diff options
Diffstat (limited to 'old/67153-0.txt')
| -rw-r--r-- | old/67153-0.txt | 8858 |
1 files changed, 0 insertions, 8858 deletions
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. Project Gutenberg is a registered trademark, -and may not be used if you charge for an eBook, except by following -the terms of the trademark license, including paying royalties for use -of the Project Gutenberg trademark. If you do not charge anything for -copies of this eBook, complying with the trademark license is very -easy. You may use this eBook for nearly any purpose such as creation -of derivative works, reports, performances and research. Project -Gutenberg eBooks may be modified and printed and given away--you may -do practically ANYTHING in the United States with eBooks not protected -by U.S. copyright law. Redistribution is subject to the trademark -license, especially commercial redistribution. - -START: FULL LICENSE - -THE FULL PROJECT GUTENBERG LICENSE -PLEASE READ THIS BEFORE YOU DISTRIBUTE OR USE THIS WORK - -To protect the Project Gutenberg-tm mission of promoting the free -distribution of electronic works, by using or distributing this work -(or any other work associated in any way with the phrase "Project -Gutenberg"), you agree to comply with all the terms of the Full -Project Gutenberg-tm License available with this file or online at -www.gutenberg.org/license. - -Section 1. General Terms of Use and Redistributing Project -Gutenberg-tm electronic works - -1.A. By reading or using any part of this Project Gutenberg-tm -electronic work, you indicate that you have read, understand, agree to -and accept all the terms of this license and intellectual property -(trademark/copyright) agreement. If you do not agree to abide by all -the terms of this agreement, you must cease using and return or -destroy all copies of Project Gutenberg-tm electronic works in your -possession. If you paid a fee for obtaining a copy of or access to a -Project Gutenberg-tm electronic work and you do not agree to be bound -by the terms of this agreement, you may obtain a refund from the -person or entity to whom you paid the fee as set forth in paragraph -1.E.8. - -1.B. "Project Gutenberg" is a registered trademark. It may only be -used on or associated in any way with an electronic work by people who -agree to be bound by the terms of this agreement. There are a few -things that you can do with most Project Gutenberg-tm electronic works -even without complying with the full terms of this agreement. See -paragraph 1.C below. There are a lot of things you can do with Project -Gutenberg-tm electronic works if you follow the terms of this -agreement and help preserve free future access to Project Gutenberg-tm -electronic works. See paragraph 1.E below. - -1.C. The Project Gutenberg Literary Archive Foundation ("the -Foundation" or PGLAF), owns a compilation copyright in the collection -of Project Gutenberg-tm electronic works. Nearly all the individual -works in the collection are in the public domain in the United -States. If an individual work is unprotected by copyright law in the -United States and you are located in the United States, we do not -claim a right to prevent you from copying, distributing, performing, -displaying or creating derivative works based on the work as long as -all references to Project Gutenberg are removed. Of course, we hope -that you will support the Project Gutenberg-tm mission of promoting -free access to electronic works by freely sharing Project Gutenberg-tm -works in compliance with the terms of this agreement for keeping the -Project Gutenberg-tm name associated with the work. You can easily -comply with the terms of this agreement by keeping this work in the -same format with its attached full Project Gutenberg-tm License when -you share it without charge with others. - -1.D. The copyright laws of the place where you are located also govern -what you can do with this work. Copyright laws in most countries are -in a constant state of change. If you are outside the United States, -check the laws of your country in addition to the terms of this -agreement before downloading, copying, displaying, performing, -distributing or creating derivative works based on this work or any -other Project Gutenberg-tm work. The Foundation makes no -representations concerning the copyright status of any work in any -country other than the United States. - -1.E. Unless you have removed all references to Project Gutenberg: - -1.E.1. The following sentence, with active links to, or other -immediate access to, the full Project Gutenberg-tm License must appear -prominently whenever any copy of a Project Gutenberg-tm work (any work -on which the phrase "Project Gutenberg" appears, or with which the -phrase "Project Gutenberg" is associated) is accessed, displayed, -performed, viewed, copied or distributed: - - 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. - -1.E.2. If an individual Project Gutenberg-tm electronic work is -derived from texts not protected by U.S. copyright law (does not -contain a notice indicating that it is posted with permission of the -copyright holder), the work can be copied and distributed to anyone in -the United States without paying any fees or charges. If you are -redistributing or providing access to a work with the phrase "Project -Gutenberg" associated with or appearing on the work, you must comply -either with the requirements of paragraphs 1.E.1 through 1.E.7 or -obtain permission for the use of the work and the Project Gutenberg-tm -trademark as set forth in paragraphs 1.E.8 or 1.E.9. - -1.E.3. If an individual Project Gutenberg-tm electronic work is posted -with the permission of the copyright holder, your use and distribution -must comply with both paragraphs 1.E.1 through 1.E.7 and any -additional terms imposed by the copyright holder. Additional terms -will be linked to the Project Gutenberg-tm License for all works -posted with the permission of the copyright holder found at the -beginning of this work. - -1.E.4. Do not unlink or detach or remove the full Project Gutenberg-tm -License terms from this work, or any files containing a part of this -work or any other work associated with Project Gutenberg-tm. - -1.E.5. Do not copy, display, perform, distribute or redistribute this -electronic work, or any part of this electronic work, without -prominently displaying the sentence set forth in paragraph 1.E.1 with -active links or immediate access to the full terms of the Project -Gutenberg-tm License. - -1.E.6. You may convert to and distribute this work in any binary, -compressed, marked up, nonproprietary or proprietary form, including -any word processing or hypertext form. However, if you provide access -to or distribute copies of a Project Gutenberg-tm work in a format -other than "Plain Vanilla ASCII" or other format used in the official -version posted on the official Project Gutenberg-tm website -(www.gutenberg.org), you must, at no additional cost, fee or expense -to the user, provide a copy, a means of exporting a copy, or a means -of obtaining a copy upon request, of the work in its original "Plain -Vanilla ASCII" or other form. Any alternate format must include the -full Project Gutenberg-tm License as specified in paragraph 1.E.1. - -1.E.7. Do not charge a fee for access to, viewing, displaying, -performing, copying or distributing any Project Gutenberg-tm works -unless you comply with paragraph 1.E.8 or 1.E.9. - -1.E.8. You may charge a reasonable fee for copies of or providing -access to or distributing Project Gutenberg-tm electronic works -provided that: - -* You pay a royalty fee of 20% of the gross profits you derive from - the use of Project Gutenberg-tm works calculated using the method - you already use to calculate your applicable taxes. The fee is owed - to the owner of the Project Gutenberg-tm trademark, but he has - agreed to donate royalties under this paragraph to the Project - Gutenberg Literary Archive Foundation. Royalty payments must be paid - within 60 days following each date on which you prepare (or are - legally required to prepare) your periodic tax returns. Royalty - payments should be clearly marked as such and sent to the Project - Gutenberg Literary Archive Foundation at the address specified in - Section 4, "Information about donations to the Project Gutenberg - Literary Archive Foundation." - -* You provide a full refund of any money paid by a user who notifies - you in writing (or by e-mail) within 30 days of receipt that s/he - does not agree to the terms of the full Project Gutenberg-tm - License. You must require such a user to return or destroy all - copies of the works possessed in a physical medium and discontinue - all use of and all access to other copies of Project Gutenberg-tm - works. - -* You provide, in accordance with paragraph 1.F.3, a full refund of - any money paid for a work or a replacement copy, if a defect in the - electronic work is discovered and reported to you within 90 days of - receipt of the work. - -* You comply with all other terms of this agreement for free - distribution of Project Gutenberg-tm works. - -1.E.9. If you wish to charge a fee or distribute a Project -Gutenberg-tm electronic work or group of works on different terms than -are set forth in this agreement, you must obtain permission in writing -from the Project Gutenberg Literary Archive Foundation, the manager of -the Project Gutenberg-tm trademark. Contact the Foundation as set -forth in Section 3 below. - -1.F. - -1.F.1. Project Gutenberg volunteers and employees expend considerable -effort to identify, do copyright research on, transcribe and proofread -works not protected by U.S. copyright law in creating the Project -Gutenberg-tm collection. Despite these efforts, Project Gutenberg-tm -electronic works, and the medium on which they may be stored, may -contain "Defects," such as, but not limited to, incomplete, inaccurate -or corrupt data, transcription errors, a copyright or other -intellectual property infringement, a defective or damaged disk or -other medium, a computer virus, or computer codes that damage or -cannot be read by your equipment. - -1.F.2. LIMITED WARRANTY, DISCLAIMER OF DAMAGES - Except for the "Right -of Replacement or Refund" described in paragraph 1.F.3, the Project -Gutenberg Literary Archive Foundation, the owner of the Project -Gutenberg-tm trademark, and any other party distributing a Project -Gutenberg-tm electronic work under this agreement, disclaim all -liability to you for damages, costs and expenses, including legal -fees. YOU AGREE THAT YOU HAVE NO REMEDIES FOR NEGLIGENCE, STRICT -LIABILITY, BREACH OF WARRANTY OR BREACH OF CONTRACT EXCEPT THOSE -PROVIDED IN PARAGRAPH 1.F.3. YOU AGREE THAT THE FOUNDATION, THE -TRADEMARK OWNER, AND ANY DISTRIBUTOR UNDER THIS AGREEMENT WILL NOT BE -LIABLE TO YOU FOR ACTUAL, DIRECT, INDIRECT, CONSEQUENTIAL, PUNITIVE OR -INCIDENTAL DAMAGES EVEN IF YOU GIVE NOTICE OF THE POSSIBILITY OF SUCH -DAMAGE. - -1.F.3. LIMITED RIGHT OF REPLACEMENT OR REFUND - If you discover a -defect in this electronic work within 90 days of receiving it, you can -receive a refund of the money (if any) you paid for it by sending a -written explanation to the person you received the work from. If you -received the work on a physical medium, you must return the medium -with your written explanation. The person or entity that provided you -with the defective work may elect to provide a replacement copy in -lieu of a refund. If you received the work electronically, the person -or entity providing it to you may choose to give you a second -opportunity to receive the work electronically in lieu of a refund. If -the second copy is also defective, you may demand a refund in writing -without further opportunities to fix the problem. - -1.F.4. Except for the limited right of replacement or refund set forth -in paragraph 1.F.3, this work is provided to you 'AS-IS', WITH NO -OTHER WARRANTIES OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT -LIMITED TO WARRANTIES OF MERCHANTABILITY OR FITNESS FOR ANY PURPOSE. - -1.F.5. Some states do not allow disclaimers of certain implied -warranties or the exclusion or limitation of certain types of -damages. If any disclaimer or limitation set forth in this agreement -violates the law of the state applicable to this agreement, the -agreement shall be interpreted to make the maximum disclaimer or -limitation permitted by the applicable state law. The invalidity or -unenforceability of any provision of this agreement shall not void the -remaining provisions. - -1.F.6. INDEMNITY - You agree to indemnify and hold the Foundation, the -trademark owner, any agent or employee of the Foundation, anyone -providing copies of Project Gutenberg-tm electronic works in -accordance with this agreement, and any volunteers associated with the -production, promotion and distribution of Project Gutenberg-tm -electronic works, harmless from all liability, costs and expenses, -including legal fees, that arise directly or indirectly from any of -the following which you do or cause to occur: (a) distribution of this -or any Project Gutenberg-tm work, (b) alteration, modification, or -additions or deletions to any Project Gutenberg-tm work, and (c) any -Defect you cause. - -Section 2. Information about the Mission of Project Gutenberg-tm - -Project Gutenberg-tm is synonymous with the free distribution of -electronic works in formats readable by the widest variety of -computers including obsolete, old, middle-aged and new computers. It -exists because of the efforts of hundreds of volunteers and donations -from people in all walks of life. - -Volunteers and financial support to provide volunteers with the -assistance they need are critical to reaching Project Gutenberg-tm's -goals and ensuring that the Project Gutenberg-tm collection will -remain freely available for generations to come. In 2001, the Project -Gutenberg Literary Archive Foundation was created to provide a secure -and permanent future for Project Gutenberg-tm and future -generations. To learn more about the Project Gutenberg Literary -Archive Foundation and how your efforts and donations can help, see -Sections 3 and 4 and the Foundation information page at -www.gutenberg.org - -Section 3. Information about the Project Gutenberg Literary -Archive Foundation - -The Project Gutenberg Literary Archive Foundation is a non-profit -501(c)(3) educational corporation organized under the laws of the -state of Mississippi and granted tax exempt status by the Internal -Revenue Service. The Foundation's EIN or federal tax identification -number is 64-6221541. Contributions to the Project Gutenberg Literary -Archive Foundation are tax deductible to the full extent permitted by -U.S. federal laws and your state's laws. - -The Foundation's business office is located at 809 North 1500 West, -Salt Lake City, UT 84116, (801) 596-1887. Email contact links and up -to date contact information can be found at the Foundation's website -and official page at www.gutenberg.org/contact - -Section 4. Information about Donations to the Project Gutenberg -Literary Archive Foundation - -Project Gutenberg-tm depends upon and cannot survive without -widespread public support and donations to carry out its mission of -increasing the number of public domain and licensed works that can be -freely distributed in machine-readable form accessible by the widest -array of equipment including outdated equipment. Many small donations -($1 to $5,000) are particularly important to maintaining tax exempt -status with the IRS. - -The Foundation is committed to complying with the laws regulating -charities and charitable donations in all 50 states of the United -States. Compliance requirements are not uniform and it takes a -considerable effort, much paperwork and many fees to meet and keep up -with these requirements. We do not solicit donations in locations -where we have not received written confirmation of compliance. To SEND -DONATIONS or determine the status of compliance for any particular -state visit www.gutenberg.org/donate - -While we cannot and do not solicit contributions from states where we -have not met the solicitation requirements, we know of no prohibition -against accepting unsolicited donations from donors in such states who -approach us with offers to donate. - -International donations are gratefully accepted, but we cannot make -any statements concerning tax treatment of donations received from -outside the United States. U.S. laws alone swamp our small staff. - -Please check the Project Gutenberg web pages for current donation -methods and addresses. Donations are accepted in a number of other -ways including checks, online payments and credit card donations. To -donate, please visit: www.gutenberg.org/donate - -Section 5. General Information About Project Gutenberg-tm electronic works - -Professor Michael S. Hart was the originator of the Project -Gutenberg-tm concept of a library of electronic works that could be -freely shared with anyone. For forty years, he produced and -distributed Project Gutenberg-tm eBooks with only a loose network of -volunteer support. - -Project Gutenberg-tm eBooks are often created from several printed -editions, all of which are confirmed as not protected by copyright in -the U.S. unless a copyright notice is included. Thus, we do not -necessarily keep eBooks in compliance with any particular paper -edition. - -Most people start at our website which has the main PG search -facility: www.gutenberg.org - -This website includes information about Project Gutenberg-tm, -including how to make donations to the Project Gutenberg Literary -Archive Foundation, how to help produce our new eBooks, and how to -subscribe to our email newsletter to hear about new eBooks. |