when the cube is at rest.</p> - -<p>But the plane creature might answer, “This orange -object is substance, solid substance, bounded completely -and on every side.”</p> - -<p>Here, of course, the difficulty comes in. His solid is our -surface—his notion of a solid is our notion of an abstract -surface with no thickness at all.</p> - -<p>We should have to explain to him that, from every point -of what he called a solid, a new dimension runs away. -From every point a line can be drawn in a direction -unknown to him, and there is a solidity of a kind greater -than that which he knows. This solidity can only be -realised by him by his supposing an unknown direction, -by motion in which what he conceives to be solid matter -instantly disappears. The higher solid, however, which -extends in this dimension as well as in those which he -knows, lasts when a motion of that kind takes place, -different sections of it come consecutively in the plane of -his apprehension, and take the place of the solid which he -at first conceives to be all. Thus, the higher solid—our<span class="pagenum" id="Page_149">[Pg 149]</span> -solid in contradistinction to his area solid, his two-dimensional -solid, must be conceived by him as something -which has duration in it, under circumstances in which his -matter disappears out of his world.</p> - -<p>We may put the matter thus, using the conception of -motion.</p> - -<p>A null point moving in a direction away generates a -yellow line, and the yellow line ends in a null point. We -suppose, that is, a point to move and mark out the -products of this motion in such a manner. Now -suppose this whole line as thus produced to move in -an upward direction; it traces out the two-dimensional -solid, and the plane being gets an orange square. The -null point moves in a red line and ends in a null point, -the yellow line moves and generates an orange square and -ends in a yellow line, the farther null point generates -a red line and ends in a null point. Thus, by movement -in two successive directions known to him, he -can imagine his two-dimensional solid produced with all -its boundaries.</p> - -<p>Now we tell him: “This whole two-dimensional solid -can move in a third or unknown dimension to you. The -null point moving in this dimension out of your world -generates a white line and ends in a null point. The -yellow line moving generates a light yellow two-dimensional -solid and ends in a yellow line, and this -two-dimensional solid, lying end on to your plane world, is -bounded on the far side by the other yellow line. In -the same way each of the lines surrounding your square -traces out an area, just like the orange area you know. -But there is something new produced, something which -you had no idea of before; it is that which is produced by -the movement of the orange square. That, than which -you can imagine nothing more solid, itself moves in a -direction open to it and produces a three-dimensional<span class="pagenum" id="Page_150">[Pg 150]</span> -solid. Using the addition of white to symbolise the -products of this motion this new kind of solid will be light -orange or ochre, and it will be bounded on the far side by -the final position of the orange square which traced it -out, and this final position we suppose to be coloured like -the square in its first position, orange with yellow and -red boundaries and null corners.”</p> - -<p>This product of movement, which it is so easy for us to -describe, would be difficult for him to conceive. But this -difficulty is connected rather with its totality than with -any particular part of it.</p> - -<p>Any line, or plane of this, to him higher, solid we could -show to him, and put in his sensible world.</p> - -<p>We have already seen how the pink square could be put -in his world by a turning of the cube about the red line. -And any section which we can conceive made of the cube -could be exhibited to him. You have simply to turn the -cube and push it through, so that the plane of his existence -is the plane which cuts out the given section of the cube, -then the section would appear to him as a solid. In his -world he would see the contour, get to any part of it by -digging down into it.</p> - - -<p><span class="smcap">The Process by which a Plane Being would gain -a Notion of a Solid.</span></p> - -<p>If we suppose the plane being to have a general idea of -the existence of a higher solid—our solid—we must next -trace out in detail the method, the discipline, by which -he would acquire a working familiarity with our space -existence. The process begins with an adequate realisation -of a simple solid figure. For this purpose we will -suppose eight cubes forming a larger cube, and first we -will suppose each cube to be coloured throughout uniformly.<span class="pagenum" id="Page_151">[Pg 151]</span> -Let the cubes in <a href="#fig_91">fig. 91</a> be the eight making a larger -cube.</p> - -<div class="figcenter illowp80" id="fig_91" style="max-width: 62.5em;"> - <img src="images/fig_91.png" alt="" /> - <div class="caption">Fig. 91.</div> -</div> - - -<p>Now, although each cube is supposed to be coloured -entirely through with the colour, the name of which is -written on it, still we can speak of the faces, edges, and -corners of each cube as if the colour scheme we have -investigated held for it. Thus, on the null cube we can -speak of a null point, a red line, a white line, a pink face, and -so on. These colour designations are shown on No. 1 of -the views of the tesseract in the plate. Here these colour -names are used simply in their geometrical significance. -They denote what the particular line, etc., referred to would -have as its colour, if in reference to the particular cube -the colour scheme described previously were carried out.</p> - -<p>If such a block of cubes were put against the plane and -then passed through it from right to left, at the rate of an -inch a minute, each cube being an inch each way, the -plane being would have the following appearances:—</p> - -<p>First of all, four squares null, yellow, red, orange, lasting -each a minute; and secondly, taking the exact places -of these four squares, four others, coloured white, light -yellow, pink, ochre. Thus, to make a catalogue of the -solid body, he would have to put side by side in his world -two sets of four squares each, as in <a href="#fig_92">fig. 92</a>. The first<span class="pagenum" id="Page_152">[Pg 152]</span> -are supposed to last a minute, and then the others to -come in in place of them, -and also last a minute.</p> - -<div class="figleft illowp50" id="fig_92" style="max-width: 25em;"> - <img src="images/fig_92.png" alt="" /> - <div class="caption">Fig. 92.</div> -</div> - -<p>In speaking of them -he would have to denote -what part of the respective -cube each square represents. -Thus, at the beginning -he would have null -cube orange face, and after -the motion had begun he -would have null cube ochre -section. As he could get -the same coloured section whichever way the cube passed -through, it would be best for him to call this section white -section, meaning that it is transverse to the white axis. -These colour-names, of course, are merely used as names, -and do not imply in this case that the object is really -coloured. Finally, after a minute, as the first cube was -passing beyond his plane he would have null cube orange -face again.</p> - -<p>The same names will hold for each of the other cubes, -describing what face or section of them the plane being -has before him; and the second wall of cubes will come -on, continue, and go out in the same manner. In the -area he thus has he can represent any movement which -we carry out in the cubes, as long as it does not involve -a motion in the direction of the white axis. The relation -of parts that succeed one another in the direction of the -white axis is realised by him as a consecution of states.</p> - -<p>Now, his means of developing his space apprehension -lies in this, that that which is represented as a time -sequence in one position of the cubes, can become a real -co-existence, <i>if something that has a real co-existence -becomes a time sequence</i>.</p> - -<p><span class="pagenum" id="Page_153">[Pg 153]</span></p> - -<p>We must suppose the cubes turned round each of the -axes, the red line, and the yellow line, then something, -which was given as time before, will now be given as the -plane creature’s space; something, which was given as space -before, will now be given as a time series as the cube is -passed through the plane.</p> - -<p>The three positions in which the cubes must be studied -are the one given above and the two following ones. In -each case the original null point which was nearest to us -at first is marked by an asterisk. In figs. 93 and 94 the -point marked with a star is the same in the cubes and in -the plane view.</p> - -<div class="figcenter illowp100" id="fig_93" style="max-width: 62.5em;"> - <img src="images/fig_93.png" alt="" /> - <div class="caption">Fig. 93.<br /> -The cube swung round the red line, so that the white line points -towards us.</div> -</div> - -<p>In <a href="#fig_93">fig. 93</a> the cube is swung round the red line so as to -point towards us, and consequently the pink face comes -next to the plane. As it passes through there are two -varieties of appearance designated by the figures 1 and 2 -in the plane. These appearances are named in the figure, -and are determined by the order in which the cubes<span class="pagenum" id="Page_154">[Pg 154]</span> -come in the motion of the whole block through the -plane.</p> - -<p>With regard to these squares severally, however, -different names must be used, determined by their -relations in the block.</p> - -<p>Thus, in <a href="#fig_93">fig. 93</a>, when the cube first rests against the -plane the null cube is in contact by its pink face; as the -block passes through we get an ochre section of the null -cube, but this is better called a yellow section, as it is -made by a plane perpendicular to the yellow line. When -the null cube has passed through the plane, as it is -leaving it, we get again a pink face.</p> - -<div class="figcenter illowp100" id="fig_94" style="max-width: 62.5em;"> - <img src="images/fig_94.png" alt="" /> - <div class="caption">Fig. 94.<br /> -The cube swung round yellow line, with red line running from left -to right, and white line running down.</div> -</div> - -<p>The same series of changes take place with the cube -appearances which follow on those of the null cube. In -this motion the yellow cube follows on the null cube, and -the square marked yellow in 2 in the plane will be first -“yellow pink face,” then “yellow yellow section,” then -“yellow pink face.”</p> - -<p>In <a href="#fig_94">fig. 94</a>, in which the cube is turned about the yellow -line, we have a certain difficulty, for the plane being will<span class="pagenum" id="Page_155">[Pg 155]</span> -find that the position his squares are to be placed in will -lie below that which they first occupied. They will come -where the support was on which he stood his first set of -squares. He will get over this difficulty by moving his -support.</p> - -<p>Then, since the cubes come upon his plane by the light -yellow face, he will have, taking the null cube as before for -an example, null, light yellow face; null, red section, -because the section is perpendicular to the red line; and -finally, as the null cube leaves the plane, null, light yellow -face. Then, in this case red following on null, he will -have the same series of views of the red as he had of the -null cube.</p> - -<div class="figcenter illowp100" id="fig_95" style="max-width: 62.5em;"> - <img src="images/fig_95.png" alt="" /> - <div class="caption">Fig. 95.</div> -</div> - -<p>There is another set of considerations which we will -briefly allude to.</p> - -<p>Suppose there is a hollow cube, and a string is stretched -across it from null to null, <i>r</i>, <i>y</i>, <i>wh</i>, as we may call the -far diagonal point, how will this string appear to the -plane being as the cube moves transverse to his plane?</p> - -<p>Let us represent the cube as a number of sections, say -5, corresponding to 4 equal divisions made along the white -line perpendicular to it.</p> - -<p>We number these sections 0, 1, 2, 3, 4, corresponding -to the distances along the white line at which they are<span class="pagenum" id="Page_156">[Pg 156]</span> -taken, and imagine each section to come in successively, -taking the place of the preceding one.</p> - -<p>These sections appear to the plane being, counting from -the first, to exactly coincide each with the preceding one. -But the section of the string occupies a different place in -each to that which it does in the preceding section. The -section of the string appears in the position marked by -the dots. Hence the slant of the string appears as a -motion in the frame work marked out by the cube sides. -If we suppose the motion of the cube not to be recognised, -then the string appears to the plane being as a moving -point. Hence extension on the unknown dimension -appears as duration. Extension sloping in the unknown -direction appears as continuous movement.</p> -<hr class="chap x-ebookmaker-drop" /> - -<div class="chapter"> -<p><span class="pagenum" id="Page_157">[Pg 157]</span></p> - -<h2 class="nobreak" id="CHAPTER_XII">CHAPTER XII<br /> - -<small><i>THE SIMPLEST FOUR-DIMENSIONAL SOLID</i></small></h2></div> - - -<p>A plane being, in learning to apprehend solid existence, -must first of all realise that there is a sense of direction -altogether wanting to him. That which we call right -and left does not exist in his perception. He must -assume a movement in a direction, and a distinction of -positive and negative in that direction, which has no -reality corresponding to it in the movements he can -make. This direction, this new dimension, he can only -make sensible to himself by bringing in time, and supposing -that changes, which take place in time, are due to -objects of a definite configuration in three dimensions -passing transverse to his plane, and the different sections -of it being apprehended as changes of one and the same -plane figure.</p> - -<p>He must also acquire a distinct notion about his plane -world, he must no longer believe that it is the all of -space, but that space extends on both sides of it. In -order, then, to prevent his moving off in this unknown -direction, he must assume a sheet, an extended solid sheet, -in two dimensions, against which, in contact with which, -all his movements take place.</p> - -<p>When we come to think of a four-dimensional solid, -what are the corresponding assumptions which we must -make?</p> - -<p>We must suppose a sense which we have not, a sense<span class="pagenum" id="Page_158">[Pg 158]</span> -of direction wanting in us, something which a being in -a four-dimensional world has, and which we have not. It -is a sense corresponding to a new space direction, a -direction which extends positively and negatively from -every point of our space, and which goes right away from -any space direction we know of. The perpendicular to a -plane is perpendicular, not only to two lines in it, but to -every line, and so we must conceive this fourth dimension -as running perpendicularly to each and every line we can -draw in our space.</p> - -<p>And as the plane being had to suppose something -which prevented his moving off in the third, the -unknown dimension to him, so we have to suppose -something which prevents us moving off in the direction -unknown to us. This something, since we must be in -contact with it in every one of our movements, must not -be a plane surface, but a solid; it must be a solid, which -in every one of our movements we are against, not in. It -must be supposed as stretching out in every space dimension -that we know; but we are not in it, we are against it, we -are next to it, in the fourth dimension.</p> - -<p>That is, as the plane being conceives himself as having -a very small thickness in the third dimension, of which -he is not aware in his sense experience, so we must -suppose ourselves as having a very small thickness in -the fourth dimension, and, being thus four-dimensional -beings, to be prevented from realising that we are -such beings by a constraint which keeps us always in -contact with a vast solid sheet, which stretches on in -every direction. We are against that sheet, so that, if we -had the power of four-dimensional movement, we should -either go away from it or through it; all our space -movements as we know them being such that, performing -them, we keep in contact with this solid sheet.</p> - -<p>Now consider the exposition a plane being would make<span class="pagenum" id="Page_159">[Pg 159]</span> -for himself as to the question of the enclosure of a square, -and of a cube.</p> - -<p>He would say the square <span class="allsmcap">A</span>, in Fig. 96, is completely -enclosed by the four squares, <span class="allsmcap">A</span> far, -<span class="allsmcap">A</span> near, <span class="allsmcap">A</span> above, <span class="allsmcap">A</span> below, or as they -are written <span class="allsmcap">A</span><i>n</i>, <span class="allsmcap">A</span><i>f</i>, <span class="allsmcap">A</span><i>a</i>, <span class="allsmcap">A</span><i>b</i>.</p> - -<div class="figleft illowp30" id="fig_96" style="max-width: 15.625em;"> - <img src="images/fig_96.png" alt="" /> - <div class="caption">Fig. 96.</div> -</div> - -<p>If now he conceives the square <span class="allsmcap">A</span> -to move in the, to him, unknown -dimension it will trace out a cube, -and the bounding squares will form -cubes. Will these completely surround -the cube generated by <span class="allsmcap">A</span>? No; -there will be two faces of the cube -made by <span class="allsmcap">A</span> left uncovered; the first, -that face which coincides with the -square <span class="allsmcap">A</span> in its first position; the next, that which coincides -with the square <span class="allsmcap">A</span> in its final position. Against these -two faces cubes must be placed in order to completely -enclose the cube <span class="allsmcap">A</span>. These may be called the cubes left -and right or <span class="allsmcap">A</span><i>l</i> and <span class="allsmcap">A</span><i>r</i>. Thus each of the enclosing -squares of the square <span class="allsmcap">A</span> becomes a cube and two more -cubes are wanted to enclose the cube formed by the -movement of <span class="allsmcap">A</span> in the third dimension.</p> - -<div class="figleft illowp30" id="fig_97" style="max-width: 34.6875em;"> - <img src="images/fig_97.png" alt="" /> - <div class="caption">Fig. 97.</div> -</div> - -<p>The plane being could not see the square <span class="allsmcap">A</span> with the -squares <span class="allsmcap">A</span><i>n</i>, <span class="allsmcap">A</span><i>f</i>, etc., placed about it, -because they completely hide it from -view; and so we, in the analogous -case in our three-dimensional world, -cannot see a cube <span class="allsmcap">A</span> surrounded by -six other cubes. These cubes we -will call <span class="allsmcap">A</span> near <span class="allsmcap">A</span><i>n</i>, <span class="allsmcap">A</span> far <span class="allsmcap">A</span><i>f</i>, <span class="allsmcap">A</span> above -<span class="allsmcap">A</span><i>a</i>, <span class="allsmcap">A</span> below <span class="allsmcap">A</span><i>b</i>, <span class="allsmcap">A</span> left <span class="allsmcap">A</span><i>l</i>, <span class="allsmcap">A</span> right <span class="allsmcap">A</span><i>r</i>, -shown in <a href="#fig_97">fig. 97</a>. If now the cube <span class="allsmcap">A</span> -moves in the fourth dimension right out of space, it traces -out a higher cube—a tesseract, as it may be called.<span class="pagenum" id="Page_160">[Pg 160]</span> -Each of the six surrounding cubes carried on in the same -motion will make a tesseract also, and these will be -grouped around the tesseract formed by <span class="allsmcap">A</span>. But will they -enclose it completely?</p> - -<p>All the cubes <span class="allsmcap">A</span><i>n</i>, <span class="allsmcap">A</span><i>f</i>, etc., lie in our space. But there is -nothing between the cube <span class="allsmcap">A</span> and that solid sheet in contact -with which every particle of matter is. When the cube <span class="allsmcap">A</span> -moves in the fourth direction it starts from its position, -say <span class="allsmcap">A</span><i>k</i>, and ends in a final position <span class="allsmcap">A</span><i>n</i> (using the words -“ana” and “kata” for up and down in the fourth dimension). -Now the movement in this fourth dimension is -not bounded by any of the cubes <span class="allsmcap">A</span><i>n</i>, <span class="allsmcap">A</span><i>f</i>, nor by what -they form when thus moved. The tesseract which <span class="allsmcap">A</span> -becomes is bounded in the positive and negative ways in -this new direction by the first position of <span class="allsmcap">A</span> and the last -position of <span class="allsmcap">A</span>. Or, if we ask how many tesseracts lie -around the tesseract which <span class="allsmcap">A</span> forms, there are eight, of -which one meets it by the cube <span class="allsmcap">A</span>, and another meets it -by a cube like <span class="allsmcap">A</span> at the end of its motion.</p> - -<p>We come here to a very curious thing. The whole -solid cube <span class="allsmcap">A</span> is to be looked on merely as a boundary of -the tesseract.</p> - -<p>Yet this is exactly analogous to what the plane being -would come to in his study of the solid world. The -square <span class="allsmcap">A</span> (<a href="#fig_96">fig. 96</a>), which the plane being looks on as a -solid existence in his plane world, is merely the boundary -of the cube which he supposes generated by its motion.</p> - -<p>The fact is that we have to recognise that, if there is -another dimension of space, our present idea of a solid -body, as one which has three dimensions only, does not -correspond to anything real, but is the abstract idea of a -three-dimensional boundary limiting a four-dimensional -solid, which a four-dimensional being would form. The -plane being’s thought of a square is not the thought -of what we should call a possibly existing real square,<span class="pagenum" id="Page_161">[Pg 161]</span> -but the thought of an abstract boundary, the face of -a cube.</p> - -<p>Let us now take our eight coloured cubes, which form -a cube in space, and ask what additions we must make -to them to represent the simplest collection of four-dimensional -bodies—namely, a group of them of the same extent -in every direction. In plane space we have four squares. -In solid space we have eight cubes. So we should expect -in four-dimensional space to have sixteen four-dimensional -bodies-bodies which in four-dimensional space -correspond to cubes in three-dimensional space, and these -bodies we call tesseracts.</p> - -<div class="figleft illowp30" id="fig_98" style="max-width: 15.625em;"> - <img src="images/fig_98.png" alt="" /> - <div class="caption">Fig. 98.</div> -</div> - -<p>Given then the null, white, red, yellow cubes, and -those which make up the block, we -notice that we represent perfectly -well the extension in three directions -(fig. 98). From the null point of -the null cube, travelling one inch, we -come to the white cube; travelling -one inch away we come to the yellow -cube; travelling one inch up we come -to the red cube. Now, if there is -a fourth dimension, then travelling -from the same null point for one -inch in that direction, we must come to the body lying -beyond the null region.</p> - -<p>I say null region, not cube; for with the introduction -of the fourth dimension each of our cubes must become -something different from cubes. If they are to have -existence in the fourth dimension, they must be “filled -up from” in this fourth dimension.</p> - -<p>Now we will assume that as we get a transference from -null to white going in one way, from null to yellow going -in another, so going from null in the fourth direction we -have a transference from null to blue, using thus the<span class="pagenum" id="Page_162">[Pg 162]</span> -colours white, yellow, red, blue, to denote transferences in -each of the four directions—right, away, up, unknown or -fourth dimension.</p> - -<div class="figleft illowp60" id="fig_99" style="max-width: 25em;"> - <img src="images/fig_99.png" alt="" /> - <div class="caption">Fig. 99.<br /> -A plane being’s representation of a block -of eight cubes by two sets of four squares.</div> -</div> - -<p>Hence, as the plane being must represent the solid regions, -he would come to by going right, as four squares lying -in some position in -his plane, arbitrarily -chosen, side by side -with his original four -squares, so we must -represent those eight -four-dimensional regions, -which we -should come to by -going in the fourth -dimension from each -of our eight cubes, by eight cubes placed in some arbitrary -position relative to our first eight cubes.</p> - -<div class="figcenter illowp80" id="fig_100" style="max-width: 50em;"> - <img src="images/fig_100.png" alt="" /> - <div class="caption">Fig. 100.</div> -</div> - -<p>Our representation of a block of sixteen tesseracts by -two blocks of eight cubes.<a id="FNanchor_3" href="#Footnote_3" class="fnanchor">[3]</a></p> - - -<div class="footnotes"><div class="footnote"> - -<p><a id="Footnote_3" href="#FNanchor_3" class="label">[3]</a> The eight cubes used here in 2 can be found in the second of the -model blocks. They can be taken out and used.</p> - -</div></div> - -<p>Hence, of the two sets of eight cubes, each one will serve<span class="pagenum" id="Page_163">[Pg 163]</span> -us as a representation of one of the sixteen tesseracts -which form one single block in four-dimensional space. -Each cube, as we have it, is a tray, as it were, against -which the real four-dimensional figure rests—just as each -of the squares which the plane being has is a tray, so to -speak, against which the cube it represents could rest.</p> - -<p>If we suppose the cubes to be one inch each way, then -the original eight cubes will give eight tesseracts of the -same colours, or the cubes, extending each one inch in the -fourth dimension.</p> - -<p>But after these there come, going on in the fourth dimension, -eight other bodies, eight other tesseracts. These -must be there, if we suppose the four-dimensional body -we make up to have two divisions, one inch each in each -of four directions.</p> - -<p>The colour we choose to designate the transference to -this second region in the fourth dimension is blue. Thus, -starting from the null cube and going in the fourth -dimension, we first go through one inch of the null -tesseract, then we come to a blue cube, which is the -beginning of a blue tesseract. This blue tesseract stretches -one inch farther on in the fourth dimension.</p> - -<p>Thus, beyond each of the eight tesseracts, which are of -the same colour as the cubes which are their bases, lie -eight tesseracts whose colours are derived from the colours -of the first eight by adding blue. Thus—</p> - -<table class="standard" summary=""> -<tr> -<td class="tdl">Null</td> -<td class="tdc">gives</td> -<td class="tdl">blue</td> -</tr> -<tr> -<td class="tdl">Yellow</td> -<td class="tdc">”</td> -<td class="tdl">green</td> -</tr> -<tr> -<td class="tdl">Red</td> -<td class="tdc">”</td> -<td class="tdl">purple</td> -</tr> -<tr> -<td class="tdl">Orange</td> -<td class="tdc">”</td> -<td class="tdl">brown</td> -</tr> -<tr> -<td class="tdl">White</td> -<td class="tdc">”</td> -<td class="tdl">light blue</td> -</tr> -<tr> -<td class="tdl">Pink</td> -<td class="tdc">”</td> -<td class="tdl">light purple</td> -</tr> -<tr> -<td class="tdl">Light yellow</td> -<td class="tdc">”</td> -<td class="tdl">light green</td> -</tr> -<tr> -<td class="tdl">Ochre</td> -<td class="tdc">”</td> -<td class="tdl">light brown</td> -</tr> -</table> - -<p>The addition of blue to yellow gives green—this is a<span class="pagenum" id="Page_164">[Pg 164]</span> -natural supposition to make. It is also natural to suppose -that blue added to red makes purple. Orange and blue -can be made to give a brown, by using certain shades and -proportions. And ochre and blue can be made to give a -light brown.</p> - -<p>But the scheme of colours is merely used for getting -a definite and realisable set of names and distinctions -visible to the eye. Their naturalness is apparent to any -one in the habit of using colours, and may be assumed to -be justifiable, as the sole purpose is to devise a set of -names which are easy to remember, and which will give -us a set of colours by which diagrams may be made easy -of comprehension. No scientific classification of colours -has been attempted.</p> - -<p>Starting, then, with these sixteen colour names, we have -a catalogue of the sixteen tesseracts, which form a four-dimensional -block analogous to the cubic block. But -the cube which we can put in space and look at is not one -of the constituent tesseracts; it is merely the beginning, -the solid face, the side, the aspect, of a tesseract.</p> - -<p>We will now proceed to derive a name for each region, -point, edge, plane face, solid and a face of the tesseract.</p> - -<p>The system will be clear, if we look at a representation -in the plane of a tesseract with three, and one with four -divisions in its side.</p> - -<p>The tesseract made up of three tesseracts each way -corresponds to the cube made up of three cubes each way, -and will give us a complete nomenclature.</p> - -<p>In this diagram, <a href="#fig_101">fig. 101</a>, 1 represents a cube of 27 -cubes, each of which is the beginning of a tesseract. -These cubes are represented simply by their lowest squares, -the solid content must be understood. 2 represents the -27 cubes which are the beginnings of the 27 tesseracts -one inch on in the fourth dimension. These tesseracts -are represented as a block of cubes put side by side with<span class="pagenum" id="Page_165">[Pg 165]</span> -the first block, but in their proper positions they could -not be in space with the first set. 3 represents 27 cubes -(forming a larger cube) which are the beginnings of the -tesseracts, which begin two inches in the fourth direction -from our space and continue another inch.</p> - -<div class="figcenter illowp100" id="fig_101" style="max-width: 62.5em;"> - <img src="images/fig_101.png" alt="" /> - <div class="caption">Fig. 101.<br /> - - -<table class="standard" summary=""> -<col width="30%" /> <col width="30%" /> <col width="30%" /> -<tr> -<td class="tdc">1</td> -<td class="tdc">2</td> -<td class="tdc">3</td> -</tr> -<tr> -<td class="tdc">Each cube is the -beginning of the first -tesseract going in the -fourth dimension. -</td> -<td class="tdc">Each cube is the -beginning of the -second tesseract. -</td> -<td class="tdc">Each cube is the -beginning of the -third tesseract. -</td> -</tr> -</table></div> -</div> - - -<p><span class="pagenum" id="Page_166">[Pg 166]</span></p> - - -<div class="figcenter illowp100" id="fig_102" style="max-width: 62.5em;"> - <img src="images/fig_102.png" alt="" /> - <div class="caption">Fig. 102.<a id="FNanchor_4" href="#Footnote_4" class="fnanchor">[4]</a></div> -<table class="standard" summary=""> -<col width="25%" /> <col width="25%" /> <col width="25%" /> <col width="25%" /> -<tr> -<td class="tdc">1</td> -<td class="tdc">2</td> -<td class="tdc">3</td> -<td class="tdc">4</td> -</tr> -<tr> -<td class="tdl">A cube of 64 cubes -each 1. in × 1 in., the beginning of a tesseract. -</td> -<td class="tdl">A cube of 64 cubes, -each 1 in. × 1 in. × 1 in. the beginning -of tesseracts 1 in. from our space -in the 4th dimension. -</td> -<td class="tdl">A cube of 64 cubes, -each 1 in. × 1 in. × 1 in. the beginning -of tesseracts 2 in. from our space -in the 4th dimension. -</td> -<td class="tdl">A cube of 64 cubes, -each 1 in. × 1 in. × 1 in. the beginning -of tesseracts 3 in. from our space -in the 4th dimension. -</td> -</tr> -</table></div> - - -<div class="footnotes"> -<div class="footnote"> - -<p><a id="Footnote_4" href="#FNanchor_4" class="label">[4]</a> The coloured plate, figs. 1, 2, 3, shows these relations more -conspicuously.</p> - -</div></div> - - -<p>In <a href="#fig_102">fig. 102</a>, we have the representation of a block of -4 × 4 × 4 × 4 or 256 tesseracts. They are given in -four consecutive sections, each supposed to be taken one -inch apart in the fourth dimension, and so giving four<span class="pagenum" id="Page_167">[Pg 167]</span> -blocks of cubes, 64 in each block. Here we see, comparing -it with the figure of 81 tesseracts, that the number -of the different regions show a different tendency of -increase. By taking five blocks of five divisions each way -this would become even more clear.</p> - -<p>We see, <a href="#fig_102">fig. 102</a>, that starting from the point at any -corner, the white coloured regions only extend out in -a line. The same is true for the yellow, red, and blue. -With regard to the latter it should be noticed that the -line of blues does not consist in regions next to each -other in the drawing, but in portions which come in in -different cubes. The portions which lie next to one -another in the fourth dimension must always be represented -so, when we have a three-dimensional representation. -Again, those regions such as the pink one, go on increasing -in two dimensions. About the pink region this is seen -without going out of the cube itself, the pink regions -increase in length and height, but in no other dimension. -In examining these regions it is sufficient to take one as -a sample.</p> - -<p>The purple increases in the same manner, for it comes -in in a succession from below to above in block 2, and in -a succession from block to block in 2 and 3. Now, a -succession from below to above represents a continuous -extension upwards, and a succession from block to block -represents a continuous extension in the fourth dimension. -Thus the purple regions increase in two dimensions, the -upward and the fourth, so when we take a very great -many divisions, and let each become very small, the -purple region forms a two-dimensional extension.</p> - -<p>In the same way, looking at the regions marked l. b. or -light blue, which starts nearest a corner, we see that the -tesseracts occupying it increase in length from left to -right, forming a line, and that there are as many lines of -light blue tesseracts as there are sections between the<span class="pagenum" id="Page_168">[Pg 168]</span> -first and last section. Hence the light blue tesseracts -increase in number in two ways—in the right and left, -and in the fourth dimension. They ultimately form -what we may call a plane surface.</p> - -<p>Now all those regions which contain a mixture of two -simple colours, white, yellow, red, blue, increase in two -ways. On the other hand, those which contain a mixture -of three colours increase in three ways. Take, for instance, -the ochre region; this has three colours, white, yellow, -red; and in the cube itself it increases in three ways.</p> - -<p>Now regard the orange region; if we add blue to this -we get a brown. The region of the brown tesseracts -extends in two ways on the left of the second block, -No. 2 in the figure. It extends also from left to right in -succession from one section to another, from section 2 -to section 3 in our figure.</p> - -<p>Hence the brown tesseracts increase in number in three -dimensions upwards, to and fro, fourth dimension. Hence -they form a cubic, a three-dimensional region; this region -extends up and down, near and far, and in the fourth -direction, but is thin in the direction from left to right. -It is a cube which, when the complete tesseract is represented -in our space, appears as a series of faces on the -successive cubic sections of the tesseract. Compare fig. -103 in which the middle block, 2, stands as representing a -great number of sections intermediate between 1 and 3.</p> - -<p>In a similar way from the pink region by addition of -blue we have the light purple region, which can be seen -to increase in three ways as the number of divisions -becomes greater. The three ways in which this region of -tesseracts extends is up and down, right and left, fourth -dimension. Finally, therefore, it forms a cubic mass of -very small tesseracts, and when the tesseract is given in -space sections it appears on the faces containing the -upward and the right and left dimensions.</p> - -<p><span class="pagenum" id="Page_169">[Pg 169]</span></p> - -<p>We get then altogether, as three-dimensional regions, -ochre, brown, light purple, light green.</p> - -<p>Finally, there is the region which corresponds to a -mixture of all the colours; there is only one region such -as this. It is the one that springs from ochre by the -addition of blue—this colour we call light brown.</p> - -<p>Looking at the light brown region we see that it -increases in four ways. Hence, the tesseracts of which it -is composed increase in number in each of four dimensions, -and the shape they form does not remain thin in -any of the four dimensions. Consequently this region -becomes the solid content of the block of tesseracts, itself; -it is the real four-dimensional solid. All the other regions -are then boundaries of this light brown region. If we -suppose the process of increasing the number of tesseracts -and diminishing their size carried on indefinitely, then -the light brown coloured tesseracts become the whole -interior mass, the three-coloured tesseracts become three-dimensional -boundaries, thin in one dimension, and form -the ochre, the brown, the light purple, the light green. -The two-coloured tesseracts become two-dimensional -boundaries, thin in two dimensions, <i>e.g.</i>, the pink, the -green, the purple, the orange, the light blue, the light -yellow. The one-coloured tesseracts become bounding -lines, thin in three dimensions, and the null points become -bounding corners, thin in four dimensions. From these -thin real boundaries we can pass in thought to the -abstractions—points, lines, faces, solids—bounding the -four-dimensional solid, which in this case is light brown -coloured, and under this supposition the light brown -coloured region is the only real one, is the only one which -is not an abstraction.</p> - -<p>It should be observed that, in taking a square as the -representation of a cube on a plane, we only represent -one face, or the section between two faces. The squares,<span class="pagenum" id="Page_170">[Pg 170]</span> -as drawn by a plane being, are not the cubes themselves, -but represent the faces or the sections of a cube. Thus -in the plane being’s diagram a cube of twenty-seven cubes -“null” represents a cube, but is really, in the normal -position, the orange square of a null cube, and may be -called null, orange square.</p> - -<p>A plane being would save himself confusion if he named -his representative squares, not by using the names of the -cubes simply, but by adding to the names of the cubes a -word to show what part of a cube his representative square -was.</p> - -<p>Thus a cube null standing against his plane touches it -by null orange face, passing through his plane it has in -the plane a square as trace, which is null white section, if -we use the phrase white section to mean a section drawn -perpendicular to the white line. In the same way the -cubes which we take as representative of the tesseract are -not the tesseract itself, but definite faces or sections of it. -In the preceding figures we should say then, not null, but -“null tesseract ochre cube,” because the cube we actually -have is the one determined by the three axes, white, red, -yellow.</p> - -<p>There is another way in which we can regard the colour -nomenclature of the boundaries of a tesseract.</p> - -<p>Consider a null point to move tracing out a white line -one inch in length, and terminating in a null point, -see <a href="#fig_103">fig. 103</a> or in the coloured plate.</p> - -<p>Then consider this white line with its terminal points -itself to move in a second dimension, each of the points -traces out a line, the line itself traces out an area, and -gives two lines as well, its initial and its final position.</p> - -<p>Thus, if we call “a region” any element of the figure, -such as a point, or a line, etc., every “region” in moving -traces out a new kind of region, “a higher region,” and -gives two regions of its own kind, an initial and a final<span class="pagenum" id="Page_171">[Pg 171]</span> -position. The “higher region” means a region with -another dimension in it.</p> - -<p>Now the square can move and generate a cube. The -square light yellow moves and traces out the mass of the -cube. Letting the addition of red denote the region -made by the motion in the upward direction we get an -ochre solid. The light yellow face in its initial and -terminal positions give the two square boundaries of the -cube above and below. Then each of the four lines of the -light yellow square—white, yellow, and the white, yellow -opposite them—trace out a bounding square. So there -are in all six bounding squares, four of these squares being -designated in colour by adding red to the colour of the -generating lines. Finally, each point moving in the up -direction gives rise to a line coloured null + red, or red, -and then there are the initial and terminal positions of the -points giving eight points. The number of the lines is -evidently twelve, for the four lines of this light yellow -square give four lines in their initial, four lines in their -final position, while the four points trace out four lines, -that is altogether twelve lines.</p> - -<p>Now the squares are each of them separate boundaries -of the cube, while the lines belong, each of them, to two -squares, thus the red line is that which is common to the -orange and pink squares.</p> - -<p>Now suppose that there is a direction, the fourth -dimension, which is perpendicular alike to every one -of the space dimensions already used—a dimension -perpendicular, for instance, to up and to right hand, -so that the pink square moving in this direction traces -out a cube.</p> - -<p>A dimension, moreover, perpendicular to the up and -away directions, so that the orange square moving in this -direction also traces out a cube, and the light yellow -square, too, moving in this direction traces out a cube.<span class="pagenum" id="Page_172">[Pg 172]</span> -Under this supposition, the whole cube moving in the -unknown dimension, traces out something new—a new -kind of volume, a higher volume. This higher volume -is a four-dimensional volume, and we designate it in colour -by adding blue to the colour of that which by moving -generates it.</p> - -<p>It is generated by the motion of the ochre solid, and -hence it is of the colour we call light brown (white, yellow, -red, blue, mixed together). It is represented by a number -of sections like 2 in <a href="#fig_103">fig. 103</a>.</p> - -<p>Now this light brown higher solid has for boundaries: -first, the ochre cube in its initial position, second, the -same cube in its final position, 1 and 3, <a href="#fig_103">fig. 103</a>. Each -of the squares which bound the cube, moreover, by movement -in this new direction traces out a cube, so we have -from the front pink faces of the cube, third, a pink blue or -light purple cube, shown as a light purple face on cube 2 -in <a href="#fig_103">fig. 103</a>, this cube standing for any number of intermediate -sections; fourth, a similar cube from the opposite -pink face; fifth, a cube traced out by the orange face—this -is coloured brown and is represented by the brown -face of the section cube in <a href="#fig_103">fig. 103</a>; sixth, a corresponding -brown cube on the right hand; seventh, a cube -starting from the light yellow square below; the unknown -dimension is at right angles to this also. This cube is -coloured light yellow and blue or light green; and, -finally, eighth, a corresponding cube from the upper -light yellow face, shown as the light green square at the -top of the section cube.</p> - -<p>The tesseract has thus eight cubic boundaries. These -completely enclose it, so that it would be invisible to a -four-dimensional being. Now, as to the other boundaries, -just as the cube has squares, lines, points, as boundaries, -so the tesseract has cubes, squares, lines, points, as -boundaries.</p> - -<p><span class="pagenum" id="Page_173">[Pg 173]</span></p> - -<p>The number of squares is found thus—round the cube -are six squares, these will give six squares in their initial -and six in their final positions. Then each of the twelve -lines of the cube trace out a square in the motion in -the fourth dimension. Hence there will be altogether -12 + 12 = 24 squares.</p> - -<p>If we look at any one of these squares we see that it -is the meeting surface of two of the cubic sides. Thus, -the red line by its movement in the fourth dimension, -traces out a purple square—this is common to two -cubes, one of which is traced out by the pink square -moving in the fourth dimension, and the other is -traced out by the orange square moving in the same -way. To take another square, the light yellow one, this -is common to the ochre cube and the light green cube. -The ochre cube comes from the light yellow square -by moving it in the up direction, the light green cube -is made from the light yellow square by moving it in -the fourth dimension. The number of lines is thirty-two, -for the twelve lines of the cube give twelve lines -of the tesseract in their initial position, and twelve in -their final position, making twenty-four, while each of -the eight points traces out a line, thus forming thirty-two -lines altogether.</p> - -<p>The lines are each of them common to three cubes, or -to three square faces; take, for instance, the red line. -This is common to the orange face, the pink face, and -that face which is formed by moving the red line in the -sixth dimension, namely, the purple face. It is also -common to the ochre cube, the pale purple cube, and the -brown cube.</p> - -<p>The points are common to six square faces and to four -cubes; thus, the null point from which we start is common -to the three square faces—pink, light yellow, orange, and -to the three square faces made by moving the three lines<span class="pagenum" id="Page_174">[Pg 174]</span> -white, yellow, red, in the fourth dimension, namely, the -light blue, the light green, the purple faces—that is, to -six faces in all. The four cubes which meet in it are the -ochre cube, the light purple cube, the brown cube, and -the light green cube.</p> - -<div class="figcenter illowp100" id="fig_103" style="max-width: 62.5em;"> - <img src="images/fig_103.png" alt="" /> - <div class="caption">Fig. 103.</div> -</div> - - -<p>The tesseract, red, white, yellow axes in space. In the lower line the three rear faces -are shown, the interior being removed.]</p> - -<p><span class="pagenum" id="Page_175">[Pg 175]</span></p> - -<div class="figcenter illowp100" id="fig_104" style="max-width: 62.5em;"> - <img src="images/fig_104.png" alt="" /> - <div class="caption">Fig. 104.<br /> -The tesseract, red, yellow, blue axes in space, -the blue axis running to the left, -opposite faces are coloured identically.</div> -</div> - -<p>A complete view of the tesseract in its various space -presentations is given in the following figures or catalogue -cubes, figs. 103-106. The first cube in each figure<span class="pagenum" id="Page_176">[Pg 176]</span> -represents the view of a tesseract coloured as described as -it begins to pass transverse to our space. The intermediate -figure represents a sectional view when it is partly through, -and the final figure represents the far end as it is just -passing out. These figures will be explained in detail in -the next chapter.</p> - -<div class="figcenter illowp100" id="fig_105" style="max-width: 62.5em;"> - <img src="images/fig_105.png" alt="" /> - <div class="caption">Fig. 105.<br /> -The tesseract, with red, white, blue axes in space. Opposite faces are coloured identically.</div> -</div> - -<p><span class="pagenum" id="Page_177">[Pg 177]</span></p> - -<div class="figcenter illowp100" id="fig_106" style="max-width: 62.5em;"> - <img src="images/fig_106.png" alt="" /> - <div class="caption">Fig. 106.<br /> -The tesseract, with blue, white, yellow axes in space. The blue axis runs downward -from the base of the ochre cube as it stands originally. Opposite faces are coloured -identically.</div> -</div> - -<p>We have thus obtained a nomenclature for each of the -regions of a tesseract; we can speak of any one of the -eight bounding cubes, the twenty square faces, the thirty-two -lines, the sixteen points.</p> -<hr class="chap x-ebookmaker-drop" /> - -<div class="chapter"> -<p><span class="pagenum" id="Page_178">[Pg 178]</span></p> - -<h2 class="nobreak" id="CHAPTER_XIII">CHAPTER XIII<br /> - -<small><i>REMARKS ON THE FIGURES</i></small></h2></div> - - -<p>An inspection of above figures will give an answer to -many questions about the tesseract. If we have a -tesseract one inch each way, then it can be represented -as a cube—a cube having white, yellow, red axes, and -from this cube as a beginning, a volume extending into -the fourth dimension. Now suppose the tesseract to pass -transverse to our space, the cube of the red, yellow, white -axes disappears at once, it is indefinitely thin in the -fourth dimension. Its place is occupied by those parts -of the tesseract which lie further away from our space -in the fourth dimension. Each one of these sections -will last only for one moment, but the whole of them -will take up some appreciable time in passing. If we -take the rate of one inch a minute the sections will take -the whole of the minute in their passage across our -space, they will take the whole of the minute except the -moment which the beginning cube and the end cube -occupy in their crossing our space. In each one of the -cubes, the section cubes, we can draw lines in all directions -except in the direction occupied by the blue line, the -fourth dimension; lines in that direction are represented -by the transition from one section cube to another. Thus -to give ourselves an adequate representation of the -tesseract we ought to have a limitless number of section -cubes intermediate between the first bounding cube, the<span class="pagenum" id="Page_179">[Pg 179]</span> -ochre cube, and the last bounding cube, the other ochre -cube. Practically three intermediate sectional cubes will -be found sufficient for most purposes. We will take then -a series of five figures—two terminal cubes, and three -intermediate sections—and show how the different regions -appear in our space when we take each set of three out -of the four axes of the tesseract as lying in our space.</p> - -<p>In <a href="#fig_107">fig. 107</a> initial letters are used for the colours. -A reference to <a href="#fig_103">fig. 103</a> will show the complete nomenclature, -which is merely indicated here.</p> - -<div class="figcenter illowp100" id="fig_107" style="max-width: 62.5em;"> - <img src="images/fig_107.png" alt="" /> - <div class="caption">Fig. 107.</div> -</div> - -<p>In this figure the tesseract is shown in five stages -distant from our space: first, zero; second, 1/4 in.; third, -2/4 in.; fourth, 3/4 in.; fifth, 1 in.; which are called <i>b</i>0, <i>b</i>1, -<i>b</i>2, <i>b</i>3, <i>b</i>4, because they are sections taken at distances -0, 1, 2, 3, 4 quarter inches along the blue line. All the -regions can be named from the first cube, the <i>b</i>0 cube, -as before, simply by remembering that transference along -the b axis gives the addition of blue to the colour of -the region in the ochre, the <i>b</i>0 cube. In the final cube -<i>b</i>4, the colouring of the original <i>b</i>0 cube is repeated. -Thus the red line moved along the blue axis gives a red -and blue or purple square. This purple square appears -as the three purple lines in the sections <i>b</i>1, <i>b</i>2, <i>b</i>3, taken -at 1/4, 2/4, 3/4 of an inch in the fourth dimension. If the -tesseract moves transverse to our space we have then in -this particular region, first of all a red line which lasts -for a moment, secondly a purple line which takes its<span class="pagenum" id="Page_180">[Pg 180]</span> -place. This purple line lasts for a minute—that is, all -of a minute, except the moment taken by the crossing -our space of the initial and final red line. The purple -line having lasted for this period is succeeded by a red -line, which lasts for a moment; then this goes and the -tesseract has passed across our space. The final red line -we call red bl., because it is separated from the initial -red line by a distance along the axis for which we use -the colour blue. Thus a line that lasts represents an -area duration; is in this mode of presentation equivalent -to a dimension of space. In the same way the white -line, during the crossing our space by the tesseract, is -succeeded by a light blue line which lasts for the inside -of a minute, and as the tesseract leaves our space, having -crossed it, the white bl. line appears as the final -termination.</p> - -<p>Take now the pink face. Moved in the blue direction -it traces out a light purple cube. This light purple -cube is shown in sections in <i>b</i><sub>1</sub>, <i>b</i><sub>2</sub>, <i>b</i><sub>3</sub>, and the farther -face of this cube in the blue direction is shown in <i>b</i><sub>4</sub>—a -pink face, called pink <i>b</i> because it is distant from the -pink face we began with in the blue direction. Thus -the cube which we colour light purple appears as a lasting -square. The square face itself, the pink face, vanishes -instantly the tesseract begins to move, but the light -purple cube appears as a lasting square. Here also -duration is the equivalent of a dimension of space—a -lasting square is a cube. It is useful to connect these -diagrams with the views given in the coloured plate.</p> - -<p>Take again the orange face, that determined by the -red and yellow axes; from it goes a brown cube in the -blue direction, for red and yellow and blue are supposed -to make brown. This brown cube is shown in three -sections in the faces <i>b</i><sub>1</sub>, <i>b</i><sub>2</sub>, <i>b</i><sub>3</sub>. In <i>b</i><sub>4</sub> is the opposite -orange face of the brown cube, the face called orange <i>b</i>,<span class="pagenum" id="Page_181">[Pg 181]</span> -for it is distant in the blue direction from the orange -face. As the tesseract passes transverse to our space, -we have then in this region an instantly vanishing orange -square, followed by a lasting brown square, and finally -an orange face which vanishes instantly.</p> - -<p>Now, as any three axes will be in our space, let us send -the white axis out into the unknown, the fourth dimension, -and take the blue axis into our known space -dimension. Since the white and blue axes are perpendicular -to each other, if the white axis goes out into -the fourth dimension in the positive sense, the blue axis -will come into the direction the white axis occupied, -in the negative sense.</p> - -<div class="figcenter illowp100" id="fig_108" style="max-width: 62.5em;"> - <img src="images/fig_108.png" alt="" /> - <div class="caption">Fig. 108.</div> -</div> - -<p>Hence, not to complicate matters by having to think -of two senses in the unknown direction, let us send the -white line into the positive sense of the fourth dimension, -and take the blue one as running in the negative -sense of that direction which the white line has left; -let the blue line, that is, run to the left. We have -now the row of figures in <a href="#fig_108">fig. 108</a>. The dotted cube -shows where we had a cube when the white line ran -in our space—now it has turned out of our space, and -another solid boundary, another cubic face of the tesseract -comes into our space. This cube has red and yellow -axes as before; but now, instead of a white axis running -to the right, there is a blue axis running to the left. -Here we can distinguish the regions by colours in a perfectly -systematic way. The red line traces out a purple<span class="pagenum" id="Page_182">[Pg 182]</span> -square in the transference along the blue axis by which -this cube is generated from the orange face. This -purple square made by the motion of the red line is -the same purple face that we saw before as a series of -lines in the sections <i>b</i><sub>1</sub>, <i>b</i><sub>2</sub>, <i>b</i><sub>3</sub>. Here, since both red and -blue axes are in our space, we have no need of duration -to represent the area they determine. In the motion -of the tesseract across space this purple face would -instantly disappear.</p> - -<p>From the orange face, which is common to the initial -cubes in <a href="#fig_107">fig. 107</a> and <a href="#fig_108">fig. 108</a>, there goes in the blue -direction a cube coloured brown. This brown cube is -now all in our space, because each of its three axes run -in space directions, up, away, to the left. It is the same -brown cube which appeared as the successive faces on the -sections <i>b</i><sub>1</sub>, <i>b</i><sub>2</sub>, <i>b</i><sub>3</sub>. Having all its three axes in our -space, it is given in extension; no part of it needs to -be represented as a succession. The tesseract is now -in a new position with regard to our space, and when -it moves across our space the brown cube instantly -disappears.</p> - -<p>In order to exhibit the other regions of the tesseract -we must remember that now the white line runs in the -unknown dimension. Where shall we put the sections -at distances along the line? Any arbitrary position in -our space will do: there is no way by which we can -represent their real position.</p> - -<p>However, as the brown cube comes off from the orange -face to the left, let us put these successive sections to -the left. We can call them <i>wh</i><sub>0</sub>, <i>wh</i><sub>1</sub>, <i>wh</i><sub>2</sub>, <i>wh</i><sub>3</sub>, <i>wh</i><sub>4</sub>, -because they are sections along the white axis, which -now runs in the unknown dimension.</p> - -<p>Running from the purple square in the white direction -we find the light purple cube. This is represented in the -<span class="pagenum" id="Page_183">[Pg 183]</span>sections <i>wh</i><sub>1</sub>, <i>wh</i><sub>2</sub>, <i>wh</i><sub>3</sub>, <i>wh</i><sub>4</sub>, <a href="#fig_108">fig. 108</a>. It is the same cube -that is represented in the sections <i>b</i><sub>1</sub>, <i>b</i><sub>2</sub>, <i>b</i><sub>3</sub>: in <a href="#fig_107">fig. 107</a> -the red and white axes are in our space, the blue out of -it; in the other case, the red and blue are in our space, -the white out of it. It is evident that the face pink <i>y</i>, -opposite the pink face in <a href="#fig_107">fig. 107</a>, makes a cube shown -in squares in <i>b</i><sub>1</sub>, <i>b</i><sub>2</sub>, <i>b</i><sub>3</sub>, <i>b</i><sub>4</sub>, on the opposite side to the <i>l</i> -purple squares. Also the light yellow face at the base -of the cube <i>b</i><sub>0</sub>, makes a light green cube, shown as a series -of base squares.</p> - -<p>The same light green cube can be found in <a href="#fig_107">fig. 107</a>. -The base square in <i>wh</i><sub>0</sub> is a green square, for it is enclosed -by blue and yellow axes. From it goes a cube in the -white direction, this is then a light green cube and the -same as the one just mentioned as existing in the sections -<i>b</i><sub>0</sub>, <i>b</i><sub>1</sub>, <i>b</i><sub>2</sub>, <i>b</i><sub>3</sub>, <i>b</i><sub>4</sub>.</p> - -<p>The case is, however, a little different with the brown -cube. This cube we have altogether in space in the -section <i>wh</i><sub>0</sub>, <a href="#fig_108">fig. 108</a>, while it exists as a series of squares, -the left-hand ones, in the sections <i>b</i><sub>0</sub>, <i>b</i><sub>1</sub>, <i>b</i><sub>2</sub>, <i>b</i><sub>3</sub>, <i>b</i><sub>4</sub>. The -brown cube exists as a solid in our space, as shown in -<a href="#fig_108">fig. 108</a>. In the mode of representation of the tesseract -exhibited in <a href="#fig_107">fig. 107</a>, the same brown cube appears as a -succession of squares. That is, as the tesseract moves -across space, the brown cube would actually be to us a -square—it would be merely the lasting boundary of another -solid. It would have no thickness at all, only extension -in two dimensions, and its duration would show its solidity -in three dimensions.</p> - -<p>It is obvious that, if there is a four-dimensional space, -matter in three dimensions only is a mere abstraction; all -material objects must then have a slight four-dimensional -thickness. In this case the above statement will undergo -modification. The material cube which is used as the -model of the boundary of a tesseract will have a slight -thickness in the fourth dimension, and when the cube is<span class="pagenum" id="Page_184">[Pg 184]</span> -presented to us in another aspect, it would not be a mere -surface. But it is most convenient to regard the cubes -we use as having no extension at all in the fourth -dimension. This consideration serves to bring out a point -alluded to before, that, if there is a fourth dimension, our -conception of a solid is the conception of a mere abstraction, -and our talking about real three-dimensional objects would -seem to a four-dimensional being as incorrect as a two-dimensional -being’s telling about real squares, real -triangles, etc., would seem to us.</p> - -<p>The consideration of the two views of the brown cube -shows that any section of a cube can be looked at by a -presentation of the cube in a different position in four-dimensional -space. The brown faces in <i>b</i><sub>1</sub>, <i>b</i><sub>2</sub>, <i>b</i><sub>3</sub>, are the -very same brown sections that would be obtained by -cutting the brown cube, <i>wh</i><sub>0</sub>, across at the right distances -along the blue line, as shown in <a href="#fig_108">fig. 108</a>. But as these -sections are placed in the brown cube, <i>wh</i><sub>0</sub>, they come -behind one another in the blue direction. Now, in the -sections <i>wh</i><sub>1</sub>, <i>wh</i><sub>2</sub>, <i>wh</i><sub>3</sub>, we are looking at these sections -from the white direction—the blue direction does not -exist in these figures. So we see them in a direction at -right angles to that in which they occur behind one -another in <i>wh</i><sub>0</sub>. There are intermediate views, which -would come in the rotation of a tesseract. These brown -squares can be looked at from directions intermediate -between the white and blue axes. It must be remembered -that the fourth dimension is perpendicular equally to all -three space axes. Hence we must take the combinations -of the blue axis, with each two of our three axes, white, -red, yellow, in turn.</p> - -<p>In <a href="#fig_109">fig. 109</a> we take red, white, and blue axes in space, -sending yellow into the fourth dimension. If it goes into -the positive sense of the fourth dimension the blue line -will come in the opposite direction to that in which the<span class="pagenum" id="Page_185">[Pg 185]</span> -yellow line ran before. Hence, the cube determined by -the white, red, blue axes, will start from the pink plane -and run towards us. The dotted cube shows where the -ochre cube was. When it is turned out of space, the cube -coming towards from its front face is the one which comes -into our space in this turning. Since the yellow line now -runs in the unknown dimension we call the sections -<i>y</i><sub>0</sub>, <i>y</i><sub>1</sub>, <i>y</i><sub>2</sub>, <i>y</i><sub>3</sub>, <i>y</i><sub>4</sub>, as they are made at distances 0, 1, 2, 3, 4, -quarter inches along the yellow line. We suppose these -cubes arranged in a line coming towards us—not that -that is any more natural than any other arbitrary series -of positions, but it agrees with the plan previously adopted.</p> - -<div class="figcenter illowp100" id="fig_109" style="max-width: 62.5em;"> - <img src="images/fig_109.png" alt="" /> - <div class="caption">Fig. 109.</div> -</div> - -<p>The interior of the first cube, <i>y</i><sub>0</sub>, is that derived from -pink by adding blue, or, as we call it, light purple. The -faces of the cube are light blue, purple, pink. As drawn, -we can only see the face nearest to us, which is not the -one from which the cube starts—but the face on the -opposite side has the same colour name as the face -towards us.</p> - -<p>The successive sections of the series, <i>y</i><sub>0</sub>, <i>y</i><sub>1</sub>, <i>y</i><sub>2</sub>, etc., can -be considered as derived from sections of the <i>b</i><sub>0</sub> cube -made at distances along the yellow axis. What is distant -a quarter inch from the pink face in the yellow direction? -This question is answered by taking a section from a point -a quarter inch along the yellow axis in the cube <i>b</i><sub>0</sub>, <a href="#fig_107">fig. 107</a>. -It is an ochre section with lines orange and light yellow. -This section will therefore take the place of the pink face<span class="pagenum" id="Page_186">[Pg 186]</span> -in <i>y</i><sub>1</sub> when we go on in the yellow direction. Thus, the -first section, <i>y</i><sub>1</sub>, will begin from an ochre face with light -yellow and orange lines. The colour of the axis which -lies in space towards us is blue, hence the regions of this -section-cube are determined in nomenclature, they will be -found in full in <a href="#fig_105">fig. 105</a>.</p> - -<p>There remains only one figure to be drawn, and that is -the one in which the red axis is replaced by the blue. -Here, as before, if the red axis goes out into the positive -sense of the fourth dimension, the blue line must come -into our space in the negative sense of the direction which -the red line has left. Accordingly, the first cube will -come in beneath the position of our ochre cube, the one -we have been in the habit of starting with.</p> - -<div class="figcenter illowp100" id="fig_110" style="max-width: 62.5em;"> - <img src="images/fig_110.png" alt="" /> - <div class="caption">Fig. 110.</div> -</div> - -<p>To show these figures we must suppose the ochre cube -to be on a movable stand. When the red line swings out -into the unknown dimension, and the blue line comes in -downwards, a cube appears below the place occupied by -the ochre cube. The dotted cube shows where the ochre -cube was. That cube has gone and a different cube runs -downwards from its base. This cube has white, yellow, -and blue axes. Its top is a light yellow square, and hence -its interior is light yellow + blue or light green. Its front -face is formed by the white line moving along the blue -axis, and is therefore light blue, the left-hand side is -formed by the yellow line moving along the blue axis, and -therefore green.</p> - -<p><span class="pagenum" id="Page_187">[Pg 187]</span></p> - -<p>As the red line now runs in the fourth dimension, the -successive sections can he called <i>r</i><sub>0</sub>, <i>r</i><sub>1</sub>, <i>r</i><sub>2</sub>, <i>r</i><sub>3</sub>, <i>r</i><sub>4</sub>, these -letters indicating that at distances 0, 1/4, 2/4, 3/4, 1 inch along -the red axis we take all of the tesseract that can be found -in a three-dimensional space, this three-dimensional space -extending not at all in the fourth dimension, but up and -down, right and left, far and near.</p> - -<p>We can see what should replace the light yellow face of -<i>r</i><sub>0</sub>, when the section <i>r</i><sub>1</sub> comes in, by looking at the cube -<i>b</i><sub>0</sub>, <a href="#fig_107">fig. 107</a>. What is distant in it one-quarter of an inch -from the light yellow face in the red direction? It is an -ochre section with orange and pink lines and red points; -see also <a href="#fig_103">fig. 103</a>.</p> - -<p>This square then forms the top square of <i>r</i><sub>1</sub>. Now we -can determine the nomenclature of all the regions of <i>r</i><sub>1</sub> by -considering what would be formed by the motion of this -square along a blue axis.</p> - -<p>But we can adopt another plan. Let us take a horizontal -section of <i>r</i><sub>0</sub>, and finding that section in the figures, -of <a href="#fig_107">fig. 107</a> or <a href="#fig_103">fig. 103</a>, from them determine what will -replace it, going on in the red direction.</p> - -<p>A section of the <i>r</i><sub>0</sub> cube has green, light blue, green, -light blue sides and blue points.</p> - -<p>Now this square occurs on the base of each of the -section figures, <i>b</i><sub>1</sub>, <i>b</i><sub>2</sub>, etc. In them we see that 1/4 inch in -the red direction from it lies a section with brown and -light purple lines and purple corners, the interior being -of light brown. Hence this is the nomenclature of the -section which in <i>r</i><sub>1</sub> replaces the section of <i>r</i><sub>0</sub> made from a -point along the blue axis.</p> - -<p>Hence the colouring as given can be derived.</p> - -<p>We have thus obtained a perfectly named group of -tesseracts. We can take a group of eighty-one of them -3 × 3 × 3 × 3, in four dimensions, and each tesseract will -have its name null, red, white, yellow, blue, etc., and<span class="pagenum" id="Page_188">[Pg 188]</span> -whatever cubic view we take of them we can say exactly -what sides of the tesseracts we are handling, and how -they touch each other.<a id="FNanchor_5" href="#Footnote_5" class="fnanchor">[5]</a></p> - -<div class="footnotes"> -<div class="footnote"> - -<p><a id="Footnote_5" href="#FNanchor_5" class="label">[5]</a> At this point the reader will find it advantageous, if he has the -models, to go through the manipulations described in the appendix.</p> - -</div></div> - -<p>Thus, for instance, if we have the sixteen tesseracts -shown below, we can ask how does null touch blue.</p> - -<div class="figcenter illowp100" id="fig_111" style="max-width: 62.5em;"> - <img src="images/fig_111.png" alt="" /> - <div class="caption">Fig. 111.</div> -</div> - -<p>In the arrangement given in <a href="#fig_111">fig. 111</a> we have the axes -white, red, yellow, in space, blue running in the fourth -dimension. Hence we have the ochre cubes as bases. -Imagine now the tesseractic group to pass transverse to -our space—we have first of all null ochre cube, white -ochre cube, etc.; these instantly vanish, and we get the -section shown in the middle cube in <a href="#fig_103">fig. 103</a>, and finally, -just when the tesseract block has moved one inch transverse -to our space, we have null ochre cube, and then -immediately afterwards the ochre cube of blue comes in. -Hence the tesseract null touches the tesseract blue by its -ochre cube, which is in contact, each and every point -of it, with the ochre cube of blue.</p> - -<p>How does null touch white, we may ask? Looking at -the beginning A, <a href="#fig_111">fig. 111</a>, where we have the ochre<span class="pagenum" id="Page_189">[Pg 189]</span> -cubes, we see that null ochre touches white ochre by an -orange face. Now let us generate the null and white -tesseracts by a motion in the blue direction of each of -these cubes. Each of them generates the corresponding -tesseract, and the plane of contact of the cubes generates -the cube by which the tesseracts are in contact. Now an -orange plane carried along a blue axis generates a brown -cube. Hence null touches white by a brown cube.</p> - -<div class="figcenter illowp100" id="fig_112" style="max-width: 62.5em;"> - <img src="images/fig_112.png" alt="" /> - <div class="caption">Fig. 112.</div> -</div> - -<p>If we ask again how red touches light blue tesseract, -let us rearrange our group, <a href="#fig_112">fig. 112</a>, or rather turn it -about so that we have a different space view of it; let -the red axis and the white axis run up and right, and let -the blue axis come in space towards us, then the yellow -axis runs in the fourth dimension. We have then two -blocks in which the bounding cubes of the tesseracts are -given, differently arranged with regard to us—the arrangement -is really the same, but it appears different to us. -Starting from the plane of the red and white axes we -have the four squares of the null, white, red, pink tesseracts -as shown in A, on the red, white plane, unaltered, only -from them now comes out towards us the blue axis.<span class="pagenum" id="Page_190">[Pg 190]</span> -Hence we have null, white, red, pink tesseracts in contact -with our space by their cubes which have the red, white, -blue axis in them, that is by the light purple cubes. -Following on these four tesseracts we have that which -comes next to them in the blue direction, that is the -four blue, light blue, purple, light purple. These are -likewise in contact with our space by their light purple -cubes, so we see a block as named in the figure, of which -each cube is the one determined by the red, white, blue, -axes.</p> - -<p>The yellow line now runs out of space; accordingly one -inch on in the fourth dimension we come to the tesseracts -which follow on the eight named in C, <a href="#fig_112">fig. 112</a>, in the -yellow direction.</p> - -<p>These are shown in C.y<sub>1</sub>, <a href="#fig_112">fig. 112</a>. Between figure C -and C.y<sub>1</sub> is that four-dimensional mass which is formed -by moving each of the cubes in C one inch in the fourth -dimension—that is, along a yellow axis; for the yellow -axis now runs in the fourth dimension.</p> - -<p>In the block C we observe that red (light purple -cube) touches light blue (light purple cube) by a point. -Now these two cubes moving together remain in contact -during the period in which they trace out the tesseracts -red and light blue. This motion is along the yellow -axis, consequently red and light blue touch by a yellow -line.</p> - -<p>We have seen that the pink face moved in a yellow -direction traces out a cube; moved in the blue direction it -also traces out a cube. Let us ask what the pink face -will trace out if it is moved in a direction within the -tesseract lying equally between the yellow and blue -directions. What section of the tesseract will it make?</p> - -<p>We will first consider the red line alone. Let us take -a cube with the red line in it and the yellow and blue -axes.</p> - -<p><span class="pagenum" id="Page_191">[Pg 191]</span></p> - -<div class="figleft illowp35" id="fig_113" style="max-width: 15.625em;"> - <img src="images/fig_113.png" alt="" /> - <div class="caption">Fig. 113.</div> -</div> - -<p>The cube with the yellow, red, blue axes is shown in -<a href="#fig_113">fig. 113</a>. If the red line is -moved equally in the yellow and -in the blue direction by four -equal motions of ¼ inch each, it -takes the positions 11, 22, 33, -and ends as a red line.</p> - -<p>Now, the whole of this red, -yellow, blue, or brown cube appears -as a series of faces on the -successive sections of the tesseract -starting from the ochre cube and letting the blue -axis run in the fourth dimension. Hence the plane -traced out by the red line appears as a series of lines in -the successive sections, in our ordinary way of representing -the tesseract; these lines are in different places in each -successive section.</p> - -<div class="figcenter illowp100" id="fig_114" style="max-width: 62.5em;"> - <img src="images/fig_114.png" alt="" /> - <div class="caption">Fig. 114.</div> -</div> - -<p>Thus drawing our initial cube and the successive -sections, calling them <i>b</i><sub>0</sub>, <i>b</i><sub>1</sub>, <i>b</i><sub>2</sub>, <i>b</i><sub>3</sub>, <i>b</i><sub>4</sub>, <a href="#fig_115">fig. 115</a>, we have -the red line subject to this movement appearing in the -positions indicated.</p> - -<p>We will now investigate what positions in the tesseract -another line in the pink face assumes when it is moved in -a similar manner.</p> - -<p>Take a section of the original cube containing a vertical -line, 4, in the pink plane, <a href="#fig_115">fig. 115</a>. We have, in the -section, the yellow direction, but not the blue.</p> - -<p><span class="pagenum" id="Page_192">[Pg 192]</span></p> - -<p>From this section a cube goes off in the fourth dimension, -which is formed by moving each point of the section -in the blue direction.</p> - -<div class="figleft illowp40" id="fig_115" style="max-width: 15.625em;"> - <img src="images/fig_115.png" alt="" /> - <div class="caption">Fig. 115.</div> -</div> - -<div class="figright illowp40" id="fig_116" style="max-width: 18.75em;"> - <img src="images/fig_116.png" alt="" /> - <div class="caption">Fig. 116.</div> -</div> - -<p>Drawing this cube we have <a href="#fig_116">fig. 116</a>.</p> - -<p>Now this cube occurs as a series of sections in our -original representation of the tesseract. Taking four steps -as before this cube appears as the sections drawn in <i>b</i><sub>0</sub>, <i>b</i><sub>1</sub>, -<i>b</i><sub>2</sub>, <i>b</i><sub>3</sub>, <i>b</i><sub>4</sub>, <a href="#fig_117">fig. 117</a>, and if the line 4 is subjected to a -movement equal in the blue and yellow directions, it will -occupy the positions designated by 4, 4<sub>1</sub>, 4<sub>2</sub>, 4<sub>3</sub>, 4<sub>4</sub>.</p> - -<div class="figcenter illowp100" id="fig_117" style="max-width: 62.5em;"> - <img src="images/fig_117.png" alt="" /> - <div class="caption">Fig. 117.</div> -</div> - -<p>Hence, reasoning in a similar manner about every line, -it is evident that, moved equally in the blue and yellow -directions, the pink plane will trace out a space which is -shown by the series of section planes represented in the -diagram.</p> - -<p>Thus the space traced out by the pink face, if it is -moved equally in the yellow and blue directions, is represented -by the set of planes delineated in Fig. 118, pink<span class="pagenum" id="Page_193">[Pg 193]</span> -face or 0, then 1, 2, 3, and finally pink face or 4. This -solid is a diagonal solid of the tesseract, running from a -pink face to a pink face. Its length is the length of the -diagonal of a square, its side is a square.</p> - -<p>Let us now consider the unlimited space which springs -from the pink face extended.</p> - -<p>This space, if it goes off in the yellow direction, gives -us in it the ochre cube of the tesseract. Thus, if we have -the pink face given and a point in the ochre cube, we -have determined this particular space.</p> - -<p>Similarly going off from the pink face in the blue -direction is another space, which gives us the light purple -cube of the tesseract in it. And any point being taken in -the light purple cube, this space going off from the pink -face is fixed.</p> - -<div class="figcenter illowp100" id="fig_118" style="max-width: 62.5em;"> - <img src="images/fig_118.png" alt="" /> - <div class="caption">Fig. 118.</div> -</div> - -<p>The space we are speaking of can be conceived as -swinging round the pink face, and in each of its positions -it cuts out a solid figure from the tesseract, one of which -we have seen represented in <a href="#fig_118">fig. 118</a>.</p> - -<p>Each of these solid figures is given by one position of -the swinging space, and by one only. Hence in each of -them, if one point is taken, the particular one of the -slanting spaces is fixed. Thus we see that given a plane -and a point out of it a space is determined.</p> - -<p>Now, two points determine a line.</p> - -<p>Again, think of a line and a point outside it. Imagine -a plane rotating round the line. At some time in its -rotation it passes through the point. Thus a line and a<span class="pagenum" id="Page_194">[Pg 194]</span> -point, or three points, determine a plane. And finally -four points determine a space. We have seen that a -plane and a point determine a space, and that three -points determine a plane; so four points will determine -a space.</p> - -<p>These four points may be any points, and we can take, -for instance, the four points at the extremities of the red, -white, yellow, blue axes, in the tesseract. These will -determine a space slanting with regard to the section -spaces we have been previously considering. This space -will cut the tesseract in a certain figure.</p> - -<p>One of the simplest sections of a cube by a plane is -that in which the plane passes through the extremities -of the three edges which meet in a point. We see at -once that this plane would cut the cube in a triangle, but -we will go through the process by which a plane being -would most conveniently treat the problem of the determination -of this shape, in order that we may apply the -method to the determination of the figure in which a -space cuts a tesseract when it passes through the 4 -points at unit distance from a corner.</p> - -<p>We know that two points determine a line, three points -determine a plane, and given any two points in a plane -the line between them lies wholly in the plane.</p> -<div class="figleft illowp40" id="fig_119" style="max-width: 18.75em;"> - <img src="images/fig_119.png" alt="" /> - <div class="caption">Fig. 119.</div> -</div> - -<p>Let now the plane being study the section made by -a plane passing through the -null <i>r</i>, null <i>wh</i>, and null <i>y</i> -points, <a href="#fig_119">fig. 119</a>. Looking at -the orange square, which, as -usual, we suppose to be initially -in his plane, he sees -that the line from null <i>r</i> to -null <i>y</i>, which is a line in the -section plane, the plane, namely, through the three -extremities of the edges meeting in null, cuts the orange<span class="pagenum" id="Page_195">[Pg 195]</span> -face in an orange line with null points. This then is one -of the boundaries of the section figure.</p> - -<p>Let now the cube be so turned that the pink face -comes in his plane. The points null <i>r</i> and null <i>wh</i> -are now visible. The line between them is pink -with null points, and since this line is common to -the surface of the cube and the cutting plane, it is -a boundary of the figure in which the plane cuts the -cube.</p> - -<p>Again, suppose the cube turned so that the light -yellow face is in contact with the plane being’s plane. -He sees two points, the null <i>wh</i> and the null <i>y</i>. The -line between these lies in the cutting plane. Hence, -since the three cutting lines meet and enclose a portion -of the cube between them, he has determined the -figure he sought. It is a triangle with orange, pink, -and light yellow sides, all equal, and enclosing an -ochre area.</p> - -<p>Let us now determine in what figure the space, -determined by the four points, null <i>r</i>, null <i>y</i>, null -<i>wh</i>, null <i>b</i>, cuts the tesseract. We can see three -of these points in the primary position of the tesseract -resting against our solid sheet by the ochre cube. -These three points determine a plane which lies in -the space we are considering, and this plane cuts -the ochre cube in a triangle, the interior of which -is ochre (<a href="#fig_119">fig. 119</a> will serve for this view), with pink, -light yellow and orange sides, and null points. Going -in the fourth direction, in one sense, from this plane -we pass into the tesseract, in the other sense we pass -away from it. The whole area inside the triangle is -common to the cutting plane we see, and a boundary -of the tesseract. Hence we conclude that the triangle -drawn is common to the tesseract and the cutting -space.</p> - -<p><span class="pagenum" id="Page_196">[Pg 196]</span></p> - -<div class="figleft illowp50" id="fig_120" style="max-width: 21.875em;"> - <img src="images/fig_120.png" alt="" /> - <div class="caption">Fig. 120.</div> -</div> - -<p>Now let the ochre cube turn out and the brown cube -come in. The dotted lines -show the position the ochre -cube has left (<a href="#fig_120">fig. 120</a>).</p> - -<p>Here we see three out -of the four points through -which the cutting plane -passes, null <i>r</i>, null <i>y</i>, and -null <i>b</i>. The plane they -determine lies in the cutting space, and this plane -cuts out of the brown cube a triangle with orange, -purple and green sides, and null points. The orange -line of this figure is the same as the orange line in -the last figure.</p> - -<p>Now let the light purple cube swing into our space, -towards us, <a href="#fig_121">fig. 121</a>.</p> - -<div class="figleft illowp40" id="fig_121" style="max-width: 21.875em;"> - <img src="images/fig_121.png" alt="" /> - <div class="caption">Fig. 121.</div> -</div> - -<p>The cutting space which passes through the four points, -null <i>r</i>, <i>y</i>, <i>wh</i>, <i>b</i>, passes through -the null <i>r</i>, <i>wh</i>, <i>b</i>, and therefore -the plane these determine -lies in the cutting space.</p> - -<p>This triangle lies before us. -It has a light purple interior -and pink, light blue, and -purple edges with null points.</p> - -<p>This, since it is all of the -plane that is common to it, and this bounding of the -tesseract, gives us one of the bounding faces of our sectional -figure. The pink line in it is the same as the -pink line we found in the first figure—that of the ochre -cube.</p> - -<p>Finally, let the tesseract swing about the light yellow -plane, so that the light green cube comes into our space. -It will point downwards.</p> - -<div class="figleft illowp40" id="fig_122" style="max-width: 21.875em;"> - <img src="images/fig_122.png" alt="" /> - <div class="caption">Fig. 122.</div> -</div> - -<p>The three points, <i>n.y</i>, <i>n.wh</i>, <i>n.b</i>, are in the cutting<span class="pagenum" id="Page_197">[Pg 197]</span> -space, and the triangle they determine is common to -the tesseract and the cutting -space. Hence this -boundary is a triangle having -a light yellow line, -which is the same as the -light yellow line of the first -figure, a light blue line and -a green line.</p> - -<p>We have now traced the -cutting space between every -set of three that can be -made out of the four points -in which it cuts the tesseract, and have got four faces -which all join on to each other by lines.</p> - -<div class="figleft illowp35" id="fig_123" style="max-width: 18.75em;"> - <img src="images/fig_123.png" alt="" /> - <div class="caption">Fig. 123.</div> -</div> - -<p>The triangles are shown in <a href="#fig_123">fig. 123</a> as they join on to -the triangle in the ochre cube. But -they join on each to the other in an -exactly similar manner; their edges -are all identical two and two. They -form a closed figure, a tetrahedron, -enclosing a light brown portion which -is the portion of the cutting space -which lies inside the tesseract.</p> - -<p>We cannot expect to see this light brown portion, any -more than a plane being could expect to see the inside -of a cube if an angle of it were pushed through his -plane. All he can do is to come upon the boundaries -of it in a different way to that in which he would if it -passed straight through his plane.</p> - -<p>Thus in this solid section; the whole interior lies perfectly -open in the fourth dimension. Go round it as -we may we are simply looking at the boundaries of the -tesseract which penetrates through our solid sheet. If -the tesseract were not to pass across so far, the triangle<span class="pagenum" id="Page_198">[Pg 198]</span> -would be smaller; if it were to pass farther, we should -have a different figure, the outlines of which can be -determined in a similar manner.</p> - -<p>The preceding method is open to the objection that -it depends rather on our inferring what must be, than -our seeing what is. Let us therefore consider our -sectional space as consisting of a number of planes, each -very close to the last, and observe what is to be found -in each plane.</p> - -<div class="figleft illowp40" id="fig_124" style="max-width: 21.875em;"> - <img src="images/fig_124.png" alt="" /> - <div class="caption">Fig. 124.</div> -</div> - -<p>The corresponding method in the case of two dimensions -is as follows:—The plane -being can see that line of the -sectional plane through null <i>y</i>, -null <i>wh</i>, null <i>r</i>, which lies in -the orange plane. Let him -now suppose the cube and the -section plane to pass half way -through his plane. Replacing -the red and yellow axes are lines parallel to them, sections -of the pink and light yellow faces.</p> - -<p>Where will the section plane cut these parallels to -the red and yellow axes?</p> - -<p>Let him suppose the cube, in the position of the -drawing, <a href="#fig_124">fig. 124</a>, turned so that the pink face lies -against his plane. He can see the line from the null <i>r</i> -point to the null <i>wh</i> point, and can see (compare <a href="#fig_119">fig. 119</a>) -that it cuts <span class="allsmcap">AB</span> a parallel to his red axis, drawn at a point -half way along the white line, in a point <span class="allsmcap">B</span>, half way up. -I shall speak of the axis as having the length of an edge -of the cube. Similarly, by letting the cube turn so that -the light yellow square swings against his plane, he can -see (compare <a href="#fig_119">fig. 119</a>) that a parallel to his yellow axis -drawn from a point half-way along the white axis, is cut -at half its length by the trace of the section plane in the -light yellow face.</p> - -<p><span class="pagenum" id="Page_199">[Pg 199]</span></p> - -<p>Hence when the cube had passed half-way through he -would have—instead of the orange line with null points, -which he had at first—an ochre line of half its length, -with pink and light yellow points. Thus, as the cube -passed slowly through his plane, he would have a succession -of lines gradually diminishing in length and -forming an equilateral triangle. The whole interior would -be ochre, the line from which it started would be orange. -The succession of points at the ends of the succeeding -lines would form pink and light yellow lines and the -final point would be null. Thus looking at the successive -lines in the section plane as it and the cube passed across -his plane he would determine the figure cut out bit -by bit.</p> - -<p>Coming now to the section of the tesseract, let us -imagine that the tesseract and its cutting <i>space</i> pass -slowly across our space; we can examine portions of it, -and their relation to portions of the cutting space. Take -the section space which passes through the four points, -null <i>r</i>, <i>wh</i>, <i>y</i>, <i>b</i>; we can see in the ochre cube (<a href="#fig_119">fig. 119</a>) -the plane belonging to this section space, which passes -through the three extremities of the red, white, yellow -axes.</p> - -<p>Now let the tesseract pass half way through our space. -Instead of our original axes we have parallels to them, -purple, light blue, and green, each of the same length as -the first axes, for the section of the tesseract is of exactly -the same shape as its ochre cube.</p> - -<p>But the sectional space seen at this stage of the transference -would not cut the section of the tesseract in a -plane disposed as at first.</p> - -<p>To see where the sectional space would cut these -parallels to the original axes let the tesseract swing so -that, the orange face remaining stationary, the blue line -comes in to the left.</p> - -<p><span class="pagenum" id="Page_200">[Pg 200]</span></p> - -<div class="figleft illowp45" id="fig_125" style="max-width: 25em;"> - <img src="images/fig_125.png" alt="" /> - <div class="caption">Fig. 125.</div> -</div> - -<p>Here (<a href="#fig_125">fig. 125</a>) we have the null <i>r</i>, <i>y</i>, <i>b</i> points, and of -the sectional space all we -see is the plane through these -three points in it.</p> - -<p>In this figure we can draw -the parallels to the red and -yellow axes and see that, if -they started at a point half -way along the blue axis, they -would each be cut at a point so as to be half of their -previous length.</p> - -<p>Swinging the tesseract into our space about the pink -face of the ochre cube we likewise find that the parallel -to the white axis is cut at half its length by the sectional -space.</p> - -<div class="figleft illowp40" id="fig_126" style="max-width: 25em;"> - <img src="images/fig_126.png" alt="" /> - <div class="caption">Fig. 126.</div> -</div> - -<p>Hence in a section made when the tesseract had passed -half across our space the parallels to the red, white, yellow -axes, which are now in our -space, are cut by the section -space, each of them half way -along, and for this stage of -the traversing motion we -should have <a href="#fig_126">fig. 126</a>. The -section made of this cube by -the plane in which the sectional -space cuts it, is an -equilateral triangle with purple, l. blue, green points, and -l. purple, brown, l. green lines.</p> - -<p>Thus the original ochre triangle, with null points and -pink, orange, light yellow lines, would be succeeded by a -triangle coloured in manner just described.</p> - -<p>This triangle would initially be only a very little smaller -than the original triangle, it would gradually diminish, -until it ended in a point, a null point. Each of its -edges would be of the same length. Thus the successive<span class="pagenum" id="Page_201">[Pg 201]</span> -sections of the successive planes into which we analyse the -cutting space would be a tetrahedron of the description -shown (<a href="#fig_123">fig. 123</a>), and the whole interior of the tetrahedron -would be light brown.</p> - -<div class="figcenter illowp100" id="fig_127" style="max-width: 50em;"> - <img src="images/fig_127.png" alt="" /> - <div class="caption">Front view. <span class="gap8l"> The rear faces.</span><br /> -Fig. 127.</div> -</div> - - -<p>In <a href="#fig_127">fig. 127</a> the tetrahedron is represented by means of -its faces as two triangles which meet in the p. line, and -two rear triangles which join on to them, the diagonal -of the pink face being supposed to run vertically -upward.</p> - -<p>We have now reached a natural termination. The -reader may pursue the subject in further detail, but will -find no essential novelty. I conclude with an indication -as to the manner in which figures previously given may -be used in determining sections by the method developed -above.</p> - -<p>Applying this method to the tesseract, as represented -in Chapter IX., sections made by a space cutting the axes -equidistantly at any distance can be drawn, and also the -sections of tesseracts arranged in a block.</p> - -<p>If we draw a plane, cutting all four axes at a point -six units distance from null, we have a slanting space. -This space cuts the red, white, yellow axes in the<span class="pagenum" id="Page_202">[Pg 202]</span> -points <span class="allsmcap">LMN</span> (<a href="#fig_128">fig. 128</a>), and so in the region of our space -before we go off into -the fourth dimension, -we have the plane -represented by <span class="allsmcap">LMN</span> -extended. This is what -is common to the -slanting space and our -space.</p> - -<div class="figleft illowp50" id="fig_128" style="max-width: 31.25em;"> - <img src="images/fig_128.png" alt="" /> - <div class="caption">Fig. 128.</div> -</div> - -<p>This plane cuts the -ochre cube in the triangle <span class="allsmcap">EFG</span>.</p> - -<p>Comparing this with (<a href="#fig_72">fig. 72</a>) <i>oh</i>, we see that the -hexagon there drawn is part of the triangle <span class="allsmcap">EFG</span>.</p> - -<p>Let us now imagine the tesseract and the slanting -space both together to pass transverse to our space, a -distance of one unit, we have in 1<i>h</i> a section of the -tesseract, whose axes are parallels to the previous axes. -The slanting space cuts them at a distance of five units -along each. Drawing the plane through these points in -1<i>h</i> it will be found to cut the cubical section of the -tesseract in the hexagonal figure drawn. In 2<i>h</i> (<a href="#fig_72">fig. 72</a>) the -slanting space cuts the parallels to the axes at a distance -of four along each, and the hexagonal figure is the section -of this section of the tesseract by it. Finally when 3<i>h</i> -comes in the slanting space cuts the axes at a distance -of three along each, and the section is a triangle, of which -the hexagon drawn is a truncated portion. After this -the tesseract, which extends only three units in each of -the four dimensions, has completely passed transverse -of our space, and there is no more of it to be cut. Hence, -putting the plane sections together in the right relations, -we have the section determined by the particular slanting -space: namely an octahedron.</p> -<hr class="chap x-ebookmaker-drop" /> - -<div class="chapter"> -<p><span class="pagenum" id="Page_203">[Pg 203]</span></p> - -<h2 class="nobreak" id="CHAPTER_XIV">CHAPTER XIV.<a id="FNanchor_6" href="#Footnote_6" class="fnanchor">[6]</a><br /> - -<small><i>A RECAPITULATION AND EXTENSION OF -THE PHYSICAL ARGUMENT</i></small></h2></div> - -<div class="footnotes"> -<div class="footnote"> - -<p><a id="Footnote_6" href="#FNanchor_6" class="label">[6]</a> The contents of this chapter are taken from a paper read before -the Philosophical Society of Washington. The mathematical portion -of the paper has appeared in part in the Proceedings of the Royal -Irish Academy under the title, “Cayley’s formulæ of orthogonal -transformation,” Nov. 29th, 1903.</p> - -</div></div> - -<p>There are two directions of inquiry in which the -research for the physical reality of a fourth dimension -can be prosecuted. One is the investigation of the -infinitely great, the other is the investigation of the -infinitely small.</p> - -<p>By the measurement of the angles of vast triangles, -whose sides are the distances between the stars, astronomers -have sought to determine if there is any deviation from -the values given by geometrical deduction. If the angles -of a celestial triangle do not together equal two right -angles, there would be an evidence for the physical reality -of a fourth dimension.</p> - -<p>This conclusion deserves a word of explanation. If -space is really four-dimensional, certain conclusions follow -which must be brought clearly into evidence if we are to -frame the questions definitely which we put to Nature. -To account for our limitation let us assume a solid material -sheet against which we move. This sheet must stretch -alongside every object in every direction in which it -visibly moves. Every material body must slip or slide -along this sheet, not deviating from contact with it in -any motion which we can observe.</p> - -<p><span class="pagenum" id="Page_204">[Pg 204]</span></p> - -<p>The necessity for this assumption is clearly apparent, if -we consider the analogous case of a suppositionary plane -world. If there were any creatures whose experiences -were confined to a plane, we must account for their -limitation. If they were free to move in every space -direction, they would have a three-dimensional motion; -hence they must be physically limited, and the only way -in which we can conceive such a limitation to exist is by -means of a material surface against which they slide. -The existence of this surface could only be known to -them indirectly. It does not lie in any direction from -them in which the kinds of motion they know of leads -them. If it were perfectly smooth and always in contact -with every material object, there would be no difference in -their relations to it which would direct their attention to it.</p> - -<p>But if this surface were curved—if it were, say, in the -form of a vast sphere—the triangles they drew would -really be triangles of a sphere, and when these triangles -are large enough the angles diverge from the magnitudes -they would have for the same lengths of sides if the -surface were plane. Hence by the measurement of -triangles of very great magnitude a plane being might -detect a difference from the laws of a plane world in his -physical world, and so be led to the conclusion that there -was in reality another dimension to space—a third -dimension—as well as the two which his ordinary experience -made him familiar with.</p> - -<p>Now, astronomers have thought it worth while to -examine the measurements of vast triangles drawn from -one celestial body to another with a view to determine if -there is anything like a curvature in our space—that is to -say, they have tried astronomical measurements to find<span class="pagenum" id="Page_205">[Pg 205]</span> -out if the vast solid sheet against which, on the supposition -of a fourth dimension, everything slides is -curved or not. These results have been negative. The -solid sheet, if it exists, is not curved or, being curved, has -not a sufficient curvature to cause any observable deviation -from the theoretical value of the angles calculated.</p> - -<p>Hence the examination of the infinitely great leads to -no decisive criterion. If it did we should have to decide -between the present theory and that of metageometry.</p> - -<p>Coming now to the prosecution of the inquiry in the -direction of the infinitely small, we have to state the -question thus: Our laws of movement are derived from -the examination of bodies which move in three-dimensional -space. All our conceptions are founded on the supposition -of a space which is represented analytically by -three independent axes and variations along them—that -is, it is a space in which there are three independent -movements. Any motion possible in it can be compounded -out of these three movements, which we may call: up, -right, away.</p> - -<p>To examine the actions of the very small portions of -matter with the view of ascertaining if there is any -evidence in the phenomena for the supposition of a fourth -dimension of space, we must commence by clearly defining -what the laws of mechanics would be on the supposition -of a fourth dimension. It is of no use asking if the -phenomena of the smallest particles of matter are like—we -do not know what. We must have a definite conception -of what the laws of motion would be on the -supposition of the fourth dimension, and then inquire if -the phenomena of the activity of the smaller particles of -matter resemble the conceptions which we have elaborated.</p> - -<p>Now, the task of forming these conceptions is by no -means one to be lightly dismissed. Movement in space -has many features which differ entirely from movement<span class="pagenum" id="Page_206">[Pg 206]</span> -on a plane; and when we set about to form the conception -of motion in four dimensions, we find that there -is at least as great a step as from the plane to three-dimensional -space.</p> - -<p>I do not say that the step is difficult, but I want to -point out that it must be taken. When we have formed -the conception of four-dimensional motion, we can ask a -rational question of Nature. Before we have elaborated -our conceptions we are asking if an unknown is like an -unknown—a futile inquiry.</p> - -<p>As a matter of fact, four-dimensional movements are in -every way simple and more easy to calculate than three-dimensional -movements, for four-dimensional movements -are simply two sets of plane movements put together.</p> - -<p>Without the formation of an experience of four-dimensional -bodies, their shapes and motions, the subject -can be but formal—logically conclusive, not intuitively -evident. It is to this logical apprehension that I must -appeal.</p> - -<p>It is perfectly simple to form an experiential familiarity -with the facts of four-dimensional movement. The -method is analogous to that which a plane being would -have to adopt to form an experiential familiarity with -three-dimensional movements, and may be briefly -summed up as the formation of a compound sense by -means of which duration is regarded as equivalent to -extension.</p> - -<p>Consider a being confined to a plane. A square enclosed -by four lines will be to him a solid, the interior of which -can only be examined by breaking through the lines. -If such a square were to pass transverse to his plane, it -would immediately disappear. It would vanish, going in -no direction to which he could point.</p> - -<p>If, now, a cube be placed in contact with his plane, its -surface of contact would appear like the square which we<span class="pagenum" id="Page_207">[Pg 207]</span> -have just mentioned. But if it were to pass transverse to -his plane, breaking through it, it would appear as a lasting -square. The three-dimensional matter will give a lasting -appearance in circumstances under which two-dimensional -matter will at once disappear.</p> - -<p>Similarly, a four-dimensional cube, or, as we may call -it, a tesseract, which is generated from a cube by a -movement of every part of the cube in a fourth direction -at right angles to each of the three visible directions in -the cube, if it moved transverse to our space, would -appear as a lasting cube.</p> - -<p>A cube of three-dimensional matter, since it extends to -no distance at all in the fourth dimension, would instantly -disappear, if subjected to a motion transverse to our space. -It would disappear and be gone, without it being possible -to point to any direction in which it had moved.</p> - -<p>All attempts to visualise a fourth dimension are futile. It -must be connected with a time experience in three space.</p> - -<p>The most difficult notion for a plane being to acquire -would be that of rotation about a line. Consider a plane -being facing a square. If he were told that rotation -about a line were possible, he would move his square this -way and that. A square in a plane can rotate about a -point, but to rotate about a line would seem to the plane -being perfectly impossible. How could those parts of his -square which were on one side of an edge come to the -other side without the edge moving? He could understand -their reflection in the edge. He could form an -idea of the looking-glass image of his square lying on the -opposite side of the line of an edge, but by no motion -that he knows of can he make the actual square assume -that position. The result of the rotation would be like -reflection in the edge, but it would be a physical impossibility -to produce it in the plane.</p> - -<p>The demonstration of rotation about a line must be to<span class="pagenum" id="Page_208">[Pg 208]</span> -him purely formal. If he conceived the notion of a cube -stretching out in an unknown direction away from his -plane, then he can see the base of it, his square in the -plane, rotating round a point. He can likewise apprehend -that every parallel section taken at successive intervals in -the unknown direction rotates in like manner round a -point. Thus he would come to conclude that the whole -body rotates round a line—the line consisting of the -succession of points round which the plane sections rotate. -Thus, given three axes, <i>x</i>, <i>y</i>, <i>z</i>, if <i>x</i> rotates to take -the place of <i>y</i>, and <i>y</i> turns so as to point to negative <i>x</i>, -then the third axis remaining unaffected by this turning -is the axis about which the rotation takes place. This, -then, would have to be his criterion of the axis of a -rotation—that which remains unchanged when a rotation -of every plane section of a body takes place.</p> - -<p>There is another way in which a plane being can think -about three-dimensional movements; and, as it affords -the type by which we can most conveniently think about -four-dimensional movements, it will be no loss of time to -consider it in detail.</p> -<div class="figleft illowp30" id="fig_129" style="max-width: 18.75em;"> - <img src="images/fig_129.png" alt="" /> - <div class="caption">Fig. 1 (129).</div> -</div> - -<p>We can represent the plane being and his object by -figures cut out of paper, which slip on a smooth surface. -The thickness of these bodies must be taken as so minute -that their extension in the third dimension -escapes the observation of the -plane being, and he thinks about them -as if they were mathematical plane -figures in a plane instead of being -material bodies capable of moving on -a plane surface. Let <span class="allsmcap">A</span><i>x</i>, <span class="allsmcap">A</span><i>y</i> be two -axes and <span class="allsmcap">ABCD</span> a square. As far as -movements in the plane are concerned, the square can -rotate about a point <span class="allsmcap">A</span>, for example. It cannot rotate -about a side, such as <span class="allsmcap">AC</span>.</p> - -<p><span class="pagenum" id="Page_209">[Pg 209]</span></p> - -<p>But if the plane being is aware of the existence of a -third dimension he can study the movements possible in -the ample space, taking his figure portion by portion.</p> - -<p>His plane can only hold two axes. But, since it can -hold two, he is able to represent a turning into the third -dimension if he neglects one of his axes and represents the -third axis as lying in his plane. He can make a drawing -in his plane of what stands up perpendicularly from his -plane. Let <span class="allsmcap">A</span><i>z</i> be the axis, which -stands perpendicular to his plane at -<span class="allsmcap">A</span>. He can draw in his plane two -lines to represent the two axes, <span class="allsmcap">A</span><i>x</i> -and <span class="allsmcap">A</span><i>z</i>. Let Fig. 2 be this drawing. -Here the <i>z</i> axis has taken -the place of the <i>y</i> axis, and the -plane of <span class="allsmcap">A</span><i>x</i> <span class="allsmcap">A</span><i>z</i> is represented in his -plane. In this figure all that exists of the square <span class="allsmcap">ABCD</span> -will be the line <span class="allsmcap">AB</span>.</p> - -<div class="figleft illowp30" id="fig_130" style="max-width: 18.75em;"> - <img src="images/fig_130.png" alt="" /> - <div class="caption">Fig. 2 (130).</div> -</div> - -<p>The square extends from this line in the <i>y</i> direction, -but more of that direction is represented in Fig. 2. The -plane being can study the turning of the line <span class="allsmcap">AB</span> in this -diagram. It is simply a case of plane turning around the -point <span class="allsmcap">A</span>. The line <span class="allsmcap">AB</span> occupies intermediate portions like <span class="allsmcap">AB</span><sub>1</sub> -and after half a revolution will lie on <span class="allsmcap">A</span><i>x</i> produced through <span class="allsmcap">A</span>.</p> - -<p>Now, in the same way, the plane being can take -another point, <span class="allsmcap">A´</span>, and another line, <span class="allsmcap">A´B´</span>, in his square. -He can make the drawing of the two directions at <span class="allsmcap">A´</span>, one -along <span class="allsmcap">A´B´</span>, the other perpendicular to his plane. He -will obtain a figure precisely similar to Fig. 2, and will -see that, as <span class="allsmcap">AB</span> can turn around <span class="allsmcap">A</span>, so <span class="allsmcap">A´C´</span> around <span class="allsmcap">A</span>.</p> - -<p>In this turning <span class="allsmcap">AB</span> and <span class="allsmcap">A´B´</span> would not interfere with -each other, as they would if they moved in the plane -around the separate points <span class="allsmcap">A</span> and <span class="allsmcap">A´</span>.</p> - -<p>Hence the plane being would conclude that a rotation -round a line was possible. He could see his square as it<span class="pagenum" id="Page_210">[Pg 210]</span> -began to make this turning. He could see it half way -round when it came to lie on the opposite side of the line -<span class="allsmcap">AC</span>. But in intermediate portions he could not see it, -for it runs out of the plane.</p> - -<p>Coming now to the question of a four-dimensional body, -let us conceive of it as a series of cubic sections, the first -in our space, the rest at intervals, stretching away from -our space in the unknown direction.</p> - -<p>We must not think of a four-dimensional body as -formed by moving a three-dimensional body in any -direction which we can see.</p> - -<p>Refer for a moment to Fig. 3. The point <span class="allsmcap">A</span>, moving to -the right, traces out the line <span class="allsmcap">AC</span>. The line <span class="allsmcap">AC</span>, moving -away in a new direction, traces out the square <span class="allsmcap">ACEG</span> at -the base of the cube. The square <span class="allsmcap">AEGC</span>, moving in a -new direction, will trace out the cube <span class="allsmcap">ACEGBDHF</span>. The -vertical direction of this last motion is not identical with -any motion possible in the plane of the base of the cube. -It is an entirely new direction, at right angles to every -line that can be drawn in the base. To trace out a -tesseract the cube must move in a new direction—a -direction at right angles to any and every line that can -be drawn in the space of the cube.</p> - -<p>The cubic sections of the tesseract are related to the -cube we see, as the square sections of the cube are related -to the square of its base which a plane being sees.</p> - -<p>Let us imagine the cube in our space, which is the base -of a tesseract, to turn about one of its edges. The rotation -will carry the whole body with it, and each of the cubic -sections will rotate. The axis we see in our space will -remain unchanged, and likewise the series of axes parallel -to it about which each of the parallel cubic sections -rotates. The assemblage of all of these is a plane.</p> - -<p>Hence in four dimensions a body rotates about a plane. -There is no such thing as rotation round an axis.</p> - -<p><span class="pagenum" id="Page_211">[Pg 211]</span></p> - -<p>We may regard the rotation from a different point of -view. Consider four independent axes each at right -angles to all the others, drawn in a four-dimensional body. -Of these four axes we can see any three. The fourth -extends normal to our space.</p> - -<p>Rotation is the turning of one axis into a second, and -the second turning to take the place of the negative of -the first. It involves two axes. Thus, in this rotation of -a four-dimensional body, two axes change and two remain -at rest. Four-dimensional rotation is therefore a turning -about a plane.</p> - -<p>As in the case of a plane being, the result of rotation -about a line would appear as the production of a looking-glass -image of the original object on the other side of the -line, so to us the result of a four-dimensional rotation -would appear like the production of a looking-glass image -of a body on the other side of a plane. The plane would -be the axis of the rotation, and the path of the body -between its two appearances would be unimaginable in -three-dimensional space.</p> - -<div class="figleft illowp30" id="fig_131" style="max-width: 18.75em;"> - <img src="images/fig_131.png" alt="" /> - <div class="caption">Fig. 3 (131).</div> -</div> - -<p>Let us now apply the method by which a plane being -could examine the nature of rotation -about a line in our examination -of rotation about a plane. Fig. 3 -represents a cube in our space, the -three axes <i>x</i>, <i>y</i>, <i>z</i> denoting its -three dimensions. Let <i>w</i> represent -the fourth dimension. Now, since -in our space we can represent any -three dimensions, we can, if we -choose, make a representation of what is in the space -determined by the three axes <i>x</i>, <i>z</i>, <i>w</i>. This is a three-dimensional -space determined by two of the axes we have -drawn, <i>x</i> and <i>z</i>, and in place of <i>y</i> the fourth axis, <i>w</i>. We -cannot, keeping <i>x</i> and <i>z</i>, have both <i>y</i> and <i>w</i> in our space;<span class="pagenum" id="Page_212">[Pg 212]</span> -so we will let <i>y</i> go and draw <i>w</i> in its place. What will be -our view of the cube?</p> - -<div class="figleft illowp30" id="fig_132" style="max-width: 18.75em;"> - <img src="images/fig_132.png" alt="" /> - <div class="caption">Fig. 4 (132).</div> -</div> - -<p>Evidently we shall have simply the square that is in -the plane of <i>xz</i>, the square <span class="allsmcap">ACDB</span>. -The rest of the cube stretches in -the <i>y</i> direction, and, as we have -none of the space so determined, -we have only the face of the cube. -This is represented in <a href="#fig_132">fig. 4</a>.</p> - -<p>Now, suppose the whole cube to -be turned from the <i>x</i> to the <i>w</i> -direction. Conformably with our method, we will not -take the whole of the cube into consideration at once, but -will begin with the face <span class="allsmcap">ABCD</span>.</p> - -<div class="figleft illowp30" id="fig_133" style="max-width: 18.75em;"> - <img src="images/fig_133.png" alt="" /> - <div class="caption">Fig. 5 (133).</div> -</div> - -<p>Let this face begin to turn. Fig. 5 -represents one of the positions it will -occupy; the line <span class="allsmcap">AB</span> remains on the -<i>z</i> axis. The rest of the face extends -between the <i>x</i> and the <i>w</i> direction.</p> - -<p>Now, since we can take any three -axes, let us look at what lies in -the space of <i>zyw</i>, and examine the -turning there. We must now let the <i>z</i> axis disappear -and let the <i>w</i> axis run in the direction in which the <i>z</i> ran.</p> - -<div class="figleft illowp30" id="fig_134" style="max-width: 18.75em;"> - <img src="images/fig_134.png" alt="" /> - <div class="caption">Fig. 6 (134).</div> -</div> - -<p>Making this representation, what -do we see of the cube? Obviously -we see only the lower face. The rest -of the cube lies in the space of <i>xyz</i>. -In the space of <i>xyz</i> we have merely -the base of the cube lying in the -plane of <i>xy</i>, as shown in <a href="#fig_134">fig. 6</a>.</p> - -<p>Now let the <i>x</i> to <i>w</i> turning take place. The square -<span class="allsmcap">ACEG</span> will turn about the line <span class="allsmcap">AE</span>. This edge will -remain along the <i>y</i> axis and will be stationary, however -far the square turns.</p> - -<p><span class="pagenum" id="Page_213">[Pg 213]</span></p> - -<div class="figleft illowp30" id="fig_135" style="max-width: 18.75em;"> - <img src="images/fig_135.png" alt="" /> - <div class="caption">Fig. 7 (135).</div> -</div> - -<p>Thus, if the cube be turned by an <i>x</i> to <i>w</i> turning, both -the edge <span class="allsmcap">AB</span> and the edge <span class="allsmcap">AC</span> remain -stationary; hence the whole face -<span class="allsmcap">ABEF</span> in the <i>yz</i> plane remains fixed. -The turning has taken place about -the face <span class="allsmcap">ABEF</span>.</p> - -<p>Suppose this turning to continue -till <span class="allsmcap">AC</span> runs to the left from <span class="allsmcap">A</span>. -The cube will occupy the position -shown in <a href="#fig_136">fig. 8</a>. This is the looking-glass image of the -cube in <a href="#fig_131">fig. 3</a>. By no rotation in three-dimensional space -can the cube be brought from -the position in <a href="#fig_131">fig. 3</a> to that -shown in <a href="#fig_136">fig. 8</a>.</p> - -<div class="figleft illowp40" id="fig_136" style="max-width: 21.875em;"> - <img src="images/fig_136.png" alt="" /> - <div class="caption">Fig. 8 (136).</div> -</div> - -<p>We can think of this turning -as a turning of the face <span class="allsmcap">ABCD</span> -about <span class="allsmcap">AB</span>, and a turning of each -section parallel to <span class="allsmcap">ABCD</span> round -the vertical line in which it -intersects the face <span class="allsmcap">ABEF</span>, the -space in which the turning takes place being a different -one from that in which the cube lies.</p> - -<p>One of the conditions, then, of our inquiry in the -direction of the infinitely small is that we form the conception -of a rotation about a plane. The production of a -body in a state in which it presents the appearance of a -looking-glass image of its former state is the criterion -for a four-dimensional rotation.</p> - -<p>There is some evidence for the occurrence of such transformations -of bodies in the change of bodies from those -which produce a right-handed polarisation of light to -those which produce a left-handed polarisation; but this -is not a point to which any very great importance can -be attached.</p> - -<p>Still, in this connection, let me quote a remark from<span class="pagenum" id="Page_214">[Pg 214]</span> -Prof. John G. McKendrick’s address on Physiology before -the British Association at Glasgow. Discussing the -possibility of the hereditary production of characteristics -through the material structure of the ovum, he estimates -that in it there exist 12,000,000,000 biophors, or ultimate -particles of living matter, a sufficient number to account -for hereditary transmission, and observes: “Thus it is -conceivable that vital activities may also be determined -by the kind of motion that takes place in the molecules -of that which we speak of as living matter. It may be -different in kind from some of the motions known to -physicists, and it is conceivable that life may be the -transmission to dead matter, the molecules of which have -already a special kind of motion, of a form of motion -<i>sui generis</i>.”</p> - -<p>Now, in the realm of organic beings symmetrical structures—those -with a right and left symmetry—are everywhere -in evidence. Granted that four dimensions exist, -the simplest turning produces the image form, and by a -folding-over structures could be produced, duplicated -right and left, just as is the case of symmetry in a -plane.</p> - -<p>Thus one very general characteristic of the forms of -organisms could be accounted for by the supposition that -a four-dimensional motion was involved in the process of -life.</p> - -<p>But whether four-dimensional motions correspond in -other respects to the physiologist’s demand for a special -kind of motion, or not, I do not know. Our business is -with the evidence for their existence in physics. For -this purpose it is necessary to examine into the significance -of rotation round a plane in the case of extensible -and of fluid matter.</p> - -<p>Let us dwell a moment longer on the rotation of a rigid -body. Looking at the cube in <a href="#fig_131">fig. 3</a>, which turns about<span class="pagenum" id="Page_215">[Pg 215]</span> -the face of <span class="allsmcap">ABFE</span>, we see that any line in the face can -take the place of the vertical and horizontal lines we have -examined. Take the diagonal line <span class="allsmcap">AF</span> and the section -through it to <span class="allsmcap">GH</span>. The portions of matter which were on -one side of <span class="allsmcap">AF</span> in this section in <a href="#fig_131">fig. 3</a> are on the -opposite side of it in <a href="#fig_136">fig. 8</a>. They have gone round the -line <span class="allsmcap">AF</span>. Thus the rotation round a face can be considered -as a number of rotations of sections round parallel lines -in it.</p> - -<p>The turning about two different lines is impossible in -three-dimensional space. To take another illustration, -suppose <span class="allsmcap">A</span> and <span class="allsmcap">B</span> are two parallel lines in the <i>xy</i> plane, -and let <span class="allsmcap">CD</span> and <span class="allsmcap">EF</span> be two rods crossing them. Now, in -the space of <i>xyz</i> if the rods turn round the lines <span class="allsmcap">A</span> and <span class="allsmcap">B</span> -in the same direction they -will make two independent -circles.</p> - -<div class="figleft illowp40" id="fig_137" style="max-width: 21.875em;"> - <img src="images/fig_137.png" alt="" /> - <div class="caption">Fig. 9 (137).</div> -</div> - -<p>When the end <span class="allsmcap">F</span> is going -down the end <span class="allsmcap">C</span> will be coming -up. They will meet and conflict.</p> - -<p>But if we rotate the rods -about the plane of <span class="allsmcap">AB</span> by the -<i>z</i> to <i>w</i> rotation these movements -will not conflict. Suppose -all the figure removed -with the exception of the plane <i>xz</i>, and from this plane -draw the axis of <i>w</i>, so that we are looking at the space -of <i>xzw</i>.</p> - -<p>Here, <a href="#fig_138">fig. 10</a>, we cannot see the lines <span class="allsmcap">A</span> and <span class="allsmcap">B</span>. We -see the points <span class="allsmcap">G</span> and <span class="allsmcap">H</span>, in which <span class="allsmcap">A</span> and <span class="allsmcap">B</span> intercept -the <i>x</i> axis, but we cannot see the lines themselves, for -they run in the <i>y</i> direction, and that is not in our -drawing.</p> - -<p>Now, if the rods move with the <i>z</i> to <i>w</i> rotation they will<span class="pagenum" id="Page_216">[Pg 216]</span> -turn in parallel planes, keeping their relative positions. -The point <span class="allsmcap">D</span>, for instance, will -describe a circle. At one time -it will be above the line <span class="allsmcap">A</span>, at -another time below it. Hence -it rotates round <span class="allsmcap">A</span>.</p> - -<div class="figleft illowp40" id="fig_138" style="max-width: 21.875em;"> - <img src="images/fig_138.png" alt="" /> - <div class="caption">Fig. 10 (138).</div> -</div> - -<p>Not only two rods but any -number of rods crossing the -plane will move round it harmoniously. -We can think of -this rotation by supposing the -rods standing up from one line -to move round that line and remembering that it is -not inconsistent with this rotation for the rods standing -up along another line also to move round it, the relative -positions of all the rods being preserved. Now, if the -rods are thick together, they may represent a disk of -matter, and we see that a disk of matter can rotate -round a central plane.</p> - -<p>Rotation round a plane is exactly analogous to rotation -round an axis in three dimensions. If we want a rod to -turn round, the ends must be free; so if we want a disk -of matter to turn round its central plane by a four-dimensional -turning, all the contour must be free. The whole -contour corresponds to the ends of the rod. Each point -of the contour can be looked on as the extremity of an -axis in the body, round each point of which there is a -rotation of the matter in the disk.</p> - -<p>If the one end of a rod be clamped, we can twist the -rod, but not turn it round; so if any part of the contour -of a disk is clamped we can impart a twist to the disk, -but not turn it round its central plane. In the case of -extensible materials a long, thin rod will twist round its -axis, even when the axis is curved, as, for instance, in the -case of a ring of India rubber.</p> - -<p><span class="pagenum" id="Page_217">[Pg 217]</span></p> - -<p>In an analogous manner, in four dimensions we can have -rotation round a curved plane, if I may use the expression. -A sphere can be turned inside out in four dimensions.</p> - -<div class="figleft illowp45" id="fig_139" style="max-width: 25em;"> - <img src="images/fig_139.png" alt="" /> - <div class="caption">Fig. 11 (139).</div> -</div> - -<p>Let <a href="#fig_139">fig. 11</a> represent a -spherical surface, on each -side of which a layer of -matter exists. The thickness -of the matter is represented -by the rods <span class="allsmcap">CD</span> and -<span class="allsmcap">EF</span>, extending equally without -and within.</p> - -<p>Now, take the section of -the sphere by the <i>yz</i> plane -we have a circle—<a href="#fig_140">fig. 12</a>. -Now, let the <i>w</i> axis be drawn -in place of the <i>x</i> axis so that -we have the space of <i>yzw</i> -represented. In this space all that there will be seen of -the sphere is the circle drawn.</p> - -<div class="figleft illowp45" id="fig_140" style="max-width: 25em;"> - <img src="images/fig_140.png" alt="" /> - <div class="caption">Fig. 12 (140).</div> -</div> - -<p>Here we see that there is no obstacle to prevent the -rods turning round. If -the matter is so elastic -that it will give enough -for the particles at <span class="allsmcap">E</span> and -<span class="allsmcap">C</span> to be separated as they -are at <span class="allsmcap">F</span> and <span class="allsmcap">D</span>, they -can rotate round to the -position <span class="allsmcap">D</span> and <span class="allsmcap">F</span>, and a -similar motion is possible -for all other particles. -There is no matter or -obstacle to prevent them -from moving out in the -<i>w</i> direction, and then on round the circumference as an -axis. Now, what will hold for one section will hold for<span class="pagenum" id="Page_218">[Pg 218]</span> -all, as the fourth dimension is at right angles to all the -sections which can be made of the sphere.</p> - -<p>We have supposed the matter of which the sphere is -composed to be three-dimensional. If the matter had a -small thickness in the fourth dimension, there would be -a slight thickness in <a href="#fig_140">fig. 12</a> above the plane of the paper—a -thickness equal to the thickness of the matter in the -fourth dimension. The rods would have to be replaced -by thin slabs. But this would make no difference as to -the possibility of the rotation. This motion is discussed -by Newcomb in the first volume of the <i>American Journal -of Mathematics</i>.</p> - -<p>Let us now consider, not a merely extensible body, but -a liquid one. A mass of rotating liquid, a whirl, eddy, -or vortex, has many remarkable properties. On first -consideration we should expect the rotating mass of -liquid immediately to spread off and lose itself in the -surrounding liquid. The water flies off a wheel whirled -round, and we should expect the rotating liquid to be -dispersed. But see the eddies in a river strangely persistent. -The rings that occur in puffs of smoke and last -so long are whirls or vortices curved round so that their -opposite ends join together. A cyclone will travel over -great distances.</p> - -<p>Helmholtz was the first to investigate the properties of -vortices. He studied them as they would occur in a perfect -fluid—that is, one without friction of one moving portion -or another. In such a medium vortices would be indestructible. -They would go on for ever, altering their -shape, but consisting always of the same portion of the -fluid. But a straight vortex could not exist surrounded -entirely by the fluid. The ends of a vortex must reach to -some boundary inside or outside the fluid.</p> - -<p>A vortex which is bent round so that its opposite ends -join is capable of existing, but no vortex has a free end in<span class="pagenum" id="Page_219">[Pg 219]</span> -the fluid. The fluid round the vortex is always in motion, -and one produces a definite movement in another.</p> - -<p>Lord Kelvin has proposed the hypothesis that portions -of a fluid segregated in vortices account for the origin of -matter. The properties of the ether in respect of its -capacity of propagating disturbances can be explained -by the assumption of vortices in it instead of by a property -of rigidity. It is difficult to conceive, however, -of any arrangement of the vortex rings and endless vortex -filaments in the ether.</p> - -<p>Now, the further consideration of four-dimensional -rotations shows the existence of a kind of vortex which -would make an ether filled with a homogeneous vortex -motion easily thinkable.</p> - -<p>To understand the nature of this vortex, we must go -on and take a step by which we accept the full significance -of the four-dimensional hypothesis. Granted four-dimensional -axes, we have seen that a rotation of one into -another leaves two unaltered, and these two form the -axial plane about which the rotation takes place. But -what about these two? Do they necessarily remain -motionless? There is nothing to prevent a rotation of -these two, one into the other, taking place concurrently -with the first rotation. This possibility of a double -rotation deserves the most careful attention, for it is the -kind of movement which is distinctly typical of four -dimensions.</p> - -<p>Rotation round a plane is analogous to rotation round -an axis. But in three-dimensional space there is no -motion analogous to the double rotation, in which, while -axis 1 changes into axis 2, axis 3 changes into axis 4.</p> - -<p>Consider a four-dimensional body, with four independent -axes, <i>x</i>, <i>y</i>, <i>z</i>, <i>w</i>. A point in it can move in only one -direction at a given moment. If the body has a velocity -of rotation by which the <i>x</i> axis changes into the <i>y</i> axis<span class="pagenum" id="Page_220">[Pg 220]</span> -and all parallel sections move in a similar manner, then -the point will describe a circle. If, now, in addition to -the rotation by which the <i>x</i> axis changes into the <i>y</i> axis the -body has a rotation by which the <i>z</i> axis turns into the -<i>w</i> axis, the point in question will have a double motion -in consequence of the two turnings. The motions will -compound, and the point will describe a circle, but not -the same circle which it would describe in virtue of either -rotation separately.</p> - -<p>We know that if a body in three-dimensional space is -given two movements of rotation they will combine into a -single movement of rotation round a definite axis. It is -in no different condition from that in which it is subjected -to one movement of rotation. The direction of -the axis changes; that is all. The same is not true about -a four-dimensional body. The two rotations, <i>x</i> to <i>y</i> and -<i>z</i> to <i>w</i>, are independent. A body subject to the two is in -a totally different condition to that which it is in when -subject to one only. When subject to a rotation such as -that of <i>x</i> to <i>y</i>, a whole plane in the body, as we have -seen, is stationary. When subject to the double rotation -no part of the body is stationary except the point common -to the two planes of rotation.</p> - -<p>If the two rotations are equal in velocity, every point -in the body describes a circle. All points equally distant -from the stationary point describe circles of equal size.</p> - -<p>We can represent a four-dimensional sphere by means -of two diagrams, in one of which we take the three axes, -<i>x</i>, <i>y</i>, <i>z</i>; in the other the axes <i>x</i>, <i>w</i>, and <i>z</i>. In <a href="#fig_141">fig. 13</a> we -have the view of a four-dimensional sphere in the space of -<i>xyz</i>. Fig. 13 shows all that we can see of the four -sphere in the space of <i>xyz</i>, for it represents all the -points in that space, which are at an equal distance from -the centre.</p> - -<p>Let us now take the <i>xz</i> section, and let the axis of <i>w</i><span class="pagenum" id="Page_221">[Pg 221]</span> -take the place of the <i>y</i> axis. Here, in <a href="#fig_142">fig. 14</a>, we have -the space of <i>xzw</i>. In this space we have to take all the -points which are at the same distance from the centre, -consequently we have another sphere. If we had a three-dimensional -sphere, as has been shown before, we should -have merely a circle in the <i>xzw</i> space, the <i>xz</i> circle seen -in the space of <i>xzw</i>. But now, taking the view in the -space of <i>xzw</i>, we have a sphere in that space also. In a -similar manner, whichever set of three axes we take, we -obtain a sphere.</p> - -<div class="figleft illowp40" id="fig_141" style="max-width: 28.125em;"> - <img src="images/fig_141.png" alt="" /> - <div class="caption"><i>Showing axes xyz</i><br /> -Fig. 13 (141).</div> -</div> - -<div class="figright illowp40" id="fig_142" style="max-width: 28.125em;"> - <img src="images/fig_142.png" alt="" /> - <div class="caption"><i>Showing axes xwz</i><br /> -Fig. 14 (142).</div> -</div> - -<p>In <a href="#fig_141">fig. 13</a>, let us imagine the rotation in the direction -<i>xy</i> to be taking place. The point <i>x</i> will turn to <i>y</i>, and <i>p</i> -to <i>p´</i>. The axis <i>zz´</i> remains stationary, and this axis is all -of the plane <i>zw</i> which we can see in the space section -exhibited in the figure.</p> - -<p>In <a href="#fig_142">fig. 14</a>, imagine the rotation from <i>z</i> to <i>w</i> to be taking -place. The <i>w</i> axis now occupies the position previously -occupied by the <i>y</i> axis. This does not mean that the -<i>w</i> axis can coincide with the <i>y</i> axis. It indicates that we -are looking at the four-dimensional sphere from a different -point of view. Any three-space view will show us three -axes, and in <a href="#fig_142">fig. 14</a> we are looking at <i>xzw</i>.</p> - -<p>The only part that is identical in the two diagrams is -the circle of the <i>x</i> and <i>z</i> axes, which axes are contained -in both diagrams. Thus the plane <i>zxz´</i> is the same in -both, and the point <i>p</i> represents the same point in both<span class="pagenum" id="Page_222">[Pg 222]</span> -diagrams. Now, in <a href="#fig_142">fig. 14</a> let the <i>zw</i> rotation take place, -the <i>z</i> axis will turn toward the point <i>w</i> of the <i>w</i> axis, and -the point <i>p</i> will move in a circle about the point <i>x</i>.</p> - -<p>Thus in <a href="#fig_141">fig. 13</a> the point <i>p</i> moves in a circle parallel to -the <i>xy</i> plane; in <a href="#fig_142">fig. 14</a> it moves in a circle parallel to the -<i>zw</i> plane, indicated by the arrow.</p> - -<p>Now, suppose both of these independent rotations compounded, -the point <i>p</i> will move in a circle, but this circle -will coincide with neither of the circles in which either -one of the rotations will take it. The circle the point <i>p</i> -will move in will depend on its position on the surface of -the four sphere.</p> - -<p>In this double rotation, possible in four-dimensional -space, there is a kind of movement totally unlike any -with which we are familiar in three-dimensional space. -It is a requisite preliminary to the discussion of the -behaviour of the small particles of matter, with a view to -determining whether they show the characteristics of four-dimensional -movements, to become familiar with the main -characteristics of this double rotation. And here I must -rely on a formal and logical assent rather than on the -intuitive apprehension, which can only be obtained by a -more detailed study.</p> - -<p>In the first place this double rotation consists in two -varieties or kinds, which we will call the A and B kinds. -Consider four axes, <i>x</i>, <i>y</i>, <i>z</i>, <i>w</i>. The rotation of <i>x</i> to <i>y</i> can -be accompanied with the rotation of <i>z</i> to <i>w</i>. Call this -the A kind.</p> - -<p>But also the rotation of <i>x</i> to <i>y</i> can be accompanied by -the rotation, of not <i>z</i> to <i>w</i>, but <i>w</i> to <i>z</i>. Call this the -B kind.</p> - -<p>They differ in only one of the component rotations. One -is not the negative of the other. It is the semi-negative. -The opposite of an <i>x</i> to <i>y</i>, <i>z</i> to <i>w</i> rotation would be <i>y</i> to <i>x</i>, -<i>w</i> to <i>z</i>. The semi-negative is <i>x</i> to <i>y</i> and <i>w</i> to <i>z</i>.</p> - -<p><span class="pagenum" id="Page_223">[Pg 223]</span></p> - -<p>If four dimensions exist and we cannot perceive them, -because the extension of matter is so small in the fourth -dimension that all movements are withheld from direct -observation except those which are three-dimensional, we -should not observe these double rotations, but only the -effects of them in three-dimensional movements of the -type with which we are familiar.</p> - -<p>If matter in its small particles is four-dimensional, -we should expect this double rotation to be a universal -characteristic of the atoms and molecules, for no portion -of matter is at rest. The consequences of this corpuscular -motion can be perceived, but only under the form -of ordinary rotation or displacement. Thus, if the theory -of four dimensions is true, we have in the corpuscles of -matter a whole world of movement, which we can never -study directly, but only by means of inference.</p> - -<p>The rotation A, as I have defined it, consists of two -equal rotations—one about the plane of <i>zw</i>, the other -about the plane of <i>xy</i>. It is evident that these rotations -are not necessarily equal. A body may be moving with a -double rotation, in which these two independent components -are not equal; but in such a case we can consider -the body to be moving with a composite rotation—a -rotation of the A or B kind and, in addition, a rotation -about a plane.</p> - -<p>If we combine an A and a B movement, we obtain a -rotation about a plane; for, the first being <i>x</i> to <i>y</i> and -<i>z</i> to <i>w</i>, and the second being <i>x</i> to <i>y</i> and <i>w</i> to <i>z</i>, when they -are put together the <i>z</i> to <i>w</i> and <i>w</i> to <i>z</i> rotations neutralise -each other, and we obtain an <i>x</i> to <i>y</i> rotation only, which -is a rotation about the plane of <i>zw</i>. Similarly, if we -take a B rotation, <i>y</i> to <i>x</i> and <i>z</i> to <i>w</i>, we get, on combining -this with the A rotation, a rotation of <i>z</i> to <i>w</i> about the -<i>xy</i> plane. In this case the plane of rotation is in the -three-dimensional space of <i>xyz</i>, and we have—what has<span class="pagenum" id="Page_224">[Pg 224]</span> -been described before—a twisting about a plane in our -space.</p> - -<p>Consider now a portion of a perfect liquid having an A -motion. It can be proved that it possesses the properties -of a vortex. It forms a permanent individuality—a -separated-out portion of the liquid—accompanied by a -motion of the surrounding liquid. It has properties -analogous to those of a vortex filament. But it is not -necessary for its existence that its ends should reach the -boundary of the liquid. It is self-contained and, unless -disturbed, is circular in every section.</p> - -<div class="figleft illowp50" id="fig_143" style="max-width: 28.125em;"> - <img src="images/fig_143.png" alt="" /> - <div class="caption">Fig. 15 (143).</div> -</div> - -<p>If we suppose the ether to have its properties of transmitting -vibration given it by such vortices, we must -inquire how they lie together in four-dimensional space. -Placing a circular disk on a plane and surrounding it by -six others, we find that if the central one is given a motion -of rotation, it imparts to the others a rotation which is -antagonistic in every two adjacent -ones. If <span class="allsmcap">A</span> goes round, -as shown by the arrow, <span class="allsmcap">B</span> and -<span class="allsmcap">C</span> will be moving in opposite -ways, and each tends to destroy -the motion of the other.</p> - -<p>Now, if we suppose spheres -to be arranged in a corresponding -manner in three-dimensional -space, they will -be grouped in figures which -are for three-dimensional space what hexagons are for -plane space. If a number of spheres of soft clay be -pressed together, so as to fill up the interstices, each will -assume the form of a fourteen-sided figure called a -tetrakaidecagon.</p> - -<p>Now, assuming space to be filled with such tetrakaidecagons, -and placing a sphere in each, it will be found<span class="pagenum" id="Page_225">[Pg 225]</span> -that one sphere is touched by eight others. The remaining -six spheres of the fourteen which surround the -central one will not touch it, but will touch three of -those in contact with it. Hence, if the central sphere -rotates, it will not necessarily drive those around it so -that their motions will be antagonistic to each other, -but the velocities will not arrange themselves in a -systematic manner.</p> - -<p>In four-dimensional space the figure which forms the -next term of the series hexagon, tetrakaidecagon, is a -thirty-sided figure. It has for its faces ten solid tetrakaidecagons -and twenty hexagonal prisms. Such figures -will exactly fill four-dimensional space, five of them meeting -at every point. If, now, in each of these figures we -suppose a solid four-dimensional sphere to be placed, any -one sphere is surrounded by thirty others. Of these it -touches ten, and, if it rotates, it drives the rest by means -of these. Now, if we imagine the central sphere to be -given an A or a B rotation, it will turn the whole mass of -sphere round in a systematic manner. Suppose four-dimensional -space to be filled with such spheres, each -rotating with a double rotation, the whole mass would -form one consistent system of motion, in which each one -drove every other one, with no friction or lagging behind.</p> - -<p>Every sphere would have the same kind of rotation. In -three-dimensional space, if one body drives another round -the second body rotates with the opposite kind of rotation; -but in four-dimensional space these four-dimensional -spheres would each have the double negative of the rotation -of the one next it, and we have seen that the double -negative of an A or B rotation is still an A or B rotation. -Thus four-dimensional space could be filled with a system -of self-preservative living energy. If we imagine the -four-dimensional spheres to be of liquid and not of solid -matter, then, even if the liquid were not quite perfect and<span class="pagenum" id="Page_226">[Pg 226]</span> -there were a slight retarding effect of one vortex on -another, the system would still maintain itself.</p> - -<p>In this hypothesis we must look on the ether as -possessing energy, and its transmission of vibrations, not -as the conveying of a motion imparted from without, but -as a modification of its own motion.</p> - -<p>We are now in possession of some of the conceptions of -four-dimensional mechanics, and will turn aside from the -line of their development to inquire if there is any -evidence of their applicability to the processes of nature.</p> - -<p>Is there any mode of motion in the region of the -minute which, giving three-dimensional movements for -its effect, still in itself escapes the grasp of our mechanical -theories? I would point to electricity. Through the -labours of Faraday and Maxwell we are convinced that the -phenomena of electricity are of the nature of the stress -and strain of a medium; but there is still a gap to be -bridged over in their explanation—the laws of elasticity, -which Maxwell assumes, are not those of ordinary matter. -And, to take another instance: a magnetic pole in the -neighbourhood of a current tends to move. Maxwell has -shown that the pressures on it are analogous to the -velocities in a liquid which would exist if a vortex took -the place of the electric current: but we cannot point out -the definite mechanical explanation of these pressures. -There must be some mode of motion of a body or of the -medium in virtue of which a body is said to be -electrified.</p> - -<p>Take the ions which convey charges of electricity 500 -times greater in proportion to their mass than are carried -by the molecules of hydrogen in electrolysis. In respect -of what motion can these ions be said to be electrified? -It can be shown that the energy they possess is not -energy of rotation. Think of a short rod rotating. If it -is turned over it is found to be rotating in the opposite<span class="pagenum" id="Page_227">[Pg 227]</span> -direction. Now, if rotation in one direction corresponds to -positive electricity, rotation in the opposite direction corresponds -to negative electricity, and the smallest electrified -particles would have their charges reversed by being -turned over—an absurd supposition.</p> - -<p>If we fix on a mode of motion as a definition of -electricity, we must have two varieties of it, one for -positive and one for negative; and a body possessing the -one kind must not become possessed of the other by any -change in its position.</p> - -<p>All three-dimensional motions are compounded of rotations -and translations, and none of them satisfy this first -condition for serving as a definition of electricity.</p> - -<p>But consider the double rotation of the A and B kinds. -A body rotating with the A motion cannot have its -motion transformed into the B kind by being turned over -in any way. Suppose a body has the rotation <i>x</i> to <i>y</i> and -<i>z</i> to <i>w</i>. Turning it about the <i>xy</i> plane, we reverse the -direction of the motion <i>x</i> to <i>y</i>. But we also reverse the -<i>z</i> to <i>w</i> motion, for the point at the extremity of the -positive <i>z</i> axis is now at the extremity of the negative <i>z</i> -axis, and since we have not interfered with its motion it -goes in the direction of position <i>w</i>. Hence we have <i>y</i> to -<i>x</i> and <i>w</i> to <i>z</i>, which is the same as <i>x</i> to <i>y</i> and <i>z</i> to <i>w</i>. -Thus both components are reversed, and there is the A -motion over again. The B kind is the semi-negative, -with only one component reversed.</p> - -<p>Hence a system of molecules with the A motion would -not destroy it in one another, and would impart it to a -body in contact with them. Thus A and B motions -possess the first requisite which must be demanded in -any mode of motion representative of electricity.</p> - -<p>Let us trace out the consequences of defining positive -electricity as an A motion and negative electricity as a B -motion. The combination of positive and negative<span class="pagenum" id="Page_228">[Pg 228]</span> -electricity produces a current. Imagine a vortex in the -ether of the A kind and unite with this one of the B kind. -An A motion and B motion produce rotation round a plane, -which is in the ether a vortex round an axial surface. -It is a vortex of the kind we represent as a part of a -sphere turning inside out. Now such a vortex must have -its rim on a boundary of the ether—on a body in the -ether.</p> - -<p>Let us suppose that a conductor is a body which has -the property of serving as the terminal abutment of such -a vortex. Then the conception we must form of a closed -current is of a vortex sheet having its edge along the -circuit of the conducting wire. The whole wire will then -be like the centres on which a spindle turns in three-dimensional -space, and any interruption of the continuity -of the wire will produce a tension in place of a continuous -revolution.</p> - -<p>As the direction of the rotation of the vortex is from a -three-space direction into the fourth dimension and back -again, there will be no direction of flow to the current; -but it will have two sides, according to whether <i>z</i> goes -to <i>w</i> or <i>z</i> goes to negative <i>w</i>.</p> - -<p>We can draw any line from one part of the circuit to -another; then the ether along that line is rotating round -its points.</p> - -<p>This geometric image corresponds to the definition of -an electric circuit. It is known that the action does not -lie in the wire, but in the medium, and it is known that -there is no direction of flow in the wire.</p> - -<p>No explanation has been offered in three-dimensional -mechanics of how an action can be impressed throughout -a region and yet necessarily run itself out along a closed -boundary, as is the case in an electric current. But this -phenomenon corresponds exactly to the definition of a -four-dimensional vortex.</p> - -<p><span class="pagenum" id="Page_229">[Pg 229]</span></p> - -<p>If we take a very long magnet, so long that one of its -poles is practically isolated, and put this pole in the -vicinity of an electric circuit, we find that it moves.</p> - -<p>Now, assuming for the sake of simplicity that the wire -which determines the current is in the form of a circle, -if we take a number of small magnets and place them all -pointing in the same direction normal to the plane of the -circle, so that they fill it and the wire binds them round, -we find that this sheet of magnets has the same effect on -the magnetic pole that the current has. The sheet of -magnets may be curved, but the edge of it must coincide -with the wire. The collection of magnets is then -equivalent to the vortex sheet, and an elementary magnet -to a part of it. Thus, we must think of a magnet as -conditioning a rotation in the ether round the plane -which bisects at right angles the line joining its poles.</p> - -<p>If a current is started in a circuit, we must imagine -vortices like bowls turning themselves inside out, starting -from the contour. In reaching a parallel circuit, if the -vortex sheet were interrupted and joined momentarily to -the second circuit by a free rim, the axis plane would lie -between the two circuits, and a point on the second circuit -opposite a point on the first would correspond to a point -opposite to it on the first; hence we should expect a -current in the opposite direction in the second circuit. -Thus the phenomena of induction are not inconsistent -with the hypothesis of a vortex about an axial plane.</p> - -<p>In four-dimensional space, in which all four dimensions -were commensurable, the intensity of the action transmitted -by the medium would vary inversely as the cube of the -distance. Now, the action of a current on a magnetic -pole varies inversely as the square of the distance; hence, -over measurable distances the extension of the ether in -the fourth dimension cannot be assumed as other than -small in comparison with those distances.</p> - -<p><span class="pagenum" id="Page_230">[Pg 230]</span></p> - -<p>If we suppose the ether to be filled with vortices in the -shape of four-dimensional spheres rotating with the A -motion, the B motion would correspond to electricity in -the one-fluid theory. There would thus be a possibility -of electricity existing in two forms, statically, by itself, -and, combined with the universal motion, in the form of -a current.</p> - -<p>To arrive at a definite conclusion it will be necessary to -investigate the resultant pressures which accompany the -collocation of solid vortices with surface ones.</p> - -<p>To recapitulate:</p> - -<p>The movements and mechanics of four-dimensional -space are definite and intelligible. A vortex with a -surface as its axis affords a geometric image of a closed -circuit, and there are rotations which by their polarity -afford a possible definition of statical electricity.<a id="FNanchor_7" href="#Footnote_7" class="fnanchor">[7]</a></p> - -<div class="footnotes"> -<div class="footnote"> - -<p><a id="Footnote_7" href="#FNanchor_7" class="label">[7]</a> These double rotations of the A and B kinds I should like to call -Hamiltons and co-Hamiltons, for it is a singular fact that in his -“Quaternions” Sir Wm. Rowan Hamilton has given the theory of -either the A or the B kind. They follow the laws of his symbols, -I, J, K.</p> - -<p>Hamiltons and co-Hamiltons seem to be natural units of geometrical -expression. In the paper in the “Proceedings of the Royal Irish -Academy,” Nov. 1903, already alluded to, I have shown something of -the remarkable facility which is gained in dealing with the composition -of three- and four-dimensional rotations by an alteration in Hamilton’s -notation, which enables his system to be applied to both the A and B -kinds of rotations.</p> - -<p>The objection which has been often made to Hamilton’s system, -namely, that it is only under special conditions of application that his -processes give geometrically interpretable results, can be removed, if -we assume that he was really dealing with a four-dimensional motion, -and alter his notation to bring this circumstance into explicit -recognition.</p> - -</div></div> - -<hr class="chap x-ebookmaker-drop" /> - -<div class="chapter"> -<p><span class="pagenum" id="Page_231">[Pg 231]</span></p> - -<h2 class="nobreak" id="APPENDIX_I">APPENDIX I<br /> - -<small><i>THE MODELS</i></small></h2></div> - - -<p>In Chapter XI. a description has been given which will -enable any one to make a set of models illustrative of the -tesseract and its properties. The set here supposed to be -employed consists of:—</p> - -<div class="blockquote"> - -<p>1. Three sets of twenty-seven cubes each.</p> - -<p>2. Twenty-seven slabs.</p> - -<p>3. Twelve cubes with points, lines, faces, distinguished -by colours, which will be called the catalogue cubes.</p> -</div> - -<p>The preparation of the twelve catalogue cubes involves -the expenditure of a considerable amount of time. It is -advantageous to use them, but they can be replaced by -the drawing of the views of the tesseract or by a reference -to figs. <a href="#fig_103">103</a>, <a href="#fig_104">104</a>, <a href="#fig_105">105</a>, <a href="#fig_106">106</a> of the text.</p> - -<p>The slabs are coloured like the twenty-seven cubes of -the first cubic block in <a href="#fig_101">fig. 101</a>, the one with red, -white, yellow axes.</p> - -<p>The colours of the three sets of twenty-seven cubes are -those of the cubes shown in <a href="#fig_101">fig. 101</a>.</p> - -<p>The slabs are used to form the representation of a cube -in a plane, and can well be dispensed with by any one -who is accustomed to deal with solid figures. But the -whole theory depends on a careful observation of how the -cube would be represented by these slabs.</p> - -<p>In the first step, that of forming a clear idea how a<span class="pagenum" id="Page_232">[Pg 232]</span> -plane being would represent three-dimensional space, only -one of the catalogue cubes and one of the three blocks is -needed.</p> - - -<h3><span class="smcap">Application to the Step from Plane to Solid.</span></h3> - -<p>Look at <a href="#fig_1">fig. 1</a> of the views of the tesseract, or, what -comes to the same thing, take catalogue cube No. 1 and -place it before you with the red line running up, the -white line running to the right, the yellow line running -away. The three dimensions of space are then marked -out by these lines or axes. Now take a piece of cardboard, -or a book, and place it so that it forms a wall -extending up and down not opposite to you, but running -away parallel to the wall of the room on your -left hand.</p> - -<p>Placing the catalogue cube against this wall we see -that it comes into contact with it by the red and yellow -lines, and by the included orange face.</p> - -<p>In the plane being’s world the aspect he has of the -cube would be a square surrounded by red and yellow -lines with grey points.</p> - -<p>Now, keeping the red line fixed, turn the cube about it -so that the yellow line goes out to the right, and the -white line comes into contact with the plane.</p> - -<p>In this case a different aspect is presented to the plane -being, a square, namely, surrounded by red and white -lines and grey points. You should particularly notice -that when the yellow line goes out, at right angles to the -plane, and the white comes in, the latter does not run in -the same sense that the yellow did.</p> - -<p>From the fixed grey point at the base of the red line -the yellow line ran away from you. The white line now -runs towards you. This turning at right angles makes -the line which was out of the plane before, come into it<span class="pagenum" id="Page_233">[Pg 233]</span> -in an opposite sense to that in which the line ran which -has just left the plane. If the cube does not break -through the plane this is always the rule.</p> - -<p>Again turn the cube back to the normal position with -red running up, white to the right, and yellow away, and -try another turning.</p> - -<p>You can keep the yellow line fixed, and turn the cube -about it. In this case the red line going out to the -right the white line will come in pointing downwards.</p> - -<p>You will be obliged to elevate the cube from the table -in order to carry out this turning. It is always necessary -when a vertical axis goes out of a space to imagine a -movable support which will allow the line which ran out -before to come in below.</p> - -<p>Having looked at the three ways of turning the cube -so as to present different faces to the plane, examine what -would be the appearance if a square hole were cut in the -piece of cardboard, and the cube were to pass through it. -A hole can be actually cut, and it will be seen that in the -normal position, with red axis running up, yellow away, -and white to the right, the square first perceived by the -plane being—the one contained by red and yellow lines—would -be replaced by another square of which the line -towards you is pink—the section line of the pink face. -The line above is light yellow, below is light yellow and -on the opposite side away from you is pink.</p> - -<p>In the same way the cube can be pushed through a -square opening in the plane from any of the positions -which you have already turned it into. In each case -the plane being will perceive a different set of contour -lines.</p> - -<p>Having observed these facts about the catalogue cube, -turn now to the first block of twenty-seven cubes.</p> - -<p>You notice that the colour scheme on the catalogue cube -and that of this set of blocks is the same.</p> - -<p><span class="pagenum" id="Page_234">[Pg 234]</span></p> - -<p>Place them before you, a grey or null cube on the -table, above it a red cube, and on the top a null cube -again. Then away from you place a yellow cube, and -beyond it a null cube. Then to the right place a white -cube and beyond it another null. Then complete the -block, according to the scheme of the catalogue cube, -putting in the centre of all an ochre cube.</p> - -<p>You have now a cube like that which is described in -the text. For the sake of simplicity, in some cases, this -cubic block can be reduced to one of eight cubes, by -leaving out the terminations in each direction. Thus, -instead of null, red, null, three cubes, you can take null, -red, two cubes, and so on.</p> - -<p>It is useful, however, to practise the representation in -a plane of a block of twenty-seven cubes. For this -purpose take the slabs, and build them up against the -piece of cardboard, or the book in such a way as to -represent the different aspects of the cube.</p> - -<p>Proceed as follows:—</p> - -<p>First, cube in normal position.</p> - -<p>Place nine slabs against the cardboard to represent the -nine cubes in the wall of the red and yellow axes, facing -the cardboard; these represent the aspect of the cube as it -touches the plane.</p> - -<p>Now push these along the cardboard and make a -different set of nine slabs to represent the appearance -which the cube would present to a plane being, if it were -to pass half way through the plane.</p> - -<p>There would be a white slab, above it a pink one, above -that another white one, and six others, representing what -would be the nature of a section across the middle of the -block of cubes. The section can be thought of as a thin -slice cut out by two parallel cuts across the cube. -Having arranged these nine slabs, push them along the -plane, and make another set of nine to represent what<span class="pagenum" id="Page_235">[Pg 235]</span> -would be the appearance of the cube when it had almost -completely gone through. This set of nine will be the -same as the first set of nine.</p> - -<p>Now we have in the plane three sets of nine slabs -each, which represent three sections of the twenty-seven -block.</p> - -<p>They are put alongside one another. We see that it -does not matter in what order the sets of nine are put. -As the cube passes through the plane they represent appearances -which follow the one after the other. If they -were what they represented, they could not exist in the -same plane together.</p> - -<p>This is a rather important point, namely, to notice that -they should not co-exist on the plane, and that the order -in which they are placed is indifferent. When we -represent a four-dimensional body our solid cubes are to -us in the same position that the slabs are to the plane -being. You should also notice that each of these slabs -represents only the very thinnest slice of a cube. The -set of nine slabs first set up represents the side surface of -the block. It is, as it were, a kind of tray—a beginning -from which the solid cube goes off. The slabs as we use -them have thickness, but this thickness is a necessity of -construction. They are to be thought of as merely of the -thickness of a line.</p> - -<p>If now the block of cubes passed through the plane at -the rate of an inch a minute the appearance to a plane -being would be represented by:—</p> - -<p>1. The first set of nine slabs lasting for one minute.</p> - -<p>2. The second set of nine slabs lasting for one minute.</p> - -<p>3. The third set of nine slabs lasting for one minute.</p> - -<p>Now the appearances which the cube would present -to the plane being in other positions can be shown by -means of these slabs. The use of such slabs would be -the means by which a plane being could acquire a<span class="pagenum" id="Page_236">[Pg 236]</span> -familiarity with our cube. Turn the catalogue cube (or -imagine the coloured figure turned) so that the red line -runs up, the yellow line out to the right, and the white -line towards you. Then turn the block of cubes to -occupy a similar position.</p> - -<p>The block has now a different wall in contact with -the plane. Its appearance to a plane being will not be -the same as before. He has, however, enough slabs to -represent this new set of appearances. But he must -remodel his former arrangement of them.</p> - -<p>He must take a null, a red, and a null slab from the first -of his sets of slabs, then a white, a pink, and a white from -the second, and then a null, a red, and a null from the -third set of slabs.</p> - -<p>He takes the first column from the first set, the first -column from the second set, and the first column from -the third set.</p> - -<p>To represent the half-way-through appearance, which -is as if a very thin slice were cut out half way through the -block, he must take the second column of each of his -sets of slabs, and to represent the final appearance, the -third column of each set.</p> - -<p>Now turn the catalogue cube back to the normal -position, and also the block of cubes.</p> - -<p>There is another turning—a turning about the yellow -line, in which the white axis comes below the support.</p> - -<p>You cannot break through the surface of the table, so -you must imagine the old support to be raised. Then -the top of the block of cubes in its new position is at the -level at which the base of it was before.</p> - -<p>Now representing the appearance on the plane, we must -draw a horizontal line to represent the old base. The -line should be drawn three inches high on the cardboard.</p> - -<p>Below this the representative slabs can be arranged.</p> - -<p>It is easy to see what they are. The old arrangements<span class="pagenum" id="Page_237">[Pg 237]</span> -have to be broken up, and the layers taken in order, the -first layer of each for the representation of the aspect of -the block as it touches the plane.</p> - -<p>Then the second layers will represent the appearance -half way through, and the third layers will represent the -final appearance.</p> - -<p>It is evident that the slabs individually do not represent -the same portion of the cube in these different presentations.</p> - -<p>In the first case each slab represents a section or a face -perpendicular to the white axis, in the second case a -face or a section which runs perpendicularly to the yellow -axis, and in the third case a section or a face perpendicular -to the red axis.</p> - -<p>But by means of these nine slabs the plane being can -represent the whole of the cubic block. He can touch -and handle each portion of the cubic block, there is no -part of it which he cannot observe. Taking it bit by bit, -two axes at a time, he can examine the whole of it.</p> - - -<h3><span class="smcap">Our Representation of a Block of Tesseracts.</span></h3> - -<p>Look at the views of the tesseract 1, 2, 3, or take the -catalogue cubes 1, 2, 3, and place them in front of you, -in any order, say running from left to right, placing 1 in -the normal position, the red axis running up, the white -to the right, and yellow away.</p> - -<p>Now notice that in catalogue cube 2 the colours of each -region are derived from those of the corresponding region -of cube 1 by the addition of blue. Thus null + blue = -blue, and the corners of number 2 are blue. Again, -red + blue = purple, and the vertical lines of 2 are purple. -Blue + yellow = green, and the line which runs away is -coloured green.</p> - -<p>By means of these observations you may be sure that<span class="pagenum" id="Page_238">[Pg 238]</span> -catalogue cube 2 is rightly placed. Catalogue cube 3 is -just like number 1.</p> - -<p>Having these cubes in what we may call their normal -position, proceed to build up the three sets of blocks.</p> - -<p>This is easily done in accordance with the colour scheme -on the catalogue cubes.</p> - -<p>The first block we already know. Build up the second -block, beginning with a blue corner cube, placing a purple -on it, and so on.</p> - -<p>Having these three blocks we have the means of -representing the appearances of a group of eighty-one -tesseracts.</p> - -<p>Let us consider a moment what the analogy in the case -of the plane being is.</p> - -<p>He has his three sets of nine slabs each. We have our -three sets of twenty-seven cubes each.</p> - -<p>Our cubes are like his slabs. As his slabs are not the -things which they represent to him, so our cubes are not -the things they represent to us.</p> - -<p>The plane being’s slabs are to him the faces of cubes.</p> - -<p>Our cubes then are the faces of tesseracts, the cubes by -which they are in contact with our space.</p> - -<p>As each set of slabs in the case of the plane being -might be considered as a sort of tray from which the solid -contents of the cubes came out, so our three blocks of -cubes may be considered as three-space trays, each of -which is the beginning of an inch of the solid contents -of the four-dimensional solids starting from them.</p> - -<p>We want now to use the names null, red, white, etc., -for tesseracts. The cubes we use are only tesseract faces. -Let us denote that fact by calling the cube of null colour, -null face; or, shortly, null f., meaning that it is the face -of a tesseract.</p> - -<p>To determine which face it is let us look at the catalogue -cube 1 or the first of the views of the tesseract, which<span class="pagenum" id="Page_239">[Pg 239]</span> -can be used instead of the models. It has three axes, -red, white, yellow, in our space. Hence the cube determined -by these axes is the face of the tesseract which we -now have before us. It is the ochre face. It is enough, -however, simply to say null f., red f. for the cubes which -we use.</p> - -<p>To impress this in your mind, imagine that tesseracts -do actually run from each cube. Then, when you move the -cubes about, you move the tesseracts about with them. -You move the face but the tesseract follows with it, as the -cube follows when its face is shifted in a plane.</p> - -<p>The cube null in the normal position is the cube which -has in it the red, yellow, white axes. It is the face -having these, but wanting the blue. In this way you can -define which face it is you are handling. I will write an -“f.” after the name of each tesseract just as the plane -being might call each of his slabs null slab, yellow slab, -etc., to denote that they were representations.</p> - -<p>We have then in the first block of twenty-seven cubes, -the following—null f., red f., null f., going up; white f., null -f., lying to the right, and so on. Starting from the null -point and travelling up one inch we are in the null region, -the same for the away and the right-hand directions. -And if we were to travel in the fourth dimension for an -inch we should still be in a null region. The tesseract -stretches equally all four ways. Hence the appearance we -have in this first block would do equally well if the -tesseract block were to move across our space for a certain -distance. For anything less than an inch of their transverse -motion we should still have the same appearance. -You must notice, however, that we should not have null -face after the motion had begun.</p> - -<p>When the tesseract, null for instance, had moved ever -so little we should not have a face of null but a section of -null in our space. Hence, when we think of the motion<span class="pagenum" id="Page_240">[Pg 240]</span> -across our space we must call our cubes tesseract sections. -Thus on null passing across we should see first null f., then -null s., and then, finally, null f. again.</p> - -<p>Imagine now the whole first block of twenty-seven -tesseracts to have moved tranverse to our space a distance -of one inch. Then the second set of tesseracts, which -originally were an inch distant from our space, would be -ready to come in.</p> - -<p>Their colours are shown in the second block of twenty-seven -cubes which you have before you. These represent -the tesseract faces of the set of tesseracts that lay before -an inch away from our space. They are ready now to -come in, and we can observe their colours. In the place -which null f. occupied before we have blue f., in place of -red f. we have purple f., and so on. Each tesseract is -coloured like the one whose place it takes in this motion -with the addition of blue.</p> - -<p>Now if the tesseract block goes on moving at the rate -of an inch a minute, this next set of tesseracts will occupy -a minute in passing across. We shall see, to take the null -one for instance, first of all null face, then null section, -then null face again.</p> - -<p>At the end of the second minute the second set of -tesseracts has gone through, and the third set comes in. -This, as you see, is coloured just like the first. Altogether, -these three sets extend three inches in the fourth dimension, -making the tesseract block of equal magnitude in all -dimensions.</p> - -<p>We have now before us a complete catalogue of all the -tesseracts in our group. We have seen them all, and we -shall refer to this arrangement of the blocks as the -“normal position.” We have seen as much of each -tesseract at a time as could be done in a three-dimensional -space. Each part of each tesseract has been in -our space, and we could have touched it.</p> - -<p><span class="pagenum" id="Page_241">[Pg 241]</span></p> - -<p>The fourth dimension appeared to us as the duration -of the block.</p> - -<p>If a bit of our matter were to be subjected to the same -motion it would be instantly removed out of our space. -Being thin in the fourth dimension it is at once taken -out of our space by a motion in the fourth dimension.</p> - -<p>But the tesseract block we represent having length in -the fourth dimension remains steadily before our eyes for -three minutes, when it is subjected to this transverse -motion.</p> - -<p>We have now to form representations of the other -views of the same tesseract group which are possible in -our space.</p> - -<p>Let us then turn the block of tesseracts so that another -face of it comes into contact with our space, and then -by observing what we have, and what changes come when -the block traverses our space, we shall have another view -of it. The dimension which appeared as duration before -will become extension in one of our known dimensions, -and a dimension which coincided with one of our space -dimensions will appear as duration.</p> - -<p>Leaving catalogue cube 1 in the normal position, -remove the other two, or suppose them removed. We -have in space the red, the yellow, and the white axes. -Let the white axis go out into the unknown, and occupy -the position the blue axis holds. Then the blue axis, -which runs in that direction now will come into space. -But it will not come in pointing in the same way that -the white axis does now. It will point in the opposite -sense. It will come in running to the left instead of -running to the right as the white axis does now.</p> - -<p>When this turning takes place every part of the cube 1 -will disappear except the left-hand face—the orange face.</p> - -<p>And the new cube that appears in our space will run to -the left from this orange face, having axes, red, yellow, blue.</p> - -<p><span class="pagenum" id="Page_242">[Pg 242]</span></p> - -<p>Take models 4, 5, 6. Place 4, or suppose No. 4 of the -tesseract views placed, with its orange face coincident with -the orange face of 1, red line to red line, and yellow line -to yellow line, with the blue line pointing to the left. -Then remove cube 1 and we have the tesseract face -which comes in when the white axis runs in the positive -unknown, and the blue axis comes into our space.</p> - -<p>Now place catalogue cube 5 in some position, it does -not matter which, say to the left; and place it so that -there is a correspondence of colour corresponding to the -colour of the line that runs out of space. The line that -runs out of space is white, hence, every part of this -cube 5 should differ from the corresponding part of 4 by -an alteration in the direction of white.</p> - -<p>Thus we have white points in 5 corresponding to the -null points in 4. We have a pink line corresponding to -a red line, a light yellow line corresponding to a yellow -line, an ochre face corresponding to an orange face. This -cube section is completely named in Chapter XI. Finally -cube 6 is a replica of 1.</p> - -<p>These catalogue cubes will enable us to set up our -models of the block of tesseracts.</p> - -<p>First of all for the set of tesseracts, which beginning -in our space reach out one inch in the unknown, we have -the pattern of catalogue cube 4.</p> - -<p>We see that we can build up a block of twenty-seven -tesseract faces after the colour scheme of cube 4, by -taking the left-hand wall of block 1, then the left-hand -wall of block 2, and finally that of block 3. We take, -that is, the three first walls of our previous arrangement -to form the first cubic block of this new one.</p> - -<p>This will represent the cubic faces by which the group -of tesseracts in its new position touches our space. -We have running up, null f., red f., null f. In the next -vertical line, on the side remote from us, we have yellow f.,<span class="pagenum" id="Page_243">[Pg 243]</span> -orange f., yellow f., and then the first colours over again. -Then the three following columns are, blue f., purple f., -blue f.; green f., brown f., green f.; blue f., purple f., blue f. -The last three columns are like the first.</p> - -<p>These tesseracts touch our space, and none of them are -by any part of them distant more than an inch from it. -What lies beyond them in the unknown?</p> - -<p>This can be told by looking at catalogue cube 5. -According to its scheme of colour we see that the second -wall of each of our old arrangements must be taken. -Putting them together we have, as the corner, white f. -above it, pink f. above it, white f. The column next to -this remote from us is as follows:—light yellow f., ochre f., -light yellow f., and beyond this a column like the first. -Then for the middle of the block, light blue f., above -it light purple, then light blue. The centre column has, -at the bottom, light green f., light brown f. in the centre -and at the top light green f. The last wall is like the -first.</p> - -<p>The third block is made by taking the third walls of -our previous arrangement, which we called the normal -one.</p> - -<p>You may ask what faces and what sections our cubes -represent. To answer this question look at what axes -you have in our space. You have red, yellow, blue. -Now these determine brown. The colours red, -yellow, blue are supposed by us when mixed to produce -a brown colour. And that cube which is determined -by the red, yellow, blue axes we call the brown cube.</p> - -<p>When the tesseract block in its new position begins to -move across our space each tesseract in it gives a section -in our space. This section is transverse to the white -axis, which now runs in the unknown.</p> - -<p>As the tesseract in its present position passes across -our space, we should see first of all the first of the blocks<span class="pagenum" id="Page_244">[Pg 244]</span> -of cubic faces we have put up—these would last for a -minute, then would come the second block and then the -third. At first we should have a cube of tesseract faces, -each of which would be brown. Directly the movement -began, we should have tesseract sections transverse to the -white line.</p> - -<p>There are two more analogous positions in which the -block of tesseracts can be placed. To find the third -position, restore the blocks to the normal arrangement.</p> - -<p>Let us make the yellow axis go out into the positive -unknown, and let the blue axis, consequently, come in -running towards us. The yellow ran away, so the blue -will come in running towards us.</p> - -<p>Put catalogue cube 1 in its normal position. Take -catalogue cube 7 and place it so that its pink face -coincides with the pink face of cube 1, making also its -red axis coincide with the red axis of 1 and its white -with the white. Moreover, make cube 7 come -towards us from cube 1. Looking at it we see in our -space, red, white, and blue axes. The yellow runs out. -Place catalogue cube 8 in the neighbourhood of -7—observe that every region in 8 has a change in -the direction of yellow from the corresponding region -in 7. This is because it represents what you come -to now in going in the unknown, when the yellow axis -runs out of our space. Finally catalogue cube 9, -which is like number 7, shows the colours of the third -set of tesseracts. Now evidently, starting from the -normal position, to make up our three blocks of tesseract -faces we have to take the near wall from the first block, -the near wall from the second, and then the near wall -from the third block. This gives us the cubic block -formed by the faces of the twenty-seven tesseracts which -are now immediately touching our space.</p> - -<p>Following the colour scheme of catalogue cube 8,<span class="pagenum" id="Page_245">[Pg 245]</span> -we make the next set of twenty-seven tesseract faces, -representing the tesseracts, each of which begins one inch -off from our space, by putting the second walls of our -previous arrangement together, and the representation -of the third set of tesseracts is the cubic block formed of -the remaining three walls.</p> - -<p>Since we have red, white, blue axes in our space to -begin with, the cubes we see at first are light purple -tesseract faces, and after the transverse motion begins -we have cubic sections transverse to the yellow line.</p> - -<p>Restore the blocks to the normal position, there -remains the case in which the red axis turns out of -space. In this case the blue axis will come in downwards, -opposite to the sense in which the red axis ran.</p> - -<p>In this case take catalogue cubes 10, 11, 12. Lift up -catalogue cube 1 and put 10 underneath it, imagining -that it goes down from the previous position of 1.</p> - -<p>We have to keep in space the white and the yellow -axes, and let the red go out, the blue come in.</p> - -<p>Now, you will find on cube 10 a light yellow face; this -should coincide with the base of 1, and the white and -yellow lines on the two cubes should coincide. Then the -blue axis running down you have the catalogue cube -correctly placed, and it forms a guide for putting up the -first representative block.</p> - -<p>Catalogue cube 11 will represent what lies in the fourth -dimension—now the red line runs in the fourth dimension. -Thus the change from 10 to 11 should be towards -red, corresponding to a null point is a red point, to a -white line is a pink line, to a yellow line an orange -line, and so on.</p> - -<p>Catalogue cube 12 is like 10. Hence we see that to -build up our blocks of tesseract faces we must take the -bottom layer of the first block, hold that up in the air, -underneath it place the bottom layer of the second block,<span class="pagenum" id="Page_246">[Pg 246]</span> -and finally underneath this last the bottom layer of the -last of our normal blocks.</p> - -<p>Similarly we make the second representative group by -taking the middle courses of our three blocks. The last -is made by taking the three topmost layers. The three -axes in our space before the transverse motion begins are -blue, white, yellow, so we have light green tesseract -faces, and after the motion begins sections transverse to -the red light.</p> - -<p>These three blocks represent the appearances as the -tesseract group in its new position passes across our space. -The cubes of contact in this case are those determinal by -the three axes in our space, namely, the white, the -yellow, the blue. Hence they are light green.</p> - -<p>It follows from this that light green is the interior -cube of the first block of representative cubic faces.</p> - -<p>Practice in the manipulations described, with a -realization in each case of the face or section which -is in our space, is one of the best means of a thorough -comprehension of the subject.</p> - -<p>We have to learn how to get any part of these four-dimensional -figures into space, so that we can look at -them. We must first learn to swing a tesseract, and a -group of tesseracts about in any way.</p> - -<p>When these operations have been repeated and the -method of arrangement of the set of blocks has become -familiar, it is a good plan to rotate the axes of the normal -cube 1 about a diagonal, and then repeat the whole series -of turnings.</p> - -<p>Thus, in the normal position, red goes up, white to the -right, yellow away. Make white go up, yellow to the right, -and red away. Learn the cube in this position by putting -up the set of blocks of the normal cube, over and over -again till it becomes as familiar to you as in the normal -position. Then when this is learned, and the corre<span class="pagenum" id="Page_247">[Pg 247]</span>sponding -changes in the arrangements of the tesseract -groups are made, another change should be made: let, -in the normal cube, yellow go up, red to the right, and -white away.</p> - -<p>Learn the normal block of cubes in this new position -by arranging them and re-arranging them till you know -without thought where each one goes. Then carry out -all the tesseract arrangements and turnings.</p> - -<p>If you want to understand the subject, but do not see -your way clearly, if it does not seem natural and easy to -you, practise these turnings. Practise, first of all, the -turning of a block of cubes round, so that you know it -in every position as well as in the normal one. Practise -by gradually putting up the set of cubes in their new -arrangements. Then put up the tesseract blocks in their -arrangements. This will give you a working conception -of higher space, you will gain the feeling of it, whether -you take up the mathematical treatment of it or not.</p> -<hr class="chap x-ebookmaker-drop" /> - -<div class="chapter"> -<p><span class="pagenum" id="Page_248">[Pg 248]</span></p> - -<h2 class="nobreak" id="APPENDIX_II">APPENDIX II<br /> - -<small><i>A LANGUAGE OF SPACE</i></small></h2></div> - - -<p>The mere naming the parts of the figures we consider -involves a certain amount of time and attention. This -time and attention leads to no result, for with each -new figure the nomenclature applied is completely -changed, every letter or symbol is used in a different -significance.</p> - -<p>Surely it must be possible in some way to utilise the -labour thus at present wasted!</p> - -<p>Why should we not make a language for space itself, so -that every position we want to refer to would have its own -name? Then every time we named a figure in order to -demonstrate its properties we should be exercising -ourselves in the vocabulary of place.</p> - -<p>If we use a definite system of names, and always refer -to the same space position by the same name, we create -as it were a multitude of little hands, each prepared to -grasp a special point, position, or element, and hold it -for us in its proper relations.</p> - -<p>We make, to use another analogy, a kind of mental -paper, which has somewhat of the properties of a sensitive -plate, in that it will register, without effort, complex, -visual, or tactual impressions.</p> - -<p>But of far more importance than the applications of a -space language to the plane and to solid space is the<span class="pagenum" id="Page_249">[Pg 249]</span> -facilitation it brings with it to the study of four-dimensional -shapes.</p> - -<p>I have delayed introducing a space language because -all the systems I made turned out, after giving them a -fair trial, to be intolerable. I have now come upon one -which seems to present features of permanence, and I will -here give an outline of it, so that it can be applied to -the subject of the text, and in order that it may be -subjected to criticism.</p> - -<p>The principle on which the language is constructed is -to sacrifice every other consideration for brevity.</p> - -<p>It is indeed curious that we are able to talk and -converse on every subject of thought except the fundamental -one of space. The only way of speaking about -the spatial configurations that underlie every subject -of discursive thought is a co-ordinate system of numbers. -This is so awkward and incommodious that it is never -used. In thinking also, in realising shapes, we do not -use it; we confine ourselves to a direct visualisation.</p> - -<p>Now, the use of words corresponds to the storing up -of our experience in a definite brain structure. A child, -in the endless tactual, visual, mental manipulations it -makes for itself, is best left to itself, but in the course -of instruction the introduction of space names would -make the teachers work more cumulative, and the child’s -knowledge more social.</p> - -<p>Their full use can only be appreciated, if they are -introduced early in the course of education; but in a -minor degree any one can convince himself of their -utility, especially in our immediate subject of handling -four-dimensional shapes. The sum total of the results -obtained in the preceding pages can be compendiously and -accurately expressed in nine words of the Space Language.</p> - -<p>In one of Plato’s dialogues Socrates makes an experiment -on a slave boy standing by. He makes certain<span class="pagenum" id="Page_250">[Pg 250]</span> -perceptions of space awake in the mind of Meno’s slave -by directing his close attention on some simple facts of -geometry.</p> - -<p>By means of a few words and some simple forms we can -repeat Plato’s experiment on new ground.</p> - -<p>Do we by directing our close attention on the facts of -four dimensions awaken a latent faculty in ourselves? -The old experiment of Plato’s, it seems to me, has come -down to us as novel as on the day he incepted it, and its -significance not better understood through all the discussion -of which it has been the subject.</p> - -<p>Imagine a voiceless people living in a region where -everything had a velvety surface, and who were thus -deprived of all opportunity of experiencing what sound is. -They could observe the slow pulsations of the air caused -by their movements, and arguing from analogy, they -would no doubt infer that more rapid vibrations were -possible. From the theoretical side they could determine -all about these more rapid vibrations. They merely differ, -they would say, from slower ones, by the number that -occur in a given time; there is a merely formal difference.</p> - -<p>But suppose they were to take the trouble, go to the -pains of producing these more rapid vibrations, then a -totally new sensation would fall on their rudimentary ears. -Probably at first they would only be dimly conscious of -Sound, but even from the first they would become aware -that a merely formal difference, a mere difference in point -of number in this particular respect, made a great difference -practically, as related to them. And to us the difference -between three and four dimensions is merely formal, -numerical. We can tell formally all about four dimensions, -calculate the relations that would exist. But that the -difference is merely formal does not prove that it is a -futile and empty task, to present to ourselves as closely as -we can the phenomena of four dimensions. In our formal<span class="pagenum" id="Page_251">[Pg 251]</span> -knowledge of it, the whole question of its actual relation -to us, as we are, is left in abeyance.</p> - -<p>Possibly a new apprehension of nature may come to us -through the practical, as distinguished from the mathematical -and formal, study of four dimensions. As a child -handles and examines the objects with which he comes in -contact, so we can mentally handle and examine four-dimensional -objects. The point to be determined is this. -Do we find something cognate and natural to our faculties, -or are we merely building up an artificial presentation of -a scheme only formally possible, conceivable, but which -has no real connection with any existing or possible -experience?</p> - -<p>This, it seems to me, is a question which can only be -settled by actually trying. This practical attempt is the -logical and direct continuation of the experiment Plato -devised in the “Meno.”</p> - -<p>Why do we think true? Why, by our processes of -thought, can we predict what will happen, and correctly -conjecture the constitution of the things around us? -This is a problem which every modern philosopher has -considered, and of which Descartes, Leibnitz, Kant, to -name a few, have given memorable solutions. Plato was -the first to suggest it. And as he had the unique position -of being the first devisor of the problem, so his solution -is the most unique. Later philosophers have talked about -consciousness and its laws, sensations, categories. But -Plato never used such words. Consciousness apart from a -conscious being meant nothing to him. His was always -an objective search. He made man’s intuitions the basis -of a new kind of natural history.</p> - -<p>In a few simple words Plato puts us in an attitude -with regard to psychic phenomena—the mind—the ego—“what -we are,” which is analogous to the attitude scientific -men of the present day have with regard to the phenomena<span class="pagenum" id="Page_252">[Pg 252]</span> -of outward nature. Behind this first apprehension of ours -of nature, there is an infinite depth to be learned and -known. Plato said that behind the phenomena of mind -that Meno’s slave boy exhibited, there was a vast, an -infinite perspective. And his singularity, his originality, -comes out most strongly marked in this, that the perspective, -the complex phenomena beyond were, according -to him, phenomena of personal experience. A footprint -in the sand means a man to a being that has the conception -of a man. But to a creature that has no such -conception, it means a curious mark, somehow resulting -from the concatenation of ordinary occurrences. Such a -being would attempt merely to explain how causes known -to him could so coincide as to produce such a result; -he would not recognise its significance.</p> - -<p>Plato introduced the conception which made a new -kind of natural history possible. He said that Meno’s -slave boy thought true about things he had never -learned, because his “soul” had experience. I know this -will sound absurd to some people, and it flies straight -in the face of the maxim, that explanation consists in -showing how an effect depends on simple causes. But -what a mistaken maxim that is! Can any single instance -be shown of a simple cause? Take the behaviour of -spheres for instance; say those ivory spheres, billiard balls, -for example. We can explain their behaviour by supposing -they are homogeneous elastic solids. We can give formulæ -which will account for their movements in every variety. -But are they homogeneous elastic solids? No, certainly -not. They are complex in physical and molecular structure, -and atoms and ions beyond open an endless vista. Our -simple explanation is false, false as it can be. The balls -act as if they were homogeneous elastic spheres. There is -a statistical simplicity in the resultant of very complex -conditions, which makes that artificial conception useful.<span class="pagenum" id="Page_253">[Pg 253]</span> -But its usefulness must not blind us to the fact that it is -artificial. If we really look deep into nature, we find a -much greater complexity than we at first suspect. And -so behind this simple “I,” this myself, is there not a -parallel complexity? Plato’s “soul” would be quite -acceptable to a large class of thinkers, if by “soul” and -the complexity he attributes to it, he meant the product -of a long course of evolutionary changes, whereby simple -forms of living matter endowed with rudimentary sensation -had gradually developed into fully conscious beings.</p> - -<p>But Plato does not mean by “soul” a being of such a -kind. His soul is a being whose faculties are clogged by -its bodily environment, or at least hampered by the -difficulty of directing its bodily frame—a being which -is essentially higher than the account it gives of itself -through its organs. At the same time Plato’s soul is -not incorporeal. It is a real being with a real experience. -The question of whether Plato had the conception of non-spatial -existence has been much discussed. The verdict -is, I believe, that even his “ideas” were conceived by him -as beings in space, or, as we should say, real. Plato’s -attitude is that of Science, inasmuch as he thinks of a -world in Space. But, granting this, it cannot be denied -that there is a fundamental divergence between Plato’s -conception and the evolutionary theory, and also an -absolute divergence between his conception and the -genetic account of the origin of the human faculties. -The functions and capacities of Plato’s “soul” are not -derived by the interaction of the body and its environment.</p> - -<p>Plato was engaged on a variety of problems, and his -religious and ethical thoughts were so keen and fertile -that the experimental investigation of his soul appears -involved with many other motives. In one passage Plato -will combine matter of thought of all kinds and from all -sources, overlapping, interrunning. And in no case is he<span class="pagenum" id="Page_254">[Pg 254]</span> -more involved and rich than in this question of the soul. -In fact, I wish there were two words, one denoting that -being, corporeal and real, but with higher faculties than -we manifest in our bodily actions, which is to be taken as -the subject of experimental investigation; and the other -word denoting “soul” in the sense in which it is made -the recipient and the promise of so much that men desire. -It is the soul in the former sense that I wish to investigate, -and in a limited sphere only. I wish to find out, in continuation -of the experiment in the Meno, what the “soul” -in us thinks about extension, experimenting on the -grounds laid down by Plato. He made, to state the -matter briefly, the hypothesis with regard to the thinking -power of a being in us, a “soul.” This soul is not accessible -to observation by sight or touch, but it can be -observed by its functions; it is the object of a new kind -of natural history, the materials for constructing which -lie in what it is natural to us to think. With Plato -“thought” was a very wide-reaching term, but still I -would claim in his general plan of procedure a place for -the particular question of extension.</p> - -<p>The problem comes to be, “What is it natural to us to -think about matter <i>qua</i> extended?”</p> - -<p>First of all, I find that the ordinary intuition of any -simple object is extremely imperfect. Take a block of -differently marked cubes, for instance, and become acquainted -with them in their positions. You may think -you know them quite well, but when you turn them round—rotate -the block round a diagonal, for instance—you -will find that you have lost track of the individuals in -their new positions. You can mentally construct the -block in its new position, by a rule, by taking the remembered -sequences, but you don’t know it intuitively. By -observation of a block of cubes in various positions, and -very expeditiously by a use of Space names applied to the<span class="pagenum" id="Page_255">[Pg 255]</span> -cubes in their different presentations, it is possible to get -an intuitive knowledge of the block of cubes, which is not -disturbed by any displacement. Now, with regard to this -intuition, we moderns would say that I had formed it by -my tactual visual experiences (aided by hereditary pre-disposition). -Plato would say that the soul had been -stimulated to recognise an instance of shape which it -knew. Plato would consider the operation of learning -merely as a stimulus; we as completely accounting for -the result. The latter is the more common-sense view. -But, on the other hand, it presupposes the generation of -experience from physical changes. The world of sentient -experience, according to the modern view, is closed and -limited; only the physical world is ample and large and -of ever-to-be-discovered complexity. Plato’s world of soul, -on the other hand, is at least as large and ample as the -world of things.</p> - -<p>Let us now try a crucial experiment. Can I form an -intuition of a four-dimensional object? Such an object -is not given in the physical range of my sense contacts. -All I can do is to present to myself the sequences of solids, -which would mean the presentation to me under my conditions -of a four-dimensional object. All I can do is to -visualise and tactualise different series of solids which are -alternative sets of sectional views of a four-dimensional -shape.</p> - -<p>If now, on presenting these sequences, I find a power -in me of intuitively passing from one of these sets of -sequences to another, of, being given one, intuitively -constructing another, not using a rule, but directly apprehending -it, then I have found a new fact about my soul, -that it has a four-dimensional experience; I have observed -it by a function it has.</p> - -<p>I do not like to speak positively, for I might occasion -a loss of time on the part of others, if, as may very well<span class="pagenum" id="Page_256">[Pg 256]</span> -be, I am mistaken. But for my own part, I think there -are indications of such an intuition; from the results of -my experiments, I adopt the hypothesis that that which -thinks in us has an ample experience, of which the intuitions -we use in dealing with the world of real objects -are a part; of which experience, the intuition of four-dimensional -forms and motions is also a part. The process -we are engaged in intellectually is the reading the obscure -signals of our nerves into a world of reality, by means of -intuitions derived from the inner experience.</p> - -<p>The image I form is as follows. Imagine the captain -of a modern battle-ship directing its course. He has -his charts before him; he is in communication with his -associates and subordinates; can convey his messages and -commands to every part of the ship, and receive information -from the conning-tower and the engine-room. Now -suppose the captain immersed in the problem of the -navigation of his ship over the ocean, to have so absorbed -himself in the problem of the direction of his craft over -the plane surface of the sea that he forgets himself. All -that occupies his attention is the kind of movement that -his ship makes. The operations by which that movement -is produced have sunk below the threshold of his consciousness, -his own actions, by which he pushes the buttons, -gives the orders, are so familiar as to be automatic, his -mind is on the motion of the ship as a whole. In such -a case we can imagine that he identifies himself with his -ship; all that enters his conscious thought is the direction -of its movement over the plane surface of the ocean.</p> - -<p>Such is the relation, as I imagine it, of the soul to the -body. A relation which we can imagine as existing -momentarily in the case of the captain is the normal -one in the case of the soul with its craft. As the captain -is capable of a kind of movement, an amplitude of motion, -which does not enter into his thoughts with regard to the<span class="pagenum" id="Page_257">[Pg 257]</span> -directing the ship over the plane surface of the ocean, so -the soul is capable of a kind of movement, has an amplitude -of motion, which is not used in its task of directing -the body in the three-dimensional region in which the -body’s activity lies. If for any reason it became necessary -for the captain to consider three-dimensional motions with -regard to his ship, it would not be difficult for him to -gain the materials for thinking about such motions; all -he has to do is to call his own intimate experience into -play. As far as the navigation of the ship, however, is -concerned, he is not obliged to call on such experience. -The ship as a whole simply moves on a surface. The -problem of three-dimensional movement does not ordinarily -concern its steering. And thus with regard to ourselves -all those movements and activities which characterise our -bodily organs are three-dimensional; we never need to -consider the ampler movements. But we do more than -use the movements of our body to effect our aims by -direct means; we have now come to the pass when we act -indirectly on nature, when we call processes into play -which lie beyond the reach of any explanation we can -give by the kind of thought which has been sufficient for -the steering of our craft as a whole. When we come to -the problem of what goes on in the minute, and apply -ourselves to the mechanism of the minute, we find our -habitual conceptions inadequate.</p> - -<p>The captain in us must wake up to his own intimate -nature, realise those functions of movement which are his -own, and in virtue of his knowledge of them apprehend -how to deal with the problems he has come to.</p> - -<p>Think of the history of man. When has there been a -time, in which his thoughts of form and movement were -not exclusively of such varieties as were adapted for his -bodily performance? We have never had a demand to -conceive what our own most intimate powers are. But,<span class="pagenum" id="Page_258">[Pg 258]</span> -just as little as by immersing himself in the steering of -his ship over the plane surface of the ocean, a captain -can lose the faculty of thinking about what he actually -does, so little can the soul lose its own nature. It -can be roused to an intuition that is not derived from -the experience which the senses give. All that is -necessary is to present some few of those appearances -which, while inconsistent with three-dimensional matter, -are yet consistent with our formal knowledge of four-dimensional -matter, in order for the soul to wake up and -not begin to learn, but of its own intimate feeling fill up -the gaps in the presentiment, grasp the full orb of possibilities -from the isolated points presented to it. In relation -to this question of our perceptions, let me suggest another -illustration, not taking it too seriously, only propounding -it to exhibit the possibilities in a broad and general way.</p> - -<p>In the heavens, amongst the multitude of stars, there -are some which, when the telescope is directed on them, -seem not to be single stars, but to be split up into two. -Regarding these twin stars through a spectroscope, an -astronomer sees in each a spectrum of bands of colour and -black lines. Comparing these spectrums with one another, -he finds that there is a slight relative shifting of the dark -lines, and from that shifting he knows that the stars are -rotating round one another, and can tell their relative -velocity with regard to the earth. By means of his -terrestrial physics he reads this signal of the skies. This -shifting of lines, the mere slight variation of a black line -in a spectrum, is very unlike that which the astronomer -knows it means. But it is probably much more like what -it means than the signals which the nerves deliver are -like the phenomena of the outer world.</p> - -<p>No picture of an object is conveyed through the nerves. -No picture of motion, in the sense in which we postulate -its existence, is conveyed through the nerves. The actual<span class="pagenum" id="Page_259">[Pg 259]</span> -deliverances of which our consciousness takes account are -probably identical for eye and ear, sight and touch.</p> - -<p>If for a moment I take the whole earth together and -regard it as a sentient being, I find that the problem of -its apprehension is a very complex one, and involves a -long series of personal and physical events. Similarly the -problem of our apprehension is a very complex one. I -only use this illustration to exhibit my meaning. It has -this especial merit, that, as the process of conscious -apprehension takes place in our case in the minute, so, -with regard to this earth being, the corresponding process -takes place in what is relatively to it very minute.</p> - -<p>Now, Plato’s view of a soul leads us to the hypothesis -that that which we designate as an act of apprehension -may be a very complex event, both physically and personally. -He does not seek to explain what an intuition -is; he makes it a basis from whence he sets out on a -voyage of discovery. Knowledge means knowledge; he -puts conscious being to account for conscious being. He -makes an hypothesis of the kind that is so fertile in -physical science—an hypothesis making no claim to -finality, which marks out a vista of possible determination -behind determination, like the hypothesis of space itself, -the type of serviceable hypotheses.</p> - -<p>And, above all, Plato’s hypothesis is conducive to experiment. -He gives the perspective in which real objects -can be determined; and, in our present enquiry, we are -making the simplest of all possible experiments—we are -enquiring what it is natural to the soul to think of matter -as extended.</p> - -<p>Aristotle says we always use a “phantasm” in thinking, -a phantasm of our corporeal senses a visualisation or a -tactualisation. But we can so modify that visualisation -or tactualisation that it represents something not known -by the senses. Do we by that representation wake up an<span class="pagenum" id="Page_260">[Pg 260]</span> -intuition of the soul? Can we by the presentation of -these hypothetical forms, that are the subject of our -present discussion, wake ourselves up to higher intuitions? -And can we explain the world around by a motion that we -only know by our souls?</p> - -<p>Apart from all speculation, however, it seems to me -that the interest of these four-dimensional shapes and -motions is sufficient reason for studying them, and that -they are the way by which we can grow into a fuller -apprehension of the world as a concrete whole.</p> - - -<h3><span class="smcap">Space Names.</span></h3> - -<p>If the words written in the squares drawn in <a href="#fig_144">fig. 1</a> are -used as the names of the squares in the positions in -which they are placed, it is evident that -a combination of these names will denote -a figure composed of the designated -squares. It is found to be most convenient -to take as the initial square that -marked with an asterisk, so that the -directions of progression are towards the -observer and to his right. The directions -of progression, however, are arbitrary, and can be chosen -at will.</p> - -<div class="figleft illowp25" id="fig_144" style="max-width: 12.5em;"> - <img src="images/fig_144.png" alt="" /> - <div class="caption">Fig. 1.</div> -</div> - -<p>Thus <i>et</i>, <i>at</i>, <i>it</i>, <i>an</i>, <i>al</i> will denote a figure in the form -of a cross composed of five squares.</p> - -<p>Here, by means of the double sequence, <i>e</i>, <i>a</i>, <i>i</i> and <i>n</i>, <i>t</i>, <i>l</i>, it -is possible to name a limited collection of space elements.</p> - -<p>The system can obviously be extended by using letter -sequences of more members.</p> - -<p>But, without introducing such a complexity, the -principles of a space language can be exhibited, and a -nomenclature obtained adequate to all the considerations -of the preceding pages.</p> - -<p><span class="pagenum" id="Page_261">[Pg 261]</span></p> - - -<p>1. <i>Extension.</i></p> - -<div class="figleft illowp35" id="fig_145" style="max-width: 15.625em;"> - <img src="images/fig_145.png" alt="" /> - <div class="caption">Fig. 2.</div> -</div> - -<p>Call the large squares in <a href="#fig_145">2</a> by the name written -in them. It is evident that each -can be divided as shown in <a href="#fig_144">fig. 1</a>. -Then the small square marked 1 -will be “en” in “En,” or “Enen.” -The square marked 2 will be “et” -in “En” or “Enet,” while the -square marked 4 will be “en” in -“Et” or “Eten.” Thus the square -5 will be called “Ilil.”</p> - -<p>This principle of extension can -be applied in any number of dimensions.</p> - - -<p>2. <i>Application to Three-Dimensional Space.</i></p> - -<div class="figleft illowp25" id="fig_146" style="max-width: 12.5em;"> - <img src="images/fig_146.png" alt="Three cube faces" /> -</div> - -<p>To name a three-dimensional collocation of cubes take -the upward direction first, secondly the -direction towards the observer, thirdly the -direction to his right hand.</p> - -<p>These form a word in which the first -letter gives the place of the cube upwards, -the second letter its place towards the -observer, the third letter its place to the -right.</p> - -<p>We have thus the following scheme, -which represents the set of cubes of -column 1, <a href="#fig_101">fig. 101</a>, page 165.</p> - -<p>We begin with the remote lowest cube -at the left hand, where the asterisk is -placed (this proves to be by far the most -convenient origin to take for the normal -system).</p> - -<p>Thus “nen” is a “null” cube, “ten” -a red cube on it, and “len” a “null” -cube above “ten.”</p> - -<p><span class="pagenum" id="Page_262">[Pg 262]</span></p> - -<p>By using a more extended sequence of consonants and -vowels a larger set of cubes can be named.</p> - -<p>To name a four-dimensional block of tesseracts it is -simply necessary to prefix an “e,” an “a,” or an “i” to -the cube names.</p> - -<p>Thus the tesseract blocks schematically represented on -page 165, <a href="#fig_101">fig. 101</a> are named as follows:—</p> - -<div class="figcenter illowp80" id="fig_147" style="max-width: 62.5em;"> - <img src="images/fig_147.png" alt="Nine cube faces" /> -</div> - -<p>2. <span class="smcap">Derivation of Point, Line, Face, etc., Names.</span></p> - -<p>The principle of derivation can be shown as follows: -Taking the square of squares<span class="pagenum" id="Page_263">[Pg 263]</span></p> - -<div class="figcenter illowp35" id="fig_148" style="max-width: 15.625em;"> - <img src="images/fig_148.png" alt="Cube face" /> -</div> -<p class="pnind">the number of squares in it can be enlarged and the -whole kept the same size.</p> - -<div class="figcenter illowp35" id="fig_149" style="max-width: 15.625em;"> - <img src="images/fig_149.png" alt="Cube face" /> -</div> - -<p>Compare <a href="#fig_79">fig. 79</a>, p. 138, for instance, or the bottom layer -of <a href="#fig_84">fig. 84</a>.</p> - -<p>Now use an initial “s” to denote the result of carrying -this process on to a great extent, and we obtain the limit -names, that is the point, line, area names for a square. -“Sat” is the whole interior. The corners are “sen,” -“sel,” “sin,” “sil,” while the lines -are “san,” “sal,” “set,” “sit.”</p> - -<div class="figleft illowp30" id="fig_150" style="max-width: 15.625em;"> - <img src="images/fig_150.png" alt="see para above" /> -</div> - -<p>I find that by the use of the -initial “s” these names come to be -practically entirely disconnected with -the systematic names for the square -from which they are derived. They -are easy to learn, and when learned -can be used readily with the axes running in any -direction.</p> - -<p>To derive the limit names for a four-dimensional rectangular -figure, like the tesseract, is a simple extension of -this process. These point, line, etc., names include those -which apply to a cube, as will be evident on inspection -of the first cube of the diagrams which follow.</p> - -<p>All that is necessary is to place an “s” before each of the -names given for a tesseract block. We then obtain -apellatives which, like the colour names on page 174, -<a href="#fig_103">fig. 103</a>, apply to all the points, lines, faces, solids, and to<span class="pagenum" id="Page_264">[Pg 264]</span> -the hyper-solid of the tesseract. These names have the -advantage over the colour marks that each point, line, etc., -has its own individual name.</p> - -<p>In the diagrams I give the names corresponding to -the positions shown in the coloured plate or described on -p. 174. By comparing cubes 1, 2, 3 with the first row of -cubes in the coloured plate, the systematic names of each -of the points, lines, faces, etc., can be determined. The -asterisk shows the origin from which the names run.</p> - -<p>These point, line, face, etc., names should be used in -connection with the corresponding colours. The names -should call up coloured images of the parts named in their -right connection.</p> - -<p>It is found that a certain abbreviation adds vividness of -distinction to these names. If the final “en” be dropped -wherever it occurs the system is improved. Thus instead -of “senen,” “seten,” “selen,” it is preferable to abbreviate -to “sen,” “set,” “sel,” and also use “san,” “sin” for -“sanen,” “sinen.”</p> -<div class="figcenter illowp100" id="fig_151" style="max-width: 62.5em;"> - <img src="images/fig_151.png" alt="See above" /> -</div> -<p><span class="pagenum" id="Page_265">[Pg 265]</span></p> - -<div class="figcenter illowp100" id="fig_152" style="max-width: 62.5em;"> - <img src="images/fig_152.png" alt="see above" /> -</div> - -<div class="figcenter illowp100" id="fig_153" style="max-width: 62.5em;"> - <img src="images/fig_153.png" alt="see above" /> -</div> - -<p><span class="pagenum" id="Page_266">[Pg 266]</span></p> - -<div class="figcenter illowp100" id="fig_154" style="max-width: 62.5em;"> - <img src="images/fig_154.png" alt="see above" /> -</div> - -<p>We can now name any section. Take <i>e.g.</i> the line in -the first cube from senin to senel, we should call the line -running from senin to senel, senin senat senel, a line -light yellow in colour with null points.</p> - -<p>Here senat is the name for all of the line except its ends. -Using “senat” in this way does not mean that the line is -the whole of senat, but what there is of it is senat. It is -a part of the senat region. Thus also the triangle, which -has its three vertices in senin, senel, selen, is named thus:</p> - - -<ul> -<li>Area: setat.</li> -<li>Sides: setan, senat, setet.</li> -<li>Vertices: senin, senel, sel.</li> -</ul> - -<p>The tetrahedron section of the tesseract can be thought -of as a series of plane sections in the successive sections of -the tesseract shown in <a href="#fig_114">fig. 114</a>, p. 191. In b<sub>0</sub> the section -<span class="pagenum" id="Page_267">[Pg 267]</span>is the one written above. In b<sub>1</sub> the section is made by a -plane which cuts the three edges from sanen intermediate -of their lengths and thus will be:</p> - - -<ul> -<li>Area: satat.</li> -<li>Sides: satan, sanat, satet.</li> -<li>Vertices: sanan, sanet, sat.</li> -</ul> - - -<p>The sections in b<sub>2</sub>, b<sub>3</sub> will be like the section in b<sub>1</sub> but -smaller.</p> - -<p>Finally in b<sub>4</sub> the section plane simply passes through the -corner named sin.</p> - -<p>Hence, putting these sections together in their right -relation, from the face setat, surrounded by the lines and -points mentioned above, there run:</p> - - -<ul> -<li>3 faces: satan, sanat, satet</li> -<li>3 lines: sanan, sanet, sat</li> -</ul> - - -<p>and these faces and lines run to the point sin. Thus -the tetrahedron is completely named.</p> - -<p>The octahedron section of the tesseract, which can be -traced from <a href="#fig_72">fig. 72</a>, p. 129 by extending the lines there -drawn, is named:</p> - -<p>Front triangle selin, selat, selel, setal, senil, setit, selin -with area setat.</p> - -<p>The sections between the front and rear triangle, of -which one is shown in 1b, another in 2b, are thus named, -points and lines, salan, salat, salet, satet, satel, satal, sanal, -sanat, sanit, satit, satin, satan, salan.</p> - -<p>The rear triangle found in 3b by producing lines is sil, -sitet, sinel, sinat, sinin, sitan, sil.</p> - -<p>The assemblage of sections constitute the solid body of -the octahedron satat with triangular faces. The one from -the line selat to the point sil, for instance, is named<span class="pagenum" id="Page_268">[Pg 268]</span> -selin, selat, selel, salet, salat, salan, sil. The whole -interior is salat.</p> - -<p>Shapes can easily be cut out of cardboard which, when -folded together, form not only the tetrahedron and the -octahedron, but also samples of all the sections of the -tesseract taken as it passes cornerwise through our space. -To name and visualise with appropriate colours a series of -these sections is an admirable exercise for obtaining -familiarity with the subject.</p> - - -<h3><span class="smcap">Extension and Connection with Numbers.</span></h3> - -<p>By extending the letter sequence it is of course possible -to name a larger field. By using the limit names the -corners of each square can be named.</p> - -<p>Thus “en sen,” “an sen,” etc., will be the names of the -points nearest the origin in “en” and in “an.”</p> - -<p>A field of points of which each one is indefinitely small -is given by the names written below.</p> - -<div class="figcenter illowp30" id="fig_155" style="max-width: 12.5em;"> - <img src="images/fig_155.png" alt="Field of points" /> -</div> - -<p>The squares are shown in dotted lines, the names -denote the points. These points are not mathematical -points, but really minute areas.</p> - -<p>Instead of starting with a set of squares and naming -them, we can start with a set of points.</p> - -<p>By an easily remembered convention we can give -names to such a region of points.</p> - -<p><span class="pagenum" id="Page_269">[Pg 269]</span></p> - -<p>Let the space names with a final “e” added denote the -mathematical points at the corner of each square nearest -the origin. We have then</p> - -<div class="figcenter illowp25" id="i_269" style="max-width: 15.625em;"> - <img src="images/i_269.png" alt="illustrating immediate text" /> -</div> -<p class="pnind">for the set of mathematical points indicated. This -system is really completely independent of the area -system and is connected with it merely for the purpose -of facilitating the memory processes. The word “ene” is -pronounced like “eny,” with just sufficient attention to -the final vowel to distinguish it from the word “en.”</p> - -<p>Now, connecting the numbers 0, 1, 2 with the sequence -e, a, i, and also with the sequence n, t, l, we have a set of -points named as with numbers in a co-ordinate system. -Thus “ene” is (0, 0) “ate” is (1, 1) “ite” is (2, 1). -To pass to the area system the rule is that the name of -the square is formed from the name of its point nearest -to the origin by dropping the final e.</p> - -<p>By using a notation analogous to the decimal system -a larger field of points can be named. It remains to -assign a letter sequence to the numbers from positive 0 -to positive 9, and from negative 0 to negative 9, to obtain -a system which can be used to denote both the usual -co-ordinate system of mapping and a system of named -squares. The names denoting the points all end with e. -Those that denote squares end with a consonant.</p> - -<p>There are many considerations which must be attended -to in extending the sequences to be used, such as -uniqueness in the meaning of the words formed, ease -of pronunciation, avoidance of awkward combinations.</p> - -<p><span class="pagenum" id="Page_270">[Pg 270]</span></p> - -<p>I drop “s” altogether from the consonant series and -short “u” from the vowel series. It is convenient to -have unsignificant letters at disposal. A double consonant -like “st” for instance can be referred to without giving it -a local significance by calling it “ust.” I increase the -number of vowels by considering a sound like “ra” to -be a vowel, using, that is, the letter “r” as forming a -compound vowel.</p> - -<p>The series is as follows:—</p> - -<table class="standard" summary=""> -<tr> -<td class="tdc" colspan="11"><span class="smcap">Consonants.</span></td> -</tr> -<tr> -<td class="tdc"></td> -<td class="tdc">0</td> -<td class="tdc">1</td> -<td class="tdc">2</td> -<td class="tdc">3</td> -<td class="tdc">4</td> -<td class="tdc">5</td> -<td class="tdc">6</td> -<td class="tdc">7</td> -<td class="tdc">8</td> -<td class="tdc">9</td> -</tr> -<tr> -<td class="tdl">positive</td> -<td class="tdc">n</td> -<td class="tdc">t</td> -<td class="tdc">l</td> -<td class="tdc">p</td> -<td class="tdc">f</td> -<td class="tdc">sh</td> -<td class="tdc">k</td> -<td class="tdc">ch</td> -<td class="tdc">nt</td> -<td class="tdc">st</td> -</tr> -<tr> -<td class="tdl">negative</td> -<td class="tdc">z</td> -<td class="tdc">d</td> -<td class="tdc">th</td> -<td class="tdc">b</td> -<td class="tdc">v</td> -<td class="tdc">m</td> -<td class="tdc">g</td> -<td class="tdc">j</td> -<td class="tdc">nd</td> -<td class="tdc">sp</td> -</tr> -<tr> -<td class="tdc" colspan="11"><span class="smcap">Vowels.</span></td> -</tr> -<tr> -<td class="tdc"></td> -<td class="tdc">0</td> -<td class="tdc">1</td> -<td class="tdc">2</td> -<td class="tdc">3</td> -<td class="tdc">4</td> -<td class="tdc">5</td> -<td class="tdc">6</td> -<td class="tdc">7</td> -<td class="tdc">8</td> -<td class="tdc">9</td> -</tr> -<tr> -<td class="tdc">positive</td> -<td class="tdc">e</td> -<td class="tdc">a</td> -<td class="tdc">i</td> -<td class="tdc">ee</td> -<td class="tdc">ae</td> -<td class="tdc">ai</td> -<td class="tdc">ar</td> -<td class="tdc">ra</td> -<td class="tdc">ri</td> -<td class="tdc">ree</td> -</tr> -<tr> -<td class="tdc">negative</td> -<td class="tdc">er</td> -<td class="tdc">o</td> -<td class="tdc">oo</td> -<td class="tdc">io</td> -<td class="tdc">oe</td> -<td class="tdc">iu</td> -<td class="tdc">or</td> -<td class="tdc">ro</td> -<td class="tdc">roo rio</td> -</tr> -</table> - - -<p><i>Pronunciation.</i>—e as in men; a as in man; i as in in; -ee as in between; ae as ay in may; ai as i in mine; ar as -in art; er as ear in earth; o as in on; oo as oo in soon; -io as in clarion; oe as oa in oat; iu pronounced like yew.</p> - -<p>To name a point such as (23, 41) it is considered as -(3, 1) on from (20, 40) and is called “ifeete.” It is the -initial point of the square ifeet of the area system.</p> - -<p>The preceding amplification of a space language has -been introduced merely for the sake of completeness. As -has already been said nine words and their combinations, -applied to a few simple models suffice for the purposes of -our present enquiry.</p> - - -<p class="center small"><i>Printed by Hazell, Watson & Viney, Ld., London and Aylesbury.</i></p> - -<div>*** END OF THE PROJECT GUTENBERG EBOOK 67153 ***</div> -</body> -</html> diff --git a/old/old-2024-12-23/67153-h/images/colop.png b/old/old-2024-12-23/67153-h/images/colop.png Binary files differdeleted file mode 100644 index 0969174..0000000 --- a/old/old-2024-12-23/67153-h/images/colop.png +++ /dev/null diff --git a/old/old-2024-12-23/67153-h/images/fig_1.png b/old/old-2024-12-23/67153-h/images/fig_1.png Binary files differdeleted file mode 100644 index 41e29c1..0000000 --- a/old/old-2024-12-23/67153-h/images/fig_1.png +++ /dev/null diff --git a/old/old-2024-12-23/67153-h/images/fig_10.png b/old/old-2024-12-23/67153-h/images/fig_10.png Binary files differdeleted file mode 100644 index b67ed05..0000000 --- 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